Combination circuits, circuits with series and parallel resistors

Many circuits have resistors in both series and parallel. To find the total current through the circuit you generally want to replace the set of resistors with its equivalent resistor. This can be done by identifying a pair of resistors that is either in series or in parallel, replacing that pair by a single equivalent resistor, and iterating until you're left with one resistor in the circuit.

This allows the total current in the circuit to be determined. The current flowing through each resistor can then be found by undoing the reduction process.



A sample combination circuit

In the circuit above, the resistors are:

R1 = 6 Ω
R2 = 36 Ω
R3 = 12 Ω
R4 = 3 Ω

What is the potential difference across each resistor? How much current passes through each resistor?

To solve this we need to find the equivalent resistance of the set of resistors. We'll contract the circuit from 4 resistors to 1.

Step 1 - R2 and R3 are in parallel. Replace this pair by a single resistor R23:

1/R23 = 1/R2 + 1/R3 = 1/36 + 1/12 = 4/36.

Therefore R23 = 36/4 = 9 Ω.

Step 2 - R4 and R23 are in series. Replace that pair by a single resistor R234 = 3 + 9 = 12 Ω.

Step 3 - R1 and R234 are in parallel. Replace that pair by a single resistor Req:

1/Req = 1/R1 + 1/R234 = 1/6 + 1/12 = 3/12

Req = 12/3 = 4 Ω.

Step 4 - Determine the current through Req.

I = ΔV / Req = 12/4 = 3A.

Now we need to expand the circuit back to the original four resistors, and determine the current through, and potential difference across, each one as we go.

Step 1 - Req represents R1 and R234 in parallel. Resistors in parallel have the same potential difference but split the current. The potential difference across each is 12 volts. The current is:

I4 = ΔV / R4 = 12/6 = 2A.

I234 = ΔV / R234 = 12/12 = 1A.

Step 2 - R234 represents R4 and R23 in series. Devices in series have the same current through them (whatever current their series combination has), so they each have 1A.

ΔV4 = I * R4 = 1 * 3 = 3 V.

ΔV23 = I * R23 = 1 * 9 = 9 V.

These add to the potential difference across the series combination.

Step 3 - R23 represents R2 and R3 in parallel. The potential difference in both cases is 9 V.

I2 = ΔV / R2 = 9/36 = 0.25 A.

I3 = ΔV / R3 = 9/12 = 0.75 A.

Step 4 - A good way to check for consistency is to label the potential at different points. Pick some point as a reference (say, 0 V at the negative terminal of the battery) and label other points relative to that. Check that the potential differences across the resistors are consistent with these potential values.