14.2 Self-Inductance and Inductors

Self-inductance describes how a changing current in a coil creates a magnetic field that opposes the change and induces a voltage. This self-induced emf explains how inductors resist sudden current changes.

Inductors use self-inductance and come in shapes such as solenoids and toroids. Their number of turns, geometry, and core material affect inductance, energy storage, and circuit behavior.

Self-Inductance and Inductors

Self-induced emf and current change

• Faraday's law of induction states that a changing magnetic flux through a loop induces an electromotive force (emf) in the loop
  • Induced emf opposes the change in magnetic flux, according to Lenz's law (back emf)
• In a circuit with an inductor, a changing current generates a changing magnetic flux
  • Changing magnetic flux induces an emf in the inductor, called the self-induced emf (voltage drop across inductor)
• Self-induced emf is proportional to the rate of change of current in the circuit
  • Constant of proportionality is the self-inductance, denoted by $L$ (henry, H)
  • Self-induced emf is given by: $\mathcal{E} = -L \frac{dI}{dt}$
    • $\mathcal{E}$ is the self-induced emf (volts, V)
    • $L$ is the self-inductance (henries, H)
    • $\frac{dI}{dt}$ is the rate of change of current (amperes per second, A/s)
• Examples of devices with self-inductance include transformers, electric motors, and solenoids

Self-inductance of cylindrical solenoids

• Self-inductance of a cylindrical solenoid depends on its physical properties
  • Number of turns, $N$
  • Length of the solenoid, $l$ (meters, m)
  • Cross-sectional area, $A$ (square meters, m²)
  • Permeability of the core material, $\mu$ (henries per meter, H/m)
• Self-inductance of a cylindrical solenoid is given by: $L = \frac{\mu N^2 A}{l}$
  • $L$ is the self-inductance (henries, H)
  • $\mu$ is the permeability of the core material (henries per meter, H/m)
    • For an air-core solenoid, $\mu = \mu_0 = 4\pi \times 10^{-7} \text{ H/m}$
  • $N$ is the number of turns
  • $A$ is the cross-sectional area (square meters, m²)
  • $l$ is the length of the solenoid (meters, m)
• Increasing the number of turns or cross-sectional area increases the self-inductance
• Increasing the length of the solenoid decreases the self-inductance
• Examples of cylindrical solenoids include electromagnets and inductors in electronic circuits
• The magnetic field inside the solenoid is directly related to its self-inductance

Self-inductance in rectangular toroids

• Toroid is a doughnut-shaped inductor with a rectangular or circular cross-section
• Self-inductance of a rectangular toroid depends on its geometry and material characteristics
  • Number of turns, $N$
  • Mean radius of the toroid, $r$ (meters, m)
  • Height of the rectangular cross-section, $h$ (meters, m)
  • Width of the rectangular cross-section, $w$ (meters, m)
  • Permeability of the core material, $\mu$ (henries per meter, H/m)
• Self-inductance of a rectangular toroid is given by: $L = \frac{\mu N^2 h w}{2\pi r}$
  • $L$ is the self-inductance (henries, H)
  • $\mu$ is the permeability of the core material (henries per meter, H/m)
  • $N$ is the number of turns
  • $h$ is the height of the rectangular cross-section (meters, m)
  • $w$ is the width of the rectangular cross-section (meters, m)
  • $r$ is the mean radius of the toroid (meters, m)
• Increasing the number of turns, height, or width of the cross-section increases the self-inductance
• Increasing the mean radius of the toroid decreases the self-inductance
• Examples of rectangular toroids include transformers and inductors in power electronics

Energy and Time Characteristics of Inductors

• Inductors store energy in their magnetic field
• The energy stored in an inductor is given by: $E = \frac{1}{2}LI^2$
  • $E$ is the energy stored (joules, J)
  • $L$ is the inductance (henries, H)
  • $I$ is the current (amperes, A)
• The inductance density of a material affects its energy storage capacity
• The time constant of an inductor in an RL circuit is given by: $\tau = \frac{L}{R}$
  • $\tau$ is the time constant (seconds, s)
  • $L$ is the inductance (henries, H)
  • $R$ is the resistance (ohms, Ω)
• Inductors with a magnetic core have higher inductance and energy storage capacity compared to air-core inductors