Microwave Physics and Techniques UCSB –June 20031
Lecture 8Resonators
A. Nassiri - ANL
June 19, 2003
Microwave Physics and Techniques UCSB –June 20032
zjcy
zjcx
zjz
z
z
z
z
eybn
xa
mA
bm
jE
eybn
xa
mA
an
jE
eybn
xa
mAH
E
β
β
β
π
ππ
ωµπ
λ−=
π
ππ
ωµπ
λ=
π
π
=
=
m
m
m
cossin
sincos
coscos
2
2
2
2
4
4
0
Field Expressions For TE Modes – Rec. WGField Expressions For TE Modes – Rec. WG
g
yx
EH
η= m
g
xy
EH
η±=
m,n = 0,1,2,… but not both zero
Microwave Physics and Techniques UCSB –June 20033
Field Expressions For TE Modes – Rec. WGField Expressions For TE Modes – Rec. WG22
21
+
µε=
bn
am
fc 222
+
=λ
bn
am
c
( ) ( )22 11 ffcc
g−
λ=
λλ−
λ=λ
( ) ( )22 1
1
1
1
ccpz
ff λλ−µε=
−µε=ν
( ) ( )22 11 ccg
ff λλ−
εµ=
−
εµ=η
Microwave Physics and Techniques UCSB –June 20034
Field Expressions For TM Modes – Rec. WGField Expressions For TM Modes – Rec. WG
zj
g
cy
zj
g
cx
zjz
z
z
z
z
eybn
xa
mA
bn
jE
eybn
xa
mA
am
jE
eybn
xa
mAE
H
β
β
β
π
ππ
πλλ
=
π
ππ
πλλ
=
π
π
=
=
m
m
m
m
m
cossin
sincos
coscos
2
2
0
2
2
g
yx
EH
η= m
g
xy
EH
η±=
m,n = 1,2,3,…
Microwave Physics and Techniques UCSB –June 20035
Field Expressions For TM Modes – Rec. WGField Expressions For TM Modes – Rec. WG22
21
+
µε=
bn
am
fc 222
+
=λ
bn
am
c
( ) ( )22 11 ffcc
g−
λ=
λλ−
λ=λ
( ) ( )22 1
1
1
1
ccpz
ff λλ−µε=
−µε=ν
( ) ( )22 11 ccg ff λλ−εµ
=−εµ
=η
Microwave Physics and Techniques UCSB –June 20036
Determine the lowest four cutoff frequencies of the dominant mode for three cases of rectangular wave guide dimensions b/a=1, b/a=1/2, and b/a =1/3. Given a=3 cm, determine the propagating mode(s) for f=9 GHz for each of the three cases.
The expression for the cutoff wavelength for the TEm n mode where m=0,1,2,3,.. and n=0,1,2,3,.. But not both m and n equal to zero and for TMmn mode where m=1,2,3,.. And n=1,2,3,.. is given by
22
22
1
+
=λ
bn
am
c
The corresponding expression for the cutoff frequency is 2
222
21
221
+
µε=
+
µε=
λν
=ba
nmab
na
mf
c
pc
Microwave Physics and Techniques UCSB –June 20037
The cutoff frequency of the dominant mode TE10 is . Hence µεa21
[ ]2
2
10
+=
ba
nmf
f
TEc
c
TE10
TE01
TM11
TE11
TE20
TE02
TM21
TM12
TE21
TE12
[ ]10TEc
cf
f1=ab
1 2 3 4 5 2 5
21
=ab
TE10
TE01
TE20
TE11
TM11
TE21
TM21
[ ]10TEc
cf
f1 2 3 4 5 5 8
31
=ab
TE10 TE20
TE30
TE01
TM11
TE11
[ ]10TEc
cf
f
1 2 3 4 5 10
Microwave Physics and Techniques UCSB –June 20038
Hence for a signal of frequency f=9GHz, all the modes for which is less that 1.8 propagate. These modes are:
TE10,TE01,TM11,TE11 for b/a=1
TE10 for b/a=1/2
TE10 for b/a=1/3
[ ]10TEc
cf
f
So for b/a≤1/2, the second lowest cutoff frequency which corresponds to that of the TE20 mode is twice of the cutoff frequency of the dominant TE10 . For this reason, the dimension b of the a rectangular wave guide is generally chosen to be less than or equal to a/2 in order to achieve single-mode transmission over a complete octave( factor of two) range of frequencies.
Microwave Physics and Techniques UCSB –June 20039
Rectangular Cavity ResonatorRectangular Cavity ResonatorAdd two perfectly conducting walls in z-plane separated by a
distance d.For B.C’s to be satisfied, d must be equal to an integer
multiple of λg/2 from the wall.Such structure is known as a cavity resonator and is the
counterpart of the low-frequency lumped parameter resonant circuit at microwave frequencies, since it supports oscillationsat frequencies foe which the foregoing condition, that is
d=l λg/2, l=1,2,3,… is satisfied.
a
bdx
y z
Microwave Physics and Techniques UCSB –June 200310
Rectangular Cavity ResonatorRectangular Cavity ResonatorSubstituting for and rearranging, we obtainλg
( )2
22
2
211
1
2
=
λ−
λ
λλ−
λ=
dp
pd
c
c
Substituting for givesλc222
2 2221
+
+
=
λ dp
bn
am 222
222
1
+
+
=λ
dp
bn
am
222
2221
+
+
µε=
λν
=dp
bn
am
f posc
Microwave Physics and Techniques UCSB –June 200311
Quality Factor QQuality Factor QThe quality factor is in general a measure of the ability of a resonator to store energy in relation to time-average power dissipation. Specifically, the Q of a resonator is defined as
wall
strPW
Q oω=π=per cycle d dissipateEnergystored energyMaximum2
mestr WWW +=Consider the TE101 mode:
+
µ=
π
π
πωµε
=ε
= ∫∫∫∫
116
44
2
22
2
0
2
00
222
d
aHabdW
dxdydzdz
ax
Ha
dvEW
e
abd
v ye
o
o sinsin
211
02
22
22101
2 adx
ax
da
a=
π
+
µεπ
=ω=ω ∫ sin and
Microwave Physics and Techniques UCSB –June 200312
The time average stored magnetic energy can be found as
dxdydzdz
ax
dz
ax
d
aH
dvHHW
abd
zv xm
π
π
+
π
πµ
=
+µ
=
∫∫∫
∫
222
0
22
2
00
2
22
4
4
sincoscossino
+
µ= 1
16 2
22
d
aHabdW m
o
Note that the the time-average electric and magnetic energies are precisely equal. This should be true in general simply follows from the complex Poyting’stheorem. Physically, the fact that energy cycles between being purely electric, partly electric and partly magnetic, and purely magnetic storage, such that on the average over a period, it is shared equally between the electric and magnetic forms. The total time-average stored energy is
+
µ=+= 1
8 2
22
d
aHabdWWW mestr
o
Microwave Physics and Techniques UCSB –June 200313
We now need to evaluate the power dissipated in the cavity walls. This dissipation will be due to the surface currents on each of the six walls as induced by the tangential magnetic fields, that is . Note that the
power dissipation is given by and that
is the surface resistance.
HnJ s ×= ˆ
ss RJ 2
21
tanHJ s =
σfR ms µπ=
+++
==
∫∫∫∫∫∫
∫
==
4444 34444 21444 3444 21444 3444 21bottomtop
a
xz
d
leftright
b
xz
d
front,back
a
zx
bs
wallswall
dxdzHHdydzHdxdyHR
dsHR
P
,,
tan
0
22
00
20
00
20
0
2
2222
2
Microwave Physics and Techniques UCSB –June 200314
After completing the integration steps, we obtain:
+
+
+
= 121
4 3
3
2
222
d
adb
d
adadHR
P swall
o
Therefore the quality factor Q, is
+
+
+
+
πµ=
ω=
121
1
3
3
2
2
2
2
101
d
adb
d
ada
d
a
DRabf
PW
Qswall
stro
Substituting for f101, gives
+
+
+
+
ηπ=
121
1
23
3
2
2
23
2
2
2
d
adb
d
ada
d
a
dRb
Qs
Microwave Physics and Techniques UCSB –June 200315
For a cubical resonator with a = b = d, we have
( )
( )[ ]21101
221011
101
33
112
121
−
−
σµπ=δδµ
µ=
πµ=
+
µε=µε=
mms
cube fa
Raf
Q
dafaf
Skin depth of the surrounding metallic walls, where µm is the permeability of the metallic walls.
Microwave Physics and Techniques UCSB –June 200316
Air-filled cubical cavityAir-filled cubical cavityWe consider an air-filled cubical cavity designed to be resonant in TE101 mode at 10 GHz (free space wavelength λ=3cm)with silver-plated surfaces (σ=6.14×107S-m-1, µm= µ0.. Find the quality factor.
( ) cmf
cf
aaf 122222
1121101101
1101 .≈
λ==
εµ=⇒µε= −
oo
At 10GHz, the skin depth for the silver is given by
( ) mµ≈×××π×××π=δ−− 6420101461041010
21779 ..
and the quality factor is
0001164203
1223
,.
.≅
µ×≅
δ=
mcma
Q
Microwave Physics and Techniques UCSB –June 200317
ObservationsObservationsPrevious example showed that very large quality factors can be achieved with
normal conducting metallic resonant cavities. The Q evaluated for a cubical cavity is in fact representative of cavities of other simple shapes. Slightly higher Q values may be possible in resonators with other simple shapes, such as an elongated cylinder or a sphere, but the Q values are generally on the order of magnitude of the volume-to-surface ratio divided by the skin depth.
( )
cavity
cavity
s
ts
v
wall
m
wall
strS
V
dsHR
dvHf
PW
PW
Qδ
≅
µπ
=ω
=ω=
∫∫ 2
2
22
22
2o
oo
Where Scavityis the cavity surface enclosing the cavity volume Vcavity.
Although very large Q values are possible in cavity resonators, disturbances caused by the coupling system (loop or aperture coupling), surface irregularities, and other perturbations (e.g. dents on the walls) in practice act to increase losses and reduce Q.
Microwave Physics and Techniques UCSB –June 200318
ObservationsObservationsDielectric losses and radiation losses from small holes may be especially important in reducing Q. The resonant frequency of a cavity may also vary due to the presence of a coupling connection. It may also vary with changing temperature due to dimensional variations (as determined by the thermal expansion coefficient). In addition, for an air-filled cavity, if the cavity is not sealed, there are changes in the resonant frequency because of the varying dielectric constant of air with changing temperature and humidity.Additional losses in a cavity occur due to the fact that at microwave frequencies for which resonant cavities are used most dielectrics have a complex dielectric constant . A dielectric material with complex permittivity draws an effective current , leading to losses that occur effectively due to
ε ′′−ε′=ε jEJeff ε ′′ω= o
*effJE ⋅
The power dissipated in the dielectric filling is
dydxdzE
dvEEdvJEP
a b dy
vv
effdielectric
2
0 0 02
21
21
∫ ∫ ∫
∫∫ε ′′ω
=
ε ′′ω⋅=⋅=
o
**
Microwave Physics and Techniques UCSB –June 200319
Dielectric LossesDielectric LossesUsing the expression for Ey for the TE101 mode, we have
ε ′′ε′
=ω=
+
µω
ε′ε ′′
=
d
strd
dielectric
PW
Q
d
aabdHP
o
oo 1
8 2
22
dvEPand
dvEWW
vydielectric
vymstr
2
2
2
22
∫∫
ε ′′ω=
ε′==
o
The total quality factor due to dielectric losses is
cd QQQ111
+=
Microwave Physics and Techniques UCSB –June 200320
Teflon-filled cavityTeflon-filled cavityWe found that an air-filled cubical shape cavity resonating at 10 GHz has a Qc of 11,000, for silver-plated walls. Now consider a Teflon-filled cavity, with ε= ε0(2.05-j0.0006). Find the total quality factor Q of this cavity.
[ ]rrr
da fc
aa
c
aff
ε′=⇒
εµ=
ε′µ≅= =
oo 22
22
12101
µr=1 for Teflon. This shows that the the cavity is smaller, or a=b=d=1.48 cm. Thus we have
Or times lower than that of the air-filled cavity. The quality factor Qd due to the dielectric losses is given by
rε′
76843
≅δ
=a
Qc
rε′
2365≅+
=cd
cdQQ
QQQ
Thus, the presence of the Teflon dielectric substantially reduces the quality factor of the resonator.
Microwave Physics and Techniques UCSB –June 200321
Cylindrical Wave FunctionsCylindrical Wave Functions
x
y
z ρ
φ
z
The Helmholtz equation in cylindrical coordinates is
011 22
2
2
2
2 =ψ+∂
ψ∂+
φ∂
ψ∂
ρ+
ρ∂ψ∂
ρρ∂∂
ρk
z
The method of separation of variables gives the solution of the form
0111 22
2
2
2
2 =+∂
+φ∂
Φ
Φρ+
ρρ
ρρk
z
ZdZ
dddR
dd
R
Microwave Physics and Techniques UCSB –June 200322
Cylindrical Wave FunctionsCylindrical Wave Functions
22
21zk
z
ZdZ
−=∂
( ) 01 2222
2=ρ−+
φ∂
ΦΦ
+ρ
ρρ
ρzkk
dddR
dd
R
22
21n
d−=
φ∂
ΦΦ
( ) 02222 =ρ−+−ρ
ρρ
ρzkkn
ddR
dd
R
Microwave Physics and Techniques UCSB –June 200323
( )[ ]
0
0
0
22
2
22
2
22
222
=+∂
=Φ+φ∂
ψ
=−ρ+ρ
ρρ
ρ
=+
ρ
ρ
ZKz
Zd
nd
RkkddR
dd
R
kkk
k
z
z
z
p satisfy toDefine
Cylindrical Wave FunctionsCylindrical Wave Functions
Microwave Physics and Techniques UCSB –June 200324
Cylindrical Wave FunctionsCylindrical Wave Functions
These are harmonic equations. Any solution to the harmonic equation we call harmonic functions and here is denoted by h(nφ)and h(kzz). Commonly used cylindrical harmonic functions are:
Where is the Bessel function of the first kind,
Is the Bessel function of the second kind, is the Hankel
function of the first kind, and is the Hankel function of the second kind.
( ) ( ) ( ) ( ) ( )ρρρρρ ρρρρρ kHkHkNkJkB nnnnn21 ,,,~
( )ρρkJ n ( )ρρkN n
( )ρρkH n1
( )ρρkH n2
Microwave Physics and Techniques UCSB –June 200325
Cylindrical Wave FunctionsCylindrical Wave Functions
Any two of these are linearly independent.
A constant times a harmonic function is still a harmonic function
Sum of harmonic functions is still a harmonic function
We can write the solution as :
( ) ( ) ( )zkhnhkB znknk zφρ=ψ ρρ ,,
Microwave Physics and Techniques UCSB –June 200326
Bessel functions of 1st kindBessel functions of 1st kind
Microwave Physics and Techniques UCSB –June 200327
Bessel functions of 2nd kindBessel functions of 2nd kind
Microwave Physics and Techniques UCSB –June 200328
Bessel functionsBessel functionsThe are nonsingular at ρ=0. Therefore, if a field is finite at
ρ=0, must be and the wave functions are
( )ρρkJ n
( )ρρkBn ( )ρρkJ n
( ) zjkjnnknk z
zeekJ φ
ρρ=ψρ ,,
The are the only solutions which vanish for
large ρ. They represent outward-traveling waves if kρ is
real. Thus must be if there are
no sources at ρ→∞. The wave functions are
( )( )ρρkH n2
( )ρρkBn( )( )ρρkH n2
( )( ) zjkjnnknk z
zeekH φ
ρρ=ψρ
2,,
Microwave Physics and Techniques UCSB –June 200329
( )( )
( )( )( )( ) ρ
ρ
−ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
jkn
jkn
n
n
ekH
ekH
kρkN
kρkJ
toous analog
togous analo
toogous anal
tologous ana
2
1
sin
cos
Bessel functionsBessel functions
Microwave Physics and Techniques UCSB –June 200330
Bessel functionsBessel functions
The and functions represent cylindrical standing waves for real k as do the sinusoidal functions. The
and functions represent traveling waves for
real k as do the exponential functions. When k is imaginary (k = -jα)it is conventional to use the modified Bessel functions:
( )ρρkJ n ( )ρρkN n
( )( )ρρkH n1 ( )( )ρρkH n
2
( ) ( )
( ) ( ) ( )( )αρ−−π
=αρΚ
αρ−=αρ
+ 212 n
nn
nn
n
Hj
jJjI
( )( ) tologous ana
tologous anaαρ−
αρ
αρΚ
αρ
e
eI
n
n
Microwave Physics and Techniques UCSB –June 200331
Circular Cavity ResonatorsCircular Cavity ResonatorsAs in the case of rectangular cavities, a circular cavity resonator can be constructed by closing a section of a circular wave guide at both ends with conducting walls.
ϕ
z
x
r
a
dThe resonator mode in an actual case depends on the way the cavity is excited and the application for which it is used. Here we consider TE011mode, which has particularly high Q.
Microwave Physics and Techniques UCSB –June 200332
Circular Cavity ResonatorsCircular Cavity Resonators
,...,,,,...;q,,,...;n,,,m
dzq
mm
ax
J mnn
TMmnq
32103213210 ===
π
φφ
ρ
=ψ
where
coscossin
,...,,,...;q,,,...;n,,,m
dzq
mm
ax
J mnn
TEmnq
3213213210 ===
π
φφ
ρ′
=ψ
where
sincossin
Microwave Physics and Techniques UCSB –June 200333
Circular Cavity ResonatorsCircular Cavity Resonators
The separation constant equation becomes
222
222
kdq
ax
kdq
ax
mn
mn
=
π
+
′
=
π
+
For the TM and TE modes,
respectively. Setting , we can solve for the resonant frequencies
µεπ= fk 2
Microwave Physics and Techniques UCSB –June 200334
Circular Cavity ResonatorsCircular Cavity Resonators
2.132.132.132.132.412.562.78
TM210
∞2.901.801.421.521.571.66
TE211
1.591.591.591.591.801.912.08
TM110
2.292.292.292.292.603.003.00
∞5.272.721.501.321.201.00
∞3.062.051.721.871.962.08
∞2.801.631.191.241.271.31
∞2.721.501.001.001.001.00
1.001.001.001.001.131.201.31
0.00.501.002.003.004.00∞
TM020TE112TM111
TE011
TM011TE111TM010d/a
antdo
mnq
r
r
f
f
min
for the circular cavity of radius a and length d
Microwave Physics and Techniques UCSB –June 200335
Circular Cavity ResonatorsCircular Cavity Resonators
8.77112.339
7.58811.06514.372
6.3809.76113.015
5.1368.41711.62014.796
3.8327.01610.17313.324
2.4055.5208.65411.792
1234
543210mn
Ordered zeros Xmn of Jn(X)
6.41610.52013.987
5.3179.28212.682
4.2018.01511.346
3.0546.7069.96913.170
1.8415.3318.53611.706
3.8327.01610.17313.324
1234
543210mn
Ordered zeros X`mn J`
n(X)
Microwave Physics and Techniques UCSB –June 200336
Circular Cavity ResonatorsCircular Cavity ResonatorsCylindrical cavities are often used for microwave frequency meters. The cavity is constructed with movable top wall to allow mechanical tuning of the resonant frequency, and the cavity is loosely coupled to a wave guide with a small aperture.
The transverse electric fields (Eρ, Eφ) of the TEmn or TMmn circular wave guide mode can be written as
( ) ( ) [ ]zjzjt
mnmn eAeAzE β−β−+ +φρ=φρ E ,,,The propagation constant of the TEnm mode is
22
′
−κ=βa
xmnmn
While the propagation constant of the TMnm mode is 2
2
−κ=β
axmn
mn
Microwave Physics and Techniques UCSB –June 200337
Circular Cavity ResonatorsCircular Cavity ResonatorsNow in order to have Et =0 at z=0, d, we must have A+= -A-, and A+sin βnmd=0 or
βmnd =lπ, for l=0,1,2,3,…, which implies that the wave guide must be an integer number of half-guide wavelengths long. Thus, the resonant frequency of the TEmnlmode is
And for TMnml mode is
22
2
π
+
′
εµπ=
dq
axc
f mn
rrmnq
22
2
π
+
εµπ=
dq
axc
f nm
rrmnq
Microwave Physics and Techniques UCSB –June 200338
Circular Cavity ResonatorsCircular Cavity ResonatorsThen the dominant TE mode is the TE111 mode, while the dominant TM mode is the TM110 mode. The fields of the TMnml mode can be written as
( )
( )
0
2
2
2
2
=
π
ρ′
′′
κη=
π
ρ′
ρ′κη
=
π
ρ′
ρ′β−
=
π
ρ′
′′
β=
πφ
ρ′
=
φ
ρ
φ
ρ
z
mnn
mn
mnn
mn
mnn
mn
mnn
mn
mnnz
E
dzq
mφa
xJ
xaHj
E
dzq
mφa
xJ
xmHaj
E
dzq
mφa
xJ
xmHa
H
dzq
mφa
xJ
xaH
H
dzq
ma
xJHH
sincos
sinsin
cossin
coscos
sincos
o
o
o
o
o
+−=εµ=η jAH 2o and
Microwave Physics and Techniques UCSB –June 200339
Circular Cavity ResonatorsCircular Cavity ResonatorsSince the time-average stored electric and magnetic energies are equal, the total stored energy is
( )
( )ρρ
ρ′
′
+
ρ′
′′πηεκ
=
φρρ+ε
==
∫
∫ ∫ ∫
=ρ
π
φρ
da
xJ
xma
ax
Jx
dHa
dzddEEWW
amn
nmn
mnn
mn
d a
e
0
22
22
2222
0
2
0 0
22
4
22
o
( )( )mnn
mnmn
xJxm
xdHa ′
′
−′
πηεκ= 2
2
2
2422
18
o
Microwave Physics and Techniques UCSB –June 200340
Circular Cavity ResonatorsCircular Cavity ResonatorsThe power loss in the conducting walls is
( ) ( )[ ]( ) ( )[ ]
( )( ) ( )
′−
′
β+
′β
+′π=
φρρ=+=+
φ=ρ+=ρ==
∫ ∫
∫ ∫∫π
=φ =ρφρ
=
π
=φφ
2
2222
222
2
0 0
22
0
2
0
222
1122
002
22
mnnmmnmnn
s
a
d
zz
s
St
sc
xm
xa
xamda
xJHR
ddzHzH
dzadaHaHR
dsHR
P
o
( )( )
( ) ( )
′−
′β
+
′
β+
′
−
′
ηκ=
ω=
2
2222
2
2
2
3
112
1
4
mnmnmn
mn
smncc
x
mxa
x
amad
xm
Rx
adaPW
Q
Microwave Physics and Techniques UCSB –June 200341
Circular Cavity ResonatorsCircular Cavity Resonators
Where is the loss tangent of the dielectric. This is the same as the result of Qd for the rectangular cavity.
( )
( )( )
δ=
ε ′′ε
=ω
=
′
′
−′
ηκε ′′ω=
ρρ
ρ′
′+
ρ′
ρ′′
πηκε ′′ω=
+
ε ′′ω=⋅=
∫
∫∫
=ρ
φρ
tan
*
1
18
4
221
22
2
2422
022
2
2
2222
22
dd
mnnmnmn
amn
nmn
nmnmn
vvd
PW
Q
xJxm
x
Ha
da
xJ
ax
Jxma
x
dHa
dvEEdvEJP
o
o
To compute the Q due to dielectric loss, we must compute the power dissipated in the dielectric. Thus,
δtan
Microwave Physics and Techniques UCSB –June 200342
Cavity wave guide mode patternsCavity wave guide mode patterns
Microwave Physics and Techniques UCSB –June 200343
Cylindrical Cavity mode patternsCylindrical Cavity mode patterns
Microwave Physics and Techniques UCSB –June 200344
Loop or Probe CouplingLoop or Probe CouplingFor a probe coupler the electric flux arriving on the probe tip furnishes the current induced by a cavity mode:
I = ωε SEwhere E is the electric field from a mode averaged over probe tip and S is the antenna area. The external Q of this simple couplerterminated on a resistive load R for a mode with stored energy Wis
In the same way for a loop coupler the magnetic flux going through the loop furnishes the voltage induced in the loop by a cavity mode: V= ωµ SH
2222
ESR
WQext
ωε=