Resonance of LC

[MrAl]

Ok, Ratch it is indeed true for your formula of Q factor is correct. So there are three formulas, one is from your textbook, one is from mine and one provided MrAI.

But I would like to know my way which is closer to me, because they are examples in this book.

MrAI, to which circuit are you reffering to. First or the second?

Hi,

I thought i already stated that it was for a lone inductor, which of course has ESR, or for an ideal inductor in series with a resistor.  In both cases we have just one resistor and one inductor.

There will be formulas for the Q and which one you use will depend on the application.  The ones we have been discussing i think fall into the category of impedances, while at least one other one:
Q=F/BW

falls into the category of filters.

The center frequency F is found and then the BW by using something like d|H(jw)| /dw=0.
We could talk about that more too.

[MrAl]

Mr. Al,

Q is a factor that can be applied to a lone inductor.  That's because of the always present ESR of the inductor.  So the circuit is really a resistor in series with an inductor, and together it has a Q factor.

Q=w*L/R

Of course.  However, nforce and I were discussing a more complicated LCR circuit involving an off resonant frequency.

Ratch

Hi again,

Yes i was replying to your statement/question in reply 96 of:
"When it comes down to it, what is Q good for other than determining the characteristics of resonant circuits? "

and wanted to broaden the field where Q might be reasonable, even in the absence of a resonance of any kind.
And then we have the Q of a low pass filter, which may not be thought of as having any resonance or may be thought of that way too.  d=1/2/Q, d the damping factor.

I think my favorite is the power and energy statement of Q, but easier to calculate i think is the reactive power over real power, for impedance elements.

[kulky64]

No, I would like to know how to determine Q factor, not resonant Q. That is other story.

Here is the wikipedia, quote:

In electrical systems, the stored energy is the sum of energies stored in lossless inductors and capacitors; the lost energy is the sum of the energies dissipated in resistors per cycle.

Here is the example from my work book:



It's not hard, it is barely one equation, but why don't we here give into account also the inductor. Wikipedia qoute: sum of energies stored in lossless inductors and capacitors.

I think you don't understand equation for calculating Q from your work book very well. Because this equation IS for calculation of Q at resonant frequency. It will not work for arbitrary frequency. And the reason why there is not explicitly calculated energy stored in inductor in numerator of your equation is because at resonance, energy stored in inductor is numerically equal to energy stored in capacitor. So you if you know for example energy stored in capacitor, you take double of that and you have sum of energy stored in capacitor and inductor.

[nForce]

kulky64, the best answer. You should come here more frequently   

Other please continue from here what kulky64 said. It is important.

kulky64, you said that it's double of energy stored in the capacitor. But in the equation in the numerator is just for one capacitor. (CU^2)/2, so it should be double of that, like this: (CU^2).

[kulky64]

In equation from your textbook, numerator and denominator is expanded by 1/2 for some unknown to me reason. If you delete this 1/2 from numerator and denominator, it will start making sense. In numerator should be only \$ C\left|U\right|^2 \$, power dissipated in resistor is \$ \frac{\left|U_R\right|^2}{R} \$, so this should be in denominator, not \$ \frac{1}{2}\frac{\left|U_R\right|^2}{R} \$.

[Ratch]

Ok, Ratch it is indeed true for your formula of Q factor is correct. So there are three formulas, one is from your textbook, one is from mine and one provided MrAI.

But I would like to know my way which is closer to me, because they are examples in this book.

MrAI, to which circuit are you reffering to. First or the second?

I have been trying to wrap my mind around what the Q with respect to a circuit driven by a non-resonant frequency means.  I don't think it has any significance.  All I can find in the literature relates the circuit Q to the resonant frequency.  At resonance, the maximum energy stored in the L and C components is the same, and the totals of the energy in the L and C components are equal  at any time.  Therefore, the Q for a series resonant circuit is the reactance of either L or C divided by the resistance.  The Q for a parallel circuit is R divided by the the reactance of either L or C.  The Q in terms of the LCR components is derived and presented in the attachment.

Ratch

[kulky64]

I think it's best to derive the expression for Q for particular circuit in hand and not fix yourself to some textbook formulas. For example if we further manipulate expression for Q from nForce example, we get:
\$ Q=\frac{1}{R}\sqrt{\frac{L}{C}-R^2} \$

[MrAl]

Ok, Ratch it is indeed true for your formula of Q factor is correct. So there are three formulas, one is from your textbook, one is from mine and one provided MrAI.

But I would like to know my way which is closer to me, because they are examples in this book.

MrAI, to which circuit are you reffering to. First or the second?

I have been trying to wrap my mind around what the Q with respect to a circuit driven by a non-resonant frequency means.  I don't think it has any significance.  All I can find in the literature relates the circuit Q to the resonant frequency.  At resonance, the maximum energy stored in the L and C components is the same, and the totals of the energy in the L and C components are equal  at any time.  Therefore, the Q for a series resonant circuit is the reactance of either L or C divided by the resistance.  The Q for a parallel circuit is R divided by the the reactance of either L or C.  The Q in terms of the LCR components is derived and presented in the attachment.

Ratch

Hi again Ratch,

in particular i am replying to the statements:
"I have been trying to wrap my mind around what the Q with respect to a circuit driven by a non-resonant frequency means.  I don't think it has any significance. "

We already talked a little about the Q of an inductor, and we could talk about the Q of a capacitor, but it woudl be a similar situation, so we know at least there is a calculation that reveals the Q of either element, taken alone by itself.
The way to proceed from here is to find a relationship between the individual Q's to other uses such as in a resonant circuit, which answers the question:
"I have two inductors one with Q=5 and one with Q=10, if i use it in a circuit with a capacitor with Q=10 which one will give me an overall higher circuit Q?"
So if you can find a way to combine the Q's of the two you could know right away which component is the better one to use with your total circuit (which also includes a cap).  That in itself without any other reasoning, gives significance to the Q of a lone element without it being in resonance.
For example, prove (or disprove) the following:
Q=1/(1/QL+1/QC)

where Q is the total circuit Q and QL and QC are the Q's of the inductor and cap respectively.

But you may not want to stop there.  You may find other relationships, but it will take some work on your part to do some calculations and see what you can come up with.  I know you do a lot of math too so you might find this interesting to do.  See what you can find :-)

The Q of an LC circuit off resonance may be interesting when the frequency is not set to the desired resonant frequency.  This information may be used to combine with other circuits as well.  You could try designing a simple resonant circuit, calculate the Q at arbitrary frequency, then see what happens when it combines with another circuit with some other arbitrary Q (at the same frequency).

[Ratch]

Hi MrAl,

I don't see any problem figuring out the Q of a resonant circuit from the Q's of the individual components.  If the resistance does not affect the frequency of circuit resonance, then the resistance of each component can be found from the Q of the component together with the reactance of the component, and combined into one resistance.  If the resistance affects the frequency, then solving the problem with nonlinear methods is in order.  I don't think trying to figure out how to combine the Q's of each component into a circuit is practical except for the simplest of circuit configurations.

I still don't think that the concept of circuit Q at an off resonant frequency means anything

Ratch

[Ratch]

I think it's best to derive the expression for Q for particular circuit in hand and not fix yourself to some textbook formulas. For example if we further manipulate expression for Q from nForce example, we get:
\$ Q=\frac{1}{R}\sqrt{\frac{L}{C}-R^2} \$

Would you be so kind to provide a post number or a circuit schematic designating which nforce example you are referencing?  I don't want to chase my tail perusing the wrong circuit.

Ratch

[kulky64]

Reply #93

[Ratch]

Reply #93

Thank you.  Usually textbooks are more accurate and knowledgeable than the folks who read them.  As you can see in the attachment, the Q formula you presented is the "inductance" Q, not the circuit Q.  If I needed to calculate the circuit Q, I would first transform the series R & L into a parallel R & L at the resonant frequency.  Then I would use R*Sqrt(C/L) formula to find the circuit Q.  I am also including an interesting link.  http://www.pronine.ca/qbw.htm

Ratch

[MrAl]

Hi MrAl,

I still don't think that the concept of circuit Q at an off resonant frequency means anything

Ratch

Hi again Ratch,

Well there is a difference between thinking and knowing.  In one case you have proof, in the other case you still dont know.  What i dont understand here though is how you could quote a passage that actually uses this concept and provides a useful result, then state that you are still thinking about if it means anything.

Q=1/(1/QL+1/QC)

is not resonant specific, and yet it provides a useful result.  Yes it could be in a resonant circuit, but each individual Q is based on the frequency of operation which may or may not be at a resonant frequency.  But even if we dont consider anything other than the resonant frequency we still are facing the concept of calculating the individual Q's at several frequencies, which we dont have to take as being any resonant point.

To state this a little more clearly:
Q(w)=1/(1/QL(w)+1/QC(w))

however we may want to know say QL(w) for several frequencies.

If we design a filter with two parts, each part may operate at a different 'resonant' frequency.  Since they are both in the same circuit however, the overall Q during either resonant point will be of concern because that will be the Q of the whole circuit, and there will always be one part that is not operating at any resonant point even when one point is.  This is partly why i suggested that you do a few numerical experiments and see what you can find out.  If you arent sure about something, go ahead and try to find out for sure.  Discovery can be as interesting as book learning, and often more.

[Ratch]

Hi MrAl,

I still don't think that the concept of circuit Q at an off resonant frequency means anything

Ratch

Hi again Ratch,

Well there is a difference between thinking and knowing.  In one case you have proof, in the other case you still dont know.  What i dont understand here though is how you could quote a passage that actually uses this concept and provides a useful result, then state that you are still thinking about if it means anything.

What reliable source did I quote?  During the discussion in this thread, we postulated about the Q at an off resonance frequency.  I stated that I could not find anything in the literature about this concept.  So please, point out the quoted passage that gives this principle legitimacy.

Q=1/(1/QL+1/QC)

is not resonant specific, and yet it provides a useful result.  Yes it could be in a resonant circuit, but each individual Q is based on the frequency of operation which may or may not be at a resonant frequency.  But even if we dont consider anything other than the resonant frequency we still are facing the concept of calculating the individual Q's at several frequencies, which we dont have to take as being any resonant point.

There might be several resonant points, and each point will have its own Q.

To state this a little more clearly:
Q(w)=1/(1/QL(w)+1/QC(w))

however we may want to know say QL(w) for several frequencies.

Fine, they can be calculated.

If we design a filter with two parts, each part may operate at a different 'resonant' frequency.  Since they are both in the same circuit however, the overall Q during either resonant point will be of concern because that will be the Q of the whole circuit, and there will always be one part that is not operating at any resonant point even when one point is.

Parts of a circuit may resonant at different frequencies.  There will be a particular Q at each resonant part of the circuit.  Therefore, there cannot be one overall Q for the whole circuit by reason that each resonant Q is only relevant at it own frequency.

This is partly why i suggested that you do a few numerical experiments and see what you can find out.  If you arent sure about something, go ahead and try to find out for sure.  Discovery can be as interesting as book learning, and often more.

Before I do anything like that, I need more guidance and purpose on what I would be trying to prove.

Ratch

[MrAl]

Hi MrAl,

I still don't think that the concept of circuit Q at an off resonant frequency means anything

Ratch

Hi again Ratch,

Well there is a difference between thinking and knowing.  In one case you have proof, in the other case you still dont know.  What i dont understand here though is how you could quote a passage that actually uses this concept and provides a useful result, then state that you are still thinking about if it means anything.

What reliable source did I quote?  During the discussion in this thread, we postulated about the Q at an off resonance frequency.  I stated that I could not find anything in the literature about this concept.  So please, point out the quoted passage that gives this principle legitimacy.

Q=1/(1/QL+1/QC)

is not resonant specific, and yet it provides a useful result.  Yes it could be in a resonant circuit, but each individual Q is based on the frequency of operation which may or may not be at a resonant frequency.  But even if we dont consider anything other than the resonant frequency we still are facing the concept of calculating the individual Q's at several frequencies, which we dont have to take as being any resonant point.

There might be several resonant points, and each point will have its own Q.

To state this a little more clearly:
Q(w)=1/(1/QL(w)+1/QC(w))

however we may want to know say QL(w) for several frequencies.

Fine, they can be calculated.

If we design a filter with two parts, each part may operate at a different 'resonant' frequency.  Since they are both in the same circuit however, the overall Q during either resonant point will be of concern because that will be the Q of the whole circuit, and there will always be one part that is not operating at any resonant point even when one point is.

Parts of a circuit may resonant at different frequencies.  There will be a particular Q at each resonant part of the circuit.  Therefore, there cannot be one overall Q for the whole circuit by reason that each resonant Q is only relevant at it own frequency.

This is partly why i suggested that you do a few numerical experiments and see what you can find out.  If you arent sure about something, go ahead and try to find out for sure.  Discovery can be as interesting as book learning, and often more.

Before I do anything like that, I need more guidance and purpose on what I would be trying to prove.

Ratch

Hi,

I think i understand now why you are not seeing this for it's simplicity.

A quick analogy would be adding series resistors to get the total resistance:
RT=R1+R2+R3

Does it make sense to do this?  Sure it does.  But what if each resistor could somehow change in value with frequency?  Still makes sense to do it:
RT(f)=R1(f)+R2(f)+R3(f)

but what stands out is that each resistance is independent of each other resistance, so we dont really have to keep adding all three if we know that two of them add up to say 5.  If we know even one of them, we dont have to keep calculating the whole bunch.  Say we only know R3 at 100Hz equals 10 ohms.  Then we reduce it to:
RT(100)=R1(100)+R2(100)+10

If we knew that R2(100) was equal to 20, we could reduce this to:
RT(100)=R1(100)+30

Maybe this doesnt work out, so we try anther resistor to replace R1:
RT(100)=R4(100)+30

The whole point here is that we did not have to keep calculating the values of R2 and R3 over and over because they are independent of one another

As for the circuit with two parts and two resonant points, if we had the SEPARATE data for each part of the circuit, maybe we could combine them without having to calculate the Q for the entire circuit twice by means of doing the entire calculation over again.  So if we knew that circuit 1 was 10 and the calculated circuit 2 as 20, we might be able to calculate the total Q for the two frequencies just knowing the Q at those two frequencies for both sections.  This means we could combine sections without constantly calculating both parts, only the part that we changed.
For a single inductor and capacitor, we have:
QT=1/(1/QL+1/QC)

and so we could try different C's for example, all the while knowing QL, and so we dont have to keep calculating the Q of the entire circuit with both elements.

What could be done is simply calculate some Q's of some circuits, then combine them, calculate the total Q, then try to see if you can find a way to combine the individual Q's in order to arrive at the total Q.
If we did this with the L and C we would end up with the formula above.  It may be more complicated with more complicated sub circuits, but you could find that out.  I realize it takes some work here, but that's what we do :-)
We could also do a search :-)

[Ratch]

Hi MrAl,

All right, let's try that idea out.  Suppose we have two coils each with a Q of 10 at a given frequency.  That means each coil has x amount of resistance and 10x amount of inductive reactance.  We put them in series.  The circuit then has 2x amount of resistance and 20x amount of reactance.  That makes a total Q of 10 for the two coils in series.  Now, you want me to figure out a universal formula that will give a Q value just by knowing only the Q of the coils?  How about if they are in parallel?  What if they are different inductance and Q values.  Isn't it just easier to compute the Q values by resistance and reactance?

Ratch

[MrAl]

Hi MrAl,

All right, let's try that idea out.  Suppose we have two coils each with a Q of 10 at a given frequency.  That means each coil has x amount of resistance and 10x amount of inductive reactance.  We put them in series.  The circuit then has 2x amount of resistance and 20x amount of reactance.  That makes a total Q of 10 for the two coils in series.  Now, you want me to figure out a universal formula that will give a Q value just by knowing only the Q of the coils?  How about if they are in parallel?  What if they are different inductance and Q values.  Isn't it just easier to compute the Q values by resistance and reactance?

Ratch

Hi again,

That's true if you only have to do circuits that are constantly so different that you never can do anything else.  But as i am sure you know, we dont always calculate things the same way.
Power=I^2*R
Power=I*E

Two different ways for the same thing.

If you are this against it, dont bother.  You wanted to know why we might consider this, so i told you.  That doesnt mean that you MUST do it, only if you want to, that's all.

[Ratch]

Hi MrAl,

All right, let's try that idea out.  Suppose we have two coils each with a Q of 10 at a given frequency.  That means each coil has x amount of resistance and 10x amount of inductive reactance.  We put them in series.  The circuit then has 2x amount of resistance and 20x amount of reactance.  That makes a total Q of 10 for the two coils in series.  Now, you want me to figure out a universal formula that will give a Q value just by knowing only the Q of the coils?  How about if they are in parallel?  What if they are different inductance and Q values.  Isn't it just easier to compute the Q values by resistance and reactance?

Ratch

Hi again,

That's true if you only have to do circuits that are constantly so different that you never can do anything else.  But as i am sure you know, we dont always calculate things the same way.
Power=I^2*R
Power=I*E

Two different ways for the same thing.

If you are this against it, dont bother.  You wanted to know why we might consider this, so i told you.  That doesnt mean that you MUST do it, only if you want to, that's all.

I don't understand the point of your answer.  I go with what is the simplest, easies,t and works.  Are you suggesting I figure out the total Q for every circuit configuration I use a lot?  OK, maybe.  However, I don't use the same circuits many times.  Something is always different.

Ratch

[MrAl]

I don't understand the point of your answer.
Ratch

That is the problem and i'm tired of trying to explain this simple concept to you.