**Combining Independent Sources**

An inspection of the KVL equations for a series circuit shows that the order in which elements are placed in a series circuit makes no difference. An inspection of the KCL equations for a parallel circuit shows that the order in which elements are placed in a parallel circuit makes no difference. We can use these facts to simplify voltage sources in series and current sources in parallel.

**Combining Independent Voltage Sources in Series**

It is not possible to combine independent voltage sources in parallel, since this would violate KVL. However, consider the series connection of two ideal voltage sources shown in (a) below:

From KVL we know that *v *= *v*1 + *v*2 , and by the definition of an ideal voltage source, this must be the voltage between nodes *a *and *b*, regardless of what is connected to them. Thus, the series connection of two ideal voltage sources is equivalent to a single independent voltage source given by:

Clearly, the obvious generalization to *N *voltage sources in series holds.

**Example**

In a previous example we determined the current *i *in the one-loop circuit shown below:

By rearranging the order in this one loop circuit (of course this does not affect *i*), we obtain the circuit shown below:

We can now combine the series independent voltage sources and the series resistors into single equivalent elements:

By Ohm’s Law:

*i *= – 24/12 = -2 A