When two first-order RC filters are cascaded, the overall transfer function is not simply the product of two individual RC filter transfer functions… WHY?
The effect of the RC-Filter transfer function for each low-pass filter is not identical due to loading effects between the stages. The second stage filter interact to the first stage filter.
Let’s analyze this case in depth:
The Ideal Case
The ideal scenario assumes that there is no interaction between both stages.
The overall system response would indeed be the product of both individual filters transfer function.
A regular first order RC low-pass filter transfer function is:
This means that the total transfer function of the ideal system without considering the loading effects is:
The Real Case
Considering a real scenario, cascading 2 first order low-pass filters introduces loading effects. The impedance of the second stage affects the behavior of the first stage.
The second RC filter acts as an additional load on the first filter stage. Therefore, the output impedance seen by the first filter stage’s output is no longer infinite (assumption of an ideal open-circuit impedance).
Transfer Function Calculation
The difference between the ideal and real case is that the first stage is slightly loaded with the second stage. The equivalent schematic of the first stage RC filter is:
The impedance equivalent to the second RC filter can be expressed as an impedance in parallel with the capacitor C1 of the first stage.
The impedance (equivalent to the second stage) attached to the first stage’s output to ground is equal to:
Simplifying for a unique resistance and capacitance value:
So, the equivalent transfer function of the first RC filter according to the previous schematic is:
Now, the TOTAL transfer function is the multiplication of the first and second stages, because we already took in account the loading effect of the second stage:
Conclusions
As explained before, the behavior of two 1st order RC Cascaded Filter differs from what intuition might think. Simply multiplying the two ideal transfer functions fails to account the loading effect between stages, leading to inaccurate results.
For circuit inspection, the direct multiplication approximation could be enough in IC circuits. Moreover, if the second section of the RC filter draws minimal current from the preceding stage, the direct multiplication remains valid.
One solution to mitigate the loading effect is to place an analog buffer between the two stages.
