Measurement of Inductance and Capacitance

Impedance-based measurement

An AC measurement determines both magnitude and phase of impedance at a specified frequency. Ideal inductive impedance is jωL and ideal capacitive impedance is 1/(jωC), where ω = 2πf. Real components have winding resistance, dielectric loss, leakage and parasitic reactance. An LCR meter reports a value using a selected series or parallel equivalent circuit. Frequency, amplitude, DC bias, fixtures and component nonlinearity can change the result. An ideal reactance formula should not be applied blindly to total impedance magnitude when resistance is appreciable.

Series models

For Z = Rs + jXs, an inductive component has Ls = Xs/ω and quality factor Q = Xs/Rs. A capacitive component has Cs = −1/(ωXs), Xs < 0, and dissipation factor D = Rs/|Xs| in the series model. Open/short fixture compensation and suitable terminal connections reduce fixture errors but do not remove all uncertainty.

Worked examples

At f = 1 kHz, an inductor has measured impedance 10 + j62.832 Ω. Its series inductance is 62.832/(2π × 1000) = 0.010 H = 10 mH. Series Q = 62.832/10 = 6.283. Using |Z|/ω would give a different and incorrect series-inductance estimate. At the same frequency an ideal capacitor has reactance −159.155 Ω. C = 1/(2π × 1000 × 159.155) = 1 μF approximately.

Common mistakes

Use hertz versus rad/s consistently. State the equivalent circuit and frequency. Discharge capacitors before connecting measurement equipment as specified by its procedure. Check that the component lies within the instrument’s range and test-level limits.

Quick check

1. Is an ideal inductor’s impedance real? No, it is positive imaginary.
2. Does a capacitor have negative reactance? Yes, under the exp(jωt) convention.
3. Can a real component’s measured value depend on frequency? Yes.