Numerical 6: Scaling

Perform scaling for R(new)=2000Ω, w=1rad/sec, C=10^5rad/sec

fig: unscaled LPF

\[
\textbf{Given:}
\]
\[
R_{\text{new}} = 2000\,\Omega,\quad \omega = 1\,\text{rad/sec},\quad C = 10^5\,\text{rad/sec}
\]

\[
\textbf{Scaling Factors:}
\]
\[
K_m = \frac{R_{\text{new}}}{R_{\text{old}}} = \frac{2000}{20} = 100
\]
\[
K_f = \frac{C}{\omega} = \frac{10^5}{1} = 10^5
\]

\[
\textbf{Resistor Transformation:}
\]
\[
\text{For } R_{\text{old}} = 1\,\Omega:\quad R_{\text{new}} = K_m \cdot R_{\text{old}} = 100 \cdot 1 = 100\,\Omega
\]

\[
\textbf{Inductor Transformation:}
\]
\[
L_{\text{new}} = L_{\text{old}} \cdot \frac{K_m}{K_f}
\]

\[
\text{For } L_{\text{old}} = 1\,H:\quad L_{\text{new}} = 1 \cdot \frac{100}{10^5} = 1\,\text{mH}
\]

\[
\text{For } L_{\text{old}} = 4\,H:\quad L_{\text{new}} = 4 \cdot \frac{100}{10^5} = 4\,\text{mH}
\]

\[
\textbf{Capacitor Transformation:}
\]
\[
C_{\text{new}} = C_{\text{old}} \cdot \frac{1}{K_m \cdot K_f}
\]

\[
\text{For } C_{\text{old}} = 0.5\,F:\quad C_{\text{new}} = 0.5 \cdot \frac{1}{100 \cdot 10^5} = 0.5\,\mu F
\]

The new circuit becomes

fig: Scaled LPF