Coil Inductance Formula

[zacky]

Coil Inductance formula is given by :

L= u.N^2.A /l

1- What is u
is it the core initial permeability or the average
permeability,
If it is the average , why the core manufacturer
gives the core initial permeability in his data
sheet ?

2-This formula is true only if there is no flux leakage
What formula can be used if there is flux leakage ?

 

[electricpete]

For #2 - yes - the formula covers only the idealized case of no flux leakage.

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[UKpete]

zacky, [μ] in the formula is the value of permeability at the working point as determined by the current. Because the core permeability varies with flux density, and flux density varies with coil current, the inductance varies as the coil current changes.

As you probably know, the magnetic characteristic for a material is usually given in the form:
B (flux density) versus H (magnetising force)

It is also possible to plot:
Relative permeability versus H (or B)
(this can be derived from the B-H plot as the permeability is the slope of the B-H curve at any point)
Note: relative permeability is the ratio of actual permeability to that of free space; hence it is dimensionless.
i.e. [μ]_r = [μ]/[μ]₀

Relative permeability for mild steel for example starts at about 2000 at zero field, increases with B to a peak of about 2700 at B=0.5T (5000G) then falls off as B is increased further and the material goes into saturation.

As all soft ferromagnetic materials are non-linear, the manufacturer can only really quote the initial or peak value of permeability, although they do publish characteristic curves. If you have a curve of permeability versus H, you can calculate the value of H for a particular current (H=NI/l where l is the length of the flux path through the core) then determine the value of permeability from the curve, use this in your formula to find the inductance at that value of current.

As for your question 2, EPete has already answered this. The formula assumes a closed path core i.e. no airgap. In this type of coil leakage generally only occurs when the core has become so heavily saturated that its permeability has become so low that flux hardly prefers it to the surrounding air, or because the flux path is relatively long (compared to the size of the coil) so that some flux is encouraged to "take a short-cut".
If you do have flux leakage the prediction is much more complex, and is usually solved by finite element analysis or by using empirical curves.

 

[zacky]

UKpete, as you indicated," u in the formula is the the value of permeability at the working point as determined by the current"
well, if the current is AC and the flux density changes from +B to -B, what would be the working point then ?
and how we can detrmine the permeability in this case from the B-H curve ?

 

[UKpete]

zacky, yes I see what you mean. In my post I was looking at the problem from a dc point of view. For ac the B-H curve tends to be a different shape anyway and is itself a function of frequency (can't explain why but published curves I have seen show it to be smaller than a dc hysteresis loop).

Because of the non-linearity of the characteristic, effectively the inductance is a function of instantaneous current i.e. it is varying through the ac cycle. So in answer to your original question, you would want an AVERAGE value of permeability. But this value would change if the amplitude or frequency of your ac (current) changes, so the manufacturer cannot be expected to quote an average value.

 

[zacky]

UKpete,thanks for your reply,
so now,if we know our current deviation (assume B goes up to a little bit before the knee-just before the saturation starts),then we know our maximum +B,-B
How then we can determine the average value of the permeabilty from the hysteresis loop - assume constant frequecny 60 Hz(or 50 Hz) and we have the hysteresis loop for that core material at that particular frequecy.

 

[UKpete]

zacky, if you restrict operation to the approximately linear part of the B-H characteristic, you would take the best-fit linear slope (delta B / delta H) of that part to give you the permeability. Yes there is hysteresis, but the demagnetization curve (going into the second quadrant) is the same slope.

What you have to bear in mind with inductance values are that they are approximate, and non-linear to a degree (unless you have an air-cored coil, or more specifically a coil without a ferromagnetic core). Of the three passive circuit element models required for mathematical analysis of electrical ciruits, namely resistance, capacitance and inductance, the latter are the most troublesome to realize physically with good accuracy and linearity. You can't buy 0.1% tolerance inductors like you can resistors.