Third Order LP Bessel Filter help

[Jadax]

Hi guys, I'm going through some preparatory work for my classes in Fall and in the main question I need to use ISIS to implement a third order low pass Bessel filter, described by the following normalised transfer funtion:

H(s) = 15/(s³ + 6s² + 15s + 15)

And with a cutoff frequency of 10kHz.

I've used the following website (**broken link removed**) to model my sallen-key circuit on and have the following values of R/C to to achieve my 10kHz cutoff frequency:

R1=9.0269k R2=14.399k R3=7.2144k C1=1.5n C2=2.2n C3=470p (this is theoretical work as of now, which is why the values are ofc not of real components)

1. The thing is, the transfer function listed on that website for the above values of R/C is not the required transfer function I noted above. I don't understand why? could someone please explain to me what I am doing wrong.

2. I need to find the differential equation relating the output voltage (Vout) and the input voltage (Vin). How would I go about doing that?

 

[_Eduardo_]

1. The thing is, the transfer function listed on that website for the above values of R/C is not the required transfer function I noted above. I don't understand why? could someone please explain to me what I am doing wrong.

Why you say that?
There are only a small difference due the rounding or perhaps a bug in the calculation.

2. I need to find the differential equation relating the output voltage (Vout) and the input voltage (Vin). How would I go about doing that?

(s³ + 6s² + 15s + 15) Vo(s) = 15 Vi(s) ==> Vout'''/ωc³ + 6 Vout''/ωc² +15 Vout'/ωc + 15 Vout = 15 Vin

 

[Jadax]

Oh I see, I guess it was naive of my to think the transfer function on that website was the exact calculation for the cutoff/r/c values I supplied.

And thanks for the clarification on the Vo/Vin - I was actually thinking in totally different terms, I knew I had to use the KVL rules to find the loop equations, but not like this.

 

[_Eduardo_]

I attach your question by PM because it is part of the thread.

Hi mate,

Thanks for your help in the thread about the 3rd order sallen-key LPF. I tried finding the differential myself using kirchoffs circuit laws, but I'm not getting the same answer as you noted.

What I did;
Split my circuit into nodes, and used kirchoffs laws to find output voltage at each node and then put them all together through substitution, but I didn't get the same answer as you.

If you could, please can you list down the main steps which allowed you to arrive at that answer.

Transfer function of 3rd order SK:

**broken link removed**

All problems of this kind are algebraically hard ==> The general shortest way is matrices & software

Then, writing the node voltage matrix G and inverting by software (if have some skill, it's only one step)

Now, with r1=9.0269k r2=14.399k r3=7.2144k c1=1.5n c2=2.2n c3=470p the filter has a 3db drop at 10kHz (no math, just watching the Derive6 graph) --> OK


With the transfer funcion of the website you should know that the transfer function given by reverse Bessel polynomials H(s)=15/(s^3+6s^2+15s+15) is normalized in gain but not in frequency.
The normalization constant for the cutoff frequency is usually chosen with the criterion of 3 dB drop, which for a 3rd order Bessel is kn=1.755672
Therefore, if you want to 10kHz cutoff frequency -> must enter as characteristic frequency 10kHz/1.755672 = 5.696kHz

However, the values r1..r3 (and H(s)!) obtained are slightly different and may be due to a different normalization constant or perhaps a bug .


Regarding normalization in Bessel filter frequency are different criteria, Matlab ie does not use the 3dB drop, it make that at low and high frequencies asymptotically approaches a Butterworth.

 

[Jadax]

Thanks, very helpful explanations :grin:

 

[Jadax]

If I am then to try to find the characteristic modes of the equation then do I do the following:

Vout = Ce^λt

so for the differential equation: (λ³ + 6λ² + 15) * Ce^λt = 0

then to determine λ: λ³ + 6λ² + 15 = 0

Then I need to solve for λ, but how do I do that?