Cavity Resonators
Microwave Engineering
EE 172
Dr. Ray Kwok
Reference: Feynman, Lectures on Physics, Vol 2
Cavity Resonators - Dr. Ray Kwok
LC Resonator (Lenzs Law)
Cavity Resonators - Dr. Ray Kwok
Helical Resonator
L
1
o =
LC
Higher frequency smaller L or C
smaller C smaller area
Just the coil itself resonate
(Helical Resonator)
Internal capacitance between turns
Cant use coil in very high frequency
L
C
R
real coil equivalent
Cavity Resonators - Dr. Ray Kwok
Cavity Resonator
L
1
o =
LC
smaller L less turns
even higher f parallel L
both E & B resonate inside?
Cavity Resonators - Dr. Ray Kwok
High frequency capacitor
B
E
dc no B
ac E & B coexist
cavity except tangential E = 0 on the walls,
more field strength at center .etc
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
Cavity Resonators - Dr. Ray Kwok
Resonate Frequencies
output
Q f / fo
Quality Factor
fo
f
many resonants
Cavity Resonators - Dr. Ray Kwok
Different modes
some require different coupling mechanisms
Cavity Resonators - Dr. Ray Kwok
Rectangular Cavity Resonators
2
m n
k 2 2 k c2 =
+
a
a
d
2
k 2 = = k c2 + 2
p g
d=
2
2 p
=
=
d
g
2
m n p
k 2mnp =
+ +
a
b d
2f mnp= ck mnp
2
For TEmnp and TMmnp modes
c m n p
f mnp=
+ +
2 a b d
air
Cavity Resonators - Dr. Ray Kwok
TEmnp modes
Ez = 0
b
a
d
From
n p
E x ~ sin y sin z
boundary conditions
b d
m p
E y ~ sin
x sin z
a
d
m & n cannot be both 0
m, n = 0,1,2....
as in the waveguide,
p = 1,2,3...
p cannot be 0 !!
First cavity mode is TE101
But a, b, d are interchangeable !!!!
So be careful when labeling the modes!!
Cavity Resonators - Dr. Ray Kwok
TMmnp modes
b
a
d
m n
E z ~ sin
x sin y
a b From
n p boundary conditions
E x ~ sin y sin z
b d
m p
E y ~ sin
x sin z
a d
m, n = 1,2,3....
p = 0,1,2,3...
p can be 0.
First cavity TM mode is TM110
Again a, b, d are interchangeable !!!!
Cavity Resonators - Dr. Ray Kwok
Example
a
6.75
inches
inches
5.6
inches
6.75
inches
inches
5.6
inches
f (GHz)
mode
f (GHz)
mode
1.093611
TE101
1.093611
TM110
1.24203
TE011
1.24203
TE101
1.370224
TM110
1.370224
TE011
1.519233
TE/TM111
1.519233
TE/TM111
1.577228
TE102
1.577228
TM210
1.683539
TE012
1.683539
TE201
1.868763
TE201
1.868763
TM120
1.897296
TE/TM112
2.042989
TE021
2.042989
TM210
2.20882
TE102
2.20882
TE021
2.283367
TE012
2.283367
TM120
2.375777
TE/TM112
Same cavity, same set of resonant frequencies. Just different notation.
Not all modes can be excited.
The probe connection dictates which orientation is correct !!
Cavity Resonators - Dr. Ray Kwok
Cylindrical Cavity Resonators
2
2
k 2 = = k c2 + 2
2 p
=
=
d
g
a
d
2
nmp
p
= k +
d
2
c
c
p
f nmp=
k c2 +
2
d
e.g. Coke can, a ~ 1.25, d ~ 5
TE111: kc = 1.8412 / 1.25 = 1.473
2
11.811
1
f 111=
(1.473) 2 + = 3.01GHz
2
5
air
Cavity Resonators - Dr. Ray Kwok
TEnmp modes
Ez = 0
p
E ~ (A cos n B sin n)J n (k c) sin z
d
p
E ~ (A cos n + B sin n)J 'n (k c) sin z
d
a
d
J 'n (k nm a ) = 0
p = 1,2,3...
From boundary conditions.
p starts from 1
First TE cavity mode is TE111.
Cavity Resonators - Dr. Ray Kwok
TMnmp modes
p
E z ~ (A cos n + B sin n)J 'n (k c) cos z
d
p
E ~ (A cos n + B sin n)J 'n (k c) sin z
d
a
d
p
E ~ (A cos n B sin n)J n (k c) sin z
d
J n (k nma ) = 0
From boundary conditions.
p = 0,1,2,3...
p begins at 0.
p = 0 means Er and E = 0 !!!
And cannot be excited with connector on the sides!
First TM cavity mode usually is TM011.
Cavity Resonators - Dr. Ray Kwok
Example
a = 1.9
d = 6.82
TM
TE
n
f (GHz)
f (GHz)
2.016756
2.379399
2.513305
2.532062
3.14312
2.942912
3.172648
3.522744
3.482614
3.888838
Again, not all modes can be excited.
Cavity Resonators - Dr. Ray Kwok
Resonant
e.g. Coke can, a ~ 1.25, d ~ 5
TE111: kc = 1.8412 / 1.25 = 1.473
2
11.811
1
f 111=
(1.473) 2 + = 3.01GHz
2
5
Cavity Resonators - Dr. Ray Kwok
Dual Mode Cavity
e.g. TE10
square waveguide
orthogonal
Cavity Resonators - Dr. Ray Kwok
Perturbation
e.g. TE10
coupled modes
Use for:
Circular polarization
Dual cavity
Cross-coupled
Cavity Resonators - Dr. Ray Kwok
Dual Mode
TE111 mode
Up to 5-modes cavity
has been demonstrated
in a spherical cavity.
