I'm currently learning about nodal analysis from a set of lecture notes. The notes are rather brief, and one question which occurred to me is whether it is possible for the admittance matrix to be singular.
That is, can we imagine a circuit that's physically possibly but whose admittance matrix is singular, so that we cannot determine the node potentials using nodal analysis?
It seems plausible that the answer would be in the negative, but I would like to see a formal argument. Note that I am not talking about issues with the admittance matrix being nearly singular, which might be problematic for numerical solution methods.
Best Answer
If you connect two nodes with a short circuit, then other things permitting you can still solve the nodal equations.
If you connect two nodes with two short circuits, then in the physical world it's not possible to determine how much current flows in each. If you set up the admittance matrix for this situation, it will be singular.
A similar situation exists if you put two open circuits in series and try to solve for the mid point voltage.
This is quite a common error in SPICE analysis, where it's all too easy to have a floating node between two capacitors, or the parallel connection of two ideal inductors. A judiciously placed very large or very small resistor is needed to bring the singular matrix back to computability.