\documentclass[10pt]{article}
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\usepackage{lipsum}
\usepackage{caption}
\usepackage{graphicx}
\usepackage{enumitem}

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  {\par\medskip\noindent\minipage{\linewidth}}
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\title{B00st converter}

\author{
    van Iterson, Arne\\
    Student Number: 1800000
    \and
    Selier, Tom\\
    Student Number: 1808444
  }

\begin{document}
\maketitle
\begin{multicols}{2}
  \section{Introduction}
    \lipsum[1-2]


  \section{Circuit Description}
    % Filler image, don't get attached
    \begin{Figure}
      \centering
      \includegraphics[scale=0.38]{SCHEMATIC_FULL.png}
      \captionof{figure}{WIP}
      \label{fig:schematic_full}
    \end{Figure}
    \lipsum[3-4]


  \section{Methodology}
    To characterize the system, several tests have been performed. The 
    characteristics of interest are the following:
    \begin{enumerate}[nosep]
        \item Efficiency
        \item Noise
        \item Ripple characteristics
        \item Transients
    \end{enumerate}
    In this section a test or measurement will be described for each of the above 
    characteristics.

    Each of the characteristics have been tested at two different output voltages
    and various load currents. The different voltages are $7V$ and $3.3V$. The 
    chosen load currents are $10$, $20$, $30$, $40$ and $50 mA$. These values 
    were chosen to give characterize the circuit over a broad range of conditions.


  \subsection{Efficiency}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.34]{SCHEMATIC_EFFICIENCY.png}
      \captionof{figure}{WIP}
      \label{fig:schematic_efficiency}
    \end{Figure}
    To measure the efficiency of the circuit, four measurements were taken.
    A current and a voltage measurement were taken at the supply and load
    respectively. The measurements were taken as shown in figure 
    \ref{fig:schematic_efficiency}. The energy used by the supply and the load 
    can be calculated using the equation \ref{eq:power}. Then, using equation
    \ref{eq:efficiency}, efficiency can be calculated.
    \begin{equation}
      \label{eq:power}
      P [W] = U[V] \cdot I[A]
    \end{equation} 
    \begin{equation}
      \label{eq:efficiency}
      \eta[\%] = \frac{P_{load}[W]}{P_{supply}[W]} \cdot 100\%
    \end{equation}
  

  \subsection{Noise}
    To measure the noice of the circuit an oscilloscope probe was placed on the
    variable resistor in figure \ref{fig:schematic_full}. Over the period of 1 
    millisecond, 20,000 points were measured. 

    Noise has several metrics in which it can be quantized. Two metrics were
    calculated, the standard devation (SD) and the peak to peak noise. 


  \subsubsection{Peak to peak}\label{section:peak_to_peak}
    Peak to peak is the simplest way to look at noise. The signal has a stationary
    mean over the period of 1 millisecond. Thus the highest measured value can be
    subtracted from the lowest measured value.


  \subsubsection{Standard Deviation}
    The second metric used to measure noise was the standard deviation. 
    Unlike, peak to peak it givesa better impression of the noise over a longer 
    signal. SD can be calculated using equation \ref{eq:sd}.

    \begin{equation}
      \label{eq:sd}
      \sigma = \sqrt{\frac{1}{N}\sum^{N-1}_{i=0}(x[i] - \mu)^2}
    \end{equation}

    Where $x[i]$ is each voltage measurement, $\mu$ is the mean of the signal and
    $N$ is the total amount of samples.

    \subsection{Ripple characteristics}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{RIPPLE.png}
      \captionof{figure}{WIP}
      \label{fig:ripple}
    \end{Figure}
    A significant source of the noise was caused by a specific ripple, shown in
    figure \ref{fig:ripple}. 
    This ripple coincided with the MOSFETs opening or closing.
    
    To further characterize this behaviour a close up measurement was taken.
    The oscilloscope was set to AC-coupling and the settigns were adjusted
    for the ripple to be full screen. Then, two additional characteristics can 
    be calculated. The ripple's peak to peak voltage and the ripple's (most prevalent) 
    frequency. The peak to peak value can be calculated using the method described in 
    section \ref{section:peak_to_peak}.

    To measure the frequency of the signal using an FFT, it had to be pre-processed 
    first using a Hamming window this eliminates sharp edges at the edge of the 
    measurement, causing unwanted frequencies to appear in the frequency domain. 
    \begin{equation} 
      \label{eq:hamming}
      % 0.54 - 0.46 * cos(2*np.pi*(n/N))
      w(i) = 0.54 - 0.46 \cdot cos \left(2 \pi \frac{i}{N} \right)
    \end{equation}
    Where $i$ is the current sample and $N$ is the total amount of samples. Each
    sample in the signal can be multiplied by the corresponding value in the window,
    preparing the signal for the FFT.

    \subsection{Transients}
    The last measurements were hocus pocus

  \section{Results}
  \lipsum[1-2]


  \section{Conclusion}
  \lipsum[3-4]
\end{multicols}
\end{document}