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Combining Independent Voltage Sources in Series

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:

Combining Independent Voltage Sources in Series

From KVL we know that v v1 + v2 , 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:

independent-voltage-sources-in-series-equation

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:

independent-voltage-sources-in-series-circuit-example

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

independent-voltage-sources-in-series-circuit-example-1

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

independent-voltage-sources-in-series-circuit-example-2

By Ohm’s Law:

i =  – 24/12 = -2 A

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