Questions about pole frequency

[alexang1983]

Dear All

It's ok that pole or zero have got real and immaginary part but if a pole has got only the real part why this value is the value of the frequency of the pole?And so what does real and immaginary part of the pole represent in frequency domain?
I hope my question is clear
thank you in advance

 

[IBO]

transfer function pole frequency

imaginary poles and zeros reresent physical frequencies

cheers

 

[LvW]

poles vs frequency

Hi alexang1983,

I´ll try to give you some background information:

For several reasons it is useful to introduce a complex frequency variable s=σ+jω.
(For example, the LAPLACE transformation requires an additional real part of the frequnecy variable).
But, at the same time, it is important to realize that in reality (measurements) only the imaginary part of s can be created as an input for real circuits.

But, nevertheless, some parameters based on the definition of the complex frequency s are easily measured (for example pole frequency fP and pole quality Qp).
Both parameters are defined in the complex frequency plane by plotting the real and imaginary parts of the "pole" Sp. This "pole" Sp is simply the s-value for which the denominator of the transfer function goes to zero.

Some textbooks and other papers present a 3D picture where the transfer function magnitude is plotted in the complex plane as a function of σ and ω.

And the connection to the real world consists of a cut through this 3D plot at σ=0 (that is along the imaginary axis). Then, you get a curve which is identical to the measured response in the frequency domain.
Any further question ?

 

[Polakrun]

frequency of a pole

Hi LvW,

I think I get what you explained about the relation between the real world and the complex frequency variable s.

However if I have a transfer function which exhibits a second order polynom D at the denominator. We suppose the roots of D are complex. If we run a "pen and paper" analysis and we find the expression of poles.

Does the complex modulus of the poles refers to the value of the frequencies that we can observe on a Bode plot ?

Thank you for your help.

 

[LvW]

pole frequency wiki

Hi LvW,
......................
Does the complex modulus of the poles refers to the value of the frequencies that we can observe on a Bode plot ?

Yes, the magnitude of the complex pole (the distance of the pole location from the center of the plane), which is the square root from (σ²+ω²) is the so called pole frequency. This parameter can be found in the Bode plot of a second order system at that point where the phase is identical to -90 deg.
This frequency is - depending on the pole Qp - not very far from the 3-dB-frequency.
For one specific case with Qp=0.7071 (Butterworth) both are identical.

 

[Polakrun]

second order rc pole frequencies

Thank you for your quick answer.

However your answer make me wonder about the position of the pole :

This parameter can be found in the Bode plot of a second order system at that point where the phase is identical to -90 deg

Isn't it when the phase is -45 deg ?
And can you confirm the following precision which I've forgotten in my first post : you need to divide the complex modulus of the pole by 2\[\pi\] if you want the frequency otherwise you have the pulsation ?

 

[LvW]

location of complex pole in frequency domain

Thank you for your quick answer.
However your answer make me wonder about the position of the pole :

This parameter can be found in the Bode plot of a second order system at that point where the phase is identical to -90 deg

Isn't it when the phase is -45 deg ?
And can you confirm the following precision which I've forgotten in my first post : you need to divide the complex modulus of the pole by 2\[\pi\] if you want the frequency otherwise you have the pulsation ?

Why do you think 45 deg ?
The proof is simple: At w=wp only the middle term of the denominator of the transfer function remains, which is pure imaginary. Thus, with A/j=-A*j >> A*exp(-90 deg).
As to your second question: The term wp normally is called pole frequency and the unit is rad/sec. Of course, if you like to have it in Hz you have to divide by 2*Pi.

 

[Polakrun]

bode plot complex pole positions

Hi LvW,

I explain you what I mean with -45° on the phase with the following example :
Let's take a second order system defined by

If I search the transfer function I find
\[\frac{1}{{s}^{2}\,C1\,C2\,R1\,R2+s\,C2\,R2+s\,C2\,R1+s\,C1\,R1+1}\]
So we have a second order transfer function and with the values specified on the schematics we get :
fpole1=1.99KHz
fpole2=3.40MHz
The Bode plot of the ac analysis give us the following trace :

(I attached the bitmap version of the trace)

When I look at the trace, the first pole occurs when the phase is -45° and the second one when the phase is -135°. Here is my justification.

If I were wrong in my analysis, don't hesitate to tell me.

Thank you.

Pawel

P.S: the simulation is run with LTspiceIV

 

[LvW]

pole frequency -3db

Hi polakrun,

it happens not very often - however, in this case both of us are right.

In your first message (25th) you spoke about a COMPLEX pole - and my answer was of course related to your question. And for a complex pole pair we have only one "pole frequency" (magnitude of the vector to the pole location); and my last answer is correct by 100%.

However, if both poles of a second order function (as in your last example) are REAL, then they appear in the BODE diagram separately and we have two DIFFERENT pole frequencies. And each pole frequency is connected with different phase shifts.
But for the first pole it is NOT 45 deg but something more - depending on the distance to the second pole. If it is very far (as in your case) it has only minor influence - and the phase is perhaps -45.5 deg. But if the 2nd pole is rather close to the first one, the phase shift gets larger. OK ?

 

[xulfee]

pole frequencies

these poles n zeros r present on real frequencies,thats y they r on frequency axis

 

[Polakrun]

magnitude of the pole frequency

Hi LvW,

I agree with the fact that

But for the first pole it is NOT 45 deg but something more - depending on the distance to the second pole. If it is very far (as in your case) it has only minor influence - and the phase is perhaps -45.5 deg. But if the 2nd pole is rather close to the first one, the phase shift gets larger. OK ?

However let me show you an other example : let's take an RC dephasor schematics

I get the following transfer function
\[\frac{1}{{s}^{3}\,C1\,C2\,C3\,R1\,R2\,R3+{s}^{2}\,C2\,C3\,R2\,R3+{s}^{2}\,C2\,C3\,R1\,R3+{s}^{2}\,C1\,C3\,R1\,R3+s\,C3\,R3+{s}^{2}\,C1\,C3\,R1\,R2+{s}^{2}\,C1\,C2\,R1\,R2+s\,C3\,R2+s\,C2\,R2+s\,C3\,R1+s\,C2\,R1+s\,C1\,R1+1}\]
We can see that this is a third order denominator. I apply the value of the componants to this transfer function and solve the third order polynom, I obtain:
s1=-4.6924e+005 +2.2296e-006i
s2=-2.4068e+007 -4.2841e-008i
s3=-9.4449e+003 -2.1867e-006i
Now I take the modulus divided by 2pi of these three complex roots, it gives:
f1=abs(s1)/s1=7.4682e+004
f2=abs(s2)/s2=3.8305e+006
f3=abs(s3)/s3=1.5032e+003
And if you look at the Bode plot of the ac analysis

I just put the cursor on the first two values and we observe again that the phase shift is not exactly -45deg and -135deg but it is around these values.

To emphasize my analysis, let's take a simple low pass filter:
The transfer function is \[H(j\omega)=\frac{1}{1+j\frac{\omega}{\omega_0}}\]. For the phase, \[\phi=-atan\frac{\omega}{\omega_0}\]. So if we are at \[\omega=\omega_0\] then \[\phi=-atan(1)=-\frac{\pi}{4}\]. Thus the frequency when a pole occurs corresponds to a phase shift of 45 deg.

Thank you for the interest you have in this topic, it makes my mind clear about some points.

Pawel