1.4 AC Circuit Impedance, Reactance & Resonance

Section 1.3 established that voltage and current stay in phase only in purely resistive AC circuits. Real electrical systems — motor windings, transformers, ballasts, power-factor-correction capacitor banks, and control-circuit wiring — contain inductance and capacitance as well as resistance, and these components oppose current flow in ways that depend on frequency and shift the phase relationship between voltage and current. This opposition is called reactance, and combined with resistance it forms impedance, the complete opposition to current flow in an AC circuit.

Inductive Reactance

An inductor (a coil, such as a motor or transformer winding) opposes any change in current by generating a back-EMF (electromotive force). Because AC current is constantly changing, an inductor continuously opposes it — this opposition is called inductive reactance (X_L), measured in ohms (Ω):

X_L = 2π f L

where f is frequency in hertz and L is inductance in henries (H). Because frequency appears directly in the formula, inductive reactance increases as frequency increases — at higher frequencies, an inductor opposes current more strongly. At DC (f = 0), X_L = 0, which is why an inductor behaves like a plain wire (ignoring its winding resistance) once a DC current has stabilized.

In an inductive circuit, current lags voltage — the current waveform reaches its peak after the voltage waveform does. In a purely inductive circuit, this lag is a full 90°.

Worked Example — Inductive Reactance

A motor winding has an inductance of 0.1 H and is connected to the 60 Hz Philippine supply. What is its inductive reactance?

X_L = 2π f L = 2π(60)(0.1) = 37.7 Ω

Capacitive Reactance

A capacitor opposes any change in voltage by storing and releasing charge. This opposition is called capacitive reactance (X_C), also measured in ohms:

X_C = 1 / (2π f C)

where C is capacitance in farads (F). Because frequency appears in the denominator, capacitive reactance decreases as frequency increases — the opposite behavior from an inductor. At very high frequencies, X_C approaches zero and a capacitor behaves almost like a short circuit; at DC (f = 0), X_C is theoretically infinite, which is why a fully charged capacitor blocks steady DC current entirely.

In a capacitive circuit, current leads voltage — the current waveform reaches its peak before the voltage waveform does, again by a full 90° in a purely capacitive circuit.

Worked Example — Capacitive Reactance

A power-factor-correction capacitor rated at 50 microfarads (µF) is connected to the 60 Hz supply. What is its capacitive reactance?

X_C = 1 / (2π f C) = 1 / (2π × 60 × 0.00005) = 1 / 0.018850 = 53.05 Ω

Impedance: Combining Resistance and Reactance

Impedance (Z), also measured in ohms, is the total opposition to current flow in an AC circuit, combining resistance (R) and net reactance (X). Because voltage across a resistor is in phase with current, while voltage across a reactance is 90° out of phase with current, resistance and reactance cannot simply be added arithmetically — they combine as vectors, using the Pythagorean relationship:

Z = √(R² + X²)

where X is the net reactance: X = X_L − X_C (inductive and capacitive reactance are 180° apart from each other, so they subtract).

Worked Example — Series RL Impedance

A series circuit has R = 30 Ω and X_L = 40 Ω (no capacitance present). What is the total impedance?

Z = √(R² + X_L²) = √(30² + 40²) = √(900 + 1600) = √2500 = 50 Ω

How Inductive and Capacitive Reactance Oppose Each Other

Because current lags voltage by 90° in a purely inductive circuit but leads voltage by 90° in a purely capacitive circuit, X_L and X_C act in directly opposite directions — a 180° phase relationship to each other. In a circuit containing both, their effects partially or fully cancel:

X_net = X_L − X_C

If X_L > X_C, the circuit behaves as net inductive (current lags voltage). If X_C > X_L, the circuit behaves as net capacitive (current leads voltage).

Resonance in a Series RLC Circuit

A circuit containing resistance, inductance, and capacitance (RLC) in series reaches resonance at the specific frequency where inductive reactance exactly equals capacitive reactance:

X_L = X_C

At this resonant frequency (f_r), the two reactances cancel completely (X_net = 0), leaving only resistance to oppose current:

Z = R (the minimum possible impedance for that circuit)

Because impedance is at its minimum, series RLC current is at its maximum at resonance, and — since X_L and X_C cancel — voltage and current return to being in phase, just as in a purely resistive circuit. The resonant frequency itself is found by setting X_L = X_C and solving for f:

f_r = 1 / (2π√(LC))

Worked Example — Resonant Frequency

A series RLC circuit has L = 0.2 H and C = 20 µF (0.00002 F). Find the resonant frequency.

f_r = 1 / (2π√(LC)) = 1 / (2π√(0.2 × 0.00002)) = 1 / (2π√0.000004) = 1 / (2π × 0.002) = 1 / 0.012566 = 79.6 Hz

Why This Matters for Motor and Control-Circuit Troubleshooting

Motor windings are heavily inductive, which is why induction motors draw a lagging current and operate at a poor power factor unless corrected. Utilities and large facilities install power-factor-correction capacitor banks specifically to offset motor inductive reactance and bring the net reactance — and therefore the phase angle between voltage and current — closer to zero, reducing the total current (and losses) the supply system must carry to deliver the same power.

However, adding capacitance to an inductive system creates the possibility of unintended resonance: if a facility's capacitor bank and the inductance of its transformers, motors, and cabling combine to produce a resonant frequency near the supply frequency (or near a harmonic of it), the resulting near-zero impedance path can allow abnormally high circulating currents and voltage magnification, damaging equipment. A licensed electrician evaluating "why does this motor circuit run hot" or "why did adding capacitors cause nuisance tripping" needs to recognize reactance and resonance as root causes, not just check for a simple overload.

Key Takeaways

• Inductive reactance (X_L = 2πfL) rises with frequency; capacitive reactance (X_C = 1/2πfC) falls with frequency — opposite behaviors.
• Impedance combines resistance and net reactance as a vector sum: Z = √(R² + X²), not simple addition.
• Current lags voltage in inductive circuits and leads voltage in capacitive circuits, each by up to 90°.
• At resonance (X_L = X_C), net reactance is zero, impedance is at its minimum (Z = R), and current is at its maximum.
• Power-factor-correction capacitors offset motor inductive reactance but can create unintended resonance if sized or applied incorrectly.