Summary of Research on Time-to-Digital Converters Summer Exchange Program 2008 Istituto Nazionale di Fisica Nucleare Rome, Italy Creative Studies Honors Program University of California Santa Barbara, California, USA Alan WahLun Mak

Introduction Through Summer Exchange Program jointly organised by the I.N.F.N. and the U.S. Department of Energy Involved in the research of I.N.F.N. Section Roma Tre from early July to late September Have been working closely with Prof. Paolo Branchini and Dr. Salvatore Loffredo Invited by Prof. Filippo Ceradini to give this talk Research on time-to-digital converters (TDCs)

Technical Background A Time-to-Digital Converter (TDC) is an electronic device that can measure the time elapsed between two incoming pulses and output the digital representation of the time interval. TDCs are deployed extensively in high energy physics experiments for particle detection. Our group implements TDCs by Field-Programmable Gate Arrays (FPGAs), and achieves a time resolution of several hundred picoseconds.

Technical Background Each TDC has a crystal oscillator as the reference clock, which oscillates at a particular frequency with long-term stability. Three TDCs considered in our laboratory: 1. Quartz as crystal oscillator; frequency 480 MHz 2. Rubidium as crystal oscillator; frequency 480 MHz 3. With high-stability Quartz oscillator; frequency 550 MHz Building blocks of the TDC: 1. Synchronous counter – for a coarse measurement 2. State machine – for a better resolution

Technical Background Figure source: Paolo Branchini, Salvatore Loffredo et al, “FPGA Implementation of a High-Resolution Time-to-Digital Converter”, 2007

Technical Background Figure source: Paolo Branchini, Salvatore Loffredo et al, “FPGA Implementation of a High-Resolution Time-to-Digital Converter”, 2007 The synchronous counter measures the time interval Δt 12 between two rising edges of the reference clock immediately following the START and STOP signals. The state machine measures Δt 1 – Δt 2, which is the phase difference between the two above-mentioned rising edges. Nutt Method: Total time elapsed between START and STOP signals = Δt 12 + Δt 1 – Δt 2

Technical Background Limitations on the measurements of time 1. Range – limited by the synchronous counter’s width 2. Resolution – limited by the reference clock’s frequency References used for the technical background: 1.Paolo Branchini, Salvatore Loffredo et al, “FPGA Implementation of a High-Resolution Time-to-Digital Converter”, Wikipedia (English-language version)

Preliminary Study The rubidium TDC of frequency 480 MHz is considered to be the most stable of the three TDCs, because of the nature of rubidium crystal and the synchronization of the Global Positioning System (GPS). Our aim is to study the performance of the quartz TDC of frequency 480 MHz, and to calibrate it with respect to the rubidium TDC. Since the two TDCs in question oscillate at the same frequency, one “count” (or one “tick”) represents equal amount of time for both TDCs. → We can use the number of counts for comparison.

Preliminary Study In the preliminary study, about 200 events are generated with a C++ program. In each event, the C++ program produces a time interval for the two TDCs to measure. The length of the time interval increases with events. In this preliminary study, only very short time intervals (< 90 picoseconds) are produced. The number of counts measured by each TDC in each event is recorded in a data file for analysis.

Preliminary Study Each count represent ~ 520 picoseconds

Preliminary Study Each count represent ~ 520 picoseconds

Further Examination The preliminary study shows a linear relation between the counts measured by the quartz TDC and the rubidium TDC. This implies that the quartz TDC maintains reasonably good stability and precision for measurements of time intervals < 90 picoseconds. Motivation: We want to examine the performance of the quartz TDC for measurements of longer time intervals events are generated. Throughout the 1000 events, the time interval to be measured increases from zero all the way up to 2 seconds.

Further Examination

Hypothesis 1 We hypothesize that the oscillatory fluctuation is due to the temperature of the TDC chips. There are several fans for cooling down the electronic devices. To check the hypothesis, we repeat the same measurements for two more times, with the speed of the fans set to maximum and minimum, respectively. Maximum fan speed = 3120 rpm → lower temperature Minimum fan speed = 1200 rpm → higher temperature A laser thermometer probe is used to verify that the temperatures of the TDC chips increase when the fan speed is switched from maximum to minimum.

Hypothesis 1 Maximum fan speed

Hypothesis 1 Minimum fan speed

Hypothesis 1 Apparently, the temperature of the TDC chips does have an effect on the discrepancy between the quartz measurements and the rubidium measurements. To pursue this further, we generate another 5000 events. Throughout the 5000 events, the time interval to be measured increases from zero up to 100 seconds. Instead of using a laser thermometer probe, this time we program the FPGAs so that the temperatures of the TDC chips are monitored continuously.

Hypothesis 1

Hypothesis 2 In the previous analysis, the temperatures of the two TDC chips vary within the range of ± 1 o C. Nonetheless, the oscillatory fluctuations in the graph of the difference between quartz count and rubidium count is very pronounced for event number > There should be something more than temperature that caused the oscillatory fluctuations. New hypothesis: The periodic synchronization of the GPS connected to the rubidium TDC is another cause of such fluctuations.

Hypothesis 2

Both temperature and GPS synchronization have some effect on the oscillatory fluctuations. More analysis is needed to find out how those factors affect the oscillatory fluctuations. Nonetheless, the 480 MHz quartz TDC can still be used for measurements of short time intervals and measurements that do not require very high precision. For measurements that require higher precision, our group has decided to replace that with a 550 MHz high-stability quartz TDC.

High-stability TDC Again, we want to calibrate the 550 MHz high-stability quartz TDC with respect to the 480 MHz rubidium TDC. Owing to the difference in the frequencies of the two crystal oscillators, we can no longer use the number of counts for direct comparison. The number of count is converted into the total amount of time that it represents.

High-stability TDC

Fine measurement The fine measurement of time involves measuring the phase difference between the START and STOP signals with the aid of the state machine. For this purpose, our group designed a delay line that consists of a train of 64 multiplexers. The time quantization step is determined by the propagation delay τ of the multiplexer. Figure source: Paolo Branchini, Salvatore Loffredo et al, “FPGA Implementation of a High-Resolution Time-to- Digital Converter”, 2007

Fine measurement Due to the short delay of the delay lines, two delay lines are required to cover the duration of a full clock period. Delay line 0 – clocked by CLK0 Delay line 1 – clocked by CLK 180 The state machine decides which delay lines are selected. A delay line has 64 multiplexers → 64 possible values for the Δt 1 (START signal); and 64 possible values for Δt 2 (STOP signal).

Fine measurement The difference between Δt 1 (START) and Δt 2 (STOP) can be any integer between –64 and bits are required to store all the possible values. (since 2 7 = 128) In the 16-bit output … … 7 least significant bits for storing the value of STOP – START Delay line used for START signal (0 or 1) Delay line used for STOP signal (0 or 1)

Fine measurement

Computer Simulations Simulate probability distributions with a built-in random number generator in ROOT. The random number generator uses the Mersenne twister algorithm, which has high efficiency and random properties. With the aid of the random number generator, generate a probability distribution for the START signal and another probability distribution for the STOP signal. Each of the two signals can take on any integer value in the closed interval [-64,64].

Computer Simulations A total of events are generated. Then, find out the probability distribution of STOP – START. Repeat these procedures for different probability distributions of START and STOP signals.

Conclusion We have examined the performance of the 480 MHz quartz TDC and the 550 MHz high-stability TDC with respect to the 480 MHz rubidium TDC. Temperature and GPS synchronization do have some effect on how the measurement of the 480 MHz quartz TDC deviate from that of the rubidium TDC. The 480 MHz quartz TDC is still good for measurements of short time intervals and measurements that do not require very high precision.

Conclusion We have verified the high stability of the 550 MHz high- stability quartz TDC, and examine its fine measurements. We have performed computer simulation to investigate the distribution of the fine measurements of the high-stability quartz TDC.

Acknowledgements

Questions? Discussions? Comments?