# 8.7 General Relations for E and H

The purpose of the next two sections is to derive expressions for E and H for the TE and TM modes in a rectangular waveguide and to examine their wave properties. We choose the coordinate system shown in Fig. 8-22, where the propagation occurs along z ˆ. For TE modes, the electric field is transverse to the direction of propagation. Hence, E may have components along x ˆ and y ˆ but not along z ˆ. In contrast, H has a z ˆ-directed component and may have components along either x ˆ or y ˆ or along both. The converse is true for TM modes.

# Figure 8-22 Waveguide coordinate system.

Our solution procedure consists of four steps:

1. Maxwell’s equations are manipulated to develop general expressions for the phasor-domain transverse field components E˜x, E˜y, H˜x, and H˜y in terms of E˜z and H˜z. When specialized to the TE case, these expressions become functions of H˜z only, and the converse is true for the TM case.

2. The homogeneous wave equations given by Eqs. (7.15) and (7.16) are solved to obtain valid solutions for E˜z (TM case) and H˜z (TE case) in a waveguide.

3. The expressions derived in step 1 are then used to find E˜x, E˜y, H˜x, and H˜y.

4. The solutions obtained in step 3 are analyzed to determine the phase velocity and other properties of the TE and TM waves.

The intent of the present section is to realize the stated goals of step 1. We begin with a general form for the E and H fields in the phasor domain:

E˜=x ˆ E˜x+y ˆ E˜y+z ˆ E˜z, (8.84a)
H˜=x ˆ H˜x+y ˆ H˜y+z ˆ H˜z. (8.84b)

In general, all six components of E˜ and H˜ may depend on (x, y, z), and while we do not yet know how they functionally depend on (x, y), our prior experience suggests that E˜ and H˜ of a wave traveling along the +z direction should exhibit a dependence on z of the form e−j β z, where β is a yet-to-be determined phase constant. Hence, we adopt the form

E˜x(x,y,z)=e˜x(x,y)e−j β z, (8.85)

where e˜x(x,y) describes the dependence of E˜x(x,y,z) on (x,y) only. The form of Eq. (8.85) can be used for all other components of E˜ and H˜ as well. Thus,

E˜=(x ˆ e˜x+y ˆ e˜y+z ˆ e˜z)e−j β z, (8.86a)
H˜=(x ˆ h˜x+y ˆ h˜y+z ˆ h˜z)e−j β z. (8.86b)

The notation is intended to clarify that, in contrast to E˜ and H˜, which vary with (x,y,z), the lower case e˜ and h˜ vary with (x,y) only.

In a lossless, source-free medium (such as the inside of a waveguide) characterized by permittivity ε and permeability μ (and conductivity σ=0), Maxwell’s curl equations are given by Eqs. (7.2b and d) with J=0,

∇×E˜=−j ω μ H˜, (8.87a)
∇×H˜=j ω ε E˜. (8.87b)

Upon inserting Eqs. (8.86a and b) into Eqs. (8.87a and b) and recalling that each of the curl equations actually consists of three separate equations—one for each of the unit vectors x ˆ, y ˆ, and z ˆ, we obtain the following relationships:

∂e˜z∂y+j β e˜y=−j ω μ h˜x, (8.88a)
−j β e˜x−∂e˜z∂x=−j ω μ h˜y, (8.88b)
∂e˜y∂x−∂e˜x∂y=−j ω μ h˜z, (8.88c)
∂h˜z∂y+j β h˜y=j ω ε e˜x, (8.88d)
−j β h˜x−∂h˜z∂x=j ω ε e˜y, (8.88e)
∂h˜y∂x−∂h˜x∂y=j ω ε e˜z. (8.88f)

Equations (8.88a-f) incorporate the fact that differentiation with respect to z is equivalent to multiplication by −j β. By manipulating these equations algebraically, we can obtain expressions for the x and y components of E˜ and H˜ in terms of their z components, namely

Here

k c 2=k 2−β 2=ω 2 μ ε−β 2, (8.90)

where k is the unbounded-medium wavenumber earlier defined as

k=ω μ ε. (8.91)

For reasons that become clear later (in Section 8-8), the constant k c is called the cutoff wavenumber. In view of Eqs. (8.89a-d), the x and y components of E˜ and H˜ now can be found readily—so long as we have mathematical expressions for E˜z and H˜z. For the TE mode, E˜z=0, so all we need to know is H˜z, and the converse is true for the TM case.