An inductor is a passive electrical component with significant inductance. Inductors are implemented by some sort of coiled conductive winding which may surround a ferromagnetic core. Large inductors used at low frequencies may have thousands of turns around an iron core; at very high frequencies a straight piece of wire (i.e., with turns and core reduced to zero) has significant inductance.

An "ideal inductor" has inductance, but no resistance or capacitance, and does not dissipate energy. A real inductor is equivalent to a combination of a significant ideal inductance, some resistance, and capacitance, usually small. The resistance, a necessary property of a wire except at superconducting temperatures, may contribute significantly to the impedance, and may dissipate significant power in some applications. At some frequency, usually much higher than the working voltage, a real inductor behaves as a resonant circuit, and can cause parasitic oscillation.

Some low-value inductors.

Symbol used to denote an inductor

Physics

Overview

Inductance (L) (measured in henrys) is an effect which results from the magnetic field that forms around a current-carrying conductor. Electric current through the conductor creates a magnetic flux proportional to the current. A change in this current creates a change in magnetic flux that, in turn, generates an electromotive force (EMF) that acts to oppose this change in current. Inductance is a measure of the amount of EMF generated for a unit change in current. For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second. The number of loops, the size of each loop, and the material it is wrapped around all affect the inductance. For example, the magnetic flux linking these turns can be increased by coiling the conductor around a material with a high permeability.

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.

Hydraulic model

Electric current can be modeled by the hydraulic analogy. The inductor can be modeled by the flywheel effect of a turbine rotated by the flow. As can be demonstrated intuitively and mathematically, this mimics the behavior of an electrical inductor; voltage is proportional to the derivative of current with respect to time. Thus a rapid change in current will cause a big voltage spike. Likewise, in cases of a sudden interruption of water flow the turbine will generate a high pressure across the blockage, etc. Magnetic interactions such as in transformers are not modeled hydraulically.

Applications

A choke with two 47mH windings, such as what might be found in a power supply.

Two (or more) inductors which have coupled magnetic flux form a transformer, which is a fundamental component of every electric utility power grid. The efficiency of a transformer decreases as the frequency increases but size can be decreased as well; for this reason, aircraft use 400 hertz alternating current rather than the usual 50 or 60 hertz, allowing a great saving in weight from the use of smaller transformers.

An inductor is used as the energy storage device in some switched-mode power supplies. The inductor is energized for a specific fraction of the regulator's switching frequency, and de-energized for the remainder of the cycle. This energy transfer ratio determines the input-voltage to output-voltage ratio. This X_L is used in complement with an active semiconductor device to maintain very accurate voltage control.

Inductors are also employed in electrical transmission systems, where they are used to depress voltages from lightning strikes and to limit switching currents and fault current. In this field, they are more commonly referred to as reactors.

As inductors tend to be larger and heavier than other components, their use has been reduced in modern equipment; solid state switching power supplies eliminate large transformers, for instance, and circuits are designed to use only small inductors, if any; larger values are simulated by use of gyrator circuits.

Inductor construction

Inductors. Major scale in centimetres.

An inductor is usually constructed as a coil of conducting material, typically copper wire, wrapped around a core either of air or of ferromagnetic material. Core materials with a higher permeability than air confine the magnetic field closely to the inductor, thereby increasing the inductance. Inductors come in many shapes. Most are constructed as enamel coated wire wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire completely in ferrite and are called "shielded". Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made with a wire passing through a ferrite cylinder or bead.

Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral pattern. Some such planar inductors use a planar core.

Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. In these cases, aluminium interconnect is typically used as the conducting material. However, practical constraints make it far more common to use a circuit called a "gyrator" which uses a capacitor and active components to behave similarly to an inductor.

In electric circuits

While a capacitor opposes changes in voltage, an inductor opposes changes in current. An ideal inductor would offer no resistance to a constant direct current; however, only superconducting inductors have truly zero electrical resistance.

In general, 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}{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 (I_P) 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 90 degrees. #

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

\ i(t) = I (1 - e^{\frac{-tR}{L}})

Laplace circuit analysis (s-domain)

When using the Laplace transform in circuit analysis, the transfer 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 opposes 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

I_0 is the initial current in the inductor.

Inductor networks

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

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.

Q factor

An ideal inductor will be lossless irrespective of the amount of current through the winding. However, typically inductors have winding resistance from the metal wire forming the coils. Since the winding resistance appears as a resistance in series with the inductor, it is often called the series resistance. The inductor's series resistance converts electrical current through the coils into heat, thus causing a loss of inductive quality. The quality factor (or Q) of an inductor is the ratio of its inductive reactance to its resistance at a given frequency, and is a measure of its efficiency. The higher the Q factor of the inductor, the closer it approaches the behavior of an ideal, lossless, inductor.

The Q factor of an inductor can be found through the following formula, where R is its internal electrical resistance and \omega{}L is Capacitive or Inductive reactance at resonance:

Q = \frac{\omega{}L}{R}

By using a ferromagnetic core the inductance is increased for the same amount of copper, raising the Q. Cores however also introduce losses that increase with frequency. A grade of core material is chosen for best results for the frequency band. At VHF or higher frequencies an air core is likely to be used. Inductors wound around a ferromagnetic core may saturate at high currents, causing a dramatic decrease in inductance (and Q). This phenomenon can be avoided by using a (physically larger) air core inductor. A well designed air core inductor may have a Q of several hundred.

An almost ideal inductor (Q approaching infinity) can be created by immersing a coil made from a superconducting alloy in liquid helium or liquid nitrogen. This supercools the wire, causing its winding resistance to disappear. Because a superconducting inductor is virtually lossless, it can store a large amount of electrical energy within the surrounding magnetic field (see superconducting magnetic energy storage).

Formulae

The table below lists some common formulae for calculating the theoretical inductance of several inductor constructions.