Calculating Lumped Parameters with Ohm’s Law
To calculate the lumped parameters, there must be at least two electrodes in the system, one of which must be grounded. Either a voltage or a current can be forced on the electrodes. After the simulation, extract the other property or the energy and use it when calculating the lumped parameter.
There are several available techniques to extract the lumped parameters. Which one to use depends on the physics interface, the parameter of interest, and how the model is solved. The overview of the techniques in this section use a 4-by-4 matrix example for the lumped parameter matrix. This represents a system of at least five electrodes, where four are used as terminals and the rest are grounded, as illustrated in Figure 3-2.
Figure 3-2: A five-electrode system with 4 terminals and one ground electrode.
If a system specifies that all electrodes are terminals, the results are redundant matrix elements. This is better understood by considering a two-electrode system. If both electrodes are declared as terminals, a 2-by-2 matrix is obtained for the system. This is clearly too many elements because there is only one unique lumped parameter between the terminals. If in addition one or more ground electrodes are declared, the system has three unique electrodes and the lumped parameter matrix becomes a 2-by-2 matrix.
Forced Voltage
If voltages are applied to the terminals, the extracted currents represent elements in the admittance matrix, Y. This matrix determines the relation between the applied voltages and the corresponding currents with the formula
so when V1 is nonzero and all other voltages are zero, the vector I is proportional to the first column of Y.
In electrostatics the current is replaced with charge and the admittance matrix is replaced with the capacitance matrix
Fixed Current
It might be necessary to calculate the Z-matrix in a more direct way. Similar to the Y calculation, the Z calculation can be done by forcing the current through one terminal at the time to a nonzero value while the others are set to zero. Then, the columns of the impedance matrix are proportional to the voltage values on all terminals:
In magnetostatics this option means that the energy method is used; see Calculating Lumped Parameters Using the Energy Method below.
Fixed Charge
The Electrostatics interface can use total charge instead of total current. This gives the inverted capacitance matrix in a similar manner as the Z and Y matrices.