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Question in Resonant Frequency

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freebird1401

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Good Day All;

Could someone help me in this question?

Calculate the resonant frequency of the circuit in Figure in the attachment.
 

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Do you know how to write an equation for the network using complex impedance?

Hint: the circuit is a voltage divider.
 
Good Day All;

Could someone help me in this question?

Calculate the resonant frequency of the circuit in Figure in the attachment.


Hello there,


There are a couple of different answers depending on how you interpret 'resonant' frequency.

The one most often quoted is the place where capacitive reactance equals inductive reactance and this is given by the well known formula:
F1=1/(2*pi*sqrt(L*C))
or you could simply solve for F1 when xL=xC.

The second point of interest that could also be called a resonant point is when the current from the source reaches max, and this second frequency would be given by:
F2=sqrt((R*sqrt(C^2*R^2+2*C*L)-L)/(C^2*L*R^2))/(2*pi)

A third point that could also be called a resonant point is when the voltage across the resistor (and capacitor) reaches a maximum amplitude, and this frequency would be given by:
F3=sqrt((2*C*R^2-L)/(2*C^2*L*R^2))/(2*pi)

This would mean the resonant frequency for your circuit could be any of (approximately, in Hertz):
F1=0.35588, or
F2=0.35567, or
F3=0.35140

In many cases they would ask for F1, but we'd really have to know where you are in your course work to be certain or what the application happens to be.
 
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Convert the parallel R-C to series equivalent R-C. Then calculate resonance of series L-C. This is maximum current resonance.

Since voltage source has zero impedance, anti-resonance (max voltage across resistor) will be parallel resonance of L-C.
 
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It is a trick question. It has no resonance; it is a low pass filter.
 

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It is a trick question. It has no resonance; it is a low pass filter.

Hi Mike,

It might just look like a low pass filter when C is equal to 0.2 femtofarads, but when C is equal to 0.2 Farads (see diagram correction) it is very resonant. In other words, get rid of the capital "F" after the "0.2" for the capacitor and run another simulation.
:)
 

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MrAl,

I agree with your analysis, but I believe F3=0.3514609010Hz. Also another definition for resonance could be the frequency where the imaginary part of the impedance of the total circuit is zero. That would be F4=0.3468701566Hz.

Ratch
 
I would use the frequency at which the phase angle difference between the input and output goes to zero.
 
I would use the frequency at which the phase angle difference between the input and output goes to zero.
I assume you mean 90 degrees?
 
Yep. It is ~90 degrees. Second plot illustrates the Real and Imaginary parts of V(out)/Vin), where V(in)=1@0degrees.
 

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MrAl,

I agree with your analysis, but I believe F3=0.3514609010Hz. Also another definition for resonance could be the frequency where the imaginary part of the impedance of the total circuit is zero. That would be F4=0.3468701566Hz.

Ratch

Hi Ratch,

Oh really...how did you arrive at F3=0.35146 ?
I'll double check my analysis for that condition (maximum Vout where Vout is the voltage across the capacitor and resistor) and get back here.

There may be other frequencies. There is a guy who redefined what resonant frequency really is, and found several candidate points for most circuits. It's based on some interesting math and it's even on the web somewhere, but i cant remember where now :) I suspect a really good search on resonant frequencies would turn it up again.
We probably just confused the OP by now anyway, as this is what usually happens when this question is asked on a web site. There are several different interpretations as well as several different candidate points for each interpretation so it gets confusing sometimes. The one the basic courses usually look for is the power related one where capacitive reactance equals inductive reactance.
 
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I would think the correct resonant frequency would be, if it were put in a feedback loop to form a oscillator, what would be the oscillation frequency? I would expect that to be where the output voltage across the capacitor and resistor is maximum.
 
I would think the correct resonant frequency would be, if it were put in a feedback loop to form a oscillator, what would be the oscillation frequency? I would expect that to be where the output voltage across the capacitor and resistor is maximum.
Not necessarily when discussing just the LC network or crystal network in the feedback path of an oscillator. I have yet to find an amplifier which has EXACTLY 0 or 180 degrees of phase shift... The actual oscillation frequency is a just as much a function of the loop phase as it is the amplitude.

Case in point; three cascaded RC networks and an amp with a gain of ~-27 will oscillate. At the oscillation frequency, the gain through the network is far from where the peak amplitude occurs.
 
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MrAl,

Oh really...how did you arrive at F3=0.35146 ?
I'll double check my analysis for that condition (maximum Vout where Vout is the voltage across the capacitor and resistor) and get back here.

I solved for the frequency where the absolute value of the impedance across L equals the absolute value of the impedance across R||C. I believe that is what you meant. Otherwise, the max voltage across R||C will occur when the frequency is at DC, because the L impedance will be zero, the C impedance will be infinite, and the resistor will 10 ohms. After you check your calculations, and you don't agree, then I will submit a detailed analysis.

There may be other frequencies. There is a guy who redefined what resonant frequency really is, and found several candidate points for most circuits. It's based on some interesting math and it's even on the web somewhere, but i cant remember where now :) I suspect a really good search on resonant frequencies would turn it up again.
We probably just confused the OP by now anyway, as this is what usually happens when this question is asked on a web site. There are several different interpretations as well as several different candidate points for each interpretation so it gets confusing sometimes. The one the basic courses usually look for is the power related one where capacitive reactance equals inductive reactance.

I believe that the frequency F4, where the reactance of the total circuit is zero, is the more "pure" definition. It is also one of the most difficult to calculate by hand. The f = 1/(2*pi*sqrt(LC)) is the easiest to find and is close to the other definitions if the Q of the circuit is not too low. If you come across that web article again, I would appreciate a link.

crutschow,

I would think the correct resonant frequency would be, if it were put in a feedback loop to form a oscillator, what would be the oscillation frequency? I would expect that to be where the output voltage across the capacitor and resistor is maximum.

I don't think the oscillator gives a damn what the voltage across the RC pair is. It will oscillate when the phase shift around the loop is 180° and the gain is > 1.

Ratch
 
Hello again,

Carl:
I hate to get into feedback if we dont have to :) We could look at that too though i guess, might be interesting. I guess we are going to assume that the circuit is going to be used as a filter where the output is across the cap and resistor.

Ratch:
Ok i checked my calculations. I used basic calculus to develop most of the formulas i posted previously (except for the well known one) but of course there is always the chance that i made an error in the math because even though it is basic there are plenty of calculations to do and get them all right. I couldnt find an error anywhere however so i did a basic maximum amplitude test using the formula for the output voltage (across cap and resistor) which isnt too hard to compute:
Vampl=R/sqrt((R-w^2*C*L*R)^2+w^2*L^2)

Using my quoted center frequency for F3 of 0.351404599520409 the maximum amplitude calculates out to be:
VamplMax=4.5003516037041

Now since i want to test this to find out if it is a true max, i look at one point slightly to the left by decrementing F3 by a tiny amount and another point slightly to the right by incrementing F3 by a tiny amount, and if the amplitude is greater than the max calculated above as VamplMax then F3 fails. If the amplitude for both these points is less than VamplMax however, then F3 passes and that means anything greater than F3 (within some small error limit) will produce less output and anything less than F3 will also produce less output. Ok, so first F3 to the left:
F_Left=F3-0.000001=0.351403599520409
and then F3 to the right:
F_Right=F3+0.000001=0.351405599520409
and note that the increment we are using is less than the difference between your quoted value for F3 and my quoted value, but just to make sure we are not missing some resonant point anyway we will define one more frequency:
F3_Test=0.35146
and that is your quoted value for F3, so we will test all three of these frequencies in the amplitude equation and see what we get.

Using all three frequencies we get three different values for w:
w_Left=2*pi*F_Left=2.20793393339665
w_Right=2*pi*F_Right=2.20794649976727
w_Test=2*pi*F_Test=2.20828830806134

and this gives us three different amplitudes but i'll repeat the max amplitude here first for quick reference:

VamplMax=4.5003516037041
Vampl_Left=4.50035160230076
Vampl_Right=4.50035160230075
Vampl_Test=4.50034729587394

Now an inspection of the max amplitude VamplMax compared to the left sample and the right sample show that both samples are less than the max so it is confirmed that VamplMax calculated above is the max amplitude so that means that F3 really does equal 0.3514045995 to 10 decimal places. Just to make sure we didnt miss anything, we also checked that fourth point and note that Vampl_Test is also smaller than VamplMax so that means the frequency of 0.35146 can not be the frequency F3 that causes a maximum output amplitude.

I think you can verify this using a circuit simulator, as long as you can zoom in on the response well enough and the output accuracy is set high enough to provide this good resolution and the built in solver method is a good one.

One last note...
It's easier to compare those amplitudes when the values are shown in code:
Code:
VamplMax=    4.5003516037041
Vampl_Left=  4.50035160230076
Vampl_Right= 4.50035160230075
Vampl_Test=  4.50034729587394
 
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MrAl,

Now that I see how you are doing it, I agree with your result. I was equating the absolute value of L with the the absolute value of R||C to get my figure. I should not have used the absolute values until I evaluated them with respect to the whole circuit. I still believe that the frequency at which the whole circuit has no reactance is the most consistent definition that defines resonance no matter what the circuit configuration.

Ratch
 
Hello again Ratch,

Oh yes, that is an interesting frequency too given by:
F4=sqrt((C*R^2-L)/(C^2*L*R^2))/(2*pi) with the restriction L<=C*R^2

That is also the frequency where the phase of the current equals 180 degrees if i remember right.
 
Correct me if I'm wrong, but I believe that, if the resistor value is zero (no dissipation), all the above definitions would amount to the same thing.

Best regards,
Tom
 
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