I'm looking for an intuitive explanation (not a proof) of why the equations produced by nodal analysis and mesh analysis are always linearly independent and always have a unique solution, if the circuit has a unique solution.
Again, please note, I'm not looking for a proof, rather an intuitive explanation.
No matter how many nodes you have, when doing nodal analysis, you describe the currents going into and out of each node. As you walk through each node, you'll end up with 1 linearly independent equation that describes all of the current going into and out of it. In nodal analysis, everything that goes into a node must come out of it.
When you finally get to the last node in your analysis, it should become obvious that none of the inputs or outputs to that node may be tweaked to your liking. Every single input (or output) to this node already has some other node determining how much current flows into or out of it. That last node can't be linearly independent because it's dependent upon all of the other nodes.
You can think of this like water pipes where voltage sources are pumps, and resistors are narrow parts of the circuit. In a circuit (i.e. closed loop), electrons can never escape from the system, they always just go in loops. The same thing would occur in a network of tubes with pumps pushing water around them with constrictions. At any joint where 3 or more tubes connect, what flows into the joint will be equal to what flows out of the joint. If you're accounting/measuring how much goes into and out of every joint, when you get to the last one, you'll realize that you don't need to measure the amount going into and out of that joint or node because you've already accounted for it because you assume that your pipes aren't leaking and you're not adding any water to the network of pipes.
That's basically all that that statement is saying. Since it's a closed loop system, you can't add or remove electrons, so the last node can't be linearly independent. It's dependent upon all of the other nodes.
It's not true that you will always end up with \$n-1\$ independent equations.
It's possible to draw ideal circuits where, for example, a node voltage is undetermined.
A trivial example:
simulate this circuit – Schematic created using CircuitLab
KCL at node 1 is
$$1A = 1A$$
Thus, \$V_1\$ is indeterminate; \$V_1\$ can be any voltage.
So, if there's a unique solution for \$n-1\$ node voltages (the remaining node voltage is 0V by definition), then it follows that there are \$n-1\$ independent equations.
However, if there is one or more floating voltage sources, ordinary node voltage analysis with KCL won't give you the full set of equations. In that case, you combine nodes into supernodes and augment the reduced set of KCL equations with KVL equations.
Why are the nodal analysis equations linearly independent?
Because each equation involves the current in the branches connected to a particular node. Since each node is connected to a different set of branches, the resulting equations will not be linear combinations of each other.
Why do the nodal analysis equations have a unique solution?
Because n independent linear equations in n unknowns have a single unique solution.
Linear systems can have infinitely many solutions (but only if the number of independent equations is less than the number of unknowns, or no solutions (but only if the number of independent equations is greater than the number of unknowns).
What about the mesh analysis?
The mesh analysis is just the dual of the nodal analysis.
Each equation involves the current potential in the branches connected to around a particular node mesh. Since each node mesh is connected to a different set of branches, the resulting equations will not be linear combinations of each other.
I'd say just draw a simple circuit with 2 nodes. Just 2 squares. Then add more and more, you might just get it "intuitively" this way.
It's the first time I hear about this "rule" and it's what I did to figure out. I don't have a real answer tho.
In a simple example, nodal analysis "encapsulates" 2 nodes per equation and one of these nodes has to be unique and the other is ground. Therefore there are N independent equations where N is the number of nodes excluding ground.
