*Note: The information below can be referenced to: Carr, J., Practical Antenna
Handbook, Tab Books, Blue Ridge Summit, PA, 1989, ISBN: 0-8306-9270-3.
Edminister, J., Electromagnetics (Schaum’s Outline), McGraw-Hill, New York, NY,
1993, ISBN: 0-07-018993-5.
Chapter 8 Waveguides
The microwave portion of the radio spectrum covers frequencies from about 900 MHz to
300 GHz, with wavelengths in free-space ranging from 33 cm down to 1 mm.
Transmission lines are used at frequencies from dc to about 50 or 60 GHz, but anything
above 5 GHz only short runs are practical, because attenuation increases dramatically as
frequency increases. There are three types of losses in conventional transmission lines:
ohmic, dielectric, and radiation. The ohmic losses are caused by the current flowing in
the resistance of the conductors making up the transmission lines. The skin effect will
increase the resistance at higher frequencies; therefore the losses tend to increase in the
microwave region. Dielectric losses are caused by the electric field acting on the
molecules of the insulator and thus, will cause heating through molecular agitation.
Radiation losses are the loss of energy as the electromagnetic wave propagates away
from the surface of the transmission line conductor.
Losses on long runs of commonly used coaxial transmission line causes concern as low
as 400 MHz. Because of the increased losses the power handling capability decreases at
higher frequencies, therefore, at higher microwave frequencies, or where long runs make
coax attenuation losses unacceptable, or where high power levels causes the coax to
overheat, waveguides are used instead of the transmission lines.
This chapter will describe the propagation characteristics in a single conductor
transmission lines referred to as waveguides. What is a waveguide? Consider the “light
pipe analogy” illustrated in Figure 8.0A. A flashlight serves as our “r-f source,” which
given that light is also an electromagnetic wave is not all that unreasonable. The source
radiates into free-space, and spreads out as a function of distance. The intensity per unit
area at the destination – a wall falls off as a function of distance (D) according to the
inverse square law (1/D2). Now consider the transmission scheme in Figure 8.0B, the
light wave still propagates over a distance D, but is now confined to the interior of a
mirrored pipe. Almost all of the energy coupled to the input end is delivered to the
output end, where the intensity is practically undiminished. The light pipe analogy may
not be the best way to explain the operation of waveguides, but rather a neat summary on
a simple level.
The internal walls of the waveguide are not mirrored surfaces, but instead electrical
conductors. Most waveguides are made of aluminum, brass, or copper. Some
waveguides internal surfaces are electroplated with either gold or silver to reduce ohmic
losses. The gold or silver have lower resistivities than most other metals.
Waveguides are hollow pipes, and may have either circular or rectangular cross sections.
Rectangular are, by far, the most common. These waveguides are used for high
frequency transmission in the gigahertz (microwave) range. The TEM mode cannot
propagate in these single conductor transmission lines. Only higher modes in the form of
transverse electric (TE) and transverse magnetic (TM) modes can propagate in the
waveguide.
Notes by: Debbie Prestridge
1
D
A
Large Diffused Beam
Figure 8.0 Waveguide analogy to light pipe.
B
Small Intense Beam
Transverse and Axial Fields
The waveguide is positioned with the longitudinal direction along the z axis.
Ø
y
r
b
0
0
a
+z
a
x
+z
(a)
(b)
Figure 8.1
Waveguide characteristics:
 guide walls have  c   (perfect conductor)
 dielectric-filled hollow has:
1.  c  0 (perfect conductor)
2.   o r
3.    o  r
4. assumed   0 (no free charge)
Notes by: Debbie Prestridge
2
The dimensions for the cross section are inside dimensions. Figure 8.1(a) is a rectangular
waveguide shown in Cartesian coordinate system; Figure 8.1(b) shows a circular or
cylindrical waveguide of radius a in a cylindrical coordinate system.
The time dependence e jt will be assumed for the electromagnetic field in the dielectric
core. The following expressions for the field vector F (which stands for either E or H),
assuming the wave is propagating in the +z direction.
Rectangular coordinates
F = F(x, y) e-jkz where:
F ( x, y)  Fx ( x, y)a x  Fy ( x, y)a y  Fz ( x, y)a z  F ( x, y)  Fz ( x, y)a z
Cylindrical coordinates
F  f (r ,  )e  jkz where:
F (r ,  )  Fr (r ,  )ar  F (r ,  )a  Fz (r ,  )a z  F (r ,  )  Fr (r ,  )a z
The wave will propagate without attenuation, because the dielectric is lossless (σ = 0).
 2 
Let k   (in rad/m) be the wave number and is constrained to be real and positive.
  
The reason for separating the field vector into a transverse vector component FT and an
axial vector component Fzaz is two-fold. The complete E & H fields in the waveguide
are known once either cartesian component Dz or Hz is known.
Transverse Components from Axial Components
Assume a rectangular coordinate system. Maxwell’s equation yields three scalar
equations:
(1a)
(1b)
(1c)
 x
y
 z
 j     y   j k   x 
x
 y  x
 j     z 

x
y
 j       x  j  k   y 
Maxwell’s equation yields three additional scalar equations with σ = 0:
(2a)
(2b)
(2c)
 x
y
 z
 j       y   j  k   x 
x
 y  x
 j       z 

x
y
 j    x  j k   y 
Notes by: Debbie Prestridge
3
Eliminate Hx between (1a) and (2b) and Hy between (1b) and (2a):
(3a)
y 
(3b)
x 
jk   z j       z

x
k c2  y
k c2
jk   z j       z

y
k c2  x
k c2
*( k c2   2       k 2 ) The parameter kc (also rad/m) functions as a critical wave
number.
Example for kc:
What is “critical” about the number kc?
For propagation through a lossless dielectric, the wave number k must be real, but
k   2       k c2  k o2  k c2
The wave number ko is of a uniform plane wave in the unbounded dielectric at the
given ω. Thus kc is a critical wave number in the sense that a guided wave’s same –
frequency “twin” must have a wave number exceeding kc. Stated otherwise, the
frequency f of the guided wave must exceed the quantity
u
o

2 k c , where uo  1
   is the wave velocity in the unbounded dielectric.
Finally, take (3b) and (3a) substitute into (2a) and (2b):
jk
k c2
jk
x  2
kc
y 
(3c)
(3d)
  z j      z

y
x
k c2
  z j        z

x
y
k c2
It is possible to force either Ez or Hz (but not both) to vanish identically. The nonvanishing axial component will determine all other components via equations (3).
Example 8.1:
Express Maxwell’s equations (1) and (2) in scalar form in cylindrical coordinate
system.
(1)   H    j    E


(2)   E    j    H
*Note: For the curl in cylindrical coordinates refer to –
Notes by: Debbie Prestridge
4
 1   z   
  r   z 
 r 
1 
A 


r   
a
 ar  
 a  
z 
r 
r  r
   z
 z
r 


Equation (1) yields (σ = 0):
j     r 
1  z
 j(k    )
r 
j        j ( k   r ) 
j     z 
 z
r
1 
1  r

r r r 
(i)
(ii)
(iii)
Equation (2) yields:
1  z
 j(k    )
r 
 z
 j          j ( k   r ) 
r
1 
1  r
 j       z 

r r r 
 j       r 
(iv)
(v)
(vi)
Example 8.2:
Using the equations of example 8.1, find all cylindrical field components in terms
of Ez and Hz.
From (i) and (v), with kc as previously defined,
j      1   z j  k    z
(1)
r  
 2
k c2 r  
kc  r
From (ii) and (iv),
r 
j    1   z j k    z
 2
k c2 r  
kc  r
(2)
From (1) and (i),
  
j    1   z j k  1   z
 2
k c2 r  r
kc r  
(3)
From (1) and (ii),
Notes by: Debbie Prestridge
5
j    1   z j    1   z

k c2 r  
k c2 r  r
Propagation Modes in Waveguide
 
(4)
In a waveguide a signal will propagate as an electromagnetic wave. Even in a
transmission line the signal propagates as a wave because the current in motion down the
line gives rise to the electric and magnetic fields that behaves as an electromagnetic field.
The transverse electromagnetic (TEM) field is the specific type of field found in
transmission lines. We also know that the term “transverse” implies to things at right
angles to each other, so the electric and magnetic fields are perpendicular to the direction
of travel. These right angle waves are said to be “normal” or “orthogonal “to the
direction of travel.
The boundary conditions that apply to waveguides will not allow a TEM wave to
propagate. However, the wave in the waveguide will propagate through air or inert gas
dielectric in a manner similar to free space propagation, the phenomena is bounded by the
walls of the waveguide and that implies certain conditions that must be met. The
boundary conditions for waveguides are:
1. The electric field must be orthogonal to the conductor in order to exist at the
surface of that conductor.
2. The magnetic field must not be orthogonal to the surface of the waveguide.
The waveguide has two different types of propagation modes to satisfy these boundary
conditions:
1. TE – transverse electric (Ez = 0)
2. TM – transverse magnetic (Hz = 0)
The transverse electric field requirement means that the E-field must be perpendicular to
the conductor wall of the waveguide. This requirement can be met with proper coupling
at the input end of the waveguide. A vertically polarized coupling radiator will provide
the necessary transverse field.
One boundary condition will require the magnetic (H) field not to be orthogonal to the
conductor surface. Since it is at right angles to the E-field, the requirement will be met.
The planes that are formed by the H-field will be parallel to the direction of propagation
and to the surface.
Waveguide Impedances
For any transverse electromagnetic wave , the wave impedance (in ohms) is defined as
being approximately equal to the ratio of the electric and magnetic fields, and converges
as a function of frequency to the intrinsic impedance of the dielectric:

Notes by: Debbie Prestridge
E
H
(4)
6
For a TE mode waveguide, (1a) & (1b) imply:
2
E
2
 x
2
 y
2
 
Or
2
2
    
 
2

   y   x   
 H
 k  
 k  
2
 
(5)
k 
Equation (4) involves only lengths of two-dimensional vectors, so η must be independent
of the coordinate system. Example 8.3 will confirm the value of ηTE by recalculating it in
cylindrical coordinates. Example 8.4 shows (using rectangular coordinates) that:
 
k 
(6)
    
Example 8.3:
Calculate ηTE from the field components in cylindrical coordinates.
Ez ≡ 0, (iv) and (v) of Example 8.1 gives:
 j       r 
1  z
 j(k    )
r 
 j          j ( k   r ) 
 z
r
H   r  
2
 
Notes by: Debbie Prestridge
(v)
2

 k  
2  k
        r
 
   
    


2
2
(iv)
E
H

2

k 
  

2
  
k 
7
Example 8.4
Calculate ηTM from the field of components in rectangular coordinates.
Hz ≡ 0, (2a) and (2b):
 x
 j    x  j k   y 
(2a)
y
 z
(2b)  j     y   j k   x 
x
2
Solution:
x
2
 y
2
2
 k 
k
2
2
      y   x  or      
  
    
 
E
H

k 
  
Determination of the Axial Fields
All that remains for a complete description of TE and TM modes is the
determination of the respective axial fields:
Fz = Hz
TE
Fz = Ez
TM
The cartesian coordinate Fz e  jkz of F (in either rectangular or cylindrical coordinates),
 2F 1  2F
must satisfy the scalar wave equation
,

 z2 u2  t 2
 2  Fz e  jkz     2      Fz e  jkz 
(7)
And the appropriate boundary conditions which are inferred from the boundary
conditions on the components of FT. *Note: Transverse components such as   e  jkz
are not cartesian components and do not obey a scalar wave equation.
Explicit Solutions for TE Modes of a Rectangular Guide
The wave equation (7) becomes:
 2 Hz  2 Hz

 k c2 H z  0
 x2
 y2
2
2
This was previously defined as k c
. Solve by using separation of
  2        k 
variables:
Notes by: Debbie Prestridge
8

 z ( x, y)    x cos k x x   x sin k x x  y cos k y y   y sin k y y

(8)
where k x2  k y2  k c2 . The separation constants kx and ky are determined by the boundary
conditions. Consider first the x-conditions  y (0, y)   y (a , y)  0 ; in view of (3a) y 
jk   z j       z
and Ez ≡ 0 these translate into:

x
k c2  y
k c2
 z
x

x0
 z
x
0
x a
Apply these conditions to: (8) -

 z ( x, y)    x cos k x x   x sin k x x  y cos k y y   y sin k y y
This will result in Bx = 0 and sin k x a  0 or k x 
m
a
m  0,1,2,...

and by
n
n  0,1,2,...
b
Each pair of nonnegative integers (m, n) –with the exception of (0, 0) which will
result in a trivial solution-identifies a distinct TE mode, indicated as TEmn. This mode
has the axial field
m x
n y
 zmn ( x , y )   mn cos
cos
(9)
a
b
And the transverse field is obtained through (3) – (refer to pages 2-3 of these notes). The
critical wave number for TEmn is:
symmetry, the boundary conditions in y force By = 0 and k y 
 m   n 
k cmn  
 

 a   b 
This is in terms of which the wave number and the wave impedance for TEmn are:
2
k mn  
mn 
2
      k
2
2
cmn
(10)
      k c2mn
(11)
 

2
*Note: m, n are integers that define the number of half wavelengths that will fit in the (a)
and (b) dimensions, respectively; a, b are the waveguide dimensions. (see Figure 8.2)
Notes by: Debbie Prestridge
9
a
b
Figure 8.2 Rectangular waveguide (end view)
Example 8.5:
This example will show for the TMmn modes of a rectangular waveguide and it will show
that kcTMmn = kcTEmn. The subscripts TE and TM can be dropped from all modal
parameters of rectangular guides save the wave impedance.
Obtain the analogues of (9) – (12) for TMmn.
Analogous to (8),

 z ( x, y)  Cx cos k x x  Dx sin k x x C y cos k y y  Dy sin k y y
2
where
k x2  k y2  k c2   2       k 
But now the boundary conditions are:
 z (0, y )   z (a , y )  0 and  z (0, y )   z ( x, b)  0
This will require that:
m
n
Cx  0 k x 
Cy  0 k y 
a
b
where m, n 1, 2, 3,.... Note that neither m nor n is zero in a TM mode.
The required formulas are:
m x
n y
 zmn ( x , y )   mn sin
sin
a
b
 m x 
 n y 
k cmn  
 
  k cmn
 a 
 b 
2
k  mn  k  mn
 
mn
Notes by: Debbie Prestridge
k 
  

(1)
2
(2)
(3)
(4)
10
Velocity and Wavelength in Waveguides:
Figure 8.3 illustrates the geometry for two wave components simplified for sake of
illustration. There are three different wave velocities to consider with respect to
waveguides: free space velocity (c), group velocity (Vg), and phase velocity (Vp).
The space velocity of propagation in unbounded free-space, i.e., the speed of light
(c = 3 * 108 m/s).
The group velocity is the straight line velocity of propagation of the wave down the
center-line (z-axis) of the waveguides. The value of Vg is always less than c, because the
actual path length taken as the wave bounces back and forth is longer than the straight
line path (i.e., path ABC is longer than path AC). The relationship between c and Vg is:
Vg = c sin a
*Note: Vg is the group velocity in (m/s), c is the free space velocity (3 * 108 m/s), and a
is the angle of incidence in the waveguide.
The phase velocity is the velocity of propagation of the spot on the waveguide wall where
the wave impinges (e.g., point “B” in Figure 8.4). This velocity is actually faster than
both the group velocity and the speed of light. The relationship between the phase and
group velocities can be seen in the “Beach analogy.” If we consider an ocean beach that
waves will arrive from offshore at an angle other than 90°, meaning the arriving wave
fronts will not be parallel to the shore. The arriving waves at Vg as it hits the shore will
strike a point down the beach first, and the “point of strike” races up the beach at a faster
phase velocity, Vp, that is faster than Vg. In a microwave waveguide the phase velocity
can be greater than c.
Antenna
a
+
b
c
This end is open!
Figure 8.3 Antenna radiator in a capped waveguide.
Notes by: Debbie Prestridge
11
B
A

4
a
a

C
Vg
0
g
4
Figure 8.4 Wave propagation in a waveguide
Mode Cutoff Frequencies
The propagation of signals in a waveguide depends in part upon the operating frequency
of the applied signal. The angle of incidence made by the plane wave to the waveguide
wall is a function of frequency. As the frequency drops, the angle of incidence increases
towards 90°.
In practice one may deal with frequencies and not wave numbers. It is desirable to
replace the concept of the critical wave number (kc) by one of the cutoff frequency (fc).
This was accomplished in the example for (kc) (refer to page 3 of these notes):
fc 
uo
k 
2 c
1
2  
(13)
kc
In terms of the cutoff frequency fc and the operating frequency f 

 fc
2
(10), (11), and (12) will become:
uo
f cmn 
2
k cmn 
2
uo
 m 2  n 
   
 a
 b
f 2  f c2mn
2
(Rectangular waveguide)
or 
nn
  mn 
Notes by: Debbie Prestridge

o
 f c mn 

1 
 f 
o
 f c mn 

1 
 f 
2
2
(10 bis)
(11 bis)
(12 bis)
12
where  o 
uo
is the wavelength of an imaginary uniform plane wave at the operating
f
frequency and where o 

is the plane wave impedance of the lossless dielectric.

The second form of (11 bis) exhibits the relation between the operating wavelength λo
and the actual guide wavelength λmn. For TMmn waves, (12 bis) is replaced by [see (6)]

mn
 f 2
  o 1   cmn 
 f 
(14)
The phase velocity of a TEmn or TMmn wave is given by:
umn  
mn
f
u0
 f cmn 
1 

 f 
2
(15)
The meaning of cutoff is made particularly clear in (15). As the operating frequency
drops to the cutoff frequency, the velocity becomes infinite. This is a characteristic, not
of wave propagation, but of diffusion (instantaneous spread of exponentially small
disturbances).
Example 8.6
Define the notion of cutoff wavelength.
The cutoff wavelength λc is the wavelength of an unguided plane wave whose
frequency is the cutoff frequency; i.e., λc* fc = uo
Is the cutoff wavelength an upper limit on the guide wavelength, just as the cutoff
frequency is a lower limit on the guide frequency?
uo
No; in fact, the formula  mn 
shows that an (m, n) mode can
2
2
f  f cmn
propagate with any guide wavelength greater than λ.
Dominant Mode
The dominant mode of any waveguide is that of the lowest cutoff frequency. Now, for a
rectangular guide, the coordinate system may always be oriented to make a ≥ b.
Notes by: Debbie Prestridge
13
2
2
 m  n 
Since f cmn       for either TE or TM, but neither m nor n can vanish in TM,
 a   b
the dominant mode of a rectangular guide is invariably TE10, with
f c10 
uo
2a
 10 
o
1  
o
2a 
2

2
k10
u`10   10 f
10 
 10

o o
From (9), Ez10 ≡ 0, and the equations (Transverse Components from Axial
Components Section):
 z10   10 cos
x
a
 x10  0
 2a 
x
  10 sin
 x10  j 
a
  10 
 2a 
x
  10 sin
 y10    10  x10   jo 
a
 o
(16)
 y10  0
For H10 real, the three nonzero field components have the time-domain expressions:
  x
 z10   10 cos 
 cos  t  k10 z 
 a 
 2a 
  x
  10 sin 
 x10   
 sin t  k10 z
 a 
  10 
(17)
 2a 
  x
  10 sin 
 y10   o 
 sin  t  k10 z
 a 
  10 
Plots of the dominant-mode fields (17) at t = 0 are given in Figs. 8.5 and 8.6.
Both  y and
 x vary as sin x a  . This is indicated in Figure 8.5 by drawing the
lines of E close together near x = a/2 and far apart near x = 0 and x = a. The lines of H
are shown evenly spaced because there is no variation with y. This same line-density
convention is used to indicated the local value of E   y in Figure 8.6(a) and of
H   2x   2z
in Figure 8.6(b). Notice that the lines of H are closed curves (div H = 0); the H field may
be considered as circulating about the perpendicular displacement current density JD.
Notes by: Debbie Prestridge
14
y
y
H
E
b
b
x
0
0
a
Figure 8.5 Transverse cross section
a
z    10 4
(  k10 z   2)
 10
 10
4
2
3  10
4
-z
0
y
H
b
E
0
 10
 10
4
2
3  10
4
a
-z
x
(b) □ y = const
(a) □ x = a/2
Figure 8.6 Longitudinal cross sections
Notes by: Debbie Prestridge
15
x
Notes by: Debbie Prestridge
16