Nodal Analysis

A circuit with $b$ branches has $2b$ unknowns since there are $b$ voltages and $b$ currents. Hence, $2b$ linear independent equations are required to solve the circuit. If the circuit has $n$ nodes and $b$ branches, it has

  • Kirchoff’s current law (KCL) equations
  • Kirchoff’s voltage law (KVL) equations
  • Characteristic equations (Ohm’s Law)

There are only $n-1$ KCLs since the nth equation is a linear combination of the remaining $n-1$. At the same time, it can be demonstrated that if we can imagine a very high number of closed paths in the network, only $b-n+1$ are able to provide independent KVLs. Finally there are $b$ characteristic equations, describing the behavior of the branch, making a total of $2b$ linear independent equations.

The nodal analysis method reduces the number of equations that need to be solved simultaneously. $n-1$ voltage variables are defined and solved, writing $n-1$ KCL based equations. A circuit can be solved using Nodal Analysis as follows

  • Select a reference node (mathematical ground) and number the remaining $n-1$ nodes, that are the independent voltage variables
  • Represent every branch current $i$ as a function of node voltage variables $v$ with the general expression $i = g(v)$
  • Write $n-1$ KCL based equations in terms of node voltage variable.

The resulting equations can be written in matrix form and have to be solved for $v$. $$\boldsymbol{Y} \boldsymbol{v} = \boldsymbol{i}$$