Importance of frequency transformation; obtaining band pass

Significance of 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.

• 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 normalized prototype low-pass filter

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

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

On the $j\omega$ axis this gives $\Omega_{LP} = \dfrac{\omega^2 - \omega_0^2}{B\omega}$. So:

• $\omega = \omega_0 \Rightarrow \Omega_{LP} = 0$ (centre of passband maps to DC of the LP);
• $\omega = \omega_1, \omega_2 \Rightarrow \Omega_{LP} = \mp1$, since $\omega_1\omega_2 = \omega_0^2$ and $\omega_2 - \omega_1 = B$;
• $\omega = 0$ and $\omega = \infty \Rightarrow \Omega_{LP} = \mp\infty$ (stopbands).

The band-pass response is geometrically symmetric about $\omega_0$, and its order is twice that of the prototype.

Element transformation:

Series inductor $L$:

$$Z = L\,\frac{s^2+\omega_0^2}{Bs} = \frac{L}{B}s + \frac{1}{\left(\dfrac{B}{L\omega_0^2}\right)s}$$

becomes a series LC: $L' = \dfrac{L}{B}$, $C' = \dfrac{B}{L\omega_0^2}$.

Shunt capacitor $C$:

$$Y = C\,\frac{s^2+\omega_0^2}{Bs} = \frac{C}{B}s + \frac{1}{\left(\dfrac{B}{C\omega_0^2}\right)s}$$

becomes a parallel LC to ground: $C' = \dfrac{C}{B}$, $L' = \dfrac{B}{C\omega_0^2}$.

Resistor $R$: unchanged (frequency independent). Every new LC pair resonates at $\omega_0$ ($L'C' = 1/\omega_0^2$).

 LP element       BP element
 series L   -->   series L'--C'
 shunt C    -->   shunt  L' || C'
 R          -->   R

Example (transfer function method): first-order prototype $T_{LP}(s) = \dfrac{1}{s+1}$. Substituting $s \to \dfrac{s^2+\omega_0^2}{Bs}$:

$$T_{BP}(s) = \frac{1}{\dfrac{s^2+\omega_0^2}{Bs} + 1} = \frac{Bs}{s^2 + Bs + \omega_0^2}$$

This is the standard second-order band-pass function with centre frequency $\omega_0$, bandwidth $B$ and $Q = \omega_0/B$. For $\omega_0 = 1000$ rad/s and $B = 100$ rad/s: $T_{BP}(s) = \dfrac{100s}{s^2 + 100s + 10^6}$, $Q = 10$.

Example (element method): the third-order Butterworth ladder (1 H, 2 F, 1 H) with $\omega_0 = 2000$ rad/s, $B = 400$ rad/s gives series arms of 2.5 mH in series with 100 μF and a shunt arm of 5 mF in parallel with 50 μH.