Reactive Power⚠ unverified
Electrical / AC · Compute the reactive power of an AC load
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| V | V | V | 1.0 | Voltage |
| I | I | A | 1.0 | Current |
| sin_phi | sin_φ | — | 1.0 | Sine of the phase angle between voltage and current, dimensionless |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | Q | ampere * volt | Reactive power, in volt-amperes reactive (VAR) |
The science & history
Understanding the Parameters
- $V$, $I$ — the RMS magnitudes, whose product is the apparent power $S = VI$ (see Apparent Power).
-
$\sin\varphi$ — projects that apparent power onto the reactive axis. $\varphi$ is the phase by which current lags (inductive) or leads (capacitive) the voltage.
-
$Q$ — positive by convention for inductive (lagging) loads that absorb reactive power, negative for capacitive (leading) loads that supply it. This sign is what makes Power Factor Correction work.
Derivation (Approaching a Proof)
For sinusoidal $v(t) = V_m\cos\omega t$ and $i(t) = I_m\cos(\omega t - \varphi)$, the instantaneous power $p(t) = v i$ expands (product‑to‑sum) into a constant term plus an oscillating term:
$$p(t) = \underbrace{V_{rms}I_{rms}\cos\varphi}_{\text{real power }P}\big(1 + \cos 2\omega t\big) + \underbrace{V_{rms}I_{rms}\sin\varphi}_{\text{reactive power }Q}\,\sin 2\omega t.$$
The first bracket averages to $P$ (net energy transfer); the second term averages to zero — it is pure energy sloshing in and out of the reactive elements at twice the line frequency. Its amplitude is
$$Q = V_{rms} I_{rms} \sin\varphi.$$
Together with real power $P = VI\cos\varphi$ and apparent power $S = VI$, it completes the power triangle $S^2 = P^2 + Q^2$, with $Q = \sqrt{S^2 - P^2}$.
History
Reactive power entered engineering practice through Charles Proteus Steinmetz's complex‑power formalism ($\mathbf S = P + jQ$) in the 1890s. As AC grids filled with inductive motors and transformers, managing $Q$ — through capacitor banks, synchronous condensers, and today power electronics — became essential to voltage stability and efficient transmission.
Related Concepts: Apparent Power, Power Factor, Power Factor Correction, Watt's Law
Notes: Registry calculator reactive-power (unverified; result unit should read var). Sign convention:
$+Q$ inductive (absorbed), $-Q$ capacitive (supplied).