LCR Series Circuits

An LCR series circuit contains three fundamental components connected in series: an inductor (L), a capacitor (C), and a resistor (R), all driven by an alternating current (AC) source. Unlike a simple DC resistor circuit, each component responds to AC in a different way — the resistor dissipates energy, the capacitor stores energy in an electric field and resists changes in voltage, and the inductor stores energy in a magnetic field and resists changes in current. Understanding how these three elements interact is one of the most elegant problems in classical physics. The key concept that unifies all three components is impedance, the AC generalisation of resistance. While a resistor's opposition to current is simply its resistance R (measured in ohms), a capacitor has capacitive reactance and an inductor has inductive reactance — both of which depend on the frequency of the AC signal. Because these reactances are frequency-dependent and also shift the phase of the current relative to the voltage, we cannot simply add them algebraically. Instead, physicists use a geometrical tool called a phasor diagram, which represents sinusoidal quantities as rotating vectors, making phase relationships visually clear. LCR circuits are not just academic exercises. They are the basis for radio tuning, audio equalizers, signal filters, and the resonant circuits inside MRI machines. When you turn a radio dial to select a station, you are physically changing the capacitance in an LCR circuit to match its resonant frequency to the broadcast frequency. Mastering LCR analysis gives you a window into how virtually all modern electronic communication technology works.

Key takeaways
  • Individual Voltage Drops
  • Reactances
  • Phasor Addition and Impedance
  • Phase Angle
  • Resonance

Individual Voltage Drops

For a series LCR circuit driven by voltage v(t) = V₀ sin(ωt), the same current i(t) = I₀ sin(ωt − φ) flows through every component. The voltage across the resistor (V_R) is in phase with the current. The voltage across the inductor (V_L) leads the current by 90°, and the voltage across the capacitor (V_C) lags the current by 90°. This means V_L and V_C point in exactly opposite directions on a phasor diagram.

Reactances

Inductive reactance is X_L = ωL, where ω = 2πf is the angular frequency. Capacitive reactance is X_C = 1/(ωC). Both are measured in ohms. X_L increases with frequency (inductors block high frequencies more strongly), while X_C decreases with frequency (capacitors block low frequencies more strongly). At high frequency, the inductor dominates; at low frequency, the capacitor dominates.

Phasor Addition and Impedance

On a phasor diagram, place V_R horizontally (the reference direction). V_L points vertically upward and V_C points vertically downward. The net reactive voltage is (V_L − V_C). The total supply voltage phasor V is the hypotenuse of the right-angled triangle formed by V_R and (V_L − V_C). Dividing every side of this triangle by the current amplitude I₀ converts the voltage triangle into an impedance triangle, with legs R and (X_L − X_C) and hypotenuse Z = √[R² + (X_L − X_C)²]. This is the circuit's impedance.

Phase Angle

The phase angle φ is the angle between the supply voltage phasor and the current phasor. From the impedance triangle: tan φ = (X_L − X_C) / R. If X_L > X_C the circuit is inductive and voltage leads current (φ > 0). If X_L < X_C the circuit is capacitive and voltage lags current (φ < 0). If X_L = X_C, then φ = 0 and the circuit is purely resistive — this is the condition for resonance.

Resonance

At resonance, X_L = X_C, so ωL = 1/(ωC), giving the resonant angular frequency ω₀ = 1/√(LC). At this frequency the impedance is at its minimum value Z = R, and the current amplitude I₀ = V₀/R is maximised. The capacitor and inductor voltages can individually be much larger than the supply voltage — this phenomenon, called voltage magnification, is quantified by the quality factor Q = ω₀L/R.

The Key Equation

Z=R2+(XL−XC)2=R2+(ωL−1ωC)2Z = \sqrt{R^2 + \left(X_L - X_C\right)^2} = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}
  • Z — Impedance (Ω (ohm)): The total opposition to AC current flow; the AC generalisation of resistance.
  • R — Resistance (Ω (ohm)): The resistor's opposition to current, independent of frequency.
  • X_L = \omega L — Inductive Reactance (Ω (ohm)): The inductor's frequency-dependent opposition; increases with frequency.
  • X_C = 1/(\omega C) — Capacitive Reactance (Ω (ohm)): The capacitor's frequency-dependent opposition; decreases with frequency.
  • \omega — Angular Frequency (rad s⁻¹): ω = 2πf, where f is the frequency of the AC source in hertz.

Worked Example

Problem. A series LCR circuit has R = 40 Ω, L = 0.10 H, and C = 100 μF. It is connected to an AC supply of peak voltage V₀ = 200 V at frequency f = 50 Hz. Find: (a) the impedance Z, (b) the peak current I₀, and (c) the phase angle φ, stating whether voltage leads or lags current.

Z≈40.0 Ω,I0=5.00 A,φ≈−0.59∘Z \approx 40.0\,\Omega,\quad I_0 = 5.00\,\text{A},\quad \varphi \approx -0.59^\circ

Answer. Z ≈ 40.0 Ω, I₀ = 5.00 A, φ ≈ −0.59° (voltage lags current; circuit is near resonance)

Figure
Diagram
LCR Phasor & Impedance DiagramPhasor DiagramV_RV_LV_CV_L−V_CVφImpedance TriangleRX_L−X_CZφZ = √[R² + (X_L − X_C)²] tan φ = (X_L − X_C)/R
Origins · 1 minCharles Proteus Steinmetz and how this was discovered
CP
Charles Proteus Steinmetz
1865–1923 · German-American

Steinmetz revolutionised AC circuit analysis by introducing the use of complex numbers (phasors) to represent sinusoidal voltages and currents. Before his 1893 paper, engineers had to solve difficult differential equations for every AC problem. By representing impedances as complex quantities Z = R + j(X_L − X_C), Steinmetz made LCR circuit analysis as straightforward as Ohm's Law for DC circuits. His method is the foundation of every AC circuit calculation taught in physics and engineering today. Steinmetz had a physical disability that caused severe curvature of the spine, yet he became one of the most celebrated engineers in American history. He worked at General Electric alongside Thomas Edison and was so valued that GE allowed him to keep a personal laboratory — complete with alligators and exotic plants — in his home.

  • Resonance in LCR Circuits — Resonance occurs in a series LCR circuit when X_L = X_C, giving minimum impedance and maximum current — a direct consequence of the impedance formula derived here.
  • Power in AC Circuits — The phase angle φ from LCR analysis determines the power factor cos φ, which governs how much electrical power is actually dissipated versus stored and returned.
  • Self-Inductance and Inductors — Understanding how an inductor's back-EMF opposes changing current explains why inductive reactance X_L = ωL increases with frequency.
  • Capacitors and Capacitance — The capacitor's behaviour in LCR circuits — blocking DC and passing AC, with reactance X_C = 1/(ωC) — builds directly on the fundamental properties of capacitors.