Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

The guide occupies $0 < x < a$ , $0 < y < b$ (Figure 7.3.1). By convention the $x$ side is the longer one, $a \geq b$ . The walls are perfect electric conductors (PEC) and the filling is a lossless simple medium with $ε = ε_{r} ε_{0}$ and $μ = μ_{r} μ_{0}$ , usually air. The wave travels along $z$ , so every field component carries the factor $e^{- j τ z}$ (and the time factor $e^{j ω t}$ , suppressed as always):

E (x, y, z) H (x, y, z) = E (x, y) e^{- j τ z}, = H (x, y) e^{- j τ z} . (7.3.1)

The rectangular waveguide: a hollow pipe with perfectly conducting walls, cross-section 0<x<a, 0<y<b with a\ge b, filled with a lossless medium \varepsilon, \mu. The wave travels along z. — **Figure 7.3.1.** The rectangular waveguide: a hollow pipe with perfectly conducting walls, cross-section $0 < x < a$ , $0 < y < b$ with $a \geq b$ , filled with a lossless medium $ε$ , $μ$ . The wave travels along $z$ .

Definition · The phase constant

τ

As in Sections 7.1 and 7.2, $τ$ is the phase constant of the wave along the guide, the quantity called $β$ for the plane waves of Chapter 6. With $\partial / \partial z \to - j τ$ the propagation constant is $γ = α + j β = j τ$ . Below cutoff $τ$ becomes imaginary, $τ = - j α$ , and $e^{- j τ z} = e^{- α z}$ decays. Written as a complex wavenumber $τ = β - j α$ , the form used for lossy media in Section 6.2, this is the case $β = 0$ . The wavenumber of the filling is $k = ω μ ε$ , its velocity $v = 1/ μ ε$ , its wavelength $λ = v / f$ and its intrinsic impedance $η = μ / ε$ .

Section 7.1 showed that in a uniform guide the four transverse components follow from the two longitudinal ones by differentiation. With

h^{2} = k^{2} - τ^{2}, (7.3.2)

they read

E_{x} E_{y} H_{x} H_{y} = - \frac{1}{h ^{2}} (j τ \frac{\partial E _{z}}{\partial x} + j ω μ \frac{\partial H _{z}}{\partial y}), = - \frac{1}{h ^{2}} (j τ \frac{\partial E _{z}}{\partial y} - j ω μ \frac{\partial H _{z}}{\partial x}), = - \frac{1}{h ^{2}} (j τ \frac{\partial H _{z}}{\partial x} - j ω ε \frac{\partial E _{z}}{\partial y}), = - \frac{1}{h ^{2}} (j τ \frac{\partial H _{z}}{\partial y} + j ω ε \frac{\partial E _{z}}{\partial x}), (7.3.3)

and $E_{z}$ , $H_{z}$ themselves obey the Helmholtz equation $\nabla^{2} ψ + k^{2} ψ = 0$ . Two families of solutions are possible in a hollow guide. In a TM wave $H_{z} = 0$ and $E_{z} \neq = 0$ , in a TE wave $E_{z} = 0$ and $H_{z} \neq = 0$ . A TEM wave ( $E_{z} = H_{z} = 0$ ) would need $h = 0$ and a transverse field that is a static field in the cross-section, which a single closed conductor cannot support.

Every mode has its own value of $h$ , fixed by the walls. Once $h$ is known, the general results of Section 7.1 apply unchanged:

f_{c} = \frac{h}{2 π μ ε}, τ = k 1 - (\frac{f _{c}}{f})^{2}, Z_{T M} = \frac{τ}{ω ε} = η 1 - (\frac{f _{c}}{f})^{2}, Z_{T E} = \frac{ω μ}{τ} = \frac{η}{1 - ( f _{c} / f ) ^{2}}, (7.3.4)

with the guide wavelength $λ_{g} = 2 π / τ$ , the phase velocity $v_{p} = ω / τ$ and the group velocity $v_{g} = d ω / d τ$ ,

λ_{g} = \frac{λ}{1 - ( f _{c} / f ) ^{2}}, v_{p} = \frac{v}{1 - ( f _{c} / f ) ^{2}}, v_{g} = v 1 - (f_{c} / f)^{2}, v_{p} v_{g} = v^{2} . (7.3.5)

Below cutoff the mode is evanescent, with $τ = - j α$ and

α = h^{2} - k^{2} = h 1 - (\frac{f}{f _{c}})^{2} (f < f_{c}) . (7.3.6)

The whole task of this section is therefore to find $h$ for every mode of the rectangular guide, and its field pattern.

7.3.1 The guide and the tools of Section 7.1