What is the difference between real, reactive, and apparent power?

Real power (P, watts) does useful work — heating, lighting, mechanical output. Reactive power (Q, VAR) oscillates between source and inductors or capacitors and does no net work. Apparent power (S, VA) is the total power delivered by the source: S = V × I, and the three are related by S² = P² + Q².

How do you calculate AC power?

Apparent power S = V × I (in VA). Real power P = V × I × cos(φ) (in watts). Reactive power Q = V × I × sin(φ) (in VAR). Together they form the power triangle: S² = P² + Q².

What is power factor?

Power factor (PF = cos φ) is the ratio of real power to apparent power: PF = P / S. A PF of 1.0 means all delivered power does useful work. A low PF means more current is required for the same useful power, which raises losses and electricity costs.

What is a leading and lagging power factor?

Lagging power factor: current lags voltage in time; the load is inductive (motors, transformers). Leading power factor: current leads voltage; the load is capacitive (capacitor banks, lightly loaded long cables).

What is the power triangle?

A right triangle where the horizontal side is real power P, the vertical side is reactive power Q, the hypotenuse is apparent power S, and the angle between P and S is the phase angle φ. Power factor equals cos(φ) = P/S.

Why does power factor matter?

Utilities deliver volt-amperes (VA), but customers pay for watts. A low power factor forces the utility to supply more current for the same useful power, increasing line losses, voltage drop, and equipment heating. Many utilities charge penalties when PF drops below about 0.9.

Is P = V × I valid for AC?

Only for purely resistive loads at unity power factor. With any reactance, the correct formula is P = V × I × cos(φ); using V × I alone gives apparent power (VA), not real power (W).

AC Power Calculator (Watts, VAR, VA & Power Factor) – Real, Reactive & Apparent Power

🔧 Input Parameters

🔋 RMS Voltage (V)

Root Mean Square

1V480V

⚡ RMS Current (I)

Root Mean Square

0.01A100A

📐 Phase Angle (φ)

Between V and I

-90° (capacitive)+90° (inductive)

🔄 Frequency

AC supply

1 Hz400 Hz

Resistive Load (φ = 0°, PF = 1.00)

📐 Power Triangle

Real Power (P)

996.2 W

Reactive Power (Q)

575.0 VAR

Apparent Power (S)

1150.0 VA

Power Factor

0.866

〰️ Voltage & Current Waveform — Phase Angle (φ) & Power Delivered

The graph below plots voltage (red) and current (blue) over time. The horizontal shift between the two waves is the phase angle φ. Yellow shaded regions show where the instantaneous product V(t)·I(t) is positive — i.e. where power is actually delivered to the load. At lagging φ > 0 current peaks after voltage (inductive); at leading φ < 0 current peaks before voltage (capacitive); at φ = 0 the two are in phase and all energy is delivered (purely resistive).

📘 AC Power Formulas

Real Power (P)

P = V × I × cos(φ)
P = S × cos(φ)
P = S² − Q² (squared)

Reactive Power (Q)

Q = V × I × sin(φ)
Q = S × sin(φ)
Q = √(S² − P²)

Apparent Power (S)

S = V × I
S = P / cos(φ)
S = √(P² + Q²)

Power Factor (PF)

PF = cos(φ)
PF = P / S
PF = R / Z

Power Triangle: S² = P² + Q². Apparent power is the hypotenuse, real power is the base, reactive power is the height. The angle between P and S is the phase angle φ.

📊 Step-by-Step Calculation

1. Apparent Power (S = V × I)

S = 230.00 × 5.00 = 1150.00 VA

2. Real Power (P = S × cos φ)

P = 1150.00 × cos(30°) = 996.19 W

3. Reactive Power (Q = S × sin φ)

Q = 1150.00 × sin(30°) = 575.00 VAR

4. Power Factor (PF = cos φ)

PF = cos(30°) = 0.866 lagging

💡 Tip: Positive φ = inductive (lagging) — current lags voltage (motors, transformers). Negative φ = capacitive (leading) — current leads voltage (capacitor banks). φ = 0° = resistive — voltage and current in phase.

AC Power Calculator: The Complete Guide to Real (W), Reactive (VAR) & Apparent (VA) Power

This AC power calculator (also called an AC wattage calculator) instantly computes real power (watts), reactive power (VAR), apparent power (VA), and power factor from RMS voltage, RMS current, and the phase angle between them. It draws a live power triangle and a voltage–current waveform with the phase angle marked, then shows the full step-by-step working. Use it for electrical design, motor and transformer sizing, power factor correction, utility billing analysis, and homework on AC circuits.

Quick Answer: The Three AC Power Formulas

Quantity	Formula	Unit	Example (230 V, 5 A, 30°)
Apparent Power (S)	S = V × I	VA (volt-amperes)	230 × 5 = 1150 VA
Real Power (P)	P = V × I × cos(φ)	W (watts)	1150 × 0.866 ≈ 996 W
Reactive Power (Q)	Q = V × I × sin(φ)	VAR	1150 × 0.5 = 575 VAR
Power Factor (PF)	PF = cos(φ) = P / S	dimensionless	cos 30° = 0.866

What Is AC Power?

In an AC (alternating current) circuit the voltage and current are sinusoidal: they swing positive and negative many times per second (50 or 60 Hz on most grids). Whenever the load contains inductance or capacitance, the current is shifted in time relative to the voltage — they are out of phase. Because power at each instant is the product V(t) × I(t), this phase shift causes part of the energy to flow back and forth between source and load without ever being consumed. AC power therefore splits into three separate quantities that must be calculated and tracked separately.

The Three Types of AC Power

Real (Active) Power — P, in watts (W)

Real power is the average power that actually does useful work: heat, light, mechanical rotation, sound. This is what your electricity meter records and what you pay for.

P = V × I × cos(φ)

V and I are RMS values, φ is the phase angle between them, and cos(φ) is the power factor.

Reactive Power — Q, in volt-amperes reactive (VAR)

Reactive power oscillates between the source and the load's reactive elements (inductors and capacitors) without doing net work. Inductors absorb reactive power to build magnetic fields; capacitors generate it by storing energy in an electric field. Although reactive power does no useful work, it must be supplied so motors and transformers can operate.

Q = V × I × sin(φ)

Q is positive for inductive (lagging) loads and negative for capacitive (leading) loads.

Apparent Power — S, in volt-amperes (VA)

Apparent power is the magnitude of the complex power vector — simply the product of RMS voltage and RMS current. It determines how much current the source, conductors, and transformer must carry. Generators, transformers, and UPS systems are always rated in kVA (apparent power), never in kW.

S = V × I
S = √(P² + Q²)

S is always ≥ P. The unit is VA, distinct from W, to emphasize that not all of it does work.

The Power Triangle

The relationship between real, reactive, and apparent power is a right triangle (shown in the live diagram above):

S² = P² + Q² (Pythagorean relation)

• Horizontal side (adjacent) = P (real power, W)
• Vertical side (opposite) = Q (reactive power, VAR)
• Hypotenuse = S (apparent power, VA)
• Angle between P and S = φ (phase angle)
• cos(φ) = P/S = Power Factor

The Phase Angle (φ) and the V–I Waveform

The phase angle φ is the time shift between the voltage and current waveforms, expressed in degrees of one cycle. In the live waveform diagram above the two sinusoids show this offset directly. When V and I have the same sign at the same instant the instantaneous power V·I is positive — energy flows from source to load. When they have opposite signs, V·I is negative and energy briefly flows back to the source. The yellow shaded regions mark the intervals where net power is delivered; as the phase angle grows, those regions shrink, which is exactly why real power drops as cos(φ) when the load becomes reactive.

φ = 0°: V and I peak together. All of V × I is positive; power delivered = V × I (purely resistive, PF = 1).
φ = 30° (lagging): Current peaks slightly after voltage; some intervals have V·I < 0. Real power drops to V × I × cos 30° ≈ 0.866 × V × I.
φ = 60°: Larger shift, only V × I × cos 60° = 0.5 × V × I of useful power delivered. Heavy reactive component.
φ = 90°: Pure inductor or capacitor — no net real power delivered at all; all energy sloshes back and forth.

Understanding Power Factor

Power factor measures how efficiently the load converts apparent power into useful real power:

PF = cos(φ) = P / S

• PF = 1.00 → purely resistive — all apparent power becomes real power
• PF = 0.95 → typical well-corrected industrial site
• PF = 0.80 → 80% useful, 20% reactive — utilities may charge a penalty
• PF = 0.50 → only half the apparent power does useful work — very inefficient
• PF = 0.00 → purely reactive — no real power, only circulating reactive current

Leading vs Lagging Power Factor

Lagging (φ > 0, inductive): Current lags voltage. Caused by motors, transformers, solenoids, fluorescent ballasts. The vast majority of industrial loads are lagging.
Leading (φ < 0, capacitive): Current leads voltage. Caused by capacitor banks, lightly loaded long underground cables, some switched-mode supplies.
Unity (φ = 0, resistive): Voltage and current in phase. Heaters, incandescent lamps, resistive loads.

kVA ↔ kW ↔ kVAR Conversions

Conversion	Formula	Example
kVA to kW	kW = kVA × PF	10 kVA × 0.9 = 9 kW
kW to kVA	kVA = kW ÷ PF	9 kW ÷ 0.9 = 10 kVA
Amps from kW (1-phase)	I = (kW × 1000) ÷ (V × PF)	9000 ÷ (230 × 0.9) ≈ 43.5 A
Amps from kVA (1-phase)	I = (kVA × 1000) ÷ V	10 000 ÷ 230 ≈ 43.5 A
kVAR from kW & PF	kVAR = kW × tan(arccos PF)	9 × tan(25.8°) ≈ 4.36 kVAR

Worked Examples

Example 1: Motor Load Analysis

Problem: A 230 V, 50 Hz supply drives a single-phase motor drawing 10 A at PF = 0.85 lagging.

φ = cos⁻¹(0.85) = 31.79°

S = 230 × 10 = 2,300 VA

P = 2,300 × 0.85 = 1,955 W

Q = 2,300 × sin(31.79°) = 1,212 VAR

Check: √(1955² + 1212²) ≈ 2,300 VA ✓

Example 2: Power Factor Correction Savings

Problem: A factory draws 50 kW at PF = 0.70. How much apparent power does the utility supply, and what happens if PF is corrected to 0.95?

Before: S = P / PF = 50 / 0.7 = 71.43 kVA, Q = √(S² − P²) = 50.97 kVAR

After: S = 50 / 0.95 = 52.63 kVA — utility supplies 26 % less current.

The factory's real power consumption (50 kW) is unchanged, but line losses and cable heating drop significantly.

Example 3: Sizing a Generator

Problem: A standby generator must supply a 40 kW load at PF = 0.8 lagging. What kVA rating is needed?

kVA = kW / PF = 40 / 0.8 = 50 kVA

The generator must be rated 50 kVA even though only 40 kW of useful work will be done — its windings have to carry the full current associated with 50 kVA.

Single-Phase vs Three-Phase AC Power

For single-phase systems use the formulas above. For balanced three-phase systems multiply by √3 (≈ 1.732):

Three-phase (line-to-line voltage V_LL):
S = √3 × V_LL × I (in VA)
P = √3 × V_LL × I × cos(φ) (in W)
Q = √3 × V_LL × I × sin(φ) (in VAR)

Practical Applications

Electricity billing & PF penalties: Utilities meter real power (kWh) but commercial customers often pay a penalty when PF falls below 0.9.
Transformer & generator sizing: These are rated in kVA because their windings must handle the total current, not only the in-phase component.
Power factor correction (PFC): Adding capacitor banks supplies reactive power locally, reducing utility current draw and saving on bills.
Cable and breaker sizing: Sized for I = S / V (not P / V), so low PF demands thicker conductors and larger breakers.
UPS & inverter ratings: Spec'd in VA — pay attention to PF if your load is reactive.
Motor commissioning: Comparing measured PF with nameplate value reveals underloaded or oversized motors.
Solar & renewable interconnection: Grid codes typically require inverters to operate at or near unity PF and may demand VAR support during faults.

Common AC Voltages and Their Use

Voltage	Frequency	Region / Use
120 V	60 Hz	North America residential
230 V	50 Hz	EU / UK / India / Asia residential
208 V / 240 V	60 Hz	North America split-phase / light commercial
400 V (3-phase L-L)	50 Hz	EU industrial
480 V (3-phase L-L)	60 Hz	US industrial

Typical Power Factors of Common Loads

Incandescent lamps / heaters: ~1.0 (purely resistive)
Modern LED lighting (good drivers): 0.90 – 0.99
Fluorescent lamps (electronic ballast): 0.85 – 0.95
Induction motors (fully loaded): 0.80 – 0.90 lagging
Induction motors (lightly loaded): 0.40 – 0.70 lagging — surprisingly poor
Welders / arc furnaces: 0.30 – 0.70 lagging
Switched-mode PSUs (no PFC): ~0.6, with non-sinusoidal current
Switched-mode PSUs (with active PFC): > 0.95

Frequently Asked Questions

Why can't I just use P = V × I for AC?

Because in AC circuits, voltage and current may be out of phase. P = V × I gives apparent power (in VA), not real power (in W). The correct formula for real power is P = V × I × cos(φ).

How do I convert kVA to kW?

Multiply by the power factor: kW = kVA × PF. For example, 10 kVA at PF 0.9 equals 9 kW.

How do I convert kW to kVA?

Divide by the power factor: kVA = kW ÷ PF. For example, 9 kW at PF 0.9 equals 10 kVA.

How many amps is 1 kW at 230 V?

It depends on the power factor. At unity PF: I = 1000 ÷ 230 ≈ 4.35 A. At 0.8 PF: I = 1000 ÷ (230 × 0.8) ≈ 5.43 A.

What is a good power factor?

Above 0.95 is excellent. 0.85–0.95 is acceptable for most industrial sites. Below 0.85 typically triggers utility penalty charges and increases line losses.

How do I improve a poor power factor?

Add power factor correction capacitors in parallel with inductive loads. Capacitors supply reactive power locally, so the utility only has to supply real current. Automatic PFC panels switch capacitor stages in and out as the load changes.

What is the difference between leading and lagging?

Lagging means the current lags the voltage (inductive load); leading means current leads voltage (capacitive load). Most real-world loads are lagging.

Does the power triangle work for non-sinusoidal currents?

The classic power triangle (S² = P² + Q²) assumes pure sinusoids. With distorted (harmonic-rich) currents, you also need to include distortion power D, giving S² = P² + Q² + D². This is common with switched-mode loads.

Related Electrical Calculators

DC Power Calculator — P = V × I and the 12-formula power wheel
Ohm's Law Calculator — V = I × R fundamentals
Inductor Calculator — inductive reactance X_L
Capacitor Calculator — capacitive reactance X_C
RLC Parallel Resonance — impedance, admittance, Q-factor
RLC Series Resonance