11.1 Self-inductance and mutual inductance

Inductance is a crucial concept in electromagnetism, describing how changing currents create magnetic fields that induce voltages. Self-inductance occurs within a single circuit, while mutual inductance involves interactions between separate circuits.

Understanding inductance is key to grasping the behavior of circuits with changing currents. This topic explores the types of inductance, their units, and the laws governing their effects, setting the stage for more complex electromagnetic phenomena.

Inductance Types

Self-Inductance

• Property of a circuit element that opposes changes in current through it
• Caused by the magnetic field created by the current in the circuit element itself
• When current changes, the magnetic field changes, inducing a voltage that opposes the change in current (Lenz's law)
• Represented by the symbol $L$ and measured in henries (H)
• Formula: $V_L = L \frac{dI}{dt}$, where $V_L$ is the induced voltage, $L$ is the self-inductance, and $\frac{dI}{dt}$ is the rate of change of current

Mutual Inductance

• Occurs when a changing current in one circuit induces a voltage in another nearby circuit
• Caused by the magnetic field created by the current in one circuit linking with the other circuit
• Mutual inductance depends on the geometry of the circuits and their relative positions
• Represented by the symbol $M$ and measured in henries (H)
• Formula: $V_2 = M \frac{dI_1}{dt}$, where $V_2$ is the induced voltage in the second circuit, $M$ is the mutual inductance, and $\frac{dI_1}{dt}$ is the rate of change of current in the first circuit

Coupling Coefficient

• Measure of the strength of the magnetic coupling between two inductors
• Ranges from 0 (no coupling) to 1 (perfect coupling)
• Depends on the physical arrangement of the inductors and their properties
• Formula: $k = \frac{M}{\sqrt{L_1 L_2}}$, where $k$ is the coupling coefficient, $M$ is the mutual inductance, and $L_1$ and $L_2$ are the self-inductances of the two inductors
• Higher coupling coefficients indicate stronger magnetic coupling and more efficient energy transfer between the inductors (transformers)

Inductance Units and Laws

Henry

• SI unit of inductance, named after Joseph Henry
• Defined as the inductance of a closed circuit in which an electromotive force of one volt is produced when the electric current in the circuit varies uniformly at a rate of one ampere per second
• Symbol: H
• Equivalent to weber per ampere ($\frac{Wb}{A}$) or volt-second per ampere ($\frac{V \cdot s}{A}$)

Faraday's Law

• States that the electromotive force (emf) induced in a closed circuit is equal to the negative of the rate of change of the magnetic flux through the circuit
• Mathematical form: $\varepsilon = -\frac{d\Phi}{dt}$, where $\varepsilon$ is the induced emf, $\Phi$ is the magnetic flux, and $t$ is time
• The negative sign indicates that the induced emf opposes the change in magnetic flux (Lenz's law)
• Fundamental principle behind the operation of transformers, generators, and inductors

Flux Linkage

• Product of the number of turns in a coil and the magnetic flux through each turn
• Represented by the symbol $\lambda$ and measured in weber-turns (Wb-turns)
• Formula: $\lambda = N\Phi$, where $N$ is the number of turns and $\Phi$ is the magnetic flux through each turn
• Related to inductance by $L = \frac{\lambda}{I}$, where $L$ is the inductance and $I$ is the current
• Flux linkage is a useful concept for analyzing inductors and transformers with multiple turns

Inductors

Coil

• Basic form of an inductor, consisting of a wire wound into a spiral or helical shape
• Inductance depends on the number of turns, the cross-sectional area, and the length of the coil
• Can be air-core or have a magnetic core material (iron, ferrite) to increase inductance
• Used in various applications, such as filters, oscillators, and energy storage

Solenoid

• Type of inductor consisting of a coil of wire wound in a cylindrical shape
• Produces a nearly uniform magnetic field inside the solenoid when current flows through the wire
• Inductance depends on the number of turns, the cross-sectional area, and the length of the solenoid
• Formula for inductance: $L = \frac{\mu N^2 A}{l}$, where $\mu$ is the permeability of the core material, $N$ is the number of turns, $A$ is the cross-sectional area, and $l$ is the length of the solenoid
• Used in applications such as electromagnets, relays, and transformers

Toroid

• Type of inductor consisting of a coil of wire wound around a donut-shaped (toroidal) core
• Magnetic field is confined within the core, minimizing external magnetic fields and reducing electromagnetic interference (EMI)
• Inductance depends on the number of turns, the cross-sectional area of the core, and the mean radius of the toroid
• Formula for inductance: $L = \frac{\mu N^2 A}{2\pi r}$, where $\mu$ is the permeability of the core material, $N$ is the number of turns, $A$ is the cross-sectional area of the core, and $r$ is the mean radius of the toroid
• Used in applications such as power supplies, transformers, and noise filters