Inductance

Why it matters

Inductance is a fundamental concept in electromagnetics that plays a crucial role in the design and functioning of electrical circuits, especially in transformers, inductors, and various types of sensors. Understanding inductance is essential for analyzing how circuits respond to changes in current and for designing systems that efficiently manage energy transfer.

Key ideas

• Inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force (EMF) in the conductor itself and in any nearby conductors by mutual inductance.
• Self-inductance occurs when a changing current in a coil induces an EMF in the same coil. It is a measure of how effectively a coil can induce voltage in itself.
• Mutual inductance is the phenomenon where a change in current in one coil induces an EMF in a nearby coil. This is the principle behind transformers.
• Inductors are passive components designed to take advantage of inductance in circuits, often used to store energy temporarily in magnetic fields.
• The unit of inductance is the henry (H), where 1 henry is the inductance of a circuit in which a change in current of 1 ampere per second induces an EMF of 1 volt.

Formulas

The N²μA/l formula assumes a long solenoid or uniform magnetic circuit with constant μ and negligible leakage/fringing. For linear inductors, stored energy is LI²/2 and passive-terminal voltage is v = L di/dt. Mutual-inductance magnitude satisfies |M| ≤ √(L1L2); polarity is determined by winding references/dots.

• Self-inductance: L = N²·μ·A / l

  • L = inductance (henry, H)
  • N = number of turns
  • μ = permeability of the core material (henry per meter, H/m)
  • A = cross-sectional area of the coil (square meters, m²)
  • l = length of the coil (meters, m)

• Mutual inductance: M = k·√(L₁·L₂)

  • M = mutual inductance (henry, H)
  • k = coupling coefficient (dimensionless)
  • L₁, L₂ = inductances of the two coils (henry, H)

Worked example

Given: A coil with 200 turns, a core permeability of 4π × 10⁻⁷ H/m, a cross-sectional area of 0.01 m², and a length of 0.5 m.

1. Calculate the self-inductance of the coil.

   Formula: L = N²·μ·A / l

   Substituting the given values:

   L = 200² × 4π × 10⁻⁷ H/m × 0.01 m² / 0.5 m

   L = 40000 × 4π × 10⁻⁷ × 0.01 / 0.5

   L = 0.00032π H

   Final Answer: L ≈ 0.001005 H (1.005 mH)

Common mistakes

• Confusing self-inductance with mutual inductance.
• Forgetting to square the number of turns in the self-inductance formula.
• Ignoring the units, especially when dealing with permeability and area.

For GATE EC

Questions on inductance often involve calculating self or mutual inductance, analyzing circuits with inductors, and understanding the effects of inductance in AC circuits. Practice problems involving transformers and energy storage in inductors.

Quick check

1. What is the unit of inductance?
2. What does mutual inductance depend on?
3. How does increasing the number of turns affect self-inductance?

Answers: 1. Henry (H), 2. Coupling coefficient and inductances of the coils, 3. Increases it.