Non linear feedback around an opamp configuration

a)

An answer:

\begin{array}{rcl} v_{o u t} & = v_{-} - v_{B E} \\ v_{+} = v_{-} = 0 \\ v_{B E} = \frac{k T}{q} \ln \frac{i_{c}}{I_{C 0}} \\ v_{o u t} & = - \frac{k T}{q} \ln \frac{v_{i n}}{I_{C 0} R} \end{array}

b)

An answer:

pict

\begin{array}{rcl} g_{B J T} & = \frac{\partial i_{C}}{\partial v_{B E}} = \frac{q}{k T} I_{C 0} e^{} & q & V_{B E} & k & T & = & \frac{q}{k T} & I_{C} \end{array}

\begin{array}{rcl} r_{o u t, c i r c u i t} & = \frac{v_{o}}{i_{o}} \\ i_{o} = \frac{v_{o}}{R_{1} + r_{b j t}} + \frac{v_{o} + A \cdot v_{o} \frac{R_{1}}{R_{1} + r_{b j t}}}{r_{o u t}} \\ = v_{o} (\frac{1}{R_{1} + r_{b j t}} + \frac{1 + A \frac{R_{1}}{R_{1} + r_{b j t}}}{r_{o u t}}) \\ r_{o u t, c i r c u i t} & = (R_{1} + r_{b j t}) ∕ ∕ \frac{r_{o u t}}{1 + A \frac{R_{1}}{R_{1} + r_{b j t}}} \\ with & r_{b j t} = \frac{k T}{q} \frac{1}{I_{C}} = \frac{k T}{q} \frac{R}{V_{I N}} \end{array}