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Lecture 10

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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 find

j Ez

1

H z

Hx

j

j H x

j

y

x

j Ez

j H z

2

2 Hx 2

y

x

or,

Hx

H z

j Ez

y

x

kc2

where kc2 k 2 2 and k 2 2 .

Similarly, we can show that

(3.5a),(7)

(3.6)

Whites, EE 481

Lecture 10

Hy

H z

j Ez

x

y

kc2

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(3.5b),(8)

Ex

j Ez

H z

y

kc2 x

(3.5c),(9)

Ey

E

H z

j

z

y

x

kc2

(3.5d),(10)

Most important point: From (7)-(10), we can see that all

transverse components of E and H can be determined from

only the axial components Ez and H z . It is this fact that allows the mode designations TEM, TE, and TM.

Furthermore, we can use superposition to reduce the complexity of the solution by considering each of these mode types

separately, then adding the fields together at the end.

TE Modes and Rectangular Waveguides

A transverse electric (TE) wave has Ez 0 and H z 0 . Consequently, all E components are transverse to the direction of propagation. Hence, in (7)-(10) with Ez 0 , then all

transverse components of E and H are known once we find a

solution for only H z . Neat!

Whites, EE 481

Lecture 10

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For a rectangular waveguide, the solutions for Ex , E y , H x , H y , and H z are obtained in Section 3.3 of the text. The solution and the solution process are interesting, but not needed in this course.

What is found in that section is that

m n

a b

2

kc ,mn

Therefore,

2

m, n 0,1,

( m n 0)

mn k 2 kc2,mn

(11)

(12)

These m and n indices indicate that only discrete solutions for the transverse wavenumber (kc) are allowed. Physically, this occurs because we’ve bounded the system in the x and y

directions. (A vaguely similar situation occurs in atoms, leading to shell orbitals.)

Notice...