## Saturday, 19 March 2011

### In electric circuits

The effect of an inductor in a circuit is to oppose changes in current through it by developing a voltage across it proportional to the rate of change of the current. An ideal inductor would offer no resistance to a constant direct current; however, only superconducting inductors have truly zero electrical resistance.

The relationship between the time-varying voltage v(t) across an inductor with inductance L and the time-varying current i(t) passing through it is described by the differential equation:

v(t) = L \frac{di(t)}{dt}

When there is a sinusoidal alternating current (AC) through an inductor, a sinusoidal voltage is induced. The amplitude of the voltage is proportional to the product of the amplitude (IP) of the current and the frequency (f) of the current.

i(t) = I_P \sin(2 \pi f t)\,

\frac{di(t)}{dt} = 2 \pi f I_P \cos(2 \pi f t)

v(t) = 2 \pi f L I_P \cos(2 \pi f t)\,

In this situation, the phase of the current lags that of the voltage by π/2.

If an inductor is connected to a direct current source with value I via a resistance R, and then the current source is short-circuited, the differential relationship above shows that the current through the inductor will discharge with an exponential decay:

\ i(t) = I e^{-(R/L)t}

Laplace circuit analysis (s-domain)

When using the Laplace transform in circuit analysis, the impedance of an ideal inductor with no initial current is represented in the s domain by:

Z(s) = Ls\,

where

L is the inductance, and
s is the complex frequency.

If the inductor does have initial current, it can be represented by:

* adding a voltage source in series with the inductor, having the value:

L I_0 \,

(Note that the source should have a polarity that is aligned with the initial current)

* or by adding a current source in parallel with the inductor, having the value:

\frac{I_0}{s}

where

L is the inductance, and
I0 is the initial current in the inductor.

Inductor networks
Main article: Series and parallel circuits

Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (Leq):

A diagram of several inductors, side by side, both leads of each connected to the same wires

\frac{1}{L_\mathrm{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}

The current through inductors in series stays the same, but the voltage across each inductor can be different. The sum of the potential differences (voltage) is equal to the total voltage. To find their total inductance:

A diagram of several inductors, connected end to end, with the same amount of current going through each

L_\mathrm{eq} = L_1 + L_2 + \cdots + L_n \,\!

These simple relationships hold true only when there is no mutual coupling of magnetic fields between individual inductors.
Stored energy

The energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current through the inductor, and therefore the magnetic field. This is given by:

E_\mathrm{stored} = {1 \over 2} L I^2

where L is inductance and I is the current through the inductor.

This relationship is only valid for linear (non-saturated) regions of the magnetic flux linkage and current relationship.