research article lossy-transmission-line analysis of...

Research ArticleLossy-Transmission-Line Analysis of Frequency ReconfigurableRectangular-Ring Microstrip Antenna

Bambang Setia Nugroho Fitri Yuli Zulkifli and Eko Tjipto Rahardjo

Department of Electrical Engineering Universitas Indonesia Kampus Baru UI Depok West Java 16424 Indonesia

Correspondence should be addressed to Bambang Setia Nugroho bambangsetiauiacid

Received 12 August 2014 Accepted 17 October 2014 Published 13 November 2014

Academic Editor Xianming Qing

Copyright copy 2014 Bambang Setia Nugroho et alThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

An analytical model for a frequency reconfigurable rectangular-ring microstrip antenna is proposedThe resonant frequencies andinput impedance of the reconfigurable antenna are analyzed using a lossy-transmission-line (LTL) model By making use of 119884-admittance matrices a formulation for the input impedance is analytically derived The structure of the frequency reconfigurableantenna consists of a rectangular-ring shaped microstrip antenna which is loaded with a rectangular patch in the middle of therectangular-ring antenna and fed by a microstrip line RF switches are applied to connect the load to the antenna in order toreconfigure the operating frequencies By modeling the antenna into a multiport equivalent circuit the total input impedanceis analytically derived to predict the resonant frequencies To verify the analysis the model input impedance and reflectioncoefficient calculation have been compared with the full-wave simulation and measurement results The proposed model showsgood agreement with full-wave simulated and measured results in the range of 1ndash3GHz

1 Introduction

Frequency reconfigurable antennas are receiving higherattention in line with the development of advanced commu-nications system Different techniques have been presentedto achieve reconfigurability of the antenna operating frequen-cies butmostly the reconfigurability is controlled bymeans ofRF switches such as PIN diodes FETs varactor diodes andRF-MEMS switches [1] Recently with the urge to integratevarious types of wireless communication systems such asUMTS LTEWiFi andWiMAX into a single device the needfor novelmultiband frequency reconfigurable antenna is evenmore important

A lot of reconfigurable antenna structures have beenwidely reported such as slot [2ndash4] monopole [5 6] square-ring [7ndash9] and rectangular-ring antenna [10ndash12] Howeverthere are only few design procedures or theories of operationfor the reconfigurable antennas that allow designers to createtheir own antennas using the same principles The reconfig-urable antennas are commonly designed and characterizedusing full-wave solver Reconfigurable slot antenna in [4]

was one that has been discussed in detail covering analyticalmodel using transverse resonant technique full-wave charac-terization and design procedure whilst squarerectangular-ring shaped reconfigurable antennas in [7ndash12] have reportedthe full-wave characterization and experimental validationbut did not include the analytical model representation

Transmission line model (TLM) is commonly used topredict the input characteristics of a microstrip antennadue to its accuracy and numerical efficiency whilst lossytransmission line (LTL) model can be utilized for modeling alossy structureThe use of the LTLmodel on a loaded square-ring antenna has been introduced by Garg and Reddy [13]Garg derived the input characteristics of a nonreconfigurablesquare-ring microstrip antenna which loaded with a singlestub However applying the LTL model to represent areconfigurable antenna has not been attempted

Therefore in this paper we extend the use of the LTLmodel and combine it with multiport network analysisto create a general model equation for the reconfigurableantenna previously reported in [10ndash12] The model could beused to explainmore detail on the effect of the switch position

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2014 Article ID 303581 10 pageshttpdxdoiorg1011552014303581

2 International Journal of Microwave Science and Technology

Ls

Wp

Lp

Wg

middot middot middot

Lg

s1 s2 s3

s4

s5

s6s7

sn

snminus1

sn switch-n where n = 1 2 10

Figure 1 Geometry of the frequency reconfigurable antenna

on the antennarsquos resonant frequency Furthermore this paperalso describes an analytical derivation of the input impedanceand prediction of the frequency alteration due to the changein switch combinations

2 The Antenna Structure

The basic antenna structure has been introduced in [10ndash12] In this paper we modify the antenna into a generalform with 119899 number of switches as shown in Figure 1 Theantenna was designed with a rectangular-ring shaped andfabricated on an FR4 substrate with a thickness of 16mmpermittivity of 44 and loss tangent of 002 The dimensionof the rectangular-ring antenna is 342mm (119882119901) times 44mm(119871119901) and the width of the ring is 67mm The antenna isfed by microstrip line with the length of 22mm (119871 119904) and thewidth of 68mm The antenna is loaded with a rectangularpatch which is positioned in the middle of the ring withdimension of 282mm times 184mmThe placement of the loadinto the rectangular ring creates a rectangular ring slot of12mm width The 119899-number RF switches are placed intothe slot to connect the load to the antenna The switchesare placed arbitrarily on each side of the rectangular slot Bychanging the position of the load connected to the antennausing different ON-switch combination different frequencybands can be produced

3 The Proposed Model Representation

31 Proposed Equivalent Circuit In this section we proposean analytical model for the reconfigurable rectangular-ringmicrostrip antenna The antenna is simplified into an equiv-alent circuit and the total input impedance is analyticallyderived to predict the reflection coefficient over a certainfrequency range

The proposed multiport equivalent circuit model of thereconfigurable antenna is shown in Figure 2 The model isbuilt using the following rules (1) the rectangular ring is

modeled as a parallel combination of several transmissionline sections (2) the RF switches locations are represented asportsWhen the switch is OFF the port ismodeled as an opencircuit port or load admittance (119884119897119899) = 0 and when a switchis ON then the associated port model is terminated by a loadadmittance 119884119871119899 (3) as shown in Figure 2 the position of port119899 coincides with the port 0 When 119904119899 is OFF the port will beopen circuit and the voltage of port 119899 119881119899 will be equal to thevoltage of port 0 1198810 Otherwise when 119904119899 is ON the port willbe terminated by load admittance119884119871119899 Hence when two portlocations coincide the ports will exhibit the same behavior asa single port So only one port is taken into account in themodel equation

The admittance matrix approach is most suitable for theanalysis of the equivalent circuitThe general model equationfor the reconfigurable antenna with 119899-switches is describedas

[[[[

[

1198680

1198681

119868119899

]]]]

]

=

[[[[

[

11988400 11988401

11988410 11988411

sdot sdot sdot 1198840119899


1198841198990 1198841198991

d

sdot sdot sdot 119884119899119899

]]]]

]

[[[[

[

1198810

1198811

119881119899

]]]]

]

(1)

The above equation can be used for arbitrary number ofswitches applied to the antenna

The port current 119868119899 is derived as

119868119899 = 119884119871119899119881119899 (2)

The input impedance of the rectangular-ring antenna119885119860as shown in Figure 2 is obtained by solving this relationship

119885119860 =1198810

1198680

(3)

Then the input impedance observed from the microstripfeeder line 119885in as depicted in Figure 2 is

119885in = 119885119888119891119885119860 + 119885119888119891 tanh 120574119891119871119891119885119888119891 + 119885119860 tanh 120574119891119871119891

(4)

International Journal of Microwave Science and Technology 3

n minus 1

0

1 2 3

4

5

678

InI0

I1 I2 I3

I4

I5

I6I7I8

+

minus

n

Vn

V(nminus1)

V1 V2 V3

V4

V5

V6V7V8

YLn

YL(nminus1)

YL1 YL2 YL3

YL4

YL5

YL6YL7YL8

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minusZA

Zin I(nminus1)

Figure 2 Proposed multiport transmission line equivalent circuit of frequency reconfigurable rectangular-ring microstrip antenna

0

1 2 3

4

5

678

9

10

L1L2 L3 L4

L5

L6L7L8L9

L10

ZAZin

Ls

Figure 3 Definition of transmission line section length 119871119899 input impedance of the rectangular-ring antenna 119885119860 and the input impedanceobserved from the microstrip feeder line 119885in

where 119885119888119891 is a characteristic impedance of line feeder 120574119891is propagation constant of feeder line and 119871119891 is feeder linelength

In the case of the reconfigurable antenna with ten RFswitches as reported in [10ndash12] 119899 in (1) equals to 0 1 2 9The port 10 is not taken into account because its locationcoincides with port 0 The ports are open circuit when theswitch is OFF and when the switch is ON the port currentwill be equal to

119868119899 = 119884119871101198810 for 119899 = 0119884119871119899119881119899 for 119899 = 1 2 9

(5)

In the following subsections the derivation of the matrixcomponents and other parameters are conducted in the caseof the reconfigurable antenna with 10 RF switches

32 Determining 119884-Matrix Components (119884119899119898) and Transmis-sion Line Parameters As seen in Figure 3 we defined thelength of the line section between the port (119899 minus 1) and

the port and by 119871119899 for example 1198711 is a line section of port0 to port 1

The 119884-admittance matrix components 119884119899119898 can bederived by using this following rule [14]

119884119899119898 =119868119899

119881119898

10038161003816100381610038161003816100381610038161003816119881119896=0 for 119896 =119898 (6)

321 Determining 11988400 To determine 11988400 all of the portsexcept port 0 are short circuited as shown in Figure 4Therefore using the transmission (ABCD) matrix [14] foreach line section 1198711 and 11987110 the admittance component 11988400can be obtained

(1198810

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(0

1198681)

1198841015840

00=1198681015840

0

1198810

=cosh 1205741199031198711119885119888119903 sinh 1205741199031198711

= 119884119888119903 coth 1205741199031198711

(7)


1 2 3

4

5

678

V0

I0

I1

I9

100

I9984000

I9984009984000

9

Figure 4 Equivalent circuit for determining 11988400 where all of the ports except port 0 are short circuited

1 2

3

4

5

678

I0

I1

100

I9984000

I9984009984000

9

V1


1 2 3

4

5

678

I0

I2

100

I9984000

I9984009984000

9

V2


The admittance1198841015840101584000is determined using the samemethod

in (7)The admittance matrix component11988400 equals1198841015840

00+11988410158401015840

00

and it is defined as

11988400 = 119884119888119903 (coth 1205741199031198711 + coth 12057411990311987110) (8)

Whereas 119884119888119903 is characteristic admittance of the ringantenna = 1119885119888119903 120574119903 is propagation constant of ring antenna

and 1198711 11987110 are line section length of port 0 to port 1 and port9 to port 0 respectively

322 Determining 11988401 To determine 11988401 all of the portsexcept port 1 are short circuited as depicted in Figure 5Therefore admittance 11988401 can be defined using transmissionmatrix equation in each transmission line section 1198711 and 11987110


Bend

T-junction

Stepand

T-junction

Figure 7 Various discontinuities in the reconfigurable antenna

05 1 15 2 25 3Frequency (GHz)

Measured

minus500

0

500

Reac

tanc

e of

(Ohm

)

Imag( ) simulatedImag( ) model

Zin

ZinZin

(a)

Resis

tanc

e of

(Ohm

)


Measured

0100200300400500600700800


MeasuredReal( ) simulatedReal( ) model

ZinZin

Zin

(b)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)

Figure 8 Case 1 results comparison between input impedance and reflection coefficient of LTL analytical model full-wave solver simulatedresult and measurement result (a) reactance (b) resistance and (c) reflection coefficient

(0

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(1198811

1198681)

1198841015840

01=1198681015840

0

1198811

=minuscosh21205741199031198711 + sinh

21205741199031198711

119885119888119903 sinh 1205741199031198711= minus119884119888119903csch1205741199031198711

(9)

Using the same method it was found that 1198841015840101584001= 0 The

admittance matrix component of 11988401 is defined as 119884101584001+ 11988410158401015840

01

and expressed as

11988401 = minus119884119888119903 (csch 1205741199031198711) (10)

323 Determining 11988402 To determine 11988402 all of the portsexcept port 2 are short circuited and the port voltages 119881119899 for119899 = 2 equal zero as described in Figure 6 Therefore

11988402 = 0 (11)

The other matrix component 1198840119899 for 119899 = 3 4 8 willbe equal to zero as well

11988402 = 11988403 = 11988404 = sdot sdot sdot = 11988408 = 0 (12)



minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)Figure 9 Case 2 results comparison between input impedance and reflection coefficient of LTL analytical model full-wave solver simulatedresult and measurement result (a) reactance (b) resistance and (c) reflection coefficient

Using the same method described above we can defineall of the 119884-admittance matrix components and they arepresented in Table 1

Transmission line parameter such as attenuation con-stant 120572 phase constant 120573 propagation constant 120574 and

characteristic admittance119884119888 are determined using themodeldeveloped in [14]

The complete model equation for the reconfigurableantenna is defined in

[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)

As can be seen in Figure 7 various types of discontinuitiesoccurred in this antenna There are bends steps and T-junctions These discontinuities were modeled by equivalentline extension

Dearnley and Barel [15] noted that the transmissionline model is also a harmonic model so it can modelthe fundamental mode and its harmonics Therefore wecombined the Dearnley model in our LTL model to create

a complete model equation of the frequency reconfigurablerectangular-ring antenna

4 Model Validation

In verifying the model presented above we have calculatedthe input impedance and the reflection coefficient of theantenna in two different cases Thereafter the calculated



minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)

Figure 10 Comparison of input impedance and reflection coefficient between LTL analytical models full-wave solver simulated results andmeasurement results when 1199049 is ON (a) reactance (b) resistance and (c) reflection coefficient

results are compared with the simulation and measurementresults to evaluate the accuracy

41 Case 1 All Switches Are OFF When all switches are OFFall of the loads on each port model are open circuit so that theports current is zero

119868119899 = 0 (14)

Substituting (14) into (13) the input impedance of therectangular-ring antenna 119885119860 can be obtained by solving themodel equation using inverse matrix

[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)

The final solution for 119885in is obtained by substituting (17)into (4)

Results of case 1 model calculation are shown in Figure 8In this figure the results are compared to full-wave simulationand measurement results to show the prediction capabilityprovided by the model It can be seen in Figure 8 that

the model agrees well with full-wave predicted input charac-teristics as well asmeasured resultsThismodel can be used topredict the input characteristics of the reconfigurable antennain the range of 1ndash3GHz

42 Case 2 One of the Switches Is ON In this case weexamine the accuracy of the model when one of the switchesis ON for example 1199041 When the 1199041 is ON then the loadadmittance in port 1 exists or 1198841198711 = 0 and its current is

1198681 = 11988411987111198811 (18)

Whereas the other ports are open circuit and the currentis equal to zero

119868119899 = 0 for 119899 = 1 (19)

The load admittance 1198841198711 is calculated by assuming thesmall patch as a microstrip line which is connected to idealswitch represented as microstrip line as well Consider

1198841198711 = 119884119888 sw119884in sp + 119884119888 sw tanh 120574sw119871 sw119884119888 sw + 119884in sp tanh 120574sw119871 sw

(20)

where 119884119888 sw is characteristic admittance of the ideal switch119884in sp is input admittance of small patch observed from



minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)


All OFFs9 ONs7 ONs2 s7 ONs1 ON

s6 s7 s8 ONs2 ON

Figure 12 The simulation results of the antenna with several different states of switches

the input of the ideal switch 120574sw is the propagation constantof the switch and 119871 sw is the length of the switch The inputadmittance of small patch 119884in sp is calculated by assumingthat the end of the small patch is open circuit So 119884in sp isdefined as

119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)

Therefore the input impedance of the rectangular-ringantenna 119885119860 is obtained by substituting (18)ndash(21) into (12)

119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)

The final solution for 119885in in case 2 is obtained by solving(4) which is substituted with (22)


Table 1 119884-matrix components

119884-matrix component Quantity11988400 119884119888119903 (coth 1205741199031198711 + coth 12057411990311987110)11988401 minus119884119888119903 (csch1205741199031198711)11988409 minus119884119888119903 (csch12057411990311987110)11988410 119884119888119903 (csch1205741199031198711)11988411 minus119884119888119903 (coth 1205741199031198711 + coth 1205741199031198712)11988412 119884119888119903 (csch1205741199031198712)11988421 119884119888119903 (csch1205741199031198712)11988422 minus119884119888119903 (coth 1205741199031198712 + coth 1205741199031198713)11988423 119884119888119903 (csch1205741199031198713)11988432 119884119888119903 (csch1205741199031198713)11988433 minus119884119888119903 (coth 1205741199031198713 + coth 1205741199031198714)11988434 119884119888119903 (csch1205741199031198714)11988443 119884119888119903 (csch1205741199031198714)11988444 minus119884119888119903 (coth 1205741199031198714 + coth 1205741199031198715)11988445 119884119888119903 (csch1205741199031198715)11988454 119884119888119903 (csch1205741199031198715)11988455 minus119884119888119903 (coth 1205741199031198715 + coth 1205741199031198716)11988456 119884119888119903 (csch1205741199031198716)11988465 119884119888119903 (csch1205741199031198716)11988466 minus119884119888119903 (coth 1205741199031198716 + coth 1205741199031198717)11988467 119884119888119903 (csch1205741199031198717)11988476 119884119888119903 (csch1205741199031198717)11988477 minus119884119888119903 (coth 1205741199031198717 + coth 1205741199031198718)11988478 119884119888119903 (csch1205741199031198718)11988487 119884119888119903 (csch1205741199031198718)11988488 minus119884119888119903 (coth 1205741199031198718 + coth 1205741199031198719)11988489 119884119888119903 (csch1205741199031198719)11988498 119884119888119903 (csch1205741199031198719)11988499 minus119884119888119903 (coth 1205741199031198719 + coth 1205741199031198711)11988490 119884119888119903 (csch1205741199031198711)Remaining components 0

After calculating all of the possible modes results of thisanalytical model are compared to the full wave simulationand measurement results as seen in Figure 9 In this figurethe model shows a good agreement with the simulated andmeasured results and it can be used to predict the resonantfrequency of the reconfigurable antenna

To show the generality of the model we present theother examples of the proposed model calculation results forthe other ON switch configurations The model calculationresults when 1199049 is ON and 1199047 is ON are depicted in Figures 10and 11 respectivelyThe results agree well with simulated andmeasured results

In Figure 12 we present the simulation results of thereconfigurable antenna with several different states of theswitches It can be seen that the reconfiguration of theantenna frequency can be achieved by changing the positionand the number of the switches

5 Conclusion

We presented modeling of a frequency reconfigurablerectangular-ring microstrip antenna using lossy-transmis-sion-line and multiport network model The model can beused to analytically derive the input characteristic of thereconfigurable antenna with arbitrary number of switchesThe results show good accuracy and agreement in a widerange of frequency for single ON-switch configurationFurthermore this analytical model can be used to predictthe appropriate switch locations in generating the desiredoperating frequency

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J T Bernhard Reconfigurable Antennas chapter 4 section 42Morgan amp Claypool 2007

[2] L Pazin and Y Leviatan ldquoReconfigurable slot antenna forswitchable multiband operation in a wide frequency rangerdquoIEEE Antennas and Wireless Propagation Letters vol 12 pp329ndash332 2013

[3] H F Abutarboush R Nilavalan S W Cheung et al ldquoAreconfigurable wideband and multiband antenna using dual-patch elements for compact wireless devicesrdquo IEEETransactionson Antennas and Propagation vol 60 no 1 pp 36ndash43 2012

[4] D Peroulis K Sarabandi and L P B Katehi ldquoDesign ofreconfigurable slot antennasrdquo IEEE Transactions on Antennasand Propagation vol 53 no 2 pp 645ndash654 2005

[5] C Zhang S Yang S El-Ghazaly A E Fathy and V KNair ldquoA low-profile branched monopole laptop reconfigurablemultiband antenna for wireless applicationsrdquo IEEE Antennasand Wireless Propagation Letters vol 8 pp 216ndash219 2009

[6] R Goncalves P Pinho and N B Carvalho ldquoCompact fre-quency reconfigurable printed monopole antennardquo Interna-tional Journal of Antennas and Propagation vol 2012 ArticleID 602780 6 pages 2012

[7] J-F Tsai and J-S Row ldquoReconfigurable square-ring microstripantennardquo IEEE Transactions on Antennas and Propagation vol61 no 5 pp 2857ndash2860 2013

[8] Y J Sung ldquoFrequency and polarisation reconfigurability froman open-loop square ring antennardquo IET Microwaves Antennasand Propagation vol 6 no 5 pp 505ndash509 2012

[9] M A Alkanhal and A F Sheta ldquoA novel dual-band reconfig-urable square-ring microstrip antennardquo Progress in Electromag-netics Research vol 70 pp 337ndash349 2007

[10] B S Nugroho F Y Zulkifli and E T Rahardjo ldquoSimplefrequency reconfigurable antenna by changing the number andposition of the switchesrdquo in Proceedings of the 12th InternationalConference on Quality in Research (QiR rsquo11) pp 428ndash433 BaliIndonesia 2011

[11] B S Nugroho F Y Zulkifli and E T Rahardjo ldquoPINdiodes slotted microstrip antenna as frequency reconfigurableantennardquo in Proceedings of the 17th International Symposiumon Antennas and Propagation pp 814ndash817 Nagoya JapanNovember 2012


[12] E T Rahardjo F Y Zulkifli and B S Nugroho ldquoMultibandreconfigurable microstrip antennardquo in Proceedings of the AsiaPacific Conference on Antennas and Propagation Chiang MaiThailand 2013

[13] R Garg and V S Reddy ldquoEdge feeding of microstrip ringantennasrdquo IEEE Transactions on Antennas and Propagation vol51 no 8 pp 1941ndash1946 2003

[14] D M Pozar Microwave Engineering chapter 3 sec 38 JohnWiley amp Sons New York NY USA 3rd edition 2005

[15] R W Dearnley and A R F Barel ldquoBroad-band transmissionlinemodel for a rectangular microstrip antennardquo IEEE Transac-tions on Antennas and Propagation vol 37 no 1 pp 6ndash15 1989

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Ls

Wp

Lp

Wg

middot middot middot

Lg

s1 s2 s3

s4

s5

s6s7

sn

snminus1

sn switch-n where n = 1 2 10

Figure 1 Geometry of the frequency reconfigurable antenna

on the antennarsquos resonant frequency Furthermore this paperalso describes an analytical derivation of the input impedanceand prediction of the frequency alteration due to the changein switch combinations

2 The Antenna Structure

The basic antenna structure has been introduced in [10ndash12] In this paper we modify the antenna into a generalform with 119899 number of switches as shown in Figure 1 Theantenna was designed with a rectangular-ring shaped andfabricated on an FR4 substrate with a thickness of 16mmpermittivity of 44 and loss tangent of 002 The dimensionof the rectangular-ring antenna is 342mm (119882119901) times 44mm(119871119901) and the width of the ring is 67mm The antenna isfed by microstrip line with the length of 22mm (119871 119904) and thewidth of 68mm The antenna is loaded with a rectangularpatch which is positioned in the middle of the ring withdimension of 282mm times 184mmThe placement of the loadinto the rectangular ring creates a rectangular ring slot of12mm width The 119899-number RF switches are placed intothe slot to connect the load to the antenna The switchesare placed arbitrarily on each side of the rectangular slot Bychanging the position of the load connected to the antennausing different ON-switch combination different frequencybands can be produced

3 The Proposed Model Representation

31 Proposed Equivalent Circuit In this section we proposean analytical model for the reconfigurable rectangular-ringmicrostrip antenna The antenna is simplified into an equiv-alent circuit and the total input impedance is analyticallyderived to predict the reflection coefficient over a certainfrequency range

The proposed multiport equivalent circuit model of thereconfigurable antenna is shown in Figure 2 The model isbuilt using the following rules (1) the rectangular ring is

modeled as a parallel combination of several transmissionline sections (2) the RF switches locations are represented asportsWhen the switch is OFF the port ismodeled as an opencircuit port or load admittance (119884119897119899) = 0 and when a switchis ON then the associated port model is terminated by a loadadmittance 119884119871119899 (3) as shown in Figure 2 the position of port119899 coincides with the port 0 When 119904119899 is OFF the port will beopen circuit and the voltage of port 119899 119881119899 will be equal to thevoltage of port 0 1198810 Otherwise when 119904119899 is ON the port willbe terminated by load admittance119884119871119899 Hence when two portlocations coincide the ports will exhibit the same behavior asa single port So only one port is taken into account in themodel equation

The admittance matrix approach is most suitable for theanalysis of the equivalent circuitThe general model equationfor the reconfigurable antenna with 119899-switches is describedas

[[[[

[

1198680

1198681

119868119899

]]]]

]

=

[[[[

[

11988400 11988401

11988410 11988411



1198841198990 1198841198991

d

sdot sdot sdot 119884119899119899

]]]]

]

[[[[

[

1198810

1198811

119881119899

]]]]

]

(1)

The above equation can be used for arbitrary number ofswitches applied to the antenna

The port current 119868119899 is derived as

119868119899 = 119884119871119899119881119899 (2)

The input impedance of the rectangular-ring antenna119885119860as shown in Figure 2 is obtained by solving this relationship

119885119860 =1198810

1198680

(3)

Then the input impedance observed from the microstripfeeder line 119885in as depicted in Figure 2 is

119885in = 119885119888119891119885119860 + 119885119888119891 tanh 120574119891119871119891119885119888119891 + 119885119860 tanh 120574119891119871119891

(4)


n minus 1

0

1 2 3

4

5

678

InI0

I1 I2 I3

I4

I5

I6I7I8

+

minus

n

Vn

V(nminus1)

V1 V2 V3

V4

V5

V6V7V8

YLn

YL(nminus1)

YL1 YL2 YL3

YL4

YL5

YL6YL7YL8

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minusZA

Zin I(nminus1)


0

1 2 3

4

5

678

9

10

L1L2 L3 L4

L5

L6L7L8L9

L10

ZAZin

Ls




119868119899 = 119884119871101198810 for 119899 = 0119884119871119899119881119899 for 119899 = 1 2 9

(5)





119884119899119898 =119868119899

119881119898

10038161003816100381610038161003816100381610038161003816119881119896=0 for 119896 =119898 (6)


(1198810

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(0

1198681)

1198841015840

00=1198681015840

0

1198810

=cosh 1205741199031198711119885119888119903 sinh 1205741199031198711

= 119884119888119903 coth 1205741199031198711

(7)


1 2 3

4

5

678

V0

I0

I1

I9

100

I9984000

I9984009984000

9


1 2

3

4

5

678

I0

I1

100

I9984000

I9984009984000

9

V1


1 2 3

4

5

678

I0

I2

100

I9984000

I9984009984000

9

V2




00+11988410158401015840

00


11988400 = 119884119888119903 (coth 1205741199031198711 + coth 12057411990311987110) (8)





Bend

T-junction

Stepand

T-junction



Measured

minus500

0

500

Reac

tanc

e of

(Ohm

)


Zin

ZinZin

(a)

Resis

tanc

e of

(Ohm

)


Measured

0100200300400500600700800



ZinZin

Zin

(b)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


(0

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(1198811

1198681)

1198841015840

01=1198681015840

0

1198811

=minuscosh21205741199031198711 + sinh

21205741199031198711

119885119888119903 sinh 1205741199031198711= minus119884119888119903csch1205741199031198711

(9)



01

and expressed as

11988401 = minus119884119888119903 (csch 1205741199031198711) (10)


11988402 = 0 (11)


11988402 = 11988403 = 11988404 = sdot sdot sdot = 11988408 = 0 (12)



minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured






[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)




4 Model Validation




minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










n minus 1

0

1 2 3

4

5

678

InI0

I1 I2 I3

I4

I5

I6I7I8

+

minus

n

Vn

V(nminus1)

V1 V2 V3

V4

V5

V6V7V8

YLn

YL(nminus1)

YL1 YL2 YL3

YL4

YL5

YL6YL7YL8

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minus

+

minusZA

Zin I(nminus1)


0

1 2 3

4

5

678

9

10

L1L2 L3 L4

L5

L6L7L8L9

L10

ZAZin

Ls




119868119899 = 119884119871101198810 for 119899 = 0119884119871119899119881119899 for 119899 = 1 2 9

(5)





119884119899119898 =119868119899

119881119898

10038161003816100381610038161003816100381610038161003816119881119896=0 for 119896 =119898 (6)


(1198810

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(0

1198681)

1198841015840

00=1198681015840

0

1198810

=cosh 1205741199031198711119885119888119903 sinh 1205741199031198711

= 119884119888119903 coth 1205741199031198711

(7)


1 2 3

4

5

678

V0

I0

I1

I9

100

I9984000

I9984009984000

9


1 2

3

4

5

678

I0

I1

100

I9984000

I9984009984000

9

V1


1 2 3

4

5

678

I0

I2

100

I9984000

I9984009984000

9

V2




00+11988410158401015840

00


11988400 = 119884119888119903 (coth 1205741199031198711 + coth 12057411990311987110) (8)





Bend

T-junction

Stepand

T-junction



Measured

minus500

0

500

Reac

tanc

e of

(Ohm

)


Zin

ZinZin

(a)

Resis

tanc

e of

(Ohm

)


Measured

0100200300400500600700800



ZinZin

Zin

(b)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


(0

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(1198811

1198681)

1198841015840

01=1198681015840

0

1198811

=minuscosh21205741199031198711 + sinh

21205741199031198711

119885119888119903 sinh 1205741199031198711= minus119884119888119903csch1205741199031198711

(9)



01

and expressed as

11988401 = minus119884119888119903 (csch 1205741199031198711) (10)


11988402 = 0 (11)


11988402 = 11988403 = 11988404 = sdot sdot sdot = 11988408 = 0 (12)



minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured






[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)




4 Model Validation




minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3

4

5

678

V0

I0

I1

I9

100

I9984000

I9984009984000

9


1 2

3

4

5

678

I0

I1

100

I9984000

I9984009984000

9

V1


1 2 3

4

5

678

I0

I2

100

I9984000

I9984009984000

9

V2




00+11988410158401015840

00


11988400 = 119884119888119903 (coth 1205741199031198711 + coth 12057411990311987110) (8)





Bend

T-junction

Stepand

T-junction



Measured

minus500

0

500

Reac

tanc

e of

(Ohm

)


Zin

ZinZin

(a)

Resis

tanc

e of

(Ohm

)


Measured

0100200300400500600700800



ZinZin

Zin

(b)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


(0

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(1198811

1198681)

1198841015840

01=1198681015840

0

1198811

=minuscosh21205741199031198711 + sinh

21205741199031198711

119885119888119903 sinh 1205741199031198711= minus119884119888119903csch1205741199031198711

(9)



01

and expressed as

11988401 = minus119884119888119903 (csch 1205741199031198711) (10)


11988402 = 0 (11)


11988402 = 11988403 = 11988404 = sdot sdot sdot = 11988408 = 0 (12)



minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured






[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)




4 Model Validation




minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Bend

T-junction

Stepand

T-junction



Measured

minus500

0

500

Reac

tanc

e of

(Ohm

)


Zin

ZinZin

(a)

Resis

tanc

e of

(Ohm

)


Measured

0100200300400500600700800



ZinZin

Zin

(b)

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


(0

1198681015840

0

) = (

cosh 1205741199031198711 119885119888119903 sinh 12057411990311987111

119885119888119903

sinh 1205741199031198711 cosh 1205741199031198711)(1198811

1198681)

1198841015840

01=1198681015840

0

1198811

=minuscosh21205741199031198711 + sinh

21205741199031198711

119885119888119903 sinh 1205741199031198711= minus119884119888119903csch1205741199031198711

(9)



01

and expressed as

11988401 = minus119884119888119903 (csch 1205741199031198711) (10)


11988402 = 0 (11)


11988402 = 11988403 = 11988404 = sdot sdot sdot = 11988408 = 0 (12)



minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured






[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)




4 Model Validation




minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus500

0

500Re

acta

nce o

f(O

hm)

Zin

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured






[[[[[[[[[[[[[[

[

1198680

1198681

1198682

1198683

1198684

1198685

1198686

1198687

1198688

1198689

]]]]]]]]]]]]]]

]

=

[[[[[[[[[[[[[[

[

11988400 11988400 0 0 0 0 0 0 0 11988409

11988410 11988411 11988412 0 0 0 0 0 0 0

0 11988421 11988422 11988423 0 0 0 0 0 0

0 0 11988432 11988433 11988434 0 0 0 0 0

0 0 0 11988443 11988444 11988445 0 0 0 0

0 0 0 0 11988454 11988455 11988456 0 0 0

0 0 0 0 0 11988465 11988466 11988467 0 0

0 0 0 0 0 0 11988476 11988477 11988478 0

0 0 0 0 0 0 0 11988487 11988488 11988489

11988490 0 0 0 0 0 0 0 11988498 11988499

]]]]]]]]]]]]]]

]

[[[[[[[[[[[[[[

[

1198810

1198811

1198812

1198813

1198814

1198815

1198816

1198817

1198818

1198819

]]]]]]]]]]]]]]

]

(13)




4 Model Validation




minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus500

0

500

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

200

400

600

800

1000


Zin

SimulatedModel

Measured

(b)

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)




119868119899 = 0 (14)


[119881] = [119884]minus1[119868] (15)

[119885] = [119884]minus1 (16)

119885119860 =1198810

1198680

= 11988500 (17)





1198681 = 11988411987111198811 (18)


119868119899 = 0 for 119899 = 1 (19)



(20)




minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











minus400

minus200

0

600

400

200

Reac

tanc

e of

(Ohm

)Z

in

SimulatedModel

Measured

(a)

Resis

tanc

e of

(Ohm

)


0

400300200100

500600700800


SimulatedModel

Measured

Zin

(b)

minus25

minus30

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)

SimulatedModel


Measured

(c)


minus30

minus25

minus20

minus15

minus10

minus5

0

Refle

ctio

n co

effici

ent (

dB)



s6 s7 s8 ONs2 ON



119884in sp = 119884119888 sp tanh 120574sp119871 sp (21)


119885119860 =1198810

1198680

= 11988500 +11988510119885011198841198711

1 minus 119885111198841198711

(22)








5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















5 Conclusion




References



















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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