What does a positive phase angle mean?

A positive phase angle means the circuit is inductive — voltage leads current. This is also called a lagging power factor because the current lags behind the voltage.

What does a negative phase angle mean?

A negative phase angle means the circuit is capacitive — current leads voltage. This is called a leading power factor.

How is phase angle related to power factor?

Power factor equals the cosine of the phase angle: PF = cos(θ). A phase angle of 0° gives PF = 1 (unity, all power is real). A phase angle of 90° gives PF = 0 (all power is reactive).

What phase angle is ideal?

For power delivery, a phase angle of 0° (PF = 1) is ideal because all power does useful work. For tuned circuits and filters, specific phase angles are deliberately designed.

How do I correct a poor power factor?

Add a capacitor bank in parallel for inductive loads (most common) or an inductor for capacitive loads. The correction component cancels the reactive power, bringing the phase angle closer to 0°. This calculator shows the required correction capacitor value.

Phase Angle Calculator

θ = arctan(X/R)

PF = cos(θ)

Xₗ = 2πfL

Xᴄ = 1/(2πfC)

Enter Circuit Values

Ω Resistance (R)

L Inductance (L)

Leave blank to omit

C Capacitance (C)

Leave blank to omit

f Frequency (f)

Optional — Power Analysis

V Source Voltage (Vs)

Enter to see current & power triangle

Enter values to calculate phase angle

Phase Angle Results

Degrees

—

Radians

—

Power Factor

—

Voltage (orange) and current (blue) phasors with phase angle θ

Phase Angle θ
—
degrees

Power Factor

—

cos(θ)

Nature

—

|Z| Impedance

—

Xₗ Inductive

—

Xᴄ Capacitive

—

Current |I|

—

Real Power P

—

Apparent S

—

Reactive Q

—

VAR

PF Correction C

—

for PF=1

Leading/Lagging

—

Figure 1: In inductive circuits voltage leads current (positive θ). In capacitive circuits current leads voltage (negative θ). At resonance θ = 0°.

Find R or X for a Target Phase Angle

Enter the target phase angle and one known value — the calculator finds the other.

θ Target Phase Angle

−90° to +90°

Ω Known R

Leave blank to find R instead

X Known X

Leave blank to find X instead

Reverse Solve Result

Required Reactance

—

Verification

—

Table of Contents

Fundamentals

What Is Phase Angle?
The Phase Angle Formula

Worked Examples

Inductive Motor at 50 Hz
RC Circuit at 1 kHz
RLC at Resonance
Power Factor Correction

Deep Dive

Reading the Phasor Diagram
Power Factor Explained
Power Factor Correction
Leading vs Lagging

Reference

Frequently Asked Questions
Related AC Circuit Calculators

What Is Phase Angle?

In an AC circuit the voltage and current do not necessarily peak at the same instant. Phase angle (θ) measures how far apart they are in time, expressed as an angle from 0° to ±90°. When θ = 0° the two waveforms are perfectly aligned and all the power delivered does useful work. As θ moves towards ±90° more of the power is reactive — stored and returned each cycle without doing anything useful.

Phase angle arises because inductors cause current to lag behind voltage, while capacitors cause current to lead voltage. In a circuit with both, the net phase angle depends on which effect dominates at the operating frequency. The Impedance Calculator shows the full complex impedance; this calculator focuses specifically on the angle and its practical consequences for power delivery.

The Phase Angle Formula

θ = arctan(X / R)
Where X = X_L − X_C (net reactance), R = resistance.
X_L = 2πfL, X_C = 1/(2πfC).
Power factor: PF = cos(θ)

When X is positive (inductive dominates) the angle is positive and voltage leads current. When X is negative (capacitive dominates) the angle is negative and current leads voltage. At resonance X = 0, θ = 0°, PF = 1.

Worked Example — Inductive Motor at 50 Hz

Given: R = 15 Ω, L = 50 mH, f = 50 Hz

Step 1 — X_L = 2π × 50 × 0.05 = 15.71 Ω

Step 2 — θ = arctan(15.71/15) = 46.3°

Step 3 — PF = cos(46.3°) = 0.691 lagging

A power factor of 0.691 means this motor draws 45% more current than an equivalent resistive load at the same real power. Electricity suppliers penalise industrial customers with a PF below 0.9, making power factor correction essential.

Worked Example — RC Circuit at 1 kHz

Given: R = 1 kΩ, C = 100 nF, f = 1 kHz

Step 1 — X_C = 1/(2π × 1000 × 100×10⁻⁹) = 1591.5 Ω

Step 2 — θ = arctan(−1591.5/1000) = −57.9°

Step 3 — PF = cos(−57.9°) = 0.531 leading

The large negative angle tells us that at 1 kHz this RC circuit is heavily capacitive. The Capacitive Reactance Calculator can show how X_C changes as you sweep the frequency.

Worked Example — RLC at Resonance

Given: R = 100 Ω, L = 50 mH, C = 10.13 µF, f = 223.6 Hz

Step 1 — X_L = 2π × 223.6 × 0.05 = 70.24 Ω

Step 2 — X_C = 1/(2π × 223.6 × 10.13×10⁻⁶) = 70.24 Ω

Step 3 — X = 70.24 − 70.24 = 0 Ω

Step 4 — θ = arctan(0/100) = 0°, PF = 1.000

At resonance the inductive and capacitive reactances cancel perfectly, leaving pure resistance. This is the principle behind the Resonant Frequency Calculator and is exploited in radio tuning, bandpass filters, and impedance matching networks.

Worked Example — Power Factor Correction

Given: Motor draws 5 A at 230 V, PF = 0.7 lagging, f = 50 Hz

Step 1 — θ = arccos(0.7) = 45.6°

Step 2 — Apparent power S = 230 × 5 = 1150 VA

Step 3 — Real power P = 1150 × 0.7 = 805 W

Step 4 — Reactive power Q = 1150 × sin(45.6°) = 821 VAR

Step 5 — Correction capacitor: C = Q/(2πfV²) = 821/(2π × 50 × 230²) = 49.4 µF

Adding a 50 µF capacitor in parallel brings the power factor to unity. The motor still draws the same real power, but the supply current drops from 5 A to 3.5 A, reducing cable losses and electricity penalties. The Power Factor Calculator provides detailed correction sizing.

Reading the Phasor Diagram

A phasor diagram represents AC quantities as rotating vectors. In this calculator, the current phasor (I) is drawn along the positive horizontal axis as the reference. The voltage phasor (V) is drawn at angle θ above (inductive) or below (capacitive) the current.

The horizontal projection of V onto I represents the resistive (real) component, and the vertical projection represents the reactive (imaginary) component. The longer the vertical projection relative to the horizontal, the worse the power factor.

Power Factor Explained

Power factor (PF) is the ratio of real power to apparent power: PF = P/S = cos(θ). A PF of 1 means every amp of current does useful work. A PF of 0.5 means you need twice the current to deliver the same real power, which doubles the I²R losses in cables and transformers.

UK electricity suppliers typically require a PF above 0.9 for industrial loads. Below that, reactive power charges apply. Residential customers are not penalised directly, but a low PF still wastes energy in the home wiring.

Power Factor Correction

Most real-world loads are inductive (motors, transformers, fluorescent lighting), so PF correction usually means adding parallel capacitors. The capacitor supplies the reactive current locally, reducing the reactive current drawn from the supply.

The correction capacitor value is: C = Q_reactive / (2πf V²). Overcorrection (adding too much capacitance) flips the phase angle to leading, which can cause voltage rise and resonance problems. Always target PF = 0.95 to 0.98 rather than exactly 1.0.

Safety note. Power factor correction capacitors store significant energy and can deliver dangerous electric shocks. Always discharge capacitors before servicing and use capacitors rated for AC power use (not general-purpose electrolytics).

Leading vs Lagging

The terms “leading” and “lagging” describe the current relative to the voltage. In an inductive circuit the current lags behind the voltage (lagging PF). In a capacitive circuit the current leads the voltage (leading PF). At resonance the two are in phase and the power factor is unity.

This distinction matters for the Three Phase Power Calculator where leading and lagging loads on different phases can partially cancel each other, and for generator synchronisation where the phase angle determines whether a generator exports or absorbs reactive power.

Frequently Asked Questions

What is the phase angle range?

For a simple series RLC circuit the phase angle ranges from −90° (purely capacitive, no resistance) to +90° (purely inductive, no resistance). In practice, real circuits always have some resistance, so θ stays between these extremes.

Is phase angle the same as power factor angle?

Yes. The power factor angle is the same as the impedance phase angle. PF = cos(θ). Some references call it φ rather than θ, but they mean the same thing in a linear circuit.

Can phase angle be exactly 90°?

Only in a theoretical circuit with zero resistance. A pure inductor has θ = +90° and a pure capacitor has θ = −90°. Real components always have some parasitic resistance, so the angle is slightly less.

How do I measure phase angle?

Use an oscilloscope with two channels (one for voltage, one for current via a current probe or shunt resistor) and measure the time difference between zero crossings. Alternatively, a power analyser or LCR meter reads phase directly.

Does phase angle change with frequency?

Yes. Reactance is frequency-dependent, so the phase angle shifts as frequency changes. The frequency sweep graph in the calculator above shows this relationship clearly. At resonance the angle crosses zero.

Why does my electricity bill mention power factor?

A low power factor means the supply must deliver more current than necessary for the real power consumed. This extra current wastes energy in the distribution network. Industrial tariffs include reactive power charges or PF penalties to encourage correction.

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

Real power (P, in watts) does useful work. Reactive power (Q, in VAR) oscillates between source and load without doing work. Apparent power (S, in VA) is the vector sum: S = √(P² + Q²). The power triangle is directly linked to the phase angle: P = S×cos(θ), Q = S×sin(θ).

Can I use this calculator for three-phase systems?

This calculator handles single-phase circuits. For three-phase analysis (including line-to-line phase shifts and unbalanced loads) use the Three Phase Power Calculator linked in the Related section below.

Browse all Electronics Calculators →

Last updated: March 2026