Inductance

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An electric current Failed to parse (Missing texvc executable; please see math/README to configure.): i

flowing around a circuit produces a magnetic field and hence a magnetic flux Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi
through the circuit. The ratio of the magnetic flux to the current is called the inductance, or more accurately self-inductance of the circuit.  The term was coined by Oliver Heaviside in February 1886[citation needed]. It is customary to use the symbol Failed to parse (Missing texvc executable; please see math/README to configure.): L
for inductance, possibly in honour of the physicist Heinrich Lenz. The quantitative definition of the inductance in SI units (webers per ampere) is

Failed to parse (Missing texvc executable; please see math/README to configure.): L= \frac{\Phi}{i}.

In honour of Joseph Henry, the unit of inductance has been given the name henry (H): 1H = 1Wb/A.

In the above definition, the magnetic flux Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi

is that caused by the current flowing through the circuit concerned. There may, however, be contributions from other circuits. Consider for example two circuits Failed to parse (Missing texvc executable; please see math/README to configure.): C_1

, Failed to parse (Missing texvc executable; please see math/README to configure.): C_2 , carrying the currents Failed to parse (Missing texvc executable; please see math/README to configure.): i_1 , Failed to parse (Missing texvc executable; please see math/README to configure.): i_2 . The magnetic fluxes Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_2
in Failed to parse (Missing texvc executable; please see math/README to configure.): C_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): C_2

, respectively, are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \Phi_1 = L_{11}i_1 + L_{12}i_2,

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \Phi_2 = L_{21}i_1 + L_{22}i_2.

According to the above definition, Failed to parse (Missing texvc executable; please see math/README to configure.): L_{11}

and Failed to parse (Missing texvc executable; please see math/README to configure.): L_{22}
are the self-inductances of Failed to parse (Missing texvc executable; please see math/README to configure.): C_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): C_2

, respectively. It can be shown (see below) that the other two coefficients are equal: Failed to parse (Missing texvc executable; please see math/README to configure.): L_{12} = L_{21} = M , where Failed to parse (Missing texvc executable; please see math/README to configure.): M

is called the mutual inductance of the pair of circuits.

Self and mutual inductances also occur in the expression

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle W=\frac{1}{2}\sum_{m,n=1}^{N}L_{m,n}i_{m}i_{n}

for the energy of the magnetic field generated by Failed to parse (Missing texvc executable; please see math/README to configure.): N

electrical circuits carrying the currents Failed to parse (Missing texvc executable; please see math/README to configure.): i_n

. This equation is an alternative definition of inductance, also valid when the currents don't flow in thin wires and when it thus is not immediately clear what the area encompassed by a circuit is and how the magnetic flux through the circuit is to be defined. The definition Failed to parse (Missing texvc executable; please see math/README to configure.): L= \Phi / i , in contrast, is more direct and more intuitive. It may be shown that the two definitions are equivalent by equating the time derivate of W and the electric power transfered to the system (see below).

Properties of inductance

The equation relating inductance and flux linkages can be rearranged as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi = Li \,

Taking the time derivative of both sides of the equation yields:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d\Phi}{dt} = L \frac{di}{dt} + i \frac{dL}{dt} \,

In most physical cases, the inductance is constant with time and so

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d\Phi}{dt} = L \frac{di}{dt}

By Faraday's Law of Induction we have:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d\Phi}{dt} = -\mathcal{E} = v

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E}

is the Electromotive force (emf) and Failed to parse (Missing texvc executable; please see math/README to configure.): v
is the induced voltage.  Note that the emf is opposite to the induced voltage.  Thus:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{di}{dt} = \frac{v}{L}

or

Failed to parse (Missing texvc executable; please see math/README to configure.): i(t) = \frac{1}{L} \int_0^tv(\tau) d\tau + i(0)

These equations together state that, for a steady applied voltage v, the current changes in a linear manner, at a rate proportional to the applied voltage, but inversely proportional to the inductance. Conversely, if the current through the inductor is changing at a constant rate, the induced voltage is constant.

The effect of inductance can be understood using a single loop of wire as an example. If a voltage is suddenly applied between the ends of the loop of wire, the current must change from zero to non-zero. However, a non-zero current induces a magnetic field by Ampère's law. This change in the magnetic field induces an emf that is in the opposite direction of the change in current. The strength of this emf is proportional to the change in current and the inductance. When these opposing forces are in balance, the result is a current that increases linearly with time where the rate of this change is determined by the applied voltage and the inductance.

Multiplying the equation for Failed to parse (Missing texvc executable; please see math/README to configure.): di/dt

above with Failed to parse (Missing texvc executable; please see math/README to configure.): Li
leads to

Failed to parse (Missing texvc executable; please see math/README to configure.): Li\frac{di}{dt}=\frac{d}{dt}\frac{L}{2}i^{2}=iv

Since iv is the energy transfered to the system per time it follows that Failed to parse (Missing texvc executable; please see math/README to configure.): \left( L/2 \right)i^2

is the energy of the magnetic field generated by the current.

Phasor circuit analysis and impedance

Using phasors, the equivalent impedance of an inductance is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): Z_L = V / I = j L \omega \,

where

Failed to parse (Missing texvc executable; please see math/README to configure.): X_L = L \omega \,

is the inductive reactance,

Failed to parse (Missing texvc executable; please see math/README to configure.): \omega = 2 \pi f \,

is the angular frequency,

L is the inductance,

f is the frequency, and

j is the imaginary unit.

Induced emf

The flux Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_i\ \!

through the i-th circuit in a set is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi_i = \sum_{j} M_{ij}I_j = L_i I_i + \sum_{j\ne i} M_{ij}I_j \,

so that the induced emf, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E} , of a specific circuit, i, in any given set can be given directly by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{E} = -\frac{d\Phi_i}{dt} = -\frac{d}{dt} \left (L_i I_i + \sum_{j\ne i} M_{ij}I_j \right ) = -\left(\frac{dL_i}{dt}I_i +\frac{dI_i}{dt}L_i \right) -\sum_{j\ne i} \left (\frac{dM_{ij}}{dt}I_j + \frac{dI_j}{dt}M_{ij} \right).

Coupled inductors

Further information: coupling (electronics)

Mutual inductance is the concept that the change in current in one inductor can induce 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): M_{21} = N_1 N_2 P_{21} \!

where

Failed to parse (Missing texvc executable; please see math/README to configure.): M_{21}

is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 to the current in coil 1.

Failed to parse (Missing texvc executable; please see math/README to configure.): N_1

is the number of turns in coil 1,

Failed to parse (Missing texvc executable; please see math/README to configure.): N_2

is the number of turns in coil 2,

Failed to parse (Missing texvc executable; please see math/README to configure.): P_{21}

is the permeance of the space occupied by the flux.

The mutual inductance also has a relationship with the coefficient of coupling. The coefficient of coupling is always between 1 and 0, and is a convenient way to specify the relationship between a certain orientation of inductor with arbitrary inductance:

Failed to parse (Missing texvc executable; please see math/README to configure.): M = k \sqrt{L_1 L_2} \!

where

k is the coefficient of coupling and 0 ≤ k ≤ 1,

Failed to parse (Missing texvc executable; please see math/README to configure.): L_1

is the inductance of the first coil, and

Failed to parse (Missing texvc executable; please see math/README to configure.): L_2

is the inductance of the second coil.

Once this mutual inductance factor M is determined, it can be used to predict the behavior of a circuit:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_1 = L_1 \frac{dI_1}{dt} + M \frac{dI_2}{dt}

where

V is the voltage across the inductor of interest,

Failed to parse (Missing texvc executable; please see math/README to configure.): L_1

is the inductance of the inductor of interest,

Failed to parse (Missing texvc executable; please see math/README to configure.): dI_1 / dt

is the derivative, with respect to time, of the current through the inductor of interest,

Failed to parse (Missing texvc executable; please see math/README to configure.): M

is the mutual inductance and 

Failed to parse (Missing texvc executable; please see math/README to configure.): dI_2 / dt

is the derivative, with respect to time, of the current through the inductor that is coupled to the first inductor.

When one inductor is closely coupled to another inductor through mutual inductance, such as in a transformer, the voltages, currents, and number of turns can be related in the following way:

Failed to parse (Missing texvc executable; please see math/README to configure.): V_s = V_p \frac{N_s}{N_p}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): V_s

is the voltage across the secondary inductor,

Failed to parse (Missing texvc executable; please see math/README to configure.): V_p

is the voltage across the primary inductor (the one connected to a power source),

Failed to parse (Missing texvc executable; please see math/README to configure.): N_s

is the number of turns in the secondary inductor, and

Failed to parse (Missing texvc executable; please see math/README to configure.): N_p

is the number of turns in the primary inductor.

Conversely the current:

Failed to parse (Missing texvc executable; please see math/README to configure.): I_s = I_p \frac{N_p}{N_s}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): I_s

is the current through the secondary inductor,

Failed to parse (Missing texvc executable; please see math/README to configure.): I_p

is the current through the primary inductor (the one connected to a power source),

Failed to parse (Missing texvc executable; please see math/README to configure.): N_s

is the number of turns in the secondary inductor, and

Failed to parse (Missing texvc executable; please see math/README to configure.): N_p

is the number of turns in the primary inductor.

Note that the power through one inductor is the same as the power through the other. Also note that these equations don't work if both transformers are forced (with power sources).

When either side of the transformer is a tuned circuit, the amount of mutual inductance between the two windings determines the shape of the frequency response curve. Although no boundaries are defined, this is often referred to as loose-, critical-, and over-coupling. 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 a critical point, the peak in the response curve begins to drop, and the center frequency will be attenuated more strongly than its direct sidebands. This is known as overcoupling.

Calculation techniques

Mutual inductance

The mutual inductance by circuit i on circuit j is given by the double integral Neumann formula

Failed to parse (Missing texvc executable; please see math/README to configure.): M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{R}_{ij}|}

The constant Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0

is the permeability of free space (4Failed to parse (Missing texvc executable; please see math/README to configure.): \pi
× 10-7 H/m), Failed to parse (Missing texvc executable; please see math/README to configure.): C_i
and Failed to parse (Missing texvc executable; please see math/README to configure.): C_j
are the curves spanned by the wires, Failed to parse (Missing texvc executable; please see math/README to configure.): R_{ij}
is the distance between two points. See a derivation of this equation.

Self-inductance

Formally the self-inductance of a wire loop would be given by the above equation with i =j. However, Failed to parse (Missing texvc executable; please see math/README to configure.): 1/R

now gets singular and the finite radius Failed to parse (Missing texvc executable; please see math/README to configure.): a
and the

distribution of the current in the wire must be taken into account. There remain the contribution from the integral over all points where Failed to parse (Missing texvc executable; please see math/README to configure.): |R| \ge a/2

and a correction term,

Failed to parse (Missing texvc executable; please see math/README to configure.): L_{jj} = L = \left (\frac{\mu_0}{4\pi} \oint_{C}\oint_{C'} \frac{\mathbf{ds}\cdot\mathbf{ds}'}{|\mathbf{R}|}\right )_{|\mathbf{R}| \ge a/2} + \frac{\mu_0}{2\pi}lY

Here Failed to parse (Missing texvc executable; please see math/README to configure.): a

and Failed to parse (Missing texvc executable; please see math/README to configure.): l
denote radius and length of the wire, and Failed to parse (Missing texvc executable; please see math/README to configure.): Y
is a constant that depends on the

distribution of the current in the wire: Failed to parse (Missing texvc executable; please see math/README to configure.): Y=0

when the current flows in the surface of the wire

(skin effect), Failed to parse (Missing texvc executable; please see math/README to configure.): Y=1/4

when the current is homogenuous across the wire. Here is a derivation of this equation.

Inductance of simple electrical circuits

The self-inductance of many types of electrical circuits can be given in closed form. Examples are listed in the table.

The constant Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0

is the permeability of free space (4Failed to parse (Missing texvc executable; please see math/README to configure.): \pi
× 10-7 H/m).

For high frequencies the electrical current flows in the conductor surface (skin effect), and depending on the geometry it sometimes is necessary to distinguish low and high frequency inductances. This is the purpose of the constant Y: Y=0 when the current is uniformly distributed over the surface of the wire (skin effect), Y=1/4 when the current is uniformly distributed over the cross section of the wire. If conductors approach each other then in the high frequency case an additional screening current flows in their surface and the expressions containing Y get invalid.

Inductance of a solenoid

A solenoid is a long, thin coil, i.e. a coil whose length is much greater than the diameter. Under these conditions, and without any magnetic material used, the magnetic flux density Failed to parse (Missing texvc executable; please see math/README to configure.): B

within the coil is practically constant and is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle B = \mu_0 Ni/l

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_0

is the permeability of free space, Failed to parse (Missing texvc executable; please see math/README to configure.): N
the number of turns, Failed to parse (Missing texvc executable; please see math/README to configure.): i
the current and Failed to parse (Missing texvc executable; please see math/README to configure.): l
the length of the coil. Ignoring end effects the magnetic flux through the coil is obtained by multiplying the flux density Failed to parse (Missing texvc executable; please see math/README to configure.): B
by the cross-section area Failed to parse (Missing texvc executable; please see math/README to configure.): A
and the number of turns Failed to parse (Missing texvc executable; please see math/README to configure.): N

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \Phi = \mu_0N^2iA/l,

from which it follows that the inductance of a solenoid is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle L = \mu_0N^2A/l.

This, and the inductance of more complicated shapes, can be derived from Maxwell's equations. For rigid air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current.

Similar analysis applies to a solenoid with a magnetic core, but only if the length of the coil is much greater than the product of the relative permeability of the magnetic core and the diameter. That limits the simple analysis to low-permeability cores, or extremely long thin solenoids. Although rarely useful, the equations are,

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle B = \mu_0\mu_r Ni/l

where Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_r

the relative permeability of the material within the solenoid,

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \Phi = \mu_0\mu_rN^2iA/l,

from which it follows that the inductance of a solenoid is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle L = \mu_0\mu_rN^2A/l.

Note that since the permeability of ferromagnetic materials changes with applied magnetic flux, the inductance of a coil with a ferromagnetic core will generally vary with current.

Inductance of a coaxial line

Let the inner conductor have radius Failed to parse (Missing texvc executable; please see math/README to configure.): r_i

and permeability Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_i

, let the dielectric between the inner and outer conductor have permeability Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_d , and let the outer conductor have inner radius Failed to parse (Missing texvc executable; please see math/README to configure.): r_{o1} , outer radius Failed to parse (Missing texvc executable; please see math/README to configure.): r_{o2} , and permeability Failed to parse (Missing texvc executable; please see math/README to configure.): \mu_o . Assume that a DC current Failed to parse (Missing texvc executable; please see math/README to configure.): I

flows in opposite directions in the two conductors, with uniform current density. The magnetic field generated by these currents points in the axial direction and is a function of radius Failed to parse (Missing texvc executable; please see math/README to configure.): r

it can be computed using Ampère's Law

Failed to parse (Missing texvc executable; please see math/README to configure.): 0 \leq r \leq r_i: B(r) = \frac{\mu_i I r}{2 \pi r_i^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): r_i \leq r \leq r_{o1}: B(r) = \frac{\mu_d I}{2 \pi r}

Failed to parse (Missing texvc executable; please see math/README to configure.): r_{o1} \leq r \leq r_{o2}: B(r) = \frac{\mu_o I}{2 \pi r} \left( \frac{r_{o2}^2 - r^2}{r_{o2}^2 - r_{o1}^2} \right)

The flux per unit length Failed to parse (Missing texvc executable; please see math/README to configure.): l

in the region between the conductors can be computed by drawing a surface with surface normal pointing axially:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d\phi_d}{dl} = \int_{r_i}^{r_{o1}} B(r) dr = \frac{\mu_d I}{2 \pi} \ln\frac{r_{o1}}{r_i}

Inside the conductors, L can be computed by equating the energy stored in an inductor, Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2}LI^2 , with the energy stored in the magnetic field:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2}LI^2 = \int_V \frac{B^2}{2\mu} dV

For a cylindrical geometry with no Failed to parse (Missing texvc executable; please see math/README to configure.): l

dependence, the energy per unit length is

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2}L'I^2 = \int_{r_1}^{r_2} \frac{B^2}{2\mu} 2 \pi r~dr

where Failed to parse (Missing texvc executable; please see math/README to configure.): L'

is the inductance per unit length. For the inner conductor, the integral on the right-hand-side is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mu_i I^2}{16 \pi}

for the outer conductor it is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mu_o I^2}{4 \pi} \left( \frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \right)^2 \ln\frac{r_{o2}}{r_{o1}} - \frac{\mu_o I^2}{8 \pi} \left( \frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \right) - \frac{\mu_o I^2}{16 \pi}

Solving for Failed to parse (Missing texvc executable; please see math/README to configure.): L'

and summing the terms for each region together gives a total inductance per unit length of:

Failed to parse (Missing texvc executable; please see math/README to configure.): L' = \frac{\mu_i}{8 \pi} + \frac{\mu_d}{2 \pi} \ln\frac{r_{o1}}{r_i} + \frac{\mu_o}{2 \pi} \left( \frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \right)^2 \ln\frac{r_{o2}}{r_{o1}} - \frac{\mu_o}{4 \pi} \left( \frac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} \right) - \frac{\mu_o}{8 \pi}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{dL}{dl} \approx \frac{\mu_d}{2 \pi} \ln\frac{r_{o1}}{r_i}

See also

References

• 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.af:Induktansie

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