Electrical Engineering Calculator

AC Power Calculator

Calculate active, reactive and apparent power for single-phase and three-phase AC circuits from voltage, current and power factor.

System Type

Select single-phase or three-phase AC

Input Parameters

Enter line-to-line voltage and line current

Line-to-Line Voltage (V_L)

Line-to-line voltage of the three-phase system.

Line Current (I_L)

RMS current drawn by the load.

Power Factor (cos φ)

—

Power factor between 0 and 1 (e.g. 0.85).

Engineering Tip

Generators and transformers are rated in kVA. Always size them according to apparent power (S), not only active power (P).

Active Power (P)

29.44kW

29,445 W

Reactive Q

18.25 kVAR

Apparent S

34.64 kVA

Power Factor

0.850

Power Triangle

S² = P² + Q²

pf = P / S = cos φ

PW / kW

Active (real) power

QVAR / kVAR

Reactive power

SVA / kVA

Apparent power

pf—

Power factor (cos φ)

Line-to-line voltage

Line current

Assumptions & Notes

Sinusoidal voltage and current
Balanced three-phase system
RMS values used
Constant power factor

Engineering Code

Reuse the AC power calculation in your own engineering workflow.

Python

import math

def ac_power(system, V, I, pf):
    """
    Calculate AC power components.

    system: 'single' or 'three'
    V: voltage (V) – line-to-line for three-phase
    I: current (A)
    pf: power factor (0 to 1)
    """

    if not (0 < pf <= 1):
        raise ValueError("Power factor must be between 0 and 1")

    phi = math.acos(pf)
    sin_phi = math.sin(phi)

    if system == 'three':
        S = math.sqrt(3) * V * I
    else:
        S = V * I

    P = S * pf
    Q = S * sin_phi

    return {
        "P_W": P,
        "Q_VAR": Q,
        "S_VA": S,
        "P_kW": P / 1000,
        "Q_kVAR": Q / 1000,
        "S_kVA": S / 1000,
        "pf": pf
    }


# Example
results = ac_power('three', 400, 50, 0.85)
for k, v in results.items():
    print(f"{k}: {v}")

MATLAB

function results = ac_power(system, V, I, pf)
    % AC power calculation
    % system: 'single' or 'three'
    % V = line-to-line voltage for three-phase

    if pf <= 0 || pf > 1
        error('Power factor must be between 0 and 1');
    end

    phi = acos(pf);
    sin_phi = sin(phi);

    if strcmp(system, 'three')
        S = sqrt(3) * V * I;
    else
        S = V * I;
    end

    results.P = S * pf;
    results.Q = S * sin_phi;
    results.S = S;
    results.P_kW = results.P / 1000;
    results.Q_kVAR = results.Q / 1000;
    results.S_kVA = results.S / 1000;
    results.pf = pf;
end

% Example
r = ac_power('three', 400, 50, 0.85);
disp(r);

Excel Formulas

S = SQRT(3)*V*I
P = S*pf
Q = S*SIN(ACOS(pf))

Example Calculation

Three-phase system: V_L = 400 V, I_L = 50 A, power factor = 0.85 (lagging):

P = √3 × 400 × 50 × 0.85 = 29,445 W ≈ 29.45 kW

S = √3 × 400 × 50 = 34,641 VA ≈ 34.64 kVA

Q = √3 × 400 × 50 × sin(φ) ≈ 18.25 kVAR

Power Factor = 0.85 lagging

Technical Explanation: AC Power in Single and Three-Phase Systems

In direct current (DC) circuits, calculating power is a straightforward geometric product of voltage and current. However, in alternating current (AC) systems, the voltage and current continuously alternate in a sinusoidal pattern. When AC circuits contain reactive load components—specifically inductive loads like electric motors and transformers, or capacitive loads—the current waveform shifts out of phase with the voltage waveform. This fundamental phase shift requires engineers to divide AC power into three distinct, interacting quantities to properly size electrical infrastructure.

The Three Components of AC Power

Active Power (P): Measured in Watts (W) or Kilowatts (kW), this is the "real" power that performs actual, useful mechanical work or generates heat. It is the only power registered by a standard residential utility meter.
Reactive Power (Q): Measured in Volt-Amperes Reactive (VAR) or kVAR. This power performs no useful work; instead, it continuously oscillates back and forth between the power source and the load. It is fundamentally required to sustain the magnetic fields in inductive equipment (like induction motors).
Apparent Power (S): Measured in Volt-Amperes (VA) or kVA, this is the vector sum of active and reactive power. It represents the total magnitude of power demanded from the electrical grid. Generators, transformers, and switchgear are always rated in kVA, not kW, because they must handle the total combined current demand.

Governing Formulas

The mathematical relationship between these quantities depends on whether the system utilizes a single-phase or a three-phase power supply. For a Single-Phase system, the equations are:

S = V × I
P = V × I × cos(φ)
Q = V × I × sin(φ)

For a balanced Three-Phase system, power is delivered across three separate conductors. The calculation utilizes the line-to-line voltage (V_L) and the line current (I_L). The formulas incorporate a √3 multiplier (approximately 1.732) to account for the phase geometry:

S = √3 × V_L × I_L
P = √3 × V_L × I_L × cos(φ)
Q = √3 × V_L × I_L × sin(φ)

Input Parameters and The Power Factor

When using this calculator, the voltage and current inputs must be the Root Mean Square (RMS) values, which are the standard values provided by multimeters and equipment nameplates.

The Power Factor (cos φ) is a dimensionless ratio ranging from 0.0 to 1.0. It defines the ratio of Active Power to Apparent Power (P / S). A power factor of 1.0 (unity) means the voltage and current are perfectly in phase, and 100% of the apparent power is being used as real work (typical of purely resistive loads like electric heaters). A low power factor (e.g., 0.70) indicates a highly inductive load. In an inductive load, the current lags the voltage. In a capacitive load, the current leads the voltage.

Engineering Interpretation: The Power Triangle

Engineers visualize these three quantities using the "Power Triangle." Because they are vectors, they form a right-angled triangle in the complex plane where Active Power (P) is the adjacent side on the real axis, Reactive Power (Q) is the opposite side on the imaginary axis, and Apparent Power (S) is the hypotenuse. The Pythagorean theorem applies directly: S² = P² + Q². The angle φ between P and S represents the phase shift angle between the voltage and current waveforms.

Assumptions, Harmonics, and Limitations

This calculator assumes perfectly sinusoidal waveforms and computes what is known as the Displacement Power Factor. In modern industrial and commercial facilities, the proliferation of non-linear loads—such as Variable Frequency Drives (VFDs), LED drivers, and switching power supplies—introduces harmonic distortion. Harmonics distort the sine wave, requiring an additional metric known as the Distortion Power Factor. True Apparent Power in a heavily non-linear system will be higher than the simplified calculation provided here.

Furthermore, the three-phase formulas assume a perfectly balanced system where the voltage and current are identical across all three phases. If a three-phase system is heavily unbalanced, power must be calculated individually for each phase and then summed together algebraically.

Practical Engineering Application

Understanding the AC power triangle is critical for power distribution design and industrial cost management. If a factory operates with a poor power factor, it draws a significantly higher line current (I) to achieve the same active power (P). This excess current causes severe I²R resistive heat losses in transmission cables and consumes the physical capacity of utility transformers. Consequently, power utilities impose steep financial penalties on industrial clients with power factors below a specific threshold (often 0.90 or 0.95). Engineers resolve this by installing capacitor banks parallel to the load, which inject leading reactive power to cancel out the lagging reactive power, effectively "correcting" the power factor back toward unity.

Design Warning: Calculating the apparent and active power correctly is only the first step in electrical sizing. An adequate kVA calculation does not automatically mean an electrical circuit is safe. Complete electrical design verification requires engineers to also evaluate conductor ampacity according to the NEC or IEC codes, verify acceptable voltage drop over long cable runs, and ensure that circuit breakers are properly sized for short-circuit fault currents.

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Real-World Engineering Cases

Industrial Plant with Low Power Factor Penalties

A manufacturing facility was paying significant monthly penalties because its overall power factor remained around 0.72. After measuring the actual active and reactive power of the largest motor loads, capacitor banks were installed and the power factor was raised above 0.95, eliminating the penalties.

Engineering Lesson

Knowing the real (kW) and reactive (kVAR) power of major loads is essential for correct power-factor correction. Simply measuring current is not enough.

Undersized Generator for a Construction Site

A temporary generator repeatedly tripped under load even though the calculated kW demand was within its rating. The loads had a low power factor (≈ 0.7), so the apparent power (kVA) exceeded the generator’s kVA rating.

Engineering Lesson

Generators and transformers are rated in kVA (apparent power). Always check both kW and kVA when sizing sources for AC loads.

Frequently Asked Questions

What is the difference between kW, kVAR and kVA?

kW (active power) is the useful power that performs work. kVAR (reactive power) is the power stored and released by inductors and capacitors. kVA (apparent power) is the vector sum of the two and determines the current drawn from the supply.

What is a good power factor?

A power factor of 0.95 or higher is generally considered good. Many utilities impose penalties below 0.90 or 0.95.

Do I use line-to-line or phase voltage for three-phase?

This calculator uses line-to-line voltage (V_L) and line current (I_L) together with the √3 factor, which is the most common practical approach.

What units should I use?

Voltage in volts (V), current in amperes (A), power factor dimensionless (0–1). Results are shown in W / kW, VAR / kVAR and VA / kVA.

Is the power factor leading or lagging?

Most industrial loads (motors, transformers) are inductive and have a lagging power factor. Capacitive loads produce a leading power factor. The calculator treats the numerical value; the sign of reactive power depends on the nature of the load.