Firstly, some basic theory….
The transfer function of a simple low pass filter is:-
This transfer function has one pole. The gain goes from 1 to almost zero as frequency increases and the phase goes from 0 to -90 deg. giving -20dB roll off at -90 deg. This transfer function is referred to as “a simple lag”. The gain magnitude Bode plot is:-
My question is - Is there any practical RC circuitry which will cancel out this pole? That is to say circuitry which has a cancelling zero of 1+Ts as its transfer function. This transfer function is referred to as “a simple lead”. The zero would have a gain increase of 20dB/decade at +90 degrees starting at the same w = 1/T frequency as the pole which it is cancelling. The overall result of cascading the low pass pole with the added circuitry would be 0dB gain and 0 deg. phase shift at the output over the full bandwidth. The transfer function of this pole cancelling circuitry would have a gain Bode plot of:-
Intuitively I would expect a simple RC high pass filter to do the job of cancelling out the pole of the simple RC low pass filter but this is not the case. A simple RC high pass filter has the transfer function:-
As can be seen, this transfer function has a zero, s in its numerator but there is also a pole in the denominator. The high pass filter gain Bode plot looks like this:-
As is apparent from the Bode plot, the high pass filter does not cancel out the low pass filter. In fact if the two filters were to be cascaded, the overall gain peaks at a maximum of ½ that of the individual filters at w = 1/T before falling away and the phase goes from +90 deg. to -90 deg. as frequency increases. (An attenuated bandpass filter).
So in a practical sense, is there any RC circuitry which will perfectly cancel out the phase and magnitude of a pole?