CBSE Class 12 › Unit IV: Electromagnetic Induction & AC

Alternating Current

NCERT Chapter 7 • LCR Circuits, Resonance & Transformers

NCERT 2025–26 Unit IV • ~8 Marks JEE Main • 2 Questions

1. AC Voltage and RMS Value

An alternating voltage varies sinusoidally with time: $v = v_m \sin \omega t$ , where $v_m$ is the peak amplitude.

Why RMS? The average value of AC over a full cycle is zero. To measure the heating effect (power), we use the Root Mean Square (RMS) value. Read detailed explanation on RMS vs Average value here.

I_{rms} = \frac{I_m}{\sqrt{2}} = 0.707 I_m

V_{rms} = \frac{V_m}{\sqrt{2}} = 0.707 V_m

Note: The voltage of household mains is 220V. This is the RMS value. The peak voltage is actually $220\sqrt{2} \approx 311$ V.

2. AC Applied to Resistor, Inductor, and Capacitor

The behavior of AC differs significantly from DC due to the phase relationship between voltage and current. For a visual comparison of these three cases, check out our guide on AC Voltage applied to R, L, and C circuits.

Phasor diagrams and waveforms for Resistor, Inductor, and Capacitor circuits. — Phase relationship between Voltage (Blue) and Current (Red) in pure R, L, and C circuits.

A. AC to Resistor

Current and voltage are in phase ( $\phi = 0$ ).

$v = v_m \sin \omega t$
$i = i_m \sin \omega t$

B. AC to Inductor

Current lags behind voltage by $\pi/2$ . The inductor opposes current change via Self-Inductance.

$i = i_m \sin (\omega t - \pi/2)$
Inductive Reactance: $X_L = \omega L = 2\pi \nu L$

C. AC to Capacitor

Current leads voltage by $\pi/2$ .

$i = i_m \sin (\omega t + \pi/2)$
Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi \nu C}$

3. Series LCR Circuit

When a Resistor (R), Inductor (L), and Capacitor (C) are connected in series, we use the Phasor Diagram method to find the net current and impedance. For the complete step-by-step math, see the Full Derivation of Series RLC Circuit Impedance.

Phasor diagram of LCR circuit and the Impedance Triangle. — The impedance Z is the vector sum of Resistance and net Reactance.

Derivation: Impedance (Z) using Phasors 5 Marks

Step 1: Phasor Setup
Since components are in series, Current $I$ is the same for all. Let $I$ be the reference phasor along x-axis.
$V_R$ is in phase with $I$ .
$V_L$ leads $I$ by 90°. $V_C$ lags $I$ by 90°.

Step 2: Net Voltage
The net reactance voltage is $V_L - V_C$ (assuming $V_L > V_C$ ).
The resultant voltage $V$ is the vector sum of $V_R$ and $(V_L - V_C)$ .
$V^2 = V_R^2 + (V_L - V_C)^2$ .

Step 3: Impedance Formula
Substituting $V=IZ$ , $V_R=IR$ , $V_L=IX_L$ , etc.:
$I^2 Z^2 = I^2 R^2 + I^2 (X_L - X_C)^2$ .
$Z = \sqrt{R^2 + (X_L - X_C)^2}$ .

Step 4: Phase Angle ( $\phi$ )
$\tan \phi = \frac{V_L - V_C}{V_R} = \frac{X_L - X_C}{R}$ .

4. Resonance

Resonance occurs when the frequency of the supply equals the natural frequency of the circuit. At this point, $X_L = X_C$ , impedance is minimum, and current is maximum.

For a deeper dive into bandwidth and sharpness of resonance (Q-factor), read our guide on Resonance in Series LCR Circuits.

Graph showing variation of current with frequency in an LCR circuit, highlighting resonance. — At resonance frequency $\omega_0$ , the current amplitude peaks.

\omega_0 = \frac{1}{\sqrt{LC}}

Impedance at Resonance: $Z = R$ (Purely Resistive)

5. Power in AC Circuits

In AC circuits, power is not just $V \times I$ because of the phase difference. The average power dissipated depends on the Power Factor ( $\cos \phi$ ).

P = V_{rms} I_{rms} \cos \phi

Power Factor: $\cos \phi = \frac{R}{Z}$

Special Cases:

Pure Resistor: $\phi = 0$ , $\cos \phi = 1$ . Power is Maximum.
Pure Inductor/Capacitor: $\phi = 90^\circ$ , $\cos \phi = 0$ . Wattless Current (No power loss).

Confused about Real vs Apparent Power? Check this detailed explanation.

6. LC Oscillations

When a charged capacitor is connected to an inductor, energy oscillates between the electric field of the capacitor ( $U_E = q^2/2C$ ) and the magnetic field of the inductor ( $U_B = Li^2/2$ ).

The frequency of oscillation is given by $\omega = \frac{1}{\sqrt{LC}}$ .

7. Transformers

A transformer is a device used to increase or decrease AC voltage. It works on the principle of Mutual Induction. For a full breakdown of efficiency and losses, see Transformer Working Principle & Formula.

Construction and working principle of a step-up and step-down transformer. — Transformers use mutual induction to transfer energy between two coils.

\frac{V_s}{V_p} = \frac{N_s}{N_p} = k

Where $k$ is the Transformation Ratio.

Ideal Transformer (100% Efficiency):
Power Input = Power Output
$V_p I_p = V_s I_s \Rightarrow \frac{I_p}{I_s} = \frac{V_s}{V_p} = \frac{N_s}{N_p}$ .

Related Concept: Displacement Current
Before moving to the next chapter (EM Waves), it is crucial to understand Displacement Current, which resolves inconsistencies in Ampere’s Law for time-varying fields.

Exam Preparation:

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