# Existence and uniqueness of solutions to Thévenin circuits

Suppose I have a circuit composed of ideal resistors, voltage sources and current sources, and Kirchoff's laws hold. The circuit can be solved using a system of linear equations derived from Kirchoff's laws. However, sometimes a solution might not exist. For example, if we short-circuit a voltage source, Kirchoff's voltage law fails on the short circuit.

As a person with a mathematical background, I am wondering the following:

1. What are the necessary and sufficient conditions for the existence of a solution?
2. Is the solution always unique?
• I don't think you should be shorting voltage sources anyway. It means that there was no resistance so equivalent resistance is 0 and equivalent voltage is the source voltage. So it kind of still applies. Commented Nov 18, 2020 at 12:00
• See if this from yesterday helps. Commented Nov 18, 2020 at 12:31
• You can't say KVL'fails' when a voltage source is shorted because all voltage sources have an internal resitance in real life. If you argue with an ideal voltage source, KVL still doesn't fail, V/R=I becomes V/0=I, which is still true for this circuit. Mathematically, division by zero is 'undefined', so is the behavior of this ideal circuit. You can't say Math 'failed'. Can you? Commented Nov 18, 2020 at 16:40
• I think you should start by providing mathematically rigorous definitions of "circuit" and "solved". It seems like you are playing pretty loose with those terms. Commented Nov 18, 2020 at 21:54
• In this model I assume that all sources are ideal. I am not concerned with the real world. I told you I have a math background! To formalize the setup, let's say that the circuit is a bipartite graph where vertices on one side represent components and vertices on the other side represent nodes (in the sense of the node method of circuit analysis). Edges connect components to the adjacent nodes. Solving a circuit means finding a set of voltages for the nodes and currents for components that are consistent with KVL and KCL and the voltage and current sources. Commented Nov 18, 2020 at 23:18