Resonance of LC

[nForce]

I have a beginners question about resonance of LC circuits:

I know that resonance frequency of just one capacitor and one inductor is sqrt(1/LC). But what if we have a network of n capacitors and m inductors? And they are all mixed connected in parallel and series. What about then?

Somewhere I have read that if we have a mixed way RLC circuit or just LC circuit we find an impedance of the whole circuit and then take just imaginary part and equate with 0. Then we solve for w (omega). But this is just an approximation.

Can someone tell more about this, and explain it? Thank you.

[rfeecs]

As the circuit gets more complicated we start talking about "poles and zeros".  Not really a beginner's concept.  But you can google it and see what you find.  Loosely speaking, poles are the frequencies where the impedance goes to infinity and zeros are the frequencies where the impedance goes to zero.  These characteristic frequencies determine a lot about the circuit behavior.

[Benta]

To get a feel for it, I'd encourage you to try it out yourself on some network with eg, two capacitors and two inductors.

Calculate like if it was four resistors, but instead of using R as impedance, use 1/sC for capacitors and sL for inductors.

When you've found the transfer function, subsistute jw for s and you can do the frequency analysis.

Have fun.

[MrAl]

I have a beginners question about resonance of LC circuits:

I know that resonance frequency of just one capacitor and one inductor is sqrt(1/LC). But what if we have a network of n capacitors and m inductors? And they are all mixed connected in parallel and series. What about then?

Somewhere I have read that if we have a mixed way RLC circuit or just LC circuit we find an impedance of the whole circuit and then take just imaginary part and equate with 0. Then we solve for w (omega). But this is just an approximation.

Can someone tell more about this, and explain it? Thank you.

Hi,

As others have pointed out, as you get into more components the analysis becomes more complicated and because of the way multiple parts work together the interpretation of 'resonance' becomes more application specific.  In fact, even a simple RLC circuit can have up to three points that are deemed the 'resonant' points although the one that is usually studied the most is the physical resonant point.

Add to that the many components may have several points that are locally a peak or dip, and so we find that we may have several points we might call 'resonant'.  Depending on the application some or all may matter, or only one.  In control theory for example often the lowest resonant point is the dominant so we may pay a lot of attention to that one while ignoring the rest.

The best bet here is to follow along the footsteps of everyone else that came though your path already, and that is to first study in detail the series RLC circuit and the parallel RLC circuit.  That will get you pretty far.  Once you get comfortable with that you may want to move on to a dual RLC circuit and see what happens.

if you know circuit analysis you can check out each circuit and try to find out the important points of each.  In particular, AC circuit analysis, which is pretty simple if you know how to analyze DC circuits with voltage sources and resistors.  A technique like Nodal Analysis is very general so you should try to learn that and then you can go a lot farther.  Of course you can always try a simulator like the free LT Spice simulator and look for peaks and dips in the frequency response.

Good luck to you.

[nForce]

As the circuit gets more complicated we start talking about "poles and zeros".  Not really a beginner's concept.  But you can google it and see what you find.  Loosely speaking, poles are the frequencies where the impedance goes to infinity and zeros are the frequencies where the impedance goes to zero.  These characteristic frequencies determine a lot about the circuit behavior.

I have another question here tho. Do we find poles and zeros just at LC circuits in rational function? What about RLC circuits, is it different? Can we also find zeros and poles at RLC circuit?

Thanks. 

[rfeecs]

I have another question here tho. Do we find poles and zeros just at LC circuits in rational function? What about RLC circuits, is it different? Can we also find zeros and poles at RLC circuit?

Yes.  RLC circuits are treated just like LC circuits as far as poles and zeros.

[Ratch]

I have a beginners question about resonance of LC circuits:

I know that resonance frequency of just one capacitor and one inductor is sqrt(1/LC). But what if we have a network of n capacitors and m inductors? And they are all mixed connected in parallel and series. What about then?

Somewhere I have read that if we have a mixed way RLC circuit or just LC circuit we find an impedance of the whole circuit and then take just imaginary part and equate with 0. Then we solve for w (omega). But this is just an approximation.

Can someone tell more about this, and explain it? Thank you.

You should specify whether you are referring to resonance in series or parallel circuits. For both series and parallel circuits, resonance occurs when the orthogonal component of the series voltage sums to zero, or parallel current orthogonal components sum to zero.  In series resonance, the lowest impedance occurs at resonance, and the voltage-current phase is zero.  In parallel resonance, there are three frequencies of note.  The frequency of resonance, the frequency of maximum impedance, and the frequency of zero voltage-current phase.  If the Q of the circuit is high (greater than 10), the three frequencies will be very close together.  It is also possible to make a parallel circuit resonant at all frequencies provided resistance is present in the circuit.  Furthermore, parallel resonance can be made to occur at two different values of L or C if resistance is present. 

Ratch

[MrAl]

As the circuit gets more complicated we start talking about "poles and zeros".  Not really a beginner's concept.  But you can google it and see what you find.  Loosely speaking, poles are the frequencies where the impedance goes to infinity and zeros are the frequencies where the impedance goes to zero.  These characteristic frequencies determine a lot about the circuit behavior.

I have another question here tho. Do we find poles and zeros just at LC circuits in rational function? What about RLC circuits, is it different? Can we also find zeros and poles at RLC circuit?

Thanks. 

Hi,

Stationary poles and zeros can be found by doing some factoring of the expression of interest to get the denominator or numerator into a certain form.  You can also do a root locus plot which tells how the poles and zeros migrate in some circuits.

[nForce]

There is an important aspect that we forget.

If I compute the impedance of the circuit containing L and C elements and equate that with zero. Will I get a current resonance or voltage?

I still don't understand, are the resonant frequencies poles or zeros in rational function?

[CraigHB]

The poles are where the transfer function tends toward infinity.  Those are the terms in the function's denominator that can go to zero.  In science a zero denominator is never considered an infinite result to an equation, but rather a failure of the equation to be meaningful.  In an actual circuit, the poles indicate where instabilities occur in the frequency response.  For a circuit on paper with zero resistance, that results in oscillations of ever increasing magnitude.  In an actual circuit there's always some resistance and a limit in voltage magnitude so it can never actually do that.  Usually it will increase in magnitude until the circuit fails or it will get to some arbitrary limit and stay there.  Can happen easily with control systems that have several poles in their functions.

[Ratch]

There is an important aspect that we forget.

If I compute the impedance of the circuit containing L and C elements and equate that with zero. Will I get a current resonance or voltage?

I still don't understand, are the resonant frequencies poles or zeros in rational function?

I don't think you read post #5 of this thread carefully.  There, I explained that resonance occurs at the frequency where the reactance is zero.  You determine resonance by by finding the frequency at which the reactance is zero.  Also, you are getting side-tracked by poles, zeros, and the transfer functions.  You seem to be confused about what resonance  is.  Why don't you post a schematic of a simple circuit and we can figure it out from there.  Ask some more questions and analyze the answers carefully.

Ratch

[rstofer]

If there is a circuit, model it in LTSpice XVII and do an AC analysis over varying frequency.  Then you will have something specific to talk about.  Otherwise, it's Laplace Transforms, poles and zeros and other truly ugly math and we still don't have a circuit to model.

[rfeecs]

Look a the original question:

I know that resonance frequency of just one capacitor and one inductor is sqrt(1/LC). But what if we have a network of n capacitors and m inductors? And they are all mixed connected in parallel and series. What about then?

I'm thinking that sounds like a filter, for example:


So what about then?  What's the resonant frequency?  How do we deal with it?

This is why I said we don't think about resonant frequency any more, but poles and zeros.

As I said, not really a beginner's concept, but if you want to go down that rabbit hole, you know what to search for.

Maybe Ratch has another idea on how to find the resonant frequencies of the above circuit.  A demonstration perhaps.

[Ratch]

Look a the original question:

I know that resonance frequency of just one capacitor and one inductor is sqrt(1/LC). But what if we have a network of n capacitors and m inductors? And they are all mixed connected in parallel and series. What about then?

I'm thinking that sounds like a filter, for example:


So what about then?  What's the resonant frequency?  How do we deal with it?

This is why I said we don't think about resonant frequency any more, but poles and zeros.

As I said, not really a beginner's concept, but if you want to go down that rabbit hole, you know what to search for.

Maybe Ratch has another idea on how to find the resonant frequencies of the above circuit.  A demonstration perhaps.

That circuit is not resonant, and does not depend on resonance to work.  High frequency is  blocked with inductance and shorted by capacitance.  Just like a non-resonant RC filter does, but this circuit is more effective. 

Ratch

[rfeecs]

That circuit is not resonant, and does not depend on resonance to work.  High frequency is  blocked with inductance and shorted by capacitance.  Just like a non-resonant RC filter does, but this circuit is more effective. 

Ratch

It depends on the definition of resonant.  That circuit is a fifth order Chebychev low pass filter.  At two points in the pass band, the insertion loss goes to zero (dB) in the ideal case, close to zero in this case.  At those two points, the input impedance is purely real.  All of the inductive and capacitive reactances cancel.  So you could say the circuit is resonant.

This fits with nForce's original procedure of finding the frequencies where the imaginary part of the impedance goes to zero.

What this has to do with poles and zeros, I don't know.  My point was just that for complex RLC circuits, people tend to focus more on the frequencies of the poles and zeros as defining circuit behavior, rather than resonant frequencies.

[Ratch]

That circuit is not resonant, and does not depend on resonance to work.  High frequency is  blocked with inductance and shorted by capacitance.  Just like a non-resonant RC filter does, but this circuit is more effective. 

Ratch

It depends on the definition of resonant.  That circuit is a fifth order Chebychev low pass filter.  At two points in the pass band, the insertion loss goes to zero (dB) in the ideal case, close to zero in this case.  At those two points, the input impedance is purely real.  All of the inductive and capacitive reactances cancel.  So you could say the circuit is resonant.

This fits with nForce's original procedure of finding the frequencies where the imaginary part of the impedance goes to zero.

What this has to do with poles and zeros, I don't know.  My point was just that for complex RLC circuits, people tend to focus more on the frequencies of the poles and zeros as defining circuit behavior, rather than resonant frequencies.

I calculated the reactance of the circuit from 0 to 2 GHz and plotted it.  I used r=50 for the source and load impedance.  It looks like resonance occurs at three points other than zero.  The reactance appears to be negative most of the time.  The calculation was too long to show here.  Unfortunately, i don't seem to be able to post the plot, but it shows reactance at zero at 0.55 Ghz, 1.02 GHz, and 1.22 GHz.

Ratch

[nForce]

So if I understand correctly, we can find resonance frequencies with transfer function or by solving Im(Z) = 0?

I don't know what I do with the "j" term? Even if I rationalize the denominator, I still have the "j" term somewhere.



And Ratch what do you mean by series or parallel circuits, that there's a difference. We can have a mixed parallel and series elements.

[rfeecs]

I calculated the reactance of the circuit from 0 to 2 GHz and plotted it.  I used r=50 for the source and load impedance.  It looks like resonance occurs at three points other than zero.  The reactance appears to be negative most of the time.  The calculation was too long to show here.  Unfortunately, i don't seem to be able to post the plot, but it shows reactance at zero at 0.55 Ghz, 1.02 GHz, and 1.22 GHz.

Ratch

Hmm.  That's not what I get.  I get zero reactance at about 81MHz, 95MHz, 162MHz, and 196MHz.  I used LTSPICE.  File is attached.  Also attached is a plot of Zin in the real-imaginary plane.

[rfeecs]

Here's the LTSPICE file.

[orolo]

I get zero reactance at about 81MHz, 95MHz, 162MHz, and 196MHz.

I agree. The impedance can be computed as a continued fraction:

\$ z \quad = \quad \displaystyle 50 + \frac{1}{\displaystyle 18\cdot 10^{-12}s + \frac{1}{\displaystyle 65\cdot 10^{-9}s + \frac{1}{\displaystyle 33\cdot 10^{-12}s + \frac{1}{\displaystyle 65\cdot 10^{-9} + \frac{1}{\displaystyle 18\cdot 10^{-12}s + \frac{1}{\displaystyle 50}}}}}} \$

Making s = 2*pi*I*f and drawing semilog plots of module and argument vs frequency to 2GHz, we get what one would expect from a Chebishev filter: 100 omhs in the passband, 50 ohms well beyond it, and a maximum near 300 ohms at almost 200MHz.

Comparting this to a resonator, maybe it looks like a very lossy and low Q one, with series resonance near 160MHz, and parallel resonance at near 200MHz. That's far fetched, after all this is a low pass filter.

[MrAl]

That circuit is not resonant, and does not depend on resonance to work.  High frequency is  blocked with inductance and shorted by capacitance.  Just like a non-resonant RC filter does, but this circuit is more effective. 

Ratch

It depends on the definition of resonant.  That circuit is a fifth order Chebychev low pass filter.  At two points in the pass band, the insertion loss goes to zero (dB) in the ideal case, close to zero in this case.  At those two points, the input impedance is purely real.  All of the inductive and capacitive reactances cancel.  So you could say the circuit is resonant.

This fits with nForce's original procedure of finding the frequencies where the imaginary part of the impedance goes to zero.

What this has to do with poles and zeros, I don't know.  My point was just that for complex RLC circuits, people tend to focus more on the frequencies of the poles and zeros as defining circuit behavior, rather than resonant frequencies.

I calculated the reactance of the circuit from 0 to 2 GHz and plotted it.  I used r=50 for the source and load impedance.  It looks like resonance occurs at three points other than zero.  The reactance appears to be negative most of the time.  The calculation was too long to show here.  Unfortunately, i don't seem to be able to post the plot, but it shows reactance at zero at 0.55 Ghz, 1.02 GHz, and 1.22 GHz.

Ratch

Hi there,

Yeah the definition of resonance is a little funny sometimes.  The way i like to explain it is it is like where one view concentrates on the energy exchange and the other view concentrates on the electrical circuit behavior as observed in a given application circuit.

The physicist guy says that resonance occurs with a certain exchange of energy, but the radio tuner guy says it occurs when his tank circuit peaks or dips.  Still yet another guy unified these two views by relating the modes of resonance to Cassini Ovals.

[nForce]

Sorry if this was already answered, but no one has answered these questions:

So if I understand correctly, we can find resonance frequencies with transfer function or by solving Im(Z) = 0?

I don't know what I do with the "j" term? Even if I rationalize the denominator, I still have the "j" term somewhere.


And Ratch what do you mean by series or parallel circuits, that there's a difference. We can have a mixed parallel and series elements.

Try answering in this manner: Yes this is true, because... (or) No, this is not true because... I don't understand these abstract answers. Thanks again.

[rfeecs]

Sorry if this was already answered, but no one has answered these questions:

So if I understand correctly, we can find resonance frequencies with transfer function or by solving Im(Z) = 0?

Yes, this seems to be one definition of resonance, because at that point the capacitive reactance and inductive reactance cancel each other.  I'm sure you saw the Wikipedia article:
https://en.wikipedia.org/wiki/RLC_circuit
It discusses a lot of terms like natural frequency and driven resonance frequency.

I don't know what I do with the "j" term? Even if I rationalize the denominator, I still have the "j" term somewhere.

I don't understand what you are asking.

[nForce]

Well with "j" I mean the imaginary unit. Because the impedance of inductor is "jwL" and impedance of capacitor is "1/(jwC).

[rfeecs]

What do you mean by

I don't know what I do with the "j" term?