7.1.3 Interpretation of the analytical model

The maxima of the cavity field inside and the radiated emission from the enclosure slot are observed at the resonance frequencies, where the denominator $k_{m}^2+k_{n}^2-k^2$ of (7.13) has its minima. At these frequencies (7.13) can be approximated by the dominating term with the minimum denominator,

$$U_{m,n}\approx\frac{j4\omega\mu h}{LW} \frac{K_{s}(x_{sp},y_{sp})K_{meas}(x,y)}{k_{m}^2+k_{n}^2-k^2},$$ (7.16)

with the source dependent term

$$K_{s}(x_{sp},y_{sp})=I_{sp}\sin(k_{m}x_{sp})\sin(k_{n}y_{sp}),$$ (7.17)

and the term,

$$K_{meas}(x,y)=\sin(k_{m}x)\sin(k_{n}y),$$ (7.18)

which depends on the position of the voltage measurement. The index of $U_{m,n}$ denotes the cavity resonance mode, which is characterized by the integer pair $m$ and $n$. Both $K_{s}(x_{sp},y_{sp})$ and $K_{meas}(x,y)$ vanish at the metallic enclosure walls. Therefore, placing a single current source closer to a metallic enclosure wall will reduce the cavity field and the emissions. Below the second resonance mode, the maximum field is in the middle of the slot at position $x=L/2,\,y=W$ and the maximum field in every enclosure cross section $y=y_{cr}\leq W$ is located at $x=L/2$. Increasing the distance of a single source from the symmetry line $x=L/2$ will reduce the emissions up to the second resonance frequency. The maxima of the enclosure field are at the enclosure slot $y=W$ for every resonance mode. According to (5.1) a trace above the ground plane of a PCB is not a single current source, because both, the source and the load currents, excite a cavity field. A short trace with negligible phase shift between the source and the load current couples to the cavity with the currents $I_{s}$ at the source position and $I_{l}=-I_{s}$ at the load position. A superposition of two terms of (7.16), one with the excitation $I_{l}$ and the other with the excitation $I_{l}=-I_{s}$ will consider the coupling from the short trace. Therefore, this trace coupling can be investigated with the derivatives of (7.17). Since the partial derivatives normal to the enclosure edges have their maxima at the metallic enclosure walls, the coupling of a differential source to the cavity is at a maximum, when it is positioned perpendicular and close to an enclosure wall. The partial derivative in $x$ direction of (7.17) vanishes in the symmetry line $x=L/2$ below the second resonance. Therefore, a symmetric placement of a trace perpendicular to this symmetry line reduces the coupling and the emissions significantly below the second resonance of the enclosure. Both partial derivatives of (7.17) vanish in the middle of the slot at position $x=L/2,\,y=W$ up to the second resonance mode of the cavity field. Moving a differential source to that position in an arbitrary direction will reduce the coupling to the cavity at the first enclosure resonance. These design guidelines have been obtained simply by a discussion of the analytical cavity model equations. Although these rules have been extracted for the rectangular enclosure in Figure 7.1, the main facts regarding the placement of sources and traces close, parallel or perpendicular to metallic walls or enclosure symmetry lines can be generalized for arbitrarily shaped enclosures.

Perfect electrically conducting planes, air in the cavity, and a perfect magnetically conducting boundary at the slot have been used to derive the cavity field formulation (7.13), neglecting any losses, which leads to significant deviations at the resonance frequencies compared to a real lossy situation. An enclosure (Figure 7.1) usually has a much higher plane separation $h$ than power-ground planes on a PCB. Therefore, the radiation loss becomes the dominant loss mechanism [45], [59], [101] and must be considered in the cavity model to obtain a reasonably good solution. The next section will consider the radiation loss in the cavity model and provide analytical expressions for the calculation of the radiated emissions from the slot. A quantitative investigation of radiated emission and coupling from sources to the enclosure will be presented based on that model. Quantitative classification of EMC design guidelines, such as placement and layout rules, is necessary to obtain information on their practical relevance for the intended application. An example for the relevance of quantitative EMC rule classification is the crosstalk from a digital signal trace to an analog circuit trace. Whether this coupling is relevant or not depends on the spectrum of the digital signal, the sensitivity of the analog circuit and the layout routing of the traces. A cost optimized design cannot be reached with global rules applied to all signals. An EMC engineer must have quantitative information, if the coupling is relevant for a decision about shielding, trace routing, and ground separation efforts.

The cavity modes depend on the cavity boundaries. Parallel rectangular planes with four open edges have been investigated for power integrity analysis purposes by [42] and [43]. They expressed the resonances of the rectangular power planes with

$$f_{r_{open}}=\frac{c_{l}}{2\pi}\sqrt{\left(\frac{m\pi}{L}\right)^2+\left(\frac{n\pi}{W}\right)^2}.$$ (7.19)

Table 7.1 lists the resonance frequencies for the first modes of rectangular power-planes with four open edges and of a rectangular enclosure with three closed edges and one open slot according to Figure 7.1, both with the same size of $L$=160mm and $W$=120mm. Since (7.17) and (7.18) vanish for all $m=0$, the enclosure resonances with $m=0$ are compensated and do not exist.

		power-planes		enclosure
$m$	$n$	$f_{r_{open}}$ (MHz)	exists	$f_{r}$ (MHz)	exists
0	0	0	no	625	no
1	0	938	yes	1127	yes
0	1	1250	yes	1875	no
1	1	1563	yes	2096	yes
2	0	1875	yes	1976	yes

Table 7.1: First resonance frequencies of rectangular parallel plane cavities with L=160mm and W=120mm and different boundaries. One with four open edges (power-planes), the other with one open slot and three closed metal edges (enclosure).

Power planes with four open edges have more resonances and different resonance frequencies than the enclosure. Resonance frequencies of the same modes are shifted some hundred MHz. In particular the first and the second resonance frequencies are interesting with respect to the previously mentioned design rules which are related to the symmetry line $x=L/2$ of the enclosure. In an enclosure with the dimensions $L$=160mm, $W$=120mm and $h=15mm$, these rules are valid up to 1976 MHz, a broad band of the 2.5GHz CISPR25 frequency range according to Table 4.1.

C. Poschalko: The Simulation of Emission from Printed Circuit Boards under a Metallic Cover