parallel plate

ECE450 – QUIZ BREAKSample problem on Parallel Plate Wave guide

Consider a parallel plate wave guide with plate separation a = 5 cm, filled with air, working at afrequency f = 10.0 GHz.

a) What is the fundamental mode of propagation and what is its cut-off frequency?

As in all parallel plate wave guides, the fundamental mode is the TEM (aka TM00) withcut-off frequency f = 0.0 Hz.

b) Find all the higher order modes of propagation.

The higher modes of propagation must have a cut-off frequency smaller than theoperating frequency. For both TM and TE modes, the cutoff frequency is obtained from

TM fc λc TE fc λcTEM or TM00 0.0 Hz ∞

TM10 3.0 GHz 0.1 [m] TE10 3.0 GHz 0.1 [m]TM20 6.0 GHz 0.05 [m] TE20 6.0 GHz 0.05 [m]TM30 9.0 GHz 0.0333 [m] TE30 9.0 GHz 0.0333 [m]

c) Determine the guide phase velocity, the group velocity, and the angles of incidence onthe plates for all the propagating modes.

mode θ vpz vg

TEM 90° 3.0×108 m/s 3.0×108 m/sTM10 & TE10 72.55° 3.1425×108 m/s 2.86×108 m/sTM20 & TE20 53.16° 3.75×108 m/s 2.4×108 m/sTM30 & TE30 25.9° 6.86×108 m/s 1.31×108 m/s

8

1/

0,1,2,3,1 3 10 3 [GHz] 2 2 0.05 1,2,3,cp

pc

cv

v TMm mf m ma TEλ

λ µε× ×= = = = =×

……

1

22

;

cos cos ( )

1 ( )1 ( )

cc

c

ppz g p c

c

f f ffvv v v f ff f

λθ θλ−= = → =

= = −−

d) Determine the guide impedance for the propagating modes

TM ηTM TE ηTETEM or TM00 376.73 Ω

TM10 359.4 Ω TE10 394.9 ΩTM20 301.5 Ω TE20 470.7 ΩTM30 164.7 Ω TE30 861.7 Ω

e) What modes are evanescent?

All TE and TM modes with cut-off frequency higher than the frequency of operation areevanescent. The cut-off frequency for TE40 and TM40 is fc = 12.0 GHz. Therefore, allmodes with index m ≥ 4 are evanescent.

f) Find the attenuation constant for the first evanescent mode

For TE40 and TM40 we have

e) What happens if the wave guide is filled with a different dielectric with relativepermittivity εr = 9.0?

The phase velocity will be decreased by a factor 3.0 and all the cut-off frequencies willbe lowered. The TE10,0 and TM10,0 modes have cut-off frequency of exactly 10.0 GHz.Therefore, modes with index up to m = 9 propagate, and with m = 10 are at cut-offcondition.

22

1 ( ) ;1 ( )

cTM TEc

f ff fηη η η= − =

−

-1

2 2 2 2 -1

-1

4 251.33 [ m ]0.05138.71 [ m ]

209.58 [ m ]

x

z x x

maj j j

π πβ

β β β β ω µεβ ω µε

= = =

=− − =− − =−= =