ENE 429 Antenna and Transmission Lines Theory Lecture 10 Waveguides

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”

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 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 or where

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 Ex, Ey, Hx, and Hy 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 Ez or Hz Transverse Magnetic (TM), non-zero Ez but Hz = 0 Transverse Electric (TE), non-zero Hz but Ez = 0

Transverse ElectroMagnetic wave (TEM) Since Ez and Hz are 0, TEM wave exists only when A single conductor cannot support TEM

Transverse Magnetic wave (TM) From We can solve for Ez and then solve for other components from (1)-(4) by setting Hz = 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) Wavelength in the guide, where u is the wavelength of a plane wave with a frequency f in an unbounded dielectric medium (, )

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

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

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

TE characteristics Cutoff frequency fc,, g, and up are similar to those in TM mode. But Propagating mode f > fc Evanescent mode f < fc

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) in the x-direction Et at the wall = 0 Ez(0,y) and Ez(a,y) = 0 and X(x) must equal zero at x = 0, and x = a. Apply x = 0, we found that C1 = 0 and X(x) = c2sin(xx). Therefore, at x = a, c2sin(xa) = 0.

Properties of wave in rectangular WGs (2) 2. in the y-direction Et at the wall = 0 Ez(x,0) and Ez(x,b) = 0 and Y(y) must equal zero at y = 0, and y = b. Apply y = 0, we found that C3 = 0 and Y(y) = c4sin(yy). Therefore, at y = a, c4sin(yb) = 0.

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

TM mode of propagation Every combination of integers m and n defines possible mode for TMmn 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- For TM mode, neither m and n can be zero otherwise Ez and all other components will vanish therefore TM11 is the lowest cutoff mode.

Cutoff frequency and wavelength of TM mode

Ex2 A rectangular wg having interior dimension a = 2 Ex2 A rectangular wg having interior dimension a = 2.3cm and b = 1cm filled with a medium characterized by r = 2.25, r = 1 Find h, fc, and c for TM11 mode 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.