Capacitance matrix calculation

Capacitance matrix provides the quantitative estimation of the mutual affect of the conductors in the electrostatic system. In practical engineering the field distribution in the system of conductors is replaced by the equivalent electric circuit consisting of capacitors.

Cable entries

line capacitance

Equivalent schema

equivalent capacitance scheme

PCB conductors

pcb capacitance system

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,

where
    U_k - k-electrode potential,
    q_k - k-electrode charge,
    a_ij - self and 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

In this system 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 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)

Coefficients c_ij are called partial capacitances.

All three matrices are symmetrical, i.e c_ij = c_ji.

QuickField's capacitance matrix calculation add-in was designed to automate the frequent task of the grounded (b) and lumped (c) capacitance matrices calculation.

You can find more information in User manual or QuickField help system.