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Kirchhof's Current Law

2 June, 2016 - 15:17

At every node, the sum of all currents entering or leaving a node must equal zero. What this law means physically is that charge cannot accumulate in a node; what goes in must come out. In the example, Figure 3.6, below we have a three-node circuit and thus have three KCL equations.

\begin{align*} -i-i_1&=0\\ i_1-i_2&=0\\ i+i_2&=0\\ \end{align*}

Note that the current entering a node is the negative of the current leaving the node.

Given any two of these KCL equations, we can find the other by adding or subtracting them. Thus, one of them is redundant and, in mathematical terms, we can discard any one of them. The convention is to discard the equation for the (unlabeled) node at the bottom of the circuit.

Figure 3.7 Simple circult
The circuit shown is perhaps the simplest circuit that performs a signal processing function. The input is provided by the voltage source vin and the output is the voltage vout across the resistor labelled R2.  

Exercise 3.4.1

In writing KCL equations, you will find that in an n-node circuit, exactly one of them is always redundant. Can you sketch a proof of why this might be true? Hint: It has to do with the fact that charge won't accumulate in one place on its own.