Calculation method

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The method for calculation of the self- and partial capacitances of the conductor system is based on the electrostatic field energy.

W = 1/2 * (q₁ * U₁ + q₂ * U₂ + .... + q_n * U_n)(4)

Use equations (2) for exclusion of the conductor charges from (4) by expressing them via potentials:

    W = 1/2 * (
        U₁ * (b₁₁ * U₁ + b₁₂ * U₂ + ... + b_1n * U_n) +
        U₂ * (b₂₁ * U₁ + b₂₂ * U₂ + ... + b_2n * U_n) +
             ........
        U_n * (b_n1 * U₁ + b_n2 * U₂ + ... + b_nn * U_n) );

Coefficients b_ji may be found by following method. First of all let us solve the series of n problems, where the test potential (100 V) is applied to only one of the conductors, and the rest of them are grounded (zero potential). Electric field energy for these cases looks like:

W_i = 1/2 · b_ii · (U₁)².

Thereby own coefficients of the electrostatic induction could be found.

Then for each pair of conductors (i, j) we should solve the problem where the test potential is applied between them, and all other conductors are grounded. Energy equation is below:

W_ij = b_ij · U_i · U_j

There will be n·(n-1)/2 such problems in total.

Thus, to find the matrix of electrostatic induction coefficients b_ij in the system of n conductors it is sufficient to solve n·(n+1)/2 problems. Each solution needs total electric field energy calculation. Computation of partial capacitances c_ij, from induction coefficients b_ij is a simple task.