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#+TITLE: Building an RC Circuit
#+AUTHOR: Lizzy Hunt
#+STARTUP: entitiespretty fold inlineimages
#+LATEX_HEADER: \usepackage{ dsfont } \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
#+LATEX:
#+OPTIONS: toc:nil
* The Experiment
The purpose of this experiment was to gain a better understanding of the effects on the voltage over a capacitor
as a time-valued function when put in a circuit in series with a resistor. To achieve
this goal - and to experimentally verify laws governing the total resistance and capacitance of
configurations of resistors and capacitors using a multimeter - we built an "RC circuit" by combining
resistors and capacitors from respective smaller-valued components.
Additionally, this lab fulfilled the requirement allowing us to play with dangerously high temperature metal equipment \smiley.
** Theory
Here we list some ideas the reader should be familiar with for reference later in the report.
*** Resistors
Consider $n$ resistors, $r_i \ni i \in [1, n]$ representing the total resistance of the i^{th} resistor,
or sub-configuration of resistors, all in parallel. Then, the total resistance, R, of the group is:
\begin{equation}
R^{-1} = \sum_{i=1}^{n}(r_i)^{-1}
\end{equation}
Consider $n$ resistors, $r_i \ni i \in [0, n]$ representing the total resistance of the i^{th} resistor,
or sub-configuration of resistors, all in series. Then, the total resistance, R, of the group is:
\begin{equation}
R = \sum_{i=1}^{n}(r_i)
\end{equation}
*** Capacitors
Total capacitance of configurations of capacitors are similar to the inversion of the laws for
resistors.
Consider $n$ capacitors, $c_i \ni i \in [1, n]$ representing the total capacitor of the i^{th} capacitor,
or sub-configuration of capacitors, all in series. Then, the total capacitance, C, of the group is:
\begin{equation}
C^{-1} = \sum_{i=1}^{n }(c_i)^{-1}
\end{equation}
Consider $n$ capacitors, $c_i \ni i \in [1, n]$ representing the total capacitor of the i^{th} capacitor,
or sub-configuration of capacitors, all in parallel. Then, the total capacitance, C, of the group is:
\begin{equation}
C = \sum_{i=1}^{n }(c_i)
\end{equation}
*** The RC Circuit
For a circuit with a resistor of resistance $R$ and capacitor with capacitance $C$ in series,
we can model the voltage over the capacitor, $V_C$, given an initial voltage $V_0$ and final
voltage $V_f$, as a function of time:
\begin{equation}
V_C(t) = (V_0 - V_f)e^{-\frac{t}{RC}} + V_f
\end{equation}
** Procedure
The given procedure to exercise our knowledge of equations (1) - (4) if to build both a relatively higher-valued
resistor, and capacitor, out of smaller-valued components - by soldering them in series / parallel configurations.
Each pair of students is to produce a resistor and capacitor at a target value (and with a 10% margin for error),
determined by seating arrangement. By happenstance, our group was chosen to build:
1. A 22 kilo-ohm resistor (22 $k \Omega$)
2. A 1.67 micro-farad capacitor ($\mu F$)
out of only 10 $k \Omega$ resistors, and 1 $\mu F$ capacitors.
I did not record the configuration we used for either. So, assume the following configurations throughout the rest of the lab (pretty sure
these were pretty close to our monstrosities):
*** Building a Resistor
Assume all resistors as $10 k \Omega$
#+attr_latex: :width 240px
[[./resistors.png]]
In theory, the total resistance measured from the leftmost point to the rightmost is 22 $k \Omega$:
\begin{align*}
R &= 10^4 \text{ (leftmost resistor in series (2))} \\
&+ 10^4 \text{ (second leftmost resistor in series (2))} \\
&+ (\frac{5}{10^4})^{-1} \text{ (5 resistors in parallel (1))} \\
&= 2.20 * 10^4 \Omega
\end{align*}
*** Building a Capacitor
#+attr_latex: :width 200px
[[./capacitors.png]]
In theory, the total capacitance measured from the leftmost point to the rightmost is 1.67 $\mu F$:
\begin{align*}
C &= 2(\frac{3}{10^-6})^{-1} \text{ (two groups of 3 1-}\mu F \text{ capacitors in series (3) in parallel with) } \\
&+ 10^{-6} \text{ (another 1-} \mu F \text{ capacitor (4)) } \\
&\approx 1.67 * 10^{-6} F
\end{align*}
*** Determining the $RC$ constant
To measure our $RC$ constant, we connected two voltage probes over $V_c$ (as shown in the diagram below) to a computer-generated
positive square wave oscillating at 0.50 Hz with an amplitude of 5V. We then record for 1.5 seconds, polling at 1 kHz, from
the time $V_C$ is at 4.95 V (the capacitor has charged) - allowing us to record at least half a second of discharge
from the capacitor.
#+attr_latex: :width 200px
[[./total_circuit.png]]
We expect to see that as it discharges, the measured voltage over the capacitor would follow an exponentially decreasing fit,
according to the $-\frac{t}{RC}$ term in (5). To find the value of $RC$ we measure the voltage at each discrete time step ($\frac{1}{1000}$ of a second)
from near the beginning of the exponential drop to where it reaches stability, and copy those values into a Magic Excel Sheet^{TM}. This
region is somewhat shown in the figure below (some values are actually truncated):
#+attr_latex: :width 200px
[[./rc-discharge.png]]
The Magic Excel Sheet^{TM} produces a good exponential fit to this data. But, it takes some manual fiddling with the $RC$ value itself
to determine the minimum sum of residuals (gradient descent inspired guess and check). The value of $RC$
producing the lowest error by this measure, is our result.
* Results
** Building a Resistor
The measured resistance (via multimeter) we obtained from our resistor was $21.68 k \Omega$.
** Building a Capacitor
The measured capacitance (via multimeter) we obtained from our resistor was $1.78 \mu F$.
** The Value Of $RC$
For our computer determined RC constant, we found it to be $3.72 * 10^{-2}$ s.
* Discussion
** Building a Resistor
Our target value was $22.00 k \Omega$, and we came out with $21.68 k \Omega$ - an error of 1.45%.
** Building a Capacitor
Our target value was $1.67 \mu F$, and we came out with $1.78 \mu F$ - an error of 6.59%.
** The Value of $RC$
If our resistor and capacitor were exactly on the target value, our $RC$ constant would be $(2.20 * 10^4 \Omega)(1.67 * 10^{-6} F) = 3.67 * 10^{-2}$ s.
The $RC$ constant from the measured resistance and capacitance would be $(2.17 * 10^4 \Omega)(1.78 * 10^{-6} F) = 3.86 * 10^-2$ s.
But, our human-gradient-descent-plus-excel-magic-thanks-computer told us it was $3.72 * 10^{-2}$ s - a 3.62% error from the
theoretical measured value, and 1.36% from the overall "target" value.
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