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# Coupling coefficient for air core inductor?

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#### MikeMl

##### Well-Known Member
Tapped RF coil, single layer, air wound, about 5uH. The tap creates a two winding auto-transformer. When modeling this in LTSpice, what value of coupling coefficient to use as in K1 L1 L2 x

What should x be?

Interesting question. I wonder if there could even be two coupling values, depending on which section of the coil is the 'primary'?
Consider a 10-turn coil with a 2-turn tap at one end. If the 8-turn section is energised, its magnetic field lines, like those of any 'long' solenoid, should be reasonably parallel and axial near the tap and so should couple well to the adjacent 2-turn section. Contrarywise, if the 2-turn section is energised, its magnetic field lines will (I think) be more divergent at the tap, so won't couple well to more than a turn or two of the 8-turn section.

I think that from what I googled recently the coupling coefficient should by the same for both directions, but it should not be a problem to verify empirically.

Agreed, the same coefficient both ways seems more plausible. My understanding of magnetics was never very good.

Measure:
L1 = Self inductance from tap to one end of the coil
L2 = Self inductance from tap to other end of the coil
L3 = Self inductance of entire coil

Then, mutual inductances between various sections of the coil are given by:

M12 = (L3 - L2 - L1)/2

Coupling coefficient:

k12 = M12/sqrt(L1 * L2) = (L3 - L2 - L1)/(2 * sqrt(L1 * L2))

BTW, coupling coefficients are not necessarily reciprocal. In other words: for two coupled coils 1 and 2, k12 is not necessarily equal to k21. However, for most coils k12 and k21 tend to be close in value. The above formulas assume that k12 and k21 are equal.

I could give an example where k12 and k21 are vastly different, but it would require me to draw a picture.

Since you're simulating this, you may not have an actual coil to measure. In that case, use an online inductance calculator to figure out the different values.
http://electronbunker.ca/eb/InductanceCalc.html
To do this, pick a reasonable coil form diameter and wire size. Feed it into the calculator, and adjust the number of turns until you get the overall value of 5uH. Suppose you end up with 36 turns. Now, decide where you want the tap located. For simplicity, let's assume you want the tap at the halfway position, 18 turns. So, using the calculator again, change the number of turns from 36 to 18, leaving the diameter, pitch and wire size the same. The new value is 2.11 uH. This is the value of self inductance from the centre tap to either end of the coil. You now have enough info to apply the above formulas and calculate the coupling coefficient.

Example 1 (long skinny coil):
36 turns #24 AWG (enamel insulation) close wound on 10mm diameter form.
L3 = 5.04 uH
Tap at half way point (18 turns)
L2 = L1 = 2.11 uH
k12=0.194

Example 2 (short fat coil):
6 turns #24 AWG (enamel insulation) close wound on 60mm diameter form.
L3 = 4.97 uH
Tap at half way point (3 turns)
L2 = L1 = 1.44 uH
k12=0.726

Edit: I revised the numbering convention for the parts of the coil and corrected the formulas. The original post had calculated the coupling coefficient from one part of the coil to the entire coil, not from one part to the other part as is required.

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BTW, coupling coefficients are not necessarily reciprocal. In other words: for two coupled coils 1 and 2, k12 is not necessarily equal to k21. However, for most coils k12 and k21 tend to be close in value. The above formulas assume that k12 and k21 are equal. I could give an example where k12 and k21 are vastly different, but it would require me to draw a picture.
Could you please do that? I still can´t wrap my head around that..

Note that I've edited my previous post. There had been an error in the coupling coefficient calculation. It's fixed now.

Could you please do that? I still can´t wrap my head around that..
This is a power transformer used to power the aircraft warning lights on a radio transmitter tower. Since the entire tower is at RF potential, it's necessary for the lighting transformer to isolate the RF from the mains power system.

The primary is tightly wound around an iron toroid, while the secondary is a large diameter loop that passes through the centre of the toroid aperture. Virtually all of the magnetic flux created by the primary winding flows through the toroidal core. As you can see the toroidal core passes well within the secondary winding. So, virtually all flux from the primary is coupled to the secondary. On the other hand, if the secondary is powered, much of the flux passes through the air, and misses the toroidal core and the primary winding. So the coupling from secondary to primary is much lower.

Now, to make things more confusing... even though the coupling coefficients k12 and k21 are not equal, the mutual inductance M12 and M21 are always equal. The apparent discrepancy is because most formulas for the coupling coefficient assume the k12=k21, and are simplified accordingly. This is the formula that you usually see in textbooks:
k=M12/sqrt(L1*L2) or M12=k*sqrt(L1*L2)
The correct formula is:
sqrt(k12*k21)=M12/sqrt(L1*L2) or M12=sqrt(k12*k21*L1*L2)

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