Inductance

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Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. The term was coined by Oliver Heaviside in February 1886. The SI unit of inductance is the henry (symbol: H). The symbol L is used for inductance, in honour of the physicist Heinrich Lenz.

The inductance has the following relationship:

[L= \frac]

::where

:::L is the inductance in henries,

:::i is the current in amperes,

:::Φ is the magnetic flux in webers

Strictly speaking, the quantity just defined is called self-inductance, because the magnetic field is created solely by the conductor that carries the current.

When a conductor is coiled upon itself N number of times around the same axis (forming a solenoid), the current required to produce a given amount of flux is reduced by a factor of N compared to a single turn of wire. Thus, the inductance of a coil of wire of N turns is given by:

:[L= \frac = N\frac]

where [\lambda] is the total 'flux linkage'.

Inductance of a solenoid

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

[ L = = \frac]

::where

:::μ₀ is the permeability of free space (4π × 10^-7 henries 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.

:::[\Phi = BA] 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.

Inductance of a circular loop

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

[ L = } - 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

Properties of inductance

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

[\lambda = Li \,]

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

[\frac = L \frac + i \frac \,]

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

[\frac = L \frac]

By Faraday's Law of Induction we have:

[\frac = -\mathcal = v ]

where [\mathcal] is the Electromotive force (emf) and [v] is the induced voltage. Note that the emf is opposite to the induced voltage. Thus:

[\frac = \frac]

or

[i(t) = \frac \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 proportionally 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 Ampere'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.

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.

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_] - the self inductance of inductor 1

[L_] - the self inductance of inductor 2

[L_ = L_] - the mutual inductance associated with both inductors

Vector field theory derivations

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_ = \frac \oint_\oint_ \frac_i\cdot\mathbf_j}> ]

See a derivation of this equation.

The mutual inductance also has the relationship:

[M_ = N_1 N_2 P_ \!]

::where

:::[M_] 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_] 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 \! ]

::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 + M \frac ]

::where

:::V is the voltage across the inductor of interest,

:::[L_1] is the inductance of the inductor of interest,

:::[dI_1 / dt] is the derivative, with respect to time, of the current through the inductor of interest,

:::[M] is the mutual inductance and

:::[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:

[V_s = V_p \frac ]

::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 ]

::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).

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_ = M_ = L_ = L_j = L = \frac \oint_\oint_ \frac\cdot\mathbf'}>]

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

Usage

The flux [\Phi_i\ \!] through the ith circuit in a set is given by:

[ \Phi_i = \sum_ M_I_j = L_i I_i + \sum_ M_I_j \,]

so that the induced emf, [\mathcal], of a specific circuit, i, in any given set can be given directly by:

[ E = -\frac = -\frac \left (L_i I_i + \sum_ M_I_j \right ) = -\left(\fracI_i +\fracL_i \right) -\sum_ \left (\frac}I_j + \fracM_ \right)]

See also

• Electromagnetic induction
• Inductor
• Dot convention
• alternating current
• electricity
• gyrator
• RLC circuit
• RL circuit
• LC circuit
• Leakage inductance
• SI electromagnetism units
• Eddy current
• Transformer

References

• Küpfmüller K., Einführung in die theoretische Elektrotechnik, Springer-Verlag, 1959.
• Heaviside O., Electrical Papers. Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429-560.

 

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