CBSE Class 12 Physics Alternating Current Notes Set 03

1. Alternating Current and Voltage

An alternating voltage (ac voltage) is one whose magnitude changes continuously with time and whose direction reverses periodically. The simplest form varies sinusoidally:
v = v_m sin ωt
where v_m is the peak value (amplitude) and ω is the angular frequency. The current it drives, i = i_m sin (ωt + φ), is called an alternating current.
ω = 2πν = \( \frac{2\pi}{T} \), where ν is the frequency and T is the time period. The domestic supply in India has ν = 50 Hz.

AC is preferred over DC for distribution because its voltage can be stepped up or down easily with transformers, so power can be sent over long distances at high voltage with small losses.

2. Mean and RMS Values

(i) Average over a full cycle: For i = i_m sin ωt, the current is positive for half the cycle and negative for the other half, so its average over one complete cycle is zero.
(ii) RMS (effective) value: The rms current is the steady (dc) current that produces the same heat in a resistor in the same time as the alternating current.
Since the average of sin² ωt over a cycle is ½, the average power in a resistor is \( \bar{P} = \frac{1}{2} i_m^2 R \) = I²R, which gives
I = I_rms = \( \frac{i_m}{\sqrt{2}} \) ≈ 0.707 i_m  and  V = V_rms = \( \frac{v_m}{\sqrt{2}} \) ≈ 0.707 v_m
1. AC ammeters and voltmeters read rms values. When we say the household supply is 220 V, this is the rms value; its peak value is 220 × √2 ≈ 311 V.
2. In terms of rms values, ac power formulae look exactly like dc ones: P = VI = I²R = V²/R for a resistor.

3. AC Voltage Applied to a Resistor

With v = v_m sin ωt across a resistor R, Kirchhoff's loop rule gives
i = \( \frac{v_m}{R} \) sin ωt = i_m sin ωt, with i_m = \( \frac{v_m}{R} \)
1. Voltage and current are in phase: both reach zero, maximum and minimum together.
2. Ohm's law holds for ac in a resistor; R does not depend on frequency.
3. Average power: \( \bar{P} = \frac{1}{2} i_m^2 R \) = I²R. The power is never negative, so a resistor always takes energy from the source and turns it into heat (Joule heating).

4. Phasors

A phasor is a vector that rotates anticlockwise about the origin with angular speed ω. Its length represents the peak value (v_m or i_m), and its projection on the vertical axis gives the instantaneous value (v or i). The angle between the voltage and current phasors is the phase difference φ. Phasor diagrams make it easy to add voltages that are not in phase.

5. AC Voltage Applied to an Inductor

For a pure inductor L (negligible resistance) connected to v = v_m sin ωt, the induced emf opposes the source: v − L \( \frac{di}{dt} \) = 0. Integrating gives
i = i_m sin \( \left(\omega t - \frac{\pi}{2}\right) \), with i_m = \( \frac{v_m}{\omega L} \)
Inductive reactance: X_L = ωL = 2πνL, SI unit ohm (Ω).
1. The current lags the voltage by π/2 (a quarter of a cycle).
2. X_L ∝ ν: an inductor offers more opposition at high frequency and zero reactance to dc (ν = 0).
3. Average power over a complete cycle is zero: energy taken from the source in one quarter-cycle is stored in the magnetic field and returned in the next.

6. AC Voltage Applied to a Capacitor

For a capacitor C connected to v = v_m sin ωt, q = Cv and i = \( \frac{dq}{dt} \), which gives
i = i_m sin \( \left(\omega t + \frac{\pi}{2}\right) \), with i_m = \( \frac{v_m}{1/(\omega C)} \)
Capacitive reactance: X_C = \( \frac{1}{\omega C} = \frac{1}{2\pi\nu C} \), SI unit ohm (Ω).
1. The current leads the voltage by π/2.
2. X_C ∝ 1/ν: a capacitor blocks dc (infinite reactance at ν = 0) but lets high-frequency ac pass easily.
3. Average power over a complete cycle is zero; energy stored in the electric field is returned to the source.

7. Series LCR Circuit (Phasor Method)

A resistor R, inductor L and capacitor C are connected in series with an ac source v = v_m sin ωt. The same current i = i_m sin (ωt + φ) flows through all three, so the current phasor is taken as the reference.

1. V_R = i_mR is in phase with I; V_L = i_mX_L leads I by 90°; V_C = i_mX_C lags I by 90°.
2. V_L and V_C are opposite, so they combine to (V_L − V_C). The source voltage is the phasor sum:
v_m² = V_R² + (V_L − V_C)² = i_m² [R² + (X_L − X_C)²]
3. Impedance: Z = \( \frac{v_m}{i_m} = \sqrt{R^2 + (X_L - X_C)^2} \), SI unit ohm (Ω). Also Z = V_rms/I_rms.
4. Phase angle between voltage and current: tan φ = \( \frac{X_C - X_L}{R} \) in the form i = i_m sin (ωt + φ). In magnitude, tan φ = \( \frac{|X_L - X_C|}{R} \).
5. If X_L > X_C, the circuit is inductive and the current lags the voltage. If X_C > X_L, it is capacitive and the current leads.
6. Impedance triangle: R, (X_L − X_C) and Z form a right triangle with Z as hypotenuse, so cos φ = R/Z.
7. Because of the phase differences, the rms voltages across R, L and C do not simply add up to the source voltage; they add as phasors.

8. Resonance

X_L = ωL increases with frequency while X_C = 1/(ωC) decreases. At one frequency they become equal; this is called resonance.
X_L = X_C  ⇒  ω₀L = \( \frac{1}{\omega_0 C} \)  ⇒  ω₀ = \( \frac{1}{\sqrt{LC}} \)  or  ν₀ = \( \frac{1}{2\pi\sqrt{LC}} \)
At resonance:
1. Z = R, its minimum value, so the current is maximum: i_m = v_m/R.
2. The voltage and current are in phase (φ = 0), and the power factor is 1.
3. V_L and V_C are equal and opposite and cancel each other, though each may be much larger than the source voltage.
4. Resonance is possible only when both L and C are present in the circuit; an RL or RC circuit cannot resonate.

Application: The tuning circuit of a radio uses series resonance. Turning the knob changes C, and when ω₀ matches the frequency of a particular station, the current from that station's signal becomes maximum and the station is heard.

9. Power in an AC Circuit: Power Factor

With v = v_m sin ωt and i = i_m sin (ωt + φ), the average power over a cycle is
P = \( \frac{v_m i_m}{2} \) cos φ = V I cos φ = I²Z cos φ
where V and I are rms values and cos φ is the power factor. For a series LCR circuit, cos φ = R/Z.
1. Pure resistor: φ = 0, cos φ = 1, power is maximum (P = VI).
2. Pure inductor or pure capacitor: φ = ±π/2, cos φ = 0, so P = 0 even though a current flows.
3. Series LCR circuit: P = I²Z cos φ = I²R. Power is dissipated only in the resistance.
4. At resonance: cos φ = 1 and P = I²R is the largest possible for the given R.
5. VI is called the apparent power (unit V A) and VI cos φ the true power (unit W).

Why a low power factor is undesirable: For a given power P = VI cos φ at fixed V, a small cos φ means a large current I, and the line losses I²R in transmission increase. Power factor is improved by connecting a suitable capacitor in parallel with an inductive load, so that the current component in quadrature is reduced.

10. Wattless Current

The current I can be resolved into two components with respect to the voltage:
1. I cos φ, in phase with the voltage. This is the active component that consumes power.
2. I sin φ, at 90° to the voltage. It contributes no average power (power factor of this part is cos 90° = 0) and is called the wattless current.
In a circuit containing only an ideal inductor or only a capacitor, the whole current is wattless.

11. Transformers

A transformer changes an alternating voltage from one value to another. It works on mutual induction: an alternating current in the primary coil sets up a changing magnetic flux in the core, which induces an emf in the secondary coil. It cannot work with steady dc.

Construction: Two coils, the primary (N_p turns) and the secondary (N_s turns), are wound on a common laminated soft-iron core, either on separate limbs or one over the other.
Working (ideal transformer): If the primary has negligible resistance and all the flux links both coils, the same rate of change of flux dφ/dt per turn acts in both coils:
ε_s = −N_s \( \frac{d\phi}{dt} \),   v_p = −N_p \( \frac{d\phi}{dt} \)  ⇒  \( \frac{V_s}{V_p} = \frac{N_s}{N_p} \)
If there is no energy loss, input power = output power: I_pV_p = I_sV_s, so
\( \frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s} \)
1. Step-up transformer: N_s > N_p, so V_s > V_p and I_s < I_p.
2. Step-down transformer: N_s < N_p, so V_s < V_p and I_s > I_p.
3. A transformer does not change the frequency, and an ideal one does not change the power.
4. Efficiency η = \( \frac{\text{output power}}{\text{input power}} \) × 100%. Large transformers reach 95% or more.

Energy losses in a real transformer:
1. Flux leakage: some primary flux does not pass through the secondary. Reduced by winding the coils one over the other.
2. Resistance of the windings: heat I²R is produced in the copper wire. Reduced by using thick wire, especially for the high-current coil.
3. Induced currents in the iron core heat it. Reduced by using a core made of thin insulated sheets (laminations).
4. Repeated magnetisation and demagnetisation of the core wastes some energy as heat. Reduced by using soft iron.

Long-distance transmission: At a generating station the voltage is stepped up (to several hundred kV), so the current in the lines is small and the I²R loss is small. Near the consumers the voltage is stepped down in stages to 220 V.

12. Quick Formula Recap