# Voltage spreading to resistors network

I want to model several electrodes (say 3) and GND on a resistive substrate.

simulate this circuit – Schematic created using CircuitLab

To characterise it, I apply a current on one electrode to GND. Then I measure the voltage on all electrodes and divide it by the current. I then repeat this for each electrode

I obtain a matrix as follow (symmetrical):

r1   r12  r13
r21  r2   r23
r31  r32  r3


Note: this matrix is not a proper matrix of resistances, it describes the interaction between electrodes.

This matrix is used to model the substrate with voltage sources as follow:

simulate this circuit

Now I would like to model it with a network of discrete resistors. The resistances R1, R2 and R3 are connected form electrode to GNDand R12, R13 and R23 are resistances between electrodes:

simulate this circuit

How can I convert my matrix into values of resistances in the network?

Note: the solution must be scalable as many more electrodes could be used.

• When you edited to change to "I apply a current on one electrode" you replaced the current source between two electrodes to: from one electrode to the GND (presumably). Is this correct? Commented Jul 29, 2019 at 15:19
• yes this is correct, I first had a reference electrode, which I changed to GND, I think it makes it simpler to understand Commented Jul 29, 2019 at 15:22
• OK. Applying currents between E1-E2, E1-E3 and E2-E3 would not be necessary too? Considering your previous diagrams. Commented Jul 29, 2019 at 15:26
• I could do these measurements too, but I think it would be redundant. The first model is complete and works well so I believe that the matrix includes all the required information to build the resistive network. Commented Jul 29, 2019 at 15:30

You have measured the impedance matrix (Z-matrix) of the device. If you invert the matrix, you will have the admittance matrix (Y-matrix) of the device. The Y-matrix will have positive entries on the diagonal and negative numbers for the off diagonal. You are familiar with resistance. The name for the inverse (1/R) of a resistor is an "admittance". Think of every row,col of the Y matrix as representing a node of a circuit. For the Y matrix, the off diagonal entries are the negative of the admittance between the nodes. The diagonal entries are the sum of the all the admittances connected to the node; (that is all the admittances between the node and the others plus the admittance from the node to ground). So, you should use MATLAB or some program to invert the matrix. Then use the off diagonal entries to find the resistors between the electrodes. Any extra admittance in the diagonal entry is a resistor from that electrode to ground.

• Are you saying that Y-matrix = inv(Z-matrix), then R12 = -1/Y12 , R13 = -1/Y13 , R23 = -1/Y23 and R1 = 1/(Y1 + Y12 + Y13) , R2 = 1/(Y12 + Y2 + Y32) , R3 = 1/(Y13 + Y23 + Y3) ? Commented Jul 29, 2019 at 16:24
• To be pedantic (which we should, since this is a teaching site), the inverse of resistance is conductance. The inverse of impedance is admittance. Commented Jul 29, 2019 at 17:01
• Form the Y-matrix. Then the resistors in your equivalent circuit can be calculated as 1/R12=-y12 , 1/R13=-y13, 1/R1=1/y11-1/y12-1/y13 and so on. You should find the Y-matrix is symmetrical, because you don't have any weird parts (parts called "non-reciprocal") in your circuit. Commented Jul 30, 2019 at 19:32

I think your basic model is flawed. If you ground one electrode and inject current at some other electrode then current will flow through many of the resistors. You have assumed that current flows through just 3 resistors. Furthermore, the various resistors are not in series as you seem to assume (when substituting with voltage sources)...the current that flows through R1, R12, and R13 is not necessarily the same, for example.

If your model is correct you should see that $$\R_{12} = -R_{21}\$$. Is that happening?

I think problems like this are generally solved by using the Van der Pauw method to measure the sheet resistance of the material, and then creating a finite-element model of the actual structure and electrodes.

• In the first model, those are not resistors, they are voltage sources, it implies that if there is a current in E2 (i2) it induces a voltage on the other electrodes too, proportionally to the element of the matrix. In other words, i2 and i3 are also present too in the branch of E1. I rather have r12 = r21. i will have a look at Van der Pauw Commented Jul 29, 2019 at 15:38
• If they are not resistors, why did you use Ohm's Law to specify the voltage? No, $R_{21} = -R_{12}$ unless you recognize that the current is flowing in the opposite direction. You really should clean up the naming and nomenclature of your diagrams...your first model shows a current of i1 injected into 3 series elements but those elements have different currents...that is invalid. Commented Jul 29, 2019 at 15:47
• I did not call it Ohm's law, It is just that the effect of one electrode on another is proportional to the current applied on the first electrode. The matrix is not resistors values, even if it has the same unit. it is a matrix of ratios between the current on one electrode and the voltage induced on the neighbors electrodes. I actually have r12 = r21, because positive current on E1 induces a positive voltage on E2 and positive current on E2 induces the same positive voltage on E1. The next 3 elements are voltage sources, you can apply any current you want in voltage sources. Commented Jul 29, 2019 at 16:00