The equations presented here make it possible to predict and analyze the resonant behavior of microwave circuits enclosed in rectangular shields.

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:

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Both *m *and *n *are integers starting at zero, and define a possible transverse mode commonly referred to as a TE* _{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.

Of practical importance is the lowest order mode, known as the dominant mode. For TM modes in a rectangular waveguide, neither *m *nor *n *can be zero due to the sine dependency of both the E- and H-fields; therefore, the dominant mode will be the TM_{11 }mode.

For TE modes, either m or n can be zero, but both cannot be zero otherwise the E- and H-fields become zero as can be seen from Eqs. 9 to 12. Therefore, the lowest TE mode is the TE_{10 }mode for the case b < a shown in Fig. 1. It should be noted that if a < b, then the dominant mode would be the TE_{01 }mode.

As deduced from the two-wire transmission line model in Fig. 2, a rectangular waveguide has a highpass frequency response determined by its dimensions. When the propagation constant is real, this corresponds to a propagating wave. When the propagation constant is imaginary, this corresponds to an exponentially decaying wave. Based on this, the cutoff frequency for a particular mode is defined when the propagation constant in Eq. 6 or 13 is 0, which means that the expression of Eq. 15 must hold true.

Solving this expression for frequency yields:

Note that the expression for the cutoff frequency is the same for TE or TM modes.

Ifω^{2 }µε(m≠/a)^{2 }+ (n≠/b)^{2}, then the propagation constant is imaginary which translates to exponentially decaying fields away from the source of excitation.

For an example, the derivations will be used to calculate the dominant mode cutoff frequency for a rectangular waveguide with dimensions, a = 2.286 cm and b = 1.016 cm. Since the dominant mode is the TE_{10 }mode, m = 1 and n = 0. The value of the permittivity, µ, in free space is 4≠ 10^{–7 }H/m while the value of ε in free space is (1/36≠) 10^{–9 }F/m. The value of a is 0.02286 m and the value of b is 0.00106 m. Substituting all of these values into Eq. 16 yields:

F_{c mm }= 1/(2{(4≠ 10^{–7})–9)>}^{0.5}) 2 + (0/0.01016)^{2}>^{0.5 }

which is:

F_{c mm }= 6.562 GHz

In the simulated S_{21 }response of the waveguide structure, signals below the cutoff frequency are attenuated, whereas signals at and above the cutoff frequency propagate with relatively low loss (Fig. 3). If a rectangular waveguide is closed on its two open sides by a conducting wall, a rectangular cavity is formed as shown in Fig. 4. Applying the new boundary conditions, the expressions for the E- and H-fields within the cavity for TM waves from Eqs. 17 to 21 hold:

Similar expressions for the E- and H-fields for the TE modes are shown in Eqs. 22 to 26:

The resonant frequencies can be calculated in a similar fashion to the cutoff frequency for a rectangular waveguide, with the result shown in Eq. 27.

Parameters m, n, and p denote the number of half-wave cycles in the x, y, and z directions, respectively. Using this fact, the location of the E- (or H- ) field maximums can be determined within the rectangular cavity.

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The lowest resonant frequency for a TM wave is the TM_{110 }mode and for a TE wave is the TE_{101 }mode. This can be seen by substituting the mode values, m, n, and p into Eqs. 17 through 26. For example, for the TM case, if either m or n is zero the expressions for the E- and H-fields collapse to zero.

For a cavity with dimensions shown in Fig. 4, the E-field and H-field hotspots for the TM_{110 }mode can be located by applying the appropriate expressions. As a second example, using Eqs. 17 to 21, for m = n = 1 and p = 0, there are three nonzero field components as shown in Eq. 28.

Only one maximum exists for this function and it occurs if (a/2,b/2,z), which is shown in Eq. 29:

The maximum occurs at (a/2,0,z) and (a/2,b,z) (as in Eq. 30):

and the maximum occurs at (0,b/2,z) and (a,b/2,z).

Since a > b, the maximums calculated by Hx are actually the true maximums and are located at (a/2,0,z) and (a/2,b,z).

By using the definition of modes, one can easily determine the maximum locations of the E- and H-fields within

a cavity. For the TM_{110 }mode, there exists one half-wave cycle in the x-axis, and one half-wave cycle in the y-axis, and no variation in the z-axis of the fields (Fig. 5). Note that the maximum occurs when x = a/2 and y = b/2 intersects, which occurs in the center of the cavity.

The location of an E-field maximum is the location of a H-field minimum and vice-versa. Using this fact, the corresponding locations for the maximum H-field can be found.

To account for the PCB effects, the rectangular cavity can be modeled as a rectangular waveguide with a dielectric slab perpendicular to the E-field with two ends closed in by conducting walls (Fig. 4). An approximate solution to the resonant frequency using this method is given below in Eq. 31 (see ref. 2)

where:

h = the thickness of the printed-circuit board;

d = the height of the shield plus the thickness of the printed circuit board; and

ε_{r }= the dielectric constant of the PCB.

As a third example, consider a PCB with shield dimension of 1.675 X 2.375 X 0.25 in., board thickness of 20 mils, and relative dielectric constant of 4.5 (ε_{r }= 4.5). The equations can be used to find the five lowest resonant frequencies and the locations of the maximum electric fields within the shield. To perform the calculations, the following parameters are known: h = 20 mils and d = 270 mils. Using Eq. 31, the TM_{110}, TM_{120},

TM_{310}, TM_{210}, and TM_{220 }modes correspond to the five lowest resonant frequencies within the shield. The lowest

TE mode is the TE_{101 }mode with a corresponding resonant frequency of over 21 GHz, which is higher than any of the TM modes listed above. The table* *lists the five lowest frequencies, as well as the corresponding locations of the maximum electric fields.

As a check of these calculations, the shield and PCB were simulated using the High-Frequency Structure Simulator (HFSS) from Ansoft Corp. (www.ansoft.com). The corresponding resonant frequencies are also provided in the table for comparison. Plots of both the E- and H-fields within the structure are also simulated with HFSS and shown in Figs. 5, 6, and 7.

For the TM_{110 }mode, it is expected that there will be a one-half-wave variation in both the x- and y-dimensions and no variation in the z-direction. This results in a maximum E-field hot spot directly in the center of the cavity (Fig. 6B). Note that where an E-field maximum occurs, there is a corresponding Hfield minimum (Fig. 6A).

For a TM_{210 }mode, it is expected that there would be two one-half-wave cycles in the x-direction, one one-half cycle in the y-direction, and no variation in the z-direction, which is shown in Fig. 7A (the H-field variation) and in Fig. 7B (the E-field variation)

For a TM_{120 }mode, it is expected that there would be a single one-half-cycle variation in the x-direction and two one half-cycle variations in the y-direction, and no variation in the z-direction which is shown in Fig. 7A (the H-field variation) and in Fig. 7B (the E-field variation).

For a TM_{310 }mode, it is expected that there would be three one-half-cycle variations in the x-direction and one one-half-cycle variation in the y-direction, and no variation in the z-direction which is shown in Fig. 7A (the H-field variation) and in Fig. 7B (the E-field variation).

For a TM_{220 }mode, it is expected that there would be two one-half-cycle variations in the x-direction and two one-half-cycle variations in the y-direction, and no variation in the z-direction which is shown in Fig. 7A (the H-field variation) and in Fig. 7B (the E-field variation).

Note that these results correspond to what is predicted in the table based on the definition of modes. It should also be noted that Eq. 28 is valid as long as there aren't large or high-profile devices attached to the PCB. Large obstacles complicate this simple computation. To account for obstacles, a full-featured three-dimensional (3D) electromagnetic (EM) solver is recommended for more complex modeling.

To excite a resonant mode, a probe with a signal at or near the resonant frequency inserted at the location of maximum electric fields will set up an E-field of considerable intensity. Similarly, a magnetic loop, inserted where the maximum H-fields are located, will set up a strong H-field of considerable intensity. Multiple probes (or loops) located in areas of maximum E-field (or H-field) intensity, can excite higher-order resonant modes.

Knowledge of the location of the maximum E- and H-fields and the corresponding resonant frequency enables a designer to avoid placement and routing of circuits that could efficiently excite these resonant modes. Note that placement and routing of circuits does not get rid of resonant modes, but can reduce the effects, which may just be the difference between a working versus a non-working design.

As an example, two filters were placed in the shield described earlier in example 1. Filter A was located in the center of the can as shown in Fig. 8. The TM_{110}, TM_{210}, and TM_{310 }modes all have Efield hot spots along this path. So, it is expect that there should be field excitations at 4.1, 7.2, and 8.3 GHz. A simulation of the frequency response using Ansoft's HFSS is shown in Fig. 9.

*See the August 2007 issue for Part 2 of this article.*