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I am currently doing a project where I need to find the capacitance of a multi-conductor system. Through my literature review, I read about the concept of a capacitance matrix. After more research, I am unable to find a closed form expression which calculates the total capacitance of the system from the capacitance matrix. I am only able to find the expression for the total capacitance for a two conductor system which is shown below. Assuming we have a capacitance matrix for a 2 conductor system.

$$ \textbf{C}= \begin{bmatrix} C_{11} & C_{12}\ C_{21} & C_{22} \end{bmatrix} $$

The total capacitance of the system is computed by:


$$ C_{total} = \frac{C_{11}C_{22}-C_{12}C_{21}}{{C_{11}} + C_{12} + C_{21} + C_{22}} $$

However, I am unable to find a closed form expression of the total capacitance of a system from the capacitance matrix with a system consisting of more than 2 conductors. Can anybody please help to explain to me how?

The total capacitance is calculated looking through the first and last conductors if it matters. If there are any resources that explain this please also let me know. Also correct me if any I am mistaken with any of the information above.

user392135
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  • Please define what you mean by "total capacitance". – DanielSank Jan 28 '24 at 18:31
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jan 28 '24 at 18:40
  • In your 2x2 example the numerator of $C_{total}$ is equal to the determinant $\mathrm{det}(\mathbf C)$ and the denominator is the sum of all capacitances, so I wonder if it would generalize to $C_{total} = \frac{\mathrm{det}(\mathbf C)}{\sum_{ij}C_{ij}}$. – Er Jio Jan 28 '24 at 22:09
  • The rank of the capacitance matrix is (n-1), i.e. the "raw" capacitance matrix is singular (See the equation (2) given here.). Thus, in your 2x2 example, the denominator of $C_\text{total}$ is zero. – HEMMI Jan 29 '24 at 05:23
  • @DanielSank Hi. If I have n conductors connected in series with one another (i.e. conductor 1 --- conductor 2 --- conductor 3 --- ... --- conductor n)(--- is a wire). What I mean is the equivalent capacitance I would measure when I have probes touching conductor 1 and n. Thanks – user392135 Jan 29 '24 at 18:31
  • Please have a look at the article "For capacitors in series" in the capacitor of wikipedia. The formula given here seems to give the result you want. In it, the non-diagonal component is small and neglected. – HEMMI Jan 30 '24 at 11:31
  • This article shows how to go from mutual capacitances to the capacitance matrix. That could be useful even though it doesn't answer your question. – DanielSank Feb 01 '24 at 07:07
  • @user392135 The expression you've written is not the series capacitance of two capacitors, so I still don't know what you mean by "total capacitance". Do you know what you mean by "total capacitance"? – DanielSank Feb 02 '24 at 05:48
  • I've asked what I think you might be trying to ask here, and I wrote an answer. – DanielSank Feb 02 '24 at 06:41

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