Exercise 8.9 An opamp circuit schematic

The figure below shows the circuit schematic of an opamp. In the circuit all BJTs are equal, with $\mathit{\beta fe}=200$ and $\mu \to \infty$. All MOS transistors are equal, with $K=-5\frac{\mathit{mA}}{V^2}$ and $\mu \to \infty$.

a)

Which one of the inputs is the non-inverting input?

b)

Derive an expression for the small-signal transfer function of the differential pair as defined in the figure, i.e. from $v_{\mathit{in}}$ to $i_x$.

c)

Derive an expression for the differential input resistance of the opamp.

d)

Derive an expression for the small-signal current transfer function from $i_x$ to $i_y$.

e)

Derive an expression for the input resistance $r_{\mathit{in}}$ of the rightmost NPN, including the load resistor $R_L$.

f)

Derive an expression for the small-signal transfer function from the voltage at the base of the rightmost BJT ($v_{B3}$) to the output, where the output is loaded by the load resistor $R_L$.

g)

Derive an expression for the output resistance $r_{\mathit{out}}$ of this opamp. Note that this expression does not contain $R_L$ as it is not a component of the opamp itself.

Given is that $I_S=0.5\mathit{mA}$.

h)

Calculate the voltage gain from $v_{\mathit{in}}$ to $v_{\mathit{out}}$ for $R_L=10k\mathrm{\Omega }$.