There is a need for a fifth-order low-pass Butterworth filter with a cutoff frequency of (100 mathrm{rad}

Question:

There is a need for a fifth-order low-pass
Butterworth filter with a cutoff frequency of \(100 \mathrm{rad} / \mathrm{s}\) to use in a medical instrumentation system with frequencies below 100 \(\mathrm{rad} / \mathrm{s}\). However, there was no power available for the filter. A young engineer went to the library and found an old textbook on passive filter design. \({ }^{6}\) Curious, he found the filter shown in Figure P14=3 8 that he felt could work.
(a) The circuit is a prototype for a 1-rad/s response. The textbook states that one can scale the frequency of the circuit by: \(L_{\text {new }}=L_{\text {old }} / k_{f}, C_{\text {new }}=C_{\text {old }} / k_{f}\), and \(R_{\text {old }}=R\) new, where \(k_{f}\) is the increase in frequency. Find the circuit's transfer function.
(b) For a fifth-order Butterworth, the nominal values of the components are: \(L_{1}=0.8944 \mathrm{H}, L_{2}=1.694 \mathrm{H}, C_{1}=\) \(0.3090 \mathrm{~F}, C_{2}=1.382 \mathrm{~F}, C_{3}=1.545 \mathrm{~F}\), and \(R_{1}=1 \Omega\). Modify the circuit for the \(100 \mathrm{rad} / \mathrm{s}\) cutoff frequency. Then scale the components to more reasonable values using \(R_{1}=100 \Omega\).
Validate, or not, the engineer's research claim using Multisim.

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