Band pass (ω1 = 100, ω2 = 10000) from normalized ladder

Frequency transformation

Frequency transformation is a change of the complex frequency variable that converts a normalized low-pass prototype filter (cutoff 1 rad/s) into a high-pass, band-pass or band-stop filter (or a low-pass with a different cutoff). The magnitude response of the prototype is mapped onto the new frequency axis, so the same attenuation values appear at the corresponding new frequencies.

Obtaining a band-pass filter from the normalized low-pass filter

Replace $s$ by $\dfrac{s^2+\omega_0^2}{Bs}$, where $\omega_0 = \sqrt{\omega_1\omega_2}$ and $B = \omega_2 - \omega_1$. Applied to each element:

• series $L$ → series $L' = L/B$ with $C' = B/(L\omega_0^2)$;
• shunt $C$ → parallel $C' = C/B$ with $L' = B/(C\omega_0^2)$;
• resistors unchanged.

This maps $\omega_0 \to 0$ and $\omega_1, \omega_2 \to \mp1$ of the low-pass prototype, so the LP passband becomes the band $\omega_1$ to $\omega_2$.

Design for $\omega_1 = 100$ rad/s, $\omega_2 = 10000$ rad/s

$$\omega_0 = \sqrt{100\times10000} = 1000\ \text{rad/s}, \quad \omega_0^2 = 10^6, \quad B = 10000 - 100 = 9900\ \text{rad/s}$$

Given ladder (4th-order Butterworth, 1 Ω terminations): series 0.7654 H, shunt 1.8485 F, series 1.8485 H, shunt 0.7654 F.

$$\begin{aligned} 0.7654\ \text{H}:&\quad L' = \frac{0.7654}{9900} = 77.313\ \mu\text{H},\quad C' = \frac{9900}{0.7654\times10^6} = 12.934\ \text{mF} \\ 1.8485\ \text{F}:&\quad C' = \frac{1.8485}{9900} = 186.72\ \mu\text{F},\quad L' = \frac{9900}{1.8485\times10^6} = 5.3557\ \text{mH} \\ 1.8485\ \text{H}:&\quad L' = 186.72\ \mu\text{H},\quad C' = 5.3557\ \text{mF} \\ 0.7654\ \text{F}:&\quad C' = 77.313\ \mu\text{F},\quad L' = 12.934\ \text{mH} \end{aligned}$$

  Rs=1  [ Z1 ]        [ Z3 ]
 o-/\/\-[    ]--+-----[    ]--+------+---o
 |              |             |      |   +
 Vs          [ Y2 ]        [ Y4 ]  RL=1  Vo
 |              |             |      |   -
 o--------------+-------------+------+---o

Check: $1/\sqrt{77.313\ \mu\text{H}\times12.934\text{ mF}} = 1000$ rad/s. These values are for $1\ \Omega$ terminations. For practical values, magnitude-scale by $k_m$ (e.g. $k_m = 1000$ for 1 kΩ terminations): multiply every L by $k_m$ and divide every C by $k_m$.