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Combination Series/Parallel Circuits

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Series and parallel in the same circuit

Real circuits often contain some parts in series and other parts in parallel. The trick is not to solve everything at once. Simplify one obvious piece at a time.

12 V R1 = 2 Ω R2 = 6 Ω R3 = 3 Ω

A useful first picture

Think of cleaning up a complicated fraction in arithmetic. Work on the part inside the parentheses first, replace it with one simpler value, and then continue.

Symbols

R1, R2, R3 are individual resistor values. RT means total resistance. IT means total current supplied by the source.

Worked example — three steps

Step 1: R2 = 6 Ω and R3 = 3 Ω are parallel. Their equivalent resistance is 2 Ω.
Step 2: That 2 Ω equivalent is in series with R1 = 2 Ω.
RT = 2 + 2 = 4 Ω.
Step 3: Find total current.
IT = 12 V ÷ 4 Ω = 3 A.

We turned one complicated-looking circuit into a simpler circuit, then used Ohm's law.

A little history

As electrical networks became larger in the 1800s, simple one-resistor relationships were not enough. Techniques developed by scientists such as Gustav Kirchhoff made it possible to analyze networks systematically instead of by guesswork.

Try these yourself

R2 = 6 Ω and R3 = 3 Ω are in parallel. What is their equivalent resistance? 12 V R1 = 2 Ω R2 = 6 Ω R3 = 3 Ω
For 6 Ω and 3 Ω in parallel, 1/R = 1/6 + 1/3 = 3/6, so R = 2 Ω.
That 2 Ω parallel equivalent is in series with R1 = 2 Ω. What is total resistance? 12 V R1 = 2 Ω R2 = 6 Ω R3 = 3 Ω
Series resistances add: R_T = 2 Ω + 2 Ω = 4 Ω.
The circuit has 12 V and R_T = 4 Ω. What is total current? 12 V R1 = 2 Ω R2 = 6 Ω R3 = 3 Ω
I_T = V/R_T = 12/4 = 3 A.

A little deeper

After finding total current, you can work backward through the circuit. First find the voltage drop across the series resistor. The voltage left over appears across the parallel network. Then use that parallel voltage to find each branch current.

This “simplify, solve, then work backward” method is one of the most useful habits in basic circuit analysis.

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