Differential Measurement Techniques Using a Programmable Josephson Voltage Standard
Estefan´ıa Luna
Guillermo Schneider
Mariano Real
Ricardo Iuzzolino
Dpto. de Metrolog´ıa Cua´ntica Dpto. de Metrolog´ıa Cua´ntica Dpto. de Metrolog´ıa Cua´ntica Dpto. de Metrolog´ıa Cua´ntica
INTI
INTI
INTI
INTI
Buenos Aires, Argentina
Buenos Aires, Argentina
Buenos Aires, Argentina
Buenos Aires, Argentina
estefanialuna@inti.gob.ar
gschneider@inti.gob.ar
mreal@inti.gob.ar
riuzzolino@inti.gob.ar
Abstract—This work presents two methods to reconstruct a signal using differential sampling measurements. These measurements were obtained by measuring an AC signal generator against a Programmable Josephson Voltage Standard (PJVS). The main objective of both methods is to eliminate the ubiquitous transients and to reconstruct the signal of the generator under test. In such direction, the Root Mean Square (RMS) value of the digitized signal was obtained and compared using both methods with uncertainties of about 0.9 µV. Therefore, both methods can be applied to measure and characterize an AC generator with a direct traceability to the voltage primary standard reducing the effect of increasing steps and uncertainties in the traceability chain.
Index Terms—Voltage measurement, Josephson array, signal synthesis, differential, sampling.
I. INTRODUCTION
T HE discovery of the Josephson effect [1] which generates voltages determined by its quantum effect [2], prompted the development of different volt realizations. The effect relies on superconducting Josephson Junctions Arrays (JJA), containing about ten thousand junctions in series connection, which are produced using thin film micro-fabrication technology. Today, Josephson voltage standards allow the generation of both DC and AC signals, enabling electrical voltages to be directly linked to the reference constants established by the International System of Units (SI) [3], according to:
II. SYSTEM OVERVIEW
Fig. 1 depicts the system configuration. For differential measurements, a custom-made sigma-delta (Σ∆) digitizer [4], performs differential sampling of two input signals, one coming from a DUT and a second being the reference signal generated by the PJVS. This configuration, called AC Quantum Voltmeter (AC-QVM), is detailed in [5]. The PC configures the digitizer through software to start the sampling measurement and gathers the digitizer readouts. The Programmable Current Source (PCS) biases the JJA segments, which is previously programmed by the PC with the desired PJVS waveform. The clock generator block synthesizes different frequencies locked to INTI’s Cesium atomic clock frequency of 10 MHz. It consists of a Digital Phase Locked Loop (DPLL) that produces 3 output frequencies, which are integer multiples; one of them is connected to the PCS and the others are the digitizer clock and trigger signals. This allows the system to run in a synchronous mode, as a common signal clock is fed to the digitizer, the PCS and the DUT. In this mode, the PJVS signal has to be in-phase with the DUT signal, to accomplish that, a phase reference signal is applied to the PCS. Fig. 2 depicts a photograph of the complete experimental setup used in the laboratory.
h
V = nf ,
(1)
2e
where n is the quantum number, f the microwave frequency
referred to a cesium atomic clock, h is the Planck constant
and e the elementary charge.
Currently, different types of Josephson systems have been
developed tailored to specific applications. One of them is
the PJVS, which can be used as a superconducting multi-bit
digital-to-analog converter (DAC). The PJVS produces stable
DC voltages or time-stepwise approximated waveforms, whose
accuracy is determined by well-defined constant voltage
steps [3], given by equation 1.
This work presents a comparison between two methods
for the reconstruction of a signal from differential sampling
measurements, using a PJVS as a reference and a sinusoidal
signal generator as a device under test (DUT).
Fig. 1. Differential measurement setup. The sigma-delta (Σ∆) ADC digitizer performs differential sampling between two input signals, a DUT and a PJVS. The PC configures the digitizer through software to start the sampling measurement. The programmable current source takes the phase reference from an external signal generator and the trigger signal from the clock generator to produce the voltages that power the JJA segments. The 10 MHz generator block distributes the system clock frequency locked to the atomic clock frequency of 10 MHz.
Fig. 2. INTI’s AC-QVM. A: Clock generator. B: Signal generators. C: DUT. D: Σ∆ digitizer. E: Programmable current source. F: Microwave generator. G: Liquid Helium dewar where the Josephson array is cooled down to 4.2 K. A and F are locked to INTI’s Cesium atomic clock frequency.
III. MEASUREMENT TECHNIQUES
The AC-QVM measures the difference between the DUT output signal and the reference signal synthesized by the PJVS as stated in section II. Then, the DUT signal is obtained by summing the sampled data plus the reference signal, this process is depicted in Fig. 3. As can be seen in Fig. 3-(b), the resulting signal has transients, due to the response of the Σ∆ digitizer’s digital 48-tap FIR filters to the transition between adjacent quantum voltage levels of the PJVS, these filters need a certain amount of samples to settle (24 in this application), where a highest accuracy level is of utmost importance condition to reach a better uncertainty level. Such transients can mislead the final result, therefore they must be removed from the resulting signal. To accomplish this, two methods were developed and compared. For which sine waveforms of 1 V amplitude with frequencies of 62.5 Hz and 31.25 Hz which are close to the industrial electrical power frequency range, were measured using the AC-QVM. The signals generated by the PJVS were synthesized with 20 steps per period, having the same amplitude and frequency as the generator (DUT) output signal, in such a way that the difference is close to tens of mV and digitized to an equivalent sampling rate of 32.5 kHz.
The following subsections describe the reconstruction of the signal and the two methods developed.
A. Method 1
The goal of these methods is to reconstruct the DUT signal while eliminating the transients and filling the time gap, in order to have a full signal according to the sampling rate and signal frequency ratio. This task is carried out in a two-step process: first, in-phase signals are sampled (signal 1 in Fig. 4-(a)); second, the unknown signal is set out-of-phase to the reference and sampled again (signal 2 in Fig. 4-(a)). This process results in two differential signals as illustrated in
Fig. 3. Comparison of the signals. (a) The pink curve represents the ideal sine wave, while the black curve shows the ideal programmed PJVS signal. The blue curve depicts the measured differential signal. In (b), the reconstructed waveform with transients is displayed. Both methods described in the text are designed to eliminate these transients.
Fig. 4-(a). Then, to each differential signal, the PJVS reference values are added, as shown in Fig. 5-(a) where the phase shift (∆ϕ) between the signals can be seen. After this stage, the transients are removed. The final step consists of the combination of both signals resulting in a final signal without any time gap, as depicted in Fig. 5-(b).
The flow diagram of Fig. 6 describes the steps carried out for the signal reconstruction, where the PJVS block corresponds to the signal generated in the PJVS being the same for both measurements.
The TE blocks perform the transients elimination process, the input signal is separated into steps and then, the transients are removed. The difference between these blocks is that one of them eliminates the counterpart of points in the other. For example, if a step of a signal has 60 points and 20 points have to be eliminated, the TE block of signal 1 eliminates 10 points from the beginning and 10 from the end of the step, resulting in 40 remaining points for each step. The TE block of signal 2 will do the opposite, eliminating 40 points of the 60, 20 from each side of the step, and 20 points will be obtained for each step of the signal. At the end of this process, two signals with gaps due to the transient elimination are obtained. The RECO block combines both signals, filling the gaps of each step from both signals.
B. Method 2
In this method, the transients elimination and signal reconstruction procedure is the same as in Method 1, but the inputs are different. Instead of shifting the phase of the unknown signal, a second Josephson signal is generated,
Fig. 4. Comparison of differential signals. Upper panel (a) shows the signals used in Method 1, a phase-shift is applied to signal 1 (blue) resulting in signal 2 (red). The lower panel (b) presents the differential signals used in Method 2, signal 2 (red) results from a new Josephson signal having the same number of steps but phase-shifted from the one used in signal 1 (blue), as described in the text.
Fig. 6. Flow diagram of Method 1. The TE blocks correspond to the transients elimination process and RECO to the recombination of the signal.
signal. The Flow diagram of this technique is shown in Fig. 7.
Fig. 5. This figure presents the signals used in Method 1. In (a) is shown the comparison between the two reconstructed signals presented in Fig. 4-(a), notice that both signals present transients. After applying the proposed method the waveform presented in (b) is obtained.
having the same number of steps as the initial signal in such a way that the position of the steps matches the necessary phase to occupy the transients positions. Then, a zero-crossing differential sampling is performed with respect to the generator
Fig. 7. Flow diagram of Method 2. The TE blocks correspond to the transients elimination process and RECO to the recombination of the signal, which are the same as Method 1.
IV. RESULTS
Having reconstructed the signal with both methods, the Root Mean Square (RMS) value was calculated and a stability analysis was carried out using Allan deviation [6]. The latter determines the minimum uncertainty that can be obtained in a given observation time or, as in this case, the number of signal periods that are necessary to perform subsequent calculations.
The time determined by the Allan deviation was 2.5 s. This translates into 156 periods for the 62.5 Hz frequency signal and 78 periods for the 31.25 Hz.
An example of the procedure carried out for a 31.25 Hz signal is shown in Fig. 8. The total input data has 500 periods, and taking as reference the Allan deviation time it was divided into 6 groups of 78 periods. Then, each group was divided again into 15 subgroups in a set of 5 periods to eliminate the 50 Hz interference on the measurements. On each set of 5 periods, the RMS value was performed. Then, over the resulting values, the average is calculated resulting in six independent RM Sk, k = 1, ..., 6 values, and its corresponding standard deviation was performed. Finally, the total RMS value RM ST was obtained by averaging the previous results (RM Sk), and the type-A uncertainty was calculated. This uncertainty value was obtained from the standard deviation of each group. All the calculations were performed with integer divisions, discarding the surplus.
about 15 mV, and the agreement between methods is about 300 µV. The type-A uncertainty of both methods was close to 0.9 µV.
Furthermore, Method 1 requires a manual adjustment for determining the phase of the signal required for the measurements, leading to an increased execution time. Instead, Method 2 requires less execution time as the generation of the PJVS is only modified once before the measurement is performed.
V. CONCLUSION
A comparison between the two methods for differential measurements was presented. Each of the methods was described and an analysis of the RMS value of 31.25 Hz and 62.5 Hz signals was performed. The results indicated that the differences concerning a nominal value of 1 V amplitude were about 15 mV, and the difference between both methods was close to 300 µV. Also, the type-A uncertainty of both was about 0.9 µV.
It is concluded that Method 2 outperforms Method 1 in terms of accuracy and efficiency.
Moreover, Method 1 requires a longer execution time of measurement, allowing extra possible errors. In contrast, Method 2 only requires a single modification of the PJVS generation before performing the measurement.
Furthermore, these methods offer an advantage over conventional techniques as they enable a more comprehensive time and frequency analysis since the resulting signal contains more information and the gaps of the transients are eliminated.
Fig. 8. Diagram of the procedure to calculate the results for a 31.25 Hz signal. The first set of 6 groups was divided taking into account the Allan deviation time and the second set of 15 subgroups correspond to the number of periods necessary to eliminate the 50 Hz interference. The final result is the total RMS value (RM ST ) and the type-A uncertainty.
The transients elimination process required removing 28 points for the 31.25 Hz signal, and only 12 for the 62.5 Hz signal.
TABLE I THIS TABLE SHOWS THE RMS VALUES CALCULATION RESULTS. NOMINAL ERROR IS THE DIFFERENCE OF EACH METHOD TO THE
REFERENCE VALUE OF A SINE WAVEFORM. METHOD ERROR CORRESPONDS TO THE DIFFERENCE BETWEEN METHODS.
Frequency 31.25 Hz
62.5 Hz
Parameters
RMS value [V] Standard deviation [µV] Uncertainty [µV] Nominal error [mV] Method error [µV] RMS value [V] Standard deviation [µV] Uncertainty [µV] Nominal error [mV] Method error [µV]
Method 1 Method 2
0.6916345 0.6919580
1.6
1.2
0.9
0.9
15.472
15.149
323.6
0.6916783 0.6919972
0.6
1.7
0.9
1.0
15.428
15.120
318.9
ACKNOWLEDGMENT
The authors thank Dr. Ralf Behr and Dr. Luis Palafox of the PTB for their constant support in the setup process of the Josephson system.
REFERENCES
[1] B. D. Josephson, “Possible new effects in superconductive tunnelling,” Physics letters, vol. 1, no. 7, pp. 251–253, 1962.
[2] J. Kohlmann and R. Behr, Development of Josephson voltage standards. Wiley-VCH, 07 2011.
[3] B. Jeanneret and S. Benz, “Application of the josephson effect in electrical metrology,” The European Physical Journal Special Topics, vol. 172, pp. 181–206, 06 2009.
[4] R. J. Iuzzolino, “Josephson waveforms characterization of a sigma-delta analog-to-digital converter for data acquisition in metrology,” Ph.D. dissertation, B-IGSM, Berlin, Aug 2011. [Online]. Available: https://publikationsserver.tu-braunschweig.de/receive/dbbs mods 00042711
[5] R. Iuzzolino, M. E. Bierzychudek, L. Palafox, R. Behr, and A. Tedesco, “On the development of an ac-quantum voltmeter,” in 2018 Conference on Precision Electromagnetic Measurements (CPEM 2018), 2018, pp. 1–2.
[6] T. Witt, “Using the allan variance and power spectral density to characterize dc nanovoltmeters,” Instrumentation and Measurement, IEEE Transactions on, vol. 50, pp. 445 – 448, 05 2001.
Table I shows the results obtained from the analysis made, where it can be seen that both methods have similar results. The differences to the nominal peak amplitude of 1 V are
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