So, we were making some prototypes for Chebyshev passive filters for the SW range, and found out that most of the time, the resonance frequencies of the individual LC circuits there are considerably lower than expected, with the apparent culprit being the inductances of the custom coils we made, which seem to be much higher (20% to 30%) in the operating range (about 25 MHz) than both the design values and the values as measured on a 200 kHz RLC meter. (Say, about 93 nH @ 25.5 MHz vs. 83 nH @ 200 kHz.)
The question is, is it some well-known effect, or am I mistaking something else for the perceived change in inductance?
(I understand that at higher frequencies the stray capacitance effectively turns a coil into a parallel LC circuit by itself, but the resulting self-resonance lies well above 100 MHz, so I guess it shouldn't make much difference at 25 MHz.)
With a vector analyzer and some Gnuplot magic [1] at hand, I was able to measure the inductances of some of our coils, the results for two of which are shown in the table below.
As it seems, the carbonyl iron shell core coil's inductance has remarkable stability over the range of interest to us, especially taking into account that our RLC meter gave it around 2.35 uH @ 200 kHz as well.
One possibility is that losses in the magnetic material (effective damping resistance) are rising with frequency, and that is doing a slight frequency-shift downward, just like if the L value was increasing. Another is that losses in the copper (skin effect) are doing the same sort of thing.
The 'error' seems small enough that either of those might be happening, but of course your air-core data shows that it's not the copper.
I know little of magnetic materials, but how thick is the wire for the coil? At HF you can get skin effects, crowding current into the edges, more resistance, which will change the freq. a bit... not 20%.
Which filter topology? Ladder network? Coupled resonators?
In the former case, you can't reasonably measure pairs of components together, except for bandpass/stop.
Coupled resonators should always measure on f_c, when they are coupled magnetically, on taps, or resistively.
If they are coupled with small capacitors or large inductors, it's detuned of course.
Hmmmm. Doesn't look very statistically significant, despite what the "error est" figure says. Each range seems to dance around, with only a modest trend between ranges.
I'd be more interested in the complex impedance, and fitting an equivalent network to that. An example here:
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Some stock Coilcraft models show inductance rising with frequency, which is unreasonable /and/ nonphysical. (Those are harder to adapt...)
Now, it could be that there is actually a third terminal in play (i.e., electric field coupled to ground), which happens to best-fit as a one-port with suspicious parameters. But that's not the model, and that's not general (electric field depends on common mode voltage and any nearby structures), so it's not a good path to follow.
FWIW, a lossy inductor necessarily has inductance decreasing with frequency, until approaching resonance, where the impedance (and inductance) spikes, then it goes resistive (at resonance) and then capacitive thereafter.
The former. Can you please point me to the relevant reading on the latter? I've found some links with a Web search, but they seem to be describe microwave filters, and our frequencies are much lower. (Yet.)
The latter is indeed our case. But we of course can tune the individual LC circuits before installing them on board. (Takes considerable time, but doable given the small-ish number of filters we have to make.)
(Not to mention that we actually have small gaps in the PCB tracks, as well as test points, so we can tune the circuits on the board before connecting them all into a filter proper. Also helps to account for the effect of the case, etc.)
?
Yes; that "error est." is straight from Gnuplot's Levenberg?Marquardt implementation, and as such (given the perhaps way too oversimplified model) is a very crude measure.
Aren't S-parameters "analogous" to (as in: interchangeable with) complex impedance? We have them, and I've been fitting (the logarithm of) the /magnitude/ of S11/S12 against a shunt series RL circuit, like [1]:
## magnitude_dB = dB * log (complex_value_squared) dB = 10 / log (10)
r_L (f) is defined to take the skin effect into account, but I'm uncertain if it's done the right way. (R_c is the characteristic impedance, 50 Ohm.)
Unfortunately, it doesn't seem possible to fit a complex value, or more than one set of scalar (float) values at a time in general, with Gnuplot, and the latter offers the most sophisticated L-M implementation I have an experience with.
ACK, thanks for the pointer.
Any specific explanation as to why is that nonphysical?
The test board we're measuring the S-parameters with has the test coil, two SMA connectors, and the tracks to connect them all. (It has a rather large "ground," however; we can try with another board, which doesn't, I suppose.) Any suggestion on how we can improve it?
I suppose we're observing exactly that at about 100 MHz or higher (depending on the coil.)
--
FSF associate member #7257 http://am-1.org/~ivan/
There used to be an excellent Java applet, modeling coupled resonators (in the abstract -- doesn't matter how you couple them, just that you get the f_o and k_{m,n}'s correct), but it died back when Java went away. Not aware of a port or replacement of it, unfortunately. :(
This is a more basic two resonator example:
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You see them a lot at microwave, of course, but also for cellular and commercial radio filters (especially diplexing, where high Q and tight frequency edges are needed). I have a rather large one in my junk bin which seems to be a VHF TV channel, with half a dozen coils inside (helical resonators). Must be very sharp; I haven't bothered to measure it yet.
They're not too common at lower frequencies, where, heck, active filters may even be easier -- but it works out the same way.
The basics for analysis are this: all f_o's are the same, and k ~= 1/Q ~= f_BW / f_o. The k's are symmetrical from end to end (k_{1,2} = k_{N-1,N}) and the variations of each pair (around the [geometric] mean 1/Q value) are determined by the filter prototype (Butterworth, etc.). To introduce zeroes, couple non-adjacent pairs (e.g., 1,3 and etc.). Which pairs are chosen, determines which side (above or below) and number of zeroes.
The input and output coupling are also abstract; typically a tapped coil is used.
The Jackson case above uses capacitor dividers for input and output coupling, and a small capacitance to couple between the top of the two resonators. This detunes the tanks downward a bit, which makes the calculation a bit crappy (all the better that they've written a calculator to do it for you!).
Magnetic coupling (inductors in proximity: shared flux) does not cause detuning, which is handy. In practice, there is some capacitance from proximity as well, which still detunes them, but not nearly as much.
The Q of individual components, of course, you want much higher than the filter Q.
The ladder configuration, by the way, is a special case of this; there, the series-parallel transformation is used on half the resonators, so that instead of parallel-resonant tanks linked "somehow", the current through the series branches is exactly the current linking parallel branches. The coupling factor is the ratio of each LC's impedance from the mean impedance (which for a RS = RL double terminated case, is just that resistance), the parallel tanks being Q times lower impedance and the series tanks being Q times higher.
Which is why the ladder configuration is preferred for large bandwidths (Q <
3, say) -- you can't really get a coupling factor that large with coupled resonators. Conversely, it's difficult to avoid parasitics (capacitance across a series inductor's terminals, and from its body to nearby ground or free space; and series inductance of parallel capacitors), so coupled resonator designs are preferred at high Q.
Ah, so there we are -- you should indeed be able to test each pair, and them coming out wrong is a bit troubling.
Explained above :)
Yes, or easily converted anyway (there is a relation between all the various port formats).
I haven't touched it, so I don't know. :^)
Would it help to set a fitness function (say, |distance| on the complex plane) and regress a given curve to match the data?
That's what I do a lot of the time, say when creating a model of an existing component. Example:
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Matches pretty closely with the datasheet value for MI1206L391R-10 (well, at zero bias). Set up the parameters, then run regression and hope it finds a good local minima.
A two-pin component is a one-port: current in one pin goes right out the other. There is only one such pair of pins, so it's a one-port.
For it to be real-valued and passive (not a generator), the impedance and phase angle must obey the Kramers-Kronig relations.
For a passive two (or more) port, a more general rule of course applies.
Dunno. Picture?
Proof is in the pudding; build the complete filter, see if it matches the design prototype, and how far off f_c is; then work backwards to determine how far off your component values were? Any means of measuring their values, near f_c, should do well enough. I guess you have a VNA, so you're way more than well enough equipped to do this! :)
The trouble with powdered iron cores is, A_L isn't very well defined, especially on lower permeability parts, and with few turns, where the winding geometry is critical. This is a bit of a protip: if you need to tune one by less than a turn's worth. (Much more info on Micrometals's website, they have good appnotes.)
I've tried to fit an RLC ((R + L) || C) model over the data, and while I'm still getting some inconsistencies with that, it appears that a 0.8 pF stray capacitance indeed explains an apparent increase of inductance (which seems to slightly decrease with frequency in this model) for one of the coils.
One problem is that the coils seemingly most affected were in parallel to rather large (no less than 470 pF) capacitors, so a stray cap of a fraction of pF shouldn't have made any difference. But perhaps our RLC meter was miscalibrated.
Now, I wonder if a 0.8 pF stray capacitance coil in series with an about 15 pF capacitor would make a resonator adequate to our needs.
--
FSF associate member #7257 http://am-1.org/~ivan/
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