EELE 3332 – Electromagnetic II Chapter 12 Waveguide Resonators

Islamic University of Gaza Electrical Engineering Department Dr. Talal Skaik

2012 1 Resonance  Resonance: The tendency of a system to oscillate at maximum amplitude at a certain frequency.  Microwave resonators are tunable circuits used in microwave oscillators, filters and frequency meters.  The operation of microwave resonators is very similar to that of the lumped-element resonators (such as parallel and series RLC resonant circuits) of circuit theory.  Transmission line sections can be used with various lengths

(nλg/2) and terminations (usually open or short circuited) to form resonators.  Resonators can be constructed from closed sections of

waveguides. Dr. Talal Skaik 2012 IUG 2

Rectangular Waveguide Cavities

Resonators can be constructed from closed sections of waveguide. Waveguide resonators are usually short circuited at both ends, thus forming a closed box or cavity. Standing waves are formed in the cavity (recall a standing wave is a combination of two waves travelling in opposite directions). Electric and magnetic energy is stored within the cavity.

Dr. Talal Skaik 2012 IUG 3 Rectangular Waveguide Cavity

To satisfy the boundary conditions, d must be equal to an integer multiple of g / 2.

A resonant wave number for the rectangular cavity can be deifned as:

2 2 2 mπ   n π   l π  kmnl         a   b   d 

The resonant frequency of the TEmnl or TM mnl mode is then given by:

2 2 2 1 m   n    l   fmnl         2  a   b   d 

Dr. Talal Skaik 2012 IUG 4 Rectangular Waveguide Cavity Waveguide (waves in one direction)

m x n y  jz H z ( x , y , z ) H0 cos cos e (TE modes) ab m x n y E( x , y , z ) E sin sin e jz (TM modes) z 0 ab

Cavity (waves in both directions, standing waves) m x n  y l  z H( x , y , z ) H cos cos sin (TE modes) z0 a b d mnl (m 0,1,2,...), ( n  0,1,2,...), ( l  1,2,3,...)

m x n  y l  z E( x , y , z ) E sin sin cos (TM modes) z0 a b d mnl (mn 1,2,3,...), ( 1,2,3,...), (l  0,1,2,...)

Dr. Talal Skaik 2012 IUG 5 Rectangular Waveguide Cavity

 A set of mnl corresponds to a mode, where the indices m,n,l refer to the number of variations in the standing wave pattern in the x,y,z directions. TE101 stands for that a rectangular waveguide cavity operating on a TE10 wave, and the length of the cavity is half of the guide wavelength.

The lowest order modes in a rectangular cavity are the TM110,

TE101, and TE011 modes. Which of these modes is the dominant mode depends on the relative dimensions of the resonator. If b

 In order to properly design the coupling and the tuning devices of the cavity, knowledge about the distribution of the fields in the cavity is required. Dr. Talal Skaik 2012 IUG 6 d z y

a The figure gives the distribution of the fields in a rectangular cavity x z x operating at the TE101 mode. b Electric field lines y Magnetic field lines

Dr. Talal Skaik 2012 IUG 7 Quality factor , Q •The cavity has walls with finite conductivity and is therefore losing stored energy.  W •The quality factor is: Q  0 P l

where 0 is the resonant angular frequency, W is the total energy, Pl is the power loss in the cavity.

The Q of the cavity with lossy conducting walls for the TE10l mode is:

3 kad b ' 1 Qc = 2 2 3 3 2 3 3 2 Rs 22l a b bd  l a d  ad 

2f 120  r 0 where k , Rs  ,  '  c 2  r

Dr. Talal Skaik 2012 IUG 8 Example : An air-filled rectangular waveguide has a=3cm , b=2cm. If a cavity resonator is to be designed, find the length of cavity so the resonant frequency is 5.44 GHz for (a) TE101 mode, (b) TE102 mode

2 2 2 1 m   n    l   fr         2  a   b   d 

2 2 2 3 108  1   0   1  For TE101 mode: fr          5.44 GHz  d  7 cm 2 3cm   2 cm   d 

2 2 2 3 108  1   0   2  For TE102 mode: fr          5.44 GHz  d  14 cm 2 3cm   2 cm   d  or : cavity resonates at lg /2.

ffcc10 g ' 1  , For TE 10l mode   g   ' 1   f 5.44 GHz cc3 108 '  , f   5 GHz,   0.139973 fa5.44 109 c10 2 g for TE dd  /2  7 cm, for TE   2 /2  14 cm 101 g 102 g 9 Example For a cavity of dimensions; 5cm x 4cm x 10 cm filled with air and 7 made of copper (c=5.8 x 10 ) Find the resonant frequency and the quality factor for the dominant mode.

2 2 2 1 mnl        fmnl         2  a   b   d 

2 2 2 3 1010  1   0   1  For TE101 mode: fr          3.354 GHz 2 5   4   10  3 kad b ' 1 2f r 0 Qc =2 2 3 3 2 3 3 , kR , s 22Rcs 22l a b bd  l a d  ad  2f  kR 70.246, 0  0.0151,  ' 120 c s 2

Qc = 14365 Dr. Talal Skaik 2012 IUG 10 Waveguide Filters

Dr. Talal Skaik 2012 IUG 11 Waveguide Filters

Filter for E-band systems (81-86 GHz)

Coupling Matrix 0 0.0589 0 0 0  0.0589 0 0.0432 0 0 k  0 0.0432 0 0.0432 0  0 0 0.0432 0 0.0589  0 0 0 0.0589 0

Dr. Talal Skaik 2012 IUG 12