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?
And more puzzlement: increasing turns count N reduces B in the voltage formula but increases it in the current formula… Why?
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.
Above is a snippet.