# 8.9 TE Modes in Rectangular Waveguide

In the TM case, where the wave has no magnetic field component along the z direction (i.e., H˜z=0), we started our treatment in the preceding section by obtaining a solution for E˜z, and then we used it to derive expressions for the tangential components of E˜ and H˜. For the TE case, the same basic procedure can be applied, except we reversed the roles of E˜z and H˜z. Such a process leads to:

E˜x=j ω μ k c 2 n π b H 0 cos m π x a sin n π y b e−j β z, (8.110a)
E˜y=−j ω μ k c 2 m π a H 0 sin m π x a cos n π y b e−j β z, (8.110b)
H˜x=j β k c 2 m π a H 0 sin m π x a cos n π y b e−j β z, (8.110c)
H˜y=j β k c 2 n π b H 0 cos m π x a sin n π y b e−j β z, (8.110d)
H˜z=H 0 cos m π x a cos n π y b e−j β z, (8.110e)

and of course, E˜z=0. The expressions for f m n, β, and u p given earlier by Eqs. (8.106), (8.107), and (8.108) remain unchanged.

Another difference between the TE and TM modes relates to the expression for the wave impedance. For TE,

Z TE=E˜x H˜y=−E˜y H˜x=η 1−(f m n/f)2. (8.111)

A summary of the expressions for the various wave attributes of TE and TM modes is given in Table 8-3. As a reference, corresponding expressions for the TEM mode on a coaxial transmission line are included as well.

# Table 8-3 Wave properties for TE and TM modes in a rectangular waveguide with dimensions a×b that are filled with a dielectric material with constitutive parameters ε and μ. The TEM case—shown here for reference—pertains to plane-wave propagation in an unbounded medium.

Table 8 3 Full Alternative Text

# Example 8-9: Cutoff Frequencies

For a hollow rectangular waveguide with dimensions a=3 cm and b=2 cm, determine the cutoff frequencies for all modes up to 20 GHz. Over what frequency range will the guide support the propagation of a single dominant mode?

Solution: A hollow guide has μ=μ 0 and ε=ε 0. Hence, u p 0=1/μ 0 ε 0=c. Application of Eq. (8.106) gives the cutoff frequencies shown in Fig. 8-24, which start at 5 GHz for the TE 10 mode. To avoid all other modes, the frequency of operation should be restricted to the 5–7.5 GHz range.

# Figure 8-24 Cutoff frequencies for TE and TM modes in a hollow rectangular waveguide with a=3 cm and b=2 cm (Example 8-9).

Figure 8 24 Full Alternative Text