Inductor

An inductor is an electrical device that consists of multiple turns of a coil around a cylindrical ferromagnetic core. They can store energy by producing a magnetic field. The inductance ($L$) is a constant defined by the current and magnetic flux through the circuit, measured in Henries. This is dependent only on the geometry of the inductor and on the permeability $\mu$.

Inductors see significantly less widespread use than capacitors. This is because they can be “large, lossy, expensive in low-frequency applications” and “hard/expensive to integrate on an integrated circuit”.

Circuit analysis

For current and voltage:

$$v(t)=L\frac{di(t)}{dt}$$ $$i(t)=\frac{1}{L}{\int }_{-\infty }^{t}v(\tau )d\tau =i(t_0)+\frac{1}{L}{\int }_{t_0}^{t}v(\tau )d\tau$$

And power and energy:

$$p(t)=L\frac{di(t)}{dt}i(t)=-\frac{\omega L{I}_A^2}{2}\sin (2\omega t)$$ $$w_L(t)=\frac{1}{2}L{i}^2(t)$$

In phasor form, both the real and imaginary parts of impedance should be positive.

Note that in transient analysis, $i$ can never change instantaneously. At DC conditions (where currents/voltages are fixed with time), inductors effectively function as short-circuits. Only when the current is changing is the voltage non-zero.

Combinations

For inductors in series:

$$L_{eq}=\sum _{j=1}^n{L}_j$$

For inductors in parallel:

$$\frac{1}{L_{eq}}=\sum _{j=1}^n\frac{1}{L_j}$$

Electromagnetism

The self-inductance is given by:

$$L=\frac{N{\Phi }_B}{i}=\frac{\Lambda }{i}$$

where $N$ is the number of turns and ${\Phi }_B$ is the magnetic flux of a single turn. For a current loop of one turn, $N=1$. Most inductors use a solenoid, $N=nl$, so this is a structure-dependent property. We define the flux linkage as:

$$\Lambda =N{\Phi }_{B,\text{1-turn}}$$

In integral form, we expand the expression for the self-inductance:

$$L=\frac{N{\iint }_S\vec{B}\cdot d\vec{S}}{i}=\frac{\mu N^2\pi a^2}{l}$$

We call this the self-inductance because the flux is calculated through the circuit that itself generated the flux (think about two distinct current loops), i.e., the flux is intercepted by the loop itself. When we discuss the flux being intercepted by another loop, we call this mutual inductance.

The current-voltage relation comes from Faraday’s law then KVL.

Circuit applications

• RL circuit
• RLC circuit
• Boost converter

See also

• Capacitor, the electric field equivalent