CBSE · Class 12 · Physics · Chapter 7

Alternating Current — Formula Sheet

Board Formulas

14 formulas

1.Alternating Voltage
$v = v_{m} sin ω t, ω = 2 π ν$
$v$ : Instantaneous voltage (V) · $v_{m}$ : Peak voltage (V) · $ω$ : Angular frequency (rad s⁻¹) · $ν$ : Frequency (Hz)
v_m is the peak value (amplitude). For Indian mains ν = 50 Hz, so ω ≈ 314 rad s⁻¹.
2.AC Through a Resistor
$i = i_{m} sin ω t, i_{m} = \frac{v _{m}}{R}$
$i_{m}$ : Peak current (A) · $R$ : Resistance (Ω)
Current and voltage are in phase. Ohm's law holds for instantaneous, peak and rms values.
3.RMS (Effective) Current and Voltage★
$I = \frac{i _{m}}{2} = 0.707 i_{m}, V = \frac{v _{m}}{2} = 0.707 v_{m}$
$I$ : rms current (A) · $V$ : rms voltage (V)
The rms value is the steady (DC) value that produces the same average power in a resistor. AC meters read rms values.
4.Average Power in a Resistor
$P ˉ = \frac{1}{2} i_{m}^{2} R = I^{2} R = V I$
The average of sin²ωt over a cycle is ½, while the average of sin ωt is zero. Unit: watt (W).
5.Inductive Reactance
$X_{L} = ω L = 2 π ν L$
$X_{L}$ : Inductive reactance (Ω) · $L$ : Inductance (henry, H)
In a pure inductor the current lags the voltage by π/2: i = i_m sin(ωt − π/2) with i_m = v_m/X_L. Unit: ohm (Ω).
6.Capacitive Reactance
$X_{C} = \frac{1}{ω C} = \frac{1}{2 π ν C}$
$X_{C}$ : Capacitive reactance (Ω) · $C$ : Capacitance (farad, F)
In a pure capacitor the current leads the voltage by π/2: i = i_m sin(ωt + π/2) with i_m = v_m/X_C. As ν → 0, X_C → ∞, so a capacitor blocks DC.
7.Power in a Pure Inductor or Capacitor
$P ˉ_{L} = P ˉ_{C} = 0$
With a phase difference of π/2, the average of sin ωt cos ωt over a cycle is zero. Energy is stored and returned every half-cycle; such a current is called wattless current.
8.Impedance of a Series LCR Circuit★
$Z = R^{2} + (X_{C} - X_{L})^{2}, i_{m} = \frac{v _{m}}{Z}$
$Z$ : Impedance (Ω)
Z plays the role of resistance for peak or rms values. Because the difference is squared, (X_C − X_L)² = (X_L − X_C)². Unit: ohm (Ω).
9.Phase Angle in a Series LCR Circuit
$tan ϕ = \frac{X _{C} - X _{L}}{R}$
$ϕ$ : Phase angle between current and voltage
With v = v_m sin ωt, the current is i = i_m sin(ωt + φ). If X_C > X_L, φ > 0 and the current leads (capacitive circuit). If X_L > X_C, φ < 0 and the current lags (inductive circuit).
10.Resonant Frequency of a Series LCR Circuit★
$ω_{0} = \frac{1}{L C}, ν_{0} = \frac{1}{2 π L C}$
$ω_{0}$ : Resonant angular frequency (rad s⁻¹) · $ν_{0}$ : Resonant frequency (Hz)
At resonance X_L = X_C, so Z = R (its minimum) and the current amplitude v_m/R is maximum. Resonance needs both L and C in the circuit.
11.Average Power in an AC Circuit★
$P = V I cos ϕ = I^{2} Z cos ϕ = I^{2} R$
$P$ : Average power (W) · $cos ϕ$ : Power factor
V and I are rms values. Only the resistance dissipates power on average.
12.Power Factor
$cos ϕ = \frac{R}{Z}$
Pure R: cos φ = 1. Pure L or pure C: cos φ = 0 (wattless current). Series LCR at resonance: cos φ = 1, maximum power.
13.Transformer Voltage Ratio★
$\frac{V _{s}}{V _{p}} = \frac{N _{s}}{N _{p}}$
$V_{p}, V_{s}$ : Primary and secondary voltages (V) · $N_{p}, N_{s}$ : Number of turns in primary and secondary
Step-up: N_s > N_p; step-down: N_s < N_p. Assumes negligible primary resistance and no flux leakage. A transformer works only with AC.
14.Ideal Transformer: Current Ratio
$V_{p} I_{p} = V_{s} I_{s}, \frac{I _{p}}{I _{s}} = \frac{N _{s}}{N _{p}}$
$I_{p}, I_{s}$ : Primary and secondary currents (A)
In an ideal (100% efficient) transformer input power equals output power. Stepping voltage up steps current down by the same ratio.