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PowerPedia:Inductance circuit

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Table of contents

1 Coupled inductors

2 Vector field theory

2.1 Mutual inductance
2.2 Self-inductance

3 Inductance for any shaped loop

3.1 Phasor circuit analysis and impedance

4 Inductance of a circular loop

5 Inductance of a solenoid

6 Related

7 References

8 See also

Inductance circuit (eg., L circuit) is an electrical network inwhich the electrical elements is primarily composed of inductors. They can both be basically simplified into power source(s) and wire(s). An electrical inductance circuit is a network that has a closed loop and, via an amount of magnetic flux surrounding the circuit, produces a given electric current. The amount of magnetic flux produced by a current depends upon the permeability of the medium surrounded by the current, the area inside the coil, and the number of turns. The greater the permeability, the greater the magnetic flux generated by a given current. Certain (ferromagnetic) materials have much higher permeability than air. If a conductor (wire) is wound around such a material, the magnetic flux becomes much greater and the inductance becomes much greater than the inductance of an identical coil wound in air.

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Coupled inductors

When the magnetic flux produced by an inductor links another inductor, these inductors are said to be coupled. Coupling is often undesired but in many cases, this coupling is intentional and is the basis of the transformer. When inductors are coupled, there exists a mutual inductance that relates the current in one inductor to the flux linkage in the other inductor. Thus, there are three inductances defined for coupled inductors:

L 11

- the self inductance of inductor 1

L 22

- the self inductance of inductor 2

L 12 = L 21

- the mutual inductance associated with both inductors

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.

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Vector field theory

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Mutual inductance

Mutual inductance is the concept that the current through 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

$M_{ij} = \frac{\mu_0}{4\pi} \oint_{C_i}\oint_{C_j} \frac{{ds}_i\cdot{ds}_j}{|{R}_{ij}|}$

See a derivation of this equation.

The mutual inductance also has the relationship:

$M_{21} = N_1 N_2 P_{21} \!$

where

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.

N 1

is the number of turns in coil 1,

N 2

is the number of turns in coil 2,

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:

$M = k \sqrt{L_1 L_2} \!$

where

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

L 1

is the inductance of the first coil, and

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:

$V = L_1 \frac{dI_1}{dt} + M \frac{dI_2}{dt}$

where

V is the voltage across the inductor of interest,

L 1

is the inductance of the inductor of interest,

d I 1 / d t

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

M

is the mutual inductance and

d I 2 / d t

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:

$V_s = V_p \frac{N_s}{N_p}$

where

V s

is the voltage across the secondary inductor,

V p

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

N s

is the number of turns in the secondary inductor, and

N p

is the number of turns in the primary inductor.

Conversely the current:

$I_s = I_p \frac{N_p}{N_s}$

where

I s

is the current through the secondary inductor,

I p

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

N s

is the number of turns in the secondary inductor, and

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).

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Self-inductance

Self-inductance, denoted L, is the usual inductance one talks about with an inductor. It is a special case of mutual inductance where, in the above equation, i =j. Thus,

$M_{ij} = M_{jj} = L_{jj} = L_j = L = \frac{\mu_0}{4\pi} \oint_{C}\oint_{C'} \frac{{ds}\cdot{ds}'}{|{R}|}$

Physically, the self-inductance of a circuit represents the back-emf described by Faraday's law of induction.

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Inductance for any shaped loop

Consider a current loop $δ S$ with current i(t). According to Biot-Savart law, current i(t) sets up a magnetic flux density at r:

${B}({r},t)= \frac{\mu_{0}\mu_{r} i(t)}{4\pi} \int_{\delta S}{\frac{d{l} \times {\hat r}}{r^2}}$

Now magnetic flux through the surface S the loop encircles is:

$\Phi(t) = \int_S{B}({r},t) \cdot d{A} = \frac{\mu_{0}\mu_{r} i(t)}{4\pi} \int_S \int_{\delta S}{\frac{d{l} \times {\hat r}}{r^2}} \cdot d{A} = Li(t)$

From where we get the expression for inductance of the current loop:

$L = \frac{\mu_{0}\mu_{r} }{4\pi} \int_S \int_{\delta S}{\frac{d{l} \times {\hat r}}{r^2}} \cdot d{A}$

where

μ₀ and μ_r are the same as above

$d l$ is the differential length vector of the current loop element

${\hat r}$ is the unit displacement vector from the current element to the field point r

$r$ is the distance from the current element to the field point r

$d A$ differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S

As we see here, the geometry and material properties (if material properties are same in surface $S$ and the material is linear) of the current loop can be expressed with single scalar quantity L.

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Phasor circuit analysis and impedance

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

$Z_L = V / I = j L \omega \,$

where

$X_L = L \omega \,$ is the inductive reactance,

$\omega = 2 \pi f \,$ is the angular frequency,

L is the inductance,

f is the frequency, and

j is the imaginary unit.

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Inductance of a circular loop

The inductance of a circular conductive loop made of a circular conductor can be determined using

$L = {r \mu_0 \mu_r \left( \ln{ \frac {8 r}{a}} - 2 \right) }$

where

μ₀ and μ_r are the same as above

r is the radius of the loop

a is the radius of the conductor

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Inductance of a solenoid

The self-inductance L of such a solenoid can be calculated from:

$L = {\mu_0 \mu_r N^2 A \over l} = \frac{N \Phi}{i}$

where

μ₀ is the permeability of free space (4π × 10^-7 henrys per metre)

μ_r is the relative permeability of the core (dimensionless)

N is the number of turns.

A is the cross sectional area of the coil in square metres.

l is the length of the coil in metres.

$Φ = B A$ is the flux in webers (B is the flux density, A is the area).

i is the current in amperes

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. However, 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.

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Main: Electromagnetic induction, Inductor, Leakage inductance, Eddy current, Transformer
Circuits: RLC circuit, RL circuit, LC circuit C circuit, R circuit
Other: SI electromagnetism units, electricity, gyrator, Dot convention, alternating current

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References

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.
Wikipedia contributors (http://en.wikipedia.org/wiki/Special:Recentchanges), Wikipedia: The Free Encyclopedia. Wikimedia Foundation. <http://en.wikipedia.org>.

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PowerPedia:Inductance circuit

From PESWiki

Coupled inductors

Vector field theory

Mutual inductance

Self-inductance

Inductance for any shaped loop

Phasor circuit analysis and impedance

Inductance of a circular loop

Inductance of a solenoid

Related

References

See also

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