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NOTES for Waveguide I
This lecture covers 7.1 and 7.
1. Rectangular waveguides
2. Transverse magnetic TM modes
A waveguide is used to transmit high-frequency EM signals.
Waveguides are better than T-lines at higher frequencies (3-
300GHz). Transmission lines become inefficient at those
frequencies due to the skin effect and due to dielectric losses. More
generally, waveguide analysis is important when wavelengths are
comparable to the cross-sectional dimension of the waveguide. In
that case a single voltage or current is not adequate to describe the
guide’s signal. Another difference between waveguides and T-
lines is that T-lines support TEM waves at any frequency. A
waveguide, in general, has a cut-off frequency (a minimum
frequency that will propagate).
1. Rectangular waveguides
The simplest waveguide is a metal rectangular waveguide.
We begin by assuming that the waveguide is filled with a
lossless material (e.g. air) as shown below.
We will (eventually) assume that the waveguide carries
energy in the z direction.
The E -field and the H -field each have three components (x,y
and z) and we will use Maxwell’s equations to describe each
of these components separately.
Waveguides: Lecture 1
We will use the phasor form of Maxwell’s equations in a
charge-free and current-free environment.:
s s
s s
H j E
E j H
The subscript s indicates a phasor.
If we write these fields in terms of their components, six
equations result. After some vector calculus we find:
zs
xs ys
ys
xs zs
xs
ys zs
zs
xs
ys
ys
xs zs
xs
ys zs
j E
y
H
x
H
j E
x
H
z
H
j E
z
H
y
H
j H
y
E
x
E
j H
x
E
z
E
j H
z
E
y
E
We can then solve for each of the four transverse components
(E
x
, E
y
, H
x and H y ) in terms of the two z-directed components
(E
z and H z ). For instance solve the fifth equation for E y and
substitute into the first equation. This will result in an
equation for H x in terms of H z and E z
If the direction of EM wave propagation is assumed along the
+z direction, it is fair to assume that all field components
follow:
j z
is it
j z
is it
H x yz H x y e
E x yz E x y e
( , , ) ( , )
( , , ) ( , )
The subscript i can be x, y or z. (Again-the subscript s
indicates phasors.) Similar to the approach we took analyzing
T-lines and UPWs, we refer to β as a propagation constant
whose value is to be determined. (The value of β will not
have the same value as that of a wave propagating in an
unbounded medium.)
This substitution leads to four equations for the transverse (x
and y) components of the electric and magnetic fields.
u
z
u
z
u
y
z
u
z
u
x
z
u
z
u
y
z
u
z
u
x
y
j H
x
j E
H
x
j H
y
j E
H
x
j H
y
j E
E
y
j H
x
j E
E
2
2 2
2
2
2 2
2
2 2 2
2
2 2 2 2
In general, we can classify EM waves from the above
expressions:
- TEM waves, where E z
=H
z =0. (These do not exist in a
metal, rectangular waveguide.)
- Transverse electric (TE) modes, where E z
=0, H
z
- Transverse magnetic (TM) modes, where H z
=0, E
z
- HE modes, where H z
0, E
z
We will be concerned with 2 and 3, TE and TM waves.
Waveguides: Lecture 1
2 2 2 2
2 2
2 2
andtherefore 1 ( / ) 1 ( / )
1
where
2
andthecut-offfrequencyisfound from
2
f f f f
c
c
b
n
a
c m
f
b
n
a
m
f
c c u c
c
u
Below the cut-off frequency f c , β is imaginary (or >0) so the
wave decays with z and is known as an evanescent wave.
The phase velocity of the guide for a given mode (m, n ) is
2
1
f
f
c
u
c
p
The intrinsic wave impedance of the mode is defined by
the ratio E x
/H
y
2 2
· 1 1
f
f
f
f
Z
c
u
c
x y
TM
mn
Update: October 12, 2006 (10:20 am)8/-
Minor typo correction, October 18, 2006
Corrections October 19, 2006, November 2, 2006, November 8, 2006, October 10, 2007
Correction October 18, 2007, minor typos corrected October 1, 2008
Waveguides: Lecture 1
