Schematics and analysis

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

$\begin{circuitikz} \ctikzset{bipoles/cisourceam/width=.4} \ctikzset{bipoles/cisourceam/height=.4} \draw (4, 4) node[op amp] (opamp) {} (0, 3.5) node[op amp] (ota) {} (opamp.out) node[anchor=south] {$v_o$} (opamp.+) node[anchor=south] {$v'_+$} (2, 4.5) to (opamp.-) (2, 4.5) to (2,5.5) (2, 5.5) to (6, 5.5) (6, 5.5) to (6, 4) (6.5, 4) to (opamp.out) (2, 3.5) to (opamp.+) (1, 3.5) to[cI] (1.25, 3.5) (1.15, 3.25) to[short] (1.15, 2.5) node[anchor=east] {$i_c$} (1.25, 3.5) to (2, 3.5) (2, 3.5) to[C, C=$C$] (2, 2.5) (2, 2.5) node[ground] {} (2, 3.5) node[anchor=south] {$i_o$} (2, 5.5) to[R, R=$R_b$] (-3, 5.5) (-3, 5.5) to[short] (-3, 4) (-3, 4) to (ota.-) (-3, 4) to[R, R=$R_s$] (-3, 2.5) node[ground] {} (-5, 5.5) node[anchor=east] {$v_i$} to[R, R=$R_b$] (-3, 5.5) (ota.+) node[anchor=south] {$v_+$} (ota.-) node[anchor=south] {$v_-$} (ota.+) to[short] (-2, 3) (-2, 3) to (-2, 2.5) node[ground] {};\end{circuitikz}$

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 a voltage follower, hence:

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

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

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