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Per the message # 17255 and those that followed.

While the algebra to the solution is a bit long, here is a simpler network that is also more realistic. It indeed points to the affect of resistance loss modification of the resonance condition. The algebra is attached.

A fun experiment is to place a high Q tuned network on the vna and then foul up the L with some R... Does it confirm???

Alan

I dislike typo's. Next to last equation, the j's should be REMOVED inside the ()... Here is a corrected.

On Fri, Sep 11, 2020 at 09:26 PM, alan victor wrote:

I dislike typo's.

Well, if you're serious, change "resonate" (in the title) to "Resonant", and leave out the ' (apostrophe) in "typo's".
Yes, that's just nitpicky, and yes, your message is clear enough as is. I'm just a bit of a 'grammar nut' and couldn't resist!

Thanks for your contribution, anyway!

Doug, K8RFT

Thank you!

Alan

On Fri, Sep 11, 2020 at 04:23 PM, alan victor wrote:

While the algebra to the solution is a bit long, here is a simpler network
that is also more realistic. It indeed points to the affect of resistance loss
modification of the resonance condition. The algebra is attached.

A fun experiment is to place a high Q tuned network on the vna and then foul
up the L with some R... Does it confirm???

I guess that I'm doing something wrong here, but LTSpice does not confirm the math :)

Just to make it very simple, I did C=10nF, L=10nH (so C/L=1), then varied RL as 1u 10m 100m 500m ohm. As expected, resonant dip becomes less prominent as R increases, but there is no "significant" change in resonant frequency as the math would suggest.

For RL=0.5 ohm, the SQRT(1-C/L*RL^2) = SQRT(0.75)=0.5625 suggests that resonant freq should shift to 56% of the "ideal" res freq, but LTSpice shows just a very minor shift of 1%

For some reason, can't attach screenshots, trying again

Excellent experiment Miro.

Take care on setting the source impedance for the AC generator. If not sufficiently large the loaded Q will be quite low and the net affect of seeing the de Q of the inductor as well the freq shift. This will be difficult. You can apply an ideal transformer to provide a step up and step down in Z level while obtaining an accurate transfer function for the network under test.

I will try the same experiment as it is a good exercise.

Here you go Miro. I set the generator to a large value, 1 Meg.

It is best to look at PHASE and the crossover at zero degree.

You will see zero degree phase point in the plot as a RED crossover X. It is centered on the ideal infinite Q point to start. Note with progressive increase in RL... the series loss of the shunt inductor, there is a significant shift LEFT as marked by the dotted sweeps of phase.

Alan

So on the plot shown you can see the migration in resonant frequency from 15.9 MHz readily down to 12 MHz and still marching downward. I said infinite Q.... nah! Actually quite finite, maybe 50 or something can't recall, but large enough that you get the point...

Alan

And of interest, peak magnitude and zero phase do not always align except for the infinite Q case... It is best to observe the definition as the imaginary part of the function goes to zero implies the phase shift of the network goes to zero. Hence, inspection of the amplitude response only is problematic.

Alan said:

Here you go Miro. I set the generator to a large value, 1 Meg.

With very large resistance you effectively have current generator so you can plot voltage on the RLC circuit :) In my simulation, I used ideal voltage source and plotted the current - "shapes are the same, but inverted :)

After reading the rest of your response

It is best to look at PHASE and the crossover at zero degree.

I redid my simulation (still with voltage gen and Rg=0), but left PHASE diagram visible - results fully resemble yours - phase lines crossing 0 are visibly shifted to the left indicating change in resonance(as the math says) :)

Honestly I did NOT expect that peaks of amplitude won't be aligned with phase crossing zero (the "real" resonance)! Well, like in some other recent threads (resonance of a dipole antenna), it's important to start with the valid definition - in both cases it's the frequency where phase crosses zero (jX=0), regardless of what amplitude (here) or SWR in the antenna cases might say :)

Great exercise!

Miro

What will happen with resonant frequency when R>C/L and value under SQRT becomes negative? Will that create a black hole and the world as we know it will collapse in it? :)

SPICE simulation shows phase approaching zero phase going to the left, but in "aggressive pace" - quite non-asymptotic

Strictly speaking, there should be bounds on the R,C,L terms so they are less than one. Otherwise a second order system, a tuned network which must posses two memory components vanishes. Consider if R is very large, L is neglected and we have a parallel RC network. No peaking in itself exists. At low frequency and C is no linger a factor and the network is dominate R. At very high frequency, C dominates and the phase shift approaches -90 degrees. So the phase response is one sided, 0 to -90. We never see the so called tuned response of a 2 - element memory (storage) electrical system.