From 4f4fd58f2629d683fbaf57e437f9fd1eae26d6e1 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?=C3=98yvind=20Skaaden?= Date: Wed, 16 Dec 2020 17:12:59 +0100 Subject: [PATCH] Finish formulas --- ntnu/tfe4152/summary/figures/active.svg | 550 ++++++++++++++++++++++++ ntnu/tfe4152/summary/figures/diode.svg | 181 ++++++++ ntnu/tfe4152/summary/figures/triode.svg | 401 +++++++++++++++++ ntnu/tfe4152/summary/summary.md | 117 ++++- 4 files changed, 1247 insertions(+), 2 deletions(-) create mode 100644 ntnu/tfe4152/summary/figures/active.svg create mode 100644 ntnu/tfe4152/summary/figures/diode.svg create mode 100644 ntnu/tfe4152/summary/figures/triode.svg diff --git a/ntnu/tfe4152/summary/figures/active.svg b/ntnu/tfe4152/summary/figures/active.svg new file mode 100644 index 0000000..6cdbd2a --- /dev/null +++ b/ntnu/tfe4152/summary/figures/active.svg @@ -0,0 +1,550 @@ + +image/svg+xml diff --git a/ntnu/tfe4152/summary/figures/diode.svg b/ntnu/tfe4152/summary/figures/diode.svg new file mode 100644 index 0000000..73721c5 --- /dev/null +++ b/ntnu/tfe4152/summary/figures/diode.svg @@ -0,0 +1,181 @@ + +image/svg+xml diff --git a/ntnu/tfe4152/summary/figures/triode.svg b/ntnu/tfe4152/summary/figures/triode.svg new file mode 100644 index 0000000..7905ed4 --- /dev/null +++ b/ntnu/tfe4152/summary/figures/triode.svg @@ -0,0 +1,401 @@ + +image/svg+xml diff --git a/ntnu/tfe4152/summary/summary.md b/ntnu/tfe4152/summary/summary.md index 4806a79..2cb2561 100644 --- a/ntnu/tfe4152/summary/summary.md +++ b/ntnu/tfe4152/summary/summary.md @@ -5,6 +5,119 @@ date: 2020-12-16 math: true --- -## Dioder +## Konstander -$$ \Phi_0 = \frac{k_B T}{q}\ln\left(\frac{N_A N_D}{n_i}\right) $$ \ No newline at end of file +| 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 $T=300\text{ K}$ | +| $\epsilon_0$ | $8.854\cdot 10^{-12}\text{F}/\text{m}$ | | +| $K_{ox (oksid)}$ | $\cong 3.9$ | | +| $K_{Si (silikon)}$ | $\cong 11.8$ | | +{: .table } + +## 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 + +![Småsignal Diode](figures/diode.svg) + +$$r_d = \frac{V_T}{I_D} $$ + +$$C_T = C_d + C_j $$ + +$$ C_d = \tau_T \frac{I_D}{V_T} $$ + +$$ C_j \approx 2 C_{j0} $$ + +$$ \tau_T = \frac{L_n^2}{D_n} $$ + + +## 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 + +![Transistor i triode](figures/triode.svg) + +$$ r_{ds} = \frac{1}{\mu_n C_{ox} \left(\frac{W}{L}\right)V_\text{eff}} $$ + +$$ C_{gd} = C_{gs} \frac{1}{2}W L C_{ox} + WL_{ov}C_{ox} $$ + +$$ C_{sb} = C_{db} = \frac{C_{j0} \left(A_s + \frac{WL}{2}\right)}{\sqrt{1 + \frac{V_{sb}}{\Phi_0}}} $$ + + +## Transistor i aktivt område + +Dette gjelder bare for $V_{GS} > V_{tn}$, $V_{DS} \geq V_\text{eff}$. + +$$ I_D = \mu C_{ox} \left(\frac{W}{L}\right) (V_{GS} - V_{tn})^2 \underbrace{\left[1 + \lambda(V_{DS} - V_\text{eff})\right]}_\text{body-effect} $$ + +$$ \lambda \propto \frac{1}{L\sqrt{V_{DS} - V_\text{eff} + \Phi_0}} $$ + +$$ V_{tn} = V_{tn\text{-}0} - \gamma\left(\sqrt{V_{SB} + 2\Phi_F} - \sqrt{2\Phi_F}\right) $$ + +$$ V_\text{eff} = V_{GS} - V_{tn} = \sqrt{\frac{2 I_D}{\mu_n C_{ox} \frac{W}{L}}} = V_{DS, \text{sat.}} $$ + + +### Småsignal for transistor i aktivt område + +![Transistor i aktivt område](figures/active.svg) + +$$\begin{aligned} + g_m &= \mu_n C_{ox} \frac{W}{L} V_\text{eff} \\ + &= \sqrt{2 \mu_n C_{ox} \frac{W}{L} I_D} \\ + &= \frac{2 I_D}{V_\text{eff}} +\end{aligned} +$$ + +$$ +\begin{aligned} + g_s &= \frac{\gamma g_m}{2 \sqrt{V_{SB} + |2\Phi_F|}}\\ + &\approx 0.2 g_m +\end{aligned} +$$ + +$$ r_{ds} = \frac{1}{\lambda I_{D\text{, sat.}}} \approx \frac{1}{\lambda I_D} $$ + +$$\lambda = \frac{k_{r_{ds}}}{2 L \sqrt{V_{DS} - V_\text{eff} + \Phi_0}} $$ + +$$ k_{r_{ds}} \sqrt{\frac{2 K_{Si} \epsilon_0}{q N_A}} $$ + +$$ C_{gs} = \frac{2}{3} W L C_{ox} + WL_{ov} C_{ox} $$ + +$$ C_{gd} = WL_{ov} C_{ox} $$