Inductor Reactor

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil.

When the current flowing through the coil changes, the time-varying magnetic field induces an electromotive force (''emf'') (voltage) in the conductor, described by Faraday's law of induction. According to Lenz's law, the induced voltage has a polarity (direction) which opposes the change in current that created it. As a result, inductors oppose any changes in current through them.

An inductor is characterized by its inductance, which is the ratio of the voltage to the rate of change of current. In the International System of Units (SI), the unit of inductance is the henry (H) named for 19th century American scientist Joseph Henry. In the measurement of magnetic circuits, it is equivalent to . Inductors have values that typically range from 1µH (10⁻⁶H) to 20H. Many inductors have a magnetic core made of iron or ferrite inside the coil, which serves to increase the magnetic field and thus the inductance. Along with capacitors and resistors, inductors are one of the three passive linear circuit elements that make up electronic circuits. Inductors are widely used in alternating current (AC) electronic equipment, particularly in radio equipment. They are used to block AC while allowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters to separate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.

The term inductor seems to come from Heinrich Daniel Ruhmkorff, who called the famous induction coil he invented in 1851 an inductorium.

Description

An electric current flowing through a conductor generates a magnetic field surrounding it. The magnetic flux linkage $\Phi_\mathbf$ generated by a given current $I$ depends on the geometric shape of the circuit. Their ratio defines the inductance $L$. Thus
:$L := \frac$.

The inductance of a circuit depends on the geometry of the current path as well as the magnetic permeability of nearby materials. An inductor is a component consisting of a wire or other conductor shaped to increase the magnetic flux through the circuit, usually in the shape of a coil or helix, with two terminals. Winding the wire into a coil increases the number of times the magnetic flux lines link the circuit, increasing the field and thus the inductance. The more turns, the higher the inductance. The inductance also depends on the shape of the coil, separation of the turns, and many other factors. By adding a "magnetic core" made of a ferromagnetic material like iron inside the coil, the magnetizing field from the coil will induce magnetization in the material, increasing the magnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor of several thousand over what it would be without it.

Constitutive equation

Any change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. By Faraday's law of induction, the voltage $\mathcal$ induced by any change in magnetic flux through the circuit is given by
:$\mathcal = -\frac.$

Reformulating the definition of above, we obtain
:$\Phi_\mathbf = LI.$
It follows that
:$\mathcal = -\frac = -\frac(LI) = -L\frac.$

for independent of time, current and magnetic flux linkage.

So inductance is also a measure of the amount of electromotive force (voltage) generated for a given rate of change of 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. This is usually taken to be the constitutive relation (defining equation) of the inductor.

The dual of the inductor is the capacitor, which stores energy in an electric field rather than a magnetic field. Its current–voltage relation is obtained by exchanging current and voltage in the inductor equations and replacing ''L'' with the capacitance ''C''.

Lenz's law

The polarity (direction) of the induced voltage is given by Lenz's law, which states that the induced voltage will be such as to oppose the change in current. For example, if the current through an inductor is increasing, the induced voltage will be positive at the current's entrance point and negative at the exit point, tending to oppose the additional current. The energy from the external circuit necessary to overcome this potential "hill" is being stored in the magnetic field of the inductor. If the current is decreasing, the induced voltage will be negative at the current's entrance point and positive at the exit point, tending to maintain the current. In this case energy from the magnetic field is being returned to the circuit.

Energy stored in an inductor

One intuitive explanation as to why a potential difference is induced on a change of current in an inductor goes as follows:

When there is a change in current through an inductor there is a change in the strength of the magnetic field. For example, if the current is increased, the magnetic field increases. This, however, does not come without a price. The magnetic field contains potential energy, and increasing the field strength requires more energy to be stored in the field. This energy comes from the electric current through the inductor. The increase in the magnetic potential energy of the field is provided by a corresponding drop in the electric potential energy of the charges flowing through the windings. This appears as a voltage drop across the windings as long as the current increases. Once the current is no longer increased and is held constant, the energy in the magnetic field is constant and no additional energy must be supplied, so the voltage drop across the windings disappears.

Similarly, if the current through the inductor decreases, the magnetic field strength decreases, and the energy in the magnetic field decreases. This energy is returned to the circuit in the form of an increase in the electrical potential energy of the moving charges, causing a voltage rise across the windings.

Derivation

The work done per unit charge on the charges passing the inductor is $-\mathcal$. The negative sign indicates that the work is done ''against'' the emf, and is not done ''by'' the emf. The current $I$ is the charge per unit time passing through the inductor. Therefore the rate of work $W$ done by the charges against the emf, that is the rate of change of energy of the current, is given by
:$\frac = -\mathcalI$
From the constitutive equation for the inductor, $-\mathcal = L\frac$ so
:$\frac= L\frac \cdot I = LI \cdot \frac$

:$dW = L I \cdot dI$

In a ferromagnetic core inductor, when the magnetic field approaches the level at which the core saturates, the inductance will begin to change, it will be a function of the current $L(I)$. Neglecting losses, the energy $W$ stored by an inductor with a current $I_0$ passing through it is equal to the amount of work required to establish the current through the inductor.

This is given by:
$W = \int_0^ L_d(I) \, I \, dI$, where $L_d(I)$ is the so-called "differential inductance" and is defined as: $L_d = \frac$.
In an air core inductor or a ferromagnetic core inductor below saturation, the inductance is constant (and equal to the differential inductance), so the stored energy is
:$\begin W &= L\int_0^ I \, dI \\ W &= \fracL ^2 \end$

For inductors with magnetic cores, the above equation is only valid for linear regions of the magnetic flux, at currents below the saturation level of the inductor, where the inductance is approximately constant. Where this is not the case, the integral form must be used with $L_d$ variable.

Voltage step response - short and long term limit

When a voltage step is applied to an inductor, its short and long-term response are easy to calculate:
* In the short-time limit, since the current cannot change discontinuously, the initial current is zero.
The short-time equivalence of an inductor is an open circuit.
* In the long-time limit, the transient response of the inductor will die out, the magnetic flux through the inductor will become constant, so no voltage would be induced between the terminals of the inductor. Therefore, the long-time equivalence of an inductor is a wire (''i.e''., a short circuit).
* To give a mathematical analysis, we should note that any practical inductor is associated with a small resistance ''R''.
Then, if the inductor ''L'' is connected to a battery of voltage ''V'' at time ''t = 0'', the circuit equation for ''t > 0'' is $V = L \dot I + I R$, whose solution is $I(t>0) = (V / R) (1 - e^)$, with limits at ''t = 0'' and $\infty$ as described in the bullets above.

Ideal and real inductors

The constitutive equation describes the behavior of an ''ideal inductor'' with inductance $L$, and without resistance, capacitance, or energy dissipation. In practice, inductors do not follow this theoretical model; ''real inductors'' have a measurable resistance due to the resistance of the wire and energy losses in the core, and parasitic capacitance due to electric potentials between turns of the wire.

A real inductor's capacitive reactance rises with frequency, and at a certain frequency, the inductor will behave as a resonant circuit. Above this self-resonant frequency, the capacitive reactance is the dominant part of the inductor's impedance. At higher frequencies, resistive losses in the windings increase due to the skin effect and proximity effect.

Inductors with ferromagnetic cores experience additional energy losses due to hysteresis and eddy currents in the core, which increase with frequency. At high currents, magnetic core inductors also show sudden departure from ideal behavior due to nonlinearity caused by magnetic saturation of the core.

Inductors radiate electromagnetic energy into surrounding space and may absorb electromagnetic emissions from other circuits, resulting in potential electromagnetic interference.

An early solid-state electrical switching and amplifying device called a saturable reactor exploits saturation of the core as a means of stopping the inductive transfer of current via the core.

''Q'' factor

The winding resistance appears as a resistance in series with the inductor; it is referred to as DCR (DC resistance). This resistance dissipates some of the reactive energy. 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 inductor. High Q inductors are used with capacitors to make resonant circuits in radio transmitters and receivers. The higher the Q is, the narrower the bandwidth of the resonant circuit.

The Q factor of an inductor is defined as

:$Q = \frac$

where $L$ is the inductance, $R$ is the DC resistance, and the product $\omega L$ is the inductive reactance

''Q'' increases linearly with frequency if ''L'' and ''R'' are constant. Although they are constant at low frequencies, the parameters vary with frequency. For example, skin effect, proximity effect, and core losses increase ''R'' with frequency; winding capacitance and variations in permeability with frequency affect ''L''.

At low frequencies and within limits, increasing the number of turns ''N'' improves ''Q'' because ''L'' varies as ''N''² while ''R'' varies linearly with ''N''. Similarly increasing the radius ''r'' of an inductor improves (or increases) ''Q'' because ''L'' varies with ''r''² while ''R'' varies linearly with ''r''. So high ''Q'' air core inductors often have large diameters and many turns. Both of those examples assume the diameter of the wire stays the same, so both examples use proportionally more wire. If the total mass of wire is held constant, then there would be no advantage to increasing the number of turns or the radius of the turns because the wire would have to be proportionally thinner.

Using a high permeability ferromagnetic core can greatly increase the inductance for the same amount of copper, so the core can also increase the Q. Cores however also introduce losses that increase with frequency. The core material is chosen for best results for the frequency band. High Q inductors must avoid saturation; one way is by using a (physically larger) air core inductor. At VHF or higher frequencies an air core is likely to be used. A well designed air core inductor may have a Q of several hundred.

Applications

Inductors are used extensively in analog circuits and signal processing. Applications range from the use of large inductors in power supplies, which in conjunction with filter capacitors remove ripple which is a multiple of the mains frequency (or the switching frequency for switched-mode power supplies) from the direct current output, to the small inductance of the ferrite bead or torus installed around a cable to prevent radio frequency interference from being transmitted down the wire. Inductors are used as the energy storage device in many switched-mode power supplies to produce DC current. The inductor supplies energy to the circuit to keep current flowing during the "off" switching periods and enables topographies where the output voltage is higher than the input voltage.

A tuned circuit, consisting of an inductor connected to a capacitor, acts as a resonator for oscillating current. Tuned circuits are widely used in radio frequency equipment such as radio transmitters and receivers, as narrow bandpass filters to select a single frequency from a composite signal, and in electronic oscillators to generate sinusoidal signals.

Two (or more) inductors in proximity that have coupled magnetic flux (mutual inductance) form a transformer, which is a fundamental component of every electric utility power grid. The efficiency of a transformer may decrease as the frequency increases due to eddy currents in the core material and skin effect on the windings. The size of the core can be decreased at higher frequencies. 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. Transformers enable switched-mode power supplies that isolate the output from the input.

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

Inductors have parasitic effects which cause them to depart from ideal behavior. They create and suffer from electromagnetic interference (EMI). Their physical size prevents them from being integrated on semiconductor chips. So the use of inductors is declining in modern electronic devices, particularly compact portable devices. Real inductors are increasingly being replaced by active circuits such as the gyrator which can synthesize inductance using capacitors.

Inductor construction

An inductor usually consists of a coil of conducting material, typically insulated copper wire, wrapped around a core either of plastic (to create an air-core inductor) or of a ferromagnetic (or ferrimagnetic) material; the latter is called an "iron core" inductor. The high permeability of the ferromagnetic core increases the magnetic field and confines it closely to the inductor, thereby increasing the inductance. Low frequency inductors are constructed like transformers, with cores of electrical steel laminated to prevent eddy currents. 'Soft' ferrites are widely used for cores above audio frequencies, since they do not cause the large energy losses at high frequencies that ordinary iron alloys do. Inductors come in many shapes. Some inductors have an adjustable core, which enables changing of the inductance. Inductors used to block very high frequencies are sometimes made by stringing a ferrite bead on a wire.

Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral

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