diff --git a/01_Passives/Resistor_Capacitor_Inductor.tex b/01_Passives/Resistor_Capacitor_Inductor.tex index d690484..c11ffaa 100644 --- a/01_Passives/Resistor_Capacitor_Inductor.tex +++ b/01_Passives/Resistor_Capacitor_Inductor.tex @@ -2,7 +2,7 @@ \subsection{Resistor:} A resistor is a two-port passive device that opposes the flow of current, we call this property resistance. The unit of resistance is Ohm (\(\Omega\)). Everything has resistance. A good way to imagine resistance is to -think about it's value as a constant of proportionality between voltage +think about its value as a constant of proportionality between voltage and current. The relationship is given by Ohm's law: \[ V = I_ \cdot R diff --git a/01_Passives/Transistor.tex b/01_Passives/Transistor.tex new file mode 100644 index 0000000..44debf9 --- /dev/null +++ b/01_Passives/Transistor.tex @@ -0,0 +1,169 @@ +\subsection{Transistor:} + +A transistor is a device that can be used either as an amplifier or +as a switch. Its three-port (has three terminals), active (requires +an external power source to operate), 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. + +The popular families are: Bipolar Junction Transistor (BJT) +and Field Effect Transistor (FET). The underlying physics is different +between them but their behaviour is similar. There are also many +other types of transistors (like IGBT, JFET, pHEMT etc.) but they are +often solutions to specific problems (like high voltage, high frequency, +low noise, etc.). + +\subsubsection{Bipolar Junction Transistor (BJT):} + +A BJT 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 has three terminals: the base (B), the collector (C) and the +emitter (E). It is often called a \emph{current controlled device} +because of its base-collector current relationship. In reality, the +base-emitter voltage controls the collector current. + +This family has two types of transistors: NPN and PNP.\ They are +(in a sense) mirror images of each other. + +\begin{circuitfig} + % Paths, nodes and wires: + \node[npn] at (5.5, 7.02){}; + \node[pnp, yscale=-1] at (9, 6.98){}; + \node[shape=rectangle, minimum width=1.465cm, minimum height=0.465cm] at (5.5, 6){} node[anchor=north, align=center, text width=1.077cm, inner sep=6pt] at (5.5, 6.25){Emitter}; + \node[shape=rectangle, minimum width=0.97cm, minimum height=0.465cm] at (4.157, 7.02){} node[anchor=north, align=center, text width=0.582cm, inner sep=6pt] at (4.157, 7.27){Base}; + \node[shape=rectangle, minimum width=0.97cm, minimum height=0.465cm] at (7.657, 6.98){} node[anchor=north, align=center, text width=0.582cm, inner sep=6pt] at (7.657, 7.23){Base}; + \node[shape=rectangle, minimum width=1.465cm, minimum height=0.465cm] at (5.5, 8.04){} node[anchor=north, align=center, text width=1.077cm, inner sep=6pt] at (5.5, 8.29){Collector}; + \node[shape=rectangle, minimum width=1.465cm, minimum height=0.465cm] at (9, 8){} node[anchor=north, align=center, text width=1.077cm, inner sep=6pt] at (9, 8.25){Collector}; + \node[shape=rectangle, minimum width=1.465cm, minimum height=0.465cm] at (9, 5.96){} node[anchor=north, align=center, text width=1.077cm, inner sep=6pt] at (9, 6.21){Emitter}; + \node[shape=rectangle, minimum width=1.965cm, minimum height=0.755cm] at (5.5, 4.855){} node[anchor=north, align=center, text width=1.577cm, inner sep=6pt] at (5.5, 5.25){\Large NPN}; + \node[shape=rectangle, minimum width=1.965cm, minimum height=0.755cm] at (9, 4.855){} node[anchor=north, align=center, text width=1.577cm, inner sep=6pt] at (9, 5.25){\Large PNP}; +\end{circuitfig} + +The behaviour of a BJT can be approximated well by the below equations. +\begin{itemize} + \item \textbf{Current \& Relations}: + The currents flowing into and out of the terminals of a BJT are + related: + \begin{itemize} + \item \textbf{Collector Current (\(I_C\))}: + The Ebers-Moll Model approximately describe the collector current as + two coupled Shockley's diode equations: + \[ + I_C = \underbrace{\alpha_F I_{FS} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right)}_{\text{Forward Junction Contribution}} - \underbrace{\alpha_R I_{RS} \left( e^{\frac{V_{BC}}{V_T}} - 1 \right)}_{\text{Reverse Junction Contribution}} + \] + \begin{itemize} + \item \(I_C\) is the collector current + \item \(I_{FS}\) is the forward saturation current + \item \(\alpha_F\) is the forward common base current gain + \item \(V_{BE}\) is the base-emitter voltage + \item \(V_T\) is the thermal voltage + \item \(I_{RS}\) is the reverse saturation current + \item \(\alpha_R\) is the reverse common base current gain + \item \(V_{BC}\) is the base-collector voltage + \end{itemize} + In forward active mode, the base-collector voltage is reverse biased + (\(V_{BC} \ll 0\)) and thus \( e^{\frac{V_{BC}}{V_T}} \approx 0 \) and the + Ebers-Moll model simplifies, + \begin{align*} + I_C &= \alpha_F I_{FS} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right) - \alpha_R I_{RS} \left( e^{\frac{V_{BC}}{V_T}} - 1 \right) \\ + I_C &= \alpha_F I_{FS} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right) - \alpha_R I_{RS}(-1) \\ + I_C &= \alpha_F I_{FS} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right) + \alpha_R I_{RS} \\ + I_C &\approx \alpha_F I_{FS} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right) + \end{align*} + Forward Saturation Current (\(I_{FS}\)) and Reverse Saturation Current (\(I_{RS}\)) + can be related to each other by defining, + \[ + \alpha_F I_{FS} = \alpha_R I_{RS} = I_S + \] + We'll use \(I_S\) as the saturation current in the rest of this document. + \item \textbf{Base Current (\(I_B\))}: + The base current is related to the collector current by the current gain + (\(\beta\)): + \[ + I_B = \frac{I_C}{\beta} + \] + \(\beta\) is not constant and varies with temperature, collector current and + collector-emitter voltage. Use Ebers-Moll model instead. + \item \textbf{Emitter Current (\(I_E\))}: + The emitter current is the sum of the collector and base currents: + \begin{align*} + I_E &= I_C + I_B \\ + I_E &= I_C + \frac{I_C}{\beta} \\ + I_E &= I_C \left( 1 + \frac{1}{\beta} \right) + \end{align*} + \end{itemize} + \item \textbf{Voltages \& Relations}: + There are several important voltages in a BJT:\ + \begin{itemize} + \item \textbf{Thermal/Boltzmann Voltage (\(V_T\))}: + The thermal voltage is given by: + \[ + V_T = \frac{kT}{q} + \] + where \(k\) is the Boltzmann constant, \(T\) is the temperature in + Kelvin and \(q\) is the charge of an electron. At room temperature, + \(V_T \approx 26\,\mathrm{mV}\): + \item \textbf{Threshold Voltage (\(V_{th}\))}: + The threshold voltage is the minimum voltage required to bias a junction + such that it conducts. For a silicon junction (most common), the + threshold voltage is \(V_{th} \approx 0.6\text{--}0.7\,\mathrm{V}\). + Since we have two junctions in a BJT, we have two threshold voltages. + We define \(V_{Fth}\) for the forward junction threshold voltage and + \(V_{Rth}\) for the reverse junction threshold voltage. + \end{itemize} + They are also related to each other: + \begin{itemize} + \item \textbf{Base-Emitter \(V_{BE}\) and Base-Collector \(V_{BC}\) Voltages}: + If you rearranged Ebers-Moll model for \(V_{BE}\) or \(V_{BC}\) after + simplifying for forward active or reverse active mode, you get: + \begin{itemize} + \item \textbf{Under Forward Active Mode}: + \begin{align*} + V_{BE} = V_T \ln \left( \frac{I_C}{I_{S}} + 1 \right) \\ + V_{BC} \ll 0 + \end{align*} + \item \textbf{Under Reverse Active Mode}: + \begin{align*} + V_{BE} \ll 0 \\ + V_{BC} = V_T \ln \left( \frac{I_C}{I_{S}} + 1 \right) + \end{align*} + \end{itemize} + \item \textbf{Collector-Emitter \(V_{CE}\) Voltage}: + The collector-emitter voltage is the voltage across the collector and + emitter terminals. It is given by: + \begin{align*} + V_{CE} &= V_{BE} - V_{BC} \\ + V_{CE} &= V_B - V_E - (V_B - V_C) \\ + V_{CE} &= V_B - V_E - V_B + V_C \\ + V_{CE} &= V_C - V_E + \end{align*} + \end{itemize} + \item \textbf{Modes of Operation}: + A BJT can operate in four different modes depending on the biasing of + the base-emitter and base-collector junctions. Practically,\\ + \begin{tabularx}{\linewidth}{|X|X|X|} \hline + Mode & Base-Emitter Voltage & Base-Collector Voltage \\ \hline + Cutoff & $V_{BE} < V_{Fth} $ & $V_{BC} < V_{Rth}$ \\ \hline + Forward Active (Amplifier) & $V_{BE} \gtrsim V_{Fth} $ & $V_{BC} < V_{Rth}$ \\ \hline + Saturation (Switch) & $V_{BE} \gtrsim V_{Fth}$ & $V_{BC} \gtrsim V_{Rth}$ \\ \hline + Reverse Active & $V_{BE} < V_{Fth}$ & $V_{BC} \gtrsim V_{Rth}$ \\ \hline + \end{tabularx} + \item \textbf{Effects}: + BJTs are affected by many non-linear effects that are not captured by + the Ebers-Moll model. Sometimes they cause hard to diagnose problems. + \begin{itemize} + \item \textbf{Base Width Modulation}: Increasing \(V_{CE}\) shrinks the base + width physically. This increases the collector current and is captured + by the Early voltage (\(V_A\)) and produces the Early Effect. + \[ + I_C = I_{S} \left( e^{\frac{V_{BE}}{V_T}} - 1 \right) \left( 1 + \frac{V_{CE}}{V_A} \right) \hspace{1cm} \text{[forward active]} + \] + \end{itemize} + \item \textbf{Noise}: + BJTs are also affected by noise namely, thermal noise, shot noise and flicker + noise. You can read more about these noises in~\ref{sec:theory_general_noise}. +\end{itemize} diff --git a/01_Passives/main.tex b/01_Passives/main.tex index f9597c3..0453f50 100644 --- a/01_Passives/main.tex +++ b/01_Passives/main.tex @@ -13,6 +13,13 @@ \section{Resistors, Capacitors and Inductors} \input{01_Passives/Resistor_Capacitor_Inductor} +\section{Transistors} +These three port devices are used almost everyday in modern analog +circuit design. They are called active devices as they require +an external power source to operate. + +\input{01_Passives/Transistor} + \section{Voltage Dividers} A voltage divider is a simple circuit that takes an input voltage and divides it into a lower output voltage. diff --git a/03_Current_Mirrors/Modified_Wilson_Mirror.tex b/03_Current_Mirrors/Modified_Wilson_Mirror.tex index 9f42451..ee13e5d 100644 --- a/03_Current_Mirrors/Modified_Wilson_Mirror.tex +++ b/03_Current_Mirrors/Modified_Wilson_Mirror.tex @@ -215,7 +215,7 @@ \subsubsection{Transfer Function} V_{z} &= V_{s} \tag{14} \end{align*} Equations (9), (10), (1), (13) and (14) can now be - solved analytically, but it's cumbersome, so I'll give the + solved analytically, but its cumbersome, so I'll give the solution directly from Octave: \begin{align*} \frac{V_{out}}{V_{s}} = \frac{-g_{m_2}(g_{m_1} + g_{o_1})}{G_5(g_{m_1} + g_{m_2} + g_{o_1} + g_{o_2}) + g_{o_2}(g_{m_1} + g_{o_1})} \tag{15} diff --git a/03_Current_Mirrors/main.tex b/03_Current_Mirrors/main.tex index 87ab34b..517946f 100644 --- a/03_Current_Mirrors/main.tex +++ b/03_Current_Mirrors/main.tex @@ -2,7 +2,7 @@ \chapter{Current Mirrors}\label{ch:Current_Mirrors} This chapter covers various types of current mirrors, which are circuits designed to copy (or ``mirror'') a current -from one it's branch to another (amplifying or attenuating +from one its branch to another (amplifying or attenuating it the process). Ideally a current mirror should have perfect current diff --git a/1X_Theory/General/Noise.tex b/1X_Theory/General/Noise.tex new file mode 100644 index 0000000..ec61fa6 --- /dev/null +++ b/1X_Theory/General/Noise.tex @@ -0,0 +1,39 @@ +\subsection{Noise}\label{sec:theory_general_noise} +Any unwanted signal that is present in a circuit is called noise. It is +usually measured in terms of its power spectral density (PSD) and then +converted to root mean square (RMS) voltage or current. Some common noises +are: +\begin{itemize} + \item Thermal Noise: This is the noise generated by the thermal fluctuations + of charge carriers (usually electrons) in a conductor or semiconductor. + It is also called Johnson-Nyquist noise, is given by: + \[ + S_n(f) = 4 k T R \hspace{1cm} \text{[in V\(^2\)/Hz; in PSD]} \\ + V_n = \sqrt{4 k T R \Delta f} \hspace{1cm} \text{[in V\(_{\text{rms}}\)]} + \] + where \(S_n(f)\) is the power spectral density of the noise, \(k\) is + Boltzmann's constant, \(T\) is the absolute temperature in Kelvin, \( + R\) is the resistance in ohms, and \(\Delta f\) is the bandwidth of the + signal. + \item Shot Noise: This is the noise generated by the discrete and probabilistic + nature of charge carriers (usually electrons) in a conductor or semiconductor. + It is given by: + \[ + S_n(f) = 2 q I \hspace{1cm} \text{[in A\(^2\)/Hz; in PSD]} \\ + I_n = \sqrt{2 q I \Delta f} \hspace{1cm} \text{[in A\(_{\text{rms}}\)]} + \] + where \(S_n(f)\) is the power spectral density of the noise, \(q\) is + the charge of an electron, \(I\) is the average current in amperes, + and \(\Delta f\) is the bandwidth of the signal. + \item Flicker Noise: This is the noise that has a power spectral density that + is inversely proportional to frequency. It is also called 1/f noise and + is given by: + \[ + S_n(f) = \frac{K}{f^\alpha} \hspace{1cm} \text{[in V\(^2\)/Hz; in PSD]} \\ + V_n = \sqrt{\frac{K}{f^\alpha} \Delta f} \hspace{1cm} \text{[in V\(_{\text{rms}}\)]} + \] + where \(S_n(f)\) is the power spectral density of the noise, \(K\) is + a constant that depends on the device and its operating conditions, \(f\) + is the frequency in hertz, and \(\alpha\) is a constant that typically + ranges from 0.5 to 2. +\end{itemize} diff --git a/1X_Theory/main.tex b/1X_Theory/main.tex index aced421..c3bf629 100644 --- a/1X_Theory/main.tex +++ b/1X_Theory/main.tex @@ -6,6 +6,7 @@ \section{General} \input{1X_Theory/General/Tolerances} \input{1X_Theory/General/Probability} \input{1X_Theory/General/Complementary} +\input{1X_Theory/General/Noise} \section{Controls} \input{1X_Theory/Control_Systems/Second_Order}