Electricity & MagnetismV = V₀ sin(ωt + φ)

AC Phasor Diagram

Visualize voltage and current phasors for resistors, inductors, and capacitors in an AC circuit. See phase relationships and impedance triangle.

Parameters

Resistance R

Inductive reactance XL

Capacitive reactance XC

Current I₀

Derived

X = XL − XC40.0 Ω

Z = √(R²+X²)72.11 Ω

φ = arctan(X/R)33.69°

pf = cos φ0.832

V_R0.00 V

V_L0.00 V

V_C0.00 V

V_total0.00 V

Z0.00 Ω

φ0.00 °

pf0.000

AC Phasor Diagrams

In AC circuits, sinusoidal voltages and currents are represented as rotating vectors (phasors). The length represents amplitude; the angle represents phase. Phasor diagrams show how voltages across R, L, and C add vectorially.

ℹPhasors rotate counterclockwise at angular frequency ω. The instantaneous value is the projection onto the horizontal axis. A static phasor diagram shows the amplitudes and relative phases at a frozen moment.

Voltage Phase Relationships

Current I is the reference phasor. V_R is in phase with I. V_L leads I by 90° (inductor voltage is ahead of current). V_C lags I by 90° (capacitor voltage is behind current). The total voltage V_total is the vector sum of V_R, V_L, and V_C.

Impedance Triangle

Dividing all voltages by current I gives the impedance triangle: R (horizontal), X = XL − XC (vertical), Z (hypotenuse). The angle φ = arctan(X/R) is the phase angle between total voltage and current.

Power Factor

Power factor = cos(φ) = R/Z. It represents the fraction of apparent power (V·I) that does real work. At resonance φ = 0, power factor = 1 (all power is real). For purely reactive loads, φ = ±90° and power factor = 0.

Phasor Relations

V_{R} = I R, V_{L} = I X_{L}, V_{C} = I X_{C}

V_{t o t a l} = V_{R}^{2} + (V_{L} - V_{C})^{2}

ϕ = arctan (\frac{V _{L} - V _{C}}{V _{R}}) = arctan (\frac{X _{L} - X _{C}}{R})

Power Factor = cos ϕ = \frac{R}{Z}

Quantity	Formula	Phase relative to I
V_R	I·R	In phase (0°)
V_L	I·XL	Leads by 90°
V_C	I·XC	Lags by 90°
V_total	√(V_R² + (V_L-V_C)²)	Leads by φ = arctan((XL-XC)/R)
Z	√(R² + (XL-XC)²)	Impedance magnitude
cos φ	R/Z	Power factor

Frequently Asked Questions

Why does V_L lead the current by 90°?

For an inductor, V = L dI/dt. If I = I₀sin(ωt), then V = LωI₀cos(ωt) = LωI₀sin(ωt+90°), leading the current by 90°.

Why does V_C lag the current by 90°?

For a capacitor, I = C dV/dt, so V = (1/C)∫I dt. If I = I₀sin(ωt), then V = -I₀cos(ωt)/(ωC) = I₀sin(ωt-90°)/(ωC), lagging current by 90°.

What is power factor and why does it matter?

Power factor cos(φ) = R/Z tells you what fraction of apparent power (V_rms·I_rms) does real work. A low power factor means current flows but little work is done — common in industrial inductive loads. Utilities charge penalties for low power factor.

Can V_L and V_C exceed the source voltage?

Yes, especially near resonance. Since V_L and V_C are 180° out of phase, they partially cancel. But each can individually be much larger than V_total. High Q circuits can develop voltages across L and C that are Q times larger than the source voltage.