Integrator,differentiator and filters

[legend killer]

Hi everyone
I am a little confused with the difference between high pass filter and differentiator,and low pass filter and integrator.The frequency response of differentiator,HPF and integrator,LPF are nearly same.
So what is the difference between them?

 

[MikeMl]

Still think they are the same?

 

[legend killer]

Thanks for replying Mike,In my book its mentioned that the gain(of integrator) is very high at low frequencies,so to control the low frequency gain,we add a resistor in parallel with the capacitor in a practical integrator,so frequency response is almost the same.I've attached the snap of the book which I am referring to

 

[MikeMl]

However, that is no longer an integrator, it has become a low pass filter with a fixed gain in the passband with the addition of the resistor in the feedback path.

I'd call it a "Low Pass Filter with a gain of xxx".

 

[legend killer]

Hi Mike,
Looking at the frequency response, cant we say that it integrates after the 3-dB cutoff frequency.

Basically this question was asked by my teacher to see how much understanding do I have on the subject.As such I could not sort out the difference,so thought of discussing it here.

Could it be the answer that the LPF after 3-dB cutoff frequency acts as integrator(Because phase angle reaches 90 degrees)

 

[crutschow]

Could it be the answer that the LPF after 3-dB cutoff frequency acts as integrator(Because phase angle reaches 90 degrees)

That's basically true for a 1st order filter, however LP filters can have higher orders and those do not act as simple integrators above the -3dB cutoff.

 

[MrAl]

Hello there,


I think the simplest way to distinguish these differences is as follows...

The transfer function for a perfect integrator is A/s, while for a first order low pass filter it is A/(s+B). Clearly, the difference is in the denominator with the constant 'B'. Now if B is small it will approximate A/s so we can say although it is not a perfect integrator it resembles one. In other words, when B is small A/(s+B) approximates A/s.

In the practical case with a given value for the feedback resistor, you can calculate B and see the difference in B when using a high value resistor and when using a low value resistor.

 

[legend killer]

Hello there,


I think the simplest way to distinguish these differences is as follows...

The transfer function for a perfect integrator is A/s, while for a first order low pass filter it is A/(s+B). Clearly, the difference is in the denominator with the constant 'B'. Now if B is small it will approximate A/s so we can say although it is not a perfect integrator it resembles one. In other words, when B is small A/(s+B) approximates A/s.

In the practical case with a given value for the feedback resistor, you can calculate B and see the difference in B when using a high value resistor and when using a low value resistor.

Hi MrAl,
I found the tranfer function and got the value as -1/(R1.Cf) and the value of B as 1/(Rf.Cf).Clearly if Rf approaches infinity,the transfer function resembles that of ideal integrator.

 

[legend killer]

Hello there,


I think the simplest way to distinguish these differences is as follows...

The transfer function for a perfect integrator is A/s, while for a first order low pass filter it is A/(s+B). Clearly, the difference is in the denominator with the constant 'B'. Now if B is small it will approximate A/s so we can say although it is not a perfect integrator it resembles one. In other words, when B is small A/(s+B) approximates A/s.

In the practical case with a given value for the feedback resistor, you can calculate B and see the difference in B when using a high value resistor and when using a low value resistor.

Hi MrAl,
I found the tranfer function and got the value of A as -1/(R1.Cf) and the value of B as 1/(Rf.Cf).Clearly if Rf approaches infinity,the transfer function resembles that of ideal integrator.

 

[MrAl]

Hi again,


You dont have to tell me twice
Yes that sounds right. As the denom goes higher and higher the fraction becomes smaller and smaller thus making B approach zero and the response approaches the ideal integrator.
Additionally, a second order LP filter can be made to approximate a double integration using the same basic idea with constants A,B,C,D where both B and D approach zero and the response approaches an ideal double integration. Following that line of reasoning further, an N^th order LP can be made to approximate N integrations with N constants approaching zero.