diff --git a/Chapter1.pdf b/Chapter1.pdf index 3fbaa2b..084ea69 100644 Binary files a/Chapter1.pdf and b/Chapter1.pdf differ diff --git a/Chapter1.tex b/Chapter1.tex index 6b60519..c688aca 100644 --- a/Chapter1.tex +++ b/Chapter1.tex @@ -222,4 +222,21 @@ \end{align*} Similarly, we find that $t_2 = -RC \ln(0.1)$. Subtracting these two gives us \[t_2 - t_1 = -RC(\ln(0.1) - \ln(0.9)) = 2.2RC\] + + \ex{1.17} + The voltage divider on the left side of the circuit can be replaced with the Th\'evenin equivalent circuit found Exercise 1.10 (c). Recall that $V_\Th = \frac{1}{2} V_\in$ and $R_\Th = 5\k\Ohm$. This gives us the following circuit. + \begin{circuit}{fig:1.17.1}{Th\'evenin equivalent circuit to Figure 1.36 from the textbook.} + (0,2) to[open, v_>=$V_\Th$, o-o] (0,0) node[ground]{} + + (0,2) to[R=$R_\Th$] (3,2) + to[C=$C$, *-] (3,0) node[ground]{} + (3,2) to[short, -o] (5,2) + to[open, v^>=$V(t)$, o-o] (5,0) node[ground]{} + \end{circuit} + + Now we have a simple RC circuit which we can apply Equation 1.21 to. The voltage across the capacitor is given by + \[V(t) = V_\text{final}(1 - e^{-t/RC}) = V_\Th (1 - e^{-t/R_\Th C} = \frac{1}{2}V_\in (1 - e^{-t\times 10^{-3}})\] + + \todo{Add graph} + \end{document} \ No newline at end of file