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Re: AC wire resistance with proximitry effects

Original poster: "Gerry  Reynolds" <gerryreynolds@xxxxxxxxxxxxx>

Hi Bart,

I finished the proximitry effects portion of Gary's paper and will share my opinion. Basically, there are two choices given in the paper - the Medhurst Resistance table and the Fraga equation. I was leaning toward the Medhurst approach until the end and after much thought I changed my mind and now think we ought to implement the Fraga equation. Here are my reasons:

MEDHURST: This method was emperically developed by Medhurst and involves a lookup table where normalized wire spacing and aspect ratio are used to find the ratio of Rm/Rac. I will use Rm to denote the Medhurst resistance and Rf to denote the Fraga resistance. The following are the conditions or assumptions Medhurst used:

a. Rm = CFm * Rac where CFm is the Medhurst correction factor, the value of which comes from his table.
b. The coils Medhurst wound were about 30-50 turns of wire.
c. Medhurst tested at a high frequency (1MHz) where the proximitry effect was fully developed (saturated).

Gary modified his equation to allow usefulness at lower frequencies for which the proximitry effect may not be fully saturated by intoducing a monotonic function "kf" that varies between 0 and 1.

Rm = (1 + kf(CFm - 1))Rac at DC kf=0, Rm=Rac=Rdc
at high freq, kf=1 and reverts to Medhurst's original equation

The down side to this approach is that the kf function has not been defined. The approach works better for coils with smaller number of turns (or small aspect ratios) than what we are use to winding and operating at frequencies perhaps higher than typical for TC's. One needs to determine the Rac with no proximitry first and then apply the proximtry correction factor CFm. The upside is provisions for space wound coils.

FRAGA: This method uses an equation (a complicated one) to predict the effective resistance Rf that includes skin effect, proximitry effect, and the degree that the proximitry effect is fully saturated. The following are the conditions or assumtions Fraga used:

a. Long coils
b. Close wound
c. Coils with negligible distributed capacitance (constant current from bottom to top)
d. Low freqency operation (wr <= sd)
e. Mulilayered coils

Rf = [2pi*N^2*a* effective_rho*(sinh(2theta)+sin(2theta)) ]  /
       [effective_sd*Lw*(cosh(2theta)-cos(2theta)) ]

[note the hyperbolic transendental functions]

where N = number of turns
         a = radius of coil (meters)
         Lw = winding length of coil (meters)

         effective_rho = 2rho(1+s/b)/sqrt(pi)
         rho = resistivity = 1.724E-8 ohm meters for copper at 20 degrees C
         b = radius of the wire copper (meters)
         s = thickness of wire insulation (meters)

         effective_sd = 0.0702*sqrt[(1+s/b)/f} (meters)
         f = frequency (Hz)

         theta = b*sqrt(pi) / effective_sd  (I'm guessing units in radians)

Downside of Fraga is it can't properly deal with short or space wound coils.

Gary had nine coils designated 12T, 14S, 14T, 16B, 18B, 18T, 20T, 22T, and 22B. The number reflects the awg wire used. T is tight (close) wound, S is space wound, and B is some sort of barrel shaped form where the form sides are not parallel and was both close and space wound. He put these coils to the test and compared Rm, Rf, and Rmeasured. All were compared at the coil's natural frequency. In addition, all but 12T were also compared with two different size top loads. Im not reporting the B coils as I was not sure what these coils are. The 12T, 14S, 22T coils also are not being reported because either they didn't use magnet wire and had thick insulation or were space wound. The remainding 14T, 18T, and 20T coils all used magnet wire, were close wound, and had an aspect ratios close to what we use. These were more of the norm and seemed more appropriate for reporting.

14T:  aspect = 6.5 and Rdc = 4.45 ohms

       Rm     = 46.4 ohms @ 251 KHz fo
       Rf       = 44.1 ohms
       Rmeas = 43.5 ohms

       Rm     = 41.7 ohms @ 211 KHz
       Rf       = 40.5 ohms
       Rmeas = 42.3 ohms

       Rm     = 35.9 ohms @ 152 KHz
       Rf       = 34.3 ohms
       Rmeas = 39.6 ohms

18T:  aspect = 4.1 and Rdc = 11.2 ohms

       Rm     = 76.5 ohms @ 237 KHz fo
       Rf       = 67.6 ohms
       Rmeas = 70.5 ohms

       Rm     = 66.5 ohms @ 176 KHz
       Rf       = 58.2 ohms
       Rmeas = 65.9 ohms

       Rm     = 57.0 ohms @ 123 KHz
       Rf       = 48.6 ohms
       Rmeas = 58.1 ohms

20T:  aspect = 4.4  and Rdc 23.4

       Rm     = 111.4ohms @ 181 KHz fo
       Rf       = 97.3 ohms
       Rmeas = 94.2 ohms

       Rm     = 98.4 ohms @ 136 KHz
       Rf       = 84.5 ohms
       Rmeas = 88.0 ohms

       Rm     = 85.0 ohms @ 94 KHz
       Rf       = 70.4 ohms
       Rmeas = 78.7 ohms

It should be noted that the Fraga prediction was always lower than the Medhurst and that the measured resistance includes all losses not just the copper loss. At first glance, it would seem that Medhurst might come closer to the measured resistance than Fraga (at least for the higher frequency coils). Gary did a more in-depth test of coil 14T where he varied the resonant frequency by changing descrete caps loading the top of the coil. With this test, one can easily see the proximitry effect weaken as the frequency is reduced. The Frada equation also predicts this weakening and agreed very well with the measurements. It was this factor that convinced me that Frada's equation may be more useful.

Gerry R.


Original poster: "Barton B. Anderson" <bartb@xxxxxxxxxxxxxxxx>

Hi Gerry,

Yes, there is much to be pondered on Gary's work. With this particular paper, I was impressed with his methods of measurement. I was satisfied that he explained the difficulties and how he overcame them. I was also very impressed with his adding of C to the topload to drop the frequency and how he went about it.

I'm not satisfied that his conclusions are right or wrong on every aspect (some are obviously correct, others require further assessment and measurement).

For the post at hand, Gary has already done the hard work for us. All we need to do is verify with a number of coils to get a good realization of our secondary losses. I don't expect Medhurst to perform well with our coils. His data just does not represent the full spectrum. It is correct for the spectrum he measured, but it would be helpful to simply perform the same with our coil specification range. If I had the time and the money, I'd love nothing more than to do just that. Such is life.

BTW, I expect your coil based on Terman at 53.92 ohms, and interestingly, Antonio's form at 61.95 ohms. Quite a difference! Also, both are quite different from the standard text book 38.24 ohms. I wonder which is right? Q will tell the tale, but that assumes Q measurement is correct, and we all know, there are a lot of "what can go wrong will go wrong" when measuring Q.

I'll be interested in what you come up with. Do you have all the goodies? Low Z amp? etc... If you need one, just ask. I can send my own your way or Terry may be able to londer his, etc. I think it's definitely worthwhile. Also, I would set up a flat ground plane for measurement somewhere where external effects are not capable of much influence. When I make Q measurements, I always do these in the backyard so that there is nothing to affect the reading. I keep the probe at least 10 feet away from the coil at center toroid height. I'll usually throw down a flat metal plate equal to the toroid OD for the ground plane. My personal coils have been rather low Q (300's range). It will be interesting if you can measure in the predicted 600's range.