About the definition of resonance frequency

[cianfa72]

TL;DR Summary

About the definition of resonance angular frequency for RLC or more complex networks

Hi, I'm confused about the meaning of resonance angular frequency for an RLC series circuit.

Call $\omega_0=\sqrt { 1/LC }$ and $Q= {\omega_0 L} /R$.
Then consider the admittance as transfer function $$Y(s) = \frac {1} {L} \frac {s} {s^2 + s \omega_0/Q + \omega_0^2}$$
This transfer function has one zero in $s=0$ and two complex coniugate poles when $Q > 1/2$.

Note that the c.c. poles define a "natural or ringing" angular frequency, i.e. $$\omega_r = \omega_0 \sqrt {1 - \frac {1} {4Q^2}}$$
So the question is: what is meant by resonance frequency of a network?

Thanks.

 

[anuttarasammyak]

I observe the denominator is$$(s+\frac{\omega_0}{2Q})^2+\omega_r^2$$
Does it help you ?

 

[marcusl]

It is well known that the resonance frequency $$\omega_r$$ of a damped RLC tank circuit differs from the undamped value $$\omega_0$$. They agree to within 1% for Q>=3.5, however, so the latter is a good approximation for most practical situations.

 

[DaveE]

TL;DR Summary: About the definition of resonance angular frequency for RLC or more complex networks

So the question is: what is meant by resonance frequency of a network?

I fear that you're going to get different responses from different engineers to this question. Networks can be complex and, IMO, might have several resonant frequencies. This term is often used in a sloppy, off hand manner. In those cases you just have to do your best to interpret what their words mean. Often it's the one at the lowest frequency or the one at the highest amplitude when there are several.

My version is to rely on the math and the transfer function of the network (including input and output definitions). Then I would say any quadratic term with complex roots (i.e. can't be easily factored into simple poles/zeros) has a resonant frequency, just as you have described it. So I would say a 5th order Butterworth filter has two resonant frequencies, even though the bode plot doesn't show them clearly.

Also, this sort of definition really only makes rigorous sense for LTI networks.

PS: oops, maybe I missed your question: Is it ω_r or ω_o? I don't know. The way I learned it was ω_o is the resonant frequency, but ω_r is the max/min gain frequency. They are close enough together that it doesn't matter in practice if you're using words and not equations. In a simple LCR circuit ω_o is the frequency where the reactive components have the same magnitude and opposite phase.

BTW, Wikipedia says I'm wrong, but I don't care enough to change what I say or how I think about it. These are just words, it's the understanding that counts.

 

[cianfa72]

My version is to rely on the math and the transfer function of the network (including input and output definitions). Then I would say any quadratic term with complex roots (i.e. can't be easily factored into simple poles/zeros) has a resonant frequency, just as you have described it. So I would say a 5th order Butterworth filter has two resonant frequencies, even though the bode plot doesn't show them clearly.

Ah ok, so in your example a 5th order Butterworth filter has 5 poles on the same half-circle in the left half complex plane of variable $s$: 1 negative real and two pairs of complex coniugate poles. Therefore each c.c. pole pair defines "its own" resonant frequency.

Also, this sort of definition really only makes rigorous sense for LTI networks.

Yes, of course.

Is it ω_r or ω_o? I don't know. The way I learned it was ω_o is the resonant frequency, but ω_r is the max/min gain frequency.

As far as I understand, $\omega_0$ is the "module/lenght" of the vector from the origin of variable $s$ complex plane up to the location of the c.c. pole pair. Conversely $\omega_r$ is the frequency on the $s=j\omega$ axis where the transfer function $F(j\omega)$ takes its max/min module value.

They are close enough together that it doesn't matter in practice if you're using words and not equations. In a simple LCR circuit ω_o is the frequency where the reactive components have the same magnitude and opposite phase.

Yes, however a simple LCR series circuit has as transfer function one zero in $s=0$ and two c.c. poles when $Q > 1/2$ (or viceversa according the transfer function picked: admittance vs impedance).

In this case, since there is also a zero $s=0$ in the admittance FDT, the max admittance's module frequency is $\omega_0$ rather than $\omega_r$.

 

[cianfa72]

Btw, given a circuit and fixed the type of inputs and outputs (i.e. current vs voltage), any transfer function of the same type (e.g. current as input and voltage as output, i.e. of type impedance) will have the same poles, right?

What is the reason behind it ?

 

[LvW]

I think, we should not mix the three termes "pole frequency", "natural frequency" and "resonant frequency".
I some simple cases, all three can be identical - however, for my opinion, we have a clear defintion for the resonance case of a frequency dependent circuit:
The frequency fo with zero phase difference between voltage across and current into the circuit is called resonant frequency. At this frequency fo the circuits input impedance is real (zero imaginary part).

 

[cianfa72]

The frequency fo with zero phase difference between voltage across and current into the circuit is called resonant frequency. At this frequency fo the circuits input impedance is real (zero imaginary part).

Ok, anyway the point is: given a circuit which is "the input" ? Actually we have multiple possible choices for "input" given the circuit.

 

[LvW]

You are mentioning "multiple possible inputs". Is this really the "normal" case?
However - where is the problem? When you have two possible inputs you can define two different input admittances and apply the definition on both.

Hi, I'm confused about the meaning of resonance angular frequency for an RLC series circuit.

Call $\omega_0=\sqrt { 1/LC }$ and $Q= {\omega_0 L} /R$.
Then consider the admittance as transfer function $$Y(s) = \frac {1} {L} \frac {s} {s^2 + s \omega_0/Q + \omega_0^2}$$
This transfer function has one zero in $s=0$ and two complex coniugate poles when $Q > 1/2$.

Why do you mention zeros and poles?
When you are interested in the resonance case, you simply must ask yourself "for which frequency is the transfer function Y(s) real?"
In this case, the imaginary part of the function is zero when both - numerator and denominator - are imaginary. Hence - the real part of the denominator must disappear.
This is the case for wo²=s².

 

[cianfa72]

You are mentioning "multiple possible inputs". Is this really the "normal" case?
However - where is the problem? When you have two possible inputs you can define two different input admittances and apply the definition on both.

Yes, however as you can see the answer to "which is the resonant frequency of a given circuit ?" is not unique.

Why do you mention zeros and poles?
When you are interested in the resonance case, you simply must ask yourself "for which frequency is the transfer function Y(s) real?"

Actually the question is: for which frequency $\omega$ is the transfer function Y(s) evaluated at $s=j\omega$ real ?

 

[LvW]

Yes, however as you can see the answer to "which is the resonant frequency of a given circuit ?" is not unique.

When we speak about a circuit and a certain signal which is injected into the circuit with a certain frequency we do this using a certain input node. This input node is - of course in conjunction with a specified output node - used for defining a transfer function. Applying this transfer function for checking the resonance case there is an unique solution. Where do you see any problem?

Actually the question is: for which frequency $\omega$ is the transfer function Y(s) evaluated at $s=j\omega$ real ?

I do not understand this question because the answer is already given. For the example under discussion the condition for resonance is at w=wo with wo=1/SQRT(LC)

 

[cianfa72]

When we speak about a circuit and a certain signal which is injected into the circuit with a certain frequency we do this using a certain input node. This input node is - of course in conjunction with a specified output node - used for defining a transfer function. Applying this transfer function for checking the resonance case there is an unique solution. Where do you see any problem?

I see no problem. My point is that the resonance frequency of a circuit isn't well-defined unless one specifies which is the input and the output.

I do not understand this question because the answer is already given. For the example under discussion the condition for resonance is at w=wo with wo=1/SQRT(LC)

Well, that is the reasonant frequency of the RLC circuit if you take as input and output the voltage and the current across the entire RLC (i.e. looking at it as a bipole).

 

[LvW]

I see no problem. My point is that the resonance frequency of a circuit isn't well-defined unless one specifies which is the input and the output.

Yes - of course. Another example: Can you define the gain (or damping) of a circuit without defing input and output?

Well, that is the reasonant frequency of the RLC circuit if you take as input and output the voltage and the current across the entire RLC (i.e. looking at it as a bipole).

Do you see any alternative method for defing the resonant case?

 

[The Electrician]

TL;DR Summary: About the definition of resonance angular frequency for RLC or more complex networks

Hi, I'm confused about the meaning of resonance angular frequency for an RLC series circuit.

Call $\omega_0=\sqrt { 1/LC }$ and $Q= {\omega_0 L} /R$.
Then consider the admittance as transfer function $$Y(s) = \frac {1} {L} \frac {s} {s^2 + s \omega_0/Q + \omega_0^2}$$
This transfer function has one zero in $s=0$ and two complex coniugate poles when $Q > 1/2$.

Note that the c.c. poles define a "natural or ringing" angular frequency, i.e. $$\omega_r = \omega_0 \sqrt {1 - \frac {1} {4Q^2}}$$
So the question is: what is meant by resonance frequency of a network?

Thanks.

These might be helpful:
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/serres.html
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/parres.html#c1

 

[cianfa72]

Do you see any alternative method for defing the resonant case?

Look at the following antiparallel RLC circuit.

Take as input the source current $I_g(s)$ and the voltage across the inductor and that across the capacitor as outputs respectively.

Both transfer functions have the same poles, however $F_1(s)$ has one zero in $s=0$ while$F_2(s)$ in $s=-R/L$.

The calculation shows that $F_1(j\omega)$ is real (zero imaginary part) at angular frequency $\omega_r=\sqrt {1/LC}$ while $F_2(j\omega)$ at $\sqrt {\frac {1} {LC} - \frac {R^2} {L^2} }$. Note that in the latter case the "resonance" doesn't exist at all whether the argument of square root is negative.

Therefore the bottom line is that doesn't exist an unique resonant frequency associated with a given circuit.

 

[LvW]

The calculation shows that $F_1(j\omega)$ is real (zero imaginary part) at angular frequency $\omega_r=\sqrt {1/LC}$ while $F_2(j\omega)$ at $\sqrt {\frac {1} {LC} - \frac {R^2} {L^2} }$. Note that in the latter case the "resonance" doesn't exist at all whether the argument of square root is negative.

Therefore the bottom line is that doesn't exist an unique resonant frequency associated with a given circuit.

No surprise - F2 has a 1st-order lowpass characteristic (real at w=0 only).
Of course, there are circuits and/or functions which cannot show a resonance effect.

 

[cianfa72]

Btw, given a circuit and fixed the type of inputs and outputs (i.e. current vs voltage), any transfer function of the same type (e.g. current as input and voltage as output, i.e. of type impedance) will have the same poles, right?

What is the reason behind it ?

Sorry, what about this post ?

 

[DaveE]

The calculation shows that F1(jω) is real (zero imaginary part) at angular frequency ωr=1/LC while F2(jω) at 1LC−R2L2. Note that in the latter case the "resonance" doesn't exist at all whether the argument of square root is negative.

I would argue (already have) that any transfer function that contains a quadratic pole or zero with complex roots is resonant. This definition is simple and precise. I haven't heard an alternate definition here yet.

 

[LvW]

Two links:
1.) https://testbook.com/electrical-engineering/resonance
"Electrical Resonance means in a circuit when the inductive reactance (XL) and capacitive reactance (XC) are equal in magnitude but opposite in phase,resulting in a purely resistive impedance at a particular frequency. "

2.) https://en.wikipedia.org/wiki/Electrical_resonance
"Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedances or admittances of circuit elements cancel each other"

 

[cianfa72]

I would argue (already have) that any transfer function that contains a quadratic pole or zero with complex roots is resonant. This definition is simple and precise. I haven't heard an alternate definition here yet.

Sorry, if you have a transfer function quadratic in the denominator with complex roots but with no zeroes (at numerator), the only angular frequency that results in zero phase (zero imaginary part) is the null frequency.

 

[cianfa72]

https://en.wikipedia.org/wiki/Electrical_resonance
"Electrical resonance occurs in an electric circuit at a particular resonant frequency when the impedances or admittances of circuit elements cancel each other"

If we assume this definition of resonance, that doesn't imply that "at resonance frequency" every transfer function is real (zero imaginary part). See for instance my example in post #15.

 

[DaveE]

Sorry, if you have a transfer function quadratic in the denominator with complex roots but with no zeroes (at numerator), the only angular frequency that results in zero phase (zero imaginary part) is the null frequency.

Where does the requirement for arg(F(s)) = 0 for resonance within F(s) come from? What about high order elliptical, Bessel, or Chebyshev filters where complex poles are located at different frequencies? Each frequency can't have zero phase for F(s). What about a high Q resonant LCR LPF cascaded with an integrator, it never has zero phase?

This is exactly why my definition focuses on complex poles and zeros only, not the whole ensemble. So, please tell us, clearly, what is your definition of resonance?

 

[DaveE]

Also consider an active state variable filter circuit. This is a single circuit that will create a HPF, BPF, and LPF. Basically, by repeated integrations of the HPF. But it is one circuit (because of the essential negative feedback), with one resonance (for high Q designs). The magnitude of the transfer functions for large Q are sketched below.

https://ocw.mit.edu/courses/2-161-s.../cce7e9ece3e590e44b219324706e3de9_lpopamp.pdf
https://www.analog.com/media/en/training-seminars/tutorials/MT-223.pdf
https://www.ti.com/lit/an/sbfa002/sbfa002.pdf?ts=1726459820870

 

[berkeman]

This is a single circuit that will create a HPF, BPF, and LPF.

Sorry, I haven't been following the thread. How does that first stage accomplish a HPF function?

 

[DaveE]

Sorry, I haven't been following the thread. How does that first stage accomplish a HPF function?

That's kind of my point. It is the whole circuit that you need to analyze. The feedback around around two integrators creates a second order system. There isn't a starting point. Like a ring of integrators, maybe? I could have just as well said that it's derivatives of the LPF. It is a resonant circuit (if you choose high Q) that has, at different points, all three filter responses.

 

[DaveE]

Btw, given a circuit and fixed the type of inputs and outputs (i.e. current vs voltage), any transfer function of the same type (e.g. current as input and voltage as output, i.e. of type impedance) will have the same poles, right?

What is the reason behind it ?

Sorry, I skipped this earlier question. This all depends on the I/O definitions of your transfer function. For example poles in the input admittance will be zeros in the input impedance. But, as I think you are implying, the basic resonance still exists (ω_o, Q) and is unchanged, you just might measure it in different ways.

 

[cianfa72]

This is exactly why my definition focuses on complex poles and zeros only, not the whole ensemble. So, please tell us, clearly, what is your definition of resonance?

Ok, suppose to breakdown a transfer function F(s) in its "factors" and consider the following factor at denominator $$s^2 + 2\xi\omega_0 s + \omega_0^2 = s^2 + s \omega_0 / {Q} + \omega_0^2$$ Its roots are written as $$\omega_0 (- \xi \pm j \sqrt {1 - \xi^2})$$ As you can see when $\xi < 1$ they are complex coniugate (c.c).

As far as I can understand your point, your definition of resonance for a given poles pair is $(\xi, \omega_0)$ or equivalently $(Q,\omega_0)$. Then the question is: what is the characteristic feature of reasonant frequency? My answer: it is the angular frequency $\omega$ such that a transfer function $F(s=j\omega)$ based on that poles pair and with a zero at the origin has zero phase (zero imaginary part).

That said, to me the concept of reasonant frequency associated with a circuit isn't well-defined. Given a circuit only if one defines which is input and which is output then one can write down the relevant F(s), calculate its poles and for each c.c. pole pair the associated reasonant frequency.

 

[DaveE]

As I originally said, resonance is a term that is usually used in a sloppy fashion, the important thing is to know and understand the math and your models. I've already told you my opinion, you can choose your own.

 

[LvW]

As I originally said, resonance is a term that is usually used in a sloppy fashion, the important thing is to know and understand the math and your models. I've already told you my opinion, you can choose your own.

OK - I agree that the term of "resonance" very often is "used in a sloppy fashion".
But does that prevent us from looking for and using an exact definition?
Because your definition "focuses on complex poles and zeros only" (pole frequencies of a filter and “high-Q” circuits) - where is the limit for “high-Q”?
Example (question): Is there any resonance effect with a second order Butterworth low pass (one complex pole pair) with a phase shift of 90deg at w=wp?

 

[LvW]

General statement:
I have checked several books and other knowledge sources on the subject of “resonance” and - without exception - the resonance case was only defined and explained for RLC circuits.

In one of these sources it even says:
"In electrical resonance, the frequency of the source is equal to the frequencyof its own circuit, which depends only on the value of inductance L and capacitance C. The prerequisite (but not sufficient) for the occurrence of electrical resonance is that the circuit have to consist both elements: capacitors and coils."

If you analyze the term “resonance” in terms of its meaning, you actually have to ask:
Which quantity can be “in resonance” with which other quantity?
And then you automatically think of the two impedances of L and C, don't you?
A similar consideration is, for example, not appropriate for the pole frequency of an RC filter, is it?

 

[DaveE]

Example (question): Is there any resonance effect with a second order Butterworth low pass (one complex pole pair) with a phase shift of 90deg at w=wp?

According to me, yes, because the roots are complex. It is a simple definition, probably not satisfying to many because there isn't any gain peaking and little overshoot in the step response. You are free to choose your own threshold. How high does Q have to be to be resonant? How do you avoid a somewhat arbitrary choice?
https://2n3904blog.com/butterworth-filter/

 

[LvW]

According to me, yes, because the roots are complex. It is a simple definition

I must admit that - up to now - I have never heard about such a definition. Is there any reference?

 

[DaveE]

I must admit that - up to now - I have never heard about such a definition. Is there any reference?

IDK of any reference for any version of definition. I probably learned it at school from R. D. Middlebrook. Please let me know if you find one. This is why I don't care much, it's just not a term I've heard used precisely in practice.

 

[LvW]

IDK of any reference for any version of definition. I probably learned it at school from R. D. Middlebrook.
Please let me know if you find one.

What about the explanations I have mentioned in my posts #19 and #30 ?

 

[DaveE]

What about the explanations I have mentioned in my posts #19 and #30 ?

#30 is just words. Although the requirement for L and C only applies to passive circuits. With gain I can make resonance with just one type.

#19 is the same as my definition, Q>0.5

edit: although #19 is more about the frequency of resonance than the amount (quality factor or damping).