Schematics and analysis

The OTA-C LP cell is a variant of this circuit.

$\begin{circuitikz} \ctikzset{bipoles/cisourceam/width=.4} \ctikzset{bipoles/cisourceam/height=.4} \draw (4, 3) node[op amp] (opamp) {} (0, 3.5) node[op amp] (ota) {} (opamp.out) node[anchor=south] {$v_o$} (opamp.-) node[anchor=south] {$v'_-$} (1, 3.5) to[cI] (1.25, 3.5) (1.15, 3.25) to[short] (1.15, 2.5) node[anchor=east] {$i_c$} (1.15, 3.5) to (opamp.-) (2.5, 3.5) to (2.5, 4.5) (2.5, 4.5) to[C, C=$C$] (6, 4.5) (6, 3) to[short] (opamp.out) (6, 3) to (6, 5.5) (opamp.+) to (2.5, 2.5) (2.5, 2.5) to (2.5, 2) node[ground] {} (2, 3.5) node[anchor=south] {$i_o$} (6, 5.5) to[R, R=$R_b$] (-3, 5.5) (-3, 5.5) to[short] (-3, 3) (-3, 3) to (ota.+) (-3, 3) to[R, R=$R_s$] (-3, 1.5) node[ground] {} (-5, 5.5) node[anchor=east] {$v_i$} to[R, R=$R_b$] (-3, 5.5) (ota.-) node[anchor=south] {$v_+$} (ota.+) node[anchor=south] {$v_-$} (ota.-) to[short] (-2, 4) (-2, 4) to (-2, 3.75) node[ground] {};\end{circuitikz}$

Kirchoff in $v_+$ yields:

$\begin{eqnarray*}\frac{1}{R_b}(v_{i}(s) - v_+(s)) + \frac{1}{R_b} (v_{o}(s) - v_+(s)) &=& \frac{1}{R_s} v_+(s) \\v_+(s) &=& \frac{R_s}{R_b + 2 R_s} (v_{i}(s) + v_{o}(s))\end{eqnarray*}$

$v_-$ is simply grounded. The current $i_o(s)$ at the output of the OTA is:

$\begin{eqnarray*}i_o(s) &=& g_m (v_+(s) - v_-(s)) \\ &=& 19.2 i_c \frac{R_s}{R_b + 2 R_s} (v_{i}(s) + v_{o}(s))\end{eqnarray*}$

The op-amp is used as an (inverted) current to voltage converter:

$\begin{eqnarray*} v_o(s) &=& - i_o(s) Z_{feedback}(s) \\ &=& -\frac{1}{Cs} i_o(s) \\ &=& -19.2 i_c \frac{R_s}{R_b + 2 R_s} (v_{i}(s) + v_{o}(s)) \frac{1}{Cs} \end{eqnarray*}$

Solving for $v_{o}(s)$ and dividing by $v_{i}(s)$ , we find the transfer function:

$H(s) = \frac{-1}{1 + \frac{(R_b + 2 R_s)}{R_s} Cs \frac{1}{19.2 i_c}}$

The pole is at the frequency $f$ such that:

$\begin{eqnarray*}0 &=& 1 + \frac{(R_b + 2 R_s)}{R_s} C 2\pi f \frac{1}{19.2 i_c} \\f &=& -\frac{19.2 i_c}{2 \pi \frac{(R_b + 2 R_s)}{R_s} C}\end{eqnarray*}$

Applications

Used in:

Roland System700 VCF (CA3080)

Patent:

US Patent 3,805,091, ARP

Other notes

Olivier Gillet: With +/-5V rails, Rb=10k, Rs=220, C=1nF, 1/2 TL072 for op-amp and LM13700, the OTA-C LP cell is unstable (crashes to one of the rails for signals peaking at more than 3.6V), but this configuration is not.