# Inductor

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An inductor is a passive electrical device employed in electrical circuits for its property of inductance. An inductor can take many forms.

## PhysicsEdit

### Overview Edit

Inductance (measured in henries) is an effect which results from the magnetic field that forms around a current carrying conductor. Electrical 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 generated emf for a unit change in current. For example, an inductor with an inductance of 1 henry produces an emf of 1 V when the current through the inductor changes at the rate of 1 ampere per second. The inductance of a conductor is increased by coiling the conductor such that the magnetic flux encloses (links) all of the coils (turns). Additionally, the magnetic flux linking these turns can be increased by coiling the conductor around a material with a high permeability.

### Stored energy Edit

The energy (measured in joules, in SI) stored by an inductor is equal to the amount of work required to establish the current flowing 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 flowing through the inductor.

### Hydraulic modelEdit

As electrical current can be modeled by fluid flow, much like water through pipes; 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; current is the integral of voltage, in cases of a sudden interruption of flow it will generate a high pressure across the blockage, etc. Magnetic interactions such as transformers, however, are not modeled.

## In electric circuits Edit

While a capacitor opposes changes in voltage, an inductor opposes changes in current. An ideal inductor would offer no resistance to direct current, however, all real-world inductors have non-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:

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

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 ($I_P$) of the current and the frequency (f) of the current.

$i(t) = I_P sin(2 \pi f t)\,$
$\frac{di(t)}{dt} = 2 \pi f I_P cos(2 \pi f t)$
$v(t) = 2 \pi f L I_P cos(2 \pi f t)\,$

Clearly, the phase of the current lags that of the voltage by 90 degrees.

### Phasor circuit analysis and impedanceEdit

Using phasors, the impedance of an inductor in ohms is given by:

$Z_L = V / I = j L \omega \,$
where
$X_L = L \omega \,$ is the inductive {{WP|Electrical reactance|reactance]],
$\omega = 2 \pi f \,$ is the angular frequency,
L is the inductance,
f is the frequency, and
j is the imaginary unit.

### Laplace circuit analysis (s-domain)Edit

When using the Laplace transform [[1]]in circuit analysis, the inductive impedance of an ideal inductor with no initial current is represented in the s domain by:

$Z(s) = s L\,$
where
L is the inductance, and
s is...

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.

In an ideal inductor, the current lags behind the voltage by 90° or π/2 radians, but since physical inductors are made from wire that has resistance, a combination resistive-inductive circuit results causing the Q of the tank to be lower.

### Inductor networks Edit

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):

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

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.

## Q Factor Edit

An ideal inductor will be lossless irrespective of the amount of current flowing through the winding. However, real 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 flowing through the coils into heat, thus causing a loss of inductive quality. This is where the quality factor is born. The quality factor (or Q) of an inductor is the ratio of its inductance 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:

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

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 SMES).

## FormulaeEdit

1. Basic inductance formula for a cylindrical coil: $L=\frac{\mu_0\mu_rN^2A}{l}$

L = Inductance in henries
μ0 = permeability of free space = 4π × 10-7 H/m
μr = relative permeability of core material
N = number of turns
A = area of cross-section of the coil in square metres (m2)
l = length of coil in metres (m)

(note: some of the following formulas were optimized to be used with imperial units)

2. Inductance of a straight wire conductor: $L = l\left(\ln\frac{4l}{d}-1\right) \cdot 200 \times 10^{-9}$

L = inductance in H
l = length of conductor in metres
d = diameter of conductor in metres

Hence a 10 mm-long conductor having 1mm diameter will have an inductance of about 5.38nH but 100 mm of the same will get about 100nH. The same formula in imperial units:

$L = 5.08 \cdot l\left(\ln\frac{4l}{d}-1\right)$

L = inductance in nH
l = length of conductor in inches
d = diameter of conductor in inches

3. Inductance of a short air core cylindrical coil in terms of geometric parameters: $L=\frac{r^2N^2}{9r+10l}$

L = inductance in µH
r = outer radius of coil in inches
l = length of coil in inches
N = number of turns

4. For a multilayer air core coil: $L = \frac{0.8r^2N^2}{6r+9l+10d}$

L = inductance in µH
r = mean radius of coil in inches
l = physical length of coil winding in inches
N = number of turns
d = depth of coil in inches (i.e., outer radius minus inner radius)

5. Inductance of a flat spiral air core coil: $L=\frac{r^2N^2}{(2r+2.8d) \times 10^5}$

L = inductance in H
r = mean radius of coil in metres
N = number of turns
d = depth of coil in metres (i.e., outer radius minus inner radius)

Hence a spiral coil with 8 turns at a mean radius of 25mm and a depth of 10mm would have an inductance of 5.13µH.

The same formula in imperial units: $L=\frac{r^2N^2}{8r+11d}$

L = inductance in µH
r = mean radius of coil in inches
N = number of turns
d = depth of coil in inches (i.e., outer radius minus inner radius)

## Inductor construction Edit

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. Small inductors can be etched directly onto a printed circuit board by laying out the trace in a spiral pattern. Small value inductors can also be built on integrated circuits using the same processes that are used to make transistors. In these cases, aluminum 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. Inductors used to block very high frequencies are sometimes made with a wire passing through a ferrite cylinder or bead.

## Applications Edit

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. This can range from the use of large inductors as chokes in power supplies, now obsolete, which in conjunction with filter capacitors remove residual hum or other fluctuations from the direct current output, to such small inductances as generated by a ferrite bead or torus 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 which have coupled magnetic flux form a transformer, which is a fundamental component of every electric utility power grid. The efficiency of a transformer increases as the frequency increases; for this reason, aircraft used 400 hertz alternating current rather than the usual 50 or 60 hertz, allowing a great savings in weight from the use of smaller transformers.

An inductor is used as the energy storage device in a switched-mode power supply. 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 intentionally depress system voltages or limit 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.