Inductance Formulas

Inductance formulas are key to understanding how coils and circuits interact with magnetic fields. These concepts, like self-inductance and mutual inductance, help explain energy storage and the behavior of inductors in various configurations, crucial for electrical engineering.

1. Self-inductance formula: L = N²μA/l

   • L represents the self-inductance of a coil, measured in henries (H).
   • N is the number of turns in the coil, indicating how tightly the wire is wound.
   • μ is the permeability of the core material, affecting the magnetic field strength.
   • A is the cross-sectional area of the coil, influencing the magnetic flux.
   • l is the length of the coil, with longer coils resulting in lower inductance.

2. Mutual inductance formula: M = k√(L₁L₂)

   • M denotes the mutual inductance between two coils, also in henries (H).
   • k is the coupling coefficient, ranging from 0 (no coupling) to 1 (perfect coupling).
   • L₁ and L₂ are the self-inductances of the individual coils.
   • Mutual inductance is crucial for understanding transformer operation and coupling between circuits.

3. Energy stored in an inductor: U = ½LI²

   • U represents the energy stored in the magnetic field of the inductor.
   • L is the inductance of the inductor, affecting how much energy can be stored.
   • I is the current flowing through the inductor, with higher currents resulting in more stored energy.
   • This formula highlights the relationship between inductance, current, and energy storage.

4. Faraday's law of induction: ε = -N(dΦ/dt)

   • ε is the induced electromotive force (emf) in volts (V).
   • N is the number of turns in the coil, amplifying the induced emf.
   • dΦ/dt represents the rate of change of magnetic flux through the coil.
   • The negative sign indicates the direction of induced emf opposes the change in flux (Lenz's law).

5. Inductance of a solenoid: L = μN²A/l

   • This formula specifically applies to solenoids, which are long coils of wire.
   • L is the inductance, influenced by the number of turns (N), area (A), and length (l).
   • The permeability (μ) of the core material significantly affects the inductance value.
   • Solenoids are commonly used in applications like electromagnets and inductors.

6. Inductance of a toroid: L = (μN²h/2π) * ln(r₂/r₁)

   • L represents the inductance of a toroidal inductor, shaped like a doughnut.
   • N is the number of turns, while h is the height of the toroid.
   • r₂ and r₁ are the outer and inner radii, respectively, affecting the magnetic path.
   • This design minimizes magnetic field leakage, making toroids efficient inductors.

7. Time constant for RL circuits: τ = L/R

   • τ (tau) is the time constant, indicating how quickly current builds up or decays in an RL circuit.
   • L is the inductance, while R is the resistance in the circuit.
   • A larger time constant means slower changes in current, affecting circuit response time.
   • Understanding the time constant is essential for analyzing transient behavior in RL circuits.

8. Inductors in series: L_total = L₁ + L₂ + L₃ + ...

   • L_total is the total inductance of inductors connected in series.
   • Each inductor's inductance (L₁, L₂, L₃, etc.) simply adds up.
   • The total inductance increases with more inductors in series, affecting circuit behavior.
   • This principle is useful in designing circuits with specific inductance requirements.

9. Inductors in parallel: 1/L_total = 1/L₁ + 1/L₂ + 1/L₃ + ...

   • L_total is the total inductance of inductors connected in parallel.
   • The reciprocal of the total inductance is the sum of the reciprocals of individual inductances.
   • This configuration results in a lower total inductance compared to individual inductors.
   • Parallel inductors can be used to achieve desired inductance values in circuits.

10. Lenz's law: ε_induced = -L(dI/dt)

• ε_induced is the induced emf in the inductor due to a change in current (I).
• L is the inductance of the inductor, influencing the magnitude of the induced emf.
• dI/dt represents the rate of change of current over time.
• The negative sign indicates that the induced emf opposes the change in current, consistent with conservation of energy.