3 Single comb-shape electrode capacitor

Next I tried a complicated case. I analyzed a capacitor of which electrode is comb-shape. Both of electrodes are not comb-shape. One electrode is comb-shape but the other one is still solid metal plane. It is like this.

Fig. 3.0

The bottom electrode has the same area to the previous analysis, 10mm*10mm. Here I changed the width of the comb teeth so as to reduce the area facing the bottom plane from 90% to 0%.

3.1 Expectation of the result

According to the analysis before, we can suppose that the capacitance would be in proportion to the area of the electrodes facing each other.
Hereafter, let's see whether it is true or not.

To evaluate the effect of the width of comb teeth I changed the place of the trace line of the upper electrode. It was placed in the outside of the bottom electrode.

Fig. 3.1

3.2 Analysis result

The cycle of the comb teeth was 2.0mm. I reduced the width of the comb teeth from 2.0mm to 0mm by 0.2mm.
Here I put the summary table.

Table 3.1

Overlapping ratio of electrodes	100%	90%	80%	70%	60%	50%
Comb teeth width and space	2.0mm-0.0mm	1.8mm-0.2mm	1.6mm-0.4mm	1.4mm-0.6mm	1.2mm-0.8mm	1.0mm-1.0mm
Figure
Estimated capacitance	9.24pF	8.81pF	8.10pF	7.30pF	6.46pF	5.59pF
Ratio of capacitance	100%	95%	88%	79%	70%	60%

Overlapping ratio of electrodes	40%	30%	20%	10%	0%
Comb teeth width and space	0.8mm-1.2mm	0.6mm-1.4mm	0.4mm-1.6mm	0.2mm-1.8mm	0.0mm-2.0mm
Figure
Estimated capacitance	4.70pF	3.80pF	2.85pF	1.92pF	0.239pF
Ratio of capacitance	51%	41%	31%	21%	3%

Fig. 3.2

The expectation was wrong. Seeing Fig. 3.2 the line of Ratio of capacitance (Blue line) is far from the expectation line (Dark green line). In 10% overlapping ratio the ratio of capacitance exceeds more than 10%.
Then I subtracted the capacitance in 0% case so as to know how much the trace line of the upper electrode affected. But the Magenta line is still far from the expected line. This difference is not a simple error.
I know that a capacitor which has this type of electrode is quite different from simple cases before.

3.3 Supposition of the reason for error

Probably, electric lines from the bottom electrode through gaps between comb tooth come back to the upper side of comb tooth. Those electric lines must increase the capacitance.
In simple cases before, we can neglect electric lines which flow out from the edge of electrodes. But in this case we can not neglect them because of many electric lines through many gaps between comb tooth.

3.4 Local Conclusion

Capacitance of a single comb-shape electrode capacitor doesn't change in proportion to the overlapping ratio of electrodes. We can not neglect electric lines which pass gaps of comb tooth.
The ratio of capacitance is more than the overlapping ratio of electrodes.