Superposition

Yet another powerful tool that can help in solving complex networks.  This method can only be applied to linear networks.  To be linear, the measured voltage or current must satisfy the properties of Additivity and Homogeneity.  These properties are mathematically described below, and provided with an example.

Additivity:

 V_x(Source₁ + Source₂ + Source₃ +….Source_N) = V_x(Source₁) + V_x(Source₂) + V_x(Source₃) +… V_x(Source_N)

Homogeneity:

 V_x(A*Source)=  A*V_x(Source)

These equations merely demonstrate that voltage at a node is equivalent to the algebraic sum of all voltages generated from each independent source.  Basically, to find the collective sum of all voltages, you must solve the circuit N times, where N is the number of independent sources (voltage or current).  When separately solving for each source, make sure to turn off remaining independent sources.  To do this we set the remaining voltage and current sources to zero.  A voltage source set to zero is the same as a short, and a current source set to zero is the same as an open circuit, as shown in Figure 1.

Figure 1. Turning off sources

Example:

In this example there are 2 independent sources, Vin and I1, therefore N=2, which will result in two sets of equations for Vx.

First we can turn off the I1 source and solve for V_X with respect to Vin.

Vx (Vin source)=

Next we solve for V_X with respect to I1, while turning the Vin source off.

Vx = (Current source)

Therefore,

Vx(total) = Vx(Vin source) + Vx(Current source)