---
abstract: 'A new resistance bridge has been built at the Laboratoire national de métrologie et d’essais (LNE) to improve the ohm realization in the *Système International* (SI) of units from the quantum Hall effect. We describe the instrument, the performance of which relies on the development of two synchronized and noise filtered current sources, an accurate and stable current divider and a cryogenic current comparator (CCC) having a low noise of $\mathrm{80~pA.t/Hz^{1/2}}$. As targeted, the uncertainty budget for the measurement of the 100 $\Omega/(R_\mathrm{K}/2)$ ratio, where $R_\mathrm{K}$ is the von Klitzing constant, amounts to a few parts in $10^{10}$ only.'
author:
- 'Wilfrid Poirier, Dominique Leprat and Félicien Schopfer[^1]'
title: 'A resistance bridge based on a cryogenic current comparator targeting sub-$10^{-9}$ measurement uncertainties'
---

Metrology, resistance, quantum Hall effect, bridge, cryogenic current comparator, SQUID.

Introduction
============

the SI[@BIPM], the ohm can be realized from the von Klitzing constant $R_\mathrm{K}=h/e^2$[@AmpereBIPM2019], where $h$ is the Planck constant and $e$ is the elementary charge, using the quantum Hall effect[@Klitzing1980]. More precisely, the quantized Hall resistance value of a quantum resistance standard (QHR), $R_\mathrm{K}/i$, where $i$ is an integer, is used as a primary reference of resistance[@Poirier2019]. The resistance unit is then disseminated by means of comparisons with this universal and reproducible reference using a resistance bridge. Comparing a resistance with the quantized Hall resistance with the lowest measurement uncertainties is challenging since the measurement current of QHR devices must remain small, i.e. a few tens of $\mu$A if based on GaAs/AlGaAs heterostructure[@Poirier2009] and a few hundreds of $\mu$A if based on graphene[@Ribeiro2015]. Thus, performing a resistance calibration with a relative uncertainty of a few parts in $10^{9}$ requires a resistance bridge very sensitive in current. The most accurate and sensitive bridge able to disseminate the resistance unit[@Poirier2009] is based on the performance of a cryogenic current comparator (CCC). The CCC[@Harvey1972] is basically a perfect transformer operating in direct current regime (dc) able to measure the ratio of the currents circulating through the two resistances to compare with a relative uncertainty below $10^{-10}$. Made of superconducting windings embedded in a superconducting shielding, its accuracy relies on the Messner effect. Its high-sensitivity comes from the flux detector equipping it, which is based on a Superconducting QUantum Interference Device (SQUID)[@Gallop2006]. The development of resistance bridge equipped with a CCC started in several national metrology institutes (NMI), including the French institute, right after the discovery of the QHE. First ones were operating in dc[@Delayahe1985; @Williams1991; @Hartland92; @Dziuba1993]. Then, other bridges adapted to measurements in the low-frequency (below 1 Hz) alternating current regime (ac) were proposed[@Delahaye1991; @Seppa1997]. Accurate operation at higher frequencies was achieved by replacing the CCC with a room-temperature current comparator using high-permeability magnetic cores[@DelahayeAC1993; @Satrapinski2017] but at the expense of larger measurement uncertainties. Since the nineties, international bilateral resistance comparisons[@BIPMEMK12], organized by the *Bureau International des Poids et Mesures* (BIPM), have demonstrated the equivalence of several NMIs using CCC-based resistance bridge (RB) for the realization of the ohm from the QHE with a relative uncertainty of a few parts in $10^9$.

The LNE has been using such a dc bridge[@Delayahe1985; @Piquemal1995] for more than thirty years to perform calibrations of wire resistors with an accuracy of a few parts in $10^9$, as about twenty NMIs do at the present time. This old instrument now suffers several limitations with regards to the current needs not only for calibrations but also for research works: few resistance ratios can be measured (1, 10, 12.9, 50, 64.5, 100, 129), the optimal resistance range extends from 1 $\Omega$ to $R_\mathrm{K}/2$ only, the type B uncertainty (of about 7 parts in $10^{10}$ in relative value for the 100 $\Omega$ calibration from the QHE) cannot be further reduced and the type A uncertainty is limited to one part in $10^{9}$ (for one hour measurement time) by the outdated performance of the radio-frequency SQUID. The improvement of digital and analog electronic components and the availability of dc SQUIDs allow the development of a resistance bridge with better performance in terms of sensitivity, accuracy and automation. Several NMIs have thus recently developed more modern and automated RB[@Sanchez2009; @Drung2009; @Gotz2009; @Williams2010].

Here, we report on a semi-automated RB designed to perform comparisons of resistances, of values ranging from $1~\Omega$ to $\mathrm{1.29~M\Omega}$, in ratios from 1 up to 1290, based on current sources delivering currents from $1~\mu$A up to 100 mA. Compared to the older LNE bridge, a strong performance upgrade is expected due to improvements not only in the technical design but also in the instruments making up the bridge. Our target combined standard uncertainty for the calibration of a $100~\Omega$ wire resistor in terms of $R_\mathrm{K}/2$ is a few parts in $10^{10}$ (k=1). The paper is organized as follows: the principle of the RCB is described in Section II, the design and the performance of the CCC are presented in Section III, Section IV describes the electronics of current sources, Section V describes current dividers used to set the fine tuning of the current ratio, Section VI presents the shielding techniques implemented, measurements of SQUID noise and of resistance ratio are presented in Section VII and finally a conclusion is made in Section VIII.

Principle of the RB
===================

![Principle of the new LNE resistance bridge based on a CCC. The figure shows the two interlocked current sources, the CCC equipped with a DC SQUID and the feedback control on the secondary current source, the current dividers injecting the in-phase, $\epsilon I_\mathrm{S}$, and the in-quadrature, $j\epsilon_{q}I_\mathrm{S}$, current fractions respectively, the null detector and the two resistances $R_\mathrm{S}$ and $R_\mathrm{P}$ to compare. The ground can be connected in position A (low potential of the secondary resistor) or B (low potential of the secondary winding).[]{data-label="fig1"}](Fig-PrincipleSchematic.pdf){width="3.5in"}

The principle of the new RB, described in fig.\[fig1\], is close to that of the older one. It is based on two synchronized sources, primary (P) and secondary (S) sources, that deliver currents $I_\mathrm{P}$ and $I_\mathrm{S}$ respectively. The primary (secondary) source supplies the resistance $R_\mathrm{P}$ ($R_\mathrm{S}$) in series with a superconducting winding of a CCC of number of turns $N_\mathrm{P}$ ($N_\mathrm{S}$). The number of turns $N_\mathrm{S}$ and $N_\mathrm{P}$ are chosen so that the ratio $N_\mathrm{S}/N_\mathrm{P}$ is close to the resistance ratio $R_\mathrm{S}/R_\mathrm{P}$. A standard current divider (SCD) is used to deviate an in-phase calibrated fraction $\epsilon$ of the current $I_\mathrm{S}$ into an auxiliary winding of number of turns $N_\mathrm{A}$. The windings of the CCC are wound according to a toroidal geometry and embedded in a superconducting shielding. Application of the Ampere’s theorem to a circulation along a cross-section of the shielding, where the magnetic flux density is zero, leads to the relationship $N_\mathrm{P}I_\mathrm{P}-(N_\mathrm{S}+\epsilon N_\mathrm{A})I_\mathrm{S}=I_\mathrm{CCC}$, where $I_\mathrm{CCC}$ is a screening current. Because the CCC shielding overlaps itself on two or three turns without electrical contact, this superconducting current circulates from the inner to the outer side of the shielding. It can be therefore detected by a pick-up coil coupled to the outer side and connected to the entry inductance of the SQUID. The secondary current source is servo-controlled by the output of the CCC SQUID electronics so that the screening current $I_\mathrm{CCC}$ (i.e. the total ampere.turn) is nulled. It results that: $$N_\mathrm{P}I_\mathrm{P}-(N_\mathrm{S}+\epsilon N_\mathrm{A})I_\mathrm{S}=0.
\label{Equation:AmpereTurns}$$ From the fraction $\epsilon_0$ setting the voltage unbalance, $R_\mathrm{S}I_\mathrm{S}=R_\mathrm{P}I_\mathrm{P}$, one obtains: $$R_\mathrm{S}/R_\mathrm{P}=(N_\mathrm{S}+\epsilon_0 N_\mathrm{A})/N_\mathrm{P}.
\label{Equation:resistance}$$ The SCD can also be inserted in the primary circuit to deviate a fraction of the current $I_\mathrm{P}$. In this case, the previous equations remain valid by simply exchanging S and P index. This is the operating mode planned for measurements involving a low resistance $R_\mathrm{S}$ (for example 1 $\Omega$) supplied by a large current $I_\mathrm{S}$ (for example 50 mA) which would lead to a too strong dissipation in the SCD if placed in the secondary circuit. Instead, the SCD inserted in the primary circuit is biased by the lower current $I_\mathrm{P}$ which is usually below 10 mA. Improvements of the bridge mainly concern i) the two current sources able to operate in DC and at very low frequencies which are both servo-controlled by a unique external voltage source, ii) the accurate and stable standard current divider used to adjust the current ratio to the resistance ratio, and iii) the new DC SQUID-based CCC.

The new bridge also includes a second current divider able to deviate an in-quadrature current fraction $j\epsilon_{q}I_\mathrm{S}$ in a fourth winding of number of turns $N_\mathrm{A}^q$. It is used to cancel the voltage overshoots occurring during current reversals at the entry of the null detector (ND) that are caused by the capacitances $C_\mathrm{P}$ and $C_\mathrm{S}$ in parallel to the resistors $R_\mathrm{P}$ and $R_\mathrm{S}$ respectively. More precisely, the master equation for ampere.turns becomes: $$N_\mathrm{P}I_\mathrm{P}-(N_\mathrm{S}+\epsilon N_\mathrm{A}+j\epsilon_{q}N_\mathrm{A}^q)I_\mathrm{S}=0.
\label{Equation:AmpereTurns}$$ Assuming first-order approximation in angular frequency $\omega$, the voltage balance condition leads to the first equation \[Equation:resistance\] and a second equation involving capacitances according to: $$(R_\mathrm{P}C_\mathrm{P}-R_\mathrm{S}C_\mathrm{S})\omega=\epsilon_{q}\frac{R_\mathrm{P}N_\mathrm{A}^q}{R_\mathrm{S}N_\mathrm{P}}.
\label{Equation:Phase}$$ The quadrature current divider (QCD) is therefore used to compensate quadrature signals caused by capacitances. By cancelling the voltage overshoots that can saturate the null detector, the reversal frequency of the current can be increased. This reduces the impact of voltage offset drift and of the $1/f$ SQUID noise on measurements. Moreover, it allows optimizing the ratio between the acquisition time and the total experience time. The calibration of the QCD fractions is not required.

Finally, careful shielding of cables and guarding of circuits are implemented to ensure the equality of the currents circulating through the resistor and the winding for better accuracy.

the CCC
=======

Design and fabrication
----------------------

![Pictures of the CCC at different stages of shielding assembly. a) The CCC alone. b) The CCC on the probe with all shields removed. c) First Pb/Brass shield of the CCC in place. d) Second Pb/Brass shield of the SQUID in place. e) External cryoperm shield in place.[]{data-label="fig2"}](Fig2.pdf){width="3.5in"}

** The cryogenic current comparator is made of 15 windings of 1, 1, 2, 4, 16, 16, 32, 64, 128, 160, 160, 1600, 1600, 2065 and 2065 turns which hold together with epoxy glue[@Soukiassian2010]. Each winding is made of superconducing and insulated $\mathrm{60~\mu m}$ diameter NbTi/Cu wire. We used optically checked 0.1 mm thick Pb sheets and Pb/Sn/Cd superconducting solder at a temperature lower than $150^{\circ}$C to realize the toroidal shielding around the windings. Our shield overlaps twice (3 layers) to prevent flux leakage. Each layer is covered with PTFE (poly-tetra-fluoro-ethylene) tape for electrical insulation. The CCC is fixed to the cryogenic probe with a piece in MACOR$^{\scriptsize\textregistered}$ material. The DC-SQUID (Quantum Design, Inc) has an input inductance of $L_i=1.8~\mu H$ and a nominal flux noise in flux-lock feedback mode of $\mathrm{3~\mu\phi_0/Hz^{1/2}}$ above 0.3 Hz. It is coupled to the CCC via a superconducting flux transformer made of a NbTi wire inserted in a lead tube. The sensitivity of the system $S_\mathrm{CCC}$ depends on the number of turns $N_\mathrm{PC}$ of the pick-up coil coupled to the CCC through the relationship:\
$$S_\mathrm{CCC}=(2/k)N_\mathrm{PC}S_\mathrm{SQ},$$ where $k$ is the coupling constant between the CCC and the pickup coil, and $S_\mathrm{SQ}$ the SQUID sensitivity (in $\mathrm{\mu A/\phi_0}$). The best sensitivity $S_\mathrm{CCC}^\mathrm{opt}$ is obtained for $N_\mathrm{PC}^\mathrm{opt}$ given by:\
$$N_\mathrm{PC}^\mathrm{opt}=\sqrt{L_i/L^\mathrm{eff}_\mathrm{CCC}},$$\
where $L^\mathrm{eff}_\mathrm{CCC}$ is the effective self inductance of the CCC taking account of the superconducting screen that isolates the SQUID from external magnetic fields (Earth’s field for instance). $L^\mathrm{eff}_\mathrm{CCC}$ and therefore $S_\mathrm{CCC}$ can be determined in a given geometry using the analytical calculation of Sesé and co-authors[@Sese1999; @Sese2003]. In our case, one calculates $L^\mathrm{eff}_\mathrm{CCC}\sim 14$ nH, $N_\mathrm{PC}^\mathrm{opt}=12$ and $S_\mathrm{CCC}^\mathrm{opt}=\mathrm{5~\mu A.t/\phi_0}$. Due to geometrical constraints, the number of turns of the pick-up coil was reduced to $N_\mathrm{PC}=6$ leading to an experimental sensitivity $S_\mathrm{CCC}=\mathrm{8~\mu A.t/\phi_0}$ close to the calculated value of $\mathrm{6~\mu A.t/\phi_0}$. Our magnetic screen is made up of 5 concentric cylinders: two Pb ones, each one embedded in a brass one, and an outer Cryoperm® cylinder. Each cylinder is closed at the top with the same material. The cryogenic probe body is made of three rods to stabilize it mechanically and prevent from vibrations.

Noise performance
-----------------

![Noise spectral density measured by the SQUID *versus* frequency *f* for the CCC alone and not connected.[]{data-label="Fig-NoiseCCCAlone"}](Fig-NoiseCCCAlone.pdf){width="3.5in"}

The CCC was firstly tested with all windings disconnected. Fig.\[Fig-NoiseCCCAlone\] shows the noise spectral density in $\mathrm{\phi_0/Hz^{1/2}}$ as a function of the frequency. The main frequency resonance due to the coupling of the large inductance of windings and the capacitance between wires is around 14 kHz. Between 6 Hz and about 2 kHz, the noise spectral density is dominated by sharp peaks with an amplitude lower than $1~\mathrm{m\phi_0/Hz^{1/2}}$ which are caused by mechanical resonances. At lower frequencies down to about 0.1 Hz, there exists a white noise regime with a constant noise spectral density of about $10~\mathrm{\mu \phi_0/Hz^{1/2}}$. Considering the CCC sensitivity of $\mathrm{8~\mu A.t/\phi_0}$, this leads to a current sensitivity of about $\mathrm{80~pA/Hz^{1/2}}$. One can guess *1/f* noise above $10~\mathrm{\mu \phi_0/Hz^{1/2}}$ at frequencies lower than 0.1 Hz what is expected if considering the 0.3 Hz corner frequency and the $3~\mathrm{\mu \phi_0/Hz^{1/2}}$ white noise level of the Quantum Design DC SQUID. We therefore conclude that the current reversal frequency of the resistance bridge should be higher than 0.1 Hz to obtain the lowest measurement noise.

Accuracy
--------

The accuracy of the CCC was tested by realising winding opposition experiments. Windings with same nominal number of turns are connected in series-opposition and supplied by a large current. The voltage at the SQUID output $V_\mathrm{Output}$ operating in internal feedback mode (MODE 5) is converted in magnetic flux $\delta_{\phi_0}$ using the SQUID gain in $V/\phi_0$. By dividing $\delta_{\phi_0}$ by the total magnetic flux $NI/S_\mathrm{CCC}$ generated by one winding, one obtains the relative error $\Delta N/N= \delta_{\phi_0} S_\mathrm{CCC}/(NI)$. We have used the very low noise current source of the RB to carry out these experiments in order to prevent from noise rectification by the SQUID. The measurement current is reversed from 100 mA to -100 mA to remove offset voltages from recorded data. For turn numbers *N* equal to 16 and 2065 which are used in the calibration of a $100~\Omega$ resistor in terms of $R_\mathrm{K}/2$, $\Delta N/N$ is found equal to $(1.9 \pm 1.2)\times 10 ^{-11}$ and $(2.5 \pm 0.04)\times 10 ^{-11}$ respectively. For all other winding opposition, turn errors are found smaller than $6\times10^{-11}$, except for 1-1 and 2-2 combinations which seem let us conclude to significant errors of  $\sim1\times10^{-9}$ and $\sim5\times10^{-10}$. But, a magnetic flux leakage caused by a hole in the toroidal shield or by a imperfect chimney would manifest by large errors for all winding opposition. Our interpretation is that these apparent errors are only caused by spurious signals coming from residual noise rectification which manifest themselves all the more as the total ampere.turn number is small, i.e. for 1-1 and 2-2 winding opposition. Further reduction of the noise emitted by the current source in the 100 mA range (which is the most noisy range) is required to refine the determination of winding errors for small turn numbers.

\[1\][&gt;m[\#1]{}]{}

[|c|M[5cm]{}|]{} **Winding combination**& **$\Delta N/N$** 1-1&$(1.29\pm 0.38)\times10^{-9}$ 2-2&$(4.64\pm 0.63)\times10^{-10}$ 16-16&$(1.9\pm 1.2)\times10^{-11}$ 16+16-32&$(2.5\pm 1.1)\times10^{-11}$ 16+16+32-64&$(0.22\pm 0.76)\times10^{-11}$ 16+16+32+64-128&$(1.0\pm 1.1)\times10^{-11}$ 160-160&$(2.5\pm 0.14)\times10^{-11}$ 160-128-32&$(0.5\pm 0.4)\times10^{-11}$ 1600-1600&$(5.6\pm 0.032)\times10^{-11}$ 2065-2065&$(2.54\pm 0.035)\times10^{-11}$\

Our conclusion is that the CCC error in the measurements of usual resistance ratios, which exploited mainly windings of number of turns 2065, 1600, 160, and 16, is of a few $10^{-11}$. The CCC contribution to the type-B uncertainty is therefore below $10^{-10}$, in relative value.

The current source electronics
==============================

Design and fabrication
----------------------

![Schemes of the electronic circuits of primary and secondary current sources. Both current sources are controlled by a unique external voltage source.[]{data-label="Fig-Circuits"}](Fig-Circuits.pdf){width="3.5in"}

The RB is based on two current sources generating the currents $I_\mathrm{P}$ and $I_\mathrm{S}$ which supply the resistors $R_\mathrm{P}$ and $R_\mathrm{S}$ respectively. In some recently developed resistance bridges, current sources are based on digital electronics connected by fiber optics to a internal micro-controller[@Drung2009] or an external PXI computer. This provides strong electrical insulation and easy automation but requires the implementation of efficient noise filtering techniques to protect the SQUID from the radio-frequency noise emitted by digital circuits.

On contrary, current sources of the LNE RB are based on linear analog circuits to avoid high-frequency noise[@Soukiassian2010]. Fig.\[Fig-Circuits\] shows the schematic of the two electronic circuits, the primary and the secondary ones. They are controlled and synchronized by a single external voltage source allowing automation of measurements. The latter can be either a dc voltage generator, like a Yogogawa 7651 for usual dc measurements or the oscillator of a lock-in detector for low-frequency ac measurements. The external reference voltage supplies the primary circuit through a high-impedance differential amplifier. A low-pass filter is then used to limit the signal bandwidth with an adjustable cutoff frequency ranging from 1 mHz to 1 kHz. After a stage summing additional voltage corrections and a division stage allowing the setting of a decimal fraction, the voltage is converted into a current in ranges extending from 1 $\mu$A up to 100 mA. This conversion is done using an amplifier inverter circuit boosted by a buffer amplifier (BUF634T) and a dividing resistor. The secondary current source, controlled by the output signal of the low-pass filter of the primary current source, is similar but the current range selected can also be multiplied by a factor 1.2906 or 1/1.2906 to adapt to measurements involving the QHR connected either to the primary or to the secondary current sources. Several additional circuits are implemented to finely adjust the current ratio $r_I=I_\mathrm{S}/I_\mathrm{P}$ to within a few parts in $10^{6}$. This is necessary to limit the ampere.turn unbalance in the CCC, not only to avoid unlocking of the SQUID feedback notably during current switching, but also to achieve the best accuracy in the $I_\mathrm{S}/I_\mathrm{P}$ ratio adjustment. Offset, in-phase and in-quadrature correction circuits are used to tune the secondary current while an asymmetry correction circuit injecting a fraction of the voltage absolute value is used in the primary circuit to compensate, to some extent, the asymmetry behaviour of operational amplifiers. Finally, the SQUID feedback voltage, after insulation by an differential amplifier, is converted into a feedback current at the last stage of the secondary circuit so that the closed-loop feedback gain remains the same as in internal feedback mode, i.e. $\sim0.75~V/\phi_0$.

Electronic circuits of each current source are integrated, but electrically isolated with PTFE material, into their own metallic box connected to ground, as shown in pictures of fig.\[FigAppendix-Sources\]. The electronic components are powered by stabilized voltages provided by a circuit itself energized by rechargeable batteries. These are also electrically isolated from the grounded metallic box in which they are placed. The only electrical link between the electronic circuits and the ground comes from the high-impedance operational amplifiers (OPA128LM) which ensure a high-isolation (in principle $\sim10^{15} ~\Omega$ resistance) from the piloting external voltage source and the SQUID feedback electronics. All these precautions aim at cancelling leakage currents.

Test of the current ratio adjustability
---------------------------------------

![Illustration of the adjustability of the current ratio $I_\mathrm{P}/I_\mathrm{S}$ using resistances $R_\mathrm{P}=\mathrm{10~k\Omega}$ and $R_\mathrm{S}=\mathrm{100~\Omega}$. The balance voltage $\Delta V$, measured by the null detector (ND) is recorded as a function of time while the current is periodically reversed (red dashed line) for different settings of the corrections circuits : adjustment of the offset correction only (blue line), adjustment of the in-quadrature correction (green line), adjustment of in-phase, in-quadrature and asymmetry corrections (deep blue line).[]{data-label="Fig-CurrentAdjustment"}](Fig-CurrentAdjustment.pdf){width="3.5in"}

Fig.\[Fig-CurrentAdjustment\] shows the experiment carried out to test the adjustability of the ratio of the two current sources[@Soukiassian2010]. Two resistors of resistance $\mathrm{10~k\Omega}$ and $\mathrm{100~\Omega}$ are fed by currents $I_\mathrm{P}$ and $I_\mathrm{S}$ respectively. The potential drop difference $\Delta V$ at the terminals of the two resistors is recorded by a null detector (nanovoltmeter EMN11). For a nominal voltage reversing from 1 V to -1 V every 20 secondes, Fig.\[Fig-CurrentAdjustment\] shows that it is possible to reduce the peak to peak $\Delta V$ amplitude to less than $\mathrm{2~\mu V}$ by optimizing the in-phase, the in-quadrature, the offset and the asymmetry corrections. Let us notably remark the effect of the in-quadrature correction in cancelling the voltage overshot caused by the fast current reversal. The current ratio can therefore be adjusted with an accuracy of 2 parts in $10^6$ even during fast current reversal which is an advantage to avoid any SQUID unlocking.

Noise optimization and filtering {#Noise optimization and filtering}
--------------------------------

![Electronic scheme of the two last stages (A and B) of the secondary current source (the primary current source does not include the SQUID feedback circuit) describing the noise filtering techniques. Picture of the common mode torus used to block the circulation of current noise towards ground. The dotted line represents the limit of the case (at ground) of the current source.[]{data-label="Fig-Filtering"}](Fig-Filtering.pdf){width="3.5in"}

Noise filtering is crucial particularly from the resonance frequency of the CCC (14 kHz) up to the operating frequency of the modulation circuit (500 kHz) not only to ensure a good working of the SQUID but also to avoid noise rectification that would alter measurement accuracy. Fig.\[Fig-Filtering\] shows the last stage of the electronic circuit of the secondary current source. The primary current source is based on a similar stage but differs by the absence of the SQUID feedback electronic circuit. In practice, the frequency bandwidth of the primary and secondary current sources was reduced to 160 Hz at the stage A and 1 kHz at the stage B of the electronic circuit using simple low-pass filters based on resistors and capacitors. By this way, the frequency bandwidth of the SQUID feedback circuit servo-controlling the secondary current is set to 1 kHz. Table.\[tableau:Range\] summarizes the capacitance $C_\mathrm{F}$ and resistance $R_\mathrm{F}$ values chosen to set the 1 kHz cut-off frequency that damps the CCC resonance for each current range (defined by the value of the resistor $R_\mathrm{C}$), considering the nominal value of the resistance in measurement $R_\mathrm{E}$.

It is also essential to avoid the circulation of the current noise coming from the capacitive coupling of the electronics circuit with ground. To cancel this noise source which renders the SQUID inoperative, a common mode torus (CMT) was introduced in the current circuit of each source (see fig.\[Fig-Filtering\]). This CMT is made of a PTFE insulated wire pair wounded about 60 times around an APERAM Nano magnetic torus (magnetic permeability of about 80000 up to a 100 kHz frequency) with a return spire. The differential inductance is around $\mathrm{3~\mu H}$ while the common mode inductance is around 0.6 H. The common mode impedance, which increases from about $200~\Omega$ at 50 Hz up to $\mathrm{150~k\Omega}$ at 1 MHz, drastically reduces the circulation of the common mode current noise. This protects the SQUID and makes it operating quite ideally whether it is a radio-frequency SQUID or a DC SQUID.

\[1\][&gt;m[\#1]{}]{}

        **Range**         **$R_\mathrm{C}$**   **$R_\mathrm{E}$**   **$C_\mathrm{F}$**   **$R_\mathrm{F}$**
  ---------------------- -------------------- -------------------- -------------------- --------------------
    1 $\mathrm{\mu A}$       5 M$\Omega$          1 M$\Omega$             300 pF            4 M$\Omega$
   10 $\mathrm{\mu A}$      500 k$\Omega$        100 k$\Omega$            300 pF           400 k$\Omega$
   100 $\mathrm{\mu A}$      50 k$\Omega$         10 k$\Omega$             3 nF             40 k$\Omega$
           1 mA              5 k$\Omega$          1 k$\Omega$             30 nF             4 k$\Omega$
          10 mA              500 $\Omega$         100 $\Omega$            300 nF            400 $\Omega$
          100 mA             50 $\Omega$          10 $\Omega$       3 $\mathrm{\mu F}$      40 $\Omega$

  : Resistance and capacitance values in the last stage of the secondary current source for the different current ranges.[]{data-label="tableau:Range"}

The SQUID feedback circuit
--------------------------

As shown in fig.\[Fig-Filtering\], the SQUID feedback voltage $V_\mathrm{SQUID}$ probed in the SQUID Quantum Design Preamplifier is sent to the secondary current source of the bridge after decoupling by a high impedance differential amplifier. A resistor is biased by the $V_\mathrm{SQUID}$ voltage to inject the feedback current in the secondary current circuit supplying the secondary winding ($N_\mathrm{S}$). The resistor value, $R_\mathrm{FB}=\mathrm{1.5~M\Omega}$, is chosen so that the closed-loop feedback gain $G_\mathrm{CLG}=R_\mathrm{FB}\times S_\mathrm{CCC}/N_\mathrm{S}$ for for $N_\mathrm{S}=16$ is the same as in the most sensitive internal feedback mode of the SQUID (mode 5 and mode 5s) equal to $0.75~V/\phi_0$. A circuit made of a small capacitance of 200 pF in series with a $\mathrm{20~k\Omega}$ resistor is connected in parallel to the $\mathrm{1.5~M\Omega}$ resistor to partially compensate the dephasing caused by the 1 kHz low-pass filter implemented in the stage B of the electronic circuit and therefore optimize the SQUID operation.

Current dividers
================

The standard (or in-phase) current divider
------------------------------------------

The standard current divider (SCD) is used to balance the voltages measured at the terminals of the two resistors by deviating a fraction of current towards the auxiliary winding. This is a key element of the RB, the accuracy of which directly impacts the uncertainty budget of the RB. An alternative technique to null the voltage measured by the detector consists in using an auxiliary current source servo-controlled by the null detector voltage output[@Williams1991; @Hartland92]. This relies on an active component, a second feedback electronics and an accurate measurement of the current delivered. On contrary, the SCD developed is a passive component that, once calibrated, can inject a known current. Moreover, it avoids the use of a second feedback electronics. However, one requirement is a good stability of the current fractions defined by the SCD. This is achieved with a design that limits the number of electrical commutations required to select the current fraction. The counterpart of these technical choices aiming at better reproducibility is that the calibration of the LNE SCD is not fully-automated contrary to that of binary compensation units[@Drung2013; @Gotz2017].

![Electrical scheme of the standard current divider (SCD). $N$, $P$, $Q$, $P'$, $Q'$ are integers defining the the setting of the SCD. A CMT is inserted between the SCD and the auxiliary winding to reduce the current noise circulation through ground.[]{data-label="fig-SCD"}](Fig-SCD.pdf){width="3.5in"}

The SCD, described in fig.\[fig-SCD\], is built to inject in the auxiliary winding fractions of a main current (lower than 10 mA) ranging from 0 to $5\times10^{-5}$ by minimal step of $5\times10^{-8}$. Made of three main series resistor networks ($10\times 20~\Omega$, $10\times 2~\Omega$, $10\times 200~\Omega$) and a large $\mathrm{4~M\Omega}$ division resistor, the SCD ratio can be adjusted using three mechanical IEC MONACO commutators with goal-coated silver contacts (see picture in fig.\[FigAppendix-SCD\] of appendix section). The current fraction is given by: $\epsilon_{(N,P,Q)}= N\times5\times10^{-6}+P\times5\times10^{-7}+Q\times5\times10^{-8}$ where *N*, *P*, *Q* are the integer values between 0 and 10 indexing the three commutator positions, respectively.

To achieve this ideal behavior, it was necessary to implement two additional compensations resistor networks in order to circumvent the non-linearities that result from the variation of the division resistance varying *P* and *Q*. Indeed, the triangle formed by resistors ($2~\Omega$, $Q\times 200~\Omega$, $(10-Q)\times 200~\Omega$) leads to the addition of a *Q* dependent resistance to the $\mathrm{4~M\Omega}$ resistance. To solve this non-linearity, a compensation resistance in series with the $\mathrm{4~M\Omega}$ resistance is selected for each *Q* value from the resistor network ($0~\Omega$, $20~\Omega$, $80~\Omega$, $180~\Omega$, $320~\Omega$, $500~\Omega$, $320~\Omega$, $180~\Omega$, $80~\Omega$, $20~\Omega$, $0~\Omega$). The variation of the division resistance changing the *P*-dependent position of the potentiometer is compensated by selecting a fraction of the resistor network ($10\times 2~\Omega$) using the index *P’*=10-*P*. Finally, the fraction of the resistor network ($10\times 20~\Omega$) selected by *N’* index adds to keep constant the total resistance of the SCD independently of the *N* index change: *N’*=10-*N*. This is useful to set the frequency bandwidth of the current source independently of the SCD setting. To achieve the best stability of the SCD fractions with time and under load, resistor networks are constituted of high stability (drift lower than $10^{-5}$/year, in relative value), low temperature coefficient ($<0.6\times10^{-6}/^{\circ}$C) and hermetic Vishay resistors (VH516-4 $20~\Omega$, VHS102 $2~\Omega$, VH518-10 $\mathrm{1~M\Omega}$, VH102K $200~\Omega$) connected in series with low electromotive force solder. Finally, a CMT is introduced that drastically reduces the current noise signals circulating from ground to the auxiliary winding through the SCD circuit.

The fractions corresponding to the three following sets of commutator positions: ($N$, $P=0$, $Q=0$) with $N$ varying from zero to ten, ($N=0$, $P$, $Q=0$) with $P$ varying from zero to ten, ($N=0$, $P=0$, $Q$) with $Q$ varying from zero to ten, were calibrated by measuring the ratio of the voltage $V_\mathrm{AB}$ to a reference bias voltage applied in place of the auxiliary winding. Calibrations performed over ten years showed that fractions have remained close to their nominal value within one part in $10^9$, drifting each by no more than 5 parts in $10^{10}$. More generally, the fraction $\epsilon_{(N,P,Q)}$ corresponding to the commutator position ($N$, $P$, $Q$) is given by : $$\begin{aligned}
\epsilon_{(N,P,Q)}&=\epsilon_{(N,P=0,Q=0)}+\epsilon_{(N=0,P,Q=0)}\\
&+\epsilon_{(N=0,P=0,Q)}-2\epsilon_{(0,0,0)}.\end{aligned}$$

The quadrature current divider
------------------------------

Capacitive leakages to ground short-circuiting the secondary resistor lead to voltage overshoots during current variations that can possibly saturate the null detector. This limits the maximum current reversing speed. To circumvent this difficulty, a second current divider (QCD) is connected in series with the SCD to cancel these voltage overshoots (see fig.\[fig1\]) by injecting an in-quadrature fraction $j\epsilon_q$ of the main current in a CCC auxiliary winding of number of turns ${N_\mathrm{A}^q}$, as highlighted by equations \[Equation:AmpereTurns\] and \[Equation:Phase\]. More precisely, the main current is flowing through a $R_\mathrm{q}$ resistor of $1~\Omega$ or $10~\Omega$ resistance value. A $100~\Omega$ potentiometer connected in parallel allows the adjustment of the voltage fraction $\alpha$ biasing a $C_\mathrm{q}=235$ nF PTFE capacitor (its parallel resistance is higher than $2\times 10^{13}~\Omega$) in series with the CCC auxiliary winding (typically ${N_\mathrm{A}^q}=1600$). The current fraction injected by the QCD is therefore $j\epsilon_q\simeq j\alpha R_\mathrm{q}C_\mathrm{q}\omega$, where $\alpha\in[0:1]$ is the potentiometer fraction. It is crucial that this current remains negligible during the data acquisition. Considering a current setting time constant of 0.5 s, one can calculate that the current fraction drops down to no more than a few parts in $10^{13}$ of the main current after a waiting time of only 8 s. Let us note that the quadrature current divider is also equipped with a CMT to reduce current noise circulation (see picture in fig.\[FigAppendix-SCD\] of appendix section).

Shielding and guarding
======================

![Left: Picture of the resistance bridge. Right: Picture of the CCC winding switching box at the top of the cryostat.[]{data-label="fig4"}](Fig4.pdf){width="3.5in"}

As shown in fig. \[fig4\], each sensitive element of the RB is carefully shielded against noise. The null detector, the current sources, the current dividers, the power supply are in grounded metallic boxes. The QHR and the CCC are each in an independent cryostat connected at ground. The continuity of the shielding between the different elements is ensured by the connection cables, the metallic sheath of which is also connected at ground as schematized in fig.\[fig1\]. This directs any leakage current between wires at different potentials towards the ground. In normal operation, the ground is connected both to the low potential of the resistance $R_\mathrm{S}$ (position A in fig.\[fig1\]) and to the case of the EM detector (there is no common mode voltage). The leakage current, $I_g$ therefore short-circuits the lowest resistance (black arrow) which reduces its impact. This grounding is usually efficient to limit the leakage current contribution to the type B uncertainty below $10^{-9}$ for the measurement of the $100~\Omega/(R_\mathrm{K}/2)$ ratio.

The ground can also be connected at the low potential of the secondary winding (position B in fig.\[fig1\]). The leakage current, $I_g$, short-circuiting both the resistor and the winding of the secondary circuit (grey arrow) in this case, the measurement accuracy of the resistance ratio is not altered at all by leakage currents. But, this is at the expense of a common mode voltage, although weak, existing between the ground and the low potential of the null detector due to the small resistance (about 2 $\Omega$) of the winding. To minimize its effect on the measurement accuracy, the case and the low potential of the null detector are short-circuited. Besides, capacitance hand-effects are cancelled because the null detector is itself placed in a grounded metallic box.

Resistance ratio Measurements
=============================

Noise spectrum and SQUID feedback stability
-------------------------------------------

![Noise spectral density measured by the SQUID *versus* frequency *f* for the measurement of the 100$\Omega$/10 k$\Omega$ ratio. SQUID operating in internal feedback mode 5 (black), in external feedback mode 5s (red), and in external feedback mode 500 (blue).[]{data-label="Fig-NoiseCCC10k100"}](Fig-NoiseCCC10k100.pdf){width="3.5in"}

The operation stability of the SQUID was demonstrated for several ratio measurements: 10 k$\Omega$/1 M$\Omega$, 100 $\Omega$/10 k$\Omega$ and 1 $\Omega$/100 $\Omega$. The following settings of the bridge were used: $N_\mathrm{P}=1600$, $N_\mathrm{S}=16$, $N_\mathrm{A}=16$ and a current divider connected in series with the secondary current source for the two first ratios, $N_\mathrm{P}=1600$, $N_\mathrm{S}=16$, $N_\mathrm{A}=1600$ and a current divider connected in series with the primary current source for the ratio 100 $\Omega$/1 $\Omega$. The quadrature divider was not used for these tests.

Fig.\[Fig-NoiseCCC10k100\] shows the noise spectral density, expressed in $\mathrm{\phi_0/Hz^{1/2}}$, determined by the Quantum Design SQUID operating in different feedback modes for the measurement of the 100 $\Omega$/10 k$\Omega$ ratio using current ranges $10$ mA/100 $\mu$A. In closed feedback mode operation, the output noise corresponds to the difference between the noises emitted by the primary and the secondary current sources since the current ratio $I_\mathrm{S}/I_\mathrm{P}$ is adjusted within $10^{-6}$ to cancel to ampere.turn unbalance of the CCC, i.e. the magnetic flux in the SQUID. So there remains only the uncorrelated noise contributions of both current sources. Let us note that the residual magnetic flux noise crossing the SQUID itself has a much lower level because of the real-time compensation by the feedback signal. It is given by the combination of the intrinsic SQUID noise (3 $\mathrm{\mu\phi_0/Hz^{1/2}}$), the environmental noise directly captured by the SQUID and the current source noise divided by the open-loop amplification gain. This latter contribution is negligible. The two others, which manifest in the CCC alone and disconnected, give a contribution of about 10 $\mathrm{\mu\phi_0/Hz^{1/2}}$ as observed in fig.\[Fig-NoiseCCCAlone\].

The internal (through the modulation coil of the SQUID) and the external (through the CCC winding) feedback mode operations mainly differ by their bandwidth as one can observe in fig.\[Fig-NoiseCCC10k100\]. In internal feedback mode 5, the bandwidth of about 20 kHz allows measuring the noise level up to the frequency resonances of the CCC. Fig.\[Fig-NoiseCCC10k100\] shows that the noise amplitude is above a noise floor level of about 140 $\mathrm{\mu\phi_0/Hz^{1/2}}$. This bottom level is notably explained by the Johnson-Nyquist noise, of 120 $\mathrm{\mu\phi_0/Hz^{1/2}}$, generated by the $R_\mathrm{C}=50$ k$\Omega$ resistor defining the 100 $\mu$A current range of the primary current source. Below 10 Hz, the noise increase is mainly caused by the $1/f$ voltage noise of the operational amplifiers (OPA111BM) which polarizes the 50 k$\Omega$ dividing resistor. The operation in external feedback mode 5s is very stable. The noise spectrum is similar but is characterized by a lower frequency bandwidth limited to about 500 Hz by the SQUID electronics. The cutoff frequency manifests itself by a peak, above which the signal then decreases of 20 dB by decade. The operation in external feedback mode 500 is also stable but with a higher cutoff frequency of about 1 kHz defined by the secondary current source filters (in fact the 200 pF capacitance of the feedback circuit (fig.\[Fig-Filtering\]) slightly extends the frequency bandwidth above 1 kHz set by the $C_\mathrm{F}$ capacitance). In some configurations, this larger frequency bandwidth can ensure a better stability of the bridge operation against higher-frequencies acoustic noises.

![Noise spectral density measured by the SQUID *versus* frequency *f* for the measurement of the 1 $\Omega$/100 $\Omega$ ratio a) and 10 k$\Omega$/1 M$\Omega$ ratio b). SQUID operating in internal feedback mode 5 (black) and in external feedback mode 5s (red).[]{data-label="Fig-NoiseCCCAutres"}](Fig-NoiseCCCAutres.pdf){width="3.5in"}

Fig.\[Fig-NoiseCCCAutres\]a) and b) demonstrate stability of operation of the external SQUID feedback in the measurements of ratios 1$\Omega$/100 $\Omega$ and 10 k$\Omega$/1 M$\Omega$ respectively. The base noise level is larger for the measurement of the 1 $\Omega$/100 $\Omega$ ratio (above $\mathrm{{2~m\phi_0/Hz^{1/2}}}$). This comes from the reduction to $R_\mathrm{C}=5$ k$\Omega$ of the resistor defining the 1 mA range used to supply the 100 $\Omega$ resistor. Conversely, the current sources are feebly noisy in the measurement configuration of the 10 k$\Omega$/1 M$\Omega$ ratio because of the $R_\mathrm{C}=5$ M$\Omega$ resistor defining the 1 $\mu$A range. One can observe in fig.\[Fig-NoiseCCCAutres\]b), a white noise level of no more than $\mathrm{20~\mu\phi_0/Hz^{1/2}}$ between 0.2 Hz and 6 Hz. This low base noise level allows observing the manifestation of moderate mechanical resonances in the range from 10 Hz and 1 kHz.

Measurement protocol and type A uncertainty
-------------------------------------------

![Measurement of the resistance ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ using the old a) and the new b) LNE bridge: relative voltage deviation $\Delta V/V$ as a function of time for several ($\mathrm{I^+}$, 0, $\mathrm{I^-}$) sequences. The signal period is about 200 s. The following settings were used: $N_\mathrm{P}=1936$, $N_\mathrm{S}=15$ and $N_\mathrm{A}=15$ for the older bridge and $N_\mathrm{P}=2065$, $N_\mathrm{S}=16$, $N_\mathrm{A}=16$ and $N_\mathrm{A}^q=1600$ for the new bridge. $\epsilon$ fractions are chosen to obtain similar deviation amplitude, $\Delta V/V$, for both bridges. $\epsilon_q$ setting of the new bridge QCD is optimized to cancel overshoots at current reversals.[]{data-label="Fig-StabilitySteps"}](Fig-StabilitySteps.pdf){width="3.5in"}

Measurements of the resistance ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ were performed using the old and the new LNE bridges. The primary current circulating through the GaAs/AlGaAs-based quantum resistance standard is set to $I_\mathrm{P}=70~\mu$A. For a $\epsilon$ fraction of the SCD which differs from $\epsilon_0$, a finite voltage $\Delta V$ can be detected by the null detector. The relative voltage $\Delta V/V$, where $V=R_\mathrm{P}I_\mathrm{P}$, is related, at the first order, to the deviation $(\epsilon-\epsilon_0)$ by: $$\Delta V/V=(\epsilon-\epsilon_0)\frac{N_\mathrm{A}}{N_\mathrm{S}}.$$ Let us note that, reversely, $\Delta V/V$ could be interpreted as a relative deviation of the ratio $r_R$ to the value $r_{R_0}$ giving $\Delta V=0$ for the fraction $\epsilon$. $\Delta V/V$ measured as a function of time during several ($\mathrm{I^+}$, 0, $\mathrm{I^-}$) sequences is reported in Fig.\[Fig-StabilitySteps\]a) and b) for the old and the new LNE bridge respectively. The signal period, of about 200 s, is imposed by the low-speed capability of the older bridge. The comparison of both data first shows the lower noise level and better stability achieved in measurements performed with the new bridge. This comes not only from the better performance of the current source electronics but also from the lower noise level of the CCC. Second, it demonstrates the efficiency and interest of the quadrature current divider which allows the cancellation of any voltage overshoot during the current switchings. This is useful to speed up the current reversal which reduces the impact of the voltage offset drift and of the $1/f$ SQUID noise.

![Measurement of the resistance ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ using the new LNE bridge with a primary current $I_\mathrm{P}=70~\mu$A: relative deviation $\Delta V/V$ as a function of time for 16 ($\mathrm{I^+}$, $\mathrm{I^-}$, $\mathrm{I^+}$) sequences and two successive settings, $\epsilon^+=-12.1\times10^{-6}$ and $\epsilon^-=+12.6\times10^{-6}$, of the SCD fractions.[]{data-label="Fig-QuickSequence"}](Fig-QuickSequence.pdf){width="3.5in"}

Fig.\[Fig-QuickSequence\] shows the typical data record for the measurement of the resistance ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ with the new bridge. It consists of two successive acquisitions of voltage measurements, $V^{\epsilon^+}$ and $V^{\epsilon^-}$, that are obtained for the two settings of the SCD fractions $\epsilon^+=-12.1\times10^{-6}$ and $\epsilon^-=+12.6\times10^{-6}$ respectively. Each acquisition is made of 16 ($\mathrm{I^+}$, $\mathrm{I^-}$, $\mathrm{I^+}$) sequences of current reversal that are used to remove voltage offsets. A mean voltage value is calculated from the average of the 16 values $[V(I^+)_1+V(I^+)_3-2V(I^-)_2]/4$, where 1,2,3 index the current state of each sequence. The resistance ratio is then obtained from the $\epsilon_0$ value calculated from the two mean voltages, $<V^{\epsilon^-}>$ and $<V^{\epsilon^+}>$, according to $\epsilon_0=\epsilon^- + (\epsilon^+-\epsilon^-)\times |<V^{\epsilon^-}>|/(|<V^{\epsilon^-}>|+|<V^{\epsilon^+}>|)$. Owing to the new bridge performances, the period of the signal was therefore reduced to about 70 s and the zero crossing step was removed. It results that the ratio between the acquisition time and the total experiment time is increased from about 50 percents with the older bridge to 75 percents with the new bridge, which is favorable to a reduction of the type A uncertainty.

![Measurement of the resistance ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ using the new LNE bridge with a primary current $I_\mathrm{P}=50~\mu$A. From an experiment described in supplementary of [@Ribeiro2015]. Standard Allan deviation of $r_R$, expressed in relative value, as a function of the acquisition time $t$ (red square). $t^{-1/2}$ adjustment of data (black dashed line). Standard deviation of the mean (blue open hexagon). Inset: ($\mathrm{I^+}$, $\mathrm{I^-}$, $\mathrm{I^+}$) sequence of $\Delta V$ measurements characterized by a period of 70 s.[]{data-label="Fig-Allan"}](Fig-Allan.pdf){width="3.5in"}

The noise performance of the new LNE bridge is demonstrated by the calculation of the Allan standard deviation[@Allan1987; @Witt2005] of $r_R=100~\Omega/(R_\mathrm{K}/2)$ from the statistical analysis of the voltage measurements performed using a primary current $I_\mathrm{P}=50~\mu$A. The evolution of this quantity, expressed in relative value, is reported in fig.\[Fig-Allan\] as a function of the experience time $t$. It follows a $t^{-1/2}$ law over measuring times longer than one hour. This shows that the white noise is dominant and that the standard deviation of the mean can be used to estimate the type A uncertainty. It follows that a type A standard uncertainty of $1.5\times10^{-10}$ can be achieved for the measurement of the ratio $r_R=100~\Omega/(R_\mathrm{K}/2)$ using a current of $50~\mu$A after an experience time of one hour. This is five time less than the best uncertainty achievable with the older bridge. This improvement relies not only on the quicker measurement protocol but also on the lower noise of the current source electronics and of the CCC. Let us remark that the contribution of the CCC to the voltage noise at the terminals of the null detector is no more than $\mathrm{0.5~nV/Hz^{1/2}}$. This is ten times lower than the EMN11 nanovoltmeter contribution, of about $\mathrm{5~nV/Hz^{1/2}}$, which limits the bridge type A uncertainty.

Preliminary uncertainty budget
------------------------------

A preliminary uncertainty budget, reported in table \[tableau:Uncertainty\], was established for the measurement of the ratio $100~\Omega/(R_\mathrm{K}/2)$. As demonstrated in the previous section, a type A uncertainty of $0.15\times10^{-9}$ can indeed be achieved for a 1 hour measurement experiment and a $I_\mathrm{P}=50~\mu$A measurement current. Further improvement would require the implementation of a lower-noise null detector.

Several components contributing to the Type B uncertainty were estimated as reported in Table \[tableau:Uncertainty\]. From the error measurements of the number of turns of windings, one can deduce a contribution of the CCC accuracy below $0.1\times10^{-9}$. However, lower-noise oppositions of windings of one and two turns are required in future to confirm the accuracy in measurements using such small turn numbers.

The accuracy of the resistance ratio measurement depends also on the SQUID feedback accuracy in cancelling the total ampere.turns number, i.e. in setting the current ratio $r_I=I_\mathrm{S}/I_\mathrm{P}$ to the target ratio given by the equation \[Equation:AmpereTurns\]. A setpoint error comes from the finite value of the open loop gain $G_\mathrm{OLG}$ of the SQUID electronics. Although the SQUID amplifier is based on an integrator which leads to an infinite gain at DC, measurements are in practice carried out with a finite time periodicity of the current reversal, typically of 70 s. The relative current ratio error is equal to $(\Delta^{adj}r_I/r_I)\times(G_\mathrm{CLG}/G_\mathrm{OLG})$, where $\Delta^{adj}r_I/r_I$ is the relative deviation between the target ratio and the preliminary adjusted current ratio. Measurements of the $100~\Omega/(R_\mathrm{K}/2)$ resistance ratio were performed for $\Delta^{adj}r_I/r_I$ values as large as a few $10^{-4}$. A relative error in the measurement of the resistance ratio lower than $10^{-11}$ is deduced in normal adjustment of the current ratio, i.e. for $\Delta^{adj}r_I/r_I\sim 10^{-6}$. This corresponds to $G_\mathrm{OLG}>7.5\times10^{4}$ V/$\phi_0$. This value is compatible with that yet determined in quantized current experiment[@Brun-Picard2016].

A main contribution comes from the calibration and stability of the standard current divider fractions. Characterizations performed over 10 years have demonstrated a very low drift of the fraction values, less than $5\times10^{-11}$/year in average value. At the present time, the calibration method of a fraction $\epsilon$ gives an uncertainty below $0.5\times10^{-9}$. A new calibration method, under development, aims at decreasing the uncertainty down to $0.3\times10^{-9}$ in order to benefit from the high-stability of the new standard current divider.

\[1\][&gt;m[\#1]{}]{}

   **Uncertainty components (k=1)**   **Contributions ($10^{-9}$)**
  ---------------------------------- -------------------------------
         **Type A (1 hour)**                    **0.15**
              **Type B**                  **0.7 (A), 0.5 (B)**
             CCC accuracy                        $<0.1$
       SQUID feedback accuracy                   $<0.01$
     Current divider calibration                 $<0.5$
          Leakage to ground              $\sim 0.5$ (A), 0.1 (B)
       **Combined Uncertainty**           **0.7 (A), 0.6 (B)**

  : Preliminary relative uncertainty budget for the $100~\Omega/(R_\mathrm{K}/2)$ ratio considering a $I_\mathrm{P}=50~\mu$A measurement current.[]{data-label="tableau:Uncertainty"}

The impact of leakage current to ground on the measurement accuracy was also estimated through several experiments. At first, no significant deviation was found within a relative uncertainty of about 0.8 part in $10^{9}$ between measurements of the resistance ratios $100~\Omega/(R_\mathrm{K}/2)$ and $200~\Omega/(R_\mathrm{K}/2)$, performed with the ground either connected in position A ($I_g$ parallel to $R_\mathrm{S}$) or position B ($I_g$ fully deviated). To obtain a better knowledge of leakage currents, comparisons were repeated with a larger secondary resistance $R_\mathrm{S}$=1 k$\Omega$ to amplify their effect. From the comparison of the two measurements of the ratio 1 k$\Omega/$1 k$\Omega$ performed with the resistors interchanged, a relative deviation of $(-3\pm 0.4)\times10^{-9}$ is found for the ground in position A which reduces to $(-0.04\pm 0.3)\times10^{-9}$ for the ground in position B. A similar discrepancy of a few parts in $10^{9}$ is also found by comparing the measurements of the ratio 1 k$\Omega/(R_\mathrm{K}/2)$ obtained for both ground positions. Moreover, it is found that the value obtained with the ground in position B agrees within 1.2 parts in $10^{9}$ with that deduced by combining the measurements of the 100 $\Omega/(R_\mathrm{K}/2)$ and 100 $\Omega/1$ k$\Omega$ ratios. One concludes that a significant discrepancy of a few parts in $10^9$ caused by leakage currents exists but can nevertheless be fully cancelled moving the ground in position B.

From these characterizations, a leakage current effect of about 0.5 part in $10^{9}$, i.e. ten times lower, can therefore be deduced for the measurement of the 100 $\Omega/(R_\mathrm{K}/2)$ ratio with the ground in position A. The contribution to the type B uncertainty budget falls below 0.1 part in $10^{9}$ by connecting the ground in position B.

Finally, no significant effect of the current reversal duration (from $I^+$ to $I^-$ and reversely) was found within a relative uncertainty of 0.35 part in $10^9$ by varying its value from 12 s to 24 s while keeping the same acquisition time.

To conclude, the total type B relative uncertainty is estimated to be either 0.7 part or 0.5 part in $10^{9}$ depending on whether the ground is connected to position A or B. The type A uncertainty being lower, the combined uncertainty is below one part in $10^{9}$. Further reduction of the measurement uncertainty will come from the improvement of the current divider calibration.

Validation of measurement accuracy
----------------------------------

The new LNE bridge was used to perform accurate universality tests of the QHE[@Lafont2015; @Ribeiro2015]. The agreement of the quantized Hall resistance, $R_\mathrm{H}$, measured in GaAs and graphene devices was demonstrated with a record[@Ribeiro2015] relative uncertainty of $8\times10^{-11}$. This result was obtained by comparing the two measurements of the ratio $100~\Omega/(R_\mathrm{H}/2)$ carried out using a 100 $\Omega$ transfer resistor. This performance therefore emphasizes the low-noise level and the reproducibility of the measurement bridge, rather than its accuracy. Besides, the capability of the resistance bridge to perform measurements at low frequency (2 Hz) allowed the determination of the temperature evolution of the quantized Hall resistance in graphene during dynamic temperature drift[@Lafont2015]. Many elements of the resistance bridge, i.e. the CCC, the current source and the current divider, were also used to build the programmable quantum current generator that allowed a practical realization of the ampere from the elementary charge with a $10^{-8}$ relative uncertainty[@Brun-Picard2016].

Table \[tableau:Comparisons\] reports on the deviations between the measurements of the ratios $100~\Omega/(R_\mathrm{K}/2)$ and $100~\Omega/10~k\Omega$ performed using the new and the old bridges. It shows that there is no significant discrepancy within a combined uncertainty below 1.5 part in $10^{9}$. Let us note that the comparison uncertainty is limited by the larger type A uncertainty of the older bridge. This agreement between the two resistance bridges, which differ not only by their electronics but also by their CCC and standard current divider, make us very confident in our measurements of these resistance ratios. It also consolidate the Type B uncertainty budget described previously in table \[tableau:Uncertainty\].

\[1\][&gt;m[\#1]{}]{}

         **Ratio**          **$100~\Omega/(R_\mathrm{K}/2)$**   **$100~\Omega/10~k\Omega$**
  ------------------------ ----------------------------------- -----------------------------
   **Relative deviation**      $(-1.4\pm1.5)\times10^{-9}$      $(-0.7\pm1.4)\times10^{-9}$

  : Relative deviations with combined standard uncertainties (k=1) between the measurements of the resistance ratio performed by the new and the older bridges.[]{data-label="tableau:Comparisons"}

Conclusion
==========

A new comparison resistance bridge based on a CCC was built at LNE. It is based on low-noise synchronized current sources that are carefully shielded and electrically isolated from ground, a new CCC with a very low-noise level of $\mathrm{80~pA.t/Hz^{1/2}}$ and a very stable standard current divider characterized by a drift of less than 0.5 part in $10^{9}$ over ten years. Stable operation of the resistance bridge, i.e. of the SQUID in external closed feedback mode, was yet demonstrated in the measurements of ratios $100~\Omega/(R_\mathrm{K}/2)$, $\mathrm{100~\Omega/10~k\Omega}$, $\mathrm{10~k\Omega/1~M\Omega}$ and $1~\Omega/100~\Omega$. The ratio $100~\Omega/(R_\mathrm{K}/2)$ ratio can be determined with a relative type A uncertainty below $0.15\times 10^{-9}$ within one hour measurement time. This performance results not only from the lower noise of the bridge and particularly of the new CCC but also from the optimization of the data acquisition thanks to the quadrature current divider which cancels voltage overshoots. Main contributions to the type B uncertainty budget have yet been estimated. They concern the standard current divider calibration, the CCC accuracy, the finite open-loop feedback gain and the electrical leakage current. The total type B uncertainty is estimated to be below 0.7 part in $10^{9}$. A further reduction to about 0.4 part in $10^{9}$ is expected using both a new calibration method of the standard current divider and the cancellation method of the leakage currents (position B). Thus, a combined standard uncertainty of 0.5 part in $10^{9}$ is expected at term.

Next development steps will consist in characterizations of the measurement of the $1~\Omega/100~\Omega$ and $\mathrm{10~k\Omega/1~M\Omega}$ resistance ratios. Preliminary experiments have yet demonstrated that the measurement of the $1~\Omega/100~\Omega$ resistance ratio with the current dividers inserted in the primary circuit works. Measurement of the $\mathrm{10~k\Omega/1~M\Omega}$ resistance ratio will require a more adapted null detector than the EMN 11 or EMN 31. The bias current noise of these devices is indeed too large. A battery-powered higher-impedance amplifier is therefore under development. Moreover, an efficient rejection of leakage currents is essential. We therefore plan to test not only the connection of the ground in position B but also the implementation of a virtual ground, as described in [@Delahaye1991].

Pictures of resistance bridge components
========================================

![Pictures (front a) and b), top c)) of the primary and secondary current sources, each one being placed in an independent box.[]{data-label="FigAppendix-Sources"}](FigAppendix-Sources.pdf){width="3.5in"}

![Pictures of the standard current divider (top) and of the quadrature current divider (bottom), each one being placed in an independent box.[]{data-label="FigAppendix-SCD"}](FigAppendix-SCD-reduc.pdf){width="3.5in"}

Acknowledgment {#acknowledgment .unnumbered}
==============

The authors would like to thank Guillaume Spengler and Laetitia Soukiassian for their work during the first stage of the development of the resistance measurement bridge that started ten years ago as well as François Piquemal for useful comments on the manuscript. The authors would like also to thank Carlos Sanchez from NRC for useful discussions about the resistance bridge operation with a ground connected to the secondary winding.

[10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{}

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[Wilfrid Poirier]{} is a graduate of the Ecole Supérieure de Physique et de Chimie Industrielles de Paris (ESPCI). He then completed his thesis on quantum electronic transport at the CEA-SPEC and received his doctorate in solid state physics in 1997. He joined LCIE in 1998 and then LNE in 2001 as head of studies on quantum resistance standards. Since then, he has devoted his research to quantum electrical metrology. These included the development of a graphene quantum resistance standard, of GaAs-based quantum Hall arrays and the realization of universality tests of the quantum Hall effect. He has also been involved in the development of precision quantum instrumentation based on SQUID technology. More recently, he has proposed and developed a quantum current generator to achieve the new definition of ampere. He obtained his Habilitation to Direct Research from the University of Paris-SUD in 2017.

[Dominique Leprat]{} worked in LNE from 2001 to 2017 as a technician in the electrical metrology department. He was first involved in the activity of traceability of alternating voltage based on thermal transfer. In 2007, he joined the quantum Hall effect team to develop metrology instrumentation.

[Félicien Schopfer]{} obtained an engineer’s degree from École Nationale Supérieure de Physique - Grenoble INP in 2001, a master’s degree in condensed matter physics from the University of Grenoble the same year. He received a PhD in Physics from the University of Grenoble in 2005, for quantum electronic transport experiments in nanostructures carried out at CNRS. He was appointed at Laboratoire national de métrologie et d’essais – LNE in 2005 to advance research in quantum electrical metrology. His research has mainly focused on the quantum Hall effect (QHE) for applications in fundamental metrology. He worked on quantum Hall arrays in GaAs/AlGaAs, co-authored reproducibility and universality tests of the quantum Hall effect with record uncertainties, and is strongly involved in graphene research, notably with important results for the development of the quantum Hall resistance standard operating under relaxed experimental conditions.

[^1]: Authors are with the Department of Fundamental Electrical Metrology, Laboratoire national de métrologie et d’essais, 78197 Trappes, France; e-mail: wilfrid.poirier@lne.fr.
