Reluctance

The term

occurs frequently enough to be assigned the name of reluctance.
Reluctance,

It is important to recognize the relationship between the reluctance of a magnetic structure and its inductance.

Magnetic Circuits

The flux density B = phi / A
The field intensity H = B / mu = phi / (A mu)
The flux density for tightly wound circular coil,
B = mu N i / l = mu H
Therefore,
H = N i / l
H = F / l
Or
F = N i = H l
F = B l / mu
F = phi l / (mu A)
Re-group the above equation as follows:
{F} = {phi} {l/(mu A)}
Compare with the Ohm's law:
V = I R

Iron-core inductor and the toroid

The flux densities for the iron-core and the toroid inductors are
B = mu N i / l
and
B = mu N i / (2 pi r2)

Magnetic variables and units

Variable Symbol Units
Current I A
Magnetic flux density B Wb/m^2 = T
Magnetic flux phi Wb
Magnetic field intensity H A/m
Electomotive force e V
Magnetomotive force F A-turns
Flux linkage lambda Wb-turns

Ampere's Law

Ampere's law states that the integral of the vector magnetic field intensity H around a closed path is equal to the total current linked by the closed path i,
int{H . dl} = sum{i}

B ---- flux density
phi -- flux
B = phi/A

H ---- magnetic field intensity
B = mu H

The magnitude of the magnetic field intensity is given by
H = i / (2 pi r)

The magnetic field intensity is unaffected by the material surrounding the conductor, but the flux density depends on the material properties, since B = mu H. Thus, the density of flux lines around the conductor would be far greater in the presense of a magnetic material than if the conductor were surrounded by air.

The product N . i is a useful quantity in electromagnetic circuits and is called the magnetomotive force, F (mmf),
F = N i A-turns

Energy stored in a magentic system

In a magnetic system, the energy stored in the magnetic field is equal to the integral of the instantenous power, which is the product of voltage and current,
Wm = int{ei dt'}

However, the voltage corresponds to the induced emf, according to Faraday's law,
e = d lambda / dt = N d phi / dt
and is therefore
Wm = int{i d lambda}

Define a fictitious quantity called co-energy, W'm.
W'm = i lambda - Wm

Electromechanics

The relationship between force and magnetic field:
f = quB sin theta
The flux is given by
phi = int{B dA}
This can be approximated by
phi = B . A

Faraday's law states that a time-varying flux causes an induced electromotive force,
e = - d phi/dt

The polarity of the induced voltage can usually be determined from physical considerations; therefore the minus sign in the above equation is usually left out.

For an N-turn coil with cross-sectional area A, we have the emf
e = N d phi/dt

The flux linkage lambda is given by
lambda = N phi

So that,
e = d lambda/dt

In addition, flux linkage and current is a linear relationship,
lambda = L i

Therefore,
v = L d i /dt where L is the ideal self-inductance

In addition to self-inductance, it is important to consider the magnetic coupling.

The magnetic coupling between the coils established by virtue of the proximity is described by a quantity called mutual inductance and defined by the symbol M. The mutual inductance is defined by the equation
v2 = M d i1 / dt

In practical electromagnetic circuits, the self-inductance of a circuit is not necessarily constant; the inductance L is not constant, in general. It will not be possible to use such a simple relationship as v = L di/dt, with L constant.

If we revisit the definition of the transformer voltage e = N d phi/dt we see that in an inductor coil, the inductance is given by
L = N phi/i = lambda/i

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