Exercise 8.8 An opamp circuit schematic

The figure below shows the circuit of a simple opamp, consisting of a differential pair, current mirror and an output stage. For all transistors it is given that if $v_{\mathit{BE}}>0.2V$ and $v_{\mathit{CE}}>0.2V:i_C=I_{C0}\cdot e^{\frac{q{v}_{\mathit{BE}}}{\mathit{kT}}}$, $\mathit{\beta fe}$ is not very large, and $r_{\mathit{ce}}\to \infty$.

a)

Draw a small-signal equivalent circuit for the differential pair and derive an expression for the small-signal transfer of the differential pair $i_{\mathit{diff}}∕v_{\mathit{in}}$ where $i_{\mathit{diff}}=i_{c,T1}-i_{c,T2}$. Define the sign of $v_{\mathit{in}}$ explicitly.

b)

Draw a small-signal equivalent circuit for the current mirror and derive an expression for the small-signal current transfer of the current mirror.

c)

Derive an expression for the small-signal transfer from $v_{\mathit{in}}$ to $i_x$. Remember that $\mathit{\beta fe}$ is not very large and hence explicitly must be taken into account.

d)

Derive an expression for the small-signal input resistance $r_{\mathit{in}}$ of the output stage.

e)

Derive an expression for the small-signal voltage gain $a_{\mathit{opamp}}=\frac{v_{\mathit{out}}}{v_{\mathit{in}}}$ of the entire opamp.

It is given that $I_S=50\mathit{\mu A}$ and the bias current through the output transistor is also $50\mathit{\mu A}$. Furthermore, $R_E=5k\mathrm{\Omega }$, $R_C=20k\mathrm{\Omega }$ and $\mathit{\beta fe}=50$.

f)

Calculate the numerical value of the gain of the opamp.

g)

Which one of the inputs (A or B) is the non-inverting input?

h)

Derive an expression for the input resistance of this opamp and calculate the numerical value.