FDNR to avoid bulky inductors

Frequency dependent negative resistor

A frequency dependent negative resistor (FDNR) is a one-port with impedance $Z(s) = \dfrac{1}{s^2D}$. At $s = j\omega$, $Z = -\dfrac{1}{\omega^2 D}$: a real, negative resistance whose size falls as $1/\omega^2$. $D$ (unit F²Ω) is its constant. It cannot be made from passive elements; a GIC with two capacitors realises it.

Avoiding bulky inductors

Inductors for audio and low frequencies are large, heavy, lossy (low Q), pick up magnetic noise and cannot be put on an IC. The FDNR method removes them:

1. Start from the doubly terminated LC ladder (it has low sensitivity to component changes).
2. Apply the Bruton transformation: divide every impedance by $s$. Then $R \to C = 1/R$, $L \to R = L$, $C \to$ FDNR $D = C$. The voltage transfer function does not change.
3. In a lowpass ladder all inductors are in series arms, so they all become ordinary resistors. The shunt capacitors become grounded FDNRs.
4. Realise each FDNR with one GIC: capacitors in positions 1 and 5, resistors in 2, 3, 4, giving $D = C^2R$.
5. Add large resistors across the terminating capacitors for a DC path.

   Iin  node 1
 o-->-----+--------------(+) A1
 Vin      |
         [Z1]
          +--- node 2 ---- output of A1
         [Z2]
          +--- node 3 ---- (-) A1 and (-) A2
         [Z3]
          +--- node 4 ---- output of A2
         [Z4]
          +--- node 5 ---- (+) A2
         [Z5]
          |
         GND

Example

Third-order Butterworth LPF at $\omega_0$ = 10,000 rad/s. Prototype: $R_s = 1$, $L_1 = 1$ H, $C_2 = 2$ F, $L_3 = 1$ H, $R_L = 1$. Without FDNR the scaled inductors (with $k_z = 10^4$) would be $L = k_zL/\omega_0$ = 1 H each, which is bulky.

With FDNR ($C_t$ = 10 nF, $k_z = 10^4\ \Omega$):

The 1 H inductors are replaced by 10 kΩ resistors and one GIC.