Sallen Key Second order low pass Active filter-Butterworth and Bessel responses-Kc values.

[sakaran51]

Hi,
I was going through a book on Electronic Principles on Active filters. While on the subject of Sallen-Key second order low pass filters,I came across the following statements:

Butterworth and Bessel Responses
"When analysing a circuit,we start by calculating Q and fp(pole frequency).If Q=0.707,we have Butterworth response and a Kc value of 1.
If Q=0.577,we have a Bessel response and a Kc value of 0.786.Next we can calculate the cut off frequency with:
fc=Kc.fp
With Butterworth and Bessel filters,the cut off frequency is always the frequency at which the attenuation is 3dB."
There is a table showing Kc values for different Q values.
Now,what is this Kc?The author does not explain this. Are Kc values the derived ones?If so,how? Why the Q and Kc values are different for Butterworth and Bessel filters?
Can anyone throw some light on this?I tried to find the answer but confusion worse confounded!
Thanks

 

[benta]

Sounds like a lousy book. I don't like using f (frequency) in analysis. angular speed is the variable to use (ω = 2πf).

Second order filters (or any other orders) are defined by their transfer function, which you should know, and this includes ω[sub]0[/sub] and Q.

Only for Butterworth LP transfer functions does the -3 dB point coincide with the ω[sub]0[/sub] term. Others (Bessel etc.) have a more complicated -3 dB frequency.
I suspect that the book tries to convey this in some way using K[sub]C[/sub].

Not a good approach IMO.

If you really want to get a picture here: make a spreadsheet with the transfer function and plot the output. It's not hard.

 

[sakaran51]

Thanks benta.
Let me first tell you that I was only trying to understand the active filter circuits. I am not familiar with filter circuits. It is a bit overwhelming. I got struck up with responses of Butterworth(if Q=0.707,then Kc=1) and Bessel (if Q=0.577,then Kc=0.786),as mentioned in the author. The author has not explained about Kc. My question is what is this Kc and from where they have come from.
Maybe the book I have come across is lousy and my question foolish! Kindly bear with me!

 

[IRstuff]

Do you have a reference to this book? Asking random people why a random author chose a particular notation isn't very productive.

TTFN (ta ta for now)
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[sakaran51]

Hi IRstuff,
Please find the link for the book I was referring. In the chapter of Active Filters, the statement mentioned in my OP,is figuring.

https://www.engineeringbookspdf.com/electronic-principles-8th-edition-albert-malvino-david-bates/

 

[IRstuff]

??? The same paragraph that talks about the Butterworth and Bessel filters having cutoff frequency equal to pole frequency defines Kc

TTFN (ta ta for now)
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[benta]

IR, you're right, but the explanations in the book is deficient, to say the least.

I've read the chapter in question, and what is an attempt from the author to "dumb down" things results in the exact opposite.
By inventing/introducing obscure terms like K[sub]C[/sub], it actually gets more difficult to understand the subject.
The author uses it to frequency-shift ω[sub]0[/sub] to coincide with the -3 dB point of the filter because it doesn't fit in a Bessel response. Complete rubbish.

This is not a good book.

OP: are you proficient in complex maths/imaginary numbers? That's the place to start if you want do do circuit analysis.

 

[sakaran51]

Thanks benta & IRstuff for saving me from any further confusion. Unfortunately I got tangled with a puzzle in a subject which is already complicated. I am not proficient in complex maths/imaginary numbers and I no have inclination for circuit analysis! I just wanted to understand the subject out of sheer interest.
Once again thank you for enlightening me.

 

[IRstuff]

Well, on the other hand, there is invariably something that acts like Kc, so you know that if you want a Bessel filter with a specific cutoff frequency, you divide by Kc to get the desired pole frequency.

TTFN (ta ta for now)
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[IRstuff]

I looked into the text some more; it's intended for a high school level student and is very reminiscent of my high school electronics text, and requires no more than trig for math background. That automatically rules out detailed understanding of anything beyond a 10,000-ft level view, since derivatives and integrals are outside the scope of the text.

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

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[sakaran51]

Of course, I went for this particular book for its conversational style and no prerequisite for higher mathematics background. Can you suggest a book or give me a link on this subject for better understanding?