ENE 429 Antenna and Transmission lines Theory Lecture 7 Waveguides DATE: 3-5/09/07

Review Impedance matching to minimize power reflection from load  Lumped-element tuners  Single-stub tuners Microstrip lines  The most popular transmission line  Knowing the characteristic impedance and the relative dielectric constant of the material helps determine the strip line configuration and vice versa. Attenuation  conduction loss  dielectric loss  radiation loss

A pair of conductors is used to guide TEM wave Microstrip Parallel plate Two-wire TL Coaxial cable

The use of waveguide Waveguide refers to the structure that does not support TEM mode, bring up “the cutoff frequency”

We can write in the instantaneous form as We begin with Helmholz’s equations: assume WG is filled in with a charge-free lossless dielectric General wave behaviors along uniform guiding structures (1) The wave characteristics are examined along straight guiding structures with a uniform cross section such as rectangular waveguides

General wave behaviors along uniform guiding structures (2) We can write and in phasor forms as and

Use Maxwell’s equations to show and in terms of z components (1) From we have

Use Maxwell’s equations to show and in terms of z components (2) We can express E x, E y, H x, and H y in terms of z-component by substitution so we get for lossless media  = j ,

Propagating waves in a uniform waveguide Transverse ElectroMagnetic (TEM) waves, no E z or H z Transverse Magnetic (TM), non-zero E z but H z = 0 Transverse Electric (TE), non-zero H z but E z = 0

Transverse ElectroMagnetic wave (TEM) Since E z and H z are 0, TEM wave exists only when A single conductor cannot support TEM

Transverse Magnetic wave (TM) From We can solve for E z and then solve for other components from (1)-(4) by setting H z = 0, then we have Notice that  or j  for TM is not equal to that for TEM.

Eigen values We define Solutions for several WG problems will exist only for real numbers of h or “eigen values” of the boundary value problems, each eigen value determines the characteristic of the particular TM mode.

Cutoff frequency From The cutoff frequency exists when  = 0 or or We can write

a) Propagating mode (1) or and  is imaginary Then This is a propagating mode with a phase constant  :

a) Propagating mode (2) where u is the wavelength of a plane wave with a frequency f in an unbounded dielectric medium ( ,  ) Wavelength in the guide,

a) Propagating mode (3) The wave impedance is then The phase velocity of the propagating wave in the guide is

b) Evanescent mode or Then Wave diminishes rapidly with distance z. Z TM is imaginary, purely reactive so there is no power flow

Transverse Electric wave (TE) From We can solve for H z and then solve for other components from (1)-(4) by setting E z = 0, then we have Notice that  or j  for TE is not equal to that for TEM.

TE characteristics Cutoff frequency f c, , g, and u p are similar to those in TM mode. But Propagating mode f > f c Evanescent mode f < f c

Ex1 Determine wave impedance and guide wavelength (in terms of their values for the TEM mode) at a frequency equal to twice the cutoff frequency in a WG for TM and TE modes.

TM waves in rectangular waveguides Finding E and H components in terms of z, WG geometry, and modes. From Expanding for z-propagating field gets where

Method of separation of variables (1) Assume where X = f(x) and Y = f(y). Substituting XY gives and we can show that

Method of separation of variables (2) Let and then we can write We obtain two separate ordinary differential equations:

General solutions Appropriate forms must be chosen to satisfy boundary conditions

Properties of wave in rectangular WGs (1) 1.in the x-direction E t at the wall = 0  E z (0,y) and E z (a,y) = 0 and X(x) must equal zero at x = 0, and x = a. Apply x = 0, we found that C 1 = 0 and X(x) = c 2 sin(  x x). Therefore, at x = a, c 2 sin(  x a) = 0.

Properties of wave in rectangular WGs (2) 2. in the y-direction E t at the wall = 0  E z (x,0) and E z (x,b) = 0 and Y(y) must equal zero at y = 0, and y = b. Apply y = 0, we found that C 3 = 0 and Y(y) = c 4 sin(  y y). Therefore, at y = a, c 4 sin(  y b) = 0.

Properties of wave in rectangular WGs (3) and therefore we can write

Every combination of integers m and n defines possible mode for TM mn mode. m = number of half-cycle variations of the fields in the x- direction n = number of half-cycle variations of the fields in the y- direction For TM mode, neither m and n can be zero otherwise E z and all other components will vanish therefore TM 11 is the lowest cutoff mode. TM mode of propagation

Cutoff frequency and wavelength of TM mode

Ex2 A rectangular wg having interior dimension a = 2.3cm and b = 1cm filled with a medium characterized by  r = 2.25,  r = 1 a) Find h, f c, and c for TM 11 mode b) If the operating frequency is 15% higher than the cutoff frequency, find (Z) TM11, (  ) TM11, and ( g ) TM11. Assume the wg to be lossless for propagating modes.