diff options
| -rw-r--r-- | execsumm/document.tex | 29 |
1 files changed, 28 insertions, 1 deletions
diff --git a/execsumm/document.tex b/execsumm/document.tex index 42265fa..9ae8838 100644 --- a/execsumm/document.tex +++ b/execsumm/document.tex @@ -34,7 +34,7 @@ The last two equations we get can be used to solve for $Z(s)$, which we find to with $b=C_1R_1\rload + R_1C_2\rload + \rload^2C_2$ to simplify notation. We can now find the partial fraction decomposition of this% -\footnote{1}{\link{Wolfram Alpha}{https://www.wolframalpha.com/input/?i% +\footnote{$^1$}{\link{Wolfram Alpha}{https://www.wolframalpha.com/input/?i% =solve+for+x1\%2Cx2\%2Cx3\%2Cx4+in+\%7B\%7B1\%2C0\%2C1\%2C0\%7D\%2C+\%7% Bb\%2C1\%2C0\%2C1\%7D\%2C+\%7BR\%2C+b\%2C+w\%5E2\%2C+0\%7D\%2C+\%7B0\%2% C+R\%2C+0\%2C+w\%5E2\%7D\%7D*\%7Bx1\%2Cx2\%2Cx3\%2Cx4\%7D+\%3D+\%7B0\%2% @@ -78,6 +78,9 @@ absorb some of the variance in current from the source, $Z(t)$ is smaller with smaller values (blue is the smallest frequency at $100Hz$). However, the initial oscillation in every curve makes a lot of sense +from a physical standpoint. With other values for the system, the shapes +of these curves vary slightly, mostly in terms of frequency with changes +of capacitance and changes of amplitude with changes in the resistance. \section{Possible Generalization} @@ -92,4 +95,28 @@ there is a general form for a sine wave and the output is proportional to the input, a Fourier transform could be used to either approximate or analytically obtain a solution for these types of waves. +\section{External Relation} + +This solution applies the Laplace transform and the inverse Laplace +transform, which is directly related to the class as a core component of +the class. We also applied, in the first iteration, an attempt at using +eigenvalues and eigenvectors to develop a solution was made. This +solution would have taken some of the similar paths as this one, +especially with the complex roots because all of the eigenvalues were +complex. However, the transition to a nonhomogenous system was virtually +intractable, which Laplace transforms helped significantly with, +especially because they directly accounted for the zero initial +current/voltage (even if there were any, these effects would die out +quickly after the initialization of the source current). + +Outside of this course, this work is likely insufficiently general to +provide any real benefit, but the principles discovered, specifically +of the gain decreasing with increasing frequencies, are highly +applicable to real-life electric systems because if they have resistors +or capacitors, they will eventually fall into a similar state if fed +by alternating current or direct current as the source. Similar +principles {\it may} be used in the design of a computational system to +discover how these effects work on a real circuit, which could have been +incorporated into modern circuit design software. + \bye |