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I am having a problem in understanding how, in a multiconductor system, the Capacitance Matrix influences the measure of voltage between two electrodes.

In the case that interest me we have 4 electrodes, 2 emitting ones and 2 receiving ones. We transmit a known sinusoidal current into the the emitting electrodes and we measure a voltage on the receiving ones.

We calculate the impedance of the quadrupole :

$$ Z= \frac{V_{r2}-V_{r1}}{i} $$

Where \$Vr\$ are the electrical potentials on the receiving electrodes, and \$i\$ is current emmited.

It is my understanding that the measured voltage is influenced by any conductor present in the medium close-by, therefore we have to determine the Capacitance Matrix to obtain the real voltage measured.

$$ V_{real} = V_{measured} \alpha C_{ij} $$

What I don't understand is the relation between the measured voltage \$V_{measured}\$, the capacitance matrix \$C_{ij}\$ and the real voltage\$V_{real}\$.

If any of you could help understand this I would be very grateful.

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I found the solution to my problem :

The equations can be written :

$$ I = V \times C_{ij} \times i\omega $$

Where I is the vector of known currents going threw the electrodes, V the potential of the electrodes and omega the rotational frequency of the current.

In this case the potential of the transmitting electrodes is known and the current of the receiving electrodes is equal to 0. We can therefore find a system of equations that will allow us to derive the potential of the receiving electrodes.

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