In the bridge network shown below:
How many nodes are there?
How many KCL equations are independent?
How many loops are there?
How many KVL equations are independent?
Notice that in any circuit there is always one more node than there are independent KCL equations. If two nodes share a branch, the current entering one node from that branch is the negative of the current entering the other node from that branch. So the sum of the currents entering two nodes does not count current going from one to the other. As a consequence, the sum of all the currents entering all but one of the nodes is the same as the current entering the remaining node. So the KCL equation for that node is the sum of the KCL equations for all the other nodes.
Also, in a circuit with more than one loop there are always more loops than KVL equations. The argument is analogous with the argument for nodes: if two loops share a branch, and we count the voltages counterclockwise in each loop, then the KVL equation for the branches containing both subloops is the sum of the KVL equations for each subloop. This bigger equation does not count the current in the shared branch, since it is of opposite sign in the two subloops.
A thought: if you have take a class in multivariate calculus, these arguments should remind you of the proofs of the divergence theorem and of Stokes's theorem.
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