WCChew

ECE Lecture Notes

Cavity Resonator

y

b Ðd

x 0 a

z

A cavity resonator is a useful microwave device If we close o two ends

of a rectangular waveguide with metallic walls we have a rectangular cavity

resonator In this case the wave propagating in the zdirection will b ounce

o the twowalls resulting in a standing waveinthezdirection For the TM

casewe have

j z j z

z z

E E sin x sin y e e

z x y

j

x z

j z j z

z z

E cos x sin y e e E

x y x

 

y x

j

y z

j z j z

z z

E E sin x cos y e e

y x y

 

y x

For the b oundary conditions to be satised we require that E z

x

E z Hence and

y

E E sin xsin y cos z

z x y z

x z

E E cos x sin y sin z

x x y z

 

x y

y z

E E sin x cos y sin z

y x y z

 

x y

Furthermore E z dE z d implying that

x y

p

p

z

d

m n

The guidance conditions for a waveguide demand that and

x y

a b

where for TM case neither m or n can be zero Now that has to satisfy

z

the TM mo de in a cavity is classied as TM mo de We note from

mnp

that p can be zero while E Hence the TM cavity mo de can exist

z mn

In order for and to b e solutions to the wave equation we require

that

  

m n p

   

x y z

a b d

For a given choice of m n and p only a single frequency can satisfy

This frequency is the resonant frequency of the cavity It is only at this

frequency that the cavity can sustain a free oscillation At other frequencies

the elds interfere destructively and the free oscillation is not sustained From

we gather that the resonant frequency for the TM mo de is

mnp



  

n p m

p

mnp

a b d

For the TE case similar derivation shows that

H H cos x cos y sin z

z x y z

j

y

H cos x sin y sin z E

x y z x

 

x y

j

x

H sin x cos y sin z E

x y z y

 

x y

Similarly the b oundary conditions require that

m n p

x y z

a b d

When p H hence TE mo de do es not exist However TE or

z mn np

TE mo des can exist The resonant frequency formula is as given in

mp

If abd the lowest resonant frequency is the TM mo de In this case





 

p



a b

and E H H E E Asketch of the eld is as shown

z x y x y

y b — H-field TM110 E-field mode

x

z a

We can decomp ose the waveinto plane waves b ouncing o the four walls

of the cavity  y b

x

0 a

As an example for a cm b cm d cm the resonant

frequency of the TM mo de is



s





p



f Hz



  

or

p

 

Hz Hz GHz f



Cavity resonators are useful as lters and tuners in microwave circuits as LC

resonators are in RF circuits Cavity resonators can also b e used to measure

the frequency of an electromagnetic signal