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\section{Modules biasing and depletion voltages}
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\label{sec:biasandvdepl}
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The LV and HV powering of modules is organized to reduce the number of
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power cables and of power supply modules on the far-end. Several
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modules are powered in parallel by the same power supply module, and
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the modules that share the same power supply are said to belong to the
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same {\em power group}. The actual layout of the power group depends
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on the layer for TIB and on the ring for TID. In TIB L1 and L2 the
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power group corresponds to all the modules connected to one mother
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cable (6 modules, 36 APVs); in TIB L3 and L4 the power group
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corresponds to the modules connected to three or four adjacent mother
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cables (9 or 12 modules, 36 or 48 APVs, respectively). As far as TID is
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concerned, for TID R1 and R2 the power group matches the mother cable
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(6 modules, 36 APVs); for R3 the power group
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corresponds to two mother cables (10 modules, 40 APVs).
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Each single power supply modules has two independent HV channels. In
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the power groups of double-sided layer/rings, i.e., TIB L1 and L2 and
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TID R1 and R2, one channel feeds $r\phi$ modules and the other stereo
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modules. In the power groups of single-sided layer/rings, the mother
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cables belonging to the power group are shared between the two
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channels, i.e., in any case the modules on the same mother cable are
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HV powered by the same HV channel.
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The bias to be applied to a given channel is roughly $V_{\rm depl}^{\rm
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max}+{\cal O}(100{\rm V})$,
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where $V_{\rm depl}^{\rm max}$ is the maximum depletion voltage of the
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sensors biased in parallel within that channel. Since depletion
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voltage distributions are quite broad, care was taken to avoid for the
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other sensors with $V_{\rm depl}\leq V_{\rm depl}^{\rm max}$ to be excessively
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over depleted. During the integration the modules were organized in
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order to have the depletion voltage as much uniform as possible within
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the power group and in particular within the same HV channel.
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The distribution of $V_{\rm depl}-V_{\rm depl}^{\rm max}$ for all the
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TIB/TID modules is shown in Figure~\ref{fig:VdeplUni}. For each
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module entry in the histogram, $V_{\rm depl}^{\rm max}$ is the maximum
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depletion voltage among the modules biased in parallel.
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\begin{figure}[h]
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\begin{center}
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\includegraphics[width=0.6\textwidth]{Figs/forIntNote_uniformity.pdf}
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\caption{Distribution of $V_{\rm depl}-V_{\rm depl}^{\rm max}$ for all the
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TIB/TID modules where $V_{\rm depl}^{\rm max}$ is the maximum
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depletion voltage among the modules biased in parallel.}
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\label{fig:VdeplUni}
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%\vskip -5mm
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\end{center}
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\end{figure}
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The difference, on average, is $\sim\!18{\rm \, V}$. $69\%$ ($78\%$) of modules
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will be biased with $<\!20{\rm \, V}$ ($<\!30{\rm \, V}$) of extra
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over depletion. Only for $0.7\%$ of TIB/TID modules the extra
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over-depletion is $>\!100{\rm \, V}$.
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The modules in the internal part of the TIB L1 and the TID R1 are the
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most subjected to the radiation with an expected fluence of
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$15.9\cdot10^{13}\,(1\!-\!{\rm MeV\, n\, eq})\cdot{\rm cm}^{-2}$ and $17.4\cdot10^{13}\,(1\!-\!{\rm MeV\, n\, eq})\cdot{\rm cm}^{-2}$, respectively, over a $\sim\!10{\rm \, y}$ LHC
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lifetime~\cite{fluencemaps}.
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\begin{figure}[b]
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\begin{center}
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\includegraphics[width=0.6\textwidth]{Figs/flu_tib_note.pdf}
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\caption{TIB fluence map.}
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\label{fig:TIBfluence}
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%\vskip -5mm
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\end{center}
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\end{figure}
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\begin{figure}[t]
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\begin{center}
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\includegraphics[width=0.6\textwidth]{Figs/flu_tid_note.pdf}
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\caption{TID fluence map.}
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\label{fig:TIDfluence}
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%\vskip -5mm
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\end{center}
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\end{figure}
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Following the method described in~\cite{migliore}, the depletion
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voltage evolution during the LHC lifetime can be estimated. The
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maximum depletion voltage to be expected as a
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function of the initial depletion voltage is shown in
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Figure~\ref{fig:vmaxestimation} for the above discussed worst cases.
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At a given fluence the evolution depends on the operating temperature
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during the data taking $T_{\rm DT}$
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and the number of times the tracker will be accessed for
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maintenance. The standard scenario consists of $T_{\rm DT}=-10^\circ
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{\rm C}$ and three intervention during the first, the third and the
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fifth winter shutdowns. Besides the standard scenario, other more
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conservative scenarios have been taken into account~\cite{migliore}:
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\begin{itemize}
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\item $T_{\rm DT}=-20^\circ {\rm C}$ and only one intervention during
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the first winter shutdown (in this scenario there is very limited
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beneficial annealing);
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\item 100\% donor removal (standard value $70\%$);
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\item neutron flux $100\%$ larger and
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\item proton flux $30\%$ larger, according to the systematic
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uncertainties to be intended on the particle flux estimation.
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\end{itemize}
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Moreover the estimation of the depletion voltage evolution itself has
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$50{\rm \, V}$ of uncertainty.
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\begin{figure}[b]
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\begin{center}
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\includegraphics[width=0.47\textwidth]{Figs/vdepmax_tib_l1_note.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.47\textwidth]{Figs/vdepmax_tid_r1_note.pdf}
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\caption{The expected maximum depletion voltage over the LHC
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lifetime as a function of the initial depletion voltage for TIB L1
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internal (left panel) and TID R1 (right panel) for various scenarios
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as discussed in the text.}
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\label{fig:vmaxestimation}
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%\vskip -5mm
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\end{center}
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\end{figure}
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The nominal module bias is $400{\rm \, V}$, although modules have been
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tested up to $450{\rm \, V}$ and power supply HV voltage channels can
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provide up to $600{\rm \, V}$. The resulting depletion voltage
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distribution for each layer, shown in Figure~\ref{fig:ModSenVdepl},
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comply with the radiation tolerance requirements due to the expected
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fluence in each module position discussed above.
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\begin{figure}[t]
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\begin{center}
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\includegraphics[width=0.32\textwidth]{Figs/TIB_L1_int.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TIB_L1_ext.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TIB_L2.pdf}
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\\
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%\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TIB_L3.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TIB_L4.pdf}
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\\
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%\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TID_R1.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TID_R2.pdf}
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\hskip 0.5mm
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\includegraphics[width=0.32\textwidth]{Figs/TID_R3.pdf}
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%\hskip 0.5mm
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\caption{Module sensors depletion voltage distributions.}
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\label{fig:ModSenVdepl}
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%\vskip -5mm
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\end{center}
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\end{figure}
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