# Inductors

### Inductors Explained

Inductance is the ability of a component with a changing current to induce a voltage across itself or a nearby circuit by means of a changing magnetic field. Inductance (measured in Henries) is an effect which results from the magnetic field that forms around a current-carrying conductor. It also can be thought of as the property of a component that opposes any change in current.

An inductor is made up of many turns of wire wrapped around a cylindrical, toroidal, or H-shaped core. The core may be empty (air) or it may contain a magnetic material (iron or ferrite) with high magnetic permeability to enhance the field.

One kind of an inductor is a solenoid which is a long, thin coil of wire. Its length is assumed to be much greater that the diameter of the coil. The magnetic flux density B within the coil is practically constant and is given by

$B = \frac{\mu_{o} \cdot N \cdot I}{L}$

where μo is the permeability of a free space, N the number of turns, I is the current and L the length of the coil. B-field has units of Tesla.

Magnetic flux $\Phi = \frac{\mu_{m} \cdot I \cdot (N) \cdot Area}{L}$ measured in units of Weber (1Wb= $10^{8}$ Maxwells) where

$Area= \frac{\pi d^{2}}{4}$ is the cross-sectional area of the coil. Magnetic permeability of free space equals

$\mu_{0} = \frac{4 \pi \cdot 10^{-7} N}{A^{2}}$

whereas the magnetic permeability of the iron (ferrite) core μm=μr∙μo may be found from tables and is usually much larger than μo. The coil is wound up around a magnetic material such as iron, iron oxide, or ferrite to increase the magnetic flux for a given current. Iron cores are usually composed of thin sheets called laminations to reduce losses due to Eddy currents. Relative permeability μr is: iron 200, nickel 100, permalloy 8,000, mumetal 20,000.

Faraday’s Law applied to a coil leads to a derivation of electrical quantity called an inductance.  As you know by now from the Faraday’s law, the induced voltage in a coil is proportional to the rate of change in magnetic flux:

$V(t) = N \cdot \frac{d\Phi}{dt}$

If the current changes with time the magnetic flux will also change in time with the same frequency:

$\Phi(t) = \frac{\mu_{m} \cdot N \cdot Area \cdot I(t)}{L}$
Combining the above two equations one gets the following:

$V(t) = N \cdot \frac{d\Phi}{dt} = \mu_{m} \cdot \frac{Area}{L} \cdot N^{2} \cdot \frac{dI}{dt} = L \cdot \frac{dI}{dt}$

where the constant in front of a derivative \$latex \frac{dI}{dt} is called an inductance

$L = \frac{\mu_{m} \cdot (Area) \cdot N^{2}}{L}$
Electric current cannot change instantly (instantaneously) in an inductor. The current resists rapid changes; think of an inductor as having a large inertia like a heavy rotating wheel. It tries to keep the current constant in a circuit

Unit of inductance are Henrys (H). One Henry of inductance generates one volt of electricity when the change in current equals 1A/s.

Energy stored in a magnetic field equals to $E = \frac{1}{2} \cdot L \cdot I_{2}$and is expressed in Joules.

Time constant: $\tau =\frac{L}{R}$ expressed in seconds, where the resistance R is the Thevenin’s equivalent resistance of the circuit as seen from the terminals of the inductor.

Real capacitors have some parasitic elements in the form of lead inductance and series resistance as well as a parallel conductance due to leakage through the dielectric.