Analyzing Parallel Resistor–Inductor AC Circuits: A Practical Guide

Using the same resistor and inductor from the series example, we now arrange them in parallel to explore AC behavior.

Parallel R‑L circuit.

Because the source frequency and component values remain unchanged, the individual impedances stay the same. We start our analysis with the following known values:

In a parallel network the voltage across each branch is identical. Transfer the total source voltage (10 V ∠ 0°) to every column:

Apply Ohm’s Law vertically (I = E / Z) to compute the branch currents:

Just as in DC, branch currents add to give the net current, consistent with Kirchhoff’s Current Law for AC:

Finally, total impedance can be found either by using Ohm’s Law (Z = E / I) in the “Total” column or by applying the reciprocal formula common to parallel impedances:

Although the reciprocal method can be calculator‑heavy, it is mathematically equivalent to the vertical calculation. Regardless of the chosen approach, the resulting total impedance is identical:

Review

• Parallel impedances combine with the reciprocal formula: Z_total = 1 / (1/Z₁ + 1/Z₂ + …).
• AC Ohm’s Law remains E = I Z; I = E / Z; Z = E / I.
• When a resistor and inductor share a parallel branch, the net impedance lies between 0° and +90°, while the circuit current falls between 0° and –90°.
• Parallel AC circuits mirror DC behavior: voltage is uniform, branch currents sum, and impedances reduce via the reciprocal rule.

Related Worksheets

• Inductors Worksheet