Numerical Related to Frequency Scaling

Scale the given circuit for ω _(new) to be 1000 

Fig: LPF

\[
W = 1, \quad \omega(C) = 1000
\]

\[
\text{We have,} \quad \omega(C) = W \times K(f) \implies K(f) = \frac{\omega(C)}{W} = \frac{1000}{1} = 1000
\]

\[
C_{\text{new}} = \frac{C_{\text{old}}}{K(f)} = \frac{1 \ \text{F}}{1000} = 1 \ \text{mF}
\]

\[
R_{\text{new}} = R_{\text{old}} = 1 \ \Omega
\]

fig: Frequency Scaled Circuit

Similarly, the transfer function for 1st figure is given by

\[
T_{\text{old}}(s) = \frac{\frac{1}{R_{\text{old}} C_{\text{old}}}}{s + \frac{1}{R_{\text{old}} C_{\text{old}}}} = \frac{1}{s + 1}
\]

\[
T_{\text{new}}(s) = \frac{\frac{1}{R_{\text{new}} C_{\text{new}}}}{s + \frac{1}{R_{\text{new}} C_{\text{new}}}} = \frac{10^{-3}}{s + 10^{-3}}
\]