I read in sci.electronics.design that John Popelish <
[email protected]>
I have seen the definition of inductance as flux times area per
current (1 henry = (1 tesla * meter^2)/amp)) but this gives my mind a
snag. Perhaps you can straighten me out.
The formula is L = N x [phi]/I, where N = number of turns, [phi] = flux
= flux density x area and I = current.
Lets say you have a large toroid core of infinite permeability with a
small air gap sawed through it, and 1 turn through the hole. I
measure the inductance and find x henries. So at 1 ampere, I have 1/2
x joules stored in the inductor and in the magnetic field in the air
volume of that gap. I then saw the gap to twice as wide and wind a
second turn through the hole, and get the same flux passing through
this thicker but same area air gap, and still measure x henries.
There are now two turns, so L = 2x. If you hadn't widened the gap, the
inductance would have been 4x.
So
by 1/2L*I^2 I still have the same energy stored in the inductor and in
the air volume in that gap, but there is now twice the original volume
of air stressed with the original flux. How can that be the same
total amount of magnetic energy?
It isn't. The inductance has doubled. From L = N[phi]/I, we get
L = NBA/I, B = induction, A = cross-sectional area
B = [mu]H, [mu] = permeability of air, H = magnetic field strength
and H = NI/l l = length of air-gap, since the rest of the circuit has
infinite permeability
So L = [mu]N^2IA/lI = [mu]N^2A/l
To get L back to its original value, with 2 turns instead of 1, we need
the gap to be 4 times longer.
If we then do a similar substitution for the stored energy:
LI^2/2 = (N[phi]I^2)/(2I) = N[phi]I/2 = N[mu]HAI/2 = N[mu](NI/l)AI/2
= [mu]/2 x N^2I^2A/l
N^2 has gone up 4 times and l has gone up 4 times, so the energy remains
the same. H, B and [phi] doubled due to the two turns, but dropped by a
factor of 4 due to the longer air-gap, giving a net halving.
