wireless thesis

8/18/2019 Wireless Thesis

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A%*nole'$ements

The a&thor o&l' li*e to )ersonall" than* Professor -e,in Wari%* for his $&i'an%e3 forthri$htness3an' %onfi'en%e that the or* )resente' herein o&l' satisf" hat I as to e4)e%t from it. It as he

ho on%e tol' me that m" &n)&!lishe' i'eas o&l' o,er%ome the threshol' of a s&%%essf&l )eer5

re,ie. -e,in ins)ire' an a))li%ation in this thesis3 the %o&)lin$ of the h&man ner,o&s s"stem to a

ma%hine &sin$ ireless )oer as a non5in,asi,e me'i&m3 folloin$ his $reat or* in Pro6e%t ("!or$.

When e ha' met at the E5Golem %onferen%e in Pra$&e S&mmer /0073 I %o&l' ha,e onl" !arel"

ima$ine' or*in$ ith him o&l' )ro'&%e s&%h s)e%ta%&lar res&lts.

The a&thor o&l' also li*e to than* Dr. William Hol'er!a&m for his %amara'erie an' 'e,otion to

ta*in$ hate,er time as nee'e' to $et o&r first )a)ers to the )&!lisher. He as instr&mental in the

steera$e in the flo of the stor" of the )a)er3 es)e%iall" in the relation of the mathemati%s an'

'ia$rams to the or*. William 'is)la"e' $reat tena%it" in $ettin$ a )oint a%ross in an otherise

&n'efine' %onte4t3 somethin$ I ha,e ta*en aa" from o&r frien'shi).

I o&l' li*e to %on,e" m" $ratit&'e to 8ames An'erson for tal*in$ )lainl" a!o&t ho one $oes a!o&t

)&rs&in$ %ommer%iall" those $en&ine i'eas from a%a'emia. It is from those e4)ressions an' ho the"

are le,era$e' to the interests of others that lies at the heart of the matter.

I o&l' also li*e to than* Professor R. Pri%e an' Professor W. DeFor' at the Uni,ersit" of Utah3 for

their ins)irational le%t&res an' free'om in resear%h )hiloso)hies that forme' the fo&n'ation of the

theor" in this thesis. Dr. #. Ar$&elles for the %o&ra$e to !e$in as*in$ the 9&estions hile aare of

their %ost an' Dr. S. Pier%e for &n'erstan'in$ the h&man %on'ition is '&e to o&r insisten%e &)on

&)hol'in$ o&r i'eals.

I o&l' li*e to %on,e" m" $ratit&'e to m" father for lettin$ me s)en' %o&ntless ho&rs in the !asement

of the famil" ho&se as a )re5teen !&il'in$ %oils an' %ir%&its tr"in$ to &n'erstan' the :in,isi!le; for%e

that is ma$netism. Witho&t his s&))ort an' o)enness to anserin$ m" relentless 9&estions3 I ne,er

o&l' ha,e !e$&n this or*.

Finall"3 I o&l' li*e to than* S. #%De,itt3 (. Smith3 R. Po)*o3 N. She!alin3 an' -. -ase" for their

restless s)irits an' ins)iration of hat this or* as to !e%ome.

The or* is 'e'i%ate' to those )ersons ho stoo' !" me as I o!sesse' o,er hat as to !e%ome this

thesis< itho&t them3 the ri%h 'i,ersit" that is this orl' o&l' ha,e ne,er !een reali=e'.

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For Daniela

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Ta!le of Fi$&res

Fi$.1.1. Tesla?s ,ision of ireless )oer. ............................................................................................ /@

Fi$./.1 The )rotot")e mo'el for the transmission of )oer !" ma$neti% resonan%e.......................... 21

Fi$././. Areas here the fiel' )&shes in the me'i&m an' fall5off ra'iation from a loo) antenna........ 2/

Fi$./.2. Ra" relationshi)s !eteen the ma$neti% fiel'3 ra'ial ,e%tor3 an' infinitesimal %&rrent.......... 2>

Fi$./.>. Geometr" of the %o&)lin$5mo'e ma$neti% fiel'...................................................................... 27

Fi$./.7. Pro6e%tion of ener$" !" a %&rrent im)resse' on the loo). ....................................................... 2B

Fi$./.C. Fiel' lines folloin$ an a4is of 'is)la%ement. ........................................................................ 2@

Fi$./.B. Fl&4 lin*a$e !eteen to in'&%ti,el"5%o&)le' %oils. ............................................................. 2

Fi$./.@. Fiel' $eometr" e4ten'in$ aa" from the )ro6e%tion............................................................... >0

Fi$./.. Pro6e%tion fiel' formin$ the mi'5fiel' re$ion !" intensit"...................................................... >0

Fi$./.10. Pro6e%tion of fiel' an' its $eometr" !eteen to in'&%ti,el"5%o&)le' %oils........................ >1

Fi$./.11. FE# mo'el or*s)a%e la"o&t............................................................................................... >>

Fi$./.1/. (al%&lator ,al&es for a s"stem resonan%e fre9&en%"3 an' ! in'&%tan%e of the transmission

loo). ...................................................................................................................................................... >@

Fi$./.12. (oil )ro)erties 'es%ri!in$ 71.............................................................................................. >

Fi$./.1>. #a$neti% %o&)lin$ of lo5κ mo'es of 'istant resonators in s)a%e an' time......................... 7/

Fi$./.17. Theoreti%al %o&)lin$ %oeffi%ient o,er a 'istan%e $i,en >7. ................................................ 72

Fi$./.1C. a The ,e%tor )otential in the ma$neti% fiel'3 ! the )otential %om)onent on the loo)

interfa%e. ............................................................................................................................................... C0

Fi$./.1B. A sin$le loo) )ro6e%tin$ a ma$neti% fiel'.............................................................................. C1

Fi$./.1@. S%hemati% 'rain$ of to %ir%&lar loo)s ith %enter 'is%s. ................................................. C/

Fi$./.1. Geometr" of a )ro6e%tion an' infinitesimal fiel' in the %ir%&lar loo). .................................. C>

C



Fi$././0. a #a$neti% fl&4 of loo) antenna3 ! 'is% a''s fl&4 stren$th.............................................. C@

Fi$././1. Pro)a$atin$ ener$" in . ....................................................................................................... C@

Fi$././/. Densit" of a=im&thal fiel'. ............................................................................................... C

Fi$././2. The re$enerati,e os%illator. ................................................................................................... B1

Fi$././>. In'&%ti,el"5%o&)le' )&sh5)&ll os%illator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................ B/

Fi$././7. The n5%hannel #OSFET str&%t&re. ....................................................................................... B2

Fi$././C. S%hemati% re)resentation of a%%eleratin$ ele%tions s)lit a%ross a tan* %ir%&it. ..................... B2

Fi$././B. Phase5shift am)lifier on %oils. ............................................................................................... B>

Fi$././@. The transmission element ith a %o&)le'5mo'e. .................................................................. B7

Fi$././. a The re%ei,in$ element ith a %o&)le'5mo'e3 ! an e4am)le re%ei,er. .......................... B7

Fi$./.20. The !roa'%astin$ mo'el for sim&lation................................................................................. BC

Fi$./.21. Sim&late' si$nal seen !" the re%ei,er. .................................................................................. BB

Fi$./.2/. Sim)lifie' %ir%&it 'ia$ram..................................................................................................... @0

Fi$./.22. The %onstr&%te' os%illator...................................................................................................... @1

Fi$./.2>. Ra'iation emissions of a ma$neti% loo) antenna................................................................... @2

Fi$./.27. Wa,e )ro)a$ation for the mo'al %o&)lin$ s%heme of ireless )oer transfer..................... @2

Fi$./.2C. S%hemati% re)resentation of a ,irt&al transmission line of an antenna to a re$ion of s)a%e. @>

Fi$./.2B. An$&lar motion of the os%illator transition )hase. ................................................................ @7

Fi$./.2@. The transmitter %oil. .............................................................................................................. @7

Fi$./.2. a S%hemati% of the re%ei,in$ loo) antenna. ! oo) antenna %onne%te' to a %a)a%itor at

the ,olta$e $a). The hite arros are s&rfa%e %&rrents........................................................................ @B

Fi$./.>0. #o'ifie' s%hemati% of RWG e'$e element an' the 'i)ole inter)retation for %&r,at&re θ. . . @

Fi$./.>1. To loo) meshes a /@@ trian$les an' ! 117/ trian$les. ................................................. @

B



Fi$./.>/. a Trian$le 9&alit" of the loo) mesh3 ! %&rrent 'ensit" on the loo).................................. 0

Fi$./.>2. Dire%tional )oer 'ensit" of fiel' on its a))roa%hin$ a,efront3 hite shos hi$her )oer.

.............................................................................................................................................................. 1

Fi$./.>>. Pro6e%tion of ma$neti% %&rrents ith hi$hest )otential %o&)lin$ )oints Hθ . ........................ 7

Fi$./.>7. A,era$e )oer5o,er5s)a%e antenna ra'iation )attern loo*in$ into the %oil. ......................... C

Fi$./.>C. S%hemati% 'ia$ram of the %ir%&it for a sin$le %o&)le'5mo'e. ............................................... B

Fi$./.>B. The )oer o,er 'istan%e shoin$ the )ro6e%tion fiel' H3 here (i ≈ ( 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . @

Fi$./.>@. The )oer o,er a lon$er 'istan%e.......................................................................................... @

Fi$./.>. The )oer o,er 'istan%e &sin$ a thi%*er antenna ire..........................................................

Fi$./.70. The )oer intensit" o,er 'istan%e shoin$ linear falloff3 here (i > ( 6. . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 0

Fi$./.71. Ra'iation )attern o!ser,e' at the re%ei,er. ......................................................................... 10/

Fi$./.7/. Protot")i%al ireless5)oer os%illator 'emonstratin$ the theoreti%al mo'el...................... 10>

Fi$./.72. Intensit" o,er 'istan%e3 '= / . r ........................................................................................... 10@

Fi$./.7>. Intensit" o,er 'istan%e3 '≫ r. ............................................................................................ 10@

Fi$./.77. E4)erimentall"5teste' orientation of m&lti)le re%ei,ers r an' s ith referen%e to ea%h

)osition at the transmitter t hen there is no an$le of rotation aa" from a4ial ali$nment art = 0. The

stan'in$ hei$ht of all %oils are e9&al. ................................................................................................. 112

Fi$./.7C. E4)erimentall" teste' orientation of m&lti)le re%ei,ers r an' s3 thro&$h ith an an$le of

rotation θ aa" from a4ial ali$nment3 ith referen%e to ea%h )osition at the transmitter t. The

stan'in$ hei$ht of all %oils is e9&al..................................................................................................... 11>

Fi$./.7B. The a% )oer intensit" seen !" a re%ei,in$ antenna ith a4ial ali$nment here (i ≈ ( 6. . 1 1 C

Fi$./.7@. The '% )oer intensit" seen !" the re%ei,in$ antenna ith a4ial ali$nment. ..................... 11B

Fi$./.7. O)erational os%illator )oerin$ m&lti)le lam)s. ................................................................ 1/1

Fi$.2.1. Teslas ireless )oer mo'el. ............................................................................................... 1/2

@



Fi$.2./. Transmitter %oil )a%*a$e. ...................................................................................................... 1/C

Fi$.2.2. Re%ei,er %oil )a%*a$e............................................................................................................ 1/C

Fi$.2.>. S%hemati% 'ia$ram of the %ir%&it for to %o&)le'5mo'es..................................................... 1/B

Fi$.2.7. The %oherent fiel' o!6e%t res&ltin$ from f&n%tions in Fi$./.1B. ............................................ 1/

Fi$.2.C. #a$neti% fiel' intensit" an' fl&4 'ensit" !eteen to se%on'ar" s)iral %oils. .................... 121

Fi$.2.B. #a$neti% fiel' intensit" an' fl&4 'ensit" at $reater 'istan%e. ............................................... 12/

Fi$.2.@. Re%ei,er )a%*a$e an' thir' %oil. (om)onent !rea'!oar' shon.......................................... 122

Fi$.2.. The effi%ien%" of the Tesla resonator o,er a 'istan%e. .......................................................... 12>

Fi$.2.10. (om)arison of the Tesla resonator ith -&rs..................................................................... 12>

Fi$.2.11. The effi%ien%" of the Tesla resonator ith enhan%e' 9&alit" fa%tor.................................... 127

Fi$.2.1/. Tesla antenna an' anti)o'es................................................................................................ 12B

Fi$.>.1. Ener$" in the 9&ant&m state. ................................................................................................. 1>1

Fi$.>./. A %ommon 'es%ri)tion of a )oint5%har$e 'i)ole. .................................................................. 1>/

Fi$.>.2. a Point5%har$es an' the a*e )attern of for%es3 ! emittin$ a ma$neti% shearin$ for%e

%on%entrate' at the ma$neti% moment m. ........................................................................................... 1>2

Fi$.>.>. Photon refle%tion an' a!sor)tion )rimiti,es.......................................................................... 1>>

Fi$.C.1. Effe%t on the sim&late' o&t)&t !ase' on %han$in$ the ,al&e of the %o&)lin$ %oeffi%ient κi6 ...

a κ = 0.00/3 ! κ = 0.00>3 % κ = 0.00@3 ' κ = 0.010. ............................................................. 17>



Ta!le of Ta!les

Ta!le /.1. oo) %oils theoreti%al s)e%ifi%ation..................................................................................... >B

Ta!le /./. Ener$"3 9&alit"3 an' loss s)e%ifi%ation. ............................................................................... 7B

Ta!le /.2. Sim&lation %ir%&it )ro)erties................................................................................................ BC

Ta!le /.>. Os%illator state !" J an' ζ . .............................................................................................. B@

Ta!le /.7. Os%illator tra,elin$ a,e )arameters. ................................................................................. @0

Ta!le /.C. (om)&tational elements from mesh to fiel' sol&tions. ....................................................... @C

Ta!le /.B. Transmission an' re%ei,in$ %oils )h"si%al s)e%ifi%ation..................................................... @B

Ta!le /.@. #atla! %om)&te' in)&t %hara%teristi% antenna 'ata............................................................. 1

Ta!le /.. E4)erimental %oils %al%&late' ele%tri%al s)e%ifi%ation I....................................................... B

Ta!le /.10. E4)erimental %oils %al%&late' ele%tri%al s)e%ifi%ation II. ..................................................

Ta!le /.11. (al%&late' antenna %hara%teristi%s !ase' on K20L. ........................................................... 10>

Ta!le /.1/. (al%&late' 9&antities in the )ro6e%te' ma$neti% fiel'...................................................... 10C

Ta!le /.12. (al%&late' 9&antities in the )ro6e%te' ma$neti% fiel'...................................................... 10C

Ta!le /.1>. E4)erimental 9&antities of the ma$neti% loo) antenna.................................................... 10B

Ta!le 2.1. Resonant %oils )h"si%al s)e%ifi%ation. ............................................................................... 1/B

Ta!le 2./. Resonant %oils %al%&late' ele%tri%al s)e%ifi%ation.............................................................. 1/@

Ta!le 2.2. Resonant %oils meas&re' ele%tri%al s)e%ifi%ation............................................................... 127

/Ta!le C.1. Val&es of fa%tor & K2/L. ................................................................................................... 172

Ta!le C./. Instr&%tions to %reate an antenna mesh file........................................................................ 177

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A!stra%t

The or* in,ol,es in,esti$ation of a t")e of ireless )oer s"stem herein its anal"sis ill "iel' the

%onstr&%tion of a )rotot")e mo'ele' as a sin$&lar te%hnolo$i%al artifa%t. It is thro&$h e4)loration of

the artifa%t that forms the intelle%t&al !asis for not onl" its )rotot")i%al forms3 !&t s&$$esti,e of

,ariant forms not "et 'is%o,ere'. Thro&$h the )ro%ess it is $reatl" %larifie' the role of the artifa%t3 its

most s&ita!le a))li%ation $i,en the %onstraints on the 'eli,er" )ro!lem3 an' o)timi=ation strate$ies to

im)ro,e it.

In or'er to im)ro,e mat&rit" an' %ontri!&te to a !o'" of *nole'$e3 this 'o%&ment )ro)oses resear%h

&tili=in$ mi'5fiel' re$ion3 effi%ient in'&%ti,e5transfer for the )&r)oses of remo,in$ ire' %onne%tions

an' ele%tri%al %onta%ts. While the 'es%ri)tion seems eno&$h to state the )&r)ose of this or*3 it 'oes

not %on,e" the %om)romises of ha,in$ to re'ra the lines of 'emar%ation !eteen near an' far5fiel'

in the tra'itional metho' of !roa'%astin$.

To stri*in$ s%enarios are a''resse' in this thesis Firstl"3 the mathemati%al e4)lanation of ireless

)oer is '&e to 8.(. #a4ell?s ori$inal e9&ations3 se%on'l"3 the !eha,ior of ireless )oer in the

%ir%&it is '&e to 8ose)h armor?s f&n'amental or*s on the '"nami%s of the fiel' %on%e)t. A mo'el of )ro)a$ation ill !e )resente' hi%h mat%hes o!ser,ations in e4)eriments. A mo'ifie' mo'el of the

'i)ole ill !e )resente' to a''ress the )henomena o!ser,e' in the theor" an' e4)eriments.

To 'istin%t sets of e4)eriments ill test the %on%e)t of sin$le an' to %o&)le'5mo'es. In a more

esoteri% %onte4t of the =ero an' first5or'er ma$neti% fiel'3 the s&$$estion of a thir' %o&)le'5mo'e is

)resente'.

Thro&$h the rema*in$ of ireless )oer in this %onte4t3 it is the intention of the a&thor to sho the

rea'er that those thin$s lost to histor"3 !o&n' to a )ath of %om)lete o!s%&rit"3 are on%e a$ain

inno,ati,e an' &sef&l i'eas.

11



De%laration of this thesis

I %onfirm that this is m" on or*

an' the &se of all material from other

so&r%es has !een )ro)erl" an' f&ll"

a%*nole'$e'.

(hristo)her Allen T&%*er

Pra$&e3 (=e%h Re)&!li%

1> 8&ne /012

1/



Ta!le of (ontents

1 Intro'&%tion .................................................................................................................................17

1.1 #a4ell an' armor ...........................................................................................................1

1./ Po"ntin$ an' Tesla ............................................................................................................../C

/ Poer transmission an' ma$neti%5resonant mo'es in an in'&%ti,el"5%o&)le' %ir%&it ................20

/.1 A sin$le %o&)le'5mo'e ith )ro6e%tion...............................................................................>1

/.1.1 The %o&)lin$ %oeffi%ient ..............................................................................................>C

/.1./ (o&)le'5mo'e )oer...................................................................................................72

/.1.2 Effi%ien%"3 9&alit"3 an' loss.........................................................................................77

/.1.> The ma$neti% )otential.................................................................................................7@

/./ The %ir%&it mo'el.................................................................................................................C

/./.1 The )&sh5)&ll os%illator ...............................................................................................B/

/././ Sim&lation an' %onstr&%tion of the %ir%&it...................................................................B7

/./.2 E9&ations of the %ir%&it mo'el.....................................................................................BB

/.2 The )ro)a$ation mo'el........................................................................................................@1

/.2.1 The re%ei,in$ antenna..................................................................................................@C

/.2./ Ra'iate' )oer ............................................................................................................/

/.2.2 Fiel' emissions ............................................................................................................7

/.> The %ir%&it e4)eriments .......................................................................................................C

/.>.1 #eas&re' )oer at a 'istan%e ith a sin$le loo) transmitter an' re%ei,er..................C

/.>./ #eas&re' )oer at a 'istan%e ith an enhan%e' ma$neti% fiel' at the )otential ......107

/.>.2 The %ir%&it e4)eriment ith s&!ten'e' an$les in h&man )ro4imit"..........................110

/.>.> O)timi=ation of the %ir%&it.........................................................................................11

2 To %o&)le'5mo'es itho&t )ro6e%tion ....................................................................................1//

2.1 The theoreti%al mo'el ........................................................................................................1/7

12



2./ The )ro)a$ation of ma$neti% %&rrents ...............................................................................120

2.2 The %ir%&it e4)eriment.......................................................................................................122

2.> O!ser,ations of !eha,iors )resent in the %ir%&it................................................................12B

> The 'i)ole %on%e)t an' 9&ant&m ma$neti% )otential ................................................................12

>.1 Wa,eform ass&m)tions .....................................................................................................1>0

>./ Attra%tion in the 'i)ole moment ........................................................................................1>1

>.2 Ga&$e fiel's.......................................................................................................................1>>

>.> The loo) ratio in 'eterminin$ 'i)ole inte$rit"...................................................................1>C

7 Thesis %on%l&sions.....................................................................................................................1>B

7.1 (o&)le'5mo'e %on%l&sions................................................................................................1>@

7./ (ir%&it mo'el an' )ro)a$ation %on%l&sions.......................................................................1>@

7.2 Antenna at the interfa%e %on%l&sions .................................................................................170

7.> To)i%s of f&t&re resear%h ...................................................................................................17/

C A))en'i4 M Ta!les3 fi$&res3 an' %o'e .......................................................................................172

B iterat&re s&r,e"........................................................................................................................17C

1>



1 Intro'&%tion

efore the s&%%ess of #ar%oni?s 10/ ra'io transmissions to hat later !e%ame the sole means of the

ireless art3 ,ario&s alternati,e metho's ere in,esti$ate' for the trans)ort of ener$" an' information

o,er lon$ 'istan%es itho&t ires. The most ell5*non )ra%ti%al e4)eriments fi,e "ears earlier ere)erforme' !" Tesla K102L3 ho ha' 'is%&sse' ireless transmission as earl" as his 1@/ le%t&res

'eli,ere' !efore the Ameri%an Instit&te of Ele%tri%al En$ineers in Ne Yor* an' the Instit&te of

Ele%tri%al En$ineers in on'on K12L. His metho's fo%&se' on non5ra'iati,e means3 in star* %ontrast to

the or* of #ar%oni K10@L. Des)ite si$nifi%ant e4)erimental s&%%esses3 the or* as not !e ta*en &)

!" the i'er s%ientifi% %omm&nit".

While #ar%oni3 s&))orte' !" the or* of Hea,isi'e3 o&l' $o on to esta!lish the )ra%ti%al :Hert=ian;

mo'el of ra'io transmission3 Tesla?s or* o&l' la" 'ormant. At the heart of Tesla?s or* as the%onsi'eration his efforts relie' hea,il" on theories es)o&se' !" Po"ntin$ an' armor. A %enter)ie%e

of this &n'erstan'in$ is the treatment of the s)a%e &se' as the %arrier me'i&m as e4ten'e' from

#a4ell?s i'ea of non5ra'iati,e ma$neti% ,orti%es K10L3 an' of a 'ist&r!an%e %ontaine' in a str&%t&re

K/2L.

Wireless ener$" transfer is most &sef&l in near or mi'5fiel' re$ions3 ar$&a!l" in some ra'iati,e form.

Near5fiel' transfer is of the t")e e4hi!ite' in the transformer effe%t an' &s&all" o!taine' thro&$h

m&t&all"5in'&%ti,e %oils an' %a)a%iti,e effe%ts. #i'5fiel' re$ions e4hi!it !eha,ior here the falloff

is ra)i'3 "et linear. It has !een )ro)ose' that 'istan%es a%hie,a!le !" this s%heme are $enerall" ,er"

short3 limite' to a fe times the 'iameter of the %oils K21L3 losses o%%&rrin$ '&e to resisti,e an'

ra'iati,e effe%ts.

Theoreti%al an' )ra%ti%al metho's of the ireless transfer of ener$"3 'istin$&ishe' !" the !roa'%ast

fre9&en%" an' %ir%&it $eometr"3 are limite' in s%o)e an' a))li%ation. Far5fiel' )oer transmission

in,ol,es hi$her fre9&en%ies here 'istan%es a%hie,e' are m&%h $reater than the ra'i&s of the %oils.

This re9&ires either the &se of mi%roa,e si$nals or o)ti%al !eamin$. In $eneral3 s&%h te%hni9&es

re9&ire !oth line5of5si$ht an' %om)le4 tra%*in$ s"stems. The )&r)ose of the resear%h 'es%ri!e' in this

thesis is to in,esti$ate effi%ient me'i&m5ran$e )oer 'eli,er" &sin$ small5s%ale antennas itho&t

tra%*in$ s"stems here the transmitter an' re%ei,er %an lie at a 'istan%e re$ar'less of )osition an'

orientation.

An im)ortant %onsi'eration to refle%t is that Po"ntin$3 armor3 an' Tesla a$ree' on to f&n'amental

)rin%i)les 1. The Earth a%ts as a %on'&%tor %orri'or3 a %har$e' shell %a)a!le as a transmission

me'i&m3 an'3 /. " 'esi$nin$ a s&ita!le %ir%&it3 )atha"s %o&l' !e %reate' alloin$ ele%tri%al

17



%&rrents to )ass !eteen 'istant lo%ations. A more s&!tle %on%l&sion is that the %oils %o&l' !e

'esi$ne' in s&%h a manner as their ma$neti% fiel's resonate3 at least in )art3 ith the S%h&mann

resonan%e K110L. Ha&s K2L o!ser,e' an' mathemati%all" in%or)orate' some of these )ossi!ilities into

his notion of e,anes%ent a,es hi%h !e%ome more )rono&n%e' at hi$her fre9&en%ies in the near5

$i$ahert= ran$e.

This thesis )resents the %onstr&%tion of hi$h5effi%ien%" resonan%e transformers hi%h fa%ilitate sta!le

transmission of )oer an' information at &sef&l 'istan%es in to 'istin%t formsone ith a sin$le set

of %oils e4hi!itin$ a )ro6e%tion fiel'3 a se%on' ith to sets of t&ne' %ir%&its e4hi!itin$ ener$"

'istri!&tion re$ar'less of %oil orientation. In !oth %ases3 the ,elo%it" of ele%trons an' their

a%%eleration on the )rimar" %oil esta!lish the relati,isti% )ara'i$m of 'ifferential time in si$nal

'eli,er". This o&l' !e in s&))ort of the remar* !" Tesla of ,elo%it"5inhi!ition an' s&$$ests these

t")es of ireless transmission 'e,i%es are !oth ra'iati,e3 in terms of a )ro6e%tion3 an' non5ra'iati,e

here the e4%han$e of ener$" is alon$ a series of stan'in$ )lanar a,es e4istin$ as a )oer series

alon$ the 'istan%e !eteen the %oils. The series is s&))orte' !" the )oer transformation in famil"

time of ma$neti% fiel's in the 'istant %ir%&its at resonan%e. Also3 that in tan'em ith these )lanar

a,es3 lie re$ions of ra'iati,e emissions here a %oil is %o&)le' to a loa'.

This thesis h")othesi=es that the s)a%e !eteen the transmittin$ an' re%ei,in$ %oils is re$ar'e' as

%ontainin$ an ele%troma$neti% fiel' o!6e%t s&s)en'e' on a stan'in$ a,e )er)et&atin$ si$nals less

than 100 #H= "et "iel'in$ smooth )oer transmission o,er lon$5ran$e 'istan%es3 here the 'istan%e

of transmission %an !e man" times lar$er than the lar$est 'imension of !oth %oils in,ol,e' in the

transfer. In this a"3 the fiel' o!6e%t is %om)ose' literall" of a series5)ro$ression of )lane a,es

!eha,in$ as a ,irt&al transmission line %onne%te' !" the for%es 'ri,en !" the in)&t os%illator or %ir%&it

the antenna are atta%he' to. Irres)e%ti,e of an" lar$er im)li%ations or 'eri,ation aa" from ma$neti%

resonan%e3 the metho' still relies on the ell5&n'erstoo' %on%e)t of in'&%ti,e %o&)lin$ an' the

transformer effe%t.

There are ,ario&s referre' metho's of ireless ener$" transfer as %ontrastin$ e4am)les to fa%ilitate the

anal"sis of the mo'el ill&strate' in this thesis hi%h 'iffers from it in f&n'amental a"s. While

similarl" )&rs&in$ in'&%ti,e %o&)lin$ of ma$neti% fiel's !eteen resonant %oils to allo the

transmission of ele%tri%al %&rrents at a 'istan%e3 this thesis follos lo$i% of the ori$inal #a4ell

e9&ations to 8ose)h armor to Ni*ola Tesla.

A)art from the sin$&lar mo'el of the ireless transfer of )oer !" )ro6e%tion3 ireless transmission

of ener$" thro&$h si$nalin$ KL is a%%om)lishe' !" the %reation of an in'&%ti,e lin* hi%h is

ma4imi=e' at a )arti%&lar resonan%e fre9&en%" ω0 $i,en the )h"si%al %hara%teristi%s of sets of t&ne'

%oils. It is %ommonl" &n'erstoo' that ener$" transfer in the transformer mo'el 'is%o,ere' !" #i%hael

1C



Fara'a"3 )ermits the transfer of ener$" !eteen %ir%&its in %lose )ro4imit". What is not so %lear is

hat s%hemes are a))ro)riate for lon$er 'istan%e transmission &sin$ the same or similar me%hanisms.

Some ill ar$&e that the near5fiel' for these s%hemes 'o not e4ist3 rather3 the ener$" is e4)resse' in

the far5fiel' e4%l&si,el"3 'e)en'ent on the t")e of antenna hi%h t")i%all" are loo)s of a fe t&rns3

re%ei,in$ ener$" from an e4ternall" 'ri,en sin&soi'al so&r%e. Ener$" is store' in the fiel' !" the

e4%han$e !eteen the in'&%ti,e an' %a)a%iti,e %om)onents %ontaine' in the %ir%&it3 thro&$h 'am)in$

of the fre9&en%". This insists that there is a %onti$&o&s fiel' o,er the 'istan%e that %annot !e sim)l"

'i,i'e' into near an' far5fiel' 'es%ri)tions. Instea'3 it ill !e 'es%ri!e' as a mi'5fiel' re$ion.

8ames (ler* #a4ell )ro)ose' mathemati%al sol&tions to the e4)erimental o!ser,ations of #i%hael

Fara'a" an' !&ilt a frameor* aro&n' hat as to !e%ome fiel' theor" in his A D"nami%al Theor" of

the Ele%troma$neti% Fiel' in 1@C7. Within thirt" "ears3 Po"ntin$ f&rther 'e,elo)e' the mathemati%al

%onte4t of the free5s)a%e fiel' %on%e)t to in%l&'e ener$" an' its 'istri!&tions in 1@@>3 Hert= first

e4)erimental %onfirme' the e4isten%e of ele%troma$neti% a,es in 1@@C3 armor 'es%ri!e' the

ele%tron an' its !eha,ior &n'er a%%eleration !" a %&rrent in 1@73 an' Tesla3 &sin$ more refine'

'e,i%es at hi$her )otentials3 'emonstrate' this or* to the )&!li% at the IEE in on'on in 1@/ an' at

the (hi%a$o Worl'?s Fair in 1@2.

In the ele%tri%al a$e of the Vi%torian Era3 the &tilit" of )oer as fl&i' an' as forme' in man"

ima$inati,e s%enarios. For the )&r)oses of this thesis3 e4)ressions of the interfa%in$ !eteen h&mans

an' te%hnolo$" are the most interestin$. In terms of ireless )oer3 some %onsiste' of a 'ire%t

interfa%in$ at the han's here ener$" as )ro6e%te' as a li$ht so that the interfa%e ma" !e o!ser,e'.

This ima$er" is similar to that a))earin$ in i=ar'r" m"thos an' not &nli*e the stories of in'i,i'&als

)&r)orte' to %omman' the en,ironment3 %astin$ th&n'er!olts as a si$n of their )oer as earl" as the

#i''le A$es. The ima$er" ser,es to ill&strate a %onsistent nee' in h&man so%iet" 1. the h&man form

is more than the s&m of its )arts3 an'3 /. !" %le,er mani)&lation3 ma%hines %o&l' !e %onstr&%te' to

%ontrol the en,ironment.

Folloin$ the First Worl' War3 from the 1/0s to the 1C0s3 the %on%e)t of si$nalin$ as isolate'

into its on %ate$or"3 '&e )rimaril" to the 'e,elo)ment of the fiel' of ra'io en$ineerin$ folloin$

#ar%oni. It a$ain !e%ame more em!e''e' in h&man a%ti,ities in the form of ireless5)oere'

im)lants an' %ell&lar )hones3 s)anin$ a )lethora of 'e,i%es in man" more forms the a&thor %alls

ar%het")es. The term is &se' )&r)osef&ll" in the 8&n$ian sense of the or'.

The in,esti$ation %on'&%te' in this thesis ill in9&ire hat %onstit&tes free5s)a%e as a transmission

me'i&m3 $i,en the set of %on'itions im)ose' &)on it. The $oal is to "iel' a mo'el of the

1B



trans)ortation of ener$" a%ross free5s)a%e !" ma$neti% resonan%e. The %on%e)t of si$nalin$ is rele,ant

to the thesis as a more 'es%ri)ti,e !o'" of theor" re$ar'in$ ra'io5a,e )ro)a$ation an'

%om)&tational strate$ies3 as o))ose' to )&rel" ele%trome%hani%al for%es s&%h as ,olta$e an' %&rrent.

The '"nami%s of the fiel' is of $reater interest than sim)le e4)ressions of )oer ratios.

The mo'el %onstr&%te' in res)onse to the in,esti$ation is str&%t&re' as

1. mo'&lar %om)onents of the mo'el ar%hite%t&re3 ea%h of the %om)onents formin$ a )art

of the hole ar$&ment3

/. in the attem)t to 'emar%ate a %ross5%om)ati!ilit" !eteen the %ir%&it an' fiel' mo'el3

2. )ro)ose metho's to 'emonstrate the so&n'ness of the mo'el in a linear $ran&larit"3

>. )ro)ose an e4)eriment to test the metho's3

7. 'is%o,er a meas&rement %riterion of )erforman%e of a )h"si%al mo'el3 an'3C. 'is%&ss the res&lts an' e4ten'e' %om)onents learne' '&rin$ the )ro%ess.

The )resentation ill !e an en$ineerin$5%entri%3 s%ientifi% in,esti$ation here the material is

or$ani=e' a))ro)riate for the s&!6e%t matter. The thesis is 'i,i'e' into three %ate$ories %ir%&its3

fiel's3 an' a,es< ea%h %ha)ter ser,es to 'isseminate the %ontri!&tion into 'istin%t3 "et inter%onne%te'3

)arts. As the a&thor is aare of the 'i,er$en%e !eteen %ir%&it an' fiel' anal"ti%s3 the %ir%&it mo'el

ill !e teste' in an e4)eriment of a or*in$ )rotot")e an' the antenna at the interfa%e ill !e here

the a&thor ill attem)t to %on,er$e the %on%e)t. This ill not onl" )ro,i'e a %on%rete e4)ression for

the theoreti%al m&sin$s3 !&t to also ma*e the %on%e)t more tan$i!le. For s&%h an a!stra%t %on%e)t3 a

%ast of the meanin$ of the e4)ression sho&l' e4hi!it !oth a form an' a f&n%tion. This )ro%ess ill

en&merate

• a %ir%&it ith antenna emittin$ ra'iation in a $eometri%all"5'efine' )attern3

• as an am!ient free5s)a%e fiel' of a &niform str&%t&re3 an'3

• a !eha,ior 'e)i%tion in terms of #a4ell3 armor3 Po"ntin$3 an' Tesla.

In an attem)t to !e %on%ise an' 'es%ri)ti,e as )ossi!le3 the remain'er of the intro'&%tion ill !e to

tra%e the s)e%ifi% )arts of #a4ell?s ori$inal e9&ations that are 'ire%tl" rele,ant to the ar$&ments

)&rs&e' in the folloin$ se%tions. The fo%&s in the short5term is then on )ast or*s to e4)lain mo'ern

)ro!lems< therefore3 the ol' masters ill then !e $i,en 'ire%t attention K1L.

1@



1.1 #a4ell an' armor

#a4ell?s e9&ations ha,e !een i'el" 'is%&sse' in the literat&re. In their %ommon an' most 'eri,e'

form3 these e9&ations 'es%ri!e fi,e ele%troma$neti% 9&antities

• The ele%tri% fiel' E3 in Vm3

• The ma$neti% fiel' 3 in T3

• The ma$neti% fiel' intensit" H3 in Am32

• The 'ensit" of ele%tri% %har$e ρ3 in (m 3/

• The 'ensit" of ele%tri% %&rrent 83 in Am 3

hi%h ha,e the ,e%tor forms3 in the mi%ros%o)i% %ase3

∂ ∇× E + =03∂t

ρ ∇⋅ E = 3

ε0 1

1 ∂E ∇× − / =µ083

% ∂t

∇⋅ =03

an' ith Am)re?s %ir%&ital la3 in the ma%ros%o)i% %ase3

∂D ∇×H=8+ 3 /

∂t

here the %onstants are tie' !" the relationshi)3 in free5s)a%e3

/ ε0µ0% =13 2

here3

• ε0 is the ele%tri% )ermitti,it" of the ,a%&&m3 in Fm3• µ0 is the ma$neti% )ermea!ilit" of the ,a%&&m3 in Hm3 an'3

• % is the s)ee' of li$ht in a ,a%&&m3 in ms.

Not i'el" 'is%&sse' is the fa%t that there e4ists to 'istin%t ,ersions of hat is %&rrentl" re)resente'

as :#a4ell?s e9&ations;. Those !" #a4ell himself an' those !" the #a4ellians K/L Hea,isi'e3

Fit=$eral'3 orent=3 an' Hert=.

1



The mo'ern an' &!i9&ito&s form of the #a4ell e9&ations are the res&lt of a ,e%tori=ation of sets of

e9&ations !" the #a4ellians to fa%ilitate a reasona!le 'es%ri)tion of mo,in$ %har$es "et i$norin$

rotations of ener$" in the fiel'. A%%o&ntin$ for ele%troma$neti% intera%tions o&tsi'e the %onte4t of the

:lo%al; fiel' as also 'is%ar'e'. While the reasons for the tr&n%ation alloe' a sim)ler re)resentation

thhi%h s&ite' the ne!orn fiel' of ra'io en$ineerin$ in the 1 (ent&r"3 s&%h a notion 'oes not nee'

to !e hel' in the )resent era. (riti%all"3 the #a4ellians !anishe' the ,e%tor an' s%alar )otential

f&n%tions from their ,ersion. As a res&lt3 the %on%e)t of the 'is)la%ement %&rrent D 3 %entral to

&n'erstan'in$ %o&)lin$ !" ma$neti% resonan%e3 as ne$le%te' K23 >L. The reason as that sin%e

neither it nor the res&ltin$ ma$neti% fiel' ere asso%iate' ith mo,in$ %har$es3 %ontrar" to the

)osition later ar$&e' !" 8.8. Thomson in his theor" of %or)&s%les3 or hat armor %alle' ele%trons3

the" sim)l" ere not ne%essar". To this 'a"3 the 'is)la%ement %&rrent is still %ontro,ersial an' rather

&n%lear K10>3 107L3 "et a %entral )art of ele%troma$neti% theor".

For the )&r)oses of this thesis an' in terms of the e4)eriment intro'&%e' later3 !oth the #a4ellian

an' #a4ell e9&ations ere not s&ffi%ient to 'es%ri!e the antenna at its interfa%e. Rather3 !e%a&se of

the %entral role of a%%elerate' ele%trons in its o)eration3 the arran$ement of the '"nami%s nee'e' to !e

&n'erstoo' from a 'ifferent )oint of ,ie3 namel" that offere' !" 8ose)h armor in a series of

)&!li%ations from 1@2 M 1@B K73 C3 >2L. In e4aminin$ !oth theoriesthe #a4ellians on one si'e3

armor on the otherone sees the )la%es here the" share %ommonalit" an' 'ifferen%es. The )oints

of ,ie3 hoe,er3 are 'istin%t an' not 'ire%tl" %om)ati!le3 therefore !" their nat&re3 'i,er$ent.

The 'i,er$en%e essentiall" re,ol,es aro&n' Thom)son?s inter)retation of the 'es%ri)tion of the

ele%tron3 hi%h armor ha' 'es%ri!e' in K7L one si'e 'isse%tin$ the )ro!lem from the )oint of ,ie

of ra'iation on the !asis of ele%tri% a%tions3 the other3 ele%tri% a%tions on the !asis of a me%hani%al

theor" of ra'iationof ra'iant str&%t&res or fiel' o!6e%ts emitte' !" the e4%han$e of ener$" at a

'istan%e !eteen to %on'&%tors. #a4ell3 et. al as to ta*e the former hile armor too* the latter.

#a4ell?s ori$inal e4)ression of ele%troma$neti% 9&antities3 folloin$ A D"nami%al Theor" of the

Ele%troma$neti% Fiel'3 %onsists of e4)ression of to s"stems A an' here one has infl&en%e on the

other an' the effe%ts are %om)o&n'e' !" %oeffi%ients of for%e. In terms of in'&%tion3 #a4ell states

that 3 #3 an' N 'e)en' on the 'istri!&tion of the ma$neti% effe%ts '&e to to %ir%&its3 their relati,e

)osition an' s&!6e%t to ,ariation $i,en its ,elo%it".

Of the ori$inal /0 #a4ell e9&ations3 onl" si4 ill !e treate'.

A The relation !eteen ele%tri% 'is)la%ement3 tr&e %on'&%tion3 an' the total %&rrent3

%om)o&n'e' of !oth as a means of re'&%e' moment&m3

/0



'( δ == Q δ 4+ Y δ "3

't

' /

Q = (( ) 9 &+ ()9,)3 >'t

' /

Y = (()9& + (9 ,)3't

here the moment&m of the s"stem referre' to as A is & + # , an' is # & + N ,.

The relation !eteen the lines of ma$neti% for%e an' the in'&%ti,e %oeffi%ients of a %ir%&it3 as

alrea'" 'e'&%e' from the las of in'&%tion an' the resisti,e for%es R an' S3 the e9&ation of

the %&rrent in 4 in A ill !e3

' ξ = + R 4 ( 4+ # ")3 7't

an'thatthe% &rrentin " in is3

' η = + S " (# 4+ N "). C

't

( The relation !eteen the stren$th of a %&rrent an' its ma$neti% effe%ts3 a%%or'in$ to the

ele%troma$neti% s"stem of meas&rement.

D The ,al&e of the ele%tromoti,e for%e in a !o'"3 as arisin$ from the motion of the !o'" in the

fiel'3 the alteration of the fiel' itself3 an' the ,ariation of ele%tri% )otential from one )art of

the fiel' to another.

E The relation !eteen ele%tri% 'is)la%ement3 an' the ele%tromoti,e for%e hi%h )ro'&%es it3 as

the or* 'one in a &nit of time arisin$ from the ,ariations of 3 # an' N of the %on'&%tin$

%ir%&its A an' 3

1' / '# 1N' /

4 + 4"+ ". B/'t 't /'t

F The relation !eteen an' ele%tri% %&rrent3 an' the ele%tromoti,e for%e hi%h )ro'&%es it.

G The relation !eteen the amo&nt of free ele%tri% it" at an" )oint3 an' the ele%tri%

'is)la%ements in the nei$h!orhoo'.

H The relation !eteen the in%rease or 'imin&tion of free ele%tri%it" an' the ele%tri% %&rrents in

the nei$h!orhoo'.

/1



Onl" those in'i%ate' a!o,e ith e9&ations3 e.$.3 A3 3 an' E3 ill !e 'is%&sse' in $reater 'etail

in their %ontri!&tions. The others ill !e a!sor!e' into the 'is%&ssion onl" here ne%essar". It is

rele,ant no to e4amine the e4)ression of the terms as 9&antities of intrinsi% ener$" of the fiel' an'

sho the a))li%a!ilit" of this metho'3 as o))ose' to stri%tl" the #a4ellian e9&ations 1 an' /3 tie'

to &nit" in 2 for a more thoro&$h mathemati%al 'es%ri)tion of the fiel' more a$reein$ in $eometr"

an' !" ins)e%tion3 hi%h has res&lte' in this )arti%&lar ireless )oer s%heme.

In or'er to 'o this3 it ill ta*e the ass&m)tion here armor an' #a4ell a$ree on the ele%tri%

'is)la%ement A an' E. armor states re$ar'in$ this term from the )oint of ,ie of the me'i&m

The )ro!lem of the ӕther has !een first 'etermine'l" atta%*e' from the si'e of ele%tri%al

)henomena !" (ler* #a4ell in 9&ite re%ent times< his $reat memoir on a D"nami%al Theor"

of the Ele%troma$neti% Fiel' is of 'ate 1@C7. It is in fa%t onl" %om)arati,el" re%entl" that theo!ser,ation of rste'3 an' the 'is%o,eries an' 'e'&%tions of Am)re3 Fara'a"3 an' Thomson

ha' a%%&m&late' s&ffi%ient material to allo the 9&estion to !e )rofita!l" atta%*e' from this

si'e. E,en as it is3 o&r notions of hat %onstit&te ele%tri% an' ma$neti% )henomena are of the

,a$&est as %om)are' ith o&r i'eas of hat %onstit&tes ra'iation3 so that #a4ells ,ies

in,ol,e 'iffi%&lties3 not to sa" %ontra'i%tions3 an' in )la%es )resent o!sta%les hi%h are to !e

s&rmo&nte'3 not !" lo$i%al ar$&ment or an" %lear re)resentation3 !&t !" the )h"si%al int&ition

of a min' sat&rate' ith this as)e%t of the )henomena. #an" of these o!sta%les ma"3 I thin*3

!e remo,e' !" !e$innin$ at the other en'3 !" e4)lainin$ ele%tri% a%tions on the !asis of a

me%hani%al theor" of ra'iation3 instea' of ra'iation on the !asis of ele%tri% a%tions. The stron$

)oint of #a4ells theor" is the ele%tromoti,e )art3 hi%h $i,es an a%%o&nt of ele%tri%

ra'iation an' of the )henomena of ele%troma$neti% in'&%tion in fi4e' %on'&%tors< an' this is

in *ee)in$ ith the remar* 6&st ma'e. The nat&re of ele%tri% 'is)la%ement3 of ele%tri% an'

ma$neti% for%es on matter3 of hat #a4ell %alls the ele%trostati% an' the ma$neti% stress in

the me'i&m3 of ele%tro%hemi%al )henomena3 are all left o!s%&re K>3 )).>>75CL.

Whi%h ass&mes a )otential ener$" str&%t&re of im)erfe%t %onsisten%" e %all free5s)a%e. The term ill

!e restri%te' to %har$e' 'iele%tri% A theoreti%al re)resentation of a s)e%ifi% t")e of )henomenon

e4tant !eteen free5s)a%e %on'&%tors on a )lanetar" s&rfa%e. It is this str&%t&re the a&thor ar$&es is an

a%%&rate re)resentation $i,en the e4)erimental e,i'en%e.

A line of a$reement !eteen armor an' #a4ell !e$ins !oth the fo&n'ation an' the an$le of

a))roa%h to the )ro!lem5s)a%e for the form&lations in this thesis3 namel" the 'is)la%ement %&rrent.

Form this !e$innin$ )oint3 the thesis ill e4)lore those tools from !oth sets of the ar$&ments to

e4)lain the )h"si%al mo'el a ireless )oer s%heme hi%h e4hi!its ma$neti% resonan%e in the mi'5

fiel' re$ion3 that is3 at the en' of the )ro6e%tion fiel' H. At )resent3 this seems the !est tool to "iel'

//



an a%%&rate mathemati%al re)resentation. This thesis ill restri%t the 'is%&ssion of this hen it

a))roa%hes 9&ant&m fiel' theor"3 'eferrin$ the mo'el to an e4am)le of s&ffi%ientl" lar$e 'iele%tri%

me'i&m $i,en a transmission fre9&en%". Sim)l"3 mo,in$ %har$es in the e4)eriment are 'es%ri!e' !"

the armor form&la an' lo%h a,es for an irre$&lar me'i&m.

Rather than &sin$ the orent=armor theor" of retar'e' time3 famil" time K/CL is &se' to 'es%ri!e

len$th %ontra%tion $i,en for%es3 ,elo%ities3 an' a,e )ro)a$ation3 $&ise' in Poin%are?s K>/L non5

E&%li'ean a))roa%h. This is )ossi!le !e%a&se the %ase is restri%te' to that of to resonant %on'&%tors

ha,in$ a %on'&%tion5%&rrent !eteen them.

Some 9&estions to anser are the folloin$

• What for%es allo the )ro)a$ation of the free5s)a%e a,e of ele%tri% an' ma$neti%

%om)onents res&ltin$ from a%tion in )arallel %on'&%tors3 or resonant o!6e%ts

• What for%es %o&ntera%t the )ro)a$ation of the free5s)a%e a,e of ele%tri% an' ma$neti%

%om)onents res&ltin$ from a%tion in )arallel %on'&%tors3 or resonant o!6e%ts

• While it is %lear s&%h o!6e%ts e4%han$e ener$" a%ross free5s)a%e3 ho 'oes the nat&re of the

'ist&r!an%e an' the 'is)la%ement ta*in$ )la%e mirror hat is !ein$ o!ser,e' in the

e4)eriment

The si$nifi%an%e of the 'is)la%ement %&rrent lies in the in%l&sion of a strain on the me'i&m3 hi%h

%o&l' !e fille' ith %har$e' matter of either a li$ht or 'ar* )re'is)osition3 'e)en'in$ on ho it is

'efine'. Parti%&larl"3 this thesis ill 'es%ri!e the me'i&m hi%h %an e4)erien%e ele%tri% strain here a

ma$neti% fiel' is im)resse'.

#a4ell?s 'eri,ation of strain is &nrelate' to the mo'ern 'a" 'eri,ation for 'is)la%ement %&rrent in

the ,a%&&m3 hi%h is !ase' on %onsisten%" !eteen Am)res la for the ma$neti% fiel' an' the

%ontin&it" e9&ation for ele%tri% %har$e. #a4ell ima$ine' the 'is)la%ement as a sea of mole%&lar

,orti%es e4istin$ in free5s)a%e an' the 'is)la%ement a strain on the me'i&m.

Hoe,er3 hen %onsi'erin$ )olari=e' %ir%&its3 the am!i$&it" in #a4ell?s 'eri,ation %lears.

The %onse9&en%es of %larit" are %osts in the form of an e4%han$e of for%es at the !o&n'ar" hose

)erio'i%it" s&$$ests the a!ilit" to %reate artifi%ial ele%tri%al str&%t&res ith a )olari=ation.

On%e #a4ells sea of mole%&lar ,orti%es ha' !een a!an'one'3 alon$ ith it the ӕther3 an

inter)retation of 'is)la%ement %&rrent e,ol,e' that treate' free5s)a%e e4)li%itl" as a se)aration of free5

/2



1s)a%e from material me'ia3 &nli*e #a4ells ori$inal %on%e)t. This is a feat&re asso%iate' ith the

)otentials3 relati,e to free5s)a%e KB3 @L an' "iel' the %&rl ill 'es%ri!e the a%t&al linear 'is)la%ement.

If the ,e%tor ( f $ 3 3 h ) 'enotes the %&rl of the me'i&m3 or ti%e the a!sol&te rotation of the )ortion of

the me'i&m at the )oint %onsi'ere'3 an' the me'i&m is s&))ose' of %r"stalline5li*e 9&alit"3 in that the

$eometr" is 'istri!&te' an' ell5or'ere' alon$ its )rin%i)al a4es3 so that the fiel' o!6e%t Ω0 o&l' !e

of the form3

1 / / / / / /

Ω 0 = ∫(a f +! $ +% h ' )υ3 @/

here 'υ is an element of ,ol&me in%l&si,e at the !o&n'ar". It follos that for an internal

e9&ili!ri&m of the str&%t&re3 it m&st ha,e3

/ / / a f ' 4 + ! $ ' " + % h ' == −' ϕ3

a %om)lete 'ifferential an' that o,er an" !o&n'ar" en%losin$ a re$ion 'e,oi' of elasti%it" the ,al&e of

ϕ m&st !e %onstant. S&%h a !o&n'ar" is the s&rfa%e of a %on'&%tor< ϕ is the ele%tri% )otential in the

fiel' '&e to %har$es on the %on'&%tors< ( f $ 3 3 h ) is the ele%tri% 'is)la%ement in the fiel'3 %ir%&ital !"

/ / /its ,er" nat&re as a rotation3 an' (a f ! 3 $ 3 % h ) is the ele%tri% for%e 'eri,e' from ϕ. The %har$e on a

%on'&%tor is the inte$ral of ∫( f $ 3 3 h ' ) So,er a s&rfa%e S en%losin$ it3 an' %annot !e altere' e4%e)t

!" o)enin$ &) a %hannel 'e,oi' of elasti%it"3 in the me'i&m3 !eteen this %on'&%tor an' another

%ontainin$ li*e )ro)erties so that ele%tri% 'is%har$e o%%&rs onl" !" the im)ression of for%e on the

elasti% 9&alit" of the me'i&m KCL.

At the interfa%e !eteen to 'iele%tri% me'ia3 ta*en to !e %r"stalline as a!o,e3 the %on'ition %omes

o&t to !e that the tan$ential ele%tri% for%e is %ontin&o&s. When the %ir%&mstan%es are those of

e9&ili!ri&m3 an' therefore an ele%tri% )otential ma" !e intro'&%e'3 this %on'ition allos 'is%ontin&it"

in the ,al&e of the )otential in %rossin$ the interfa%e3 !&t 'eman's that the amo&nt of this

'is%ontin&it" shall !e the same all alon$ the interfa%e< these are )re%isel" the %ir%&mstan%es of the

o!ser,e' )henomena of ,oltai% )otential 'ifferen%es. The %om)onent3 normal to the interfa%e3 of the

1 The final 'isen$a$ement of :,a%&&m; from real me'ia o%%&rre' ith the international a$reement to &se the material5

&nrelate' terms ele%tri% %onstant an' ma$neti% %onstant to re)la%e the seemin$l" material5relate' terms )ermitti,it" of

,a%&&m an' )ermea!ilit" of ,a%&&m. These %onstants ha,e 'efine' not meas&re' ,al&es that refer to free s)a%e3 hi%h is

,iee' as an &nattaina!le i'eali=ation< not as a real3 o!ser,a!le me'i&m3 not e9&i,alent e,en to a 9&ant&m ,a%&&m.

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ele%tri% 'is)la%ement is of %o&rse ala"s %ontin&o&s3 from the nat&re of that ,e%tor as a fl&4. This ill

!e &se' to im)l" that %al%&lation of the fiel' at the interfa%e alon$ ith its )ro6e%tion H3 $i,en D

ill 'es%ri!e the t")e of emitte' fiel' an' its relati,e $eometr".

The ma$neti% fiel'3 %al%&late' as a f&n%tion of its intensit"3 is s&!6e%t to a )ro)ert" %alle' relati,isti%5

time3 or ,elo%it"5inhi!ition. In free5s)a%e3 it is3

1/0πµ0 8α × r

=∇× A= ∫ 2 'υ3>π 0 r

10∂D > 2

H = ε0 ⋅ π r 3∂t 2

here the ener$" re,eale' at the !o&n'ar" of the antenna an' the free5s)a%e !o&n'ar" is $i,en !" H3

,α = t − a relati,isti%5time fa%tor3 also %alle' famil" time K/CL3 'υ an element of ,ol&me. In area

%

en%lose' !" H3 εφ is free5ele%tri%it" here the fiel' is %onsi'ere' am!ient< its motion is $i,en !" the

'ire%tion of A. The a))li%ation of this h")othesis is ill&strate' !" %hara%teristi%s of store' an'

'istri!&te' ener$" )ro)erties shon in or'inar" $eometries of a sim)le loo) antenna3 the restri%tion of

s&%h )ro)erties to this arran$ement e4%l&si,el".

armor states the ener$" $i,en off !" the a%%eleration of an ele%tron %an transform its ener$" at the

interfa%e of an antenna to s&stain a ma$neti% fiel'. When a%%eleratin$ or 'e%eleratin$ the ele%tron

releases the for%e a))lie' !" a%%eleration3 the ener$" ra'iate' is $i,en !"3

/ /e a

P = 2 . 11Cπε0%

Gi,en in terms of the s%alar an' ,e%tor )otential3 a''resse' in /.1.>3

∂Α E =−∇ϕ − . 1/

∂t

E9&ation 1/ ill&strates the e4)e%tation that the ener$" in the fiel' in '&e )rimaril" to !oth )otentials

at the interfa%e of the antenna to free5s)a%e. Also3 that the transmission )ath hi%h "iel's the ener$"

m&st %ontain some inherent str&%t&re for transfer to o%%&r. This ill !e *non as the :,irt&al

transmission line; later in the thesis.

/7



1./ Po"ntin$ an' Tesla

Folloin$ the or* of #a4ell3 8.H. Po"ntin$ o&l' 'es%ri!e the ener$" in the fiel'< folloin$ the

or* of #a4ell3 Po"ntin$3 an' armor3 N. Tesla o&l' e4)erimentall"5,erif" these mathemati%al

thor*s K3 123 1>3 173 7/L. " the en' of the 1 (ent&r"3 the sta$e as set for later e4)erimenters an'

in,entors to %reate hat !e%ame *non as :the ireless art; or ra'io.

Po"ntin$ %ommente' KL on #a4ell?s theor" hi%h all&'es to later %ommentaries !" armor KCL

I am not s&re that there has hitherto !een an" 'istin%t theor" of the a" in hi%h the ener$"

'e,elo)e' in ,ario&s )arts of the %ir%&it has fo&n' its a" thither3 !&t there is3 I !elie,e3 a

)re,ailin$ an' somehat ,a$&e o)inion that is some a" it has !een %arrie' alon$ ith the

%on'&%tor !" %&rrent. Pro!a!l" #a4ell?s &se of the term 'is)la%ement? to 'es%ri!e one of the

fa%tors of the ele%tri% ener$" of the me'i&m has ten'e' to s&))ort this notion. It is ,er" 'iffi%&lt to

*ee) %learl" in min' that this 'is)la%ement? is3 as far as e are "et arrante' in 'es%ri!in$ it3

merel" somethin$ ith 'ire%tion3 hi%h has some of the )ro)erties of an a%t&al 'is)la%ement in

in%om)ressi!le fl&i's or soli's. When e learn that the 'is)la%ement? in a %on'&%tor ha,in$ a

%&rrent in it in%reases %ontin&all" ith time3 it is almost im)ossi!le to a,oi' )i%t&rin$ somethin$

mo,in$ alon$ the %on'&%tor3 an' it then seems onl" nat&ral to en'o this somethin$ ith ener$"5

%arr"in$ )oer. Of %o&rse3 it ma" t&rn o&t that there is an a%t&al 'is)la%ement alon$ the lines of

ele%tromoti,e intensit". &t it is 9&ite as li*el" that the ele%tri% 'is)la%ement? is onl" a f&n%tion

of the tr&e 'is)la%ement3 an' it is %on%ei,a!le that man" theories ma" !e forme' hi%h this is the

%ase3 hile the" ma" all a%%o&nt for the o!ser,e' fa%ts. It seems to me then that o&r &se of the

term is somehat &nfort&nate3 as s&$$estin$ to o&r min's so m&%h that is &n,erifie' or false3

hile it is so 'iffi%&lt to !ear in min' ho little it reall" means.

I ha,e therefore $i,en se,eral %ases in %onsi'era!le 'etail of the a))li%ation of the mo'e of

transfer of ener$" in %&rrent5!earin$ %ir%&its a%%or'in$ to the la $i,en a!o,e3 as I thin* it is

ne%essar" that e sho&l' reali=e thoro&$hl" that if e a%%e)t #a4ell?s theor" of ener$" resi'in$

in the me'i&m3 e m&st no lon$er %onsi'er a %&rrent as somethin$ %on,e"in$ ener$" alon$ the

%on'&%tor. A %&rrent in a %on'&%tor is rather to !e re$ar'e' as %onsistin$ essentiall" of a%on,er$en%e of ele%tri% an' ma$neti% ener$" from the me'i&m &)on the %on'&%tor an' its

transformation there into other forms. The %&rrent thro&$h a seat of so5%alle' ele%tromoti,e for%e

%onsists essentiall" of a 'i,er$en%e of ener$" from the %on'&%tor into the me'i&m. The ma$neti%

lines of for%e are relate' to the %ir%&it in the same a" thro&$ho&t3 hile the lines of ele%tri% for%e

are in o))osite 'ire%tions in the to )arts of the %ir%&itith the so5%alle' %&rrent in the

%on'&%tor3 a$ainst it in the seat of ele%tromoti,e for%e. It follos that the total E.#.I. ro&n' the

%ir%&it ith a stea'" %&rrent is =ero3 or the or* 'one in %arr"in$ a &nit of )ositi,e ele%tri%it"

ro&n' the %ir%&it ith the %&rrent is =ero. For or* is re9&ire' to mo,e it a$ainst the E.#.I. in the

/C



seat of ener$"3 this or* sen'in$ ener$" o&t into the me'i&m3 hile an e9&al amo&nt of ener$"

%omes in the rest of the %ir%&it here it is mo,in$ ith the E.#.I. This mo'e of re$ar'in$ the

relations of the ,ario&s )arts of the %ir%&it is3 I am aare3 ,er" 'ifferent from that &s&all" $i,en3

!&t it seems to me to $i,e &s a !etter a%%o&nt of the *non fa%ts.

It ma" seem at first si$ht that e o&$ht to ha,e ne e4)erimental in'i%ations of this sort of

mo,ement of ener$"3 if it reall" ta*es )la%e. We sho&l' loo* for )roofs at )oints here the ener$"

is transforme' into other mo'ifi%ations3 that is3 in %on'&%tors. No in a %on'&%tor3 hen the fiel'

is in a stea'" state3 there is no ele%tromoti,e intensit"3 an' therefore no motion an' no

transformation of ener$". The ener$" merel" streams ro&n' the o&tsi'e of the %on'&%tor3 if in

motion at all in its nei$h!orhoo'. If the fiel' is %han$in$3 ener$" %an )ass into the %on'&%tor3 as

there ma" !e tem)orar" E.#.I. set &) ithin it3 an' there ill !e a transformation. &t e alrea'"

*no the nat&re of this transformation3 for it %onstit&tes the in'&%e' %&rrent. In'ee'3 the

f&n'amental e9&ation 'es%ri!in$ the motion of ener$" is onl" a 'e'&%tion from #a4ell?s

e9&ations3 hi%h are forme' so as to e4)ress the e4)erimental fa%ts as far as "et *non. Amon$

these are the las of in'&%tion in se%on'ar" %ir%&its3 an' the" m&st therefore a$ree ith the la of

transfer. We %an har'l" ho)e3 then3 for an" f&rther )roof of the la !e"on' its a$reement ith the

e4)eriments alrea'" *non &ntil some metho' is 'is%o,ere' of testin$ hat $oes on in the

'iele%tri% in'e)en'entl" of the se%on'ar" %ir%&it.

The a!o,e3 in the o)inion of the a&thor3 is tr"in$ to %om)artmentali=e that the si=e3 sha)e3 an' s)a%e

ta*en !" an ele%tromoti,e for%e a))earin$ in free5s)a%e as a fiel'3 is holl" 'e)en'ent &)on the

a))arat&s3 or antenna3 )ro6e%tin$ it. Po"ntin$?s %omments infer that )lane a,es )ro)a$ate on a

transmission line 'ri,en !" the %&rrents an' its fiel' is 'istri!&te' alon$ a s&rfa%e3 e,en is the line an'

s&rfa%e are ,irt&al. The len$th of transmission is 'e)en'ent &)on the intensit" at the fiel' an' the

)ermitti,it" of the a,e$&i'e. In free5s)a%e at /0( an' one atmos)here3 atten&ation of the fiel' )ast

the mi'5fiel' re$ion3 'es%ri!e' in 'etail in Se%tion /3 o%%&rs at a ratio of the %oil ra'i&s to the 'istan%e3

t")i%all" si4teen times K10L. The n&m!er is a %onser,ati,e estimate. A sim)le e4)eriment shoin$ a

meas&ra!l"5)oere' state of a loo) an' re%ei,er ill&strates the )ro)a$ation of )hotons in an emission

)attern %onstit&tin$ a free5s)a%e fiel' %ontainin$ a )erio'i% str&%t&re %reate' !" the )resen%e of the

%oils K11L an' that the emitte' fiel' is 9&anti=e' K1/L.

Tesla &n'ertoo* e4)erimental ,erifi%ation of these theories as earl" as 1@/ K12L3 an' )resente' his

res&lts in 100 K102L. Tesla as to fo%&s his efforts on ireless )oer transmission3 first thro&$h

stationar" a,es then !" si$nalin$3 sen'in$ )oer to remotel" li$ht !&l!s an' t&rn motors. In a !o'"

of resear%h o,er a )erio' of "ears from 1@> to 10@ at his la!oratories in Ne Yor* an' (olora'o

S)rin$s. Tesla?s metho' as to )ro6e%t a,es !" ma$neti%all" %o&)lin$ %oils a%tin$ as an interfa%e

!eteen the os%illator %ir%&it an' free5s)a%e. It is Tesla?s metho' that is the fo&n'in$ )rin%i)le of this

/B



or*. Hoe,er3 in this thesis3 it is also interestin$ to &n'erstan' hat is $oin$ on in the 'iele%tri% of

free5s)a%e !" &sin$ :in'ire%t; metho's. The reason for this is that it is the a&thor?s o)inion that ith

the a,aila!le reso&r%es3 a 'ire%t test %an onl" !e &sef&l hen there is a frameor* 'es%ri!in$ a

'efinite me'i&m3 i.e.3 the ӕther or 'ar* matter.

Tesla?s ma%ros%o)i% ,ision as to &se stationar" a,es to !roa'%ast !eteen anti)o'es on the s&rfa%e

of the )lanet as a means to sen' )oer3 hi%h 'iffere' from the ra'iati,e mo'el %onstr&%te' !" Hert=.

In !rief3 the 'ifferen%e is s&!tle< Tesla es)o&se' the transfer of ener$" at a 'istan%e itho&t ires !"

stan'in$ a,es3 Hert= es)o&se' the transfer of ener$" in the same manner ,ia ra'iation.

Tesla?s notion of ireless )oer transfer as to &se the )lanet as !oth so&r%e an' %on'&%tor ith

toers at )ositions of an anti)o'e to e4%ite *ineti% ener$" 'is)erse' in the ionos)here hile the a,e

transmitte' alon$ the s&rfa%e of the )lanet. Re%ei,ers ere %onstr&%te' ith sim)le aerials hi%h

o&l' a!sor! these %&rrents. In terms of )ro)a$ation3 Tesla?s main em)hasis as the &se of stan'in$

a,es3 similar to those o!ser,e' in a,e$&i'es to a%%om)lish )oer transmission. He note' the

a',anta$e of these a,es is their omni)resen%e on the )lanet s&rfa%e an' lon$ a,elen$th< %o&)le'

ith the &se of the $ro&n'in$ )lane3 it is )ossi!le in this metho' to &se sin$le5ire re%ei,ers. The

)lanet %an !e %onsi'ere' as a resonant %a,it" '&e to the hi$h %on'&%ti,it" !eteen altit&'es. In

)ra%ti%al terms3 this means an an"thin$ ith an antenna %an a!sor! %&rrents here the ire len$th is

'ire%tl" )ro)ortional to the ma$nit&'e. Tesla intro'&%e' a )rofo&n' %on%e)t that to this 'a" has !een

lar$el" &ne4)lore'. If histor" ha' &nfol'e' as Tesla ha' en,isione'3 the mo'ern ele%tri%al $ri' o&l'

loo* li*e that re)resente' in Fi$.1.1.

Fi$.1.1. Tesla?s ,ision of ireless )oer.

After e4)erimental trials at War'en%l"ffe from a )erio' of 1@ M 1023 some s)ora'i% or* as late as

1113 Tesla?s or* ent &nfinishe'. The toer as 'emolishe' '&rin$ the First Worl' War.

/@



Afterar's3 Tesla a!an'one' his stan'in$5a,e h")othesis an' e4%l&si,el" e4)erimente' ith

ra'iati,e3 or ra'iation5!ase' emission3 mo'els K1>L. At the time3 )re,ailin$ attention as toar'

#ar%oni?s !roa'%astin$ s"stem an' not the transmission of )oer. After the )&!li%ation of the or*

on the Ya$i antenna K17L in 1/C an' the 'e,elo)ment of shorta,e K1CL3 ireless )oer o&l' fa'e

from ,ie.

Fort" "ears )asse' !efore interest in ireless )oer res&rfa%e'. In the 1C0s3 short5a,elen$th

transmission e4)eriments K1BL an' the mo'elin$ of )rotot")i%al forms shoe' )ositi,e res&lts K1@L. A

rethin*in$ of in'&%ti,e transfer !e$an to s&rfa%e in sol,in$ )ro!lems asso%iate' ith s&!%&taneo&s

im)lants K1L. While man" 'i,er$ent t")es of ireless )oer transmission ha,e res&lte'3 es)e%iall"

sin%e /00>3 this thesis ill fo%&s on a ra'iation5!ase' emission of )oer in the form of )lane a,es

from loo) antenna o,er 'istan%es of one meter o)eratin$ at fre9&en%ies &) to 7 #H=.

This thesis ill e4)lore to 'istin%t e4)ressions of ireless )oer That of a sin$le )rimar" loo)

transmitterre%ei,er s"stem3 an' that of a )rimar"se%on'ar" loo) )air transmitterre%ei,er s"stem.

The former ill e4hi!it the a&thor?s notion of the mi'5fiel' re$ion of the ra'iation mo'el of a sin$le

%o&)le'5mo'e3 hile latter ill e4hi!it Tesla?s stationar" or stan'in$ a,es an' t&na!le a,e %a,it"

of to %o&)le'5mo'es.

A !rief o&tline of hat is to !e )resente' in this thesis is the folloin$.

(ha)ter / ill 'es%ri!e the theoreti%al an' mathemati%al mo'el of a %ir%&lar loo) antenna %onsistin$

of to )rimar" %oils. It ill !e shon that the %ir%&it has a s)e%ifi% !eha,ior $i,en the nat&re of its

arran$ement. Se%tion /.1 ill 'is%&ss a %o&)le'5mo'e ith a )ro6e%tion an' the for%es that are

%onti$&o&s. Se%tion /./ ill 'is%&ss the !eha,ior as also em!e''e' in the &ni9&e %ir%&it hi%h 'ri,es

the %o&)le'5mo'e. Se%tion /.2 ill 'is%&ss the )ro)a$ation mo'el an' the e9&ations hi%h are

%a)a!le of 'es%ri!in$ it. Se%tion /.> ill 'is%&ss se,eral sets of e4)eriments3 rele,ant to a sin$le

%o&)le'5mo'e3 in ,ario&s forms. In li$ht of hat is 'es%ri!e' in the intro'&%tion re$ar'in$ h&mans

an' ireless )oer K/0L3 !iome'i%al im)lants ill !e one s&%h e4)eriment. This se%tion ill also

'is%&ss o)timi=ation of the mo'el.

(ha)ter 2 ill 'is%&ss to %o&)le'5mo'es in terms of an e4istin$ %ir%&it3 namel"3 that of Ni*ola Tesla

in his U.S. Patent C>737>C of 100.

(ha)ter > ill 'is%&ss a no,el mo'el of the 'i)ole3 $i,en the theor" an' e4)erimental e,i'en%e

)resente' in the )re,io&s %ha)ters.

/



(ha)ter 7 ill )resent %on%l&sions an' s&$$estions for f&t&re resear%h.

An im)ortant %ontri!&tion of this thesis is3 %ontrastin$l" to the &se of the #a4ell5Hea,isi'e ,ersion

of #a4ell?s e9&ations in the literat&re s&r,e"3 &sin$ the 1@C7 ,ersions to !e$in the formation of the

theoreti%al frameor* from here the ar$&ments an' anal"ti%s are 'eri,e' from it. Tra%in$ thro&$h

three3 sele%te' !e%a&se of their a))ro)riateness t o t he s&!6e%t The relation !eteen ele%tri%

'is)la%ement3 %on'&%tion an' %&rrent3 !eteen the lines of ma$neti% for%e an' in'&%ti,e for%es3 an'

!eteen ele%tri% 'is)la%ement an' ele%tromoti,e for%e. It ill !e shon in the thesis the no,el

%ontri!&tion is frame' in the e4)ression of these #a4ell e9&ations.

(ontrastin$l" to other literat&re3 this thesis ill 'is%&ss the $eometr" an' am!ien%e of the ma$neti%

thfiel' in terms of 8ose)h armor %on%e)tion of o!6e%ts. O&tsi'e of some or*s in the earl" /0

(ent&r"3 it seems most ha,e i$nore' hat is otherise a ,er" )oerf&l set of tools to 9&antif" the

for%es res)onsi!le for ma$neti% resonant %o&)lin$. This thesis ill %hoose a set of e9&ations an'

for%e5re)resentations that ill !e &se' to e4)lain h" the )henomenon is at it a))ears $i,en

e4)erimental e,i'en%e.

/ Poer transmission an' ma$neti%5resonant mo'es in an in'&%ti,el"5%o&)le' %ir%&it

The theor" an' those s)e%ifi% ori$inal #a4ell e9&ations from (ha)ter 1 ill no !e a))lie' to a

)h"si%al mo'el. A %ir%&it %onsists of to 'istin$&isha!le )arts a transmitter an' a re%ei,er. The

transmitter is %om)rise' of a re$&late' am)lifi%ation an' timin$ %om)onents %onne%te' to a loo)

antenna an' a %a)a%itor. The am)lifier is %onne%te' to a so&r%e of 'ire%t %&rrent '% )oer. The

re%ei,er is %om)rise' of a loo) antenna3 a %a)a%itor3 an' a loa' s&%h as a lam) or motor. The re%ei,er

is )la%e' at a 'istan%e aa" from the transmitter. The %ir%&it ill transmit ra'io5fre9&en%" ele%tri%al

ener$" into free5s)a%e the form of )hotons e4%han$e' !eteen to or more %o&)le' resonators3

'e)en'in$ ho the" are %haine' to$ether K10CL. At %ertain 'istan%es an' antenna orientations3 or*

)erforme' !" the ener$" transfer %an !e o!ser,e' at a re%ei,er !" an ill&minate' lam) or t&rnin$

motor. A )air of %ir%&lar loo)s of a s&ffi%ientl" similar $eometr" set at a 'istan%e form a %lose'

in'&%tion %ir%&it ill&strate' in Fi$./.1. These loo)s form a resonant %ir%&it3 a a,e$&i'e3 an' a ,irt&al

transmission5line sim&ltaneo&sl".

20



Fi$./.1 The )rotot")e mo'el for the transmission of )oer !" ma$neti% resonan%e.

On e4amination of Fi$./.13 %&rrent flos from so&r%e to os%illator to %oil loo) i 3 a%%eleratin$

ele%trons at a rate $i,en loo) %&r,at&re an' ratio of loo) %ir%&mferen%e to a,elen$th< tor9&e is $i,en

as a f&n%tion of the %&rrent I0. In this mo'el3 %&rrent at s&ffi%ient le,els3 that is3 as a )ro)ert" of the

%om)onents &se' to %onstr&%t the os%illator3 there is eno&$h tor9&e so that )hotons are )ro6e%te' into

free5s)a%e in the form of ra'iation at a %hara%teristi% an$&lar fre9&en%" ω0 .

The 'es%ri)tion of the n&m!er of )hotons emitte' !" a $i,en arran$ement is shon !" the ener$"3 γ03

of the )hotons emitte' !" the antenna is $i,en !"3

γ ω 0= ℏ 03

12

an' the n&m!er of )hotons )er se%on'3 γt 3 is $i,en !" the )resen%e of %ir%&ital )oer 80 3

80

γt = . 1>γ0

At a 'istan%e3 %oil loo) 6 a!sor!s the )hotons %on,ertin$ them !a%* into ele%tron %&rrents ,ia a

,olta$e $a). The %&rrent is then %o&)le' to )erform or* )oerin$ a loa'. The )ro%ess in,ol,es

1. The transfer of ener$" !eteen in)&t %&rrent %ontaine' in the os%illator3 a%%elerate'

$i,en a am)lit&'e im)resse' as a ma$nit&'e on the %oil ire of the transmission

antenna3

/. the transit of )hotons3 γ 0 3 a%ross free5s)a%e3 an'3

2. the transfer of ener$" !eteen free5s)a%e an' the %onstit&tion of ele%tri%al %&rrents in

the %oil ire of the re%ei,in$ antenna.

21



The flo )attern in time o%%&rs at lo%ations here the ener$" is not %ontaine' in the fiel'3 (Eθ 3 Hθ )3

here the ma$neti% fl&4 Φ → # 0. At these )oints3 the fiel' manifests as a )ro6e%tion en%om)assin$

an area H hi%h e4ten's at a len$th from the antenna3 i 3 to a s&rfa%e in free5s)a%e3 Sθ 3 at its

!o&n'ar". The !o&n'ar" is here there is e9&ili!ri&m in the for%e e4erte' !" the 'is)la%ement3 D3

an' that e4erte' !" the me'i&m. H 'enotes the mi'5fiel' re$ion. At Sθ 3 the far fiel' !e$ins. The

)arameters ε03 µ03 an' σ0 are in,ariant ith )osition e4%e)t at Sθ K/1L.

Fi$././. Areas here the fiel' )&shes in the me'i&m an' fall5off ra'iation from a loo) antenna.

#For the %onser,ation of ele%tri% %har$e3 a 'ifferen%e in the fl&4 at the s&rfa%e normal3 n3 to the

antenna m&st im)l" a %han$e in the total ele%tri% %har$e 9 %ontaine' ithin the ,ol&me 'enote' !" Sθ

∂9as . Poer in the form of )hotons )ro6e%te' toar' the !o&n'ar" "iel's the 'is)la%ement of ener$"

∂t

from the loo) in %om!ination ith its %ir%&it. (on%&rrin$ ith armor3 the !o&n'ar" has elasti%it" an'

is s&s%e)ti!le to strain. In a''ition3 %on%&rrin$ ith the ori$inal #a4ell3 the !o&n'ar" is the )oint of

'is)ersion. For lineari=ation3 the fiel' is %onsi'ere' homo$eno&s a%ross the s)a%e ith %har$es

%onsi'ere' e,enl" 'istri!&te' e4)erien%in$ a &niform motion. The antenna therefore %reates a

'ist&r!an%e as a 'eformation of the me'i&m. At the !o&n'ar" of the 'is)la%ement3 the rotational

)ro)ert" of Sθ or instantaneo&s tor9&e3 τ03 is $i,en !"3

τ0 =∇×D3 17

!e%a&se of Fit=$eral'?s )ro)osition of the i'entit" of ele%troma$neti% ra'iation as an e4%han$e of

for%e K//L3

∇× F X ∇ ∇⋅ F 5 δ F3 1C

2/



hen3

ρ = 03 an' 8 = 0. 1B

E9&ation 11 ill "iel' a a,e e9&ation in the form of3

/1 ∂ ψ

=∆ψ . 1@/ /

% ∂t

The fiel' is therefore s&!6e%t to &niform e4ternal for%es 'e)en'ent &)on %on'itions of tem)erat&re an'

)ress&re an' has a s&s%e)ti!ilit"3 hi%h is e4)e%te'. Therefore3 ener$ies an' fre9&en%ies limite' !" a

$i,en $eometr" share a %ommon state of e9&ili!ri&m. The effe%t of the e9&ili!ri&m is analo$o&s to

inflatin$ a !alloon. The tension at the s&rfa%e hol'in$ the ener$" insi'e is $i,en !" the shearin$ for%es

!eteen the fiel' at its !or'er an' the me'i&m. Ga&ss?s theorem transforms the s&rfa%e into a%han$in$ fl&4 of the ele%tri% fiel' thro&$h the %lose' s&rfa%e forme' !" the ,ol&me of o)en

s&rfa%es. #ore )re%isel"3 an" :missin$; fl&4 of the %&rrent 'ensit" 8 is a!sor!e' from the fl&4 of the

∂E,e%tor ε0 or at the )otential ,e%tor A.

∂t

In terms of ma$neti% fl&43 an ener$i=e' %ir%&lar loo) emits a ma$neti% fiel' of a $i,en intensit"

an' $eometr" a%%or'in$ to the iot5Sa,art an' Ne&mann )rin%i)les. oth )ro)ose that the

%ontri!&tion of ea%h )ie%e of the ire %ontaine' in the loo) ,aries in,ersel" as thes9&areof the

'istan%e. O,er the entire len$th of the ire3 ea%h %ontri!&tion a''s &) to a total fiel' that ,aries

in,ersel" as the 'istan%e from the loo). The %ontri!&tion to the ma$netostati% fiel' at the ori$in of a

%&rrent element3δ I3 at )osition r is3

µ 0r ×δ I δ = 2 3 1

>πr

$i,en !" the relationshi) of its ra'ial len$th an' the infinitesimal of the ma$neti% fiel'3 shon in

Fi$./.2.

22



Fi$./.2. Ra" relationshi)s !eteen the ma$neti% fiel'3 ra'ial ,e%tor3 an' infinitesimal %&rrent.

The 9&antit"3 δ I3 is the %&rrent m&lti)lie' !" the len$th of tra,el res&ltin$ from inte$ratin$ the

%&rrent 'ensit"3 83 alon$ the s&rfa%e o,er a small element of ,ol&me @. In )arti%&lar3 for a thin ire

hose len$th element δ s has a total %&rrent I0 tra,ersin$ it in the 'ire%tion of δs3 δ δ I = I s3

an'

ne$ati,el" otherise. The )ro)osition hol's for mono5'ire%tional %&rrents onl". In the !i5'ire%tional

%ase3 an os%illator %reates a rotational for%e thro&$h π )hases3 the 'ire%tion of the %&rrent %han$es

ti%e in a %"%le thro&$ho&t all of δ s. The 'is)la%ement at the mole%&lar le,el of the %on'&%tor is

$i,en !" #a4ell in >.

The mo'el of ireless )oer transfer3 ill&strate' in Fi$./.13 %onsists of to 'istin%t ,ies of

o)eration3 that of a s"stem of %o&)le' mo'es an' that of )ro6e%tion. The first ,ie is e4)resse' !" the

&tili=ation of a )air of %oils3 one a%tin$ as a transmitter or )rimar" %oil3 the other as a re%ei,er or

se%on'ar" %oil. The %om!ination of these to3 or more K10CL3 antenna is a %o&)le'5mo'e K/>L. The

se%on' ,ie as a transmission antenna )ro6e%tin$ a for%e.

Some of the )ertinent 9&estions in this %ase ill in,ol,e reali=in$ the $eometr" of the s%heme3 the

manner of ener$" stora$e3 an' its !o&n'ar" %on'itions. To a''ress the latter 9&estion re9&ires a *e"

ass&m)tion that the me'i&m is not 'e,oi' of neither )arti%les nor ener$".

In the mo'el3 the ma$neti% fiel' is ass&me' to ha,e a sim)le s"mmetr" as in the %ase of free5s)a%e

fiel's. Am)re?s la "iel's the ,al&e of the ma$neti% fiel' thro&$ho&t the s)a%e itho&t %om)le4

inte$rations3 s&%h as those &se' hen sol,in$ for m&t&al in'&%tan%e. Ga&ss? theorem ill 'es%ri!e the

sha)e of the ma$netostati% fiel' in %ases ith s)heri%al s"mmetries3 $enerall" a))lie' here. For the

s)e%ial %ase of ma$neti% in'&%tion on the a4is of a %ir%&lar %&rrent loo)3 se,eral ass&m)tions are

'is%&sse' in the remain'er of this se%tion of i an' 6.

2>



The arran$ement ill&strate' in Fi$./.1 is analo$o&s to a )air of Helmholt= %oils. Hoe,er3 ea%h half of

the %ir%&it is not %onne%te' to the other !" ire3 rather3 &tili=e the 'iele%tri% of free5s)a%e as a means

of %onne%tion.

The first ,ie3 of a %o&)le'5mo'e3 is manifest as a fl&45lin*a$e !eteen s"mmetri%al antennas. It is

!" 'esi$n 'e)en'ent &)on the fiel'. Geometr" of the %o&)lin$ mo'e is shon in Fi$./.>.

Fi$./.>. Geometr" of the %o&)lin$5mo'e ma$neti% fiel'.

For a %ir%&lar %&rrent loo)3 all ra'ial %ontri!&tions ill %an%el o&t< the res&ltin$ ma$neti% in'&%tion

is oriente' alon$ the a4is here X =. e%a&se of the s"mmetr" of the trian$les %onsistin$ the

interior of the loo)3 shon in Fi$./.>3 the %ontri!&tion δ is r times 13 as3= '

r µ0I δ= = / δs. /0

'>π'

All the elements3 δs3 a'' &) to the %ir%&mferen%e3

/r µ0I 1 µ0I r

= = /π r = 3 /1/ 2

'>π ' / '

hile the fiel' at the %enter of the loo) is3

µ0I = 3 ///r

/r

at )ea* ,al&e here = = / .'

What is essential to &n'erstan' ne4t is ho the me'i&m ill !e treate' l"in$ !eteen a set of loo)s in

a %o&)le'5mo'e. To a%%om)lish the treatment3 #a4ell?s ori$inal e9&ation 7 ill !e %on,erte' into

a 'ifferential e9&ation 'es%ri!in$ a mo'e of )ro)a$ation in a a,e$&i'e.

27



A a,e$&i'e3 for the )&r)oses here3 is %onsi'ere' as an infinite s"stem of transmission lines an' the

a,es tra,elin$ alon$ these lines mo,e aa" alon$ the ,e%tor normal from the so&r%e at a %onstant

fre9&en%"3 as ill&strate' in Fi$././. The a,efront resem!les an infinite series of )arallel )lanes of

%onstant )ea*5to5)ea* am)lit&'e normal to the )hase ,elo%it" ,e%tor.

Analo$o&s to a )air of Helmholt= %oils3 %onsi'er to loo)s shon in Fi$./.1 sharin$ the same

hori=ontal a4is3 or in a4ial ali$nment. et their res)e%ti,e latit&'es !e +a an' −a3 ma$neti%

in'&%tion is alon$ the a4is of a4ial ali$nment at 'istan%e =.

1 / −2/ / −2// / /

= µ0Ir r + ( ) a − = + r + ( ) a + = . /2 /

The se%on' 'eri,ati,e of /2 ith res)e%t to = at = = 0 3 is

/ / / / / −B/

!! ( ) 0 = 2µ0Ir (>a − r )(r + a ) . />

1The ,al&e a= r is the lar$est ,al&e here the ma$neti% in'&%tion has a sin$le ma4im&m ,al&e

/

alon$ the a4is. When ,al&es of a are lar$e3 !! is )ositi,e at the %enter here = = 0 in'i%atin$ a

minim&m ,al&e )resent. The %onfi$&ration here the se)aration !eteen the to loo)s is e9&al to

their ra'i&s3 /a= r3 "iel's a ma$neti% fiel'3 hi%h is almost &niform3 near the %enter of the loo)3 as3

2/ > µ0I µ0I

= ≈ .0B177> . /7 7 r r

The s%alar )otential of the ma$neti% fiel' at the %enter of the loo)3 an' here it is e9&i'istant !eteen

to loo)s are e4am)les of a %&rrent5free re$ion. It ta*es the form of a m&lti,al&e' f&n%tion hose

$ra'ient is the ma$netostati% in'&%tion. For a sim)l"5%onne%te' re$ion &n'er %o&)le'5mo'es3 s&%h a

)otential is ell5'efine' &) to a &niform a''iti,e %onstant. Otherise3 an am!i$&it" arises hene,er

the re$ion %ontains loo)s hi%h are interlo%*e' ith loo)s of o&tsi'e %&rrent. In that %ase a

%ontin&o&s )otential %an onl" !e 'efine' ith a mo'&lo o)eration for a %ertain n&m!er of 'is%rete

9&antities3 ea%h of hi%h %orres)on's to one )ie%e of the o&tsi'e %&rrent.

The metho' 'es%ri!e' a!o,e fits to 1 that in stati% 'istri!&tions3 all %&rrents m&st %ir%&late in

%lose' loo)s here ∇×8 =0. E9&ation 1 %annot e4)ress ith '"nami% 'istri!&tions here lo%al

ele%tri% %har$es ma" ,ar" a%%or'in$ to the in!o&n' fl&4 of %&rrent. A more so)histi%ate' metho' is

re9&ire'.

2C



" the -el,in5Sto*es form&la3 the %ir%&lation of a ,e%tor aro&n' an oriente' loo) is e9&al to the fl&4

of its rotational element3 or %&rl3 thro&$h an" smooth oriente' s&rfa%e !or'ere' !" that loo) "iel's

Am)re?s la in inte$ral form3

µ µ 0I " 0 8 ⋅'S3 /C∫∫S

hi%h is the same as the $eometr" on the )lane3

⋅'r. /B∫∂S

The sim)lest an' most f&n'amental 'ire%t a))li%ation of Am)re?s la is to first form&late &n'er the

iot5Sa,art an' Ne&mann )rin%i)les3 then e4ten' the flo of ma$neti% in'&%tion analo$o&s to astrai$ht ire or a a,e$&i'e *ee)in$ in min' the intensit" is in,ersel" )ro)ortional to the 'istan%e.

Fi$./.7. Pro6e%tion of ener$" !" a %&rrent im)resse' on the loo).

(onsi'er a %ir%&lar loo) of ra'i&s r hose a4is is a strai$ht ire %arr"in$ an instantaneo&s %&rrent

I0. The ma$neti% in'&%tion at the loo) is tan$ential< the )ro6e%tion of the motion is %onstant as

lon$ as I t ( ) is. The ma$neti% %ir%&lation an' Am)re?s la then $i,es3

/π r = µ0I 3 /@

or e9&i,alentl"3

µ0I = . /

/π r

#a4ell amen'e' the stati% la of Am)re into the folloin$ $enerali=ation that hol's in all %ases

in%l&'in$ %han$in$ %har$e 'istri!&tions K/2L3

2B



1 ∂E ∇× − / =µ083 20

% ∂t

hi%h a$rees ith the ,e%tori=e' ,ersion of Fara'a"?s la3

∂ ∇× E + =0. 21

∂t

The %&rl of 3 $i,en 203 is the %hara%teristi% of a rotatin$ %om)onent of . The %&rl is ill&strate' in

Fi$./.C.

Fi$./.C. Fiel' lines folloin$ an a4is of 'is)la%ement.

" the a%tion of the %&rl3 the )ro)a$ation ta*es the form of the 'is)la%ement D. The 'is)la%ement has

a %om)onent3 %alle' 'ire%tional %&rrent3 or $ro&) ,elo%it" in terms of a,es3 alon$ the ma$neti% fl&4

∂lines. The intensit" of D linearl" 'e%reases ith 'istan%e. %o&)les to )erform or* here %an

∂t

flo in a se%on' %oil. The meas&rement of the intensit" D ill !e 'etermine' !" the ,elo%it" of fiel'

flo. For e4am)le3 let υ !e the ,e%tor fiel' of a fl&i' flo. At a $i,en )oint in the flo3 a small

heel ith !la'es hose a4is is oriente' in the 'ire%tion of ∇× υ at an" )oint is )la%e' in the flo in

or'er to meas&re ra'ial intensit". Then the an$&lar ,elo%it" ω1 of the heel?s rotation from the a%tion

∇× υof the %&rrent ill ma4imi=e3 here its ,al&e ill !e e9&al to . Ass&min$ υ has ar!itrar"

/

%oor'inates P 4 " ( 3 3 =)3 J(4 " = 3 3 )3 an' R 4 " ( 3 3 =)3 then ∇× υ has the %oor'inates3

∂ ∂ R J ∂ ∂ P R ∂ ∂ J P − 3 − 3 − . 2/

∂ ∂ " = ∂ ∂ = 4 ∂ ∂ 4 "

The ar%s %ast !" /7 form !o&n'aries in three5'imensions of ∇× υ at the interfa%e ith the antenna

an' fee'!a%* from the re%ei,in$ o!6e%t3 ill&strate' in Fi$./.B.

2@



Fi$./.B. Fl&4 lin*a$e !eteen to in'&%ti,el"5%o&)le' %oils.

The ma$neti% s%alar )otential3 ϕ03 for %reate' !" the loo)s ill&strate' in Fi$./.B is )ro)ortional to

the %&rrents3 Ii 6 3 3 in the loo) an' to the soli' an$le5s)a%e3 Ω03 s&!ten'e' !" the si'e of the loo)

fa%in$ the o!ser,er3 as3

H =−∇ 3 ϕ0

µ I 220ϕ =− Ω0.

>π

An am!i$&it" arises in the soli' an$le !e%a&se there is a mo'&lo o)eration of > < π the an$le is an

a))ro4imation that !e%omes loses )re%ision ea%h time the ,al&e is ta*en. The si$n %on,ention is s&%h

that the si'e of a small loo) is seen at a soli' an$le3 hi%h e4%ee's a m&lti)le of >π !" a small

)ositi,e 9&antit".

The se%on' ,ie3 of )ro6e%tion3 is manifest as a a,efront from a sin$le transmission antenna. The

len$th an' intensit" of the )ro6e%tion3 H3 is '&e to !oth the )ro)erties of the antenna an' the )oer

a))lie' !" the %ir%&it. #o'ifi%ation of Fi$./.> to in%l&'e )ro6e%tion is shon in Fi$./.@.

2



Fi$./.@. Fiel' $eometr" e4ten'in$ aa" from the )ro6e%tion.

The )ro6e%tion is the formation of the a,e$&i'e3 %arr"in$ ener$" a%ross free5s)a%e 'is)la%in$ it as it

e4hi!its motion aa" from the antenna in !oth the )ositi,e an' ne$ati,e 'ire%tions of =. For

%on,enien%e3 it is 'i,i'e' &) into am)lit&'e =ones3 Aλ 3 here meas&rements are ta*en to o!tain the

$eometr" of the emission !" ma))in$ its intensit". The )ro6e%tion of the loo) is ill&strate' in Fi$./..

Fi$./.. Pro6e%tion fiel' formin$ the mi'5fiel' re$ion !" intensit".

A %om)osite of !oth the %o&)le'5mo'e an' )ro6e%tion is shon in Fi$./.10.

>0



Fi$./.10. Pro6e%tion of fiel' an' its $eometr" !eteen to in'&%ti,el"5%o&)le' %oils.

The PDE 'es%ri)tion of the s"stem ill !e elli)ti%3 here its s)atial sol&tions ta*e a h")er!oli% form

%onsistin$ of sinh an' %osh f&n%tions. J&anti=ation of the fiel' %om)onents an' manifestin$ them in

an e4)eriment ill !e the aims of the folloin$ se%tions in this 'o%&ment.

/.1 A sin$le %o&)le'5mo'e ith )ro6e%tion

The relationshi) !eteen the loo)s an' the amo&nt of ener$" trans)orte' !eteen them is 'es%ri!e'

!" ho the" are %o&)le'. (o&)lin$ ill onl" refer to the relationshi) !eteen loo)s an' not to free5

s)a%e. Ener$" a%ross free5s)a%e an' its im)e'an%e is not !" the )henomenon of %o&)lin$3 rather3 it is

the %onse9&en%e of %o&)lin$ !eteen the o!6e%t hi%h allos the transit of ener$". #a$neti% loo)s of

a relati,el" small si=e an' n&m!er of t&rns e4hi!it a ma$neti% fiel' 'ensit" s&ffi%ient to !e an

e4%ellent mo'el for e4)erimentation.

A mo'e 'es%ri!es the ener$" in )atterns of ele%tri%3 ma$neti%3 an' for%e $enerate' !" the loo)s. The

ma$neti% fiel' intensit" H3 is relate' to the ma$neti=ation3 #3 of the s&rro&n'in$ s)a%e on the

antenna !" s&s%e)ti!ilit"3 χυ. For the 'i)ole moment in the %on'&%tor3

# = χυH. 2>

The !o&n'ar" %on'ition ill !e the s&rfa%e Sθ . Analo$o&s to the %ase of the thin %&rrent sheet the

ass&m)tion is ta*en that the thi%*ness of the loo) str&%t&re is ne$li$i!le. Therefore3 Am)re?s la ill

"iel' the instantaneo&s )oer 'ensit" on an :a))ro4imatel"5%lose'; %onto&r3

2 / 8H = H A r i + Hiℓi 3 27

λ

>1



If the me'i&m is ass&me' homo$eno&s an' isotro)i% an' the s&rfa%e im)e'an%e of the !o&n'ar"

linearl" a))roa%hes =ero3 it is %onsi'ere' a )rimiti,e a,e$&i'e. An" 'is%ontin&it" in a a,e$&i'e

fa%ilitates a %o&)lin$ !eteen some or all 'esira!le mo'es K/>L in ,ar"in$ ma$nit&'es.

To ele%tri%al o!6e%ts3 %onsistin$ of in'&%ti,e an' %a)a%iti,e %om)onents of relati,e ,al&e to ea%h

other3 form a %lose' in'&%tion %ir%&it. The lin*in$ is '&e to attra%ti,e resonators in a a,e$&i'e of

%hara%teristi% im)e'an%e is im!&e'3 )&rel" !" int&ition3 as sharin$ )ro)erties ith transmission5line

resonator ith small losses K/>3 /73 /CL. A sin$le %o&)le'5mo'e %an !e i'entifie' !" )la%in$ a

re%ei,in$ loo) at an am)lit&'e =one Aλ.

i is a %ir%&lar loo) ith a ,olta$e $a) no $reater than the si=e of its ra'i&s. e%a&se of its lo

ra'iation resistan%e an' hi$h rea%tan%e3 im)en'en%e mat%hin$ is 'iffi%&lt to a sin&soi'al os%illator.

Therefore3 i is %onne%te' in )arallel to a %a)a%itor )oere' !" a sit%hin$ am)lifier. (om)ara!l"3

the loo) antenna 6 %an also !e %onne%te' in )arallel to a %a)a%itor hose ,al&e is no $reater than

that of i.

Wa,e im)e'an%e is the ratio of the trans,erse %om)onents of the ele%tri% E0 an' ma$neti% H0 fiel's.

For a trans,erse5ele%tri%5ma$neti% TE# )lane a,e tra,elin$ thro&$h a nearl"5homo$eneo&s

me'i&m at a $i,en tem)erat&re an' )ress&re3 a,e im)e'an%e is e,er"here e9&al to the intrinsi%

im)e'an%e of the me'i&m. The a,e im)e'an%e3 in terms of fiel's is

−E E0 ( 4)

Z = 3 Z0 = − . 2CH H0 ( ) 4

Disassem!lin$ the %om)onents of 2C into finer )arameters3 the free5s)a%e im)e'an%e is

6ω0µ0

Z0 = 3 2Bσ ω 0+ 6 0ε 0

In 2B3 6 is an ima$inar" &nit3 an' ω0 is the an$&lar fre9&en%". In the %ase of a 'iele%tri% here the

%on'&%ti,it" is =ero3 the e9&ation re'&%es to3

ε0

Z0 = . 2@µ0

(al%&late' at /0( an' one atmos)here3

>/



Z0 = 2BC.B2Ω 2

=11./πΩ 3

isolate' from the %on'&%ti,it" of free5s)a%e for the ,al&e %onsi'ere' in the )resent 'efinition of the

&nit of the Am)re. For an" a,e$&i'e in the form of a hollo metal t&!e of %on'&%ti,it" σ03 s&%h as

%ir%&lar $&i'e3 the a,e im)e'an%e of a tra,elin$ a,e is 'e)en'ent on the resonan%e fre9&en%" f 03

!&t is ass&me' &niform thro&$ho&t the $&i'e. For trans,erse ele%tri% TE mo'es of )ro)a$ation the

a,e im)e'an%e is

Z0

Ze = 3 >0/

f %1−

f 0

here f % is the %&t5off fre9&en%" of the mo'e. For trans,erse ma$neti% T# mo'es of )ro)a$ation

this ,al&e is

/

f % Zm = Z0 1− . >1

f 0

A!o,e the %&t5off ( f 0 > f % )3 the im)e'an%e is real resisti,e an' the a,e %arries ener$". elo %&t5

off the im)e'an%e is ima$inar" rea%ti,e an' the a,e is e,anes%ent3 the latter shon to !e an

effe%ti,e s%heme K21L. These e4)ressions ne$le%t the effe%t of resisti,e loss in the alls of the

a,e$&i'e3 or at the s&rfa%e !o&n'ar" Sθ . The )resen%e of a 'iele%tri% res&ltin$ from atmos)heri%

)henomena %haoti%all" shifts in im)e'an%e thro&$h π mo'if"in$ f %.

To *ee) sele%ti,it" as narro as )ossi!le onl" the emission of a sin$le fre9&en%" is 'is%&sse'. The

%hoi%e %omes at a %ost im)e'an%e is not a f&n%tion of the )ro)erties the %ir%&it3 rather3 %om!ine' ith

that of free5s)a%e. The ra'iation )attern e4hi!its a to)olo$i%al s&rfa%e from an ener$i=e' os%illator.

The fiel' is 9&antifia!le at %ertain len$ths from the antenna !&t has fra%tal 'is)ersion $eometr"3

similar to tas* of %al%&latin$ the len$th of a shoreline3 so hat %onstit&tes the o!ser,a!le !o&n'ar" is

relati,e to the te%hni%al so)histi%ation of the la!orator" of the e4)erimenter. Generall"3 the o!ser,a!le

free5s)a%e ele%troma$neti% fiel' has its !o&n'ar" at an e'$e here the ener$" store' at the hori=on

,anishes3 as Φ → # 0. De)en'in$ on the $eometr" of the %oils an' the intensit" of ener$" store'3 the

!o&n'ar" shifts.

>2



Some a))li%ations here the a,e$&i'e or transmission5line %ontainin$ more than one t")e of

'iele%tri% me'i&m o%%&rs3 s&%h as %om)le4 mi%rostri) K/BL3 in terms of the !o&n'ar" %on'itions3 the

a,e im)e'an%e ill in $eneral ,ar" o,er the %ross5se%tion of the line3 as a a,e$&i'e transient K/@L.

A similar notion3 in,esti$ate' !" Aharono,5ohm K/L3 s&$$ests s&%h a s%heme at loer tem)erat&res

re,eals interestin$ %on%l&sions a!o&t the nat&re of Sθ . The 'i)ole3 as the s&))ortin$ )henomenon of a

%o&)le'5mo'e in !oth its real an' ima$inar" ,al&e' term3 intera%ts ith the s)a%e s&rro&n'in$ the

loo) %a&sin$ the 'is)la%ement D.

Referrin$ to Fi$./.13 %onsi'er a s"stem of resonators %onsistin$ of a )air of ma$neti% 'i)ole loo)

antenna i3 6 se)arate' !" a 'istan%e3 hi%h %ontain a sin$le %o&)le'5mo'e K/>L. i3 6 are set on a

or*5)lane as to %ontain a )erio'i% str&%t&re. To )erform an anal"sis3 a softare )a%*a$e %alle'

(omsol #&lti)h"si%s is &se' to sim&late the ele%troma$neti% %hara%teristi%s of the s%heme. The or*5

)lane %onsists of to %ir%&lar loo)s of a material ith %on'&%ti,it" σm an' ra'i&s r i. These loo)s are

ali$ne' in s&%h a a" so that the %enter of their ra'ii shares a %ommon a4is or an a4ial ali$nment. The

FE# la"o&t is ill&strate' in Fi$./.11.

Fi$./.11. FE# mo'el or*s)a%e la"o&t.

The in,esti$ation is intereste' in %onsi'erin$ small antenna si=es ith a hi$h effi%ien%". In the mo'el3

the loo) si=e is the most %riti%al )ro)ert". If %oil loo) is o&n' at a s)e%ifi% ratio of %ir%&mferen%e to

a,elen$th3 it is e9&i,alent to an antenna %onne%te' to an ele%tri%all"5short transmission line. To

ma4imi=e the emission of ra'iation3 the ma$neti% loo) sho&l' !e at a ratio of its %ir%&mferen%e to its

9&arter a,elen$th. S&%h a ma$neti% loo) ill !e shon to ha,e the i'eal ratio3 03 in terms of

ma4im&m transmission effi%ien%"3 is hen this ratio falls on an e4)onential s%ale3

>>



/π r i= 3

0./7λ4

0 =10 3 >/

4=−13 −/3 −2...n.

This allos the mo'el to allo loo)s ith a ra'i&s of 2 %entimeters that are set a)art at a 'istan%e of C

%entimeters. The loo) s&rfa%e lies on the 43 " )lane hile = !eteen the loo)s forms the a4ial line.

In %onsi'eration of the effi%ien%" of ener$" transfer3 the re%ei,in$ antenna %olle%ts ele%troma$neti%

)hotons3 the amo&nt $i,en the area its s&rfa%e o%%&)ies an' the infl&en%e3 in the form of fee'!a%*3 it

im)oses on the s)a%e s&rro&n'in$ it K20L. With the intro'&%tion of a ,olta$e $a)3 ele%tri%al ener$" is

e4tra%te' in the form of )oer an' or* is )erforme' !" this metho'.

In %onsi'eration of the s&rro&n'in$ me'i&m of free5s)a%e3 the arran$ement of the antenna in s)a%e is

6 t ωa )erio'i% lo%all"5a))ro4imate' a,e$&i'e ith im)li%it time5'e)en'en%e e < &n%o&)le' s)atial

− 6 = β1 − 6β/='e)en'en%es are e an' e . In a %o&)le'5mo'e3 the s)atial e9&ations K/>L are3

' /π =ai =− 6βiai − 6/κi6 % os a 6 3

'= l >2

' /π =a 6 =− 6β 6a 6 − 6/κ 6i % os ai3

'= l

$i,en their %o&)lin$ ith a ea* time 'e)en'en%e3

'ai

= 6 a ω i i + 6 a κi6 6 3't

>>'a 6

= 6ω 6a 6 + 6κ 6iai3't

relati,e to the %o&)le' state of the resonators3

#i6

κ κ i6 = = 6i 3 >7i 6

as a ea*l"5%o&)le' s"stem3 here κi6 $ 0.01. This is in %ontrast to a :stron$l"5%o&)le'; s%heme

K21L. The attention on ea*5%o&)lin$ is referrin$ to the ea* time 'e)en'en%e >> for a s"stem

ta*in$ an ass&m)tion lo %o&)lin$ %oeffi%ient κi6 affe%ts effi%ien%" ηi6 ea*l" an' )ro)erties of the

%ir%&it affe%t ηi6 stron$l".

>7



/.1.1 The %o&)lin$ %oeffi%ient

In %onsi'eration of the $eometr" of the s%heme )resente' in the )re,io&s se%tions3 the ,aria!les to !e

sol,e' are the %o&)lin$ %oeffi%ient κi6 3 the m&t&al in'&%tan%e #i6 3 the 9&alit" fa%tors Ji an' J 6

e4)resse' !" %oils i an' 6 3 !" the %ontri!&tions of their fiel's E0 an' H0.

To &n'erstan' the effe%ti,eness of the s%heme an' the or* 'one at a 'istan%e3 the ener$"5stora$e

a!ilit" of the s"stem is 'etermine' !" the %o&)lin$ %oeffi%ient in transfer mo'e. The intensit" of flo

of the ener$" a%ross the 'istan%e is 'etermine' !" κi6 !eteen %oils i an' 6 at a 'istan%e '. When

a %&rrent I0 is a))lie' to %oil i 3 the )oer transfer flos from %oils i to 6 3 an' 6 to i.

The %o&)lin$ %oeffi%ient in >7 is sol,e' for a series of %oils of %hara%teristi% si=e an' $eometr". Thelin*a$e !eteen the series of ma$neti% resonant %oils is 'efine' at ea%h %o&)lin$ here #i6 is the

m&t&al in'&%tan%e3 i an' 6 are the self5in'&%tan%es of the %oils. The in'&%tan%e %an !e %al%&late'

sin%e its si=e an' $eometr" are alrea'" s)e%ifie'3 !" the a))ro4imation of the in'&%tan%e of a %ir%&lar

loo) K2/3 22L

/ @r / @r 6 i >C

i ≈ µ µ 0 r ni r i ln − / + Y 3 6 ≈ µ µ 0 r n 6 r 6 ln − / + Y 3a a i 6

Where r r i3 6 is the loo) ra'i&s3 a a i3 6 the ire ra'i&s3 n n i3 6 the n&m!er of t&rns3 an' Y the

flo

%onstant of the s*in5effe%t of the emitte' ra'iation3 $i,en the resonan%e fre9&en%"3 hi%h is the

a))ro4imate' to !e the same at the s&rfa%e of !oth %oils. The resonan%e fre9&en%" of the s"stem3

1 f 0 = ( ) f i + f 6 3 >B

/

is %om)rise' of the resonan%e fre9&en%" f f i3 3 6 . . . f n of ea%h of the %oils3

1 f i 6 3 = 3 >@

/π ( i 6 3 i6

an' the fre9&en%" error !eteen ea%h half of the %ir%&it3

f e= f i − f 6 . >

>C



A %onsi'eration of the 'esi$n is not onl" to *ee) the antenna small !&t also to *ee) the resonan%e

fre9&en%" f 0 lo eno&$h as to not to !e 'eleterio&s to h&man tiss&e. St&'ies ha,e shon that a

fre9&en%" s)e%tr&m !eteen 100 *H= an' > #H= is s&ita!le K2>L. The theoreti%al s)e%ifi%ation of the

antenna is %al%&late' here3

n = 23

r = 20mm3

a = 0.>mm3 −1/ −1 70

ε0 = @.@7⋅ 10 F ⋅ m 3−B 51

µ0 = >π ⋅ 10 H ⋅ m 3

µr = 0.>3

Y = 0.

The material )ro)erties of the %oil is for an ins&late' %o))er ire o&n' in %on%entri% loo)s so the

total loo) s&rfa%e is %om)rise' of a n&m!er of t&rns. E9&al to a it= in'in$3 ea%h t&rn ins&late' from

the other re'&%es the %ontri!&tion of the s*in effe%t Y.

The in'in$ has le$s as to allo the loo) to !e mo&nte' into a %ir%&it !oar' an' )ro6e%te' alon$ a

)lane linear to the earth. The a''itional in'&%tan%e of the le$s is ta*en into %onsi'eration. A (.NET

)ro$ram %alle' (ir%&it %al%&lator3 !&ilt s)e%ifi%all" for )erformin$ an' ,is&ali=in$ the relationshi)s

of the ,al&es of ireless )oer !" ma$neti% resonan%e3 %al%&lates the theoreti%al s)e%ifi%ation of the

mo'el. Gi,en >@3 a %a)a%itan%e of 100nF is %hosen< the %al%&lation se9&en%e is shon in Fi$./.1/.

The ta!le of ,al&es for the mo'el loo)s i 6 3 an' resonan%e fre9&en%ies f i 6 3 is liste' in Ta!le /.1.

Ta!le /.1. oo) %oils theoreti%al s)e%ifi%ation.

>B



In terms of the )rotot")e3 a fre9&en%" of a))ro4imatel" >00 *H= is e4)e%te' to !e the resonan%e

fre9&en%" for the )h"si%al loo) antenna. #&t&al in'&%tan%e is ne4t to !e sol,e'.

Fi$./.1/. (al%&lator ,al&es for a s"stem resonan%e fre9&en%"3 an' ! in'&%tan%e of the transmission loo).

#&t&al in'&%tan%e

The m&t&al in'&%tan%e !eteen to %oils is 'e)en'ent )&rel" on the $eometr" of the s%heme. An

ener$i=e' loo) affe%ts a%tion at a 'istan%e. Its attra%tion3 or m&t&al in'&%tan%e3 is $i,en !" the

Ne&mann form&la K27L3

µ s s i ⋅ 6 0 71#i6 = %%∫∫ .

>π ( ( 'i 6

The infinitesimal )ositions si an' s 6 in 71 are 'eri,e' &n'er the ass&m)tion that the ma$neti% fiel'

is e,enl" 'istri!&te'3 more %on%entrate' toar' the %enter of the loo) than in other )la%es3 an' in a

mostl" linear 'istri!&tion. In this form3 the loo) ire is %onsi'ere' to !e %ontaine' of %oa4ial %ir%&lar

filaments K2/L.

>@



Fi$./.12. (oil )ro)erties 'es%ri!in$ 71.

#&t&al in'&%tan%e is e4%l&si,el" the )ro)ert" of %o&)lin$ !eteen %oils i an' 6. The ma$neti%

fl&4 inter%e)te' !" one %oil is analo$o&s to a &nit of %&rrent floin$ in the other K2CL3 the m&t&alth th

in'&%tan%e of the i an' 6 elements an' the inte$ration o,er the s&rfa%e Sθ sho the m&t&al

in'&%tan%e is also3

& n ⋅ 'S∫∫

S 6

#i6 = 3 7/I

i

here is the fl&4 inter%e)te' !" the re%ei,er %oil3 I

is the %&rrent )assin$ thro&$h the first %oil3i

n = (n n n 43 "3 =) is the &nit s&rfa%e normal ,e%tor3 an' 'S is an infinitesimal area element of the lo%al

a))earan%e of Sθ as S 6. It is )ossi!le to transform the s&rfa%e inte$ral in 7/ into a sim)ler line

inte$ral !" &sin$ the ma$neti% ,e%tor )otential3

=∇× A3 72

to$ether ith Sto*es theorem3 hi%h states that a s&rfa%e inte$ral of the %&rl of a fiel' e9&als the

%lose' line inte$ral o,er the rim of the s&rfa%e3

( ∇× A)⋅n 'S A t ⋅ 'l∫∫ %∫ 6 Sθ

#i6 = = 3 7>I I

i i

>



here t = (t t t 43 3 " =) is the &nit tan$ent ,e%tor of the %&r,e Sθ an' 'l is an infinitesimal line

element.

r i 'The m&t&al in'&%tan%e3 then3 'e)en's &)on the )arameters 3 3 an' its )ro)ortionalit" to the ra'ii3

r 6 r 6

as3

/ / α

#i6 = µ0 r ir 6 1− -( ) α − E( ) α 3 77α /

here - (α ) an' E(α ) are elli)ti% inte$rals an' α is o!taine' !"3

>rr i 6

α = / . 7C/( ) r i + + r 6 '

Usin$ K2/L to fin' the ,al&e here3

r i #i6 = α r ir 6 = α r 6 ( ) µH 3 7B

r 6

Where α is 'etermine' as 'e)en'in$ on the ,aria!le3

/ /r i '

1− + / r r / 6 6

& = / 3 7@ /

r i '1+ + / r r 6 6

from 72. A %al%&lation as )erforme' in (ir%&it %al%&lator for the m&t&al in'&%tan%e #i6 !ase' on

7B an' 7@ from the s)e%ifi%ations liste' in Ta!le /.13

/& = 0.73

7#i6 = 0.00>/77@µH3

here the res&lt is &se' to %al%&late the 'is%rete ,al&e of

κi6 = 0.00/2>C. C0

70



The %o&)lin$ stren$th !eteen resonators is 'es%ri!e' !" the %o&)lin$ %oeffi%ient. esi'es hat has

alrea'" !een 'es%ri!e' as metho's for %al%&latin$ the %o&)lin$ stren$th3 another metho' K2BL is to

%onsi'er the !an'i'th of the transmission3 as

/( f i − f 6 ) !an'i'th = 3 C1

f i − f 6

Where f i an' f 6 are the s)lit resonan%e fre9&en%ies of the resonators3 hi%h are most effe%ti,e &sin$

a narro !an'i'th. In the sense of the arran$ement ill&strate' in Fi$./.273 there is shon a

se)aration of the resonators into their ele%tri% an' ma$neti% %om)onents. To maintain %ohesion in the

!eam of ener$" e4%han$e' !eteen i an' 6 3 insi$hts into the h" the %o&)lin$ is so si$nifi%ant to

ma$neti% resonant )oer transmission are im)ortant to as%ertain. A no,el a))roa%h K2@L %alle' the

)ert&r!ation metho'3 stri,es to $lean insi$hts a!o&t those %om)onents hi%h most affe%t the o,erall

%o&)lin$. The folloin$ e9&ation3 mo'ifie'3

+ [ −µ H0 ⋅H0'ν − εE0⋅H0'ν∫ ∫

i i

κi6 = / 3 C/ε E0 'ν∫

6

−im)lies that the moti,ation to &n'erstan'in$ the %o&)lin$ %oeffi%ient is here E0 ma*es the stron$est

+%ontri!&tion toar' the intensit" of H0 an' hat %ir%&ital %om)onents o&l' s&))ort the transfer. It

also s&$$ests that in%reasin$ the amo&nt of ma$neti% fiel' at the loo) o&l' res&lt in a''itional

im)ro,ements. In its most !asi% sense3 anserin$ these s&!tle 9&estions ta*es time an' e4)erien%e

an' the" are a''resse' as moti,ations in this thesis< the" ill irre,o%a!l" remain a most im)ortant

to)i% of f&t&re resear%h. Hoe,er3 the a&thor %an i'entif" that it is the %ontri!&tion of )otential

ma$neti% ,e%tor 'e%om)osition on the interfa%e3 hi%h is of $reatest im)ortan%e to $ras) if

&n'erstan'in$ of the s%heme is to a',an%e.

The 'es%ri)tion of the interfa%e is that of the loo) is in sit& ith free5s)a%e. The transmission loo) an'

the :amo&nt of ma$neti% lines; %rossin$ the re%ei,in$ loo) is a t")i%al 'es%ri)tion of a ra'iate'

ma$neti% fiel' an' its effe%t on 'istant o!6e%ts. An e4am)le is shon in Fi$./.1>.

71



Fi$./.1>. #a$neti% %o&)lin$ of lo5κ mo'es of 'istant resonators in s)a%e an' time.

As note' in K2L3 the 'istan%e 'oes not affe%t mo'al %o&)lin$3 rather3 s)atial e4tensions in the =

'ire%tion relate more %losel" to the trans)ort of ele%tri%al )oer '&e m&t&al in'&%tan%e3 e4)resse' in

the E an' H fiel's. In s&%h effe%ts3 !roa'%ast )oer is s&!6e%t to atten&ation o,er the im)e'an%e of

free5s)a%e if the )otential is ass&me' to %ontri!&te minimall" to the fiel's. Ne,ertheless3 it is

im)ortant to *ee) in min' that the !o&n'ar" 'oes not 'es%ri!e an' is in'e)en'ent of the sin&soi' as

ell as its a,e f&n%tion. Therefore3 the )resen%e of the antenna an' the )resen%e of )h"si%al free5

s)a%e %onstit&te the entiret" of the si$nal. If one %onsi'ers the ar$&ment that the a,eform is &ni9&e

from its %on'&%torin the manner of in'&%tion 'es%ri!e' !" Fara'a" as ell as the ori$inal e9&ations

of #a4ellthe 'ist&r!an%e an' the 'is)la%ement o)en the 'oor to 9&estions of hat in free5s)a%e is

alloin$ the e9&ili!ri&m of the for%es of the %&rrents 'ri,in$ ra'iation an' its infl&en%e !eteen

antenna at 'istan%es $reater than those 'es%ri!e' for the s*in5effe%t.

To mo'el the relati,e )osition an' motion of the ener$" in terms of the e4%han$e of )hotons3 the )ath

an' intensit" is 'es%ri!e' !" the 'ensit" of ma$neti% lines3 ill&strate' in Fi$./.1>.

The antenna3 !" the nat&re of its si=e3 $eometr"3 an' s"mmetr" %o&)les the ma$neti% %om)onent of

the ener$" on the %oil5ire s&rfa%e to an interfa%e K20L. Un'er %loser e4amination an' %onsistent iththe %ir%&it ar%hite%t&re3 the )resen%e of the im)resse' a,e is holl" '&e to the )ro)erties of the

in'&%tor i3 6 ith the )otential of free5s)a%e A $i,en its im)e'an%e Z0. The ma$neti% loo)3 at a

ratio to its 9&arter a,elen$th3 %ontains the effe%ts of a f&ll a% %"%le im)resse' on it !" the in%i'ent

a,e in terms of transmission or re%e)tion.

7/



Fi$./.17. Theoreti%al %o&)lin$ %oeffi%ient o,er a 'istan%e $i,en >7.

The 'ata in Fi$./.17 is 'eri,e' from the %al%&lation of %o&)lin$3 shon in >7 in %onte4t of antenna

as a f&n%tion of transmitte' )oer3 in the manner of free5s)a%e inter)retation of >@3 7/3 7>. The

res&lt is the ,al&e of κi6 atten&ate' !" free5s)a%e. As is shon3 the sol&tion of this t")e of 'es%ri)tion

s&$$ests that a 'ifferent anal"ti%al tool is re9&ire' to sho the me%hanisms res)onsi!le for ener$"

transmission. As the )&r)ose of the arran$ement is to trans)ort ener$" effi%ientl"3 the n&meri%al

sol&tion of the %ir%&it )oer is inten'e' to meas&re the 9&antit" of ener$" store' !eteen the

resonators an' the 'issi)ate' )oer at the re%ei,er a,aila!le to a loa'. Therefore3 %o&)le'5mo'es in

s)a%e 'es%ri!e the 'istri!&tion of an ener$i=e' fiel' hile %o&)le'5mo'es in time "iel' the rate at

hi%h ener$" is trans)orte'.

/.1./ (o&)le'5mo'e )oer

The ener$" store' an' 'issi)ate' !eteen to %o&)le' o!6e%ts 'es%ri!es a )oer 'istri!&tion s"stem.

While the )&rs&it of a ire' analo$ is 'esira!le to some a))li%ations K10L3 for the )&r)oses here3 it is

more effe%ti,e to %onsi'er the %o&)lin$ in terms of ener$" trans)ort !" the ,irt&al a,e$&i'e. It isar$&e' !" the a&thor that this )arti%&lar metho'olo$" ill "iel' $reater insi$hts not onl" a!o&t the

mo'el &n'er in,esti$ation3 !&t also a!o&t the en,ironment here the mo'el is o)eratin$.

Ener$" transfer is an o!ser,a!le res&lt in the %onte4t of for%es mo,in$ %har$es. In the 'ri,en %ase3 one

%an 9&antif" for%es '&e to am)lifi%ation. In more s&!tle %ases3 the 'istri!&tion of for%es on %har$es is

'es%ri!e' !" the orent=5Fit=$eral' transformation K>03 >1L &se' in s)e%ial relati,it" an' the so&r%e of

#in*os*i li$ht %ones an' other inno,ati,e s)atial $eometries ith mo,in$ for%es K>/3 >2L.

72



The armor for%e transfers ener$" !eteen the fiel' an' the %har$es. The )oer F , ⋅ of the for%e

e9&ation is 9E ⋅ ,3 th&s the )oer re%ei,e' !" the ele%tri% %har$es )er &nit of ,ol&me is E 8 ⋅ . (har$e

%arriers %on,ert )oer at the re%ei,er from the lo%al ele%troma$neti% fiel' into other forms of ener$"3

in%l&'in$ the *ineti% ener$" of )arti%les. (on,ersel"3 E 8 ⋅ %an !e ne$ati,e3 in hi%h %ase there is a

transfer of ener$" from the %har$e %arriers to the fiel'. One )ro%ess is as a time re,ersal of the

other. In this3 it is )referre' to retain the famil"5time sol&tion of #a4ells e9&ations to the motion of

the so&r%es an' %han$es in the fiel' that are reti%ent.

The $eometr" of a %ir%&lar loo) is s&%h that the ra'iation ta*es the )attern of a h")er!oli% f&n%tion3 a

transmitter of a )rimar" loo) in an e4%han$e of ener$" ith a )rimar" loo) of a re%ei,er in a

theoreti%al %onte4t is shon in Fi$./.B.

In terms of the total ma$neti% intensit"3 H3 %om)rise' of the )ro6e%tion H3 an' the fl&4 'ensit" as

0= µ0H0 emitte' !eteen antenna. Sol,in$ ∇× υi6 for all 'esire' meas&rement )oints of H0 ill

in'i%ate the amo&nt of ener$" the fiel' store' at the $i,en )oint. To %al%&late this ,al&e relia!l"3 it

ill re9&ire a fitte' sol&tion to Po"ntin$?s Theorem.

It is 'esira!le to e4)ress the ener$" as a )ro)ert" of the ele%tri% fiel' an' %&rrent 'ensit" an' %om)are

this ith the ,al&e of ∇× υi6 for an ar!itraril"5$i,en H0. The 9&antit" E 8 ⋅ is e4)resse' in terms of

the ele%troma$neti% fiel's !" ta*in$ the 'ot )ro'&%t of the %&rrent 'ensit" into −E on !oth si'es of the

Am)re5#a4ell e9&ation3

∂H ∂E − ⋅ E ( ∇ × H)+ µ0H + ε0E⋅ = − ⋅ E 8. C2

∂t ∂t

The i'entit"3

E⋅ ∇ ( ×H) =H⋅ ∇ ( ×E) − ∇ ( ⋅E)×H3 C>

an' Fara'a"?s la "iel'3

∂H0

− ⋅ E ( ∇ × H)= H0 ⋅ + ( ) ∇ ⋅ E × H. C7∂t

The )oint here this o%%&rs is at a )otential ,al&e for e9&ili!ri&m of the ener$" e4%han$e !eteen H0

an' the %&rrent 'ensit" at an ar!itrar" )oint3

7>



/ /∂ ε0E \H

( ) ∇⋅ E ] H+ =−E 8 ⋅ 3 CC∂t /

"iel's Po"ntin$?s theorem aro&n' that )oint.

SXE]H3 CB

shoin$ the 'ensit" of )oer is emitte' is a relati,e )ro)ert" of the antenna at the interfa%e K10BL. At

an" )oint in the ,ol&me3 $i,en the )ossi!le lo%ations shon in Fi$./.B3 the time5a,era$e' ener$"

store' is3

1 / / /

ε0 (E + % H ). C@/

E9&ation C@3 a 'eri,ation of the Po"ntin$ theorem3 ill sho that a ,ariation of ener$" in a $i,en

,ol&me %omes from )oer that is either 'eli,ere' 'ire%tl" !" so&r%es or ra'iate' a%ross the s&rfa%e

as fl&4 floin$ alon$ a %a,it" or in this )arti%&lar mo'el3 a ,irt&al a,e$&i'e. The im)li%ation is that

a )erfe%t ele%tri%all"5short loo) antenna at a fa,ora!le ratio o&l' ha,e effi%ien%" of ra'iation at

ma4im&m3 in'i%atin$ a relate' amo&nt of )oer transfer an' ener$" stora$e in the ,ol&me.

The )h"si%al mo'el is for the %ase of %ir%&lar loo)s in /.2.1. These %a,ities3 hile homo$eno&s in

free s)a%e3 maintain a total internal refle%tion K@>L. To s&stain the state of e9&ili!ri&m of the fiel'3 an

o))osin$ for%e is alloin$ the s&s)ension. This im)lies a for%e5'ri,in$ element3 a))earin$ in the

%ir%&it as the a%%eleration ,al&e of ele%trons floin$ in the %on'&%tor of the antenna.

The o!ser,a!le !eha,ior in the )h"si%al mo'el ill !e 'is%&sse' firstl" from the )oint5of5,ie of the

ma$neti% )otential Α as =∇× A3 then to the transition to %ir%&ital mathemati%al )ro)erties.

/.1.2 Effi%ien%"3 9&alit"3 an' loss

To &n'erstan' the !eha,ior of %o&)le'5mo'e %hara%teristi%s3 'es%ri!e' in /.1.13 these are &se' in

real5orl' )ro!lems. The orthiness of the mo'el is in terms of its o,erall effi%ien%"3 that is3 !" a

%om)arison !eteen the ener$" store' at the in)&t to the ener$" 'issi)ate' in%l&'in$ those in losses.

The effi%ien%" of the mo'el is meas&re' !" the e9&ation K>>L3

/κi6 JJi 6

ηi6 = / 3 C1+ κi6 JJi 6

77



Where ηi6 is 'e)en'ent on the %o&)lin$ %oeffi%ient κi6 3 !&t also !" the 9&alities J i an' J 6 of the

%oils i an' 6. The 9&alit" of a %oil K>7L is o!taine'3

ω ω0 60 i

Ji = 3 J 6 = 3 B0Ri R 6

as a )ro)ert" of the ener$" store' in the an$&lar fre9&en%" ω0 the in'&%tan%e 3 an' the free5s)a%e

th th)ro6e%tion H0 ith the 'issi)ation !" the resistan%e R of the i an' 6 %oils. The 9&alit" fa%tor is

then 'etermine' !"3

1 1 i 1 i 1 1 = + = + B1

Ji Ri (i R)i (i JR Ji i

th th%om)rise' of the in'&%tan%e 3 resistan%e R3 an' %a)a%itan%e (3 an' of the i an' 6 %oil in the

os%illator an' re%ei,er %ir%&its. The s%heme is 'efine' so that the s"stem is )erformin$ or*3 the

&nloa'e' 9&alit" J is ne$le%te'. The 9&alit" fa%tors then !e%ome3i

1 i ( 6

Ji = 3 J 6 = R 6 3 B/Ri (i 6

for !oth %ases of a series an' )arallel %onfi$&ration in an i'eal %ir%&it arran$ement. The %oils arearran$e' ithin their res)e%ti,e %ir%&its as to ma4imi=e the amo&nt of ener$" store' !eteen the

in'&%tan%es an' the %a)a%itan%es. The ma4im&m ener$" store' in i 6 3 an' (i 6 3 is3

/ /&i = I i 03 &(i = (Vi 0 3

B2/ /

& 6 = I 6 R3 &( 6 = ( 6VR 3

hi%h is 'is%&sse' in $reater 'etail in /.2.1. The loo) %an !e treate' as a stan'ar' os%illator in

%o&)le' mo'e 'efinin$ >2>> as

i

a t i ( ) = I t 0 ( ) . B>/

It is ell5&n'erstoo' that 9&alit" fa%tor has a 'ire%t %onse9&en%e &)on the effi%ien%" of an ele%tri%al

transmission %ir%&it.

7C



The remainin$ %om)onent rele,ant to the )ro)erties of %o&)le'5mo'es is the losses in the s"stem

res&ltin$ from the materials &se' in the %onstr&%tion of the loo)s. osses in the s"stem are o!ser,e' as

'issi)ation of the ener$" store'. For the losses '&e to ohmi% resistan%e K7@L3

ni n 6µω ℓi 6 30 0

Ri 6 3 = '' 3 B7/σ m i= = 1 6 1>π r i36

Where σm is the %on'&%ti,it" of the material3 n the n&m!er of t&rns3 ℓ the ire len$th of the %oil3

th than' r i 6 3 the ire ra'i&s of the i an' 6 %oils. The %al%&lation of the 9&alit" fa%tor an' the losses for

the s"stem are shon in Ta!le /./.

Ta!le /./. Ener$"3 9&alit"3 an' loss s)e%ifi%ation.

(ol&mn m in Ta!le /./ is )ro,i'e' for the o!6e%t im)resse' on free5s)a%e as a f&n%tion of the

in'&%tan%e an' resisti,it" of the me'i&m. A,era$e'3 the )otential effi%ien%" is ,er" hi$h for the

%o&)le'5mo'e3 altho&$h it ill !e shon in e4)eriments that it is transforme' solel" on the )ro)erties

of the material %onsistin$ the antenna. In lo%al s)a%e3 the effi%ien%" is o!ser,e' to !e ,er" hi$h !&t it

refers to the a!ilit" of the loo) to transform the a%%eleration of ele%trons into )hotons3 not in%l&'in$

an" of those 9&ant&m %onsi'erations here. Effi%ien%" of the transmission o,er 'istan%e is the linear

'e%a" of )oer o,er 1 . This is 'is%&sse' in terms of ra'iate' )oer of the antenna in /.2./.r

Ener$" stora$e in Ta!le /./ is shon )er %"%le.

With these !aseline %riteria for the e4)e%te' )ro)erties of the antenna as a f&n%tion of the materials

an' $eometr" of the ire esta!lishe'3 the ne4t %ha)ter ill a''ress the %ir%&it mo'el that %an s&stain

the os%illations. The %riteria is the means !" hi%h 'ifferentiatin$ mo'els ill !e 6&'$e' to !e &se' in

%ommon)la%e a))li%ation3 s&%h as that in /.>. The s"stem &n'er %onsi'eration is 'esi$ne' ith

%om)a%tness in min' an' ea%h %om)onent %hosen for an e4)e%te' le,el of )erforman%e. The

)&r)osef&l %hoi%e of &sin$ the t")e of os%illator 'es%ri!e' herein is not !" a%%i'ent< it is %hosen for its

sim)li%it" an' si=e.

7B



/.1.> The ma$neti% )otential

The ma$neti% ,e%tor 'e%om)osition is &sef&l to 'es%ri!e the nat&re of the fiel' o!6e%t !eteen the

resonators as a res&lt of the for%es that sha)es the o!6e%t. The fiel' mo'el is %entere' aro&n' the

ass&m)tion that the $eometr" %an !e &n'erstoo' from to)olo$i%al ar$&ments of the for%es in there$ion.

The mo'el &ses the ma$neti% ,e%tor )otential A = ( A43 A"3 A=) to %onstr&%t a finite sol&tion !" %astin$

a )ro6e%tion At as the 'ot )ro'&%t of the tan$ent ,e%tor an' the ma$neti% )otential3 as3

At= At ⋅ t 4+ At ⋅ t "+ At ⋅ t =. BC4 " =

BC is the res)onse of the )otential to the 'ist&r!an%e of the a,e. Its ma$nit&'e is relati,e to the

ener$" store' in the fiel' o!6e%t. While not a))earin$ insi'e the fiel'3 it o%%&rs at the limit of the fiel'

at its !o&n'ar" an' is s"mmetri%al alon$ the s)a%e !eteen %o&)le'5mo'e o!6e%ts. The fiel' has

'is)ersion 'e)en'ent on the $eometr" of the antenna< the transmitter an' re%ei,er )la%e' at

se)arations in all three (artesian 'ire%tions. The &nit" of the )otential in %onte4t ith the fiel' o!6e%t

is o!ser,a!le &n'er meas&rement of an &n)oere' antenna an' is e4tant at the n&lls to the left an'

ri$ht minor lo!es. #o'al ma$neti% resonan%e %o&)lin$ is more effe%ti,e at 'es%ri!in$ ma$nit&'es of

ma$neti% fl&4 'ensit" in terms of the )otential3 one s&%h is 'es%ri)tion is in terms of 'is)la%ement

%&rrents.

The antenna loo) %arries a %om)le4 %&rrent3

I t ( ) = I %osωt 3 BB

hi%h is %onfine' to the loo) near its s&rfa%ethe s&rfa%e of the %on'&%tor %onsistin$ of it= ire.

The ,e%tor )otential from the loo) is e4)resse' as3

− 6*r !µ0I e A ( ) r = 'l!3 B@%∫ℓ !>π r

here I = I%osωt. To sim)lif" the a))earan%e of the harmoni% %&rrent in the folloin$ e9&ations3 it

is e4)resse' as

I X & γ . B

7@



/ − 6*r µ 0 (I a π κ ) e

A r ( ) ≈ 6 sinθ&γ . @C>π r

E9&ation @C s&$$ests there are o!ser,a!le effe%ts at the !o&n'ar" !eteen the fiel' o!6e%t an' the

)otential. The res&lt as sim&late' an' is shon in Fi$./.1C.

Fi$./.1C. a The ,e%tor )otential in the ma$neti% fiel'3 ! the )otential %om)onent on the loo) interfa%e.

The sim&lation shos that the ele%troma$neti% )otential in enmeshe' in the fiel'3 as )re'i%te'3 !&t

also is in motion at the !o&n'ar" !eteen the fiel' an' free5s)a%e folloin$ the 4− " )lane of the

loo)s. What is interestin$ is that motion is %losel" %onne%te' to the interfa%in$ of the loo) an' the

me'i&m an' that it o%%&rs in 'ifferential 'e$rees of ma$nit&'e. What is most interestin$ is that if the

)otential %an infl&en%e the line inte$ral at the interfa%e. If it %an si$nifi%antl" mo,e it aa" from =ero3it o&l' sho a se%on' %o&)le' mo'e insi'e the s%heme.

To 9&estions as*e' are is the )otential3 A3 a %om)onent of the entirel" %ontaine' ithin the

ma$neti% intensit" H 3 else a %o&)lin$ to Is it a )ro)ert" of the me'i&m inferrin$ a transfer of

for%e !eteen an' A The &se of the #a4ell 'is)la%ement %&rrent D is the form of the

'is)la%ement !e%a&se of the for%es !eteen antenna an' a fiel' o!6e%t K>BL3 as3

µ0 6 r × µ0 (∂ ∂ D t )× r =∇× A= ∫ 2 'υ + ∫ 2 'υ. @B

>π r >π r

The se9&en%e of sol&tions to fin' the )otential3 in terms of a %ir%&it3 is 'iffi%&lt hen tr"in$ to

'e%om)ose the )otential ,e%tor on the interfa%e to 9&antif" the 'e)en'en%e of the )otential on

%o&)lin$< this is es)e%iall" 'iffi%&lt hen there is no 'efinition3 o&tsi'e of the im)e'an%e3 of free5

s)a%e. We ha,e free5s)a%e in the sense of that on a )lanet3 e ha,e free5s)a%e in the sense of the

:,a%&&m; of o&ter s)a%e3 an' e ha,e the %ontention that there is 'ar* matter fillin$ the a))arentl"

em)t" s)a%e. (an there !e a &nifi%ation K/73 >@L !" &sin$ the #in*os*i %ones an' time 'ilation of

C0



Einstein?s s)e%ial theor" Are ele%troma$neti% for%es an' the %har$es that )o)&late them stri%tl"

'e)en'ent on the 9&ant&m effe%ts3 i.e.3 the '&alit" of the ele%tron K>L an' its esoteri% )ro)erties (an

a,e5)arti%le '&alit" !e e4)resse' in s&%h ener$i=e' str&%t&res The a&thor %onten's this )ro,i'es

s&ffi%ient material to s&))ort the notion of free5s)a%e a,e$&i'es3 %onstr&%te' !" the s%heme3 an'

ener$" %an !e transmitte' a%ross them. It %o&l' ,er" ell in'i%ate e,i'en%e of )ro)erties onl" ,isi!le

in the 9&ant&m &ni,erse3 are also a,aila!le to the ma%ro orl' K70L. The mo'el o&l' then !e

'es%ri!e' to)olo$i%all" as a a,e$&i'e fiel' K71L3 WG F3 an' the )rimar" %ontri!&tor to the

%los&re of %&rrents in an o)en %ir%&it a%ross free5s)a%e.

The s&!stanti,e %laim )resente' in this thesis is of an in'e)en'en%e of the ma$neti% fiel' from its

ele%tri% %o&nter)art in terms of for%e an' attra%tion'es%ri!e' in terms of the ma$neti% )otential at

the interfa%e of the antenna. It is most o!ser,a!le &n'er )arti%&lar %on'itions in the antenna here the

ra'i&s of the antenna is e4)resse' as a ratio3 3 of the loo) %ir%&mferen%e to its a,elen$th as an

e4)onential f&n%tion3 shon in >/3 hose fiel' ta*es the form of /1 an' //3 &)on a))li%ation

of a tr&e %on'&%tion %&rrent the fiel' lines e4ten' from the %enter of the loo) form a h")er!oli% )lane

alon$ the = an' −= 'ire%tions3 as shon in Fi$./.1B.

Fi$./.1B. A sin$le loo) )ro6e%tin$ a ma$neti% f iel'.

The )resen%e of the fiel' in free5s)a%e emitte' !" a ma$neti% loo) is o!taina!le !" 'is%rete

tri$onometri% f&n%tions3 $i,en its )ro6e%tion K2 0L. The effe%ti,e area3 here for%es 'eform the

isotro)" into a 'is%erni!le sha)e3 transforms into a %om)osite of s&ffi%ient fl&4 'ensit" )ro,i'in$ the

a))earan%e of a %lose' s&rfa%e. The e9&ili!ri&m of the area is 'e)en'ent &)on the ma$neti% ener$"

%ontaine' ithin an' here the !o&n'ar" is 'is%erne'. Altho&$h in fa%t the !o&n'ar" is in'eterminate

at e4treme )ositions alon$ the %&r,at&re of fiel' )ro)a$ation3 here the re%ei,e' )oer a))roa%hes

=ero is s&%h a !o&n'ar". Des%ri)tion of this !o&n'ar" is $i,en !" the infinitesimal of the ma$neti%

fiel'. The s&rro&n'in$ free5s)a%e is %onsi'ere' as a )erio'i% lo%all"5a))ro4imate' a,e$&i'e K10BL

C1



6 t ω − 6β1= − 6β/=ith im)li%it time5'e)en'en%e e that3 &n%o&)le'3 ha,e the s)atial 'e)en'en%es e an' e 3

res)e%ti,el" here β β 1≈ / are )ro)a$ation %onstants. For the $eometr" shon in Fi$./.@ ith a

)erio' Λ3 the %o&)lin$ %oeffi%ient ta*es the form /κ % os /π = . The s)atial e9&ations >2 $i,enTR ( Λ)

their ea* time 'e)en'en%e of am)lit&'es aT an' aR >>3 an' %o&)le' state >73 hen e9&i,alent to

the relationshi) of the store' ener$" at resonan%e ,ia %on'&%tan%e G an' resistan%e R atten&atin$ the

transmission3 %ontri!&tin$ sim&ltaneo&sl" to a %om)le4 im)e'an%e Z0 an' a'mittan%e Y03 %onstr&es

that the for%es e4%han$e ener$" in %onstr&%ti,e an' 'estr&%ti,e interferen%e )atterns as a $lo!al

os%illator" for%e re)resentin$ the a,e$&i'e K/C3 10B3 111L.

It is )ro)ose' that if a s)e%ial %ir%&it %o&l' a%%elerate ele%trons in an antenna3 the emitte' ra'iation

%o&l' !e %ontrolle' an' 'ire%te' !" mani)&latin$ the )resen%e of the ma$neti% %om)onent. An

in%rease of ill "iel' a $reater m&t&al in'&%tan%e 7>. With the a''ition of metalli% a%%retion

'is%s3 that is3 :a%%retion; meanin$ 'is%s formin$ a &niform %on'&%ti,e s&rfa%e3 l"in$ in the 4 5 "

)lane an' )ro6e%tin$ alon$ = so that a %orona effe%t o%%&rs in the $a) !eteen the 'is% an' the inner

loo) s&rfa%e. S&%h 'is%s are ali$ne' ith their %enters at the same )oint as the loo) %enter3 ill&strate'

in Fi$./.1@.

Fi$./.1@. S%hemati% 'rain$ of to %ir%&lar loo)s ith %enter 'is%s.

Un'er normal atmos)heri% )ress&res an' tem)erat&res3 a ma$neti% fiel' has a )otential fl&4 'ensit"

hi%h is e,enl" 'istri!&te'3 is %on%entrate' toar' the %enter of a loo) rather than in other )la%es3 an'

%an !e mo'ifie' to in'&%e %&rrents at a 'istan%e K10BL. An ener$i=e' %ir%&lar loo) antenna emits a

ma$neti% fiel' of a $i,en intensit" hi%h is a )ro'&%t of the %&rrent 'e)en'ent &)on a s)heri%al

C/



s&rfa%e an' the $eometr" e4)resse' !" the antenna. The ra'i&s r i of the loo) i is $i,en as a ratio3 i3

of loo) %ir%&mferen%e to resonan%e a,elen$th3 λi 3 e4)resse' as an e4)onential )oer f&n%tion ,3

/π r i

i = 30.10./7λi

, @@ i =10 3

, =−13 −/3 −2...n.

With res)e%t to the loo) 'es%ri!e' !" 1/>3 the ma$neti% fiel' in the 4 5 " )lane ta*es the folloin$

form3

µ0I 0 µ0I 0

4" = ⋅/π r i = . @/

>π r i /r i

here I03 is the %&rrent a))lie' to the loo). (ontrastin$l"3 the fiel' at a 'istan%e ' alon$ the = 5a4is

from the %enter of the loo) i is3

/r i µ0I0 1 µ0I r 0 i

= = /π r i = 3 0/ 2

'>π ' / / / /(' + r i )

in hi%h ' 3 is the or'inal fiel' 'istan%e. When a %&rrent3 I03 is a))lie' to the loo)3 the ma$neti%=

fiel' lines e4ten' from the %enter of the loo) an' form a %&r,at&re as a h")er!oli% )lane alon$ the =

an' −= 'ire%tions3 as ill&strate' in Fi$./.1B.

The )resen%e of the fiel' in free5s)a%e emitte' !" a ma$neti% loo) is o!taina!le !" 'is%rete

tri$onometri% f&n%tions3 $i,en its )ro6e%tion. The effe%ti,e area3 here for%es 'eform the isotro)" into

a 'is%erni!le sha)e3 transforms into a %om)osite of s&ffi%ient fl&4 'ensit" )ro,i'in$ the a))earan%e of

a %lose' s&rfa%e.

The e9&ili!ri&m of the area is 'e)en'ent &)on the ma$neti% ener$" %ontaine' ithin the area an'

here the !o&n'ar" is 'is%erne'. Altho&$h the !o&n'ar" is in'eterminate at e4treme )ositions alon$

the %&r,at&re of fiel' )ro)a$ation3 here the re%ei,e' )oer a))roa%hes =ero is s&%h a !o&n'ar".

Des%ri)tion of the moment&m of ener$" )ro)a$ation is $i,en !" the infinitesimal of the ma$neti%

fiel'.

C2



Fi$./.1. Geometr" of a )ro6e%tion an' infinitesimal fiel' in the %ir%&lar loo).

Fi$./.1 ill&strates the $eometr" of the ener$" )attern emitte' from a %ir%&lar loo) !" its intensit" in

motion a%ross free5s)a%e3 'is)la%in$ the s)a%e $i,en its ma$nit&'e3 H^δ = 3 in !oth the )ositi,e an'

ne$ati,e 'ire%tions of =. The )henomenon of )ro6e%tion3 H3 is manifest as a a,efront from a

transmission antenna here an a%%eleration for%e has !een a))lie' to ele%trons on its %on'&%ti,e

interior. The len$th an' intensit" of the )ro6e%tion is '&e sim&ltaneo&sl" to the $eometr" of the

antenna an' the )oer a))lie' to it.

In Fi$./.13 the len$th of transmission alon$ the =5a4is is a %om)onent %onsistin$ of the )ro6e%tion

fiel' len$th 'H an' the or'inal fiel' 'istan%e ' 3 $i,en its ra'ial si=e3 r 3 an' the infinitesimal fiel'= r

δ . The len$th of & is $i,en !" the )ersisten%e of am)lit&'es to the limit of their am!ien%e3 manifest

as $i,en in !oth 'ire%tions of )ro)a$ation in =. " %al%&latin$ the an$le θ an' inte$ratin$ aro&n'

the loo)3 one %an 'etermine the amo&nt of ener$" )resent in .

The ma$neti% fl&4 'ensit" is 'e)en'ent &)on the %&rrent a))lie' to the loo) antenna. An in%rease of

E or ill 9&asi5linearl" in%rease the fl&4 'ensit" of the fiel'3 $i,en thermo'"nami% las. The

ass&m)tion is that an in%rease' ill "iel' an e9&al in%rease' amo&nt of E in the mo'el.

The mo'el &n'er %onsi'eration is a ea*l"5%o&)le'3 lo κ3 s"stem at resonan%e !eha,in$

analo$o&sl" to a lo%all"5a))ro4imate' transmission5line resonator K213 111L. Un'er s&%h %on'itions3

the %om)le4 %&rrent an' ,olta$e o)eratin$ on the loo)s ta*e the form3

π =I = ( ) = I 0 %os 3

l 1

π =V( ) = =− 6 Z0I 0sin 3

l

C>



here I0 is the %&rrent in the short5%ir%&it an' l is the len$th of the attra%ti,e transmission )ath. The

sim)le im)e'an%e3 Z03 is that of free5s)a%e3 _2BBΩ. The a,era$e store' ma$neti% ener$" Wm in the

loo) at resonan%e ω0 is3

l l1 / 1 / / π =

Wm ω = ∫ I = '== I 0 ∫%os '=0 > > l

0 0 /

1 /

= l I0 .@

The a,era$e store' ele%tri%al ener$" 'oes not nee' to !e %al%&late' se)aratel" sin%e the s"stem is at

resonan%e. i.e.3 We = Wm as3ω0 ω0

1 / Wtotal = + We Wm = /Wm = l I 0 . 2ω0 ω0 ω0 >

The s&rro&n'in$ free5s)a%e is %onsi'ere' as a )erio'i% lo%all"5a))ro4imate' a,e$&i'e K/>3 10BL

6 t ω − 6β1= − 6β/=ith im)li%it time5'e)en'en%e e that3 &n%o&)le'3 has the s)atial 'e)en'en%es e an' e 3

res)e%ti,el"3 here β β 1≈ / are the )ro)a$ation %onstants. For the $eometr" shon in Fi$. 2 ith a

)erio' Λ3 the %o&)lin$ %oeffi%ients κ transmitter an' κ re%ei,er ta*e the form /κ % os /π = .T R TR ( Λ)

Therefore3 the s)atial e9&ations are3

' /π =aT =− 6βTaT − 6/κT % os aR3

'= l >

' /π =aR =− 6βRaR − 6/κR% os aT3

'= l

$i,en a ea* time 'e)en'en%e of the am)lit&'es aT an' aR to resonan%e fre9&en%ies ωT an' ωR 3

'aT = 6ω TaT + 6κTaR3

't 7

' aR = 6ω RaR + 6κ RaT3't

relati,e to the %o&)le' state of the resonatin$ elements i an' 6 3

#i6

κ κ T ≈ R = . C i 6

C7



th thThe m&t&al in'&%tan%e of the i an' 6 elements an' the inte$ration o,er the s)heri%al s&rfa%e S 6

from Fi$./.1 is therefore3

&∫∫ n ⋅ 'S

S 6

# i6 = 3 BI

i

)ro)ortional to the %&rrent I0 of the o))osin$ resonator i3 "iel'in$ the )ro)ert" that ener$" %an !e

a))lie' to the s"stem to a ma4im&m )oint that the a,e$&i'e %an s&stain. The 'issi)ate' )oer3

hile a )ro)ert" of the loa' )resent3 is also a )ro)ert" of the )oer s&staine' in the ma$neti% fiel'3 as3

1 / 1 /

)' = R I = ( ) + G V( ) =/ /

l l @1 / / π = 1 / / π =

= R I 0 ∫%os '=+ G V 0 ∫%os '=3/ 0 l / 0 l

shon in 12> to !e e9&i,alent to the relationshi) of the store' ener$" at resonan%e ,ia %on'&%tan%e

G an' resistan%e R atten&atin$ the transmission3 %ontri!&tin$ to !oth a %om)le4 im)e'an%e Z0 an'

a'mittan%e Y0. This is s&$$esti,e of an os%illator" for%e a%tin$ in a a,e$&i'e.

On%e the ele%trons ithin the %on'&%ti,e material are a%%elerate'3 the" s&!se9&entl" ra'iate ener$" in

a 9&anti=e' form &)on 'e%eleration K7L. The ener$" emitte' in the form of ra'iation from the %har$e

9 ith a%%eleration a is $i,en !"3

/ /9 a

)e = 3 2

Cπε0%

here % is the s)ee' of li$ht. The emission of ener$"3 in a 'i)olar form is3

/ )γ = µ0 sin θ 3 100

here θ is the an$le !eteen the 'ire%tion of a%%eleration an' the emission. The ra'iation is

)olari=e' s&%h that the emission fiel' is )arallel to the a%%eleration for%e on the ele%tron.

It is )ro)ose' here that if a %ir%&it %o&l' a%%elerate ele%trons in an antenna3 the emitte' ra'iation %o&l'

!e %ontrolle' an' 'ire%te' !" mani)&latin$ the )resen%e of the ma$neti% %om)onent. An in%rease of

CC



o&l' then "iel' a $reater ,al&e of #i6 ith the a''ition of metalli% a%%retion 'is%s3 that is3

:a%%retion; meanin$ 'is%s formin$ a &niform %on'&%ti,e s&rfa%e3 l"in$ in the 4 5 " )lane so that a

thermo'"nami% e,ent o%%&rs in the $a) !eteen the 'is% an' the inner loo) s&rfa%e %onne%tin$ the

to as a %o&)le' )otential. S&%h 'is%s %an !e ali$ne' ith their %enters at the same )oint as the loo)

%enter.

The ma$nit&'e of )ro6e%tion at a 'istan%e from the loo) is $i,en !" the 9&antities of %osh a an'

sinh a3 as shon in Fi$./.1B3 !eteen the limit at the transient =one3 at )oint !?3 an' the %enter. Point

a? is here the infl&en%e of the ma$neti% fiel' is at a ma4im&m3 s&%h that a se%on' loo) a!sor!s a

ma4im&m 9&antit" of transmitte' )oer "iel'e' !" the first loo). The )oer 'ensities in the 'ar*er

$re" =ones3 la!ele' a an' ! 3 !ase' on their orientation on the 4 an' = 5a4is3 are e9&al &nless/ /

there is an a=im&thal fiel' )resent. The )oer 'ensit" in these =ones is 'e)en'ent &)on the %&rrent I0

a))lie' to the loo).

If a loo) is set at a 'istan%e ith a se%on' loo) of the same n&m!er of t&rns3 loo) ra'i&s3 an' ire

'iameter to a reasona!le 'e$ree of e9&al a))ro4imation the fiel' :lines; e4ten' the h")er!oli% )lane

from (403 "03 = 0) to (4 " = 13 3 1 1) ith less %&r,at&re θ than that o!ser,e' in the %ase of a sin$&lar

loo).

The limit !eteen the 'istan%e3 $i,en !" (41− 403 "1− "03 =1− =0)3 an' the fall5off of the fiel'3 $i,en !"

1 3 is asserte' here as !ein$ '&e to the %&r,at&re of the )ro)a$ation an' the s&!stanti,e a,e$&i'e/

r

an' is not ne%essaril" 'e)en'ent &)on the si=e of the loo).

The a''ition of a %on'&%ti,e me'ia l"in$ alon$ the 4 5 " )lane3 )er)en'i%&lar to = 3 ill %ontort the

=ones. If a flat metalli% sheet is )la%e' in the )lane of a len$th an' hei$ht $reater than the 'iameter of

the loo)s3 then ener$" is a!sor!e' on its s&rfa%e.

(ontrastin$l"3 the intro'&%tion of a metal 'is% at the %enter of the re%ei,in$ loo) ill in%rease the

ma$nit&'e of . The )ro)ensit" of the 'is%s to enhan%e the ma$neti% fiel' has a $reater infl&en%e on

the re%ei,er as o))ose' to the transmitter.

CB



Fi$././0. a #a$neti% fl&4 of loo) antenna3 ! 'is% a''s fl&4 stren$th.

The ma$neti% loo)3 at a ratio to its 9&arter a,elen$th3 %ontains the effe%ts of a f&ll alternatin$5%&rrent

A( %"%le im)resse' on it !" an in%i'ent a,e K20L. The rate of e4%han$e of ener$i=e' )arti%les is

then re)resente' !" the 'ensit" of :ma$neti% lines;3 ill&strate' on the left si'e of Fi$././1. The ri$ht

si'e of Fi$././1 meanhile shos the same loo) s"stem ith an a%%retion 'is% a''e' at the %enter of

the transmission loo). The more attra%tion !eteen the loo)s3 so the more effi%ientl" the ener$" is

transferre'.

The lines in Fi$././1a an' Fi$. /./1! sho the ma$neti% fl&4 'ensit" hile the sli%es at the en's sho

ma$neti% fiel' intensit" at the loo) at norm . For the )&r)oses of the )a)er3 the n&m!er or :'ensit";

of fiel' lines )re%l&'es a hi$her ma$neti% fl&4 in the re$ion hile a sha'e from 'ar* to li$ht 'enotes

$reater intensit".

The stren$th of the ma$neti% fiel' $enerate' !" the loo) ma" !e in%rease' !" a''in$ t&rns n of

%on'&%tin$ ire3 instea' of a sin$le t&rn an' in%reasin$ the len$th of the le$s m shon in Fi$./.1@.

In terms of the e4)eriments hi%h ha,e !een )erforme' an' are re)orte' on in this )a)er 3 the loo)

as %onstr&%te' here n = 2 an' m = >%m. In terms of %al%&latin$ the ma$nit&'e3 it is )erha)s more

a))ro)riate to %onsi'er the ma$neti% fiel' in terms of H here = µH.

Fi$././1. Pro)a$atin$ ener$" in .

C@



A sin$le 'is% is )la%e' insi'e the re%ei,in$ loo) so that it lies on the 4 5 " )lane hile its %enter lies

alon$ the =5a4is3 as shon in Fi$./.7C3 here the ma$nit&'e of is 'ire%te' thro&$h its %enter. With

the a''ition of the 'is%3 'istortions are )resent !" im!alan%es in the fiel'3 here a''itional

transforme' ener$"3 8φ 3 a))ears in the a=im&thal fiel'3 shon in Fi$././2 as3

8φ = f r ( 3 θ φ ) . 101

A 'is%3 a))ro4imate' as a %onse%&ti,e se9&en%e of ire5t&rns in the form of a s)iral here the i'th

!eteen the in'in$ arms a))roa%hes ti%e the ire ra'i&s3 as3 / 3 a has a s&rfa%e ra'i&s of 1/

mm. It is e4)e%te' the 'is% ill in%rease the ma$nit&'e of H relati,e to its si=e ith the a''ition of

the ne ,e%tor &. The a=im&thal ma$neti% fiel' is e4)resse' as a se%on' or'er )henomenon. An

in%rease in ener$" in the fiel' is o!ser,e' !" an in%rease' %on,ersion of ele%tri% %&rrent into H.

Fi$././/. Densit" of a=im&thal fiel'.

Distan%e 'oes not 'ire%tl" affe%t mo'al %o&)lin$ KCL3 rather3 s)atial e4tensions in the =5'ire%tion relate

more %losel" to the trans)ort of ele%tri%al )oer3 e4)resse' in terms of the E an' fiel's3 atten&ate'

alon$ its len$th. Therefore3 !roa'%ast )oer is s&!6e%t to an atten&ation rate 'e)en'ent &)on a ,al&e

of im)e'an%e. The a))earan%e of an a=im&thal %om)onent hen the 'is% is )la%e' insi'e the loo) is

li*el" '&e to shearin$ effe%ts3 or im!alan%es in the e9&ili!ri&m state of the ma$neti% fiel' %onsistin$

the a,e$&i'e as it transforms from isotro)" toar' anisotro)" in the 4 5 " )lane.

E4)erimental or* on this as)e%t of the theor" is )resente' in /.>./.

/./ The %ir%&it mo'el

As it is )rimaril" inten'e' that the s"stem transmit a,es3 &n'er s&%h %on'itions the os%illator is

%onsi'ere' to !e the 'ri,in$5for%e of the s"stem. Witho&t os%illations3 a,es o&l' not )ro)a$ate.

C



For %om)arati,e )&r)oses3 if the antenna as %har$e' ith a non5os%illatin$ %&rrent s&%h as that

)ro,i'e' !" a 'ire%t %&rrent '% )oer s&))l"3 the re%ei,er o&l' not see the si$nal an' ma$neti%

ener$" stati%all" emitte' into s)a%e3 $i,en the area3 'A3 of the %on'&%tor. There o&l' !e no e4)li%itl"

'efine' %o&)le'5mo'e an' the si$nal o&l' not %o&)le to a 'istant loa'. As far as the fiel' o!6e%t is

%on%erne'3 it is ran'oml" $enerate' ith no %ohesion in the am!ien%e.

The tas* of this se%tion is to 'es%ri!e the %onstr&%tion of a %ir%&it3 hi%h ill&strates the 'ia$rammati%

re)resentation of the so&r%e3 the os%illator3 an' the loo)s ill&strate' in Fi$./.1. The %ir%&it mo'el ill

also attem)t to satisf" the mathemati%al 'es%ri)tions of the )re,io&s se%tions hile intro'&%e a

re%on%iliation of the %ir%&it ,erses fiel' mo'el 'is%ontin&ities K7/L onl" in the %onfines of an

a$reement of the s&))osition of resonant %ir%&it5as5antenna %orrelation.

To satisf" these aims3 a %ir%&it hi%h %onsists of a transmitter an' a re%ei,er is re9&ire' %onsistin$ of

a %om)a%t os%illator 'ri,in$ a ma$neti%5resonant loo) antenna. The %o&)lin$ of the so&r%e an'

os%illator o!6e%ts are holl" res)onsi!le for the %hara%ter of the manifest fiel' !eteen the antennas3

as 'es%ri!e' in /.1.

The most imme'iate an' o!,io&s 9&estion is hat *in' of os%illator to &se %om)a%t3 )oerf&l3 an'

the a!ilit" to im)ress the e4)e%te' %&rrents on the l oo). Effi%ien%" is im)ortant here as the

transmission s%heme has a %riterion of :orthiness;< it is ta*en into %onsi'eration hen %hoosin$ the

os%illator t")e.

In the earl" 'a"s of ra'io3 n&mero&s t")es of os%illators for si$nalin$ ere !ase' &)on the s)ar*5$a)

transmitter first &se' !" Hert= K723 7>L. As the transmission of ra'io a,es e,ol,e' from tele$ra)h"

into ,oi%e3 in%reasin$l" the os%illator nee'e' to !e free from noise for !roa'%astin$. As the s)ar*5$a)

starte' to fall o&t of fa,or3 sin&soi'al os%illators a!le to %ontain mo'&lation ere )ro)ose' in the form

of the re$enerati,e %ir%&it !" Armstron$ K77L. The &ni,ersalit" of the s%heme shoe' that an" ra'io5

fre9&en%" os%illator as a to)olo$" %ontainin$ fee'!a%*< this to)olo$" %o&l' !e mo'ifie' to )ro,i'e a

%ontrolla!le re'&%tion in fee'!a%* loo) %o&)lin$ on the antenna %ir%&it. If the metho' of %o&)lin$ asto s)lit the %hara%teristi% im)e'an%e of the %ir%&it thro&$h an am)lifi%ation sta$e3 it f&n%tions as a

%om!ination of an os%illator an' mi4er3 hi%h %on,erts the mo'&lation 'ire%tl" to the !ase!an'.

B0



Fi$././2. The re$enerati,e os%illator.

On e4amination of Fi$././23 the 'ri,in$ for%e of the ,a%&&m t&!e U1 is %arrie' a%ross a )air of

in'&%tan%es3 a %o&)lin$ %oil3 /3 an' ti%*ler %oil3 23 alon$ ith t&nin$ %a)a%itor3 (1. The %a)a%itors

(/ ser,es to s&))ress lea*a$e from the $ri' of U13 %a)a%itor (2 for fre9&en%" !")ass. The a))li%ation

of '% )oer to the trio'e am)lifies the si$nal in%omin$ to terminals 81 an' 8/ treatin$ the !eat

fre9&en%" of the antenna s%heme an' )oer im)resse' on it as if the entire )a%*a$e as a

%om!ination of so&r%e an' os%illator am)lif"in$ the resonant fre9&en%" of the antenna. If the %ir%&it is

mo'ifie' to s)lit the antenna into to e9&al )hases an' the 'ri,in$ am)lit&'e )oerin$ ea%h 1@0*

)hase5shift3 a %ir%&it ith ti%e as m&%h a))lie' )oer %o&l' !e %onstr&%te'. S&%h a %ir%&it is the)&sh5)&ll os%illator.

The )&sh5)&ll os%illator is a t")e of ele%troni% %ir%&it that %an 'ri,e !oth a )ositi,e an' a ne$ati,e

%&rrent )hase5shift into a loa'. A))earin$ in the earl" 1/0s3 later 'es%ri!e' in 'etail !" Hoa$ K7CL3

the mo'el is a hi$hl"5effi%ient os%illator hi%h %ontains a %om)lementar" )air of )oer5'ri,ers3 one

'issi)atin$ or sin*in$ %&rrent from the loa' to the ne$ati,e )oer s&))l"3 an' the other s&))l"in$ or

so&r%in$ %&rrent to the loa' from a )ositi,e )oer s&))l" o,er a %enter5ta))e' %oil '&rin$ a %om)lete

'&t" %"%le. e%a&se of the hi$h5effi%ien%" an' the sim)li%it" of the mo'el3 it is i'eal for ireless)oer transmission shon in Fi$./.1. Sin%e the s%heme is !ase' on in'&%tion3 the os%illator ill ha,e

to !e mo'ifie' to also )hase5shift the antenna in'&%tan%e thro&$h the %"%le.

An in'&%ti,el"5%o&)le' )&sh5)&ll os%illator is shon in Fi$././>. When ,olta$e is first a))lie' !" the

'% so&r%e3 V13 the fl&%t&ation of the ,olta$e ma*es the $ri' of U1 )ositi,e an' the $ri' of U/

ne$ati,e. e%a&se the %&rrent is la$$in$ !ehin' the ,olta$e3 it is e4)resse' as %han$es in the )late

%&rrent. The la$ of the %&rrent is an im)ortant feat&re3 hi%h is 'ire%tl" %a)t&re' in this t")e of

os%illator3 the 'i)ole is %reate' on the )late. The resonan%e !eteen the in'&%tan%e an' %a)a%itan%e in

B1



the )late %ir%&it %a&ses ,olta$e %han$es on the )late3 hi%h fe' thro&$h the )late to $ri' %a)a%itan%e

or !" ma$neti% %o&)lin$ !eteen the to tan* %ir%&its3 re,erses the )olarit" of the $ri's3 sit%hin$

them e4a%tl" o))osite. As this %ontin&es3 the tan* %ir%&it im)resses the ,olta$e %han$es of the $ri's

an' %&rrent on the )lates3 so that os%illations of in%reasin$ ma$nit&'e rise. The ma$nit&'e rises to the

limit im)ose' !" the ma$nit&'e of the so&r%e ,olta$e an' the ener$" is s)rea' e,enl" a%ross the

%ir%&it $i,en its s"mmetr". E,ent&all"3 the rise rea%hes a ma$nit&'e here the 'i)ole !rea*s an'

ener$" is release'.

/./.1 The )&sh5)&ll os%illator

The )&sh5)&ll os%illator is a !alan%e' os%illator. This t")e of %ir%&it is a!le to 'eli,er lar$er o&t)&ts

ith less 'istortion than sin$le5trio'e am)lifiers. The o&t)&t is a sin&soi' relatin$ the o))osin$ for%es

of a!sor)tion an' 'issi)ation in the tan* %ir%&it %reate' !" the %o&)lin$ of %oils an' %a)a%itors a%rossthe am)lifiers.

Fi$././>. In'&%ti,el"5%o&)le' )&sh5)&ll os%illator.

e%a&se the os%illator in Fi$././> is s&ita!le to 'ri,e the s%heme3 it is ne%essar" to re%ast the %ir%&it

sin%e the anti9&it" an' )oer nee's of the ,a%&&m5t&!e trio'es ma*e re%reatin$ the %ir%&it in its e4a%t

form im)ra%ti%al. What is more realisti% is to re)la%e U1 an' U/ ith somethin$ li*e a semi%on'&%tor

trio'e3 hi%h has the a!ilit" to im)ress a hi$h le,el of am)lit&'e ith fast sit%hin$ s)ee's. S&%h an

item that is ,er" a',anta$eo&s for this )&r)ose is the n5#OSFET transistor.

Sin%e this is a fiel'5effe%t transistor3 'e)en'ent &)on hi$h5s)ee'3 ele%tri% fiel' sit%h5transitions3 the

n5#OSFET has %o&)le'5mo'es. The mo'e rele,ant here is the trio'e mo'e or linear re$ion3 also

*non as the ohmi% mo'e K7BL. The ar%hite%t&re of an n5%hannel #OSFET shon in Fi$././7. A

metal5o4i'e5semi%on'&%tor fiel'5effe%t transistor has three terminals the so&r%e3 the $ate3 an' the

'rain. In an n5#OSFET3 !oth the so&r%e an' the 'rain are n5t")e an' the s&!strate !eteen them is )5

B/



t")e3 shon in Fi$././7. A thin la"er of sili%on 'io4i'e ins&lates the $ate an' the )5t")e s&!strate.

D&e to this ins&lation3 there is no $ate %&rrent to either the so&r%e or the 'rain.

Fi$././7. The n5%hannel #OSFET str&%t&re.

The n5#OSFET %an !e e9&ate' ith a ,olta$e5%ontrolle' sit%h. When a s&ffi%ient ,olta$e VGS is

a))lie' !eteen the $ate an' the so&r%e3 the )ositi,e )otential at the $ate ill in'&%e eno&$h ele%trons

from the )5t")e s&!strate to form an ele%troni% %hannel %reatin$ an in,ersion la"er !eteen the so&r%e

an' the 'rain3 an' a %&rrent IDS is forme'. The !eha,ior of a n5#OSFET %an !e 'es%ri!e' !" the

f&n%tion I DS = f V ( GS 3VDS ) ith a threshol' ,olta$e VT . These )ro)erties %an !e ta*en a',anta$e of

in the ireless )oer s%heme !" )la%in$ the n5#OSFET in the %ir%&it in s&%h a a" as to s)lit

os%illations into to $ro&)s those of the transmission antenna an' those !eteen the )air3 ill&strate'

in Fi$././C3 an' the ste)s thro&$h ea%h )hase3 ill&strate' in Fi$././B.

Fi$././C. S%hemati% re)resentation of a%%eleratin$ ele%tions s)lit a%ross a tan* %ir%&it.

Fi$././C is inten'e' to sho ho a sin$le #OSFET in the a))li%ation a))lies an a%%eleration for%e3 as

a )ro)ert" of the )otential ,olta$e an' in'&%tan%e here the os%illation is s&staine' in the antenna.

The a!ilit" to )hase5shift !eteen ea%h )art of the %"%le s"n%hroni=es ith the shiftin$ )hases of the

)otential ill %lose the %ir%&it )erio'i%all" an' allos ener$" to transfer. Free5s)a%e a,e )ro6e%tion

B2



is )ossi!le hen the s"n%hroni=ation is ith an" one of the rest5state )otentials em!e''e' in the

)hase. Altho&$h the s)e%ifi%ation of these )otentials is o&tsi'e the s%o)e of this thesis3 s&ffi%e to sa"

the s"n%hroni=ation satisfies the )resen%e of the antenna as analo$o&s to %o&)le'5mo'es3 merel" !"

ins)e%tion. It is at the shiftin$ !eteen )hases here intera%tion ith the interfa%e ta*es )la%e.

Fi$././B. Phase5shift am)lifier on %oils.

A 'ee)er anal"sis of )hase5shiftin$ on %ir%&its is also o&t of the s%o)e of this thesis. What is

reasona!le is that a )oer n5#OSFET is a s&ita!le %om)onent to re)la%e U1 an' U/ of Fi$././>. The

arran$ement of Fi$././> is altere' to Fi$././@

• (oil / !e%omes the antenna3 an' is %onne%te' a%ross the 'rain of J1 an' J/ to s)lit in half

the antenna %ir%&it in%l&'in$ the fee' from the )oer s&))l"3 )olarit" o))osite3

• )late terminals U1 an' U/ are tie' to$ether an' %onne%t to the ne$ati,e fee' from the )oer

s&))l"3 an'3

• %oil > is the re%ei,er antenna3 shon in Fi$././.

A s%hemati% of the os%illator to 'ri,e the %ir%&it is ill&strate' in Fi$././@.

B>



Fi$././@. The transmission element ith a %o&)le'5mo'e.

This mo'el %ontains onl" a sim)le 'es%ri)tion of the 'i)ole !&t it is s&s)e%te' s&%h a 'es%ri)tion

%o&l' $o 'ee)er3 !riefl" e4amine' in the A))en'i4. Re$ar'less3 the %ir%&it ill %reate an' 'estro" its

'i)ole in a %"%le releasin$ its ener$". Sin%e 'estr&%tion of the 'i)ole is o%%&rrin$ !" the %&rrent

ret&rnin$ to the so&r%e in the form of fee'!a%* )oer from the antenna3 e4%ess ener$" ill !lee' off

as heat. The %om)onent that emits the heat '&e to internal losses an' fee'!a%* of ener$" %o&)lin$ at

the interfa%e is the %a)a%itor (1.

At hi$her so&r%e le,els3 heatin$ of J1 o%%&rs< heat5sin*s for !oth J1 an' J/ an' a metali=e'%a)a%itor ill *ee) the %ir%&it from !&rnin$ o&t. The a&thor stron$l" s&$$ests f&t&re resear%h !e

%on'&%te' on 8o&le heatin$ in the %ir%&it.

/././ Sim&lation an' %onstr&%tion of the %ir%&it

The %ir%&it %o&)lin$ 'ia$ram in the sim&lator )ro$ram3 National Instr&ments #&ltiSim3 is shon in

Fi$././a.

Fi$././. a The re%ei,in$ element ith a %o&)le'5mo'e3 ! an e4am)le re%ei,er.

B7



Usin$ a s&ita!l" %onstr&%te' re%ei,er3 ill&strate' in Fi$././!3 a sim&lation of the %ir%&it is %on'&%te'.

The 'ri,in$ f&n%tion of the %ir%&it in the sim&lation ill !e a )&lse. Hert= )ro)oses that a :rin$er; is

re9&ire' to start the os%illator K72L. A )&lse %&rrent I )&lse ill !e &se' in s&!stit&tion of V13 shon in

Fi$./.20. This is similar to the &se of the s)ar*5$a). One of the am)lifiers m&st !e shorte' hile

)oer is a))lie' in or'er to rin$ the resonator stron$l" eno&$h for os%illations to !e s&staine'. A

momentar" short is )la%e' !eteen the $ate an' the 'rain of J1. The os%illator is self5s&ffi%ient

re9&irin$ onl" )hase offset am)lifiers to s&stain the os%illations thro&$h the %"%les thereafter. This is

an i'eal re)resentation of a *i%*5start !e%a&se hen the )&lse %&rrent is a''in$ ener$"3 it forms a

%lose' %ir%&it. Sim)l" state'3 the os%illator re9&ires )rimin$. To sim&late a momentar" !&rst of ener$"

to the os%illator3 the settin$s of the )&lse %&rrent are liste' in Ta!le /.2.

Ta!le /.2. Sim&lation %ir%&it )ro)erties.

The !roa'%astin$ mo'el these settin$s are a))lie' to is shon in Fi$./.20.

Fi$./.20. The !roa'%astin$ mo'el for sim&lation.

BC



The fre9&en%" %o&nter rea's >// *H=< the si$nal at the os%illos%o)e3 / ,olts )er 'i,ision on the "5

s%ale3 ith a time !ase of /00 µS )er 'i,ision on the 45s%ale is shon in Fi$./.21.

Fi$./.21. Sim&late' si$nal seen !" the re%ei,er.

The sim&lation refers to !oth the transmitter an' re%ei,er elements of the theoreti%al mo'el first

ill&strate' in Fi$./.1. As the sim&lation is ta*en in %onte4t to in'&%ti,el"5%o&)le' elements !ase' on

%ir%&ital las3 the %om)onents %onsistin$ the !roa'%ast mo'el ha,e the )ro)erties

• (om)onents 13 /3 23 an' (1 sim&late the one resonatorthe transmission antenna ill&strate'

in 'etail in Fi$./.2@< li*eise3 > an' (/ sim&late a se%on' resonator3 the" are %o&)le' a%ross

free5s)a%e at the %al%&late' %o&)lin$ %oeffi%ient from >73

• J13 J/3 (23 an' (> %onstit&te the os%illator 'ri,en !" ,olta$e so&r%e V13

• The s"stem transmits ener$" $i,en the theoreti%al )re'i%tions. Variation of the %o&)lin$

%oeffi%ient κi6 shos the ,ario&s %ir%&it res)onses3 &sin$ the same 4 an' "5s%ale 'i,isions of

Fi$./.213 shon in Fi$.C.1 in the A))en'i4.

• 2 %ontri!&tes ea*l" to the in'&%tan%e of the loo)3

(om)onent 2 is )rimaril" the )ositi,e %onne%tion to the am)lifiers in the %ir%&it3 as shon in

Fi$./.20. Sin%e it is onl" a one5half t&rn on the loo)3 it is e4)e%te' the in'&%tan%e of 2 is m&%h

smaller than the in'&%tan%e of the loo) re)resente' !eteen %om)onents 1 an' /.

/./.2 E9&ations of the %ir%&it mo'el

It is e4)e%te' that a sin&soi' to !e emitte' from the transmission antenna ith !oth a )ositi,e an'

ne$ati,e $ro&) ,elo%it" o&tar' alon$ = from the loo) s&rfa%e l"in$ in the 43 " )lane. Gro&) ,elo%it"

is 'efine' as a $ro&) of am)lit&'es &se' to 'istin$&ish emanate' a,es in !oth )ositi,e an' ne$ati,e

'ire%tions of =3 )ro6e%te' !" the antenna. If onl" one si'e is %onsi'ere'3 then the effi%ien%" is 'i,i'e'

!" the missin$ $ro&) a,es3 at a %om)le4 ,elo%it" sin%e it %annot !e ass&me' that ea%h )hoton in the

ener$" is mo,in$ at a %onstant ,elo%it" thro&$ho&t the )ro6e%tion. This ass&m)tion is !e%a&se the

BB



,elo%it" is '&e to the ma$nit&'e of for%e of a%%eleration on the ele%tron in the %on'&%tor. The

ass&m)tion is %om)ati!le ith the %onsi'eration of the a,e en,elo)e3 hoe,er3 ith a sto%hasti%

'istri!&tion of ener$ies.

Ali$nment of loo) %enters3 or a4ial ali$nment3 ill "iel' the hi$hest ma$nit&'e of ener$" transmitte'.

Performan%e in the %ir%&it seems i'eal hen no'e ( is )la%e' at the )oint one5half of the len$th of

A. It o&l' !e interestin$ for f&t&re resear%h if this ire )la%ement 'iffers in )erforman%e on the

len$th. The %"%li% 9&alit"3 J t ( )3 is rele,ant to 'es%ri!in$ the internal 9&alit" of the os%illator at an"

)arti%&lar time t0 hen3

ω0e

J t ( ) = t0. 10/Re

E9&ation 10/ infers a %om)le4 set of am)lit&'es )resent a%ross the fiel' o!6e%t. The effe%ti,e 9&alit"

fa%tor Je alon$ ith the effe%ti,e 'am)in$ fa%tor ζe satisfies the tra,elin$ a,e sol&tion. These

)ro)erties are !ase' on the states of the os%illator for ea%h instan%e of the antenna elements as3

4

'J J J i3 6 3 *3 . . . 3 Jm 3m=1

1024

'ζ ζ ζ i3 6 3 *3 . . . 3 ζ n3n=1

for ea%h of the antenna elements i 6 3 3 3 * . . . 3 . n The states 'es%ri!e !eha,ior of the s"stem hen there are

small %han$es in Re3 $i,en B/3 shon in Ta!le /.>.

Ta!le /.>. Os%illator state !" J an' ζ .

De)en'in$ on the ,al&es of the effe%ti,e 'am)in$ ζ e an' the effe%ti,e 9&alit" fa%tor Je in !oth the

antenna an' the os%illator3 the emitte' a,eform ill !e one of the three t")es. The meanin$

re)resente' in the 'ifferent states of os%illation is '&e to the 'is)la%ement of a a,e in s)a%e $i,en its

)erio'i% motion. In or'er to satisf" )erio'i%it"3 the os%illator m&st !e elasti%in the sense of

ret&rnin$ )re%isel" to its ori$inal %onfi$&ration after 'eformation this re9&ires a *non )osition of

B@



e9&ili!ri&m an' a %o&nter for%e hi%h restores e9&ili!ri&m. The os%illator m&st !e a!le to )ro'&%e a

s&staine' sin$le fre9&en%" on a so&r%e o!6e%t ith a toleran%e small eno&$h that an o!6e%t 'esi$ne' to

a!sor! the os%illations ill 'o so if it has a))ro4imatel" the same )arameters. The antenna ill

%om)rise the ,i!rator hile the os%illator ill )ro,i'e s"n%hrono&s am)lit&'e.

In the tra,elin$ a,e %ase for ra'iati,e ener$" emitte' from the %o&)le' antenna3 the sol&tion is $i,en

!" the a,e e9&ation3 transforme' from 7 &sin$ 3

/ / /∂ ∂ & & 1 ∂ &

/ + = / / /3 10>∂ ∂ 4 " , ∂ t

here , is the )hase ,elo%it". As the a,e )ro)a$ates in !oth the += an' −= 'ire%tions)ositi,e

/π,an' ne$ati,e $ro&) ,elo%it"the 'es%ri)tion of the trans,erse motion = /π ω f =3 hereλ

, = f λ allos the 'es%ri)tion to !e3

/π" 4 ( ) 3 t = A si n ( ) 4− ,t

λ'" /π

, " ( ) 4 t 3 = = ω A% os ( ) 4− , t 107

't λ/

' " / / /π

a" ( ) 4 t 3 = = / − ω "= − ω Asi n( ) 4− , t .'t λ

The folloin$ e9&i,alent )arametri% forms of the a,e sol&tion of 107 are3

" 4 ( 3 t ) = A sin(* 4 − ωt )3

/π " 4 ( ) 3 t = A sin ( ) 4 − , t 3 10C

λ

4 t " 4 ( ) 3 t = A sin/π − .

λ T

The 'etermination of the ,elo%it"3 ,3 an' its inhi!ition3 ,3 is 'e)en'ent &)on the )ro)erties of the

me'i&m in tan'em ith the %o&)le' in'&%tan%e hi%h forms an a))ro4imation $i,en the a,e

relationshi). The )arameters are shon in Ta!le /.7.

B



Ta!le /.7. Os%illator tra,elin$ a,e )arameters.

e%a&se3 in the ireless )oer %ase3 the tra,elin$ a,e emitte' !" the antenna is a f&n%tion of the

am)lit&'e $enerate' in the os%illator3 is %onfine' to one )lane in s)a%e an' ,aries sin&soi'all" in !oth

s)a%e an' time an' e4)resse' in %om!inations of3

"= Asin(*4− ωt )3 10B

"= A%os(*4− ωt )3

an' of the %om)le4 form3

i*( 4−ωt) "= Ae . 10@

As an E&ler i'entit"3 the form ta*es3

iθ e = %osθ + isinθ. 10

Alterations to the %ir%&it o&l' affe%t stron$ %han$es to its state. The transmission %ir%&it is a series

%ir%&it an' the a''ition of Re o&l' ma*e Je smaller. The re%ei,in$ %ir%&it is arran$e' in )arallel so

that %han$es in Re ill ma*e Je lar$er.

Fi$./.2/. Sim)lifie' %ir%&it 'ia$ram.

ase' holl" on these theoreti%al res&lts3 a %ir%&it is %onstr&%te' !ase' on the s%hemati% &se' in the

sim&lation herein to test the h")othesis. The transmission %ir%&it in%l&'in$ the antenna 'is%&sse' in

/.2.13 is shon in Fi$./.22. #eas&re' ith a fre9&en%" %o&nter3 the !roa'%astin$ fre9&en%" is

@0



f 0 = >/@ *H=. It is e4)e%te' that if the 'i)ole o&l' !e )reser,e' after ea%h %"%le3 that effi%ien%"

o&l' in%rease as ell as the )otential at the !o&n'ar".

Fi$./.22. The %onstr&%te' os%illator.

The ne4t se%tion a''resses the 'etails of the )ro)a$ation mo'el )ro)ose' in this se%tion.

/.2 The )ro)a$ation mo'el

The trans)ortation of ener$" a%ross free5s)a%e o%%&rs in a )air of t&ne' %ir%&its3 s"mmetri% an' set a

'istan%e a)art3 hen one of the %ir%&its is ele%tri%all" stim&late'. " am)lifi%ation3 the resonan%e

fre9&en%" %an !e &se' to a%%elerate ele%trons in a %oil.

In )ra%ti%e3 this means an ele%tri% %&rrent ill !e in'&%e' in an" 'istant %ir%&it here the ma$neti%

fl&4 interse%ts ith an a))ro)riate %on'&%tor in the )ath of the os%illations. The )la%ement of the

resonant o!6e%ts in the s%heme ill !e in the )ath !ase' on the )ro6e%tions of their antenna3 %o&)le' to

their resonant %ir%&its.

The 9&arter5a,elen$th ma$neti% loo) antenna is a &sef&l 'e,i%e to transmit si$nifi%ant 9&antities of

ele%tri%al %&rrent at reasona!le effi%ien%". For sim)li%it" an' !re,it"3 this is a f&n'amental artifa%t of

the s%heme )resente' in this thesis. The %om)lete ran$e of !eha,ior a,aila!le is e4%l&si,el" '&e to the

interfa%e !eteen a set of ele%tri%al %har$es an' an am!ient str&%t&re of free5s)a%e at normal

tem)erat&res an' )ress&res3 for the )&r)oses here of atmos)heri% air ith %hara%teristi% im)e'an%e.

The mathemati%al %om)onents nee'e' to 'es%ri!e the )ro!lem at han' are %&rrents in the %on'&%ti,e

me'ia an' %&rrents in the 'iele%tri%. If the me'i&m insi'e the a,e$&i'e is homo$eno&s an' isotro)i%

an' if the s&rfa%e im)e'an%e at the !o&n'ar" is =ero3 the metho' of se)aratin$ ,aria!les o!tains a set

of normal3 &n%o&)le' mo'es of )ro)a$ation. An" irre$&larit" or 'is%ontin&it" in the a,e$&i'e

@1



)ro,i'es a %o&)lin$ !eteen some3 or all3 mo'es of )ro)a$ation K/>L3 trans)orte' on a transmission

element hose ra'i&s is smaller than the 'iele%tri% s)a%e3 hose a,e$&i'e is s&ffi%ientl" infinite3 to

im)ose i'eal )lanetar" ele%tri%al %hara%teristi%s.

A ne%essaril" narro!an' antenna ith stee) n&lls is re9&ire' to allo the )oer to transmit in the

si$nal hile re6e%tin$ s)&rio&s noise an' other transient )henomena the ma$neti% fiel's %o&l' !e

s&!6e%t to3 in%l&'in$ solar ra'iation. The 9&arter5a,elen$th ma$neti% loo) antenna has ,er" shar)

n&lls at ri$ht an$les to the )lane of the loo).

These n&lls %an !e ,er" &sef&l for s&))ressin$ ra'io interferen%e. A rotation of the loo) ill re'&%e

noise. e%a&se the n&lls are shar)3 small %han$es in antenna orientation %an ma*e a lar$e %han$e in

si$nal or noise re%ei,e' from the 'ire%tion of the n&lls alloin$ the interferen%e to !lee' into the fiel'.

The n&lls are shar)est on the smallest loo)s at a )erimeter a))ro4imatel" 0.07λ hi%h linearl"

in%reases to 0.1λ at the )eri)her" K20L.

The )ro)a$ation mo'el3 for reasons state' in the )re,io&s se%tions3 an' in 'etail in /.13 is %on%erne'

)rimaril" ith small antenna si=es ith a hi$h effi%ien%". In the )ro)ose' mo'el3 the loo) si=e is the

most %riti%al )ro)ert". If %oil loo) is o&n' at a s)e%ifi% ratio of %ir%&mferen%e to a,elen$th3 it is

e9&i,alent to %on'&%tor %onne%te' to an ele%tri%all"5short transmission line ith %o&)le' mo'es K203

10BL. To ma4imi=e the emission of ra'iation3 the ma$neti% loo) sho&l' !e at a ratio of its

%ir%&mferen%e to its 9&arter a,elen$th. S&%h a ma$neti% loo) ill !e shon to ha,e the i'eal ratio3

in terms of ma4im&m transmission effi%ien%"3 is hen this ratio falls on an e4)onential s%ale3 shon

in >/. (onsi'er a %ir%&it %onsistin$ of a sin$le loo) of ins&late' ire o&n' in s&%h a a" as to

%reate a %ir%&lar loo) of a fe t&rns. This loo) an' its %a)a%itor !e%ome one5half of a resonant %ir%&it

formin$ a transmitter T4. The len$th of this ire loo)3 ℓt 3 is then '&)li%ate' to %onstr&%t a similar

loo) of i'enti%al %hara%teristi%s3 ℓr 3 alon$ ith a %a)a%itor formin$ the se%on' half of a resonant

%ir%&it formin$ a re%ei,er R4.

A %&rrent is a))lie' to this %ir%&it3 in'&%es an ele%tri%al fiel' in the self5in'&%tan%e of loo) ℓt 3

%reatin$ a ma$neti% fiel' hi%h is %o&)le' at the interfa%e. " )la%in$ a loa' in )arallel to the ,olta$e

$a) in loo)3 ma$neti% ener$" is 'ran into the $a) an' then %on,erte' into ele%tri%al %&rrent.

In%l&si,e in the loo) is a ,olta$e $a) of i'th no $reater than the loo) ra'i&s3 so that $ r. The

%oil ℓt is )ositione' in s&%h a a" that the 4− " a4es form the loo) s&rfa%e an' the =5a4is the

tra6e%tor" o&t from the loo) in the 'ire%tions of the $ro&) ,elo%it" of the ra'iation)ositi,el" for

ri$ht5han'e'3 ne$ati,el" for left5han'e'. This relationshi) is ill&strate' in Fi$./.2>.

@/



The %hosen fre9&en%" %om)&tes 2 to !e !eteen >1/ an' 7/C *H=.

Fi$./.2>. Ra'iation emissions of a ma$neti% loo) antenna.

The antenna loo) is e4%ite' !" the a%%eleration of ele%trons on its %&r,at&re3 'ri,en !" the %&rrent I0

+from the am)lifiers in the %ir%&it. The )ositi,e $ro&) ,elo%it" emits ra'iation in the form H0 3 the

−ne$ati,e $ro&) ,elo%it" emits ra'iation in the form H0 3 an' γθ 3 the 9&anti=e' ener$" '&e

e4%l&si,el" to )hotons3 ill&strate' in Fi$./.27.

Fi$./.27. Wa,e )ro)a$ation for the mo'al %o&)lin$ s%heme of ireless )oer transfer.

An ass&m)tion of s"mmetr" ill re,eal the mo'el of ma$neti% a,es3 or %&rrents3 in the form of

+)hotons of a $i,en an$&lar fre9&en%" ω0 flo )ositi,el" at a $i,en %oor'inate (γθ 3 H0 ) from %oils

−ℓ ℓ t r an' ne$ati,el" at (−γθ 3 H0 ) from %oils ℓ ℓ t r ith an effi%ien%" η. There is a'istin%tion

in the sol&tions later that an' not ener$" e4%han$e !eteen loo)s sin%e %&rrents are &se' for

ele%troma$neti% 'es%ri)tions at ea%h half of the %ir%&it.

@2



From the %ir%&it )oint of ,ie3 antennas a))ear as a resistan%e Rr %alle' ra'iation resistan%e. It is not

relate' to an" resistan%e in the antenna itself3 rather3 a resistan%e %o&)le' from s)a%e to the antenna

terminals. A%ti,e ra'iation raises the am!ient tem)erat&re of Rr . This tem)erat&re is e9&i,alent

a%ross the %ir%&it !e%a&se the tem)erat&re of the 'istant antenna res)e%ti,el" from the transmitter%an onl" see its s"mmetri% analo$&e '&e to the nat&re of in'&%ti,e %o&)lin$ in that the )hotons e4tant

in the ma$neti% fiel' onl" rea%t ith li*e or other resonant o!6e%ts K213 7@L. Un'er s&%h %on'itions Rr

is 9&antifie' !" the re$ion of s)a%e insi'e the loo) as a f&n%tion of the ,irt&al transmission line

lin*in$ the antenna to the 'istant re$ion3 shon in Fi$./.2C.

Fi$./.2C. S%hemati% re)resentation of a ,irt&al transmission line of an antenna to a re$ion of s)a%e.

The loo) ill o)erate &sin$ a sin$le5!an' transmission ith re$ar' to the antenna im)e'an%e3'ire%ti,it"3 an' $ain. The antenna anal"sis is essentiall" e9&i,alent to that for a %&r,e' 'i)ole an'

)ro6e%te' ma$neti% fiel' $i,en 1. The %om)&tation of the antenna ill tar$et ele%tri%all" small loo)

antennas in the normal an' a4ial mo'e of o)eration. Ra'ial mo'e is an a',an%e' to)i%. #&lti)le

mo'es are e4hi!ite' in /.>.

Folloin$ the 'is%&ssion of the %ir%&it )ro)erties in /./3 an' the s%hemati% shon in Fi$././@3 the

transmission antenna %oil m&st !e 'esi$ne' to %onform to the shiftin$ )hase of the am)lifiers.

Folloin$ armor?s theorem3 'is%&sse' in 1.1 an' /.1./3 it is i'eal if an ele%tri%al for%e %an !e

a))lie' to a %on'&%tor in s&%h a a" as to a%%elerate ele%trons !eteen one half of the %on'&%tor an'

the other3 similar to a ire treate' as a 'i)ole. The mo,ement is treate' as the )hase.

Phase is 'efine' as re)resentin$ mo,ement alon$ the ar% of the antenna thro&$h 'i,isions /π thro&$h

the %ir%le?s %ir%&mferen%e here the ra'i&s r is the ma$nit&'e of the ele%tri% fiel'3 Eφ an' υ3 a

%om)onent of ,elo%it" of the shiftin$ )hase re)resentin$ the ,elo%it" of the ele%trons. A s%hemati%

re)resentation is shon in Fi$./.2B.

@>



Fi$./.2B. An$&lar motion of the os%illator transition )hase.

The a%%eleration for%es mo,e thro&$h π on the loo)3 shiftin$ 'ire%tion !eteen %om)onents 1 A

an' 2 (3 an' / an' 2 ( hi%h are ne$ati,e an' )ositi,e ,olta$es res)e%ti,el".

Theoreti%all"3 the for%e %ontaine' in the %&rrent a%%elerates the ele%trons thro&$h one )hase of π then

sit%hes 'ire%tion thro&$h the other )hase of π. At ea%h sit%h in 'ire%tion3 the ele%trons in the

%on'&%tor 'e%elerate an' release their ener$" in the form of )hotons alon$ the 'ire%tion of the

ma$neti% fiel'.

The material of the %oil sho&l' s&stain the an$&lar ,elo%it" of the %&rrents3 ω03 os%illatin$ thro&$h

inte$er m&lti)les of π. (o))er is s&ffi%ient3 artifi%ial materials %o&l' s&!stantiall" im)ro,e the

rea%tan%e. An ins&late' len$th of ire3ℓ

3 $i,en !" A3 3 is o&n' into a loo) of a %hosens)e%ifi%ation. It has a !rea* %&t into its ins&lation at the mi')oint in the loo)3 here (= 1 ℓ3 an' a

/

thir' ire follos the loo) formin$ an e4tra n&m!er of t&rns3 n = 0.73 an' an e4tra stan'in$ len$th3 or

le$3 ill&strate' in Fi$./.2@.

Fi$./.2@. The transmitter %oil.

@7



The transmission antenna in no a three terminal %oil3 antenna A3 3 ( %orres)on's to 13 23 /

res)e%ti,el"3 ith 13 23 / !eha,in$ as an os%illator hen %onne%te' 'ire%tl" to an am)lifier.

Terminals A 1 an' / are %onne%te' a%ross a %a)a%itor to the %orres)on'in$ 'rains of to n5

#OSFETs hose so&r%es are %onne%te' to the ne$ati,e terminal of the )oer s&))l". Terminal (

2 is %onne%te' to the )ositi,e terminal of the )oer s&))l".

A $enerali=a!le %om)&tation3 the metho' of moments3 ill !e fitte' to sol,e the )ro!lem of the

%ir%&lar loo) antenna as a f&n%tion of the s%atterin$ of the fiel' emitte' !" the transmitter. This

$eneral sol&tion is sol,e' &sin$ the softare )a%*a$e #atla! an' files from K7L. Pro)erties of the

loo) antenna %an then e )&t into the )ro$ram (ir%&it %al%&lator to arri,e at the theoreti%al ,al&es.

The %om)&tational str&%t&res ill !e these s%ri)ts< their )&r)ose is liste' in Ta!le /.C.

Ta!le /.C. (om)&tational elements from mesh to fiel' sol&tions.

Ea%h of the element files ere %om)ile' a$ainst the mesh files %reate' in ea%h )ro$ressi,e ste). The

antenna as sol,e' for an' its )ro)erties %om)&te'. These theoreti%al res&lts an' %om)arison to the

meas&re' res&lts ill !e 'is%&sse'. The )h"si%al %hara%teristi%s of the antenna ill !e 'is%&sse' in the

ne4t se%tion.

The antenna is the sole artifa%t of ireless )oer transmission an' relies on no other analo$&e. If

ireless ener$" transfer is to !e%ome more &sef&l3 it is ne%essar" to esta!lish a set of $enerali=ations

re$ar'in$ s)e%ifi% %on'itions of o)eration.

/.2.1 The re%ei,in$ antenna

A re%ei,in$ antenna ma" !e ,iee' as an" metal o!6e%t that s%atters an in%i'ent ele%troma$neti% fiel'

from a transmittin$ antenna3 hi%h )ro)a$ates a sin$le5!an' si$nal. e%a&se of s%atterin$3 an ele%tri%

%&rrent a))ears on the antenna?s s&rfa%e. That %&rrent in t&rn %reates a %orres)on'in$ ele%tri% fiel'. If

a %a)a%itor is atta%he' to the free en's of the loo)3 as shon in Fi$./.2!3 a ,olta$e 'ifferen%e a))ears

@C



at the terminals. The terminal ,olta$e %onstit&tes the re%ei,e' si$nal< )oer is e4tra%te' thro&$h the

,olta$e an' fe' to an in%an'es%ent lam)3 shon in Fi$././. Anal"ti%all" s)ea*in$3 it is i'eal if the

s%heme %o&l' !e a''resse' as s&rfa%e %&rrent 'istri!&tion o,er the antenna as a three5'imensional

o,al s&rfa%e &sin$ its e'$es as )ro)a$ation of the ma$neti% fiel' e4)e%te' in a olt=mann 'istri!&tion.

The 'es%ri)tion here ill !e for a to5'imensional flat s&rfa%e antenna.

Fi$./.2. a S%hemati% of the re%ei,in$ loo) antenna. ! oo) antenna %onne%te' to a %a)a%itor at the ,olta$e

$a). The hite arros are s&rfa%e %&rrents.

(onsi'erin$ solel" the ,ie)oint of ener$" transfer3 an antenna in the re%ei,in$ mo'e %olle%ts

ele%troma$neti% ener$" o,er a %ertain area3 shon in Fi$./.2a3 e4tra%ts the %a)t&re' )oer hile

!oth a!sor!in$ an' refle%tin$ the ener$" !a%* into free s)a%e3 !eha,in$ li*e a si$nal re)eater. The

s)e%ifi%ations of the %oils3 as ill&strate' in Fi$./.13 are shon in Ta!le /.B.

Ta!le /.B. Transmission an' re%ei,in$ %oils )h"si%al s)e%ifi%ation.

The metho' of moments is sol,e' for the antenna s&rfa%e &sin$ Rao5Wilton5Glisson RWG e'$e

elements. A short 'is%&ssion of the 'etail of the RWG al$orithm is re9&ire' to &n'erstan' ho thee'$e elements are &se' to a))ro4imate the fiel' sol&tion KC7L. The s&rfa%e of an ins&late' metal loo)

antenna 'i,i'e' into se)arate trian$les as shon in Fi$./.>0a. Ea%h )air of trian$les3 ha,in$ a

%ommon e'$e3 %onstit&tes a %orres)on'in$ e'$e element3 see Fi$./.>0!. One of the trian$les has a

)l&s si$n an' the other a min&s si$n in'i%atin$ the 'ire%tion of the %&rrent. A ,e%tor !asis f&n%tion3

@B



l + +

+ ρ ( ) r 3 rinT /A

l − −

f ( ) r = − ρ ( ) r 3 rinT 110 /A

03 otherise3

is assi$ne' to ea%h e'$e element here l is the e'$e len$th an' A is the area of trian$le T . Ve%tor

+ρ %onne%ts the free ,erte4 of the )l&s trian$le to the o!ser,ation )oint r3 shon in Fi$./.>0!.

−Ve%tor ρ %onne%ts the o!ser,ation )oint to the free ,erte4 of the min&s trian$le KC0L.

The ,e%tori=e' s&rfa%e ele%tri% %&rrent on the antenna s&rfa%e3 shon in Fi$./.>0a3 is a s&m of the

%ontri!&tions o,er all e'$e elements ith &n*non %oeffi%ients. These %oeffi%ients are fo&n' from the

moment e9&ation 110. The s&rfa%e %&rrent 'ensit" on a s&rfa%e Sθ of a )erfe%tl" %on'&%tin$

str&%t&re is $i,en !" an e4)ansion into RWG !asis f&n%tions o,er m e'$e elements. The moment

e9&ations are a s"stem of linear e9&ations ith the im)e'an%e Z. The !asis f&n%tion of the e'$e

%− %+element %orres)on's to a small !&t finite ele%tri% 'i)ole of len$th ' = r − r shon in Fi$./.>0!

an' K/1L. In'e4 % 'enotes the %enter of trian$le T for the 'i,ision of a loo) antenna str&%t&re into

elementar" ele%tri% 'i)oles3 shon in Fi$./.>0'.

The a))ro4imation here is similar a))roa%h to 'es%ri!e the %&rrents a%%or'in$ to #a4ell?s e9&ations

KC1L in terms of those s)e%ifie' at in 1.1. The %&rrent 'ensit"3 83 an' the antenna im)e'an%e Z is

re)resente' on s&rfa%e Sθ . If Sθ is o)en3 8 is re$ar'e' as the ,e%tor s&m of s&rfa%e %&rrents on

o))osite si'es of Sθ . As3

l + +m + ρm ( ) r 3 rinT

/Am

#

l − − 8= ' I m f m 3 f m = − ρm ( ) r 3 rinT 111m=1 /Am

03 otherise3

Where Z e4)resses the ,e%tor ,olta$e from e4)ansion %oeffi%ients Im formin$ the im)e'an%e

moment Z I ⋅ = V. The antenna ,olta$e is e4)resse' !" KC0L3

@@



%+ %− + ρm − ρm in% % Vm = lm Em ⋅ + Em ⋅ 3 Em = E ( ) r m 3 m =13 . . . 3 # 3 11/

/ /

in%

here E is the ele%tri% fiel' on an in%i'ent ele%troma$neti% si$nal o,er the len$th3 lm

3 of theantenna. The ,olta$e e4%itation ,e%tor is analo$o&s to the %ir%&it ,olta$e. The s&rfa%e ele%tri% %&rrent

on the antenna s&rfa%e is a s&m of the %ontri!&tions 111 o,er all e'$e elements.

Fi$./.>0. #o'ifie' s%hemati% of RWG e'$e element an' the 'i)ole inter)retation for %&r,at&re θ.

Sim)l" state The im)e'an%e Z 'es%ri!es the intera%tion !eteen nei$h!orin$ elementar" 'i)oles

$i,en o,er the len$th of the %on'&%tor. If e'$e elements m an' n are 'i)oles3 the element Zmn

'es%ri!es the %ontri!&tion of 'i)ole n3 thro&$h the ra'iate' fiel'3 to the ele%tri% %&rrent of 'i)ole m.

The si=e of Z is e9&al to the n&m!er of e'$e elements. This %ontri!&tion is %al%&late' &sin$ the

ele%tri%al fiel' inte$ral e9&ation KC/3 C2L.

The antenna as %onstr&%te' an' sim&late' &sin$ #atla! hile theoreti%al e9&ations ere %om)&te'

&sin$ the (ir%&it %al%&lator. Usin$ #atla! ith the PDE tool!o43 the s&rfa%e mesh is 'esi$ne' !"

%reatin$ a mo'el of the loo) an' %reatin$ RWG trian$les. Vis&all"3 one %an %reate the meshes !ase'

on a /5D mo'el a))ro4imation of a loo) s&rfa%e3 shon in Fi$./.>1a for 1@0 no'es an' /@@ trian$les

an' Fi$./.>1! for C>@ no'es an' 117/ trian$les. A s"stemati% 'es%ri)tion is %ontaine' in Ta!le C./.

a !Fi$./.>1. To loo) meshes a /@@ trian$les an' ! 117/ trian$les.

@



D&rin$ the )ro%ess of %reatin$ the mesh3 %he%*in$ the trian$le 9&alit" ill )ro,i'e an a,era$e

%hara%teristi% !efore %ontin&in$ ith the RWG anal"sis. The trian$le 9&alit" of !oth meshes shon in

Fi$./.>/a hile the e4)e%te' %&rrent 'ensit"3 shon in Fi$./.>/!3 is %onfine' to the inner e'$e of the

loo). The RWG e'$e elements are more a',anta$eo&s than the sim)le finite 'i)oles. In )arti%&lar3

the" s&))ort a &niform a4ial ele%tri% %&rrent alon$ a thin metal stri)3 a))ro4imatin$ for %&r,at&re KC>L.

For hel) %al%&latin$ in #atla! for the 'esire' antenna %hara%teristi%s from the %onstr&%te' meshes3

%o'e from KC7L ill !e &se' to )erform the %om)&tations.

Fi$./.>/. a Trian$le 9&alit" of the loo) mesh3 ! %&rrent 'ensit" on the loo).

The .mat files %ontainin$ the %om)ile' mesh of the antenna str&%t&re are &se' as in)&t to the e'$e

$enerator r$1.m an' to the rest of the %o'e se9&en%e liste' in Ta!le /.C. The PDE tool is &se'

%reate' %om)ile' mesh :ma$neti%oo)#esh.mat; ith /@@ trian$les.

Usin$ the in)&t %hara%teristi% antenna 'ata

• >1/ *H= , , f 0 7/C *H=3

• o!ser,ation )oint 70 %entimeters on the = a4is3

• ra'iation s)here ra'i&s 100 %entimeters3

• fee'in$ e'$e K513 03 0L3 'i)ole3

res&lte' the se9&en%e for o&t)&t 'ata is liste' in Ta!le /.@.

0



Ta!le /.@. #atla! %om)&te' in)&t %hara%teristi% antenna 'ata.

The %om)&tation of the mo'el s&$$ests that the str&%t&re of the ra'iation is e4%l&si,el" a )ro)ert" of

the $eometr" of the %oil. A s)e%ifi% )attern of ra'iation emission is o!ser,a!le !" shoin$ intensit" of

)ro6e%tion fiel' H3 o&t from the 'i)ole at the %enter of the loo)3 shon in Fi$./.>2.

Fi$./.>2. Dire%tional )oer 'ensit" of fiel' on its a))roa%hin$ a,efront3 hite shos hi$her )oer.

" e4amination of Fi$./.>23 the fiel' emanatin$ from the loo) antenna3 the hiter toar' the %enter

of the loo) is the )lane of )ro6e%tion3 relati,e to the loo) aro&n' the s)here. The )oer si$nat&re is

'&e to the 'i)ole releasin$ ener$" an' the for%es &)on the ele%trons in the %on'&%tor. This is $i,en in

1



more 'etail in (ha)ter >. The hei$ht of 'is)la%ement alon$ the "5a4is is the !eam i'th. It is

interestin$ to no 'es%ri!e the %om)&tations hi%h 'etermine' the emitte' )oer an' ra'iation

resistan%e of the fiel' $i,en the 'istan%e !eteen the antenna !ase' on K20L.

/.2./ Ra'iate' )oer

In terms of the %ir%&it mo'el of the )h"si%al antenna3 ener$" an' )oer are %om)&te' &sin$ the (

)ro$ram (ir%&it %al%&lator. The ,olta$e an' %&rrent are relate' ,ia the im)e'an%e of the loo). In the

%ase of ele%tri%all"5small loo)s3 im)e'an%e is the series %om!ination at the an$&lar fre9&en%"3 ω03 of

the rea%tan%e of the e4ternal in'&%tan%e3 e3 an' ra'iation resistan%e Rr 3 an' ohmi% resistan%e3 Ro3

of the %on'&%tor3

Z0= + + Ro Rr 6ω0(e+ i )3 112

an' terms o)eratin$ on the loo) ha,e the forms3

π =I = ( ) = I 0 %os 3

ℓ 11>

π =V( ) = =− 6 Z 0I 0sin 3

ℓ

here I0 is the %&rrent in the short5%ir%&it at = = 0 an' ℓ is the len$th of the attra%ti,e transmission

)ath. The a,era$e store' ma$neti% ener$" in the loo) in its sin$le mo'e at resonan%e is3

l1 /

Wm = ∫ I = '=ω0 > 0

l1 / / π =

= I 0 ∫ %os '= 117> 0 ℓ

1 /

= ℓ I 0 .@

The a,era$e store' ele%tri%al ener$" 'oes not nee' to !e %al%&late' se)aratel" sin%e the s"stem is at

resonan%e3 We = Wm 3 so3ω0 ω0

Wtotal = + We Wmω0 ω0

1 11C/

= /Wm = ℓ I 0 .ω0 >

/



One inter)retation that %an !e 'ran from the mathemati%s is ener$" %an !e a))lie' to the s"stem to

the )oint as the a,e$&i'e %an s&stain.

A 'is%&ssion of the %al%&lation of the )otential of free5s)a%e is ell o&tsi'e the s%o)e of this thesis3

hoe,er3 s&ffi%e it to sa"3 it is e4)e%te' this n&m!er to !e ,er" lar$e $i,en the ra'i&s of the %oils. Thisis a ,er" im)ortant feat&re3 hi%h %an !e e4)lore' !" this metho'3 an' it is s&$$este' to !e %on'&%te'

in f&t&re resear%h.

The 'issi)ate' )oer3 hile a )ro)ert" of the loa' )resent3 is also a )ro)ert" of the )oer s&staine' in

the ma$neti% fiel'. A resonant o!6e%t ill allo )oer to !e seen at the re%ei,er as3

1 / 1 /

)' = R I =( ) + G V( ) =/ /

l

1 / / π = = R I0 %os '= 11B∫/ 0 ℓ

l1 / / π =

+ G V0 ∫%os '=./ 0 ℓ

Its im)e'an%e "iel's the store' ener$" at resonan%e ithin the transmitte' a,e. For a non5isotro)i%

ra'iator if the emitte' )oer is Pra'W at a 'istan%e ' from the so&r%e3 the ma$nit&'e of the a,e?s

Po"ntin$ ,e%tor )oer )er &nit area is3

Pra'

Pt = / . 11@>π '

The ra'iate' )oer Pra' is re)la%e' !" PGt t3 here Pt is the )oer 'eli,ere' to the transmitter

antenna !" the am)lif"in$ sit%hes in the %ir%&it an' Gt is the transmitter antenna $ain. The ele%tri%

fiel' intensit"3 the res&lt of a%ti,e %o&)lin$3 at the antenna interfa%e is3

Eθ = Z0P3 t 11

here Z0 is the im)e'an%e of free5s)a%e 2. At a relati,el" lar$e 'istan%e from a non5isotro)i%

ra'iator3 the ele%tri% fiel' intensit"3 or the sim)le ma$nit&'e of the 'ist&r!an%e3 is3

PGt t 20PGt t

IE = 1/0π ⋅ / = Vm. 1/0θ >π ' '

If the ele%tri% fiel' intensit" an' )oer at the re%ei,in$ antenna are Ere% an' Pre% 3 res)e%ti,el"3 the

ma4im&m )oer a!le to )erform or* inter%e)te' !" the re%ei,er is3

2



/λ

Pr = ⋅ Pre%. 1/1>π

For a non5isotro)i% antenna of re%ei,er )oer $ain3 G 3 r an' $i,en 11B3 the re%ei,e' )oer is3

/ / / /

λ λ Ere% Ere%λ Gr

Pr = ⋅ Pre%Gr = ⋅ ⋅ Gr = ⋅ . 1//>π >π Z0 /π 1/0

The a,era$e ra'iate' )oer %an !e %onsi'ere' lost as far as the so&r%e os%illator is %onsi'ere'<

therefore3 the antenna !eha,es similar to a resistor as 'issi)atin$ )oer from the so&r%e. The ra'iation

resistan%e Rr is 'efine' as an a,era$e 9&antit"3

/

I0 1 Rr = 3 1/2/ P

here I0 is the in)&t %&rrent to the transmitter %oil3 see Fi$./.1 an' Fi$./.27. This is im)l"in$ that the

Am)re &nit here is re)resentati,e of a for%e. #o'if"in$ 1/23 ass&min$ a &niform %&rrent restri%te'

to a olt=mann 'istri!&tion3

/ >

Rr = /0π ( ) *a 3 1/>

here *a = /π a Sol,in$ for ra'iation resistan%e of the antenna3λ

/I0 > / /

Rr = = 10β A I 0 3 1/7/

here I0 is the )ea* am)lit&'e a))lie' to the antenna terminals. The ratio of se)aration !eteen the

%oils an' the $eometri% mean of the transmitter an' re%ei,er %oil ra'ii r i6 = r i 6 r ill !e the meas&re

of this ma4im&m $ra'ient alon$ the len$th aa" from the %oil alon$ the =5a4is3 ill&strate' in Fi$./.>>.

>



Fi$./.>>. Pro6e%tion of ma$neti% %&rrents ith hi$hest )otential %o&)lin$ )oints Hθ .

In the s)a%e %ontainin$ the ener$"3 the intensit" of the ma$neti% fiel' %an !e *non in a',an%e $i,en

the 'istan%e !eteen the %oils an' the ra'i&s of the )rimar" %oil. In a m&lti)le5t&rn it=5o&n' %oil of

n t&rns ith ra'i&s r i 3 the ma$neti% fiel' stren$th alon$ = from the %enter of the %oil alon$ the a4is

%an !e ritten as KCCL3

/µ0I n r ⋅ ⋅ i

H( ) =3 r i = . 1/C/ / 2

/ ( ) r i + =

Differentiatin$ ith res)e%t to r i shos that for r i = = /3 Hθ ill !e ma4imi=e' KCBL.

The ratio im)lies a $eometr" of the emitte' fiel's3 to e4)lore this3 it is rele,ant to 'is%&ss ma))in$ the

fiel' to 'etermine here ener$" is store' in $reatest 9&antities an' in hat forms. The am!ien%e of

the fiel' emission3 or fl&43 is 'es%ri!e' as a 'ire%t %onse9&en%e of the %ir%&it?s ele%tri%al )ro)erties. It

ill !e shon that for an antenna of a $i,en loo) ratio3 the fl&4 as ell as the intensit" are %al%&late'

to !e ma4im&m in terms of the e4%han$e of ener$" !eteen the transmitter an' the re%ei,er. As

'is%&sse' in /.1./3 the amo&nt of )oer transmitte' is %losel" relate' to the %o&)lin$ !eteen the

loo) %oils. The )otential %o&)lin$3 here ψ Hθ =ψ Hϕ 3 shifts the f&rther the 'istan%e aa" as a

meas&re of %oor'inates alon$ it.

/.2.2 Fiel' emissions

Res&lts from the sim&lation sho the emission )attern is a f&n%tion of the antenna $eometr"3 as

ill&strate' in Fi$./.>7.

7



Fi$./.>7. A,era$e )oer5o,er5s)a%e antenna ra'iation )attern loo*in$ into the %oil.

The res&lt of the %om)&tational )attern a$rees ith hat is e4)e%te'3 $i,en K20L3 an' ill&strates that

the )attern is a !eam of a )arti%&lar i'th.

It is no rele,ant to %reate a set of e4)eriments3 $i,en in the sim&lation res&ltin$ from the theoreti%al

frameor*3 here this !eam )attern is %onfirme' an' shon in hat ma$nit&'e $i,en a 'istan%e

aa" from its so&r%e. There ill !e to areas of ill&stration an' 'is%&ssion

1. At 'istan%es here 'istan%e is less than the ra'i&s of the %oils3 '$ r3 an'3

/. at 'istan%es here the 'istan%e is $reater than or e9&al to the ra'i&s of the %oils3 '- r.

It is an interestin$ to)i% of f&t&re resear%h to 'o a ,er" 'etaile' ma) of the ener$" 'istri!&te' in the

fiel' !eteen to resonant loo)s.

/.> The %ir%&it e4)eriments

Three 'istin%t sets of e4)eriments ere )erforme' ith the %ir%&it an' antenna mo'el 'es%ri!e'

thro&$ho&t /. These $ro&)s of e4)eriments are

1. Usin$ a sin$le transmittin$ an' re%ei,in$ %oil3

/. &sin$ a sin$le transmittin$ an' re%ei,in$ %oil3 &sin$ a %on'&%ti,e 'is% at the %enter of one or

!oth %oils3 an'3

2. &sin$ a )air of transmittin$ an' re%ei,in$ %oils.

Of ea%h of the e4)eriments3 'ifferent sets of )ro)erties are mani)&late' to sha)e an &n'erstan'in$ of

the !eha,ior of the arran$ement3 $i,en the theoreti%al im)li%ations in the mo'el.

/.>.1 #eas&re' )oer at a 'istan%e ith a sin$le loo) transmitter an' re%ei,er

The first e4)eriment as to test the theoreti%al frameor*3 the sim&lation3 an' %ir%&it mo'el3 set&)

shon in the form of Fi$./.1. Some ass&m)tions are the &se of off5the5shelf r&!!er5ins&late' %o))er

C



ire of fairl" hi$h 9&alit" to a))l" as m&%h of an o)timi=ation as )ossi!le. If o)timi=ation is *e)t in

min' '&rin$ e4)erimentation3 it is !&t a linear in%rease of )oer transmission ith more 'eli!eratel"

ma'e materials3 s&%h a nio!i&m5tin3 a to)i% of f&t&re resear%h. A s%hemati% of the %ir%&it is shon in

Fi$./.>C.

Fi$./.>C. S%hemati% 'ia$ram of the %ir%&it for a sin$le %o&)le'5mo'e.

Usin$ the %oil s)e%ifi%ation liste' in Ta!le /.B3 ith sin$le transmittin$ an' re%ei,in$ it= ire

antennas in%l&'in$ le$s3 shon in Fi$./.2@ an' Fi$./.23 of n&m!er i = 2 an' 6 = / of to

%entimeters of the total in'&%tan%es i3 6 3 at a transmission 'istan%e ,arie' !eteen %onta%t an' si4

%entimeters3 e4)eriments ere %on'&%te' to test the )ro)erties of the %oils an' ener$" transfer $i,en

the theoreti%al frameor*. The %al%&late' s)e%ifi%ation is liste' in Ta!le /..

Ta!le /.. E4)erimental %oils %al%&late' ele%tri%al s)e%ifi%ation I.

(a)a%itan%e ( ( i3 6 is set to as e9&al as )ossi!le3 100 nF. With an in)&t at the os%illator of >.7 ,olts

an' 1700 mA from an e4ternal D( s&))l"3 )oer meas&rements ere ta*in$ !" meas&rin$ the

,olta$e an' %&rrent )resent at the re%ei,er at ste)s !eteen %lose5)ro4imit" of the %oils an' 'istan%es

to si4 %entimeters.

The res&lts of )oer meas&rements at a 'istan%e are shon in Fi$./.>B.

B



Fi$./.>B. The )oer o,er 'istan%e shoin$ the )ro6e%tion fiel' H3 here (i ≈ ( 6.

In anal"=in$ the 'ata3 it is a))arent there is an area3 as 'es%ri!e' in the theor"3 hat is a )ro6e%tion

here the ener$" is nearl" %onstant a%ross the in%reasin$ 'istan%e. At the en' of the )ro6e%tion3 hi%h

%ontains a )ea* ,al&e3 transmission )oer o,er 'istan%e falls off linearl"3 as3 1 ./

r

The )attern is )resent hen the %a)a%itor ,al&e &se' ith the transmitter %oil is nearl" e9&al3 (i ≈ ( 6.

It is h")othesi=e' that the 'ri,in$ for%e of the )ro6e%tion is fee'!a%* )oer from the re%ei,er to the

transmitter a%ross the s)a%e. It is e4)e%te' that the )ro6e%tion o&l' !e less intense the more ea%h half

of the %ir%&it 'rifts aa" from resonan%e ith the other.

A se%on' set of e4)eriments ere %on'&%te' to test the )oer o,er a 'istan%e of 100 %entimeters. The

res&lts of )oer meas&rements at a lon$er 'istan%e are shon in Fi$./.>@.

Fi$./.>@. The )oer o,er a lon$er 'istan%e.

@



The !eha,ior ill&strate' in Fi$./.>@ is3 hile after the )ro6e%tion there is a falloff of 1 3 near to the/

r

)oint of 'istan%e of ten %entimeters3 falloff transitions to a rate of 1 an' is %ontin&o&s &ntil the noiser

floor of the en,ironment. A s&$$estion for f&t&re resear%h is to o!tain a more e4a%tin$ set of 'ata

%o&l' !e o!taine' !" &sin$ a s)e%tr&m anal"=er ith an atten&ator to )re,ent o,erloa'in$ at 'istan%esless than si4 %entimeters.

A thir' set of e4)eriments ere %on'&%te' &sin$ a thi%*er ire to see if )erforman%e %o&l' !e

im)ro,e'. It is e4)e%te' that a hi$her amo&nt of )oer o&l' !e o!ser,e' at ea%h meas&rement )oint

from the )re,io&s set of e4)eriments. The %oil )ro)erties are liste' in Ta!le /.10.

Ta!le /.10. E4)erimental %oils %al%&late' ele%tri%al s)e%ifi%ation II.

It as o!ser,e' '&rin$ e4)eriments that the %o&)lin$ %oeffi%ient in%reases ith a thi%*er ire3 hile

the m&t&al in'&%tan%e !eteen the %oils sta"s the same. To %on'itions ere a))lie' to the

e4)eriment one here the %a)a%itor at the transmitter an' re%ei,er are e9&al3 an' one here the

%a)a%itors are not e9&al.

The res&lts of )oer meas&rements of a thi%*er ire at a 'istan%e are shon in Fi$./.>.

Fi$./.>. The )oer o,er 'istan%e &sin$ a thi%*er antenna ire.

It is %onfirme' that the ass&m)tion of more )oer transmitte' )er 9&otient of 'istan%e in%reases. It is

also note' that the )oer seems3 for la%* of a more )re%ise or'3 :smoother; &sin$ the thi%*er )ie%e



of ire. There is a )ro)ert" of %on'&%tor thi%*ness $i,en ele%tron a%%eleration at the antenna hi%h is

an interestin$ )oint to st&'" in f&t&re resear%h. It is e4)e%te' a similar !eha,ior3 e4)e%t a $reater le,el

of )oer 'eli,er" in the same manner as Fi$./.> ill manifest o,er the len$th of 100 %entimeters as

in the )re,io&s set of e4)eriments.

A fo&rth set of e4)eriments ere %on'&%te' %han$in$ the ,al&e of the %a)a%itor !eteen the

transmitter an' the re%ei,er3 %om)onent (1 of Fi$././@ an' (> Fi$././ res)e%ti,el". In the )re,io&s

set of e4)eriments3 the ,al&e of %a)a%itors as set to !e as e9&al as )ossi!le3 $i,en the limitations of

the %om)onents an' the la!orator" meas&rement e9&i)ment ith 'ifferen%es no $reater than 10 )F. In

this set of e4)eriments3 the ,al&e of the %a)a%itan%e at the re%ei,er as re'&%e' in%rementall" to =ero3

or no %a)a%itor )resent. The res&lt of meas&rements o,er the 'istan%e is shon in Fi$./.70.

eteen the %ases hen (i ≈ ( 6 an' (i > ( 6 3 a %ontrastin$l" 'ifferent !eha,ior is o!ser,e'. It is

h")othesi=e' that the reason for the )ro6e%tion fiel' is )rimaril" a fee'!a%* of )oer from the

re%ei,er to the transmitter hi%h a))ears as a ma4im&m in %lose )ro4imit" to the transmittin$

antenna3 or hat this thesis %alls the )ro6e%tion fiel' H. From 'ata in the e4)eriments3 it is e4)e%te'

that the )ro6e%tion o&l' !e more intense the %loser ea%h half of the %ir%&it is at resonan%e ith the

other.

A%%or'in$ the 'ata from all e4)eriments3 the amo&nt of re%ei,e' )oer is '&e more stron$l" to the

in'&%ti,e %om)onent than the %a)a%iti,e %om)onent. This is %on%l&'e' sin%e the lam) is lit e,en if no

%a)a%itor is &se' at the re%ei,er.

Fi$./.70. The )oer intensit" o,er 'istan%e shoin$ linear falloff3 here (i > ( 6.

100



A fifth set of e4)eriments ere %on'&%te' hen (i ≈ ( 6 &sin$ a non5&niform it= loo) in'in$. At

the meas&rement )oints $i,en in the )re,io&s e4)eriments3 ener$" is still transmitte'3 altho&$h ith a

far less intensit"3 %al%&late' to !e at a!o&t //` of either set of e4)eriments &sin$ the 'ifferent ire

thi%*nesses. The )ro6e%tion fiel'3 H3 is also )resent. It is note' that the ener$" re%e)tion is tie' more'ire%tl" to the total ire len$th of the re%ei,in$ antenna than that of the $eometr" of the loo).

In this %ase hen (i ≈ ( 6 3 the ener$" %ontaine' in the )ro6e%tion3 H3 as ell as the area of the

#transmission here falloff is at a rate of 1 3 %alle' the mi'5fiel' re$ion3 is '&e to the in%i'ent E

/ 1r

#an' refle%te' E1! a,es3 as3

# 6ωt * = − # 6ωt * = + Ee 0 3 Ee! 0 3 1/B

1 1

are $i,en as os%illator )ro)erties3 here *0 is the ,a%&&m a,e n&m!er. The %al%&late' ,al&e is

shon in Ta!le /.7. There are )oints in the fiel' here there is %onstr&%ti,e an' 'estr&%ti,e

interferen%e in the !eam inferrin$ a )olari=ation in the )lane5a,es. These fiel's are trans,erse an'

onl" the tan$ential %om)onents e4ist at the !o&n'ar" )lane of = = 0. E9&ation 1/B re)resents the

ele%tri% fiel' )resent of the in%i'ent an' refle%te' a,es %onstr&%ti,el" interferin$. The 'etermination

of the stren$th of the ele%tri% fiel' at the %oil in'in$ is 'e)en'ent &)on the %&rrent at the it=

in'in$3

/ /sin β sin n∂

I E = I 0 sin / / 3 1/@θ β sin ∂

here3

π θ sin π ' sinθ βE = 3 ∂ = . 1/

θλ θ

The 9&antit" IE is the ele%tri% fiel' intensit" on the ire3 ' is the 'iameter3 an' θ is the in'e4 of θ

refra%tion. The )rofile of the antenna an' the )attern of the transmission ener$" are ill&strate' in

Fi$./.71.

101



Fi$./.71. Ra'iation )attern o!ser,e' at the re%ei,er.

The )lot to the left of Fi$./.71 shos the ra'iation )attern o!ser,e' from the re%ei,er. The )lot to the

ri$ht of Fi$./.71 shos a hi$her resol&tion !an'i'th. Folloin$ the theoreti%al 'is%&ssion of the

e4)e%te' !eamin$ )attern in /.2.23

The ra'iation )attern of Fi$./.71 is o!ser,e' here r3 )ro)ortional to the fiel' intensit"3 is

a))roa%hin$ ma4im&m at a 'istan%e from the transmitter. The )attern has its main5lo!e ma4im&m in

the = 'ire%tion (θ = 0) ith minor lo!es to the left an' the ri$ht in other 'ire%tions. eteen the lo!es

are n&lls in the 'ire%tions of =ero or minim&m ra'iation.

To s)e%if" the ra'iation )attern ith res)e%t to the fiel' intensit" an' )olari=ation re9&ires three

−1feat&res the θ %om)onent of the ele%tri% fiel' Eθ (θ φ 3 )(Vm )3 the φ %om)onent of the ele%tri%

−1fiel' Eφ (θφ 3 )(Vm )3 an' the )hases of these fiel's as f&n%tions of the an$les θ an' φ or δ θ θ (3φ )

an' δ θ φ ( 3 φ ). Di,i'in$ a fiel' %om)onent !" its ma4im&m ,al&e ill re,eal a normali=e' fiel'

)attern3 a 'imensionless n&m!er ith a ma4im&m ,al&e of &nit". The normali=e' fiel' )attern for the

θ %om)onent of the ele%tri% fiel' is3

Eθ (θ φ 3 ) Eθ ( ) θ φ 3 = . 120N Eθ ( ) θ φ 3

ma4

At 'istan%es relati,e to the si=e of the antenna an' small %om)are' to the a,elen$th3 the fiel' )attern

is e4)resse' in terms of )oer )er &nit area3 or the Po"ntin$ ,e%tor S (θ φ 3 ). Normali=in$ the )oer

ith res)e%t to its ma4im&m ,al&e3 as in the ele%tri% fiel' e4am)le3 "iel's a normali=e' )oer )attern

as a f&n%tion of an$le3 hi%h is a 'imensionless n&m!er3 ith a ma4im&m ,al&e of &nit". The

normali=e' )oer )attern is3

10/



S (θ φ 3 ) Pn ( ) θφ 3 = 3 121

S ( ) θ φ 3ma4

here S (θφ 3 )3 the Po"ntin$ ,e%tor e9&als3

/ /Eθ (θφ3 ) + Eφ (θφ3 ) −/

Wm 3 12/Z0

e4)resse' in re%tan$&lar %oor'inates on a 'e%i!el s%ale3

' =10lo$10 Pn (θφ3 )3 122

shon in Fi$./.71. The effi%ien%" of the transmitte' )oer in the )rimar" mo'e is $i,en !" the ratio

of its total !eam area3 Ω A3 %onsistin$ of the main5lo!e area3 Ω# 3 )l&s the minor5lo!e area Ωm. The

ratio of the main !eam area is the !eam effi%ien%"3

Ω#

η# = 3 12>Ω A

hile the ratio of the minor5lo!e area to the total !eam area is the stra" fa%tor3

Ωm

ξ# = 3 127Ω A

ith a total effi%ien%" of3

η ξ # + = # 1. 12C

In e4)eriments3 this ,al&e is shon to !e 0.BC@. The %al%&late' ,al&es of the ra'iate' )oer

%hara%teristi%s for the antenna are liste' in Ta!le /.11. These ere %al%&late' in the (ir%&it %al%&lator

softare.

102



Ta!le /.11. (al%&late' antenna %hara%teristi%s !ase' on K20L.

The o)erational %ir%&it in%l&'in$ the antenna )oerin$ a lam) a 'istan%e is shon in Fi$./.7/. The

fiel' shos a %ontin&o&s flo of ener$" from transmitter to re%ei,er is ta*in$ )la%e on s)atial an'

tem)oral ,e%tors. (on%e)tion of the ener$" in the fiel' as a %om!ination of the %&rrent at the s&rfa%e

of the antenna an' the 'ensit" of )ro)a$ate' ener$" a%ross free5s)a%e. Dire%tional fin'in$ KC@L is

)ossi!le in the s%heme !&t as not e4)lore' in 'e)th. Information transfer3 altho&$h not 'ire%tl"

e4amine' is also )ossi!le !" means of the same %ir%&itr"3 $i,en the sin&soi'al nat&re of the )oer

transmission. A''itions to the am)lifier %ir%&it o&l' hoe,er !e re9&ire' here a mo'&lation si$nal

is intro'&%e' to the %arrier. #o'ifi%ations to the re%ei,er o&l' hoe,er also !e re9&ire'. For

sim)li%it"3 the a,e is &n'erstoo' as a %hara%teristi% of its )otential )oer3 $i,en the %ir%&ital

%hara%teristi%s alrea'" 'is%&sse'.

Fi$./.7/. Protot")i%al ireless5)oer os%illator 'emonstratin$ the theoreti%al mo'el.

The ra'iation an' )attern as a f&n%tion of the fiel' emission is o!ser,a!le 'ire%tl" as a f&n%tion of

$eometr"3 an' mat%h the sim&late' )attern shon in Fi$./.>7. Or$ani% material 'oes not 'estr&%ti,el"

interfere ith the transmission. A s&$$estion for f&t&re resear%h is to &se metal )lates to %reate e''"

%&rrents in'&%e' !" a %han$in$ ma$neti% fiel' to le,era$e the si$nal to a h&man %onne%tion3 s&%h as

10>



the %o&)lin$ of the ner,o&s s"stem to an e4ternal %om)&ter ,ia ma$neti% %o&)lin$ in%l&'in$

mo'&lation as an information %arrier.

What has !een 'is%&sse' in this se%tion is for a sin$le loo) )oer s%enario e4%l&si,el")rimar" %oil

to )rimar" %oil %o&)lin$ of a sin$le mo'e. Intro'&%tion of metho's to enhan%e the ma$neti% fiel'

%&lminatin$ at the %enter of the loo) are treate' in the ne4t se%tion.

/.>./ #eas&re' )oer at a 'istan%e ith an enhan%e' ma$neti% fiel' at the )otential

This se%tion )ro)oses a metho' to enhan%e the ma$neti% fiel's in an in'&%ti,e5%o&)le'3 resonant5lin*

mo'el hi%h reali=es an im)ro,ement in )oer re%ei,e' at a 'istan%e.

The theoreti%al 'es%ri)tion of the im)ro,ement &tili=es a sin$le mo'e of )ro)a$ation K/>L3 an isotro)i%

a,e$&i'e transforme' anisotro)i%all" !" the motion of a ma$neti% fiel' !eteen a transmittin$ an'

re%ei,in$ element res&ltin$ from an a%%eleration for%e )resent in the ele%tri% fiel' at the so&r%e.

4 "Isotro)" in this %ase im)lies that %han$es to the ma$neti% fiel' are )resent e9&all" in !oth the 5

an' = )lane. The or*in$ 'efinition of ra'iation emitte' in a lo%all"5a))ro4imate' isotro)i%

a,e$&i'e is an e4%han$e of 'is%reti=e' ener$" hose ,elo%it" is atten&ate' !" a %hara%teristi%

resisti,e 9&alit" of its 'is)la%ement in the me'ia. S&%h resistan%e is also '&e to the ma$neti% fiel'

'ensit" an' %o&)lin$ errors e4)resse' !" 'rift of the resonant5lin* !eteen the elements. The

e4%han$e of ele%troma$neti% for%es in%reases anisotro)" an' forms the a,e$&i'e into a sha)e ith a

)arti%&lar %&r,at&re3 "iel'in$ a state of e9&ili!ri&m here %&rrents are o!ser,e' floin$ from the

transmitter to )erform a 9&antit" of or* at the re%ei,er.

A $oal is to e4amine a te%hni9&e of in%reasin$ the real5,al&e' fl&4 'ensit" an' !in' the %&r,at&re of

)ro)a$atin$ %har$es a%ross a %ir%&it %onsistin$ of to loo)s o)eratin$ at a %onstant resonant

fre9&en%". It is )roffere' that if the fle4ion of %&r,at&re is re'&%e' in the a,e$&i'e !" the

intro'&%tion of %on'&%tin$ 'is%s3 then the re%ei,e' )oer is in%rease' o,er lon$er 'istan%es. The

res&lt she's li$ht on the 'etails of the theoreti%al frameor* &n'er)innin$ the h")othesis that ireless

ener$" transfer in %o&)le'5mo'es is 'e)en'ent &)on the state an' or'er of the ma$neti% fiel'.

Theoreti%al an' e4)erimental set&) is shon in Fi$./.1@.

Fo&r sets of e4)eriments ere )erforme' a strai$ht forar' )oer transmission itho&t 'is%s3

transmission ith one 'is% in the transmittin$ loo)3 transmission ith one 'is% in the re%ei,in$ loo)3

an' finall" ith one 'is% in !oth the transmittin$ an' re%ei,in$ loo)s sim&ltaneo&sl".

107



The e4)eriment %onsiste' of a %ir%&it of to loo)s 'ri,en to a fre9&en%" of >70 *H= at a )oer le,el

of >.7 ,olts an' 1.1 am)eres as &se' to test the theoreti%al frameor*.

In or'er to sho the )re'i%tion of an in%rease in re%ei,e' ener$" '&e to the a''ition of a metal 'is%3 a

)osition is 'etermine' so that mo,ements aa" 'emonstrate the re'&%tion in ele%tri%al )otential an'

)oer linearl" as 1 3 affe%tin$ the matri4 !" intro'&%in$ a ,e%tor &./

r

The relati,e 'ifferen%e !eteen to loo)s set at a 'istan%e of a ra'i&s len$th3 2 %m3 at a )otential of

>.0 ,olts "iel's )oer meas&re' &sin$ a s)e%tr&m anal"=er of 11./C'm at a %al%&late' ,al&e of

>C0Ω3 altho&$h it is anti%i)ate' that it %o&l' !e a m&%h hi$her loa' im)e'an%e K20L3 $i,en its

resonan%e fre9&en%" ω0.

" mo,in$ the re%ei,er aa" from the transmitter3 so the less li*el" it is to interse%t ith the

h")er!oli% fiel'3 as as 'is%&sse' in the theoreti%al frameor*3 $i,en the %&r,at&re of the fiel'

esta!lishe' !" the %o&)lin$. It is shon here that the a''ition of the 'is% in%reases H3 or3 re'&%es the

%&r,at&re of the h")er!oli% f&n%tion here more :fiel' lines; ill interse%t ith it.

D&rin$ the e4)eriment3 the ele%tri% fiel' as meas&re' !" a )ro!e an' e4hi!ite' as a f&n%tion of the

loo) antenna l"in$ in the 4 5 " )lane an' %al%&late' in 8o&les< the ma$neti% fiel' as meas&re' !" a

Hall )ro!e an' e4hi!ite' as a f&n%tion of its )ro6e%tion alon$ the =5a4is an' %al%&late' in Teslas ith

the 'ominant a=im&thal ma$neti% fiel'ne$le%tin$ %ontri!&tions of minor a=im&thal fiel'sin

Teslas at a )ress&re of one atmos)here an' a tem)erat&re of /0 (.

Ta!le /.1/. (al%&late' 9&antities in the )ro6e%te' ma$neti% fiel'.

Ta!le /.12. (al%&late' 9&antities in the )ro6e%te' ma$neti% fiel'.

10C



E4amination of the res&lts %ontaine' in Ta!les /.1/ an' /.12 sho that an in%rease in ma$neti% fiel'

"iel's an in%rease in 9&anti=e' ener$" 'ensit". ensin$ as not %onsi'ere' in the e4)eriment3 onl"

ho ener$" transformation as more effi%ient &sin$ the 'is%s. A h")othesis for the in%rease'

effi%ien%" is that a %on,ersion of ener$" )ro6e%te' from the loo) is ta*in$ )la%e at a len$th %loser to

the re%ei,er hen it %ontains a 'is%. The in%i'ent a,es3 hose ma$nit&'e is nearer to here the

)oer is !ein$ meas&re'3 are more effi%ientl" %on,erte' into ele%tri% %&rrent.

Gi,en the $eometri% )atterns of Fi$./.1B an' Fi$./.1 of at the )oint here δ ma*es a stron$er

%ontri!&tion3 fiel' moment&m is h")er!oli% in its 'istri!&tion on a %ir%&larl" )erfe%t antenna s&rfa%e

an' the %on%entration of ener$" is %onfine' to the area insi'e the loo) alon$ its transmission a4es.

Hen%e3 it is more ill&strati,e $i,en the e4)erimental set&)3 to %al%&late fiel' 'e%a" o,er a 'istan%e

folloin$ H as %on,er$in$ to a )oint at the %enter of the re%ei,er.

In Ta!le /.1>3 the im)e'an%e is %al%&late' $i,en the )oer meas&re' an' that a,aila!le to the

s)e%tr&m anal"=era ran$e !eteen 70 an' C00Ω.

Ta!le /.1>. E4)erimental 9&antities of the ma$neti% loo) antenna.

The res&lts liste' in Ta!le /.1> sho the e9&i,alent n&m!ers in terms of the e9&i)ment meas&rin$ the

fiel's. The o!ser,ation )oint as %onsi'ere' to !e at the %enter of the loo)3 here the %oor'inate

03 03 0 as fo&n'. An" mo,ements in the )oint of o!ser,ation in,ol,e' tra6e%tories alon$ the =5a4is3

altho&$h 'efle%tions in the 4 5 " )lane ere also )ossi!le.

In the )ro6e%tion H3 the ma$neti% fiel' e4)erien%es minim&m 'e%a"< o&tsi'e the )ro6e%tion area3 the

)oer transfer falls at a rate of 1 &ntil a 'istan%e of / 3 r shon in Fi$./.723 then at a rate of 1/ r

r

&ntil )oer transfer falls into the noise floor shon in Fi$./.7>.

10B



" %onsi'erin$ the $eometr" of a ma$neti% fiel' as a ra'iation5o!6e%t3 it e4ten's !asi% %on%e)ts to

in%l&'e a 'efinition of ener$" e4%han$e !eteen resonant antenna. An e4tension is the %on%e)t of

%o&)le'5mo'es is s&%%essf&ll" a))lie' to ener$"5e4%han$e o!6e%t. This e4tension !ro&$ht into the

mo'el those )ro)erties of mo'al5mo'es as a h")otheti%al e4)lanation of hi$hl"5effi%ient ireless

ener$" transmission !" ma$neti% resonan%e hi%h has not !een a''resse' a'e9&atel" in the literat&re.

It has !een shon that the 'es%ri)tion in K/>L is s&ita!le to 'es%ri!e %o&)le'5mo'es.

Also3 to ta*e into a%%o&nt motions of intensit" alon$ the 4 5 " )lane3 it is s&$$este' !" the

e4)erimental e,i'en%e that altho&$h the theoreti%al frameor* is so&n'3 it is too sim)listi% a mo'el in

its )resent form to %a)t&re the f&ll %om)le4it" of the ener$" transformation in the fiel's.

Therefore3 a re%ommen'ation for f&t&re resear%h is to a''ress in more 'etail the 'ee)er as)e%ts of

ener$" e4%han$e an' transformation in the )olari=ation !eteen the loo) antennas manifestin$

ireless )oer transfer.

In this se%tion an in,esti$ation of %o&)le' mo'es3 ener$" e4%han$e3 an' ener$" transformation

!eteen to 'istant %ir%&lar loo) antennas as &n'erta*en. We ha,e intro'&%e' a no,el %on%e)t of

enhan%ement of the resonantl"5%o&)le' ma$neti% fiel' !" the a''ition of a%%retion 'is%s there!"

in%reasin$ the )otential of the antennas at the interfa%e of free5s)a%e. " the %onsi'erin$ the $eometr"

of a ma$neti% fiel' as 9&asi5str&%t&re' ra'iation folloin$ from a 9&antit" of for%e3 it has e4ten'e' the

!asi% %on%e)ts of antenna en$ineerin$ !" intro'&%in$ a 'efinition of ener$" e4%han$e !eteen

resonant loo)s as a f&n%tion of )ro6e%tion an' os%illation to ma4im&m entro)i% states.

Altho&$h it is ell5in,esti$ate' loo) antenna for ireless )oer transfer o,er short 'istan%es3 hat

as 'is%&sse' in this e4)eriment as the a''ition of a%%retion 'is%s to in%rease the ma$neti% ener$"

'ensit". What as also 'is%o,ere' as that s&%h 'is%s hel) to 'ire%t the ma$neti% ener$" alon$ the

resonant tra6e%tor" toar' the other resonant element. What as somehat of a s&r)rise as that the

'is%s an' their )resen%e in the fiel' seeme' to s&$$est that a )ossi!le e4)lanation for the a''ition of

ma$neti% ener$" as an attra%ti,e for%e !eteen the resonators in their )arti%&lar %o&)le'5mo'e.

A h")othesis to e4)lain the o!ser,ation is that the %on'&%ti,e material in the 'is% is res)onsi!le for

the ener$" shifts. This o&l' !e )ossi!le if the 'is%s %reate a first or'er ma$neti% fiel' K/>3 /CL<

altho&$h not 'ire%tl" o!ser,e'3 the ma$neti% fiel' e4)erien%es little falloff in the )ro6e%tion area H3

then %ontin&es falloff at a rate of 1 3 here the fiel' e4)resses an infinitesimal 3 then ass&mes a/

r

falloff of 1 . This is o!ser,e' !" in%reasin$ the 'istan%e !eteen the resonant elements an'r

10



meas&rin$ the ener$". eam 'i,er$en%e as not ta*en into a%%o&nt in this e4)eriment3 onl" the

attra%ti,e for%e !eteen ma$neti% fiel's3 as )resente' in /.>.1. A 'etaile' e4amination of !eamin$3

'i,er$en%e an' if s&%h a)ert&res %an !e mani)&late' is a rele,ant to)i% of f&t&re resear%h.

If f&rther e4)eriments %o&l' reinfor%e the 'etermination that the fiel' is transformin$ a%%or'in$l"3 it

o&l' s&$$est that in e4ten'in$ the )resen%e of the ma$neti% fiel' from the mi'5fiel' re$ion to the

far5fiel' im)lies a &sa$e of !oth Einstein?s s)e%ial relati,it"3 hi%h 'oes not in%l&'e an ӕther !&t

mo,in$ frames an' a%%eleratin$ ele%tro'"nami%s KCL hi%h in%l&'es an ӕther. Altho&$h on the

s&rfa%e it o&l' seem there is a %ontra'i%tion in the re)resentation of ,elo%ities3 the a&thor offers that

)erha)s there are more )oerf&l %ontri!&tions of the notion of time 'ilation to ele%tro'"nami%s than

meets the e"e.

/.>.2 The %ir%&it e4)eriment ith s&!ten'e' an$les in h&man )ro4imit"

One as)e%t of the &tili=ation of ireless )oer3 as shon in the intro'&%tion3 as the form atta%he' to

the h&man !o'". This se%tion ill 'es%ri!e an a))li%ation of the os%illator an' mo'el3 ill&strate' in

'etail in /3 for )oerin$ s&!%&taneo&s im)lants an' %om)onents3 irelessl" %onne%tin$ the h&man

ner,o&s s"stem to a ma%hine.

Poer 'eli,er" for !iome'i%al im)lants is a ma6or %onsi'eration in their 'esi$n for !oth meas&rement

an' stim&lation. When )erforme' !" a ireless te%hni9&e3 transmission effi%ien%" is %riti%all"

im)ortant not onl" !e%a&se of the %osts asso%iate' ith an" losses !&t also !e%a&se of the nat&re of

those losses3 e.$. e4%essi,e heat %an !e &n%omforta!le for the in'i,i'&al in,ol,e'. In this se%tion a

metho' an' means of ireless )oer transmission s&ita!le for !iome'i%al im)lants is !oth 'is%&sse'

an' e4)erimentall" e,al&ate'. The )ro%e'&re initiate' is %om)ara!le in si=e an' sim)li%it" to those

metho's alrea'" em)lo"e'< hoe,er3 some of Tesla?s f&n'amental i'eas ha,e !een in%or)orate' in

or'er to o!tain a si$nifi%ant im)ro,ement in effi%ien%". This se%tion %ontains a theoreti%al !asis for

the a))roa%h ta*en altho&$h the em)hasis is on )ra%ti%al e4)erimental anal"sis.

Im)lanta!le !iome'i%al 'e,i%es ha,e3 )arti%&larl" in re%ent "ears3 re%ei,e' m&%h attention ith re$ar'

to their a))li%ation for a ,ariet" of &ses in,ol,in$ !oth stim&lation an' monitorin$. In 'ire%t %ontrast

to eara!le monitorin$ health%are s"stems KCL3 one of the *e" iss&es for im)lante' 'e,i%es is the

satisfa%tor" )ro,ision of )oer on an on$oin$ !asis. For short term e4)erimentation it is 9&ite

)ossi!le for s&ffi%ient trans%&taneo&s )oer to !e )ro,i'e'3 a $oo' e4am)le of this !ein$ in the 2

month im)lantation testin$ of the Utah Arra" KB0L. For lon$5term im)lantation hoe,er )oer nee's

%an ,ar" %onsi'era!l" KB13 B/L.

110



Im)lants s&%h as the Utah Arra" ha,e in fa%t !een &se' for a ,ariet" of a))li%ations3 in%l&'in$ an

alternati,e sensor" in)&t an' ne&ral %ontrol of )rostheses as ell as a ne means of %omm&ni%ation

KB2L. Trans%&taneo&s )oer 'eli,er" for an" lon$5term em)lo"ment ill hoe,er ala"s %arr" ith it

the %han%e of infe%tion for the re%i)ient as ell as the )ossi!ilit" of me%hani%al le,era$in$. (learl" an

on!oar'on %hi) )oer )i%* &) 'e,i%e that a,oi's trans%&taneo&s )oer s&))l" is an attra%ti,e

alternati,e hi%h ill most li*el" res&lt in more i'es)rea' &se of the te%hnolo$" KB>L.

Some 'e,i%es3 hi%h re9&ire se,eral milliam)s of stim&latin$ %&rrent3 s&%h as those &se' in the 'ee)

!rain stim&latin$ ele%tro'es for the treatment of Par*inson Disease3 nee' f&ll !atter" im)lantation

KB7L. This te%hni9&e then s&ffers from the re9&irement of )erio'i% !atter" re)la%ement. On the other

han'3 it is )ossi!le to %onsi'er ener$" har,estin$ ithin the !o'". This a))roa%h is hoe,er still in its

infan%" an' its )ra%ti%al &sef&lness is "et to !e f&ll" reali=e' KBCL.

Wireless )oer 'eli,er" offers the a',anta$es that it re'&%es the ris*s )arti%&larl" '&e to infe%tion

asso%iate' ith either !atter" re)la%ement s&r$er" or a trans%&taneo&s s&))l". In'&%ti,e %o&)lin$ %an

!e em)lo"e' for s&%h )oer transfer3 !&t the effi%ien%" of transfer is a ,er" sensiti,e f&n%tion of

%oil 'imensions an' the 'istan%e !eteen them. Res&ltant effi%ien%ies for !iome'i%al im)lants are3 as

a res&lt3 $enerall" ,er" lo KBBL3 )arti%&larl" so in a )ra%ti%al3 or*in$ en,ironment. The most

attra%ti,e s%heme is ar$&a!l" therefore %o&)lin$ !eteen ma$neti%all" resonant o!6e%ts KB@L.

While the i'ea of ireless )oer has !een e4)lore' e4tensi,el" in the literat&re ith se,eral

%om)etin$ )oer 'eli,er" te%hni9&es !ein$ %onsi'ere'3 the most 'ire%tl" rele,ant are those in hi%h

)oer is not 'ire%te'3 !&t rather is a!sor!e' KB3 @0L. In essen%e3 the re9&ire' loa' or 'ra5'on

%&rrent is 'etermine' !" the o)erational %onstraints an' not !" the !eamin$ metho' em)lo"e'. This

feat&re allos ma$neti% %&rrents to e4ist in a )assi,e mo'e3 i.e.3 the ener$" 'oes not )ersist in the

en,ironment %ontin&o&sl" !&t rather is ta))e' into on5'eman'. As a %onse9&en%e of this3 less ener$"

is %ons&me' to 'ri,e the %ir%&its. What is ma'e a))arent here is the small si=e an' relati,el" fe

%om)onents re9&ire' in the metho' 'es%ri!e' as %om)are' to the relati,el" lar$e an' effi%ient amo&nt

of )oer transmitte'.

The transmission fre9&en%" sele%te' for the s"stem is >70 *H=3 '&e )rimaril" to the fa%t that that no

a',erse !iolo$i%al effe%ts ha,e th&s far !een re)orte' at this fre9&en%" K2>L. Ne,ertheless3 in the

s%heme of ma$neti% resonan%e for )oer transmission3 it has alrea'" !een 'etermine' that the

transmission 'oes not intera%t ith off5resonant o!6e%ts K21L. Usin$ an on5!oar' miniat&re soli'5state

hi$h5)oer am)lifier3 e 'emonstrate here a )rotot")e %a)a!le of 'eli,erin$ relia!le o)eratin$

%hara%teristi%s s&ita!le for )ra%ti%al im)lants re9&irin$ a stea'" )oer s&))l" of an"thin$ from >.7 to

1/ ,olts '%.

111



(ontrar" to the )&r)oses of other im)lementations K@13 @/3 @23 @>3 @7L3 of s)e%ifi% interest here is to

9&estion hether a s%heme %o&l' !e a%%om)lishe' &sin$ small %oils of a fe t&rns 'ri,en !" a sim)le

am)lifi%ation %ir%&it. The $oal of this e4er%ise is therefore to esta!lish a 'ri,in$ %ir%&it ith a

minim&m amo&nt of %om)onents3 there!" re'&%in$ 'e,i%e %om)le4it".

As a )h"si%al 'emonstration of the o)eratin$ %hara%teristi%s of the metho' 'es%ri!e'3 a set

re9&irement is to maintain a s&ffi%ientl" hi$h5)oere' si$nal relia!l" )oerin$ a lam) an' motor.

The reasonin$ !ein$ that if the te%hni9&e %an f&n%tion ell in terms of s&%h e4ternal re9&irements3 it

ill %ertainl" )erform a'e9&atel" in the %ase of an im)lant s)e%ifi%ation. In this re$ar'3 it has not !een

the $oal to 'eli,er )oer at the sort of 'istan%es re)orte' on in s&%h as K@/3 @2L here effi%ien%ies of

>0` are )erha)s the &))er tar$et. Rather here to attem)t to %om)are 'ire%tl" ith transmission o,er

relati,el" short 'istan%es a fe %entimeters3 as re)orte' on in s&%h as K@CL3 ith a hi$h effi%ien%" of

transmission o,er B7` !ein$ the tar$et.

An o,erri'in$ aim is to o,er%ome )oer s&))l" iss&es3 eminentl" a))arent in the st&'" of !iome'i%al

im)lants3 !" reali=in$ a ireless s%heme hi%h is s&ffi%ientl" )oerf&l s&%h that an im)lant %an

relia!l" re%ei,e its )oer remotel". Hen%e3 'ire%tion of )oer transmission as ell as the si=e of

te%hnolo$" in,ol,e' has !een im)ortant in this st&'".

The %on%e)t of ireless )oer transfer3 first 'es%ri!e' !" 8ose)h armor an' Po"ntin$3 an'

e4)erimentall" ,erifie' !" Tesla3 has !een ill&strate' in the literat&re as a ,ia!le metho' to trans)ortele%tri%al %&rrent !eteen 'istant )oints K/>3 B@3 @C3 @B3 @@3 @L. The e4tension of this metho' herein

maintains a si$nifi%ant amo&nt of &sef&l )oer transmitte' at intensities of less than 17 ,olts in

hi%h3 a%ross a ,ol&me of air3 ma$neti% a,es are e4%han$e' !eteen to or more %o&)le'

resonators.

(onsi'er a %ir%&it %onsistin$ of a sin$le loo) of ins&late' ire o&n' in s&%h a a" as to %reate a

%ir%&lar loo) of a fe t&rns. This loo)3 %onne%te' in )arallel to a %a)a%itor3 !e%omes one5half of a

resonant %ir%&it. This is 'esi$nate' as the transmitter t. The len$th of the ire loo) in t is thenre)li%ate' to %onstr&%t a similar loo) )la%e' on an in'e)en'ent %ir%&it !oar' )ositione' a 'istan%e

aa" an' hi%h is %onne%te' in )arallel ith a %a)a%itor of the same rea%ti,e ,al&e an' a loa'. In its

entiret" this !e%omes the se%on' half of a resonant %ir%&it3 'esi$nate' the re%ei,er r.

Ea%h half of the resonant %ir%&it is )la%e' a 'istan%e from the other. The transmitter is %onne%te' to a

)oer so&r%e s&%h that the ( %ir%&it is e4%ite'. e%a&se of the s"mmetr" !eteen ea%h half of the

%ir%&it3 ma$neti% a,es flo from transmitter to re%ei,er at resonan%e fre9&en%" f 0 ith effi%ien%"

11/



η. When ener$i=e'3 this %ir%&it en$a$es an ele%tri%al fiel' in the loo) in t3 %reatin$ a ma$neti% fiel'3

hi%h is %o&)le' to the loo) in r. " )la%in$ the loa' in )arallel to the loo) in r3 ma$neti% ener$" is

%on,erte' into ele%tri%al %&rrent. The arran$ement is ill&strate' in Fi$./.1.

A n&m!er of re%ei,ers %an !e fe' from one %ommon transmitter3 there!" o)enin$ &) the )otential to

)oer mo!ile ro!oti% )latforms !" this means. A theoreti%al %onfi$&ration of the %ir%&it in a ro!oti%

im)lementation is a sin$le fi4e'5)osition transmitter t 'eli,erin$ ener$" to 'istant re%ei,ers r an' s.

S&%h a %ir%&it has !een %onstr&%te' an' teste' for the intensit" of the in'&%e' %&rrent at 'ifferent

'istan%es aa" from the transmitter. The re%ei,ers are $ro&)e' into to %ate$ories

1. a% mo'e li$htin$ an in%an'es%ent lam)3 an'3

/. '% mo'e t&rnin$ a motor.

Either of the re%ei,er r or s is a lam) or a motor< these !e%ome s"non"mo&s )latforms in here the"

are re9&ire'. The e4)erimentall"5teste' ran$e of this metho' is ill&strate' in Fi$./.77 an' Fi$./.7C.

Fi$./.77. E4)erimentall"5teste' orientation of m&lti)le re%ei,ers r an' s ith referen%e to ea%h )osition at the

transmitter t hen there is no an$le of rotation aa" from a4ial ali$nment art = 0. The stan'in$ hei$ht of all

%oils are e9&al.

112



Fi$./.7C. E4)erimentall" teste' orientation of m&lti)le re%ei,ers r an' s3 thro&$h ith an an$le of rotation θ

aa" from a4ial ali$nment3 ith referen%e to ea%h )osition at the transmitter t. The stan'in$ hei$ht of all %oils is

e9&al.

When &sin$ to or more re%ei,ers3 J ill !e a''iti,e to the sol&tion for ea%h re%ei,er from the

)ers)e%ti,e of the entire effi%ien%" %al%&lation. If intera%tion !eteen the re%ei,ers is %onsi'ere'3

mo'if"in$ C then $i,es KB@L

/ / /(κi6JJi 6 )(κ 6*J J 6 * )(κ*lJ J* l ) ηi 6 * 3 3 3 ...n= . 12B / / / / /(1+ κi6JJi 6 )(1+ κ*lJ J* l )+ κ 6*J J 6 * 1+ κ 6*J J 6 * + κ*lJ J* l

If it is 'esira!le to %onsi'er the intera%tion of the ma$neti% fiel's ea%h of the elements transmitter3

re%ei,er one3 re%ei,er to3 or more o)eratin$ on ea%h other. If the %onsi'eration is $i,en that the

o)eration is limite' to !eteen a transmitter an' re%ei,er one3 re%ei,er to e4%l&si,el"3 i.e.3 not

intereste' in the %o&)lin$ !eteen ea%h re%ei,er !&t noti%in$ the in%rease in κ. The total is then

$i,en K>7L !"3

1 η = 3 12@

i 6 * 3 3 3 ...n 1 1 1 1 1 1 1 R R 6 *1 + / + + + 1+ 1+

κ J R J R J R J J

i6 i i 6 6 * * 6 *

here the loa'e' 9&alities3 Ji3J 6 are a )ro)ert" of the %a)a%itan%es ( ( i3 6 3 resistan%es R R i3 6 3 an'

in'&%tan%es i3 6 of the %ir%&its %ontainin$ the %oils are 'efine' in B/. O,erall hoe,er3 altho&$h

the se)aration 'istan%e e4amine' here as relati,el" short3 al!eit a))ro)riate for !iome'i%al im)lants.

Gi,en the ra'ii of the %oils3 a hi$h amo&nt of )oer is 'eli,ere' at hi$h effi%ien%".

11>



The !eha,ior of the re%ei,e' %&rrent as %ontrastin$l" 'ifferent !eteen ea%h mo'e. In a% mo'e3

there is a meas&re of fee'!a%* refle%te' a%ross the %ir%&it3 hi%h in this %ase3 o&l' initiall" s&$$est

an im)ro,ement of )erforman%e3 as ill&strate' in Fi$./.7@.

Hoe,er3 this %omes at a %ost. The 'istan%e from the transmitter as re'&%e' to B %entimeters in a%

mo'e hile it as hel' at ten %entimeters in '% mo'e. Usin$ the mo'es ill&strate' in the ne4t se%tion

'es%ri!e the o)erational free'om of the mo!ile re%ei,er.

a% mo'e

It %an !e seen3 from Fi$./.7B3 that hen &sin$ one a% mo'e3 si$nal )erforman%e of a re%ei,er r as

ma4im&m hen its 'istan%e 'tr from t as to %entimeters an' its a4ial 'ifferen%e art as no

$reater than three %entimeters to the left or to the ri$ht of t3 as as ill&strate' in Fi$./.77. #eanhilethe ma4im&m 'istan%e for the re%ei,er hen the lam) as a!sor!in$ eno&$h )oer to i$nite its

filament3 as o!ser,e' at si4 %entimeters.

Fi$./.7B. The a% )oer intensit" seen !" a re%ei,in$ antenna ith a4ial ali$nment here (i ≈ ( 6.

When &sin$ to a% mo'es meanin$ the &se of to re%ei,ers in a% mo'e M essentiall" to lam)s

o)eratin$ sim&ltaneo&sl"3 the si$nal )erforman%e of one re%ei,e r3 hile %om)ara!le to one A(

mo'e3 %ons&me' )oer rele,ant to its )ro4imit" ith t3 as e4)e%te'3 !&t atten&ate' !" a 'istant

re%ei,er s. In e4)eriments the ma4im&m 'istan%e hen the lam)s ere a!sor!in$ eno&$h )oer to

i$nite their filaments3 as o!ser,e' at si4 %entimeters. Hoe,er3 !" linin$ &) r an' s ith minim&m

a4ial 'is)la%ement3 the %&rrents %onstr&%ti,el" interfere' %reatin$ a solenoi' str&%t&re. This metho'

e4ten'e' the ran$e of &sef&l in'&%e' %&rrent to ten %entimeters.

11C



When &sin$ one a% mo'e an' rotatin$ at an an$le θ as ill&strate' in Fi$./.7C3 the si$nal )erforman%e

of a re%ei,er r as o!ser,e' hen θ as !eteen >0 an' 0 'e$rees offset from t. In this ran$e3 the

)oer falls off an' remains stea'" at θ = 0. The )ea* seen at to %entimeters is '&e to fee'!a%* of

ener$" hen the %ir%&it is near o)timal t&nin$ !eteen the %oils.

At a 'istan%e 'tr 3 θ = C03 the lam) as at )ea* !ri$htness ithin 1.7 %entimeters3 mo,ement alon$

the a4ial len$th re'&%e' the si$nal to its minim&m at three %entimeters. Hoe,er3 mo,ement alon$

the tra6e%tor" of θ re'&%e' the si$nal to a minim&m at to %entimeters. E4a%tl" the same !eha,ior

as o!ser,e' for re%ei,er s. Tentati,e res&lts s&$$est that the fiel' is %onsistent a%ross its manifol'3

i.e.3 %onsistin$ of a finite s)atial $eometr" s"mmetri% alon$ its a4ial len$th.

When &sin$ to a% mo'es an' rotatin$ at an an$le θ 3 si$nal )erforman%e of the re%ei,er r3 hile

%om)ara!le to one a% mo'e3 it as fo&n' that for se$ments of its rotation thro&$h θ ener$" as

e4%han$e' !eteen the to re%ei,ers.

When intro'&%in$ a thir' an' fo&rth a% mo'e3 e.$.3 a''in$ a thir' an' fo&rth re%ei,er3 the $eometri%

)ositionin$ of the intensit" of the ma$neti% fiel' o!6e%t remaine' %onstant an'3 rele,ant to the )osition

of the other re%ei,ers3 more ener$" as a!sor!e' at the )ea*3 as alrea'" 'es%ri!e'.

'% mo'e

When e4)erimentin$ ith one '% mo'e3 &sin$ a $ermani&m re%tifier !ri'$e3 the si$nal )erforman%e of

a re%ei,er r as ma4im&m hen the 'istan%e 'tr as at its minim&m )ro4imit" to t an' the a4ial

'ifferen%e art as no $reater than 12 of the ra'i&s to the left or to the ri$ht of t3 as ill&strate' in

Fi$./.7@.

Fi$./.7@. The '% )oer intensit" seen !" the re%ei,in$ antenna ith a4ial ali$nment. 11B



In the e4)eriment3 motor )erforman%e as o!ser,e' at the re%ei,er ith s&ffi%ient tor9&e to 'ri,e the

heel of a ro!ot hen the re%ei,in$ %oil as %losest to t3 ith the )oer in'&%e' fallin$ off stea'il"

&) to a 'istan%e of fo&r %entimeters3 here it as 6&st s&ffi%ient to 'o the same. When the motor as

not loa'e'3 it e4hi!ite' a similar )erforman%e &) to a 'istan%e of nine %entimeters.

The ne4t e4)eriment in,ol,e' &sin$ to '% mo'es meanin$ the &se of to re%ei,ers in '% mo'e M

essentiall" to motors o)eratin$ sim&ltaneo&sl" in a $ermani&m re%tifier !ri'$e. Res&lts ere fo&n'

to !e %om)ara!le to one '% mo'e in terms of )oer %ons&m)tion rele,ant to its )ro4imit" ith t3 as

e4)e%te'3 !&t the effe%ts of the 'istant re%ei,er atten&ate' the )oer 'eli,ere' at s.

The ma4im&m 'istan%e3 hen there as fo&n' to !e eno&$h )oer to 'ri,e a loa'e' motor3 as

o!ser,e' at %entimeters. " linin$ &) r an' s ith minim&m a4ial 'is)la%ement3 the %&rrents ere

ma'e to %onstr&%ti,el" interfere3 %reatin$ a solenoi' str&%t&re. In this a"3 s&ffi%ient in'&%e' )oer to

'ri,e a motor %o&l' !e e4ten'e' to a 'istan%e 1/ %entimeters.

When &sin$ one '% mo'e an' rotatin$ at an an$le θ 3 as ill&strate' in Fi$./.7C3 si$nal )erforman%e of

a re%ei,er r as o!ser,e' hen θ as !eteen >0 an' 0 'e$rees offset from t. In this ran$e3 the

l&mines%en%e fell off3 remainin$ stea'" at θ = 0.

When &sin$ to '% mo'es an' rotatin$ at an an$le θ 3 si$nal )erforman%e of the re%ei,er r3 hile

%om)ara!le to one '% mo'e3 e4hi!ite' se$ments in its rotation thro&$h θ here ener$" ase4%han$e' !eteen the to re%ei,ers.

When &sin$ one a% an' one '% mo'e3 si$nal )erforman%e of the re%ei,ers as fo&n' to !e %om)ara!le

in ea%h %hara%teristi% )erforman%e !ase' on e4)erimental o!ser,ations. Ener$" is e4%han$e' so that it

is 'istri!&te' !" the )osition of the re%ei,er rele,ant to its m&lti)le !ase' on the total ener$" a,aila!le

in the %ir%&it. " in%reasin$ )oer to the so&r%e3 hi$her intensit" %&rrents are a,aila!le to ea%h

re%ei,er.

While 'emonstratin$ the transmitter $enerates ma$neti% a,es hi%h %an )ass thro&$h alls an' 'oes

not intera%t ith off5resonant o!6e%ts s&%h as h&mans or animals3 there are reasons here it is

&n'esira!le to e4%han$e ener$" ith a %ir%&it hi%h shares %hara%teristi% )ro)erties of the resonant

%ir%&it. In s&%h %ases3 shiel'in$ of a resonant o!6e%t is )ossi!le !" en%losin$ the ma%hine in thin

metalli% foil. It as o!ser,e' ma$neti% a,es )resent in the s"stem %annot )enetrate s&%h a s&!stan%e.

11@



/.>.> O)timi=ation of the %ir%&it

On e4amination of B/3 an' 'is%&sse' in /./3 it %an !e seen that it is )ossi!le to affe%t the 9&alit"

fa%tor of the %oil !" a''in$ resistan%e into the re%ei,er %ir%&it. This is a%hie,a!le )arti%&larl" hen

&sin$ hi$h5atta$e resistors at lo 9&antities of resistan%e.

For e4am)le3 !" a''in$ a 700 Ω resistor in )arallel ith the loa' this raises the 9&alit" fa%tor of the

%ir%&it there!" %a&sin$ the lam) to $i,e off a !ri$hter l&mines%en%e. In 'oin$ so it also im)ro,es the

effi%ien%" of the %ir%&it. It has to !e sai' hoe,er that the a''ition of resistan%e into a '% %ir%&it as

not teste' here. A''in$ series resistan%e is s&))orte' !" or* in K@/L.

" in%reasin$ the len$th of ire after the re%tifier %ir%&it3 the &sef&l ran$e of the '% si$nal as

e4ten'e' o&t to tel,e %entimeters. At hi$her ,olta$es3 i.e.3 those at nine ,olts an' $reater3 the heat

$i,en off as fo&n' to !e minimal in the %ir%&its teste'. What heat there as fo&n' to !e %on%entrate'

aro&n' the %a)a%itor in )arallel ith the loo) an' as linearl" relate' to the in)&t %&rrent.

The ma4im&m )oer as transmitte' in this e4)eriment !" 'ri,in$ the am)lifier %ir%&it at tel,e

,olts. At this le,el3 there as fo&n' to !e si$nifi%ant heatin$ of the transmitter?s %a)a%itor. (han$in$

the in)&t ,olta$e altere' the o&t)&t ,olta$e3 i.e. there as no a))arent stora$e of ma$neti% ener$" in

the ma$neti% fiel'.

An" in%rease or 'e%rease in )oer s&))lie' 'oes not alter the lo%ations here the %oils a%hie,e a

ma4im&m or minim&m in their resonan%e %o&)lin$3 rather it in%reases or 'e%reases res)e%ti,el" the

a,aila!le )oer at those lo%ations.

The ma4im&m ener$" transmitte' in e4)eriments as 2C *8 o,er the )erio' of one ho&r. The a))arent

limitin$ fa%tor in this s%heme as the n&m!er of am)lifiers e em)lo"e'. It o&l' !e of interest to

a))l" f&rther am)lifiers3 there!" in%reasin$ the )oer in the s"stem3 in or'er to &n'erstan' ho m&%h

%&rrent %an !e )ra%ti%all" trans)orte' in the $eometri% s)a%e.

Perha)s the most si$nifi%ant res&lts o!taine' from e4)eriments ere those relatin$ to the effi%ien%" of

)oer transfer. Here the aim as to &se small3 sim)le an' effi%ient means3 at 'ire%tl" %om)ara!le

'istan%es to those re)orte' in K>>3 B@3 @CL. The main $oal of the resear%h )ro$ram !ein$ to &ltimatel"

'eli,er )oer for !iome'i%al im)lants M as 'es%ri!e' in those or*s.

In this %ase3 ith to %oils3 !oth of ra'i&s three %entimeters3 set si4 %entimeters a)art3 a meas&re'

)oer transfer effi%ien%" of BC.@/` as a%hie,e'. This %om)ares 'ire%tl" an' e4tremel" fa,ora!l"

ith the @/` re)orte' in KB@L an' the @0` re)orte' in K>>L. It is orth a''in$ that it %om)ares e,en11



more fa,ora!l" ith the >7` re)orte' in K@CL altho&$h it is orth stressin$ that the $oal in that

se%tion as not merel" hi$h effi%ien%".

As an interestin$ asi'e3 it is also a))arent that the same metho' of )oer 'eli,er"3 as teste' in

e4)eriments3 %an also !e em)lo"e' in the %ase of m&lti)le re%ei,ers. With the a''ition of a''itional

re%ei,ers3 the transfer effi%ien%" remaine' at a minim&m of @/`3 hilst 'i,i'in$ the )oer M this

!ase' on the )osition of the re%ei,ers relati,e to the transmitter. (learl"3 this %an mean that the

metho' intro'&%e' here %o&l' !e &sef&l to )oer se,eral meas&rement an'or stim&latin$ )oints

ithin one !o'"3 itho&t si$nifi%ant )oer 'e$ra'ation3 from one sin$le e4ternal transmission so&r%e.

Poer iss&es fa%e' !" K0L 'eal ith s"stem %onsistin$ of mo'esa%ti,e an' slee)an' are

%on%erne' ith )oer %ons&m)tion of a %ir%&itr" at ran'om inter,als3 s&%h as hen a ,i!ration

tri$$ers the )ie=oele%tri% res)onse to $enerate ,olta$es an' %&rrents. Re$ar'in$ K1L3 the notion of

!ea%on5'elimite'3 &ltra5lo !o'" netor* is %om)ara!le to the resear%h here in that the" share to

%ommon themes

1. sen'in$ )oer for me'i%al im)lants or !o'" monitorin$ a))arat&s3

/. sen'in$ information at an a))re%ia!le 'ata rate3 hoe,er3 the %&rrent resear%h onl"

a%*nole'$es the a))arat&s has the a!ilit" to transmit 'ata $i,en the a,eform is a

stea'"5state sin&soi'.

e%a&se of the im)ortan%e of the s&!6e%t area3 a !o'" of resear%h !oth theoreti%al an' e4)erimental

has a))eare'3 )arti%&larl" in the last 'e%a'e3 on this to)i%. As a res&lt e ha,e attem)te' to ma*e it

%lear ho the or* %om)ares fa,ora!l" ith that of others in terms of the res&lts o!taine'. It is

im)ortant hoe,er to also $i,e an in'i%ation of ho these im)ro,ements in res&lts ha,e !een

a%hie,e' thro&$h ori$inal 'ifferen%es in f&n'amental 'esi$n.

It m&st !e reali=e' that the !asi% i'eas &n'erl"in$ the 'ifferen%es ha,e !een s)ar*e' !" ret&rnin$ to

some of the f&n'amental or* of Tesla3 armor3 an' Po"ntin$3 stemmin$ from the ori$inal #a4ell

e9&ations. As s&%h it is ma'e %lear in this se%tion that the $eometr" of the ma$neti%all" %o&)le' %oils

e em)lo"e' as not stan'ar'? !&t rather e4hi!ite' 'ifferen%es to that &se' in other or* KB@3 B3

@03 @73 @@3 /L. Here is a list of 'ifferen%es.

Firstl"3 the antenna ma'e &se of a it= in'in$3 not o&n' in an" )arti%&lar orientation. This same

a))roa%h as in fa%t &se' in KB@L3 !&t not so in K>>3 @CL. Se%on'l"3 the ra'i&s of the re%ei,er an'

transmitter %oils in e4)eriments as the same three %entimeters3 others meanhile &se' one ra'i&s

1/0



for the transmitter an' another for the re%ei,er K>>3 B3 @03 /L. On to) of this there ere ,er" fe

t&rns on !oth %oils3 here n X 23 an' no &se of a se%on'ar" %oil. This is in 'ire%t %ontrast to KB@3 @CL.

Also 'i,er$in$ ith )re,io&s or*3 the )rotot")e &se' an analo$ os%illator3 inte$rate' ith the loo)s3

i.e. the in'&%tor an' %a)a%itor forme' a resonant %ir%&it. It also &se' )lain3 %ir%&lar loo)s3 hereas

K>>3 B/3 BBL all &se' flat %oils hilst in KB@L the s%heme &se' a sta$$ere' arran$ement. In this a"3 the

loo)s %ons&me the entire a% %"%le. The )rotot")e also !roa'%ast a loer resonant fre9&en%"3 i.e.

lon$est a,elen$th3 than that in )re,io&s resear%h.

O,erall3 the *e" to the a))roa%h as sim)li%it". The mo'el &se'3 !" 9&ite some a"3 the feest

n&m!er of %om)onents 12 in total for the os%illator an' "et $enerate' a %lean sin&soi'al a,eform at

>>0 *H=. On to) of this3 the re%ei,er %ontaine' onl" three %om)onents. " %ontrast3 in K>>L the

a&thors em)lo"e' a nois" 'i$ital 'ri,er3 hi%h also re9&ire' s&))lemental %ir%&itr".

The o)erational %ir%&it in%l&'in$ the antenna )oerin$ m&lti)le lam)s a 'istan%e in a))ro4imate'

rotations aa" from a4ial ali$nment3 is shon in Fi$./.7.

Fi$./.7. O)erational os%illator )oerin$ m&lti)le lam)s.

E4)erimental res&lts sho that si$nifi%ant im)ro,ements in terms of )oer transfer effi%ien%" are

a%hie,e' !" 'ire%tl" %onne%tin$ the ( %ir%&it to an am)lifier %ir%&it instea' of e4%itation !ein$

a%hie,e' from an e4ternal sin&soi'al so&r%e.

The meas&re' res&lts ere fo&n' to !e in ,er" $oo' a$reement ith the theoreti%al mo'els. The ne4t

ste) %learl" ill nee' to in,ol,e tests in,ol,in$ a%t&al !iolo$i%al tiss&e. Altho&$h3 !" anal"=in$

res&lts from )re,io&s %om)ara!le st&'ies KB@3 >>L3 it is not anti%i)ate' that there ill !e an" iss&es of

1/1



si$nifi%an%e3 ne,ertheless s&%h a st&'" is a ne%essit" !efore a%t&al )ra%ti%al a))li%ation of the

)ro%e'&re %an $o ahea' in sit&.

In tests it as fo&n' that the effi%ien%" of the ener$" transfer s"stem %an !e im)ro,e' !" in%reasin$

the 9&alit" fa%tor3 J 3 of the %oils. The )oer transfer s"stem a%hie,es at least fo&r times more

effi%ien%" an' )oer 'ensit" in atts )er %entimeter3 $i,en its small si=e3 %om)are' to )rior

in'&%ti,e5lin* s%hemes.

Reasons !ehin' the im)ro,e' res&lts3 as re)orte' in this thesis3 in %om)arison ith )re,io&s or* are

not )arti%&larl" '&e to the antenna t")e or 9&alit" fa%tor of the %oils3 as one mi$ht e4)e%t. The nat&re

of the loo) antenna has !een thoro&$hl" in,esti$ate' in the literat&re< rather3 it is a )ro)ert" of the

os%illator 'ri,in$ the s"stem &n'er resonan%e. In the 'esi$n3 e ha,e &se' a ma$neti% loo)3 hi%h has

the a',anta$e hen its %ir%&mferen%e is at a ratio of the 'ri,in$ fre9&en%" 9&arter5a,elen$th. The

os%illator fre9&en%" has then !een set so the loo) %an !e small an' the ra'iation non5to4i% in h&man

)resen%e. It is the manner of ho the 'ri,in$ fre9&en%" is reali=e'3 ho %lean the os%illator in

s"n%hroni=in$ %&rrents in )hase3 that 'etermines the 9&alit" of the ener$" e4hi!ite' in the re%ei,er.

As re)orte' in the theoreti%al se%tion3 effi%ien%" res&lts of BB` in )oer transfer o,er a ran$e of si4

%entimeters !eteen transmitter an' re%ei,er %oils are e4tremel" en%o&ra$in$. For !iome'i%al

im)lants3 this then starts to loo* at the )ossi!ilit" of remote ireless )oer )ro,ision3 in hi%h %ase

the o))ort&nit" for m&lti)le re%i)ients also !e%omes rele,ant.

The 'is%&ssion in this %ha)ter %entere' aro&n' a sin$le %o&)le'5mo'e ith se,eral e4)eriments to

)ro,e the theor". Gi,en the &tilit" of the %on%e)t3 it is interestin$ to s&))lement the 'is%&ssion ith an

a''ition of a se%on' %o&)le'5mo'e $i,en an alternati,e %ir%&it arran$ement.

2 To %o&)le'5mo'es itho&t )ro6e%tion

The last %ha)ter 'is%&sse' the theoreti%al im)li%ations an' frameor* of a sin$le %o&)le'5mo'e

ma$neti% resonant transmission in%l&'in$ e4)erimental e,i'en%e %onfirmin$ the theor". This %ha)ter

ill e4ten' the or* from the last3 a''in$ a se%on' %o&)le'5mo'e to the arran$ement. In this form3 it

is %on%e)t&all" %on,enient to 'is%&ss it in terms of a %ir%&it hi%h has alrea'" !een %reate'. The

%on%e)t of to %o&)le'5mo'es ill !e 'is%&sse' an' ill&strate' in terms of a re%reate' mo'el of

Tesla?s %ir%&it ill&strate' in his 100 )atent A S"stem of Transmission of Ele%tri%al Ener$"3 nota!l"3

the arran$ement of fo&r t&ne' %ir%&its e4hi!ite' as to %on%entri% s)irals.

Three %ontri!&tions are )ro)ose' in this %ha)ter

1//



1. An anal"sis of the to %o&)le'5mo'e mo'el of ireless )oer transmission an'

%onstr&%tin$ a %ir%&it<

/. %om)arin$ the )erforman%e of the %ir%&it ith the those 'es%ri!e' one in the literat&re3

K10L< an'3

2. intro'&%in$ an inno,ation herein mo'ifi%ation of the %ir%&it lin*s its mo'el ith thefiel' mo'el3 in stron$er s&))ort of the mo'el3 )arti%&larl" the %on%e)t of the %onti$&o&s

,irt&al homo$eno&s a,e$&i'e of free5s)a%e 'es%ri!e' in in /.

Tesla )resente' a mo'el of ireless )oer transmission that he %laime' in U.S. Patent C>737>C %o&l'

!e &se' to sen' ireless si$nals to an"here on the )lanet. A s%hemati% of his set&) is shon in

Fi$.2.1.

Fi$.2.1. Teslas ireless )oer mo'el.

The Tesla %ir%&it %om)rises fo&r t&ne' %ir%&its a transmission an' re%ei,in$ )air of %ir%&lar %oils ea%h

%ontainin$ a thi%*5ire )rimar" of a fe t&rns an' a thin5ire se%on'ar" of man" t&rns. Ill&strate' in

Fi$.2.13 ( re)resents the )rimar" %oil A the se%on'ar" %oil3 one free en' of the se%on'ar" %oil3

an' D the other free en' of the se%on'ar" %oil of the transmitter. For the re%ei,er3 these i'enti%al

%om)onents are la!ele' ith )rimes so as to 'istin$&ish the transmission si'e from the re%ei,er si'eof the %om)lete %ir%&it.

The %ir%&it in'in$s3 sensiti,el" 'e)en'ent &)on the )osition of the %oils relati,e to ea%h other3

)erform i'eall" hen the 'istan%e !eteen the )rimar" an' se%on'ar" in'in$ is no $reater than the

thi%*ness of the )rimar"3 similarl" to the transformer effe%t at %lose )ro4imit". This ens&res ma4im&m

%&rrent 'is)la%ement in the o&ter in'in$ of the se%on'ar". When a %onstant %&rrent is maintaine' on

the se%on'ar"3 it is )ossi!le to enhan%e the %hara%teristi% of this %oil. An inno,ati,e enhan%ement ill

!e 'is%&sse' later in this %ha)ter hi%h intro'&%es a %onne%tion of a thir' %oil ith )assi,e

1/2



%om)onents s&%h as %a)a%itors an' resistors !eteen an' D3 an' !eteen ′ an' D′ &se' to

mani)&late the or'er of the ma$neti% fiel'3 %om)o&n'in$ the )oer series3 )ro6e%tin$ aa" from the

se%on'ar" %oil. A))l"in$ a sin&soi'al %&rrent G to the en's of the )rimar" %oil at the transmitter3 the

in)&t %&rrent is o!ser,e' at the en's of the )rimar" %oil at the re%ei,er. The %&rrent %an !e &tili=e' ,ia

a loa' re)resente' as lam)s an' motors # an' o!ser,e' on a s)e%tr&m anal"=er hen

transmittin$ at lo )oer le,els or &sin$ atten&ators.

The in'in$s3 so %onstr&%te' an' oriente' as 'es%ri!e'3 ea%h form one5half of a t&ne' %ir%&it. "

)la%in$ ea%h half of the %ir%&it at a 'istan%e $reater than the ra'i&s of its se%on'ar" %oils3 the total

%ir%&it e4)resses %a,it" effe%ts o%%&rrin$ alon$ the tra6e%tor" of ener$" transfer. It is ass&me' the

o!ser,e' !eha,ior an' the emission of )lane a,es is &niforml" )resent3 tho&$h )erha)s in 'ifferent

9&antities3 ith %ir%&its of 'ifferent si=es !&t ha,in$ the same relati,e $eometr". It is a fairl" tri,ial

e4er%ise to artifi%iall" %reate resonan%e %a,ities3 m a$net i %al l " 5 %o&)l e' h er ei n ener $" an'

information are transmitte' !i5'ire%tionall". These %a,ities3 hile homo$eno&s in free5s)a%e3 maintain

a total internal refle%tion im)l"in$ ea%h in'i,i'&all" or m&lti)l"5%onne%te' %a,it" is a holomor)hi%

manifol' K11L. In )lain terms3 it 'emonstrates that at the e4treme limit of mi'5fiel' transfer3 the

transmitter an' re%ei,er %an !e )la%e' at se)arations in all three (artesian 'ire%tions an' %an !e

'esi$ne' in s&%h a manner as to !e a )lanetar" transmitter3 hi%h is not &nli*e Tesla?s ori$inal

'es%ri)tion.

(onsi'erin$ the ra'i&s of the se%on'ar" %oil an' the len$th of the %on'&%tor %onsistin$ the s)iral3 a

)ea*5transmission fre9&en%" is esta!lishe' hen !oth hal,es of the %ir%&it are !&ilt to e4a%tin$

s)e%ifi%ations "et o&n' in the o))osite 'ire%tions3 hi%h3 for the )&r)oses herein3 is /B.70 #H= b

7`. " a))l"in$ an e4ternal sin&soi'al stea'"5state si$nal to the )rimar" %oil ( at G3 the i'enti%al

si$nal is o!ser,e' at the )rimar" %oil (′ at an' # itho&t 'rift or 'istortion. " ins)e%tion3 it is

%lear the %ir%&it has a %om)le4 s"mmetr"< a))l"in$ e4ternal sin&soi's to either halfat the

transmitter or re%ei,erthe si$nal is o!ser,e' at the other half. E4)eriments ha,e shon that in

$eneral that the hi$her 'e$ree of s"mmetr" in the %ir%&it3 the $reater the resonan%e an' the !etter the

)erforman%e.

The to )airs of self5resonant %oils allo the trans)ortation of 9&antities of ener$" at si$nifi%ant

'istan%es %om)ara!le to the si=es of the in'in$s in,ol,e'. Rather than the &se of lar$e %oils relati,e

to the 'istan%e of transmission hi%h rel" on fi4e' %oil 9&alities3 small %oils are &se' ith a t&nin$

a))arat&s hi%h %ontrols the t&ne' state of the resonator %a,it" thro&$h or'ere' ma$neti% fiel's.

1/>



2.1 The theoreti%al mo'el

The effi%ien%"3 η 3 is a meas&re of )erforman%e of a %ir%&it to %atalo$ the s&%%ess of this )arti%&lar

e4)erimental 'esi$n. It is e4)resse' as a fra%tional 9&antit"3 as3

Usef&l )oer o&t)&t η = . 12

Total )oer in)&t

Another meas&re of )erforman%e is the ener$" store' in the ma$neti% fiel' H0 &tili=e' for ma$neti%5

resonant %o&)lin$. The %ir%&it %onsists of an os%illator 'ri,in$ a loo) of ire T %o&)le' to a

%a)a%itor (T an' a resistor RT at the %ir%&it?s resonan%e fre9&en%" ω0 re)resentin$ the transmitter< a

se%on' loo) of ire R of e9&al ra'i&s %o&)le' to a %a)a%itor (R an' a resistor RR %onne%te' to a

li$ht5!&l! re)resentin$ the re%ei,er. T an' R are )la%e' at a 'istan%e a)art from ea%h other. The

)ro6e%tion H is not seen in this mo'el3 as the ener$" is a!sor!e' into the se%on'ar" %oil.

In terms of the )ro)erties of the ra'io5fre9&en%" a,es transmitte' !" the %ir%&it3 the effi%ien%" η is

first $i,en in terms of the 'e$ree of %o&)lin$ !eteen ea%h half of the %ir%&it as

/*TR JTJR

η = / 1>01+ *TR JTJR

here *TR is the %o&)lin$ %oeffi%ient3 JT 3 JR are the 9&alit" fa%tor of the %oils T an' R 'ri,en at

resonan%e fre9&en%" ω0 .

The %riterion of effi%ien%" in the s%heme is )arameteri=e' !" the %o&)lin$ %oeffi%ient !eteen the

se%on'ar" %oils in ea%h half of the %ir%&it an' their 9&alit" fa%tor "iel'in$ the ener$" store'. "

%reatin$ a $eneri% mo'el of the means of transmission3 e %an $lean insi$hts a!o&t the effi%ien%" η of

the s%heme an' ho to im)ro,e it. Ea%h )art of the Tesla resonator is 'is%&sse' in the folloin$

se%tions< it is a',anta$eo&s to 'is%ern the s"stem in terms of its $eometr" an' %onstr&%t the fiel'

mo'el aro&n' it.

The transmitter %oils

There are to sets of %oils for ea%h half of the %ir%&it3 the" s&!sist of a )air of %o&)le' %ir%&its in the

form of a s)iral. The arran$ement of the transmitter %oils is shon in Fi$.2./. There is a )rimar" %oil

of a fe t&rns an' a se%on'ar" %oil of man" t&rns. A ti$ht %o&)lin$ *a! !eteen the )rimar"?s

1/7



in'&%tan%e a an' the se%on'ar"?s in'&%tan%e ! 3 s&$$ests the %a)a!ilit" to ha,e a hi$h m&t&al

in'&%tan%e #a! 3 hi%h $reatl" ai's more )oer to !e transmitte'.

Fi$.2./. Transmitter %oil )a%*a$e.

The re%ei,er %oils

e%a&se of the s"mmetr" a ti$ht %o&)lin$ *%' !eteen the se%on'ar"?s in'&%tan%e % an' the

)rimar"?s in'&%tan%e ' is also o!ser,e' "iel'in$3 a$ain3 a hi$h m&t&al in'&%tan%e #%' . The

arran$ement of the re%ei,er %oils is shon in Fi$.2.2. Note the sole 'ifferen%e is the 'ire%tion of the

in'in$ it is o))osite to that of the transmitter.

Fi$.2.2. Re%ei,er %oil )a%*a$e.

The )h"si%al )ro)erties of the %oils are shon in Ta!le 2.1.

1/C



Ta!le 2.1. Resonant %oils )h"si%al s)e%ifi%ation.

The %o&)lin$ %oeffi%ient

The %o&)lin$ %oeffi%ient *!% 3 m&t&al in'&%tan%e #!% !eteen 'istant %oils ! an' % 3 an' the 9&alit"

fa%tors of all the %oils are the si$nifi%ant )h"si%al )ro)erties in this arran$ement. The m&t&al

relationshi) an' the %o&)lin$ !eteen ea%h of the %oils is ill&strate' in Fi$.7. To &n'erstan' the

effe%ti,eness of the s%heme an' the or* 'one at a 'istan%e3 the ener$"5stora$e a!ilit" of the s"stem

is 'etermine' !" %o&)lin$ %oeffi%ients in transfer mo'e. The flo of the ener$" E0 in 6o&les a%ross

the 'istan%e is 'etermine' !" *!% !eteen %oils ! an' % < *a! an' *%' are relati,el" %onstant as the

'istan%e !eteen %oils a an' ! 3 an' %oils % an' ' 'o not mo,e ith res)e%t to ea%h other. If the

resonan%e fre9&en%" ω0 is *e)t %onstant3 the in'&%tan%es of %oil a an' %oil ' are small hile the

'istan%es !eteen %oils a an' ' 3 %oils a an' % 3 an' %oils ! an' ' are relati,el" lar$e. Th&s3

*a' 3 *a% 3 an' *!' are relati,el" small an' hen%e %an !e ne$le%te'. When %&rrent I0 is a))lie' to %oil

a 3 the )oer transfer flos from %oils a to ! 3 ! to % 3 an' % to ' .

Fi$.2.>. S%hemati% 'ia$ram of the %ir%&it for to %o&)le'5mo'es.

Referrin$ to the s%heme ill&strate' in Fi$.2.>3 the %o&)lin$ %oeffi%ients !eteen the si$nifi%ant

ma$neti%all"5%o&)le' %oils are 'efine' as3

#a! #%' #!%

*a! = 3 *%' = 3 an'*!% = 3 1>1 a ! % ' ! %

1/B



here #a! 3 #%' 3 an' #!% are the m&t&al in'&%tan%es3 a 3 ! an' % 3 ' an' ! 3 % are the self5

in'&%tan%es of the %oils. Sin%e e %annot %al%&late the %o&)lin$ %oeffi%ient 'ire%tl"3 it is ne%essar" to

%al%&late the in'&%tan%es an' the m&t&al in'&%tan%e of the 'istant %oils. For the %oils a an' ' 3 e

ill &se the a))ro4imation for the in'&%tan%e of a %ir%&lar loo)3

/ @r a / @r ' a ≈ µ µ 0 r nar a ln − / + Y 3 ' ≈ µ µ 0 r n'r ' ln − / + Y 3 1>/

Ra R'

here r a 3 r ' is the loo) ra'i&s an' Ra 3 R' is the ire ra'i&s3 na 3 n' is the n&m!er of t&rns3 an' Y

is the flo %onstant of the s*in5effe%t of the emitte' ra'iation. For the %oils ! an' % 3 e ill &se

the a))ro4imation for the in'&%tan%e of an air5%ore flat s)iral %oil K20L3

/ / / /r n ! ! r n % %

! ≈ µ µ 0 r 3 % ≈ µ µ 0 r 3 1>2@r !−11R!+ ! @r %−11R%+ %

here r ! 3 r % is the loo) ra'i&s an' R! 3 R% is the ire ra'i&s3 n! 3 n% is the n&m!er of t&rns3 an' ! 3

% is the i'th !eteen ea%h t&rn of the in'in$s. Ta!le 2./ shos the %al%&late' ,al&es for the

%oils.

Ta!le 2./. Resonant %oils %al%&late' ele%tri%al s)e%ifi%ation.

Determination of m&t&al in'&%tan%e in the mo'el

The m&t&al in'&%tan%e #!% is %al%&late' !"3

&∫∫ n ⋅ 'S

%

#!% = 3 1>>I

!

here % is the area of the re%ei,er se%on'ar" %oil3 is the ma$neti% fiel'3 I is the %&rrent )assin$!

&thro&$h the transmitter se%on'ar" %oil an' n is the ,e%tor normal to the ener$" a%ross free5s)a%e. In

e4aminin$ m&t&al in'&%tan%e an' the 'istan%e3 the s)a%e !eteen is fille' ith a ra'iate' fiel' o!6e%t

res&ltin$ from the e4%han$e of ener$" an' fee'!a%* !eteen the %oils3 hose si=e an' sha)e is1/@



'e)en'ent &)on %oil $eometr". As s&%h3 e %an e4)e%t the s)a%e to !e a me'i&m o%%&)ie' !" a

%har$e' fiel' e4hi!itin$ an ener$" flo.

An a%%&rate %al%&lation of m&t&al in'&%tan%e 'e)en's on the ass&m)tion of ener$" 'istri!&tion

o%%&)"in$ the free5s)a%e !eteen the 'istant %oils. It is ass&me' a olt=mann 'istri!&tion of the

ma$neti% %&rrents alon$ the a,eform limite' at its !o&n'ar" or other non5lo%al )ositions3 $i,en

#a4ell?s notion of the )otential. The mo'el still hoe,er ass&mes a smooth !o&n'ar". Usin$ these

ass&m)tions3 the %al%&late' %o&)lin$ %oeffi%ients an' in'&%tan%es are also shon in Ta!le 2./.

The e4)li%it inno,ation here is that the %oni%al %oil allos the ma$neti=ation to !e mani)&late'. The

)olari=ation of the fiel's a%ross the s)a%e is s&%h that it ma$nifies a ma$neti% mono)ole in the re$ion

hi%h allos the stron$er transmission of Hφ . " mani)&latin$ this ma$neti=ation3 the fiel' %an !e

t&ne' ith the intro'&%tion of )assi,e %om)onents.

Fi$.2.7. The %oherent fiel' o!6e%t res&ltin$ from f&n%tions in Fi$./.1B.

Vis&all"3 it is e4)e%te' the s)a%e to !e o%%&)ie' !" a h")er!oli%5sha)e' fiel' o!6e%t in three

'imensions sat&rate' ith %har$e an' flo3 ill&strate' in Fi$.2.7.

On 9&alit" an' loss

The 9&alit" fa%tors of the %oils loa'e' !" a )arallel %ir%&it %ontainin$ a %a)a%itan%e3 (i 3 an' a li$ht

!&l! are %al%&late'. The arran$ement to !e %al%&late' is shon in Fi$./. Un'er s&%h $eometr"3 the

9&alit" fa%tor is %al%&late' the same manner for ea%h of the fo&r %oils. In the s%heme3 the %om)onents

of the %ir%&it are in )arallel ith referen%e to ea%h other !eteen the transmitter an' re%ei,er

se%on'ar" %oils. The o,erall loa'e' 9&alities3 Ji 3 are 'efine' ,ia the relation3

1/



(i (i

Ji = Ro + Rr 3 1>7i i

here the 9&alit" Ji re)resents the 9&alit" fa%tor of the resonant %ir%&it either '&e to ohmi% resistan%e

Ro an' ra'iation resistan%e Rr . The losses are 'i,i'e' into the losses '&e to ohmi% resistan%e3

/µ0ω0 ℓ

Ro = 3 1>C/σ π > R

here σ is the %on'&%ti,it" of the material3 ℓ the ire len$th of the %oil an' R is the ire ra'i&s3

an' the losses '&e to ra'iation resistan%e

> /µ0 π / ω r / h

Rr = n + 2 3 1>Bε0 1/ % 2π %

here n is the n&m!er of t&rns of the %oil an' % is the s)ee' of li$ht. " e4amination of 1>73

in%reasin$ either the %a)a%itan%e or %ir%&it resistan%e ill in%rease the 9&alit" fa%tor of the %oils.

Gi,en 1>0 a small in%rease of the 9&alit" fa%tor of the %oils ill in%rease the amo&nt of ener$"

trans)orte' im)ro,in$ the effe%ti,e effi%ien%". The )oer transfer effi%ien%" of the fo&r5%oil s"stem

%al%&late' from the 9&alit" fa%tors an' %o&)lin$ %oeffi%ients from a!o,e is3

/ / /(*J J a! a ! )(*J J!% ! % )(*J J%' % ' ) ηTotal = . 1>@

/ / / / /(1+ *J J a! a ! )(1+ *J J%' % ' )+ *J J!% ! % 1+ *J J!% ! % + *J J%' % '

2./ The )ro)a$ation of ma$neti% %&rrents

In %onsi'eration of the or* of Po"ntin$3 it is interestin$ in )lottin$ the ma$neti% fiel' intensit" an'

the ma$neti% fl&4 'ensit" of the ma$neti% %&rrents emanatin$ !eteen the %oils ! an' % . It is

insi$htf&l to ma) these ma$neti% fiel's !eteen %oils ! an' % hen a so&r%e of 1 am)ere is a))lie'.

Usin$ a %om)&tational softare )a%*a$e (omsol #&lti)h"si%s3 e are a!le to )lot the ma$neti% fl&4

'ensit" an' ma$neti% intensit" !eteen the 'istan%e %oils. The )lot is shon in Fi$.2.C.

120



Fi$.2.C. #a$neti% fiel' intensit" an' fl&4 'ensit" !eteen to se%on'ar" s)iral %oils.

In or'er to %al%&late the ma$neti% fiel' stren$th at a 'istan%e from the %oils3 it is ne%essar" to tr&n%ate

the %om)&tational 'omain3 !e%a&se the finite element %al%&lation re9&ires a finite5si=e' mesh. The

s)here3 S 3 hi%h ser,es as the %om)&tational !o&n'ar" of the 'omain3 has a ra'i&s of @0 %entimeters.

The )lot 'is)la"s a %al%&lation of ma$neti% fiel' stren$th the o,als on to) an' !ottom an' the

ma$neti% fl&4 'ensit" the lines )assin$ thro&$h an' aro&n' the %oils. The %olors from li$ht to 'ar*

sho the in%reasin$ intensit" of the ma$neti% fiel' emitte' !" the %oils< the )ea* stren$th 'ar*est

%olor in the %enter lies 1.7 times the ra'i&s of the %oils. The minim&m stren$th lies at 17 times the

ra'i&s of the %oils. The ma$neti% fl&4 'ensit" the f iel' lines e4ten's the fiel' intensit"a))ro4imatel" @ times the ra'i&s of the %oils or halfa" !eteen the ma4im&m an' minim&m fiel'

stren$th. As shon in Fi$.2.C3 the ma$neti% fiel' intensit" is stron$ in the free s)a%e !eteen the %oils

hi%h s&$$est this $eometr"3 $i,en the stren$th of the fiel's to )erform or* at a reasona!le

effi%ien%"3 is s&ita!le for transmittin$ ele%tri%al %&rrents of s&ffi%ient ma$nit&'e.

121



Fi$.2.B. #a$neti% fiel' intensit" an' fl&4 'ensit" at $reater 'istan%e.

It is also interestin$ to *no if the mo'el hol's at 'istan%es aa" from the near5fiel' re$ion. Fi$.2.B is

a )lot of the same se%on'ar" %oils ! an' % emittin$ a ma$neti% fiel' at a 'istan%e of B7 %entimeters3

the !o&n'ar" of the %om)&tation 'omain has a ra'i&s of 1/0 %entimeters. The intensit" remains

&niform a%ross the 'istan%e. The limit at the !o&n'ar" is o!taine' !" the ass&m)tion of ho the fiel's

are %om)&te' an' the ass&m)tions a!o&t free5s)a%e. De)en'in$ on the ass&m)tions %onstr&%tin$ the

metho'3 the limit for me'i&m5ran$e transfer %an !e e4ten'e'.

One metho' to e4ten' the ran$e of ener$" trans)ort is to in%rease the 'ensit" of a %om)onent of the

ma$neti% fiel'3 Hφ . This t")i%all" in,ol,es in%reasin$ the in'&%tan%e of the %oils as to satisf" the

relationshi) of the store' ener$" W to in'&%tan%e an' the fiel's3 as3

1 / 1 W = I3 W = ∫ H ⋅ ',3 1>

/ /

or in%reasin$ the 9&alit" fa%tor of the %oils themsel,es. A''in$ a thir' %oni%al %oil of a ra'i&s r e at a

'istan%e of 10 %entimeters alon$ the = − a4is of the se%on'ar" %oil an' %onne%tin$ at terminal 3 the

free en' of the se%on'ar" %oil3 in%reases the ,al&e of ! linearl". The ,al&e of H is also in%rease'3 it

is $i,en the len$th of ire ℓe %ontaine' on the %&r,at&re3 θe 3 of the %oni%. We a,era$e' the ,al&e of

H !" settin$ the %oni% an$le θ = π alon$ a len$th here r , r an' therein "iel'e' a si$nifi%ante C e !

in%rease of ma$neti% )otential Aµ . An in%rease of Hφ is o!ser,e' !" a))l"in$ the for%e &)on the

'i)ole resi'in$ at the %enter of the se%on'ar" %oil. We h")othesi=e the motion of the ma$neti% fiel'

from the %oni%al %oil %onstr&%ti,el" interferes ith the ma$neti% fiel' l"in$ on the s)iral %oil $i,en the

to %oils are sharin$ the same ele%tri%al fiel' E an' ele%tri% )otential ϕ . " a''in$ %a)a%iti,e an'

12/



resisti,e %om)onents a%ross !oth %oils ! an' % 9&alit" an' %o&)lin$5%oeffi%ient ,al&es are 'ire%tl"

a%%esse'. The thir' %oil arran$ement is shon in Fi$.2.@.

Fi$.2.@. Re%ei,er )a%*a$e an' thir' %oil. (om)onent !rea'!oar' shon.

The )ossi!ilit" of %al%&latin$ %o&)lin$ effe%ts3 nota!l" the %o&)lin$ %oeffi%ient !eteen the 'istant

s)iral %oils 'iffers from normal mo'els in that the s)iral lies insi'e of the loo) %oil %ontainin$ the

$enerator an' therefore is ass&me' hi$hl" effi%ient folloin$ the ell5&n'erstoo' transformer effe%t.

2.2 The %ir%&it e4)eriment

A re%onstr&%tion of the Tesla resonator3 ill&strate' in Fi$.2.13 as !&ilt ith the )h"si%al

s)e%ifi%ations shon in Ta!le 2.1. The in'&%tan%es ere meas&re' on a Helett5Pa%*ar' h)>1/A

into 70 Ω3 shon in Ta!le 2.2. The first test as to )lot the res&lts of a))l"in$ a stea'"5state

sin&soi'al so&r%e at the )ea* resonan%e fre9&en%" of /B.70 #H= an' %hart the meas&re' 'ifferential

)oer !eteen in)&t an' o&t)&t. The effi%ien%"3 η 3 as a f&n%tion of the 'istan%e !eteen the %oils is

shon in Fi$.2..

122



Fi$.2.. The effi%ien%" of the Tesla resonator o,er a 'istan%e.

To &n'erstan' ho this mo'el )erforme' a$ainst a %om)arati,e ireless )oer s"stem3 res&lts ere%ontraste' ith K10L. Altho&$h %ontainin$ a 'ifferent orientation an' $eometr"3 '&e to similar so&r%es

of )rior art s&ffi%ient share' f&n'amentals e4ist. Fi$.2.10 shos a %ontrast of the to s%hemes.

Fi$.2.10. (om)arison of the Tesla resonator ith -&rs.

The re%onstr&%te' Tesla resonator3 in its ori$inal form itho&t mo'ifi%ation3 )erforms !etter ith

se%on'ar" %oils of more in's "iel'in$ more ma$neti% fiel' )ro)erties. (onsi'erin$ ma$neti%

resonan%e %o&)lin$ s%hemes st&'ie' in this %ha)ter insist a hi$her ma$nit&'e of ma$neti% fl&4 'ensit"

as 'ire%tl" translatin$ into $reater effi%ien%" o,er mi'5fiel' 'istan%es3 the o!ser,ation of a linear

relationshi) in for%e to ener$" 'ensit" of the ma$neti% fiel' is not s&r)risin$. What is interestin$ is

that ener$" transmission %an !e a%hie,e' here the mo'el 'oes not rel" on a4ial ali$nment.

12>



Ta!le 2.2. Resonant %oils meas&re' ele%tri%al s)e%ifi%ation.

The se%on' test as to a'' )assi,e %om)onents to ,erif" if3 as a%%or'in$ to 1>73 the 9&alit" fa%tor of

the s%heme %o&l' !e im)ro,e'. " 'oin$ so3 the effi%ien%" shon in 1>@ is im)ro,e'. (a)a%itors

a''e' in )arallel to the transmission %oil3 hile resistors a''e' in )arallel to the re%ei,er %oil o&l'

sho an im)ro,e' effi%ien%" in the transmission. S&%h 'ire%t %onne%tions !eteen the %ir%&it an' the

fiel' are ai'e' !" the a''ition of the thir' %oil in'&%ti,el" %o&)le' to the transmission fiel'. Val&es of

a''e' or s&!tra%te' %a)a%itan%e an' resistan%e affe%te' a'mittan%e an' %ir%&it im)e'an%e !" alterin$

the )hase of the ener$" at the 'istant se%on'ar" %oil. A series of fi,e tests ere %on'&%te' a''in$

%a)a%itan%e an' resistan%e as ,al&e5!ase' )arameters to %oils ! an' % %o&)le' thro&$h e .

As ill&strate' in Fi$.2.113 the %o&)lin$ %oeffi%ient *!% %an !e affe%te' !" not onl" re'&%in$ the losses3

!&t also affe%tin$ the 9&alit" fa%tor !" a''in$ ,al&es of %a)a%itan%e an' resistan%e. T&nin$ the

resonant fre9&en%" to the )ea* ma$neti% fl&4 'ensit" of the %ir%&it o&l' f&rther im)ro,e

)erforman%e.

Fi$.2.11. The effi%ien%" of the Tesla resonator ith enhan%e' 9&alit" fa%tor.

This im)lies the %a,it" is %onti$&o&s an' has a )ro)ert" inherent to it3 s&s)en'e' insi'e free5s)a%e.

127



Dis%&ssion of the e4)eriment

Folloin$ the theoreti%al frameor* of to %o&)le'5mo'es3 the e4)erimental re%onstr&%tion of

Tesla?s si$nalin$ a))arat&s from his 100 )atent A S"stem of Transmission of Ele%tri%al Ener$"3

ill&strate' in Fi$.2.13 is also e4amine' in terms %ontraste' ith other attem)ts %laimin$ a Tesla

ireless )oer s"stem. In K10L the a&thor %on%erne' himself ith Tesla?s 11> )atent A))arat&s for

Transmittin$ Ele%tri%al Ener$" fo%&sin$ )rimaril" on the transmission of ele%tri%al %&rrents $i,en

lar$e %oil loo)s of a fe t&rns. Here3 e ha,e attem)te' to 'emonstrate an alternati,e a))roa%h &sin$

%oils si$nifi%antl" smaller "et a%hie,in$ !etter res&lts for effi%ien%". One stri*in$ 'ifferen%e is that the

$eometr" an' orientation of the Tesla resonator e4amine' in this %ha)ter 'oes not rel" on an a4ial

ali$nment of the %oils< rather3 the transmitter5re%ei,er )air %an !e )la%e' li!erall" at a 'istan%e relati,e

to ea%h other itho&t an" si$nifi%ant a''itional losses. A%%or'in$ to K10L3 the limit of this t")e of

transfer is @ times the 'iameter of the %oils3 the ele%tri%al intensit" !eteen the %oils 'iminishin$ at a

rate 1 . E4)erimental res&lts ha,e in'i%ate' a $reater lin* in the ma$neti% fiel' 'ensit" a))ears at/

r

the se%on'ar" %oils in this resonator5%o&)le' mo'el. The e4treme 'istan%e meas&re' in the e4)eriment

as fo&r meters.

The last ste) as to 'is%ern if the re%onstr&%te' resonator %o&l' !e mo'ifie' to in%rease ener$"

transmission an' intensit" of the ma$neti% fiel's in the ma$neti%5resonant mo'el. What !e%ame

imme'iatel" a))arent as !" a''in$ )assi,e %om)onent %ir%&itr"3 the fiel' o!6e%t !eha,e' as a %a,it"

not &nli*e a transmission5line resonator. The sharin$ of those !asi% %hara%teristi%s in o&r mo'el $i,es

a ireless transmission5line resonator. #ore resear%h on this formalism is s&$$este'.

It has !een o!ser,e' that the resonator?s )erforman%e is sensiti,el" 'e)en'ent &)on its im)e'an%e3 as

e4)e%te'. The si$nal $enerator &se' '&rin$ e4)eriments o&t)&ts into 70Ω3 im)en'en%e at the re%ei,er

as meas&re' to !e 7@Ω. T&nin$ the resonator3 e ere a!le to more %losel" mat%h the im)e'an%e of

the %ir%&it alloin$ more of the )oer seen at the so&r%e to !e 'issi)ate' in the loa'. From this

a))roa%h3 !" a''in$ an' s&!tra%tin$ im)e'an%e3 effi%ien%ies o,er lon$er 'istan%es are 'ramati%all"

im)ro,e'. Altho&$h3 not 'ire%tl" a''resse' in the e4)eriments in this %ha)ter3 the a))earan%e of

stan'in$ a,es in the resonator s&$$ests a )ossi!le e4)lanation of ,olta$e ma$nifi%ation len'in$

%re'en%e to Tesla?s 'es%ri)tion of ,elo%it"5inhi!ition.

The i'ea of ,elo%it"5inhi!ition is relate' to the armor %on%e)t of a%%eleration3 &se' here in this

thesis3 an' 'es%ri!e' in 1.1. What is f&rther interestin$ is that Tesla 'es%ri!e' a metho' of anti)o'es

K110L3 or )ositions on the )lanet here it o&l' !e i'eal to )la%e transmitters for orl' ireless

transmission. A 'rain$ of this i'ea is shon in Fi$.2.1/.

12C



Fi$.2.1/. Tesla antenna an' anti)o'es.

The a''ition of a thir' %oil on the = − a4is alloe' the mani)&lation of the ,aria!les manifestin$ the

ma$neti% fiel's at the se%on'ar" s)iral %oils "iel'in$ an im)ro,ement in )erforman%e. The )resen%e of

the %oil e in the fiel' alon$ the tra6e%tor" %o&)les3 thro&$h a hi$her5or'er ma$neti% fiel' K/CL3 to the

transmission %ir%&it of ! an' % 3 shon hen there is an' not a %onne%tion at an' ′. ase'

&)on this o!ser,ation3 it is of $reat interest in f&t&re e4)eriments to in,esti$ate hat a%%ess there is to

the ma$neti% )otential Aµ.

2.> O!ser,ations of !eha,iors )resent in the %ir%&it

The metho' here 'es%ri!es me'i&m5ran$e ireless )oer transmission !" means of lo5fre9&en%"

ra'iatin$ a,es3 as o))ose' to either near5fiel' in'&%ti,e 'ire%t %o&)lin$ or far5fiel' mi%roa,e

transmission. We ha,e )ro)ose' a t")e of ireless )oer transmission !ase' on the notion of a

transmission5line resonator itho&t an" )h"si%al %ontainment other than the fiel's themsel,es. It

s&$$ests that the e4isten%e of a )air of t&ne' %ir%&its3 as ill&strate' in Fi$.2.1 an' s%hemati%all"

'emonstrate' in Fi$./.13 e4hi!its s&%h )ossi!ilities hen meas&rin$ &n)oere' o&t)&ts at !oth the

re%ei,er as ell as the transmittera %a,it" )resen%e is ,isi!le hether e4ternal )oer is a))lie' or

not.

The o!ser,ations lea' to an im)li%ation of a resonator %a,it" ith %hara%teristi% )ea*s in the

fre9&en%" s)e%tr&m hi%h $o &n%han$e' re$ar'less of the 'istan%e !eteen transmitter an' re%ei,er.

It is this &ni9&eness that ma*es the %ir%&it orth" of 'ee)er st&'". Poer effi%ien%ies re%or'e' are felt

to !e res)e%ta!le for resear%h at this sta$e3 !ein$ si$nifi%antl" hi$her than those )re,io&sl" re)orte'

for mi'5fiel' re$ions3 itho&t the nee' for the e4tremel" lar$e 'ire%te' antenna.

12B



A 'etaile' an' 9&antitati,e anal"sis of the effe%t of e4ternal3 non5resonant o!6e%ts on the transmission

%ir%&it is %om)ati!le here as it is ith the metho' 'is%&sse' in the other )arts of the thesis.

It is orth notin$ that the )oer transfer is not affe%te' if h&mans or ,ario&s e,er"'a" o!6e%ts3 !oth

lar$e an' small3 are )la%e' !eteen the re%ei,er an' transmitter. This in%l&'es %ases here o!6e%ts

%om)letel" o!str&%t the line of si$ht or lie i th in a fe %entimeters of the %oils. W hile the

transmission fre9&en%" is somehat o&tsi'e that 'eeme' safe for h&man e4)os&re K1@L3 %o&)lin$ is

onl" )ossi!le if a re%ei,er %ontains a %oil s"mmetri% to the transmitter3 !&t o))ositel" o&n'. It is

s&s)e%te' the ra'io5fre9&en%" fiel's rea%t ea*l" ith off5resonant o!6e%ts.

D&rin$ the %o&rse of %on'&%tin$ e4)eriments3 ,olta$e ma$nifi%ations ere o!ser,e' hen a))l"in$

resistan%es to the re%ei,in$ %ir%&it. #a$nifi%ations as m&%h as 70 times ere o!ser,e' hi%h manifest

themsel,es as %&rrent s)i*es refle%tin$ !a%* a%ross the resonator. Altho&$h note'3 the" ere not

'ire%tl" anal"=e' '&rin$ the in,esti$ation for this %ha)ter. Ne,ertheless3 it is a))arent that Tesla?s

resonator has )ro)erties hi%h a))ear either &nreali=e' or &n'er5in,esti$ate' as s&%h3 altho&$h the"

ere )artiall" s*et%he' o&t in Tesla?s ori$inal or*s K13 /3 23 /73 /CL. It is therefore s&$$este' that a

'etaile' !o'" of resear%h !e %on'&%te' on this to)i%. T&nin$ te%hni9&es an' strate$ies3 fiel'5or'er

%o&)lin$ are also %onsi'ere' ,er" fr&itf&l )aths.

Whilst the res&lts )resente' in this %ha)ter are of interest in themsel,es3 the" also thro &) a n&m!er

of intri$&in$ 9&estions hi%h nee' to !e resear%he'. For e4am)le3 in terms of )erforman%e o,er 'istan%e3 ho im)ortant are the sha)e an' 'ire%tional settin$s of the in'in$s an' the amo&nt of

)oer 'ran !" the loa' ith re$ar' to the transmission effi%ien%" What remains %onsistent is that

an im)ro,ement of 9&alit" lea's 'ire%tl" to an im)ro,ement of )erforman%e not onl" at the 'istan%es

shon in the fi$&res3 !&t also at $reater 'istan%es3 as e4)e%te'.

Gi,en the a))earan%e of more %om)le4 a,e )atterns in the transmission3 it is h")othesi=e' the

ma$neti% fiel's of the first5or'er are3 at least in )art3 ith the S%h&mann resonan%e K110L. This is

s&))orte' !" e4)eriments %on'&%te' ith the thir' %oil )resent in !oth the transmission an' re%ei,in$%oils3 intro'&%in$ a thir' %o&)le'5mo'e.

(om)are' to the sin$le %o&)le'5mo'e mo'el from the )re,io&s %ha)ter3 this mo'el shares man" of

the %hara%teristi%s of the former. (o&)lin$ !eteen %lose5)ro4imit" %oils3 here the 'istan%e is m&%h

smaller than the ra'i&s3 resem!les the %ommonl"5&n'erstoo' transformer mo'el. (o&)lin$ !eteen

'istant %oils3 s&%h as the %o&)lin$ !eteen ea%h se%on'ar" %oil3 resem!les the theoreti%al frameor*

of the )re,io&s %ha)ter in se,eral a"s. First3 interse%tion of :fl&4 lines; is 'eem)hasi=e' an' the

notion of the ,irt&al homo$eno&s a,e$&i'e is stren$thene'. (ontraril"3 it is also %on%e)t&all" %orre%t

12@



to %onsi'er the interse%tion of :fl&4 lines; as the lin*in$ for%e %onstit&tin$ the str&%t&re of the

a,e$&i'e. Se%on'3 altho&$h the os%illator in this %ase is not ti$htl" inte$rate' into the )rimar" %oil

antenna an' the %ir%&it res)on's more to rin$in$ than a%%eleration3 it is 'istin%tl" a))arent that the

)ro6e%tion fiel'3 H3 from the )re,io&s %ha)ter is )resent !e%a&se of the fa%t that the thir' %oil

e4)erien%es a ma$nit&'e of ener$" alon$ the = 5a4is. Thir'3 the enhan%e' ma$neti% fiel' ma$nit&'e3

a$ain e4)erien%e' !" the thir' %oil3 is s&))orte' !" the or* of /.>./. The theor" is f&rther e4ten'e'

in this %ha)ter !" the %on%e)t of the or'ere' ma$neti% fiel'=ero an' first or'erfolloin$ from the

theoreti%al 'is%&ssions in K/13 /CL. Personall" s)ea*in$3 I !elie,e the 'emonstration of )h"si%al

e,i'en%e of the first or'er ma$neti% fiel' is a ,er" )oerf&l %laim an' 'eser,es serio&s %onsi'eration

in f&t&re resear%h.

This %ha)ter 'is%&sse' to %o&)le'5mo'es !" ill&stratin$ a re%onstr&%tion of a Tesla s"stem of

ireless )oer3 %ontraste' to the sin$le %o&)le'5mo'e of the )re,io&s %ha)ter. The &tilit" of the

s%hemes ill&strate' in !oth %ha)ters )resent man" )ossi!ilities for the f&t&re of effi%ient ireless

ener$" transfer in an in'&strial %a)a%it".

> The 'i)ole %on%e)t an' 9&ant&m ma$neti% )otential

This %ha)ter ill 'is%&ss a mo'ifie' mo'el of the 'i)ole. The 'i)ole %on%e)t an' 9&ant&m ma$neti%

)otential are f&n'amental %on%e)ts 'es%ri!in$ the motion an' for%es of se)aration of ele%tri% %har$es

on an antenna. D&rin$ Hert=? resear%h3 it as 'is%o,ere' some )e%&liar effe%ts o%%&r in %ertain

antennas K1C3 203 BC3 2L s&%h as the %ir%&lar loo)3 e,en at those a,elen$ths &tili=e'. While a

%om)rehensi,e e4)lanation is o&t of s%o)e3 it is interestin$ ne,ertheless to e4amine the motions of

se)aration of to %har$es in the ,ie)oint of 'e ro$lie a,es K>L as ell as their 'es%ri)tion !"

ohm KCL to form a &nifie' mo'el !eteen 9&ant&m an' %lassi%al 'es%ri)tions of the mo'el in this

thesis.

It is ortho'o4" in ele%tri%al en$ineerin$ an' a !asi% tenet of )h"si%s3 that or* is 'one in ele%tri%al

s"stems !" the se)aration of %har$es an' ener$" is release' hen those same %har$es %ome !a%*to$ether3 or an os%illation ta*es )la%e alon$ a line of tra6e%tor". (onser,ation las 'i%tate that the

or* 'one to se)arate %har$es is e9&al to the ener$" that is o!ser,e'. While this &n'erstan'in$ is

%on,enient to fin'in$ sol&tions in :%ommon; )ro!lems an' has shon itself to !e a rather &sef&l tool

as s&%h3 in itself3 it )orten's the ele%tron re)resentin$ the %har$e3 $i,en orent=3 is in'e)en'ent of the

s)a%e here it is lo%ate'. If e are to a%%e)t the notion that %har$es e4ist !e%a&se of matter3 the

se)aration of %har$es in the %on'&%tor )ro6e%t ener$" onto the interfa%e here the" lie. The interfa%e

has an interestin$ feat&re in that the la$ of %&rrent from ,olta$e allos the 'i)ole to s&stain lon$

12



Fi$.>.1. Ener$" in the 9&ant&m state.

S&%h an analo$&e o&l' allo the sele%tion of %har$es of e9&al ma$nit&'e "et o))osite %har$e an'

)olari=ation !ase' on the )hase herein a for%e is a))lie' to se)arate them into $a&$e $ro&)s. The

%har$es are still ran'om in the statisti%al sense3 "et 'ran into s&s)ension !" sim)le for%es of

attra%tion an' re)&lsion3 %onstraine' to )erform or*.

The ma$neti% fiel' of a 'i)ole is %al%&late' at the limit of a %&rrent loo) s)lit into to )hases< a )air

of %har$es as the so&r%e shrin*s to a )oint hile *ee)in$ the ma$neti% moment3 m3 %onstant K@L. For

the %&rrent loo)3 this limit is most easil" 'eri,e' for the ,e%tor )otential. O&tsi'e of the so&r%e re$ion3

this )otential is3

µ0 m]r A r ( ) = 2 3 170

>π r

an' the ma$neti% fl&4 'ensit" is3

µ0 2rm( ⋅r ) m r ( ) =∇× A = 7 − 2 . 171

>π r r

The s%alar )otential from the mono)ole limit is3

m ⋅r ϕ ( ) r = 2 3 17/

>π r

an' the ma$neti% fiel' stren$th at a )oint %har$e is3

1 2rm( ⋅r ) m H( ) r =−∇ϕ = 7 − 2 = . 172>π r r µ0

The ma$neti% fiel' is ass&me' s"mmetri% &n'er rotations a!o&t the a4is of the ma$neti% moment m.

>./ Attra%tion in the 'i)ole moment

Fi$.>./ shos a )oint5%har$e emittin$ a fiel' into free5s)a%e3 ill&stratin$ the a4ial an' lateral emission

of ener$" ele%tri%al for an ele%tri% )oint5%har$e3 ma$neti% for a ma$neti% )oint5%har$e.

1>1



Fi$.>./. A %ommon 'es%ri)tion of a )oint5%har$e 'i)ole.

While the )attern of the fiel' emission is &niform o,er the a4es of )ro)a$ation3 the 'e)i%tion im)lies

the )oint5%har$e is not mo,in$. Sin%e the Hert= antenna is a 'i)ole thro&$h the se)aration of %har$es3

the $enerali=ation im)lie' allos a f&rther s&$$estion of the ener$eti% state of a )oint5%har$e. Sin%e it

is '&e to the %onser,ation of ener$" an' Ga&ss? l a t hat the len$th of )ro)a$ation ,aries

)ro)ortionall" to the ma$nit&'e of the emission an' that the len$th %annot e4ten' fore,er3 it is

rele,ant to )oint o&t there is a formal !o&n'ar" to the fiel's in !oth the a4ial an' lateral 'ire%tions. It

is also rele,ant to %onsi'er that the ener$" mo,in$ toar' the !o&n'ar" 'oes not ne%essaril" 'is)erse

from the )oint5%har$e3 if it is the ass&m)tion that the )oint5%har$e onl" %ontains ener$". If this

%on'ition ere tr&e3 the )oint5%har$e o&l' !e e4ha&ste' after a $i,en time. Sin%e it 'oes not e4ha&st3

ill&strate' !" the fa%t that ele%tri%al %ir%&its still f&n%tion3 its internal mo'&lationor the fre9&en%" !"

hi%h the ener$" shifts !eteen mo,in$ toar' an' aa" from the !o&n'ar"is the attri!&te

'efinin$ its state. Otherise3 an &nseen for%e is a))l"in$ ener$" to the )oint5%har$e s&stainin$ its state

fee'in$ it as a for%e hol'in$ the e9&ili!ri&m3 so that the fiel' is %onser,ati,e. A mo'ifie' 'es%ri)tion

of the 'i)ole is ill&strate' of the for%es on the se)aration of %har$es $i,en their m&t&al attra%tion is

shon in Fi$.>.2.

1>/



Fi$.>.2. a Point5%har$es an' the a*e )attern of for%es3 ! emittin$ a ma$neti% shearin$ for%e %on%entrate' at

the ma$neti% moment m.

Fi$.>.2a shos )oint5%har$es + 93− 9 %reatin$ the a*e )attern lines of attra%tion at the ma$neti%

moment m 'is)ersin$ ener$" at ri$ht an$les to the )oint %har$es hen or* is a))lie' se)aratin$

them3 ,is&ali=e' in Fi$.>.1. In Fi$.>.2!3 for%es !eteen the )oint %har$es re,eal a %ir%&latin$ ma$neti%&

shearin$ for%e F analo$o&s to the armor for%es KCL. The se)aration of %har$es3 the ri))lin$ of the

%har$es3 !e%a&se of the for%es of se)aration3 the a*e %a&ses the elli)ti%al feat&re of the effe%t of

)oint %har$es on free5s)a%e. The ener$" from the a*e is e4)resse' at ri$ht5an$les as the ma$neti%

fiel'.

The !en'in$ of the for%e lines ill&strates the %ir%&lation of for%e a%ross the moment m. The tension

!eteen the %enters of for%e ill&strates the res&ltin$ fiel' e4%han$e !eteen the to )oint %har$es an'

a 9&anti=e' %om)onent of ener$" tra,elin$ alon$ the a4is of orientation !eteen the )oints. The

re)&lsi,e for%es !eteen the )oints are !alan%e' !" the attra%ti,e for%es fol'e' in the motion of the

)assin$ ener$" %reatin$ e9&ili!ri&m in the s"stem. Sin%e the s"stem sim)l" either refle%ts or refra%ts

the )assin$ ma$nit&'e of ener$"3 there is no fiel' intera%tion !eteen the )oints3 onl" the orent= an'

armor $a&$e.

The 'i)ole 'es%ri)tion has !een 'eri,e' e4%l&si,el" from the $eometr" of the antenna &se' in the

os%illator3 theoreti%all" s&))orte' !" %o&)le'5mo'e theor". The %on%e)t ill&strate' here for the 'i)ole

is to em)hasi=e the e4%han$e of ener$" !eteen itselfas a f&n%tion of the se)aration of %har$es

an' other ,e%tor5!ase' %om)onents it %o&l' )ossi!l" intera%t ith as a Riemann s&rfa%e. Greater

ener$ies in hi$her %on%entrations a%t to infl&en%e other 'i)oles not &nli*e $ra,itational for%es !eteen

1>2



%elestial !o'ies. As s&%h3 the ar%hite%t&re of the ele%tri%al )arti%les is 'efine' !" a!sor)tions an'

refle%tions of 9&anti=e' ener$"3 in the form of ele%troma$neti% )hotons. Point5%har$es on the

manifol' 'efine ea%h other !" the e4%han$e of )hotons3 otherise3 the" o&l' !e inert an' &nseen3

)erha)s %onf&se' for the ,a%&&m. The 'efinition of o!6e%ts on the manifol' an' ho the" e4%han$e

ener$" $i,es rise to this 'es%ri)tion of the ele%troma$neti% s)e%tr&m. Here3 the ele%troma$neti%

s)e%tr&m is 'efine' thro&$h to 'ominant )ro%esses )oint5%har$es refle%tin$ a )hoton an' )oint5

%har$es a!sor!in$ a )hoton.

Fi$.>.>. Photon refle%tion an' a!sor)tion )rimiti,es.

Fi$.>.> shos a a )hoton refra%tion !" the 'i)ole fiel'3 ! )hoton a!sor)tion !" the 'i)ole fiel'. On

ea%h s)atial s&rfa%e3 a )hoton %arr"in$ ele%troma$neti% ener$"3 γ03 is re)resente' as a )a%*et of

ener$" mo,in$ ith a %onstant ma$nit&'e an' 'ire%tion3 !ase' on the emission from the transmitter.

" ins)e%tion3 γ0 has the )otential to transfer ener$" to the 'i)ole or to !e emitte' from it !" the

a%%eleration of ele%trons in the %on'&%tor. The )oint here the )hoton %han$es 'ire%tion !" a fa%tor of

refle%tion or refra%tion in'i%ates a ra'ial )oint of the intensit" of ener$" !" its shift in 'ire%tion $i,in$

rise to the ma$nit&'e of the free5s)a%e %&rrents IE .θ

>.2 Ga&$e fiel's

The intera%tion of the stream of )arti%les ith the ra'iate' fiel' ill res&lt in a $ra'&ate' elli)soi'al

fiel' $roin$ aa" from the emission )oint. Instea' of a stati% 9&antit"3 the moment&m of the )arti%le

stream %reates an elli)ti%al area on the manifol' here the relationshi) !eteen the to %har$es3 the

,irt&al an' the )assin$ fiel'3 are in resonan%e ith res)e%t to ea%h other3 %reatin$ a more 'ense form

on the manifol'. Some )assin$ )arti%les3 hi%h %o&l' %ontain masses3 are tem)oraril" tra))e' in the

sin* an' or!it the %entral a4is of the fiel'. The ra'i&s of the fiel' an' the e9&ili!ri&m of the s"stem are

'e)en'ent on the ma$nit&'e of ener$" store' as a resonan%e !eteen the transmission an' re%ei,in$

loo) antenna. This 9&otient of ener$" is tra))e' in the sin* as lon$ as the s"stem remains sta!le in

terms of its $eometr" an' the resonan%e )atterns %an hol' it to$ether. The form o&l' hol' &nless

!ro*en !" a stron$er e4ternal ele%tri% fiel'. As a series of 'i)oles3 analo$o&s in the %ase of a short

'i)ole3 a %hainin$ of 'i)oles "iel's the )olari=ation of the )ro)a$atin$ a,eformthe rise an' fall of

1>>



%&rrent from one )olarit" to the othera%ross free5s)a%e3 ith a $i,en a'mittan%e an' rea%tan%e

'e)en'ent &)on the tem)erat&re an' )ress&re.

What is rele,ant to &n'erstan' at the f&n'amental le,el are that the notions of fl&4 an' )ro)a$ation

are a!sor!e' into the s%alar3 ,e%tor3 an' 9&aternion re)resentations of the )otential of the a,e$&i'e

str&%t&re3 hi%h %onsists an' s&!sists the s)a%e !eteen an' of the antenna 'is%&sse' thro&$ho&t this

thesis. As the )henomenon is )h"si%all" o!ser,a!le3 it in'i%ates that the )otential Aµ fiel's3 here

µ = 03 13 /3 23... 'o )ossess a )h"si%al si$nifi%an%e as $lo!al5to5lo%al o)erators or the %ase of $a&$e

fiel's3 s&!6e%t to )re%isel" %onstraine' to)olo$ies K71L. The a,aila!le %onstraine' to)olo$ies in the

%ase of the mo'el 'es%ri!e' in this thesis are

• The %ase of lo%ali=e' a,e$&i'e3 an'3

• %lassi%all"5'efine' ele%tro5state %har$e manifol's3

here !oth are limite' in ma$nit&'e an' statisti%al olt=mann 'istri!&tion !" a tem)erat&re5

'e)en'ent ele%tron %oheren%e len$th $i,en the antenna ar%hite%t&re. In the %ase of the lo%ali=e'

a,e$&i'e %har$e' manifol'3 t")e is i'entifie' !ase' on the 'efinitions of

• fiel's free ele%tron (F FE )3

• fiel's %on'&%tin$ ele%tron (F (E )3

• a,e$&i'e fiel' (WG F)3 an'3

• %on'&%tin$ ele%tron fiel's ((E F)3

)ro,i'e' !" the

1. Aharono,5ohm an' Altsh&ler5Arono,5S)i,a* effe%ts F FE an' F (E 3

/. the to)olo$i%al )hase effe%ts of err"3 Aharono,3 Anan'an3 Pan%haratnam3 (hiao an' W&

(WG F)3

2. the 8ose)hson effe%t ((E F) in the sense of the ,aria!le )hase of Aµ 3

>. an' the De Haas5Van Al)hen effe%t (F (E ) in the sense of the )erio'i%it" of os%illations

in the effe%t 'etermine' !" Aµ )otential 'e)en'en%" an' $a&$e in,arian%e.

1>7



In ea%h of these %ir%&mstan%es3 the Aµ )otentials are 'es%ri!e' as )h"si%all" meanin$f&l %onstr&%ts at

the %lassi%al le,el3 n&m!ers / an' 2 an' at lon$ ran$e in the %ase of effe%t /. (oheren%e len$th

limitation is note' in effe%ts 13 2 an' >. What this im)lies is that a )ro)erl" %onstr&%te' an' ell5

'efine' to)olo$i%al o!6e%t %an manifest3 alon$ ith ma$neti% fl&43 sets of )otentials alon$ its len$th

an' tra6e%tor" %on'itionall" 'e)en'ent &)on the antenna %reatin$ the fiel' o!6e%t.

It o&l' !e interestin$ in f&t&re resear%h to e4)lore those to)olo$ies as 9&anti=ation of the fiel' o!6e%t

here a))ro)riate.

Finall"3 H.F. Harm&th )ro)ose' an amen'ment to #a4ell?s e9&ations K3 100L. The s&!6e%t ha,in$

$one thro&$h m&%h 'is%&ssion K101L3 )resents elo9&entl" an ar$&ment of )ossi!ilities of e4)ansion in

the inter)retation of the more a!stra%t )arts of the #a4ell e9&ations. One %onse9&en%e the

a))li%ation of for%e to se)arate the %har$es !" the a))li%ation of %&rrent3 the %&rrent la$s !ehin' the

,olta$e )otential3 the )oint %har$es are !oth )arti%les an' a,es K10/L. It is anti%i)ate' that ste)s

toar' a )roof of the notions 'is%&sse' in this se%tion %an !e forme' th&sl".

An ele%tri%all"5%har$e' short antenna has a tri,ial %hara%teristi% im)e'an%e hen %om)are' loa'

hose im)e'an%e 'ominates the %ir%&it3 'esira!le for sen'in$ )oer o,er 'istan%es< e.$.3 the )oer is

in'&%e' to tra,erse the free5s)a%e !eteen the transmitter an' re%ei,er to the loa' here the

im)e'an%e is lo%ate'3 a )rimiti,e form of 'ire%tional fin'in$ KC@L.

>.> The loo) ratio in 'eterminin$ 'i)ole inte$rit"

T")i%all"3 small loo) antennas ha,e a )oor effi%ien%" &nless the" are %onstr&%te' at a ratio of their

%ir%&mferen%e to the a,elen$th of the transmission fre9&en%". (onsi'erin$ the ratio h")othesis3 for

f&rther si=e re'&%tion3 the loo) %an also ha,e similar )oer transmission effi%ien%" if the ratio is also

one5tho&san'th relati,e to the 9&arter a,elen$th ith the %ir%&mferen%e. It is s&$$este' it o&l' also

!e effe%ti,e at one5ten tho&san'th $i,en the e4)onential )ro$ression3

/πr i 0.01@@>777m1 = = =0.0100@0B20/3

0.010./7λ 0.>1@B//@m 17>

r i = 0.002m.

A))lie' to the )h"si%al antenna mo'el3 the antenna %hara%teristi%s are tr&n%ate' to a 9&arter5a,e

λantenna3 as3 3 the e4)erimental set&) of the antenna *ee)s it mo&nte' )er)en'i%&lar to the earth an'

>

not %onne%te' to a $ro&n' )lane. The $ro&n' )lane is i$nore'. The resonator floatin$ a!o,e $ro&n' is

1>C



s&s%e)ti!le to the for%es at the interfa%e< the )hase shiftin$ of the am)lifier o,er the s&rfa%e of the

antenna thro&$h π rotations in the im)e'an%e of free5s)a%e to see*s a %lose' %ir%&it %onne%tion.

It is this see*in$ effe%t that "iel's some 9&alifie' e,i'en%e that there is a )ossi!ilit" of a strainin$ of

the s)a%e !eteen the antenna %ite' the )attern of re)etitio&s )ro)erties not &nli*e a set of harmoni%sa%ross s)a%e. It is s&s)e%te' the o!ser,a!le strain is analo$o&s to that 'es%ri!e' !" #a4ell.

7 Thesis %on%l&sions

The literat&re hi%h %ontains 'es%ri)tions of )h"si%al 'e,i%es ten's to a''ress t")i%al )ro!lems in

ireless )oer transferantennas3 in'&%tan%e3 %o&)lin$3 an' ener$" transfer3 !" %lear mention of

ma$neti% resonan%e or notre$ar'less of %oil si=e or 'esi$n of e4)erimental set&). It %an !e s&rmise'

!" this fa%t alone that the )henomenon of ma$neti% resonan%e ireless )oer is s&%h that it matters

little of the si=e of the %oils or the 'istan%e a)art or the )attern an' n&m!er of antennas or the means

!" hi%h the ener$" is &tili=e'< hat matters is that the )henomenon is %ommon)la%e la of )h"si%s3

altho&$h not e4)ressl" state' as s&%h. e%a&se of the %on%l&sion of the similarities in anal"ti%s of

other resear%hers3 it !e%omes a))arent that the ele%tron affe%ts other ele%trons at a 'istan%e !" a

%onne%tion e4)laine' !" 9&ant&m me%hani%s. Hoe,er3 in none of the literat&re s&r,e"e' for this

thesis has the mathemati%al tools of 9&ant&m me%hani%s !een a))lie' to ma$neti% resonant %o&)lin$.

This thesis attem)ts to !e$in the frameor* !" inte$ratin$ $a&$e fiel' theor" to sha)e an

&n'erstan'in$ of the 9&ant&m me%hani%al im)li%ations an' a thoro&$h anal"sis of the ma$neti%

)otential !eteen the %o&)lin$ an' free5s)a%e re,ealin$ str&%t&ral e9&ations hi%h )re'i%t the

rea%ti,e for%es an' s&!se9&ent o!ser,a!le )ro)erties.

The or* )resente' in this thesis attem)ts to a''ress ma$neti% resonant %o&)lin$ form a

)henomenolo$i%al )oint5of5,ie. That is3 as*in$ 9&estions an' formin$ h")otheses on o!ser,ations of

to 'istin%tl" 'ifferent t")es of ireless )oer s%hemes. In a''ition3 the thesis tries to str&%t&re its

anal"ti%al ar$&ments !ase' on non5isolation of the )henomenon< meanin$ the )henomenon 'oes not

e4ist in e4%l&sion of other for%es3 rather3 !e%a&se of them. A no,el %ontri!&tion the thesis is

)resentin$ is e4a%tl" this )oint that for%es a))lie' to an ele%tron emittin$ ra'iation im)ress a )attern

onto free5s)a%e an' that this )attern e4ists onl" !e%a&se there are %o&nterfor%es sim)l" !" the

%onne%tion !eteen the ori$inal for%e an' %onse9&ential for%e. It insists there is a stron$ %onne%tion

!eteen the a%%elerate' ele%tron an' the im)ression of its ra'iation on non5%har$e' s)a%es.

In this thesis3 the metho' an' means of ireless )oer !" ma$neti% resonan%e has !een 'es%ri!e'.

The fo%&s of this resear%h as on the 'es%ri)tion an' e4)erimental testin$ of an effi%ient ireless

)oer mo'el to !e &se' in se,eral a))li%ation areas3 nota!l" those for in a h&man )ro4imit".

1>B



Hoe,er3 the metho' of )oer 'eli,er" is $eneral an' %an !e a))lie' to other a))li%ations hi%h

re9&ire ireless )oer transfer on s%ales !oth lar$e an' small K213 @@L as ell as man" others.

7.1 (o&)le'5mo'e %on%l&sions

While the seat of the ar$&ments for the metho' of h" for ma$neti% resonan%e lie e4%l&si,el" ith

K/>L3 it is &se' ,er!osel" !" man" other e4am)les of ireless )oer transfer. While the a&thor %an

%ite ho %o&)le'5mo'e theor" o&l' a))l" to s%hemes s&%h as the one ill&strate' in this thesis3 it is

still not %lear ho the %o&)le'5mo'es manifest $i,en the sim)listi% mo'el of the 'i)ole )&r,e"e' in

s%holasti% )h"si%s te4ts. For %o&)le'5mo'e to !e s&%%essf&l3 it seems a))arent that the 'imensionalit"

of the 'i)ole is !ro&$ht into 9&estion. In terms of si$nalin$3 as it has !een &n'erstoo' sin%e the 'a"s

of #ar%oni3 the sin$le 'imensionalit" is satisf"in$. In terms of ireless )oer transfer3 hi%h is a

form of si$nalin$ in the s%heme of ra'iant transfer3 the 'imensionalit" fails to 'es%ri!e the s"stem. In

or'er to attem)t to %ome to terms ith the 'is%re)an%"3 > intro'&%es a 'imensionall"5e9&i,alent

'es%ri)tion of the ener$ies o!ser,e' in the s%heme. It is anti%i)ate' that 'es%ri)tions in this

ar%hite%t&re %an she' li$ht onto %&rrent 'ilemmas in ele%troma$netism. Re$ar'less of intention or

'is%re)an%"3 %o&)le5mo'e theor" is a &sef&l tool to 'es%ri!e mathemati%all" ireless )oer s%hemes.

7./ (ir%&it mo'el an' )ro)a$ation %on%l&sions

The %ir%&it mo'el is an attem)t to )resent the ar$&ments for %o&)le'5mo'es into a )h"si%al )ara'i$m.

The mo'el )ro)ose' the em!o'iment of %o&)le' mo'es an' ener$" e4%han$e )ro)erties !eteen to

'istant %ir%&lar loo)s. " %onsi'erin$ the $eometr" of the loo)s as a f&n'amental 'es%ri)tion of the

manifest ele%tri% an' ma$neti% fiel's %o&)le' to its ra'iation3 it has hel)e' to e4ten' the !asi%

%on%e)ts of a 'efinition of !roa'%ast )oer !eteen 'istant antenna formin$ a %lose' in'&%tion

%ir%&it. The e4tension !&n'le' into mo'el )ro)erties an' a))lie' o)timi=ations of %o&)le'5mo'es as a

h")otheti%al e4)lanation of effi%ient ireless ener$" transmission.

If the loo)s are ma$neti% 'i)ole antenna of a si=e relati,e to their 9&arter a,elen$th3 a'6&stment

alon$ a 'es%en'in$ lo$10 s%ale ill "iel' ma4im&m )oer 'issi)ation in the free5s)a%e !eteen the

antenna. In s&%h a manner3 the e4tant ele%troma$neti% fiel' %reates a s)heri%al shell hose !o&n'ar"

%on'itions intera%t intrinsi%all" ith the %on'&%ti,e )ro)erties of the me'i&m. e%a&se of s&%h a

seemin$l" hi$h %on'&%ti,it" in the s%heme3 here there )erha)s sho&l' not !e $i,en the im)e'an%e of

free5s)a%e3 these res&lts lea' the a&thor to h")othesi=e f&rther that at the !o&n'ar"3 the %on'&%ti,e

me'i&m on 9&ant&m s%ales %onsists of some interfa%e of &nit" in the notion of s)a%e5time itself. This

%o&l' !e the ӕther or 'ar* matter. What is %lear is that somethin$ is $oin$ on at the !o&n'ar" !eteen

the ener$i=e' fiel' an' the me'i&m. 1>@



The mo'el in its sim)li%it" )ro,i'es star* insi$hts that the ener$ies that )ermeate the fiel' %an !e

e4)ressl" 9&antifie'3 $i,en eno&$h so)histi%ation in the metho's. It is therefore interestin$ to e4)lore

the 9&antifi%ations an' to hat f&rther insi$hts %an !e $leane'. Altho&$h it is ,er" interestin$ to

&n'erstan' the )h"si%alit" of hat %onsists a %o&)le' ma$neti% resonant mo'e3 %ontrastin$l"3 hat

rea%ti,e )henomena are res)onsi!le for )ro,i'in$ the s&stainin$ %o&nter5for%e $i,in$ it s&%h a stron$

linear e9&ili!ri&m is e9&all" so. The a&thor s&$$ests f&rther resear%h nee's to !e %on'&%te' on hat

%onsists the str&%t&re of :em)t" s)a%e;.

Whilst the res&lts )resente' are of interest in themsel,es3 the" also thro &) a n&m!er of intri$&in$

9&estions that nee' to !e resear%he'. For e4am)le3 in terms of )erforman%e o,er 'istan%e3 ho

im)ortant are the sha)e an' 'ire%tional settin$s of the in'in$s an' the amo&nt of )oer 'ran !"

the loa' ith re$ar' to the transmission effi%ien%" What *in' of im)ro,ement of the 9&alit"5fa%tor

lea's 'ire%tl" to an im)ro,ement of )erforman%e not onl" at the 'istan%es shon in the fi$&res3 !&t

also at $reater 'istan%es What is the relationshi) of the ar%hite%t&re of the antenna to fa%ilitatin$

hi$her ma$nit&'es at those 'istan%es

(learl"3 the a))roa%h 'es%ri!e' here has enormo&s )otential for ireless )oer transfer from a %entral

transmitter !ase to remote re%ei,er stations3 itho&t in%&rrin$ the losses '&e to ire resistan%e an' the

inherent %osts of !&il'in$ )h"si%al transmission lines. The a''e' a',anta$e is that mo!ile 'e,i%es

ithin that en,ironment 'o not nee' to %arr" their on )oer %ells !&t rather %an )i%* &) )oer

irelessl" from the transmittin$ station. These mo!ile 'e,i%es3 hi%h %o&l' !e a&tonomo&s or sarm

mi%ro5ro!ots3 %an also !e 'ri,en hen o!str&%tions are in their )ath s&%h as alls an' other o!6e%ts.

With in%rease' )oer transmitte' o,er $reater 'istan%es this metho' %o&l' )otentiall" also !e &se' as

a )oer so&r%e for e4ternal ,ehi%les3 ea%h merel" nee'in$ to ho&se its on re%ei,er &nit in or'er to

)i%* &) )oer from a 'istant station. (ontrastin$l"3 its small si=e %o&l' !e e4)loite' to ma*e

miniat&re 'e,i%es re%ei,in$ )oer an' instr&%tions. In all these a"s 'is%o,erin$ hat me'i&m5ran$e

an' )erha)s lon$er 'istan%es !" re)etition are a%hie,a!le ith this metho' of )oer transfer o&l' !e

of $reat interest to 'e,elo)in$ so%ieties an' hi$hl" a',an%e' te%hnolo$i%al )ara'i$ms. The &tilit" of

the s%heme ill&strate' in this thesis )resents man" )ossi!ilities for the f&t&re of effi%ient ireless

ener$" transfer.

Asi'e from the a))li%ation a',anta$es3 it is e9&all" )oi$nant to e4amine other %om)&tational metho's

to 'es%ri!e the ele%troma$neti% )ro!lem3 $i,en the &ni9&eness of the mo'el. Ho 'oes the translation

in mathemati=in$ the mo'el from %ir%&it theor" to fiel' theor" hel) to ease $eneti% falla%ies K>BL

1>



)re,alent in ele%troma$netism (an the more ar!itrar" %om)onents of #a4ell?s theor" !e $i,en

s&!stan%e3 es)e%iall" those %onsi'ere' as )&rel" anal"ti%al 'e,i%es s&%h as the )otential

The resear%h o)ens the 'oor to man" 9&estions !" a,ailin$ a means to anser them in a )h"si%al

e4)eriment. It allos the in,esti$ation of the )ro)erties of hat is %o&)le' to it as ell as some

masses %ontri!&te' !" free5s)a%e< this is the most interestin$ ta*e5aa" *nole'$e from the resear%h

)ro%ess.

7.2 Antenna at the interfa%e %on%l&sions

It is a ,ali' h")othesis3 $i,en the e,i'en%e in this or*3 that it is )ossi!le the metho' is 'ete%tin$

somethin$ at the interfa%e3 s&%h as the 'ar* matterin the )resent a%%e)te' 'es%ri)tion of hat in

anti9&it" as %alle' the ӕther. The %onfi$&ration of the %ir%&itr" 'es%ri!e' for a )rotot")e e4hi!ite'

in the mo'el is the %hara%teristi% ar%het")e of all of the %lasses for ireless )oer transmission

metho'olo$". Nearl" all other s%hemes of %om)ara!le metho'olo$ies are 'eri,e' from it.

The )resen%e of %&rrents on the ire $i,e rise to the ma$neti% fiel's )ro)a$ate' into free5s)a%e

!e%a&se the loo) antenna is a hi$h emitter of ele%troma$neti% )hotons. An interestin$ o!ser,ation is it

this la%* of mass of the )hotons an' their a!ilit" to 'o or*. Hoe,er3 !e%a&se at some ener$" le,el3

a motor is t&rne'3 some mass is enterin$ into the s"stem. Wire' s"stems rel" on ele%tron masses to

)erform or* at a 'istan%e3 ireless s"stems rel" on )hotons to )erform or* at a 'istan%e.

The &nit" of the )otential in %onte4t ith the fiel' o!6e%t is o!ser,a!le &n'er meas&rement of an

&n)oere' antenna an' is e4tant at the n&lls to the left an' ri$ht minor lo!es. It is ass&me' this is a

loose meas&re of %onfirmation of the e4isten%e of a str&%t&re of free5s)a%e that ill ine,ita!l" lea' to

a more 'etaile' f&t&re 'es%ri)tion. A''itional la!orator" reso&r%es o&l' !e re9&ire'3 more than the

!asi% e9&i)ment a,aila!le a mem!er of the )&!li%.

The mo,ement of the os%illation thro&$h π 3 in %onsi'eration of ho the antenna is %o&)le' to an

o)en s"stem thro&$h the interfa%e3 s&$$ests im)li%ations on the im)e'an%e of free5s)a%e of 1/0πΩ.

Sin%e ea%h )hase mo,es thro&$h π 3 a total of C0 %"%les o&l' s&m &) the im)ression of a sin$le

%om)le4 a,e on free5s)a%e. Does the interfa%e s&))ort a lo$arithmi%all"5!ase' %o&)lin$ metho'

For the )&r)oses of the thesis3 the !eha,ior that is interestin$ to the rele,ant 'is%&ssion is that the n5

#OSFET is a hi$h5s)ee' sit%h at a s&ita!le le,el of )oer3 that is3 a!o&t 100 atts.

170



Altho&$h 9&ant&m Hall effe%ts are !est e4hi!ite' in #OSFETs ith $alli&m arseni'e GaAs %hannels3

%onsi'ere' the :hol" $rail; in semi%on'&%tor resear%h3 the os%illator 'is%&sse' in this thesis mi$ht !e

e4hi!itin$ some of these )ro)erties. It is s&$$este' a%%ess to the 9&anti=e' Hall states is a%%essi!le

thro&$h the n5#OSFET ar%hite%t&re hi%h are !ein$ mani)&late' on the antenna interfa%e.

The a&thor h")othesi=es an os%illator" rea%tion for%e )ermeates the a,e$&i'e. The in%rease of

ma$neti% )otential is the means !" hi%h to &n'erstan' the for%e.

This or* has 'es%ri!e' the )rin%i)le of ireless ener$" transmission !" ma$neti% resonan%e.

JED

171



7.> To)i%s of f&t&re resear%h

Thro&$ho&t this thesis3 to)i%s for f&t&re resear%h ha,e !een note'. Generall" s)ea*in$3 it o&l' !e of

f&rther interest

To st&'" the )h"si%al $eometr" of the s%heme the a''ition of a se%on'ar" %oil hose in'in$ is

e9&i'istant on the 4 " 3 )lane an' the a''ition of re%ei,ers to :%hain; the )oer si$nal o,er lon$er

'istan%es.

To re,isit the Aharono,5ohm e4)eriments K/L3 an' fla,ors of s&!se9&ent attem)ts to see if the

)ro!lem %an !e mo'ele' on lar$er s%ales3 as r π 3 an' at hi$her tem)erat&res.

To e4)eriment ith 'ifferent len$ths of %om)onent 2 of the transmitter %oil an' its )osition alon$

the len$th A. Is there a 'ifferen%e in )ro6e%te' fiel' $eometr" or )oer

To %reate antenna from metamaterials< the metamaterial %on%e)t has %a)t&re' the ima$ination of a

si$nifi%ant an' $roin$ )ool of resear%hers from a i'e ran$e of fiel's3 ith the mo,e toar'

)ra%ti%al a))li%ations )erha)s most e,i'ent in the area of antennas an' ireless ener$" )ro)a$ation. It

is interestin$ to &se %olle%tions of s&!5a,elen$th elements in a str&%t&re' me'i&m of ,ol&metri%

metamaterials three5'imensions3 meta5s&rfa%es to 'imensions3 an' meta5ires one 'imension

for %ontrol o,er the !roa'%ast an' re%ei,in$ )ro)erties of the antenna. Th&sl"3 the me'i&m ithin the

loo) %an !e %aref&ll" %ontrolle' an' its e4oti% )ro)erties re,eale' %ontrolle' in that its %ommonl"

&n'erstoo' )ro)erties ith &n%ommon %ontrol o,er s)atial ,ariation3 )olari=ation sensiti,it"3 an'

fre9&en%" 'is)ersion< E4oti% in that the ansers to the reasons of its !eha,ior ere not forth%omin$.

Ne,ertheless3 the me'i&m is %&rrentl" ina%%essi!le3 the e4)ansion of !asi% )arameters an' more finite

$ran&lar %ontrol o,er the antenna an' its infl&en%e at the interfa%e o&l' $reatl" ai' in,esti$ation in

s&!se9&ent )ro)erties of the me'i&m of free5s)a%e !e"on' the interfa%e. Of all the )ossi!le s&$$este'

resear%h3 not one is more )rofo&n' than the a))li%ation of metamaterials to the in,esti$ation.

To st&'" the )arameters of a %om)&tational )ara'i$m aro&n' the mo'el to %a)t&re the %om)le4

$eometr" an' )otentials of the fiel's.

astl"3 'oes the a))arat&s 'ete%t 'ar* matter

17/



C A))en'i4 M Ta!les3 fi$&res3 an' %o'e

This se%tion %ontains the lar$e ta!les an' fi$&res as ell as the !asi% #atla! %o'e to %reate an antenna

mesh as the one 'es%ri!e' in this thesis an' rea'" for &se ith a sol,er of "o&r %hoi%e. I &se' the

metho' of moments #o#.

The ,aria!le names in Ta!le C.1 ha,e !een %han$e' for %onsisten%" in the te4t. The names affe%te'

/ /are *! & 3 f α3 a r i3 A r 6.

/Ta!le C.1. Val&es of fa%tor & K2/L.

172



Fi$.C.1. Effe%t on the sim&late' o&t)&t !ase' on %han$in$ the ,al&e of the %o&)lin$ %oeffi%ient κi6 a

κ = 0.00/3 ! κ = 0.00>3 % κ = 0.00@3 ' κ = 0.010.

NOTE The 4 an' "5s%ales for ea%h a,eform are /00 µS an' / ,olts )er 'i,ision3 res)e%ti,el".

17>



This is a set of instr&%tions to &se #atla! ith the PDE tool!o4 to '&)li%ate the mesh &se' to %om)&te

the antenna.

Ta!le C./. Instr&%tions to %reate an antenna mesh file.

Instr&%tions to %reate an antenna file str&%t&re.

1. O)en #atla!3 t")e )'etool3/. (li%* O)tionsA4es E9&al3

2. Dra an" elli)se3 'o&!le5%li%* on its s&rfa%e3a. Set Q5%enter to 03

!. Set Y5%enter to 03%. Set A5semia4es to 2 for a ra'i&s of 2 &nits3

'. Set 5semia4es to 2 for a ra'i&s of 2 &nits3>. Dra a se%on' elli)se3 'o&!le5%li%* on its s&rfa%e3

a. Set Q5%enter to 03!. Set Y5%enter to 03

%. Set A5semia4es to /.B7 for a ire thi%*ness of 0./7 &nits3

'. Set 5semia4es to /.B7 for a ire thi%*ness of 0./7 &nits3

7. Set form&la E15E/3

C. (li%* #eshParameters...a. Set #a4im&m e'$e si=e to Inf3

B. Initiali=e the mesh3

@. Refine the mesh3a. Refinin$ the mesh ti%e3 %he%* trian$le 9&alit" !" %li%*in$ #eshTrian$le 9&alit"3

. The a,era$e 9&alit" sho&l' !e 0.7 a%ross the s&rfa%e for an i'eal mesh3

10. Sa,e the file3

11. E4)ort the mesh !" %li%*in$ #eshE4)ort #esh3a. ea,e the ,aria!le names for mesh 'ata )oints3 e'$es3 trian$les ) e t3

!. The Wor*s)a%e no %ontains the 'ata3

1/. The arra" of !o&n'ar" e'$es e is not nee'e'3

a. t")e )23X0<12. Sa,e the %om)ile' mat file3

a. Sa,e filename ) t

1>. The antenna str&%t&re file has !een %reate'.

The meshes are &se' as antenna $eometr" to r&n the #o# sol&tion. Usin$ a sam)le sol&tion KC7L3 r&n

it in the or'er liste' in Ta!le /.C to $enerate the 'ata 'is%&sse' in /.2.

177



B iterat&re s&r,e"

P&!li%ations res&ltin$ from this thesis

(.A. T&%*er3 :#a$neti% resonant mo'es in a ireless5)oere' %ir%&it3; 1th Tele%omm&ni%ations for&m TEFOR3 )).BB5

@03 No,. /011.

(.A. T&%*er3 -. Wari%*3 an' W. Hol'er!a&m3 :Effi%ient ireless )oer 'eli,er" for !iome'i%al im)lants3; IET Wireless

Sensor S"stems3 Vol. /3 Iss. 23 ))./@B5C3 A&$. /01/.

(.A. T&%*er3 -. Wari%*3 an' W. Hol'er!a&m3 :A %ontri!&tion to the ireless transmission of )oer3; International

8o&rnal of Ele%tri%al Poer an' Ener$" S"stems3 Vol. >B3 ))./275>/3 #a" /012.

K1L G. Re!er3 En'less3 o&n'less3 Sta!le Uni,erse3 Uni,ersit" of Tasmania3 Tasmania3 1BB.

K/L . H&nt3 The #a4ellians3 (ornell Uni,ersit" Press3 Ne Yor*3 /007.

K2L F. War!&rton3 :Dis)la%ement (&rrent3 A Useless (on%e)t3; Ameri%an 8o&rnal of Ph"si%s3 Vol. //3 Iss. 73 ))./52073

O%t. 17>.

K>L W. Rosser3 :Dis)la%ement %&rrent an' #a4ell?s e9&ations3; Ameri%an 8o&rnal of Ph"si%s3 Vol. >23 Iss. C3 )).@0B5@3

8&n. 1B7.

K7L 8. armor3 :A '"nami%al theor" of the ele%tri% an' l&minifero&s me'i&m3; Philoso)hi%al Transa%tions of the Ro"al

So%iet"3 Vol. 103 >2@5C13 De%. 1@2.

KCL 8. armor3 :A '"nami%al theor" of the ele%tri% an' l&minifero&s me'i&m3 Part II3 theor" of ele%trons3; Philoso)hi%al

Transa%tions of the Ro"al So%iet"3 Vol. 103 )). C75B>23 8&n 1@7.

KBL P. Nahin3 Oli,er Hea,isi'e The ife3 Wor*3 an' Times of an Ele%tri%al Geni&s of the Vi%torian A$e3 8ohns Ho)*ins

Uni,ersit" Press3 #ar"lan'3 /00/.

K@L V. Ste)in3 Theoreti%al -nole'$e3 S)rin$er3 Ne Yor*3 /00/.

KL 8. Po"ntin$3 :On the Transfer of Ener$" in the Ele%troma$neti% Fiel'3; Phil. Trans. 1B73 )).2>25C13 1@@>.

K10L A. -&rs3 Poer Transfer Thro&$h Stron$l" (o&)le' Resonan%es3 #asters of S%ien%e thesis3 #IT3 Se). /00B.

K11L -. -o'aira3 (om)le4 #anifol's an' Deformation of (om)le4 Str&%t&res3 Ne Yor*3 S)rin$er3 Ne Yor*3 )). /@573

1@1.

K1/L R. Pfeifer3 T. Nieminen3 N. He%*en!erf3 an' H. R&!ins=tein5D&nlo)3 :#oment&m of an ele%troma$neti% a,e in

'iele%tri% me'ia3; Re,ies of #o'ern Ph"si%s3 Vol. B3 )).11B51/1C3 O%t. /00B.

K12L N. Tesla3 :E4)eriments With Alternate (&rrents of Hi$h Potential an' Hi$h Fre9&en%"3; IEE A''ress3 on'on3

Fe!r&ar" 23 1@/3 %ontaine' in In,entions3 Resear%hes an' Writin$s of Ni*ola Tesla3 D. ,an Nostran' (om)an"3 Ne

Yor* 3 )).1@C513 1@>.

K1>L N. Tesla3 A))arat&s for the Utili=ation of Ra'iant Ener$"3 U.S. Patent No. C@737B3 File' /1 #a" 1013 Aar'e' 7

No,. 101.

K17L H. Ya$i3 :eam transmission of &ltra5short a,es3; Pro%ee'in$s of the Instit&te of Ra'io En$ineers3 Vol. 1C3 Iss. C3

)).B175>03 1/@.

K1CL E. Sierin$5St&m)f3 :Shorta,e DQ Han'!oo* Yo&r $atea" to orl'i'e DQin$3; Dar%5Verla$3 erlin3 /007.

K1BL W. Tei%h an' W. ron3 :Desi$n of (W am)litrons for s)a%e5!orne telemetr" a))li%ations3; IEEE Transa%tions on

Ele%tron De,i%es3 Vol. 103 Iss. /3 ). 10@3 1C2.

K1@L W. ron3 :E4)eriments in,ol,in$ a mi%roa,e !eam to )oer an' )osition a heli%o)ter3; IEEE Transa%tions on

Aeros)a%e an' Ele%troni% S"stems3 Vol. AES573 Iss. 73 )).C/5B0/3 1C.

K1L 8. S%h&'er3 :Ener$" transfer into a %lose' %hest !" means of stationar" %o&)lin$ %oils an' a )orta!le hi$h5)oer

os%illator3; Ameri%an So%iet" for Artifi%ial Internal Or$ans3 Vol. B3 )).2/B5213 1C1.

17C



K/0L #. Soma3 D. Gal!raith3 an' R. White3 :Ra'io5fre9&en%" %oils in im)lanta!le 'e,i%es #isali$nment anal"sis an' 'esi$n

)ro%e'&re3; IEEE Transa%tions on iome'i%al En$ineerin$3 #E52>>3 )). /BC5@73 A)r. 1@B.

K/1L R. Fano3 . (h&3 an' R. Al'er3 Ele%troma$neti% Fiel's3 Ener$"3 an' For%es3 8ohn Wile" Sons3 Ne Yor*3 1C.

K//L G. Fit=$eral'3 :The Ether an' the Earths Atmos)here3; S%ien%e3 Vol. 1/3 Iss. 2/@3 )).203 #a" 1@@.

K/2L 8. #a4ell3 On )h"si%al lines of for%e3 on'on Ta"lor an' Fran%is3 1@C1.

K/>L H. Ha&s an' W. H&an$3 :(o&)le'5mo'e theor"3; Pro%ee'in$s of the IEEE3 Vol. B3 No. 103 )).170751@3 O%t. 11.

K/7L S. S%hel*&noff3 :General Tele$ra)hist?s E9&ations3; The ell S"stem Te%hni%al 8o&rnal3 Vol. 1113 ).B@>5@013 8&l. 17/.

K/CL R. Al'er3 . (h&3 an' R. Fano. Ele%troma$neti% Ener$" Transmission an' Ra'iation3 8ohn Wile" Sons3 In%.3

(am!ri'$e3 1C.

K/BL S. Ilić3 #.T. Perić3 S.R. Ale*sić3 an' N.. Raič,ić3 :J&asi TE# anal"sis of s"mmetri%al %o&)le' stri) line &sin$ ne

h"!ri' !o&n'ar" element metho'3; 1th Tele%omm&ni%ations for&m TEFOR3 )).100@5113 No,. /011.

K/@L S. #anić3 S. Sa,ić3 #. Ilić3 an' . Notaroc. :(om!inin$ finite element metho' an' Fo&rier transform to anal"=ea,e$&i'e transients3; 1th Tele%omm&ni%ations for&m TEFOR3 )).100>5B3 No,. /011.

K/L Y. Aharono, an' D. ohm3 :Si$nifi%an%e of Ele%troma$neti% Potentials in the J&ant&m Theor"3; Ph"s. Re,.3 Vol. 1173

Iss. 23 A&$. 17.

K20L 8. -ra&s3 Antennas for All A))li%ations3 2r' E'ition3 #%Gra5Hill3 oston3 /001.

K21L A. -aralis3 8. 8oanno)o&los an' #. Sol6ačić3 :Effi%ient ireless non5ra'iati,e mi'5ran$e ener$" transfer3; Annals of

Ph"si%s3 Vol. 2/23 )).2>5>@3 A)r. /00@.

K2/L F. Gro,er3 In'&%tan%e %al%&lations Wor*in$ form&las an' ta!les3 Ne Yor*3 D. ,an Nostran' (om)an"3 1>C.

K22L R. Elliott3 Ele%troma$neti%s. Ne Yor*3 IEEE Press3 12.

K2>L #. Soma3 D. Gal!raith3 an' R. White3 :Transmission of time ,ar"in$ ma$neti% fiel' thro&$h !o'" tiss&e3; 8o&rnal of

iolo$i%al Ph"si%s3 Vol. 23 )).7510/3 8&n. 1B7.

K27L F. Ne&mann3 :All$emeine Geset=e 'er in'&%irten ele*tris%hen Strdme3; A!han'l&n$en 'er -dni$li%hen A*a'emie 'er

Wissens%haften =& erlin3 1@>7.

K2CL . Hanna*am3 :ere%hn&n$ 'er Ge$enin'&*ti,itat a%hsen)aralleler Z"lin'ers)&len3; Ar%hi,fiir Ele*trote%hni*3 Vol. 713

No. 23 )).1>157>3 1CB.

K2BL I. Aai3 Y. Zhan$3 T. -omori3 an' T. Ishi=a*i3 :(o&)lin$ %oeffi%ient of s)iral resonators &se' for ireless )oer

transfer3; Pro%ee'in$s of Asia5Pa%ifi% #i%roa,e (onferen%e3 )).12/@5213 De%. /010.

K2@L I. Aai an' Y. Zhan$3 :Se)aration of %o&)lin$ %oeffi%ient !eteen resonators into ele%tri% an' ma$neti% %ontri!&tions

toar' its a))li%ation to DF 'e,elo)ment3; (hina58a)an 8oint #i%roa,e (onferen%e3 )).C1573 Se). /00@.

K2L V. (hristianto an' F. Smaran'a%he3 :A Note on (om)&ter Sol&tion of Wireless Ener$" Transmit ,ia #a$neti%

Resonan%e3; Pro$ress in Ph"si%s3 Vol. 13 8an. /00@.

K>0L H. orent=3 :Attem)t of a theor" of ele%tri%al an' o)ti%al )henomena in mo,in$ !o'ies3; E.8. rill3 ei'en3 The

Netherlan's3 1@7.

K>1L H. orent=3 :Ele%troma$neti% )henomena in a s"stem mo,in$ ith an" ,elo%it" smaller than that of li$ht3; Pro%e'&res

of A%a'emi% S%ien%e Amster'am3 Vol. 13 )).>/B5>23 10>.

K>/L H. Poin%ar3 :On the '"nami%s of the ele%tron3; Wee*l" re)orts of meetin$s of the A%a'em" of S%ien%es3 Vol. 1>03

)).170>5@3 107.

K>2L 8. armor3 :On a '"nami%al theor" of the ele%tri% an' l&minifero&s me'i&m3 Part III3 relations ith material me'ia3;

Philoso)hi%al Transa%tions of the Ro"al So%iet"3 Vol. 103 /075200.

K>>L E. Ho%hmair3 :S"stem o)timi=ation for im)ro,e' a%%&ra%" in trans%&taneo&s si$nal an' )oer transmission3; IEEE

Transa%tions on iome'i%al En$ineerin$3 ,ol. 213 )). 1BB51@C3 Fe!. 1@>.

17B



K>7L (. Zierhofer3 :Geometri% A))roa%h for (o&)lin$ Enhan%ement of #a$neti%all" (o&)le' (oils3; IEEE Transa%tions on

iome'i%al En$ineerin$3 Vol. >23 No. B3 )). B0@51>3 8&l. 1C.

K>CL H. S%he"3 Di, Gra' (&rl an' all that An informal te4t on ,e%tor %al%&l&s3 W.W. Norton (om)an"3 1B.

K>BL 8. Ro%he3 :The )resent stat&s of #a4ell?s 'is)la%ement %&rrent3; E&r. 8. Ph"s.3 Vol. 13 )).1775CC3 1@.

K>@L E. Post. :Interferometri% )ath5len$th %han$es '&e to motion3; 8o&rnal of the O)ti%al So%iet" of Ameri%a3 Vol. C/3 Iss. /3

))./2>53 1B/.

K>L . 'e ro$lie3 :Resear%hes on the 9&ant&m theor"3; Ann&als of Ph"si%s3 Vol. 23 No. //3 1/>.

K70L Y. (o&'er an' E. Fort3 :Sin$le5)arti%le 'iffra%tion an' interferen%e at a ma%ros%o)i% s%ale3; Ph"si%al Re,ie etters3

Vol. B3 Iss. 173 O%t. /00C.

K71L T. arrett. To)olo$i%al Fo&n'ations of Ele%troma$netism. Worl' S%ientifi% P&!lishin$ (om)an". /00@.

K7/L -. (or&m an' 8. (or&m3 ;RF (oils3 Heli%al Resonators an' Volta$e #a$nifi%ation !" (oherent S)atial #o'es;3 IEEE

#i%roa,e Re,ie3 Vol. B3 No. /3 )). 2C5>73 Se). /001.

K72L H. Hert=. :A!o&t ,er" ra)i' ele%tri%al os%illations3; Annalen 'er Ph"si*3 Vol. /CB3 No. B3 )).>/15>@3 #a" 1@@B.

K7>L H. Hert=. :On the )ro)a$ation ,elo%it" of the ele%tro'"nami% effe%ts3; Annalen 'er Ph"si*3 Vol. /B03 No. B3 )).7715C3

#a" 1@@@.

K77L E. Armstron$3 :O)eratin$ Feat&res of the A&'ion3; Annals of the Ne Yor* A%a'em" of S%ien%es3 Vol. /B3 No. 13

))./17M>23 A&$. 11B.

K7CL 8. Hoa$3 asi% Ra'io The Essentials of Ele%tron T&!es an' their (ir%&its3 Ne Yor*3 D. Van Nostran' (om)an"3

1>/.

K7BL N. #ali*3 Ele%troni% %ir%&its anal"sis3 sim&lation3 an' 'esi$n3 Prenti%e Hall3 Ne 8erse"3 )).217MC3 17.

K7@L A. -&rs3 A. -aralis3 R. #offatt3 8. 8oanno)o&los3 P. Fisher3 an' #. Sol6ači%3 :Wireless Poer Transfer ,ia Stron$l"

(o&)le' #a$neti% Resonan%es3; S%ien%e3 Vol. 21B3 )).@25@C3 8&l. /00B.

K7L S. #a*aro,3 :#o# antenna sim&lators ith #atla! RWG !asis f&n%tions3; IEEE Antennas an' Pro)a$ation

#a$a=ine3 Vol. >23 )).1005B3 O%t. /001.

KC0L S. Rao3 D. Wilton3 an' A. Glisson3 :Ele%troma$neti% s%atterin$ !" s&rfa%es of ar!itrar" sha)e3; IEEE Transa%tions on

Antennas an' Pro)a$ation3 Vol. 203 No. 23 )).>051@3 #a" 1@/.

KC1L D. Sar*ar3 Ve%tor !asis f&n%tion sol&tion of #a4ell?s e9&ations3 Ph.D. 'issertation3 Ri%e Uni,ersit"3 TQ3 USA3 #a"

1B.

KC/L . raaten3 R. Nelson3 an' #. #ohamma'3 :Ele%tri% fiel' inte$ral e9&ations for ele%troma$neti% s%atterin$ )ro!lems

ith ele%tri%all" small an' ele%tri%all" lar$e re$ions3; IEEE Transa%tions on Antennas an' Pro)a$ation3 Vol. 7C3 Iss. 13

)).1>/5703 8an. /00@.

KC2L A. Peterson3 S. Ra"3 an' R. #ittra3 (om)&tational #etho's for Ele%troma$neti%s3 IEEE Press3 Pis%ataa"3 N83 1@.

KC>L 8. Shin3 #o'elin$ of ar!itrar" %om)osite o!6e%ts ith a))li%ations to 'iele%tri% resonator antennas3 Ph.D. 'issertation3

The Uni,ersit" of #ississi))i3 #S3 USA3 A&$. /001.

KC7L S. #a*aro,3 Antenna an' E# mo'elin$ ith #atla!3 Wile"5Inters%ien%e3 /00/.

KCCL R. Pa&l3 Ele%tri%al fiel's an' sim)le %ir%&its3 2r' E'ition3 S)rin$er5Verla$3 erlin3 12.

KCBL A. Gar%a3 8. (arras%o3 8. Soto3 F. #a$anto3 an' (. #orgn3 :A metho' for %al%&latin$ the ma$neti% fiel' )ro'&%e' !" a

%oil of an" sha)e3; Ph"si%al Sensors an' A%t&ators A3 Vol. 13 Iss. /3 ))./205/3 /001.

KC@L R. Smith5Rose an' H. Ho)*ins3 :The a))li%ation of &ltra5short5a,e 'ire%tion fin'in$ to ra'io so&n'in$ !alloons3;

Pro%ee'in$s of the Ph"si%al So%iet"3 Vol. 7@3 No. /3 #ar. 1>C.

KCL A. Pantelo)o&los an' N. o&r!a*is3 :A s&r,e" on eara!le sensor5!ase' s"stems for health monitorin$ an' )ro$nosis3;

IEEE Transa%tions on S"stems3 #an3 an' ("!erneti%s3 Vol. >03 Iss .13 )).151/3 8an. /010.

17@



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Im)lant Te%hnolo$" for ("!erneti% S"stems3; asi% S%ien%e Seminars in Ne&rolo$"3 Vol. C03 No. 103 )).12C5B23 O%t.

/002.

KB1L N. Neihart an' R. Harrison3 :#i%ro)oer %ir%&its for !i'ire%tional ireless telemetr" in ne&ral re%or'in$ a))li%ations3;IEEE Transa%tions on iome'i%al En$ineerin$3 Vol. 7/3 )).170M73 No,. /007.

KB/L #. Gho,anloo an' -. Na6afi3 :A ireless im)lanta!le m&lti%hannel mi%rostim&latin$ s"stem5on5a5%hi) ith mo'&lar

ar%hite%t&re3; IEEE Transa%tions on Ne&ral S"stems an' Reha!ilitation En$ineerin$3 Vol. 173 )).>>M7B3 Se). /00B.

KB2L -. Wari%*3 #. Gasson3 . H&tt3 I. Goo'he3 P. -"!er'3 H. S%h&l=rinne3 an' Q. W&3 :Tho&$ht (omm&ni%ation an'

(ontrol A First Ste) Usin$ Ra'iotele$ra)h"3; IEEE Pro%ee'in$s on (omm&ni%ations3 Vol. 1713 Iss. 23 )).1@753 8&n.

/00>.

KB>L #. Gasson3 . H&tt3 I. Goo'he3 P. -"!er'3 an' -. Wari%*3 :In,asi,e Ne&ral Prosthesis for Ne&ral Si$nal Dete%tion

an' Ner,e Stim&lation3; International 8o&rnal of A'a)ti,e (ontrol an' Si$nal Pro%essin$3 Vol. 13 Iss. 73 )).2C75B73

8&n. /007.

KB7L P. imo&sin3 P. -ra%*3 P. Polla*3 A. ena==o&=3 (. Ar'o&in3 D. Hoffmann3 an' A5. ena!i'3 :Ele%tri%al Stim&lation

of the S&!thalami% N&%le&s in A',an%e' Par*insons Disease3; Ne En$lan' 8o&rnal of #e'i%ine3 Vol. 223 )).11075

113 O%t. 1@.

KBCL E. Romero3 R. Warrin$ton3 an' #. Ne&man3 :Ener$" s%a,en$in$ so&r%es for !iome'i%al sensors3; Ph"siolo$i%al

#eas&rement3 Vol. 203 )).R275RC/3 /00.

KBBL #. a*er an' R. Sar)esh*ar3 :Fee'!a%* anal"sis an' 'esi$n of RF )oer lin*s for lo5)oer !ioni% s"stems3; IEEE

Transa%tions on iome'i%al (ir%&its an' S"stems3 Vol. 13 ))./@M2@3 #ar. /00B.

KB@L A. RamRa*h"ani3 S. #ira!!asi3 an' #. (hiao. :Desi$n an' O)timi=ation of Resonan%e5ase' Effi%ient Wireless

Poer Deli,er" S"stems for iome'i%al Im)lants3; IEEE Transa%tions on iome'i%al (ir%&its an' S"stems3 Vol. 73

No. 13 )). >@5C23 Fe!. /011.

KBL 8. Gao3 :In'&%ti,e )oer transmission for &ntethere' mi%ro5ro!ots3; IEEE In'&strial Ele%troni%s So%iet" (onferen%e3

)).CM113 No,. /007.

K@0L T. De"le an' #. Re"nol's3 :S&rfa%e ase' Wireless Poer Transmission an' i'ire%tional (omm&ni%ation for

A&tonomo&s Ro!ot Sarms3; IEEE International (onferen%e on Ro!oti%s an' A&tomation3 )).102C5>13 #a" /00@.

K@1L Z. o3 R. (hin$a3 R. Tsen$3 an' 8. in3 :Desi$n an' test of hi$h5)oer hi$h5effi%ien%" loosel" %o&)le' )lanar

ireless )oer transfer s"stem3; IEEE Transa%tions on In'&strial Ele%troni%s3 Vol. 7C3 No. 73 #a" /00.

K@/L 8. #&r5#iran'a3 G. Fanti3 Y. Fen$3 -. Omana*&ttan3 R. On$ie3 A. Set6oa'i3 an' N. Shar)e3 :Wireless )oer transfer

&sin$ ea*l" %o&)le' ma$netostati% resonators3; IEEE Ener$" (on,ersion (on$ress an' E4)osition3 )).>1B5@C3 Se).

/010.

K@2L -. O?rien3 G. S%hei!le3 an' H. G&el'ner3 :#a$neti% Fiel' Generation in an In'&%ti,el" (o&)le' Ra'io5Fre9&en%"

Poer Transmission S"stem3; IEEE Poer Ele%troni%s S)e%ialists (onferen%e3 )).15B3 8&n. /00C.

K@>L -. (or&m an' 8. (or&m3 ;RF (oils3 Heli%al Resonators an' Volta$e #a$nifi%ation !" (oherent S)atial #o'es3;Tele%omm&ni%ations in #o'ern Satellite3 (a!le an' roa'%astin$ Ser,i%e3 Vol. 13 )).225>@3 Se). /001.

K@7L . (annon3 8. Ho!&r$3 D. Stan%il3 an' S. Gol'stein3 :#a$neti% Resonant (o&)lin$ as a Potential #eans for Wireless

Poer Transfer to #&lti)le Small Re%ei,ers3; IEEE Transa%tions on Poer Ele%troni%s3 Vol. />3 No. B3 )).1@15/73

8&l. /00.

K@CL . W&3 Z. Yan$3 E. asham3 an' W. i&3 :An effi%ient ireless )oer lin* for hi$h ,olta$e retinal im)lant3; IEEE

iome'i%al (ir%&its an' S"stems (onferen%e3 )). 1015>3 No,. /00@.

K@BL . Rin'orf3 . a'in$3 an' O. rein!6er$3 :Resonantl" %o&)le' antennas for )assi,e sensors3; in Pro%. IEEE Sensors3

)). 1C11M1C1>3 O%t. /C5/ /00@.

K@@L 8. Shi)le". In%or)oratin$ Wireless Poer Transfer in an ED i$htin$ A))li%ation3 #aster?s Thesis3 ri$ham Yo&n$

Uni,ersit"3 Utah3 A&$. /00C.

17



K@L N. Hem%he an' A. 8aafari3 :Wireless transmission of )oer &sin$ a P( transformer ith mo!ile se%on'ar"3; IEEE

Ele%trote%hni%al (onferen%e3 )).C/52>3 #a" /00@.

K0L S. ar*er3 D. rennan3 N.G. Wri$ht3 an' A.. Horsfall3 :Pie=oele%tri%5)oere' ireless sensor s"stem ith

re$enerati,e transmit mo'e3; IET Wireless Sensor S"stems3 Vol. 13 Iss. 13 )).215@3 #ar. /011.

K1L N. Timmons an' W. S%anlon3 :Im)ro,in$ the &ltra5lo )oer )erforman%e of IEEE @0/.17.C !" a'a)ti,e

s"n%hroni=ation3; IET Wireless Sensor S"stems3 Vol. 13 Iss. 23 )).1C15B03 A)r. /011.

K/L (. Sa&er3 #. Stana%e,i%3 G. (a&en!er$hs an' N. Tha*or3 :Poer har,estin$ an' telemetr" in (#OS for im)lante'

'e,i%es3; IEEE Transa%tions on (ir%&its an' S"stems3 Vol. 7/3 Iss. 1/3 ))./C075123 De%. /007.

K2L H. Ha&s3 Wa,es an' Fiel's in O)toele%troni%s3 Prenti%e5Hall3 Ne 8erse"3 1@>.

K>L R. Fe"nman3 JED The Stran$e Theor" of i$ht an' #atter3 Prin%eton Uni,ersit" Press3 1@@.

K7L D. Griffiths3 Intro'&%tion to J&ant&m #e%hani%s3 Prenti%e Hall3 oston3 1>.

KCL D. ohm3 J&ant&m #e%hani%s3 Ne 8erse"3 Prenti%e5Hall3 171.

KBL R. St&t=3 Toar's meas&rin$ the ele%tron ele%tri% 'i)ole moment &sin$ tra))e' mole%&lar ions3 Ph.D. thesis3 Uni,ersit"of (olora'o3 /010.

K@L T. (ho3 Intro'&%tion to ele%troma$neti% theor" a mo'ern )ers)e%ti,e3 Ne Yor* 8ones artlett P&!lishers3 /00C.

KL H. Harm&th. :(orre%tion for #a4ell?s e9&ations for si$nals I3; IEEE Trans. Ele%troma$neti% (om)ati!ilit"3 Vol. /@3

Iss. >3 ))./705@3 No,. 1@C.

K100L H. Harm&th. :(orre%tion for #a4ell?s e9&ations for si$nals II3; IEEE Trans. Ele%troma$neti% (om)ati!ilit"3 Vol. /@3

Iss. >3 ))./75CC3 No,. 1@C.

K101L (omments an' re)lies to K3 100L.

K10/L #. A!i'3 . An'reotti3 S. Do&a'" an' (. Nore3 :Os%illatin$ str&%t&res in a stret%he'5%om)resse' ,orte43; 8o&rnal of

Fl&i' #e%hani%s3 >703 ))./0B5123 #a" /001.

K102L N. Tesla3 (olora'o S)rin$s Notes3 Nolit3 el$ra'e3 Ser!ia3 1.

K10>L -. Sel,an an' S. Ehsan3 :A re,isitin$ of s%ientifi% an' )hiloso)hi%al )ers)e%ti,es on #a4ell?s 'is)la%ement %&rrent3;

IEEE Antennas an' Pro)a$ation3 Vol. 713 No. 23 )). 2C5>C3 /00.

K107L 8. Arth&r3 :An elementar" ,ie of #a4ell?s 'is)la%ement %&rrent3; IEEE Antennas an' Pro)a$ation3 Vol. 713 No. C3

)). 7@5C@3 /00.

K10CL (.A. T&%*er3 -. Wari%*3 an' W. Hol'er!a&m3 :Effi%ient ireless )oer 'eli,er" for !iome'i%al im)lants3; IET

Wireless Sensor S"stems3 Vol. /3 Iss. 23 ))./@B5C3 A&$. /01/.

K10BL (.A. T&%*er3 :#a$neti% resonant mo'es in a ireless5)oere' %ir%&it3; 1th Tele%omm&ni%ations for&m TEFOR3

)).BB5@03 No,. /011.

K10@L N. Tesla3 #" In,entions The A&to!io$ra)h" of Ni*ola Tesla3 en 8ohnston e'.3 E4)erimenter P&!lishin$ (om)an"3

In%.3 Ne Yor*3 11.

K10L N. Pa)ani%olao& an' T. Tomaras3 :D"nami%s of ma$neti% ,orti%es3; N&%lear Ph"si%s 3 Vol. 2C03 Iss. /M23 )). >/75C/3

A&$. 11.

K110L N. Gerson3 8. Hen$en3 R. Pi))3 8. We!ster3 :Ra'io5a,e )ro)a$ation to the anti)o'e3; (ana'ian 8o&rnal of Ph"si%s3

Vol. >B3 No. /03 )). /1>2573 O%t. 1C.

K111L (.A. T&%*er3 -. Wari%*3 an' W. Hol'er!a&m3 :A %ontri!&tion to the ireless transmission of )oer3; International

8o&rnal of Ele%tri%al Poer an' Ener$" S"stems3 Vol. >B3 ))./275>/3 #a" /012.

1C0



With s)e%ial than*s to S"'ran'ria

wireless thesis

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