Series RLC circuit analysis

RLC Phase Angle Calculator

Calculate phase angle, resistance, inductive reactance, or capacitive reactance for a series RLC circuit. Review impedance, power factor, net reactance, and current behavior with each result.

Series RLC circuit analysis

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Voltage and current timing

What is phase angle in an RLC circuit?

Phase angle describes the angular difference between the voltage and current waveforms in an alternating-current circuit.

In a series RLC circuit, the phase angle depends on the difference between inductive reactance and capacitive reactance, compared with the circuit resistance.

Main equation

RLC phase angle formula

Phase angleφ = tan⁻¹((Xₗ − Xc) ÷ R)

  • φ is the phase angle in degrees.
  • R is resistance in ohms.
  • Xₗ is inductive reactance in ohms.
  • Xc is capacitive reactance in ohms.

Worked example

Calculate the phase angle

Suppose a series RLC circuit has a resistance of 30 Ω, an inductive reactance of 50 Ω, and a capacitive reactance of 10 Ω.

  1. Calculate net reactance: 50 − 10 = 40 Ω.
  2. Divide net reactance by resistance: 40 ÷ 30.
  3. Apply the inverse tangent.
  4. φ = tan⁻¹(1.3333).
  5. The phase angle is approximately 53.13°.

Because the result is positive, the circuit is inductive and current lags voltage.

Solve missing circuit values

Rearranged phase angle equations

ResistanceR = (Xₗ − Xc) ÷ tan(φ)

Inductive reactanceXₗ = Xc + R tan(φ)

Capacitive reactanceXc = Xₗ − R tan(φ)

These rearranged equations allow the calculator to solve for any one of the four principal variables when the remaining values are known.

Circuit classification

Positive, negative, and zero phase angles

  • Positive angle: Xₗ is greater than Xc, so the circuit is inductive.
  • Negative angle: Xc is greater than Xₗ, so the circuit is capacitive.
  • Zero angle: Xₗ equals Xc, so the circuit is at resonance.

Current relationship

Does current lead or lag voltage?

In an inductive circuit, current lags voltage. In a capacitive circuit, current leads voltage. At resonance, current and voltage are in phase.

Total AC opposition

Phase angle and impedance

Net reactanceX = Xₗ − Xc

ImpedanceZ = √(R² + X²)

Impedance combines resistance and net reactance. As the magnitude of net reactance increases, the phase angle moves farther from zero.

Real and apparent power

Phase angle and power factor

Power factorPF = cos(φ) = R ÷ Z

A phase angle near zero produces a power factor near 1. Larger positive or negative phase angles produce a lower power factor.

Balanced reactance

Phase angle at resonance

At resonance, inductive reactance and capacitive reactance are equal. Their effects cancel, leaving zero net reactance.

The circuit impedance equals its resistance, the phase angle is zero, and the ideal power factor is 1.

Model assumptions

Calculator assumptions and limitations

  • The equations model a series RLC circuit.
  • Resistance and reactance values are treated as ideal, steady-state quantities.
  • Phase angles must remain between −90° and 90°.
  • Real components may include parasitic resistance, capacitance, and inductance.

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