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feat: add diode and fix devices #8

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feat: add diode and fix devices
is-it-ayush Aug 15, 2026
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chore: fix ci
is-it-ayush Aug 15, 2026
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14 changes: 6 additions & 8 deletions 01_Devices/Actives/BJT.tex
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\subsection{Bipolar Junction Transistor (BJT)}

A BJT is three-port (has three terminals), active, non-linear (the
relationship between the voltage and current is not a straight line)
and time-variant (behaviour changes as their parameters drift due
to temperature, age, etc.) device. It is physically built from two
PN junctions that share a single region, with the two junctions
oriented oppositely. This shows up in its governing model. It is
called a \emph{bipolar} device because both electrons and holes
are involved in the conduction process.
A BJT is three-port, active, non-linear and time-variant device.
It is physically built from two PN junctions that share a single
region, with the two junctions oriented oppositely. This shows
up in its governing model. It is called a \emph{bipolar} device
because both electrons and holes are involved in the conduction
process.

The three terminals are the base (B), the collector (C) and the
emitter (E). It is often called a \emph{current controlled device}
Expand Down
144 changes: 91 additions & 53 deletions 01_Devices/Passives/Capacitor.tex
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\subsection{Capacitor}
A capacitor is a two-port passive device that stores energy in the form of
an electric field. The property of a capacitor is called capacitance and is
measured in Farads (F). The core relationship between charge, capacitance
and voltage is given by:
\begin{align*}
Q = C \cdot V \\
\frac{dQ}{dt} &= C \cdot \frac{dV}{dt} \hspace{1cm} \text{(assuming \(C\) is constant)} \\
I &= C \cdot \frac{dV}{dt} \hspace{1cm} [I = \frac{dQ}{dt}]
\end{align*}
This is the fundamental relationship between current and voltage in a
capacitor or if you may call this the capacitor's version of Ohm's law.
The inverse of capacitance is called elastance (\(S\)) and is measured in Farads
inverse (F\(^{-1}\)).
An ideal capacitor is a two-port, passive, linear, time-invariant,
non-dissipative device that stores energy (\& thus charge) in the
form of an electric field.

The impedance of a capacitor is given by:
\[
Z_C = \frac{1}{j\omega C} = \frac{1}{j 2 \pi f C} = \frac{1}{s C}
\]
The inverse of impedance is called admittance (\(Y\)) and is measured in
Siemens (S). The relationship between impedance and admittance is:
\[
Y = \frac{1}{Z} = \frac{1}{\frac{1}{sC}} = sC
\]
You'll notice when we apply KCL to circuits within this book, we'll mostly
use admittance \(Y\) to express the impedance, we do this as it simplifies
the analysis a lot. The admittance version of Ohm's law is:
\[
I = V \cdot Y
\]
The impedance of a capacitor is inversely proportional to frequency. This
means that at low frequencies, the impedance of a capacitor is very high and
at frequencies, the impedance of a capacitor is very low. This is why
capacitor effects are more pronounced at high frequencies as their
impedance drops enough to load the circuit. This is also why capacitors are
used for decoupling (they act as a short circuit at high frequencies).
\begin{itemize}
\item \textbf{Capacitance:}
The relationship between charge and
voltage is expressed by a proportionality factor called capacitance
(\(C\)) and is measured in Farads (F). It also gives us the
capacitor's version of Ohm's law:
\begin{align*}
Q = C \cdot V \\
\frac{dQ}{dt} &= C \cdot \frac{dV}{dt} \hspace{1cm} \text{(assuming \(C\) is constant)} \\
I &= C \cdot \frac{dV}{dt} \hspace{1cm} [I = \frac{dQ}{dt}]
\end{align*}
The inverse of capacitance is called elastance (\(S\)) and is measured in Farads
inverse (F\(^{-1}\)).
\item \textbf{Impedance:}
Unlike resistor where a proportionality factor directly relates
voltage and current, a capacitor is different, the relationship
between voltage and current is related instead by a differential
equation. We thus can not describe resistance of a capacitor in the
same way we do for a resistor. We must somehow trick the above
capacitor relationship into giving us a proportionality factor
between voltage and current. Lapalace Transform can be used to
achieve this,
\begin{align*}
I &= C \cdot \frac{dV}{dt} \\
I(s) &= C \cdot s V(s) \\
V(s) &= \frac{1}{sC} I(s)
\end{align*}
We can now define the impedance of a capacitor as the ratio
of voltage to current in the Laplace domain:
\[
Z_C = \frac{1}{s C} = \frac{1}{j\omega C} = \frac{1}{j 2 \pi f C}
\]
\item \textbf{Admittance:}
The inverse of impedance is called admittance (\(Y\)) and
is measured in Siemens (S). The relationship between impedance
and admittance is:
\[
Y = \frac{1}{Z} = \frac{1}{\frac{1}{sC}} = sC
\]

An ideal capacitor is a linear (that is, the relationship between voltage and
current is a straight line) and time-invariant (that is, the relationship
between voltage and current does not change with time) of the three passive
components. It is also the only device that in its ideal form does not
dissipate energy.

A capacitor can be formed anywhere two conductors are separated by an
insulator. The charges are stored on the surface of the conductors and the
energy is stored in the electric field between the conductors. The energy
stored in a capacitor is given by:
\begin{align*}
I &= C \cdot \frac{dV}{dt} \\
P &= V \cdot I \\
P &= V \cdot C \cdot \frac{dV}{dt} \\
P &= C \cdot V \cdot \frac{dV}{dt} \\
P &= C \cdot \frac{1}{2} \cdot \frac{d(V^2)}{dt} \hspace{1cm} [\frac{d(V^2)}{dt} = 2V \cdot \frac{dV}{dt}] \\
\int P dt &= \int C \cdot \frac{1}{2} \cdot \frac{d(V^2)}{dt} dt \\
E &= \frac{1}{2} C V^2
\end{align*}
You'll notice when we apply KCL to circuits within this book,
we'll mostly use admittance \(Y\) to express the impedance, we
do this as it simplifies the analysis a lot. The admittance
version of Ohm's law is:
\[
I = V \cdot Y
\]
\item \textbf{Frequency Response:}
The impedance of a capacitor is inversely proportional to
frequency. This means that at low frequencies, the impedance
of a capacitor is very high and at frequencies, the impedance
of a capacitor is very low. This is why at high frequencies,
the capacitor starts to load the circuit which is often a
very desirable effect in many applications. This is also why
capacitors are used for decoupling (they act as a short
circuit at high frequencies).
\item \textbf{Energy and Power:}
The energy and power stored in a capacitor is given by:
\begin{align*}
I &= C \cdot \frac{dV}{dt} \\
P &= V \cdot I \\
P &= V \cdot C \cdot \frac{dV}{dt} \\
P &= C \cdot V \cdot \frac{dV}{dt} \\
P &= C \cdot \frac{1}{2} \cdot \frac{d(V^2)}{dt} \hspace{1cm} [\frac{d(V^2)}{dt} = 2V \cdot \frac{dV}{dt}] \\
\int P dt &= \int C \cdot \frac{1}{2} \cdot \frac{d(V^2)}{dt} dt \\
E &= \frac{1}{2} C V^2
\end{align*}
\item \textbf{Types of Capacitors}:
There are many types of capacitors, some of the most common are:
\begin{itemize}
\item \textbf{Ceramic Capacitor}: Used for decoupling and filtering.
They are very useful in power supply circuits. Examples include
0805, 0603, 0402, etc.
\item \textbf{Electrolytic Capacitor}: Used for bulk capacitance.
They are very useful in power supply circuits. Examples include
10uF, 100uF, 1000uF, etc.
\item \textbf{Tantalum Capacitor}: Used for bulk capacitance.
They are very useful in power supply circuits. Examples include
10uF, 100uF, 1000uF, etc.
\item \textbf{Film Capacitor}: Used for decoupling and filtering.
They are very useful in power supply circuits. Examples include
10nF, 100nF, 1uF, etc.
\end{itemize}
\item \textbf{They're everywhere in the real world}:
A capacitor exists anywhere where you have two conductors
separated by an insulator. The charges are stored on the
surface of the conductors and the energy is stored in the
electric field between the conductors.
\end{itemize}
171 changes: 166 additions & 5 deletions 01_Devices/Passives/Diode.tex
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Expand Up @@ -2,8 +2,169 @@ \subsection{Diode}

A diode is a two-terminal, active, non-linear and time-variant
device that allows current to flow in one direction while blocking
it in the opposite direction. The most common type of diode is the
semiconductor diode, which is made from a piece of semiconductor
material (usually silicon) that has been doped with impurities to create
a p-n junction. Other types of diodes include Schottky diodes, Zener
diodes, light-emitting diodes (LEDs) and photodiodes.
it in the opposite direction.

It is constructed from a semiconductor material where one side is doped
with a material that has an excess of electrons and the other side
is doped with a material that has a slightly less electrons that
the first side. The region where the two side meet is called the
PN junction \& due to recombinations there are no free charge
carriers. For a voltage to cross the depletion region, it must be
of greater than a known magnitude called the threshold voltage
(determined by the material, doping and the depletion width) and
is typically around 0.7V for silicon diodes and 0.3V for germanium
diodes.

\begin{enumerate}
\item \textbf{Types of Diodes}:
There are many types of diodes, some of the most common are:
\begin{itemize}
\item \textbf{Rectifier Diode}: Used to convert AC to DC.\ They
are very useful in power supply circuits. Examples include 1N4001,
1N4007, 1N5408, etc.
\item \textbf{Zener Diode}: Used to regulate voltage. They exploit
the reverse breakdown region of the diode to maintain a
constant voltage across the diode. Examples include the
1N4733A (5.1\,V), 1N4742A (12\,V), and the BZX79 series.
\item \textbf{Light Emitting Diode (LED)}: Used to emit
light when current flows through it. Under the photovoltaic
effect, they can also be used to convert light into electrical
current. Although they aren't as efficient as photodiodes.
Examples include the 5\,mm red T-1 3/4 (e.g.\ Kingbright L-53LID),
and high-power types such as the Cree XLamp XP-G3 and Lumileds
Luxeon Rebel.
\item \textbf{Schottky Diode}: Used for fast switching
applications. The Schottky diode has a lower forward voltage
drop than a regular diode, which allows it to switch on and
off faster. Examples include the 1N5817, 1N5819, and the
SS14 (surface-mount).
\item \textbf{Photodiode}: Used to convert light into
electrical current. Examples include the BPW34, BPW21, and
the SFH 203 series.
\item \textbf{Varactor Diode}: Used to vary capacitance in
a circuit. The varactor diode is a reverse-biased diode that
has a variable capacitance that changes with the applied voltage.
Examples include the BB135 and the 1SV149, commonly used in
RF tuning circuits and voltage-controlled oscillators (VCOs).
\item \textbf{TVS Diode}: Used to protect circuits from voltage
spikes. The TVS diode is a bidirectional diode that can clamp
voltage spikes to a safe level. These are often used to protect
sensitive electronics from electrostatic discharge (ESD) and
other transient voltage events. Examples include the P6KE
series (unidirectional/bidirectional, 600\,W) and the
SMBJ series (surface-mount).
\end{itemize}
\item \textbf{Diode Parameters}:
The parameters of a diode are:
\begin{itemize}
\item \textbf{Forward Voltage Drop (\(V_F\))}: The voltage drop
across the diode when it is forward biased.
\item \textbf{Reverse Breakdown Voltage (\(V_{BR}\))}: The voltage
at which the diode will conduct in reverse bias.
\item \textbf{Reverse Saturation Current (\(I_S\))}: The current
that flows through the diode when it is reverse biased.
\item \textbf{Maximum Forward Current (\(I_F\))}: The maximum
current that can flow through the diode when it is forward biased.
\item \textbf{Maximum Reverse Current (\(I_R\))}: The maximum
current that can flow through the diode when it is reverse biased.
\end{itemize}
\item \textbf{Diode Models}:
The models of a diode are:
\begin{itemize}
\item \textbf{Ideal Diode Model}: The ideal diode model assumes
that the diode has zero forward voltage drop and infinite reverse
resistance. It is a good approximation for low current applications.
It is given by the following equation:
\[
I = \begin{cases}
0 & \text{if } V_D < 0 \\
\infty & \text{if } V_D > 0
\end{cases}
\]
\item \textbf{Shockley Diode Model}: The Shockley diode model
is a more accurate model that takes into account the forward voltage
drop and the reverse saturation current. It is a good approximation
for high current applications.
\[
I = I_S \left( e^{\frac{V_D}{n V_T}} - 1 \right)
\]
where:
\begin{itemize}
\item \(I\) is the current through the diode.
\item \(I_S\) is the reverse saturation current.
\item \(V_D\) is the voltage across the diode.
\item \(n\) is the ideality factor (typically between 1
and 2).
\item \(V_T\) is the thermal voltage, given by \(V_T = \frac{kT}{q}\).
\end{itemize}
This model does not account for the reverse breakdown conduction of
the diode, nor does it account for the series or breakdown resistance
of the diode.
\item \textbf{Piecewise Linear Model}: The piecewise linear model
is a more accurate model that approximates the current-voltage
relationship as a series of linear segments. It is obtained by
asymptotically solving the Shockley Diode Model in two separate
voltage regimes: forward, cutoff. The breakdown regime is added as
an independent term to account for reverse breakdown conduction.
The model also includes the effects of series resistance and breakdown
resistance.
\begin{enumerate}
\item \textbf{Series Resistance:}
The Shockley Diode Model treats the entire applied voltage \(V_D\)
as appearing across the ideal exponential junction. In a real diode,
part of \(V_D\) drops across the bulk and contact series resistance
\(R_S\), so only the remainder, the junction voltage
\(V_j = V_D - I R_S\), drives the exponential term:
\[
I = I_S \left( e^{\frac{V_D - I R_S}{n V_T}} - 1 \right)
\]
This equation is transcendental in \(I\) and has no closed-form
solution. The piecewise linear model is obtained by solving it
asymptotically in three separate voltage regimes.
\item \textbf{Forward Regime:}
For \(I \gg I_S\), the \(-1\) term is negligible and
\(I \approx I_S e^{V_j / (nV_T)}\), so
\[
V_j \approx n V_T \ln\!\left(\frac{I}{I_S}\right)
\]
Because the logarithm varies slowly, \(V_j\) changes by only
\(n V_T \ln(10) \approx 60\,\text{mV}\) per decade of current, so
across a diode's typical operating range \(V_j\) is well
approximated as constant. Fixing \(V_j \approx V_{TH}\), the
junction voltage at some nominal operating current \(I_0\), and
substituting into \(V_j = V_D - I R_S\) gives
\[
I \approx \frac{V_D - V_{TH}}{R_S} \hspace{1cm} \text{if } V_D \geq V_{TH}
\]
\item \textbf{Cutoff Regime:}
For \(-V_{BR} \leq V_D < V_{TH}\), the current is small enough that
\(I R_S\) is negligible, so \(V_j \approx V_D\), and \(V_D\) is not
large enough for the exponential term to matter. The Shockley model
then reduces directly to
\[
I = I_S \left( e^{V_D / (nV_T)} - 1 \right) \approx -I_S \approx 0
\]
since \(I_S\) is typically \(10^{-9}\) to \(10^{-15}\,\text{A}\).
\item \textbf{Breakdown Regime:}
Reverse breakdown arises from avalanche multiplication or Zener
tunneling, mechanisms entirely absent from the minority-carrier
diffusion current that the Shockley equation models. No asymptotic
limit of the exponential term can reproduce this behavior, so it
must be introduced as an independent term. Modeled with the same
linear structure as the forward regime, using breakdown resistance
\(R_{ZK}\):
\[
I \approx \frac{V_D + V_{BR}}{R_{ZK}} \hspace{1cm} \text{if } V_D < -V_{BR}
\]
\item \textbf{Assembled Model}
\[
I \approx
\begin{cases}
\dfrac{V_D + V_{BR}}{R_{ZK}} & \text{if } V_D < -V_{BR} \\[2mm]
0 & \text{if } -V_{BR} \leq V_D < V_{TH} \\[2mm]
\dfrac{V_D - V_{TH}}{R_S} & \text{if } V_D \geq V_{TH}
\end{cases}
\]
\end{enumerate}
\end{itemize}
\end{enumerate}
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