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Is this really a 100uF cap or marketing BS
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T3sl4co1l:

--- Quote from: magic on March 29, 2020, 09:19:30 am ---L shows up in your expression because you use the RLC model with constant R, L and C to derive a C or RC model with frequency-dependent R and C, which mimics the way naive capacitance/ESR meters work.

The C in your C model is not the same as the C in your RLC model.
--- End quote ---

Correct, I separated out C as the LF-asymptotic, nominal or rated value, and Co as the apparent (measured) value.

An RLC model is just an approximation to a real component, and we can apply higher order approximations if we so desire.  You are welcome to solve the algebra for such a case; the RLC case however proves that a more complicated model is not necessary to explain the phenomenon, and indeed it is the least complicated model possible to show this effect. :-+



--- Quote ---Philosophically, we can argue which C is the "true" C of the capacitor ;) I'm going to say it's the RLC C because the RLC model more closely reflects the physical reality, which is plain magnetic inductance existing in the capacitor rather than some magical property of the dielectric or frequency-dependent variation in electrode spacing.
--- End quote ---

Again, yeah, C is nominal, Co is measured.  Apologies if that was not clear in the derivation (I thought I had added a paragraph to that effect, perhaps it was not well marked).



--- Quote ---By transient conditions I mean startup, shutdown or abrupt frequency change, when resonance gets out of whack and sudden peaks or dips are possible due to the L. Yes, the models are theoretically equivalent so you could model it in frequency domain by integrating the variable capacitance over all frequencies present in the transient, but in practice it would be a rather masochistic approach.

--- End quote ---

A much simpler model applies to these cases:

Because C varies so much with signal level and bias, it would be, at the least not very meaningful, and at worst rather disingenuous, to apply a high-order linear model.

Instead, the design should be guardbanded: assume minimum and maximum possible values for the nonlinear elements in question.  If the circuit works at both extremes, and can also be shown to work inbetween*, that's it -- you don't need any higher order models, don't need complicated impedances, it works for a whole space of variation.

*For example, solving for the Q of the PDN (power distribution network) and checking that its maxima is either one or the other endpoints, or no more than a definite maximum value inbetween.


This is somewhat aside, but incidentally -- frequency step change is easily modeled, for a linear system anyway.  Take this example:
https://www.seventransistorlabs.com/Images/Induction725.jpg
LLC (series driven parallel resonant) tank, driven with a FSK waveform.  The envelope seems to resonate at different frequencies.  It turns out, these are actually the difference between F_driven and Fo.

This is easily understood from diff eq -- an inhomogeneous LTI system (i.e., one with an independent driving term, and only linear time-invariant dependent terms) has two responses: the homogeneous solution (the diff eq's natural solutions, in whatever proportion follows from initial conditions), and the driven solution (proportional to the driven source).

Also, that we can separate the driving source into two components: a tone burst at F1, interleaved with a tone burst at F2.  These signals have complementary windows, so that their sum is apparently a continuous FSK signal.

When we evaluate the system for a transient start or stop of the driving waveform at F1, we observe basically the first envelope.  A signal which in turn is composed of the transient (homogeneous) solution, essentially the system's impulse or step response, plus the driven solution, essentially the system's filtering effect upon the driven signal (a phase shift and gain term).

When we evaluate the system at F2, we get the second solution.  Sum them together and you get the whole repeating waveform.

A less formal way to look at it, but just as valid (and accurate!): suppose there's a source, with frequency at F1, which simply stops, drops to zero completely.  Well, the tank will ring down some.  Over a time constant of Q / Fo, more or less.  Also, it had been driven at F1, but the tank doesn't know that, it just rings down at Fo, its characteristic frequency.  Suppose we superimpose that response on the response of F2 suddenly starting, which will be a ramp up at the same time constant, at the different frequency, and reaching a different final amplitude.  What does their sum look like?  Well, the tank rings down at Fo.  We take the envelope of the sum.  What's the envelope over two different frequency components?  It's a DC offset plus their difference frequency.  And what should the envelope look like, given that one of those components is decaying and the other is increasing?  Well, it'd be a bounce of some sort.  And this is exactly what we observe in the waveform!  Finally, we can calculate the expected magnitude of those peaks or dips, simply from the resonant parameters of the circuit.

The application for this image was induction heating, so, amplitude control is relevant; this shows just how challenging the control loop will be for a frequency-shift type control.  But it's also useful knowledge for PLLs, communications signals (e.g. the required bandwidth of a FSK channel) and so on.

Tim
trobbins:

--- Quote from: JustMeHere on March 28, 2020, 09:39:27 pm --- SMPS at 500k might work very well with these.
--- End quote ---
Yes I'd expect like many very high-density multilayer ceramics, there are quite a few material and performance tradeoffs in the quest for ever higher capacitance density, and material non-linearities start to really stand out.  SMPS may be one application, although the ability to put bulk caps right at IC/load device terminals, with no track meandering out to some convenient place to put a bulk cap, can be very beneficial.
Conrad Hoffman:
IMO, the dielectric properties are terrible, but the real question is what can be accomplished in a given volume and is it adequate for what the circuit needs. For many low voltage applications the high capacity MLCC is the best available choice, even if not a "pure" and mathematically elegant component.
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