Power Factor⚠ unverified
Electrical / Basic · Compute the power factor from real and apparent power
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| real_power | realpower | W | 1.0 | Real (active) power |
| apparent_power | apparentpower | VA | 1.0 | Apparent power |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | PF | watt / volt_ampere | Dimensionless power factor (ratio of real to apparent power) |
The science & history
Understanding the Parameters
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Real power $P$ — the average of instantaneous power over a cycle; the energy per second that does useful work or heats the load. See Watt's Law.
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Apparent power $S$ — what the source "sees": $S = V_{rms}I_{rms}$. It sets the current the wiring and transformers must carry, regardless of how much of it does useful work.
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$\mathrm{PF}$ — the fraction of apparent power that is real. Lagging PF (inductive loads: motors, transformers) and leading PF (capacitive loads) both describe current out of phase with voltage.
Derivation (Approaching a Proof)
For sinusoidal voltage and current with phase difference $\varphi$, the instantaneous power $p(t)=v(t)i(t)$ averages over a cycle to
$$P = V_{rms} I_{rms} \cos\varphi,$$
while the apparent power is $S = V_{rms} I_{rms}$. Their ratio is the displacement power factor:
$$\mathrm{PF} = \frac{P}{S} = \cos\varphi.$$
The three powers form a right triangle — $S^2 = P^2 + Q^2$, where $Q = V_{rms}I_{rms}\sin\varphi$ is the reactive power (VAR) that sloshes in and out of inductors and capacitors without net transfer. When current is distorted (non‑sinusoidal, e.g. rectifier loads), the true power factor $P/S$ also includes a distortion factor, so $\mathrm{PF} = \cos\varphi \times (\text{distortion factor})$.
Why it matters. For a fixed real power, low PF forces higher current ($I = P/(V\,\mathrm{PF})$), which raises $I^2R$ losses and consumes transformer/line capacity — hence utilities bill large customers on PF and engineers add power‑factor‑correction capacitors to cancel inductive $Q$.
History
Power factor became central with the spread of AC power in the 1890s–1900s. Charles Proteus Steinmetz's complex‑power formalism ($\mathbf{S} = P + jQ$) gave real, reactive, and apparent power a unified phasor treatment, making PF a design and billing quantity for the growing motor and lighting loads of early electrical grids.
Related Concepts: Apparent Power, Reactive Power, Watt's Law, Power Factor Correction, Three Phase Power
Notes: Registry calculators basic-power-factor (Electrical / Basic) and the Electrical / Power
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