Inductor

A selection of low-value inductors
Type Passive
Working principle Electromagnetic induction
Electronic symbol

An inductor, also called a coil or reactor, is a passive two-terminal electrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored in a magnetic field in the coil. When the current flowing through an inductor changes, the time-varying magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic induction, which by Lenz's law opposes the change in current that created it.

An inductor is characterized by its inductance, the ratio of the voltage to the rate of change of current, which has units of henries (H). 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 electric circuits. Inductors are widely used in alternating current (AC) electronic equipment, particularly in radio equipment. They are used to block the flow of AC current 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.

## Overview

Inductance (L) results from the magnetic field around a current-carrying conductor; the electric current through the conductor creates a magnetic flux proportional to the current. Any change in the current creates a voltage across the conductor which opposes the current change. The voltage v in volts across the terminals of an inductor is given by

$v = L \frac{di}{dt} \qquad \qquad \qquad \qquad \text{(1)} \,$

where i is the current through the inductor in amperes and L is the inductance in henrys. Inductance is a measure of the amount of electromotive force (EMF) generated per 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.

Inductance is a geometrical property of a circuit which is determined by how much magnetic flux is created by a given current. Any wire or other conductor will generate a magnetic field when current flows through it, so every conductor has some inductance. In inductors the conductor are shaped to increase the magnetic field. Winding the wire into a coil increases the number of times the magnetic flux lines link the circuit, increasing the inductance. The more turns, the higher the inductance. By winding the coil on a "magnetic core" made of a ferromagnetic material like iron, 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.

### Ideal and real inductors

In circuit theory, inductors are idealized as obeying the mathematical relation (1) above precisely. An "ideal inductor" has inductance, but no resistance or capacitance, and does not dissipate or radiate energy. However real inductors have side effects which cause their behavior to depart from this simple model. They have resistance (due to the resistance of the wire and energy losses in core material), and parasitic capacitance (due to the electric field between the turns of wire which are at slightly different potentials). At high frequencies the capacitance begins to affect the inductor's behavior; at some frequency, real inductors behave as resonant circuits, becoming self-resonant. Above the resonant frequency the capacitive reactance becomes the dominant part of the impedance. Energy is dissipated by the resistance of the wire, and by any losses in the magnetic core due to hysteresis. At high currents, iron core inductors also show gradual departure from ideal behavior due to nonlinearity caused by magnetic saturation. At higher frequencies, resistance and resistive losses in inductors grow due to skin effect in the inductor's winding wires. Core losses also contribute to inductor losses at higher frequencies. An inductor may radiate a part of energy processed into surrounding space and circuits, and may absorb electromagnetic emissions from other circuits, taking part in electromagnetic interference. Circuits and materials close to the inductor will have near-field coupling to the inductor's magnetic field, which may cause additional energy loss. Real-world inductor applications may consider the parasitic parameters as important as the inductance.

## Applications

An inductor with two 47mH windings, as may be found in a power supply.

Inductors are used extensively in analog circuits and signal processing. Inductors in conjunction with capacitors and other components form tuned circuits which can emphasize or filter out specific signal frequencies. Applications range from the use of large inductors in power supplies, which in conjunction with filter capacitors remove residual hums known as the mains hum or other fluctuations 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. Smaller inductor/capacitor combinations provide tuned circuits used in radio reception and broadcasting, for instance.

Two (or more) inductors that have coupled magnetic flux 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.[1] The principle of coupled magnetic fluxes between a stationary and a rotating inductor coil is also used to produce mechanical torque in induction motors, which are widely used in appliances and industry. The energy efficiency of induction motors is greatly influenced by the conductivity of the winding material.

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 XL 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.

Larger value inductors may be 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 or ferrimagnetic material. Core materials with a higher permeability than that of air increase the magnetic field and confine 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. Most are constructed as enamel coated wire (magnet wire) wrapped around a ferrite bobbin with wire exposed on the outside, while some enclose the wire completely in ferrite and are referred to as "shielded". 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 cylinder or bead on a wire.

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. Aluminium interconnect is typically used, laid out in a spiral coil pattern. However, the small dimensions limit the inductance, and it is far more common to use a circuit called a "gyrator" that uses a capacitor and active components to behave similarly to an inductor.

## Types of inductor

### Air core inductor

The term air core coil describes an inductor that does not use a magnetic core made of a ferromagnetic material. The term refers to coils wound on plastic, ceramic, or other nonmagnetic forms, as well as those that have only air inside the windings. Air core coils have lower inductance than ferromagnetic core coils, but are often used at high frequencies because they are free from energy losses called core losses that occur in ferromagnetic cores, which increase with frequency. A side effect that can occur in air core coils in which the winding is not rigidly supported on a form is 'microphony': mechanical vibration of the windings can cause variations in the inductance.

At high frequencies, particularly radio frequencies (RF), inductors have higher resistance and other losses. In addition to causing power loss, in resonant circuits this can reduce the Q factor of the circuit, broadening the bandwidth. In RF inductors, which are mostly air core types, specialized construction techniques are used to minimize these losses. The losses are due to these effects:

• Skin effect: The resistance of a wire to high frequency current is higher than its resistance to direct current because of skin effect. Radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface. Therefore, in a solid wire, most of the cross sectional area of the wire is not used to conduct the current, which is in a narrow annulus on the surface. This effect increases the resistance of the wire in the coil, which may already have a relatively high resistance due to its length and small diameter.
• Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy currents in the wire of the coil, which causes the current in the conductor to be concentrated in a thin strip on the side near the adjacent wire. Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance.
• Parasitic capacitance: The capacitance between individual wire turns of the coil, called parasitic capacitance, does not cause energy losses but can change the behavior of the coil. Each turn of the coil is at a slightly different potential, so the electric field between neighboring turns stores charge on the wire, so the coil acts as if it has a capacitor in parallel with it. At a high enough frequency this capacitance can resonate with the inductance of the coil forming a tuned circuit, causing the coil to become self-resonant.

To reduce parasitic capacitance and proximity effect, RF coils are constructed to avoid having many turns lying close together, parallel to one another. The windings of RF coils are often limited to a single layer, and the turns are spaced apart. To reduce resistance due to skin effect, in high-power inductors such as those used in transmitters the windings are sometimes made of a metal strip or tubing which has a larger surface area, and the surface is silver-plated.

• Honeycomb coils: To reduce proximity effect and parasitic capacitance, multilayer RF coils are wound in patterns in which successive turns are not parallel but crisscrossed at an angle; these are often called honeycomb or basket-weave coils.
• Spiderweb coils: Another construction technique with similar advantages is flat spiral coils. These are often wound on a flat insulating support with radial spokes or slots, with the wire weaving in and out through the slots; these are called spiderweb coils. The form has an odd number of slots, so successive turns of the spiral lie on opposite sides of the form, increasing separation.
• Litz wire: To reduce skin effect losses, some coils are wound with a special type of radio frequency wire called litz wire. Instead of a single solid conductor, litz wire consists of several smaller wire strands that carry the current. Unlike ordinary stranded wire, the strands are insulated from each other, to prevent skin effect from forcing the current to the surface, and are braided together. The braid pattern ensures that each wire strand spends the same amount of its length on the outside of the braid, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.

### Ferromagnetic core inductor

Ferromagnetic-core or iron-core inductors use a magnetic core made of a ferromagnetic or ferrimagnetic material such as iron or ferrite to increase the inductance. A magnetic core can increase the inductance of a coil by a factor of several thousand, by increasing the magnetic field due to its higher magnetic permeability. However the magnetic properties of the core material cause several side effects which alter the behavior of the inductor and require special construction:

• Core losses: A time-varying current in a ferromagnetic inductor, which causes a time-varying magnetic field in its core, causes energy losses in the core material that are dissipated as heat, due to two processes:
• Eddy currents: From Faraday's law of induction, the changing magnetic field can induce circulating loops of electric current in the conductive metal core. The energy in these currents is dissipated as heat in the resistance of the core material. The amount of energy lost increases with the area inside the loop of current.
• Hysteresis: Changing or reversing the magnetic field in the core also causes losses due to the motion of the tiny magnetic domains it is composed of. The energy loss is proportional to the area of the hysteresis loop in the BH graph of the core material. Materials with low coercivity have narrow hysteresis loops and so low hysteresis losses.
For both of these processes, the energy loss per cycle of alternating current is constant, so core losses increase linearly with frequency. Online core loss calculators[2] are available to calculate the energy loss. Using inputs such as input voltage, output voltage, output current, frequency, ambient temperature, and inductance these calculators can predict the losses of the inductors core and AC/DC based on the operating condition of the circuit being used.[3]
• Nonlinearity: If the current through a ferromagnetic core coil is high enough that the magnetic core saturates, the inductance will not remain constant but will change with the current through the device. This is called nonlinearity and results in distortion of the signal. For example, audio signals can suffer intermodulation distortion in saturated inductors. To prevent this, in linear circuits the current through iron core inductors must be limited below the saturation level. Using a powdered iron core with a distributed air gap allows higher levels of magnetic flux which in turn allows a higher level of direct current through the inductor before it saturates.[4]

#### Laminated core inductor

Low-frequency inductors are often made with laminated cores to prevent eddy currents, using construction similar to transformers. The core is made of stacks of thin steel sheets or laminations oriented parallel to the field, with an insulating coating on the surface. The insulation prevents eddy currents between the sheets, so any remaining currents must be within the cross sectional area of the individual laminations, reducing the area of the loop and thus reducing the energy losses greatly. The laminations are made of low-coercivity silicon steel, to reduce hysteresis losses.

#### Ferrite-core inductor

For higher frequencies, inductors are made with cores of ferrite. Ferrite is a ceramic ferrimagnetic material that is nonconductive, so eddy currents cannot flow within it. The formulation of ferrite is xxFe2O4 where xx represents various metals. For inductor cores soft ferrites are used, which have low coercivity and thus low hysteresis losses. Another similar material is powdered iron cemented with a binder.

#### Toroidal core inductor

In an inductor wound on a straight rod-shaped core, the magnetic field lines emerging from one end of the core must pass through the air to reenter the core at the other end. This reduces the field, because much of the magnetic field path is in air rather than the higher permeability core material. A higher magnetic field and inductance can be achieved by forming the core in a closed magnetic circuit. The magnetic field lines form closed loops within the core without leaving the core material. The shape often used is a toroidal or doughnut-shaped ferrite core. Because of their symmetry, toroidal cores allow a minimum of the magnetic flux to escape outside the core (called leakage flux), so they radiate less electromagnetic interference than other shapes. Toroidal core coils are manufactured of various materials, primarily ferrite, powdered iron and laminated cores.[5]

### Variable inductor

A variable inductor can be constructed by making one of the terminals of the device a sliding spring contact that can move along the surface of the coil, increasing or decreasing the number of turns of the coil included in the circuit. A disadvantage of this type is that the contact usually short-circuits one or more turns. These turns act like a short-circuited transformer secondary winding, with large currents that cause power losses. A more widely-used construction method is to use a moveable ferrite magnetic core, which can be slid in or out of the coil. Moving the core farther into the coil increases the permeability, increasing the inductance. Many inductors used in radio applications (usually less than 100 MHz) use adjustable cores in order to tune such inductors to their desired value, since manufacturing processes have certain tolerances (inaccuracy). Sometimes such cores for frequencies above 100 MHz are made from highly conductive non-magnetic material such as aluminum. They decrease the inductance because the magnetic field must bypass them.

Another method to control the inductance without any moving parts requires an additional DC current bias winding which controls the permeability of an easily saturable core material. See Magnetic amplifier.

## 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.

\begin{align} 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) \end{align}

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:

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

### 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:
$L I_0 \,$

where

$L$ is the inductance, and
$I_0$ is the initial current in the inductor.

(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

$I_0$ is the initial current in the inductor.
$s$ is the complex frequency.

### Inductor networks

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

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

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:

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

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

### Stored energy

Neglecting losses, 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. In general if one decides to find the energy stored in a LTI inductor that has initial current in a specific time between $t_0$ and $t_1$ can use this:

$E = \int_{t_0}^{t_1} \! P(t)\,dt = \frac{1}{2}LI(t_1)^2 - \frac{1}{2}LI(t_0)^2$

That we have

\begin{align} P(t) &= V(t)I(t) = I_L(t)L\frac{dI_L(t)}{dt} \\ I_L(t) &= I_L(\infty) + [I_L(t_0) - I_L(\infty)]e^{-\frac{1}{\tau}(t - t_0)} \end{align}

where

$\tau = \frac{L}{R}$

## 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 electric 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 (Series Model) electrical resistance and $\omega{}L$ is the inductive reactance at resonance:

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

By using a ferromagnetic core, the inductance is greatly increased for the same amount of copper, multiplying up 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). Bear in mind that for inductors with cores, core losses still exist.

## Inductance formulae

The table below lists some common simplified formulas for calculating the approximate inductance of several inductor constructions.

Construction Formula Dimensions Notes
Cylindrical air-core coil[6] $L = \frac{1}{l} \mu_0 K N^2 A$
• L = inductance in henries (H)
• μ0 = permeability of free space = 4$\pi$ × 10−7 H/m
• K = Nagaoka coefficient[6]
• N = number of turns
• A = area of cross-section of the coil in square metres (m2)
• l = length of coil in metres (m)
Straight wire conductor[7] $L = \frac{\mu_0}{2\pi} \left( l \ln\left[\frac{1}{c}\left(l + \sqrt{l^2 + c^2}\right)\right] - \sqrt{l^2 + c^2} + c + \frac{l}{4 + c \sqrt{\frac{2}{\rho}\omega\mu}} \right)$
• L = inductance
• l = cylinder length
• μ0 = vacuum permeability = 4$\pi$ × 10−7 H/m
• μ = conductor permeability
• p = resistivity
• ω = phase rate
Exact if ω = 0 or ω = ∞
$L = \frac{1}{5} l \left[\ln\left(\frac{4l}{d}\right) - 1\right]$
• L = inductance (nH)[8][9]
• l = length of conductor (mm)
• d = diameter of conductor (mm)
• f = frequency
• Cu or Al (i.e., relative permeability is one)
• l > 100 d[10]
• d2 f > 1 mm2 MHz
$L = \frac{1}{5} l \left[\ln\left(\frac{4l}{d}\right) - \frac{3}{4}\right]$
• L = inductance (nH)[11][9]
• l = length of conductor (mm)
• d = diameter of conductor (mm)
• f = frequency
• Cu or Al (i.e., relative permeability is one)
• l > 100 d[10]
• d2 f < 1 mm2 MHz
Short air-core cylindrical coil[12] $L = \frac{r^2N^2}{9r + 10l}$
• L = inductance (µH)
• r = outer radius of coil (in)
• l = length of coil (in)
• N = number of turns
Multilayer air-core coil[citation needed] $L = \frac{4}{5} \cdot \frac{r^2N^2}{6r + 9l + 10d}$
• L = inductance (µH)
• r = mean radius of coil (in)
• l = physical length of coil winding (in)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (in)
Flat spiral air-core coil[13][citation needed] $L = \frac{r^2N^2}{20r + 28d}$
• L = inductance (µH)
• r = mean radius of coil (cm)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (cm)
$L = \frac{r^2N^2}{8r + 11d}$
• L = inductance (µH)
• r = mean radius of coil (in)
• N = number of turns
• d = depth of coil (outer radius minus inner radius) (in)
• accurate to within 5 percent for d > 0.2 r.[14]
Toroidal core (circular cross-section)[15] $L = 0.01595 N^2 \left(D - \sqrt{D^2 - d^2}\right)$
• L = inductance (µH)
• d = diameter of coil winding (in)
• N = number of turns
• D = 2 * radius of revolution (in)
$L \approx 0.007975 {d^2 N^2 \over D}$
• L = inductance (µH)
• d = diameter of coil winding (in)
• N = number of turns
• D = 2 * radius of revolution (in)
• approximation when d < 0.1 D
Toroidal core (rectangular cross-section)[14] $L = 0.00508 N^2 h \ln\left({\frac{d_2}{d_1}}\right)$
• L = inductance (µH)
• d1 = inside diameter of toroid (in)
• d2 = outside diameter of toroid (in)
• N = number of turns
• h = height of toroid (in)

## Notes

1. ^ "Aircraft electrical systems". Wonderquest.com. Retrieved 2010-09-24.
2. ^ Vishay. "Products - Inductors - IHLP inductor loss calculator tool landing page". Vishay. Retrieved 2010-09-24.
3. ^ View: Everyone Only Notes. "IHLP inductor loss calculator tool". element14. Retrieved 2010-09-24.
4. ^ "Inductors 101". vishay. Retrieved 2010-09-24.
5. ^ "Inductor and Magnetic Product Terminology". Vishay Dale. Retrieved 2012-09-24.
6. ^ a b Nagaoka, Hantaro (1909-05-06). The Inductance Coefficients of Solenoids 27. Journal of the College of Science, Imperial University, Tokyo, Japan. p. 18. Retrieved 2011-11-10.
7. ^ Rosa, Edward B. (1908). "The Self and Mutual Inductances of Linear Conductors". Bulletin of the Bureau of Standards 4 (2): 301–344. doi:10.6028/bulletin.088
8. ^ Rosa 1908, equation (11a), subst. radius ρ = d/2 and cgs units
9. ^ a b Terman 1943, pp. 48–49, convert to natural logarithms and inches to mm.
10. ^ a b Terman (1943, p. 48) states for l < 100 d, include d/2l within the parentheses.
11. ^ Rosa 1908, equation (10), subst. radius ρ = d/2 and cgs units
12. ^ ARRL Handbook, 66th Ed. American Radio Relay League (1989).
13. ^ For the second formula, Terman 1943, p. 58 which cites to Wheeler 1938.
14. ^ a b Terman 1943, p. 58
15. ^ Terman 1943, p. 57