To calculate reactive power, match the formula to the readings you already have: use Q = P × tan(arccos(PF)) when kilowatts and power factor are known, or take the missing triangle leg with Q = √(S² − P²) when kilovolt-amperes and kilowatts are known. This page walks those two plant paths, then the one-phase versus three-phase trap, then what to do with the kvar number.
Which Meter Inputs You Need Before Any Formula
Pick the formula from the columns on the logger or bill, not from a memorized “one true equation.”
| What you already have | Formula family to use | What you are solving for |
|---|---|---|
| Active power P (kW) and power factor | Q = P × tan(arccos(PF)) | Present reactive power |
| Apparent power S (kVA) and active power P (kW) | Q = √(S² − P²) | Present reactive power |
| RMS voltage, RMS current, and PF (or φ) | Q = V I sinφ (1φ) or √3 U I sinφ (3φ) | Present reactive power from electricals |
If the feeder is rich in harmonics, a nameplate cosφ is a weak stand-in for measured Q. Use a power-quality meter that reports kvar directly when the waveform is not a clean sine.
For the meaning of the P / Q / S columns themselves, see active vs reactive power. This page stays on the arithmetic.
Write the boundary on the worksheet before the numbers: main switchboard, MCC bus, or a single large motor. Mixing those boundaries is how a clean formula produces a useless kvar.
Calculate Q from kW and Power Factor
When the logger already prints kilowatts and power factor, use the tan path.
Power factor is active power divided by apparent power. For sinusoidal displacement, that ratio is cos φ, so φ = arccos(PF) and Q = P × tan φ.
Worked plant-style arithmetic: a motor feeder draws 56 kW at a power factor of 0.86. Then φ = arccos(0.86), tan φ ≈ 0.59, and Q ≈ 56 × 0.59 ≈ 33 kvar.
The same route works for a whole bus if P and PF are the values that belong to that bus, not a single motor nameplate while other loads are online.
Keep the sign in mind. Lagging inductive loads take positive Q on the usual plant convention. Leading capacitive current subtracts.
If the only printout is kilowatts and a PF trend from last month’s utility summary, treat that PF as a monthly average, not a peak-feeder snapshot. Recalculate with a logged interval that matches the decision you are about to make.
Calculate Q from kVA and kW
When the transformer or UPS is rated in kVA and the process meter shows kilowatts, use the power triangle.
Apparent power is the hypotenuse. Reactive power is the remaining leg: Q = √(S² − P²), provided S ≥ P.
Using the same feeder: S = 65 kVA and P = 56 kW gives Q = √(65² − 56²) ≈ 33 kvar. That matches the tan-path result and is a useful cross-check.
If S is smaller than P on paper, the inputs do not belong to the same boundary. Fix the metering point before blaming the formula.
A short reactive power formula set is enough: Q from sin φ when you have V and I, Q from tan φ when you have P and PF, Q from the triangle when you have S and P.
Apparent power on a UPS nameplate is a ceiling, not a continuous load reading. Pair it with measured process kilowatts only when that UPS is the boundary you intend to correct.
Single-Phase Versus Three-Phase Line Factors
The √3 three-phase factor appears only when you build Q from line voltage and line current on a balanced three-phase circuit.
| Circuit | Reactive-power form (sinusoidal) |
|---|---|
| Single-phase (phase–neutral) | Q = V I sin φ |
| Single-phase (phase–phase) | Q = U I sin φ |
| Three-phase balanced (3-wire or 3-wire + N) | Q = √3 U I sin φ |
U is line-to-line voltage. V is line-to-neutral. I is line current. φ is the phase angle between voltage and current on that definition.
Do not multiply a single-phase V I sinφ result by √3 “to make it three-phase.” Start from the three-phase row instead.
Nameplate current on a three-phase motor is a line current. Pair it with line-to-line voltage and the three-phase row, or use the kW/PF path and skip V·I altogether.
In short form for a single-phase phase-to-neutral circuit, Q equals V times I times sin φ. For a balanced three-phase circuit the line form is Q equals the square root of three times U times I times sin φ.
A panel builder who copies a single-phase homework formula onto a three-phase feeder without the three-phase factor will understate Q and undersize every capacitor conversation that follows.
Mistakes That Inflate or Cancel Q on Mixed Loads
The algebra fails in the field when loads with different power factors are combined the wrong way.
You cannot add two apparent-power magnitudes and treat the sum as a path to total Q. Apparent power is not a scalar you stack when angles differ.
من الميدان: على Electrical Engineering Stack Exchange, answers call out the same exam trap: adding apparent-power magnitudes gets you nowhere, and leading Q cancels lagging Q. Sum the active powers and the signed reactive powers, then rebuild the apparent-power magnitude from the power triangle if you need total kilovolt-amperes.
Dividing kilowatts by power factor returns kilovolt-amperes, not kvar. If you need Q after that step, still take √(S² − P²) or P × tan(arccos(PF)).
Capacitor Q is not a special third physics. For a capacitor branch, Q = V I sinφ with a leading current, or Q = I² Xc with the capacitor reactance. The plant question is usually how much leading kvar the bank injects at rated voltage, which is a datasheet kvar rating at a stated voltage.
After You Have Q: Choose Capacitor ΔQ or an SVG Solution
Present Q answers “how many kvar is this boundary asking for right now?” It is not yet a bill of materials.
If the goal is to raise a fairly steady lagging power factor, the next arithmetic is the difference between present Q and target Q. That ΔQ problem is covered in capacitor bank sizing for power factor correction, and the hardware families sit on the مُعَوِّض القدرة غير الفعالة hub.
If Q swings with welders, cranes, or batch lines, a switched bank may step too slowly. Then the conversation moves to SVG static var generators.
The SVG product page lists 230–690 V service, a response below 10 ms, a compensation factor above 95%, and efficiency above 97%. Those figures describe that product page. They do not replace a site study or the Q calculation above.
Use the calculated kvar to brief the vendor. Do not skip the input check and ask for a cabinet size from a single motor nameplate while the rest of the bus is ignored. Re-run the same arithmetic after a process change before anyone frees a purchase order.
الأسئلة الشائعة
How to find reactive power?
Read which of P, S, PF, V, and I you already trust at one boundary. Then use the matching row in the input table above.
If the meter already prints kvar, treat that as the measurement and use the formulas only as a cross-check.
Which formula calculates reactive power?
Common plant forms are Q = P × tan(arccos(PF)), Q = √(S² − P²), and Q = V I sinφ (with √3 on balanced three-phase line quantities).
They are the same power triangle written for different knowns.
What is reactive power?
It is the oscillating component that sustains magnetic or electric fields and is reported in var or kvar. For a plain-language comparison with watts and volt-amperes, use the active vs reactive power article.
How do I calculate the reactive power of a capacitor?
At a known RMS voltage and current with a 90° lead, Q = V I. With reactance, Q = I² Xc or Q = V² / Xc for an ideal capacitor.
Bank nameplates usually state kvar at a rated voltage; derate if the actual bus voltage differs.
What is the reactive power formula in 3 phase?
For balanced three-phase line quantities, Q = √3 × U × I × sinφ, with U the line-to-line voltage.
Many plants skip this and compute Q from three-phase kW and PF instead.
Can I add kVA from two loads to get total Q?
No. Add active powers and signed reactive powers. Rebuild apparent power from the totals only when you need |S|.
Mixed leading and lagging loads partially cancel in Q even when every |S| looks large.
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