Question about the definition of Quality factor

Hello everyone,

In Yue's journal in 1998 "on-chip spiral inductors with patterned ground shields for Si-based RFIC's".

The definition of the quality factor is:

Q=2*pi* (energy stored)/ (energy loss in one oscillation cycle).

My question is, what is the purpose to add a scalar 2*Pi here ? Does it have any physical meaning ?

Thank you.

TWang

Although the definition for the quality factor Q is an arbritary one - any definition should make sense.
In this case, it is the task of the factor 2*Pi to make the quality factor Q=1
(a) at the corner frequency of a first order network (imag. part=real part)
Hint: Don´t mix this definition with the pole-Q of a filter.
(b) or, for example, for an impedance R||C resp. RL-series. (also Im=Re).

In other words: The factor Q connects the physical world with some known electronic parameters (impedance).

The right answer is in the words "one cycle". Cycle means period of the signal or time for one energy exchange. This is why 2*pi coefficient is used. This is not arbitrary definition, this is exact definition of the Q. Beside it three more way to determine Q-factor are exist. This parameter can be determine through bandwidth, through phase or through group delay. These methods used because it easier to measure required parameters: bandwidth, phase or group delay than energy. All three methods suitable for determination of Q-factor for resonant circuit. For non-resonant circuits determination of Q-factor is more complicated because the reactance slopes are not equal. Differential equations usually used in this case.

Best regards,
RF-OM

RF-OM said:

This is not arbitrary definition, this is exact definition of the Q.

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Just for clarification: Each definition is an arbitrary one ! That is the nature of a definition.
And, more than that, each definition is an "exact" one. Otherwise, it would be not a definition.
By the way, that is not playing with words. Instead, it is the necessary understanding of the meaning of words with the aim to avoid misunderstandings.

For non-resonant circuits: For my knowledge, there is only one definition for Q - and that is the so called "pole-Q". This pole-Q describes the location of a pole pair in the complex s-plane. And, more than that, for the bandpass case this pole-Q is identical to the "bandwirdth-Q" (Q=resonant frequency/bandwidth).
For a lowpass response these Q values are given in tables for each pole-pair (example: BUTTERWORTH response with Q=0.7071)

To LvW,

I am definitely not an expert in English. I use word “exact just because definition of the Q-factor through energy exchange is indeed exact one. Three other are not exact definitions, they are the way how to determine the Q value and they have more narrow meaning than definition through energy exchange. Therefore, I think that word arbitrary is more suitable for them, but again, I am not an expert in English.

Regarding determination of non-resonant circuit Q-factor. I talked only about determination, not a definition. Definition of Q-factor does not depend on resonance property of the circuit.

Best regards,
RF-OM

To LvW,

I want to add a little bit to clarification as well. If we check the definition for word “arbitrary” with academic dictionary we will get: 1. Not fixed by rules but left to one’s judgment or choice. 2. Based on one’s preference, notion , whim, etc. Definitely, we cannot use such kind of statements to scientific definition of natural phenomenon like discussed above definition of Q-factor. This is classical, well known definition that widely used around the world. I discussed it with my coworkers who are old and very experienced native English speaking engineers and one of them university professor and Senior Member of IEEE. I believe we may trust their opinion. After all discussions we decided that arbitrary definition is not for our case. The sample of arbitrary definition is, for example, speed limit on the roads. It depends on somebody decision, preference, and actual variable situation, and so on. But we have scientific statement that describes natural phenomenon which does not depend on humans at all. This definition of Q-factor is exactly describes the phenomenon of energy exchange into non-reactance-free networks when dissipation exceeds resistive losses.

I hope these arguments will be helpful,
RF-OM

Quote: I want to add a little bit to clarification as well. If we check the definition for word “arbitrary” with academic dictionary we will get: 1. Not fixed by rules but left to one’s judgment or choice. 2. Based on one’s preference, notion , whim, etc.

Of course, completely agreed ! That is exactly what has happened to define the quality factor Q ("one´s judgement ore choice" resp. "one´s preference").

Quote: Definitely, we cannot use such kind of statements to scientific definition of natural phenomenon like discussed above definition of Q-factor.

Why not ? Of course, we can! Somebody has decided some time ago - and it was a decision which makes sense - to use the word "quality factor Q" for a certain mathematic expression (see below).

Quote: This is classical, well known definition that widely used around the world.


I never have argued against it.

Quote:... After all discussions we decided that arbitrary definition is not for our case...... But we have scientific statement that describes natural phenomenon which does not depend on humans at all. This definition of Q-factor is exactly describes the phenomenon of energy exchange into non-reactance-free networks when dissipation exceeds resistive losses.

Yes, that is completely right. But nevertheless, it is arbitrary !
To be specific, Q is defined as follows:
Q= 2*Pi*energy stored/energy dissipated per cycle.

But this of course is arbitrary (although it makes sense). Another choice could have been
(a) just half a cycle in the above denominator or
(b) the use of any other factor instead of 2*Pi.

But both alternatives (a) resp. (b) are of less sense, because the agreed definition has the advantage, that it corresponds with some circuit properties which easily can be measured (for example: real and imaginary parts of the impedance).
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Final comment: I know, it is "only" a discussion about the meaning of words, but - as an engineer - I know about the importance to discriminate between a natural law (like the characteristic of gravity .......,which cannot be explained or justified) and some other rules which are based on definitions. And it is the nature of a definition (and the meaning of the word "definition") to be arbitrary. Otherwise it would be no "definition" but a property of a physical phenomenon.
Example: time is a natural phenomenon, but the duration of a second is an arbitrary definition. Agreed ?

Let’s divide the problem into two parts. The first is wording. I think that we may call the quality factor as we want. If other people will understand what we are trying to say it is okay. And from this point of view our terminology is arbitrary. But since everybody in the world use the same term, when we change it arbitrarily (by our own discretion) it will sound strange for other people. There is high probability that they will reject our new terminology. The same is true for any other term in our discussion: energy, cycle and so on. But changing the word will not change the physical meaning of the phenomenon.

The second problem is the natural issue of physical process. The Earth, as the planet, is so called closed physical system. It means that the major parameters that describe this system CANNOT BE CHANGED. They are not arbitrary. I disagree when you are talking about duration of the second. Time as system parameter is independent of our judgment and we cannot change it. This is the basic parameter in SI system. We have science as the model that describes our physical system and this model is not human product, it is something that was given by the Nature. The PI as a number is also from the Nature.

Next point is about half of the cycle. We cannot consider a half of the cycle because the energy itself and rate of energy exchange (or slope) is not necessary the same in different parts of the cycle. This is why we use integration to calculate the RMS value. When we need to determine Q-factor for non-resonant system we use differential equations instead of simple summation or superposition because the slopes are different.

Wording is just a kind of sophistic, it is the art. But Physics is Physics, it is the exact science.

I also have an idea about the starting point of our discussion. Q-factor is important parameter in engineering and young guys need to understand it well. What I feel after spending some time here, is that many of guys here do not have clear understanding of quality factor. You are the lecturer and have a good language. May be you can write some short tutorial about Q-factor, what it is and how to use it right, three ways of determination of Q-factor and so on. Many of my students and coworkers asked me to help with this issue and I explained it for them many times, but it was done in other language. I am sure people here need such information. What do think about it?

Best regards,
RF-OM

It's obvious, that quality factor has a well-defined meaning in high-frequency engineering. You may say it's arbitrary, but only in very general sense. For a student in a RF test, it's absolutely not arbitrary, he's expected to know it. He must not necessarily think about the Q defintion of communication engineering guys (pole Q), if the two definitions are related, probably identical, whatsoever.

I also completely agree with RF-OM, that the Q factor is an important property in RF engineering. It can give an intuitive understanding of circuit behaviour to my opinion.

We have to take for granted, that individual engineering fields have their specific theory, methods and definitions. They can be corresponding at best but may be contradicting sometimes.

Good explanation in easy to read language! I think that Q-factor is important at any frequency. Even at 50 or 60 Hz AC power lines. It become more and more pronounced at high frequencies, especially at GHz range. The effect of not-high-enough Q can be disastrous and engineer need to know how to predict it and how to do design that counts all these parasitic.

Best regards,
RF-OM

Quote FvM: It's obvious, that quality factor has a well-defined meaning in high-frequency engineering. You may say it's arbitrary, but only in very general sense.

I can agree to this compromizing formulation. Nevertheless, the word “definition” implies that somebody anywhen in the past has proposed to call a certain mathematical combination of physical parameters “quality factor”. And that is not a natural law, but an (arbitrary) definition. Of course, there was a common and worldwide approval upon this. But for my opinion, it is very important for an engineer to know that such a parameter is “just a definition” (based on an agreement) and – more important – that perhaps there are other definitions with the same name.
1st example: Pole-Q of a filter circuit is (arbitrarily) defined via eigenvalues in the s-plane.
2nd example: Bandwidth of a lowpass can be defined with the help of 3-dB-points, but not necessarily ! In some cases (Tschebyscheff responses) another definition is more appropriate and is in use therefore.

Quote FvM: We have to take for granted, that individual engineering fields have their specific theory, methods and definitions. They can be corresponding at best but may be contradicting sometimes.

Agreed ! Why they can be “contradicting” sometimes ? Because they are no natural laws, but made by men based on convenient agreements. And that´s exactly the meaning of the word “definition” in a technical sense.

Now I agree with you. You are talking about wording. It may be arbitrary. My point is that Q-factor definition through energy is not arbitrary. You may have any circuit, say just the black box. If you know the transfer function and therefore phase characteristic you may determine Q-factor for this black box. It is not hard to find some references into books to confirm it.

Best regards,
RF-OM

Hi Guys, LvW and RF_OM . TWang the guy who started this thread vanished or might have got afraid seeing all these. Hope his doubt is clarified .

Greate disussion. Healthy one too.

The whole discussion was useful. Now I understood how important the Language is even in Engineering.

Thanks
GV

never lose your scientific temper

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I agree. There is more sophistic than real engineering. Definitely here is not the best place for such discussions, but,from the other hand, this site is not only for beginners. Everybody has right to say something.

I hope that our discussion was helpful for all guys.

Best regards,
RF-OM

RF-OM said:

You may have any circuit, say just the black box. If you know the transfer function and therefore phase characteristic you may determine Q-factor for this black box.
RF-OM

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To RF-OM: I wonder if this is possible. Please, can you give an example using a transfer function of at least 3rd order ? If this topic is not the right place, please send it via PM or e-mail. Thank you.

Okay. I will try to do it tomorrow.

A gute nacht!

RF-OM

Hi Both,

If this topic is not the right place, please send it via PM or e-mail. Thank you.

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The topic is in the right place. Please dont take this offline, some, (atleast myself is) are intrested in your discussions .

Regards
GV

To LvW,

As you asked I simulated the 3rd order Butterworth today and got the numbers (bandwidth, low and high cut-off frequencies, central Fo, phase angles for cut-offs and so on). Then I use these numbers to determine Q-factor by two ways: through bandwidth and through phase slope. I got very close numbers. So it works. But, frankly speaking, I do not like this method because when Q is high the phase slope is also steep. Taking tan from steed slope is not a big fun and there is easy to make mistake. Definitely I used in the past other formulas. They were based on phase slope, but calculations were more robust. I will try to find these old worksheets that we used before electronic calculators become available. It will be very interesting to see them again. Now we use simulators for all tasks and completely forgot the old design methods.

I also found one good book with very interesting chapter that describes how to use filter’s phase slope to determine Q-factor of the filter. It is not what I had in old worksheets, but looks close. Unfortunately, I forgot my file with all the numbers and this book information on my office desk. Tomorrow I will be able to send it to you.

Best regards,
RF-OM

RF-OM said:

As you asked I simulated the 3rd order Butterworth today and got the numbers (bandwidth, low and high cut-off frequencies, central Fo, phase angles for cut-offs and so on). Then I use these numbers to determine Q-factor by two ways: through bandwidth and through phase slope. I got very close numbers. So it works.
RF-OM

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I doubt that "it works". You only have made some tests on only one specific filter circuit - and this leads you to a general rule ?

I have simulated several bandpass responses of 4th order (Butterworth, Bessel, Tschebyscheff, elleiptic) - all with the same 3-dB-bandwidth.
And - as a result - all phase slopes at the central frequency are different !!

Summary: There is no relationship between phase slope and bandwidth for transfer functions with an order higher than two

Yes, a few simulations (both my and yours) cannot be considered as a general rule, and I do not try to establish any rules. I just think that phase slope closely related to Q-factor. You did not convince me that this is not true. My major goal is to find (or recall) how to use phase slope in my everyday work and it is normal to expect that other people will help with this task. Why you so aggressively try to do the opposite even after I sent you a reference to a book that clearly shows that phase slope is closely related to the Q-factor?
Let’s consider this point. Filter is reflective device. It works because it reflects the RF energy away. In order to reflect the RF energy, filter must have reactance, which means phase shift. A reflective property of any passive device is closely related to the quality factor of this device. This is easy to understand and this is not in contrary with your theory. There are not only resonant networks in RF engineering and methods of Q-factors description are different depends on actual network. It looks that you are trying to transfer some theory to all cases.
Let’s say you have a multistage non-resonant network and this network has 100 components. It is not difficult to show equation that will provide us with calculation of Q-factor for this network. It is not easy, but differential equation. It is exist and well known. And this equation counts slopes!
Best regards,
RF-OM