What is capacitance matrix?

In the system consisting of several charged electrodes the potential of every electrode depends upon all the electrode charges. If the media properties are independent from the field (this condition is always supposed in the Electrostatic QuickField problems) this function may be presented as a linear equation:

U₁ = a₁₁ * q₁ + a₁₂ * q₂ + ... + a_1n * q_n
U₂ = a₂₁ * q₁ + a₂₂ * q₂ + ... + a_2n * q_n
......
U_n = a_n1 * q₁ + a_n2 * q₂ + ... + a_nn * q_n,

(1)

where
    U_k - k-electrode potential,
    q_k - k-electrode charge,
    a_ij - potential coefficients.

Constants in the equations (1) are called potential coefficients.

Coefficients a_ii, located at the matrix diagonal designate the contribution of i-conductor charge to its own potential. They are called self-capacitance coefficients. Coefficients not on the diagonal a_ij (then i ≠ j) correspond to j-conductor contribution into i-conductor potential. They are called mutual potential coefficients.

Quite often the inverse problem-calculation of charges basing on the known potentials distribution arises. Resolving the system of equations (1) for charges (q₁, q₂, ... q_n) yields to:

q₁ = b₁₁ * U₁ + b₁₂ * U₂ + ... + b_1n * U_n
q₂ = b₂₁ * U₁ + b₂₂ * U₂ + ... + b_2n * U_n
......
q_n = b_n1 * U₁ + b_n2 * U₂ + ... + b_nn * U_n

(2)

In the system (2) coefficients b_ij have dimensions of capacitance. They are usually called electrostatic induction coefficients or partial capacitances relative to ground. Coefficients b_ij may be positive or negative.

It is often required in practice to replace the conductors system by their equivalent schema, where each pair of conductors is presented by the capacitors with specially fitted capacitances. This form corresponds to the system of equations (3) where the conductor charges are expressed through the potential differences between the conductor and other conductors, including earth:

q₁ = c₁₁ * (U₁ - 0) + c₁₂ * (U₁ - U₂) + ... + c_1n * (U₁ - U_n)
q₂ = c₂₁ * (U₂ - U₁) + c₂₂ * (U₂ - 0) + ... + c_2n * (U₂ - U_n)
......
q_n = c_n1 * (U_n - U₁) + c_n2 * (U_n - U₂) + ... + c_nn * (U_n - 0)

(3)

This form (3) is convenient because c_ij is always positive and allows natural interpretation as capacitances of the equivalent schema. c_ii coefficient corresponds to the contribution to the conductor charge caused by its own potential, that is self-capacitance. Coefficients c_ij where i and j are different correspond to the part of the conductor i charge caused by the potential difference between this and j conductor, which is equivalent to the capacitance of the capacitor formed by electrodes i and j. They are called partial capacitances.

All three matrices are symmetrical, i.e a_ij = a_ji.