First Order Notch Filter: Building Guide

[Bromio]

Hello.

Is there any way to build a band reject filter ('notch') whose transfer function, H(s), has only two complex zeros and only one real pole?

For example:

H(s) = \displaystyle\frac{s^2+4}{s+1000}

 

[MisterX]

Here is the problem with this.

The pole would determine the where magnitude response starts to decrease significantly at the lower frequencies of the trough. One zero would determine around what frequencies the the magnitude response would become approximately level (the trough of the magnitude response), and the other zero would determine at what frequency the magnitude response would increase significantly, at the higher frequencies of the trough.

One more pole would be required to level off this increase. Otherwise the magnitude response may look something like the attached image.

 

[rbj]

we generally have a small problem (the nasty differentiator) when the order of the numerator is higher than that of the denominator in a transfer function.

 

[Bromio]

I understand everything you say. However, there is an issue that I'd like to be clarified.

As you can see in the image attached two posts above, if the system input is limited, |x(t)| < B (B real), then the system output won't be limited too, because the magnitude response will tend to infinity at high frequencies. However, this electric circuit has only a pole, which is in the negative real axis, so the system is stable.

How can these facts both agree?

 

[Bromio]

I've been thinking of a circuit whose transfer function has the same format as that written above. I've attached its diagram.

H(s) = -\displaystyle\frac{L}{R}\displaystyle\frac{s^2+ \displaystyle\frac{1}{LC_2}}{s+\displaystyle\frac{1}{RC_1}}

Is it incorrect?

Thank you.