Cascadability

[Devrra]

Fig 1 shows an RC-circuit. This block can act as a LTI system (say S).
The transfer function of S = H1(s) = 1/(1+RCs).
From basic concepts of LTI systems, if we cascade 2 blocks of S, then resultant transfer function should be H2(s) = 1/(1+RCs)².

But, the transfer function of circuit in Fig 2 is, H3(s) = 1/((2+RCs)(1+RCs)+1) \[ \neq \] H2(s).

And, the transfer function of circuit in Fig 3 is, H4(s) = 1/(1+RCs)²= H2(s).

Does there exist any condition for cascadibility?

 

[Audioguru]

By simply adding -3dB filters, the result is a very droopy filter. Two stages make the cutoff down at -6dB, four stages make it down at -12dB. Lower frequencies are also reduced.
Instead an active Butterworth filter using an opamp is used to boost the level at the cutoff frequency creating a sharper cutoff and less reduction at lower frequencies.

 

[BradtheRad]

With each RC stage you get increased phase shift. The phase shift is inseparable from the attenuation of high frequencies.

By cascading two stages you get 2x as much phase shift. Amplitude is likewise reduced. It's mathematically unavoidable.

 

[Easy peasy]

To avoid loading effects the 2nd stage resistor should be 10x the 1st stage and the cap 10x less - this gives nearer the ideal cascaded filter, as the 2nd stage is not loading the 1st stage so much, but the RC time constant of each stage is the same
then buffer with op-amp. ....