Alternating voltage
Most electricity is supplied as alternating current (AC), whose value varies sinusoidally with time:
v = v_m sin ωt
where v_m is the peak value and ω = 2πν the angular frequency. In India, ν = 50 Hz.
AC is preferred because its voltage can be stepped up and down easily with transformers, which makes long-distance transmission efficient.
RMS values
The average of a sinusoidal current over a full cycle is zero, yet it heats a resistor. To compare AC with DC, we use the root mean square (rms) or effective value: the steady DC current that would produce the same heating.
I_rms = I_m ⁄ √2 ≈ 0.707 I_m and V_rms = V_m ⁄ √2
When we say mains voltage is 220 V, that is the rms value; the peak is 220√2 ≈ 311 V. AC meters read rms values.
AC through single elements
Phasors are rotating vectors whose projection on the vertical axis gives the instantaneous value. They make phase relationships easy to see.
Element | Opposition to current | Phase of current relative to voltage | Average power |
|---|---|---|---|
Resistor R | resistance R | in phase | V_rms I_rms |
Inductor L | inductive reactance X_L = ωL | current lags voltage by π/2 | zero |
Capacitor C | capacitive reactance X_C = 1/ωC | current leads voltage by π/2 | zero |
X_L increases with frequency: an inductor blocks high-frequency AC and passes DC (ω = 0, X_L = 0).
X_C decreases with frequency: a capacitor blocks DC (X_C → ∞) and passes high-frequency AC.
A memory aid: in an inductor (L), voltage leads current (ELI: E before I); in a capacitor (C), current leads voltage (ICE).
The series LCR circuit
For a resistor, inductor and capacitor in series with an AC source, the voltages across them are out of phase, so they add as phasors. The total opposition is the impedance:
Z = √(R² + (X_L − X_C)²), and I_m = V_m ⁄ Z
The phase angle φ between voltage and current is given by tan φ = (X_L − X_C) ⁄ R.
If X_L > X_C, the circuit is inductive: current lags voltage.
If X_C > X_L, it is capacitive: current leads voltage.
Resonance
When X_L = X_C, the reactances cancel, the impedance is minimum (Z = R), and the current is maximum. This happens at the resonant frequency:
ω₀ = 1 ⁄ √(LC), or ν₀ = 1 ⁄ (2π√(LC))
The sharpness of the resonance is measured by the quality factor: Q = ω₀L/R = 1/(ω₀CR). A small R gives a sharp, tall peak.
Application: tuning a radio. Adjusting the capacitor changes the resonant frequency to match one station's broadcast, so that station's signal produces a large current while others don't.
Example. L = 0.1 H and C = 10 μF: ω₀ = 1/√(0.1 × 10⁻⁵) = 1/√(10⁻⁶) = 1000 rad/s.
Power in AC circuits
The average power consumed is
P = V_rms I_rms cos φ
where cos φ is the power factor.
Pure resistor: φ = 0, cos φ = 1, maximum power.
Pure inductor or capacitor: φ = ±π/2, cos φ = 0, no power is consumed. The current through them is called wattless current.
At resonance, cos φ = 1.
A low power factor means a larger current for the same useful power, wasting energy in transmission lines. Capacitors are connected in industry to improve the power factor.
LC oscillations
A charged capacitor connected to an inductor (with no resistance) discharges through it. Energy passes back and forth between the capacitor's electric field and the inductor's magnetic field, and the charge oscillates at
ω₀ = 1/√(LC)
This is like a mass on a spring: L plays the role of mass and 1/C of the spring constant. In real circuits, resistance damps the oscillations.
Transformers
A transformer changes an alternating voltage using mutual induction. It has two coils, the primary (N_p turns) and secondary (N_s turns), wound on a laminated soft iron core.
For an ideal transformer:
V_s ⁄ V_p = N_s ⁄ N_p = I_p ⁄ I_s
Step-up transformer: N_s > N_p, voltage increases, current decreases.
Step-down transformer: N_s < N_p, voltage decreases, current increases.
Electricity is transmitted at very high voltages (hundreds of kilovolts) so the current is small, reducing I²R heat loss in the wires. It is stepped down near homes to 220 V.
Energy losses in real transformers come from flux leakage, resistance of the windings, eddy currents (reduced by a laminated core), and hysteresis (reduced by using soft iron). Transformers work only with AC, because a changing flux is needed.
Summary
v = v_m sin ωt; V_rms = V_m/√2; mains 220 V is rms.
X_L = ωL (current lags by 90°); X_C = 1/ωC (current leads by 90°).
Z = √(R² + (X_L − X_C)²); resonance at ω₀ = 1/√(LC), where Z = R.
P = V_rms I_rms cos φ; pure L or C consume no average power.
Transformers: V_s/V_p = N_s/N_p; high-voltage transmission reduces losses.