Exercise 5.10 An NPN amplifier driven from a current source

Given is the circuit below. For the BJT, operation active forward operation may be assumed where $i_C=I_{C0}\cdot e^{\frac{q\cdot v_{\mathit{BE}}}{\mathit{kT}}}$; the $\mathit{\beta fe}$ is finite and cannot be neglected. At signal frequencies, the capacitors can be assumed to be low-impedant and the inductors can be assumed to be high-impedant.

Note that the input signal source is a current source (and note it’s direction). For voltage gain derivations this source is allowed to be replaced by a voltage source. For this circuit, the sign of $I_1$ should be included properly (KCL).

a)

derive a proper small signal equivalent for the transistor in the circuit using the specified element equations AND derive (not: only state) the small signal parameters as function of e.g. the bias current of the transistor.

b)

derive an expression for the input impedance $z_{\mathit{in}}$ as seen by the signal source.

c)

derive an equation for the required value of resistor $R_1$ to get a collector bias current equal to $I_C$, as a function of all relevant component values/properties.

d)

derive an expression for the voltage gain $a_v=v_{\mathit{out}}∕v_{\mathit{in}}$. If in the schematic your circuit is driven by a current source, for this specific question you may replace that one by a signal voltage source between the same nodes in the schematic.

e)

derive an expression for the output impedance $z_{\mathit{out}}$.