Electrical Model for Magnetic Circuits
At
Alevel, we have compared electric fields with gravity fields, as the
two are analogous. This tutorial shows how magnetic fields can be
modelled using electrical principles. A model
in this case is a way of explaining difficult concepts in simple terms
(and nothing to do with model trains or model boys and girls in the
fashion industry).
Although
magnetism has been a subject of study for many centuries, the
association with electricity is more recent. The detailed study of
magnetism is quite complex. Some physicists have used electrical
circuits as models for magnetic circuits. This can be useful.
Let's look at the two:
The electrical circuit has a source of voltage, called
the electromotive force and given the physics code
E (curly E).
The units are Volts (V). The magnetic circuit has a
source of magnetism, which is shown as the current carrying coil of
wire. The current I flows through the coil of N
turns. The result of this is magnetomotive force, F_{m},
which is the product of the current and number of turns:
F_{m} = NI
The units for magnetomotive force are amp turns (At).
Since turns is not a unit with dimensions, we can correctly describe the
unit as amps (A).
When a voltage is applied across a circuit, a current,
I amps flows. This is representative of the number of
electrons flowing every second.
When a magnetic field is applied to a magnetic circuit,
there is a magnetic flow, flux, of F
Weber (Wb). This is the number of field lines in the circuit.
In magnetic fields, the more commonly used quantity is
flux density, B Tesla (T. 1 Weber per square metre (Wb m^{2})
= 1 T).
This is equivalent of current density, J amp per
square metre, where:
Reluctance
With an electric circuit there is resistance, R
ohm (W) which is defined as the ratio
between the voltage and the current:
In the same way there is a magnetic resistance called
reluctance. It is defined as the ratio of magnetomotive
force to flux. It is given the physics code S and has
the units "per Henry" (H^{1}) or amp (turns) per Weber (A Wb^{1}).
We know that the magnetic field strength is given by:
We also know that:
F = BA
So we can
write:
We can rewrite this equation as:
We know that:
and we can write:
If there is a magnetic material of relative permeability
m_{r}, we know that:
and the final relationship becomes:
Question 1 
A toroidal (doughnut shaped)
piece of magnetic material has a relative permeability of 200.
It is 7.5 cm in radius and has a a circular cross section of
radius 1.0 cm. Calculate the reluctance. 

Question 2 
The coil providing the magnetic
field has 200 turns. The magnetic flux is 0.14 T.
(a) Calculate the magnetic
field strength.
(b) Calculate the current. 

We can
work out the reluctance if we know the area, the length, the number of
turns, and the current. All of these are easily measured. We
know that for a material of relative permeability
m_{r}:
Then we can work out the flux:
F = BA
From that
we can work out the reluctance:
Question 3 
A current of 2.0 A flows through
a coil that has 150 turns. The coil is made on a former of
square section 2.0 cm × 2.0 cm, and 10 cm long. In the
coil there is a material of relative permeability 150.
Calculate the reluctance. 

Ferromagnetic materials tend to have a low value of reluctance.
Reluctance has applications in transformers and some types of induction
motors.
Permeance
This is a
term that is used in magnetic circuits, and is analogous to
conductance. We know from basic electricity that
conductance, physics code G, is the reciprocal to resistance
and has the units Siemens (S) (or mho):
The
permeance, Physics code P, is the reciprocal to reluctance:
Question 4 
What is the permeance of the coil
in Question 3? 

Absolute Permeability
We have come across the permeability of free space
m_{0} and the relative
permeability, m_{r}.
The
product of these two quantities gives as the absolute permeability,
m. The units for
m are Henry per metre (H m^{1})
m =
m_{0}m_{r}
So we can rewrite our equation for reluctance as:
We have already compared magnetic reluctance with
electrical resistance. In electrical circuits the resistance
depends on:

The length of the conductor;

The area of the conductor;

The material of the conductor.
The material has a property of resistivity, which
is defined as the resistance of a conductor of unit crosssectional
area and unit length. The physics code is
r (rho, a Greek letter 'r') and the units are Ohm
metres. The formula for resistance in terms of resistivity is:
In the same way, we can say that the reluctance of a
magnetic material depends on:

The length;

The area;

The magnetic properties.
Both are directly proportional to the length and
inversely proportional to the area. The constant of
proportionality in resistance is r,
while the constant of proportionality in reluctance is 1/m.
The electrical analogy for absolute permeability,
m, is conductivity,
s.
