Notes 5 ECE 5317-6351 Microwave Engineering Waveguides Part 2: Fall 2011 Prof. David R. Jackson Dept. of ECE Notes 5 Waveguides Part 2: Parallel Plate Waveguide

Field Equations (from Notes 4) Summary These equations will be useful to us in the present discussion.

Parallel-Plate Waveguide Both plates assumed PEC w >> d,  Neglect x variation, edge effects The parallel-plate stricture is a good 1ST order model for a microstrip line.

TEM Mode To solve for TEM mode: Parallel-plate waveguide 2 conductors  1 TEM mode To solve for TEM mode: for Boundary conditions:

TEM Mode (cont.) where

TEM Mode (cont.) Recall For a wave prop. in + z direction Time-ave. power flow in + z direction:

TEM Mode (cont.) Transmission line voltage Transmission line current Characteristic Impedance (Assume + z wave) Phase Velocity (lossless case) Note: c = 2.99792458 108 m/s

TEM Mode (cont.) For wave propagating in + z direction Time-ave. power flow in +z direction: (calculated using the voltage and current) Recall that we found from the fields that: same This is expected, since a TEM mode is a transmission-line type of mode, which is described by voltage and current.

TEM Mode (cont.) We can view the TEM mode in a parallel-plate waveguide as a “piece” of a plane wave. y PEC PMC x E H The PEC and PMS walls do not disturb the fields of the plane wave.

TMz Modes (Hz = 0) Recall where subject to B.C.’s Ez = 0 @ y = 0, d

TMz Modes (cont.) Recall: No x variation

TMz Modes (cont.) Summary TMn mode Note: Each value of n corresponds to a unique TM field solution or “mode.” TMn mode Note: (In this case, we absorb the An coefficient with the kc term.)

TMz Modes (cont.) Lossless Case Fields decay exponentially  evanescent fields  “cutoff” mode

Frequency that defines border between cutoff and propagation TMz Modes (cont.) Frequency that defines border between cutoff and propagation (lossless case): fc  cutoff frequency cutoff frequency for TMn mode

TMz Modes (cont.) Time average power flow in z direction (lossless case): Real for f > fc Imaginary for f < fc

TEz Modes Recall where subject to B.C.’s Ex = 0 @ y=0, d

TEz Modes (cont.) Recall: No x variation

TEz Modes (cont.) TEn mode Cutoff frequency Summary Each value of n corresponds to a unique TE field solution or “mode.” TEn mode Cutoff frequency

All Modes For all the modes of a parallel-plate waveguide, we have The mode with lowest cutoff frequency is called the “dominant” mode of the wave guide.

Power in TEz Mode Time average power flow in z direction (lossless case): n = 1,2,….. Real for f > fc Imaginary for f < fc

Field Plots TEM TM1 TE1 y x