How transformation reduces design steps; band stop design

How frequency transformation reduces design steps

Without frequency transformation, a band-stop filter would need its own approximation (finding a suitable $T(s)$ with the required notch band) and its own synthesis, which is long and error-prone. With frequency transformation:

1. The band-stop specification is converted to an equivalent normalized low-pass specification.
2. The order and the ladder element values are read directly from the standard low-pass table (no new approximation).
3. Each element of the low-pass ladder is replaced by a simple LC pair using fixed formulas (no new synthesis).

So one set of low-pass tables, plus a few substitution rules, covers HP, BP and BS designs.

Design of the band-stop filter

The LP → BS transformation is $s \to \dfrac{Bs}{s^2+\omega_0^2}$. Element rules:

• series $L$ → parallel tank in the series arm: $L' = \dfrac{LB}{\omega_0^2}$, $C' = \dfrac{1}{LB}$;
• shunt $C$ → series LC in the shunt arm: $L' = \dfrac{1}{CB}$, $C' = \dfrac{CB}{\omega_0^2}$.

Prototype from the table ($n = 3$, $R_s = R_L = 1\ \Omega$): series $L_1 = 1$ H, shunt $C_2 = 2$ F, series $L_3 = 1$ H.

$$\begin{aligned} L_1 = 1\ \text{H}:&\quad L_1' = \frac{L_1B}{\omega_0^2} = \frac{400}{4\times10^6} = 0.1\ \text{mH},\quad C_1' = \frac{1}{L_1B} = \frac{1}{400} = 2.5\ \text{mF} \\ C_2 = 2\ \text{F}:&\quad L_2' = \frac{1}{C_2B} = \frac{1}{800} = 1.25\ \text{mH},\quad C_2' = \frac{C_2B}{\omega_0^2} = \frac{800}{4\times10^6} = 0.2\ \text{mF} \\ L_3 = 1\ \text{H}:&\quad L_3' = 0.1\ \text{mH},\quad C_3' = 2.5\ \text{mF} \end{aligned}$$

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

Check: $1/\sqrt{0.1\text{ mH}\times2.5\text{ mF}} = 1/\sqrt{1.25\text{ mH}\times0.2\text{ mF}} = 2000$ rad/s. Band edges: $\omega_{1,2} = \sqrt{\omega_0^2 + (B/2)^2} \mp B/2 = 1810.0$ and $2210.0$ rad/s (3 dB points).

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 $k_m$.