Frequency 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 $k_m$.

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-pass transformation

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

$$\begin{aligned} L_1 = 1:&\quad L_1' = \frac{1}{400} = 2.5\ \text{mH},\quad C_1' = \frac{400}{1\times4\times10^6} = 100\ \mu\text{F} \\ C_2 = 2:&\quad C_2' = \frac{2}{400} = 5\ \text{mF},\quad L_2' = \frac{400}{2\times4\times10^6} = 50\ \mu\text{H} \\ L_3 = 1:&\quad L_3' = 2.5\ \text{mH},\quad C_3' = 100\ \mu\text{F} \end{aligned}$$

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

Check: $1/\sqrt{2.5\text{ mH}\times100\ \mu\text{F}} = 2000$ rad/s. The 3 dB band edges are $\omega_{1,2} = \sqrt{\omega_0^2+(B/2)^2} \mp B/2 = 1810.0$ and $2210.0$ rad/s, so $\omega_2 - \omega_1 = 400$ rad/s. The result is a 6th-order band-pass filter.

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 $k_m$.

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. For a band-stop filter, $s \to \dfrac{Bs}{s^2+\omega_0^2}$.

Design of the band-stop filter

Prototype (table of doubly terminated Butterworth values, $n = 3$, $R_s = R_L = 1\ \Omega$): series $L_1 = 1$ H, shunt $C_2 = 2$ F, series $L_3 = 1$ H.

Element rules for LP → BS:

• 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: $L' = \dfrac{1}{CB}$, $C' = \dfrac{CB}{\omega_0^2}$.

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

  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 $k_m$.

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.

Frequency transformation and its applications

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.

Applications in filter design:

• 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.).

High-pass filter from a normalized low-pass filter

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

$$s \;\rightarrow\; \frac{\omega_c}{s} \quad (\text{normalized: } s \to 1/s)$$

On the $j\omega$ axis $\Omega_{LP} = -\omega_c/\omega$: low frequencies map to the LP stopband and high frequencies to the LP passband, and $\omega = \omega_c$ maps to the LP band edge $\Omega = 1$. So the attenuation the LP has at $\Omega$ appears in the HP at $\omega_c/\Omega$.

Element transformation (normalized, $\omega_c = 1$):

• inductor $L$: $Z = sL \to L/s = \dfrac{1}{s(1/L)}$, i.e. a capacitor $C' = 1/L$;
• capacitor $C$: $Y = sC \to C/s = \dfrac{1}{s(1/C)}$, i.e. an inductor $L' = 1/C$;
• resistors unchanged.

Then frequency-scale to $\omega_c$ and magnitude-scale to the required impedance.

Example: third-order Butterworth low-pass prototype ($R_s = R_L = 1\ \Omega$; series $L_1 = 1$ H, shunt $C_2 = 2$ F, series $L_3 = 1$ H). Required: high-pass, $\omega_c = 1000$ rad/s, terminations 1 kΩ.

1. LP → HP (normalized): series $C_1 = 1/1 = 1$ F, shunt $L_2 = 1/2 = 0.5$ H, series $C_3 = 1$ F.
2. Scale with $k_f = 1000$, $k_m = 1000$ ($C' = C/(k_mk_f)$, $L' = Lk_m/k_f$):

$$C_1' = C_3' = \frac{1}{10^3\times10^3} = 1\ \mu\text{F}, \qquad L_2' = \frac{0.5\times10^3}{10^3} = 0.5\ \text{H}$$

  Rs=1k   1uF            1uF
 o-/\/\---||-----+--------||-----+-----o
 |               |               |     +
 Vs            0.5 H           RL=1k   Vo
 |               |               |     -
 o---------------+---------------+-----o

Transfer function check: $T_{LP}(s) = \dfrac{1}{s^3+2s^2+2s+1}$ (for equal terminations the passband gain is 1/2, omitted here); $s \to 1000/s$ gives $T_{HP}(s) = \dfrac{s^3}{s^3 + 2000s^2 + 2\times10^6 s + 10^9}$, a third-order Butterworth high-pass with 3 dB frequency 1000 rad/s.

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 $k_m$.

Frequency transformation is a substitution for the complex frequency variable $s$ that converts a normalized low-pass prototype filter (passband edge 1 rad/s) into a low-pass filter of another cutoff, or into a high-pass, band-pass or band-stop filter, while keeping the same type of response (Butterworth, Chebyshev, etc.).

Importance in filter design:

• Only the low-pass approximation problem has to be solved; tables of normalized LP poles and ladder element values serve all filter types.
• It can be applied to the transfer function or element by element to an LC ladder (e.g. a series L becomes a series LC in a band-pass filter), so circuits are obtained directly.
• It saves design effort and reduces errors.

Example: $T_{LP}(s) = \dfrac{1}{s+1}$ with $s \to \omega_c/s$ gives $T_{HP}(s) = \dfrac{s}{s+\omega_c}$.

Use the low-pass to band-pass transformation $s \to \dfrac{s^2+\omega_0^2}{Bs}$ on the normalized prototype $T(s) = \dfrac{1}{s^2+1.414s+1}$ (second-order Butterworth).

Specifications in rad/s

$$\begin{aligned} \omega_0 &= 2\pi\times10^4 = 62\,831.85\ \text{rad/s}, \quad \omega_0^2 = 3.9478\times10^9 \\ B &= 2\pi\times4000 = 25\,132.74\ \text{rad/s}, \quad B^2 = 6.3165\times10^8 \end{aligned}$$

Substitution

$$T_{BP}(s) = \frac{1}{\left(\dfrac{s^2+\omega_0^2}{Bs}\right)^2 + 1.414\dfrac{s^2+\omega_0^2}{Bs} + 1} = \frac{B^2s^2}{(s^2+\omega_0^2)^2 + 1.414Bs(s^2+\omega_0^2) + B^2s^2}$$

Expanding the denominator:

$$s^4 + 1.414B\,s^3 + (2\omega_0^2 + B^2)s^2 + 1.414B\omega_0^2\,s + \omega_0^4$$ $$T_{BP}(s) = \frac{6.3165\times10^8\, s^2}{s^4 + 3.5538\times10^4 s^3 + 8.5273\times10^9 s^2 + 1.4030\times10^{14}s + 1.5585\times10^{19}}$$

Check: at $s = j\omega_0$, $|T_{BP}| = 1$ (0 dB at 10 kHz), and the 3 dB points are $\Omega_{LP} = \pm1$, i.e. $4$ kHz apart.

Realization as two cascaded biquads

The LP poles $-0.707 \pm j0.707$ each map to a pair of BP poles. Factoring the denominator:

$$T_{BP}(s) = \frac{25\,132.7\,s}{s^2 + 15\,256.8s + 2.9699\times10^9}\cdot\frac{25\,132.7\,s}{s^2 + 20\,280.9s + 5.2479\times10^9}$$

Note $f_{01}f_{02} = (10\text{ kHz})^2$. Each section can be built as a band-pass biquad (e.g. MFB or Tow-Thomas) and the two are cascaded with buffering.

Answer: a fourth-order band-pass filter with the transfer function above, centre frequency 10 kHz, 3 dB bandwidth 4 kHz.

Use the low-pass to band-pass transformation $s \to \dfrac{s^2+\omega_0^2}{Bs}$ with $\omega_0 = 1500$ rad/s ($\omega_0^2 = 2.25\times10^6$) and $B = 300$ rad/s.

Prototype (table of doubly terminated Butterworth values, $n = 4$, $R_s = R_L = 1\ \Omega$, taken as 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.

Element rules:

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

$$\begin{aligned} L_1:&\quad L_1' = \frac{0.7654}{300} = 2.5513\ \text{mH},\quad C_1' = \frac{300}{0.7654\times2.25\times10^6} = 174.20\ \mu\text{F} \\ C_2:&\quad C_2' = \frac{1.848}{300} = 6.16\ \text{mF},\quad L_2' = \frac{300}{1.848\times2.25\times10^6} = 72.150\ \mu\text{H} \\ L_3:&\quad L_3' = \frac{1.848}{300} = 6.16\ \text{mH},\quad C_3' = \frac{300}{1.848\times2.25\times10^6} = 72.150\ \mu\text{F} \\ C_4:&\quad C_4' = \frac{0.7654}{300} = 2.5513\ \text{mF},\quad L_4' = \frac{300}{0.7654\times2.25\times10^6} = 174.20\ \mu\text{H} \end{aligned}$$

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

Each LC pair resonates at 1500 rad/s. The result is an 8th-order band-pass filter with 3 dB edges at $\omega_{1,2} = \sqrt{1500^2+150^2} \mp 150 = 1357.5$ and $1657.5$ 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 $k_m$.

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.

Importance in filter design

• 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.).

Design of the band-pass filter

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

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

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

$$\begin{aligned} L_1 = 1:&\quad L_1' = \frac{1}{400} = 2.5\ \text{mH},\quad C_1' = \frac{400}{1\times4\times10^6} = 100\ \mu\text{F} \\ C_2 = 2:&\quad C_2' = \frac{2}{400} = 5\ \text{mF},\quad L_2' = \frac{400}{2\times4\times10^6} = 50\ \mu\text{H} \\ L_3 = 1:&\quad L_3' = 2.5\ \text{mH},\quad C_3' = 100\ \mu\text{F} \end{aligned}$$

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

Check: $1/\sqrt{2.5\text{ mH}\times100\ \mu\text{F}} = 2000$ rad/s. The 3 dB band edges are $\omega_{1,2} = \sqrt{\omega_0^2+(B/2)^2} \mp B/2 = 1810.0$ and $2210.0$ rad/s, so $\omega_2 - \omega_1 = 400$ rad/s. The result is a 6th-order band-pass filter.

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 $k_m$.

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 $k_m$.

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.

Obtaining a band-pass filter from the normalized low-pass filter

Replace $s$ by $\dfrac{s^2+\omega_0^2}{Bs}$, where $\omega_0 = \sqrt{\omega_1\omega_2}$ and $B = \omega_2 - \omega_1$. Applied to each element:

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

This maps $\omega_0 \to 0$ and $\omega_1, \omega_2 \to \mp1$ of the low-pass prototype, so the LP passband becomes the band $\omega_1$ to $\omega_2$.

Design for $\omega_1 = 100$ rad/s, $\omega_2 = 10000$ rad/s

$$\omega_0 = \sqrt{100\times10000} = 1000\ \text{rad/s}, \quad \omega_0^2 = 10^6, \quad B = 10000 - 100 = 9900\ \text{rad/s}$$

Given ladder (4th-order Butterworth, 1 Ω terminations): series 0.7654 H, shunt 1.8485 F, series 1.8485 H, shunt 0.7654 F.

$$\begin{aligned} 0.7654\ \text{H}:&\quad L' = \frac{0.7654}{9900} = 77.313\ \mu\text{H},\quad C' = \frac{9900}{0.7654\times10^6} = 12.934\ \text{mF} \\ 1.8485\ \text{F}:&\quad C' = \frac{1.8485}{9900} = 186.72\ \mu\text{F},\quad L' = \frac{9900}{1.8485\times10^6} = 5.3557\ \text{mH} \\ 1.8485\ \text{H}:&\quad L' = 186.72\ \mu\text{H},\quad C' = 5.3557\ \text{mF} \\ 0.7654\ \text{F}:&\quad C' = 77.313\ \mu\text{F},\quad L' = 12.934\ \text{mH} \end{aligned}$$

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

Check: $1/\sqrt{77.313\ \mu\text{H}\times12.934\text{ mF}} = 1000$ 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 $k_m$.

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. For band-pass: $s \to \dfrac{s^2+\omega_0^2}{Bs}$.

Band-pass filter from the given low-pass filter

Given: $\omega_0 = 10^4$ rad/s ($\omega_0^2 = 10^8$), $B = 9.9\times10^4$ rad/s (values used as given; they correspond to band edges of about $\omega_1 \approx 1000$ and $\omega_2 \approx 10^5$ rad/s, since $\omega_1\omega_2 = 10^8$ and $\omega_2 - \omega_1 = 9.9\times10^4$).

Prototype (Figure 1): $R_1 = 1\ \Omega$, series $L_1 = 2.024$ H, shunt $C_1 = 0.994$ F, series $L_2 = 2.024$ H, $R_2 = 1\ \Omega$.

Element rules: series $L$ → series $L/B$ and $B/(L\omega_0^2)$; shunt $C$ → parallel $C/B$ and $B/(C\omega_0^2)$.

$$\begin{aligned} L_1 = 2.024:&\quad L_1' = \frac{2.024}{9.9\times10^4} = 20.444\ \mu\text{H},\quad C_1' = \frac{9.9\times10^4}{2.024\times10^8} = 489.13\ \mu\text{F} \\ C_1 = 0.994:&\quad C' = \frac{0.994}{9.9\times10^4} = 10.040\ \mu\text{F},\quad L' = \frac{9.9\times10^4}{0.994\times10^8} = 995.98\ \mu\text{H} \\ L_2 = 2.024:&\quad L_2' = 20.444\ \mu\text{H},\quad C_2' = 489.13\ \mu\text{F} \end{aligned}$$

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

Check: $1/\sqrt{20.444\ \mu\text{H}\times489.13\ \mu\text{F}} = 10^4$ 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 $k_m$.

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 $k_m$.