Cavity Resonators
Microwave Engineering EE 172
Dr. Ray Kwok

Reference: Feynman, Lectures on Physics, Vol 2

Cavity Resonators - Dr. Ray Kwok

LC Resonator (Lenz’s Law)

Cavity Resonators - Dr. Ray Kwok

Helical Resonator
L C

1 ωo = LC
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

Cavity Resonators - Dr. Ray Kwok

Cavity Resonator
L C

1 ωo = LC

smaller L → less turns

even higher f → parallel L

both E & B resonate inside?

Cavity Resonators - Dr. Ray Kwok

High frequency capacitor dc – no B ac – E & B coexist

B E

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 b  mπ   nπ  2 k 2 − β2 ≡ k c =   +  a   b    2π  2 k 2 =   = k c + β2  λ  pλ g d= 2 2 π pπ = β= d λg
2

2

2

a d

 mπ   nπ   pπ  k2 =   +  +  mnp a   b   d   2πf mnp= ck mnp
2 2

2

2

2

For TEmnp and TMmnp modes

c  mπ   nπ   pπ  f mnp=   +  +  2π  a   b   d 

2

air

Cavity Resonators - Dr. Ray Kwok

TEmnp modes
Ez = 0 b a d From  nπ   pπ  E x ~ sin  y  sin  z  boundary conditions  b   d   mπ   pπ  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

 mπ   nπ  E z ~ sin  x  sin  y   a   b  From  nπ   pπ  boundary conditions E x ~ sin  y  sin  z   b   d   mπ   pπ  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 b d m 1 0 1 1 1 0 2 1 2 0 1

6.75 5.6 9 n 0 1 1 1 0 1 0 1 1 2 2

inches inches inches p 1 1 0 1 2 2 1 2 0 1 0 f (GHz) 1.093611 1.24203 1.370224 1.519233 1.577228 1.683539 1.868763 1.897296 2.042989 2.20882 2.283367 mode TE101 TE011 TM110 TE/TM111 TE102 TE012 TE201 TE/TM112 TM210 TE021 TM120

a b d m 1 1 0 1 2 2 1 0 1 0 1

9 6.75 5.6 n 1 0 1 1 1 0 2 2 0 1 1

inches inches inches p 0 1 1 1 0 1 0 1 2 2 2 f (GHz) 1.093611 1.24203 1.370224 1.519233 1.577228 1.683539 1.868763 2.042989 2.20882 2.283367 2.375777 mode TM110 TE101 TE011 TE/TM111 TM210 TE201 TM120 TE021 TE102 TE012 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
 2π  2 k 2 =   = k c + β2  λ  2π pπ β= = λg d d 2

a

k

2 nmp

 pπ  = k +   d 
2 c

2

c  pπ  2 f nmp= kc +   2π  d 
e.g. Coke can, a ~ 1.25”, d ~ 5” TE111: kc = 1.8412 / 1.25 = 1.473
2

2

air

11.811  1π  f 111= (1.473) 2 +   = 3.01GHz 2π 5

Cavity Resonators - Dr. Ray Kwok

TEnmp modes
Ez = 0  pπ  E ρ ~ (A cos nφ − B sin nφ)J n (k cρ) sin  z   d   pπ  E φ ~ (A cos nφ + B sin nφ)J 'n (k cρ) sin  z   d  d a

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
 pπ  E z ~ (A cos nφ + B sin nφ)J 'n (k cρ) cos z   d   pπ  E ρ ~ (A cos nφ + B sin nφ)J 'n (k cρ) sin z   d   pπ  E φ ~ (A cos nφ − B sin nφ)J n (k cρ) sin  z   d  J n (k nma ) = 0 From boundary conditions. p = 0,1,2,3...

a d

p begins at 0.

p = 0 means Er and Eρ = 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” TE n 1 1 2 1 2 m 1 1 1 1 1 p 1 2 1 3 2 f (GHz) 2.016756 2.513305 3.14312 3.172648 3.482614

TM n 0 0 0 0 1 m 1 1 1 1 1 p 0 1 2 3 1 f (GHz) 2.379399 2.532062 2.942912 3.522744 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

11.811  1π  f 111= (1.473) 2 +   = 3.01GHz 2π 5

2

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.