How can the pulling force of an electromagnet be calculated? To help answer this question we'll consider a concrete example of an ordinary electromechanical relay; but keep in mind that the procedure outlined here is applicable generally to other problems involving linear motion caused by a current carrying coil.
Our objective is to find the mechanical force, f, exerted on the armature. Two principles form the basis of our approach. The first is the principle of virtual work. That is, we assume that the armature undergoes an infinitesimally small displacement to reduce the air gap. We then consider how the electrical energy supplied to the coil is distributed between the magnetic field and the relay spring using the second principle: energy balance. We represent this balance by an equation -
This is saying that (during movement of the armature) the electrical
power supplied at the coil terminals, W_{e}, is converted into
mechanical work, W_{m}, and an increase in the magnetic field
energy by an amount, W_{f}. We derive each term
separately.
Assume that -
Our assumption of zero ohmic losses in the coil will not invalidate the result because we can pretend that the armature displacement is sufficiently large or sufficiently quick that the inductive 'kick' from the coil will represent an indefinitely greater power. Faraday's Law says that the change in flux will give a constant voltage across the coil, V,
The electrical power input is V × I or
and since this power is applied for a time dt the total electrical
energy input is then
From the definition of inductance
In our case I and N are constant, so
Substituting into eqn. LMD -
As the gap is reduced the reluctance decreases. If the coil current is held constant then the flux will increase and so will the field energy. This is true even though there is a smaller volume for the field because the energy is proportional to B^{2}. The standard relationship between inductance and stored energy is
The coil current has not changed so the increase in field energy must
represent an increase in coil inductance, dL. So -
Where W_{f} is the increase in field energy (not its total).
The standard equation for mechanical work is simply
The form that this mechanical energy takes is immaterial. It could be
compression of a spring, movement against friction or an increase in
momentum. Since there isn't actually going to be any motion we don't
care; it's only virtual work that we consider. If it helps, you
can pretend that the iron has no mass or momentum; it won't alter the
validity of the result.
We can now substitute the results of equations LMG, LMI and LMJ into equation LMA
Which is easily solved for the force -
From this we draw the conclusions -
So, the mechanical work done is equivalent to the change in field energy, and the force is equal to the rate of change of the field energy with distance.
Now, although the resulting equation LML is very neat, it says nothing
about how the inductance of any particular device may be calculated.
Inductance (as well as field energy) is determined by the geometry
of the coil and all the iron structures within the field. Finding an
approximate expression for the inductance is sometimes possible in
simple cases such as the relay but next to impossible for others such as
lifting magnets.
One way round this problem might be to use field analysis software to integrate the field energy density throughout the whole volume and thus derive the inductance indirectly. There should be more on this topic within a year.
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E-mail:
R.Clarke@surrey.ac.uk
Last revised: 2006 January 21st.