The circuit in Figure P4=37 has a diode in its feedback path and is called a "log-amp" because its output is proportional to the natural \(\log\) of the input.

(a) Show that \({ }^{v_{\mathrm{O}}=-V_{\mathrm{T}} \ln \left(1+\frac{v_{\mathrm{S}}}{R_{\mathrm{S}} \mathrm{S}_{\mathrm{O}}}ight)}\) if the \(i-v\) characteristics of the diode is \(i_{\mathrm{D}}=I_{\mathrm{O}}\left(e^{v_{\mathrm{D}} / V_{\mathrm{T}}}-1ight)\).
(b) Using MATLAB plot \(v_{\mathrm{O}}\) versus \(v_{\mathrm{S}}\) for \(R_{\mathrm{S}}=15 \mathrm{k} \Omega, I_{\mathrm{O}}=\) \(3 \times 10^{-14} \mathrm{~A}\), and \(V_{\mathrm{T}}=0.026 \mathrm{~V}\). Plot your results on a semilog plot for \(10^{-6} \mathrm{~V} \leq v_{\mathrm{S}} \leq 100 \mathrm{~V}\).

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