481 lecture10

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 z-
direction:
y
Metal walls

b

μ, ε
x
z a

and that the wave is propagating in z as e − j β 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),

© 2011 Keith W. Whites


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

Proceeding from the Maxwell curl equations:
ˆ
x yˆ ˆ
z
∂ ∂ ∂
∇ × E = − jωμ H ⇒ = − jωμ H
∂x ∂y ∂z
Ex Ey Ez
∂Ez ∂E y
or ˆ
x: − = − jωμ H x
∂y ∂z
⎛ ∂E ∂E ⎞
y : − ⎜ z − x ⎟ = − jωμ H y
ˆ
⎝ ∂x ∂z ⎠
∂E y ∂Ex
ˆ
z: − = − jωμ H z
∂x ∂y

However, the spatial variation in z is known so that
∂ ( e− jβ z )
= − j β ( e− jβ z )
∂z
Consequently, these curl equations simplify to
∂Ez
+ j β E y = − jωμ H x (3.3a),(1)
∂y
∂E
− z − j β Ex = − jωμ H y (3.3b),(2)
∂x
∂E y ∂Ex
− = − jωμ H z (3.3c),(3)
∂x ∂y


We can perform a similar expansion of Ampère’s equation
∇ × H = jωε E to obtain
∂H z
+ j β H y = jωε Ex (3.4a),(4)
∂y
∂H z
− jβ H x − = jωε E y (3.4b),(5)
∂x
∂H y ∂H x
− = jωε 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 ⎞
Hx = + jβ Ey ⎟
ωμ ⎜ ∂y
⎝ ⎠
Substituting for Ey from (5) we find
j ⎡ ∂Ez 1 ⎛ ∂H z ⎞ ⎤
Hx = ⎢ + jβ ⎜ − jβ H x − ⎟⎥
ωμ ⎣ ∂y jωε ⎝ ∂x ⎠ ⎦
j ∂Ez β2 j β ∂H z
= + 2 Hx − 2
ωμ ∂y ω με ω με ∂x
j ⎛ ∂Ez ∂H z ⎞
or, Hx = ⎜ ωε −β ⎟ (3.5a),(7)
kc2 ⎝ ∂y ∂x ⎠
where kc2 ≡ k 2 − β 2 and k 2 = ω 2 με . (3.6)

Similarly, we can show that


j ⎛ ∂Ez ∂H z ⎞
Hy = − ωε +β
kc2 ⎜ ∂y ⎟
(3.5b),(8)
⎝ ∂x ⎠
− j ⎛ ∂Ez ∂H z ⎞
Ex = ⎜ β + ωμ ⎟ (3.5c),(9)
kc2 ⎝ ∂x ∂y ⎠
j ⎛ ∂E ∂H z ⎞
Ey = − β z + ωμ
kc2 ⎜ ∂x ⎟
(3.5d),(10)
⎝ ∂y ⎠

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!


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
⎛ mπ ⎞ ⎛ nπ ⎞
2 2
m, n = 0,1,…
kc ,mn = ⎜ ⎟ +⎜ ⎟ (11)
⎝ a ⎠ ⎝ b ⎠ ( m = n ≠ 0)

Therefore, β = β mn = k 2 − kc2,mn (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 something important. From (11), we find that m = n = 0
means that kc ,00 = 0 . In (7)-(10), this implies infinite field
amplitudes, which is not a physical result. Consequently, the
m = n = 0 TE or TM modes are not allowed.

One exception might occur if Ez = H z = 0 since this leads to
indeterminate forms in (7)-(10). However, it can be shown that
inside hollow metallic waveguides when both m = n = 0 and


Ez = H z = 0 , then E = H = 0 . This means there is no TEM
mode.

Consequently, EM waves will propagate only when the
frequency is “large enough” since there is no TEM mode.
Otherwise β will be imaginary ( β → − jα ), leading to pure
attenuation and no propagation of the wave e − j β z → e −α z .

This turns out to be a general result. That is, for a hollow
conductor, EM waves will propagate only when the frequency is
large enough and exceeds some lower threshold. This minimum
propagation frequency is called the cutoff frequency f c ,mn .

It can be shown that guided EM waves require at least two
distinct conductors in order to support wave propagation all the
way down to 0+ Hz.

The cutoff frequencies for TE modes in a rectangular waveguide
are determined from (12) with β = 0 to be
1 ⎛ mπ ⎞ ⎛ nπ ⎞
2 2
m, n = 0,1,…
f c ,mn = ⎜ ⎟ +⎜ ⎟ (3.84),(13)
2π με ⎝ a ⎠ ⎝ b ⎠ (m = n ≠ 0)
In other words, these are the frequencies where β mn = 0 and
wave propagation begins when the frequency slightly exceeds
f c ,mn .


For an X-band rectangular waveguide, the cross-sectional
dimensions are a = 2.286 cm and b = 1.016 cm. Using (13):

TEm,n Mode Cutoff Frequencies
m n fc,mn (GHz)
1 0 6.562
2 0 13.123
0 1 14.764
1 1 16.156

In the X-band region ( ≈ 8.2-12.5 GHz), only the TE10 mode can
propagate in the waveguide regardless of how it is excited.
(We’ll also see shortly that no TM modes will propagate either.)
This is called single mode operation and is most often the
preferred application for hollow waveguides.

On the other hand, at 15.5 GHz any combination of the first
three of these modes could exist and propagate inside a metal,
rectangular waveguide. Which combination actually exists will
depend on how the waveguide is excited.

Note that the TE11 mode (and all higher-ordered TE modes)
could not propagate. (We’ll also see next that no TM modes will
propagate at 15.5 GHz either.)


TM Modes and Rectangular Waveguides

Conversely to TE modes, transverse magnetic (TM) modes have
H z = 0 and Ez ≠ 0 .

The expression for the cutoff frequencies of TM modes in a
rectangular waveguide
⎛ mπ ⎞ ⎛ nπ ⎞
2 2
1
f c ,mn = ⎜ ⎟ +⎜ ⎟ m, n = 1, 2… (14)
2π με ⎝ a ⎠ ⎝ b ⎠
is very similar to that for TE modes given in (13).

It can be shown that if either m = 0 or n = 0 for TM modes,
then E = H = 0 . This means that no TM modes with m = 0 or
n = 0 are allowable in a rectangular waveguide.

For an X-band waveguide:

TMm,n Mode Cutoff Frequencies
m n fc,mn (GHz)
1 1 16.156
1 2 30.248
2 1 19.753

Therefore, no TM modes can propagate in an X-band
rectangular waveguide when f < 16.156 GHz.


Dominant Mode

Note that from 6.56 GHz ≤ f ≤ 13.12 GHz in the X-band
rectangular waveguide, only the TE 10 mode can propagate. This
mode is called the dominant mode of the waveguide.

See Fig 3.9 in the text for plots of the electric and magnetic
fields associated with this mode.

TEM Mode

The transverse electric and magnetic (TEM) modes are
characterized by Ez = 0 and H z = 0 .

In order for this to occur, it can be shown from (3.4) and (3.5)
that it is necessary for f c = 0 . In other words, there is no cutoff
frequency for waveguides that support TEM waves.

Rectangular, circular, elliptical, and all hollow, metallic
waveguides cannot support TEM waves.

It can be shown that at least two separate conductors are
required for TEM waves. Examples of waveguides that will


allow TEM modes include coaxial cable, parallel plate
waveguide, stripline, and microstrip.

Microstrip is the type of microwave circuit interconnection that
we will use in this course. This “waveguide” will support the
“quasi-TEM” mode, which like regular TEM modes has no non-
zero cutoff frequency.

However, if the frequency is large enough, other modes will
begin to propagate on a microstrip. This is usually not a
desirable situation.

481 lecture10

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