Design Space Exploration

In this subsection, the proposed stochastic Williamson model is applied to conduct a design space exploration of a signal via in a multilayer PCB. To get a first impression of the range of possible transmission and reflection characteristics, the via radius, antipad radius, cavity height, and distance of the surrounding ground vias are varied. Consider the case of a twelve layer PCB and a signal via surrounded by four ground vias of a radius of 5 mil.

The substrate is a lossless dielectric with ε_r = 4.4. The via radius is a = 5÷20 mil, the antipad radius is b = a+ 2÷17 mil, the cavity height is d = 4÷60 mil, and the distance of the ground vias isd_g = 40÷160 mil. Here, ÷denotes the range. All parameters are uniformly distributed in the stated range. Figure 5.10 shows the mean and the 99%

confidence interval when all parameters are varied in the stated range. The simulation was conducted using the proposed stochastic Williamson model with P = 3 and considering stochastic via and antipad radii. The other variables were varied using MCS. Here, this scheme is selected because it allows the representation of the results that is discussed in

0 -20 -40 -60

|S|(dB)11 -80

0 5 10 15 20 25

Frequency (GHz) 0.0

-0.2 -0.4 -0.6 -0.8 -1.0

|S21|(dB) MCS (FEM)

PCE MCS

Figure 5.9: Magnitude ofS₁₁and S₁₂ as a function of frequency for a via interconnect in a 12 layer PCB under variation of the via and antipad radius. The shaded areas are 99%

confidence intervals and the central line represents the mean. FEM results were obtained with [161] using MCS with 15 samples. Figure adapted and caption taken from [11].

the next paragraph. Looking at Figure 5.10 reveals that the variation has a significant impact on the transmission and reflection. Transmission is not much affected for frequencies below 5 GHz, above 5 GHz the range of possible values increases and finally occupies the complete range between −100 dB and 0 dB for frequencies above 15 GHz. The possible outcomes of the reflection stay within a range of 30 to 40 dB. This analysis shows that the range of possible realizations is very large.

To draw conclusions for practical designs, the information that a large range of realizations are possible, is not valuable on its own. Usually, one is only interested in the best possible realization under given circumstances. We assume that the via radius and the antipad radius are variables that can be adjusted in a way to get the optimal performance. Now, the question is how good is the optimal performance under given circumstances. How do early design decisions, like the number of layers and cavity height, affect the ability to design a good via interconnect. Let us assume the number of layers N_L, number of ground vias N_g, cavity height d, and distance of ground viasd_g are set: then, the optimal choice of the via radius a and the antipad radius b for the best possible transmission is chosen.

Good transmission is understood as the minimum transmission in the considered frequency

5.2 Stochastic Model of a Signal Via

0 -20 -40 -60

|S|(dB)11 -80

0 5 10 15 20 25

Frequency (GHz) -100

-80 -60 -40 -20 0

|S21|(dB)

Figure 5.10: Magnitude of S₁₁ andS₁₂ as a function of frequency for a via interconnect embedded in a 12 layer PCB and surrounded by four ground vias. The via and antipad radius, cavity height, and the distance of the surrounding ground vias are uniformly distributed in a given range. The shaded areas are 99% confidence intervals and the central line is the mean.

Figure adapted and caption taken from [11].

range being as high as possible. Mathematically, this constraint may be written as

maxa,b min_ω |S₂₁(ω;a, b, d, d_g, N_g, N_L)|. (5.17)

The resulting transmission minimum over a considered frequency range of 0 to 5 GHz and 0 to 25 GHz are illustrated on the left and right in Figures 5.11 and 5.12, respectively. The results are visualized as contour plots for the case of four and twelve layer PCBs (upper and lower plots, respectively) and two (Figure 5.11a), four (Figure 5.11b), six (Figure 5.12a), and eight (Figure 5.12b) ground vias. The cavity height and distance of the ground vias to the signal via were evaluated for 50 values each. The optimum combination of the via and antipad radii was found by selecting the best according to (5.17) from 25,000 samples generated from the PCE representation. Selecting a point on one of the maps in Figures 5.11 and 5.12 provides the best possible transmission for the selected frequency range, number and distance of ground vias, as well as number and individual height of cavities. All values assume a ground via radius of 5 mil and a lossless dielectric of ε_r = 4.4. The values in Figures 5.11 and 5.12 should be seen as upper bounds, as all actual realizations will be

40

Figure 5.11: Minimal transmission of the best realization for frequencies below 5 GHz (left side) and 25 GHz (right side) as a function of the cavity height dand the distance of the ground viasd_g. For a four (upper plot) and twelve (lower plot) layer PCBs and two (a) and four (b) surrounding ground vias. Figure adapted and caption taken from [11].

5.2 Stochastic Model of a Signal Via

Figure 5.12: Minimal transmission of the best realisation for frequencies below 5 GHz (left side) and 25 GHz (right side) as a function of the cavity height dand the distance of the ground viasd_g. For a four (upper plot) and twelve (lower plot) layer PCBs and six (a) and eight (b) surrounding ground vias. Figure adapted and caption taken from [11].

below due to losses.

The representation of the results of the design space exploration in Figures 5.11 and 5.12 are convenient to draw conclusions for practical design of via interconnects. It is observed that the possibility to make a good via design decreases with the highest considered frequency.

For frequencies up to 5 GHz a transmission of above −1 dB can be achieved in all cases, except for the case of 2 ground vias with large distances and a high cavity. For the case of frequencies up to 5 GHz, a low cavity height and close ground vias are preferable. Generally, using more ground vias increases the chance of obtaining a good transmission. The largest difference is observed when increasing the number of ground vias from two to four. For frequencies up to 25 GHz the behavior changes dramatically. For two ground vias, the best possibility to achieve good transmission is given, when the cavities are as thin as possible. For the case of four ground vias, a good transmission is achieved when placing the ground vias close to the signal via. This effect is also observed for the case of six and eight ground vias. This effect can be explained by the geometry: the ground vias are circularly arranged around the signal via and form a circular resonator. When the ground vias are close to the signal via the cutoff frequency of this resonator is high. If the distance is larger the resonance frequency is below the upper bound of 25 GHz, the excited resonance deteriorates the transmission. The lowest resonance frequency of a circular cavity is given by [113, Chaper 6.4]

fc= j_1,0 2π√

µεd_g. (5.18)

The resonant frequency only depends on the distance of the ground vias and does not depend on any other geometric parameter. For the given dielectric, the resonance frequency multiplied by the distance of the ground vias can be written as

f_cd_g = j_1,0 2π√

µε₀ε_r ≈ 114.7

√ε_r GHz mm≈ 4517

√ε_rGHz mil

εr=4.4

= 54.7GHz mm≈2154GHz mil.

(5.19)

For the resonance frequency above 25 GHz, this yields a distance of d_g < 86 mil. This confirms the results shown in the plots and gives a physical interpretation.

The findings of the design space exploration can be utilized to suggest a design guideline:

• use as many ground vias as possible,

• the distance must be smaller than 4517 mil/(√

ε_rf_max,GHz) with the maximum fre-quencyf_max,GHz in GHz,

• low cavity heights are preferable.