How to calculate inductor flux density?

[reson8r]

There are two ways to approach calculating B: by inductor voltage E or by current I. The voltage formula is B = E / (k * Ae * N * f) and the current formula involves calculating H = (N * I) / (path length) and determining B from H * mu or a material hysteresis curve.

The problem is these often lead to very different results. For example, consider a simple L-C lowpass filter passing significant current well below its cutoff frequency. The voltage across the inductor in this case is very low which indicates low B from the first formula, but the current is high indicating high B from the second.

The two formulas depend on different physical inductor characteristics, too: Ae in one and path length in the other. Increasing path length, for example, doesn't reduce B in the first formula but does in the second.

Can anyone shed any light on this? Thanks in advance!

 

[reson8r]

And more puzzlement: increasing turns count N reduces B in the voltage formula but increases it in the current formula!

 

[LiteYear]

The problem is that those formulas have been developed with some assumptions that aren't necessarily compatible. As long as you maintain the assumptions, things work out. For example, you've assumed inductor voltage and current is sinusoidal and the peak values are E and I respectively. So for a fixed frequency, greater E means greater B from the 1st formula and greater H means greater I from the 2nd formula, which is as you would expect for an inductor at a fixed frequency. The problem you have is that your relative measurements of "low" and "high" are comparing different things. It's the low 'f' that bumps up B in the 1st formula, giving you the high I in the 2nd.

On the physical parameters, the "path length" in the secondary formula is very specific and probably misleadingly labelled. The field strength right at the centre of a loop of conductor carrying current I, with loop diameter l, is I/l. If you imagine there are several loops right on top of each other, it will be N*I/l. Notice how "path length" is actually related to Ae (not to the length of the inductor), so changing one changes the other.

Finally, increasing N in the first formula also changes E, because E is proportional to N^2. So you'll see by cancelling the N's that both formulas are proportional to N.

 

[reson8r]

Yes, I'm assuming sinusoidal E and I. I'm not sure I follow the rest of your first paragraph. E and I are the voltage and current in the inductor at a given frequency and operating condition, and calculating B from one or the other gives different results.

H in the middle of a loop is indeed I/l as you've defined l, but the path length l in N*I/l is measured along the axis of the stacked loops of your example. In a toroid path length l is the mean circumference, and in a solenoid it's the length of the solenoid. So l is perpendicular to the cross sectional area Ae.

Changing N doesn't change E. E is the voltage applied across the inductor of turns N.

 

[reson8r]

Well, I'll be darned! I set up a hypothetical inductor in MathCad and calculated B both ways, and it comes out the same when I vary E, N, or f. In this example I'm applying E across the inductor and calculating I, but it would work the other way around too. There is a slight numerical discrepancy between the two results, but that's due to redundancy in the core constants and isn't important.

LiteYear, I think this is what you were saying. I haven't worked out the whole linkage between the two formulas, but I bet it depends on the definitions of the core constants Ae, AL, l, and mu.

 

[LiteYear]

Oh, I see what you mean about the path length now - you're considering the infinite length solenoid approximation. Okay, different explanation then:

Increasing path length, for example, doesn't reduce B in the first formula but does in the second.

That's because the formulae assume constant reluctance, R.

R = path length /(u * Ae)

or

path length = R * u * Ae

so to maintain constant reluctance, Ae goes as path length goes.

Changing N doesn't change E. E is the voltage applied across the inductor of turns N.

You've got the cause and effect back to front. Voltage only forms across an inductor due to the change in flux. If you increase the number of turns and want to keep the voltage the same, you'll have to lower the current. If you keep the same current, voltage will go up, in proportion to N^2.

I set up a hypothetical inductor in MathCad and calculated B both ways, and it comes out the same when I vary E, N, or f.

That might be a bit less surprising than you think. Because you're using the manufacturer's inductance factor (A_L), you're effectively cancelling the other constants. Your two B formulas are identical, provided (A_L * I) / (u_i * A_e) is 1. It's actually 0.97, hence the small difference in result.

But yes, I guess the equivalence of the formulae was what I was trying to show in the first place...

 

[reson8r]

You've got the cause and effect back to front. Voltage only forms across an inductor due to the change in flux.

I don't quite agree with this. Faraday's Law describes a relationship between voltage and flux; it doesn't decree which comes first. If you apply a voltage across an inductor a flux will result, and vice versa.

Anyway, I ground through the algebra and satisfied myself that the two expressions for flux density are the same. I wrote it up here. I used Faraday's Law to derive B as a function of E and the relationship of B = mu * H for B as a function of I, but you can also derive them both from Faraday's Law.

I got hung up on current transformers for a while till I realized that very little of the primary current actually goes to creating flux in the core. Most of it is transformed to the secondary. This is true of transformers in general, actually, but I didn't think of current transformers that way.

 

[LiteYear]

You've got the cause and effect back to front. Voltage only forms across an inductor due to the change in flux.

I don't quite agree with this. Faraday's Law describes a relationship between voltage and flux; it doesn't decree which comes first. If you apply a voltage across an inductor a flux will result, and vice versa.

Quite right. I was specifically addressing your assertion that "Changing N doesn't change E". I meant to point out that changing N does indeed change E, if everything else stays the same. The E is a result of the inductor's properties and the change in flux. Or equivalently, as you point out, the change in flux is a result of the applied E and the inductor's properties.

Thanks for publishing your results. Very useful to see your treatment of the science.

 

[reson8r]

Yes, it all depends on what you're driving. I asserted that "changing N doesn't change E" because I was assuming E was the fixed source. In that case, changing N changes B instead.

Thanks for your help & insight!