Second order analog filter

[zionico90]

Hi, guys

in my first control design work, I must realize an analog filter (second order) with this characteristics:

- two c.c poles with wn=100 (is the nyquist frequency equal to ws/2?) and a dumping factor equal to 0.1
- no zero
- unitary steady state gain

Which is the correct starting point?

 

[Debdut]

Take a look at this

 

[FvM]

Take a look at this

Instead of a bandpass + integrator combination, we would primarly think of a low-pass structure...

 

[zionico90]

Thank you for the paper.

In according to this i understand that the tf is:

G(s)=a0/(s^2+s*Wn/Q+Wn^2)

When a0 is equal to the DC gain time Wn*2 and in this particular case (unitary gain) a0=Wn^2

What is Q in this formula?The dumping factor?

 

[LvW]

Which is the correct starting point?

I think a good starting point for you would be first to become familiar with filter parameters.
Do you know the meaning of wn and the damping factor?
Do you want a lowpass or a band pass (you speak of a "filter" only).
Do you intend to realize a passive or an active filter? (I suppose active because of the gain requirement).

Remark: The mentioned damping factor d=0.1 is connected with the quality factor Qp by the relation Q=1/(2*d).
From this, I derive Q=1/0.2=5 (not 500 as mentioned by debdut)

 

[Debdut]

Once again I'm uploading an image
View attachment Untitled2.jpg

 

[LvW]

Hi debdut, may I place some comments?

1.) It is not quite correct that "Q determines the distance of the poles from the jw axis". It is not the distance but the ANGLE of the vector connecting the origin with the poles.
The length of this connecting line gives the pole frequency wp and Qp is defined as is Qp=1/2cos(phi) with phi=angle between this line and the neg. real axis. Thus Qp=0.5 for a real pole (phi=0).

2.) There is no necessity to "judge" about the value of Qp. The relation between the damping factor d and Qp is defined as Qp=1/2d. Therefore we have also: d=cos(phi).
More than that, the (false) method as given in post#2 gives a Q value with a unit "rad/s". This cannot be correct.

3.) You are proposing a second order lowpass (you shouldn`t call it "resonator") . Why? Does Zionico90 ask for a lowpass? I think a lowpass with a pole Q of Qp=5 makes not much sense. Rather I would gues that he needs a bandpass.
But his answer is still pending.

 

[zionico90]

Thank you for the answer but unfortunately i don't have any information about the type of the filter. I think that is a LPF because the characteristics are: 2cc poles and no zero but i'm not sure.
If Qp=1/2d than the tf is equal to: a0/s^2+s*(Wn/Q)+Wn^2) where:

Wn is the nyquist frequency equal to 2pi/Ts
a0=AM(DC gain)*wn^2
Q=1/2d

With my parameters I should obtain:

Steady state Gain=1 ----> a0=Wn^2

G(s)= Wn^2/(s^2+s*2d*Wn*Wn^2)
G(s)= 10000/(s^2+20s+10000)

and if I verify the tf I obtain:

2 poles c.c
No zero
and the dc gain lim(s->0)(G(s))=1

Is this the right way to define a tf for a LPF filter?

Thank you

 

[LvW]

Yes - the transfer function is correct.
1.) However, please note that the parameter called wn is NOT the Nyquist frequency but the POLE FREQUENCY, which gives - as mentioned before - the length of the connecting line beween the origin and the pole location in the s-plane.

2.) What is the meaning of your sentence "2cc poles and no zero but i'm not sure"?
Not sure about the zero requirement? This is important because this is related to the question: Low pass or bandpass?
A lowpass has a zero at w>>infinite and a bandpass has a zero at w=0 and at w>>infinite.

3.) Be aware that a lowpass with a damping factor of 0.1 (Qp=5) has a gain peaking up to A(w=wp)=5. Therefore my remark that it makes not much sense to design a lowpass for Qp=5.
That means, that your lowpass starts (at w=0) with a gain of A(w=0)=1 and peaks as high as A=5.
For comparison: A classical maximally flat lowpass (Butterworth) has a pole Q of Qp=0,707 with no peaking at all.

 

[zionico90]

Thank you for your patience!:grin:

I'm sure that the text required 2 c.c poles and no zero. I think that this request means LP filter. It's correct?
Another question, the pole frequency is equal to 2*ws? Because the second step is to write the equivale digital filter with a matlab command "c2s" with the right Ts.

 

[LvW]

Another question, the pole frequency is equal to 2*ws?

As you have seen, I have used no symbol without explaining what it means.
Therefore: What is ws?

EDIT: I really cannot imagine that a lowpass with a pole Q of Qp=5 is desired. Do you know how it magnitude looks like?

Perhaps "no zeros" means: No zeros for any frequency (except the origin w=0) ?

 

[zionico90]

For me Ws is the sampling frequency used to determine the sampling interval request in the c2s function.

 

[LvW]

OK - but such a question comes up after the analog prototyp is realized.

Again: I still cannot imagine that a lowpass with a pole Q of Qp=5 is desired. Do you know how the magnitude with a 500% peaking looks like?
Perhaps "no zeros" means: No zeros for any finite frequency (except the origin w=0) ? And this would be a band pass.

 

[zionico90]

Yes, the request it's very strange. I write an e-mail to request other information and let you know.

Thank you

- - - Updated - - -

Yes, the filter si a band pass filter!Not a low pass filter!

 

[LvW]

Yes, the filter si a band pass filter!Not a low pass filter!

OK - this makes sense.
In this case, wn=wp is identical to the center frequency wo of the bandpass.
And the pole Q is identical to the classical bandpass quality factor Qp=Q=fo/B (fo=wo/2Pi and B=3-dB bandwidth in Hz).

 

[zionico90]

And why I have the dumping factor?

 

[LvW]

Without any damping (what is not possible because of losses in L and C) you would have a bandpass that is a "peak" only (zero bandwidth).
Thus, the damping factor d=1/2Q determines the bandwidth of the bandpass.
In your case: 2Pi*B=wo/Q=100/5=20 rad/s.

Note that - up to now - I have assumed that the term wn (as mentioned by you from the beginning) is identical to the pole frequency wp because you have mentioned it in context with the c.c. poles.
Because YOU have used this term it is YOU who should know it`s meaning.

 

[FvM]

Yes, the filter si a band pass filter!Not a low pass filter!

Then you'll have a zero in the transfer function, in constrast to the initial post.

 

[LvW]

Then you'll have a zero in the transfer function, in constrast to the initial post.

Yes - that`s true (see my post#13). Perhaps the task description means: No zero for s=finite value (also a lowpass has a zero far, far away).