1. Cavity resonators confine electromagnetic waves inside hollow structures such as rectangular boxes or cylindrical cans through resonance. 2. The resonant modes inside the cavity depend on its geometry and are determined by solving Maxwell's equations with the appropriate boundary conditions. 3. Common modes include TE and TM, where the electric and magnetic fields are transverse to the axis of propagation. 4. Coupling mechanisms such as wires or loops are used to input and output power to selectively excite specific resonant modes within the cavity.

1. Cavity Resonators
Microwave Engineering
EE 172
Dr. Ray Kwok
Reference: Feynman, Lectures on Physics, Vol 2

2. Cavity Resonators - Dr. Ray Kwok
LC Resonator (Lenz’s Law)

3. Cavity Resonators - Dr. Ray Kwok
Helical Resonator
L C LC
1
o =ω
Higher frequency → smaller L or C
L C smaller C → smaller area
Just the coil itself – resonate
(Helical Resonator)
Internal capacitance between turns
Can’t use coil in very high frequency
L
C
R
real coil equivalent

4. Cavity Resonators - Dr. Ray Kwok
Cavity Resonator
L C LC
1
o =ω
smaller L → less turns
even higher f → parallel L
both E & B resonate inside?

5. Cavity Resonators - Dr. Ray Kwok
High frequency capacitor
dc – no B
ac – E & B coexist
E
B
cavity – except tangential E = 0 on the walls,
more field strength at center ….etc

6. Cavity Resonators - Dr. Ray Kwok
Coupling in and out of cavity
Wire / connector
Couple E-field
Capacitive coupling
Line up with concentrated E-field to induced V
Wire / loop
Couple H-field
Inductive coupling
Loop thru H-field to induced current

7. Cavity Resonators - Dr. Ray Kwok
Resonate Frequencies
f
output
fo
Q ≡ ∆f / fo
Quality Factor
many resonants

8. Cavity Resonators - Dr. Ray Kwok
Different modes
some require different coupling mechanisms

9. Cavity Resonators - Dr. Ray Kwok
Rectangular Cavity Resonators
a
b
d
222
mnp
mnpmnp
222
2
mnp
g
g
22
c
2
2
22
2
c
22
d
p
b
n
a
m
2
c
f
ckf2
d
p
b
n
a
m
k
d
p2
2
p
d
k
2
k
b
n
a
m
kk





 π
+




 π
+




 π
π
=
=π





 π
+




 π
+




 π
=
π
=
λ
π
=β
λ
=
β+=





λ
π
=





 π
+




 π
=≡β−
airFor TEmnp and TMmnp modes

10. Cavity Resonators - Dr. Ray Kwok
TEmnp modes
a
b
d
...3,2,1p
....2,1,0n,m
z
d
p
sinx
a
m
sin~E
z
d
p
siny
b
n
sin~E
0E
y
x
z
=
=





 π





 π





 π





 π
=
m & n cannot be both 0
as in the waveguide,
p cannot be 0 !!
First cavity mode is TE101
But a, b, d are interchangeable !!!!
So be careful when labeling the modes!!
From
boundary conditions

11. Cavity Resonators - Dr. Ray Kwok
TMmnp modes
a
b
d
...3,2,1,0p
....3,2,1n,m
z
d
p
sinx
a
m
sin~E
z
d
p
siny
b
n
sin~E
y
b
n
sinx
a
m
sin~E
y
x
z
=
=





 π





 π





 π





 π





 π





 π
p can be 0.
First cavity TM mode is TM110
Again a, b, d are interchangeable !!!!
From
boundary conditions

12. Cavity Resonators - Dr. Ray Kwok
Example
TM1202.283367021
TE0212.20882120
TM2102.042989012
TE/TM1121.897296211
TE2011.868763102
TE0121.683539210
TE1021.577228201
TE/TM1111.519233111
TM1101.370224011
TE0111.24203110
TE1011.093611101
modef (GHz)pnm
inches9d
inches5.6b
inches6.75a
TE/TM1122.375777211
TE0122.283367210
TE1022.20882201
TE0212.042989120
TM1201.868763021
TE2011.683539102
TM2101.577228012
TE/TM1111.519233111
TE0111.370224110
TE1011.24203101
TM1101.093611011
modef (GHz)pnm
inches5.6d
inches6.75b
inches9a
Same cavity, same set of resonant frequencies. Just different notation.
Not all modes can be excited.
The probe connection dictates which orientation is correct !!

13. Cavity Resonators - Dr. Ray Kwok
Cylindrical Cavity Resonators
2
2
cnmp
2
2
c
2
nmp
g
22
c
2
2
d
p
k
2
c
f
d
p
kk
d
p2
k
2
k





 π
+
π
=





 π
+=
π
=
λ
π
=β
β+=





λ
π
=
air
a
d
e.g. Coke can, a ~ 1.25”, d ~ 5”
TE111: kc = 1.8412 / 1.25 = 1.473
GHz01.3
5
1
)473.1(
2
811.11
f
2
2
111 =




 π
+
π
=

14. Cavity Resonators - Dr. Ray Kwok
TEnmp modes
...3,2,1p
0)ak(J
z
d
p
sin)k(J)nsinBncosA(~E
z
d
p
sin)k(J)nsinBncosA(~E
0E
nm
'
n
c
'
n
cn
z
=
=





 π
ρφ+φ





 π
ρφ−φ
=
φ
ρ
a
d From boundary conditions.
p starts from 1
First TE cavity mode is TE111.

15. Cavity Resonators - Dr. Ray Kwok
TMnmp modes
...3,2,1,0p
0)ak(J
z
d
p
sin)k(J)nsinBncosA(~E
z
d
p
sin)k(J)nsinBncosA(~E
z
d
p
cos)k(J)nsinBncosA(~E
nmn
cn
c
'
n
c
'
nz
=
=





 π
ρφ−φ





 π
ρφ+φ





 π
ρφ+φ
φ
ρ
a
d
From boundary conditions.
p begins at 0.
First TM cavity mode “usually” is TM011.
p = 0 means Er and Eρ = 0 !!!
And cannot be excited with connector on the sides!

16. Cavity Resonators - Dr. Ray Kwok
Example
3.482614212
3.172648311
3.14312112
2.513305211
2.016756111
f (GHz)pmn
a = 1.9”
d = 6.82”
TE
3.888838111
3.522744310
2.942912210
2.532062110
2.379399010
f (GHz)pmn
TM
Again, not all modes can be excited.

17. Cavity Resonators - Dr. Ray Kwok
Resonant
e.g. Coke can, a ~ 1.25”, d ~ 5”
TE111: kc = 1.8412 / 1.25 = 1.473
GHz01.3
5
1
)473.1(
2
811.11
f
2
2
111 =




 π
+
π
=

18. Cavity Resonators - Dr. Ray Kwok
Dual Mode Cavity
e.g. TE10
orthogonal
square waveguide

19. Cavity Resonators - Dr. Ray Kwok
Perturbation
e.g. TE10
coupled modes
Use for:
Circular polarization
Dual cavity
Cross-coupled

20. Cavity Resonators - Dr. Ray Kwok
Dual Mode
TE111 mode
Up to 5-modes cavity
has been demonstrated
in a spherical cavity.