Inductor

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Inductor
Electronic component inductors.jpg
A selection of low-value inductors
Type Passive
Working principle Electromagnetic induction
First production Michael Faraday (1831)
Electronic symbol
Inductor.svg

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. [1] An inductor typically consists of an insulated wire wound into a coil around a core.

Terminal (electronics) Connection point in electronic circuits

A terminal is the point at which a conductor from a component, device or network comes to an end. Terminal may also refer to an electrical connector at this endpoint, acting as the reusable interface to a conductor and creating a point where external circuits can be connected. A terminal may simply be the end of a wire or it may be fitted with a connector or fastener.

Electronic component basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields

An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are not to be confused with electrical elements, which are conceptual abstractions representing idealized electronic components.

Magnetic field Spatial distribution of vectors allowing the calculation of the magnetic force on a test particle

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

Contents

When the current flowing through an inductor changes, the time-varying magnetic field induces an electromotive force (e.m.f.) (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.

Electromotive force scalar physical quantity

Electromotive force, abbreviated emf, is the electrical action produced by a non-electrical source. A device that converts other forms of energy into electrical energy, such as a battery or generator, provides an emf as its output. Sometimes an analogy to water "pressure" is used to describe electromotive force.

Voltage difference in the electric potential between two points in space

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.

Faradays law of induction Basic law of electromagnetism of magnetic fields inducing a potential difference

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

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 weber/ampere. Inductors have values that typically range from 1 µH (10−6 H) to 20 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 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.

Inductance electrical property

In electromagnetism and electronics, inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current through a conductor creates a magnetic field around the conductor, whose strength depends on the magnitude of the current. A change in current causes a change in the magnetic field. From Faraday's law of induction, any change in magnetic field through a circuit induces an electromotive force (EMF) in the conductors; this is known as electromagnetic induction. So the changing current induces a voltage in the conductor. This induced voltage is in a direction which tends to oppose the change in current, so it is called a back EMF.

International System of Units a system of units of measurement for base and derived physical quantities

The International System of Units is the modern form of the metric system and is the most widely used system of measurement. It comprises a coherent system of units of measurement built on seven base units, which are the second, metre, kilogram, ampere, kelvin, mole, candela, and a set of twenty prefixes to the unit names and unit symbols that may be used when specifying multiples and fractions of the units. The system also specifies names for 22 derived units, such as lumen and watt, for other common physical quantities.

Henry (unit) SI derived unit of inductance

The henry is the SI derived unit of electrical inductance. If a current of 1 ampere flowing through the coil produces flux linkage of 1 weber turn, the coil has a self inductance of 1 henry.‌ The unit is named after Joseph Henry (1797–1878), the American scientist who discovered electromagnetic induction independently of and at about the same time as Michael Faraday (1791–1867) in England.

Description

An electric current flowing through a conductor generates a magnetic field surrounding it. The magnetic flux linkage generated by a given current depends on the geometric shape of the circuit. Their ratio defines the inductance . [2] [3] [4] [5] Thus

Electrical conductor object or material which permits the flow of electricity

In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge in one or more directions. Materials made of metal are common electrical conductors. Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases.

Magnetic flux physical quantity

In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B passing through that surface. The SI unit of magnetic flux is the weber (Wb), and the CGS unit is the maxwell. Magnetic flux is usually measured with a fluxmeter, which contains measuring coils and electronics, that evaluates the change of voltage in the measuring coils to calculate the measurement of magnetic flux.

.

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. 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.

Helix Type of smooth space curve

A helix, plural helixes or helices, is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helices are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved".

Electromagnetic coil electrical component

An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil, spiral or helix. Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in devices such as electric motors, generators, inductors, electromagnets, transformers, and sensor coils. Either an electric current is passed through the wire of the coil to generate a magnetic field, or conversely an external time-varying magnetic field through the interior of the coil generates an EMF (voltage) in the conductor.

Magnetization physical quantity, density of magnetic moment per unit volume

In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector M.

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 induced by any change in magnetic flux through the circuit is given by [5]

.

Reformulating the definition of above, we obtain [5]

.

It follows, that

.

for independent of time.

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. [6] 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. [7] [8] [9] 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 . The negative sign indicates that the work is done against the emf, and is not done by the emf. The current is the charge per unit time passing through the inductor. Therefore the rate of work done by the charges against the emf, that is the rate of change of energy of the current, is given by

From the constitutive equation for the inductor, so

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 . Neglecting losses, the energy stored by an inductor with a current passing through it is equal to the amount of work required to establish the current through the inductor. This is given by:

.

In an air core inductor or a ferromagnetic core inductor below saturation, the inductance is constant, so the stored energy is

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 variable.

Ideal and real inductors

The constitutive equation describes the behavior of an ideal inductor with inductance , 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. [10] [11]

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, where L is the inductance, R is the DCR, and the product ωL 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 N2 while R varies linearly with N. Similarly increasing the radius r of an inductor improves (or increases) Q because L varies as r2 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

Example of signal filtering. In this configuration, the inductor blocks AC current, while allowing DC current to pass. InductorSignalFilter1.png
Example of signal filtering. In this configuration, the inductor blocks AC current, while allowing DC current to pass.
Example of signal filtering. In this configuration, the inductor decouples DC current, while allowing AC current to pass. InductorSignalFilter2.png
Example of signal filtering. In this configuration, the inductor decouples DC current, while allowing AC current to pass.

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. [12] 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

Common mode choke 2A with 20mH inductance.jpg
A ferrite core inductor with two 20 mH windings.
Ferrite bead no shell.jpg
A ferrite "bead" choke, consisting of an encircling ferrite cylinder, suppresses electronic noise in a computer power cord.
Drosselspule im Umspannwerk Bisamberg.jpg
Large 50 Mvar three-phase iron-core loading inductor at a utility substation

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 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 interconnects. 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. Regardless of the design, because of the low inductances and low power dissipation on-die inductors allow, they're currently only commercially used for high frequency RF circuits.

Shielded inductors

Inductors used in power regulation systems, lighting, and other systems that require low-noise operating conditions, are often partially or fully shielded. [13] [14] In telecommunication circuits employing induction coils and repeating transformers shielding of inductors in close proximity reduces circuit cross-talk.

Types

Air-core inductor

An antenna tuning coil at an AM radio station. It illustrates high power high Q construction: single layer winding with turns spaced apart to reduce proximity effect losses, made of silver-plated tubing to reduce skin effect losses, supported by narrow insulating strips to reduce dielectric losses. Antenna tuning coil - station WOR.jpg
An antenna tuning coil at an AM radio station. It illustrates high power high Q construction: single layer winding with turns spaced apart to reduce proximity effect losses, made of silver-plated tubing to reduce skin effect losses, supported by narrow insulating strips to reduce dielectric losses.

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.

Radio-frequency inductor

Collection of RF inductors, showing techniques to reduce losses. The three top left and the ferrite loopstick or rod antenna, bottom, have basket windings. Hf spoler og transformatorer.jpg
Collection of RF inductors, showing techniques to reduce losses. The three top left and the ferrite loopstick or rod antenna, bottom, have basket windings.

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. For example, at 6 MHz the skin depth of copper wire is about 0.001 inches (25 µm); most of the current is within this depth of the surface. Therefore, in a solid wire, the interior portion of the wire may carry little current, effectively increasing its resistance.
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.
Dielectric losses
The high frequency electric field near the conductors in a tank coil can cause the motion of polar molecules in nearby insulating materials, dissipating energy as heat. So coils used for tuned circuits are often not wound on coil forms but are suspended in air, supported by narrow plastic or ceramic strips.
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.
High Q tank coil in a shortwave transmitter Transmitter tank inductor.jpg
High Q tank coil in a shortwave transmitter
Spider coil.jpg
Kreuzwickelspule.png
(left) Spiderweb coil (right) Adjustable ferrite slug-tuned RF coil with basketweave winding and litz wire

To reduce parasitic capacitance and proximity effect, high Q 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.

Basket-weave coils
To reduce proximity effect and parasitic capacitance, multilayer RF coils are wound in patterns in which successive turns are not parallel but criss-crossed at an angle; these are often called honeycomb or basket-weave coils. These are occasionally wound on a vertical insulating supports with dowels or slots, with the wire weaving in and out through the slots.
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 a number of 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 twisted or braided together. The twist pattern ensures that each wire strand spends the same amount of its length on the outside of the wire bundle, so skin effect distributes the current equally between the strands, resulting in a larger cross-sectional conduction area than an equivalent single wire.
Axial Inductor

Small inductors for low current and low power are made in molded cases resembling resistors. These may be either plain (phenolic) core or ferrite core. An ohmmeter readily distinguishes them from similar-sized resistors by showing the low resistance of the inductor.

Ferromagnetic-core inductor

A variety of types of ferrite core inductors and transformers Aplikimi i feriteve.png
A variety of types of ferrite core inductors and transformers

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.
Core loss is non-linear with respect to both frequency of magnetic fluctuation and magnetic flux density. Frequency of magnetic fluctuation is the frequency of AC current in the electric circuit; magnetic flux density corresponds to current in the electric circuit. Magnetic fluctuation gives rise to hysteresis, and magnetic flux density causes eddy currents in the core. These nonlinearities are distinguished from the threshold nonlinearity of saturation. Core loss can be approximately modeled with Steinmetz's equation. At low frequencies and over limited frequency spans (maybe a factor of 10), core loss may be treated as a linear function of frequency with minimal error. However, even in the audio range, nonlinear effects of magnetic core inductors are noticeable and of concern.
Saturation
If the current through a magnetic core coil is high enough that the core saturates, the inductance will fall and current will rise dramatically. This is a nonlinear threshold phenomenon 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. Some laminated cores have a narrow air gap in them for this purpose, and powdered iron cores have a distributed air gap. This allows higher levels of magnetic flux and thus higher currents through the inductor before it saturates. [19]
Curie point demagnetization
If the temperature of a ferromagnetic or ferrimagnetic core rises to a specified level, the magnetic domains dissociate, and the material becomes paramagnetic, no longer able to support magnetic flux. The inductance falls and current rises dramatically, similarly to what happens during saturation. The effect is reversible: When the temperature falls below the Curie point, magnetic flux resulting from current in the electric circuit will realign the magnetic domains of the core and its magnetic flux will be restored. The Curie point of ferromagnetic materials (iron alloys) is quite high; iron is highest at 770 °C. However, for some ferrimagnetic materials (ceramic iron compounds – ferrites) the Curie point can be close to ambient temperatures (below 100 °C).[ citation needed ]

Laminated-core inductor

Laminated iron core ballast inductor for a metal halide lamp Vorschaltdrossel Kvg.jpg
Laminated iron core ballast inductor for a metal halide lamp

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-conductivity silicon steel to further reduce eddy current 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.

Powdered-iron-core inductor

Another material is powdered iron cemented with a binder.

Toroidal-core inductor

Toroidal inductor in the power supply of a wireless router 3Com OfficeConnect ADSL Wireless 11g Firewall Router 2012-10-28-0869.jpg
Toroidal inductor in the power supply of a wireless router

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 re-enter 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 and is a source of electromagnetic interference. 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. [20]

Variable inductor

Ferrite slug tuned inductor with pot core.JPG
Variometer.jpg
(left) Inductor with a threaded ferrite slug (visible at top) that can be turned to move it into or out of the coil, 4.2 cm high. (right) A variometer used in radio receivers in the 1920s
A "roller coil", an adjustable air-core RF inductor used in the tuned circuits of radio transmitters. One of the contacts to the coil is made by the small grooved wheel, which rides on the wire. Turning the shaft rotates the coil, moving the contact wheel up or down the coil, allowing more or fewer turns of the coil into the circuit, to change the inductance. Rollspule.jpg
A "roller coil", an adjustable air-core RF inductor used in the tuned circuits of radio transmitters. One of the contacts to the coil is made by the small grooved wheel, which rides on the wire. Turning the shaft rotates the coil, moving the contact wheel up or down the coil, allowing more or fewer turns of the coil into the circuit, to change the inductance.

Probably the most common type of variable inductor today is one with a moveable ferrite magnetic core, which can be slid or screwed in or out of the coil. Moving the core farther into the coil increases the permeability, increasing the magnetic field and 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. [21] They decrease the inductance because the magnetic field must bypass them.

Air core inductors can use sliding contacts or multiple taps to increase or decrease the number of turns included in the circuit, to change the inductance. A type much used in the past but mostly obsolete today has a spring contact that can slide along the bare surface of the windings. The disadvantage of this type is that the contact usually short-circuits one or more turns. These turns act like a single-turn short-circuited transformer secondary winding; the large currents induced in them cause power losses.

A type of continuously variable air core inductor is the variometer. This consists of two coils with the same number of turns connected in series, one inside the other. The inner coil is mounted on a shaft so its axis can be turned with respect to the outer coil. When the two coils' axes are collinear, with the magnetic fields pointing in the same direction, the fields add and the inductance is maximum. When the inner coil is turned so its axis is at an angle with the outer, the mutual inductance between them is smaller so the total inductance is less. When the inner coil is turned 180° so the coils are collinear with their magnetic fields opposing, the two fields cancel each other and the inductance is very small. This type has the advantage that it is continuously variable over a wide range. It is used in antenna tuners and matching circuits to match low frequency transmitters to their antennas.

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.

Choke

An MF or HF radio choke for tenths of an ampere, and a ferrite bead VHF choke for several amperes. Two ferrite beads.jpg
An MF or HF radio choke for tenths of an ampere, and a ferrite bead VHF choke for several amperes.

A choke is an inductor designed specifically for blocking high-frequency alternating current (AC) in an electrical circuit, while allowing DC or low-frequency signals to pass. It usually consists of a coil of insulated wire wound on a magnetic core, although some consist of a donut-shaped "bead" of ferrite material strung on a wire. Like other inductors, chokes resist changes in current passing through them increasingly with frequency. The difference between chokes and other inductors is that chokes do not require the high Q factor construction techniques that are used to reduce the resistance in inductors used in tuned circuits.

Circuit analysis

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:

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.

In this situation, the phase of the current lags that of the voltage by π/2 (90°). For sinusoids, as the voltage across the inductor goes to its maximum value, the current goes to zero, and as the voltage across the inductor goes to zero, the current through it goes to its maximum value.

If an inductor is connected to a direct current source with value I via a resistance R (at least the DCR of the inductor), 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:

Reactance

The ratio of the peak voltage to the peak current in an inductor energised from an AC source is called the reactance and is denoted XL.

Thus,

Reactance is measured in ohms but referred to as impedance rather than resistance; energy is stored in the magnetic field as current rises and discharged as current falls. Inductive reactance is proportional to frequency. At low frequency the reactance falls; at DC, the inductor behaves as a short-circuit. As frequency increases the reactance increases and at a sufficiently high frequency the reactance approaches that of an open circuit.

Corner frequency

In filtering applications,with respect to a particular load impedance, an inductor has a corner frequency defined as:

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:

where

is the inductance, and
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:

    where

    is the inductance, and
    is the initial current in the inductor.
    (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:

    where

    is the initial current in the inductor.
    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):

Inductors in parallel.svg

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:

Inductors in series.svg

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

Mutual inductance

Mutual inductance occurs when the magnetic field of an inductor induces a magnetic field in an adjacent inductor. Mutual induction is the basis of transformer construction. M=(L1×L2)^(1/2) where M is the maximum mutual inductance possible between 2 inductors and L1 and L2 are the two inductors. In general M<=(L1×L2)^(1/2) as only a fraction of self flux is linked with the other. This fraction is called "Coefficient of flux linkage" or "Coefficient of coupling". K=M÷((L1×L2)^0.5)

Inductance formulas

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

ConstructionFormulaNotes
Cylindrical air-core coil [22] Calculation of Nagaoka’s coefficient (K) is complicated; normally it must be looked up from a table. [23]
Straight wire conductor [24] ,

where:

  • L = inductance
  • = cylinder length
  • r = cylinder radius
  • μ0 = permeability of free space = 4 × 10−7 H/m
  • μ = conductor permeability
  • ρ = resistivity
  • ω = phase rate
  • = 0.2 µH/m, exactly.
Exact if ω = 0, or if ω = ∞.

The term B subtracts rather than adds.

(when d² f ≫ 1 mm² MHz)

(when d² f ≪ 1 mm² MHz)

  • L = inductance (nH) [25] [26]
  • = length of conductor (mm)
  • d = diameter of conductor (mm)
  • f = frequency
  • = 0.2 µH/m, exactly.
Requires  > 100 d [27]

For relative permeability μr = 1 (e.g., Cu or Al).

Small loop or very short coil [28]
  • L = inductance in the same units as μ0.
  • D = Diameter of the coil (conductor center-to-center)
  • d = diameter of the conductor
  • N = number of turns
  • f = operating frequency (regular f, not ω)
  • σ = specific conductivity of the coil conductor
  • μr = relative permeability of the conductor
  • Total conductor length should be roughly 110 wavelength or smaller. [29]
  • Proximity effects are not included: edge-to-edge gap between turns should be 2×d or larger.
  • = 0.2 µH/m, exactly.
Conductor μr should be as close to 1 as possible copper or aluminum rather than a magnetic or paramagnetic metal.
Medium or long air-core cylindrical coil [30]
  • L = inductance (µH)
  • r = outer radius of coil (in)
  • = length of coil (in)
  • N = number of turns
Requires cylinder length  > 0.4 r: length must be at least 15 of the diameter. Not applicable to single-loop antennas or very short, stubby coils.
Multilayer air-core coil [31]
  • L = inductance (µH)
  • r = mean radius of coil (in)
  • = 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 [32] [33]
  • L = inductance (µH)
  • r = mean radius of coil (cm)
  • N = number of turns
  • d = depth of coil (outer radius minus inner radius) (cm)
  • 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. [34]
Toroidal core (circular cross-section) [35]
  • L = inductance (µH)
  • d = diameter of coil winding (in)
  • N = number of turns
  • D = 2 * radius of revolution (in)
  • 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) [34]
  • L = inductance (µH)
  • d1 = inside diameter of toroid (in)
  • d2 = outside diameter of toroid (in)
  • N = number of turns
  • h = height of toroid (in)

See also

Notes

  1. Nagaoka’s coefficient (K) is approximately 1 for a coil which is much longer than its diameter and is tightly wound using small gauge wire (so that it approximates a current sheet).

Related Research Articles

Transformer electrical device that transfers energy through electromagnetic induction

A transformer is a passive electrical device that transfers electrical energy between two or more circuits. A varying current in one coil of the transformer produces a varying magnetic flux, which, in turn, induces a varying electromotive force across a second coil wound around the same core. Electrical energy can be transferred between the two coils, without a metallic connection between the two circuits. Faraday's law of induction discovered in 1831 described the induced voltage effect in any coil due to changing magnetic flux encircled by the coil.

Inductive coupling

In electrical engineering, two conductors are said to be inductively coupled or magnetically coupled when they are configured such that a change in current through one wire induces a voltage across the ends of the other wire through electromagnetic induction. A changing current through the first wire creates a changing magnetic field around it by Ampere's circuital law. The changing magnetic field induces an electromotive force in the second wire by Faraday's law of induction. The amount of inductive coupling between two conductors is measured by their mutual inductance.

Alternating current electric voltage which periodically reverses direction; form in which electric power is delivered to businesses and residences; form of electrical energy that consumers typically use when they plug electric appliances into a wall socket

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

Solenoid Invention by André-Marie Ampère

A solenoid is a type of electromagnet, the purpose of which is to generate a controlled magnetic field through a coil wound into a tightly packed helix. The term was invented in 1823 by André-Marie Ampère to designate a helical coil.

Skin effect

Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor, and decreases with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. The skin effect causes the effective resistance of the conductor to increase at higher frequencies where the skin depth is smaller, thus reducing the effective cross-section of the conductor. The skin effect is due to opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller. Increased AC resistance due to the skin effect can be mitigated by using specially woven litz wire. Because the interior of a large conductor carries so little of the current, tubular conductors such as pipe can be used to save weight and cost.

Gyrator analog circuit

A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

A magnetic circuit is made up of one or more closed loop paths containing a magnetic flux. The flux is usually generated by permanent magnets or electromagnets and confined to the path by magnetic cores consisting of ferromagnetic materials like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as electric motors, generators, transformers, relays, lifting electromagnets, SQUIDs, galvanometers, and magnetic recording heads.

A magnetic core is a piece of magnetic material with a high magnetic permeability used to confine and guide magnetic fields in electrical, electromechanical and magnetic devices such as electromagnets, transformers, electric motors, generators, inductors, magnetic recording heads, and magnetic assemblies. It is made of ferromagnetic metal such as iron, or ferrimagnetic compounds such as ferrites. The high permeability, relative to the surrounding air, causes the magnetic field lines to be concentrated in the core material. The magnetic field is often created by a current-carrying coil of wire around the core.

Bifilar coil

A bifilar coil is an electromagnetic coil that contains two closely spaced, parallel windings. In engineering, the word bifilar describes wire which is made of two filaments or strands. It is commonly used to denote special types of winding wire for transformers. Wire can be purchased in bifilar form, usually as different colored enameled wire bonded together. For three strands, the term trifilar coil is used.

The search coil magnetometer or induction magnetometer, based on an inductive sensor, is a magnetometer which measures the varying magnetic flux due to Lenz's law. An inductive sensor connected to a conditioning electronic circuit constitutes a search coil magnetometer. It is a vector magnetometer which can measure one or more components of the magnetic field. A classical configuration uses three orthogonal inductive sensors. The search-coil magnetometer can measure magnetic field from mHz up to hundreds of MHz.

Choke (electronics) A type of electronic inductor

In electronics, a choke is an inductor used to block higher-frequency while passing direct current (DC) and lower-frequencies of alternating current (AC) in an electrical circuit. A choke usually consists of a coil of insulated wire often wound on a magnetic core, although some consist of a doughnut-shaped "bead" of ferrite material strung on a wire. The choke's impedance increases with frequency. Its low electrical resistance passes both AC and DC with little power loss, but its reactance limits the amount of AC passed.

An inductive sensor is a device that uses the principle of electromagnetic induction to detect or measure objects. An inductor develops a magnetic field when a current flows through it; alternatively, a current will flow through a circuit containing an inductor when the magnetic field through it changes. This effect can be used to detect metallic objects that interact with a magnetic field. Non-metallic substances such as liquids or some kinds of dirt do not interact with the magnetic field, so an inductive sensor can operate in wet or dirty conditions.

Transformer types

A variety of types of electrical transformer are made for different purposes. Despite their design differences, the various types employ the same basic principle as discovered in 1831 by Michael Faraday, and share several key functional parts.

Proximity effect (electromagnetism)

In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions. The resulting current crowding is termed the proximity effect. This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.

Toroidal inductors and transformers

Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal shape. They are passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated iron, iron powder, or ferrite, around which wire is wound.

The article Ferromagnetic material properties is intended to contain a glossary of terms used to describe ferromagnetic materials, and magnetic cores.

The gyrator–capacitor model - sometimes also the capacitor-permeance model - is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

References

  1. Alexander, Charles; Sadiku, Matthew. Fundamentals of Electric Circuits (3 ed.). McGraw-Hill. p. 211.
  2. Singh, Yaduvir (2011). Electro Magnetic Field Theory. Pearson Education India. p. 65. ISBN   978-8131760611.
  3. Wadhwa, C. L. (2005). Electrical Power Systems. New Age International. p. 18. ISBN   978-8122417227.
  4. Pelcovits, Robert A.; Josh Farkas (2007). Barron's AP Physics C. Barron's Educational Series. p. 646. ISBN   978-0764137105.
  5. 1 2 3 Purcell, Edward M.; David J. Morin (2013). Electricity and Magnetism. Cambridge Univ. Press. p. 364. ISBN   978-1107014022.
  6. Shamos, Morris H. (2012-10-16). Great Experiments in Physics: Firsthand Accounts from Galileo to Einstein. Courier Corporation. ISBN   9780486139623.
  7. Schmitt, Ron (2002). Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics. Elsevier. pp. 75–77. ISBN   978-0080505237.
  8. Jaffe, Robert L.; Taylor, Washington (2018). The Physics of Energy. Cambridge Univ. Press. p. 51. ISBN   978-1108547895.
  9. Lerner, Lawrence S. (1997). Physics for Scientists and Engineers, Vol. 2. Jones and Bartlet Learning. p. 856. ISBN   978-0763704605.
  10. Bowick, Christopher (2011). RF Circuit Design, 2nd Ed. Newnes. pp. 7–8. ISBN   978-0080553429.
  11. Kaiser, Kenneth L. (2004). Electromagnetic Compatibility Handbook. CRC Press. pp. 6.4–6.5. ISBN   978-0849320873.
  12. "Aircraft electrical systems". Wonderquest.com. Retrieved 2010-09-24.
  13. Ott, Henry W. (2011). Electromagnetic Compatibility Engineering. John Wiley and Sons. p. 203. ISBN   978-1118210659.
  14. Violette, Norman (2013). Electromagnetic Compatibility Handbook. Springer. pp. 515–516. ISBN   978-9401771443.
  15. "An Unassuming Antenna – The Ferrite Loopstick". Radio Time Traveller. January 23, 2011. Retrieved March 5, 2014.
  16. Frost, Phil (December 23, 2013). "What's an appropriate core material for a loopstick antenna?". Amateur Radio beta. Stack Exchange, Inc. Retrieved March 5, 2014.
  17. Poisel, Richard (2011). Antenna Systems and Electronic Warfare Applications. Artech House. p. 280. ISBN   978-1608074846.
  18. Yadava, R. L. (2011). Antenna and Wave Propagation. PHI Learning Pvt. Ltd. p. 261. ISBN   978-8120342910.
  19. "Inductors 101" (PDF). vishay. Retrieved 2010-09-24.
  20. "Inductor and Magnetic Product Terminology" (PDF). Vishay Dale. Retrieved 2012-09-24.
  21. "page with aluminum cores" (PDF). Coilcraft catalog. Retrieved 10 July 2015.
  22. 1 2 Nagaoka, Hantaro (1909-05-06). "The Inductance Coefficients of Solenoids" (PDF). 27. Journal of the College of Science, Imperial University, Tokyo, Japan: 18. Retrieved 2011-11-10.Cite journal requires |journal= (help)
  23. Kenneth L. Kaiser, Electromagnetic Compatibility Handbook, p. 30.64, CRC Press, 2004 ISBN   0849320879.
  24. Rosa, Edward B. (1908). "The Self and Mutual Inductances of Linear Conductors" (PDF). Bulletin of the Bureau of Standards. 4 (2): 301–344. doi:10.6028/bulletin.088.
  25. Rosa 1908 , equation (11a), subst. radius ρ = d/2 and cgs units
  26. Terman 1943 , pp. 48–49, convert to natural logarithms and inches to mm.
  27. Terman (1943 , p. 48) states for  < 100 d, include d/2 within the parentheses.
  28. Burger, O. & Dvorský, M. (2015). Magnetic Loop Antenna. Ostrava, Czech Republic: EDUCA TV o.p.s.
  29. Values of up to 13 wavelength are feasible antennas, but for windings that long, this formula will be inaccurate.
  30. ARRL Handbook, 66th Ed. American Radio Relay League (1989).
  31. Wheeler, H.A. (October 1928). "Simple Inductance Formulas for Radio Coils". Proceedings of the Institute of Radio Engineers. 16 (10): 1398. Retrieved 22 June 2015.
  32. For the second formula, Terman 1943 , p. 58 which cites to Wheeler 1928.
  33. "A Magnetic Elevator for Neutral Atoms into a 2D State-dependent Optical Lattice Experiment". Uni-Bonn. Retrieved 2017-08-15.
  34. 1 2 Terman 1943 , p. 58
  35. Terman 1943 , p. 57

Further reading

General