Bruton transformation; FDNR 4th order LPF at 2000 rad/s

Bruton transformation

The Bruton transformation scales every impedance of an LC ladder by $1/s$ (i.e. by $k/s$). The transfer function (a ratio of impedances) does not change, but the element types change:

The FDNR (frequency-dependent negative resistance) has impedance $Z = \dfrac{1}{s^2D}$, which is $-\dfrac{1}{\omega^2D}$ on the $j\omega$ axis. All inductors become resistors and the shunt capacitors become grounded FDNRs, which are easy to build with a GIC.

Design: 4th-order Butterworth LPF, $\omega_c = 2000$ rad/s

Prototype (table, $R_s = R_L = 1\ \Omega$): series $L_1 = 0.7654$, shunt $C_2 = 1.848$, series $L_3 = 1.848$, shunt $C_4 = 0.7654$.

After the Bruton transformation (normalized):

• $R_s = 1\ \Omega \to C_s = 1$ F; $R_L = 1\ \Omega \to C_L = 1$ F
• $L_1 \to R_1 = 0.7654\ \Omega$; $L_3 \to R_3 = 1.848\ \Omega$
• $C_2 \to D_2 = 1.848$; $C_4 \to D_4 = 0.7654$

Scaling: $k_f = 2000$, choose $k_m = 5000$ (so $C_s$ becomes 0.1 μF):

$$\begin{aligned} C' &= \frac{C}{k_fk_m} = \frac{1}{2000 \times 5000} = 0.1\ \mu\text{F} \\ R_1' &= 0.7654 \times 5000 = 3.827\ \text{k}\Omega, \quad R_3' = 1.848 \times 5000 = 9.24\ \text{k}\Omega \\ D' &= \frac{D}{k_f^2k_m}: \quad D_2' = \frac{1.848}{2\times10^{10}} = 9.24\times10^{-11},\quad D_4' = 3.827\times10^{-11}\ \text{F·s} \end{aligned}$$

FDNR by GIC: in $Z_{in} = \dfrac{Z_1Z_3Z_5}{Z_2Z_4}$, make $Z_1 = \dfrac{1}{sC_1}$, $Z_3 = \dfrac{1}{sC_3}$ and the rest resistors:

$$Z_{in} = \frac{R_5}{s^2C_1C_3R_2R_4} \;\Rightarrow\; D = \frac{C_1C_3R_2R_4}{R_5}$$

Take $C_1 = C_3 = 0.1\ \mu$F and $R_2 = R_5 = 10\ \text{k}\Omega$, so $D = C^2R_4$:

$$R_4 = \frac{D}{C^2}: \quad D_2' \Rightarrow R_4 = \frac{9.24\times10^{-11}}{10^{-14}} = 9.24\ \text{k}\Omega, \quad D_4' \Rightarrow R_4 = 3.827\ \text{k}\Omega$$

Vin --||--[3.827k]--+--[9.24k]--+----+---- Vout
     0.1uF          |           |    |
                  [D2]        [D4]  ===  0.1uF
                    |           |    |
GND ----------------+-----------+----+

DC path: the series 0.1 μF source capacitor blocks DC bias for the op-amps, so add large bleeding resistors (e.g. 1 MΩ) across the source capacitor and from output to ground. With equal values they keep the DC gain at about 0.5, as in the original ladder.

Answer: the inductorless filter has half-power frequency 2000 rad/s, a 4th-order Butterworth response and only practical R, C and op-amp elements.