When does function order matter?

2nd Order RC Passive Filter Transfer Function?

• Hi, I know that 2 first order filters produce a 2nd order filter, but pls check this diagram: http://i51.tinypic.com/29m1vlf.jpg [both Rs and both Cs have equal values] Two scenarios are shown, first (above) one, where the transfer function would be calculated together for that circuit once. (Vout/Vin) Second (below) one where two 1st order filters are cascaded, their individual transfer functions will be multiplied. (Hence, TF = Vout of the 2nd one / Vin of the 1st one) Would the overall Transfer functions of BOTH be "exactly" the same? If yes, should Zeta always be equal to 1 ? if not, why not ? Thanks !

• Answer:

  No, they are not the same. In the first configuration, the second filter loads down the first filter, so you don't have the two ideal filter responses of the second configuration. In the second configuration, if you placed an ideal unity-gain op amp between the stages so that there is no loading on the input or output, then cascading the two identical frequency responses is correct. You will recall that the corner frequency is defined to be the frequency where the response is down 3dB (ie, the voltage gain is square root of 1/2). If you cascade two identical filter stages, each with the corner frequency at f1, then in the combined filter response, the gain is 1/2, not the square root of 1/2, at frequency f1 (because root 1/2 times root 1/2 equals 1/2). So there will be a new corner frequency (higher than f1 in the case of a low-pass filter) at which the frequency response is down 3db. 20 log ( (1/2)^(1/2)) = -3.01db