Years ago my guru/mentor showed me mathematically how the resonant frequency of an LC was changed by the Loss resistance. I think if the losses of the L and the C are equal then there is no effect, but usually the L has more losses. Can someone show me the math in it's simplest form to prove this.
Thanks, Mikek
Is this a parallel L-C-G, then solve for zeroes of the total conductance S^2 x C + S x G + 1/L by completing the square and you can see how G pulls the roots- quickly from quadratic formula (-G +/- sqrt( G^2-4xCx1/L))/2 x C
Sorry about the private response.
Yes parallel LC, but I work with series resistance. ;-/ Can you do that in ohms instead of mhos, and skip the quadratic. It's been 41 years and I didn't get it then.
And a followup, My Q meter only measures to a Q of 625. I add a series 1 ohm resistor to read higher Q's and back out the one ohm. How does that one ohm affect resonate frequency? It's not a loss in either L or C. Does it have no effect on Resonant frequency?
Thanks, Mikek
It all falls out of the equation for the harmonic oscillator. It doesn't matter where the loss is. (L or C) And it only has a significant effect for low Q circuits.
(I searched a little, but didn't find a good web page.) well this is OK.. omega_sub_1
George H.
Is your definition of resonance where the impedance is real, or the magnitude of impedance is maximum? ...Jim Thompson
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When the Q meter gives up you can try other methods, for instance measuring half power bandwidth (3dB down or 71% voltage) and using Q=Fo/BW.
Example: your LC peaks at 1.5Mhz and you find is 3dB down at 1.5015 and
1.4985 then BW is 3kHz so Q=500piglet
Doesn't make any difference, a series R can always be transformed into an equivalent parallel R that is frequency independent about resonance.
I used this method with good results in the past (way past):
Just to help me understand this stuff, I solved the equations for resonance once. You end up with the R of the components in the equation so the R value *will* affect the resonant frequency. I wrote it on a white board and it's gone so I cant' share that with you. But it isn't that hard to derive.
Start with the condition that the magnitude of the impedance of the two components is equal. Include the loss resistance in each component as complex numbers and solve. Duck soup!
Off the top of my head, the loss resistance of the inductor will increase the impedance at a given frequency with phase shift. The same loss resistance in the capacitor will increase the impedance as well but with the opposite phase shift. So I think you are right that with equal losses the frequency does not change.
But if the inductor has more loss resistance than the capacitor (the typical case) the impedances will have equal magnitudes at a lower frequency.
-- Rick C
Until you ask I didn't know there was difference. I have a Boonton 260A Q meter, I'm guessing it measures Magnitude of impedance at maximum. My new problem today, my Q meter only measures up to 625 for Q. So, I added a series 2.24 ohm resistor. I then measured an inductor with and without the resistor. Then subtracted the 2.24 ohms from the loss resistance. Tuesday I had a maximum difference of 0.08 ohms across the AM band, (ie perfect) today it is not linear with up to 24% error. I don't know what I did different. Same coil, same resistor. Gotta work the rest of the week, maybe next Tuesday I'll get a handle on it.
Mikek
So what definition would you like to pick? Because there are three from a
3rd order system.An ideal series-parallel transformation, with these three conditions coinciding, is ONLY possible when the Q is infinite (i.e., in the limit as ESR goes to zero / EPR goes to infinity).
Tim
-- Seven Transistor Labs, LLC Electrical Engineering Consultation and Contract Design Website: http://seventransistorlabs.com
Yes, I spent several years using the 3db method measuring potcore inductors I wound. I always peaked with 7 units on the scope and down 3db on 5 units.
5 / 7 = 0.714 vs 0.707, as close as I could read on a scope. Sometimes I did it with drive level, other times I used the variable gain knob to get 7 units.Mikek
Is this what you had in mind...
?? ...Jim Thompson
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| James E.Thompson | mens |
| Analog Innovations | et |
| Analog/Mixed-Signal ASIC's and Discrete Systems | manus |
| STV, Queen Creek, AZ 85142 Skype: skypeanalog | |
| Voice:(480)460-2350 Fax: Available upon request | Brass Rat |
| E-mail Icon at http://www.analog-innovations.com | 1962 |
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I can work with that, thanks, Can it be mathematically shown what the optimum spacing of wire is for optimum Q? I'm in the process of having threads cut in polystyrene pipe couplers to compare 12 TPI, 10 TPI and 9, 8, 7 TPI. 10TPI has better Q than
12TPI. I hope 8 is better that 7, otherwise my pipe couplers are too short. If 7 is better, then I'll need to make a 6TPI. Using 660/46 Litz (diameter 0.0693") a 6.5" Diameter coupler. Mikek
I have not a clue.
...Jim Thompson
--
| James E.Thompson | mens |
| Analog Innovations | et |
| Analog/Mixed-Signal ASIC's and Discrete Systems | manus |
| STV, Queen Creek, AZ 85142 Skype: skypeanalog | |
| Voice:(480)460-2350 Fax: Available upon request | Brass Rat |
| E-mail Icon at http://www.analog-innovations.com | 1962 |
I'm looking for work... see my website.
Q is the inductor impedance divided by the loss resistance. If you are making a coil from a fixed length of wire the winding spacing that gives you maximum impedance will give you the highest Q. If you are working with a fixed inductance, the winding spacing that gives you that inductance with the fewest turns will have the lowest resistance and the highest Q. I would mention the skin effect and proximity effect, but since you are using Litz wire, I'm not sure how that would be impacted, but my first guess is the same as a single conductor.
So in short, look at the formula for impedance of a coil and see how impedance changes with turn spacing. Good luck picking one formula out of the many. I eventually found Lundin's which seems to be the best overall. But it is complex to see the effect of one variable. Wheeler's formula is much simpler and shows the inductance in the denominator and so is inversely proportional to the inductance. A shorter coil (all other factors being the same) should have a higher inductance and a higher Q. Are you using a fixed number of turns or reducing the number of turns to maintain a constant inductance? Either way I would expect a coil with smaller spacing to have higher Q.
Is it possible the proximity effect is increasing the loss resistance faster than the shorter coil increases the inductance?
One point I've never been clear on is what is meant by coil length, exactly. Is it the center to center spacing or the outside edges of the wire? The derivation of all inductance formula seem to start with a current sheet model. So I would expect the coil length to include the wire at each end. But some references seem to include the wire diameter and others don't. I've also seen references that seem to include the spacing of N turns when in fact everywhere except for the point where the two ends of the coil line up, there are only N-1 spacings. 1 turn = no spacing and 2 turns = 1 spacing. I've never found a reference that addresses any of this clearly.
-- Rick C
Probably. You'd want to ask this guy:
Usual rule is pitch twice the diameter. I don't know where the actual Q maximum is, and probably it varies with aspect ratio and number of turns.
If you're doing a pancake coil, turns in the center contribute little to the inductance, while adding the same wire resistance -- plus eddy currents, because of induction from all the surrounding turns, too! Not that the wire length used in the center is very much (the circumference tapers down to zero per turn!), but it's diminishing returns.
I would suggest an inner diameter (i.e., of the innermost turn) about half the OD.
Likewise, solenoids have poorer Q when they are very long (there's little mutual inductance between the turns at either end of the coil). Best is near square (length over diameter about 1).
Tim
-- Seven Transistor Labs, LLC Electrical Engineering Consultation and Contract Design Website: http://seventransistorlabs.com
You seem to be assuming... a constant (DC) resistance?
Unfortunately, AC resistance goes up with frequency, and geometrically with proximity.
Inductance goes up geometrically with proximity as well (on the order of N^2), so it's not clear at all whether one or the other should win out. That is, higher order effects dominate.
This is the best calculator I know of:
The physical reality of even a simple helix, is that it is analytically intractible (if not necessarily insolvable). The effect of any given variable _should be expected_ to be complex!
Most solutions are approximated, with a combination of empirical and theoretical treatments. The above is within 1% of reality for everything I've seen, which is better than I can measure anything to.
Bingo!
The drawing above, shows center to center distances. Other formulae may use different definitions. You'd have to find the original papers.
The calculator complains about length, even for one turn, so I don't think it can be used to calculate a flat loop. Obviously, in practice, a low-pitch loop will be very similar to a flat loop, so it's not a problem.
The calculator also does fractional turns, for which it's not obvious how the leads exit the solenoid (they surely wouldn't arc around to the same parallel axes that a whole-turn coil can, but they also wouldn't simply stretch out to infinity, either..). But it still does a good job, even for such poorly-defined things.
Tim
-- Seven Transistor Labs, LLC Electrical Engineering Consultation and Contract Design Website: http://seventransistorlabs.com
Assuming that the formula is correct w = sqrt((L-R^2C)/(CL^2))
can be linearised to avoid the sqrt using the rational approx which holds provided that R^2C < 2L and is accurate for small deviations
w = w0(1 - 2R^2C/(4L-R^2C))
where w0 = sqrt(1/LC) the resonance when R=0
Basically from sqrt(1+x) ~= 1 + 2x/(4+x)
which isn't bad for -0.5 < x < 1
Only for ideal solenoids ignoring edge effects and other local coil turn interactions. You are into trial and error guided by models to get the optimum Q for a given inductance and length of wire.
I suspect that optimum Q requires a coil former with the wires slightly non uniformly spaced to get most possible L for the least total R. Possible to do today for a CNC machine if you are hand winding it.
-- Regards, Martin Brown
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