# Bode Plot First-order Butterworth Filter

#### SpNw

Joined May 27, 2020
13
Hi, I need some help,

I want to know if someone can explain to me how do I elaborate a theoretical bode plot of a first-order Butterworth Filter (Band pass).

Imagine my input is 0.06V vp and the pass band frequency should be between (Low frequency = 20Hz, High frequency = 500Hz) (Gain = 10)

Thanks everyone!

#### LvW

Joined Jun 13, 2013
1,749
A Butterworth response has a complex pole pair.
However, in your realization there are two real poles only. Hence, the transfer function cannot have a Butterworth response.

#### Papabravo

Joined Feb 24, 2006
21,029
A Butterworth response has a complex pole pair.
However, in your realization there are two real poles only. Hence, the transfer function cannot have a Butterworth response.
There is a 1st order Butterworth response with a pole on the negative real axis having a Q =0.5

#### LvW

Joined Jun 13, 2013
1,749
There is a 1st order Butterworth response with a pole on the negative real axis having a Q =0.5
It is just a name and, therefore, not too important. But there is a fixed definition for a Butterworth lowpass (which can be transformed to a corresponding bandpass): Maximum flat magnitude with a pole-Q of Qp=0.7071 (corresponding to a complex pole-pair).

#### Papabravo

Joined Feb 24, 2006
21,029
It is just a name and, therefore, not too important. But there is a fixed definition for a Butterworth lowpass (which can be transformed to a corresponding bandpass): Maximum flat magnitude with a pole-Q of Qp=0.7071 (corresponding to a complex pole-pair).
All of that has absolutely nothing to do with the mathematics that allows for a 1st order lowpass and a first order highpass. The inability to specify a 1st order bandpass does not obviate the existence the first order sections. In fact, the first order sections are maximally flat with a Q of 0.5.

#### LvW

Joined Jun 13, 2013
1,749
In fact, the first order sections are maximally flat with a Q of 0.5.
I think this is a kind of misconception.
Let me explain:
For first-order sections we do not have the choice (as with second order structures) between "more or less flat".
There is only one type of transfer function with the pole quality Qp=0.5.
Therefore, no name is assigned to these first order sections, because there are no alternatives.
The name "Butterworth" is commonly used only for functions of at least second order with a specific pole distribution (Qp=0.7071 in case of 2nd-order).
(The term "maximally" flat is reserved for functions which result from a certain mathematical optimization technique. When this method is applied to the phase function we get a "maximally flat" phase response ("Bessel"-filter).

#### Papabravo

Joined Feb 24, 2006
21,029
I think you're wrong about that and fortunately the authors of major textbooks agree on this matter. I see no point in further engagement on this matter.

#### Audioguru again

Joined Oct 21, 2019
6,619
Your first-order filters have very gradual slopes and their cutoff frequencies are very far apart so the filter does not do much filtering.
Here is a simulation. Add as much gain as you want and add an output opamp if you want.

#### Attachments

• 26.1 KB Views: 10

#### LvW

Joined Jun 13, 2013
1,749
I think you're wrong about that and fortunately the authors of major textbooks agree on this matter. I see no point in further engagement on this matter.
Too bad - no factual arguments showing/explaining that (and why) I am wrong.

#### Papabravo

Joined Feb 24, 2006
21,029
Too bad - no factual arguments showing/explaining that (and why) I am wrong.
That's correct. Why should I waste my time constructing an argument your ego will not allow you to accept. I don't care what you believe or don't believe. Someone else may care enough, but not me.

#### LvW

Joined Jun 13, 2013
1,749
Why should I waste my time constructing an argument your ego will not allow you to accept.
I am surprised - are such personal attacks really necessary and adequate? Why?
Didn`t I try to technically explain my sight in post #5 and #7? Does it not deserve a factual answer?

#### BobTPH

Joined Jun 5, 2013
8,682
This is certainly a matter of semantics. All single pole filters have the same response shape, therefore all are maximally flat. Butterworth response is used to minimize ripples in the bandpass, first order filters have no ripples to start with.

#### Audioguru again

Joined Oct 21, 2019
6,619
The single pole filter response is the same as one of the two 2-pole filters in a Bessel filter type, droopy.
A Butterworth filter type uses some positive feedback to have a flat response then a sharp corner.

#### crutschow

Joined Mar 14, 2008
34,063
bode plot of a first-order Butterworth Filter (Band pass).
Since you are already using op amps, you should convert them to 2-pole or 3-pole active filters to get a much steeper roll-off between the stop-band and and the pass-band.