11.5 Compound Direct Current Circuits

Series and parallel structure

A compound direct current circuit contains multiple resistors arranged in combinations of series and parallel branches. In series, the current is the same through each element and the potential differences add. In parallel, the potential difference is the same across each branch and the currents add. These two ideas let you reduce many circuits step by step to find equivalent resistance and then work backward to determine branch currents and voltages. The same topic also includes ideal measurement tools: an ideal ammeter is placed in series and has zero resistance, while an ideal voltmeter is placed in parallel and has infinite resistance.

The standard equivalent-resistance formulas are

and

The parallel formula can be derived from conservation of current. Since the same voltage V appears across each branch, the total current is

But also ${I = V/R_{\text{eq}}}$, so