Inductance

Inductance is the tendency of an electrical conductor

to oppose a change in the electric current flowing Inductance
through it. The electric current produces a magnetic Common symbols L
field around the conductor. The magnetic field strength SI unit henry (H)
depends on the magnitude of the electric current, and In SI base units kg⋅m2⋅s−2⋅A−2
therefore follows any changes in the magnitude of the
Derivations from L=V/(I/t)
current. From Faraday's law of induction, any change other quantities
in magnetic field through a circuit induces an L=Φ/I
electromotive force (EMF) (voltage) in the conductors, Dimension M1·L2·T−2·I−2
a process known as electromagnetic induction. This
induced voltage created by the changing current has the effect of opposing the change in current. This is
stated by Lenz's law, and the voltage is called back 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 the magnetic permeability of the conductor and nearby materials.[1] An electronic component
designed to add inductance to a circuit is called an inductor. It typically consists of a coil or helix of wire.

The term inductance was coined by Oliver Heaviside in May 1884, as a convenient way to refer to
"coefficient of self-induction".[2][3] It is customary to use the symbol for inductance, in honour of the
physicist Heinrich Lenz.[4][5] In the SI system, the unit of inductance is the henry (H), which is the
amount of inductance that causes a voltage of one volt, when the current is changing at a rate of one
ampere per second.[6] The unit is named for Joseph Henry, who discovered inductance independently of
Faraday.[7]

History
The history of electromagnetic induction, a facet of electromagnetism, began with observations of the
ancients: electric charge or static electricity (rubbing silk on amber), 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 by Michael Faraday in 1831.[8][9] 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 a galvanometer, 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.[10] This current was induced by the
change in magnetic flux that occurred when the battery was connected and disconnected.[11] 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").[12]
Source of inductance
A current flowing through a conductor generates a magnetic field around the conductor, which is
described by Ampere's circuital law. The total magnetic flux 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, the magnetic flux through the circuit changes. By Faraday's law of
induction, any change in flux through a circuit induces an electromotive force (EMF, ) in the circuit,
proportional to the rate of change of flux

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 called Lenz's law. The potential is therefore called a back 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, is the ratio between the induced
voltage and the rate of change of the current

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 the SI system is the henry (H), named
after Joseph Henry, which is the amount of inductance that generates a voltage of one volt when the
current is changing at a rate of one ampere 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 the
magnetic permeability of nearby materials; ferromagnetic materials with a higher permeability like iron
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 of magnetic flux to current[13][14][15][16]

An inductor is an electrical 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 a coil or helix. 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 multiple flux linkages. The inductance is proportional to the square of the
number of turns in the coil, assuming full flux linkage.
The inductance of a coil can be increased by placing a magnetic core of ferromagnetic material in the
hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning its magnetic
domains, and the magnetic field of the core adds to that of the coil, increasing the flux through the coil.
This is called a ferromagnetic core inductor. A magnetic core can increase the inductance of a coil by
thousands of times.

If multiple electric 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 be inductively coupled. Due to Faraday'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 the
mutual inductance of circuit and circuit as the ratio of voltage induced in circuit to the rate of
change of current in circuit . This is the principle behind a transformer. The property describing the
effect of one conductor on itself is more precisely called self-inductance, and the properties describing the
effects of one conductor with changing current on nearby conductors is called mutual inductance.[17]

Self-inductance and magnetic energy

If the current through a conductor with inductance is increasing, a voltage 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 time the power
flowing into the magnetic field, which is equal to the rate of change of the stored energy , is the
product of the current and voltage across the conductor[18][19][20]

From (1) above

When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive
losses, the energy (measured in joules, in SI) stored by an inductance with a current 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:

If the inductance is constant over the current range, the stored energy is[18][19][20]
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.

If ferromagnetic materials are located near the conductor, such as in an inductor with a magnetic core, the
constant inductance equation above is only valid for linear regions of the magnetic flux, at currents below
the level at which the ferromagnetic material saturates, 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
When a sinusoidal alternating current (AC) is passing through
a linear inductance, the induced back-EMF is also sinusoidal.
If the current through the inductance is ,
from (1) above the voltage across it is

The voltage ( , blue) and current ( , red)

where is the amplitude (peak value) of the sinusoidal waveforms in an ideal inductor to which
current in amperes, is the angular frequency of the an alternating current has been applied.
The current lags the voltage by 90°
alternating current, with being its frequency in hertz, and
is the inductance.

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

Inductive reactance is the opposition of an inductor to an alternating current.[21] It is defined analogously

to electrical resistance in a resistor, as the ratio of the amplitude (peak value) of the alternating voltage to
current in the component
Reactance has units of ohms. It can be seen that inductive reactance of an inductor increases
proportionally with frequency , 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 are out of phase; the voltage peaks occur earlier in each cycle than the
current peaks. The phase difference between the current and the induced voltage is radians or
90 degrees, showing that in an ideal inductor the current lags the voltage by 90°.

Calculating self inductance

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, with skin effect, the
surface current densities and magnetic field may be obtained by solving the Laplace 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

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
For derivation of the formulas below, see Rosa (1908).[22] The total low frequency inductance (interior
plus exterior) of a straight wire is:

where

is the "low-frequency" or DC inductance in nanohenry (nH or 10−9H),

is the length of the wire in meters,
is the radius of the wire in meters (hence a very small decimal number),
the constant is the permeability of free space, commonly called , divided by ; in
the absence of magnetically reactive insulation the value 200 is exact when using the
classical definition of μ0 = 4π × 10−7 H/m, and correct to 7 decimal places when using the
2019-redefined SI value of μ0 = 1.256 637 062 12(19) × 10−6 H/m.
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 radius is much less than the length , 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, is then given by a
very similar formula:

where the variables and are the same as above; note the changed constant term now 1, from 0.75
above.

For example, a single conductor of a lamp cord 10 m long, made of 18 AWG (1.024 mm) wire, would
have a low frequency inductance of about 19.67 μH, at k=0.75, if stretched out straight.

Wire loop
Formally, the self-inductance of a wire loop would be given by the above equation with
However, here becomes infinite, leading to a logarithmically divergent integral.[a] This
necessitates taking the finite wire radius 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,[23]

where

and are distances along the curves and respectively

is the radius of the wire
is the length of the wire
is a constant that depends on the distribution of the current in the wire:
when the current flows on the surface of the wire (total skin effect),
when the current is evenly over the cross-section of the wire.
is an error term whose size depends on the curve of the loop:
when the loop has sharp corners, and
when it is a smooth curve.
Both are small when the wire is long compared to its radius.

Solenoid
A solenoid 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, the magnetic flux density within the coil is
practically constant and is given by
where is the magnetic constant, the number of turns, the current and the length of the coil.
Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density
by the cross-section area :

When this is combined with the definition of inductance , it follows that the inductance of a
solenoid is given by:

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

Coaxial cable
Let the inner conductor have radius and permeability , let the dielectric between the inner and outer
conductor have permeability , and let the outer conductor have inner radius , outer radius , and
permeability . However, for a typical coaxial line application, we are interested in passing (non-DC)
signals at frequencies for which the resistive skin effect cannot be neglected. In most cases, the inner and
outer conductor terms are negligible, in which case one may approximate

Multilayer coils
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
Many inductors include a magnetic core at the center of or partly surrounding the winding. Over a large
enough range these exhibit a nonlinear permeability with effects such as magnetic 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:

The differential or small-signal inductance, on the other hand, is used in calculating voltage. It is defined
as:
The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by
Faraday's Law and the chain rule of calculus.

Similar definitions may be derived for nonlinear mutual inductance.

Mutual inductance

Definition of Mutual induction or Coefficient of mutual induction

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

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

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 circuit on a filamentary circuit is given by the double
integral Neumann formula[24]
where

and are the curves followed by the wires.

is the permeability of free space (4π×10−7 H/m)
is a small increment of the wire in circuit Cm
is the position of in space
is a small increment of the wire in circuit Cn
is the position of in space.

Derivation

where

is the current through the th wire, this current creates the magnetic flux through the
th surface
is the magnetic flux through the ith surface due to the electrical circuit outlined by :[25]

where

is the curve enclosing surface ; and is any arbitrary orientable area with edge
is the magnetic field vector due to the -th current (of circuit ).
is the vector potential due to the -th current.
Stokes' theorem has been used for the 3rd equality step. For the last equality step, we used the retarded
potential expression for 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 symbol M.

Derivation of mutual inductance

The inductance equations above are a consequence of Maxwell's equations. For the important case of
electrical circuits consisting of thin wires, the derivation is straightforward.
In a system of wire loops, each with one or several wire turns, the flux linkage of loop , , is given
by

Here denotes the number of turns in loop ; is the magnetic flux through loop ; and are
some constants described below. This equation follows from Ampere's law: magnetic fields and fluxes are
linear functions of the currents. By Faraday's law of induction, we have

where denotes the voltage induced in circuit . This agrees with the definition of inductance above if
the coefficients are identified with the coefficients of inductance. Because the total currents
contribute to it also follows that is proportional to the product of turns .

Mutual inductance and magnetic field energy

Multiplying the equation for vm above with imdt and summing over m gives the energy transferred to the
system in the time interval dt,

This must agree with the change of the magnetic field energy, W, caused by the currents.[26] The
integrability condition

requires Lm,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,

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. A mechanical analogy in the K = 1 case
with magnetic field energy (1/2)Li2 is a body with mass M, velocity u 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).
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 which transformers work, but it
can also cause unwanted coupling between conductors in a
circuit.

The mutual inductance, , is also a measure of the coupling

between two inductors. The mutual inductance by circuit on
circuit is given by the double integral Neumann formula, see
calculation techniques Circuit diagram of two mutually
coupled inductors. The two vertical
The mutual inductance also has the relationship: lines between the windings indicate
that the transformer has a
ferromagnetic core . "n:m" shows the
where ratio between the number of windings
of the left inductor to windings of the
is the mutual inductance, and the subscript right inductor. This picture also shows
specifies the relationship of the voltage induced in coil the dot convention.
2 due to the current in coil 1.
is the number of turns in coil 1,
is the number of turns in coil 2,
is the permeance of the space occupied by the flux.
Once the mutual inductance is determined, it can be used to predict the behavior of a circuit:

where

is the voltage across the inductor of interest;

is the inductance of the inductor of interest;
is the derivative, with respect to time, of the current through the inductor of interest,
labeled 1;
is the derivative, with respect to time, of the current through the inductor, labeled 2,
that is coupled to the first inductor; and
is the mutual inductance.
The minus sign arises because of the sense the current has been defined in the diagram. With both
currents defined going into the dots the sign of will be positive (the equation would read with a plus
sign instead). [27]

Coupling coefficient
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 one magnetic 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:
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

thus,

where

is the coupling coefficient,

is the inductance of the first coil, and
is the inductance of the second coil.
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 as , but some[28] define it as
. Allowing negative values of captures phase inversions of the coil connections and the
direction of the windings.[29]

Matrix representation
Mutually coupled inductors can be described by any of the two-port network parameter matrix
representations. The most direct are the z parameters, which are given by[30]

The y parameters are given by

Where is the complex frequency variable, and are the inductances of the primary and secondary
coil, respectively, and is the mutual inductance between the coils.
Multiple Coupled Inductors
Mutual inductance may be applied to multiple inductors simultaneously. The matrix representations for
multiple mutually coupled inductors are given by[31]

Equivalent circuits

T-circuit
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.[32]

This can be analyzed as a two port network. With the output

terminated with some arbitrary impedance , the voltage gain
, is given by: T equivalent circuit of mutually coupled
inductors

where is the coupling constant and is the complex frequency variable, as above. For tightly coupled
inductors where this reduces to

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:

For this reduces to

Thus, current gain is not independent of load unless the further condition

is met, in which case,

and

π-circuit
Alternatively, two coupled inductors can be modelled using a
π equivalent circuit with optional ideal transformers at each
port. While the circuit is more complicated than a T-circuit, it
can be generalized[33] to circuits consisting of more than two
coupled inductors. Equivalent circuit elements , have
physical meaning, modelling respectively magnetic
π equivalent circuit of coupled inductors
reluctances of coupling paths and magnetic reluctances of
leakage paths. For example, electric currents flowing through
these elements correspond to coupling and leakage magnetic fluxes. Ideal transformers normalize all self-
inductances to 1 Henry to simplify mathematical formulas.

Equivalent circuit element values can be calculated from coupling coefficients with

where coupling coefficient matrix and its cofactors are defined as

and
For two coupled inductors, these formulas simplify to

and

and for three coupled inductors (for brevity shown only for and )

and

Resonant transformer
When a capacitor is connected across one winding of a transformer, making the winding a tuned circuit
(resonant circuit) it is called a single-tuned transformer. When a capacitor is connected across each
winding, it is called a double tuned transformer. These resonant transformers can store oscillating
electrical energy similar to a resonant circuit and thus function as a bandpass filter, allowing frequencies
near their resonant frequency to pass from the primary to secondary winding, but blocking other
frequencies. The amount of mutual inductance between the two windings, together with the Q 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 the coupling
coefficient . 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 for wireless power transfer between devices in the mid
range distances (up to two metres).[34] Strong coupling is required for a high percentage of power
transferred, which results in peak splitting of the frequency response.[35][36]

Ideal transformers
When , the inductor is referred to as being closely coupled. If in addition, the self-inductances go to
infinity, the inductor becomes an ideal transformer. In this case the voltages, currents, and number of
turns can be related in the following way:

where

is the voltage across the secondary inductor,

is the voltage across the primary inductor (the one connected to a power source),
is the number of turns in the secondary inductor, and
 is the number of turns in the primary inductor.
Conversely the current:

where

is the current through the secondary inductor,

is the current through the primary inductor (the one connected to a power source),
is the number of turns in the secondary inductor, and
is the number of turns in the primary inductor.
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

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 radius is much smaller than the
dimensions of the shape, and if no ferromagnetic materials are nearby (no magnetic core).
 Self-inductance of thin wire shapes
Explanation of
Type Inductance
symbols

Wheeler's approximation formula for current-sheet model

air-core coil:[37][38] : inductance in μH
(10−6 henries)
Single : number of turns
layer (inches) (cm) : diameter in
solenoid (inches) (cm)
: length in (inches)
This formula gives an error no more than 1% (cm)
when

: Outer conductor's
inside radius
: Inner conductor's
Coaxial radius
cable (HF)
: Length
: see table
footnote.

: Loop radius
Circular : Wire radius
loop[39] : see table
footnotes.

: Side lengths
Rectangle
from
round : Wire radius
wire[40] : see table
footnotes.

: Wire radius
: Separation
Pair of distance,
parallel
wires : Length of pair
: see table
footnotes.

: Wire radius
: Separation
Pair of distance,
parallel : Length (each) of
wires (HF) pair
: see table
footnote.
 is an approximately constant value between 0 and 1 that depends on the distribution of the current in
the wire: when the current flows only on the surface of the wire (complete skin effect),
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:[22]

where

is the angular frequency, in radians per second;

is the net magnetic permeability of the wire;
is the wire's specific conductivity; and
is the wire radius.
is represents small term(s) that have been dropped from the formula, to make it simpler. Read the
term as "plus small corrections that vary on the order of " (see big O notation).

See also
Electromagnetic induction
Gyrator
Hydraulic analogy
Leakage inductance
LC circuit, RLC circuit, RL circuit
Kinetic inductance

Footnotes

a. The integral is called "logarithmically divergent" because for ,

hence it approaches infinity like a logarithm whose argument approaches infinity.

References
1. Serway, A. Raymond; Jewett, John W.; Wilson, Jane; Wilson, Anna; Rowlands, Wayne
(2017). "Inductance". Physics for global scientists and engineers (2 ed.). Cengage AU.
p. 901. ISBN 9780170355520.
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 (https://archive.org/details/electricalpaper00heavgoog/page/353/
mode/2up).
4. Elert, Glenn. "The Physics Hypertextbook: Inductance" (http://physics.info/inductance/).
Retrieved 30 July 2016.
 5. Davidson, Michael W. (1995–2008). "Molecular Expressions: Electricity and Magnetism
Introduction: Inductance" (http://micro.magnet.fsu.edu/electromag/electricity/inductance.htm
l).
6. The International System of Units (https://www.bipm.org/documents/20126/41483022/SI-Bro
chure-9-EN.pdf) (PDF) (9th ed.), International Bureau of Weights and Measures, Dec 2022,
ISBN 978-92-822-2272-0, p. 160
7. "A Brief History of Electromagnetism" (http://web.hep.uiuc.edu/home/serrede/P435/Lecture_
Notes/A_Brief_History_of_Electromagnetism.pdf) (PDF).
8. Ulaby, Fawwaz (2007). Fundamentals of applied electromagnetics (https://books.google.co
m/books?id=-1GDkgEACAAJ) (5th ed.). Pearson / Prentice Hall. p. 255. ISBN 978-0-13-
241326-8.
9. "Joseph Henry" (https://web.archive.org/web/20131213121232/http://www.nas.edu/history/m
embers/henry.html). Distinguished Members Gallery, National Academy of Sciences.
Archived from the original (http://www.nas.edu/history/members/henry.html) on 2013-12-13.
Retrieved 2006-11-30.
10. Pearce Williams, L. (1971). Michael Faraday: A Biography. Simon and Schuster. pp. 182–
183. ISBN 9780671209292.
11. Giancoli, Douglas C. (1998). Physics: Principles with Applications (https://archive.org/detail
s/physicsprinciple00gian) (Fifth ed.). pp. 623–624 (https://archive.org/details/physicsprincipl
e00gian/page/623).
12. Pearce Williams, L. (1971). Michael Faraday: A Biography. Simon and Schuster. pp. 191–
195. ISBN 9780671209292.
13. Singh, Yaduvir (2011). Electro Magnetic Field Theory (https://books.google.com/books?id=0
-PfbT49tJMC&pg=PA65). Pearson Education India. p. 65. ISBN 978-8131760611.
14. Wadhwa, C.L. (2005). Electrical Power Systems (https://books.google.com/books?id=Su3-0
UhVF28C&pg=PA18). New Age International. p. 18. ISBN 8122417221.
15. Pelcovits, Robert A.; Farkas, Josh (2007). Barron's AP Physics C (https://books.google.com/
books?id=yON684oSjbEC&pg=PA646). Barron's Educational Series. p. 646. ISBN 978-
0764137105.
16. Purcell, Edward M.; Morin, David J. (2013). Electricity and Magnetism (https://books.google.
com/books?id=A2rS5vlSFq0C&pg=PA364). Cambridge Univ. Press. p. 364. ISBN 978-
1107014022.
17. Sears and Zemansky 1964:743
18. Serway, Raymond A.; Jewett, John W. (2012). Principles of Physics: A Calculus-Based Text,
5th Ed (https://books.google.com/books?id=egmU-OumDAgC&pg=PA802). Cengage
Learning. pp. 801–802. ISBN 978-1133104261.
19. Ida, Nathan (2007). Engineering Electromagnetics, 2nd Ed (https://books.google.com/book
s?id=2CbvXE4o5swC&pg=PA572). Springer Science and Business Media. p. 572.
ISBN 978-0387201566.
20. Purcell, Edward (2011). Electricity and Magnetism, 2nd Ed (https://books.google.com/book
s?id=Z3bkNh6h4WEC&pg=PA285). Cambridge University Press. p. 285. ISBN 978-
1139503556.
21. Gates, Earl D. (2001). Introduction to Electronics (https://books.google.com/books?id=IwC5
GIA0cREC&pg=PA153). Cengage Learning. p. 153. ISBN 0766816982.
22. Rosa, E.B. (1908). "The self and mutual inductances of linear conductors" (https://doi.org/1
0.6028%2Fbulletin.088). Bulletin of the Bureau of Standards. 4 (2). U.S. Bureau of
Standards: 301 ff. doi:10.6028/bulletin.088 (https://doi.org/10.6028%2Fbulletin.088).
23. Dengler, R. (2016). "Self inductance of a wire loop as a curve integral". Advanced
Electromagnetics. 5 (1): 1–8. arXiv:1204.1486 (https://arxiv.org/abs/1204.1486).
Bibcode:2016AdEl....5....1D (https://ui.adsabs.harvard.edu/abs/2016AdEl....5....1D).
doi:10.7716/aem.v5i1.331 (https://doi.org/10.7716%2Faem.v5i1.331). S2CID 53583557 (htt
ps://api.semanticscholar.org/CorpusID:53583557).
24. Neumann, F.E. (1846). "Allgemeine Gesetze der inducirten elektrischen Ströme" (https://zen
odo.org/record/1423608) [General rules for induced electric currents]. Annalen der Physik
und Chemie (in German). 143 (1). Wiley: 31–44. Bibcode:1846AnP...143...31N (https://ui.ad
sabs.harvard.edu/abs/1846AnP...143...31N). doi:10.1002/andp.18461430103 (https://doi.or
g/10.1002%2Fandp.18461430103). ISSN 0003-3804 (https://search.worldcat.org/issn/0003-
3804).
25. Jackson, J. D. (1975). Classical Electrodynamics (https://archive.org/details/classicalelectro
00jack_0). Wiley. pp. 176 (https://archive.org/details/classicalelectro00jack_0/page/176),
263. ISBN 9780471431329.
26. The kinetic energy of the drifting electrons is many orders of magnitude smaller than W,
except for nanowires.
27. Nahvi, Mahmood; Edminister, Joseph (2002). Schaum's outline of theory and problems of
electric circuits (https://books.google.com/books?id=nrxT9Qjguk8C&pg=PA338). McGraw-
Hill Professional. p. 338. ISBN 0-07-139307-2.
28. Thierauf, Stephen C. (2004). High-speed Circuit Board Signal Integrity (https://archive.org/d
etails/highspeedcircuit00thie_269). Artech House. p. 56 (https://archive.org/details/highspee
dcircuit00thie_269/page/n70). ISBN 1580538460.
29. Kim, Seok; Kim, Shin-Ae; Jung, Goeun; Kwon, Kee-Won; Chun, Jung-Hoon (2009). "Design
of a Reliable Broadband I/O Employing T-coil" (http://ocean.kisti.re.kr/downfile/volume/ieek/
E1STAN/2009/v9n4/E1STAN_2009_v9n4_198.pdf) (PDF). Journal of Semiconductor
Technology and Science. 9 (4): 198–204. doi:10.5573/JSTS.2009.9.4.198 (https://doi.org/1
0.5573%2FJSTS.2009.9.4.198). S2CID 56413251 (https://api.semanticscholar.org/CorpusI
D:56413251). Archived (https://web.archive.org/web/20180724115136/http://ocean.kisti.re.k
r/downfile/volume/ieek/E1STAN/2009/v9n4/E1STAN_2009_v9n4_198.pdf) (PDF) from the
original on Jul 24, 2018 – via ocean.kisti.re.kr.
30. Aatre, Vasudev K. (1981). Network Theory and Filter Design (https://archive.org/details/netw
orktheoryfil0000aatr/page/n1/mode/2up). US, Canada, Latin America, and Middle East:
John Wiley & Sons. pp. 71, 72. ISBN 0-470-26934-0.
31. Chua, Leon O.; Desoer, Charles A.; Kuh, Ernest S. (1987). Linear and Nonlinear Circuits (htt
ps://archive.org/details/linearnonlinearc0000leon). McGraw-Hill, Inc. p. 459. ISBN 0-07-
100685-0.
32. Eslami, Mansour (May 24, 2005). Circuit Analysis Fundamentals (https://archive.org/details/
circuitanalysisf0000esla/mode/2up). Chicago, IL, US: Agile Press. p. 194. ISBN 0-9718239-
5-2.
33. 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 (https://ui.adsabs.
harvard.edu/abs/2012IJSSC..47.2484R). doi:10.1109/JSSC.2012.2204545 (https://doi.org/1
0.1109%2FJSSC.2012.2204545). S2CID 29266328 (https://api.semanticscholar.org/CorpusI
D:29266328).
34. 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 (https://ui.adsabs.harvard.edu/abs/2007Sci...31
7...83K). CiteSeerX 10.1.1.418.9645 (https://citeseerx.ist.psu.edu/viewdoc/summary?doi=1
0.1.1.418.9645). doi:10.1126/science.1143254 (https://doi.org/10.1126%2Fscience.114325
4). PMID 17556549 (https://pubmed.ncbi.nlm.nih.gov/17556549). S2CID 17105396 (https://
api.semanticscholar.org/CorpusID:17105396).
35. 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 (http
s://doi.org/10.1109%2FTIE.2010.2046002). S2CID 14721 (https://api.semanticscholar.org/C
orpusID:14721).
36. 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 (http
s://doi.org/10.1109%2FMEMS51670.2022.9699458). ISBN 978-1-6654-0911-7.
S2CID 246753151 (https://api.semanticscholar.org/CorpusID:246753151).
37. Wheeler, H.A. (1942). "Formulas for the Skin Effect". Proceedings of the IRE. 30 (9): 412–
424. doi:10.1109/JRPROC.1942.232015 (https://doi.org/10.1109%2FJRPROC.1942.23201
5). S2CID 51630416 (https://api.semanticscholar.org/CorpusID:51630416).
38. Wheeler, H.A. (1928). "Simple Inductance Formulas for Radio Coils". Proceedings of the
IRE. 16 (10): 1398–1400. doi:10.1109/JRPROC.1928.221309 (https://doi.org/10.1109%2FJ
RPROC.1928.221309). S2CID 51638679 (https://api.semanticscholar.org/CorpusID:516386
79).
39. Elliott, R.S. (1993). Electromagnetics. New York: IEEE Press. Note: The published constant
−3⁄ in the result for a uniform current distribution is wrong.
2
40. Grover, Frederick W. (1946). Inductance Calculations: Working formulas and tables. New
York: Dover Publications, Inc.

General references
Frederick W. Grover (1952). Inductance Calculations. Dover Publications, New York.
Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.) (https://archive.org/detail
s/introductiontoel00grif_0). 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 (https://archive.org/stre
am/bitsavers_rcaRadiotr1954_94958503/Radiotron_Designers_Handbook_1954#page/n46
9/mode/2up), 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 1964 University Physics: Third Edition (Complete
Volume), Addison-Wesley Publishing Company, Inc. Reading MA, LCCC 63-15265 (no
ISBN).

External links
Clemson Vehicular Electronics Laboratory: Inductance Calculator (https://web.archive.org/w
eb/20171115094017/http://www.cvel.clemson.edu/emc/calculators/Inductance_Calculator/in
dex.html)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Inductance&oldid=1286420024"