Alternating Current
'DC goes one way — AC alternates. And that alternation makes ALL the difference for power transmission and circuit behaviour.'
1. Chapter Overview
Alternating Current (AC) is the form of electricity used in homes and industry. Topics include: RMS VALUE of AC, PURELY RESISTIVE/INDUCTIVE/CAPACITIVE circuits, the LCR SERIES CIRCUIT (and its IMPEDANCE), RESONANCE in AC circuits, POWER IN AC circuits (real power, reactive power, power factor), and the TRANSFORMER.
2. AC Voltage and Current
- AC voltage: V = V₀ sin(ωt). V₀ = peak voltage.
- AC current: I = I₀ sin(ωt + φ). φ = phase difference between V and I.
RMS Values
- V_rms = V₀/√2. I_rms = I₀/√2.
- 'RMS gives the DC-equivalent value — the AC voltage that produces the SAME heating effect as an equivalent DC voltage.'
Average Power
- P_avg = V_rms × I_rms × cos φ. 'Only the component of current IN PHASE with voltage contributes to average power.'
3. Purely Resistive, Inductive, and Capacitive Circuits
| Circuit Type | V and I Phase | Impedance | Power Factor | Average Power |
|---|---|---|---|---|
| Resistive (R) | In phase (φ=0°) | R | 1 (unity) | V_rms × I_rms |
| Inductive (L) | I LAGS V by 90° | X_L = ωL | 0 | 0 |
| Capacitive (C) | I LEADS V by 90° | X_C = 1/ωC | 0 | 0 |
- 'In pure L or C, average power is ZERO — energy is alternately stored and returned, never dissipated.'
4. LCR Series Circuit
Impedance (Z)
- Z = √[R² + (X_L − X_C)²].
- Phase angle: tan φ = (X_L − X_C)/R.
Resonance
- Resonance condition: X_L = X_C ⇒ ωL = 1/ωC ⇒ ω₀ = 1/√(LC).
- At resonance: Z = R (minimum). I₀ = V₀/R (maximum).
- Sharpness of resonance: Q-factor = (ω₀L)/R = 1/(ω₀RC). 'Higher Q means SHARPER resonance — a more frequency-selective circuit.'
- 'At resonance, an LCR circuit behaves as a PURE RESISTOR — voltage and current are in phase.'
Worked Example 1
Problem: An LCR circuit has R = 10 Ω, L = 0.1 H, C = 100 μF. Find the resonant frequency. Solution: ω₀ = 1/√(LC) = 1/√(0.1 × 100×10⁻⁶) = 1/√(10⁻⁵) = 316.2 rad/s. f₀ = ω₀/(2π) ≈ 50.3 Hz.
5. Power in AC Circuits
- True power: P = V_rms I_rms cos φ. 'The power actually DISSIPATED.'
- Apparent power: S = V_rms I_rms. 'The product of RMS V and I.'
- Power factor: cos φ = R/Z. 'How much of the apparent power is actual power?'
- 'A LOW power factor means HIGH current for the same real power — causing higher losses in transmission lines.'
Wattless Current
- I_rms sin φ — component of current that does NOT contribute to power.
6. Transformer
- Principle: Mutual induction between two coils wound on a common core.
- Voltage ratio: V_s/V_p = N_s/N_p.
- Current ratio: I_s/I_p = N_p/N_s (for ideal transformer).
- Power: P_in = P_out (ideal). P_out = P_in − losses (real).
Types
| Type | N_s/N_p | V_s | I_s | Use |
|---|---|---|---|---|
| Step-up | N_s > N_p | V_s > V_p | I_s < I_p | Power transmission (increase voltage, decrease current → reduce I²R loss) |
| Step-down | N_s < N_p | V_s < V_p | I_s > I_p | Home appliances (reduce voltage from transmission lines) |
Energy Losses in a Transformer
- Copper loss: I²R in the windings.
- Eddy current loss: Laminated core reduces this.
- Hysteresis loss: Use soft iron core (low hysteresis loop area).
- Flux leakage: Not all flux from primary reaches secondary.
7. Comparison Table: DC vs AC
| Feature | DC | AC |
|---|---|---|
| Direction | Constant (one direction) | Reverses periodically |
| Frequency | 0 | 50 Hz (India) / 60 Hz (USA) |
| Generation | Battery, cell | Generator, alternator |
| Transmission | High loss over distance | Can be stepped up/down — LOW loss |
| Capacitor | Blocks DC | Passes AC (impedance depends on frequency) |
| Inductor | Zero impedance (steady state) | Impedance X_L = ωL |
8. Common Mistakes
- RMS vs peak: V_rms = V₀/√2. I_rms = I₀/√2. Many students use these interchangeably.
- Phase in LCR: tan φ = (X_L − X_C)/R. If X_L > X_C, φ > 0 (voltage leads). If X_C > X_L, φ < 0 (current leads).
- Resonance misunderstanding: At resonance, Z = R is MINIMUM, and I is MAXIMUM. NOT the other way around.
- Transformer equations: V_s/V_p = N_s/N_p = I_p/I_s. Note the INVERSE relationship for current.
9. CBSE Exam Focus
- RMS value — definition and calculation for sinusoidal waveforms
- Phasor diagrams — R, L, C, and LCR series circuits
- Impedance and phase angle — Z = √(R² + (XL − XC)²), tan φ = (XL − XC)/R
- Resonance — ω₀ = 1/√(LC), Q-factor
- Power in AC — P = V_rms I_rms cos φ, power factor
- Transformer — voltage/current ratios, step-up/step-down, losses
10. Self-Test
Q1: A 220 V AC supply has a peak voltage of? A1: V₀ = V_rms × √2 = 220 × 1.414 = 311 V.
Q2: An LCR circuit has R = 20 Ω, XL = 30 Ω, XC = 10 Ω. Find impedance and phase angle. A2: Z = √(20² + (30−10)²) = √(400 + 400) = √800 = 28.28 Ω. tan φ = (30−10)/20 = 20/20 = 1 ⇒ φ = 45° (voltage LEADS current).
Q3: A 0.1 H inductor and 50 μF capacitor are in series. Find the resonant frequency. A3: ω₀ = 1/√(0.1 × 50×10⁻⁶) = 1/√(5×10⁻⁶) = 447.2 rad/s. f₀ = 447.2/2π ≈ 71.2 Hz.
Q4: A step-down transformer converts 220 V to 11 V. If the primary has 1000 turns, find secondary turns. A4: N_s/N_p = V_s/V_p ⇒ N_s/1000 = 11/220 = 0.05 ⇒ N_s = 50 turns.
Q5: In an LCR circuit, V = 200 V, I = 5 A, and the power consumed is 800 W. Find the power factor. A5: P = V_rms I_rms cos φ ⇒ 800 = 200×5×cos φ ⇒ cos φ = 800/1000 = 0.8 (lagging or leading).
11. Conclusion
AC power is the BACKBONE of modern electrical systems:
- RMS: 'The DC-equivalent — what you actually pay for on your electricity bill.'
- REACTANCE: 'Inductors and capacitors have frequency-dependent resistance — they STORE and RELEASE energy.'
- RESONANCE: 'The magic frequency where L and C cancel — current MAXIMISES, impedance MINIMISES.'
- POWER FACTOR: 'How efficiently you use the current drawn from the supply.'
- TRANSFORMER: 'The reason AC won over DC — voltage can be transformed UP for efficient transmission.'
'AC is the universal language of power distribution — its ability to transform voltage makes it indispensable.'
