Data Source & Methodology

This calculator uses the fundamental principles of AC power analysis for balanced three-phase systems. The formulas are standard in electrical engineering and are universally taught.

  • Authoritative Source: Fundamentals of Electric Circuits, 7th Edition by Charles K. Alexander and Matthew N. O. Sadiku.
  • Reference: Chapter 12, "AC Power Analysis".

All calculations are based strictly on the fundamental AC power formulas described in this source, including the use of $\sqrt{3}$ for balanced three-phase systems.

The Formulas Explained: The Power Triangle

In AC circuits, power isn't a single value. It's a vector relationship between three types of power, known as the power triangle.

[Image of the power triangle]
  • $P$ - Real Power (kW): The "working" power that does useful work, like turning a motor or lighting a lamp.
  • $S$ - Apparent Power (kVA): The "total" power in the circuit, which is the vector sum of Real and Reactive power. Utility transformers and wiring are rated in kVA.
  • $Q$ - Reactive Power (kVAR): The "wasted" or "magnetic" power required to build magnetic fields in motors and transformers. It does no useful work but still uses circuit capacity.
  • $\cos \phi$ - Power Factor (PF): The ratio of Real Power to Apparent Power ($P/S$). It's a measure of efficiency.

Three-Phase Formulas (Line-to-Line vs. Line-to-Neutral)

The key to 3-phase calculation is using the correct voltage. This calculator handles both.

1. Using Line-to-Line Voltage ($V_{LL}$): This is the most common formula, where $V_{LL}$ is the voltage between two phases (e.g., 480V) and $I_L$ is the current in one phase.

$$ S = \sqrt{3} \times V_{LL} \times I_L $$
$$ P = \sqrt{3} \times V_{LL} \times I_L \times \cos(\phi) $$

2. Using Line-to-Neutral Voltage ($V_{LN}$): This formula uses the voltage from one phase to the neutral (e.g., 277V in a 480/277V system). The relationship is $V_{LL} = \sqrt{3} \times V_{LN}$. The power is simply 3x the power of a single phase.

$$ P = 3 \times V_{LN} \times I_L \times \cos(\phi) $$

This tool automatically uses the correct formula based on your "Voltage Type" selection.

Glossary of Variables

  • $P$ (Real Power): Measured in Watts (W) or Kilowatts (kW).
  • $S$ (Apparent Power): Measured in Volt-Amps (VA) or Kilovolt-Amps (kVA).
  • $Q$ (Reactive Power): Measured in Volt-Amps Reactive (VAR) or Kilovolt-Amps Reactive (kVAR).
  • $V_{LL}$ (Line-to-Line Voltage): Voltage measured between two power lines (phases).
  • $V_{LN}$ (Line-to-Neutral Voltage): Voltage measured between one power line and the neutral.
  • $I_L$ (Line Current): The current flowing in a single power line (phase).
  • $\cos(\phi)$ (Power Factor): A dimensionless ratio between 0 and 1.
  • $\phi$ (Power Factor Angle): The angle between Real Power (P) and Apparent Power (S).

How It Works: A Step-by-Step Example

Let's find the Real Power (kW) and Apparent Power (kVA) for a three-phase motor.

  • Goal: Calculate Power
  • Voltage: 480V (Selected as Line-to-Line, $V_{LL}$)
  • Current: 25A (Line Current, $I_L$)
  • Power Factor: 0.85
  1. Identify Formula: We use the Line-to-Line formulas.
    $S = \sqrt{3} \times V_{LL} \times I_L$
    $P = S \times \cos(\phi)$
  2. Calculate Apparent Power (S):
    $S = \sqrt{3} \times 480 \text{ V} \times 25 \text{ A}$
    $S \approx 1.732 \times 480 \times 25 = 20,784 \text{ VA}$
    $S = 20.78 \text{ kVA}$
  3. Calculate Real Power (P):
    $P = 20,784 \text{ VA} \times 0.85$
    $P = 17,666 \text{ W}$
    $P = 17.67 \text{ kW}$
  4. Calculate Reactive Power (Q) (Bonus):
    $Q = \sqrt{S^2 - P^2} = \sqrt{20784^2 - 17666^2} \approx 10,934 \text{ VAR}$
    $Q = 10.93 \text{ kVAR}$

Frequently Asked Questions (FAQ)

Why is the square root of 3 (√3) used in three-phase power?

The square root of 3 (approximately 1.732) is the factor that relates Line-to-Line voltage ($V_{LL}$) and Line-to-Neutral voltage ($V_{LN}$) in a balanced Wye (Star) connected system: $V_{LL} = V_{LN} \times \sqrt{3}$. The formula $P = \sqrt{3} \times V_{LL} \times I_L \times PF$ is the universal formula for any balanced 3-phase system (Wye or Delta) and is derived from the per-phase power ($P_{phase} = V_{LN} \times I_L \times PF$) multiplied by 3.

What is the difference between Line-to-Line and Line-to-Neutral voltage?

Line-to-Line ($V_{LL}$) is the voltage measured between two power lines (e.g., 480V in a 480/277V system). Line-to-Neutral ($V_{LN}$) is the voltage measured between one power line and the neutral wire (e.g., 277V). Using the wrong one in a formula will result in an error by a factor of $\sqrt{3}$.

What is the difference between kW and kVA?

kW (Kilowatts) is Real Power (P): the actual work-performing power used by a load (e.g., to create heat, light, or mechanical motion).
kVA (Kilovolt-Amps) is Apparent Power (S): the total power drawn by the circuit ($S = \sqrt{P^2 + Q^2}$). It is the vector sum of Real Power (kW) and Reactive Power (kVAR).

What is Power Factor (PF)?

Power Factor (PF) is the ratio of Real Power (kW) to Apparent Power (kVA). It's a measure of efficiency from 0 to 1. A purely resistive load (like a heater) has a PF of 1.0. A motor (an inductive load) might have a PF of 0.85. A lower PF means more 'wasted' reactive power (kVAR) is being drawn, which utility companies may charge for.

What is a typical power factor for a motor?

A typical induction motor running at full load will have a power factor around 0.85 to 0.90. This value can drop significantly (to 0.20-0.40) if the motor is lightly loaded. This is why a PF of 0.85 is a common default for calculations.

Does this calculator work for Delta ($\Delta$) and Wye ($Y$) systems?

Yes. The formulas $S = \sqrt{3} \times V_{LL} \times I_L$ and $P = \sqrt{3} \times V_{LL} \times I_L \times PF$ are universal for any balanced three-phase system, regardless of whether it's a Wye or Delta configuration. This calculator assumes a balanced system (where currents and voltages are equal across all three phases).

Tool developed by Ugo Candido.
Engineering content reviewed by the CalcDomain Editorial Board for accuracy.
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