this is for holding javascript data
Lucy Liang added To_analyze_this_data_we__1.tex
over 8 years ago
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To analyze this data, we made a plot of resistance (ohms) versus $V^2$ so that we could analyze the slope. The equation used to understand why we did this is shown in Equation \ref{eq:Equatoin} ($V_{\textrm{mean square ac voltage }} = = (4 R \Delta f) k_B T$).
where $k_B$ is the Boltzmann Constant we are looking for, $T$ is the temperature in Kelvin, $R$ is the resistance in ohms, and $\Delta f$ is the ``equivalent noise bandwidth'' (ENBW) that we varied by changing the values on the low and high pass filters. In order to find the $k_B$, we will be plotting the data in order to find the slope is that it is proportional to $k_B$ or, if you prefer, you can create a plot where the slope is equal to $k_B T \Delta f$, than solve for $k_B$. Personally, we found it easier to plot the data so that the slope was proportional to $k_B$.
In order to find the bandwidth, we had to use values that came from the Noise Fundamentals Test Data (\ref{table:Johnson2}), which gave us the measured values from the filters. You can see the values we used in \ref{table:Johnson2}, and as you can see, the measured values are different from the nominal values. After using the measured values from the Noise Fundamentals test data, we had to calculate the Equivolent Noise Bandwidth (ENBW) in order to find $\Delta f$. To do this, we used an equation:
\begin{equation}
\label{eq:ENBW}
ENBW = \left(\frac{\pi}{2} \ Q\right) f_c
\end{equation}
Where $f_c$ is the High Frequency filter values ($f_2$) and Low Frequency filter values ($f_1$) subtracted shown in \ref{table:True_Bandwidth1} and \ref{table:True_Bandwidth2}. The equation for finding $f_c$ is: $f_c = f_2 - f_1$
