Maharashtra State Board · Class 12 · Physics · Chapter 13

AC Circuits — Formula Sheet

Board Formulas

14 formulas

1.Alternating EMF and Current
$e = e_{0} sin ω t, i = i_{0} sin (ω t - ϕ), ω = 2 π f = \frac{2 π}{T}$
$e_{0}$ : Peak emf (V) · $i_{0}$ : Peak current (A) · $ω$ : Angular frequency (rad s⁻¹) · $f$ : Frequency (Hz) · $ϕ$ : Phase angle by which the current lags the emf (rad)
e₀ and i₀ are peak values. Here φ is the angle by which the current lags the emf; a negative φ means the current leads (capacitive circuit). Indian mains: f = 50 Hz, ω = 100π ≈ 314 rad s⁻¹.
2.Mean (Average) Value over Half a Cycle
$i_{a v} = \frac{2 i _{0}}{π} \approx 0.637 i_{0}, e_{a v} = \frac{2 e _{0}}{π}$
Taken over a half cycle. Over a full cycle the mean value of a sinusoidal current is zero, because the positive and negative halves cancel.
3.RMS (Effective) Value★
$i_{r m s} = \frac{i _{0}}{2} \approx 0.707 i_{0}, e_{r m s} = \frac{e _{0}}{2}$
The steady (DC) current that would produce the same heat in a resistor in the same time. AC ammeters and voltmeters read rms values.
4.AC Through a Pure Resistor
$i = \frac{e _{0}}{R} sin ω t, i_{0} = \frac{e _{0}}{R}$
Current and emf are in phase (φ = 0). Ohm's law holds for instantaneous, peak and rms values.
5.Inductive Reactance
$X_{L} = ω L = 2 π f L, i = i_{0} sin (ω t - \frac{π}{2})$
$X_{L}$ : Inductive reactance (Ω) · $L$ : Self-inductance (H)
In a pure inductor the current lags the emf by π/2. X_L grows with frequency: an inductor blocks high-frequency AC but passes DC freely. Unit: Ω.
6.Capacitive Reactance
$X_{C} = \frac{1}{ω C} = \frac{1}{2 π f C}, i = i_{0} sin (ω t + \frac{π}{2})$
$X_{C}$ : Capacitive reactance (Ω) · $C$ : Capacitance (F)
In a pure capacitor the current leads the emf by π/2. X_C falls as frequency rises: a capacitor blocks DC (f = 0, X_C infinite) but passes high-frequency AC. Unit: Ω.
7.Impedance of a Series LCR Circuit★
$Z = R^{2} + (X_{L} - X_{C})^{2}, i_{0} = \frac{e _{0}}{Z}, i_{r m s} = \frac{e _{r m s}}{Z}$
$Z$ : Impedance (Ω) · $R$ : Resistance (Ω)
Impedance is the total opposition to AC. Resistance and the net reactance (X_L − X_C) add like perpendicular vectors, not like numbers. Unit: Ω.
8.Phase Angle in a Series LCR Circuit
$tan ϕ = \frac{X _{L} - X _{C}}{R}$
φ is the angle by which the emf leads the current. X_L > X_C: inductive, current lags. X_L < X_C: capacitive, current leads. X_L = X_C: resonance, φ = 0.
9.Average Power in an AC Circuit★
$P_{a v} = e_{r m s} i_{r m s} cos ϕ$
Only the component of current in phase with the emf delivers power. The product e_rms·i_rms (without cos φ) is the apparent power. Unit: W.
10.Power Factor
$cos ϕ = \frac{R}{Z}$
Pure resistor: cos φ = 1 (maximum power). Pure inductor or capacitor: cos φ = 0 (no power). Series LCR at resonance: cos φ = 1.
11.Wattless Current
$i_{w a ttl ess} = i_{r m s} sin ϕ$
The textbook defines wattless (idle) current as the current through a pure inductor or ideal capacitor, which consumes no power (φ = 90°). In a general circuit, the component i_rms sin φ, 90° out of phase with the emf, is the wattless part; the in-phase component i_rms cos φ is the one that does work.
12.Resonant Frequency of a Series LCR Circuit★
$f_{r} = \frac{1}{2 π L C}, ω_{r} = \frac{1}{L C}$
At resonance X_L = X_C, so Z = R (minimum), current is maximum and in phase with the emf: an acceptor circuit. In a parallel LC circuit (negligible resistance) the same frequency gives maximum impedance and minimum line current: a rejector circuit.
13.Quality Factor (Sharpness of Resonance)
$Q = \frac{ω _{r}}{ω _{1} - ω _{2}} = \frac{ω _{r}}{2Δ ω} = \frac{ω _{r} L}{R} = \frac{1}{R} \frac{L}{C}$
$Q$ : Quality factor (no unit) · $ω_{r}$ : Resonant angular frequency (rad s⁻¹) · $ω_{1}, ω_{2}$ : Half-power angular frequencies; ω₁ − ω₂ = 2Δω is the bandwidth (rad s⁻¹)
Textbook definition: resonant frequency divided by the bandwidth ω₁ − ω₂, where ω₁ and ω₂ are the frequencies on either side of resonance at which the current falls to 1/√2 of its maximum (half-power points). For a series LCR circuit the bandwidth equals R/L, which gives Q = ω_rL/R. Q also equals the ratio of the voltage across L (or C) at resonance to the applied voltage. Small R gives large Q and a sharp resonance curve: good selectivity, as in a radio tuning circuit. Q has no unit.
14.LC Oscillations
$q = q_{0} cos ω_{0} t, ω_{0} = \frac{1}{L C}, U = \frac{q _{0}^{2}}{2 C} = \frac{1}{2} L i_{0}^{2}$
$q_{0}$ : Initial (maximum) charge on the capacitor (C) · $U$ : Total energy of the circuit (J)
A charged capacitor discharging through an ideal inductor. Energy passes back and forth between the electric field of C and the magnetic field of L; the total stays constant if R = 0. Peak current i₀ = ω₀q₀.