Abstract

A planar capacitor--a parallel-plate capacitor is seen in a textbook of electromagnetics. But I have scarcely seen modified ones other than the basic type. For example, electrodes of a capacitor are horizontally shifted or have different size or shape from the other one. In textbooks such problems may be left to readers as an exercise.

Here I used an electromagnetic simulator and analyzed the variation of capacitance of a planar capacitor. I used Sonnet Lite of Sonnet Software Inc. here as well as my touch sensor analysis.

These results show that electric lines which flow out from the edge of electrodes are not negligible in a capacitor which has a complicated-shape electrode such as a comb or a mesh, and that estimation of its capacitance is not simple.

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Contents

1 Shifted-electrodes capacitor

First I analyzed a planar capacitor of which both of electrodes are the same size. How much does the overlapping ratio of a pair of electrodes affect the capacitance?

1.1 Base point--the nornal case

First I analyzed a basic shape capacitor.
 
Fig. 1.1

The descriptions are:

Area of an electrode	10mm*10mm
Gap height between electrodes	0.1mm
Dielectric in the gap	air
Overlapping ratio of electrodes	100%

Both of electrodes were connected by an ungrounded internal port on layer 0. With this port I analyze impedance of a capacitor. The height to the ceiling and floor of Sonnet box is 10mm.

I got 1734 ohms at 10MHz. The capacitance was 9.18pF. This is the base point data.

1.2 75% overlappping capacitor

Second I analyzed a capacitor of which electrodes overlapped 75%.

Fig. 1.2

I got 2242 ohms at 10MHz. The capacitance was 7.10pF.

1.3 50% overlappping capacitor

Third I analyzed a capacitor of which electrodes overlapped 50%.

Fig. 1.3

I got 3251 ohms at 10MHz. The capacitance was 4.90pF.

1.4 25% overlappping capacitor

Fourth I analyzed a capacitor of which electrodes overlapped 25%.


Fig. 1.4

I got 5925 ohms at 10MHz. The capacitance was 2.69pF.

1.5 0% overlappping capacitor

Fifth I analyzed a capacitor of which electrodes overlapped 0%.

Fig. 1.5
  
I got 34.008k ohms at 10MHz. The capacitance was 0.468pF.

1.6 -25% overlappping capacitor

Sixth I analyzed a capacitor of which electrodes overlapped -25%. This is just for a reference.
Fig. 1.6

I got 46.417k ohms at 10MHz. The capacitance was 0.343pF.

1.7 Capacitance vs. overlapping ratio

Here I put the summary table.

Table 1.1

Overlapping ratio of electrodes	100%	75%	50%	25%	0%
Figure
Estimated capacitance	9.18pF	7.10pF	4.90pF	2.69pF	0.468pF
Ratio of capacitance	100%	77%	53%	29%	5%
Error to the overlapping ratio	N/A	2%	3%	4%	5%

We understand that the capacitance of a planar capacitor changes in proportion to the overlapping ratio of electrodes. But some error exists.
Then I subtracted the number of the capacitance in 0% from other cases.  Then the table changed like below:

Table 1.2

Overlapping ratio	100%	75%	50%	25%	0%
Capacitance subtracted 0.468pF	8.71pF	6.63pF	4.43pF	2.22pF	N/A
Ratio of capacitance	100%	76%	51%	26%	N/A

The error reduced to 1%. These numbers are well fit to the overlapping ratio.


Fig. 1.7

Actually we have electric field which flows out from the perimeter of electrodes. So we have 0.468pF even though the overlapping ratio is 0%.
That is, in each case the electric field which flows out from the edge of the electrodes almost equals to the capacitance in the 0%-overlapped case.

1.8 Local Conclusion

Capacitance of a planar capacitor changes in proportion to the overlapping ratio of electrodes.