Getting resonant frequency in LC circuit with L as air core coil

Firstly, there is an error in the dot notation - scenario 4 provides the highest inductance yet the dot notation implies that if the two inductors were perfectly coupled, the net inductance would be 0. Because scenario 4 gives the highest value of inductance it can be concluded that it really has the dot notation of scenario 3.

Secondly, you have not considered that the two inductors may not be 100% coupled.

Next is to work out the coupling and a bit of math in my head tells me it's about 70%. Individually each winding has about 600 uH and in series aiding this rises to about 1800uH. If the two windings were 100% coupled they would produce a total inductance of 2400uH when connected in series.

So if 70% of each winding is perfectly coupled then the total inductance is: -

(4 x 0.7 x 600 uH) + (2 x 0.3 x 600 uH) = 2040 uH. OK my head-guess was a little optimistic on coupling. 50% coupling realizes an aiding inductance of 1800 uH.

When put series opposing, 50% of the coupled inductance totally cancels leaving a net inductance of about 2 x 300 uH.

Near enough.

EDIT to explain my math

The standard formula for coupled inductors is: -

\$L_{EQ} = L_1 + L_2 + 2k\sqrt{L_1L_2}\$ and, when both inductors are the same value this results in: -

\$L_{EQ} = L + L + 2kL\$ and, when k=1 (100% coupling), equals 4L

If a fraction (70%) of L1 is 100% coupled to L2, the fraction produces an inductance of 0.7 X 4 L.

The remaining uncoupled parts of L1 and L2 do not interact and are just additive i.e. (1-0.7) X 2 L.

Hope this makes sense.

The reactance of the coil will change and there are two opposing mechanisms at play: -

• The permeability of the magnet will tend to increase inductance
• The conductivity of the magnet will cause eddy currents that tend to decrease inductance

To what extent that either increases or decreases the coil's inductance is guesswork and will vary at different frequencies. I've seen situations where reactance/inductance doesn't change at all.

Another effect is that the eddy currents in the magnetic material will cause the magnet to take power from the AC circuit i.e. losses will increase.

At high frequencies the magnet will act like a plate of a capacitor and thus there will be a resonance effect. Again, without detail of the experiment this is hard to quantify.