All, I posted this back in March, but am still stuck.
I'm doing an analysis / simulation of a tube Hartley oscillator. (full circuit from Sept. 92 issue of QST, also in _QRP_Power_)
My problem is that I don't know how to treat the tapped coil, mathematically. I was rereading my textbooks and found in the derivation of the usual formula for a toroid's inductance that we need to assume the wire was uniformly wrapped about the core. Also, that only assumes a _single_ current flowing, whereas my circuit has a current source on one of the taps.
What I need is the basic model / equations that will allow me to relate current and voltage in and across the various sections of the coil.
Many thanks, Ross Tucker, NS7F Arizona State University
That should be straightforward, so long as you are dealing with linear models. It will be self-inductances representing the turns between adjacent taps (and the coil ends, of course), and a mutual inductance between each pair of those self-inductances. Equivalently, the mutual inductances may be represented by coupling coefficients.
You may determine the mutual inductances by measurement; you can measure the inductance of one of the self-inductances and then short one of the others and re-measure the inductance. In the case of a single tap, there would be only two self-inductances and one mutual inductance.
You can think of the mutual inductance as an extension of the mathematical model for self-inductance: v=L*di/dt goes to v = L1*di(L1)dt + M21*di(L2)/dt + ... +Mn1*di(Ln)/dt. Then the vector of inductor voltages is the matrix of self- and mutual-inductances times the vector of di/dt's.
A formula for finding the mutual inductances from core permeability and winding spacings would probably be difficult to find for a general case for toroid cores, but for single-layer air core solenoid coils, it's not too difficult. Assuming that the old standby formula
L = r^2*n^2/(9*r+10*l)
where L is the self-inductance in microhenries, r is the coil radius in inches, n is the number of turns, and l is the coil length in inches -- assuming that's accurate, you can find, for example, the self-inductances of say the five turns from one end to a tap, and the ten turns from the tap to the other end, and the whole fifteen-turn coil. From those, you can find the mutual inductance and/or the coupling coefficient. I've found that inductance formula to be accurate to better than 5% for practical RF coils in open air, and the error changes rather gradually with r/l ratio, so I think it's reasonable to expect the mutual inductance calculated that way to be within 5% also. It can get tricky to measure RF inductances much more accurately than that anyway.
LTSpice is easy to use, free, and lets you simulate coupled inductors. Just add text to the schematic, like:
M1 L1 L2 .95
As for the coupling coefficient...did you mean k=1? I usually see M meaning mutual inductance, which has the units of henries, and will depend on the values of the self-inductances. M = k*sqrt(L1*L2)
I think k=1 for typical powedred iron toroids is rather too high. I've been winding some recently, and find that I can adjust the inductance tens of percent by changing the spacing around the core, assuming the wire size lets me do that. That tells me that the coupling is not very tight among the turns.
Total inductance of two inductances in series (i.e. a tapped coil) is given by L1+L2+2*M, or L1+L2+2*k*sqrt(L1*L2). We can use that and an inductance formula for air core coils to estimate the mutual inductance or coupling coefficient between the two inductances composing a tapped coil. For example, my coil software says a 1" ID coil of 16AWG wire at
10 turns per inch will be 3104nH for 15 turns, 1839nH for 10 turns, and
692nH for 5 turns. So if 15 turns is a 5-turn coil in series with a
10-turn coil, the mutual inductance must be (3140-1839-692)/2 =
304.5nH, and the coupling coefficient k must be 304.5/sqrt(692*1839) =
0.27. Note that's not really very tight coupling. I expect a winding on a typical RF powdered iron toroid will be quite a bit higher, but still far from unity.
I expect that, for comparison, the coupling coefficient between the two windings of a twisted-pair transmission line on a ferrite toroid core (with much higher permeability than the powedered iron mentioned above) will be greater than .99, if it's done carefully, and in fact a little simulation in Spice shows that it's critical to have very high coupling if you want a broadband transformer. The high end falls off with coupling lower than unity.
As others have noted, this can be modelled in SPICE using coupled inductors. If you want to know what coupling factor to use, then if you have the time to create a file describing the geometry of your coil, the free program FastHenry (from MIT) will calculate the inductance, RF resistance (including skin effect), mutual inductance etc. of all the windings for you. Of course if you actually have the coil then you could also measure it which is probably quicker.