Resonance of LC

[Ratch]

rfeecs,

Here is the plot I made of the first circuit you posted earlier.  I suppressed the output of the calculations after the first two components because the output contained too
many terms to display.  After the calculation was complete, there were over 100 terms for the j-component alone. That circuit must be very sensitive because just rounding off the terms caused the plot to vary significantly.  The "%" sign in the calculations is the symbol for the previous result.  I made two plots of the reactance, 0-1GHz and 0-2Ghz. As you can see, I started from the right alternating between parallel and series.  I hope I did it right.

Ratch

[Ratch]

To the Ineffable All,

The two attached files contain two pages from the Schaum's Outline Series, Electric Circuits.  It contains a concise description of parallel resonance.  It also shows how two different values of a L or C component can have the same resonant frequency, and how the circuit can be resonant at all frequencies or not resonant at any frequency.  Enjoy.

Ratch

[The Electrician]

rfeecs,

Here is the plot I made of the first circuit you posted earlier.  I suppressed the output of the calculations after the first two components because the output contained too
many terms to display.  After the calculation was complete, there were over 100 terms for the j-component alone. That circuit must be very sensitive because just rounding off the terms caused the plot to vary significantly.  The "%" sign in the calculations is the symbol for the previous result.  I made two plots of the reactance, 0-1GHz and 0-2Ghz. As you can see, I started from the right alternating between parallel and series.  I hope I did it right.

Ratch

It looks to me like your results are in radians/sec, not Hz.

[MrAl]

Just to add more confusion, the attached circuit clearly has a resonance, but the impedance is purely real, equal to 50 ohms.

So finding resonances by looking at the impedance doesn't always work.

That is an example of a circuit which is resonant at all frequencies.  I mentioned such a circuit in post #34 of this thread.

Ratch

How is it resonant at all frequencies?  Explain that one.

Oh, because the reactance is zero at all frequencies, maybe?  Then a resistor is resonant at all frequencies.

I don't get it.  I think it has a very definite resonance at one frequency, around 159MHz.

OK, I guess I see it now.  If you look at the two main arms of the circuit, the capacitive and inductive susceptance are equal and opposite at all frequencies and they cancel out.

I guess you could say that is resonating at all frequencies.  Very strange way to look at it.  But OK.

Hi,

Which circuit are you talking about here?

[Ratch]

rfeecs,

Here is the plot I made of the first circuit you posted earlier.  I suppressed the output of the calculations after the first two components because the output contained too
many terms to display.  After the calculation was complete, there were over 100 terms for the j-component alone. That circuit must be very sensitive because just rounding off the terms caused the plot to vary significantly.  The "%" sign in the calculations is the symbol for the previous result.  I made two plots of the reactance, 0-1GHz and 0-2Ghz. As you can see, I started from the right alternating between parallel and series.  I hope I did it right.

Ratch

It looks to me like your results are in radians/sec, not Hz.

Correct, I probably should have plotted the curve in hertz.

Ratch

[Ratch]

Just to add more confusion, the attached circuit clearly has a resonance, but the impedance is purely real, equal to 50 ohms.

So finding resonances by looking at the impedance doesn't always work.

That is an example of a circuit which is resonant at all frequencies.  I mentioned such a circuit in post #34 of this thread.

Ratch

How is it resonant at all frequencies?  Explain that one.

Oh, because the reactance is zero at all frequencies, maybe?  Then a resistor is resonant at all frequencies.

I don't get it.  I think it has a very definite resonance at one frequency, around 159MHz.

OK, I guess I see it now.  If you look at the two main arms of the circuit, the capacitive and inductive susceptance are equal and opposite at all frequencies and they cancel out.

I guess you could say that is resonating at all frequencies.  Very strange way to look at it.  But OK.

Hi,

Which circuit are you talking about here?

The one depicted in posts #12 and #13.

Ratch

[MrAl]

Just to add more confusion, the attached circuit clearly has a resonance, but the impedance is purely real, equal to 50 ohms.

So finding resonances by looking at the impedance doesn't always work.

That is an example of a circuit which is resonant at all frequencies.  I mentioned such a circuit in post #34 of this thread.

Ratch

How is it resonant at all frequencies?  Explain that one.

Oh, because the reactance is zero at all frequencies, maybe?  Then a resistor is resonant at all frequencies.

I don't get it.  I think it has a very definite resonance at one frequency, around 159MHz.

OK, I guess I see it now.  If you look at the two main arms of the circuit, the capacitive and inductive susceptance are equal and opposite at all frequencies and they cancel out.

I guess you could say that is resonating at all frequencies.  Very strange way to look at it.  But OK.

Hi,

Which circuit are you talking about here?

The one depicted in posts #12 and #13.

Ratch

Hi,

No the circuit that you said was "resonant at all frequencies".
Where is that one?

[The Electrician]

Just to add more confusion, the attached circuit clearly has a resonance, but the impedance is purely real, equal to 50 ohms.

So finding resonances by looking at the impedance doesn't always work.

That is an example of a circuit which is resonant at all frequencies.  I mentioned such a circuit in post #34 of this thread.

Ratch

How is it resonant at all frequencies?  Explain that one.

Oh, because the reactance is zero at all frequencies, maybe?  Then a resistor is resonant at all frequencies.

I don't get it.  I think it has a very definite resonance at one frequency, around 159MHz.

OK, I guess I see it now.  If you look at the two main arms of the circuit, the capacitive and inductive susceptance are equal and opposite at all frequencies and they cancel out.

I guess you could say that is resonating at all frequencies.  Very strange way to look at it.  But OK.

Hi,

Which circuit are you talking about here?

The one depicted in posts #12 and #13.

Ratch

Hi,

No the circuit that you said was "resonant at all frequencies".
Where is that one?

Reply #39

[MrAl]

Hi,

Thanks, and i wont quote all that text again here :-)

Yeah that circuit is interesting from a theoretical standpoint, but we should probably look at the practical significance also.  I have a feeling the practical circuit when in use for a real application will have a different overall effect.  I have to say though that of course i cant think of every possible use under the sun.  For example, if someone wants to create a very complicated resistor then i cant stop them from doing so :-)  Of course to the contrary is if there is an output that would mean constant loading to the previous stage, which would of course be more than very practical.

[snarkysparky]

resonance is where there are complex conjugate pole pairs in the denominator of the transfer function.  The resonant points are I think the real part of the pole.

[MrAl]

resonance is where there are complex conjugate pole pairs in the denominator of the transfer function.  The resonant points are I think the real part of the pole.

Hi,

Well i think what some people in this thread have been doing is just looking at the imaginary part of the transfer function and finding when that goes to zero, because that indicates physical resonance.
What i was doing was trying to point out that the resonance found that way may not be the only resonance and other methods may have to be employed.  In fact, that view may end up being the most insignificant relative to the operation of the circuit when used in an application.  Also, we run into special theoretical examples where the outcome is far from a practical one.  This happens more than we might like because theory does not transfer immediately to the practical with even the tiniest imperfection sometimes.  There's no doubt though that theory may uncover a good way to do things when we take measures to reduce the imperfections or use it in a way that takes advantage of that theoretical aspect.

[rfeecs]

Of course to the contrary is if there is an output that would mean constant loading to the previous stage, which would of course be more than very practical.

Yes, one application for that type of circuit is a reflectionless filter:
https://www.microwaves101.com/15-homepage/1317-reflection-less-filters

While most filters reflect power in the stopband, this type absorbs the stopband power and presents a constant impedance.

[rfeecs]

To the Ineffable All,

The two attached files contain two pages from the Schaum's Outline Series, Electric Circuits.  It contains a concise description of parallel resonance.  It also shows how two different values of a L or C component can have the same resonant frequency, and how the circuit can be resonant at all frequencies or not resonant at any frequency.  Enjoy.

Ratch

The admittance of LC circuit presented there (attached) in general has two poles and two zeros.  The poles are at -1/RC and -L/R.  To make it "resonant at all frequencies", the R,L,C are chosen to set the frequencies of the zeros to be the same as the poles.  So all frequency dependent terms cancel.  But that also means that all the poles and zeros are real (no complex conjugate pairs).  In the time domain response, you just have decaying exponential responses, no sinusoids with energy bouncing back and forth.

It seems to me this another example of satisfying a mathematical definition of resonance, but not really what most people think of a resonant circuit.

It is also an example where looking at the poles and zeros provides a better insight into what the circuit is really doing.

[Ratch]

resonance is where there are complex conjugate pole pairs in the denominator of the transfer function.  The resonant points are I think the real part of the pole.

Hi,

Well i think what some people in this thread have been doing is just looking at the imaginary part of the transfer function and finding when that goes to zero, because that indicates physical resonance.
What i was doing was trying to point out that the resonance found that way may not be the only resonance and other methods may have to be employed.  In fact, that view may end up being the most insignificant relative to the operation of the circuit when used in an application.  Also, we run into special theoretical examples where the outcome is far from a practical one.  This happens more than we might like because theory does not transfer immediately to the practical with even the tiniest imperfection sometimes.  There's no doubt though that theory may uncover a good way to do things when we take measures to reduce the imperfections or use it in a way that takes advantage of that theoretical aspect.

As I mentioned in post #6 of this thread, some folks think that besides zero reactance, an alt right definition of parallel resonance is the frequency where the phase is zero or the impedance is maximum.  At high Q levels, these three frequencies are quite close to each other.  I believe that zero reactance is the proper definition and the other two are spurious.  There is no ambiguity in series resonance because all definitions have the same frequency.

Ratch

[kulky64]

I manually derived transfer function for circuit in Reply #39:
\$ Z(s)=\frac{V(s)}{I(s)}=\frac{R_1L_1C_1L_2C_2s^4+(R_1R_2L_1C_1C_2+L_1L_2C_2)s^3+(R_1L_1C_1+R_1L_2C_2+R_2L_1C_2)s^2+(R_1R_2C_2+L_1)s+R_1}{L_1C_1L_2C_2s^4+(R_1+R_2)L_1C_1C_2s^3+(L_1C_1+L_1C_2+L_2C_2)s^2+(R_1+R_2)C_2s+1} \$ ,
and then in Matlab calculated poles, zeros and gain for component values as given in Reply #39. From there it is clear why this circuit behaves as pure resistance at all frequencies. It has two equal pairs of complex conjugate poles and exactly the same zeros. So zeros in numerator cancel out with poles in denominator, remains only real 50 ohms.

[MrAl]

Hi,

Well one striking characteristic of the circuit as discussed so far is the criterion:
1/sqrt(L1*C1)=1/sqrt(L2*C2) [which of course simplifies]

Change that relationship a little and see what happens :-)

[kulky64]

Only L1*C1=L2*C2 is not sufficient condition for this circuit to work, there are total 4 conditions that must be met:
L2=4*L1
C2=(1/4)*C1
R1=2*sqrt(L1/C1)
R2=R1

So you can choose arbitrarily only value of L1 and C1, other component values must be calculated according to these conditions. Break any one of them and this circuit will no longer be resistive for all frequencies.

[MrAl]

Hi,

Perhaps so, but what jumps right out at you is L1*C1=L2*C2.  Right off we see someone was up to something here :-)

There could be other interesting things too.

[nForce]

I think, it's better if I give a simple example of a circuit, and my attempt to solve:

http://shrani.si/f/k/oB/4jLK5yDq/circuit.jpg

I don't know how to insert Latex code here, so I have photograph the "solution". How to continue here?

I think you have a sign error.  An additional problem with that circuit is that the resonance occurs where the impedance goes to infinity.

Sorry, but I have to get to the bottom of this. How do you know for my circuit that the resonance is when the impedance goes to infinity? Please do explain.

And which case is in my example? Because we have parallel and series connected components. It can't be just series resonance or parallel resonance which I read on the net.

Thank you.

[rfeecs]

You circuit has two resonances.  It has a parallel resonance where the impedance goes to infinity, and a series resonance where the impedance goes to zero.  Plots of the impedance are attached showing the two resonance points.

[nForce]

Hey, thanks for these graphs.

How did you make them? Did you calculate the transfer function by hand, and then plot in Octave/Matlab or I don't know what is this.

Can you show me how to continue calculating the impedance...

[rfeecs]

I used a circuit simulator, Keysight ADS, I had handy at work.  You could also use LTSPICE, which is free.  LTSPICE has a .net directive that lets you calculate Z parameters.

Octave or Matlab would also work if you work out the impedance equation by hand.  Any plotting program, even Excel would work in that case.

[Richard Crowley]

Not being a great wiz at math, I always appreciated this Frequency-Reactance Nomograph.
I first saw it on the OpAmp Labs website, but now it is available at: RF Cafe...

http://www.rfcafe.com/references/electrical/frequency-reactance-nomograph.htm

[kulky64]

You circuit has two resonances.  It has a parallel resonance where the impedance goes to infinity, and a series resonance where the impedance goes to zero.  Plots of the impedance are attached showing the two resonance points.

How did you get these plots? Phase in this circuit can be only +90 deg or -90 deg. Nothing in between.

[rfeecs]

You circuit has two resonances.  It has a parallel resonance where the impedance goes to infinity, and a series resonance where the impedance goes to zero.  Plots of the impedance are attached showing the two resonance points.

How did you get these plots? Phase in this circuit can be only +90 deg or -90 deg. Nothing in between.

I plotted phase of S11 instead of Z11.  Attached I have added phase of Z11.  You are right.  It only takes values of +90 or -90.