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Inductance
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An electric current i flowing around a circuit produces a magnetic field and hence a Magnetic Flux \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 . It is customary to use the symbol L for inductance, possibly in honour of the physicist Heinrich Lenz . The quantitative definition of the inductance is therefore
:L= rac{\Phi}{i}.

It follows that the SI units for inductance are Webers per Ampere . 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 \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 C_1, C_2, carrying the currents i_1, i_2. The magnetic fluxes \Phi_1 and \Phi_2 in C_1 and C_2, repectively, are given by
:\displaystyle \Phi_1 = L_{11}i_1 + L_{12}i_2,
:\displaystyle \Phi_2 = L_{21}i_1 + L_{22}i_2.
According to the above definition, L_{11} an L_{22} are the self-inductances of C_1 and C_2, repectively. It can be shown (see below) that the other two coefficients are equal: L_{12} = L_{21} = M, where M is called the mutual inductance of the pair or circuits.


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 B within the coil is practically constant and is given by
:\displaystyle B = \mu_0 Ni/l
where \mu_0 is the Permeability of free space (4π × 10-7 H/m), N the number of turns, i the current and l the length of the coil. Ignoring end effects the magnetic flux through the coil is obtained by multiplying the flux density B by the cross-section area A and the number of turns N:
:\displaystyle \Phi = \mu_0N^2iA/l,
from which it follows that the inductance of a solenoid is given by:
:\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,
:\displaystyle B = \mu_0\mu_r Ni/l
where \mu_r the relative permeability of the material within the solenoid,
:\displaystyle \Phi = \mu_0\mu_rN^2iA/l,
from which it follows that the inductance of a solenoid is given by:
:\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 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{ rac {8 r}{a}} - 2 + Y ight) }
where
μ

r

a

Y



INDUCTANCE FOR ANY SHAPED LOOP


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)= rac{\mu_{0}\mu_{r} i(t)}{4\pi} \int_{\delta S}{ rac{d\mathbf{l} imes \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} = rac{\mu_{0}\mu_{r} i(t)}{4\pi} \int_S \int_{\delta S}{ rac{d\mathbf{l} imes \mathbf{\hat r}}{r^2}} \cdot d\mathbf{A} = Li(t)

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

:L = rac{\mu_{0}\mu_{r} }{4\pi} \int_S \int_{\delta S}{ rac{d\mathbf{l} imes \mathbf{\hat r}}{r^2}} \cdot d\mathbf{A}
where
μ

d\mathbf{l}

\mathbf{\hat r}

r

d\mathbf{A}


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


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) = rac{\mu_i I r}{2 \pi r_i^2}
:r_i \leq r \leq r_{o1}: B(r) = rac{\mu_d I}{2 \pi r}
:r_{o1} \leq r \leq r_{o2}: B(r) = rac{\mu_o I}{2 \pi r} \left( rac{r_{o2}^2 - r^2}{r_{o2}^2 - r_{o1}^2} ight)

The flux per unit length l in the region between the conductors can be computed by drawing a surface with surface normal pointing axially:

: rac{d\phi_d}{dl} = \int_{r_i}^{r_{o1}} B(r) dr = rac{\mu_d I}{2 \pi} \ln rac{r_{o1}}{r_i}

Inside the conductors, L can be computed by equating the energy stored in an inductor, rac{1}{2}LI^2, with the energy stored in the magnetic field:

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

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

: rac{1}{2}L'I^2 = \int_{r_1}^{r_2} rac{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 rac{\mu_i I^2}{16 \pi}; for the outer conductor it is rac{\mu_o I^2}{4 \pi} \left( rac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} ight)^2 \ln rac{r_{o2}}{r_{o1}} - rac{\mu_o I^2}{8 \pi} \left( rac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} ight) - rac{\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' = rac{\mu_i}{8 \pi} + rac{\mu_d}{2 \pi} \ln rac{r_{o1}}{r_i} + rac{\mu_o}{2 \pi} \left( rac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} ight)^2 \ln rac{r_{o2}}{r_{o1}} - rac{\mu_o}{4 \pi} \left( rac{r_{o2}^2}{r_{o2}^2 - r_{o1}^2} ight) - rac{\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

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


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

In most physical cases, the inductance is constant with time and so
: rac{d\lambda}{dt} = L rac{di}{dt}

By Faraday's Law of Induction we have:
: rac{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:
: rac{di}{dt} = rac{v}{L}
or
:i(t) = rac{1}{L} \int_0^tv( au) d au + 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


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_{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


Mutual inductance










SEE ALSO



REFERENCES