Frequency Scaling

It scales the frequency without affecting the magnitude of the impedance. It only changes the frequency dependent component such as Capacitor ‘C’ and Induction ‘L’ but not Resistor ‘R’. With Frequency Scaling, transfer function also changes and values are represented by new frequency domain. Frequency scaling is done by a factor of K(f).

fig: Frequency scaling

When,

K(f) > 1, then it is called expansion scaling

K(f) < 1, then it is called compression scaling

\[
\begin{aligned}
&\text{For Resistor:} \\
&\quad \text{Since resistor is not frequency dependent, it is not affected:} \\
&\quad R_{\text{new}} = R_{\text{old}}
\end{aligned}
\]

\[
\begin{aligned}
&\text{For Inductor:} \\
&\quad X_L = sL \\
&\quad X_L = j \omega L \\
&\quad \text{Multiply and divide by } K(f), \text{ we have:} \\
&\quad X_L = j \omega L \times \frac{K(f)}{K(f)} = (j \omega K(f)) \left(\frac{L}{K(f)}\right) \\
&\quad X_L = j \omega(L) \cdot L_{\text{new}} \\
&\quad \text{where} \quad \omega(L) = \omega \times K(f) \quad \text{and} \quad L_{\text{new}} = \frac{L}{K(f)}
\end{aligned}
\]

\[
\begin{aligned}
&\text{Similarly, for Capacitor:} \\
&\quad X_C = \frac{1}{Cs} \\
&\quad X_C = \frac{1}{j \omega C} \\
&\quad \text{Multiply and divide by } K(f), \text{ we have:} \\
&\quad X_C = \frac{1}{j \omega C} \times \frac{K(f)}{K(f)} = \frac{1}{j \omega K(f) \cdot \frac{C}{K(f)}} \\
&\quad \text{where} \quad X_C = \frac{1}{j \omega(C) \cdot C_{\text{new}}}, \quad \omega(C) = \omega \times K(f), \quad C_{\text{new}} = \frac{C_{\text{old}}}{K(f)}
\end{aligned}
\]

\[
\text{Old Transfer function } T(j \omega) \text{ is replaced by new transfer function } T(j \omega_c K(f))
\]

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

\[
L_{\text{new}} = \frac{L_{\text{old}}}{K(f)}
\]

\[
C_{\text{new}} = \frac{C_{\text{old}}}{K(f)}
\]