Mixed networks

To find the equivalent resistance of a mixed network, replace each parallel or series group by its equivalent, working step by step from the farthest end towards the source. To find individual currents and voltages, work backwards: total current → voltage across each group → current in each branch.

A reliable method:

1. Identify the groups that are clearly series (the same current, with no junction between them) or clearly parallel (the same two end points).
2. Replace the innermost group first and redraw a simpler circuit.
3. Repeat until one resistor is left; that is R.
4. Find the main current from I = V / R (or I = V_B / (R + r) if the source has internal resistance).
5. Go back through the diagram: voltage across each equivalent resistor = I × its value; then split the current between parallel branches.
6. Check by adding the branch currents and by checking that the voltages around the circuit add up to the battery voltage.