Microwave circuits are generally enclosed in rectangular shields before integration into a larger system. Unfortunately, when the shield cover goes on, it can cause unexpected results, such as the oscillation of "unconditionally stable" amplifiers, an increase in transmission-line losses, and unwanted coupling. Essentially, the presence of the shielded enclosure can throw off all those advanced computer-aided-engineering (CAE) predictions. And, because it is late in the design cycle, the only recourse may be the addition of RF absorbers and gasket material to the enclosure. But the effects of a shielded enclosure in high-frequency printed-circuit boards (PCBs) can be minimized by properly predicting the frequency, location, and nature of these enclosure-induced resonant modes.

A rectangular shield can be considered a rectangular waveguide with two of its open sides enclosed by a conducting wall. To better understand the behavior of the resonant modes in a rectangular cavity, it might make sense to review some of the fundamental relationships of rectangular waveguide theory.

A rectangular waveguide can be considered a hollow rectangular tube that supports the propagation of electromagnetic (EM) waves. Figure 1 shows a rectangular waveguide with dimensions a, b, and l. Note that a > b. The two types of EM waves supported in a rectangular waveguide are the transverse-electric (TE) waves and transverse-magnetic (TM) waves. TE waves do not contain an electricfield (E-field) component in the direction of propagation while TM waves do not contain a magnetic-field (H-field) component in the direction of propagation.

A simple way to understand how an EM wave can propagate in a rectangular waveguide can be deduced starting with the transmission line model of Fig. 2. It shows a two-wire transmission line with quarter-wave shorted stubs attached across it. The shorted stubs have no effect on the propagation of a signal on the two-wire line (at the quarter-wave frequency). If quarter-wave shorted stubs were added with infinitesimally small spacing between them, the structure would assume the behavior of a rectangular waveguide transmission line (Fig. 3).

In Fig. 3, the larger cross-sectional diameter is one-half wavelength while the shorter dimension is the spacing of the original two-wire line. This configuration is the smallest cross-section that can be used to efficiently propagate a signal of a given wavelength.

If the wavelength of the signal is larger in comparison with the cross-sectional dimensions of the line, the signal will be significantly attenuated as it propagates down the waveguide. If the wavelength of the signal is shorter in comparison with the cross-sectional dimensions of the line, then other modes of propagation may occur.³ These conditions can be modeled as the superposition of two plane waves reflecting and re-reflecting down the line. The plane waves set up different mode patterns and propagation characteristics, which have been reproduced in equation form (See ref. 1) below for both TE and TM waves.¹

For TM waves, the E- and H-fields as a function of position along the waveguide are given by Eqs. 1-5:

where:
a and b = box dimensions as oriented in Fig. 1 and = a propagation constant given by Eq. 6:

and

Both m and n are integers starting at zero, and define a possible transverse mode commonly referred to as a TM_mn mode. The first subscript denotes the number of half-cycle variations of the fields in the x-direction, and the second subscript denotes the number of half-cycle variations of the fields in the y-direction. It is evident that there are infinite modes that can exist based on the dimensions of the waveguide.

Similar expressions for TE waves are given by Eqs. 8 through 14: