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© 2011 Keith W. Whites
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:
x
y
z (^) a
b μ , ε
Metal walls
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),
- Transverse Electric (TE), or
- Transverse Magnetic (TM).
Proceeding from the Maxwell curl equations:
ˆ ˆ ˆ
x y z
x y z
E j H j H x y z E E E
∇ × = − ωμ ⇒ ∂^ ∂^ ∂ = − ωμ
or x ˆ : z y x
E E
j H y z
y^ ˆ : E^ z Ex j Hy x z
− ⎛⎜^ ∂ − ∂ ⎞⎟= − ωμ
⎝ ∂^ ∂ ⎠
z^ ˆ : y^ x z
E E
j H x y
However, the spatial variation in z is known so that
j z e (^) j e j z z
β
− ∂ (^) = − − ∂ Consequently, these curl equations simplify to
z y x
E (^) j E j H y
(3.3a),(1)
z x y
E (^) j E j H x
(3.3b),(2)
y (^) x z
E E
j H x y
(3.3c),(3)
2 y^ z^ z c
j E H H k x y
= − ⎛⎜^ ωε ∂^ +β ∂ ⎞⎟
⎝ ∂^ ∂ ⎠
(3.5b),(8)
2 x^ z^ z c
j E H E k x y
= −^ ⎛⎜^ β ∂^ +ωμ∂ ⎞⎟
⎝ ∂^ ∂ ⎠
(3.5c),(9)
2 y^ z^ z c
j E H E k y x
= ⎛⎜^ − β ∂^ +ωμ∂ ⎞⎟
⎝ ∂^ ∂ ⎠
(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 E (^) z 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 E (^) z = 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 2 2 ,
c mn ( 0)
m n m n k a b m^ n
= ⎛^ π^ ⎞^ +⎛^ π^ ⎞ =
Therefore, β^ =^ β mn =^ k^^2 −^ kc mn^2 , (12)
These m and n indices indicate that only discrete solutions for the transverse wavenumber ( k (^) c ) 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 E (^) z = Hz = 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
For an X-band rectangular waveguide, the cross-sectional dimensions are a = 2.286 cm and b = 1.016 cm. Using (13):
TE m , n Mode Cutoff Frequencies m n fc , mn (GHz) 1 0 6. 2 0 13. 0 1 14. 1 1 16.
In the X-band region (≈ 8.2-12.5 GHz), only the TE 10 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 TE 11 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 E (^) z ≠ 0.
The expression for the cutoff frequencies of TM modes in a rectangular waveguide 2 2 ,
c mn 2 f m^ n m n a b
= ⎛^ ⎞^ + ⎛^ ⎞ =
⎜⎝ ⎟⎠ ⎜⎝ ⎟⎠ …^ (14)
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:
TM m , n Mode Cutoff Frequencies m n fc , mn (GHz) 1 1 16. 1 2 30. 2 1 19.
Therefore, no TM modes can propagate in an X-band rectangular waveguide when f < 16.156 GHz.
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.
