Exercise 8.1 A differential pair

a)

An answer:

$$\begin{array}{rcll}{i}_{C1}=\frac{I_{\mathit{TAIL}}}{1+e^{-\frac{q}{\mathit{kT}}{v}_{\mathit{in}}}},i_{C2}=\frac{I_{\mathit{TAIL}}}{1+e^{\frac{q}{\mathit{kT}}{v}_{\mathit{in}}}}& & & \end{array}$$

b)

An answer:

c)

An answer:

$$\begin{array}{rcll}{g}_m=\frac{q}{\mathit{kT}}\frac{I_{\mathit{TAIL}}}{2}=20\frac{\mathit{mA}}{V}& & & \end{array}$$

d)

An answer:
This would make the transconductance $g_m$ independent of the temperature.

e)

An answer:
A common-voltage variation of $2V$ will give a current variation of $\frac{2V}{100k\mathrm{\Omega }}=0.02\mathit{mA}$ . This is a variation of $2\%$. The variation in $g_m$ will also be $2\%$.

f)

An answer:

$$\begin{array}{rcll}& v_{\mathit{in}}& \le (1+\mathit{A\beta })v_{\mathit{diff},\mathit{max}}& \\ & & =(1+\mathit{A\beta })\cdot 0.4\cdot \frac{\mathit{kT}}{q}& \\ & & \approx 1.01V& \end{array}$$