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Consider the circuit given in the figure is using an ideal operational amplifier.



The characteristics of the diode are given by the relation $\mathrm{I}=\mathrm{I}_{\mathrm{S}}\left(\frac{\mathrm{V}}{e^{\mathrm{kT}}}-1\right)$ where $V$ is the forward voltage across the diode.

  1. Express $V_{u}$ as function of $V_{i}$ assuming $V_{1}>0$
  2. If $\mathrm{R}=100 \; k \Omega, I_{s}=1 \; \mu \mathrm{A}$ and $\frac{k \mathrm{T}}{q}=25 \mathrm{~mV}$, find the input voltage $V_{i}$, for which $V_{0}=0$.
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