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\section{Diode applications}
\subsection{Small signal}
If $i\ll I$ and $v\ll V$, then we can use the small signal approximation.
\begin{equation}
\begin{split}
I+i &= I_0\left(\exp\frac{q(V+v)}{kT}-1\right) \\
&=I_0\left(\exp\frac{q(V)}{kT}\exp\frac{q(v)}{kT}-1\right)\\
&\approx I_0\left(\exp\frac{q(V)}{kT}\left(1+\frac{qv}{kT}\right)-1\right)\\
&= I_0\left( \exp\frac{qV}{kT} - 1 \right) + I_0 \left( \exp\left(\frac{qV}{kT}\right)\frac{qv}{kT} \right)
\end{split}
\end{equation}
Which gives us the smallsignal current
\begin{align}
i & = \frac{q\left(I+I_0\right)}{kT}v = g_d v \\
g_d & = \frac{q\left(I+I_0\right)}{kT}
\end{align}
\subsubsection{Capacitances of small-signal model}
\begin{figure}[h]
\centering
\caption*{Small signal model for diode}
\begin{circuitikz}
\draw (-3,2) to [Do] (-3,0);
\draw[->] (-2,1) -- (-1,1);
\draw (0,0) to [R,l=$g_d$] (0,2);
\draw (2,0) to [C,l=$C_j$] (2,2);
\draw (4,0) to [C,l=$C_d$] (4,2);
\draw (0,0) to [short] (4,0);
\draw (0,2) to [short] (4,2);
\draw (2,2) to [short] (2,2.5);
\draw (2,-0.5) to [short] (2,0);
\end{circuitikz}
\end{figure}
\begin{align}
C_j & = \frac{C_{j0}}{\sqrt{1-\frac{V}{\phi_B}}} \\
C_d & = \frac{q}{kT}\tau_T I \\
\tau_T & \equiv \text{equivalent transit time of carriers}
\end{align}
Where $C_j$ dominates in reverse bias and small forward bias,
and $C_d$ dominates in large forward bias ($V>\frac{\phi_B}{2}$).
\subsection{Large signal}
\subsubsection{Rectifier}
This is pretty much trivial.
\begin{center}
\begin{circuitikz}
\draw (0,-2) to [sV,l=$V_{in}$] ++(0,4)
to [short] ++(2,0)
to [Do,v=$V_D$] ++(0,-2)
to [R,l=$R_L$,v=$V_o$] ++(2,0)
to [Do] ++(0,-2)
to [short] (0,-2);
draw (4,0) to [Do] ++(0,2)
to [short] ++(-2,0);
\draw (2,-2) to [Do] ++(0,2);
\draw (4,0) to [Do] ++(0,2)
to [short] ++(-2,0);
\end{circuitikz}
\end{center}
\begin{equation}
V_o = V_{in}-2V_D
\end{equation}
\subsubsection{Voltage regulator}
\begin{center}
\begin{circuitikz}
\draw (0,0) to [V,v<=$10\pm1\ V$] (0,8)
to [short] ++(2,0)
to [R,l=$R_1$] ++(0,-2)
to [short] ++(0,-2)
to [Do] ++(0,-1)
to [Do] ++(0,-1)
to [Do] ++(0,-1)
to [short] ++(0,-1)
to [short] (0,0);
\draw (2,5) to [nos] ++(2,0)
to [R,l=$R_2$] ++(0,-5)
to [short] (0,0);
\end{circuitikz}
\end{center}
How to go about it:
\begin{align}
I & = \frac{V_{in}-\sum V_D}{R_1} \\
r_d & =\left[\frac{\mathrm{d}I_D}{\mathrm{d}V_d}\right]^{-1} =\frac{nV_t}{I_0} \\
r & = \sum r_d \\
\Delta V_o & = \Delta V_{in}\frac{r}{r+R_2} \\[1em]
n & \equiv\text{non-ideality factor} \\
V_t & = \frac{kT}{q}
\end{align}
Once the load is connected and draws current, we have a further small variation:
\begin{align}
I_{load} & = \frac{\sum V_D}{R_2} \\
\Delta V_o & = I_{load}r
\end{align}
\subsection{Special diode types}
\subsubsection{Zener}
Is heavily doped making the depletion layer extremely thing, and thus allowing for QM tunneling in reverse biased diode.
(In this case known as band-to-band tunneling.)
\begin{figure}[h]
\centering
\caption[short]{Band-bending zener diode}
\includegraphics[width=.75\textwidth]{imgs/zener_diode_band_bending.png}
\end{figure}
The voltage at which the diode starts conducting is called the zener voltage $V_Z$.
The diode then has a low resistance $R_Z$.
\subsubsection{Esaki}
Heavily doped with the tunneling effect in forward bias.
\begin{figure}[h]
\centering
caption{Esaki tunnel diode}
\includegraphics[width=.75\textwidth]{imgs/esaki_tunnel_diode.png}
\end{figure}
\subsubsection{Schottky}
A schottky diode has $I_0$ $10^3$ to $10^8$ times bigger than a PN diode.
Preferred in low voltage high current applications.
\subsubsection{Photodiodes}
\begin{equation}
I = I_0\left( e^{\frac{qV}{kT}} - 1\right)-I_{photo}
\end{equation}