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+\newcommand{\tans}[1]{\framebox{\textbf{#1}}} + +% Make easier use of EE order of magnitude notation +\renewcommand{\k}{\ensuremath{\text{k}}} % kilo +\newcommand{\M}{\ensuremath{\text{M}}} % Mega + +\newcommand{\V}{\ensuremath{\text{V}}} +\newcommand{\Ohm}{\ensuremath{\Omega}} +\newcommand{\W}{\ensuremath{\text{W}}} + +\newcommand{\Th}{\ensuremath{\text{Th}}} % Thevenin +\renewcommand{\in}{\ensuremath{\text{in}}} % In (for V_{In}) +\newcommand{\out}{\ensuremath{\text{out}}} + +\newenvironment{circuit} + {\begin{center} + \begin{circuitikz}[american]\draw + } + {; + \end{circuitikz} + \end{center} + } + +% Set first level of list to (a) format +\setlist[enumerate,1]{label=(\alph*)} + +% Be sure to use \boxed{} to box the answers and maintain + +\begin{document} + + \ex{1.1} + \begin{enumerate} + \item + $R = 5\k + 10\k = \mans{15\k\Omega}$ + + \item + $R = \dfrac{R_1 R_2}{R_1 + R_2} = \dfrac{5\k \cdot 10\k}{5\k + 10\k} = \mans{3.33\k\Omega}$ + + \end{enumerate} + + \ex{1.2} + $P = IV = \left(\dfrac{V}{R}\right)V = \dfrac{(12\V)^2}{1\Ohm} = \mans{144\W}$ + + \ex{1.3} + \todo{Solve this problem} + + \ex{1.4} + \todo{Solve this problem} + + \ex{1.5} + Given that $P = \dfrac{V^2}{R}$, we know that the maximum voltage we can achieve is 15V and the smallest resistance we can have across the resistor in question is $1\k\Ohm$. Therefore, the maximum amount of power dissipated can be given by \[P = \frac{V^2}{R} = \frac{(15\V)^2}{1\k\Ohm} = \mans{0.225\W}\] + This is less than the 1/4W power rating. + + \ex{1.10} + \begin{enumerate} + \item + With two equal-value resistors, the output voltage is half the input voltage. + \[V_{out} = \frac{1}{2}V_{in} = \frac{30\V}{2} = \mans{15\V}\] + + \item + To treat $R_2$ and $R_{load}$ as a single resistor, combine the two resistors which are in parallel to find that the combined (equivalent) resistance is $5\k\Ohm$. Now, we have a simple voltage divider with a $10\k\Ohm$ resistor in series with the $5\k\Ohm$ equivalent resistor. The output voltage is across this equivalent resistance. The output voltage is given by + \[V_{out} = V_{in} \frac{5\k\Ohm}{10\k\Ohm + 5\k\Ohm} = \frac{30\V}{3} = 10\V \] + + \item + We can redraw the voltage divider circuit to make the ``port'' clearer. + \begin{circuit} + (0,2) to[V=$V_{\in}$] (0,0) + to[short] (2,0) + to[R=$R_2$] (2,2) + to[R=$R_1$](0,2) + (2,0) to[short, *-o] (3,0) + (2,2) to[short, *-o] (3,2) + (3,0) to[open, v_<=$V_\out$] (3,2) + \end{circuit} + + We can find $V_{\Th}$ by leaving the ports open (open circuit) and measuring $V_\out$, the voltage across $R_2$. This comes out to be half the input voltage when $R_1 = R_2$, so $V_\out = 15\V$. Thus $V_{\Th} = 15\V$. + + To find the Th\'evinen resistance, we need to find the short circuit current, $I_{SC}$. + + \todo{finish} + + \begin{circuit} + (0,2) to[V=$V_{\Th}$] (0,0) + to[short, -o] (3,0) + (0,2) to[R=$R_{\Th}$, -o] (3,2) + ; + \end{circuit} + \end{enumerate} + % \begin{circuitikz} \draw + % (0,0) to[battery] (0,4) + % to[ammeter] (4,4) + % to[american inductor] (4,0) + % to[lamp] (0,0) + % ; + % \end{circuitikz} +\end{document} \ No newline at end of file