# cascading two single pole low pass filters

Discussion in 'Homework Help' started by PG1995, Jun 2, 2012.

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1. ### lebevtiMember

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[MODNOTE]Deleted Off Topic[/MODNOTE]

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2. ### PG1995Active Member

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Thank you very much, Winterstone.

Seriously I still don't see how I can reach this result. Which result of these, "1" or "2", do you think is similar to the Wikipedia result? Please let me know. Thank you.

Regards
PG

Last edited: Jun 26, 2012
3. ### WinterstoneBanned

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Hi PG1995,
Now I believe to understand your problem.
At first, you must know that there are two different ways to arrive at a 2nd-order transfer function:
1.) Knowing that a 2-nd order function of our interest has two complex poles in the LHP of the s-plane, one can plot these poles and define two important parameters:
The pole frequency wp and the pole quality factor Qp (wp=length of the vector to the pole position, Qp=1/(2cos(phi)) with phi=angle beween the negative real axis and the vector).
Then, the transfer function can be written as a function of wp and Qp.
This is the general form of the transfer function - applicable to ALL 2nd-order low pass functions.
2.) Independent on this approach, you have a 2nd-order circuit with feedback, that is able to produce a complex pole pair. Now you can calculate the transfer function, which belongs to this circuit only!
3.) As a 3rd step you can compare both functions (if the have the same standard form). This gives you two equations saying how to select the parts values in order to enable the desired value for wp and Qp.
_____________

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5. ### MrAlWell-Known MemberMost Helpful Member

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Hi,

What you are looking at seems to be two different forms of the same thing. Equations come in various forms which all mean the same thing in general but they show different detail characteristics. Some forms are better than others for understanding certain things about a problem or circuit.

For example, an equation with the "Q" in it would be much better for understanding what the Q is in the circuit, obviously better than an equation without it. Often we want to rearrange the equation to explicitly *show* this information rather than something else or some random form that we happen to have come up with because of the analysis approach. This means we might start with an equation without the Q in it and try to rearrange it so that the Q appears somewhere in it so we can later look at the circuit in terms of its Q rather than just the R and C component values. This often helps enormously in some problems and without this ability to rearrange we would be lost.

Another good example is the poles. Often we want to know the poles and zeros so we arrange it so we can see them more clearly.

A real simple example is this:
Fo=1/(w*sqrt(LC))

but for our design application we really wanted to be able to look at the function Fo in terms of the frequency f in Hertz rather than in angular frequency w, so we simply solve for w and insert the result and we get:
Fo=1/(2*pi*f*sqrt(LC))

and there we have it. It could have come at us the other way around too, where we already had the latter equation but we wanted to know how it looked in terms of w, so we would have simply made 2*pi*f equal to w and that would give us the new view we wanted.

Equations are not just formulas. They relate variables to each other. By contrast, formulas are used to calculate something but equations show the relationship between several variables to each other.

Last edited: Jun 29, 2012
6. ### WinterstoneBanned

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Fo=1/(w*sqrt(LC))

7. ### MrAlWell-Known MemberMost Helpful Member

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Hi there Winterstone,

Ah haaaaaa...Caught ya!

You didnt read the last paragraph of my previous post..."Equations are not formulas".

The equation:
Fo=1/(w*sqrt(LC))

is not the same as:
w=1/(sqrt(LC))

I cant blame you too much i guess though because it does sort of look like the formula for the resonate frequency of an LC circuit. It's good to try to stay on our toes and keep an eye out for things that look almost the same but are not the same. I wish i could say that i do this all the time too but unfortunately i dont either
It's is also good that you check for things like this because typo's are all too frequent.

Last edited: Jun 28, 2012
8. ### WinterstoneBanned

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Surprisingly, I did.
Nevertheless - equations or formulas, both have to fulfill the rules of eqal dimensions/units on both sides (sorry for this bad expression, hope you know what I mean) : [1/sec]=[1/sec].
By the way - I could imagine that it was a typo. But also typo's should be corrected, should they not?
W.

9. ### MrAlWell-Known MemberMost Helpful Member

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Hi again,

Yeah you're trying to force your own definition on my own words and then state that it is not correct. You dont have to do that, it's ok the way it is. I never suggested that there were any particular dimensions either.

You're thinking of:
freq=1/(2*pi*sqrt(LC))

and trying to force MY equation to be that. It does not have to be that and the dimensions do not have to be what you think they should be.

You're certainly correct though when you say that the dimensions of an equation should be ok too. But remember we also have K=5cm/6cm.
For example, 1=1/(2*pi*f*sqrt(LC))
with dimensions just fine.

Last edited: Jun 28, 2012
10. ### WinterstoneBanned

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Thank you, MrAl, for the lesson.
Now I have learned how to justify my own typos (see your post'66).

Apparently, I was in error assuming that the following sentence would contain a parameter Fo with the dimension Hz=1/sec (although you call it "frequency").

Finally, whenever I had produced a typo - or even a logical error - I had no problems to say "sorry" and to correct it.
Kind regards
W.

Quote: "It does not have to be that and the dimensions do not have to be what you think they should be."
Would you please so kind and tell me what the dimensions in your equation are?

Final remark (excuse): By writing line 2 and 3 of my reply I couldn't resist to be a bit ironical.

Last edited: Jun 29, 2012
11. ### MrAlWell-Known MemberMost Helpful Member

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Hello again,

You're right in that i probably shouldnt have said "frequency" but that still doesnt mean that i cant write an 'equation' to illustrate how to look at it one way and convert it to another way. That was the main point of the post but you dont really care about that do you.

But since you're never going to let up until this equation is 'corrected' (and by 'corrected' i mean it has to be written YOUR way and ONLY your way), here ya go:

Fo(w)=1/(w*sqrt(LC))

So i hope that satisfies you, now Fo is a function which takes 'w' as an argument.
And also the original text was altered to read 'function' instead of 'frequency' so the units work out to your desired specification.
A brief 'thank you' is in order here so i'll be waiting

Last edited: Jun 29, 2012
12. ### WinterstoneBanned

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MrAl, are you joking?
I am sure, all forum members - including me - reading this equation would like to know what the meaning of this "funny" equation is.
Please, can you instruct us accordingly? Didn't you say in your post#64 that Fo is an example for the pole frequency?

Of course, your expression does NOT "satisfy" me, since it is now - in contrast to your post #64 - nothing else than an abbreviation for a frequency ratio: Fo(w)=1/(w*sqrt(LC))=wo/w .

If you really are interested in "my way" of writing the formula for the resonance frequency of a tank (which seems to be also an equation) - see my posting #65.
MrAl, excuse me - but it is really not easy to discuss with you since you are not able to correct a simple error. Sorry.
I propose to end this "discussion" (?) - no, it was something else.
W.

13. ### MrAlWell-Known MemberMost Helpful Member

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Hello again,

Oh ok i see you are starting to speak for "all forum members" now, so i guess you PM'd or email'd thousands of members to solicit an opinion. That's very nice of you to take all that time just for this one little equation

You're starting to be just a little too argumentative over this so and refuse to see any other point than your own. Thus, i wont reply to this particular 'equation' again, be it a 'correct' equation or an 'incorrect' equation. No matter what i say you'll keep insisting that it is 'incorrect' in some way which you dont like, so i propose that you write your 'own' equation exactly the way you'd like to see it written and end this silly back and forth.
As i said though, i wont reply to this equation again unless i see it as constructive in some beneficial way. There's just no point.

On the plus side, i do appreciate your interest in these matters.

Last edited: Jun 29, 2012
14. ### PG1995Active Member

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Thank you very much, Winterstone, MrAl, for all the help. I understand it now.

Regards
PG

15. ### WinterstoneBanned

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Hi PG1995,
thank you for finalizing this (funny) thread.

I suppose, you have learned
* something about active analog filters, and
* how a technical discussion should NOT come to an end.

What has happened? Something which is quite normal in human life:
Somebody has made a simple error in writing (post#64) and I have tried to correct this error with my post#65.
That’s all..
I really don’t know what MrAl was aiming at by starting then a discussion (post#66) in order to justify this error (instead to admit a simple typo).
However, that's life - today, I regret to have participated.
Regards to you
W.

16. ### WinterstoneBanned

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PG1995,

1.) If you have found the transfer function - in terms of parts values and gain - you can compare this function with the general second order function in terms of Qp and wp.
As a result, you get two (three) equations for Qp and wp (and gain), respectively, which lead to new formulas to design the filter according to your requirements.
In this context, of course, some values can be freely chosen because you have only two (three) equations but at least 4 or 5 parts.

2.) The mentioned filter requirements are normally given in terms of cut-off frequency (very often, but not always, identical to the 3dB points) and kind of filter approximation (Butterworth, Chebyshev,...).
There are tables (textbooks, internet) which relate these requirements to the needed Qp and wp values.

3.) Thus, you can fix all element values.
_______________

Hope this helps.
Don't hesitate to ask further questions (filters, oscillators) - but open a new thread!

Regards
W.

17. ### MrAlWell-Known MemberMost Helpful Member

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Hello again PG and Wintystone,

I am happy you got something out of this thread. It's unfortunate that Winterstone insists that there is an error in an issue that has no right or wrong, and even after i invited him to post his own version of what it 'should' be.

Winterstone:
After all i have told you if you still feel there is some 'error' then you have every right to post your own version of what you 'think' it should be. But i dont believe you are socially correct in trying to make a joke out of something based on your own misconceptions. POST YOUR OWN VERSION and lets see what it looks like and stop mocking things that you dont understand. It's not fair to the other members.
POST YOUR OWN VERSION that's all it takes.

Last edited: Jun 30, 2012
18. ### WinterstoneBanned

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Dear Forum member Mral !

It was not my intention to continue this unworthy dispute. However, your personal attacks on me have changed my mind. Do you really think that makes your "arguments" more credible?
My experience up to now regarding discussions on technical subjects with other people: As soon as somebody starts personal attacks he fails to have technical arguments.
For my opinion, it is the aim of this forum to allow exchange of technical arguments and information. If you like to attack me personally, use personal messages.

Two times you have asked for my "OWN VERSION". Version for what? Didn't you recognize that we do not speak about "versions"?
The subject under discussion is simply a formula - right or wrong?
I gave you my VERSION already in post#65. If this is false - correct me! I really don't know what you are aiming at.

Have you forgotten your own words? Here is a reminder:

Quote MrAl (post#64): Another good example is the poles. Often we want to know the poles and zeros so we arrange it so we can see them more clearly.
A real simple example is this:
Fo=1/(w*sqrt(LC))
but for our design application we really wanted to be able to look at the function Fo in terms of the frequency f in Hertz rather than in angular frequency w

Five posts later - and for my opinion not correct because of some replies in between - you have modified (post#70) your own post post#64 (term exchange):

Quote MrAl (post#70): And also the original text was altered to read 'function' instead of 'frequency'

From post#64 (in its original version with the term "frequency") I have deduced that the symbol Fo should represent nothing else than the pole resp. resonance frequency of an LC tank circuit.

Thus, it was clear for me that the symbol w in the denominator was simply a typo - and I have corrected it in my reply post#65:

w=1/(sqrt(LC))

Dear MrAl - I repeat: This is my "VERSION" of a formula that relates the angular pole frequency to the elements L and C of a tank circuit.
If you are still not satisfied, you should be a bit more specific what kind of "VERSION" you are referring to.

Respectfully
W.

Last edited: Jul 1, 2012
19. ### MrAlWell-Known MemberMost Helpful Member

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Hello again Winterstone,

Yeah, you're just reading too much into this. It's not as difficult as you are trying to make it. And i dont appreciate your personal attacks toward my screen name. But i'll give you the benefit of the doubt anyways and repost the intended dialog here now...

The original text was intended to provide insight into the idea of isolating a variable to allow a quicker examination or understanding from the equation.

A common simple example is when we convert a transfer function into it's pole zero form.

[Start of new paragraph]
A real simple example of an equation is this:
Y=1/(w*sqrt(A*B))

where
w=2*pi*f

Writing it as:
Y=1/(w*sqrt(A*B))

allows us to see it as a function of w so we can quickly see the influence of w on Y. But writing it as:
Y=1/(2*pi*f*sqrt(A*B))

allows us to see it as a function of f so we can quickly see the influence of f on Y.

We may even decide that sqrt(A*B) is too much information for this equation as it may have less meaning at the time so we may choose to write it simply as K which makes our equation even simpler:
Y=1/(2*pi*f*K)

or the alternate view:
Y=1/(w*K)

So the way we write the equation can make a profound difference on how we understand it and it's relationship to other concepts associated with the overall problem at hand. We might have other equations in 'w' we want to compare with it later.

Last edited: Jul 1, 2012
20. ### WinterstoneBanned

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I have no problems to follow your lesson above. But it is another - although a related - story than the subject of this thread.

Why don't you use clear words/formulation ?
What is correct and what is false in my arguments?
What do you mean with "difficult as you are trying to make it" ?
At which point I am "reading too much into this" ?
Is my formula post #65 correct yes/no ?
Why don't you answer my questions?
Was the symbol Fo in your post#64 representing the pole frequency yes/no ?
If not - do you confess that all of your formulations (terms: pole, frequency) gave occasion to assume that you mean "pole frequency"?
Don't you understand that - in this case - a correction was unavoidable?

In the past, I have told all my students that the main difference between technical and other (social, medical, ...) sciences is the following:
It is always (perhaps not always, but by far in most cases) possible to proof if a claim (formula, sentence, criterion,...) is correct or not.
I think, also in this case it would be wise to restrict ourself on technical arguments - without using weak formulations as contained in the cited line above.
Otherwise misunderstandings are unavoidable (as we have experienced).