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), in honor of Joseph Henry. The symbol L is used for inductance, possibly in honour of the physicist Heinrich Lenz.

The inductance has the following relationship:

L= \frac{\Phi}{i}

where

L is the inductance in henrys,
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{\lambda}{i} = N\frac{\Phi}{i}

where

\lambda is the total 'flux linkage'.

Inductance of a solenoid[edit]

The amount of magnetic flux produced by a current flowing through a coil 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 = {\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)
μr is the relative permeability of the core (unitless)
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 (NOT the wire) 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[edit]

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 + Y\right) }

where

μ0 and μr are the same as above
r is the radius of the loop
a is the radius of the conductor
Y is a constant. Y=0 when the current flows in the surface of the wire (skin effect), Y=1/4 when the current is homogeneous across the wire.

Inductance for any shaped loop[edit]

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

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

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

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

where

μ0 and μr are the same as above
d\mathbf{l} is the differential length vector of the current loop element
\mathbf{\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\mathbf{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.


Inductance of a coaxial line[edit]

Let the inner conductor have radius r_i and permeability \mu_i, let the dielectric between the inner and outer conductor have permeability \mu_d, and let the outer conductor have inner radius r_{o1}, outer radius r_{o2}, and permeability \mu_o. Assume that a DC current 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 r; it can be computed using Ampère's Law:

0 \leq r \leq r_i: B(r) = \frac{\mu_i I r}{2 \pi r_i^2}
r_i \leq r \leq r_{o1}: B(r) = \frac{\mu_d I}{2 \pi r}
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 l in the region between the conductors can be computed by drawing a surface with surface normal pointing axially:

\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, \frac{1}{2}LI^2, with the energy stored in the magnetic field:

\frac{1}{2}LI^2 = \int_V \frac{B^2}{2\mu} dV

For a cylindrical geometry with no l dependence, the energy per unit length is

\frac{1}{2}L'I^2 = \int_{r_1}^{r_2} \frac{B^2}{2\mu} 2 \pi r~dr

where L' is the inductance per unit length. For the inner conductor, the integral on the right-hand-side is \frac{\mu_i I^2}{16 \pi}; for the outer conductor it is \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}{16 \pi}

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

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}{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

\frac{dL}{dl} \approx \frac{\mu_d}{2 \pi} \ln\frac{r_{o1}}{r_i}

Properties of inductance[edit]

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{d\lambda}{dt} = L \frac{di}{dt} + i \frac{dL}{dt} \,

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

\frac{d\lambda}{dt} = L \frac{di}{dt}

By Faraday's Law of Induction we have:

\frac{d\lambda}{dt} = -\mathcal{E} = v

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

\frac{di}{dt} = \frac{v}{L}

or

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.

Phasor circuit analysis and impedance[edit]

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

X_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[edit]

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.

Vector field theory derivations[edit]

Mutual inductance[edit]

The circuit diagram representation of mutually inducting 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 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{\mathbf{ds}_i\cdot\mathbf{ds}_j}{|\mathbf{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,
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{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).

Self-inductance[edit]

Self-inductance, denoted L, is the usual inductance one talks about with an inductor. Formally the self-inductance of a wire loop would be given by the above equation with i =j. However, 1/|\mathbf{R}| now gets singular and the finite radius 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 |\mathbf{R}| \ge a/2 and a correction term,

 M_{ij} = M_{jj} = L_{jj} = L_j = 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 a and l denote radius and length of the wire, and Y is a constant that depends on the distribution of the current in the wire: Y=0 when the current flows in the surface of the wire (skin effect), Y=1/4 when the current is homogenuous across the wire.

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

Usage[edit]

The flux \Phi_i\ \! through the i-th circuit in a set is given by:

 \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, \mathcal{E}, of a specific circuit, i, in any given set can be given directly by:

 \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)

See also[edit]

References[edit]

  • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X. 
  • Wangsness, Roald K. (1986). Electromagnetic Fields (2nd ed. 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.