Band pass (2000 rad/s, 400 rad/s) from 4th order

Importance of frequency transformation

Frequency transformation converts a normalized low-pass prototype into HP, BP or BS filters by a change of variable, so:

• 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 fourth-order Butterworth low-pass filter

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

Prototype (Table 2, $n = 4$, $R_s = R_L = 1\ \Omega$, starting with a series inductor): $L_1 = 0.7654$ H, $C_2 = 1.848$ F, $L_3 = 1.848$ H, $C_4 = 0.7654$ F.

$$\begin{aligned} L_1:&\quad L_1' = \frac{0.7654}{400} = 1.9135\ \text{mH},\quad C_1' = \frac{400}{0.7654\times4\times10^6} = 130.65\ \mu\text{F} \\ C_2:&\quad C_2' = \frac{1.848}{400} = 4.62\ \text{mF},\quad L_2' = \frac{400}{1.848\times4\times10^6} = 54.113\ \mu\text{H} \\ L_3:&\quad L_3' = 4.62\ \text{mH},\quad C_3' = 54.113\ \mu\text{F} \\ C_4:&\quad C_4' = 1.9135\ \text{mF},\quad L_4' = 130.65\ \mu\text{H} \end{aligned}$$

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

All LC pairs resonate at 2000 rad/s; the 3 dB band edges are 1810.0 and 2210.0 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 $k_m$.