Exercise 8.12 A more elaborate opamp circuit schematic

The figure below shows the circuit schematic of an op amp. For the NMOS transistors $V_T=1V$ while for the PMOS transistors $V_T=-1V$; all transistors can be assumed to operate in strong inversion, saturation: there the square law ($i_D]1∕2\cdot K{(v_{\mathit{GS}}-V_T)}^2$) holds.

The two output transistors $M_{o1}$ and $M_{o2}$ are equal, with $K=K_1$. All other NMOS transistors are also equal, with $K=K_2=0.1\mathit{mA}∕V^2$, and the PMOS transistors are the same with $K=K_3$.

The current source $I_{\mathit{BIAS}}$ is $0.2\mathit{mA}$. The dashed resistor $r_{\mathit{ds}}$ indicates the combined small-signal drain-source resistance of the 3 transistors connected with their drain to node $C$.

a)

Derive the bias currents in all transistors, for $v_A=v_B$, expressed in $I_{\mathit{BIAS}}$.

b)

Which input is the non-inverting input and which input is the inverting input?

c)

Give an expression for the small-signal output resistance $r_{\mathit{out}}$ of this opamp.

d)

Calculate the required value for $K_1$ of the output transistors so that the output resistance $r_{\mathit{out}}=500\mathrm{\Omega }$.

e)

Derive an expression for the small-signal transfer function of the differential pair, from ($v_A-v_B$) to $i_x$. After that, express this also as a function of e.g. $K_1$, $K_2$, $I_{\mathit{BIAS}}$, $r_{\mathit{ds}}$.

f)

Derive an expression for the small-signal transfer function from $i_x$ to $i_y$, as a function of e.g. $K_1$, $K_2$, $I_{\mathit{BIAS}}$, $r_{\mathit{ds}}$.

g)

Derive an expression for the small-signal transfer function from $i_y$ to the voltage at node $C$, as a function of e.g. $K_1$, $K_2$, $I_{\mathit{BIAS}}$, $r_{\mathit{ds}}$.

h)

Derive an expression for the small-signal transfer function from the voltage at node $C$ to the output voltage $v_{\mathit{OUT}}$, as a function of e.g. $K_1$, $K_2$, $I_{\mathit{BIAS}}$, $r_{\mathit{ds}}$.

i)

Derive an expression for the total small-signal voltage gain of this opamp $a_v=\frac{v_{\mathit{out}}}{v_A-v_B}$, as a function of e.g. $K_1$, $K_2$, $I_{\mathit{BIAS}}$, $r_{\mathit{ds}}$.