In electromagnetism and electronics, inductance
describes the tendency of an electrical conductor,
such as coil, to oppose a change in the electric
current through it. The change in current
induces a reverse electromotive force (voltage).
When an electric current flows through a conductor,
it creates a magnetic field around that conductor.
A changing current, in turn, creates a changing
magnetic field, the surface integral of which
is known as magnetic flux. From Faraday's
law of induction, any change in magnetic flux
through a circuit induces an electromotive
force (voltage) across that circuit, a phenomenon
known as electromagnetic induction. Inductance,
L
{\displaystyle L}
, is defined as the ratio between this induced
voltage,
v
{\displaystyle v}
, and the rate of change of the current
i
(
t
)
{\displaystyle i(t)}
in the circuit.
L
:=
−
v
(
d
i
d
t
)
−
1
⇒
v
=
−
L
d
i
d
t
{\displaystyle L:=-v\left({di \over dt}\right)^{\!-1}\;\;\Rightarrow
\;\;v=-L{di \over dt}}
This proportionality factor L does not depend
on electromagnetical quantities, but depends
only on the geometric setting of the conductors
and the material properties of the regions
crossed by the fields. The voltage created
is actually self-induced when the magnetic
field of a current-carrying conductor acts
back on the conductor itself. From Lenz's
law, this induced voltage, or "back EMF",
will be in 
a direction so as to oppose the change in
current which created it. Thus, self-inductance
measures the property of a specific geometric
arrangement of a conductor which opposes any
change in current through the conductor. An
inductor is the electrical component which
adds inductance to a circuit. It typically
consists of a coil or helix of wire.
The term inductance was coined by Oliver Heaviside
in 1886. It is customary to use the symbol
L
{\displaystyle L}
for inductance, in honour of the physicist
Heinrich Lenz.
In the SI system, the unit of inductance is
the henry (H), which is the amount of inductance
which causes a voltage of 1 volt when the
current is changing at a rate of one ampere
per second. It is named for Joseph Henry,
who discovered inductance independently of
Faraday.
== 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 began in the late 18th
century.
Electromagnetic induction was first described
by Michael Faraday in 1831. 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. This current was induced
by the change in magnetic flux that occurred
when the battery was connected and disconnected.
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").
== Source of inductance ==
A current
i
{\displaystyle i}
flowing through a conductor generates a magnetic
field around the conductor, which is described
by Ampere's circuital law. If the magnetic
field is uniform across and perpendicular
to the area bounded by the circuit, the total
magnetic flux through a circuit
Φ
{\displaystyle \Phi }
is equal to the product of the magnitude of
the magnetic field and the area of the surface
spanning the current path. If the current
varies, the magnetic flux
Φ
{\displaystyle \Phi }
through the circuit changes. By Faraday's
law of induction, any change in flux through
a circuit induces an electromotive force (EMF)
or voltage
v
{\displaystyle v}
in the circuit, proportional to the rate of
change of flux
v
(
t
)
=
−
d
Φ
(
t
)
d
t
{\displaystyle v(t)=-{d\Phi (t) \over dt}\,}
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,
L
{\displaystyle L}
is the ratio between the induced voltage and
the rate of change of the current
v
(
t
)
=
L
d
i
d
t
(
1
)
{\displaystyle \;v(t)=L{di \over dt}\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 the
SI system is the henry (H), named after American
scientist Joseph Henry, which is the amount
of inductance which 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
L
=
Φ
(
i
)
i
{\displaystyle L={\Phi (i) \over i}}
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.
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
M
l
,
k
{\displaystyle M_{l,k}}
of circuit
l
{\displaystyle l}
and circuit
k
{\displaystyle k}
as the ratio of voltage induced in circuit
k
{\displaystyle k}
to the rate of change of current in circuit
l
{\displaystyle l}
. This is the principle behind a transformer
with several secondaries.
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 are called mutual
inductances. The notion of inductance is especially
handy for dealing with discrete, concentrated
components (wires, coils, transformers, ...) at
low frequencies. Electric circuits which are
located sufficiently close together, so the
magnetic field created by the current in one
passes through the other, are said to be inductively
coupled. A change in current in one circuit
will cause the magnetic flux through the other
circuit to vary, which will induce a voltage
in the other circuit, by Faraday's law. The
ratio of the voltage induced in the second
circuit to the rate of change of current in
the first circuit is called the mutual inductance
M
{\displaystyle M}
between the circuits. It is also measured
in the unit henry. The sections below will
describe self-inductance, the effect of inductance
in a single conductor or circuit. Mutual inductance,
inductance between circuits, is described
in the section at the end.
== Self-inductance and magnetic energy ==
If the current through a conductor with inductance
is increasing, a voltage
v
(
t
)
{\displaystyle v(t)}
will be induced across the conductor with
a polarity which opposes the current, as described
above (this is in addition to any voltage
drop caused by the conductor's resistance).
The charges flowing through the circuit lose
potential energy moving from the higher voltage
to the lower voltage end. The energy from
the external circuit required to overcome
this "potential hill" is being stored in the
increased magnetic field around the conductor.
Therefore, any inductance with a current through
it stores energy in its magnetic field. At
any given time
t
{\displaystyle t}
the power
p
(
t
)
{\displaystyle p(t)}
flowing into the magnetic field, which is
equal to the rate of change of the stored
energy
U
{\displaystyle U}
, is the product of the current
i
(
t
)
{\displaystyle i(t)}
and voltage
v
(
t
)
{\displaystyle v(t)}
across the conductor
p
(
t
)
=
d
U
d
t
=
v
(
t
)
i
(
t
)
{\displaystyle p(t)={dU \over dt}=v(t)i(t)\,}
From (1) above
d
U
d
t
=
L
(
i
)
i
d
i
d
t
{\displaystyle {dU \over dt}=L(i)i{di \over
dt}\,}
d
U
=
L
(
i
)
i
d
i
{\displaystyle dU=L(i)idi\,}
When there is no current, there is no magnetic
field and the stored energy is zero. Neglecting
resistive losses, the energy
U
{\displaystyle U}
(measured in joules, in SI) stored by an inductance
with a current
I
{\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
=
∫
0
I
L
(
i
)
i
d
i
{\displaystyle U=\int _{0}^{I}L(i)idi\,}
If the inductance
L
(
I
)
{\displaystyle L(I)}
is constant over the current range, the stored
energy is
U
=
L
∫
0
I
i
d
i
{\displaystyle U=L\int _{0}^{I}idi}
U
=
1
2
L
I
2
{\displaystyle U={1 \over 2}LI^{2}}
So therefore inductance is also proportional
to how much energy is 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
will decrease, 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 will
return 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 will begin 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 will also be sinusoidal.
If the current through the inductance is
i
(
t
)
=
I
p
sin
⁡
(
ω
t
)
{\displaystyle i(t)=I_{p}\sin(\omega t)}
, from (1) above the voltage across it will
be
v
(
t
)
=
L
d
i
d
t
=
L
d
d
t
[
I
p
sin
⁡
(
ω
t
)
]
=
ω
L
I
p
cos
⁡
(
ω
t
)
=
ω
L
I
p
sin
⁡
(
ω
t
+
π
2
)
{\displaystyle {\begin{aligned}v(t)&=L{di
\over dt}=L{d \over dt}[I_{p}\sin(\omega t)]\\&=\omega
LI_{p}\cos(\omega t)=\omega LI_{p}\sin(\omega
t+{\pi \over 2})\end{aligned}}}
where
I
p
{\displaystyle I_{p}}
is the amplitude (peak value) of the sinusoidal
current in amperes,
ω
=
2
π
f
{\displaystyle \omega =2\pi f}
is the angular frequency of the alternating
current, with
f
{\displaystyle f}
being its [frequency]] in hertz, and
L
{\displaystyle L}
is the inductance.
Thus the amplitude (peak value) of the voltage
across the inductance will be
V
p
=
ω
L
I
p
=
2
π
f
L
I
p
{\displaystyle V_{p}=\omega LI_{p}=2\pi fLI_{p}}
Inductive reactance is the opposition of an
inductor to an alternating current. 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
X
L
=
V
p
I
p
=
2
π
f
L
{\displaystyle X_{L}={V_{p} \over I_{p}}=2\pi
fL}
Reactance has units of ohms. It can be seen
that inductive reactance of an inductor increases
proportionally with frequency
f
{\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 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
ϕ
=
π
/
2
{\displaystyle \phi =\pi /2}
radians or 90 degrees, showing that in an
ideal inductor the current lags the voltage
by 90°.
== Calculating 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.
=== Inductance of a straight single wire ===
A straight single wire has some inductance,
which in our ordinary experience is intangible
because it is negligibly small so it can't
readily be measured at low frequencies, and
its effect is not detectable. A long straight
wire like an electric transmission line has
substantial inductance that reduces its capacity,
and there is no problem at all measuring it.
As a practical matter, longer wires have more
inductance, and thicker wires have less, analogous
to their electrical resistance, though the
relationships aren't linear nor are they the
same relationships as those quantities bear
to resistance. As an essential component of
coils and circuits, understanding what the
inductance of a wire is, is essential. Yet,
there is no simple answer.
There is no unambiguous definition of the
inductance of a straight wire. If we consider
the wire in isolation we ignore the question
of how the current gets to the wire. That
current will affect the flux which is developed
in the vicinity of the wire. But this flux
is a part of the definition. A consequence
of Maxwell's equations is that we cannot define
the inductance of only a portion of a circuit,
we can only define the inductance of a whole
circuit, which includes how the current gets
to the wire and how it returns to the source.
The magnetic flux incident to the whole circuit
determines the inductance of the circuit and
of any part of it. The magnetic flux is an
indivisible entity, yet we wish to consider
only a part of it, the part incident to the
wire, between whatever we define to be the
"ends" of the wire.
The total low frequency inductance (internal
plus external) of a straight wire is:
L
d
c
=
2
l
⋅
[
ln
⁡
(
2
l
/
r
)
−
0.75
]
{\displaystyle L_{dc}=2l\cdot \left[\ln(2l/r)-0.75\right]}
where
L
d
c
{\displaystyle L_{dc}}
is the "low-frequency" or DC inductance in
nanohenries (nH or 10−9H),
l
{\displaystyle l}
is the length of the wire in cm
r
{\displaystyle r}
is the radius of the wire in cm.This result
is based on the assumption that the radius
r
{\displaystyle r}
is much less than the length
l
{\displaystyle l}
, which is commonly true.
For sufficiently high frequencies skin effects
cause the internal inductance to go to zero
and the inductance becomes:
L
a
c
=
2
l
⋅
[
ln
⁡
(
2
l
/
r
)
−
1.00
]
{\displaystyle L_{ac}=2l\cdot \left[\ln(2l/r)-1.00\right]}
See.These inductances are often referred to
as "partial inductances" to indicate that
they must be used with care.
In an everyday notion, one conductor of a
100m 18gauge lamp cord, stretched out straight,
would have inductance of about 0.24mH.
=== Mutual inductance of two parallel straight
wires ===
There are two cases to consider: current travels
in the same direction in each wire, and 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
2-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. None-the-less, the error terms,
which are not included in the integral will
only be small if the geometries of the loops
are mostly smooth and convex: they do not
have too many kinks, sharp corners, coils,
crossovers, parallel segments, concave cavities
or other topological "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
m
{\displaystyle m}
on a filamentary circuit
n
{\displaystyle n}
is given by the double integral Neumann formula
L
m
,
n
=
μ
0
4
π
∮
C
m
⁡
∮
C
n
⁡
d
x
m
⋅
d
x
n
|
x
m
−
x
n
|
{\displaystyle L_{m,n}={\frac {\mu _{0}}{4\pi
}}\oint _{C_{m}}\oint _{C_{n}}{\frac {d\mathbf
{x} _{m}\cdot d\mathbf {x} _{n}}{|\mathbf
{x} _{m}-\mathbf {x} _{n}|}}}
where
Cm and Cn are the curves spanned by the wires.
μ
0
{\displaystyle \mu _{0}}
is the permeability of free space (4π × 10−7
H/m)
d
x
m
{\displaystyle d\mathbf {x} _{m}}
is a small increment of the wire in circuit
Cm
x
m
{\displaystyle \mathbf {x} _{m}}
is the position of
d
x
m
{\displaystyle d\mathbf {x} _{m}}
in space
d
x
n
{\displaystyle d\mathbf {x} _{n}}
is a small increment of the wire in circuit
Cn
x
n
{\displaystyle \mathbf {x} _{n}}
is the position of
d
x
n
{\displaystyle d\mathbf {x} _{n}}
in space
=== Derivation ===
M
i
j
=
d
e
f
Φ
i
j
I
j
{\displaystyle M_{ij}\ {\stackrel {\mathrm
{def} }{=}}\ {\frac {\Phi _{ij}}{I_{j}}}}
where
Φ
i
j
{\displaystyle \Phi _{ij}\ \,}
is the magnetic flux through the ith surface
due to the electrical circuit outlined by
Cj
I
j
{\displaystyle I_{j}}
is the current through the jth wire, this
current creates the magnetic flux
Φ
i
j
{\displaystyle \Phi _{ij}\ \,}
through the ith surface.
Φ
i
j
=
∫
S
i
B
j
⋅
d
a
=
∫
S
i
(
∇
×
A
j
)
⋅
d
a
=
∮
C
i
⁡
A
j
⋅
d
s
i
=
∮
C
i
⁡
(
μ
0
I
j
4
π
∮
C
j
⁡
d
s
j
|
s
i
−
s
j
|
)
⋅
d
s
i
{\displaystyle \Phi _{ij}=\int _{S_{i}}\mathbf
{B_{j}} \cdot \mathbf {da} =\int _{S_{i}}(\nabla
\times \mathbf {A_{j}} )\cdot \mathbf {da}
=\oint _{C_{i}}\mathbf {A_{j}} \cdot \mathbf
{ds_{i}} =\oint _{C_{i}}\left({\frac {\mu
_{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathbf
{ds} _{j}}{|\mathbf {s_{i}-s_{j}} |}}\right)\cdot
\mathbf {ds} _{i}}
where
Ci is the enclosing curve of Si; and Si is
any arbitrary orientable area with boundary
Ci
Bj is the magnetic field vector due to the
j-th current (of circuit Cj).
Aj is the vector potential due to the j-th
current.Stokes' theorem has been used for
the 3rd equality step.
For the last equality step, we used the Retarded
potential expression for Aj 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.
=== Self-inductance of a wire loop ===
Formally, the self-inductance of a wire loop
would be given by the above equation with
m
{\displaystyle m}
=
n
{\displaystyle n}
. However, here
1
/
|
x
−
x
′
|
{\displaystyle 1/|\mathbf {x} -\mathbf {x}
'|}
becomes infinite, leading to a logarithmically
divergent integral. 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,
L
=
μ
0
4
π
[
∮
C
⁡
∮
C
′
⁡
d
x
⋅
d
x
′
|
x
−
x
′
|
]
+
μ
0
4
π
l
Y
+
O
{\displaystyle L={\frac {\mu _{0}}{4\pi }}\left[\oint
_{C}\oint _{C'}{\frac {d\mathbf {x} \cdot
d\mathbf {x} '}{|\mathbf {x} -\mathbf {x}
'|}}\right]+{\frac {\mu _{0}}{4\pi }}lY+O}
for
|
s
−
s
′
|
{\displaystyle |\mathbf {s} -\mathbf {s} '|}
>
a
/
2
{\displaystyle a/2}
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 l}
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 (total skin effect),
Y
=
1
/
2
{\displaystyle Y=1/2}
when the current is homogeneous over the cross-section
of the wire.
O
{\displaystyle O}
is an error term
O
(
μ
0
a
)
{\displaystyle O(\mu _{0}a)}
when the loop has sharp corners, and
O
(
μ
0
a
2
/
l
)
{\displaystyle O\left(\mu _{0}a^{2}/l\right)}
when it is a smooth curve. These are small
when the wire is long compared to its radius.
=== Inductance of a 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
B
{\displaystyle B}
within the coil is practically constant and
is given by
B
=
μ
0
N
i
l
{\displaystyle \displaystyle B={\frac {\mu
_{0}Ni}{l}}}
where
μ
0
{\displaystyle \mu _{0}}
is the magnetic constant,
N
{\displaystyle N}
the number of turns,
i
{\displaystyle i}
the current and
l
{\displaystyle l}
the length of the coil. Ignoring end effects,
the total magnetic flux through the coil is
obtained by multiplying the flux density
B
{\displaystyle B}
by the cross-section area
A
{\displaystyle A}
:
Φ
=
μ
0
N
i
A
l
,
{\displaystyle \displaystyle \Phi ={\frac
{\mu _{0}NiA}{l}},}
When this is combined with the definition
of inductance
L
=
N
Φ
i
{\displaystyle \displaystyle L={\frac {N\Phi
}{i}}}
, it follows that the inductance of a solenoid
is given by:
L
=
μ
0
N
2
A
l
.
{\displaystyle \displaystyle L={\frac {\mu
_{0}N^{2}A}{l}}.}
Therefore, for air-core coils, inductance
is a function of coil geometry and number
of turns, and is independent of current.
=== Inductance of a coaxial cable ===
Let the inner conductor have radius
r
i
{\displaystyle r_{i}}
and permeability
μ
i
{\displaystyle \mu _{i}}
, let the dielectric between the inner and
outer conductor have permeability
μ
d
{\displaystyle \mu _{d}}
, and let the outer conductor have inner radius
r
o
1
{\displaystyle r_{o1}}
, outer radius
r
o
2
{\displaystyle r_{o2}}
, and permeability
μ
o
{\displaystyle \mu _{o}}
. 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
L
′
=
d
L
d
l
≈
μ
d
2
π
ln
⁡
r
o
1
r
i
{\displaystyle L'={\frac {dL}{dl}}\approx
{\frac {\mu _{d}}{2\pi }}\ln {\frac {r_{o1}}{r_{i}}}}
=== Inductance of 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:
L
s
(
i
)
=
d
e
f
N
Φ
i
=
Λ
i
{\displaystyle L_{s}(i)\ {\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:
L
d
(
i
)
=
d
e
f
d
(
N
Φ
)
d
i
=
d
Λ
d
i
{\displaystyle L_{d}(i)\ {\overset {\underset
{\mathrm {def} }{}}{=}}\ {\frac {d(N\Phi )}{di}}={\frac
{d\Lambda }{di}}}
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.
v
(
t
)
=
d
Λ
d
t
=
d
Λ
d
i
d
i
d
t
=
L
d
(
i
)
d
i
d
t
{\displaystyle v(t)={\frac {d\Lambda }{dt}}={\frac
{d\Lambda }{di}}{\frac {di}{dt}}=L_{d}(i){\frac
{di}{dt}}}
Similar definitions may be derived for nonlinear
mutual inductance.
== Mutual inductance ==
=== 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 K wire loops, each with one
or several wire turns, the flux linkage of
loop m, λm, is given by
λ
m
=
N
m
Φ
m
=
∑
n
=
1
K
L
m
,
n
i
n
.
{\displaystyle \displaystyle \lambda _{m}=N_{m}\Phi
_{m}=\sum \limits _{n=1}^{K}L_{m,n}i_{n}.}
Here Nm denotes the number of turns in loop
m; Φm, the magnetic flux through loop m;
and Lm,n are some constants. 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
v
m
=
d
λ
m
d
t
=
N
m
d
Φ
m
d
t
=
∑
n
=
1
K
L
m
,
n
d
i
n
d
t
,
{\displaystyle \displaystyle v_{m}={\frac
{d\lambda _{m}}{dt}}=N_{m}{\frac {d\Phi _{m}}{dt}}=\sum
\limits _{n=1}^{K}L_{m,n}{\frac {di_{n}}{dt}},}
where vm denotes the voltage induced in circuit
m. This agrees with the definition of inductance
above if the coefficients Lm,n are identified
with the coefficients of inductance. Because
the total currents Nnin contribute to Φm
it also follows that Lm,n is proportional
to the product of turns NmNn.
=== 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,
∑
m
K
i
m
v
m
d
t
=
∑
m
,
n
=
1
K
i
m
L
m
,
n
d
i
n
=
!
∑
n
=
1
K
∂
W
(
i
)
∂
i
n
d
i
n
.
{\displaystyle \displaystyle \sum \limits
_{m}^{K}i_{m}v_{m}dt=\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}di_{n}{\overset
{!}{=}}\sum \limits _{n=1}^{K}{\frac {\partial
W\left(i\right)}{\partial i_{n}}}di_{n}.}
This must agree with the change of the magnetic
field energy, W, caused by the currents. The
integrability condition
∂
2
W
∂
i
m
∂
i
n
=
∂
2
W
∂
i
n
∂
i
m
{\displaystyle \displaystyle {\frac {\partial
^{2}W}{\partial i_{m}\partial i_{n}}}={\frac
{\partial ^{2}W}{\partial i_{n}\partial i_{m}}}}
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,
W
(
i
)
=
1
2
∑
m
,
n
=
1
K
i
m
L
m
,
n
i
n
.
{\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. 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, M, is also a measure
of the coupling between two inductors. The
mutual inductance by circuit i on circuit
j is given by the double integral Neumann
formula, see calculation techniques
The mutual inductance also has the relationship:
M
21
=
N
1
N
2
P
21
{\displaystyle M_{21}=N_{1}N_{2}P_{21}\!}
where
M
21
{\displaystyle M_{21}}
is the mutual inductance, and the subscript
specifies the relationship of the voltage
induced in coil 2 due to the current in coil
1.
N1 is the number of turns in coil 1,
N2 is the number of turns in coil 2,
P21 is the permeance of the space occupied
by the flux.Once the mutual inductance, M,
is determined, it can be used to predict the
behavior of a circuit:
v
1
=
L
1
d
i
1
d
t
−
M
d
i
2
d
t
{\displaystyle v_{1}=L_{1}{\frac {di_{1}}{dt}}-M{\frac
{di_{2}}{dt}}}
where
v1 is the voltage across the inductor of interest,
L1 is the inductance of the inductor of interest,
di1/dt is the derivative, with respect to
time, of the current through the inductor
of interest,
di2/dt is the derivative, with respect to
time, of the current through the inductor
that is coupled to the first inductor, and
M is the mutual inductance.The minus sign
arises because of the sense the current i2
has been defined in the diagram. With both
currents defined going into the dots the sign
of M will be positive (the equation would
read with a plus sign instead).
=== Coupling coefficient ===
The coupling coefficient is the ratio of the
open-circuit actual voltage ratio to the ratio
that would obtain if all the flux coupled
from one 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 2-port matrix the open-circuit voltage
ratio is found to be:
V
2
V
1
(
open circuit
)
=
M
L
1
{\displaystyle {V_{2} \over V_{1}}({\text{open
circuit}})={M \over L_{1}}}
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
V
2
V
1
(
max coupled
)
=
L
2
L
1
{\displaystyle {V_{2} \over V_{1}}({\text{max
coupled}})={\sqrt {L_{2} \over L_{1}}}}
thus,
M
=
k
L
1
L
2
{\displaystyle M=k{\sqrt {L_{1}L_{2}}}}
where
k is the coupling coefficient,
L1 is the inductance of the first coil, and
L2 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 0 ≤ k < 1,
but some define it as −1 < k < 1. Allowing
negative values of k captures phase inversions
of the coil connections and the direction
of the windings.
=== 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
[
z
]
=
s
[
L
1
M
M
L
2
]
{\displaystyle [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\
M\\M\ L_{2}\end{bmatrix}}}
where s is the complex frequency variable,
L1 and L2 are the inductances of the primary
and secondary coil, respectively, and M is
the mutual inductance between the coils.
=== 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.
This can be analyzed as a two port network.
With the output terminated with some arbitrary
impedance, Z, the voltage gain, Av, is given
by,
A
v
=
s
M
Z
s
2
L
1
L
2
−
s
2
M
2
+
s
L
1
Z
=
k
s
(
1
−
k
2
)
L
1
L
2
Z
+
L
1
L
2
{\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}}}}\,}}}
where k is the coupling constant and s is
the complex frequency variable, as above.
For tightly coupled inductors where k = 1
this reduces to
A
v
=
L
2
L
1
{\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,
Z
i
n
=
s
2
L
1
L
2
−
s
2
M
2
+
s
L
1
Z
s
L
2
+
Z
=
L
1
L
2
Z
(
1
1
+
(
Z
s
L
2
)
)
(
1
+
(
1
−
k
2
)
(
Z
s
L
2
)
)
{\displaystyle Z_{\mathrm {in} }={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}={\frac
{L_{1}}{L_{2}}}\,Z\,{\biggl (}{\frac {1}{1+\left({\frac
{Z}{\,sL_{2}\,}}\right)}}{\biggr )}{\Biggl
(}1+{\frac {\left(1-k^{2}\right)}{\left({\frac
{Z}{\,sL_{2}\,}}\right)}}{\Biggr )}}
For k = 1 this reduces to
Z
i
n
=
s
L
1
Z
s
L
2
+
Z
=
L
1
L
2
Z
(
1
1
+
(
Z
s
L
2
)
)
{\displaystyle Z_{\mathrm {in} }={\frac {sL_{1}Z}{sL_{2}+Z}}={\frac
{L_{1}}{L_{2}}}\,Z\,{\biggl (}{\frac {1}{1+\left({\frac
{Z}{\,sL_{2}\,}}\right)}}{\biggr )}}
Thus, the current gain, Ai is not independent
of load unless the further condition
|
s
L
2
|
≫
|
Z
|
{\displaystyle |sL_{2}|\gg |Z|}
is met, in which case,
Z
i
n
≈
L
1
L
2
Z
{\displaystyle Z_{\mathrm {in} }\approx {L_{1}
\over L_{2}}Z}
and
A
i
≈
L
1
L
2
=
1
A
v
{\displaystyle A_{\mathrm {i} }\approx {\sqrt
{L_{1} \over L_{2}}}={1 \over A_{\mathrm {v}
}}}
==== 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 to circuits
consisting of more than two coupled inductors.
Equivalent circuit elements Ls, Lp have physical
meaning, modelling respectively magnetic 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 H to simplify mathematical formulas.
Equivalent circuit element values can be calculated
from coupling coefficients with
L
S
i
j
=
det
(
K
)
−
C
i
j
{\displaystyle L_{S_{ij}}={\dfrac {\det(\mathbf
{K} )}{-\mathbf {C} _{ij}}}}
L
P
i
=
det
(
K
)
∑
j
=
1
N
C
i
j
{\displaystyle L_{P_{i}}={\dfrac {\det(\mathbf
{K} )}{\sum _{j=1}^{N}\mathbf {C} _{ij}}}}
where coupling coefficient matrix and its
cofactors are defined as
K
=
[
1
k
12
⋯
k
1
N
k
12
1
⋯
k
2
N
⋮
⋮
⋱
⋮
k
1
N
k
2
N
⋯
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
}
and
C
i
j
=
(
−
1
)
i
+
j
M
i
j
.
{\displaystyle \quad \mathbf {C} _{ij}=(-1)^{i+j}\,\mathbf
{M} _{ij}.}
For two coupled inductors, these formulas
simplify to
L
S
12
=
−
k
12
2
+
1
k
12
{\displaystyle L_{S_{12}}={\dfrac {-k_{12}^{2}+1}{k_{12}}}\quad
}
and
L
P
1
=
L
P
2
=
k
12
+
1
,
{\displaystyle \quad L_{P_{1}}=L_{P_{2}}\!=\!k_{12}+1,}
and for three coupled inductors (for brevity
shown only for Ls12 and Lp1)
L
S
12
=
2
k
12
k
13
k
23
−
k
12
2
−
k
13
2
−
k
23
2
+
1
k
13
k
23
−
k
12
{\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
}
and
L
P
1
=
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
.
{\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 ===
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 narrower 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 k. When two tuned
circuits are loosely coupled through mutual
inductance, the bandwidth will be 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.
=== Ideal transformers ===
When k = 1, 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:
V
s
=
N
s
N
p
V
p
{\displaystyle V_{\text{s}}={\frac {N_{\text{s}}}{N_{\text{p}}}}V_{\text{p}}}
where
Vs is the voltage across the secondary inductor,
Vp is the voltage across the primary inductor
(the one connected to a power source),
Ns is the number of turns in the secondary
inductor, and
Np is the number of turns in the primary inductor.Conversely
the current:
I
s
=
N
p
N
s
I
p
{\displaystyle I_{\text{s}}={\frac {N_{\text{p}}}{N_{\text{s}}}}I_{\text{p}}}
where
Is is the current through the secondary inductor,
Ip is the current through the primary inductor
(the one connected to a power source),
Ns is the number of turns in the secondary
inductor, and
Np 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
a
{\displaystyle {\boldsymbol {a}}}
is much smaller than the dimensions of the
shape, and if no ferromagnetic materials are
nearby (no magnetic core).
The symbol μ0 denotes the magnetic constant
(4π×10−7 H/m) in SI units.
Y
{\displaystyle Y}
is a constant between 0 and 1 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 (total skin effect),
Y
=
1
{\displaystyle Y=1}
when the current is homogeneous over the cross-section
of the wire (direct current).
O
(
x
)
{\displaystyle O(x)}
is represents small term(s) that have been
dropped from the formula, to make it simpler.
Read the symbol “
+
O
(
x
)
{\displaystyle +O(x)}
” as “plus small corrections on the order
of”
x
{\displaystyle x}
. See also Big O notation.
== See also ==
Electromagnetic induction
Gyrator
Hydraulic analogy
Leakage inductance
LC circuit, RLC circuit, RL circuit
Kinetic inductance
