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Looking for proof on difficult RLC problem

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lammas

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Hello,


the problem is associated to the fact that when resistance in RLC is not zero, then capacitor's voltage and inductor's voltage reach resonance at different frequency(right around the area of current's resonance frequency).

I know the equations for calculating them:
Volatage resonance for inductor at:
ω=1/(LC-(R^2*C^2)/2)
Voltage resonance for capacitor at:
ω=sqrt( 1/(LC) - R^2/(2*L^2) )

It is easy to see that if R>0, then voltage resonance for inductor comes into being at higher resonance, for capacitor at lower.
What I need is an explanation, not a mathematical proof, but a physical explanation of it. WHY?

Thanks!
 
Hi lammas,

at first: there is only one single "resonance" effect at one frequency only - and that happens between both elements L and C. With other words: The voltage maximum you can observe across L or across C is not a kind of "resonance", it is simply a maximum.

Here is my attempt for an explanation: I think, for a simple explanation (without formulas) you can consider two cases: R very large and R very small.

1.) For R very small (nearly zero) we have an ideal LC circuit, which is driven by a voltage source - of course with one single current maximum only (resonance).
2.) For R very large we have an ideal LC circuit, which is now driven by a constant current source. It is obvious that the voltage across L goes up with the frequency (and goes down across the capacitor).
Thus, both maxima are very far from each other (remember case 1: one single common maximum)
3.) Now - for a resistor with a finite value between these both cases, we have a situation in between with two different maxima which have a distance proportional to the resistor value.
 
Hello,


the problem is associated to the fact that when resistance in RLC is not zero, then capacitor's voltage and inductor's voltage reach resonance at different frequency(right around the area of current's resonance frequency).

I know the equations for calculating them:
Volatage resonance for inductor at:
ω=1/(LC-(R^2*C^2)/2)
Voltage resonance for capacitor at:
ω=sqrt( 1/(LC) - R^2/(2*L^2) )

It is easy to see that if R>0, then voltage resonance for inductor comes into being at higher resonance, for capacitor at lower.
What I need is an explanation, not a mathematical proof, but a physical explanation of it. WHY?

Thanks!


Hi,


I think Winterstone gave a very good explanation of what we have here. The only question i have for you is are you considering a series RLC circuit or a parallel RLC circuit, and is there any drive voltage or current with it or has it been excited previously with the excitation source now removed?
 
The system has a driver. I am not specified with the output voltage nor current of the system, but I conducted measurments around calculated resonance (or maximum as if it was mentioned above). I was given the value of L, C and had to measure different values of the system around resonance

It is a series RLC circuit.


The explanation is somewhat abstract. I need an explanation that actually is able to describe what is happening in the system.

What I came up is this:
For the inductor :
Greater R causes slower current change, which causes less selfinduction in the inductor. Current is at resonance at ω[SUB]r[/SUB]. In the presence of the resistor (R>0) less selfinduction voltage is produced(compared to if R=0) atω[SUB]r[/SUB], which means in order to regain it, we need to increase the speed of current change, which can be done with the increase in frequency.
For the capacitor:
Greater R causes condensator to charge slower => if R>0, then condensator is charged slower, which in essence means that less charge is able to settle on the plates of the condensator before it is forced to move away, which means in order to gain prior situation (where more charge was able to settle on the plates), we need to increase the time condensator can charge up, which means we need to reduce the frequency.

comments, please
 
lammas,

It is a series RLC circuit.

Ah, you finally revealed what you should have done in your first post. Namely, that it was a series circuit.

What makes you think the resistance of a series circuit influences its resonant frequency? It doesn't. Show why you think differently, and we can go on from there.

Ratch
 
There are three frequencies discussed:

1)ω[SUB]r[/SUB] - resonance frequency for the current (obviously the same for the whole series)
2)ω[SUB]L[/SUB] - resonance frequency for the inductor's voltage
3)ω[SUB]C[/SUB] - resonance frequency for the capacitor's voltage

The first one doesn't change, the two latter do as R goes up.

I've thought about it more and it comes down to this question:
Why by increasing frequency from ω[SUB]r[/SUB] a little bit (when R>0), the effect of selfinduction increasing (due to the fact that current change becomes faster)is proportionally greater to inductor's voltage change rather than the effect of pulling away charge faster from the capacitor's plates? And viceversa...
 
lammas,

There are three frequencies discussed:

1)ωr - resonance frequency for the current (obviously the same for the whole series)
2)ωL - resonance frequency for the inductor's voltage
3)ωC - resonance frequency for the capacitor's voltage

The first one doesn't change, the two latter do as R goes up.

I have no idea what those three terms listed mean. Could you provide some link or other material that discusses what they are? You see, resistance has no resonant frequency. Neither does L or C unless they are considered together.

Ratch
 
lammas,
I have no idea what those three terms listed mean. Could you provide some link or other material that discusses what they are? You see, resistance has no resonant frequency. Neither does L or C unless they are considered together.
Ratch

Yes - that`s what I have explained to lammas in my post#2: There can be two different MAXIMA (wL and wC) - but that`s not any resonance effect at all.
 
Hello again,


I think what he is talking about are the resonant peaks, which are sometimes called the "resonance" just like the more familiar resonance which just involves L and C.

There are three types of resonance for a circuit like this, but unfortunately i cant remember if this is the same thing as what he is talking about, and those three (he may not be talking about) are related by a mathematical formulation. But what he is *actually* asking for is simple to calculate really, by just analyzing the circuit excited by an arbitrary amplitude voltage source. In this way, we can calculate three different resonant peaks as requested:
one for the current through the circuit,
one for the voltage across the capacitor,
and one for the voltage across the inductor.

The one for the current through the circuit is the more familiar one that only involves L and C we'll call F0 (F zero).
The one for the capacitor is slightly lower than that we'll call FC.
The one for the inductor is slightly higher than F0, well call FL.

So for a network with 1 ohm, 1mH, and 10uf, we'll see an F0 around 1592Hz (calculate this yourself to get more accurate), and for FC we'll see a few Hertz lower, and for FL we'll see a few Hertz higher. Note this should show up in a simulation but may be hard to spot because the three frequencies may be quite close together, especially for some choices of R, L, and C.

So what this means for an application that has to be tuned for the resonant peak of the capacitor voltage is that the usual calculation using just L and C will not provide the true accurate highest value of peak voltage across the cap. It's usually close but it depends on the circuit values.

The "proof" however does not come from intuition about how the circuit works, it comes from a down to earth circuit analysis. You have to get your circuit analysis tools out to get to the real truth here, that's what they are made for. There can be many analogies that will still not really explain how it works. For example, the capacitor is an open circuit to DC and short to infinite frequency AC, so the natural response is to have more voltage across it for lower frequencies and less for higher frequencies. The inductor is just the opposite: more voltage for higher frequencies and less for lower frequencies. But does this really help? Not too much but you be the judge.
 
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Hello again,

...knowing the terms and definitions can always help to avoid misunderstandings and misinterpretations.
Therefore - what is the thing called "resonance"?

According to my understanding a frequency dependent electronic circuit shows a "resonance effect" at that frequency for which the resulting impedance is REAL.
With other words: Phase shift of 0 deg between voltage and current - leading to a "resonant peak" because imaginary parts cancel out each other.
However, this has nothing to do with other effects, which may lead to a local voltage or current maximum.
Example: Gain peaking for an opamp with feedback and a phase margin lower than 60 deg.
 
Mr Al. understands the problem best. I am not familiar with the terminology in english so sorry about that.

If it is really about analysis with formulas then I am back where I started - have to do math to get there. But that doesn't actully explain it, it just describes when things happen. The formulas in my first post are the result of analysis, but they don't tell us a clear story which we could put into words and visualize clearly. At least I'm not able to...
 
lammas,

don`t forget that there are good reasons for describing electronic-based effects with formulas.
There are many special effects, which cannot be explained sufficiently with words and/or visualizations only.
For my opinion, this is true in particular if phase excursions are involved - as is the case in your example.
You have a circuit with one real (R) and two imaginary impedances (L,C) - and the magnitudes as well as the phase excursions of both imag. units behave differently (contrarotating).
Therefore, to understand quantitatively what`s really going on (peak yes/no and where?) we cannot avoid to use some mathematical descriptions (formulas).

Are YOU able to realize by thinking only how different phase and magnitude values are to be combined - dependent on frequency?

In summary, I understand and appreciate your desire to explain some effects using words/visualizations only. However, this seems to be not always possible.
In some cases this might work assuming extreme conditions. This was my approach as mentioned in post#2.
 
Mr Al. understands the problem best. I am not familiar with the terminology in english so sorry about that.

If it is really about analysis with formulas then I am back where I started - have to do math to get there. But that doesn't actully explain it, it just describes when things happen. The formulas in my first post are the result of analysis, but they don't tell us a clear story which we could put into words and visualize clearly. At least I'm not able to...

Hi again,


There's a Phd paper on the web somewhere that explains the three resonant points but it's been a long time since i read it (maybe 10 years) so you'll have to do a search if you want to look at that method. It's a bit different than just calculating the three different circuit responses though.

The resonant point that we usually think of where the reactances cancel is usually referred to as the "physical resonance". But the others have significance too because after all we often want to know the frequency where we get the maximum response at a point in a circuit, and then we may not care whether or not the reactances have canceled or not, we just want that maximum response. Low power filters would be one example. The paper above shows how these resonant points are related.

But what you are looking for is intuition, and i think you can get a really good idea about this by following Winterstones suggestions which means thinking about the circuit as one or more components change value. In fact, if you analyze the circuit mathematically more carefully (check your results from the first post where you may be missing a square root in one of your results and possibly a sign change) you can then start to compute the resonant points as the resistance R changes from zero to some reasonable value like 10 ohms or maybe even higher. Dont be afraid to plug in some values for L and C and then let R change as you plot the three resonant points. You'll see how the points change as R changes and that will in fact give you great insight into how R changes things.
As to understanding the basic underlying reasons, i think you were correct when you suggested that it would be a good idea to consider only two components at a time rather than all three at once: R with L, then R with C, then L with C, and see what the difference is. One of the main resonant points is when L and C cancel which means we only have R left in the circuit.
This is often a good idea with other circuits too and is often done in order to figure out if something very undesirable might happen in a circuit if a component goes to one extreme of it's tolerance. Then we call it a "sensitivity analysis" where we usually only let the component value vary a fraction of it's original value, but it's the same idea we just let the component under question (here it is R) change by a much larger amount.
Try this and see how it helps. You already (almost) have the formulas, so why not use them. That's part of how the design process works anyway.
 
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To the Ineffable All,

Only in parallel resonant circuits are there three resonance points. They are the frequency where the inductive and capacitive reactance are equal, the frequency where the impedance is the greatest, and the frequency where the voltage and current are in phase with each other. If the Q of the parallel resonant circuit is greater than 10, then those three frequency points are quite close together. Series resonant circuits, as the OP has said he is inquiring about, have only one resonant point. Unless the OP can produce some documentation that shows multifrequency resonance for series circuits, everyone is spinning their wheels postulating about it.

Ratch
 
To the Ineffable All,

Only in parallel resonant circuits are there three resonance points. They are the frequency where the inductive and capacitive reactance are equal, the frequency where the impedance is the greatest, and the frequency where the voltage and current are in phase with each other. If the Q of the parallel resonant circuit is greater than 10, then those three frequency points are quite close together. Series resonant circuits, as the OP has said he is inquiring about, have only one resonant point. Unless the OP can produce some documentation that shows multifrequency resonance for series circuits, everyone is spinning their wheels postulating about it.

Ratch


Hi Ratch,


You're kidding right? :)

Both the OP and myself have calculated the three resonant peaks and that's exactly what the OP wanted. We've come up with formulas (similar to the OP original post) and calculated three frequencies that are different from each other. I've also included a numerical example which helps to illustrate this. If you took the time to do the calculations you'd have to find the same thing too.

You are stating here that there is only one resonant point in a series circuit yet you are willing to accept further documentation that shows multi-frequency resonance for series circuits, so apparently you are not completely sure to begin with. To be sure, simply do a circuit analysis and compute the voltage across all three elements excited by an AC voltage source. You'll definitely see the voltages across each element peak at three different frequencies. We've done this already so that's where we get the result (note one of the OP's original formulations is not correct however and needs adjustment, but then it definitely works).

If i remember right for the example i provided, i got frequencies of 1587, 1592, and 1595 Hertz, or thereabouts.

Here are the more exact numerical calculations and formulas:
w for vC max: w=sqrt((2*L-C*R^2)/(2*C*L^2)), f=1587.565571276712
w for vR max: w=sqrt(1/(L*C)), f=1591.549430918954
w for vL max: w^2=-2/(C^2*R^2-2*L*C), f=1595.543287715288
 
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Hi Ratch and MrAl,

as both of you speak about more than only one single "resonant point" would you kindly comment my claims as contained in post#10?
I must confess that I still think that any RLC circuitry (series oder parallel) can have only ONE point of resonance - based on the common definition of resonance ("hopelessly pedantic").
As my example shows (opamp amplifier with amplitude peaking in the vicinity of loop gain 0 dB) a local maximum is not always identical to a resonant point.
W.

by the way: Merry Christmas to both of you.
 
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MrAL,

You're kidding right? :)

Nope.

Both the OP and myself have calculated the three resonant peaks and that's exactly what the OP wanted. We've come up with formulas (similar to the OP original post) and calculated three frequencies that are different from each other. I've also included a numerical example which helps to illustrate this. If you took the time to do the calculations you'd have to find the same thing too.

Were the three resonant points for a series circuit? I have calculated plenty of series resonant circuits. There can only be one series resonant frequency.

You are stating here that there is only one resonant point in a series circuit yet you are willing to accept further documentation that shows multi-frequency resonance for series circuits, so apparently you are not completely sure to begin with.

Wait a minute. Which post did I say I accepted mult-frequency resonance for series circuits? In fact, I specifically said otherwise.

To be sure, simply do a circuit analysis and compute the voltage across all three elements excited by an AC voltage source. You'll definitely see the voltages across each element peak at three different frequencies. We've done this already so that's where we get the result (note one of the OP's original formulations is not correct however and needs adjustment, but then it definitely works).

We must be talking about cross topics here, although I don't see how. Would you attach a schemat or a good verbal description of a series circuit, and include your calculations for multi-resonant points?

Ratch
 
Winterstone,

...knowing the terms and definitions can always help to avoid misunderstandings and misinterpretations.
Therefore - what is the thing called "resonance"?

My understanding for series resonance is when the voltage due to the sum of the inductive reactance is equal to the sum of the capacitive reactance. There is only one frequency in a series circuit where than can happen.

My understanding for a parallel resonance is, for a three branch circuit each containing only L,C, and R, when the current due to the sum of the inductive reactances is equal to the current due to the sum of the capacitive reactance. Some texts define the frequency of highest impedance or the frequency of in-phase voltage and current as two other resonance points. I really don't agree with those later two definitions. If the resistance is high enough so that the Q is ten or greater, then all three frequencies are quite close together. If a two branch circuit contains inductance in series with resistance in one branch, and capacitance in series with resistance in the other branch, then the values can be adjusted so that the circuit is resonant at all frequencies.

I must confess that I still think that any RLC circuitry (series oder parallel) can have only ONE point of resonance - based on the common definition of resonance ("hopelessly pedantic").
As my example shows (opamp amplifier with amplitude peaking in the vicinity of loop gain 0 dB) a local maximum is not always identical to a resonant point.

By the definition of resonance being the cancellation of reactances, I agree with you. The mult-frequency points are due to series-parallel circuit topology.

Ratch
 
Because inductors tend to be substantially more lossy than capacitors, parallel resonant circuits are often shown with a resistor in series with the inductor, but not the capacitor. See the first circuit in this link:

https://www.opamp-electronics.com/tutorials/resonance_in_series_parallel_circuits_2_06_05.htm

But one sometimes finds a resistor in series with the capacitor as well (as shown in the second circuit of the link). And, there can be other topologies as well.

If the circuit is in a black box with only two terminals, only the behavior at the terminals is accessible. It wouldn't be possible to measure the voltage across the capacitor or inductor unless the topology is such that the particular component is across the black box terminals. It would be reasonable then to choose definitions of resonance that involved only what could be measured at the terminals.

In the case of a purely series resonant circuit with a single resistor, capacitor and inductor in series, the frequency of minimum impedance (maximum current) is the same as the frequency of zero phase angle between applied voltage and current, so there is only one resonance frequency.

In the case of the second circuit in the link, the frequency of maximum impedance can be different from the frequency of zero phase angle (as measured at the black box terminals), so we could say that there can be two resonance frequencies, using these two common definitions of resonance.
 
...... so we could say that there can be two resonance frequencies, using these two common definitions of resonance.

Two definitions? Is each impedance maximum/minimum - according to your definition - always a resonance point? Or only in some specific cases?
What about a Chebyshev filter response with gain peaking? Resonance?
 
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