Oppsumering av TFE4152

Konstander

Symbol	Verdi	Kommentar
$q$	$1.602\cdot 10^{-19}\text{C}$
$k$	$1.38\cdot 10^{-23}\text{J}\cdot\text{K}^{-1}$
$n_i$	$1.1\cdot 10^{16}\text{bærere}/\text{m}^3$	Ved $T=300\text{ K}$
$\epsilon_0$	$8.854\cdot 10^{-12}\text{F}/\text{m}$
$K_{ox (oksid)}$	$\cong 3.9$
$K_{Si (silikon)}$	$\cong 11.8$

Revers-forspent diode

\[C_j = \frac{C_{j0}}{\sqrt{1+\frac{V_R}{\Phi_0}}}\] \[Q = 2 C_{j0} \Phi_0 \sqrt{1 + \frac{V_R}{\Phi_0}}\] \[C_{j0} = \sqrt{\frac{q K_{Si} \epsilon_0}{2 \Phi_0} \frac{N_D N_A}{N_D + N_A}} \] \[C_{j0} = \sqrt{\frac{q K_{Si} \epsilon_0 N_D}{2 \Phi_0}}, \text{ hvis } N_A \gg N_D\] \[\Phi_0 = \frac{k_B T}{q}\ln\left(\frac{N_A N_D}{n_i}\right)\]

Normalt forspent diode

\[ I_D = I_S \exp{\frac{V_D}{V_T}} \] \[I_D = A_D q n_i^2 \left(\frac{D_n}{L_n N_A}+\frac{D_p}{L_p N_D}\right)\] \[V_T = \frac{k T}{q} \approx 26\text{mV, ved } T=300\text{ K}\]

Småsignal for forspent diode

\[r_d = \frac{V_T}{I_D}\] \[C_T = C_d + C_j\] \[C_d = \tau_T \frac{I_D}{V_T}\] \[C_j \approx 2 C_{j0}\] \[\tau_T = \frac{L_n^2}{D_n}\]

Transisor i triodeområdet

Dette gjelder for $V_{GS} > V_{tn}$, $V_{DS} \leq V_\text{eff}$.

\[I_D = \mu C_{ox} \left(\frac{W}{L}\right) \left[(V_{GS} - V_{tn})V_{DS} - \frac{V_{DS}^2}{2}\right]\] \[V_\text{eff} = V_{GS} - V_{tn} \] \[V_{tn} = V_{\text{tn-}0} + \gamma\left(\sqrt{V_{SB} + 2\Phi_F} - \sqrt{2\Phi_F}\right)\] \[\Phi_F = \frac{k T}{q}\ln\left(\frac{N_A}{n_i}\right)\] \[\gamma = \frac{\sqrt{2 q K_{Si} \epsilon_0 N_A}}{C_{ox}}\] \[C_{ox} = \frac{K_{ox} \epsilon_0}{t_{ox}}\]

Småsignal av transistor i triodeområdet

\[r_{ds} = \frac{1}{\mu_n C_{ox} \left(\frac{W}{L}\right)V_\text{eff}}\] \[C_{gd} = C_{gs} \frac{1}{2}W L C_{ox} + WL_{ov}C_{ox}\] \[C_{sb} = C_{db} = \frac{C_{j0} \left(A_s + \frac{WL}{2}\right)}{\sqrt{1 + \frac{V_{sb}}{\Phi_0}}}\]

Transistor i aktivt område

Dette gjelder bare for $V_{GS} > V_{tn}$, $V_{DS} \geq V_\text{eff}$.

\[I_D = \frac{1}{2}\mu C_{ox} \left(\frac{W}{L}\right) (V_{GS} - V_{tn})^2 \underbrace{\left[1 + \lambda(V_{DS} - V_\text{eff})\right]}_\text{body-effect}\] \[\lambda \propto \frac{1}{L\sqrt{V_{DS} - V_\text{eff} + \Phi_0}}\] \[V_{tn} = V_{tn\text{-}0} - \gamma\left(\sqrt{V_{SB} + 2\Phi_F} - \sqrt{2\Phi_F}\right)\] \[V_\text{eff} = V_{GS} - V_{tn} = \sqrt{\frac{2 I_D}{\mu_n C_{ox} \frac{W}{L}}} = V_{DS, \text{sat.}} \]

Småsignal for transistor i aktivt område

\[\begin{aligned} g_m &= \mu_n C_{ox} \frac{W}{L} V_\text{eff} \\ &= \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} \\ &= \frac{2 I_D}{V_\text{eff}} \end{aligned}\] \[\begin{aligned} g_s &= \frac{\gamma g_m}{2 \sqrt{V_{SB} + |2\Phi_F|}}\\ &\approx 0.2 g_m \end{aligned}\] \[r_{ds} = \frac{1}{\lambda I_{D\text{, sat.}}} \approx \frac{1}{\lambda I_D}\] \[\lambda = \frac{k_{r_{ds}}}{2 L \sqrt{V_{DS} - V_\text{eff} + \Phi_0}}\] \[k_{r_{ds}} \sqrt{\frac{2 K_{Si} \epsilon_0}{q N_A}}\] \[C_{gs} = \frac{2}{3} W L C_{ox} + WL_{ov} C_{ox}\] \[C_{gd} = WL_{ov} C_{ox}\]