improvements in the high precision TDC (8~ps RMS ) developed before
\cite{tdc_ieee}, as well as the novel way of measuring ToT using a single TDC
channel, while still achieving high precision (as low as 11.7~ps RMS). The
-effect of voltage, generated by a DC-Dc converter, over the precision is also
+effect of voltage, generated by a DC-DC converter, over the precision is also
discussed. Finally, the outcome of the temperature change over the pulse width
measurement is shown and a correction method is suggested to limit the
degradation.}
figure~\ref{fig:tdcArch}. The start signal for the delay line is a digital
signal from the front-end electronics (FEE). The propagation of the signal is
stopped with the next rising edge of the 200~MHz clock signal. The thermometer
-code output of the delay line is then sent to the decoder and converted to
+code\footnote{Thermometer code shows a natural number, \textit{n}, with a
+series of \textit{n} ones (zeros) followed by zeros (ones).} output of the
+delay line is then sent to the decoder and converted to
binary code. This data is saved in a ring buffer together with the timestamps
from the coarse and epoch counters (5~ns granularity). The readout and slow
control of the TDC are based on the TRBNet protocol \cite{trbnet} developed for
-TDC Readout Boards (TRB) \cite{trb3}.
+Trigger and Readout Boards (TRB) \cite{trb3}.
% As explained in our previous paper the architecture of the TDC is based on the
% interpolation method \cite{kalisz}, where the interpolator is built as a tapped
\begin{figure}[tbp]
\centering
- \includegraphics[width=.5\textwidth]
+ \includegraphics[width=.45\textwidth]
{figures/stretcher_semiAsync.pdf}
\caption{The semi-asynchronous stretcher logic stretches the trailing edge
of the input signal ($hit$) beyond the clock period. The stretched
longer than the intrinsic TDC dead time are achievable.
Using this circuitry allows us to measure both edges of pulses shorter than the
-dead time of the TDC on a single channel. The disadvantage of this method is the
-extra dead time after the measurement of the trailing edge, which limits the
-measurement of double pulse which are separated in time less than the total dead
-time (pulse length + stretcher delay + conversion dead time). Also, the
+dead time of the TDC on a single channel. The disadvantage of this method is
+the extra dead time after the measurement of the trailing edge, which limits
+the measurement of double pulses separated in time less than the total dead time
+(pulse length + stretcher delay + conversion dead time). Also, the
non-deterministic stretching time between the channels and the design
compilations has to be tackled to get the real time information.
\begin{figure}[tbp]
\centering
- \includegraphics[width=.48\textwidth]
+ \includegraphics[width=.45\textwidth]
{figures/tdiff_8ps.pdf}
%{figures/9ps_res.pdf}
\caption{The leading edge measurement precision of a sample channel. (The mean
The measured ToT values with both methods do not match the generated pulse width
(3.32~ns) because of the offset between the channels and the offset induced by
-the stretcher, which have not yet been accounted for. In order to test the
+the stretcher, which have not yet been taken into account. In order to test the
correctness of the ToT measurement a pulse width sweep is applied and the
results are recorded with both an oscilloscope and the TDC
(figure~\ref{fig:totSweep}). The maximum deviation in the ToT value relative to
\subsection{Effect of DC-DC converters over wide pulses}
-We also tested the quality of the long time interval measurements as this is
-both important for the long time of flight (ToF) and ToT measurements. The test
-setup consists of the FPGA-TDC and a pulse generator (Tektronix AWG7000) with
-two outputs controlled over a general purpose interface bus (GPIB). The two
-outputs of the pulser are sent to two different channels of the TDC, where the
-time interval between the outputs is measured. Starting with 0~s time interval
-one output is shifted with 1~ns after each successful measurement until a time
-interval of 1~us is reached. The precision is recorded for each measurement.
-The calibration of the channels is done only once at the beginning of the test.
-
-In figure~\ref{fig:rmsWITHdcdc} the precision of a channel as a function of the
-measured interval is shown. It was observed that over a microsecond time
-interval the precision value oscillates with an amplitude of 48~ps and this
-effect was thought to be from the DC-DC converters. Next, the board was
-stripped down of the converters and the FPGA was powered with a linear power
-supply (HMP4040). The test was repeated to be noted that the oscillation
-amplitude is improved by a factor of $\sim$12. The exposed secondary oscillation
-has an amplitude of $\sim$3~ps and a frequency of 25~MHz
-(figure~\ref{fig:rmsNOdcdc}). This additional oscillation is not further
-investigated as the amplitude is negligible.
-
-\begin{figure}[tbp]
+We also tested the quality of the long time interval measurements, as this is
+important for both time of flight (ToF) and ToT measurements. Tektronix AWG7000
+is used as the pulse generator with two outputs controlled over a general
+purpose interface bus (GPIB). Signals from two outputs are sent to two different
+channels of the TDC, where the time interval between the signals is measured.
+Starting with 0~s, time interval is incremented with 1~ns granularity until
+1~us. The precision is recorded for each measurement.
+
+% We also tested the quality of the long time interval measurements as this is
+% both important for the long time of flight (ToF) and ToT measurements. The test
+% setup consists of the FPGA-TDC and a pulse generator (Tektronix AWG7000) with
+% two outputs controlled over a general purpose interface bus (GPIB). The two
+% outputs of the pulser are sent to two different channels of the TDC, where the
+% time interval between the outputs is measured. Starting with 0~s time interval
+% one output is shifted with 1~ns after each successful measurement until a time
+% interval of 1~us is reached. The precision is recorded for each measurement.
+% The calibration of the channels is done only once at the beginning of the test.
+
+\begin{figure}[bp]
\begin{subfigure}{.5\textwidth}
\centering
- \includegraphics[width=.8\linewidth]
+ \includegraphics[width=.86\linewidth]
{figures/rms_trb3_with_dcdc.pdf}
\caption{With DC-DC converters.}
\label{fig:rmsWITHdcdc}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
- \includegraphics[width=.8\linewidth]
+ \includegraphics[width=.86\linewidth]
{figures/rms_trb3_without_dcdc.pdf}
\caption{With a linear power supply.}
\label{fig:rmsNOdcdc}
\end{subfigure}%
\caption{The precision of the measurements is recorded over a microsecond time
interval between two outputs of a pulse generator. The oscillation amplitude is
-reduced to 3~ps by powering the FPGA with a linear power supply instead of DC-DC
+reduced to 3~ps by powering the FPGA with a linear power supply instead of
+DC-DC
converters.}
\label{fig:rmsVSdcdc}
\end{figure}
+In figure~\ref{fig:rmsWITHdcdc} the precision as the function of the measured
+interval is shown. It was observed that over a microsecond time interval the
+precision value oscillates with an amplitude of 48~ps and this effect was
+thought to be from the DC-DC converters. Next, the board was stripped down of
+the converters and the FPGA was powered with a linear power supply (HMP4040).
+The test was repeated to be noted that the oscillation amplitude was improved
+by a factor of $\sim$12. The exposed secondary oscillation has an amplitude of
+$\sim$3~ps and a frequency of 25~MHz (figure~\ref{fig:rmsNOdcdc}). This
+additional oscillation is not further investigated as the amplitude is
+negligible.
+
\subsection{Effect of temperature over ToT}
The effect of the temperature on the ToT measurements are observed by building a
measured by a temperature sensor on the board close to the FPGA. 14 different
measurements are done in the temperature range of 36.8$^{\circ}$C and
42.8$^{\circ}$C. The mean of the ToT measurements are plotted as a function of
-temperature (figure~\ref{fig:totVStemp}). In order to see the change the ToT
-values at the initial temperature are subtracted from the measured values.
+temperature (see figure~\ref{fig:totVStemp}). In order to examine the change the
+ToT values at the initial temperature are subtracted from the measured values.
From the plot it can be seen that there are two different disturbances to be
corrected. One is the change in the ToT on a single channel with the changing
calculated as $\sim$0.4$\%/^{\circ}$C.
The temperature correction coefficient is calculated iteratively until the
-smallest ToT deviation among every channel at the measured temperatures is
+smallest ToT deviation among all channels at the measured temperatures is
reached.
The secondary effect of the temperature change is corrected by applying a linear
the maximum temperature (42.8$^{\circ}$C) (figure~\ref{fig:totVSoffset}). The
slope of the linear fit is used as a correction coefficient.
-Both corrections can be written in a single equation as:
-
-\begin{equation}
-\label{eq:temp_correction}
-ToT' = ToT_m - k_T*\Delta T*k_O*ToT_i
-\end{equation}
-where $ToT'$ is the corrected ToT, $ToT_m$ is the measured ToT, $k_T$ is the
-temperature coefficient, $\Delta T$ is the change in temperature, $k_O$ is the
-offset coefficient and $ToT_i$ is the ToT at the initial temperature. Knowing
-these two coefficient values one can correct the ToT values by using the change
-in the temperature and the initial ToT value, independent of the channel.
-
-Applying only the temperature correction of the above equation, the maximum ToT
-shift can be corrected by a factor of $\sim$4
-(figure~\ref{fig:totVStemp_tempCorr}). This correction factor increases to
-$\sim$10 limiting the ToT shift with $\sim$65~ps over 6$^{\circ}$C temperature
-change, when both - temperature and offset - corrections are applied.
\begin{figure}[tbp]
\begin{subfigure}{.5\textwidth}
\label{fig:temp}
\end{figure}
+
+Both corrections can be written in a single equation as:
+
+\begin{equation}
+\label{eq:temp_correction}
+ToT' = ToT_m - k_T*\Delta T*k_O*ToT_i
+\end{equation}
+where $ToT'$ is the corrected ToT, $ToT_m$ is the measured ToT, $k_T$ is the
+temperature coefficient, $\Delta T$ is the change in temperature, $k_O$ is the
+offset coefficient and $ToT_i$ is the ToT at the initial temperature. Knowing
+these two coefficient values one can correct the ToT values by using the change
+in the temperature and the initial ToT value, independent of the channel.
+
+Applying only the temperature correction of the above equation, the maximum ToT
+shift can be corrected by a factor of $\sim$4. This correction factor increases
+to $\sim$10, limiting the ToT shift with $\sim$65~ps over 6$^{\circ}$C
+temperature change, when both -- temperature and offset -- corrections are
+applied. In figure~\ref{fig:temp2} the results of the alone temperature
+correction as well as temperature and offset correction with the uncorrected
+values are shown.
+
+
\begin{figure}[tbp]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=1\linewidth]
{figures/temp_vs_tot_with_bothCorrection.pdf}
\caption{Correction with temperature \& offset coefficients.}
- \label{fig:totVStempCorrected2}
+ \label{fig:totVStemp_bothCorr}
\end{subfigure}%
-\caption{The shift of ToT is corrected by a factor of $\sim$10 over 6$^{\circ}$C
+\caption{The shift of ToT is corrected by a factor of $\sim$10 over
+6$^{\circ}$C
temperature change.}
\label{fig:temp2}
\end{figure}
\section{Conclusion}
In this paper we presented our novel way of measuring ToT on an FPGA TDC using a
-single channel. Based on the conducted tests precision of the leading edge
+single channel. Based on the conducted tests, precision of the leading edge
measurement is recorded as low as 8~ps, resulting in 5.66~ps error on a single
-channel. The precisions for ToT measurements with the conventional and novel
-methods are recorded as 10.3~ps and 11.7~ps respectively. The novel method is
-investigated further to find out, that the ToT value differs maximum 38~ps from
-the oscilloscope measurements once the stretcher offset is eliminated. It is
-also discovered that the deterioration in the long time interval measurement
-precision can be limited to 3~ps, if the FPGA is powered with a linear power
-supply.
+channel. The precisions for ToT measurements with the conventional method (using
+two TDC channels for two edges) and novel method (using a single TDC channel for
+both edges) are recorded as 10.3~ps and 11.7~ps respectively. The ToT values,
+measured with the novel method, differ maximum 38~ps from the oscilloscope
+measurements, once the stretcher offset is eliminated. It is also discovered
+that the deterioration in the long time interval measurement precision can be
+limited to 3~ps, if the FPGA is powered with a linear power supply.
The effect of the temperature change on the ToT measurement is also assessed and
the average relative effect of the temperature on delay line is calculated as
-$\sim$0.4$\%/^{\circ}$C. This result shows that the with the increasing length
-of the delay line and temperature difference the shift in the mean value
-increases. This degeneration is corrected by a factor of $\sim$10 and limited
-to $\sim$65~ps with a correction model.
-
-With correct environmental conditions and controlling, the temperature change
-can be minimised to less then 1$^{\circ}$C. Although in many high energy physics
-applications high precision ToT measurements are not required, if necessary, the
-conventional method should be used, as the temperature effect plays a minimal
-role in short routing lines.
+$\sim$0.4$\%/^{\circ}$C. These results show, that the temperature effect is
+larger for the longer delay lines. This effect can be compensated by a factor of
+$\sim$10 and limited to $\sim$65~ps with the described correction model.
+
+For many practical cases the temperature effect can be neglected, as it is
+possible to control the temperature change below 1$^{\circ}$C. Although in many
+high energy physics applications the provided precision for ToT measurements is
+sufficient, if necessary for other applications, the conventional method should
+be used, as the temperature effect plays a minimal role in short routing lines.
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