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Inductance

From Wikipedia, the free encyclopedia
Property of electrical conductors
Inductance
Common symbols
L
SI unithenry (H)
InSI base unitskg⋅m2⋅s−2⋅A−2
Derivations from
other quantities
DimensionM1·L2·T−2·I−2
Electromagnetism
Solenoid

Inductance is the tendency of anelectrical conductor to oppose a change in theelectric current flowing through it. The electric current produces amagnetic field around the conductor. The magnetic field strength depends on the magnitude of the electric current, and therefore follows any changes in the magnitude of the current. FromFaraday's law of induction, any change in magnetic field through a circuit induces anelectromotive force (EMF) (voltage) in the conductors, a process known aselectromagnetic induction. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated byLenz's law, and the voltage is calledback EMF.

Inductance is defined as the ratio of the induced voltage to the rate of change of current causing it.[1] It is a proportionality constant that depends on the geometry of circuit conductors (e.g., cross-section area and length) and themagnetic permeability of the conductor and nearby materials.[1] Anelectronic component designed to add inductance to a circuit is called aninductor. It typically consists of acoil or helix of wire.

The terminductance was coined byOliver Heaviside in May 1884, as a convenient way to refer to "coefficient of self-induction".[2][3] It is customary to use the symbolL{\displaystyle L} for inductance, in honour of the physicistHeinrich Lenz.[4][5] In theSI system, the unit of inductance is thehenry (H), which is the amount of inductance that causes a voltage of onevolt, when the current is changing at a rate of oneampere per second.[6] The unit is named forJoseph Henry, who discovered inductance independently of Faraday.[7][8]

History

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Main article:History of electromagnetic theory

The history of electromagnetic induction, a facet ofelectromagnetism, began with observations of the ancients: electric charge or static electricity (rubbing silk onamber), electric current (lightning), and magnetic attraction (lodestone). Understanding the unity of these forces of nature, and the scientific theory of electromagnetism was initiated and achieved during the 19th century.

Electromagnetic induction was first described byMichael Faraday in 1831.[9][10] In Faraday's experiment, he wrapped two wires around opposite sides of an iron ring. He expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. Using agalvanometer, he observed a transient current flow in the second coil of wire each time that a battery was connected or disconnected from the first coil.[11] This current was induced by the change inmagnetic flux that occurred when the battery was connected and disconnected.[12] Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk").[13]

Source of inductance

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A currenti{\displaystyle i} flowing through a conductor generates amagnetic field around the conductor, which is described byAmpere's circuital law. The totalmagnetic fluxΦ{\displaystyle \Phi } through a circuit is equal to the product of the perpendicular component of the magnetic flux density and the area of the surface spanning the current path. If the current varies, themagnetic fluxΦ{\displaystyle \Phi } through the circuit changes. ByFaraday's law of induction, any change in flux through a circuit induces anelectromotive force (EMF,E{\displaystyle {\mathcal {E}}}) in the circuit, proportional to the rate of change of flux

E(t)=−ddtΦ(t){\displaystyle {\mathcal {E}}(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)}

The negative sign in the equation indicates that the induced voltage is in a direction which opposes the change in current that created it; this is calledLenz's law. The potential is therefore called aback EMF. If the current is increasing, the voltage is positive at the end of the conductor through which the current enters and negative at the end through which it leaves, tending to reduce the current. If the current is decreasing, the voltage is positive at the end through which the current leaves the conductor, tending to maintain the current. Self-inductance, usually just called inductance,L{\displaystyle L} is the ratio between the induced voltage and the rate of change of the current

v(t)=Ldidt(1){\displaystyle v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;}

Thus, inductance is a property of a conductor or circuit, due to its magnetic field, which tends to oppose changes in current through the circuit. The unit of inductance in theSI system is thehenry (H), named afterJoseph Henry, which is the amount of inductance that generates a voltage of onevolt when the current is changing at a rate of oneampere per second.

All conductors have some inductance, which may have either desirable or detrimental effects in practical electrical devices. The inductance of a circuit depends on the geometry of the current path, and on themagnetic permeability of nearby materials;ferromagnetic materials with a higher permeability likeiron near a conductor tend to increase the magnetic field and inductance. Any alteration to a circuit which increases the flux (total magnetic field) through the circuit produced by a given current increases the inductance, because inductance is also equal to the ratio ofmagnetic flux to current[14][15][16][17]

L=Φ(i)i{\displaystyle L={\Phi (i) \over i}}

Aninductor is anelectrical component consisting of a conductor shaped to increase the magnetic flux, to add inductance to a circuit. Typically it consists of a wire wound into acoil orhelix. A coiled wire has a higher inductance than a straight wire of the same length, because the magnetic field lines pass through the circuit multiple times, it has multipleflux linkages. The inductance is proportional to the square of thenumber of turns in the coil, assuming full flux linkage.

The inductance of a coil can be increased by placing amagnetic core offerromagnetic material in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning itsmagnetic domains, and the magnetic field of the core adds to that of the coil, increasing the flux through the coil. This is called aferromagnetic core inductor. A magnetic core can increase the inductance of a coil by thousands of times.

If multipleelectric circuits are located close to each other, the magnetic field of one can pass through the other; in this case the circuits are said to beinductively coupled. Due toFaraday's law of induction, a change in current in one circuit can cause a change in magnetic flux in another circuit and thus induce a voltage in another circuit. The concept of inductance can be generalized in this case by defining themutual inductanceMk,ℓ{\displaystyle M_{k,\ell }} of circuitk{\displaystyle k} and circuitℓ{\displaystyle \ell } as the ratio of voltage induced in circuitℓ{\displaystyle \ell } to the rate of change of current in circuitk{\displaystyle k}. This is the principle behind atransformer. The property describing the effect of one conductor on itself is more precisely calledself-inductance, and the properties describing the effects of one conductor with changing current on nearby conductors is calledmutual inductance.[18]

Self-inductance and magnetic energy

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If the current through a conductor with inductance is increasing, a voltagev(t){\displaystyle v(t)} is induced across the conductor with a polarity that opposes the current—in addition to any voltage drop caused by the conductor's resistance. The charges flowing through the circuit lose potential energy. The energy from the external circuit required to overcome this "potential hill" is stored in the increased magnetic field around the conductor. Therefore, an inductor stores energy in its magnetic field. At any given timet{\displaystyle t} ,p(t){\displaystyle p(t)} is thepower flowing into the magnetic field, which is equal to the rate of change of the stored energyU{\displaystyle U}, and to the product of the currenti(t){\displaystyle i(t)} and voltagev(t){\displaystyle v(t)} across the conductor[19][20][21]

p(t)=dUdt=v(t)i(t){\displaystyle p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)}

From (1) above

dUdt=L(i)ididtdU=L(i)idi{\displaystyle {\begin{aligned}{\frac {{\text{d}}U}{{\text{d}}t}}&=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}\\[3pt]{\text{d}}U&=L(i)\,i\,{\text{d}}i\end{aligned}}}


When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive losses, theenergyU{\displaystyle U} (measured injoules, inSI) stored by an inductance with a currentI{\displaystyle I} through it is equal to the amount of work required to establish the current through the inductance from zero, and therefore the magnetic field. This is given by:

U=∫0IL(i)idi{\displaystyle U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,}

If the inductanceL(i){\displaystyle L(i)} is constant over the current range, the stored energy is[19][20][21]

U=L∫0Iidi=12LI2{\displaystyle {\begin{aligned}U&=L\int _{0}^{I}\,i\,{\text{d}}i\\[3pt]&={\tfrac {1}{2}}L\,I^{2}\end{aligned}}}

Inductance is therefore also proportional to the energy stored in the magnetic field for a given current. This energy is stored as long as the current remains constant. If the current decreases, the magnetic field decreases, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This returns stored magnetic energy to the external circuit.

Ifferromagnetic materials are located near the conductor, such as in an inductor with amagnetic core, the constant inductance equation above is only valid forlinear regions of the magnetic flux, at currents below the level at which the ferromagnetic materialsaturates, where the inductance is approximately constant. If the magnetic field in the inductor approaches the level at which the core saturates, the inductance begins to change with current, and the integral equation must be used.

Inductive reactance

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The voltage(v{\displaystyle v}, blue) and current(i{\displaystyle i}, red) waveforms in an ideal inductor to which an alternating current has been applied. The current lags the voltage by 90°

When asinusoidalalternating current (AC) is passing through a linear inductance, the inducedback-EMF is also sinusoidal. If the current through the inductance isi(t)=Ipeaksin⁡(ωt){\displaystyle i(t)=I_{\text{peak}}\sin \left(\omega t\right)}, from (1) above the voltage across it isv(t)=Ldidt=Lddt[Ipeaksin⁡(ωt)]=ωLIpeakcos⁡(ωt)=ωLIpeaksin⁡(ωt+π2){\displaystyle {\begin{aligned}v(t)&=L{\frac {{\text{d}}i}{{\text{d}}t}}=L\,{\frac {\text{d}}{{\text{d}}t}}\left[I_{\text{peak}}\sin \left(\omega t\right)\right]\\&=\omega L\,I_{\text{peak}}\,\cos \left(\omega t\right)=\omega L\,I_{\text{peak}}\,\sin \left(\omega t+{\pi \over 2}\right)\end{aligned}}}

whereIpeak{\displaystyle I_{\text{peak}}} is theamplitude (peak value) of the sinusoidal current in amperes,ω=2πf{\displaystyle \omega =2\pi f} is theangular frequency of the alternating current, withf{\displaystyle f} being itsfrequency inhertz, andL{\displaystyle L} is the inductance.

Thus the amplitude (peak value) of the voltage across the inductance is

Vp=ωLIp=2πfLIp{\displaystyle V_{p}=\omega L\,I_{p}=2\pi f\,L\,I_{p}}

Inductivereactance is the opposition of an inductor to an alternating current.[22] It is defined analogously toelectrical resistance in a resistor, as the ratio of theamplitude (peak value) of the alternating voltage to current in the component

XL=VpIp=2πfL{\displaystyle X_{L}={\frac {V_{p}}{I_{p}}}=2\pi f\,L}

Reactance has units ofohms. It can be seen thatinductive reactance of an inductor increases proportionally with frequencyf{\displaystyle f}, so an inductor conducts less current for a given applied AC voltage as the frequency increases. Because the induced voltage is greatest when the current is increasing, the voltage and current waveforms areout of phase; the voltage peaks occur earlier in each cycle than the current peaks. The phase difference between the current and the induced voltage isϕ=12π{\displaystyle \phi ={\tfrac {1}{2}}\pi }radians or 90 degrees, showing that in an ideal inductor,the current lags the voltage by 90°.

Calculating self-inductance

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In the most general case, inductance can be calculated from Maxwell's equations. Many important cases can be solved using simplifications. Where high frequency currents are considered, withskin effect, the surface current densities and magnetic field may be obtained by solving theLaplace equation. Where the conductors are thin wires, self-inductance still depends on the wire radius and the distribution of the current in the wire. This current distribution is approximately constant (on the surface or in the volume of the wire) for a wire radius much smaller than other length scales.

Straight single wire

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As a practical matter, longer wires have more inductance, and thicker wires have less, analogous to their electrical resistance (although the relationships are not linear, and are different in kind from the relationships that length and diameter bear to resistance).

Separating the wire from the other parts of the circuit introduces some unavoidable error in any formulas' results. These inductances are often referred to as “partial inductances”, in part to encourage consideration of the other contributions to whole-circuit inductance which are omitted.

Practical formulas

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For derivation of the formulas below, see Rosa (1908).[23]The total low frequency inductance (interior plus exterior) of a straight wire is:

LDC=200 nHmℓ[ln⁡(2ℓr)−0.75]{\displaystyle L_{\text{DC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-0.75\right]}

where

The constant 0.75 is just one parameter value among several; different frequency ranges, different shapes, or extremely long wire lengths require a slightly different constant (see below). This result is based on the assumption that the radiusr{\displaystyle r} is much less than the lengthℓ{\displaystyle \ell }, which is the common case for wires and rods. Disks or thick cylinders have slightly different formulas.

For sufficiently high frequencies skin effects cause the interior currents to vanish, leaving only the currents on the surface of the conductor; the inductance for alternating current,LAC{\displaystyle L_{\text{AC}}} is then given by a very similar formula:

LAC=200 nHmℓ[ln⁡(2ℓr)−1]{\displaystyle L_{\text{AC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-1\right]}where the variablesℓ{\displaystyle \ell } andr{\displaystyle r} are the same as above; note the changed constant term now 1, from 0.75 above.

For example, a single conductor of a lamp cord10 m long, made of 18 AWG (1.024 mm) wire, would have a low frequency inductance of about19.67 μH, at k=0.75, if stretched out straight.

Wire loop

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Formally, the self-inductance of a wire loop would be given by the above equation with m=n .{\displaystyle \ m=n\ .} However, here 1/|x−x′| {\displaystyle \ 1/\left|\mathbf {x} -\mathbf {x} '\right|\ } becomes infinite, leading to a logarithmically divergent integral.[a]This necessitates taking the finite wire radius a {\displaystyle \ a\ } and the distribution of the current in the wire into account. There remains the contribution from the integral over all points and a correction term,[24]

L=μ04π[ ℓ Y+∮C∮C′dx⋅dx′ |x−x′|  ]+Obend for |s−s′|>12a {\displaystyle L={\frac {\mu _{0}}{4\pi }}\left[\ \ell \ Y+\oint _{C}\oint _{C'}{\frac {\mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} '}{\ \left|\mathbf {x} -\mathbf {x} '\right|\ }}\ \right]+{\mathcal {O}}_{\mathsf {bend}}\quad {\text{ for }}\;\left|\mathbf {s} -\mathbf {s} '\right|>{\tfrac {1}{2}}a\ }

where

 s {\displaystyle \ \mathbf {s} \ } and s′ {\displaystyle \ \mathbf {s} '\ } are distances along the curves C {\displaystyle \ C\ } and C′ {\displaystyle \ C'\ } respectively
 a {\displaystyle \ a\ } is the radius of the wire
 ℓ {\displaystyle \ \ell \ } is the length of the wire
 Y {\displaystyle \ Y\ } is a constant that depends on the distribution of the current in the wire:
 Y=0 {\displaystyle \ Y=0\ } when the current flows on the surface of the wire (totalskin effect),
 Y=12 {\textstyle \ Y={\tfrac {1}{2}}\ } when the current is evenly over the cross-section of the wire.
 Obend {\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}\ } is an error term whose size depends on the curve of the loop:
 Obend=O(μ0a) {\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}(\mu _{0}a)\ } when the loop has sharp corners, and
 Obend=O(μ0a2/ℓ) {\textstyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}{\mathord {\left({\mu _{0}a^{2}}/{\ell }\right)}}\ } when it is a smooth curve.
Both are small when the wire is long compared to its radius.

Solenoid

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Asolenoid is a long, thin coil; i.e., a coil whose length is much greater than its diameter. Under these conditions, and without any magnetic material used, themagnetic flux densityB{\displaystyle B} within the coil is practically constant and is given byB=μ0Niℓ{\displaystyle B={\frac {\mu _{0}\,N\,i}{\ell }}}

whereμ0{\displaystyle \mu _{0}} is themagnetic constant,N{\displaystyle N} the number of turns,i{\displaystyle i} the current andl{\displaystyle l} the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux densityB{\displaystyle B} by the cross-section areaA{\displaystyle A}:Φ=μ0NiAℓ,{\displaystyle \Phi ={\frac {\mu _{0}\,N\,i\,A}{\ell }},}

When this is combined with the definition of inductanceL=NΦi{\displaystyle L={\frac {N\,\Phi }{i}}}, it follows that the inductance of a solenoid is given by:L=μ0N2Aℓ.{\displaystyle L={\frac {\mu _{0}\,N^{2}\,A}{\ell }}.}

Therefore, for air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.

Coaxial cable

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Let the inner conductor have radiusri{\displaystyle r_{i}} andpermeabilityμi{\displaystyle \mu _{i}}, let the dielectric between the inner and outer conductor have permeabilityμd{\displaystyle \mu _{d}}, and let the outer conductor have inner radiusro1{\displaystyle r_{o1}}, outer radiusro2{\displaystyle r_{o2}}, and permeabilityμ0{\displaystyle \mu _{0}}. However, for a typical coaxial line application, we are interested in passing (non-DC) signals at frequencies for which the resistiveskin effect cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate

L′=dLdℓ≈μd2πln⁡ro1ri{\displaystyle L'={\frac {{\text{d}}L}{{\text{d}}\ell }}\approx {\frac {\mu _{d}}{2\pi }}\ln {\frac {r_{o1}}{r_{i}}}}

Multilayer coils

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Most practical air-core inductors are multilayer cylindrical coils with square cross-sections to minimize average distance between turns (circular cross -sections would be better but harder to form).

Magnetic cores

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Many inductors include amagnetic core at the center of or partly surrounding the winding. Over a large enough range these exhibit a nonlinear permeability with effects such asmagnetic saturation. Saturation makes the resulting inductance a function of the applied current.

The secant or large-signal inductance is used in flux calculations. It is defined as:

Ls(i)=defN Φi=Λi{\displaystyle L_{s}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {N\ \Phi }{i}}={\frac {\Lambda }{i}}}

The differential or small-signal inductance, on the other hand, is used in calculating voltage. It is defined as:

Ld(i)=defd(NΦ)di=dΛdi{\displaystyle L_{d}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {{\text{d}}(N\Phi )}{{\text{d}}i}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}}

The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and thechain rule of calculus.

v(t)=dΛdt=dΛdididt=Ld(i)didt{\displaystyle v(t)={\frac {{\text{d}}\Lambda }{{\text{d}}t}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}{\frac {{\text{d}}i}{{\text{d}}t}}=L_{d}(i){\frac {{\text{d}}i}{{\text{d}}t}}}

Similar definitions may be derived for nonlinear mutual inductance.

Mutual inductance

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Further information:Inductive coupling

Definition of Mutual induction or Coefficient of mutual induction

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The mutual inductance or the coefficient of mutual induction of two magnetically linked coils is equal to the flux linkage of one coil per unit current in the neighboring coil. OR

The mutual inductance or the coefficient of mutual induction of two magnetically linked coils is numerically equal to the emf induced in one coil (secondary) per unit time rate of change of current in the neighboring coil (primary).

Mutual inductance of two parallel straight wires

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There are two cases to consider:

  1. Current travels in the same direction in each wire, and
  2. current travels in opposing directions in the wires.

Currents in the wires need not be equal, though they often are, as in the case of a complete circuit, where one wire is the source and the other the return.

Mutual inductance of two wire loops

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This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents. Nonetheless, the error terms, which are not included in the integral are only small if the geometries of the loops are mostly smooth and convex: They must not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities, or other topologically "close" deformations. A necessary predicate for the reduction of the 3-dimensional manifold integration formula to a double curve integral is that the current paths be filamentary circuits, i.e. thin wires where the radius of the wire is negligible compared to its length.

The mutual inductance by a filamentary circuitm{\displaystyle m} on a filamentary circuitn{\displaystyle n} is given by the double integralNeumann formula[25]

Lm,n=μ04π∮Cm∮Cndxm⋅dxn |xm−xn|  ,{\displaystyle L_{m,n}={\frac {\mu _{0}}{4\pi }}\oint _{C_{m}}\oint _{C_{n}}{\frac {\mathrm {d} \mathbf {x} _{m}\cdot \mathrm {d} \mathbf {x} _{n}}{\ \left|\mathbf {x} _{m}-\mathbf {x} _{n}\right|\ }}\ ,}

where

Cm{\displaystyle C_{m}} andCn{\displaystyle C_{n}} are the curves followed by the wires.
μ0{\displaystyle \mu _{0}} is thepermeability of free space (4π×10−7 H/m)
dxm{\displaystyle \mathrm {d} \mathbf {x} _{m}} is a small increment of the wire in circuitCm
xm{\displaystyle \mathbf {x} _{m}} is the position ofdxm{\displaystyle \mathrm {d} \mathbf {x} _{m}} in space
dxn{\displaystyle \mathrm {d} \mathbf {x} _{n}} is a small increment of the wire in circuitCn
xn{\displaystyle \mathbf {x} _{n}} is the position ofdxn{\displaystyle \mathrm {d} \mathbf {x} _{n}} in space.

Derivation

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Mij=defΦijIj{\displaystyle M_{ij}\mathrel {\stackrel {\mathrm {def} }{=}} {\frac {\Phi _{ij}}{I_{j}}}}

where

Φij=∫SiBj⋅da=∫Si(∇×Aj)⋅da=∮CiAj⋅dsi=∮Ci(μ0Ij4π∮Cjdsj|si−sj|)⋅dsi{\displaystyle \Phi _{ij}=\int _{S_{i}}\mathbf {B} _{j}\cdot \mathrm {d} \mathbf {a} =\int _{S_{i}}(\nabla \times \mathbf {A_{j}} )\cdot \mathrm {d} \mathbf {a} =\oint _{C_{i}}\mathbf {A} _{j}\cdot \mathrm {d} \mathbf {s} _{i}=\oint _{C_{i}}\left({\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathrm {d} \mathbf {s} _{j}}{\left|\mathbf {s} _{i}-\mathbf {s} _{j}\right|}}\right)\cdot \mathrm {d} \mathbf {s} _{i}}

where

Stokes' theorem has been used for the 3rd equality step. For the last equality step, we used theretarded potential expression forAj{\displaystyle A_{j}} and we ignore the effect of the retarded time (assuming the geometry of the circuits is small enough compared to the wavelength of the current they carry). It is actually an approximation step, and is valid only for local circuits made of thin wires.

Mutual inductance is defined as the ratio between the EMF induced in one loop or coil by the rate of change of current in another loop or coil. Mutual inductance is given the symbolM.

Derivation of mutual inductance

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The inductance equations above are a consequence ofMaxwell's equations. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward.

In a system ofK{\displaystyle K} wire loops, each with one or several wire turns, theflux linkage of loopm{\displaystyle m},λm{\displaystyle \lambda _{m}}, is given byλm=NmΦm=∑n=1KLm,n in.{\displaystyle \displaystyle \lambda _{m}=N_{m}\Phi _{m}=\sum \limits _{n=1}^{K}L_{m,n}\ i_{n}\,.}

HereNm{\displaystyle N_{m}} denotes the number of turns in loopm{\displaystyle m};Φm{\displaystyle \Phi _{m}} is themagnetic flux through loopm{\displaystyle m}; andLm,n{\displaystyle L_{m,n}} are some constants described below. This equation follows fromAmpere's law:magnetic fields and fluxes are linear functions of the currents. ByFaraday's law of induction, we have

vm=dλmdt=NmdΦmdt=∑n=1KLm,ndindt,{\displaystyle \displaystyle v_{m}={\frac {{\text{d}}\lambda _{m}}{{\text{d}}t}}=N_{m}{\frac {{\text{d}}\Phi _{m}}{{\text{d}}t}}=\sum \limits _{n=1}^{K}L_{m,n}{\frac {{\text{d}}i_{n}}{{\text{d}}t}},}

wherevm{\displaystyle v_{m}} denotes the voltage induced in circuitm{\displaystyle m}. This agrees with the definition of inductance above if the coefficientsLm,n{\displaystyle L_{m,n}} are identified with the coefficients of inductance. Because the total currentsNn in{\displaystyle N_{n}\ i_{n}} contribute toΦm{\displaystyle \Phi _{m}} it also follows thatLm,n{\displaystyle L_{m,n}} is proportional to the product of turnsNm Nn{\displaystyle N_{m}\ N_{n}}.

Mutual inductance and magnetic field energy

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Multiplying the equation forvm above withimdt and summing overm gives the energy transferred to the system in the time intervaldt,∑mKimvmdt=∑m,n=1KimLm,ndin=!∑n=1K∂W(i)∂indin.{\displaystyle \sum \limits _{m}^{K}i_{m}v_{m}{\text{d}}t=\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}{\text{d}}i_{n}\mathrel {\overset {!}{=}} \sum \limits _{n=1}^{K}{\frac {\partial W\left(i\right)}{\partial i_{n}}}{\text{d}}i_{n}.}

This must agree with the change of the magnetic field energy,W, caused by the currents.[27] Theintegrability condition

∂2W∂im∂in=∂2W∂in∂im{\displaystyle \displaystyle {\frac {\partial ^{2}W}{\partial i_{m}\partial i_{n}}}={\frac {\partial ^{2}W}{\partial i_{n}\partial i_{m}}}}

requiresLm,n = Ln,m. The inductance matrix,Lm,n, thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents,W(i)=12∑m,n=1KimLm,nin.{\displaystyle \displaystyle W\left(i\right)={\frac {1}{2}}\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}i_{n}.}

This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. Amechanical analogy in theK = 1 case with magnetic field energy (1/2)Li2 is a body with massM, velocityu and kinetic energy (1/2)Mu2. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage).

Circuit diagram of two mutually coupled inductors. The two vertical lines between the windings indicate that the transformer has aferromagnetic core . "n:m" shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows thedot convention.

Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by whichtransformers work, but it can also cause unwanted coupling between conductors in a circuit.

The mutual inductance,Mij{\displaystyle M_{ij}}, is also a measure of the coupling between two inductors. The mutual inductance by circuiti{\displaystyle i} on circuitj{\displaystyle j} is given by the double integralNeumann formula, seecalculation techniques

The mutual inductance also has the relationship:M21=N1 N2 P21{\displaystyle M_{21}=N_{1}\ N_{2}\ P_{21}\!}where

Once the mutual inductanceM{\displaystyle M} is determined, it can be used to predict the behavior of a circuit:v1=L1 di1dt−M di2dt{\displaystyle v_{1}=L_{1}\ {\frac {{\text{d}}i_{1}}{{\text{d}}t}}-M\ {\frac {{\text{d}}i_{2}}{{\text{d}}t}}}where

The minus sign arises because of the sense the currenti2{\displaystyle i_{2}} has been defined in the diagram. With both currents defined going into thedots the sign ofM{\displaystyle M} will be positive (the equation would read with a plus sign instead).[28]

Coupling coefficient

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The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from onemagnetic circuit to the other. The coupling coefficient is related to mutual inductance and self inductances in the following way. From the two simultaneous equations expressed in the two-port matrix the open-circuit voltage ratio is found to be:

V2V1open circuit=ML1{\displaystyle {V_{2} \over V_{1}}_{\text{open circuit}}={M \over L_{1}}}where

while the ratio if all the flux is coupled is the ratio of the turns, hence the ratio of the square root of the inductances

V2V1max coupling=L2L1 {\displaystyle {V_{2} \over V_{1}}_{\text{max coupling}}={\sqrt {L_{2} \over L_{1}\ }}}

thus,

M=kL1 L2 {\displaystyle M=k{\sqrt {L_{1}\ L_{2}\ }}}where

The coupling coefficient is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance. Most authors define the range as0≤k<1{\displaystyle 0\leq k<1}, but some[29] define it as−1<k<1{\displaystyle -1<k<1\,}. Allowing negative values ofk{\displaystyle k} captures phase inversions of the coil connections and the direction of the windings.[30]

Matrix representation

[edit]

Mutually coupled inductors can be described by any of thetwo-port network parameter matrix representations. The most direct are thez parameters, which are given by[31]

[z]=s[L1 MM L2].{\displaystyle [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}.}

They parameters are given by

[y]=1s[L1 MM L2]−1.{\displaystyle [\mathbf {y} ]={\frac {1}{s}}{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}^{-1}.} Wheres{\displaystyle s} is thecomplex frequency variable,L1{\displaystyle L_{1}} andL2{\displaystyle L_{2}} are the inductances of the primary and secondary coil, respectively, andM{\displaystyle M} is the mutual inductance between the coils.

Multiple Coupled Inductors

[edit]

Mutual inductance may be applied to multiple inductors simultaneously. The matrix representations for multiple mutually coupled inductors are given by[32][z]=s[L1M12M13…M1NM12L2M23…M2NM13M23L3…M3N⋮⋮⋮⋱M1NM2NM3N…LN]{\displaystyle {\begin{aligned}&[\mathbf {z} ]=s{\begin{bmatrix}L_{1}&M_{12}&M_{13}&\dots &M_{1N}\\M_{12}&L_{2}&M_{23}&\dots &M_{2N}\\M_{13}&M_{23}&L_{3}&\dots &M_{3N}\\\vdots &\vdots &\vdots &\ddots \\M_{1N}&M_{2N}&M_{3N}&\dots &L_{N}\\\end{bmatrix}}\\\end{aligned}}}

Equivalent circuits

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T-circuit

[edit]
T equivalent circuit of mutually coupled inductors

Mutually coupled inductors can equivalently be represented by a T-circuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the step-down side may take on a negative value.[33]

This can be analyzed as a two port network. With the output terminated with some arbitrary impedanceZ{\displaystyle Z}, the voltage gainAv{\displaystyle A_{v}}, is given by:

Av=sMZs2L1L2−s2M2+sL1Z=ks(1−k2)L1L2Z+L1L2{\displaystyle A_{\mathrm {v} }={\frac {sMZ}{\,s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z\,}}={\frac {k}{\,s\left(1-k^{2}\right){\frac {\sqrt {L_{1}L_{2}}}{Z}}+{\sqrt {\frac {L_{1}}{L_{2}}}}\,}}}

wherek{\displaystyle k} is the coupling constant ands{\displaystyle s} is thecomplex frequency variable, as above.For tightly coupled inductors wherek=1{\displaystyle k=1} this reduces to

Av=L2L1{\displaystyle A_{\mathrm {v} }={\sqrt {L_{2} \over L_{1}}}}

which is independent of the load impedance. If the inductors are wound on the same core and with the same geometry, then this expression is equal to the turns ratio of the two inductors because inductance is proportional to the square of turns ratio.

The input impedance of the network is given by:

Zin=s2L1L2−s2M2+sL1ZsL2+Z=L1L2Z(11+ZsL2)(1+1−k2ZsL2){\displaystyle Z_{\text{in}}={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)\left(1+{\frac {1-k^{2}}{\frac {Z}{\,sL_{2}\,}}}\right)}

Fork=1{\displaystyle k=1} this reduces to

Zin=sL1ZsL2+Z=L1L2Z(11+ZsL2){\displaystyle Z_{\text{in}}={\frac {sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)}

Thus, current gainAi{\displaystyle A_{i}} isnot independent of load unless the further condition

|sL2|≫|Z|{\displaystyle |sL_{2}|\gg |Z|}

is met, in which case,

Zin≈L1L2Z{\displaystyle Z_{\text{in}}\approx {L_{1} \over L_{2}}Z}

and

Ai≈L1L2=1Av{\displaystyle A_{\text{i}}\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\text{v}}}}

π-circuit

[edit]
π equivalent circuit of coupled inductors

Alternatively, two coupled inductors can be modelled using aπ equivalent circuit with optional ideal transformers at each port. While the circuit is more complicated than a T-circuit, it can be generalized[34] to circuits consisting of more than two coupled inductors. Equivalent circuit elementsLs{\displaystyle L_{\text{s}}},Lp{\displaystyle L_{\text{p}}} have physical meaning, modelling respectivelymagnetic reluctances of coupling paths andmagnetic reluctances ofleakage paths. For example, electric currents flowing through these elements correspond to coupling and leakagemagnetic fluxes. Ideal transformers normalize all self-inductances to 1 Henry to simplify mathematical formulas.

Equivalent circuit element values can be calculated from coupling coefficients withLSij=det(K)−CijLPi=det(K)∑j=1NCij{\displaystyle {\begin{aligned}L_{S_{ij}}&={\frac {\det(\mathbf {K} )}{-\mathbf {C} _{ij}}}\\[3pt]L_{P_{i}}&={\frac {\det(\mathbf {K} )}{\sum _{j=1}^{N}\mathbf {C} _{ij}}}\end{aligned}}}

where coupling coefficient matrix and its cofactors are defined as

K=[1k12⋯k1Nk121⋯k2N⋮⋮⋱⋮k1Nk2N⋯1]{\displaystyle \mathbf {K} ={\begin{bmatrix}1&k_{12}&\cdots &k_{1N}\\k_{12}&1&\cdots &k_{2N}\\\vdots &\vdots &\ddots &\vdots \\k_{1N}&k_{2N}&\cdots &1\end{bmatrix}}\quad } andCij=(−1)i+jMij.{\displaystyle \quad \mathbf {C} _{ij}=(-1)^{i+j}\,\mathbf {M} _{ij}.}

For two coupled inductors, these formulas simplify to

LS12=−k122+1k12{\displaystyle L_{S_{12}}={\frac {-k_{12}^{2}+1}{k_{12}}}\quad } andLP1=LP2=k12+1,{\displaystyle \quad L_{P_{1}}=L_{P_{2}}\!=\!k_{12}+1,}

and for three coupled inductors (for brevity shown only forLs12{\displaystyle L_{\text{s12}}} andLp1{\displaystyle L_{\text{p1}}})

LS12=2k12k13k23−k122−k132−k232+1k13k23−k12{\displaystyle L_{S_{12}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{13}\,k_{23}-k_{12}}}\quad } andLP1=2k12k13k23−k122−k132−k232+1k12k23+k13k23−k232−k12−k13+1.{\displaystyle \quad L_{P_{1}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{12}\,k_{23}+k_{13}\,k_{23}-k_{23}^{2}-k_{12}-k_{13}+1}}.}

Resonant transformer

[edit]
Main article:Resonant inductive coupling

When a capacitor is connected across one winding of a transformer, making the winding atuned circuit (resonant circuit) it is called a single-tuned transformer. When a capacitor is connected across each winding, it is called adouble tuned transformer. Theseresonant transformers can store oscillating electrical energy similar to aresonant circuit and thus function as abandpass filter, allowing frequencies near theirresonant frequency to pass from the primary to secondary winding, but blocking other frequencies. The amount of mutual inductance between the two windings, together with theQ factor of the circuit, determine the shape of the frequency response curve. The advantage of the double tuned transformer is that it can have a wider bandwidth than a simple tuned circuit. The coupling of double-tuned circuits is described as loose-, critical-, or over-coupled depending on the value of thecoupling coefficientk{\displaystyle k}. When two tuned circuits are loosely coupled through mutual inductance, the bandwidth is narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond the critical coupling, the peak in the frequency response curve splits into two peaks, and as the coupling is increased the two peaks move further apart. This is known as overcoupling.

Stongly-coupled self-resonant coils can be used forwireless power transfer between devices in the mid range distances (up to two metres).[35] Strong coupling is required for a high percentage of power transferred, which results in peak splitting of the frequency response.[36][37]

Ideal transformers

[edit]

Whenk=1{\displaystyle k=1}, the inductor is referred to as being closely coupled. If in addition, the self-inductances go to infinity, the inductor becomes an idealtransformer. In this case the voltages, currents, and number of turns can be related in the following way:

Vs=NsNpVp{\displaystyle V_{\text{s}}={\frac {N_{\text{s}}}{N_{\text{p}}}}V_{\text{p}}}where

Conversely the current:

Is=NpNsIp{\displaystyle I_{\text{s}}={\frac {N_{\text{p}}}{N_{\text{s}}}}I_{\text{p}}}where

The power through one inductor is the same as the power through the other. These equations neglect any forcing by current sources or voltage sources.

Self-inductance of thin wire shapes

[edit]
See also:Inductor § Inductance formulas

The table below lists formulas for the self-inductance of various simple shapes made of thin cylindrical conductors (wires). In general these are only accurate if the wire radiusa{\displaystyle a} is much smaller than the dimensions of the shape, and if no ferromagnetic materials are nearby (nomagnetic core).

Self-inductance of thin wire shapes
TypeInductanceExplanation of symbols
Single layer
solenoid

Wheeler's approximation formula for current-sheet model air-core coil:[38][39]

L=N2D218D+40ℓ{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{18D+40\ell }}} (inches)       L=N2D245.72D+101.6ℓ{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{45.72D+101.6\ell }}} (cm)

This formula gives an error no more than 1%whenℓ>0.4D .{\displaystyle \ell >0.4\,D~.}

Coaxial
cable (HF)		L=μ02πℓln⁡(ba){\displaystyle {\mathcal {L}}={\frac {\mu _{0}}{2\pi }}\ell \ln \left({\frac {b}{a}}\right)}
Circular loop[40]L=μ0 r [ln⁡(8ra)−2+14Y+O(a2r2)]{\displaystyle {\mathcal {L}}=\mu _{0}\ r\ \left[\ln \left({\frac {8r}{a}}\right)-2+{\tfrac {1}{4}}Y+{\mathcal {O}}\left({\frac {a^{2}}{r^{2}}}\right)\right]}
Rectangle from
round wire[41]

L=μ0π [ ℓ1ln⁡(2ℓ1a)+ℓ2 ln⁡(2ℓ2a)+2ℓ12+ℓ22 −ℓ1 sinh−1⁡(ℓ1ℓ2)−ℓ2sinh−1⁡(ℓ2ℓ1)−(2−14Y )(ℓ1+ℓ2) ]{\displaystyle {\begin{aligned}{\mathcal {L}}={\frac {\mu _{0}}{\pi }}\ {\biggl [}\ &\ell _{1}\ln \left({\frac {2\ell _{1}}{a}}\right)+\ell _{2}\ \ln \left({\frac {2\ell _{2}}{a}}\right)+2{\sqrt {\ell _{1}^{2}+\ell _{2}^{2}\ }}\\&-\ell _{1}\ \sinh ^{-1}\left({\frac {\ell _{1}}{\ell _{2}}}\right)-\ell _{2}\sinh ^{-1}\left({\frac {\ell _{2}}{\ell _{1}}}\right)\\&-\left(2-{\tfrac {1}{4}}Y\ \right)\left(\ell _{1}+\ell _{2}\right)\ {\biggr ]}\end{aligned}}}

Pair of parallel
wires		L= μ0π ℓ [ln⁡(sa)+14Y]{\displaystyle {\mathcal {L}}={\frac {\ \mu _{0}}{\pi }}\ \ell \ \left[\ln \left({\frac {s}{a}}\right)+{\tfrac {1}{4}}Y\right]}
Pair of parallel
wires (HF)

L=μ0π ℓ cosh−1⁡(s2a)=μ0π ℓ ln⁡(s2a+s24a2−1)≈μ0π ℓ ln⁡(sa){\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {\mu _{0}}{\pi }}\ \ell \ \cosh ^{-1}\left({\frac {s}{2a}}\right)\\&={\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{2a}}+{\sqrt {{\frac {s^{2}}{4a^{2}}}-1}}\right)\\&\approx {\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{a}}\right)\end{aligned}}}

Y{\displaystyle Y} is an approximately constant value between 0 and 1 that depends on the distribution of the current in the wire:Y=0{\displaystyle Y=0} when the current flows only on the surface of the wire (completeskin effect),Y=1{\displaystyle Y=1} when the current is evenly spread over the cross-section of the wire (direct current). For round wires, Rosa (1908) gives a formula equivalent to:[23]

Y≈11+a 18μσω{\displaystyle Y\approx {\frac {1}{\,1+a\ {\sqrt {{\tfrac {1}{8}}\mu \sigma \omega \,}}\,}}}

where

O(x){\displaystyle {\mathcal {O}}(x)} is represents small term(s) that have been dropped from the formula, to make it simpler. Read the term+O(x){\displaystyle {}+{\mathcal {O}}(x)} as "plus small corrections that vary on the order ofx{\displaystyle x}" (seebig O notation).

See also

[edit]

Footnotes

[edit]
  1. ^The integral is called "logarithmically divergent" because ∫1x dx=ln⁡(x) {\displaystyle \ \int {\frac {1}{x}}\ \mathrm {d} x=\ln(x)\ } for x>0 {\displaystyle \ x>0\ }, hence it approaches infinity like a logarithm whose argument approaches infinity.

References

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  2. ^Baker, Edward Cecil (1976).Sir William Preece, F.R.S.: Victorian Engineer Extraordinary. Hutchinson. p. 204.ISBN 9780091266103..
  3. ^Heaviside, Oliver (1894). "The induction of currents in cores".Electrical Papers, Vol. 1. London: Macmillan. p. 354.
  4. ^Elert, Glenn."The Physics Hypertextbook: Inductance". Retrieved30 July 2016.
  5. ^Davidson, Michael W. (1995–2008)."Molecular Expressions: Electricity and Magnetism Introduction: Inductance".
  6. ^The International System of Units(PDF), V3.01 (9th ed.), International Bureau of Weights and Measures, Aug 2024,ISBN 978-92-822-2272-0, p. 160
  7. ^"A Brief History of Electromagnetism"(PDF).
  8. ^Note also that if the voltage across a one Henry inductor is changed in a step from zero to 1 one volt, then, by this same definition, the current will increase by one Amp per second; at least in theory, since there is always some natural limit to the current increasing.
  9. ^Ulaby, Fawwaz (2007).Fundamentals of applied electromagnetics (5th ed.). Pearson / Prentice Hall. p. 255.ISBN 978-0-13-241326-8.
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  18. ^Sears and Zemansky 1964:743
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  21. ^abPurcell, Edward (2011).Electricity and Magnetism, 2nd Ed. Cambridge University Press. p. 285.ISBN 978-1139503556.
  22. ^Gates, Earl D. (2001).Introduction to Electronics. Cengage Learning. p. 153.ISBN 0766816982.
  23. ^abRosa, E.B. (1908)."The self and mutual inductances of linear conductors".Bulletin of the Bureau of Standards.4 (2).U.S. Bureau of Standards: 301 ff.doi:10.6028/bulletin.088.
  24. ^Dengler, R. (2016). "Self inductance of a wire loop as a curve integral".Advanced Electromagnetics.5 (1):1–8.arXiv:1204.1486.Bibcode:2016AdEl....5....1D.doi:10.7716/aem.v5i1.331.S2CID 53583557.
  25. ^Neumann, F.E. (1846)."Allgemeine Gesetze der inducirten elektrischen Ströme" [General rules for induced electric currents].Annalen der Physik und Chemie (in German).143 (1). Wiley:31–44.Bibcode:1846AnP...143...31N.doi:10.1002/andp.18461430103.ISSN 0003-3804.
  26. ^Jackson, J. D. (1975).Classical Electrodynamics. Wiley. pp. 176, 263.ISBN 9780471431329.
  27. ^The kinetic energy of the drifting electrons is many orders of magnitude smaller than W, except for nanowires.
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  30. ^Kim, Seok; Kim, Shin-Ae; Jung, Goeun; Kwon, Kee-Won; Chun, Jung-Hoon (2009)."Design of a Reliable Broadband I/O Employing T-coil"(PDF).Journal of Semiconductor Technology and Science.9 (4):198–204.doi:10.5573/JSTS.2009.9.4.198.S2CID 56413251.Archived(PDF) from the original on Jul 24, 2018 – via ocean.kisti.re.kr.
  31. ^Aatre, Vasudev K. (1981).Network Theory and Filter Design. US, Canada, Latin America, and Middle East: John Wiley & Sons. pp. 71, 72.ISBN 0-470-26934-0.
  32. ^Chua, Leon O.; Desoer, Charles A.; Kuh, Ernest S. (1987).Linear and Nonlinear Circuits. McGraw-Hill, Inc. p. 459.ISBN 0-07-100685-0.
  33. ^Eslami, Mansour (May 24, 2005).Circuit Analysis Fundamentals. Chicago, IL, US: Agile Press. p. 194.ISBN 0-9718239-5-2.
  34. ^Radecki, Andrzej; Yuan, Yuxiang; Miura, Noriyuki; Aikawa, Iori; Take, Yasuhiro; Ishikuro, Hiroki; Kuroda, Tadahiro (2012). "Simultaneous 6-Gb/s Data and 10-mW Power Transmission Using Nested Clover Coils for Noncontact Memory Card".IEEE Journal of Solid-State Circuits.47 (10):2484–2495.Bibcode:2012IJSSC..47.2484R.doi:10.1109/JSSC.2012.2204545.S2CID 29266328.
  35. ^Kurs, A.; Karalis, A.; Moffatt, R.; Joannopoulos, J. D.; Fisher, P.; Soljacic, M. (6 July 2007). "Wireless Power Transfer via Strongly Coupled Magnetic Resonances".Science.317 (5834):83–86.Bibcode:2007Sci...317...83K.CiteSeerX 10.1.1.418.9645.doi:10.1126/science.1143254.PMID 17556549.S2CID 17105396.
  36. ^Sample, Alanson P.; Meyer, D. A.; Smith, J. R. (2011). "Analysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power Transfer".IEEE Transactions on Industrial Electronics.58 (2):544–554.doi:10.1109/TIE.2010.2046002.S2CID 14721.
  37. ^Rendon-Hernandez, Adrian A.; Halim, Miah A.; Smith, Spencer E.; Arnold, David P. (2022). "Magnetically Coupled Microelectromechanical Resonators for Low-Frequency Wireless Power Transfer".2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS). pp. 648–651.doi:10.1109/MEMS51670.2022.9699458.ISBN 978-1-6654-0911-7.S2CID 246753151.
  38. ^Wheeler, H.A. (1942). "Formulas for the Skin Effect".Proceedings of the IRE.30 (9):412–424.doi:10.1109/JRPROC.1942.232015.S2CID 51630416.
  39. ^Wheeler, H.A. (1928). "Simple Inductance Formulas for Radio Coils".Proceedings of the IRE.16 (10):1398–1400.doi:10.1109/JRPROC.1928.221309.S2CID 51638679.
  40. ^Elliott, R.S. (1993).Electromagnetics. New York: IEEE Press. Note: The published constant−3⁄2 in the result for a uniform current distribution is wrong.
  41. ^Grover, Frederick W. (1946).Inductance Calculations: Working formulas and tables. New York: Dover Publications, Inc.

General references

[edit]
  • Frederick W. Grover (1952).Inductance Calculations. Dover Publications, New York.
  • Griffiths, David J. (1998).Introduction to Electrodynamics (3rd ed.). Prentice Hall.ISBN 0-13-805326-X.
  • Wangsness, Roald K. (1986).Electromagnetic Fields (2nd ed.). Wiley.ISBN 0-471-81186-6.
  • Hughes, Edward. (2002).Electrical & Electronic Technology (8th ed.). Prentice Hall.ISBN 0-582-40519-X.
  • Küpfmüller K.,Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.
  • Heaviside O.,Electrical Papers. Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429-560.
  • Fritz Langford-Smith, editor (1953).Radiotron Designer's Handbook, 4th Edition, Amalgamated Wireless Valve Company Pty., Ltd. Chapter 10, "Calculation of Inductance" (pp. 429–448), includes a wealth of formulas and nomographs for coils, solenoids, and mutual inductance.
  • F. W. Sears and M. W. Zemansky 1964University Physics: Third Edition (Complete Volume), Addison-Wesley Publishing Company, Inc. Reading MA, LCCC 63-15265 (no ISBN).

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