WCChew ECE Lecture Notes Cavity Resonator y b –d 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 .