add bjt

08_bjt.tex

+\section{Bipolar junction transistor (BJT)}
+\begin{figure}[h]
+    \centering
+    \caption{BJT}
+    \includegraphics[width=.75\textwidth]{imgs/bjt_terminals_and_functioning.png}
+\end{figure}
+
+But what's going on?
+If $V_{BE}>0$ injection of electrons from E to B, of holes from B to E.
+If  $V_{BC}<0$  extraction of electrons from B to C, of holes from C to B.
+
+
+\subsection{BJT characteristics}
+\begin{align}
+    I_E & = -I_C-I_B \\
+    \begin{split}
+        \beta &= \frac{I_C}{I_B}
+        =\frac{n_{pB_0}\frac{D_n}{W_B}}{p_{nE_0}\frac{D_p}{W_E}}\\
+        &=  \frac{N_{dE} D_n W_E}{N_{aB} D_p W_B}
+    \end{split}
+\end{align}
+
+Collector current,
+focus on electron diffusion in base:
+\begin{align}
+    n_{pB}(0) & =n_{pB_0}e^{\frac{qV_{BE}}{kT}}                                                     \\
+    n_{pB}(x) & =n_{pB}(0)(1-\frac{x}{W_B})                                                         \\[1em]
+    \begin{split}
+        J_{nB} &= qD_n\frac{\mathrm{d} n_{pB}}{\mathrm{d}x}\\
+        &= -qD_n\frac{n_{pB}(0)}{W_B}
+    \end{split} \\
+    \begin{split}
+        I_C &=-J_{nB}A_E\\
+        &=qA_E\frac{E_n}{W_B}n_{pB_0}e^{\frac{qV_{BE}}{kT}}
+    \end{split}                        \\
+    I_C       & = I_Se^{\frac{qV_{BE}}{kT}}
+\end{align}
+Base current,
+focus on hole injection and recombination in emitter:
+\begin{align}
+    p_{nE}(-x_{BE})     & =p_{nE_0}e^{-\frac{qV_{BE}}{kT}}                                                                                        \\
+    p_{nE}(-W_E-x_{BE}) & =p_{nE_0}                                                                                                               \\
+    p_{nE}(x)           & =\left[ p_{nE}(-x_{BE}-p_{nE_0}) \right]\left( 1+\frac{x+x_{BE}}{W_E} \right)+P_{nE_0} & \leftarrow \text{Hole Profile} \\[1em]
+    \begin{split}
+        J_{pE}&=-qD_p\frac{\mathrm{d}p_{nE}}{\mathrm{d}x}\\
+        &=-qD_p\frac{p_{nE(-x_{BE})-p_{nE_0}}}{W_E}
+    \end{split}                                                                                              \\
+    \begin{split}
+        I_B&=-J_{pE}A_E\\
+        &=qA_E\frac{D_p}{W_E}p_{nE_0}\left( e^{\frac{qV_{VE}}{kT}} -1 \right)
+    \end{split}                     \\
+    I_B                 & =\frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}}-1\right)                                                                 \\
+    I_B\approx\frac{I_C}{\beta}
+\end{align}
+
+\subsubsection{`Good' transistor}
+We want  collector and emitter current to be identical and so we define $\alpha$ as measurement of how close we are:
+\begin{align}
+    I_C   & =-\alpha I_E                \\
+          & =\alpha\left(I_B+I_C\right) \\
+          & =\frac{\alpha}{1-\alpha}I_B \\
+          & =\beta I_B                  \\
+    \beta & =\frac{\alpha}{1-\alpha}
+\end{align}
+
+\subsection{Summary forward active}
+\begin{align}
+    I_C & = I_Se^{\frac{qV_{BE}}{kT}}                              \\
+    I_B & = \frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}}-1\right) \\
+    I_E & = -I_C-I_B
+\end{align}
+
+For reverse, it is the same but $\beta_R\approx [0.1,5]\ll\beta$.
+
+\subsection{Summary cut-off}
+\begin{alignat}{2}
+    I_{B1} & = -\frac{I_S}{\beta}  &  & =-I_E \\
+    I_{B2} & =-\frac{I_S}{\beta_R} &  & =-I_C
+\end{alignat}
+
+\subsection{Summary saturation}
+\begin{align}
+    I_C & =I_S\left(e^{\frac{qV_{BE}}{kT}} - e^{\frac{qV_{BC}}{kT}}\right)-\frac{I_S}{\beta_R}\left( e^\frac{qV_{BC}}{kT} - 1 \right)  \\
+    I_B & =\frac{I_S}{\beta}\left( e^{\frac{qV_{BE}}{kT}}-1 \right)+\frac{I_S}{\beta_R}\left( e^{\frac{qV_{BC}}{kT}} -1 \right)        \\
+    I_E & =\frac{I_S}{\beta}\left(e^{\frac{qV_{BE}}{kT}} - 1\right) - I_S\left( e^{\frac{qV_{BE}}{kT}} -e^{\frac{qV_{BC}}{kT}} \right)
+\end{align}
+
+\subsection{Ebers-Moll model}
+\begin{center}
+    \begin{circuitikz}
+        \draw (0,0) node[left] {B} to [short,*-] ++(1,0)
+        to [Do,l=$\frac{I_S}{\beta_R}\left( e^{\frac{qV_{BC}}{kT}} -1 \right)$] ++(0,2)
+        to [short] ++(2,0)
+        to [I,l=$I_S\left( e^{\frac{qV_{BE}}{kT}} - e^{\frac{qV_{BC}}{kT}} \right)$,i=$$] ++(0,-4)
+        to [short] ++(-1,0);
+        \draw (1,0) to [Do,l_=$\frac{I_S}{\beta}\left( e^{\frac{qV_{BE}}{kT}}-1 \right)$] ++(0,-2)
+        to [short] ++(1,0)
+        to [short,-*] ++(0,-1) node [below] {E};
+        \draw (2,2) to [short,-*] ++(0,1) node[above] {C};
+    \end{circuitikz}
+\end{center}
+
+\subsection{Early effect}
+With increasing $V_{CE}$, the depletion region inceases.
+To not have to deal with that, we introduce a correction factor
+\begin{equation}
+    I_C = I_S e^{\frac{V_{BE}}{V_{th}}}\left(1+\frac{V_{CE}}{V_A}\right)
+\end{equation}
+
+\subsection{Transfer characteristics}
+We evaluate the transistor at its operating point ($OP$ or $Q=(V_{BE},V_{CE})$) to find the transconductance $g_m$.
+\begin{equation}
+    \label{label:eq:bjt_transconductance}
+    g_m = \left. \frac{\partial i_C}{\partial V_{BE}} \right|_{OP} = \frac{qI_C}{kT}
+\end{equation}
+

semiconductor_summary.tex

@@ -45,4 +45,5 @@
 \include{05_pn_junction_bias}
 \include{06_pn_junction_diode}
 \include{07_diode_applications.tex}
+\include{08_bjt}
 \end{document}