How to compute series RLC circuit AC impedance

Compute impedance Z, phase shift phi, Q factor and resonance frequency f_0 of series RLC AC circuit. XL XC reactances, active/reactive/apparent power, power factor. 60 Hz US examples.

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How to compute series RLC circuit impedance: in sinusoidal steady-state, complex impedance Z = R + jX where net reactance X = X_L - X_C, with X_L = omega*L (inductive) and X_C = 1/(omega*C) (capacitive), omega = 2*pi*f. Magnitude |Z| = sqrt(R^2 + X^2) (Ohms), phase shift phi = arctan(X/R) (degrees). Resonance frequency f_0 = 1/(2*pi*sqrt(L*C)) where X_L = X_C, Z minimum = R, current maximum. Quality factor Q = (1/R)*sqrt(L/C); bandwidth BW = f_0/Q. Power factor cos(phi) measures AC load energy efficiency. Fundamental tool for filter design, power supplies, oscillators, motors, 60 Hz US grid, RF telecom, EMC compliance.

RLC circuit formulas

Reactances: X_L = omega*L = 2*pi*f*L (Ohms), proportional to frequency. X_C = 1/(omega*C) = 1/(2*pi*f*C) (Ohms), inversely proportional. At 60 Hz, 100 mH inductor has X_L = 37.7 Ohms; 10 uF capacitor has X_C = 265 Ohms. At 1 kHz: X_L = 628 Ohms, X_C = 15.9 Ohms.

Series impedance: Z = R + j*(X_L - X_C). Magnitude |Z| = sqrt(R^2 + (X_L - X_C)^2). Phase phi = arctan((X_L - X_C)/R). When X_L > X_C inductive (phi > 0, voltage leads current). X_L < X_C capacitive (phi < 0). X_L = X_C resonance (phi = 0, Z = R, PF = 1).

Resonance frequency: f_0 = 1/(2*pi*sqrt(L*C)) Hz. For L=100 mH and C=10 uF: f_0 = 1/(2*pi*sqrt(0.1*10*10^-6)) = 159 Hz. For 1 MHz AM radio: typically L = 250 uH, C = 100 pF. NFC 13.56 MHz: L=2 uH, C=68 pF.

Quality factor: Q = (1/R)*sqrt(L/C) = omega_0*L/R = 1/(omega_0*R*C). High Q = selective filter (FM radio Q ≈ 100, quartz oscillator Q = 10,000-100,000). Low Q = wide band (audio filter Q ≈ 0.7 Butterworth). Bandwidth: BW = f_0/Q. -3 dB cutoff frequencies at f_0 +/- BW/2.

AC powers: P_active (W) = V*I*cos(phi) = I^2*R, dissipated in resistance. Q_reactive (VAR) = V*I*sin(phi), oscillates between L/C, no net work. S_apparent (VA) = V*I = sqrt(P^2 + Q^2). Power factor PF = P/S = cos(phi). Industrial loads with induction motors low PF 0.7-0.8: capacitor compensation to bring above 0.9 (reduces utility penalties, line losses, IEC 61000-3-2 compliance).

History and context

AC circuit theory foundations were developed by Charles Proteus Steinmetz (General Electric) between 1893-1898 with complex number (j-operator) concept for impedances. Oliver Heaviside (1880s, UK) introduced operational calculus and terms "impedance", "inductance", "permittance" (now "capacitance"). Edwin Howard Armstrong (1912) invented superheterodyne and regenerative receiver, exploiting RLC resonance for radio. Ralph V. L. Hartley (1915 Western Electric) and Edwin Henry Colpitts (1919 Western Electric) invented LC oscillators with positive feedback. Regulatory standards: IEC 60038 standardizes nominal voltages (120 V US, 230 V Europe, 100 V Japan), IEEE 519 harmonics, IEEE 1547 distributed resources. Frequencies: 60 Hz Americas (Westinghouse, Edison), 50 Hz Europe/Asia/Africa/Australia (Tesla, Westinghouse 1888). Global electrical equipment industry > $1.5 trillion/year: utilities (Duke Energy, ConEd, Iberdrola), industrial (GE, Schneider, Eaton), consumer electronics (Apple, Samsung), electric vehicles (Tesla, GM Ultium).

Typical RLC applications table

  • US power grid 60 Hz: 120/240 V split-phase, 480 V three-phase, NEC NFPA 70
  • PC power supply filter: LC 100 uH/470 uF cutoff ~24 kHz
  • AM Radio: f0 540-1700 kHz, L ≈ 250 uH, variable C 10-365 pF, Q 50-100
  • FM Radio: f0 88-108 MHz, Q ≈ 100, BW ≈ 200 kHz
  • Wi-Fi 2.4 GHz: RF filters Q ≈ 50-100, LC antenna match
  • NFC 13.56 MHz: L=2 uH, C=68 pF, Q=10-30
  • Quartz crystal oscillator: Q = 10,000-100,000, drift < 50 ppm
  • SMPS PFC: boost converter PF correction 0.99 IEC 61000-3-2
  • Induction motor: PF 0.7-0.85 (capacitor correction C > 0.9)
  • Fluorescent lamp: LC ballast, PF 0.5 uncorrected, 0.9 corrected
  • Capacitor PFC penalties: utility surcharges below 0.9 PF commercial

Worked numerical examples

Example 1 - 60 Hz US industrial circuit. Load R=100 Ohm, L=50 mH, C=10 uF, V=120 V (US residential NEC). omega = 2*pi*60 = 377 rad/s. X_L = 377*0.05 = 18.85 Ohm. X_C = 1/(377*10*10^-6) = 265 Ohm. X = 18.85 - 265 = -246 Ohm (capacitive dominant). |Z| = sqrt(100^2 + 246^2) = 265.5 Ohm. phi = arctan(-246/100) = -67.9°. PF = cos(-67.9°) = 0.376. I = 120/265.5 = 0.452 A. P_active = 120*0.452*0.376 = 20.4 W. Load reflects much reactive power.

Example 2 - series circuit resonance. R=10 Ohm, L=100 mH, C=100 uF. f_0 = 1/(2*pi*sqrt(0.1*100*10^-6)) = 1/(2*pi*0.00316) = 50.3 Hz. Q = (1/10)*sqrt(0.1/100*10^-6) = (1/10)*31.6 = 3.16. BW = 50.3/3.16 = 15.9 Hz (cutoff 42-58 Hz). At resonance X_L = X_C = 31.4 Ohm, |Z| = R = 10 Ohm, PF = 1. Band-pass filter near 50/60 Hz.

Example 3 - SMPS power supply low-pass filter. R=1 Ohm output, L=100 uH, C=470 uF. f_cutoff = 1/(2*pi*sqrt(L*C)) = 1/(2*pi*sqrt(100*10^-6 * 470*10^-6)) = 734 Hz. At 100 kHz switching frequency: attenuation (100k/734)^2 = 18,500 (-44 dB), effectively reduces ripple. UL 60950 IT equipment safety.

Example 4 - 480 V three-phase induction motor. 5 kW motor PF=0.75 lagging. S = 5000/0.75 = 6.67 kVA. Q_reactive = sqrt(6670^2 - 5000^2) = 4400 VAR. PUC penalties below PF 0.9: PFC capacitor Q_C = 4400 - 5000*tan(arccos(0.9)) = 4400 - 2421 = 1979 VAR. C = Q_C/(omega*V^2) = 1979/(377*480^2) = 22.7 uF three-phase 480 V. Reduces grid penalties, line I^2*R losses, cable size.

Example 5 - 100 MHz FM antenna. Quarter-wave antenna 75 cm long. Model: L=0.5 uH, C=5 pF (parasitic), R=50 Ohm match coaxial RG-58. f_0 = 1/(2*pi*sqrt(0.5*10^-6 * 5*10^-12)) = 100.6 MHz. Q = (1/50)*sqrt(0.5/5*10^-6) = 6.3. BW = 100/6.3 = 15.8 MHz: covers full FM band 88-108 MHz. FCC Part 15 Subpart B unlicensed RF.

Limits and considerations

  1. Valid only sinusoidal steady-state; transient requires differential equations (Laplace transform).
  2. Ideal components assumed: real inductors have parasitic R (DCR), capacitors ESR (equivalent series resistance).
  3. High frequencies: parasitics become significant (skin effect resistance, parasitic inductor capacitance).
  4. Non-linear: ferromagnetic core inductors saturate above B_sat (1.5 T silicon iron).
  5. EMC: harmonics generated by non-linear loads (rectifiers, SMPS) above 50/60 Hz fundamental (IEC 61000-3-2).
  6. Unwanted resonance in power lines can cause overvoltage, IEEE 1159 limits +/- 10%.
  7. PFC capacitors require design: oscillatory discharge with line inductor can create resonance.
  8. Frequency tolerance: 60.0 Hz US +/- 0.5 Hz normal, +/- 1% degraded (NERC).
  9. SMD vs through-hole components: ESL (equivalent series inductance) changes dramatically with package.
  10. For 3-phase analysis use positive/negative/zero sequence phasors (Fortescue).

Common errors

  1. Confusing reactance X (imaginary) with impedance Z (complex).
  2. Adding X_L and X_C linearly: correct X = X_L - X_C (opposite signs, vectors).
  3. Computing Q as V_C/V_R without considering frequency (Q depends on omega).
  4. Ignoring real transformer/inductor DCR (can drastically degrade Q).
  5. Confusing cos(phi) (PF) with efficiency eta = P_out/P_in.
  6. Computing P*I^2*R with peak I instead of I_RMS.
  7. Saturating inductor with DC bias current (inductance collapses, X_L drops).
  8. Exact resonance depends on L*C, but components have 5-20% tolerances.
  9. Ignoring temperature effect on L and C (typically +/- 2%/40°C).
  10. Using series formulas for parallel circuits (different Y parameters).

Practical applications

  1. Power grid: utilities Duke Energy, NERC FERC, IEC 60038/61000.
  2. Power supplies: SMPS PFC ATX, IEC 61000-3-2, Energy Star.
  3. Radio/TV: AM/FM/HD Radio tuner, Q factor selectivity, FCC.
  4. Wi-Fi/Bluetooth/NFC: LC matching network, FCC Part 15.
  5. Industrial motors: capacitor compensation utility penalties, NEMA MG-1.
  6. Lighting: fluorescent ballasts, LED drivers, ENERGY STAR.
  7. Audio filters: equalizers, crossovers, Butterworth/Chebyshev, AES.
  8. Instrumentation: oscilloscopes, multimeters, UL 61010 safety.
  9. EV charging: AC-DC converter, SAE J1772 Level 2, CCS DC fast.
  10. Electric vehicles: 3-phase AC motors, Tesla/GM/Ford inverters.

Support

Questions: info@calcoi.com. Educational tool for AC circuits. For professional applications consult Sedra-Smith "Microelectronic Circuits" 8th ed (Oxford 2020), Hayt-Kemmerly "Engineering Circuit Analysis" 9th ed (McGraw-Hill 2018), IEC 60038 nominal voltages, IEC 61000-3-2 harmonics, IEEE 519, SPICE/LTspice/PSpice software.

Updated July 2026

Frequently Asked Questions

What does reactance X measure?

Reactance measures opposition to AC current due to electric fields (capacitive X_C) or magnetic fields (inductive X_L). X_L = omega*L grows with frequency (inductors block high freq). X_C = 1/(omega*C) decreases with freq (capacitors block low freq, pass high). Pure reactance does not dissipate energy (ideal): power oscillates between L or C and source. Resistance R dissipates: P = I^2*R. Impedance Z = R + jX combines both into complex number.

What does resonance mean in RLC circuit?

Frequency f_0 = 1/(2*pi*sqrt(L*C)) where X_L = X_C, reactances cancel: Z = pure R, minimum. Current maximum I_max = V/R, phase phi = 0, PF = 1. In series: voltage across L or C can exceed source V by Q factor. In parallel: maximum impedance at resonance. Applications: radio tuning filters, LC oscillators, antenna matching, MRI proton magnetic resonance.

Difference between high and low Q factor?

Q = (1/R)*sqrt(L/C) measures how many free oscillations decay by 1/e. High Q (>10) = narrow resonance curve, reduced BW = selective filter (FM radio Q≈100, quartz Q=10,000-100,000). Low Q (0.5-2) = wide curve, large BW = audio filter (Butterworth Q=0.707 max flat). Q < 0.5 overdamped (no oscillation). Q infinite = ideal no R (impossible practical).

Power factor PF: why does it matter?

PF = cos(phi) = P_active/S_apparent measures energy efficiency. PF=1 ideal: all current does useful work. PF=0.7 induction motor: 30% current does no work but loads lines. PUC penalizes commercial users with PF < 0.90. Solution: parallel power factor correction capacitors that compensate motor XL. IEEE 519 limits harmonics, IEC 61000-3-2 to reduce grid total harmonic distortion (THD).

Series vs parallel resonance: differences?

SERIES RLC: at resonance Z = R minimum, I maximum, V_L and V_C in opposition cancel (but individually |V_L| = Q*V > source V). PARALLEL RLC (tank circuit): at resonance Z = QR maximum (theoretically infinite ideal), source I minimum but I_L and I_C circulate in inner loop with Q*I_source amplitude. Series application: trap selective filters. Parallel: oscillators, radio tuners.

How to choose R for desired Q?

From Q = (1/R)*sqrt(L/C), for target Q given L and C: R = sqrt(L/C)/Q. Example: L=100 mH, C=10 nF, Q=10 -> R = sqrt(0.1/10*10^-9)/10 = 316 Ohm. For FM radio: Q=100, L=100 nH, C=100 pF -> R=3.16 Ohm (low, careful with copper winding DCR). For audio: Q=0.7, L=10 mH, C=10 uF -> R=22 Ohm. Consider inductor DCR (parasitic series resistance) and capacitor ESR, reduce effective Q.

Unwanted resonance in fluorescent lamps?

Old magnetic ballasts (L inductor) create PF 0.5 lagging. Adding parallel capacitor C to improve PF creates series LC resonance with power line: at 60 Hz if f_0 ≈ 300-600 Hz (5th-10th harmonic), SMPS supply harmonics undergo Q amplification. Solution: electronic ballast SMPS with active PFC, IEC 61000-3-2 compliance. Modern LED lamps have integrated PFC reducing problem.

Formula

X_L = omega*L, X_C = 1/(omega*C), omega = 2*pi*f
|Z| = sqrt(R^2 + (X_L - X_C)^2)
phi = arctan((X_L - X_C)/R)
f_0 = 1/(2*pi*sqrt(L*C))
Q = (1/R)*sqrt(L/C)
BW = f_0 / Q
I = V / |Z|
P_attiva = I^2*R, S_apparente = V*I, PF = cos(phi)
Steinmetz 1893; Heaviside 1880s; IEC 60038; IEEE 519

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