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6th-Semester-Spring-2024/DSP/Notes/dsp-notes.aux

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6th-Semester-Spring-2024/DSP/Notes/dsp-notes.log

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6th-Semester-Spring-2024/DSP/Notes/dsp-notes.tex

@@ -643,4 +643,95 @@ where $r$ is the magnitude and $\theta$ is the phase angle.
 \noindent
 If a region of convergence (ROC) does not include the unit circle, then its DTFT does not exist except for special case signals. If the ROC \emph{does} contain the unit circle, then the DTFT can be found by substituting $z = e^{j\omega}$.
 
+\section{Z Transform Transfer Functions}
+Recall that the impulse response of an LTI system completely characterizes the system. It can be used to determine responses of other systems via convolution, and it is a useful tool in determining the BIBO stability of the system.
+\\
+\\
+Given an impulse response to a discrete LTI system, the Z-transform of the impulse response is called the \emph{transfer function}
+$$H(z) = Z[h[n]] = \sum_n h[n]z^{-n}$$
+An LTI system is causal if $h[n]$ is a causal signal. This is the same as if the ROC of $H(z)$ includes $z=\infty$. These conditions are equivalent because the ROC of a Z-transform only contains $z=\infty$ for causal signals.
+\\
+\\
+Additionally, an LTI system is BIBO stable if $h[n]$ is absolutely summable.
+$$\sum_n |h[n]| < \infty$$
+The equivalent condition for $H(z)$ is the ROC of $H(z)$ must include the unit circle. These conditions are equivalent because the DTFT exists when the unit circle is in the ROC, and the DTFT only exists when the signal is absolutely summable.
+\nt{
+	For causal systems only, BIBO stability is achieved if all the poles of $H(z)$ are inside the unit circle. Since non-causal systems do not include $z=\infty$ in the ROC, this test will not hold for non-causal systems.
+}
+\noindent
+If the impulse response $h[n]$ has finite support, then the output, $y[n]$, given an input, $x[n]$, can be found by convolution.
+$$y[n] = h[n] * x[n] = \sum_{k=0}^{N-1} h[k] x[n-k]$$
+However, if the impulse response $h[n]$ has infinite support, the output, $y[n]$, is given by a constant coefficient difference equation.
+$$\sum_{j=0}^N a[j]y[n-j] = \sum_{k=0}^M b[k]x[n-k]$$
+where $a[j]$ are the constant coefficient of the output, and $b[k]$ are the constant coefficients of the input.
+
+\ex{}
+{
+	Given the infinite impulse response:
+	$$h[n] = a^n u[n]$$
+	The transfer function of the system is:
+	$$H(z) = {z \over z - a}$$
+	with a ROC of $|z| > |a|$. Since this includes $z=\infty$, the system is causal. If $|a| < 1$, then the system is also BIBO stable.
+
+	$$H(z) = {z \over z-a} = {Y(z) \over X(z)}$$
+	$${Y(z) \over X(z)} = {1 \over 1-az^{-1}}$$
+	$$Y(z)-az^{-1}Y(z) = X(z)$$
+	Taking the inverse Z-transform:
+	$$y[n] - a y[n-1] = x[n]$$
+	$$y[n] = x[n] + ay[n-1]$$
+	Since the output relies on both the input and the previous output, the recursive implementation of the input-output relationship is used.
+}
+
+\ex{}
+{
+	Given the impulse response,
+	$$h[n] = na^n u[n]$$
+	the corresponding transfer function is
+	$$H(z) = {az \over (z-a)^2}$$
+	with ROC, $|z| > |a|$. Since the impulse response is causal, the system is causal. Equivalently, since the ROC includes $z=\infty$, the system is causal.
+	Given this region of convergence, the system is BIBO stable if and only if $|a| < 1$.
+	$$H(z) = {az\over (z-a)^2} = {Y(z) \over X(z)}$$
+	$$H(z) = {az \over z^2 - 2az + a^2} = {az \over z^2(1 -2az^{-1} + a^2z^{-2})}$$
+	$$az^{-1}X(z) = Y(z) - 2az^{-1}Y(z) + a^2z^{-2}Y(z)$$
+	Taking the inverse Z-transform to find the input-output relationship:
+	$$x[n-1] = y[n] - 2ay[n-1] + a^2y[n-2]$$
+	$$y[n] = x[n-1] + 2ay[n-1] - a^2y[n-2]$$
+}
+
+\ex{}
+{
+	$$h[n] = u[n] - u[n-4]$$
+	$$h[n] = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, 0 \le n \le 3$$
+	$$H(z) = \sum_n h[n] z^{-n} = 1 + z^{-1} + z^{-2} + z^{-3}$$
+	For finite support signals, the Z-transform is the entire Z-plane with the possible exception of $z=0$ and $z=\infty$. Checking $z=0$ reveals a pole, but $z=\infty$ works, so the system is causal. All finite support impulse responses (FIR) are BIBO stable.
+	$$y[n] = \sum_{k=0}^3 h[k]x[n-k]$$
+	$$y[n] = h[0]x[n] + h[1]x[n-1] + h[2]x[n-2] + h[3]x[n-3]$$
+}
+
+\section{The Inverse Z-Transform}
+Given the causal signal $x[n] = a^n u[n]$, its Z-transform is $X(z) = {z \over z - a}$ with region of convergence $|z| > |a|$. Given an anti-causal signal $x[n] -a^n u[-n-1]$, its Z-transform is $X(z) = {z \over z - a}$ with region of convergence $|z| < |a|$.
+\\
+\\
+Consider the following second-order LTI system whose transfer function is given by:
+$$H(z) = {z (z - 0.2) \over (z-0.8)(z-0.9)}$$
+If the system is causal, then the ROC includes $\infty$, so the ROC must be $|z|>0.9$. In this case the system would be BIBO stable. However, if the system is anti-causal, then the ROC must include 0 and therefore would be $|z|<0.8$. In this case, the system would not be BIBO stable. If the system is two-sided, then the ROC would be $0.8 < |z| < 0.9$. In this case the system would also not be BIBO stable.
+\\
+\\
+Find the partial fraction expansion of $H(z) \over z$:
+$${H(z) \over z} = {z - 0.2 \over (z-0.8)(z-0.9)} = {A \over z - 0.9} + {B \over z - 0.8}$$
+$$z - 0.2 = (z-0.8)A + (z-0.9)B$$
+$$B = -6, A = 7$$
+$${H(z) \over z} = {7 \over z-0.9} - {6 \over z-0.8}$$
+$$H(z) = {7z \over z-0.9} - {6z \over z-0.8}$$
+Apply the inverse Z-transform on both sides. If the system is causal:
+$$h[n] = 7(0.9)^n u[n] - 6(0.8)^n u[n]$$
+If the system is anti-causal:
+$$h[n] = -7(0.9)^n u[-n-1] + 6(0.8)^n u[-n-1]$$
+If the system is two-sided:
+$$h[n] = -7(0.9)^n u[-n-1] - 6(0.8)^n u[n]$$
+
+\nt
+{
+	For FIR systems, if the Z-transform does not converge at $|z|= 0$ or $|z| = \infty$, they are not considered poles, because only IIR systems can have poles.
+}
 \end{document}

6th-Semester-Spring-2024/DSP/Notes/dsp-notes.toc

@@ -31,4 +31,6 @@
 \contentsline {subsection}{\numberline {3.1.2}Properties of the Discrete Time Fourier Transform}{17}{subsection.3.1.2}%
 \contentsline {subsection}{\numberline {3.1.3}Special Cases}{18}{subsection.3.1.3}%
 \contentsline {chapter}{\numberline {4}The Z-Transform}{19}{chapter.4}%
+\contentsline {section}{\numberline {4.1}Z Transform Transfer Functions}{20}{section.4.1}%
+\contentsline {section}{\numberline {4.2}The Inverse Z-Transform}{22}{section.4.2}%
 \contentsfinish 

6th-Semester-Spring-2024/EnI/Group Work/low-res-prototyping.wiki

@@ -0,0 +1,19 @@
+= Low-Res Prototyping =
+- Proof of concept
+- For an app or website use storyboarding
+	- For websites
+		- Slideshow
+		- Google Sites
+		- Wix
+		- GoDaddy
+	- For apps
+		- Keynotopia.com
+		- MarvelApp
+		- Figma
+		- Glide
+- Look for possible problems
+- Prioritize features or functionality
+- Get the general idea across
+
+== Going from low-res to high-res ==
+- OnShape for 3D modelling

6th-Semester-Spring-2024/EnI/Notes/EnI-notes.aux

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6th-Semester-Spring-2024/EnI/Notes/EnI-notes.tex

@@ -91,4 +91,25 @@ A business model canvas has nine segments in four core areas.
 	Meets needs and requirements of the CVP while mitigating risk
 }
 
+\chapter{Customers and Markets}
+\dfn{Customer}
+{
+	Someone who wants to purchase a product or service.
+}
+\dfn{Market}
+{
+	A place where people can sell their product or service.
+}
+\dfn{Demand}
+{
+	A \emph{need} of the market.
+}
+\dfn{End-User Profile}
+{
+	A description of the demographics and psychographics of the customer.
+}
+
+\chapter{Midterm Review}
+Entrepreneurship, intrapreneurship, innovation, creativity, startup vs. corp. vs. LLC, types of entrepreneurship, different types of mindset, effectuation vs. self-leadership, elements and characteristics of an opportunity, perception vs. alertness vs. intuition vs. instinct, pattern recognition and opportunity recognition, process of identifying opportunity, design thinking and its elements, feasibility vs. viability vs. durability vs. desireability, implementation phase and its purpose, four parts of a business model, business model canvas parts related to customers, segments of business model canvas, chain of customers, importance of buyer profile and steps, process of market sizing, empathy map
+
 \end{document}

6th-Semester-Spring-2024/EnI/Notes/EnI-notes.toc

@@ -2,4 +2,6 @@
 \contentsline {chapter}{\numberline {2}}{3}{chapter.2}%
 \contentsline {section}{\numberline {2.1}Business Model}{3}{section.2.1}%
 \contentsline {subsection}{\numberline {2.1.1}Business Model Canvas}{3}{subsection.2.1.1}%
+\contentsline {chapter}{\numberline {3}Customers and Markets}{4}{chapter.3}%
+\contentsline {chapter}{\numberline {4}Midterm Review}{5}{chapter.4}%
 \contentsfinish 

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6th-Semester-Spring-2024/Power-Electronics/Homework/Homework-3/homework3.tex

@@ -0,0 +1,23 @@
+\documentclass{article}
+\usepackage{graphicx}
+
+\begin{document}
+\begin{figure}
+	\center
+	\includegraphics[width=0.5\textwidth]{ChargingCircuit.png}
+	\caption{Charging circuit with 50v DC source as load}
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{iL50v.png}
+	\caption{Current through inductor with 50v DC source as load}
+\end{figure}
+\begin{figure}
+	\includegraphics[width=\textwidth]{iL100v.png}
+	\caption{Current through inductor with 100v DC source as load}
+\end{figure}
+\begin{figure}
+	\includegraphics[width=\textwidth]{iL150v.png}
+	\caption{Current through inductor with 150v DC source as load}
+\end{figure}
+\end{document}

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6th-Semester-Spring-2024/Power-Electronics/Notes/power-electronics-notes.tex

@@ -115,5 +115,42 @@ The term, $a_0 \over 2$ is the \emph{DC offset} of the signal. While the values
 $$f(t) = A_0 + \sum_{n=1}^\infty A_n \cos(n \omega t + \varphi_n)$$
 where $A_n = \sqrt{a_n^2 + b_n^2}$ and $\varphi_n = -\arctan\lt({b_n \over a_n}\rt)$
 
+\chapter{Three-Phase Power}
+\section{Y and $\Delta$ Configurations of AC Voltage Sources}
+\subsection{The Y Configuration}
+The Y configuration is shaped like the letter Y with a neutral connection in the center.
+$$V_a = V_an = V_s \sin(\omega t)$$
+$V_b$ lags by 120$^\circ$.
+$$V_b = V_bn = V_s \sin(\omega t - {2\pi \over 3})$$
+$V_c$ lags by 240$^\circ$.
+$$V_c = V_cn = V_s \sin(\omega t - {4\pi \over 3})$$
+
+\subsection{The $\Delta$ Configuration}
+The $\Delta$ configuration is shaped like the letter $\Delta$ with no neutral connection.
+$$V_ab = V_a - V_b = V_s \lt[\sin(\omega t) - \sin(\omega t - {2\pi \over 3})\rt] = \sqrt{3} V_s \sin(\omega t + {\pi \over 6})$$
+$$V_bc = V_b - V_c = \sqrt{3}V_s \sin(\omega t - {\pi \over 2})$$
+$$V_ca = V_c - V_a = \sqrt{3}V_S \sin(\omega t + {5\pi \over 6})$$
+
+\subsection{Y Configuration Power}
+If a Y configuration of AC sources is connected to a Y configuration of resistors, the power through to each load resistor is:
+$$P_a = {V_s^2 \over 2R_a} \lt[1 - \cos(2\omega t)\rt]$$
+$$P_b = {V_s^2 \over 2R_a} \lt[1 - \cos(2\omega t - {4\pi \over 3})\rt]$$
+$$P_c = {V_s^2 \over 2R_c} \lt[1 - \cos(2\omega t + {4\pi \over 3})\rt]$$
+The total power transfer is:
+$$P = 3{V_s^2 \over 2R}$$
+where $R = R_a = R_b = R_c$.
+\\
+\\
+\noindent
+One benefit of using three-phase is constant power from the source. Another advantage of using three-phase power is that any harmonics divisible by either 2 or 3 cancel out. The first harmonic after the fundamental is the 5$^\text{th}$ harmonic, followed by the 7$^\text{th}$, 11$^\text{th}$, and 13$^\text{th}$. Harmonics are non-zero for $6n \pm 1, n \in \mathbb{Z}$.
+
+\chapter{Silicon Controlled Rectifier (SCR)}
+The SCR will be off even when $V_{AC} > 0$ until a pulse is applied to the gate. Once the pulse is applied, the device will stay on as long as $i_F > 0$.
+
+\nt
+{
+	The voltage across the SCR can go negative as long as the current remains positive.
+}
+
 
 \end{document}

6th-Semester-Spring-2024/Power-Electronics/Notes/power-electronics-notes.toc

@@ -4,4 +4,10 @@
 \contentsline {section}{\numberline {1.3}Full Bridge with Resistive Load and Inductive Filter}{3}{section.1.3}%
 \contentsline {section}{\numberline {1.4}Periodic Steady State}{3}{section.1.4}%
 \contentsline {section}{\numberline {1.5}Trigonometric Fourier Series}{4}{section.1.5}%
+\contentsline {chapter}{\numberline {2}Three-Phase Power}{5}{chapter.2}%
+\contentsline {section}{\numberline {2.1}Y and $\Delta $ Configurations of AC Voltage Sources}{5}{section.2.1}%
+\contentsline {subsection}{\numberline {2.1.1}The Y Configuration}{5}{subsection.2.1.1}%
+\contentsline {subsection}{\numberline {2.1.2}The $\Delta $ Configuration}{5}{subsection.2.1.2}%
+\contentsline {subsection}{\numberline {2.1.3}Y Configuration Power}{5}{subsection.2.1.3}%
+\contentsline {chapter}{\numberline {3}Silicon Controlled Rectifier (SCR)}{7}{chapter.3}%
 \contentsfinish