microwave electronics - slac.stanford.edu · microwave electronics ☞maxwell's equations...
TRANSCRIPT
Microwave Electronics
Maxwell's Equations & Modes in a Guide Equivalent Circuit for Waveguide Modes Modes of a Cavity Cavity with a Port & External Q Microwave Networks Slater's Perturbation Theorem
An Introduction to the Notion of Equivalent Circuit
Starting from Maxwell’s Equations
in six lectures
Microwave Electronics I
Maxwell's Equations and
Waveguide Modes
① Lorentz Force Law② Maxwell’s Equations③ Skin Depth ④ Orthogonal Modes⑤ Phase & Group Velocity Quiz
①Lorentz Force Law
Newtons per Coulomb=V/m“electric field strength”
Hendrik Antoon Lorentz b. July 18, 1853, Arnhem, Netherlandsd. Feb. 4, 1928, Haarlem
defines the fields & abstracts them from the sourcesdescribes “test particle” motiondescribes response of media
r r r rF q E v B= + ×( )
r
E
rB Newtons per Ampere-meter=T=Wb/m2
”magnetic flux density” or “magnetic induction”
Example: Conductivity
P. Drude, 1900
+ +
+ +
+ +
+ +
+ +
+ +
+ +
+ +
Georg Simon Ohm(b. March 16, 1789, Erlangen, Bavaria --d. July 6, 1854, Munich)
m
dvdt
qE mv
r r r= −
τ
r rv
qm
E= τ
J nqv
nqm
E
E
=
=
=
r
r
r
2τ
σ
σ σ= ≈ ×" " . /DC mho m5 8 107 for Cu
②Maxwell’s Equations
Charges repel or attract Current carrying wires repel or attract Time-varying currents can induce
currents in surrounding media
Electricity & Magnetism before Maxwell...
After Maxwell...
Light is an electromagnetic phenomenon
Nature is not Galilean Thermodynamics applied to
electromagnetic fields gives divergent results
Matter appears not to be stable Questions arise concerning
gravitation...
Microwave Electronics
“Ordinary” Electronics
•voltages vary slowly on the scale of thetransit time = circuit size / speed of light•circuit size small compared to wavelength•voltage between two points independent of path•may treat elements as “lumped”•unique notion of impedance of an element•bring a multimeter
•circuit size appreciable compared to a wavelength•voltage between two points depends on path•elements are “distributed”, spatial phase-shiftsoccur between them•if the word “impedance” is used, you may alwaysask how it was defined...•if any result of test & measurement is quoted, you may always ask how the equipment was calibrated•bring crystal detectors, filters, mixers, a signal generator, a spectrum analyzer, and, if you have them a network analyzer, calibration kit, vector voltmeter
HistoryHenry Cavendish(b. Oct. 10, 1731, Nice, France--d. Feb. 24, 1810, London, Eng.)
Charles-Augustin de Coulomb (b. June 14, 1736, Angoulême, Fr.--d. Aug. 23, 1806, Paris)
André-Marie Ampere(b. Jan. 22, 1775, Lyon, France--d. June 10, 1836, Marseille)
Karl Friedrich Gauss(b. April 30, 1777, Brunswick--d. 1855)
Hans Christian Ørsted(b. Aug. 14, 1777, Rudkøbing, Den.--d. March 9, 1851, Copenhagen)
Siméon-Denis Poisson(b. June 21, 1781, Pithiviers, Fr.--d. April 25, 1840, Sceaux)
Michael Faraday(b. Sept. 22, 1791, Newington, Surrey --d. August 25, 1867, Hampton Court)
James Clerk Maxwell(b. June 13 or Nov. 13, 1831, Edinburgh--d. Nov. 5, 1879, Glenlair )
Heinrich (Rudolf) Hertz(b. Feb. 22, 1857, Hamburg--d. Jan. 1, 1894, Bonn)
Guglielmo Marconi(b. April 25, 1874, Bologna, Italy--d. July 20, 1937, Rome)
Gauss’s Law
Ampere’s Lawbefore Maxwell
r rD dS dV
V V•∫ = ∫
∂ρ
r r r rH dl J dS
S S∂∫ ∫• = •
electric displacementor electric flux density
r rD D nA A2 1−( ) • =ˆ Σ
Medium 2
Medium 1+ + + + + + + + + + + + + + ++
surface charge density Σpillbox area A
n
Medium 2
Medium 1
surface current densitycontour side L
n
rK
l
D C m⇔ ⇔Σ / 2
r r rH H l K n l2 1−( ) • = × •ˆ ˆ ˆ
n H H K× −( ) =r r r
2 1
H K A m⇔ ⇔ /
magnetic field strength or magnetic flux
In Vacuum...In Vacuum...
µ π074 10= × − henry per meter
ε0128 85 10≈ × −. farad per meter
12 9979 10
0 0
8
ε µ= = ×c m s. /
µε
0
0377= Ω
or
r rD E= ε0
r rH B= / µ0
In Media...In Media...
magnetic dipole moment density
r r r r
D E P E= + =ε ε0
electric dipole moment density r r
P Ee= χ ε0
r r
M Hm= χ
Finer Points:•These are really frequency domain expressions•In general ε,µ are tensors• µ may be non-linear & biased by a DCfield•H,D depend on your point of view
r r r r
H B M B= − =/ /µ µ0
Faraday’s Law
r rr
rE dl
Bt
dSS S
•∫ = − •∫∂
∂∂
Before Maxwell, Ampere’s Law was inconsistent with conservation of chargeAfter Maxwell, the fields didn’t need charge to support them, they could propagate on their own
Of course no one believed Maxwell, but the fieldsdidn’t mind
Charge Conservation
∂ρ∂t
J+ ∇ • =r
0
∂∂
ρ∂t
dV J dSV V∫ ∫= − •
r ror
∂ρ∂t
J H= −∇ • = −∇ • ∇ × =r r
0
Ampere’s Law (before Maxwell’s additionof Displacement Current) implied
...actually not a bad approximation in conductors, or a dense plasma..excellent for electrostatics, magnetostatics
Gauss’s Law
Ampere’s LawwithMaxwell’s Displacement Current
Faraday’s Law
r r∇ • =D ρ
r rr
∇ × = −EBt
∂∂
r r rr
∇ × = +H JDt
∂∂
∇ • =rB 0No Magnetic Charge
Maxwell’s Equations
James Clerk Maxwellb. June 13 or Nov. 13, 1831, Edinburghd. Nov. 5, 1879, Glenlair
Medium 2
Medium 1+ + + + + + + + + + + + + + ++
surface charge density Σpillbox area A
n
Medium 2
Medium 1
surface current densitycontour side L
n
rK
l
Boundary Conditions
r r
r rD D n
B B n
2 1
2 1 0
−( ) • =
−( ) • =
ˆ
ˆ
Σ
ˆ
ˆ
n H H K
n E E
× −( ) =
× −( ) =
r r r
r r2 1
2 1 0
apply Maxwell’s equations in integral form...
Example to Illustrate ε:Unmagnetized Plasma
rE Ee Ee E ej t j t j t= ℜ( ) = +( )−˜ ˜ ˜ *ω ω ω1
2
m
dvdt
qE mv
r r r
= −τ
˜ ˜ ˜ ˜ ˜vj
qm
E J nqvj
EDC=+( )
⇒ = =+( )
11 1ωτ
τ σωτ
Apply charge conservation & Gauss’s Law...
∂ρ∂
ωρt
J j J+ ∇ • = ⇒ + ∇ • =r
0 0˜ ˜
∇ • = = − ∇ • = − ∇ •+( )
ε ρω ω
σωτ0
1 11
˜ ˜ ˜ ˜Ej
Jj j
EDC
∇ • ++( )
= ⇒ = ++( )
εω
σωτ
ε εω
σωτ0 0
11
01
1j jE
j jDC DC˜
N B withnqm
pp. . lim
τ
εε
ωω
ωε→∞
= − =0
2
22
2
01
Evidently...
•electric permittivity is a frequency-domain concept...
ε εω
σωτ
ε ω= ++( )
= ( )01
1j jDC
•electric displacement depends on what you consider to be the “external” circuit, electric field does not
View #1
View #2
∇ • = +
=
ε ρ ρ
ε0
0
˜ ˜ ˜
˜ ˜
E
D E
plasma other
∇ • =
=
ε ρ
ε
˜ ˜
˜ ˜
E
D E
other
Polarization in the time domain...
r
r
r
P td
e P
de E
de
dte E t
dt G t t E t
j t
j te
j te
j t
( ) ˜
˜
= ∫ ( )
= ∫ ( ) ( )
= ∫ ( ) ′ ′( )∫
= ′ − ′( ) ′( )∫
−∞
+∞
−∞
+∞
−∞
+∞−
−∞
+∞
−∞
+∞
ωπ
ω
ωπ
ε χ ω ω
ωπ
ε χ ωπ
ε
ω
ω
ω ω
2
2
2 2
0
0
0
G t d e j te( ) = ∫ ( )
−∞
+∞12π
ω χ ωω
where the Green’s function is
χ ω ωe
j tdt e G t( ) = ∫ −
−∞
+∞( )
Example...unmagnetized plasma...
G t d e
d ej j
e H t
j te
j t p
pt
( )
( )/
= ∫ ( )
= ∫ +( )= −( )
−∞
+∞
−∞
+∞
−
12
12 1
1
2
2
πω χ ω
πω
ω τω ωτ
ω τ
ω
ω
τ
ω-plane
pole at ω=0
pole at ω=j/τ
for t<0, contourmay be closedat Imω→−∞so that G=0
contour
for t>0, contourmay be closedat Imω→+∞with contributionsfrom two poles
Susceptibility in the High Frequency Limit
when the Green’s function is analytic near t>0,
χ ω
∂∂ω
∂∂ω ω
ω
ω
ω
ω
ej t
j tn
n
n
n
n
nj t
n
n
n
dt e G t
dt e H tG
nt
Gn
j dt e
Gn
jj
jG
( ) = ∫
= ∑∫
= ∑
∫
= ∑
= −
−
−∞
+∞
−
=
∞
−∞
+∞
=
∞ −+∞
=
∞
( )
( )( )!
( )!
( )!
( )
( )
( )
( )
0
0
0 1
0
0
0 0
0
−− +GjG( ) ( )( ) ( )1
2
2
30 0
ω ωK
Example: unmagnetized plasma
χ ωωω
ωω τe
p pj( ) = − − +2
2
2
3 K
Kramers-Kronig Relations
Since G is causal χe must be analyticin the Imω<0 half-plane, so that
for points ω,ω’ and contour in the lower half-plane. Let the contour liejust below the real axis and use
χ ωπ
χ ωω ωe
e
j( ) = ′( )
′ −∫1
2
limε ω ω ε ω ω
π δ ω ω→ ′ − −
=′ −
+ ′ −( )
0
1 1j
P j
then
χ ωπ
χ ωω ωe
e
jP( ) = ′( )
′ −∫−∞
+∞1
ℜ ( ) = ℑ ′( )′ −∫
ℑ ( ) = − ℜ ′( )′ −∫
−∞
+∞
−∞
+∞
χ ωπ
χ ωω ω
χ ωπ
χ ωω ω
ee
ee
P
P
1
1
relatesdispersion&absorption
Scalar & Vector Potentials
Maxwell’s Equations...
r r r r r
r v rr
r r rr
∇ • = ∇ • ∇ × =
∇ × = ∇ × − − ∇
= − ∇ × = −
B A
EAt t
ABt
0
∂∂
ϕ ∂∂
∂∂
r r r r r r r
r rr
r
rr r
∇ × = ∇ × ∇ ×( ) = ∇ ∇ •( ) − ∇
= ∇ × = +
= − − ∇
+
µ
µ µ ∂ε∂
µ
µε ∂∂
∂∂
ϕ µ
H A A A
HEt
J
tAt
J
2
r r r r rr
r∇ • = ∇ • = ∇ • − − ∇
=D EAt
ε ε ∂∂
ϕ ρ
r r rB A= ∇ ×
vr
rE
At
= − − ∇∂∂
ϕ
r r rA A
t
→ − ∇
→ +
ψ
ϕ ϕ ∂ψ∂
Gauge Invariance
leaves E,B unchanged
Lorentz Gauge ∇ • + =
rA
tµε ∂ϕ
∂0
∇ − = −2
2
2
rr
rA
At
Jµε ∂∂
µ ∇ − = −22
2ϕ µε ∂ ϕ∂
ρ εt
/
evidently the characteristic speedof propagation in the medium is v = ( )−µε 1 2/
Coulomb Gauge r r∇ • =A 0
∇ − = − + ∇2
2
2
rr
r rA
At
Jt
µε ∂∂
µ µε ∂ϕ∂
∇ = −2ϕ ρ ε/
∇ = −
2ψ µε ∂ϕ∂t Lorentz Gauge
related to the Lorentz Gauge potentials via
Hertzian Potentials
in a homogeneous, isotropic source-free region...
Magnetic Hertzian potential
r r∇ • = ∇ • =˜ ˜E H 0
E j m= − ∇ ×ωµr
Π
choice of gauge
∇ + =202 0Π Πm mk
∇ × = = ∇ × =
⇒ = + ∇
∇ × = − ∇∇ • − ∇( )= − = − + ∇( )
˜ ˜
˜
˜
˜
H j E k with k
H k
E j
j H j k
m
m
m m
m
ωε µεω
ψ
ωµ
ωµ ωµ ψ
02
02 2
02
2
02
r
r
r r
r
Π
Π
Π Π
Π
ψ ψ ψ= ∇ • ⇒ ∇ + =r
Πm k202 0
Electric Hertzian potential H j e= ∇ ×ωεr
Π
∇ + =202 0Π Πe ek E ke e= ∇∇ • +
r rΠ Π0
2
H km m= ∇∇ • +r r
Π Π02
Energy ConservationEnergy Conservation
In a linear medium, with ε and µ independent of frequency:
− • =
= − ∇ ×
•
= • −∇ × • + ∇ × • − ∇ × •
= • + ∇ • ×( ) +
∇• ×( )
r r
rr r r
rr r r r r r r1 24444 34444
r r r
rr r r r
r
r r r
J E rate of work done on fields
Dt
H E
Dt
E H E E H E H
Dt
E E HB
E H
∂∂
∂∂
∂∂
∂∂tt
H•r
r r rS E H Poynting Flux= × =
− • = ∇ • +
r r r rJ E S
ut
∂∂
where
u E D B H field energy density= • + • =r r r r
When ε and µ are not independent of frequency we should work in the frequency domain...
rE Ee Ee E ej t j t j t= ℜ( ) = +( )−˜ ˜ ˜ *ω ω ω1
2and similarly for J, H, etc...let us compute averagesover the rapid rf oscillation...can show that
S E H= ℜ ×( )∗12
˜ ˜
somewhat more challenging is the calculation of therate of change of field energy density
Questions arise...is this integrable?
r
Dt
EBt
H= • + •∂∂
∂∂
rr
rr
rut
= =∂∂
?
and if so, what is the average stored energy density ?u
To address this problem we first compute
take
and compute P...
then
∂∂
rrD
tE•
r
r r
E t E t e E E t e
D t E t P t e
j t j t
j t
( ) ˜( , ) ˜ ( ) ˜ ( )
( ) ( ) ˜( , )
= ℜ = ℜ +( ) = + ℜ
ω ω ω
ε ω
ω ω
ω
0 1
0
or
˜( , ) ( ) ˜ ( ) ˜ ( )
˜ ( ) ( ) ˜ ( ) ( )
˜ ( )
P t e dt G t t E E t e
E dt G t t e Ej
dt G t t e
E
j t j t
j t j t
ω ε ω ω
ε ω ε ω ∂∂ω
ε ω χ
ω ω
ω ω
= ′ − ′ + ′( )
= ′ − ′ + ′ − ′
=
−∞
+∞′
−∞
+∞′
−∞
+∞′
∫
∫ ∫
0 0 1
0 0 0 1
0 0 eej t
ej te E
jeω ε ω ∂
∂ωχ ωω ω( ) + ( )0 1
˜ ( )
˜( , ) ˜ ( ) ˜ ( )
˜( , ) ˜ ( )
P t E E t j
E t j E
e ee
ee
ω ε ω χ ω ε ω χ ∂χ∂ω
ε χ ω ω ε ∂χ∂ω
ω
= ( ) + −
= ( ) −
0 0 0 1
0 0 1
∂∂
ε ∂∂
ε ω ω ∂∂
ε ∂∂
ε ω χ ω ω ∂χ∂ω
ω χ ω ∂ ω∂
ω ω
ω ω
r r
r
Dt
Et
j P t e ePt
Et
j E t j E e eE t
t
j t j t
ee j t j t
e
= + ℜ +
= + ℜ ( ) −
+ ( )
0 0
0 0 1
˜( , )˜
˜( , ) ˜ ( )˜( , )
= ℜ + +
= ℜ ( ) +
eE t
tj E t E
eE t
tj E t
j t e
j t
ω
ω
ε ∂ ω∂
εω ω ε ω ω ∂χ∂ω
∂ ω∂
∂ ωε∂ω
εω ω
˜( , ) ˜( , ) ˜ ( )
˜( , ) ˜( , )
0 1
∂∂
∂ ωε∂ω
∂ ω∂
ω εω ωr
rDt
EE t
tE t j E t• = ℜ ( ) • +
∗12
2˜( , ) ˜ ( , ) ˜( , )
finally
∂∂
∂ ωε∂ω
∂∂
ω ε ω ω∂ ωε
∂ω∂ ω
∂ω
rrD
tE
tE t E t
E tt
E tri
i• = ( ) − − ( ) ℑ •
∗1
42
2 2˜( , ) ˜( , )˜( , ) ˜ ( , )
Finally
so that, in the absence of losses,
where the field energy density is
Energy conservation takes the form
S E H= ℜ ×( )∗12
˜ ˜
with time-averaged Poynting flux
r
Dt
EBt
Hut
= • + • =∂∂
∂∂
∂∂
rr
rr
u E t H t= ( ) + ( )
14
2 2∂ ωε∂ω
ω ∂ ωµ∂ω
ω˜( , ) ˜ ( , )
− • = ∇ • +
r rJ E S
ut
∂∂
③Skin DepthStart from Maxwells Equations
Fields ∝ exp(jωt)
Reduce to an equation for H
δµω σµ
= = −
≈
2
2 1
" "skin depth
m at GHz in Cu
r
r
r r
∇ × = + ≈ =
∇ × = − = −
∇ • = ∇ • =
˜˜
˜ ˜ ˜
˜˜
˜
˜ ˜
HDt
J J E
EBt
j B
E B
∂∂
σ
∂∂
ω
0
r r r r
r∇ × ∇ ×( ) = ∇ ∇ •( ) − ∇ = −∇
= ∇ × ( ) = − = −
˜ ˜ ˜ ˜
˜ ˜ ˜
H H H H
E j B j H
2 2
σ ωσ µωσ
∇ − =22
20˜ sgn ˜H j Hω
δ
ξ
CONDUCTORVACUUMt angent ial H
normal H
Let’s solve for fields in conductor...
Zj
R js s= + = +( )11
sgnsgn
ωσδ
ω
select coordinate alongoutward normal
“impedance boundary condition”
R
at GHz in Cu
s = =
≈
1
1σδ
surface resistance
8.3mΩ
dd
H j Ht t
2
2 22
0ξ
ωδ
˜ sgn ˜− =
˜ ˜ exp ( sgn )H H jt tξ ξδ
ω( ) = ( ) − +
0 1
˜ ˜ ˆ
˜ˆ ˜E H n
HZ n Hs t= ∇ × ≈ × = − ×1 1
σ σ∂∂ξ
r
Power per m2 into the conductor...
S n E H n
Z n H H n
Z n H
R n H R K
s t
s t
s s
• = ℜ × •( )= − ℜ ×( ) × •( )= ℜ ×
= × =
∗
∗
ˆ ˜ ˜ ˆ
ˆ ˜ ˜ ˆ
ˆ ˜
˜ ˜
12
12
1212
12
2
2 2
˜ ˜ ˆ ˜K Jd n H= = − ×∫ ( )∞
ξ0
0
where in the last line we have made useof the result for surface current density,
④Orthogonal Modes(Uniform Waveguide)
From Maxwell’s Equations, with fields
r r
r
r
∇ • = ∇ • =
∇ × = −
∇ × =
˜ ˜
˜ ˜
˜ ˜
E B
E j B
H j D
0
ω
ω
r r
r∇ × ∇ ×( ) = −∇
= ∇ × = −( )=
˜ ˜
˜ ˜
˜
H H
j D j j B
H
2
2
ω ωε ω
ω εµ
we can see that E & H satisfy the wave equation,
and similarly for E, so that
where k02 2= ω εµ
∝ −e j t jk zzω
∇ +( ) =
∇ +( ) =
⊥
⊥
2 2
2 2
0
0
k H
k E
c
c
˜
˜
k k kc z2
02 2= −
The divergence conditions take the form
˜ ˜Ejk
Ezz
= ∇ •⊥ ⊥1 ˜ ˜H
jkHz
z= ∇ •⊥ ⊥
1
so that the longitudinal field components may be determined from the transverse components. In addition,we may write the curl equations
to express the transverse components in termsof the longitudinal
˜ ˜ ˜ ˜Ejk
Z H Z Hjk
E Z= ∇ × = − ∇ × =1 1
00 0
00
r r µε
− = − × ∇ − ×
= − × ∇ − ×⊥ ⊥ ⊥
⊥ ⊥ ⊥
jk Z H z E jk z E
jk E z Z H jk z Z H
z z
z z
0 0
0 0 0
˜ ˆ ˜ ˆ ˜
˜ ˆ ˜ ˆ ˜
r
r
or
˜ ˆ ˜ ˜
˜ ˆ ˜ ˜
Ekjk
z Z Hkk
E
Z Hkjk
z Ekk
Z H
cz
zz
cz
zz
⊥ ⊥ ⊥
⊥ ⊥ ⊥
= − × ∇ − ∇
= − − × ∇ − ∇
02 0
0
002
00
r r
r r
TE Modes Ez=0transverse fields may be determined from Hz
Z Hkjk
Z H
Ekjk
z Z Hkk
z Z H
z
cz
cz
z
0 2 0
02 0
00
˜ ˜
˜ ˆ ˜ ˆ ˜
⊥ ⊥
⊥ ⊥ ⊥
= ∇
= − × ∇ = − ×
r
r
0 00 2 0= • = • ∇ =⊥ ⊥ˆ ˜ ˆ ˜ . .
˜n Z H
kjk
n Z H i eHn
z
cz
zr ∂
∂
∇ +( ) =⊥2 2 0k Hc z
˜
longitudinal field satisfies
with boundary condition
∇ +( ) =⊥2 2 0k Ec z
˜
TM Modes Hz=0transverse fields may be determined from Ez
˜ ˜
˜ ˆ ˜ ˆ ˜
Ekjk
E
Z Hkjk
z Ekk
z E
z
cz
cz
z
⊥ ⊥
⊥ ⊥ ⊥
= ∇
= × ∇ = ×
2
002
0
r
r
longitudinal field satisfies
with boundary condition Ez = 0
Cut-Off & Characteristic Impedance Zc
In general for a given mode we have
Z H z Ec˜ ˆ ˜⊥ ⊥= ×
where
Boundary conditions restrict the permissible valuesof cut-off wavenumber kc to a discrete set. Each modehas a corresponding minimum wavelength λc beyond which it is “cut-off” in the waveguide.
λ πc ck= 2 /
The guide wavelength is
λ π λ λ λg z ck= = −2 10 02 2/ / /
λ π0 02= / k
Z
Zkk
Z TE e
Zkk
Z TM ec
z
g
z
g
==
=
00
00
00
00
λλλλ
mod
mod
Modal Decomposition
A general solution for a given geometry may berepresented as a sum over modes
where
and we adopt the normalization
˜ ( , )
˜ ( , )
E E r V z
H H r I z Z
t aa
a
t aa
a ca
= ( )∑
= ( )∑ ( )
⊥ ⊥
⊥ ⊥
r
r
ω
ω ω
Z H z Eca a a⊥ ⊥= ׈
d r E r E ra a2 1⊥ ⊥ ⊥ ⊥ ⊥∫ ( ) • ( ) =
r r
where the integral is over the waveguide cross-section.We choose the sign of Z for positive kz.The coefficients V,I take the forms
V z V e V e
Z I z V e V e
a ajk z
ajk z
ca a ajk z
ajk z
za za
za za
( , )
( , )
ω
ω
= +
= −
+ − −
+ − −
This requires Green’s Theorem,
d r E r E r
Z Z d r H r H r
d r z E H Z
a b ab
ca cb a b ab
a b ab ca
2
2
2 1
⊥ ⊥ ⊥ ⊥ ⊥
⊥ ⊥ ⊥ ⊥ ⊥
⊥ ⊥ ⊥−
∫ ( ) • ( ) =
∫ ( ) • ( ) =
∫ • ×( ) =
r r
r rδ
δ
δˆ
One can also show, for non-degenerate modes, that
and the eigenvalue equations for Hz & Ez
ψ ψ ψ ψ ψ ∂ψ
∂12
2 1 22
12∇ + ∇ • ∇( )∫ = ∫
r rd r
ndl
Relation between Power, V & I
As a result, one may express the power flow in thewaveguide, in terms of V & I according to
P d r E H
V I
t t
aa a
= ∫ ℜ ×( )= ∑ ℜ( )
∗
∗
2 12
12
˜ ˜
Meaning of V,I
Given the orthogonality relations, one can determine V,I from the transverse fields at a point z
V z d r E r z E r
I z Z d r H r z H r
a t a
a ca t a
( , ) ˜ ,
( , ) ˜ ,
ω
ω
= ∫ ( ) • ( )= ∫ ( ) • ( )
⊥ ⊥ ⊥ ⊥
⊥ ⊥ ⊥ ⊥
2
2
r r
r r
and this is enough to determine the solutioneverywhere in the uniform guide, since thisfixes the right & left-going amplitudes.
Given the uniqueness of V,I, theirrelation to power, and the units(volts, amperes) it is natural to referto them as voltage & current.
It is important to keep in mind however that they appear as complex mode amplitudes,not work done on a charge or time rate of change of charge.
at the same time, for particular geometries and applications, V & I can often be related to these more conventional concepts
⑤Phase & Group Velocityconsider a narrow-band drive at z=0
V t f t e
Vdt
f t e
f td
V e
j t
j t
j t
( , ) ( )
˜( , ) ( )
( ) ˜( , )
0
02
20
0
0
0
=
= ∫
= ∫
−∞
∞ −( )
−∞
∞ −( )
ω
ω ω
ω ω
ωπ
ωπ
ω
compute the voltage down-range
V t zd
V e
dV e
ed
V e
j t jk z
j t jk z jdkd
z
j t jk z j tdk
z
zz
z
( , ) ˜( , )
˜( , )
˜( , )
= ∫
≈ ∫
= ∫
−∞
∞ −
−∞
∞ −( ) − ( ) − ( ) −( )
− ( )−∞
∞ −( ) −
ωπ
ω
ωπ
ω
ωπ
ω
ω
ω ω ωω
ω ω ω
ω ω ω ω
20
20
20
0 0 0 0
0 00
zz
z
dz
j t jk z ze f tdkd
z
ω
ω ω
ω
− ( )= −
0 0
can see that constant phase-frontstravel at
vk
phase velocityz
ϕω= = −
while the modulation f travels at
vddk
group velocitygz
= = −ω
For a mode in uniform guide,
vk k
c
k k
c
vc k
kc
k kc
c c
gz
c
ϕω
µε µε
µε µε µε
=−
=−
>
= = − <
02 2 2
02
0
202
1
1
1
/
/
Summary
Lorentz Force Law Maxwell’s Equations Skin Depth δ Modes in a Waveguide Phase Velocity vφ & Group
Velocity vg
Acknowledgements
Encyclopedia Brittanica http://www.eb.com
100 Years of Radiohttp://www.alpcom.it/hamradio/
Dept. Materials Science, MIThttp://tantalum.mit.edu/
Dept. of Physics and Astronomy, Michigan State UniversityUniversity of Guelph
Varian Associateshttp://www.varian.com/
Special Thanks to Prof. Shigenori Hiramatsu, KEK and Prof. Perry Wilson, SLAC for introducing me to this subject
Nikola Tesla’s Home Pagehttp://www.neuronet.pitt.edu/~bogdan/tesla/
Vocabulary
Electric Field E Magnetic Field H Energy U, Power P Frequency f, or Angular Frequency ω Conductivity, σ or Resistivity ρ Phase Velocity vφ, Group Velocity vg
Distributed vs Lumped Elements E & H Fields, Charged Particles Behavior of Fields in Media
For More Information...
W3 Virtual Library of Beam Physics
http://beam.slac.stanford.edu/
links to all accelerator labs on the planet...conferences...schools...news...jobs...
companies...vendors...databases...researchers...preprints...
Recommended Reading
RF Engineering for Accelerators, Turner, (CERN 92-03)
An introduction to RF as applied to accelerators.
Microwave Electronics, Slater The classic introduction to microwave electronics.
Classical Electrodynamics, Jackson A graduate level electrodynamics text.
Field Theory of Guided Waves, Collin A modern introduction to microwave electronics.
Foundations for Microwave Engineering, Collin An overview of the elements of microwave electronics.
Microwave Measurements, Ginzton An introduction to practical microwave work.
Principles of Microwave Circuits, Montgomery, Dicke, PurcellAn introduction to common network elements.
Waveguide Handbook, MarcuvitzAnalysis of the circuit parameters for network elements.
Related TextsRelated Texts
Handy NumbersHandy Numbers
Copper
10 1 2 3
10 1 3 5
20 0 99 0 1
1 0
10
10
10
log /
log /
log . .
( ) ≈ −( ) ≈ −( ) ≈ −
≡
dB
dB
dB
mW dBm
κ
α
ρ
≈ °≈ °
≈ × °
≈ × −
−
−
401
385
1 7 10
1 56 10
5
8
W K m
C J K kG
K
m
/
/
. /
. Ω
RF Engineering for Particle
Accelerators
UnderstandInventDesignBuildOperate
To
RF Systems
Joint Accelerator School
Outline for Morning Lectures
1 Microwave Electronics 1 -Maxwell's Equations & Modes in a Guide 2 M.E. 2 - Equivalent Circuit Representation for Modes in a Guide3 M.E. 3 - Modes of a Cavity4 Cavity Design5 M.E. 4 - Cavity with a Port & External Q 6 M.E. 5 - Microwave Networks7 M.E. 6 - Slater's Perturbation Theorem8 Superconducting Cavities9 Beam-Cavity Interaction, Beam-Loading10 Klystron 1 - Space-Charge Limited Flow, Guns11 Structure 1-Standing-Wave12 SLED Pulse Compression13 Wakefields 1 - Fundamentals14 Klystron 2 - Bunching, Space-Charge15 Structure 2-Travelling Wave16 Ferrite Loaded Cavity 117 Wakefields 2 - in SW & TW Structures18 Klystron 3 - Simulation19 Structure 3-Fabrication and Conditioning20 Structure 4 -Surface fields, Breakdown, Multipactor, Dark Current20 Wakefields 3 - Other Sources of Impedance21 Other RF Sources22 High Gradients in Superconducting Cavities23 Modulators24 Windows & High-Power Transmission25 Ferrite Loaded Cavity 226 Design for System Stability - Heavy Beam Loading
Outline for Morning Lectures
1 Microwave Electronics 1 -Maxwell's Equations & Modes in a Guide 2 M.E. 2 - Equivalent Circuit Representation for Modes in a Guide3 M.E. 3 - Modes of a Cavity4 Cavity Design5 M.E. 4 - Cavity with a Port & External Q 6 M.E. 5 - Microwave Networks7 M.E. 6 - Slater's Perturbation Theorem8 Superconducting Cavities9 Beam-Cavity Interaction, Beam-Loading10 Klystron 1 - Space-Charge Limited Flow, Guns11 Structure 1-Standing-Wave12 SLED Pulse Compression13 Wakefields 1 - Fundamentals14 Klystron 2 - Bunching, Space-Charge15 Structure 2-Travelling Wave16 Ferrite Loaded Cavity 117 Wakefields 2 - in SW & TW Structures18 Klystron 3 - Simulation19 Structure 3-Fabrication and Conditioning20 Structure 4 -Surface fields, Breakdown, Multipactor, Dark Current20 Wakefields 3 - Other Sources of Impedance21 Other RF Sources22 High Gradients in Superconducting Cavities23 Modulators24 Windows & High-Power Transmission25 Ferrite Loaded Cavity 226 Design for System Stability - Heavy Beam Loading
Microwave ElectronicsMicrowave Electronics
Maxwell's Equations & Modes in a Guide Equivalent Circuit for Waveguide Modes Modes of a Cavity Cavity with a Port & External Q Microwave Networks Slater's Perturbation Theorem
An Introduction to the Equivalent Circuit
Starting from Maxwell’s Equations
in six lectures
Microwave Electronics
Maxwell's Equations & Modes in a Guide Equivalent Circuit for Waveguide Modes Modes of a Cavity Cavity with a Port & External Q Microwave Networks Slater's Perturbation Theorem
An Introduction to the Equivalent Circuit
Starting from Maxwell’s Equations
in six lectures
Microwave Electronics
Maxwell's Equations & Modes in a Guide Equivalent Circuit for Waveguide Modes Modes of a Cavity Cavity with a Port & External Q Microwave Networks Slater's Perturbation Theorem
An Introduction to the Equivalent Circuit
Starting from Maxwell’s Equations
in six lectures
Why are you here?
•Distributed vs. Lumped Elements•The Meaning of Current & Voltage•Transit Time & Retardation
What do you want?
Understanding of fundamentals...
Familiarity with the language...
•V, I, Z, δ, Rs, Qw, Qe, R/Q...• π mode, Travelling Wave,...•Tee, Load, Circulator, 3dB Coupler...
Ability to Solve Problems...
•How to design, build & tune my cavity?•What is the right power source to use?•My system isn’t working, what to do?