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\author{Carlos Gil Soriano\\BE-CO-HT\\
\href{mailto:carlos.gil.soriano@cern.ch}{\textbf{\textit{carlos.gil.soriano@cern.ch}}}}
\title{\textbf{{\LARGE Flyback circuit outputting Standard Blocking for CTDAH}}}
\date{\today}

\begin{document}

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\pagebreak

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\paragraph{System Description and Purpose}
A flyback circuit topology is exposed as a feasible solution for the Pulse
Converter Unit. The document shows the theoretical analysis, simulation and
measurements done of a flyback circuit to be employed in the Pulse Converter
Unit.
The present solution offered takes a starting point from Level Conver board by
C. Dehavay \cite{ClaudeDoc} and improves it so as to comply with
Standard Blocking requirements \cite{StandardBlocking}.
\pagebreak

{\Large \textbf{History of changes}}\\

This document version has been checked by:\\

This document version has been approved by:\\
\begin{center}
\begin{tabular}{|p{3.5cm}|p{2.5cm}|p{5cm}|}
\hline
\textbf{Date} & \textbf{Pages} & \textbf{Changes}\\ 
\hline
\hline
September 16, 2011 & All & Initial submission\\
\hline
September 21, 2011 & All & Revised to comply with Standard
Blocking Definition\cite{StandardBlocking}\\
\hline
&&\\
\hline
\end{tabular}
\end{center}

\pagebreak

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\pagebreak

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\section{Introduction}
A flyback circuit configuration consist on a Power MOSFET that can open or
shortcuit one pole of the primary winding of a transformer to ground. By doing
this, the primary inductance of the transformer experiences a voltage virtually
copied from the voltage supply it has in the other pole of the primary
winding.\\

When the Power MOSFET is conducting, the secondary winding of the transformer
outputs the voltage to the load. Thus, the load forces a current
along each of the windings of the transformer. Apart from this current, a magnetizing
current is present in the primary winding. The magnetizing current ramps up with
a factor scale dependent on the primary inductance and the voltage along it.
Depending on the R$_{ON}$ value of the Power MOSFET, the voltage along the
primary inductance can experience a small tilt, yielding that the
magnetizing current will not built up linearly. We will assume now on that
R$_{ON}$ effect is negligible on the magnetizing current ramp --then, it will
increase linearly.\\

Attention should be paid to the saturation current of the
transformer to avoid irreversible damage. Because of this, it will be clearly
stated in this document the maximum pulse width it can be outputted and the time
it will take to the recovery phase to drain enough magnetizing current.\\

\begin{figure}[htb]
    \begin{center}
      \includegraphics[scale=0.25, keepaspectratio]{IntroFlyback}
      \caption{Basic Flyback circuit}
    \end{center}
\end{figure}

Furthermore, the Power MOSFET should comply with power requirements as well as
a slow rise and fall times to avoid reflections. This can be easily achived for
both rise and fall edges with an appropiate triggering subcircuit and an addition of a
parallel diode with a resistor, serially connected to the ground gate of the
Power MOSFET.
\pagebreak
\section{Theoretical analysis}
\subsection{On phase}
During the on phase, the Power MOSFET is conducting and the transformer
experience a voltage near to $V_P$. The equivalent circuit is:\\
\begin{figure}[htb]
    \begin{center}
      \includegraphics[scale=0.25, keepaspectratio]{ONstate.png}
      \caption{Equivalent ON state circuit}
    \end{center}
\end{figure}
\begin{description}
\item \textbf{Kirchoff's equations}\\
The electrical relationships in the circuit are:
\begin{equation}
V_P - V_{R_{on}} = L \frac{di_L(t)}{dt}\\
\end{equation}
\begin{equation}
V_{R_{on}} = R_{on} i_L(t) \nonumber
\end{equation}
\item \textbf{Differential equation}\\
which form a non-homogeneous linear differential
equation:
\begin{equation}
\frac{di_L(t)}{dx}L+i_L(t)R_{on} = V_P
\end{equation}
\\Let's assume that at the beginning of the ON state, the inductor has no
remaining current. The initial conditions are:
$$i_L(0^+) = 0$$
$$i_L(+\infty) = \frac{V_P}{R_{on}} $$
\\and the solution is:
\begin{equation}
i_L(t) = \frac{V_P}{R_{on}}(1-e^{-\frac{t}{L/R_{on}}}) 
\end{equation}
\\It will always be, at the beginning of the ON state, a small
remaining current in the inductance, $I_{L_R}$:
$$i_L(0^+) = I_{L_R} $$
\\A more accurate solution is:\\
\begin{equation}
  i_L(t) = \frac{V_P}{R_{on}}(1-e^{-\frac{t}{L/R_{on}}}) + I_{L_R}
  e^{-\frac{t}{L/R_{on}}}
\end{equation}
\\$I_{L_R}$ can be treated as zero if it is small.
A dangerous situation is met when $I_{L_R}$ reaches a level of saturation:
$$i_L(t_{SAT}) = I_{L_{SAT}} $$
\item \textbf{Results}\\
 The time-to-saturation under no load or \textbf{\textit{maximum pulse width
 under no load}} is:
 $$t_{P_{max}}_{oc} = t_{SAT} = \frac{L}{R_{on}}ln(\frac{ \frac{V_P}{R_{on}} -
 I_{L_R}}{ \frac{V_P}{R_{on}} - I_{SAT} })$$
however, if a load is connected in the second winding of the transformer, a
I$_{R_{load}}$ is added to $i_L(t)$. Then, the time-to-saturation under load
or \textbf{\textit{maximum pulse width under load}} is:
$$ t_{P_{max}}_{loaded} = t_{SAT} = \frac{L}{R_{on}}ln(\frac{ \frac{V_P}{R_{on}} -
 I_{L_R}}{ \frac{V_P}{R_{on}} - (I_{SAT} - I_{load}) })$$
The most restrictive maximum pulse width occurs when there is a load. Thus, the
 \textbf{\textit{maximum pulse width}} corresponds to:
\begin{equation}
t_{P_{max}} = t_{P_{max}}_{loaded}
\end{equation}
\end{description}
\pagebreak
\subsection{Recovery phase}
When the Power MOSFET is open, the equivalent circuit is:\\

\begin{figure}[htb]
    \begin{center}
      \includegraphics[scale=0.25, keepaspectratio]{Snubber.png}
      \caption{Equivalent circuit in Recovery Phase}
    \end{center}
\end{figure}
which has two snubber nets. This snubber has two behaviours when draining the
current off the inductance:
\begin{itemize}
  \item Linear: the two snubber nets are active.
  \item Exponential: the resistive net is the only one active.\\
\end{itemize}

\subsubsection{Linear Decrease}\\
At the beginning of the recovery phase the current of the inductance is, from
the on state:
$$ i_L(0^-) = i_L(t_P)= I_{L_0}$$
where we have changed the temporal reference to the beginning of the recovery
phase for easing notation.\\
\item \textbf{Kirchoff's equations}\\
As the two net are conducting:
\begin{equation}
i_L(t) = I_{L_0} - (\frac{V_{D_Z} + V_{D_1}}{L} + \frac{V_{D_Z}}{R_{sn}}) t
\end{equation}
this stage ends when zenner diode is not conducting, hence the boundary
condition: $$ i_{L_{Lin,end}} = \frac{V_{D_Z}}{R_{sn}} $$ 
\item \textbf{Results}\\
The linear decrease ends at $t_{L_{end}}$ from the beginning of the recovery
phase:\\
\begin{equation}
t_{L_{end}} = (I_{L{_0}} - \frac{V_{D_Z}}{R_{sn}})(\frac{V_{D_Z} + V_{D_1}}{L} +
\frac{V_{D_Z}}{R_{sn}})^{-1}
\end{equation}
\subsubsection{Exponential Decrease}\\
In this case the net formed by the zenner diode, $D_Z$ and $D_1$, is
open. Then, due to the input capacitance of the primary winding of the
transformer, $C_1$, a parallel RLC circuit appears.\\
\item \textbf{Kirchoff's equations}\\
The highest value of $R_{sn}$ to avoid overshoot is:\\
$$ R_c = \frac{1}{2}\sqrt{\frac{L}{C_1}} $$
As it was done in \cite{BlockingDocumentation} thanks to \cite{NormanIEEPaper},
given a targeted remaining current -$I_{L_0}$- in the inductance, a maximum
value for the recovery time can be expressed.

\item \textbf{Results}\\
The \textbf{\textit{maximum recovery time}} from the beginning of the recovery
phase is:
\begin{equation}
t_{recovery} = f(I_{L_0}) < \frac{V_P}{V_{D_Z}}t_P +
\frac{L}{R_{sn}}[ln(\frac{V_P}{R_{sn}I_R})-1] = t_{recovery,max}
\end{equation}

\subsubsection{Off state}
The circuit is considered to enter in the off state when it is able to be
triggered again without risk of damage. This happens when the recovery time
ends, which means that the remaining current flowing through
the magnetizing inductance is less than $I_{L_0}$.
\pagebreak
\subsection{Trigger circuit}
A trigger circuit based on the shown below is used.\\
\begin{figure}[htb]
    \begin{center}
      \includegraphics[scale=0.25, keepaspectratio]{TriggerFast.png}
      \caption{Fast trigger circuit}
    \end{center}
\end{figure}

The voltage of the BJT base with regard to ground in a steady state is:\\
\begin{equation}
 V_{{base, GND}_{steady}} = v_i \frac{R_1}{R_1+R_2} + 5\frac{R_2}{R_1+R_2}
\end{equation}\\
\subsubsection{Valid resistance values}
This circuit is activated when the input signal goes to ground. It must be
satisfied that when the input is high in a steady-state, the transistor is not
conducting. When it is low in steady-state -ground-, the transistor is
conducting.
The following table shows the formal relations and the last column displays the
valid resistance relation for $V_{base,GND} = 0.6V$

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{BJT condition} & \textbf{$V_{base,GND}$} & \textbf{$v_i$} &
\textbf{Circuit parameters}\\
\hline
NO CONDUCTION & $> 5 - V_{EB,cond}$ & $v_{i,H} = 3.3V$ & $R_2 > 1.83 R_1$ \\
\hline
NO CONDUCTION & $> 5 - V_{EB,cond}$ & $v_{i,H} = 2.5V$ & $R_2 > 3.16 R_1$\\
\hline
\hline
CONDUCTION &  $< 5 - V_{EB,cond}$ & GND & R_2 < 7.3R_1\\
\hline
\end{tabular}
\end{center}

By fixing $V_{EB,cond}$ to 0.6V, we can use an input signal of different
voltages if properly selected $R_2$ and $R_1$ :
\begin{center}
\begin{tabular}{|p{2.5cm}|p{5cm}|}
\hline
$\textbf{v_{i,H}}$ & \textbf{Valid resistance ratios} \\
\hline
\hline
3.3V & R_2 \in ( 1.83$R_1$, 7.3$R_1$) \\
\hline
2.5V & R_2 \in ( 3.16$R_1$, 7.3$R_1$) \\
\hline
\end{tabular}
\end{center}

\subsubsection{Transients}
\item \textbf{High to Low}\\
The transient equation of the voltage in the base of the BJT referred to ground
when there is a high to low change is:
\begin{equation}
V_{base,GND}(t) = v_{i,H} \frac{R_1}{R_1+R_2}e^{-\frac{t}{RC}} +
5\frac{R_2}{R_1+R_2}
\end{equation}
where:
$$R = R_1//R_2$$

\item \textbf{Low to High}\\
The transient equation of the voltage in the base of the BJT referred to ground
when there is a low to change is:
\begin{equation}
V_{base,GND}(t) = v_{i,H} (1-\frac{R_1}{R_1+R_2}e^{-\frac{t}{RC}}) +
5\frac{R_2}{R_1+R_2}
\end{equation}
where:
$$R = R_1//R_2$$

\subsubsection{Modified trigger}
Figure 4 showed a common trigger circuit for Power MOSFET. One of the most
important capabilities of the Power MOSFET is its sharp rise and fall edges if
properly trigger. In the case of this design, a sharp rise and fall edge is a
drawback because of reflections. Hence, a solution for effectively slowing down
the rise edge \cite{grantBook} -the fall slope is slow enough- is achieved by
serially adding a resistance in parallel to a diode to the base of the Power MOSFET:

\begin{figure}[htb]
    \begin{center}
      \includegraphics[scale=0.25, keepaspectratio]{TriggerSlow.png}
      \caption{Slowed rising edge trigger circuit}
    \end{center}
\end{figure}

An analysis of the switching features are explained in the next section.

\pagebreak
\subsection{Switching}
The switching analysis of a Power MOSFET is divided in four stages. Due to the
excesive complication of the switching behaviour, an in-depth demonstration of
these stages should be directly consulted in \cite{grantBook}.
The key parameter of the switching phase is the $R_{slow}$ resistance value
which modifies the slope of the rising edge. Experimental measures are taken to
select the value that better suit to produce a Standard
Blocking compliant \cite{StandardBlocking} signal when a 50 $\Omega$ load is
connected to the second winding of the transformer.

\pagebreak
\section{Circuit proposed}
\subsection{Schematics}
A schematic of the circuit used for the testing board can be found attached at
the end of this document.
\pagebreak
\section{Improvements over Level Converter board}
\subsection{Snubber Circuit}
\begin{description}
\item[Previous] The snubber circuit is a 1K$\Omega$ resistance in
parallel to the second winding. This kind of snubber recovers slow and
dissipates considerable power at the beginning, increasing the
duration of the recovery phase and being candidate of damage,
respectively. In case the resistor breaks, it would burnt the transformer is
more pulses are received --because there is only a residual resistance for 
eliminating the remaining current.
\item[Current] The snubber circuit is formed by a zenner diode, a resistance
activated by a diode and an optional parallel resistance to the second winding
of the transformer. As it was explained beforehand, the first two nets help in
draining the magnetizing current faster.
\item[Improvement] The magnetizing current is more rapidly drained off. The
circuit is able to reduce the time between regenerated output pulses --the
recovery phase is quicker. Apart from this, as more nets take part in draining
the current in the inductance, the circuit is less liable to failure.
\end{description}
\subsection{Adjustable Fall and Rise Time}
\begin{description}
\item[Previous] The trigger circuitry is directly connected to the ground gate
of the Power MOSFET. By doing this, the fall and rise time are fixed.
\item[Current] The trigger circuitry is not directly connected to the ground
gate of the Power MOSFET. In between the trigger and the ground gate a parallel
connection formed by a diode and a resistor is added. The aim of this dipole
is offering a resistance for the rising edge of the output pulse and a
shortcircuit for the falling one. Thus, the rising time is configurable by
changing one resistor.
\item[Improvement] Slowing down the rising edge reduces reflections.
\end{description}
\subsection{Less Tilt in the Output Pulse}
\begin{description}
\item[Previous] The maximum R$_{ON}$ value of the BUZ32 Power MOSFET is 400
$m\Omega$.
\item[Current]  The R$_{ON}$ value of the IRLML0060TRPBF Power MOSFET is 92
$m\Omega$. A lower value reduces the voltage losses in the Power MOSFET that 
translates to a reduction of the tilt.
\item[Improvement] Tilt is hardly noticeable when compared to previous version.
\end{description}
\pagebreak
\section{Simulation and Measurements}
Simulations have been done to verify the following parameters:
\begin{itemize}
  \item Top voltage of the output pulse.
  \item Rise time of the output pulse.
  \item Fall time of the output pulse.
  \item Smoke parameters of the circuit.
\end{itemize}
\subsection{Simulations}
Plots are separately attached to this document.
\subsection{Measurements}
Measurements of the flyback converter were taken with a LeCroy Waverunner LT364L
oscilloscope. A Farnell instruments PDD3010A dual power supply was used to
provide 30V and 5V. The input signal for the trigger subcircuit was taken from CTVR VME board and connected directly to
the flyback converter through a 10ns cable.

\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\textbf{Input signal} & \textbf{Low time} & \textbf{Period} \\
\hline
inverted-TTL & 10 $\mu{}s$ & 1 $ms$  \\
\hline
\end{tabular}
\end{center}

The output pulse was measured:

\begin{center}
\begin{tabular}{|p{5cm}|p{2cm}|}
\hline
\textbf{R$_L$} & 50 $\Omega$\\
\hline
\textbf{Initial top voltage value} & 24V \\
\hline
\textbf{Tilt} & $\leq$ 0.5V \\
\hline
\textbf{Floor voltage value} & 0V \\
\hline
\textbf{Pulse width} & 10 $\mu{}s$ \\
\hline
\textbf{Rise Time} & 1.2 $\mu{}s$ \\
\hline
\textbf{Fall Time} &  2.49 $\mu{}s$\\
\hline
\end{tabular}
\end{center}

\pagebreak
\bibliographystyle{unsrt}
\bibliography{FlybackTech}
\end{document}