Class 12 Physics · Chapter 7 NotesAlternating Current

When a sinusoidal voltage v = vₘ sin ωt is applied across a resistor R, Kirchhoff's loop rule gives vₘ sin ωt = iR, so the current is i = iₘ sin ωt with iₘ = vₘ/R. This is Ohm's law and it works for ac just as it does for dc. The important point is that voltage and current reach zero, minimum and maximum at exactly the same instants, so they are in phase. Even though the average current over a full cycle is zero, the average power is not zero, because Joule heating depends on i², which is always positive. The instantaneous power is p = iₘ²R sin²ωt, and its average over a cycle works out to (1/2)iₘ²R.

• v = vₘ sin ωt, i = iₘ sin ωt, iₘ = vₘ/R
• Voltage and current are in phase for a pure resistor
• Average power P = (1/2) iₘ²R

To write ac power in the same form as dc power, we define the root mean square or effective current I = iₘ/√2 = 0.707 iₘ, and similarly the rms voltage V = vₘ/√2 = 0.707 vₘ. Using these, the average power becomes P = I²R = IV, and the relation between current and voltage becomes V = IR, exactly like the dc case. This is why rms values are used to specify ac quantities. For example, the household supply of 220 V is an rms value; its peak value is vₘ = √2 × 220 V ≈ 311 V. The rms current is the equivalent dc current that would produce the same average power loss.

• I = iₘ/√2 ≈ 0.707 iₘ
• V = vₘ/√2 ≈ 0.707 vₘ
• P = I²R = IV and V = IR
• 220 V mains is an rms value; peak is about 311 V

A phasor is a vector that rotates about the origin with angular speed ω. Its magnitude represents the amplitude (peak value) of the alternating quantity, and its vertical component at any instant gives the instantaneous value. Phasor diagrams make it easy to see the phase relationship between voltage and current in a circuit. For a pure resistor, the voltage and current phasors point in the same direction, showing zero phase difference. Phasors are not vectors in the true sense because voltage and current are scalar quantities; they are used only because harmonically varying scalars combine in the same way as the projections of rotating vectors.

• Phasor magnitude = amplitude of the quantity
• Vertical component = instantaneous value
• For a resistor, V and I phasors are in the same direction

For a purely inductive circuit, Kirchhoff's loop rule gives v − L(di/dt) = 0. Integrating, the current is i = iₘ sin(ωt − π/2), so the current lags the voltage by π/2, or one-quarter of a cycle. The quantity ωL plays the role of resistance and is called inductive reactance, X_L = ωL, measured in ohm. The current amplitude is iₘ = vₘ/X_L. Inductive reactance increases with frequency and with inductance, so a high-frequency ac is opposed more strongly by an inductor. The instantaneous power is p = −(iₘvₘ/2) sin 2ωt, whose average over a complete cycle is zero, so an ideal inductor consumes no net power.

• X_L = ωL, unit ohm
• iₘ = vₘ/X_L
• Current lags voltage by π/2
• Average power over a cycle is zero

When an ac source is connected to a capacitor, the capacitor charges and discharges alternately, so current flows continuously, unlike in a dc circuit where the current stops once the capacitor is fully charged. The current is i = iₘ sin(ωt + π/2), so the current leads the voltage by π/2. The quantity 1/ωC is called capacitive reactance, X_C = 1/ωC, and the current amplitude is iₘ = vₘ/X_C. Capacitive reactance decreases as frequency or capacitance increases. As with the inductor, the average power supplied to a pure capacitor over a complete cycle is zero, because the average of sin 2ωt over a cycle is zero.

• X_C = 1/ωC, unit ohm
• iₘ = vₘ/X_C
• Current leads voltage by π/2
• Average power over a cycle is zero

In a series LCR circuit the same current flows through all three elements. Using phasors, V_R is in phase with I, V_L leads I by π/2 and V_C lags I by π/2. Combining these, the source voltage amplitude satisfies vₘ² = v_Rm² + (v_Cm − v_Lm)², which gives iₘ = vₘ/Z, where Z = √(R² + (X_C − X_L)²) is the impedance of the circuit. The phase angle is given by tan φ = (X_C − X_L)/R. If X_C > X_L the circuit is predominantly capacitive and current leads the voltage; if X_C < X_L it is predominantly inductive and current lags the voltage. The impedance diagram is a right triangle with Z as the hypotenuse.

• Z = √(R² + (X_C − X_L)²)
• iₘ = vₘ/Z
• tan φ = (X_C − X_L)/R
• Capacitive if X_C > X_L; inductive if X_C < X_L

Resonance occurs when the inductive and capacitive reactances become equal, X_C = X_L, so the impedance is minimum and equal to R. The frequency at which this happens is the resonant frequency ω₀ = 1/√(LC). At resonance the current amplitude is maximum, iₘ = vₘ/R, and the voltages across L and C cancel each other because they are out of phase. Resonance is possible only when both L and C are present; an RL or RC circuit cannot show resonance. This principle is used in the tuning circuit of a radio or TV, where the capacitance is varied so that the circuit resonates at the frequency of the desired station.

• Resonant frequency ω₀ = 1/√(LC)
• At resonance Z = R and iₘ = vₘ/R
• Voltages across L and C cancel
• Used in radio and TV tuning circuits

For a series LCR circuit driven by v = vₘ sin ωt, the current is i = iₘ sin(ωt + φ). The instantaneous power is p = (vₘiₘ/2)[cos φ − cos(2ωt + φ)], and the average power over a cycle is P = (vₘiₘ/2) cos φ = V I cos φ. The quantity cos φ is called the power factor. For a pure resistor φ = 0, so cos φ = 1 and power dissipation is maximum. For a purely inductive or capacitive circuit φ = π/2, so cos φ = 0 and no net power is dissipated; such a current is called wattless current. In an LCR circuit, power is dissipated only in the resistor. A low power factor in transmission means a larger current for the same power, increasing I²R losses.

• P = V I cos φ
• cos φ is the power factor
• cos φ = 1 for pure R; cos φ = 0 for pure L or C
• Wattless currentno net power dissipation

A transformer changes an alternating voltage from one value to another using mutual induction. It has a primary coil of N_p turns and a secondary coil of N_s turns wound on a soft-iron core. For an ideal transformer, V_s/V_p = N_s/N_p and I_s/I_p = N_p/N_s. If N_s > N_p the voltage is stepped up and the current is reduced; if N_s < N_p the voltage is stepped down and the current is increased. In practice, energy losses occur due to flux leakage, resistance of windings, eddy currents and hysteresis, and these are reduced by careful winding, thick wire, laminated cores and low-hysteresis material. For transmission, voltage is stepped up to reduce current and hence I²R loss, then stepped down near consumers.

• V_s/V_p = N_s/N_p and I_s/I_p = N_p/N_s
• Step-up transformerN_s > N_p
• Step-down transformerN_s < N_p
• Lossesflux leakage, winding resistance, eddy currents, hysteresis