Alternating Current — Notes

A sinusoidal supply: $v = V_0\sin\omega t$, $\omega = 2\pi f$ (India: 50 Hz, ω ≈ 314 rad s⁻¹). The average current over a cycle is zero, but the average of $\sin^2$ is 1/2, so power is delivered.

rms value: the steady value giving the same average heating: $\bar P = I_\text{rms}^2R = \tfrac12 I_0^2R$, so $I_\text{rms} = I_0/\sqrt2 \approx 0.707I_0$ and $V_\text{rms} = V_0/\sqrt2$. AC meters read rms; “220 V mains” is rms (peak ≈ 311 V).

• Inductor: $L\,di/dt = V_0\sin\omega t$ gives $i = \dfrac{V_0}{\omega L}\sin(\omega t - \pi/2)$.
• Capacitor: $i = C\,dv/dt = \omega CV_0\cos\omega t = \dfrac{V_0}{1/\omega C}\sin(\omega t + \pi/2)$.
• $X_L \propto f$ (blocks high frequencies, passes DC); $X_C \propto 1/f$ (blocks DC, passes high frequencies).
• Phasors: arrows of length equal to the peak value rotating anticlockwise at ω; projections give instantaneous values; angles give phase differences.

Same current through all; voltages add as phasors: $V = \sqrt{V_R^2 + (V_L - V_C)^2}$.

$Z = \sqrt{R^2 + (X_L - X_C)^2},\qquad I = V/Z,\qquad \tan\phi = \dfrac{X_L - X_C}{R}$ $X_L > X_C$: inductive (V leads I); $X_C > X_L$: capacitive (I leads V).

Resonance: $X_L = X_C$ at $\omega_0 = 1/\sqrt{LC}$, $f_0 = 1/2\pi\sqrt{LC}$. Then $Z = R$ (minimum), $I = V/R$ (maximum), φ = 0. Smaller R gives a sharper resonance — used in radio tuning. Voltages across L and C at resonance can be much larger than the supply voltage.

Average power $P = V_\text{rms}I_\text{rms}\cos\phi$; power factor $\cos\phi = R/Z$ (series LCR).

• Pure R: cos φ = 1. Pure L or C: cos φ = 0 (energy stored and returned each cycle). At resonance: 1.
• Only resistance dissipates: $P = I_\text{rms}^2R$.
• Wattless current: the component $I\sin\phi$, 90° out of phase with V, delivers no average power but still causes $I^2R$ loss in the supply lines. Capacitors are added to inductive loads (motors) to raise the power factor.

AC generator: a coil (N turns, area A) rotates at ω in field B: $\Phi = NBA\cos\omega t$, $\varepsilon = NBA\omega\sin\omega t$, peak $\varepsilon_0 = NBA\omega$ (largest when the coil’s plane is parallel to B). Slip rings and brushes connect the coil to the circuit. Mechanical energy → electrical energy.

Transformer: primary and secondary on a laminated soft-iron core; works by mutual induction, AC only.

• Ideal: $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} = \dfrac{I_p}{I_s}$; power in = power out.
• Step-up ($N_s > N_p$) for transmission; step-down for distribution and chargers.
• Losses: copper ($I^2R$, thick wires), flux leakage (interleaved windings), hysteresis (soft iron), heating of the core by induced currents (laminated core).