Power in AC Circuits

In a direct-current (DC) circuit, power is simply P = IV — voltage times current, and every joule delivered by the source does useful work. Alternating-current (AC) circuits are more subtle. Because voltage and current both oscillate sinusoidally and can be out of phase with each other, not all of the power that sloshes back and forth between source and load actually gets converted into heat, light, or mechanical work. Understanding how power behaves in AC circuits is essential for everything from designing home wiring to running the national electricity grid efficiently. AC circuits distinguish three related but distinct power quantities. Real power (P), measured in watts, is the average rate at which energy is permanently transferred — the part that actually does work. Reactive power (Q), measured in volt-amperes reactive (VAR), represents energy that oscillates back and forth between the source and reactive elements (inductors and capacitors) without ever being consumed. Apparent power (S), measured in volt-amperes (VA), is simply the product of the RMS voltage and RMS current and sets the size of cables, transformers, and generators that must be installed. The ratio of real power to apparent power is called the power factor (pf = P/S). A power factor of 1 means all delivered power does useful work; a power factor near zero means enormous currents flow while very little work is done — a nightmare for utility companies. By connecting capacitors or inductors to counteract unwanted reactance, engineers can 'correct' the power factor back toward unity, saving money and reducing carbon emissions from wasted electrical energy.

Key takeaways

Instantaneous Power and Phase Angle
Real Power P
Reactive Power Q
Apparent Power S and the Power Triangle
Power Factor Correction

Instantaneous Power and Phase Angle

In an AC circuit, voltage and current are v(t) = V_m cos(ωt) and i(t) = I_m cos(ωt − φ), where φ is the phase by which current lags voltage (positive for inductive loads). Instantaneous power is p(t) = v(t)·i(t). Using the product-to-sum identity this expands to p(t) = (V_m I_m / 2)[cos φ + cos(2ωt − φ)]. The first term is a constant — it is the average real power. The second term oscillates at twice the supply frequency and averages to zero over a full cycle.

Real Power P

Averaging over one period, the oscillating term vanishes and we get P = (V_m I_m / 2) cos φ = V_rms I_rms cos φ. This is the power actually consumed — turning electrical energy irreversibly into heat in resistors, light in bulbs, or torque in motors. The factor cos φ is the power factor. For a purely resistive load φ = 0, so cos φ = 1 and P = V_rms I_rms. For a purely reactive load (ideal inductor or capacitor) φ = ±90°, so cos φ = 0 and P = 0 — no energy is consumed at all.

Reactive Power Q

The energy that sloshes back and forth with inductors and capacitors is quantified by reactive power: Q = V_rms I_rms sin φ, measured in VAR. Inductors absorb positive Q (they store energy in a magnetic field each half-cycle and return it the next). Capacitors absorb negative Q (they store energy in an electric field). This is why capacitor banks are placed next to inductive factory loads — the capacitor supplies the reactive power the inductor demands locally, preventing large reactive currents from flowing all the way from the generator.

Apparent Power S and the Power Triangle

Apparent power S = V_rms I_rms (in VA) is what an electrician measures with a voltmeter and ammeter without knowing the phase angle. The three quantities form a right triangle: S² = P² + Q². The angle of this triangle is φ, so P = S cos φ and Q = S sin φ. Think of S as the hypotenuse of a right triangle with legs P (horizontal) and Q (vertical). This 'power triangle' is a direct geometric expression of the Pythagorean theorem applied to AC power.

Power Factor Correction

If a factory runs large inductive motors (lagging power factor, say pf = 0.7), the utility must supply S = P/0.7 ≈ 1.43 P of apparent power to deliver P of real power. Every transmission cable must be sized for that larger current, causing extra I²R losses. Adding a capacitor in parallel reduces |Q|, shrinks S toward P, and raises pf toward 1. The formula for the required capacitance is C = Q_C / (ω V_rms²), where Q_C is the reactive power the capacitor must supply to cancel the inductive reactive power.

The Key Equation

S = \sqrt{P^2 + Q^2}, \quad P = S\cos\varphi, \quad Q = S\sin\varphi, \quad \text{pf} = \frac{P}{S} = \cos\varphi

S — Apparent Power (VA (volt-ampere)): Product of RMS voltage and RMS current; sets the required ratings of electrical equipment.
P — Real (Active) Power (W (watt)): Average rate of energy consumption; the only part that does useful work.
Q — Reactive Power (VAR (volt-ampere reactive)): Power oscillating between source and reactive elements; positive for inductive loads, negative for capacitive.
\varphi — Phase Angle (rad or degrees): Angle by which current lags (or leads) voltage; determines how much power is reactive vs. real.
\text{pf} — Power Factor (dimensionless (0 to 1)): Fraction of apparent power that is real; equals 1 for purely resistive loads, 0 for purely reactive loads.

Worked Example

Problem. An industrial motor is connected to a 240 V RMS, 50 Hz supply and draws a current of 15 A RMS at a lagging power factor of 0.80. (a) Find the real power P, reactive power Q, and apparent power S consumed by the motor. (b) What capacitance connected in parallel would correct the power factor to unity?

S = 3600\text{ VA},\; P = 2880\text{ W},\; Q = 2160\text{ VAR};\quad C \approx 119\,\mu\text{F}

Answer. S = 3600 VA, P = 2880 W, Q = 2160 VAR (inductive); correction capacitor ≈ 119 μF.

Figure

Diagram

Origins · 1 minCharles Proteus Steinmetz and how this was discovered

Charles Proteus Steinmetz

1865–1923 · German-American

Steinmetz developed the phasor (complex-number) method for analysing AC circuits, published in 1893. By representing sinusoidal voltages and currents as complex numbers (phasors), he transformed AC circuit analysis from laborious trigonometry into straightforward algebra. His framework makes power factor analysis, impedance calculations, and the concept of reactive power mathematically transparent — it is still the universal language of electrical engineering today. Steinmetz had severe kyphoscoliosis and stood only about 1.4 m tall, yet General Electric hired him specifically to work on AC power systems. His colleagues called him 'The Wizard of Schenectady,' and Thomas Edison — a sworn DC advocate — visited him seeking advice on AC power, a remarkable tribute to his genius.

Impedance and Phasors — Impedance (Z = R + jX) determines the phase angle φ between voltage and current, which directly sets the power factor and the split between real and reactive power.
Resonance in RLC Circuits — At resonance, inductive and capacitive reactances cancel exactly (Q_L = Q_C), so reactive power is zero and power factor equals 1 — the ideal condition for maximum power delivery.
Transformers — Transformers are rated in VA (apparent power) rather than watts because they must handle the full current demanded by reactive loads, regardless of whether that current does useful work.
RMS Values in AC Circuits — Real, reactive, and apparent power are all defined using RMS voltage and current, so understanding RMS is a prerequisite for all AC power analysis.