Inductors Notes

Inductors

Inductor Basics

• An inductor is a device designed with large self-inductance, useful in circuits, especially AC circuits.
• Achieving large inductance:
  • Use many turns of wire (like a solenoid) to create a concentrated magnetic field.
  • Amplify the field by adding a ferromagnetic material like iron oxide.
• Examples include toroidal coils, cylindrical coils with adjustable iron slugs, and laminated iron frame coils.
  • Lamination prevents eddy currents.

Self-Inductance Calculation

• Consider a cylindrical solenoid with an iron core, characterized by magnetic permeability $\mu$.
• The magnetic field inside the solenoid is given by $B = \mu n I$, where:
  • $n$ is the number of turns per unit length ($\frac{N}{L}$).
  • $I$ is the current.
• Magnetic flux is $\phi = B \cdot A = \mu \frac{N}{L} I A$.
• Self-inductance $L$ is calculated as: $L = \frac{N \phi}{I} = \frac{N^2 \mu A}{l}$.

Inductor Behavior in Circuits

• Symbol for an inductor: a loop of wire.
• Induced EMF opposes the rate of change of current (Faraday's Law).
• If current is increasing, the induced EMF opposes this increase, leading to a voltage drop.
• If current is decreasing, the induced EMF reinforces the decrease, leading to a voltage increase.

Inductors in Series

• Total inductance for inductors in series (assuming no mutual inductance) is the sum of individual inductances: $L<em>{total} = L</em>1 + L<em>2 + L</em>3 + …$
• The EMFs are additive when inductors are in series.

Inductors in Parallel

• Total inductance for inductors in parallel is calculated by summing the inverse of individual inductances:
  $\frac{1}{L<em>{total}} = \frac{1}{L</em>1} + \frac{1}{L<em>2} + \frac{1}{L</em>3} + …$
• The voltage across inductors in parallel is the same.

Energy Storage in an Inductor

• The energy is stored in a magnetic field.
• The power required to increase the current is $P = -EMF \cdot I$.
• Total energy stored in an inductor: $U = \frac{1}{2} L I^2$
• Alternative formulas:
  • $U = \frac{N^2 \phi^2}{2L}$
  • $U = \frac{1}{2} I N \phi$

Magnetic Energy Density

• For a solenoid, the total magnetic energy is $U = \frac{1}{2} \frac{B^2}{\mu_0} A l$.
• The magnetic energy density is $u_m = \frac{U}{Volume} = \frac{1}{2} \frac{B^2}{\mu}$.
• Total energy density with both electric and magnetic fields: $u<em>{total} = u</em>m + u<em>e = \frac{1}{2} \epsilon</em>0 E^2 + \frac{1}{2} \frac{B^2}{\mu}$.