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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.

Physics

Overview

Inductance (measured in henrys) is an effect which results from the magnetic field that forms around a current carrying conductor. Current flowing through the inductor creates a magnetic field which has an associated electromotive field which opposes the applied voltage. This counter electromotive force (emf) is generated which opposes the change in voltage applied to the inductor and current in the inductor resists the change but does rise. This is known as inductive reactance. It is opposite in phase to capacitive reactance. Inductance can be increased by looping the conductor into a coil which creates a larger magnetic field.

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 flowing through the inductor, and therefore the magnetic field. This is given by:

${\displaystyle E_{\mathrm {stored} }={1 \over 2}LI^{2}}$

where L is inductance and I is the current flowing through the inductor.

Hydraulic model

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

While a capacitor resists changes in voltage, an inductor resists 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:

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

When a sinusoidal alternating current (AC) flows through an inductor, a sinusoidal alternating voltage (or electromotive force (emf) ) is induced. The amplitude of the emf is equal to the amplitude of the current and to the frequency of the sinusoid by the following equation. The phase of the current lags that of the voltage by 90 degrees. In a capacitor the current leads voltage by 90 degrees. When the inductor is combined with a capacitor, in series or parallel, an LC circuit is formed with a specific resonant frequency:

${\displaystyle V=I\times \omega L\,}$

where ω is the angular frequency of the sinusoid defined in terms of the frequency F as:

${\displaystyle \omega =2\pi F\,}$

Inductive reactance, Xl, is defined as:

${\displaystyle X_{L}=\omega L=2\pi FL\,}$

where X_L is the inductive reactance, ω is the angular frequency, F is the frequency in hertz, and L is the inductance in henries.

Inductive reactance is the positive component of impedance. It is measured in ohms. The impedance of an inductor (inductive reactance) is then given by:

${\displaystyle Z=j\omega L=j2\pi FL=jX_{L}\,}$

where X_L is in ohms.

When using the Laplace transform in circuit analysis, the inductive impedance is represented in the s domain by:

${\displaystyle Z(s)=sL\,}$

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

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

${\displaystyle {\frac {1}{L_{\mathrm {eq} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}$

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:

${\displaystyle L_{\mathrm {eq} }=L_{1}+L_{2}+\cdots +L_{n}\,\!}$

These relationships hold true only in the limit that they are in magnetically decoupled environments.

Q Factor

There has not been an ideal inductor created to-date, the nearest approximation being a supercooled inductor (for example, one cooled with liquid nitrogen or a similar supercooled substance). In the real world inductors have a series resistance created by the copper or other electrically conductive metal wire forming the coils. This series resistance converts the 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 is a ratio of the inductance to the resistance.

The quality factor of an inductor can be found through this formula, where R is its internal electrical resistance:

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

Formulae

1. Basic inductance formula: ${\displaystyle L={\frac {\mu _{0}\mu _{r}N^{2}A}{l}}}$

L = Inductance in henries

μ₀ = permeability of free space = 4π × 10 H/m

μ_r = relative permeability of core material

N = number of turns

A = area of cross-section of the coil in square metres (m)

l = length of coil in metres (m)

2. Inductance of a straight wire conductor: ${\displaystyle L=5.081\left(\ln {\frac {4l}{d}}-1\right)}$

L = inductance in nH

l = length of conductor

d = diameter of conductor in the same units as l

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

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

L = inductance in μH

r = outer radius of coil in inches

l = length of coil in inches

N = number of turns

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

L = inductance in μH

r = mean radius of coil in inches

l = length of coil in inches

N = number of turns

d = depth of coil in inches

5. Inductance of a spring coil: ${\displaystyle L={\frac {r^{2}N^{2}}{6r+11d}}}$

L = inductance in μH

r = mean radius of coil in inches

N = number of turns

d = depth of coil in inches

Inductor construction

An inductor is usually constructed as a coil of conducting material, typically copper wire, wrapped around a core either of air or of ferrous 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

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 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 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.

See also

• Capacitor
• Resistor
• Electricity
• Electronics
• Gyrator
• Inductance (including mutual inductance)
• induction coil
• Induction loop
• Saturable reactor
• Transformer

Synonyms

• coil
• choke
• reactor

External links

General

• Spiral inductor models. Good article on inductor characteristics and modeling.
• AC circuits
• The Inductor Example of commercial use of inductors to create "flameless" heat.
• Magnetic Cores Good link to magnetic cores.
• About Reactors

Patents

• U.S. patent 2,415,688 -- "Induction device"

• Electromagnetic components