$2$ pts

Simon Sze: Chapter $6$ Question $1$

Derive $(23)$ from $(20)$ and $(22)$

$|Q_n(y)|=[V_G-V_{F{B}^{-}}{}_i(y)-2{}_B]C_{ox}-\sqrt[\phantom{2}]{2{}_{s}q{N}_A[{}_i(y)+2{}_B]}$

$I_D=\frac{Z}{L}{\displaystyle \underset{0}{\overset{L}{}}}|Q_n(y)|v(y)dy$

$I_D=\frac{Z{}_n}{L}{\displaystyle \underset{0}{\overset{L}{}}}|Q_n(y)|E(y)dy=\frac{Z{}_n}{L}{\displaystyle \underset{0}{\overset{L}{}}}|Q_n(y)|\frac{d{}_i(y)}{dy}dy=\frac{Z{}_n}{L}{\displaystyle \underset{0}{\overset{V_D}{}}}|Q_n({}_i)|d{}_i$

$=\frac{Z}{L}{}_n{C}_{ox}\{(V_G-V_{FB}-2{}_B-\frac{V_D}{2})V_D-\frac{2}{3}\frac{\sqrt[\phantom{2}]{2{}_{s}q{N}_A}}{C_{ox}}[(V_D+2{}_B{)}^{\frac{3}{2}}-(2{}_B{)}^{\frac{3}{2}}]\}$