What is a voltage phasor?

A phasor is a complex number representing the magnitude and phase angle of a sinusoidal voltage. It simplifies AC circuit analysis by converting differential equations into algebraic ones.

What is the difference between polar and rectangular form?

Polar form uses magnitude and angle: V∠θ. Rectangular form uses real and imaginary parts: Re + jIm. They represent the same quantity. Polar is easier for multiplication/division; rectangular is easier for addition/subtraction.

How do I add phasors?

Convert each phasor to rectangular form, add the real parts and imaginary parts separately, then convert back to polar if needed. This calculator does this automatically.

What does a negative angle mean?

A negative angle means the phasor lags the reference (0°). Positive angle means it leads. In a three-phase system, the three phases are typically at 0°, -120° and +120°.

Can I subtract phasors?

Yes. Use the +/- selector next to each phasor to toggle between addition and subtraction. Subtraction is used to find voltage differences across components.

Why is the sum of three-phase phasors zero?

In a balanced three-phase system, the three voltages are equal in magnitude and 120° apart. Their rectangular components cancel perfectly, giving a resultant of zero. Try the 3-Phase preset to see this.

Voltage Phasor Calculator

|V| = √(Re²+Im²)

θ = arctan(Im/Re)

Re = |V|cosθ

Im = |V|sinθ

Enter Phasors (Polar Form)

Enter at least one phasor

Resultant Phasor

Polar Form

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Rectangular Form

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Phasor sum diagram

Figure 1: A phasor can be expressed in polar form (magnitude and angle) or rectangular form (real and imaginary parts). The real part is the horizontal projection and the imaginary part is the vertical projection.

Table of Contents

Fundamentals

What Are Voltage Phasors?
Polar vs Rectangular Form

Worked Examples

Adding Two Series Voltages
Three-Phase Balance Check
Voltage Across a Capacitor

Deep Dive

Phasor Operations
Phasors in Three-Phase Systems

Reference

Frequently Asked Questions
Related AC Circuit Calculators

What Are Voltage Phasors?

A phasor is a complex number that represents the magnitude and phase angle of a sinusoidal AC voltage at a specific frequency. Instead of working with time-varying sine waves, phasor analysis converts everything to vectors that can be added, subtracted, multiplied and divided using algebra — much simpler than solving differential equations.

The phasor V = 230 ∠ 0° means a voltage with an RMS magnitude of 230 V at 0° phase. The RMS Voltage Calculator converts between RMS and peak values if needed.

Polar vs Rectangular Form

Polar: V = |V| ∠ θ (magnitude and angle)
Rectangular: V = Re + jIm (real and imaginary parts)
Convert: Re = |V|cosθ, Im = |V|sinθ | |V| = √(Re²+Im²), θ = arctan(Im/Re)

Use rectangular form for adding and subtracting phasors (just add the real and imaginary parts separately). Use polar form for multiplying and dividing (multiply magnitudes, add/subtract angles). The calculator above handles both forms and converts between them automatically.

Worked Example — Adding Two Series Voltages

Given: V₁ = 100 V ∠ 0°, V₂ = 50 V ∠ 60°

Step 1 — V₁ rectangular: 100 + j0

Step 2 — V₂ rectangular: 50cos(60°) + j50sin(60°) = 25 + j43.3

Step 3 — Sum: (100+25) + j(0+43.3) = 125 + j43.3

Step 4 — Resultant: |V| = √(125² + 43.3²) = 132.3 V ∠ 19.1°

The resultant is not simply 100 + 50 = 150 V because the two voltages are out of phase. The phase difference reduces the effective sum. The Impedance Calculator uses the same phasor arithmetic internally.

Worked Example — Three-Phase Balance Check

Given: V_A = 230 ∠ 0°, V_B = 230 ∠ −120°, V_C = 230 ∠ +120°

Sum: (230+0) + (−115 − j199.2) + (−115 + j199.2) = 0 + j0 = 0 V

In a perfectly balanced three-phase system the three voltages sum to exactly zero. Any non-zero resultant indicates an imbalance. The Three Phase Power Calculator handles full three-phase power analysis.

Worked Example — Voltage Across a Capacitor

Given: V_source = 12 ∠ 0°, V_R = 9.6 ∠ 0° (measured across resistor)

V_C = V_source − V_R = (12 − 9.6) + j0 = 2.4 V ∠ 0°

In practice V_R and V_C are not in phase, so subtraction must account for the angle difference. This is where phasor subtraction is essential — scalar subtraction would give the wrong answer.

Phasor Operations

Addition/Subtraction: Convert to rectangular, add or subtract real and imaginary parts, convert back. This calculator does this automatically for up to 5 phasors with selectable +/− operations.

Multiplication: In polar form, multiply magnitudes and add angles. V₁×V₂ = |V₁||V₂| ∠ (θ₁+θ₂). Used when multiplying voltage by impedance to find current.

Division: Divide magnitudes and subtract angles. V₁/V₂ = |V₁|/|V₂| ∠ (θ₁−θ₂). The Complex Impedance Calculator handles these more advanced operations.

Phasors in Three-Phase Systems

Three-phase power systems use three voltage phasors separated by 120°. Phasor analysis is essential for calculating line voltages from phase voltages (√3 × V_phase), neutral currents in unbalanced loads, and sequence components for fault analysis. The symmetrical component method decomposes unbalanced phasors into positive, negative, and zero sequence sets.

Frequently Asked Questions

Do phasors only work for sine waves?

Phasor analysis works for any sinusoidal signal at a single frequency. For non-sinusoidal waveforms, you must decompose into harmonics (Fourier analysis) and analyse each harmonic separately as a phasor.

What is the j operator?

j is the imaginary unit (√−1), used instead of i to avoid confusion with current. It represents a 90° rotation in the complex plane. j² = −1.

Can phasors represent current as well as voltage?

Yes. Current phasors work identically to voltage phasors. In fact, I = V/Z in phasor form. This calculator focuses on voltage, but the mathematics applies to any AC quantity.

Why does phasor addition give a smaller result than scalar addition?

When phasors are out of phase, their components partially cancel. Only phasors at 0° (in phase) add to their full scalar sum. At 180° (antiphase) they subtract completely.

What is the reference phasor?

The reference is typically chosen as 0° — all other phasors are measured relative to it. The choice of reference does not affect the results; it just simplifies the numbers. In three-phase systems, phase A is usually the reference.

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Last updated: March 2026