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CISCEISC (Science) ISC Class 12
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पाठ्यक्रम

Types of Ac Circuits > Circuit Containing Inductance Only

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Estimated time: 8 minutes
CISCE: Class 12

Introduction

When a pure inductor (having negligible resistance) is connected across an AC source, the current in the circuit does not stay "in step" with the voltage. Understanding this phase mismatch is essential for analyzing AC circuits, power calculations, and impedance in more complex LCR networks.

CISCE: Class 12

Circuit Setup

  • A pure inductor L is connected directly across an alternating voltage source.
  • The source voltage varies sinusoidally: V = V0 sin ⁡ωt
  • The circuit has no resistance or capacitance.

CISCE: Class 12

Derivation: Current in a Pure Inductive Circuit

Step 1: Apply Kirchhoff's voltage law:
The self-induced EMF must balance the applied voltage:

  • L\[\frac {dI}{dt}\] = V0 sin⁡ ωt

Step 2: Integrate both sides:

  • dI = \[\frac {V_0}{L}\]sin⁡ ωt dt
  • I = −\[\frac {V_0}{ωL}\]cos⁡ ωt + C

Step 3: Express in standard sinusoidal form:

  • I = \[\frac{V_0}{\omega L}\sin\left(\omega t-\frac{\pi}{2}\right)\]

Step 4: Identify peak current:

  • I0 = \[\frac {V_0}{ωL}\]

Final result:

  • I = \[{I_0\sin\left(\omega t-\frac{\pi}{2}\right)}\]
CISCE: Class 12

Phasor Representation

Analogy: Think of voltage as a "leader" walking a quarter-turn ahead and current as a "follower" always a quarter-cycle behind — they never overlap, but move at the same speed (same frequency).

CISCE: Class 12

Inductive Reactance

Inductive reactance is the AC equivalent of resistance for an inductor:

  • XL = ωL = 2πfL
Quantity Symbol Formula Unit
Inductive reactance XL 2πfL ohm (Ω)
Peak current I0 V0/XL ampere (A)
Phase angle ϕ 90° (current lags) degree/radian

Behaviour of XL​ with Frequency

  • XL ∝ f — reactance increases linearly with frequency.
  • Graph of XL​ vs f is a straight line passing through the origin.
  • For DC supply: f = 0 ⇒ XL = 0 (inductor offers no opposition to steady current, behaves as a plain conductor).

CISCE: Class 12

Comparison: DC vs AC Response of Inductor

Aspect DC Circuit AC Circuit
Frequency f = 0 f > 0
Reactance XL​ 0 2πfL
Current behavior Steady, unopposed after initial transient Continuously opposed, phase-shifted
Phase relation No phase difference Current lags voltage by 90°
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