An amplfier with a single NPN

a)

An answer:

pict

b)

An answer:

\begin{array}{rcl} a_{v} & = \frac{v_{out}}{v_{in}} \\ v_{out} = - g_{m} v_{be} R_{2} \\ v_{be} = v_{in} - v_{e} \\ v_{e} = g_{m} v_{be} (1 + \frac{1}{βfe}) R_{4} \\ v_{be} = v_{in} - g_{m} v_{be} (1 + \frac{1}{βfe}) R_{4} \\ = \frac{v_{in}}{1 + g_{m} (1 + \frac{1}{βfe}) R_{4}} \\ a_{v} & = - \frac{g_{m} R_{2}}{1 + g_{m} (1 + \frac{1}{βfe}) R_{4}} \end{array}

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