Chapter 7: Alternating Current | Summary

Phasors are rotating vectors used to represent sinusoidally varying quantities in ac circuits. A phasor rotates about the origin with angular speed ω, and its vertical component at any instant gives the instantaneous value of the quantity. The magnitude of a phasor represents the amplitude (peak value) of the quantity.

Phasor diagrams are particularly useful for showing the phase relationship between voltage and current in ac circuits. For a pure resistor, the voltage and current phasors point in the same direction, confirming they are in phase. Phasors, though represented as vectors, are not true vectors — they represent scalar quantities whose amplitudes and phases combine mathematically like rotating vectors.

When ac voltage v = vm sin(wt) is applied to a pure inductor L, the current is i = im sin(wt − π/2), meaning the current lags the voltage by π/2 (one quarter cycle). The quantity XL = ωL is called inductive reactance, measured in ohms. It limits current in an inductive circuit just as resistance limits current in a resistive circuit, and is directly proportional to both frequency and inductance.

The average power supplied to a pure inductor over one complete cycle is zero. The instantaneous power alternates between positive and negative values, meaning energy is alternately stored in and returned from the magnetic field of the inductor. No net energy is dissipated.

When ac voltage v = vm sin(wt) is applied to a pure capacitor C, the current is i = im sin(wt + π/2), meaning the current leads the voltage by π/2. The capacitive reactance XC = 1/(ωC) plays the role of resistance and is inversely proportional to both frequency and capacitance. In a dc circuit, a fully charged capacitor blocks current entirely; in an ac circuit, it allows current to flow by alternately charging and discharging each half cycle.

As with a pure inductor, the average power supplied to a pure capacitor over a complete cycle is zero. The energy is alternately stored in and released from the electric field of the capacitor. The current through a capacitor leads the voltage by π/2, which is opposite to the behavior of an inductor.

In a series LCR circuit driven by v = vm sin(wt), the current is i = im sin(wt + φ), where im = vm/Z and Z = √[R² + (XC − XL)²] is the impedance. The phase angle φ = tan⁻¹[(XC − XL)/R]. When XC > XL, φ is positive and the circuit is capacitive (current leads voltage); when XC < XL, φ is negative and the circuit is inductive (current lags voltage).

The phasor diagram solution gives a steady-state result. In the LCR phasor diagram, VR is parallel to current phasor I, VC lags I by π/2, and VL leads I by π/2. Since VC and VL are in opposite directions, they partially cancel, and the net voltage phasor is the vector sum of VR and (VL − VC). Voltages across individual components cannot be added algebraically because they are out of phase.

Resonance in a series LCR circuit occurs when XC = XL, i.e., when ω₀ = 1/√(LC). At this resonant frequency, the impedance is minimum (Z = R) and the current amplitude is maximum (im = vm/R). The voltages across L and C cancel each other since they are equal in magnitude and opposite in phase.

Resonance is used in practical applications such as radio and TV tuning. By varying the capacitance, the resonant frequency of the tuning circuit is made equal to the frequency of the desired station, maximizing the current for that signal. Resonance can only occur when both L and C are present in the circuit; RL or RC circuits alone cannot exhibit resonance.

The average power dissipated in an ac circuit is P = VI cos φ, where cos φ is the power factor. In a purely resistive circuit, φ = 0 and cos φ = 1, giving maximum power dissipation. In purely inductive or capacitive circuits, φ = π/2 and cos φ = 0, so no power is dissipated even though current flows — this current is called wattless current. In a series LCR circuit, power is dissipated only in the resistor.

At resonance in an LCR circuit, φ = 0 and power dissipated is maximum: P = I²R. The power factor can be improved (made closer to 1) by connecting a capacitor in parallel to neutralize the lagging wattless component of current. A low power factor in power transmission lines implies high current for a given power delivery, leading to large I²R losses.

A transformer works on the principle of mutual induction and is used to change ac voltage from one value to another. It consists of a primary coil (Np turns) and a secondary coil (Ns turns) wound on a soft-iron core. The voltage ratio is Vs/Vp = Ns/Np, and for an ideal transformer, the current ratio is Ip/Is = Ns/Np. If Ns > Np, voltage is stepped up (step-up transformer); if Ns < Np, voltage is stepped down (step-down transformer).

In practice, transformers have small energy losses due to flux leakage (reduced by winding coils over each other), resistance of windings (minimized using thick wire for high-current windings), eddy currents in the iron core (reduced by using a laminated core), and hysteresis loss (minimized by using low-hysteresis magnetic materials). A well-designed transformer has efficiency exceeding 95%. Transformers enable long-distance power transmission by stepping up voltage (reducing current and hence I²R losses) and stepping down at the consumer end.

Chapter 7 on Alternating Current establishes the foundational principles governing ac circuits. The key insight is that resistors, inductors, and capacitors respond differently to ac: resistors dissipate power with voltage and current in phase; inductors cause current to lag voltage by π/2 with zero average power; capacitors cause current to lead voltage by π/2 also with zero average power. In series LCR circuits, impedance combines these effects, and at resonance (ω₀ = 1/√LC), the circuit draws maximum current with all source voltage appearing across the resistance. The power factor (cos φ) governs actual power consumption, and transformers — exploiting mutual induction — enable efficient long-distance power transmission by voltage conversion.

From the CBSE board exam perspective, this chapter is highly important and regularly tested. Students must be able to derive expressions for reactance, impedance, and phase angle; draw and interpret phasor diagrams; apply the resonance condition and explain its applications; calculate rms values, power, and power factor; and explain transformer working with energy loss mechanisms. Numerical problems involving LCR circuits, resonance frequency calculation, and transformer turns ratio are standard board-level questions and require careful formula application.