AC Power: Real (P), Reactive (Q), Apparent (S), Power Factor

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In DC, power is one number: $P = VI$. In AC, one number is not enough. The same 230 V outlet delivering 10 A can be transferring 2300 W to a heater or 0 W to an ideal inductor. The current is identical in both cases — the wiring, the breaker and the transformer feel the same stress — but only one of them does work.

Four quantities describe this properly. Getting them straight is the difference between a power supply that passes EN 61000-3-2 and one that doesn't, between a meter that reads correctly and one that under-reports by 40 %, and between a plant that pays a reactive energy penalty and one that doesn't.

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1. Start from instantaneous power

Instantaneous power is never ambiguous. It is always the product of the instantaneous values:

$p(t) = v(t)\,i(t)$

Take a sinusoidal voltage and a current lagging it by phase angle $\varphi$:

$v(t) = V_m\cos(\omega t), \qquad i(t) = I_m\cos(\omega t - \varphi)$

Multiply and apply the product-to-sum identity:

$p(t) = \frac{V_m I_m}{2}\Big[\cos\varphi + \cos(2\omega t - \varphi)\Big]$

Two terms, and the physical interpretation of the whole topic sits in them:

• A constant term, $\frac{V_m I_m}{2}\cos\varphi$. This is net energy transfer, one-way, from source to load.
• An oscillating term at $2\omega$ (100 Hz on a 50 Hz line) with zero average. Energy flows into the load for part of the cycle and back out for the rest. Nothing is consumed; something is being stored and returned — a magnetic field in an inductor, an electric field in a capacitor.

That double-frequency ripple is also why single-phase inverters and single-phase PFC stages need a bulk capacitor sized for 100/120 Hz energy buffering, and why single-phase motors vibrate at twice the line frequency.

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2. The four definitions

Using RMS values ($V = V_m/\sqrt{2}$, $I = I_m/\sqrt{2}$):

In complex form, with $\mathbf{V}$ and $\mathbf{I}$ as RMS phasors:

$\mathbf{S} = \mathbf{V}\,\mathbf{I}^{*} = P + jQ, \qquad |\mathbf{S}| = \sqrt{P^2 + Q^2}$

The conjugate on the current is not cosmetic. It is what makes $Q$ come out positive for an inductive load under the standard convention.

Sign conventions worth memorizing

• Inductive load — current lags voltage, $\varphi > 0$, $Q > 0$. The load absorbs reactive power. Called lagging PF. Motors, transformers, solenoids, chokes.
• Capacitive load — current leads voltage, $\varphi < 0$, $Q < 0$. The load supplies reactive power. Called leading PF. Capacitor banks, long lightly-loaded cables, some LED drivers.
• $P$ is always positive for a passive consumer. Negative $P$ means export — a PV inverter, a regenerating drive.

The power triangle follows directly: $P$ on the horizontal, $Q$ vertical, $S$ the hypotenuse, and $\varphi$ the angle between $P$ and $S$.

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3. Power factor is not the same thing as cos φ

This is the single most common error in the field, and it is a firmware-era problem: it appears exactly when your load is a switch-mode converter rather than a motor.

Power factor is defined as $\mathrm{PF} = P/S$. Always. That definition never breaks.

$\mathrm{PF} = \cos\varphi$ is a special case that holds only when both voltage and current are pure sinusoids at the same frequency. A rectifier drawing 3 ms current spikes out of a 20 ms line cycle violates that assumption completely.

For a sinusoidal voltage and a distorted current, split the current into its fundamental and its harmonics:

$P = V I_1 \cos\varphi_1$

Only the fundamental current component produces real power, because harmonic currents have no voltage component at their frequency to multiply against. But the RMS current — the thing that heats your wiring — includes everything:

$I = \sqrt{I_1^2 + I_2^2 + I_3^2 + \dots}$

So:

$\mathrm{PF} = \frac{P}{S} = \underbrace{\frac{I_1}{I}}_{\text{distortion factor}} \times \underbrace{\cos\varphi_1}_{\text{displacement factor}}$

And since $\mathrm{THD}_i = \sqrt{I_2^2 + I_3^2 + \dots}\,/\,I_1$:

$\mathrm{PF} = \frac{\cos\varphi_1}{\sqrt{1 + \mathrm{THD}_i^2}}$

True PF = displacement factor × distortion factor. A load can be perfectly in phase — $\cos\varphi_1 = 1$ — and still have a terrible power factor purely from waveform shape.

The canonical example

A bridge rectifier feeding a bulk capacitor conducts only near the AC peaks. Typical current THD is 100–150 %, which gives:

$\mathrm{PF} = \frac{1}{\sqrt{1 + 1.2^2}} \approx 0.64$

with a displacement factor essentially equal to 1. Adding a capacitor across the input does nothing — the problem is not phase shift.

Non-sinusoidal apparent power

When harmonics are present, $S^2 = P^2 + Q^2$ no longer holds with $Q$ as the only non-active term. Budeanu's decomposition adds a distortion power $D$:

$S^2 = P^2 + Q^2 + D^2$

This decomposition is convenient but theoretically contested — IEEE 1459 defines a more rigorous framework separating fundamental and non-fundamental apparent power. For practical work, the point to internalize is: any "reactive power" number your meter reports for a non-sinusoidal load is almost certainly $\sqrt{Q^2 + D^2}$, not true reactive power.

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4. Why anyone cares (the money and the copper)

Current is set by apparent power, not real power:

$I = \frac{S}{V} = \frac{P}{V \cdot \mathrm{PF}}$

Conductor loss scales with the square of that:

$P_{\text{loss}} = I^2 R = \left(\frac{P}{V\cdot \mathrm{PF}}\right)^2 R$

Drop the power factor from 1.0 to 0.7 while delivering the same useful watts and your I²R losses increase by a factor of $(1/0.7)^2 \approx 2$. That loss is in the utility's transformer and the building's wiring — not in the load — which is why it becomes a billing and compliance issue rather than the load owner's efficiency problem.

Three consequences follow:

1. Equipment is rated in VA, not W. Transformers, UPSs, generators and breakers are sized by apparent power because that's what determines their copper and core stress. A 1 kVA UPS driving a 0.6 PF load delivers only 600 W.
2. Reactive energy is metered and penalized. Industrial tariffs commonly apply a penalty when inductive reactive energy exceeds roughly 20 % of active energy over the billing period (the threshold used in Türkiye, for example; capacitive is limited more tightly). This is what drives capacitor bank installations.
3. Harmonic current is regulated. IEC/EN 61000-3-2 sets per-harmonic current limits for equipment up to 16 A per phase; for Class D equipment (PCs, monitors, TVs) the limits bite above 75 W input power. This is why active PFC exists in every modern PC supply — the standard limits harmonics, and high PF is the byproduct.

Correction: match the mechanism to the cause

• Displacement problem (motors): install shunt capacitors. To move from $\varphi_1$ to $\varphi_2$ at constant $P$:

  $Q_C = P\,(\tan\varphi_1 - \tan\varphi_2)$

  A 50 kW load at 0.75 PF (φ₁ = 41.4°) taken to 0.95 PF (φ₂ = 18.2°) needs $50(0.882 - 0.329) \approx 27.7$ kvar. Watch out for resonance with system inductance and for over-correction at light load, which pushes you into capacitive penalty territory.

• Distortion problem (rectifiers): capacitors make it worse by forming a resonant tank with line inductance. You need active PFC, an LCL/passive filter, or a multi-pulse/interleaved topology.

Three-phase, in passing

$S = \sqrt{3}\,V_{LL} I_L, \qquad P = \sqrt{3}\,V_{LL} I_L \cos\varphi$

Balanced three-phase has no double-frequency power ripple at all — instantaneous power is constant. That is the reason three-phase inverters need far less DC-link buffering than single-phase ones.

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5. Measuring it correctly

This is where embedded engineers get burned. The formula you cannot use:

// WRONG — this is |S|, not P. It reads 2300 VA for a 0 W inductive load.
P = Vrms * Irms;

The correct approach accumulates the product of simultaneous samples:

// Per-sample accumulation over an integer number of line cycles
acc_p    += (int64_t)v_sample * i_sample;   // for real power
acc_vsq  += (int64_t)v_sample * v_sample;   // for Vrms
acc_isq  += (int64_t)i_sample * i_sample;   // for Irms

// At the end of the window (N samples):
P    = (float)acc_p / N * k_p;                 // real power, W
Vrms = sqrtf((float)acc_vsq / N) * k_v;
Irms = sqrtf((float)acc_isq / N) * k_i;
S    = Vrms * Irms;                            // apparent power, VA
PF   = P / S;                                  // true power factor, signed
Q_ap = sqrtf(fmaxf(S*S - P*P, 0.0f));          // NOTE: this is sqrt(Q^2+D^2)

Practical constraints that decide whether this works:

• Simultaneous sampling. V and I must be sampled at the same instant, or the phase error becomes a direct PF error. A 1-sample skew at 4 kHz on a 50 Hz line is 4.5° — enough to turn a real 0.999 PF into a reported 0.92. Use a dual simultaneous-sampling ADC, or compensate the known delay in firmware.
• Integer cycle windows. Accumulate over a whole number of line cycles (zero-crossing-locked or PLL-tracked), otherwise the 2ω term leaks into your average and P wanders with the window phase.
• Bandwidth. To capture up to the 40th harmonic on 50 Hz you need meaningful response to 2 kHz, so sample at ≥ 8–10 kHz with an anti-alias filter matched to it. Undersampling makes a distorted load look clean and inflates your PF reading.
• Sign of Q. The magnitude method above loses it. To get lagging vs leading you need either the Hilbert/quarter-cycle-shifted voltage method ($Q = \overline{v(t-T/4)\,i(t)}$) or an FFT/Goertzel at the fundamental.
• Consider a metering IC. ADE7953/ADE9000 (Analog Devices) or STPM32 (ST) implement all of the above in hardware with calibrated accuracy classes, and hand you P, Q, S, PF and THD over SPI. For anything that has to meet a metering standard, this is almost always the faster and cheaper path than an MCU + discrete AFE.

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6. Worked example

A single-phase load on 230 V, 50 Hz draws 12 A RMS. Measured real power is 1980 W. Current THD is 65 %.

• $S = 230 \times 12 = 2760\ \text{VA}$
• $\mathrm{PF} = 1980 / 2760 = 0.717$
• Non-active power: $\sqrt{S^2 - P^2} = \sqrt{2760^2 - 1980^2} = 1923\ \text{VA}$
• Distortion factor: $1/\sqrt{1 + 0.65^2} = 0.838$
• Displacement factor: $\cos\varphi_1 = 0.717 / 0.838 = 0.856 \Rightarrow \varphi_1 = 31.1°$
• True reactive power: $Q = V I_1 \sin\varphi_1$, with $I_1 = 0.838 \times 12 = 10.1\ \text{A}$, so $Q = 230 \times 10.1 \times 0.516 \approx 1195\ \text{var}$
• Distortion power: $D = \sqrt{1923^2 - 1195^2} \approx 1508\ \text{VA}$

The instructive part: the 1923 VA of non-active power is mostly distortion, not reactive. A capacitor bank sized against that 1923 figure would massively over-correct and could excite a resonance. Sized against the true 1195 var, it works.

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7. Key takeaways

• $p(t) = v(t)i(t)$ always. Everything else is an average or a decomposition of it.
• $P$ does work (W), $Q$ shuttles energy into storage (var), $S$ sizes the hardware (VA), PF $= P/S$ tells you how efficiently the VA is used.
• $\mathrm{PF} = \cos\varphi$ only for pure sinusoids. True PF = displacement × distortion factor.
• Capacitors fix displacement. They do not fix distortion — active PFC does.
• Never compute P as $V_{rms} \times I_{rms}$. Accumulate sample products over integer line cycles with simultaneous V/I sampling.
• Any single "reactive power" figure from a meter on a switching load is $\sqrt{Q^2+D^2}$ unless the datasheet says otherwise.

Further reading

• IEEE Std 1459 — Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions
• IEC/EN 61000-3-2 — harmonic current emission limits
• Erickson & Maksimović, Fundamentals of Power Electronics, chapters on input-current shaping and PFC
• Analog Devices AN-1268 / ADE9000 datasheet — practical metering architecture and calibration