Uncertainty analysis for ac-dc di erence measurements with

Uncertainty analysis for ac-dc difference measurements withthe AC Josephson Voltage Standard

Jason M Underwood

National Institute of Standards and Technology, 100 Bureau Drive, MS 8172, Gaithersburg, MD20899-8172, USA

E-mail: [email protected]

Abstract. A detailed analysis of the uncertainties obtained in ac-dc difference measurementswith an AC Josephson Voltage Standard (ACJVS) is presented. For audio frequencies and forvoltages less than 200 mV, ac-dc transfers with the ACJVS may reduce the combined uncertaintyby factors of 2 to 10, compared with conventional methods based on thermal converters. Type Auncertainties are predominantly limited by the thermal transfer standard (TTS), or the digitalvoltmeter used to acquire the output voltage from the TTS. In agreement with earlier work, thetransmission line is the primary contributor to Type B errors for frequencies above 10 kHz. AMonte Carlo sensitivity analysis is used to demonstrate how the uncertainties of transmission lineimpedance and on-chip inductance impact the accuracy of the rms amplitude conveyed to theTTS.

ACJVS uncertainty analysis 2

1. Introduction

The forthcoming redefinition of the SI is moti-vated in part by the success of quantum-basedelectrical standards, such as those based on theJosephson effect [1]. The appeal of standardswith realizable physical quantities, which aretraceable to fundamental constants, invariantwith respect to time and environment, and read-ily disseminated cannot be overstated. Over aperiod of roughly two decades, the conventionalJosephson voltage standard (CJVS) reduced dcvoltage measurement uncertainty by about fourorders of magnitude over the Weston cell stan-dards of the early 20th century.

With the advent of the pulse-driven Josephsonvoltage standard in 1996, similar improvementsin ac voltage uncertainty were expected [2]. Af-ter decades of development (see e.g., [3] and ref-erences therein), the pulse-driven JVS, hereinreferred to as the AC Josephson Voltage Stan-dard (ACJVS)‡, has indeed improved uncertain-ties for some regions of voltage-frequency space.However, the frequency response of the inter-connection between the ostensibly perfect volt-age on-chip and the device under test (DUT)presents significant challenges to scaling beyondaudio frequencies. The fundamental issue is thatthe transmission line cannot be sufficiently char-acterized to provide a suitably-accurate correc-tion without resorting to calibration with artifactstandards [4].

The National Institute of Standards and Tech-nology (NIST) has been actively disseminatingJosephson Voltage Standards under its StandardReference Instrument program § to U.S. primarystandards laboratories, as well as to NationalMetrology Institutes in other countries. Thisprogram now includes the ACJVS, and thus acomprehensive uncertainty analysis is warrantedin order to help end users understand the limi-

‡ The pulse-driven JVS is alternately known as theJosephson Arbitrary Waveform Synthesizer (JAWS).§ NIST standard reference instruments: https://www.

nist.gov/sri.

tations of such instrumentation. Presently, themost common application for the ACJVS isac-dc difference calibration of commercial ther-mal transfer standards (TTS). As a result, suchmeasurements are the primary focus in this re-port.

2. Operating Principle of theACJVS

The operating principle of all Josephson voltagestandards is the inverse Josephson effect, inwhich the Josephson junction functions as anideal frequency-to-voltage converter. For a dcoutput voltage, this relationship is expressedas,

Vdc =nNhf

2e, (1)

where n is an integer representing the Shapirostep, or spike, to which the junctions are biased,N is the number of junctions, h is the Planckconstant, e is the charge of an electron, andf is the frequency of the applied microwavebias.

For the ACJVS, it is the pulse area quantizationbehavior of the Josephson junctions that yields apractical ac voltage standard. This follows fromthe Josephson relationship between the instanta-neous voltage v(t) and the phase difference φ(t)across the junction,

v(t) =~2e

dφ

dt. (2)

If the junction is biased with a dc current belowits critical current threshold Ic, φ is constant andthe junction produces zero voltage. If the dc biasI > Ic, φ(t) will evolve monotonically in time.Now, if a current pulse is applied to the junctionin such a way that the net change in φ is 2π wehave, ∫

v(t)dt =~2e

∫dφ

dtdt =

h

2e, (3)

and thus the junction produces an outputpulse with a voltage-time area of one flux

https://www.nist.gov/sri

https://www.nist.gov/sri


quantum, Φ0 = h/2e. In order to synthesize asinusoid, for example, a sequence of flux quantaare generated in which the interval betweenpulses varies inversely with the voltage of thesine wave. Such pulse-density encoding ofthe waveform is performed by a delta-sigmamodulator (described below).

As the pulse duration and spacing is oforder 10 ps, the instantaneous output voltageis extremely broadband, containing spectralcontent from below the audio band to severaltens of gigahertz. For rms measurements, thisraw waveform must be appropriately lowpassfiltered, so that the power in the higher frequencybands is insignificant relative to the power in thetone itself.

2.1. Delta-Sigma modulation

In order to synthesize a pulse pattern appro-priate for a specific waveform, an oversampleddelta-sigma modulator is employed that convertsthe mathematically-defined waveform into a cor-responding pulse pattern. This initial conver-sion occurs entirely in the digital realm, but isan analog-to-digital transformation in the sensethat the initial waveform samples are representedin double precision and the pulse pattern sam-ples are represented by only 2 to 3 discrete lev-els (e.g., −1, 0, +1 for a bipolar waveform).Once the pulse pattern is created, it is then up-loaded to a pattern generator, which delivers themicrowave pulse sequence to the ACJVS. TheJosephson junction array(s), in turn, transformthe incoming “digital” pulse pattern into an ana-log waveform whose voltage is traceable to fun-damental constants. A representative plot of thespectral density of the pulse pattern is shown inFig. 1. The encoded waveform is a 10 kHz si-nusoid with an amplitude that is −1 dB withrespect to the full scale amplitude of a givenJosephson junction array‖. The in-band part ofthe spectrum is shown with white background,

‖ Because of dynamic response limitations of theJosephson junctions, the actual pulse patterns used withthe ACJVS are often limited to about 90 % of full scale.

while the typical out-of-band span is depictedwith a gray background. The in-band signal-to-noise ratio (SNR) is 166 dB. As a point ofreference, a commercial 24-bit delta-sigma au-dio digital-to-analog converter (DAC) of com-parable bandwidth has a typical SNR of 80 dBto 100 dB. Likewise, the spurious free dynamicrange (SFDR) of the ACJVS pulse pattern for a1 kHz sine wave is 210 dB (not shown), whereasa commercial audio DAC might have an SFDRas low as 120 dB.

2.2. Typical performance operating space

As with the other JVSs, many thousandsof junctions are needed to generate voltagesrelevant for ac metrology. Pushing the upperamplitude beyond 1 V is an active researcheffort [6], and faces integration challenges bothon-chip and in terms of bias electronics. Forexample, a 1 V ACJVS requires at least twoseparate microwave biases and four separate low-frequency compensation biases. An rms voltageof 2 V has been achieved without increasing thenumber of microwave lines, through the use ofadditional on-chip Wilkinson dividers [7]. But,such a device would require 8 compensation lines,or the number of junctions per array would needto be doubled.

At present, the ACJVS frequency range spansthe space between 1 Hz to more than 10 MHz,as well as dc. Note that while the ACJVS iscapable of operating in this space, the respectiveuncertainties may vary greatly. The outputbandwidth is predominantly limited by thepattern generator memory for low frequencies,and by the transmission line interconnect at highfrequencies. The maximum possible voltage ina waveform scales with the update, or symbol,rate fs of the pattern generator. The symbolrate is usually fixed at an upper limit, contingentupon the capabilities of the pattern generatoror the characteristic frequency of the Josephsonjunctions. The symbol rate for ACJVS systemsat the United States National Institute ofStandards and Technology (NIST) is typically


104 106 108 1010-250

-200

-150

-100

-50

0

Figure 1: Calculated power spectrum of the delta-sigma pulse sequence for a 10 kHz sine wave. Theraw spectrum is shown in black, the response of a 2nd-order low-pass filter is shown by the blue curve,and the filtered spectrum is depicted by the red dashed line. The white area indicates the region overwhich the signal-to-noise ratio is calculated. The input pulse sequence was processed with a Hannwindow prior to computing the spectrum. The power spectrum is sinewave-scaled and the ordinateunits are decibels with respect to modulator full-scale per noise bandwidth (cf. Ref. [5] for details.)

(10 to 20) GHz.

Commercial instrumentation is available thatcan generate pulse patterns at update rates near100 GHz and store pulse sequences larger than4 GSa (1 GSa = one billion samples), althoughboth of these capabilities may not exist in thesame instrument. For a given symbol rate andpulse pattern length L, we can determine thepattern repetition rate f0 = fs/L. The largepattern memory allows waveforms with periodsof up to 1 s. Naturally, such patterns requirecorrespondingly long wait times to upload thesequence to the generator and to verify properquantum operation. By way of comparison,the Programmable Josephson Voltage Standard(PJVS) can be configured to provide time-

dependent waveforms with very high accuracyfor frequencies below 100 Hz with no appreciabletime penalty [8]. However, such piecewise-defined waveforms would not be suitable for rmsmeasurements.

3. Experimental Configuration

For evaluation of Type A uncertainties andfor validation of certain Type B error models,we used a specific experimental configuration(summarized in Table 1) involving an ACJVSdie with two Josephson junction arrays anda Fluke 792A thermal transfer standard ¶.

¶ Commercial instruments are identified in this paper inorder to adequately specify the experimental procedure.Such identification does not imply recommendation or


Additional details concerning this experimentalarrangement are discussed in 6.2.4 and asimplified schematic is given in Figure 2.

The dip probe was configured differently forsingle-array and dual-array measurements. Formeasurements at an rms voltage of 200 mV, acopper jumper wire was used at the cryopack-age to connect the two arrays and a separatetransmission line was run from the cryopackageto a separate output port at the room tempera-ture end of the probe. For all other voltages, thearrays were configured with independent outputlines. The experimental parameter space spansfrequencies from 100 Hz to 100 kHz, and rmsvoltages from 2 mV to 200 mV.

4. Limitations and Comparison to EarlierWork

Systematic errors specific to the ACJVS havebeen discussed extensively in [4, 9, 10, 11, 12,13]. Uncertainty analyses for the ACJVS thatconsider most, if not all, known error sourceshave been presented in [14, 15], as well as ininternational comparisons using a Fluke 792A asthe transfer standard [16, 17]. In Refs. [9, 14], itwas highlighted that several systematic errors,such as those due to pulse bias feedthroughand compensation bias, can be largely mitigatedthrough optimized device layout/fabrication andimproved experimental bias hardware. TheACJVS system used in this work incorporatesmany of these advancements, including (1)refined on-chip low-pass filter design [18, 19]and (2) integrated bias electronics that combineboth the bitstream generator, CW microwavegenerator, and compensation generator into asingle enclosure with very good control oversynchronization [20].

In cases where systematic errors cannot be sat-isfactorily reduced through engineering, proce-dures have been developed to measure and cor-

endorsement by the National Institute of Standardsand Technology, nor does it imply that the equipmentidentified is necessarily the best available for the purpose.

rect for errors related to compensation bias [9],dc blocks and feedthrough in the pulse bias [9,10, 11], and the output transmission line [10,11, 4, 12, 13]. The utility of these correc-tion procedures depends on the particular er-ror being considered. Unfortunately, in someinstances corrections are used without consid-ering the uncertainty involved in their deriva-tion or measurement. For example, using room-temperature measurements of ACJVS transmis-sion line parameters to correct for the line’s re-sponse neglects temperature-dependent phenom-ena like the skin effect, which can be significantabove 20 kHz.

The focus in this report is to assist non-expertusers of the ACJVS to better understand itsmost significant limitations and to develop theirown uncertainty budgets. For reasons relatedto inexperience or limited time, we find thatcalibration staff usually wish to avoid the tediousmeasurements or simulations required for propercorrection of ACJVS systematic errors. On theother hand, calibration staff are keenly interestedin a quantitative assessment of the uncorrecteduncertainties. As a result, this report doesnot present the lowest attainable uncertaintiesfor any given ACJVS, nor does it present theworst-case uncertainties of all ACJVS systems.Rather, we outline the mechanisms responsiblefor the known systematics and provide nominalestimates of their relative contributions to theoverall uncertainty budget. For compensationand transmission line systematics, particularattention has been given to estimating theuncertainty involved if a user wishes to performtheir own correction.

Two uncertainty summaries are provided below:one assuming a user makes no corrections tosystematic errors (“Uncorrected”) and anotherwhere it is assumed that a user properly cor-rects for the transmission line and quadratureeffects (“Corrected”). If it is preferable, thereader may interpret the corrected uncertaintiesas “achievable” uncertainties. The only uncer-tainties claimed or realized in this report are


Equipment Make/Model/Parameter CommentsDevice/cryopackage NIST Superconductive

Electronics GroupDip probe is wired withseparate output transmissionlines for each array, and forthe two arrays in series.

Number of arrays 2 Arrays connected in series oncryopackage with Cu wire forV = 200 mV.

Number of JJs per array 6400Critical current (nominal) 5 mAPulse pattern generator HSCC ABG-2Digitizer NI PXI-5922Compensation bias amplifier VMetrix A-200 Isolation from ground is

approximately 1 MΩ‖40 pF.Inner/Outer DC Blocks Cinch DCB-3511 Cutoff frequency ≈ 500 MHz

Table 1: Parameters of the Josephson junction devices and experimental equipment used to determineuncertainties in this report.

those of the uncorrected category. Type A un-certainties obtained from measurements with aTTS are only included in the final, overall uncer-tainties. All other uncertainties are considered ofType B, even if statistical methods were involvedin their estimation (as in the Monte Carlo simu-lations for the transmission line). This is becauseeven though the parameter values were chosenrandomly, the distribution is assumed and themodel is perfectly deterministic.

The majority of the analysis presented is fora single array. Measurements at 200 mVrequired the operation of two arrays in series.As a result, we have included an additionaluncertainty term for these points that representsthe combined effects of the outer dc blockson the microwave lines and the parasiticcapacitance to ground of the isolation amplifiers.Errors due to low-frequency feedthrough inthe pulse bias (which can drive the arrayinductance) are not considered here becausesuch feedthrough is adequately filtered up to100 kHz (the upper frequency limit of ourmeasurements) by the inner dc blocks. Thisis confirmed by measurements when the pulsebias amplitude is below the threshold for

driving, or pulsing, the Josephson junctions (seee.g., [3, 9]). Although we include discussionsof errors due to electromagnetic interference(EMI) and connection repeatability, such errorsare not assessed in our overall uncertaintyestimates.

5. Type A Uncertainty Analysis

Type A uncertainties for an ac-dc transferwith the ACJVS are predominantly limitedby the noise inherent in the transfer standarditself. During ac-dc transfers on the 792A,the dc measurement points often show muchmore variability than the ac measurementpoints. Thus, the uncertainties depicted inTable 2 mostly represent the stability of repeatedmeasurements of dc input voltages with the792A. Deficiencies in the experimental setup(e.g., ground loops, electromagnetic interference)may give the appearance of statistical variabilityin the ACJVS’ output. Therefore, the usershould make every effort to eliminate oraccount for such effects prior to performing atransfer.

The increased variability observed for dc points


I/O DC blockspulse pattern generator

floating compensation source

DUT or digitizer input impedance (typically floating)

termination resistor

coax cable

twisted pair cable (inside dip probe or cryostat)

inter-array wire jumper (may be on-chip, on cryopackage, or wired to outside of cryostat)

JJ array with intrinsic inductance

array 1

array 2

Figure 2: Schematic of the dual-array ACJVS network used in this report, showing the arrangementof bias sources, Josephson junction arrays, and their interconnection. The connection shown herecorresponds to that used when both arrays are operated in series to obtain an rms voltage of 200 mV.For rms voltages less than or equal to 100 mV, each array was used separately, and the jumper removed.

can be attributed to the 1/f noise of the thermalsensor and associated electronics within the792A. The 1/f noise corner frequency variesfrom unit to unit, but is in the range of 200 Hzdown to 2 Hz for the thermal sensor itself.Correspondingly, ac-ac transfers for frequenciesabove the 1/f corner (e.g., 1 kHz vs 100 kHz)exhibit reduced variability.

5.1. Data Collection and Analysis

In conventional ac-dc difference measurements, δis defined as,

δ ≡ Qac −Qdc

Qdc

∣∣∣∣Eac=Edc

, (4)

where Qac is the ac input quantity (e.g., rms

current or voltage) that produces the sameresponse Edc at the output of the transferstandard as the dc input quantity Qdc. There aredifferent methodologies to perform the transfer,depending on the accuracy desired and the typeof standard under test. One example is thenull method, wherein the operator first appliesan ac input Qac and notes the response Eac.Next, a positive dc input is applied and thenadjusted until the response Edc+ is equal to Eac.After achieving a null, the input dc value isthen noted as Qdc+. In order to correct forreversal error and thermovoltages, the last stepis repeated with the polarity of the dc inputreversed to obtain Qdc−. Thus, the operatorobtains three input quantities: Qac, Qdc+, Qdc−,


such that their respective responses are equal:Eac = Edc+ = Edc−. In this methodology, thedc input reference value is taken to be the meanof the two dc polarities,

Qdc =Qdc+ +Qdc−

2. (5)

An equivalent relationship is assumed forEdc. To calibrate a transfer standard inthis arrangement requires that either (1) theapplied inputs are known (e.g., using an idealsource), or (2) that the transfer standard iscalibrated against a reference device with aknown ac-dc characteristic (e.g., using a thermalconverter).

For the ACJVS, ac-dc difference measurementsof the 792A are typically performed using thedeflection method. The voltages applied withthe ACJVS are known to high accuracy ataudio frequencies. Thus, rather than adjust theapplied dc voltage to achieve the same outputresponse, the relevant metric is the differencein output response to dc and ac inputs of thesame rms amplitude. In order to provide someconnection to the conventional definition abovefor δ, it is assumed that an incremental relativechange in the input to a thermal converterproduces a proportional relative change in itsoutput [21],

∆E

E= n

∆Q

Q. (6)

In other words, the response function follows apower law E(Q) = kQn, where k is a constant.We then obtain the following relationshipbetween δ and the output responses,

δ ≈ δdefl. =Edc − Eac

nEdc

∣∣∣∣Qac=Qdc

.(7)

The 792A is sufficiently linear (n ≈ 1)that errors in δ due to residual nonlinearityare correspondingly small. For example, anonlinearity of 0.5 % (greater than typical) yieldsa relative error in δ of about the same magnitude,and thus we assume n is unity for measurementsof the 792A. For thermal converters, however,

1.6 < n < 2 and it becomes more critical tocarefully evaluate n.

In this work, the data used to derive Type Auncertainties were obtained by (1) performingmultiple ac-dc transfers at 1 kHz for eachvoltage and range listed, and (2) multiple ac-ac transfers against 1 kHz for the frequenciesand voltages outlined below. The ac-dc andac-ac measurements were performed in noparticular order. But, ac-dc measurementswere periodically interspersed over time to checkfor any long-term drift or otherwise anomalousbehavior. Short term drift — that is, statisticallysignificant deviations over timescales comparableto the measurement of each point of an ac-dc transfer — was a feature of most dcmeasurements. As noted above, such drift is dueto the 1/f noise of the 792A’s thermal sensorand input amplifier stages.

The uncertainties in Table 2 represent the aver-age of the daily, pooled sample standard devi-ation of numerous ac-dc transfer measurementsover the course of months on one of NIST’s setof 792A transfer standards. The pooled standarddeviation is given by,

s2x,pooled =

1

MN − 1

(M∑i=1

(N − 1)s2

xi+Nx2

i

−MNx2

), (8)

where M is the number of transfers in a givensequence, N is the number of acquired readingsof the output voltage of the TTS during eachtransfer, sxi and xi are the standard deviationand mean, respectively, of the ith transfer, andx is the pooled sample mean.

It should be emphasized that we do notuse the standard deviation of the mean (or,standard error SE(x)) to represent our Type Auncertainties. This is for two reasons. First, theSE(x) is an inferential statistic that describeshow well the mean x of a single sample (ofN individuals, or digital multimeter readings)estimates the population mean µ,

SE(x) = sx =σ√N, (9)


where σ is the population standard deviation.While it is true that we are primarily interestedin how well our sample mean of δ estimatesthe true mean, we must first have a reliablevalue for σ. If σ is not known a priori,then the best estimate is the pooled samplestandard deviation. Furthermore, all reporteduncertainties would be based upon a particularchoice for N , and would need to be scaled tofacilitate comparison with other measurements.The pooled sample standard deviation shouldbe approximately fixed for a given 792A, underconditions of repeatability.

Secondly, the representation of uncertainty withthe standard error is only valid when thesampling distribution of the transfer means isapproximately normal (i.e., when the centrallimit theorem assumptions are valid). If δis indeed drifting between transfers, then theunderlying population is not stationary andthe use of SE(x) is inappropriate. In ac-dc measurements, deviations from normality inthe sampling distribution of transfers may bechecked with a sufficiently powerful test, suchas the Anderson-Darling test. However, eventhe most powerful test statistics require manytransfers (i.e., M > 50) to facilitate a gooddecision.

6. Type B Uncertainty Analysis

6.1. Intrinsic errors

Assuming perfect quantization of the inputpulse bias sequence, the ACJVS has twoinstrinsic errors that derive from its basicoperating principle as a quantum voltage source.These include phase noise from the pulsepattern generator and the delta-sigma conversionerror.

6.1.1. Phase noise All Josephson voltage stan-dards are based on the inverse Josephson effect,wherein highly accurate time and frequency con-trol is leveraged to generate accurate voltages.Variability in the bias frequency (or pulse density

per unit time for the ACJVS) that results fromphase noise in the microwave bias electronics willbe directly converted into errors in voltage. Thephase noise error is typically much smaller thanextrinsic systematics for the ACJVS. As a re-sult, our goal here is simply to estimate an upperbound on the phase noise for a typical system.For a more detailed analysis of this topic, the in-terested reader is referred to the recent study byDonnelly et al. [22].

First, we consider the relative voltage error dueto fluctuations in the pulse density p(t),

σV (τ)

V (τ)=σp(τ)

p(τ)= σy(τ) , (10)

where τ is the epoch over which the measurementis completed, the denominators of the leftequation are averages over that epoch, and σy(τ)is the Allan deviation of fractional frequencyfluctuations y = δν/ν0. The Allan deviation canbe estimated by integration of the phase noisespectral density (typically represented as L(f)in units of [dBc/Hz]) [23, 24],

σy(τ) =4

(πτν0)2

∫ fH

0

10L(f)/10 sin4(πτf)df , (11)

where ν0 is the oscillator frequency at whichL(f) is given, and fH is an upper frequencycutoff beyond which the phase noise is negligible.For example, since the on-chip filters have abandwidth on the order of 100 MHz, it isreasonable to assume that fH ≤ 100 MHz.

Because the instantaneous output voltage of theACJVS directly depends on the pulse densityfunction p(t), any errors in the relative timingof pulses will lead to errors in voltage, just aswith other JVSs. However, for dc, the voltagemeasurement bandwidth can be made arbitrarilysmall by sampling for a very long time. Forac waveform synthesis, the bandwidth dependson the transmission line, or at a minimum, onthe on-chip filters. Another way to frame therelationship is to ask how well we need to knowa particular property of a waveform in the timeinterval τ? If we are only concerned about the


Type A Standard Uncertainty of AC-DC Difference, uA / (µV V−1)

792A Range(mV)

RMS Voltage(mV)

100 Hz 400 Hz 1 kHz 2 kHz 5 kHz 10 kHz 20 kHz 50 kHz 100 kHz

22 2 62.5 58.9 55.8 64.4 59.8 59.7 57.7 59.0 57.422 6 20.2 19.4 18.9 19.3 19.3 19.2 19.3 19.2 19.222 10 12.9 12.4 12.2 12.4 12.5 12.5 12.7 12.8 12.622 20 6.8 6.6 6.5 6.6 6.6 6.6 6.6 6.6 6.6220 20 24.7 24.5 22.1 24.6 25.2 25.2 25.8 25.4 25.6220 60 4.3 4.0 3.2 4.0 4.0 4.0 4.0 4.3 4.2220 100 2.3 2.0 1.6 2.0 2.1 1.9 2.1 1.9 2.1220 200 0.93 0.93 0.86 0.95 0.90 0.91 0.91 0.90 0.92

Table 2: Type A standard uncertainties (k = 1) for AC-DC difference measurements of a 792Athermal transfer standard with the ACJVS. Listed uncertainties are typical. Actual uncertainties willvary with the particular 792A used, as well as the experimental setup. Deviations as large as 30 %from the tabulated uncertainties were found for certain 792A instruments within otherwise identicalexperimental arrangements.

overall rms content of a sine wave, then τ isjust an integer multiple of the period of thatwaveform. In that case, the impact of generatorphase noise is quite small indeed. However, if wewant an estimate of the waveform’s value at agiven instant in time, then τ may approach thelimit imposed by the on-chip filters.

With those caveats in mind, we can begin toestimate σy(τ). It is assumed that the patterngenerator is phase locked to a precision externalfrequency reference. If the phase-locked loopis functioning properly, the phase noise of thegenerator should resemble that of the reference,and be flat above about 1 kHz. If we furthertake the worst case scenario that τ−1 is given bythe bandwidth of the on-chip elements (severalhundred MHz), then we can work backwards todetermine the phase noise performance neededto achieve a given voltage uncertainty at anupdate rate of about 15 GHz. For a relativevoltage error less than 5 parts in 107, the patterngenerator needs to have a phase noise better than-150 dBc/Hz, which is not trivial to achieve inpractice. However, if our concern is the rmsamplitude of a 1 kHz sinusoid, then the relativeerror falls to 0.2 nV/V with a phase noise of just-120 dBc/Hz.

For the purposes of this report, it is assumed that

the worst-case uncertainty due to phase noise is1 nV/V. This value is based on a measurementepoch τ = 30 s and L(f) = −80 dBc/Hz,independent of frequency. The basis for thisdecision is that commercially available frequencyreferences are readily available with phase noiselower than L(f) = −80 dBc/Hz and therelative contribution to the overall uncertaintyfor this choice is already negligible. For thereader’s convenience, the 1/τ scaling of σV /V fordifferent values of L(f) is depicted graphically inFigure 3.

6.1.2. Delta-Sigma conversion error A delta-sigma converter takes advantage of oversamplingand noise shaping to shift in-band quantizationnoise power (IQNP ) to frequencies outside theband of interest. This is what gives rise tothe rapid upward increase of the quantizationnoise with frequency shown in Fig. 1. Thehigh out-of-band QNP is low-pass filteredby on-chip superconductive filters, and to alesser extent, frequency-dependent loss in theoutput transmission line. If the outputs ofthe Josephson junction arrays are adequatelyfiltered (or the transmission line and/or TTS arethemselves band-limited), then the out-of-bandQNP will have negligible impact on rms voltagecalibrations at audio frequencies.


10-6 10-4 10-2 100 10210-8

10-6

10-4

10-2

100

102

104

106

Figure 3: Relative uncertainty in parts per million of the output voltage of the ACJVS due tofrequency reference phase noise, in the form of an Allan deviation plot. The curves were generatedwith the assumption that the one-sided phase noise spectral densities L(f) are white, with the valuesindicated in the legend. The horizontal line at 1 µV V−1 is a guide for the eye.

The delta-sigma conversion error is typicallydwarfed by other, external systematic errors.The error for a given delta-sigma pattern isalso fully deterministic: in the absence ofintentionally added noise (e.g., dithering), thesame modulator input parameters (i.e., delta-sigma settings, tone amplitude and frequency)will always yield the same error. So, while itis not possible to predict in advance the errorfor a given pattern, it can be calculated aftersynthesis and then used as a correction factorin subsequent measurements. However, if suchcorrection is not performed then the end-usermust include the error discussed here as anadditional Type B component.

To determine the delta-sigma error, we begin byestimating IQNP of the delta-sigma modulator.The simplest approximation for the delta-sigma

modulator is a linearized z-domain model thatsubstitutes the nonlinear effects of the quantizingelements with additive noise. In this linearizedapproximation, the IQNP for an order Lmodulator (specifically, the IQNP of its noisetransfer function) is given by [5],

σ2q =

π2L∆2

12(2L+ 1)(OSR)2L+1, (12)

where ∆ is the step size of the quantizer andOSR is the oversampling ratio of the modulator,defined as the ratio of the sampling frequencyfs to the Nyquist frequency 2 fB (twice thespecified bandwidth). This formula assumesthat fs >> fB and that the quantization noiseis uncorrelated and its spectrum is white. Inthe delta-sigma data converter literature, ∆ istypically taken equal to 2, in order to havea unity transfer characteristic for a variety of


modulator types. For the ACJVS, and a three-level, bipolar pulse pattern (i.e., −1, 0,+1), ∆is the Shapiro step voltage for the Josephsonjunction array(s). Note that the IQNP givenby the above equation may deviate significantlyfrom the actual IQNP , which can only beobtained by analyzing the spectrum of thegenerated pulse pattern.

In the convention of Ref. [5], the full-scale rangeof the quantizer FSR = M∆, where M is thenumber of quantizer steps, or increments. Thenumber of quantizer levels is nlev = M + 1. Thepeak-to-peak amplitude of an input sinusoid canbe expressed in terms of ∆ as,

Vpk−pk = kVFS = kM∆ , (13)

where k is a dimensionless constant between 0and 1 that represents the amplitude relative toFSR. The power of the sinusoid is,

Psignal = V 20 =

1

2

(kM∆

2

)2

, (14)

and the signal-to-quantization noise ratio (SQNR)is then,

SQNR ≡ Psignal

σ2q

=3k2(2L+ 1)(OSR)2L+1(nlev − 1)2

2π2L.(15)

As a first, highly-simplified approximation, weassume that the amplitude of the pure toneis reproduced perfectly in the encoded pulsepattern, and that the error in the voltage is dueonly to the quantization noise in band,

V =√V 2

0 + σ2q . (16)

The relative rms amplitude error is,

δV

V0=

√V 2

0 + σ2q − V0

V0=√

1 + (SQNR)−1−1 ≈ 1

2(SQNR), (17)

assuming of course that SQNR >> 1. Justas the above linear model for the IQNPallows us to derive an upper bound for theachievable SQNR of a second-order modulator,this expression serves as a lower bound on theuncertainty of the encoded tone.

This lower bound is often much, much lower thanthe actual rms error of a given pulse pattern(the reader is reminded that we are referringhere to the error in the digital representationitself, before the pulses ever propagate downthe microwave cable). The reason for thisdiscrepancy is that it is not possible to predicta priori the impact of the conversion processon the frequency bin in which the tone resides.In addition to harmonic distortion, noise in thefundamental bin will add directly to the tone,rather than in quadrature. Thus, dependingon the transfer characteristic, or the relativephase of the input signal and noise, the signalpower in that bin may be slightly lower orhigher than desired. The most accurate way toassess this error is simply to encode the patternin the digital domain and then analyze theresultant pulse pattern. Representative delta-sigma conversion errors are listed in Table 3.Note that the errors are signed, and thus thesynthesized voltage may be slightly less than orgreater than that desired.

Finally, although the modulator order can beincreased to achieve a smaller error, high-ordermodulators may become unstable when theout-of-band gain is too high and/or the inputsignal approaches fullscale. The susceptibilityof a modulator to such overloading is alsodependent on the waveform applied. Thereare also practical limitations: (1) high-ordermodulators require correspondingly longer timesto encode the waveforms, and (2) several otherACJVS error sources dwarf the already low rmserror obtained with second-order delta-sigmamodulators.

6.2. Extrinsic errors

6.2.1. Transmission line response and loadingTransmission line errors are the most significantpart of the overall error budget for the ACJVSat frequencies above 10 kHz. At 100 mVand 100 kHz, for example, the deviation fromflatness for the probe used in this work isof order 100 µV V−1, much greater than a


RMS Voltage Frequency

(mV) (dBFS) 100 Hz 1 kHz 10 kHz 100 kHz 1 MHz

2 -37 4.6× 10−08 5.1× 10−08 5.5× 10−08 2.2× 10−07 4.4× 10−06

6 -27 1.8× 10−09 2.0× 10−09 4.2× 10−09 2.0× 10−07 1.8× 10−05

10 -23 4.2× 10−10 5.2× 10−10 1.8× 10−09 1.4× 10−07 1.4× 10−05

20 -17 5.3× 10−11 6.9× 10−11 7.8× 10−10 6.9× 10−08 6.8× 10−06

60 -7.0 2.6× 10−12 4.2× 10−12 1.1× 10−10 9.9× 10−09 1.0× 10−06

100 -2.6 7.5× 10−13 7.0× 10−13 -3.1× 10−11 -3.3× 10−09 -3.4× 10−07

125 -0.65 4.4× 10−13 7.9× 10−14 -6.6× 10−11 -6.7× 10−09 -6.7× 10−07

Table 3: Relative rms voltage error for delta-sigma conversion. Parameters used to derive thistable are: Modulator order, 2; Sampling frequency, 14.4 GHz; Number of Josephson junctions, 6400;Bandwidth, 10 MHz; noise transfer function (NTF) maximum out-of-band gain, 1.2. A subset of theNTF zeros were optimized.

typical multijunction thermal converter (MJTC)uncertainty at the same point. Unless acorrection is made, the operator may choose tosimply never use the ACJVS above the audioband, or to only use it at the lowest voltages,where an uncertainty of ±100 µV V−1 is stilllower than the conventional method. Below,we discuss the issues involved with attemptingto flatten, or to correct for, the transmissionline response, as well as provide estimates fora particular means of correction.

The transmission line comprises on-die elementsof the ACJVS chip, wirebonds joining the chipto the carrier, signal traces on the cryopackage’sprinted circuit board, the wiring within thedip probe or cryostat, and any external cablingbetween the ACJVS and the TTS. Furthermore,the observed response of the TTS will dependon its input impedance (about 10 MΩ‖40 pFfor the active ranges of the Fluke 792A; thesymbol ‖ means “in parallel with”). If additionalequipment is connected to the transmission line,this can result in additional loading or straycurrent injection into the measurement setup.If two or more arrays are series-connected toincrease output voltage, then the choice ofouter dc block on the microwave bias is alsorelevant [11]. As a result, the quantum-accuracythat exists on-chip is diminished, and the

challenge becomes one of classical transmissionline impedance analysis.

Although a Josephson array presents zero sourceresistance when dc-biased under the critical cur-rent threshold, its effective source impedanceduring waveform synthesis is difficult to analyzebecause of its nonlinear dependence on manyfactors, such as the voltage and frequency ofthe waveform being synthesized, the parametersof the Josephson arrays, and the particular mi-crowave biasing technique being used. A rea-sonable approximation is to represent the arraysas a series combination of an inductance L andan impedance consisting of the Josephson super-conducting element shunted by its normal stateresistance RN . The total inductance is com-prised of (in decreasing order of significance):the finite wiring inductance between individualJosephson junctions and between separate ar-rays, the Josephson inductance, and the kineticinductance of the supercurrent. For the SNSjunctions used in NIST JVS systems, the wiringinductance is approximately 3 pH per junction,while that of the on-chip lowpass filters is in therange of (50 to 100) nH per filter, and there aretwo such filters per array. As a result, the effec-tive source inductance will be greater for chipswith more junctions per array and/or when morearrays are connected in series to increase the


overall output voltage.

At frequencies such that the wavelength is muchlonger than the electrical cable length, thetransmission line may be crudely approximatedas a lumped element RLC circuit. The RLCcircuit (cf. Figure 4) is a series connection ofR, L, and C, and functions as a lowpass filter(LPF). The transfer function |H(f)| = Vo/Vi

Vi VoC

RL

Figure 4: Lowpass RLC filter schematic,which may be used as a simplified approxima-tion for the response of the ACJVS transmis-sion line. The input is applied across all threeelements and the output is taken across the ca-pacitor C.

for the RLC LPF is second order, and thus itsresponse can be parameterized by the naturalfrequency,

ω0 =1√LC

, (18)

and the damping parameter,

ζ =R

2

√C

L=

R

2Z0, (19)

where Z0 is the characteristic impedance ofthe transmission line. For our purposes it isthe flatness of the response that is relevant,and the maximally flat Butterworth responseis obtained when ζ =

√2. Although

the ACJVS is comprised of one or moredistributed transmission lines, the crude RLCapproximation serves to qualitatively illustratethe dependence of the line’s response on certainof its parameters.

In an effort to flatten the transmission line

response, sometimes a series resistance isinserted at some point along the line [4, 9, 13].However, as depicted in Fig. 5, the flatness of theRLC-approximated line’s response is extremelysensitive to the choice of R. For R = 0, thedeviation from a perfectly flat response becomesrelatively large: 10 µV V−1 at around 100 kHz.+

If R is carefully chosen to achieve a small ac-dcdifference at a high frequency (say 1 MHz forthe case in Fig. 5), then the transmission lineerror at lower frequencies can be reduced. ForR less than some threshold value (≈ 70 Ω inFigure 5) and for f ω0/2π, the response isroughly quadratic and we obtain a reduction inuncertainty as,

u(fl) ∼(flfh

)2

u(fh) , (20)

where fl and fh are the low and high frequencies,respectively∗. This approximate scaling is alsovalid for distributed transmission lines when theproduct of the propagation constant and linelength is much less than unity. However, thereare drawbacks to this approach. Namely,

(i) The presence of a relatively large seriesresistance means that loading effects wouldbe significant. A model would needto be developed in order to make avalid comparison between ac-dc data fromconventional calibrations and those with theACJVS.

(ii) The presence of a finite R in the ACJVSoutput circuit will make the measurementsetup more susceptible to stray bias currentsand to the effects of electromagnetic inter-ference.

(iii) The sensitivity of the response to the valueof R places severe constraints on the charac-teristics of the passive components, such as

+ Note that the deviation at 100 kHz for a typical ACJVSsetup is closer to 100 µV V−1, because of the additionalline length and on-chip inductance.∗ The same scaling concept was leveraged in Ref. [4], butusing a thermal transfer standard to fix the line responseat the high frequency.


0

0.5

1

1.5

102 103 104 105 106 107 108-50

0

50

Figure 5: Magnitude response versus frequency of an RLC approximation to a 50 Ω, 1 m longtransmission line for selected resistances R. The frequency axis is the same for upper and lower plots.The upper plot shows the raw magnitude response. The lower plot shows a zoomed-in portion of theresponse deviation δ|H(f)| = |H(f)| − |H(0)|.

parasitic reactances and temperature coeffi-cient. Metal foil surface mount resistors cansatisfy some of these requirements, but theyare obviously not adjustable. If a trimpotis used, its backlash, stability, and induc-tance will become limiting factors. While itmay be possible to place a metal foil resistorat 4 K that accounts for the majority of Rand a trimmer at room temperature, smallvalue trimmers often exhibit poorer charac-teristics than their high-value counterparts.

(iv) Even if R could be fashioned with perfectstability, the rest of the transmission linecomponents may vary over time, due toeffects such as the liquid helium level(assuming a dip probe is used), connectionrepeatability, flexure, and changing loads.These effects imply that R would need to

be periodically verified and adjusted.

(v) This pseudo-matching technique is based onthe assumption that we can accurately mea-sure the voltage output at high frequencies.Or, at least accurately compare a voltageat high frequency with that at a lower fre-quency. In other words, if an accuracy ofbetter than 10 µV V−1 at 100 kHz is re-quired, then we need a detector that hasa traceable flatness better than 0.1%, or0.01 dB, for frequencies near 1 MHz. Ther-mal transfer standards are arguably the onlydevices that can achieve that requirement atthat frequency. Thus, this arrangement setsup a circular condition wherein the ACJVSis calibrated with the device it was intendedto calibrate.

An alternative approach to reducing the trans-


mission line error involves carefully characteriz-ing its physical properties, such as impedanceor the on-chip inductances [10]. The challenge,however, is that much of the transmission lineis inaccessible when the chip is cold. Evenif good, representative measurements could beperformed, there is a limit to the accuracy ofsuch measurements, due to the underlying accu-racy of the instrumentation (e.g., LCR meters,impedance analyzers). The best LCR meter un-certainties at audio frequencies for Z ≈ (50 to100) Ω and L ≈ 100 nH are 0.1 % to 1 %. Thatis for 4-wire components measured at the frontpanel of the LCR meter; the addition of adaptersor fixtures quickly inflates the uncertainty bud-get.

Assuming that such an approach is indeedfeasible, it is instructive to model the effectsof such variability on the transmission lineresponse. Figures 6–8 show the expectedresponse of a composite transmission linemodel for the ACJVS with uncertainty inthe measurement of the in-probe or in-vacuumtwisted pair impedance Z and in the on-chip inductance L. The variability of eachparameter is modeled in Monte Carlo fashionwith values chosen from a normal distributioncentered about the nominal value. We usedcommercial RF simulation software (MathworksRF Toolbox) to model the entire structure forthe composite line, including: (1) intra-array andon-chip filter inductance, (2) series impedance(if applicable), twisted-pair line within the dipprobe, (3) coaxial cable outside the dip probe(including common-mode choke), and (4) theinput impedance of any connected equipment,such as the TTS. The model accounts for skineffects in the coaxial cable, but not in thetwisted-pair line.

Figure 6 shows the variability in transmissionline response for parameter uncertainties corre-sponding to measurements at the front panel ofa precision impedance analyzer, σL,Z = 0.2 %(based on the uncertainty equations given in theuser manual for a Keysight E4990A). Note that

the uncertainty of the transmission line at 1 MHzis comparable to that for a multijunction ther-mal converter for voltages at or above 100 mV.This suggests that minimizing the transmissionline error at 1 MHz and for 1 m line lengthsto that of thermal converters would place strin-gent — likely unrealistic — requirements on thecharacterization of the ACJVS inductance andimpedance.

Measurements of a cold dip probe with a4-terminal pair impedance analyzer (KeysightE4990A) suggest realistically achievable param-eter uncertainties of σL = 12 % and σZ = 6 %.Figure 7 shows the variability in transmissionline response corresponding to these relaxed pa-rameter estimates. The roughly 100 µV V−1 er-ror at 100 kHz can be corrected down toward±10 µV V−1, much closer to that obtained withMJTCs. While the 1 % error at 1 MHz can becorrected down to roughly 350 µV V−1, this isstill much greater than the 10 µV V−1 uncertain-ties (k = 1) attainable with MJTCs.

As mentioned, the model does not account forskin and proximity effects in the twisted-pairline. Measurements of the copper twisted-pairline indicate that the total inductance changesby about 15 % to 20 % over the range of frequen-cies accessible with the ACJVS. Furthermore,since the resistivity of copper is temperature-dependent, the temperature distribution of theline (e.g., how much of it is submerged in liq-uid helium) will influence the spatial distributionand mean of the line’s respective electrical pa-rameters. Comparing measurements of the lineat room temperature versus immersion in liquidhelium indicate inductance variations of order10 %, depending on frequency. Such frequency-and temperature-dependent effects also impactthe accuracy with which the on-chip characteris-tics can be measured in-situ, although they maybe partially mitigated through the use of low-RRR (residual resistivity ratio) conductors, suchas phosphor bronze. However, as outlined abovethe finite resistance can create additional mea-surement challenges.


Figure 6: The frequency response of a composite transmission line model for a single, 6400-junctionarray of the ACJVS for σL = 0.2 % and σZ = 0.2 %. Frequency axis is the same for upper and lowerplots. The model includes the effects of source resistance (when applicable), on-chip lowpass filters,twisted-pair wire in the probe (1.5 m), coaxial cable outside the probe (1.5 m), and the typical loadpresented by the Fluke 792A on its active ranges. The top plot shows the overall transfer function ofthe composite line and the parameters used in the model. The black dashed curve in the bottom plotis the nominal deviation of the response from its value at DC. The remaining solid green curves aredeviations from that nominal response due to variability in L and Z. Of the roughly one thousandcurves generated, only those that lie within ±1 standard deviation of the nominal response are shown.

For the Monte Carlo simulations used inFigures 6–7, we assumed zero additional seriesimpedance in the line (i.e., beyond the smallresistance of the copper in the transmissionline components). One conjecture is thatuncertainties could be improved through acombination of flattening the line response with aseries resistor, as well as characterizing elementsof the line itself. However, as depicted in Fig. 8,choosing the optimal resistor value (which wehave the luxury of doing in a simulation) doesnot reduce the sensitivity of the response tovariability in the line’s elements. This resultraises questions about so-called “matching” or

“tuning” procedures, such as that in [13].While the approach outlined in [13] has theadvantage of not relying on lumped elementapproximations, it does not represent a true“impedance matching” procedure because onlyone component of the complex impedance isbeing matched. More significantly, however,is that the procedure depends on criticallyadjusting a series resistor and line length toobtain a cancellation of several terms in theline’s response. If the critically-tuned equation(their Equation A.7) is taken as a startingpoint for an uncertainty calculation, it willyield unreasonably small uncertainties for this


Figure 7: The frequency response of a composite transmission line model for a single, 6400-junctionarray of the ACJVS for σL = 12 % and σZ = 6 %. Model parameters are otherwise identical to thosein Fig. 6. Additional details are in the text and the caption for Figure 6.

correction. This is because although the termsmay cancel perfectly in the response equation,their respective uncertainty contributions do not.The model used in this report fully reproducesthe critically-tuned response given in [13], butit goes further by showing that such proceduresdo not circumvent the critical dependence of theresponse on the line’s parameters.

6.2.2. Quadrature and compensation bias errorsThe cryopackaged die of the ACJVS containsone or more arrays of thousands of Josephsonjunctions connected in series. Each arrayexhibits an inductance of tens of nH. Undera current bias, the intra-array inductance maycontribute a voltage VQ that is in quadraturewith the desired output tone V as,

Vout =√V 2 + V 2

Q =√V 2 + (ωLI)2 , (21)

where ω is the angular frequency of the tone,L is the the array inductance, and I is thecurrent through the array at the frequencyω. It should be noted that this effect islimited to biasing arrangements in which themicrowave pulse sequence reaching the array hassignificant power at the frequency ω, or whena compensation bias is used. If there is noappreciable current flowing through the array atthe output tone(s) of interest, the quadratureeffect may be disregarded.

For compensation biasing at audio frequencies,the quadrature error is typically less than 0.1 µVV−1, which is much less than the typical TypeA uncertainties for an ac-dc transfer. However,at higher frequencies the error can become large(e.g., 0.05 µV V−1 at 100 kHz versus 130 µVV−1 at 5 MHz). Additionally, since the criticalcurrent Ic of the junctions sets the overall scale


0

0.5

1

1.5

102 103 104 105 106 107 108-100

-50

0

50

100

Figure 8: The frequency response of a composite transmission line model for a single, 6400-junctionarray of the ACJVS with a non-zero source resistance, the value of which is chosen to optimize theflatness of the composite line. σL = 12 % and σZ = 6 %, as in the previous plot. Model parameters areotherwise identical to those in Fig. 6. Additional details are in the text and the caption for Figure 6.

for the current bias, the quadrature error will begreater for arrays with larger Ic.

The quadrature error may be characterized byanalyzing both the in-phase and quadraturecomponents during operation. If such vector-based measurements are done across frequency,then the inductance L can be known to relativelyhigh precision. Likewise, if non-compensatedbias arrangements are used, such as withcareful shaping of the input microwave pulses,then the quadrature error can be substantiallyreduced. However, compensation biasing withan additional lower-speed generator is requiredin order to synthesize the largest rms voltagewith the ACJVS. In this situation, phasemisalignment between the low-speed generatorand the pulse pattern generator can contributean additional error that is closely related to the

quadrature error.

To illustrate, consider the array as a seriescombination of the ideal Josephson junctions andthe array inductance L. If we assume that thecompensation generator’s phase is delayed by φ(relative to the waveform that is output by theideal JJ array), then we can rewrite the aboveequation for Vout as,

Vout =√V 2I + V 2

Q =√

(V + ωLI sinφ)2 + (ωLI cosφ)2 , (22)

where VI represents the in-phase component ofthe measured output tone. If we normalizethis expression to the desired output voltage,we obtain an expression for the relative rms er-ror due to the combined effects of array induc-tance (quadrature) and compensation phase er-


FrequencyUncorrected(µV V−1)

Corrected, uL = 12%,uZ = 6% (µV V−1)

100 Hz 0 0400 Hz 0.001 01 kHz 0.009 02 kHz 0.037 0.0015 kHz 0.23 0.0110 kHz 0.93 0.0320 kHz 3.7 0.150 kHz 23 0.8100 kHz 93 3.41 MHz 9400 344

Table 4: Transmission line systematic errorsfor a single, 6400-junction array of the ACJVSand a transmission line analysis based onthe composite model discussed in the text.Uncorrected values represent the deviation froma perfectly flat response (|H(f)| = 1) for thechosen nominal parameters: Rsource = 0 Ω,Zload = 10 MΩ‖40 pF, L = 160 nH, Ztp =100 Ω, and `tp = `coax = 1.5 m. Thecorrected values represent 1σ uncertainties aftera realistic correction based on that nominalresponse has been applied. A “0” indicates thatthe actual value is less than 0.001 µV V−1.

ror,

uquad/comp ≡Vout − V

V=√

(1 + β sinφ)2 + (β cosφ)2−1 , (23)

where β = ωLI/V . Since the ACJVS waveformsare derived by way of pulse-density modulation,the current I is proportional to the synthesizedvoltage V ,

I ∼ IcV

Vfs, (24)

where Vfs is the full scale voltage of the array. Wethen have β ≈ ωLIc/Vfs, which only depends onthe junction array parameters and the frequency.When cast in this form, it is evident that β ispractically always less than unity for realisticdevice parameters. Furthermore, although thequadrature component itself depends linearly onfrequency, its impact will scale quadraticallywith frequency in ac-dc difference measurements,

which depend on the rms content of thewaveform. A plot of uquad/comp vs. φ, fordifferent β, is shown in Fig. 9. A plot ofuquad/comp vs. frequency, for specific types ofarrays relevant to the NIST ACJVS, is given inFig. 10.

A potential source of confusion is that the com-pensation generator phase that maximizes theACJVS operating margin] may not necessarilycorrespond to φ = 0 (see, e.g., Section 4.1 ofRef. [4]). However, the correct phase can beobtained accurately by the procedure outlinedin [9], or by explicitly measuring the in-phaseand quadrature components of the output andmaximizing the quadrature part.

As written above, uquad/comp > 0 when thequadrature error increases the rms voltage fromits nominal value. An increasing ac rmsamplitude implies a decreasing AC-DC differenceδ. Thus, if the quadrature error is smalland can be estimated with reasonable precision,the end user may correct for it by simplyadding it to the AC-DC difference for theTTS. Such a correction is reasonable when (a)TTS nonlinearity and the net change in rmsvoltage due to the quad error are sufficientlysmall that the nonlinear component of the ac-dc difference (voltage coefficient times voltagechange) is smaller than uquad/comp, and (b) theuncertainty in the estimate of uquad/comp is lessthan, or approximately equal to, that of the AC-DC difference δ. Under these conditions (andwhen β or φ are both much less than unity), thesensitivity of uquad/comp to fluctuations in β or

] Some authors may use the phrase “quantum lockingrange” instead of “operating margin”. In both cases,the metric is the same: the dc offset current bias rangewithin which a property of the output waveform exhibitsminimal deviation. The property may be the residualfrom a sinewave fit, the total harmonic distortion (THD),or some other relevant parameter. The threshold fordetermining what is an acceptable “minimal deviation” issomewhat arbitrary, but strict enough that the operatorhas confidence that the output waveform is quantum-accurate over that range.


10-2 10-1 100 101 10210-3

10-2

10-1

100

101

102

103

104

Figure 9: Error in the rms output of a single, 6400-junction array of the ACJVS vs. the phase offsetbetween the pulse pattern and the low-frequency compensation bias.

φ is approximately the same,

∂u

∂β≈ ∂u

∂φ≈ β . (25)

From the above we can derive an estimatefor the uncertainty of the quadrature errorwith the usual linear propagation of errormethodology,

σu ≈ β√σ2β + σ2

φ . (26)

For example, if β is of order 10−3, and theuncertainties for both parameters are of the sameorder, then the uncertainty of the correctionfor the quad error is roughly 1 µV V−1, whichis about the same as the quad error itself,so a correction is not going to be particularlyhelpful in improving the uncertainty budget. Onthe other hand, if β is known to a reasonableaccuracy (say σβ/β ≈ 10−2), and σφ canbe ignored, then σu ≈ 0.01 µV V−1, and a

correction is warranted.

As mentioned above, a properly-configured biasarrangement will enforce φ = 0. However,the compensation generator DAC used in thiswork is clocked at 100 MSa/s (megasamplesper second), and delays are limited to integermultiples of the sample period. As a result, theavailable phase resolution is highly discretized.For a 1 MHz tone synthesized using a single,6400-junction array with Ic = 5 mA (β ≈ 0.003),the phase increment will be about 4. Theinability to precisely adjust φ translates to anerror in uquad/comp of order 100 µV V−1. Unlessthe vector components of the output waveformare measured, the end user can only assume thatthe quad error is bounded to within ±100 µVV−1.

At high frequencies, the compensation genera-tor’s jitter may also increase the uncertainty in φ.


103 104 105 106 10710-6

10-4

10-2

100

102

104

Figure 10: Error in the rms output of a single array of the ACJVS due to the quadrature error vs.frequency. The horizontal line at 1 µV V−1 is a guide for the eye.

However, if the source was manufactured in thelast decade or so, an rms jitter less than 10 ps isattainable, which corresponds to σφ ≈ 60 micro-radians at 1 MHz. Thus, with the exception ofhardware-limited phase control, the uncertaintyin the quad error will be dominated by the un-certainty in β. When φ is indeed zero, β is equalto the ratio of the quadrature and in-phase com-ponents of the voltage output by the ACJVS. Ifthese components are measured directly for thepurposes of correction, we can re-write the ex-pression for uquad/comp as,

uquad/comp = sec(θ)− 1 , (27)

where θ = tan−1(VQ/VI). If the phase offsetφ is nonzero but small, then the deviationfrom the ideal voltage V cannot be determinedunambiguously from measurements of only VIand VQ; the phase offset must also be measured.If we assume for now that φ = 0, then

the uncertainty of the quad correction isapproximated by,

σu ≈ θσθ . (28)

Thus, the determination of the quad erroruncertainty is reduced to estimating θ, or therelative magnitudes of VI and VQ. Sinceoperating conditions are often such that VQ VI , the uncertainty of the quad error dependson the nonlinearity and gain error (if notalready calibrated) of the device (e.g., digitizer,lock-in amplifier) used to measure the vectorcomponents. At 1 MHz, θ ≈ β = 3.3 × 10−3,uquad/comp ≈ 5 µV V−1, and if we assume thata total harmonic distortion of −90 dBc for theADC is a reasonable proxy for its nonlinearityat 1 MHz, then σθ ≈ 30 µV V−1 and thus σu ≈0.1 µV V−1. Quadrature/compensation errorsfor the different scenarios considered here areprovided in Table 5.


In summary, the quadrature error in the ACJVSis sufficiently small (relative to other effects)at audio frequencies that it can be ignored forthe majority of ac-dc difference measurements.Starting around 100 kHz, the magnitude ofthe error approaches 1 µV V−1, and so itmay be warranted to measure the in-phaseand quadrature components of the waveformin order to apply a correction. However,in the majority of scenarios, the transmissionline error will exceed that of the combinedquadrature/compensation effects.

Table 5: Quadrature/compensation system-atic errors for a single, 6400-junction array ofthe ACJVS. The “DAC-Limited Phase” columnrefers to the case when the compensation phaseresolution is limited by the DAC sample pe-riod. Here, a sampling frequency of 100 MSa/sis assumed. The “Intrinsic” column is the resid-ual quadrature error when compensation phasealignment is perfect (i.e., φ = 0, as defined inthe text). The “Measured, Corrected” columnassumes that φ = 0 and that the vector compo-nents of the output waveform have been mea-sured with a spectrum analyzer or digitizer withperformance comparable to the NI 5922. Thevalues in this column are the 1σ uncertainties inthe ac-dc difference due to the quad error afterthe correction is applied. A “0” indicates thatthe actual value is less than 0.001 µV V−1.

Relative RMS Error, uquad/comp / (µV V−1)

Frequency β = ωLIV DAC-

limitedphase

IntrinsicMeasured,cor-rected

5 kHz 1.6× 10−05 5.3×10−03

0 0

10 kHz 3.3× 10−05 0.0210 020 kHz 6.6× 10−05 0.0852.2×

10−030

50 kHz 1.6× 10−04 0.53 0.0141.2×10−03

100 kHz 3.3× 10−04 2.1 0.0542.9×10−03

1 MHz 3.3× 10−03 210 5.4 0.10

6.2.3. Thermovoltage errors Errors arisingfrom thermal electromotive forces (EMFs) arein most cases negligible for ac-dc differencemeasurements with the ACJVS. However, forcompleteness we will briefly discuss them here.The thermal noise of the ACJVS is vanishinglysmall, since the source resistance is small— typically just the series resistance of thetransmission line. The noise floor of room-temperature instrumentation will most often bethe limiting factor.

However, there will be an additional voltagenoise due to fluctuations of thermal EMFs in thetransmission line, since it spans the temperaturegradient between 4 K and 300 K. A constantthermovoltage will be automatically correctedfor as a result of the way ac-dc transfers areperformed. But, the dc+ and dc− voltages aremeasured at different instances in time, and ifthese times are well separated, then drift in thethermovoltage can contribute a small error to theac-dc difference.

This error can be quantified by connecting ananovoltmeter to the output of the ACJVSand measuring the instantaneous voltage withthe ACJVS off. Typical values for the meanthermovoltage for a dip probe are (10 to 100) nV,with rms fluctuations of about (10 to 20) nV.However, these can be larger if the transmissionline wiring is not optimized for these effects, or ifthe wiring within the ACJVS was not completelythermally stabilized (e.g., starting measurementssoon after the dip probe or cryostat are cooled).Estimates for the uncertainty contribution ofsuch fluctuations are given in Table 6.

6.2.4. Electromagnetic Interference: Groundloops and common mode noise In many pre-cision metrological applications, it is advanta-geous or simply mandatory that the measure-ment apparatus be isolated from interferenceoriginating from external equipment. Electro-magnetic interference (EMI) may couple into asystem capacitively via unshielded cabling or un-grounded/floating instrumentation. EMI may


RMS Voltage(mV)

Thermovoltage Error(µV V−1)

2 106 3.310 220 160 0.3100 0.2200 0.1

Table 6: Uncertainty contribution due tothermovoltage fluctuations. The values listedassume an rms thermal noise voltage Vth =20 nV, which should not depend strongly onthe details of the experimental configuration.Note that this error only impacts dc and verylow frequency measurements. It should not beincluded in the uncertainty for AC-AC transfersat frequencies well above 10 Hz.

also couple inductively if ground loops are inad-vertently created through the use of unoptimizedwiring routes.

Given the intricate biasing arrangement of theACJVS, there are numerous experimental setupsthat may result in difficult-to-detect systematicerrors. Here, we enumerate the details of oneparticular setup using the equipment outlinedin Table 1 that has (after many iterations)shown itself to be optimal for ac-dc transfermeasurements. Even so, the reader is cautionedthat the performance of this configurationmay improve or degrade, depending upon howsignificantly future ACJVS-specific hardwaredeviates from that described in this report.For this reason, and the fact that the relativecontribution of such EMI-induced errors dependon the specific value for δ, we do not include sucherrors in our overall uncertainty budget.

The configuration used for this report isdepicted in Fig. 11, and may be summarized asfollows:

(i) A single, EMI-filtered power strip is at-tached to the equipment rack and powers

the pattern generator, PXI chassis (contain-ing digitizer, etc.), and digital multimeter(DMM) for measuring the output of the792A. The power strip serves as the primarygrounding point in our configuration.

(ii) The pattern generator is connected tothe controlling computer via an unshieldedethernet cable through a network switch.Since the termination points of the ethernetconnection are already balun-isolated, thereis no need to use optical links or mediaconverters. In fact, the switching powersupplies of the media converters are oftena significant source of EMI, unless they areplaced as far as possible from the ACJVS.

(iii) When needed, external baluns are used tobreak ground loop and common mode loopcoupling that may occur in the distributionof the 10 MHz reference clock. Asnoted above, for certain instrumentation thebaluns are internal.

(iv) The PXI chassis containing the NI 5922digitizer is connected to the computer via anoptical link. The digitizer input is typicallyconfigured as “single ended”, which meansthe shell of the BNC connector is groundedto the chassis. The DMM for measuring the792A output, which is also under computercontrol, is likewise galvanically isolated fromthe computer.

(v) Battery-powered isolation amplifiers areused between the pattern generator com-pensation outputs and the cryostat/dipprobe. The amount of isolation is primar-ily determined by the input bias resistors ofthe amplifiers, typically 1 MΩ to 10 MΩ atdc. The isolation vs frequency of these am-plifiers was measured and corresponds to acapacitance of order 40 pF.

(vi) The guard terminal of the 792A is shortedto its ground at the banana jacks near theoutput terminals. The low terminal of the792A is not connected to guard or ground.The ground of the 792A is then tied back to


the power strip.

(vii) A common mode choke, comprised of a46 cm length of coaxial cable threadedseveral times through a toroidal, high-permeability core, is inserted between theoutput of the dip probe and a BNC teeconnector, which is in turn connected to theinput of the 792A.

(viii) The output of the 792A is connected to theDMM via a shielded, twisted pair cable.The shield of that cable is grounded at the792A end to the 792A ground; the shield isnot connected to the DMM.

(ix) The other port of the tee mentioned inItem vii is connected to the digitizerwith a 46 cm coaxial cable. This cableis disconnected at the tee during ac-dctransfers.

(x) For cryopackages that do not have on-chipinner/outer DC blocks, all input/outputports of the dip probe are isolated fromthe probe body (e.g., using isolated groundpanel-mount feedthroughs). The body ofthe probe and the dewar are all groundedto the 792A ground. Inner/outer DCblocks (fc ≈ 500 MHz) are inserted betweenthe microwave feedthroughs at the dipprobe and the microwave cabling from thebitstream generator.

When attempting to troubleshoot EMI-relatedproblems, one should keep in mind thatestablished techniques used to isolate equipmentat dc are not necessarily good ac techniques.When floating equipment on batteries, theungrounded metal components become effectiveentry points for broadband noise and the impactof this noise on measurements will depend on thephysical arrangement of nearby equipment andcabling. From the standpoint of repeatabilityand reproducibility, it is preferable to establisha well-defined ground, as well as the impedancesbetween signal leads and that ground.

The internal guard of the 792A affords some

protection from high-frequency common modenoise††. But, it can be easily defeated if otherequipment without such protection is connectedto the measurement network. For example, ifthe digitizer is powered from the mains andconnected to the 792A, then the low terminalbecomes grounded and the voltage measurementis effectively single-ended. If instead thedigitizer is powered from a battery, then thedigitizer chassis and the additional cablingincrease the susceptibility of the system tocommon mode noise through stray capacitanceto ground.

Because the 792A responds to frequencies wellbeyond its specified 1 MHz limit (its −3 dBbandwidth is around 10 MHz, depending oninput range), it is preferable to disconnectthe digitizer prior to performing a transfer(once ACJVS operating margins have beenestablished). Ideally, the disconnection pointshould be as close as possible to the transferstandard, so as to minimize the overall lengthof the transmission line (cf. Item ix inthe list above). In situations where thedigitizer must be spliced into the network (e.g.,for automation), a common-mode choke, orcurrent equalizer, is usually required to obtainrepeatable results.

In alternate ACJVS configurations with rela-tively short transmission lines (< 0.5 m) andwithout the digitizer connected, the author hasfound that common-mode noise is sufficientlysmall that the choke is unnecessary. This is im-portant because the effective length of the trans-mission line depends on whether or not a chokeis inserted into the network. The impact of thechoke on the line’s response at 1 kHz is negligible,but may be significant at high frequencies. Forthis report, we do not consider this interactiondirectly since we used a choke for all experimen-tal results, and assumed a choke was present in

††Here, the phrase “common mode” refers to EMI thatis injected into the output voltage leads from externalsources, and is common to both the high and low voltageleads.


I/O DC blocks (here or on-chip)

pulse pattern generator

floating compensation source

common-mode choke

cryostat

G

Fluke 792A (battery pack unplugged)

Ground

Guard

Low

High

DMM

Low

High

Digitizer (remove during transfer, if possible)

+-

Earth (lab) ground

equipment chassis ground

Arbitrary Bitstream Generator

Figure 11: Simplified schematic of the ACJVS grounding arrangement.

the simulations for the transmission line in Sec-tion 6.2.1.

Measurements of the 792A’s response tocommon-mode noise arising from the ACJVSnetwork are shown in Table 7. The largeshifts in the 792A’s output voltage, depend-ing on whether the common-mode choke ispresent, suggest that there is significant broad-band, common-mode noise on the output leadsof the ACJVS. Note that this noise is not due tothe rapidly rising noise floor of the delta-sigmaconversion, because shifts of a similar scale areobserved when the ACJVS is not running (butstill connected to the 792A). The broadband rmsmagnitude of the common mode noise is esti-mated from spectral measurements to be of or-der 1 mV without the choke and of order 100 µVwith the choke.

Fortuitously, errors arising from common-modenoise largely cancel out because the noise ispresent for every voltage of an ac-dc transfermeasurement. If the TTS is sufficiently linear,then the presence of large broadband noise

RMS Input(mV)

792A Range(mV)

792A Voltage Output (V) Relative Shift(µV V−1)with choke without choke

20 22 1.829 103 1.835 400 3443100 220 0.908 186 0.908 371 204200 220 1.816 520 1.816 615 52

Table 7: The effect of common-mode noiseon the measured output of the 792A thermaltransfer standard for a 1 kHz sinusoidal input.The presence of the choke changes the effectivetransmission line length, but the change inthe line’s response at 1 kHz is negligiblecompared to the observed shifts in outputvoltage. Additional details are given in themain text.

simply shifts the operating point of the transfer.There is a limit to this, of course — at the lowestvoltages the common mode noise is of the sameorder as the signal itself. Based on measurementsat audio frequencies, it is estimated that EMI-induced errors affect the ac-dc difference at alevel below 1 % — even at the lowest voltages.


In other words, an EMI-related error would shiftthe measured value of δ by less than 1 %. Thecorresponding errors are given in Table 8.

Common Mode Error (% of δ)

RMS Voltage(mV)

Vn = 100 µV Vn = 1 mV

2 0.25 206 0.03 2.710 0.01 1.020 2.5× 10−3 0.2560 2.8× 10−4 0.03100 1× 10−4 0.01200 2.5× 10−5 2.5× 10−3

Table 8: Estimated errors due to common-mode noise. The common-mode noise isassumed to be broadband and its rms amplitudenearly constant, and so will add in quadraturewith the primary tone. As a result, the errorscales with the value of the ac-dc differenceδ, and so the uncertainty is expressed relativeto δ. The two columns for the common-modeerror correspond to the effective broadbandvoltage noise with and without the common-mode choke in the transmission line.

6.2.5. Connection repeatability In additionto exercising the usual care when handlingRF connectors, the impact of connector lossand repeatability should be considered duringcalibration with the ACJVS. Unfortunately,connector repeatability data in the span of audiofrequencies, or even from 100 kHz to 10 MHz,is not readily available. In order to estimatethis effect, one may perform a statistical analysisof many ac-dc transfers in which the cable’sorientation is reversed between transfers.

Although not included as part of our overalluncertainty estimates, an example of such ananalysis is provided in Table 9 for an rms voltageof 200 mV. It is instructive to compare themagnitude of the measured reversal error againstthe Type A uncertainty u1σ of the sampleddataset for δ. At each frequency the observed

reversal error is comparable to the statisticaluncertainty of the measurement itself. At largervoltages or smaller load impedances, the effectmay become more pronounced.

7. Uncertainty Summary

A representative uncertainty budget for V =100 mV and f = 100 kHz is providedin Table 10. The combined uncertainty isthe root-sum-square of the individual errorcontributions. Type B uncertainty matrices arepresented in Tables 11 (uncorrected) and 12(corrected). These latter tables give a clearerpicture of the underlying systematic errors of theACJVS before being combined with the TypeA uncertainties obtained when measuring theTTS. Corresponding combined uncertainties forcalibration of a Fluke 792A transfer standardwith the ACJVS are given in Tables 13 and14.

Reasonable agreement is found when comparingthe overall uncertainties tabulated here withac-dc calibrations and intercomparisons of theFluke 792A with the ACJVS. For example,the uncertainties presented here for 100 mVand 10 kHz (4 µV V−1, k = 2, regardlessof correction) are in line with that reportedin [15, 10, 16, 17] (5 to 7) µV V−1, k = 2. Forthe same voltage, but at 100 kHz, the correcteduncertainty (7 µV V−1, k = 2) is slightly lowerthan, but still in rough agreement with, otherexperimental results in which a correction wasapplied [4, 17] (9 to 16) µV V−1, k = 2.

Recent results using a cryostat with a shorttransmission line (approximately 70 cm) showeda flatness deviation of 510 µV V−1 at 1 MHz [12],which is much smaller than the typical 1 % de-viation obtained with dip probes (no compensa-tion bias was used, so the output voltage waslimited to 20 mV). However, according to ta-ble 4, significant improvements can still be ob-tained for conventional dip probes by performingthe corrections outlined in 6.2.1, with a post-correction uncertainty of about 350 µV V−1 at


Cable Forward Cable Reversed

fδ

(µV V−1)u1σ

(µV V−1)δ

(µV V−1)u1σ

(µV V−1)Rev. Error

(µV V−1)Rel. Rev. Error

(%)

100 Hz 12.50 0.07 12.44 0.13 0.06 0.471 kHz 5.25 0.09 5.25 0.09 0.00 -0.0410 kHz -7.92 0.10 -7.69 0.12 -0.24 3.01100 kHz -122.93 0.33 -122.78 0.29 -0.15 0.121 MHz -10776 15 -10792 37 16 -0.15

Table 9: Observed errors due to reversing the cable orientation between the ACJVS and the Fluke792A thermal transfer standard between measurements. Although by no means rigorous, such errorsmay be used to provide qualitative estimates of connection repeatability. Applied rms voltage was200 mV and transfer standard input range was 220 mV. An RG-58 coaxial cable (30.5 cm length,BNC-type connectors) was used to connect the transfer standard to the output of the ACJVS.

Standard Uncertainty (µV V−1)

Error Contribution Uncorrected Corrected

∆Σ conversion 0.003 —Phase noise 0.001 0.001Transmission line 93.4 3Quadrature error 2.1 0.003Thermovoltage fluctuations 0.2 0.2Transfer measurements (Type A) 2.1 2.1

Overall, combined (k = 2) 187 7

Table 10: Uncertainty budget for V = 100 mV and f = 100 kHz, based on the single-array, 6400-junction model. Since the ∆Σ error can be estimated with very high precision, it is not included in thecorrected budget. The difference in the assigned uncertainties for the transmission line and quadratureerrors are based on the discussion in 6.2.1 and 6.2.2. As discussed in 4, the following sources are notincluded in the overall uncertainty budget: low-frequency pulse bias feedthrough, electromagneticinterference, and connection repeatability.

1 MHz (k = 1). Although this is still muchlarger than what can be obtained using MJTCsfor 100 mV (11 µV V−1, k = 1), it suggests thatthe combined strategies of shortening the trans-mission line and improving its parameterizationmay increase the usable upper frequency rangeof the ACJVS in the future.


Uncorrected Type B Standard Uncertainty, uB / (µV V−1)

RMS Voltage(mV)

100 Hz to 2 kHz 5 kHz 10 kHz 20 kHz 50 kHz 100 kHz 1 MHz

2 20 20 20 21 51 188 18 8446 7 7 7 10 47 187 18 84410 4 4 4 8 47 187 18 84420 2 2 3 8 47 187 18 84460 0.7 0.8 2 8 47 187 18 844100 0.4 0.6 2 7 47 187 18 844200 0.2 0.8 3 12 77 307 31 123

Table 11: Uncorrected, expanded (k = 2) Type B uncertainty matrix for the ACJVS. Values representthe combined errors due to: transmission line, quadrature/compensation, delta-sigma conversion, andphase noise. Note that two, 6400-junction arrays are required to produce 200 mV. For those points,an additional error contribution is included due to the parasitics related to the outer dc blocks andthe isolation amplifiers.

Corrected Type B Standard Uncertainty, uB / (µV V−1)

RMS Voltage(mV)

100 Hz to 10 kHz 20 kHz 50 kHz 100 kHz 1 MHz

2 20 20 20 21 6876 7 7 7 9 68710 4 4 4 8 68720 2 2 3 7 68760 0.7 0.7 2 7 687100 0.4 0.5 2 7 687200 0.2 0.6 3 14 1410

Table 12: Corrected, expanded (k = 2) Type B uncertainty matrix for the ACJVS. Values representthe combined errors due to: transmission line after determination of the characteristic impedance Ztp

and on-chip inductance L (cf. 6.2.1), and phase noise. Note that two, 6400-junction arrays are requiredto produce 200 mV. For those points, an additional error contribution is included due to the parasiticsrelated to the outer dc blocks and the isolation amplifiers.

8. Conclusion

A detailed uncertainty analysis for ac-dc dif-ference calibrations with an ACJVS was devel-oped and presented. The tabulated uncertaintieswere compared against prior calibrations and in-tercomparisons involving the ACJVS, and goodagreement was obtained for most of the param-eter space explored.

Type B uncertainties for the ACJVS suggest

that, compared to conventional calibrationsbased on MJTCs, substantial improvements inac-dc measurement uncertainty can be obtainedfor low voltages in the audio band, withoutapplying corrections. Modest uncertaintyreductions can be obtained up to 100 kHz if areasonably accurate assessment of transmissionline parasitics is available. For frequencieswell above several hundred kilohertz, the resultssuggest that uncertainty reductions are unlikely


without major modifications to, or painstakingcharacterization of, both the on-chip elementsand the transmission line parameters.

9. Acknowledgments

The author would like to thank ThomasLipe, Nathan Shattuck, and Stefan Cular forvaluable discussions related to ac-dc difference,Bryan Waltrip and Nathan Flowers-Jacobs fortheir input on the manuscript, and NIST’sSuperconductive Electronics Group for theirsupport with the bias electronics and ACJVSdevices. This work was supported in partby interagency agreement #WOH9AA-15109-852.


Uncorrected, Combined Standard Uncertainty, uC / (µV V−1)

792A Range(mV)

RMS Voltage(mV)


22 2 127 119 113 130 121 121 117 128 22022 6 41 39 38 39 39 39 40 61 19122 10 26 25 25 25 25 25 27 53 18922 20 14 13 13 13 13 14 15 49 187220 20 49 49 44 49 50 50 52 69 194220 60 9 8 6 8 8 8 11 47 187220 100 5 4 3 4 4 4 9 47 187220 200 2 2 2 2 2 4 12 77 307

Table 13: Combined, expanded (k = 2) uncertainty for calibration of a Fluke 792A thermal transferstandard with the ACJVS. Values represent the combined errors due to: Type A uncertainties frommeasurement and estimated Type B uncertainties when no correction is applied. Note that two, 6400-junction arrays are required to produce 200 mV. For those points, an additional error contribution isincluded due to the parasitics related to the outer dc blocks and the isolation amplifiers.

Corrected, Combined Standard Uncertainty, uC / (µV V−1)

792A Range(mV)

RMS Voltage(mV)


22 2 127 119 113 130 121 121 117 120 11722 6 41 39 38 39 39 39 39 39 4022 10 26 25 25 25 25 25 26 26 2622 20 14 13 13 13 13 13 13 13 15220 20 49 49 44 49 50 50 52 51 52220 60 9 8 6 8 8 8 8 9 11220 100 5 4 3 4 4 4 4 4 8220 200 2 2 2 2 2 2 2 4 14

Table 14: Combined, expanded (k = 2) uncertainty for calibration of a Fluke 792A thermal transferstandard with the ACJVS. Values represent the combined errors due to: Type A uncertainties frommeasurement and estimated Type B uncertainties when the correction described in 6.2.1 is applied.Note that two, 6400-junction arrays are required to produce 200 mV. For those points, an additionalerror contribution is included due to the parasitics related to the outer dc blocks and the isolationamplifiers.

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Uncertainty analysis for ac-dc di erence measurements with