Editing Inductor (section)

==Circuit analysis==
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 [[superconductor|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]]:

:<math>v(t) = L \frac{di(t)}{dt}</math>
 
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 (<math>I_P</math>) of the current and the angular frequency (<math>\omega</math>) of the current.

:<math>\begin{align}
              i(t) &= I_\mathrm P \sin(\omega t) \\
  \frac{di(t)}{dt} &= I_\mathrm P \omega \cos(\omega t) \\
              v(t) &= L I_\mathrm P \omega \cos(\omega t)
\end{align}</math>

In this situation, the [[Phase (waves)|phase]] of the current lags that of the voltage by π/2 (90°). For sinusoids, as the voltage across the inductor goes to its maximum value, the current goes to zero, and as the voltage across the inductor goes to zero, the current through it goes to its maximum value.

If an inductor is connected to a direct current source with value ''I'' via a resistance ''R'' (at least the DCR of the inductor), 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]]:

:<math>i(t) = I e^{-\frac{R}{L}t}</math>

===Reactance===
The ratio of the peak voltage to the peak current in an inductor energised from an AC source is called the [[electrical reactance|reactance]] and is denoted ''X''<sub>L</sub>.

:<math>X_\mathrm L = \frac {V_\mathrm P}{I_\mathrm P} = \frac {\omega L I_\mathrm P}{I_\mathrm P} </math>

Thus,

:<math>X_\mathrm L = \omega L </math>

where ''ω'' is the [[angular frequency]].

Reactance is measured in ohms but referred to as ''impedance'' rather than resistance; energy is stored in the magnetic field as current rises and discharged as current falls.  Inductive reactance is proportional to frequency.  At low frequency the reactance falls; at DC, the inductor behaves as a short circuit.  As frequency increases the reactance increases and at a sufficiently high frequency the reactance approaches that of an open circuit.

===Corner frequency===
In filtering applications, with respect to a particular load impedance, an inductor has a [[corner frequency]] defined as:

:<math>f_\mathrm{3\,dB} = \frac{R}{2\pi L}</math>

===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:
:<math>Z(s) = Ls\, </math>

where
: ''<math>L</math>'' is the inductance, and
: ''<math>s</math>'' is the complex frequency.

If the inductor does have initial current, it can be represented by: {{bulleted list
| adding a voltage source in series with the inductor, having the value:
:<math> L I_0 \,</math>

where
: ''<math>L</math>'' is the inductance, and
: ''<math>I_0</math>'' is the initial current in the inductor.

(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:
:<math> \frac{I_0}{s} </math>

where
: ''<math>I_0</math>'' is the initial current in the inductor.
: ''<math>s</math>'' is the complex frequency.
}}

===Inductor networks===
{{main|Series and parallel circuits}}
Inductors in a parallel configuration each have the same potential difference (voltage). To find their total equivalent inductance (''L''<sub>eq</sub>):

: [[Image:inductors in parallel.svg|A diagram of several inductors, side by side, both leads of each connected to the same wires]]

:<math> L_\mathrm{eq} = \left(\sum_{i=1}^n{1\over L_i}\right)^{-1} = \left({1\over L_1} + {1\over L_2} + \dots + {1\over L_n}\right)^{-1}.</math>

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:

: [[Image:inductors in series.svg|A diagram of several inductors, connected end to end, with the same amount of current going through each]]

:<math> L_\mathrm{eq} = \sum_{i=1}^n L_i = L_1 + L_2 + \cdots + L_n.\,\! </math>

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

====Mutual inductance====
Mutual inductance occurs when the magnetic field of an inductor induces a magnetic field in an adjacent inductor.  Mutual induction is the basis of transformer construction.
:<math> M = \sqrt{L_1L_2} </math>
where M is the maximum mutual inductance possible between 2 inductors and L<sub>1</sub> and L<sub>2</sub> are the two inductors.
In general 
:<math> M \leq \sqrt{L_1L_2} </math>
as only a fraction of self flux is linked with the other. 
This fraction is called "Coefficient of flux linkage (K)" or "Coefficient of coupling".
:<math> M = K\sqrt{L_1L_2} </math>