Low pass to band stop transformation

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.

Low-pass to band-stop transformation

Transformation: replace $s$ in the low-pass prototype by

$$s \;\rightarrow\; \frac{B\,s}{s^2 + \omega_0^2}, \qquad \omega_0 = \sqrt{\omega_1\omega_2},\quad B = \omega_2 - \omega_1$$

It is the band-pass transformation applied to the high-pass ($s \to 1/s$) version. On the $j\omega$ axis, $\Omega_{LP} = \dfrac{B\omega}{\omega_0^2 - \omega^2}$, so:

• $\omega = 0$ and $\omega = \infty \Rightarrow \Omega_{LP} = 0$ (passbands at low and high frequencies);
• $\omega = \omega_0 \Rightarrow \Omega_{LP} = \infty$ (maximum attenuation, the notch);
• $\omega = \omega_1, \omega_2 \Rightarrow |\Omega_{LP}| = 1$ (band edges).

Element transformation:

Series inductor $L$:

$$Z = \frac{LBs}{s^2+\omega_0^2} \;\Rightarrow\; Y = \frac{s}{LB} + \frac{\omega_0^2}{LBs}$$

becomes a parallel LC tank in the series arm: $C' = \dfrac{1}{LB}$, $L' = \dfrac{LB}{\omega_0^2}$.

Shunt capacitor $C$:

$$Y = \frac{CBs}{s^2+\omega_0^2} \;\Rightarrow\; Z = \frac{s}{CB} + \frac{\omega_0^2}{CBs}$$

becomes a series LC in the shunt arm: $L' = \dfrac{1}{CB}$, $C' = \dfrac{CB}{\omega_0^2}$.

Resistors are unchanged; every LC pair resonates at $\omega_0$. At $\omega_0$ the series-arm tanks are open and the shunt-arm series LCs are shorts, so no signal reaches the load.

 LP element       BS element
 series L   -->   series arm: L' || C'
 shunt C    -->   shunt arm:  L'--C' in series
 R          -->   R

Example: third-order Butterworth prototype ($R_s = R_L = 1\ \Omega$, $L_1 = 1$ H, $C_2 = 2$ F, $L_3 = 1$ H) converted to a band-stop filter with $\omega_0 = 2000$ rad/s and $B = 400$ rad/s ($\omega_0^2 = 4\times10^6$).

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

  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 $k_m$.