Waveguide and Cavity Resonators

Assuming z-directed propagation

satisfying

then

From and , we have

and

Solving in terms of and , we have

where

1. TEM waves: , .2. TM waves: , .3. TE waves: , .

TEM waves

, not dispersion.

To support TEM waves, two conductors are necessary.

TM waves

Cutoff frequency occurs when for a given .

For , , propagation mode.

Guided wavelength

Phase velocity

Group velocity

Impedance

, pure real.

For , , evanescent mode.

Impedance

, pure imag.

TE modes

Cutoff frequency occurs when for a given .

For , , propagation mode.

Guided wavelength, Phase velocity, Group velocity same as TM mode.Impedance

, pure real.

For , , evanescent mode.

Impedance

, pure real.

Parallel-Plate Waveguide

Assume no variation in the -direction.

TM modes

Wave equation:

B. C.:

Therefore,

Note: when , TM becomes TEM mode.

TE Modes

Wave equation:

B. C.:

Therefore,

Rectangular Waveguide

TM modesAssume

Let

then.

To satisfy boundary condition

then

and

Fundamental mode:

TE modesAssume

Let

then.

To satisfy boundary condition

then

and

Fundamental mode:

Approximation of Attenuation Constant for Low Loss Casesdielectric loss:

conductor loss:

For mode, see Eq. (10.187).

total:

Discontinuities in Rectangular Waveguides

Series Capacitive iris. Pure imaginary series load.

Series Inductive iris. Pure imaginary series load.

Post or screw in a waveguide.

Rectangular CavitiesTM modesAssume

Let

then.

To satisfy boundary condition

then

and

Fundamental mode: .

Circular Waveguides

Let

then

In general,

Due to the boundary condition, choose

TM modes

TE modes

and for TM modes, for TE modes.

Circular Cavity Resonator

First mode: for .

for .

Dielectric Waveguides

Assume two dimensional TM case.

In the dielectric

In the air

Odd TM modes

In the dielectric

Outside dielectric, plus sign for , minus sign for

must be continuous,

or

also

or

and

Surface impedance

Similarly, for even mode

and

Cutoff frequency occurs at For odd mode

For even mode

Assume two dimensional TE case.

In the dielectric

In the air

Odd TE modes

In the dielectric

Outside dielectric, plus sign for , minus sign for

must be continuous,

also

and

Surface impedance

Similarly, for even mode

and

Cutoff frequency occurs at For odd mode

For even mode

T-junction Magic T

Twist Bend Directional Coupler

Waveguide Components

Waveguide Slot-Line Waveguide Line Section

Waveguide Match Load Coaxial toWaveguide Adapter

Double Ridge Waveguide