added lots of words about "external relation"HEADmaster

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