Calibration of a precision current measurement system for high AC voltages using an AC quantum voltmeter

The PS-HVAC measuring system consists of an operational amplifier (OPA)-based current-to-voltage converter, precision resistors in each of its seven measurement ranges (marked as '100 Ω' up to '10 kΩ'), a digitizer, and a laptop running a LabVIEW program, forming a measurement unit (figure 2). The PS-HVAC is based on the Chubb–Fortescue method, but the loading current of the high voltage capacitor is converted into a proportional voltage. Since the current of the capacitor is a direct derivative of the high voltage, the measurement signal must be integrated. Here, a software-based integration is used with careful attention to the initial conditions.

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This system has been developed for calibrations of currents in the milliampere range at a frequency of 50 Hz. However, the system was characterized for measurements of signals with frequencies from 15 Hz to 400 Hz using a sinusoidal current in the range of 0.5 mA to 50 mA, which is the typical loading current of compressed gas high voltage capacitors with values up to 500 pF [1]. The targeted overall relative uncertainty of the entire system, including the HV capacitor, is 25 μV V⁻¹ so the requirement is to make current calibration with a relative uncertainty better than 10 μA A⁻¹. Since the low voltage side of HV capacitor is directly connected to the input of OPA, the influence of the connecting coaxial cable should be negligible on current measurement. The characterization of the individual components has been described in detail [1, 2].

Calibration of the PS-HVAC system means applying a precisely known reference current, measured by AC-QVM, and comparing the current reading on the display of the system (i.e., measured current). Internally measured current is calculated from the voltage drop (and the associated ADC voltage reading) on the precision resistance (with the known nominal value for each range). Therefore, calibration means the determination of the relative correction for measured current c_I , expressed in μA A⁻¹ (see results given in table 3) and to be stored into the control LabVIEW program, until it is equal to the reference current. The same procedure should be repeated for each of seven measurement ranges (i.e., at least seven corrections c_I should be determined), while the calibration at one particular range can be done with different currents (see results given in table 3). In ideal case, the relative correction for one range should remain constant regardless of the current level used.

The AC-QVM and its extension to a quantum calibrator are discussed in detail in [5, 6], while the schematic of this floating device is given in figure 3, in a setting for AC measurements. The essential part, the PJVS, can generate stable and quantized DC voltages, or stepwise approximated sine waves for frequencies up to the kilohertz range. For DC voltage measurements the AC-QVM is used as common DC Josephson array voltage standard and generates quantized DC voltage up to 10 V. The voltage difference between the voltage generated by the PJVS and the voltage drop on the resistor is measured by an Agilent 3458A ³ voltmeter on its lowest measuring range 100 mV (the resistor and the voltmeter 3458A are not shown in figure 3).

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For AC voltage measurements, there are two important features which are essential for the correct use of AC-QVM. The first feature is the synchronization between synthesized waveform (stepwise approximation of sine wave), the measured AC voltage and sampler, while the other one is the grounding point defined at the outer BNC connector of the input of a fast sampler (PXI NI 5922 ³ ) which operates on power mains. This sampler is used to digitize the difference voltage and it operates with the 1 MΩ differential input. The 48-tap standard finite impulse response filter is selected. A Keithley 3390 50 MHz arbitrary waveform generator is used to set the phase shift between the synthesized waveform and the measured AC voltage by locking its output for the chosen phase difference (details are described in [5]). For AC measurements, the following setting is made (expressed for the completeness of information and chosen as an optimal set-up): a frequency of 62.5 Hz, 20 Josephson steps per period and 4 MSa s⁻¹ sample rate for the sampler. To cut out the transients from the synthesized waveform 200 readings are deleted on both sides of the transients. Typically, 30 periods are measured in a loop (0.48 s) for one RMS value, and then averaging RMS values during one to two minutes is performed. Further details are described in [5].

The DC set-up is given in figure 4. During DC measurements a Fluke 5720A calibrator ³ (DCCS) is set on voltage output (as we only investigate small currents) and is left floating, while the grounding point is set in the middle point of two resistors.

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For this resistance comparison at DC current, the AC-QVM is used as common DC quantum standard, and it measures the voltage U_DC on the reference resistor R_DC, and after that the voltage U_Sh on the shunt resistors R_Sh, in a time series with manual reconnection, including four sequences altogether with different polarity of the PJVS voltage and current flow. Finally, the DC resistance of the AC shunt R_Sh is obtained:

$$\begin{equation}{R}_{\text{Sh}}={U}_{\text{Sh}}{R}_{\text{DC}}/{U}_{\text{DC}}.\end{equation} \tag{ 1 }$$

The AC shunt R_Sh (in our case a Fluke A40B ³ for 1 mA, 10 mA, and 20 mA, as appropriate; see data in table 1) [13] is compared to reference resistors R_DC (Fluke 742A ³ −10 k, −100 k or −1 k). These reference standards were chosen due to their robustness (because they are rigid and compact for easy handling), loading capabilities (to handle the relatively high currents) and very small temperature coefficients (which means they could be used at room temperatures without insertion in specially developed thermostat); their characteristics are given in table 2. In the application of R_DC in this experiment, the current level for chosen particular reference resistor was held below the maximal DC current to maintain the specified accuracy value, I_DCACC, to avoid additional uncertainty contribution due to its self-heating; the exception is only one measuring point at 6.5 mA when Fluke 742-100 is used (which is relatively little higher than 5 mA).

The frequency dependence of the shunt resistance is almost negligible for such coaxial current shunts. For the frequency range from DC up to 62.5 Hz it can be assumed that it is <0.3 μA A⁻¹, as shown by the experimental results given in [7], but also from other analysis even for higher currents and/or higher frequencies [14–18]. Therefore, the same shunt can be used for AC measurements, using the value R_Sh obtained in (1) as its resistance value (figure 5). The AC-QVM measures the voltage U_Sh, while the current I_AC for calibrating the PS-HVAC is equal to:

$$\begin{equation}{I}_{\text{AC}}={U}_{\text{Sh}}/{R}_{\text{Sh}}.\end{equation} \tag{ 2 }$$

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The outer pin of the digitizer input connector defines the grounding point of the measurement set-up as the digitizer. In this configuration the Fluke 5720A calibrator (ACCS) is set on the current output. For simultaneous measurements of the AC current, it is essential that the connection of the shunt R_Sh and the PS-HVAC in series does not cause a problem for the AC voltage measurement by the AC-QVM due to grounding or synchronisation. To verify influences of the PS-HVAC on the AC-QVM reading and cable reflections due to a 25 m long connecting cable measurements with connection schemes B and C (see figure 5) have been made and are discussed in section 3.3. The best calibration results were obtained, and were possible, with simultaneous AC current measurements by both AC-QVM and PS-HVAC, and the control notebook of the PS-HVAC powered on battery. Such setting was not possible with the shunt Fluke A40B-1mA which uses an amplifier that must be powered all the time. In that case the measurements were done successively.

The reference resistors R_DC used at DC measurements are Fluke type 742A (table 2) with their predicted resistance values. These standards were calibrated before this measurement campaign against a quantum Hall resistance using a cryogenic current comparator (see figure 1) with a relative uncertainty of 0.15 μΩ Ω⁻¹ [19–21], that can be assigned to such a type of resistor. To verify our DC calibration procedure (figure 4) and the calibration values for these resistance standards we performed DC comparisons of them. An example of such a 1:10 ratio measurement using Fluke 742A-100 and 742A-1k resistors is given in figure 6. Each measurement point represents one complete measurement set during a time interval of 5 min i.e., each measurement contains four 1 min sequences with different PJVS voltage and current polarities. A ratio can be measured with a type-A uncertainty of 0.2 × 10⁻⁶ for a current of 0.9 mA. If the current is halved to 0.45 mA, the type-A increases to about 0.45×10⁻⁶. It can be seen from the results that the stability of the determined ratio of about 0.15 × 10⁻⁶ for the same current can be expected without many measurement repetitions.

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In figure 7 the results of the DC calibration of a Fluke A40B-10 mA shunt are shown (as relative errors to a nominal value of 80 Ω) for currents 1.6 mA, 3.3 mA and 6.5 mA using a Fluke 742A-100 as reference. It is obvious from the results that the current dependence is an issue which must be taken into account, i.e., the calibration should be performed at the same current level used for the calibration of the PS-HVAC. The relative experimental standard deviation of the measured shunt resistance of one measuring sequence is 0.26 μΩ Ω⁻¹ at 1.6 mA, 0.13 μΩ Ω⁻¹ at 3.3 mA and 0.07 μΩ Ω⁻¹ at 6.5 mA.

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For comparison, when calibrating a Fluke A40B-20 mA shunt, it can be as low as 0.24 μΩ Ω⁻¹ for a 15 mA current, while for a Fluke A40B-1 mA shunt as low as 1.1 μΩ Ω⁻¹ for a 0.35 mA current.

The stability of an AC current measurement at 62.5 Hz using a Fluke A40B-10mA AC shunt at 3.3 mA is given in figures 8(a) and (b) and in different ways, as explained below.

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In figure 8(a) the differences (as relative errors) between a measured value and the mean value of the 30 min interval are given and marked with ΔI. Each point represents the result of a one-minute measurement started every minute again. Average values for five repeated points obtained in that way have a typical standard deviation of 0.5 μA A⁻¹, while for continuous measurement (without starting the measurement after every minute again) could be even lower. This means that the calibration of the AC current can be done within the time interval of a few minutes with satisfying uncertainty, and this is pointed out in figure 8(b), where the relative Allan deviation is given for such measurement sequence, showing that, after about two to 5 minutes, a relative uncertainty of (0.2 to 0.3) μV V⁻¹ can be achieved.

As an example, the calibration of the PS-HVAC calibration on the 2 kΩ range at 1.6 mA is described here, and summarised results are given in table 3. In the first two columns, the reference AC current I_AC, measured by AC-QVM according to (2), and associated relative standard deviation of the mean s_x are shown; in the last four columns there are data relevant to the PS-HVAC: measured AC current I_HVAC, associated relative standard deviation of the mean s_x , relative error e_I of the measured current. In the last column, the relative correction for the measured current c_I is given. This is the correction which should be applied to the PS-HVAC as the value obtained through this calibration process; all relative values are expressed in μA A⁻¹. A Fluke A40B-10mA shunt, that is calibrated at DC current with a Fluke 742A-100 reference standard, is used. After the first measurement, the relative correction c_I is calculated and adjusted, and such iterative process is repeated until the error e_I falls within ±0.5 μA A⁻¹. Before continuing with the next calibration point, usually two more readings were taken to confirm that the satisfied stability is ensured. The final correction c_I at this calibration point is 43.0 μA A⁻¹; this result is given in table 4, too.

The overall calibration results for the PS-HVAC on all ranges are given in table 4, where the procedure to obtain the relative correction for measured current c_I at one calibration point is the same as explained for the results presented in table 3. In the first column of table 4 the reference resistor of type Fluke 742 used at DC measurements is given, in the second column the AC shunt of type Fluke A40B used for AC measurements is given, while in the third column the reference AC current used for calibration is shown. In the fourth, fifth and sixth columns the data related to PS-HVAC are given: selected range, utilization (expressed as percentage of the full scale), and determined c_I , respectively. The presented results are very satisfactory and representative even though obtained within the limited time and as first experience. We expect that a repetition of such exercise could lead to even better results.

In ideal case the values of c_I should be zero. If that is not the case, as obvious from the results given in table 4, this can be expressed as 'gain error' of the PS-HVAC on its selected range. Furthermore, the values of c_I obtained at different current for the same range (for instance, at 30%, 50% and 100% of full scale) should be the same. If this is not the case, the differences can be expressed as 'linearity error'. The obtained results show that such linearity error (if it exists) is comparable to the measurement uncertainty (see table 5). It is also worth to notice that the calibration on the 2 kΩ range, where the same values of c_I are obtained at 50% and 100% of full scale, were achieved by using two different reference resistors for the DC measurements and two different shunts for the AC measurements. Such an agreement raises the overall confidence in the calibrations and the PS-HVAC traceability chain. Furthermore, it emphasizes the possibility of the application of the AC-QVM for fast, accurate and traceable precision DC to AC measurements.

The results presented in tables 3 and 4 were obtained when the reference current feeds directly to the input of the PS-HVAC system. However, for the use in the high voltage set-up, it is necessary to connect the PS-HVAC with the 25 m long coaxial cable, and the evaluation of the influence of such cable on the calibration results is necessary. Furthermore, the influence of the PS-HVAC on the calibration using the AC-QVM must be evaluated (see connections schemes B and C of figure 5), too.

Thus, a measurement of the frequency dependence for the Fluke 5720A 3.3 mA output current on the ACI range from 31.25 Hz to 1 kHz using the shunt Fluke A40B-10mA is presented in figure 9. ∆I is the relative difference for the two cases: (A) with and (B) without the connection of the PS-HVAC system in series with R_Sh (see connections in figure 5). An additional load could cause a linear frequency behaviour which is fitted by the black dashed line in figure 9. The differences at 62.5 Hz and 125 Hz are at a level of −0.1 μA A⁻¹ and −0.2 μA A⁻¹, respectively.

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The error of the output current ΔI, measured with the PS-HVAC system with (C) and without (A) the 25 m long coaxial cable connected in series, is performed, and their differences are also presented in figure 9. The frequency was varied from 31.25 Hz to 125 Hz, again with Fluke 5720A calibrator used as AC current source. As the differences of both measurements agree well within 1.1 μA A⁻¹ we can conclude that the connection of PS-HVAC and the long cable, cases (B–A) and (C–A), have a negligible influence at 50 Hz. The blue dashed line shown in figure 9 indicates a quadratic fit which is well-known from cable error investigations in a measurement loop. The fit is based on typical numbers from [23] and by taking the 25 m long cable into account. The result shows that such an error should be very small at a frequency of 50 Hz.

In table 5, the input quantities and analysis of their uncertainty contributions are shown, and the relative combined standard uncertainty of the calibrated current is calculated as follows:

$$\begin{equation}{u}_{\text{cr}}\left({I}_{\text{CAL}}\right)=\sqrt{{u}_{\text{r}}^{2}\left(\delta {I}_{\text{ref}}\right)+{u}_{\text{r}}^{2}\left(\delta {I}_{\text{Sh}}\right)+{u}_{\text{r}}^{2}\left(\delta {I}_{\text{ACDC}}\right)+{u}_{\text{r}}^{2}\left(\delta {I}_{\text{RMS}}\right)+{u}_{\text{r}}^{2}\left(\delta {I}_{\text{CAL}}\right)}.\end{equation} \tag{ 3 }$$

The input quantities are: δI_ref—comparison of the reference resistance standards at DC current, which includes the uncertainty of the used reference and its time stability; δI_Sh—calibration of the AC shunt at DC current; δI_ACDC—AC–DC difference for the used AC shunt; δI_RMS—measurement of the AC current using an AC shunt and the AC-QVM; δI_CAL—residual contribution of the PS-HVAC calibration due to the set-up and measurement procedure. All given standard uncertainties and combined uncertainty are expressed as relative values in μA A⁻¹. Further uncertainties from the AC-QVM are at a level of 10⁻⁸ for AC calibrations [24] and even better for DC calibrations [5, 6] and can be neglected here.