GIC current relation and grounded FDNR

GIC circuit

The generalized impedance converter (GIC) (Antoniou circuit) uses two op-amps and five impedances in a chain from the input to ground:

   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

Relation between input and output current

Treat node 1 as port 1 ($V_1$, $I_1$) and node 5 as port 2, where the load $Z_5$ draws current $I_2$ ($V_2 = I_2 Z_5$). Ideal op-amps force node 1, node 3 and node 5 to the same voltage and draw no input current, so $V_1 = V_2$. Let $V_{N2}$ and $V_{N4}$ be the voltages of nodes 2 and 4 (op-amp outputs).

$$\begin{aligned} \text{Node 5: } & V_{N4} = V_2 + Z_4 I_2 \\ \text{Node 3: } & \frac{V_{N2} - V_1}{Z_2} = \frac{V_1 - V_{N4}}{Z_3} = -\frac{Z_4}{Z_3}I_2 \\ \text{Node 1: } & I_1 = \frac{V_1 - V_{N2}}{Z_1} = \frac{Z_2 Z_4}{Z_1 Z_3}\, I_2 \end{aligned}$$

So the GIC passes voltage unchanged and scales current:

$$V_1 = V_2, \qquad I_1 = \frac{Z_2Z_4}{Z_1Z_3}\, I_2, \qquad Z_{in} = \frac{V_1}{I_1} = \frac{Z_1Z_3}{Z_2Z_4}\,Z_5 = k(s)\,Z_5$$

Grounded FDNR

Put capacitors in positions 1 and 5 ($Z_1 = 1/sC_1$, $Z_5 = 1/sC_5$) and resistors in 2, 3, 4:

$$Z_{in} = \frac{R_3}{s^2 C_1 C_5 R_2 R_4} = \frac{1}{s^2 D}, \qquad D = \frac{C_1 C_5 R_2 R_4}{R_3}$$

With $C_1 = C_5 = C$ and $R_2 = R_3 = R_4 = R$: $D = C^2 R$. At $s = j\omega$, $Z_{in} = -1/(\omega^2 D)$, a negative resistance whose value depends on frequency: an FDNR (also called a D-element or supercapacitor).

Practical design: choose $C$ (e.g. 10 nF), then $R = D/C^2$. Example: $D = 10^{-12}\ \text{F}^2\Omega$ with $C$ = 10 nF gives $R = 10^{-12}/10^{-16}$ = 10 kΩ. Because port 1 is referred to ground, this FDNR is grounded, which is exactly what the Bruton-transformed lowpass ladder needs (shunt capacitors become grounded FDNRs).