AC series and parallel circuits

In AC circuits with inductors and capacitors, current and voltage are generally out of phase, so power is no longer simply V × I. Some power does useful work (active), some oscillates back and forth between source and reactive elements (reactive), and the product of RMS voltage and current (apparent power) sets equipment ratings. Power factor ties these together and has a direct effect on losses and electricity bills. Resonance — when inductive and capacitive reactances cancel — is the second theme of this topic.

Key concepts

• Impedance in series circuits: $Z = R + j(X_L - X_C)$, $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$, $X_L = \omega L$, $X_C = 1/\omega C$. Phase angle $\phi = \tan^{-1}\frac{X_L - X_C}{R}$; current lags if $X_L > X_C$, leads if $X_C > X_L$.
• Parallel circuits: easier with admittance $Y = 1/Z = G + jB$ (siemens); branch currents add as phasors.
• Active (real/true) power P: $P = VI\cos\phi = I^2R$, measured in watt (W). It is the average power actually converted to heat, light or mechanical work. Only resistance consumes active power; pure L and C consume none on average.
• Reactive power Q: $Q = VI\sin\phi = I^2X$, in volt-ampere reactive (VAr). Average value zero; it is energy exchanged each half-cycle with magnetic and electric fields. By convention inductive loads absorb (positive) Q and capacitors supply (negative) Q.
• Apparent power S: $S = VI$, in volt-ampere (VA). Complex power $\bar S = \bar V\bar I^* = P + jQ$. Transformers, generators and cables are rated in kVA because their heating depends on current, regardless of pf.
• Power triangle: $S^2 = P^2 + Q^2$; angle between S and P is φ.
• Power factor: $\text{pf} = \cos\phi = P/S = R/|Z|$. Lagging for inductive loads (motors, transformers, fluorescent chokes), leading for capacitive loads, unity for pure resistance.
• Effects of low power factor: larger current for the same real power → higher $I^2R$ losses, larger voltage drop, bigger conductors and equipment kVA, reduced capacity of generators/transformers, penalty charges by utilities.
• Power factor improvement: shunt capacitor banks (static, most common), synchronous condensers, phase advancers, and automatic pf correction panels in industry. Capacitor kVAr required: $Q_C = P(\tan\phi_1 - \tan\phi_2)$.
• Series resonance (RLC): occurs when $X_L = X_C$, at $f_r = \frac{1}{2\pi\sqrt{LC}}$.
  • Impedance minimum and purely resistive (Z = R); current maximum (V/R); pf = 1.
  • Voltages across L and C can be much larger than the supply — voltage magnification = Q factor.
  • Called an acceptor circuit; used in tuning radio receivers.
  • Quality factor $Q = \frac{\omega_rL}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$; bandwidth $BW = f_r/Q = R/(2\pi L)$; half-power frequencies $f_1, f_2$ with $f_r = \sqrt{f_1f_2}$ (geometric mean).
• Parallel resonance (tank circuit): L (with resistance R) in parallel with C.
  • Line current minimum, impedance maximum (dynamic impedance $Z_D = L/(CR)$), pf = 1.
  • Current magnification — circulating current in the tank is Q times the line current.
  • Called a rejector circuit; used in oscillators, filters, wave traps.
  • Resonant frequency $f_r = \frac{1}{2\pi}\sqrt{\frac{1}{LC} - \frac{R^2}{L^2}}$ (reduces to $1/2\pi\sqrt{LC}$ when R is negligible).
• Below resonance a series RLC circuit is capacitive; above resonance it is inductive (the opposite for the parallel circuit).

High-yield points

• $P = VI\cos\phi$ (W), $Q = VI\sin\phi$ (VAr), $S = VI$ (VA), $S^2 = P^2 + Q^2$.
• pf = $\cos\phi = P/S = R/|Z|$.
• Series resonance: $f_r = 1/(2\pi\sqrt{LC})$, minimum Z = R, maximum current, voltage magnification Q.
• $Q = \omega_rL/R = (1/R)\sqrt{L/C}$; $BW = f_r/Q$.
• Parallel resonance: dynamic impedance $L/(CR)$, minimum line current, current magnification.
• Capacitor kVAr for pf correction: $P(\tan\phi_1 - \tan\phi_2)$.

Formulas

$P = VI\cos\phi,\quad Q = VI\sin\phi,\quad S = VI = \sqrt{P^2+Q^2}$

$f_r = \frac{1}{2\pi\sqrt{LC}},\qquad Q = \frac{\omega_rL}{R},\qquad BW = \frac{f_r}{Q}$

$Q_C = P(\tan\phi_1 - \tan\phi_2),\qquad C = \frac{Q_C}{2\pi fV^2}$

(V = voltage across the capacitor; for a three-phase bank use per-phase $Q_C$ and per-phase voltage.)

Worked example

A series RLC circuit has R = 10 Ω, L = 0.1 H, C = 10 µF on a 100 V variable-frequency supply. Find $f_r$, current at resonance, Q factor and voltage across C.

1. $f_r = \frac{1}{2\pi\sqrt{0.1\times10^{-5}}} = \frac{1}{2\pi\times10^{-3}} = 159.2$ Hz.
2. $I = 100/10 = 10$ A.
3. $X_L = 2\pi\times159.2\times0.1 = 100\ \Omega$; $Q = 100/10 = 10$.
4. $V_C = IX_C = 10\times100 = 1000$ V — ten times the supply (voltage magnification).

Exam tips

• Active power in W, reactive in VAr, apparent in VA; machines are rated in kVA.
• Series resonance: minimum Z, maximum current; parallel resonance: maximum Z, minimum line current.
• Bandwidth = $f_r/Q$: a higher Q means a sharper, more selective circuit.
• Capacitors supply reactive power and improve lagging pf.
• At resonance pf = 1 in both series and parallel circuits.

Exam-style questions

From Calmon's question bank, which has 24 questions on this topic.

1. The power factor of a purely inductive circuit is:

• A. Zero leading
• B. Zero lagging
• C. Unity
• D. 0.707 lagging

Answer: B. In a pure inductor current lags voltage by $90^\circ$, so $\cos90^\circ = 0$ (lagging).

2. A series RLC circuit has $R = 10\,\Omega$, $L = 0.1$ H and $C = 10\,\mu F$. Its quality factor at resonance is:

• A. 1000
• B. 10
• C. 100
• D. 1

Answer: B. $Q = \frac{1}{R}\sqrt{L/C} = \frac{1}{10}\sqrt{0.1/10^{-5}} = 100/10 = 10$.

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