Band pass (2000 rad/s, 400 rad/s) from given ladder

Importance of frequency transformation

• Only one set of approximation tables (low-pass poles and ladder element values) is needed; HP, BP and BS filters are derived from it.
• The difficult approximation step (finding $T(s)$ for given $\alpha_{max}$, $\alpha_{min}$) is done once, for the simpler low-pass case; specifications of the other types are converted to an equivalent LP specification.
• It can be applied either to the transfer function (substitute for $s$) or directly to each element of an LC ladder, so the circuit is obtained without re-deriving it.
• It reduces design time and errors, and keeps the properties of the prototype (equiripple, maximally flat, etc.).

Band-pass filter from the given low-pass filter

LP → BP: $s \to \dfrac{s^2+\omega_0^2}{Bs}$ with $\omega_0 = 2000$ rad/s ($\omega_0^2 = 4\times10^6$), $B = 400$ rad/s.

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

Given ladder: shunt $C_1 = 2.0237$ F, series $L_2 = 0.9941$ H, shunt $C_3 = 2.0237$ F, 1 Ω terminations.

$$\begin{aligned} C_1:&\quad C_1' = \frac{2.0237}{400} = 5.0593\ \text{mF},\quad L_1' = \frac{400}{2.0237\times4\times10^6} = 49.414\ \mu\text{H} \\ L_2:&\quad L_2' = \frac{0.9941}{400} = 2.4853\ \text{mH},\quad C_2' = \frac{400}{0.9941\times4\times10^6} = 100.59\ \mu\text{F} \\ C_3:&\quad C_3' = 5.0593\ \text{mF},\quad L_3' = 49.414\ \mu\text{H} \end{aligned}$$

  Rs=1         [ Z2 ]
 o-/\/\--+-----[    ]-----+------+---o
 |       |                |      |   +
 Vs   [ Y1 ]           [ Y3 ]  RL=1  Vo
 |       |                |      |   -
 o-------+----------------+------+---o

Each LC pair resonates at 2000 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 $k_m$.