LPF to HPF Transformation

fig: LPF to HPF transformation

 

\[
\text{For old low-pass: cut-off frequency } \omega \text{ for } s = \omega
\]

\[
\text{For new high-pass: cut-off frequency } \Omega
\]

\[
\text{Replace } s \text{ by } \Omega \, \omega s
\]

\[
\text{For normalized value } \omega = 1 \text{ rad/sec, then } s' = \Omega s
\]

\[
\text{This maps the interval from } j \Omega \text{ to } +\infty \text{ in s-plane to interval } -j \omega \text{ to } 0 \text{ in s'-plane, and vice versa.}
\]

\[
\text{Frequency } 0 \text{ to } \infty \text{ for low-pass is transformed to } \infty \text{ to } 0 \text{ for high-pass.}
\]

\[
\text{Passband of low-pass } 0 < |\omega| \le 1 \text{ is mapped to passband } 1 \le |\omega| \le \infty \text{ of high-pass.}
\]

\[
\text{New components:}
\]

\[
\text{Resistor: No change, frequency independent}
\]

\[
\text{Inductor: } X_L = L s, \quad \text{replace } s \text{ by } \Omega s: X'_L = L \Omega s
\]

\[
\text{Comparing } L \Omega s \text{ with } \frac{1}{C s}, \text{ inductor is replaced by capacitor: } 
C' = \frac{1}{L \Omega}
\]

fig: Inductor and its representation incase of HPF

\[
\text{For Capacitor: } X_C = \frac{1}{C s}
\]

\[
\text{Replace } s \text{ by } \frac{\Omega}{s}: X'_C = \frac{1}{C} \cdot \frac{\Omega}{s} = \frac{\Omega}{C s}
\]

\[
\text{Comparing } \frac{\Omega}{C s} \text{ with } L s, \text{ the capacitor is replaced by inductor, such that: } L' = \frac{1}{C} \Omega
\]

\[
\text{Capacitor } C \text{ is replaced by Inductor, where } \Omega \text{ is the cut-off frequency for High Pass.}
\]

fig: Capacitor and its representation incase of HPF