Where do µ’ and µ” come from?

Lets say we measure the impedance of a 1t wind on a FT240 size core to have Z=13.6+j19.1Ω @ 7.31MHz.

But it has a resistive component, it is not an ideal or lossless inductor.

Nevertheless, we can consider that Z=j*2*pi*f*L, and since Z is complex, a complex values is implied for L.

In the general sense, Z is a function of frequency, and so is L.

By rearranging we get L=Z/(j*2*pi*f) and we can calculate for the above example that L=0.429-j0.305µH.

Now, in terms of the physical parameters of the inductor, L=µ*N^2*(A/l) where:

  • µ is the permeability, 4*π*1e-7*µr;
  • N=number of turns; and
  • A/l is effective area divided by the effective length.

We can make µr the subject; µr=L/(4*π*1e-7*N^2*(A/l)) and solve. Solving, we get µr=313-j223.

This complex relative permeability is often expressed in the form of u’ as the real part , and the negative of the imaginary part as u”, µr=µ’-jµ”

In this case, u’=313, u”=223.

We can use those values to calculate the impedance of the inductor. In this case, working the example case:

Above, the results reconcile reasonably well. There is small error because the calculator above ignores the rounded corners on the core section, so it calculates a value for A/l slightly higher than the datasheet, and of course some rounding error. The error is trivial in terms of the 20% tolerance of these cores.

The important thing is that the use of the complex permeability allows calculation of the RF impedance of the inductor, including the inductive reactance as the imaginary part and equivalent core loss resistance as the real part.

So, repeating this procedure over a range in frequencies, you should obtain something similar to the following.

Above, Fair-rite’s curves for #43 material which was used for the example.