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}(Source_{1}+ Source_{2}+ Source_{3}+….Source_{N}) = V_{x}(Source_{1}) + V_{x}(Source_{2}) + V_{x}(Source_{3}) +… 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

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)**

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