Waveguide

Whites, EE 481

Lecture 10

Page 1 of 10

Lecture 10: TEM, TE, and TM Modes for
Waveguides. Rectangular Waveguide.
We will now generalize our discussion of transmission lines by considering EM waveguides. These are “pipes” that guide EM waves. Coaxial cables, hollow metal pipes, and fiber optical cables are all examples of waveguides.
We will assume that the waveguide is invariant in the zdirection: y Metal walls b ,  x z

a

and that the wave is propagating in z as e  j  z . (We could also have assumed propagation in –z.)

Types of EM Waves
We will first develop an extremely interesting property of EM waves that propagate in homogeneous waveguides. This will lead to the concept of “modes” and their classification as
 Transverse Electric and Magnetic (TEM),
© 2012 Keith W. Whites

Whites, EE 481




Lecture 10

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Transverse Electric (TE), or
Transverse Magnetic (TM).

Proceeding from the Maxwell curl equations:
ˆ
ˆ
ˆ
x y z
  E   j H 

or


x
Ex


y
Ey


  j H
z
Ez

Ez E y

  j H x
y
z
 E E 
ˆ
y :   z  x    j H y
z 
 x
E y Ex
ˆ

  j H z z: x
y

ˆ x: However, the spatial variation in z is known so that
  e j z 
  j   e j z 
z
Consequently, these curl equations simplify to
Ez
 j  E y   j H x
y
E
 z  j  Ex   j H y
x
E y Ex

  j H z
x
y

(3.3a),(1)
(3.3b),(2)
(3.3c),(3)

Whites, EE 481

Lecture 10

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We can perform a similar expansion of Ampère’s equation
  H  j E to obtain
H z
 j  H y  j Ex
(3.4a),(4)
y
H z
 j H x 
 j E y
(3.4b),(5)
x
H y H x

 j Ez
(3.5c),(6)
x
y
Now, (1)-(6) can be manipulated to produce simple algebraic equations for the transverse (x and y) components of E and H .
For example, from (1):

j  Ez
 j Ey 
Hx 
  y


Substituting for Ey from (5) we