Factors affecting inductance?

Initiative

There are four basic factors of inductor construction determining the amount of inductance created. These factors all dictate inductance by affecting how much magnetic field flux will develop for a given amount of magnetic field force (current through the inductor’s wire coil)

NUMBER OF WIRE WRAPS, OR “TURNS” IN THE COIL

 All other factors being equal, a greater number of turns of wire in the coil results in greater inductance; fewer turns of wire in the coil results in less inductance.

Explanation 

More turns of wire means that the coil will generate a greater amount of magnetic field force (measured in amp-turns!), for a given amount of coil current.

COIL AREA 

All other factors being equal, greater coil area (as measured looking lengthwise through the coil, at the cross-section of the core) results in greater inductance; less coil area results in less inductance.

Explanation

Greater coil area presents less opposition to the formation of magnetic field flux, for a given amount of field force (amp-turns).

COIL LENGTH 

All other factors being equal, the longer the coil’s length, the less inductance; the shorter the coil’s length, the greater the inductance.

Explanation

 A longer path for the magnetic field flux to take results in more opposition to the formation of that flux for any given amount of field force (amp-turns).

CORE MATERIAL

 All other factors being equal, the greater the magnetic permeability of the core which the coil is wrapped around, the greater the inductance; the less the permeability of the core, the less the inductance.

Explanation

A core material with greater magnetic permeability results in greater magnetic field flux for any given amount of field force (amp-turns).

An approximation of inductance for any coil of wire can be found with this formula:

L =  N^2 u A /I

u = ur*u0

Inductance

Scientists

In electromagnetism and electronics, inductance is the property of an electrical conductor by which a change in current through it induces an electromotive force in both the conductor itself and in any nearby conductors by mutual inductance.

These effects are derived from two fundamental observations of physics: a steady current creates a steady magnetic field described by Oersted's law and a time-varying magnetic field induces an electromotive force (EMF) in nearby conductors, which is described by Faraday's law of induction. According to Lenz's law,a changing electric current through a circuit that contains inductance induces a proportional voltage, which opposes the change in current (self-inductance). The varying field in this circuit may also induce an EMF in neighbouring circuits (mutual inductance).

The term inductance was coined by Oliver Heaviside in 1886. It is customary to use the symbol L for inductance, in honour of the physicist Heinrich Lenz. In the SI system, the measurement unit for inductance is the henry, with the unit symbol H, named in honor of Joseph Henry, who discovered inductance independently of, but not before, Faraday.

Circuit analysis

An electronic component that is intended to add inductance to a circuit is called an inductor. Inductors are typically manufactured from coils of wire. This design delivers two desired properties, a concentration of the magnetic field into a small physical space and a linking of the magnetic field into the circuit multiple times.

The relationship between the self-inductance, L, of an electrical circuit, the voltage, v(t), and the current, i(t), through the circuit is

{\displaystyle \displaystyle v(t)=L\,{\frac {di}{dt}}} {\displaystyle \displaystyle v(t)=L\,{\frac {di}{dt}}}.
A voltage is induced across an inductor (back EMF), that is equal to the product of the inductor's inductance and the rate of change of current through the inductor.

All circuits have, in practice, some inductance, which may have beneficial or detrimental effects. For a tuned circuit, inductance is used to provide a frequency-selective circuit. Practical inductors may be used to provide filtering, or energy storage, in a given network. The inductance per unit length of a transmission line is one of the properties that determines its characteristic impedance; balancing the inductance and capacitance of cables is important for distortion-free telegraphy and telephony. The inductance of long AC power transmission lines affects the power capacity of the line. Sensitive circuits, such as microphone and computer network cables, may utilize special cabling construction, limiting the inductive coupling between circuits.

The generalization to the case of K electrical circuits with currents, {im}, and voltages, {vm}, reads

{\displaystyle \displaystyle v_{m}=\sum \limits _{n=1}^{K}L_{m,n}\,{\frac {di_{n}}{dt}}.} {\displaystyle \displaystyle v_{m}=\sum \limits _{n=1}^{K}L_{m,n}\,{\frac {di_{n}}{dt}}.}
Here, inductance L is a symmetric matrix. The diagonal coefficients Lm,m are called coefficients of self-inductance, the off-diagonal elements are called coefficients of mutual inductance. The coefficients of inductance are constant, as long as no magnetizable material with nonlinear characteristics is involved. This is a direct consequence of the linearity of Maxwell's equations in the fields and the current density. The coefficients of inductance become functions of the currents in the nonlinear case.

Derivation from Faraday's law of inductance

The inductance equations above are a consequence of Maxwell's equations. There is a straightforward derivation in the important case of electrical circuits consisting of thin wires.

In a system of K wire loops, each with one or several wire turns, the flux linkage of loop m, λm, is given by

{\displaystyle \displaystyle \lambda _{m}=N_{m}\Phi _{m}=\sum \limits _{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

{\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}},} {\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.

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,

{\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}.} {\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.[9] The integrability condition

{\displaystyle \displaystyle {\frac {\partial ^{2}W}{\partial i_{m}\partial i_{n}}}={\frac {\partial ^{2}W}{\partial i_{n}\partial 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,

{\displaystyle \displaystyle W\left(i\right)={\frac {1}{2}}\sum \limits _{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).

Coupled inductors and mutual inductance

Further information: Coupling (electronics)

The circuit diagram representation of mutually coupled inductors. The two vertical lines between the inductors indicate a solid core that the wires of the inductor are wrapped around. "n:m" shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows the dot convention.
Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by 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:

{\displaystyle M_{21}=N_{1}N_{2}P_{21}\!} M_{21}=N_{1}N_{2}P_{21}\!
where

{\displaystyle M_{21}} 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:

{\displaystyle v_{1}=L_{1}{\frac {di_{1}}{dt}}-M{\frac {di_{2}}{dt}}} 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).

Matrix representation

The circuit can be described by any of the two-port network parameter matrix representations. The most direct are the z parameters, which are given by

{\displaystyle [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}} [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}
where s is the complex frequency variable.

Coupling coefficient[edit]
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:

{\displaystyle {V_{2} \over V_{1}}({\text{open circuit}})={M \over 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

{\displaystyle {V_{2} \over V_{1}}({\text{max coupled}})={\sqrt {L_{2} \over L_{1}}}} {\displaystyle {V_{2} \over V_{1}}({\text{max coupled}})={\sqrt {L_{2} \over L_{1}}}}
thus,

{\displaystyle M=k{\sqrt {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.

Equivalent circuit

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

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,

{\displaystyle A_{\mathrm {v} }={\frac {sMZ}{s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}}} A_{\mathrm {v} }={\frac {sMZ}{s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}}
For tightly coupled inductors where k = 1 this reduces to

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

{\displaystyle Z_{\mathrm {in} }={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}} Z_{\mathrm {in} }={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}
For k = 1 this reduces to

{\displaystyle Z_{\mathrm {in} }={\frac {sL_{1}Z}{sL_{2}+Z}}} Z_{\mathrm {in} }={\frac {sL_{1}Z}{sL_{2}+Z}}
Thus, the current gain, Ai is not independent of load unless the further condition

{\displaystyle |sL_{2}|\gg |Z|} |sL_{2}|\gg |Z|
is met, in which case,

{\displaystyle Z_{\mathrm {in} }\approx {L_{1} \over L_{2}}Z} Z_{\mathrm {in} }\approx {L_{1} \over L_{2}}Z
and

{\displaystyle A_{\mathrm {i} }\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\mathrm {v} }}} A_{\mathrm {i} }\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\mathrm {v} }}...Courtesy of wikipedia.....