Exercise 8.10 An opamp circuit schematics

The figure below shows the circuit schematic of an opamp. All BJTs have the same (absolute value of) $I_{C0}$, have identical $\mathit{\beta fe}=200$ and $r_{\mathit{ce}}\to \infty$. Furthermore, $\mathit{kT}∕\mathit{q\cong }25\mathit{mV}$ as the circuit operates at room temperature.

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 $g_{\mathit{diffpair}}=\frac{i_x-i_y}{v_{\mathit{in}}}$.

c)

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

d)

Derive an expression for the small-signal current $i_z$ due to $i_x$ and due to $i_y$. Note that this leads to 2 current transfer functions $i_z∕i_x$ and $i_z∕i_y$.

e)

From the derivations to a previous question, it appeared that $i_x=-i_y$. Now, derive the current transfer from the differential current $(i_x-i_y)$ to $i_z$

f)

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

g)

Derive an expression for the input resistance $r_{\mathit{in}}$ of the output BJT, for an unloaded opamp which is equivalent to $R_L\to \infty$.

h)

Derive an expression for the small-signal voltage gain from the voltage at the base of the rightmost transistor to the output, again including the load resistor $R_L$.

i)

Derive an expression for the output resistance of this opamp (this expression does not contain $R_L$).

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

j)

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