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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} \label{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 Start up
    \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} \label{section: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.

\begin{lstlisting}[language=Python, caption={Peak to peak function}]
def PK_PK(all_data, ch):
  output_pk = []
  for data in all_data:
      maximum = max(data[ch + 1])
      minimum = min(data[ch + 1])
      pk_pk = maximum-minimum
      output_pk.append(pk_pk)
  return output_pk
\end{lstlisting}


  \subsubsection{Standard Deviation}\label{section:standard_devation}
    The second metric used to measure noise was the standard deviation (SD). 
    Unlike peak to peak, it gives a better impression of the average noise over 
    a longer period. 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}

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

\begin{lstlisting}[language=Python, caption={SD function}]
def SD(all_data, ch):
  output_SD = []
  for data in all_data:
      N = len(data[ch + 1])
      MU = sum(data[ch + 1])/N
      x = 0
      for val in data[ch + 1]:
          x += (val-MU)**2
      SD = sqrt((1/N) * x)
      output_SD.append(SD)
  return [output_load, output_SD]
\end{lstlisting}


  \subsection{Ripple characteristics} \label{section:ripple}
    \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.

\begin{lstlisting}[language=python, caption={Hamming function}]
def window(y):
  N = len(y)
  hamming = []
  for n in range(N):
      hamming.append(
        0.54 - 0.46 * cos(2*np.pi*(n/N)))
  y = [hamming[i]*x for i, x in enumerate(y)]
  return y
\end{lstlisting}

\begin{lstlisting}[language=python, caption={FFT function}]
def Freq(all_data, ch):
output_freq = []
for data in all_data:
    # FFT
    # using realfft, 
    # because the signal only has real parts 
    y = np.fft.rfft(data[ch + 1])

    # Window
    y = window(y)

    # Calculate the bins
    dt = data[1][1] - data[1][0]
    x = np.fft.rfftfreq(len(data[ch+1]), dt)
    
    # find maximum, max() and np.argmax() 
    # are not playing nice with 
    # imaginary numbers
    maximum = 0
    max_idx = 0
    offset = 1  # Skip the first bin, DC offset
    for idx, val in enumerate(y[offset:]):
        mag = sqrt(val.real**2 + val.imag**2)
        if mag > maximum:
            maximum = mag
            max_idx = idx + offset
    
    # get the frequency of the maximum
    output_freq.append(x[max_idx])

return [output_load, output_freq]
\end{lstlisting}

  \subsection{Start up} \label{section:start_up}
    The last characteristics is the start up, specifically the different rise times
    under load. The voltage was measured at the output as the supply was turned on.

    Different rise times can be defined. First off, $\tau$ and $2 \tau$ were
    defined as $63\%$ and $95\%$ respectively. Further more, 'rise time' was defined
    as $90\%$, a metric used often in control theory. 

    One problem that occured during the measurements, is that the aforementioned 
    ripples and noise would cause erroneous readings. As such, the signal was
    filtered using a low pass filter, reducing the high frequencies from the 
    measurement. 

\begin{lstlisting}[language=python, caption={LPF snippet}]
initial = data[3][0]
dt = data[1][1] - data[1][0]

x_filter = [initial]
for i in range(1, len(data[3])):
    x_dot_filter = \
      (data[3][i] - x_filter[i-1])/0.0002
    x_filter.append(
      x_filter[i-1] + x_dot_filter*dt)
\end{lstlisting}


  \section{Results}
    In this section the results from section \ref{section:methodology} will be 
    discussed, as well as discuss some probable causes for unknown or unintended 
    results.


  \subsection{Efficiency}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{EFFICIENCY_PERCENTAGE.jpg}
      \captionof{figure}{WIP}
      \label{fig:efficiency}
    \end{Figure}
    \noindent The results for the efficiency measurements, as described in section 
    \ref{section:efficiency} are displayed in figure \ref{fig:efficiency}. 
    The $7V$ measurements follow a predictable curve, however, the $3.3V$ makes
    an unexplained jump back to a higher percentage.


  \subsection{Noise} \label{section:result_noise}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{SNR_LOADVSPKPK.jpg}
      \captionof{figure}{WIP}
      \label{fig:noise_pkpk}
    \end{Figure}
    \noindent The results for the noise measurements, as described in section 
    \ref{section:peak_to_peak} are displayed in figure \ref{fig:noise_pkpk}.
    The peak to peak voltage is a significant fraction of the output voltage,
    with $3V$ peaking at $33\%$. It seems there is a relation between peak to
    peak voltage and the output voltage as well, as $7V$ has more noise than
    $3.3V$
    
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{SNR_LOADVSSD.jpg}
      \captionof{figure}{WIP}
      \label{fig:noise_sd}
    \end{Figure}
    \noindent The results for the noise measurements, as described in section 
    \ref{section:standard_devation} are displayed in figure \ref{fig:noise_sd}.
    Although the voltage peaks are high, the standard deviation of the noise is 
    in the range of millivolts. The trend that a higher output voltage has more 
    noise is continued in this graph.


  \subsection{Ripple}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{RIPPLE_LOADVSPKPK.jpg}
      \captionof{figure}{WIP}
      \label{fig:ripple_pkpk}
    \end{Figure}
    \noindent The results for the ripple measurements, as described in section 
    \ref{section:ripple} are displayed in figure \ref{fig:ripple_pkpk}. The 
    voltage level in the graph  seems to confirm that the peak to peak noise, 
    seen in section \ref{section:result_noise} is caused by the ripple. 

    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{RIPPLE_LOADVSFREQ.jpg}
      \captionof{figure}{WIP}
      \label{fig:ripple_freq}
    \end{Figure}
    \noindent The frequency of the ripple is roughly $38 MHz$ and independant of
    the load. To figure out if this ripple is caused by the combination of the 
    inductor and the capactitor the following equation can be used.
    \begin{equation}
      f = \frac{1}{2 \pi \sqrt{LC}}
    \end{equation}
    Using the values from figure \ref{fig:schematic_full}, the resonating frequency
    of the circuit should be around $27KHz$. Thus, this cannot be the cause of
    the high frequency. As the frequency of the ripple is magnitudes higher
    than the LC-circuit's resonant frequency, what is seen is most likely the
    Self Resonating Frequency (SRF) of the inductor. Typically the SRF is
    $>10 MHz$, so that could be a probable source of the high frequencies.
    
    \subsection{Start Up}
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{TRANSIENT_RISE_10_MA.jpg}
      \captionof{figure}{WIP}
      \label{fig:start_10}
    \end{Figure}
    
    \begin{Figure}
      \centering
      \includegraphics[scale=0.5]{TRANSIENT_RISE_50_MA.jpg}
      \captionof{figure}{WIP}
      \label{fig:start_50}
    \end{Figure}

    \begin{center}
    \captionof{table}{$10 mA$}
    \label{table:start_10}
    \begin{tabular}{llll}
    Metric  & $\tau$ & $2\tau$ & Rise time \\
    Percentage [$\%$] & 63 & 95 & 90 \\
    Time [$s$] & 0.031 & 0.075 & 0.053 \\
    \end{tabular}
    \end{center}
    
    \begin{center} 
    \captionof{table}{$50 mA$}
    \label{table:start_50}
    \begin{tabular}{llll}
    Metric  & $\tau$ & $2\tau$ & Rise time \\
    Percentage [$\%$] & 63 & 95 & 90 \\
    Time [$s$] & 0.033 & 0.048 & 0.043 \\
    \end{tabular}
    \end{center}
    
    \noindent The results for the start up  measurements, as described in section 
    \ref{section:start_up} are displayed in figure \ref{fig:start_10} and 
    \ref{fig:start_50}, and table \ref{table:start_10} and \ref{table:start_50}.

    Counterintuitively, the rise time is shorter with a higher load. 
      

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