401 lines
14 KiB
TeX
401 lines
14 KiB
TeX
\documentclass[10pt]{article}
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\usepackage[english]{babel}
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% \usepackage[a4paper,top=2cm,bottom=2cm,left=2cm,right=2cm,marginparwidth=1.75cm]{geometry}
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\usepackage[a4paper, total={7in, 10in}]{geometry}
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\usepackage{multicol}
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\usepackage{lipsum}
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\usepackage{caption}
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\usepackage{graphicx}
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\usepackage{enumitem}
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\usepackage{listings}
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\usepackage{xcolor}
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\newenvironment{Figure}
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{\par\medskip\noindent\minipage{\linewidth}}
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{\endminipage\par\medskip}
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\lstdefinestyle{mystyle}{
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backgroundcolor=\color{backcolour},
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commentstyle=\color{codegreen},
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keywordstyle=\color{magenta},
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numberstyle=\tiny\color{codegray},
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stringstyle=\color{codepurple},
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basicstyle=\ttfamily\footnotesize,
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breakatwhitespace=false,
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captionpos=b,
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numbersep=5pt,
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showspaces=false,
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showstringspaces=false,
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showtabs=false,
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tabsize=2
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}
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\lstset{style=mystyle}
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\graphicspath{{images/}}
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\title{B00st converter}
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\author{
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van Iterson, Arne\\
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Student Number: 1800000
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\and
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Selier, Tom\\
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Student Number: 1808444
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}
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\begin{document}
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\maketitle
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\begin{multicols}{2}
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\section{Introduction}
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\lipsum[1-2]
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\section{Circuit Description}
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% Filler image, don't get attached
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\begin{Figure}
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\centering
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\includegraphics[scale=0.38]{SCHEMATIC_FULL.png}
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\captionof{figure}{WIP}
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\label{fig:schematic_full}
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\end{Figure}
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\lipsum[3-4]
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\section{Methodology} \label{section:methodology}
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To characterize the system, several tests have been performed. The
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characteristics of interest are the following:
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\begin{enumerate}[nosep]
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\item Efficiency
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\item Noise
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\item Ripple characteristics
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\item Start up
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\end{enumerate}
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In this section a test or measurement will be described for each of the above
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characteristics.
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Each of the characteristics have been tested at two different output voltages
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and various load currents. The different voltages are $7V$ and $3.3V$. The
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chosen load currents are $10$, $20$, $30$, $40$ and $50 mA$. These values
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were chosen to characterize the circuit over a broad range of conditions.
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For all tests, the data was handled in a simular way. For test 2 through 4,
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an oscilloscope was set up on the output voltage. The probe was set to
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10x attenuation to minimize it's influence on the circuit. The oscilloscope's
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settings are set to get the signal full screen, and the acquire settings were
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adjusted so that it would store 20,000 points. Then, at each measurement
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.csv (comma seperated values) was stored on a USB drive.
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After the measurements were collected, they were processed using a python
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script. The major functions are listed at each test.
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\subsection{Efficiency} \label{section:efficiency}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.34]{SCHEMATIC_EFFICIENCY.png}
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\captionof{figure}{WIP}
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\label{fig:schematic_efficiency}
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\end{Figure}
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To measure the efficiency of the circuit, four measurements were taken.
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A current and a voltage measurement were taken at the supply and load,
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respectively. The measurements were taken as shown in figure
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\ref{fig:schematic_efficiency}. The energy used by the supply and the load
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can be calculated using equation \ref{eq:power}. Then, using equation
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\ref{eq:efficiency}, efficiency can be calculated.
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\begin{equation}
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\label{eq:power}
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P [W] = U[V] \cdot I[A]
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\end{equation}
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\begin{equation}
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\label{eq:efficiency}
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\eta[\%] = \frac{P_{load}[W]}{P_{supply}[W]} \cdot 100\%
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\end{equation}
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\subsection{Noise}
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To measure the noice of the circuit an oscilloscope probe was placed on the
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variable resistor in figure \ref{fig:schematic_full}. Over the period of 1
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millisecond, 20,000 points were measured.
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Noise has several metrics in which it can be quantized. Two metrics were
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calculated, the standard devation (SD) and the peak to peak noise.
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\subsubsection{Peak to peak}\label{section:peak_to_peak}
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Peak to peak is the simplest way to look at noise. The signal has a stationary
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mean over the period of 1 millisecond. Thus, the highest measured value can be
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subtracted from the lowest measured value.
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\begin{lstlisting}[language=Python, caption={Peak to peak function}]
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def PK_PK(all_data, ch):
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output_pk = []
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for data in all_data:
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maximum = max(data[ch + 1])
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minimum = min(data[ch + 1])
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pk_pk = maximum-minimum
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output_pk.append(pk_pk)
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return output_pk
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\end{lstlisting}
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\subsubsection{Standard Deviation}\label{section:standard_devation}
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The second metric used to measure noise was the standard deviation (SD).
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Unlike peak to peak, it gives a better impression of the average noise over
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a longer period. SD can be calculated using equation \ref{eq:sd}.
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\begin{equation}
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\label{eq:sd}
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\sigma = \sqrt{\frac{1}{N}\sum^{N-1}_{i=0}(x[i] - \mu)^2}
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\end{equation}
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\noindent Where $x[i]$ is each voltage measurement, $\mu$ is the mean of the
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signal and $N$ is the total amount of samples.
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\begin{lstlisting}[language=Python, caption={SD function}]
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def SD(all_data, ch):
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output_SD = []
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for data in all_data:
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N = len(data[ch + 1])
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MU = sum(data[ch + 1])/N
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x = 0
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for val in data[ch + 1]:
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x += (val-MU)**2
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SD = sqrt((1/N) * x)
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output_SD.append(SD)
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return [output_load, output_SD]
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\end{lstlisting}
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\subsection{Ripple characteristics} \label{section:ripple}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{RIPPLE.png}
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\captionof{figure}{WIP}
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\label{fig:ripple}
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\end{Figure}
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A significant source of the noise was caused by a specific ripple, shown in
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figure \ref{fig:ripple}.
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This ripple coincided with the MOSFETs opening or closing.
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To further characterize this behaviour a close up measurement was taken.
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The oscilloscope was set to AC-coupling and the settigns were adjusted
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for the ripple to be full screen. Then, two additional characteristics can
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be calculated. The ripple's peak to peak voltage and the ripple's (most prevalent)
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frequency. The peak to peak value can be calculated using the method described in
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section \ref{section:peak_to_peak}.
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To measure the frequency of the signal using an FFT, it had to be pre-processed
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first using a Hamming window, this eliminates sharp edges at the edge of the
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measurement, causing unwanted frequencies to appear in the frequency domain.
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\begin{equation}
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\label{eq:hamming}
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% 0.54 - 0.46 * cos(2*np.pi*(n/N))
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w(i) = 0.54 - 0.46 \cdot cos \left(2 \pi \frac{i}{N} \right)
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\end{equation}
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Where $i$ is the current sample and $N$ is the total amount of samples. Each
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sample in the signal can be multiplied by the corresponding value in the window,
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preparing the signal for the FFT.
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\begin{lstlisting}[language=python, caption={Hamming function}]
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def window(y):
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N = len(y)
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hamming = []
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for n in range(N):
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hamming.append(
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0.54 - 0.46 * cos(2*np.pi*(n/N)))
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y = [hamming[i]*x for i, x in enumerate(y)]
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return y
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\end{lstlisting}
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\begin{lstlisting}[language=python, caption={FFT function}]
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def Freq(all_data, ch):
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output_freq = []
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for data in all_data:
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# FFT
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# using realfft,
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# because the signal only has real parts
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y = np.fft.rfft(data[ch + 1])
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# Window
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y = window(y)
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# Calculate the bins
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dt = data[1][1] - data[1][0]
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x = np.fft.rfftfreq(len(data[ch+1]), dt)
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# find maximum, max() and np.argmax()
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# are not playing nice with
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# imaginary numbers
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maximum = 0
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max_idx = 0
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offset = 1 # Skip the first bin, DC offset
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for idx, val in enumerate(y[offset:]):
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mag = sqrt(val.real**2 + val.imag**2)
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if mag > maximum:
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maximum = mag
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max_idx = idx + offset
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# get the frequency of the maximum
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output_freq.append(x[max_idx])
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return output_freq
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\end{lstlisting}
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\subsection{Start up} \label{section:start_up}
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The last characteristics is the start up, specifically the different rise times
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under load. The voltage was measured at the output as the supply was turned on.
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Different rise times can be defined. First off, $\tau$ and $2 \tau$ were
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defined as $63\%$ and $95\%$ respectively. Further more, 'rise time' was defined
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as $90\%$, a metric used often in control theory.
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One problem that occured during the measurements, is that the aforementioned
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ripples and noise would cause erroneous readings. As such, the signal was
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filtered using a low pass filter, reducing the high frequencies from the
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signal.
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\begin{lstlisting}[language=python, caption={LPF snippet}]
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initial = data[3][0]
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dt = data[1][1] - data[1][0]
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x_filter = [initial]
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for i in range(1, len(data[3])):
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x_dot_filter = \
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(data[3][i] - x_filter[i-1])/0.0002
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x_filter.append(
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x_filter[i-1] + x_dot_filter*dt)
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\end{lstlisting}
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\section{Results}
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In this section the results from section \ref{section:methodology} will be
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discussed, as well as discuss some probable causes for unknown or unintended
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results.
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\subsection{Efficiency}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{EFFICIENCY_PERCENTAGE.jpg}
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\captionof{figure}{WIP}
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\label{fig:efficiency}
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\end{Figure}
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\noindent The results for the efficiency measurements, as described in section
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\ref{section:efficiency} are displayed in figure \ref{fig:efficiency}.
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The $7V$ measurements follow a predictable curve, however, the $3.3V$ makes
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an unexplained jump back to a higher percentage.
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\subsection{Noise} \label{section:result_noise}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{SNR_LOADVSPKPK.jpg}
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\captionof{figure}{WIP}
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\label{fig:noise_pkpk}
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\end{Figure}
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\noindent The results for the noise measurements, as described in section
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\ref{section:peak_to_peak} are displayed in figure \ref{fig:noise_pkpk}.
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The peak to peak voltage is a significant fraction of the output voltage,
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with $3V$ peaking at $33\%$. It seems there is a relation between peak to
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peak voltage and the output voltage as well, as $7V$ has more noise than
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$3.3V$
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{SNR_LOADVSSD.jpg}
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\captionof{figure}{WIP}
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\label{fig:noise_sd}
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\end{Figure}
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\noindent The results for the noise measurements, as described in section
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\ref{section:standard_devation} are displayed in figure \ref{fig:noise_sd}.
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Although the voltage peaks are high, the standard deviation of the noise is
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in the range of millivolts. The trend that a higher output voltage has more
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noise is continued in this graph.
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\subsection{Ripple}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{RIPPLE_LOADVSPKPK.jpg}
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\captionof{figure}{WIP}
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\label{fig:ripple_pkpk}
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\end{Figure}
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\noindent The results for the ripple measurements, as described in section
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\ref{section:ripple} are displayed in figure \ref{fig:ripple_pkpk}. The
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voltage level in the graph seems to confirm that the peak to peak noise,
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seen in section \ref{section:result_noise} is caused by the ripple.
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{RIPPLE_LOADVSFREQ.jpg}
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\captionof{figure}{WIP}
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\label{fig:ripple_freq}
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\end{Figure}
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\noindent The frequency of the ripple is roughly $38 MHz$ and independant of
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the load. To figure out if this ripple is caused by the combination of the
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inductor and the capactitor the following equation can be used.
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\begin{equation}
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f = \frac{1}{2 \pi \sqrt{LC}}
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\end{equation}
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Using the values from figure \ref{fig:schematic_full}, the resonating frequency
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of the circuit should be around $27KHz$. Thus, this cannot be the cause of
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the high frequency. As the frequency of the ripple is magnitudes higher
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than the LC-circuit's resonant frequency, what is seen is most likely the
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Self Resonating Frequency (SRF) of the inductor. Typically the SRF is
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$>10 MHz$, so that could be a probable source of the high frequencies.
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\subsection{Start Up}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{TRANSIENT_RISE_10_MA.jpg}
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\captionof{figure}{WIP}
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\label{fig:start_10}
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\end{Figure}
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\begin{Figure}
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\centering
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\includegraphics[scale=0.5]{TRANSIENT_RISE_50_MA.jpg}
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\captionof{figure}{WIP}
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\label{fig:start_50}
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\end{Figure}
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\begin{center}
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\captionof{table}{$10 mA$}
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\label{table:start_10}
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\begin{tabular}{llll}
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Metric & $\tau$ & $2\tau$ & Rise time \\
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Percentage [$\%$] & 63 & 95 & 90 \\
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Time [$s$] & 0.031 & 0.075 & 0.053 \\
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\end{tabular}
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\end{center}
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\begin{center}
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\captionof{table}{$50 mA$}
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\label{table:start_50}
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\begin{tabular}{llll}
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Metric & $\tau$ & $2\tau$ & Rise time \\
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Percentage [$\%$] & 63 & 95 & 90 \\
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Time [$s$] & 0.033 & 0.048 & 0.043 \\
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\end{tabular}
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\end{center}
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\noindent The results for the start up measurements, as described in section
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\ref{section:start_up} are displayed in figure \ref{fig:start_10} and
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\ref{fig:start_50}, and table \ref{table:start_10} and \ref{table:start_50}.
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Counterintuitively, the rise time is shorter with a higher load.
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\section{Conclusion}
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\lipsum[3-4]
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\end{multicols}
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\end{document} |