14.2 Self-Inductance and Inductors

Self-inductance is a crucial concept in electromagnetic circuits. When current changes in a coil, it creates a magnetic field that opposes the change, inducing a voltage. This self-induced emf is key to understanding how inductors work.

Inductors, devices that exploit self-inductance, come in various shapes like solenoids and toroids. Their properties, such as number of turns and core material, affect their inductance and energy storage capacity. Understanding these factors is essential for circuit design and analysis.

Self-Inductance and Inductors

Self-induced emf and current change

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• 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 $[L](https://www.fiveableKeyTerm:L)$ (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