Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges
ABSTRACT A new traceability chain for the derivation of the farad from dc quantum Hall effect has been implemented at INRIM. Main components of the chain are two new coaxial transformer bridges: a resistance ratio bridge, and a quadrature bridge, both operating at 1541 Hz. The bridges are energized and controlled with a polyphase directdigitalsynthesizer, which permits to achieve both main and auxiliary equilibria in an automated way; the bridges and do not include any variable inductive divider or variable impedance box. The relative uncertainty in the realization of the farad, at the level of 1000 pF, is estimated to be 64E9. A first verification of the realization is given by a comparison with the maintained national capacitance standard, where an agreement between measurements within their relative combined uncertainty of 420E9 is obtained. Comment: 15 pages, 11 figures, 3 tables
 [Show abstract] [Hide abstract]
ABSTRACT: We present here the concept of threearm current comparator impedance bridge, which allows comparisons among three unlike impedances. Its purpose is the calibration of impedances having arbitrary phase angles, against calibrated nearlypure impedances. An analysis of the bridge optimal setting and proper operation is presented. To test the concept, a two terminalpair digitallyassisted bridge has been realized; measurements of an aircore inductor and of an RC network versus decade resistance and capacitance standards, at kHz frequency, have been performed. The bridge measurements are compatible with previous knowledge of the standards' values with relative deviations in the 10^5  10^6 range.08/2014;  Measurement 11/2013; 46(9):37013707. · 1.53 Impact Factor

Conference Paper: Digitally assisted coaxial bridge for automatic quantum Hall effect measurements at audio frequencies
[Show abstract] [Hide abstract]
ABSTRACT: This paper describes the principle of a new fully automatic digitally assisted coaxial bridge having a large bandwidth ranging from 100 Hz to 20 kHz. The bridge is characterized by making a 1:1 comparison between calculable ac resistors. The agreement between the calculated and the measured frequency dependence of the resistors is better than 10^7 over its entire bandwidth. Such a bridge is a perfect tool to start investigating the ac transport properties of graphene in the quantum Hall regime.CPEM 2014; 01/2014
Page 1
arXiv:1003.1582v1 [physics.insdet] 8 Mar 2010
Realization of the farad from the dc quantum Hall
effect with digitallyassisted impedance bridges
Luca Callegaro† §, Vincenzo D’Elia†, and Bruno Trinchera†
† Istituto Nazionale di Ricerca Metrologica (INRIM), Str. delle Cacce 91, 10135
Torino, Italy
Abstract.
A new traceability chain for the derivation of the farad from dc quantum
Hall effect has been implemented at INRIM. Main components of the chain are
two new coaxial transformer bridges: a resistance ratio bridge, and a quadrature
bridge, both operating at 1541Hz. The bridges are energized and controlled with
a polyphase directdigitalsynthesizer, which permits to achieve both main and
auxiliary equilibria in an automated way; the bridges and do not include any
variable inductive divider or variable impedance box. The relative uncertainty in
the realization of the farad, at the level of 1000pF, is estimated to be 64 × 10−9.
A first verification of the realization is given by a comparison with the maintained
national capacitance standard, where an agreement between measurements within
their relative combined uncertainty of 420 × 10−9is obtained.
§ Corresponding author (l.callegaro@inrim.it)
Page 2
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges2
1. Introduction
A number of National Metrology Institutes work on measurement setups to trace
the farad to the representation of the ohm given by the dc quantum Hall effect
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] . The traceability chains employed involve a number of
experimental steps, usually requiring not less than three different coaxial ac bridges.
The bridges are complex networks of passive electromagnetic devices [12]: some of
such devices are fixed (transformers and singledecade inductive voltage dividers),
and several are variable (resistance and capacitance boxes, multidecadic inductive
voltage dividers). Bridges are balanced by operating on the variable devices to reach
equilibrium; that is, the detection of zero voltage or current on a number of nodes in
their electrical networks. Variable devices, especially inductive voltage dividers, are
typically manually operated; only a few models have been described [13, 14, 7, 15]
? which permit remote control. Consequently, a large part of existing bridges are
manually operated.
In most cases, the role of variable devices in a bridge is to synthesize signals
(voltages or currents) to be injected in the bridge network to bring a detector position
to zero. The signals are isofrequential with the main bridge supply, and can be adjusted
in their amplitude and phase relationships. The amplitude and phase of one (or more)
of the signals enters the measurement model equation, and must be calibrated (main
balance), but the others (auxiliary balances) don’t need a calibration.
In this view, it is straightforward to consider a substitution in the bridge network
of most, or all, variable passive devices with a corresponding number of active sinewave
generators, locked to the same frequency but adjustable in amplitude and phase
independently of each other. Direct digital synthesis (DDS) of sinewaves [16] is a
wellestablished technique that permit the realization of such generators; hence, in
this sense, we may speak of digitallyassisted impedance bridges when DDS generators
are used. Digitallyassisted bridges impedance have been considered both theoretically
[17, 18] and in a number of implementations [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29];
some commercial impedance meters are also digitallyassisted.
In this paper, we consider for the first time the feasibility of a complete ohm
to farad traceability chain based on digitallyassisted bridges.
resistance ratio bridge and a quadrature bridge, to measure capacitance (at the level
of 1000pF) in terms of dc quantum Hall resistance (at the level of RK/2 ≈12906.4Ω,
where RK is the Von Klitzing constant). The bridges are automated, and a single
measurement can be conducted in minutes. The estimated relative uncertainty of the
capacitance determination related to the traceability chain is 64 × 10−9.
This accuracy claim hasn’t yet been verified with a comparison with other farad
realization. However, by completing the chain with an older manual transformer ratio
bridge [30], we performed a comparison between the new realization and the present
national capacitance standard, maintained as a group value at the level of 10pF with
a relative uncertainty of 400 × 10−9· The measurement results of the comparison are
compatible within the relative compound uncertainty.
We constructed a
? A commercial item is the Tegam mod. PRT73.
Page 3
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges3
2. The traceability chain
The traceability chain which has been set up, including the steps for the comparison
(Sec. 6.5) with the maintained capacitance standard is shown in graphical form in
Fig. 1. Its steps are here summarized and will be described in more detail in the
course of the paper.
• The quantum Hall effect is employed to calibrate a resistor RQhaving the nominal
value RK/2 ≈ 12906.4Ω and a calculable frequency performance;
• RQ is employed in a 8:1 resistance ratio bridge to calibrate two resistance
standards R1,2, with nominal value 4 × RK ≈ 103.251kΩ, at the frequency
f ≈1541.4Hz;
• R1,2are employed in a quadrature bridge to calibrate the product C1×C2of two
capacitance standards C1,2with nominal value of 1000pF;
• in order to perform the comparison with the maintained capacitance standard,
maintained at the level of 10pF, a capacitance ratio bridge is employed to perform
a scaling up to 1000pF and the measurement of C1and C2;
• a small calculated frequency correction is applied to permit the comparison.
3. Impedance standards
The standards employed in the traceability chain are:
RQ a quadrifilar resistance standard, having a nominal value RQ = RK/2.¶
frequency performance and reactive parameter are calculable from geometrical
dimensions [31]: the result of the calculation is shown in Fig. 2. The standard
is thermostated to improve its stability. The original standard has an inductance
of ≈3µH; in order to reduce its phase angle, a small gasdielectric capacitor has
been added in parallel to its current terminals.
R1,2 two resistance standards with nominal value R1,2 = 4 × RK. Presently, two
thin film resistors+encased in a metal shield and defined as two terminalpair
standards are employed. The casings are within a single air bath, having 1mK
temperature stability.Two new standards with independent thermostats are
under construction.
C1,2 two gasdielectric capacitance standards C1,2=1nF are constructed from General
Radio 1404A standards, reencased in a thermostated bath at 23◦C with 1mK
stability and redefined as two terminalpair impedance standards. Ref. [7] gives
a detailed description of the construction and characterization.
Its
4. Digitallyassisted coaxial bridges
The digitallyassisted bridges developed are a 8:1 resistance ratio bridge, and a
quadrature bridge. The coaxial schematics can be seen in Fig.
respectively. The bridges are based on the same design concept and share common
instrumentation: the polyphase generator (Sec. 4.1), the impedance standards (Sec.
3), and the detector. A photo of both bridges is shown in Fig. 5.
3 and Fig.4
¶ NL engineering Type QF, serial 1294.
+Vishay mod.VHA512 bulk metal foil precision resistors, ±0.001% tolerance, 0.6ppm/◦C
temperature coefficient.
Page 4
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges4
acdc?calculation
dc?potentiometer
acdc?correction?to
1541?Hzf
RR
QQ
=R
Q
acdc ac/dc
+ ?
2resistors
103.251?k
1541?Hz
?
?
R
f
1,2
resistance?ratio?bridge
2
1?nF?gasdielectric
1541?Hzf
capacitors
?
C
1,2
frequency?corrections
100?pF?at?1592?Hz
10?pF?at?1592?Hz
two?terminalpair?ac?bridge
two?terminalpair?ac?bridge
Quantum?Hall?effect
in?dc
1
R
resistor
12906.4=/2
dc?calibration
?
?
quadrifilar
R
KQ
dc
quadrature?bridge
2
1?nF?gasdielectric
f1592?Hz
capacitors
?
C1,2
comparison
Figure 1. Graphical representation of the traceability chain for the realization
of the farad unit from the quantum Hall effect.
4.1. Polyphase generator
Both bridges are energized (one at a time) by a polyphase DDS generator; the
schematic diagram is reported in Fig. 6. The core of the generator is a commercial
digitaltoanalogue (DAC) board∗.The board is programmed for a continuous
∗National Instruments mod. NiDaq6733 PCI board, 8 DAC outputs, variable reference input, 16
bit resolution, maximum sampling rate 1MSs−1, voltage span ±10V.
Page 5
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges5
?
???
????????
??
?
?
?
?
?
?
?
??
Figure 2. Frequency performance of the RQresistor.
D
D
Va
Cw
Tw
R1,2
RQ
Vw
Vb
Vc
Tc
Tb
T
R
Figure 3. Simplified diagram of the 8:1 resistance ratio bridge. Black rings along
the mesh are current equalizers.
generation of sinewaves; each wave can be updated without stopping the generation
(large amplitude or phase changes are gradually achieved to avoid steps in the output).
Since the sinewaves are represented by an integer number of samples (presently
628), the output frequency is finely tuned by changing the common DAC update
clock frequency, typically ranging between 950kHz and 1MHz. The clock is given by
a commercial synthesizer♯ connected to the DAC board by an optical fibre link [32]
to minimize highfrequency interferences. The synthesizer is in turn locked to INRIM
10MHz timebase; hence, the frequency uncertainty of the polyphase generator is better
than 1 × 10−10.
Five DAC channels are used. Four are employed on the bridge network, the fifth
gives a reference signal for the lockin amplifier which acts as zero detector. The four
channels enter a purposelybuilt analog electronics which include, for each channel, a
line receiver (which decouple the computer ground from the bridge ground), a 200kHz
♯ Stanford Research System mod. DS345.
Page 6
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges6
CW
Va
Vmag
Tw
Vc
Cc
R1
C1
C2
VQ
Vw
N
T
+E
E
R2
D
A2
D
A1
RW
Figure 4. Simplified diagram of the twoterminalpair quadrature bridge.
Figure 5. A photo of the two bridges. On the left the quadrature bridge; on the
right the resistance ratio bridge. The instruments in the middle are common to
both bridges.
low pass twopole Butterworth filter to reduce the quantization noise, and a buffer
amplifier with automatic control of dc offset [33] to avoid possible magnetizations
of the electromagnetic components. The analog gain of each channel can be finely
trimmed.
4.2. Resistance ratio bridge
A simplified coaxial diagram is shown in Fig. 3. Output Vaof the polyphase generator
energizes the main isolation transformer T, which has two secondary windings: one
supplies the measurement current, the other energizes the magnetizing winding of the
main ratio divider R. RQis defined as a four terminalpair impedance, whereas R1,2
is defined as two terminalpair impedance. Output Vbof the generator, with injection
transformer Tb, adjusts the current in RQ. Output Vc, and injection transformer
Tc having ratio Dc, provide the main balance by adjusting R ratio. Output VW,
with transformer TWand injection capacitance CW, provides Wagner balance. The
Page 7
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges7
Vmag
V V
b
Vc
Vw
Q
,
Vref
PC
DAC?Board
GPIB
IN
A
Va
GPIB
Lock?in?Detector
Primary
timebase
10?MHz
Synthesized
Generator
Optical?link1?MHz
~
D
Gnd
Figure 6. Schematic diagram of the sinewave synthesizer, see Sec. 4.1 for details.
detector D is the input of the lockin amplifier (in floating mode), manually switched
between detection points.
The result of measurement R1,2/RQ≈ 8 can be expressed as
R1,2
RQ
= 8
?
1 +10
8
·
1
Dc
·Vc
Vacos[arg(Vc) − arg(Va)] −81
8ǫph
?
,
(1)
Eq. 1 takes into account also the complex deviation ǫ of the ratio k of R from its
nominal value 1/9, expressed as k = 1/9+ǫph+jǫqd. R is calibrated using a bootstrap
technique [34] validated in an international intercomparison [35].
4.3. Quadrature bridge
The quadrature bridge is shown in Fig. 4. The bridge measure the product C1C2in
terms of R1R2; it is an evolution of a similar bridge presented in [36].
The main ratio transformer T has a magnetizing winding (driven by generator
output Vmag) and a primary winding (driven by output Va). The secondary winding
is a centertapped bifilar winding providing two nominally equal outputs +E and E.
The double equilibrium of the quadrature bridge is obtained by adjustments of
the quadrature voltage (provided by generator output VQ) and of a balancing current
(provided by output Vcand an injection capacitor Cc).
A fixed combining network N decouples the adjustments; detector points are
monitored with lownoise amplifiers A1 and A2 and the lockin amplifier, manually
switched between the two detection points ††.
Output VW, with transformer TW and injection network CWRW, provides
Wagner balance.
The reading of the quadrature bridge can be expressed as (see Ref. [12], ch. 6.2.2)
ω2R1R2C1C2= 1 + δ
(2)
††The notch filter for harmonics rejection, commonly employed in other setups [12, 11], has proven
unnecessary because of the high harmonic rejection (−90dB) of the digital lockin amplifier employed,
Stanford Research Systems model SR830, and of A1and A2. The residual effect has been considered
as an uncertainty contribution, see Sec. 7.
Page 8
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges8
The real part Re[δ] of δ, which links principal values of R1,R2,C1,C2 (the
imaginary part Im[δ] links resistance time constants to capacitance losses) is given
by the expression
Re[δ] =Vc
VQω CcR2sin[arg(Vc) − arg(VQ)]
(3)
where Vc and VQ are the amplitudes of phasors Vcand VQ, and arg(Vc), arg(VQ)
are their phases.
Eq. 3 does not take into account possible asymmetries of transformer T; however,
these are compensated by exchanging the connections of the outputs of T to the bridge
network, and by averaging the two values of Re[δ] obtained with the two equilibria.
4.4. Bridge operation
The bridges are operated in a similar way, with the same control program. The user
interface permits to set the amplitude and relative phase of each generator output;
to achieve equilibrium, an automated procedure [37] is implemented, resulting an
increased speed and ease of operation. Presently the detector input must be manually
switched between different detection points; despite this, equilibrium is reached from
an arbitrary setting in a few minutes; if the if the bridge is already near equilibrium
condition the procedure is faster.
5. Maintained capacitance unit
The Italian capacitance national standard is presently maintained as the group value
of several 10pF quartzdielectric capacitors [30].
periodically monitored, and the group value is updated by drift prediction and by
participating to international comparisons [38]. The scaling from 10pF to 1000pF
is performed with a manual two terminalpair coaxial ratio bridge [30] and a step
up procedure which permits to compensate for possible deviations of the transformer
ratio from its nominal value.
The capacitance differences are
6. Results
6.1. Measurements of RQ
The representation of the ohm at INRIM is given [39] by the dc quantum Hall effect
on the i = 2 step, RK/2 ≈ 12906.4Ω. A dc potentiometer [40] performs calibrations
of resistance standards. A time series of measurements of RQis shown in Fig. 7: a
significant, but predictable, drift of 5nΩΩ−1d−1is estimated.
6.2. Characterization of the polyphase generator
As shown in Sec. 4.2 and 4.3, the reading of each bridge is given by a mathematical
expression whose input quantity is the complex ratio of the nominal settings of two
generators (Vc/Vafor the ratio bridge, Vc/VQfor the quadrature bridge). The tracking
of the different outputs of the generator (under proper loading conditions) has been
adjusted and calibrated; the deviations from nominal values are within a few parts in
104. Since the impedance standards deviate from their nominal values by less than a
Page 9
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges9
??
??
?
?
?
?
??
??
Figure 7. Time drift of the quadrifilar resistor RQ.
few parts in 10−5, and their relative phases have been adjusted, the contribution to
final accuracy of each bridge caused by the generators can be kept near 1 × 10−8.
The stability and noise of the polyphase generator can be inferred from drifts of
detector readings of the bridge after an equilibrium. Fig. 8 show the time evolution
of the detector reading at the combining network of the quadrature bridge, which is
affected by all generator output drifts.
?
?
?
?
?
?
?
?
Figure 8.
quadrature components) at the combining network detection point of the
quadrature bridge, after an equilibrium operation (for t = 0).
Typical time evolution of the detector reading (inphase and
6.3. Ratio bridge measurements
Fig. 9 shows the measurement of R2/RQwith the resistance ratio bridge (the result of
R1/RQ, not shown, is very similar) over a period of more than one year of operation.
Page 10
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges10
The ratio drift is caused by the compound drift of both R2and RQ. The inset of Fig.
9 shows the repeatability of measurements over a few days.
?
?
?
?
??
?
?
?
?
?
??
Figure 9. Results of measurement of R2/RQwith the 8:1 resistance ratio bridge
over a period of 400 days. Data is expressed as relative deviation (in parts per
106) from the nominal ratio [R2/RQ]nominal= 8.
6.4. Quadrature bridge measurements
Fig. 10 shows the measurement of δ (see Eq. 2) with the quadrature bridge over the
same time period of Fig. 9. The drift is the compound drift of the standards R1, R2,
C1, C2. The inset of Fig. 9 shows the repeatability of measurements over a few days.
?
??
?
?
?
?
?
?
?
?
???
Figure 10. Results of measurement of δ ≡ ω2R1R2C1C2−1 with the quadrature
bridge over a period of 400 days. The inset shows the repeatability of the
measurement over a few days.
Page 11
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges11
6.5. Final result, and comparison with the maintained capacitance standard
Fig. 11 shows the geometric mean of C1and C2from the nominal (1000pF) value. The
result is computed from all data previously described. The observed drift is attributed
to coumpond drift of C1and C2.
In the same figure, the result (with uncertainty bars) given by a stepup
measurement from the maintained national capacitance standard is shown; a visual
agreement can be appreciated. This measurement is performed at the frequency of
1592Hz and should be corrected to 1541Hz because of the frequency dependence of
C1and C2. Indirect measurements of such dependence, performed with the socalled
Smatrix method [41] give a correction below 1 × 10−9. An uncertainty contribution
associated with the correction has been nevertheless added to the uncertainty budget
(see Sec. 7).
?
?
?
?
????
??
?
?
?
?
?
Figure 11. Comparison between quadrature bridge and step up procedure.
7. Uncertainty
Tables 1–3 give the uncertainty expression corresponding to the measurements
described in Sec. 6:
• Tab. 1 lists the uncertainty contributions related to the various measurements
and standards employed in the new traceability chain;
• Tab. 2 gives the uncertainty budget for the measurement of the geometric mean
(C1C2)1/2of the 1000pF capacitance standards C1and C2in terms of the INRIM
representation of the ohm given by the quantum Hall effect in dc regime;
• Tab. 3 gives the uncertainty budget for the comparison described in Sec. 6.5 and
shown in graphic form in Fig. 11.
8. Conclusions
The paper described a new traceability chain for the realization of the farad from
the quantum Hall effect, which include two bridges, a resistance ratio bridge and a
Page 12
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges12
Table 1.
uncertainty expression (contributions and rootsumsquare RSS)
Source of uncertainty
1: Calibration of RQ@ 1541Hz
DC calibration of RQ
phase correction of RQ
Frequency dependence
Shortterm stability
RSS
Measurement steps of the ohmfarad traceability chain:relative
Type
ur
Note
nΩ · Ω−1
25
12
3
10
30
nΩ · Ω−1
10
12
10
1
1
10
45
5
50
nF · F−1
20
0
12
6
3
1
1
5
25
A, B
B
B
B
Calibration with dc QHE
1 × 10−5loss angle of 10pF capacitor
10% of calculated frequency deviation
Estimated drift is 5 × 10−9/day
2: Resistance ratio bridge
Noise
Main balance injection
4TP impedance definition
4TP cable corrections
2TP contact resistance repeatability
Residual loading on main IVD
Main IVD ratio
Noncoaxiality
RSS
A
B
B
B
B
B
B
B
Std of the mean of 10 measurements
5 × 10−4of a 25 × 10−6injection
See [12]
BPO repeatability, 100µΩ
Bootstrap calibration [34]
Calibration of R1 and R2
3: Quadrature bridge
Noise
Frequency
Main balance injection
Distortion
Residual offset after inversion
2TP contact resistance repeatability
2TP capacitance repeatability
Noncoaxiality
RSS
A
B
B
B
B
B
B
B
Std of the mean of 10 measurements
Lock to INRIM 10MHz frequency standard
5 × 10−4of a 25 × 10−6injection
Harmonic amplitude and lockin rejection ratio
Average of difference of direct and reverse meas.
BPO repeatability, 100µΩ
Calibration of (C1C2)1/2
Table 2.
standards from dc quantum Hall effect with the new traceability chain:
uncertainty expression.
Source of uncertainty
ur× 10−9
Calibration of RQ @ 1541Hz
25
Resistance ratio bridge
50
Shortterm stability of R1 and R2
10
Quadrature bridge
25
RSS
64
Measurement of the geometric mean (C1C2)1/2of the capacitance
Note
Tab. 1, #1.
Tab. 1, #2.
TC of 2 × 10−6K−1; 5mK std over 1h
Tab. 1, #3.
quadrature bridge, based on a polyphase sinewave generator. The bridges do not
contain variable passive components like multidecadic inductive voltage dividers or
impedance decadic boxes; the equilibrium is obtained by direct digital synthesis of
the necessary signals. In the present implementation the bridge operation is semi
automated and the equilibrium is reached in short time.
The total relative uncertainty of the traceability chain is estimated to be
Page 13
Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges13
Table 3. Comparison between the measurement with the new traceability chain,
and the maintained capacitance national standard.
Source of uncertainty
1: Calibration of RQ@ 1541Hz
10pF maintained group value at 1592Hz
Capacitance bridge, 10pF to 100pF stepup
Capacitance bridge, 100pF to 1000pF stepup
Shortterm stability of 1000pF capacitors
Frequency correction
New traceability chain, (C1C2)1/2
RSS
Type
ur
Note
nF · F−1
400
40
100
4
5
64
419
B
B
B
B
B
B
2 measurements
TC 4 × 10−6K−1; 1mK controller stability
1541Hz to 1592Hz, gasdielectric
Tab. 2
64 × 10−9at the level of 1000pF, therefore adequate for a national metrology institute.
A first verification of the realization accuracy is given by a comparison with the
maintained capacitance national standard, but an international comparison is being
planned in the next months.
Future improvements of the implementation will include the installation of
individually thermostatted resistance standards for R1 and R2, and the complete
automation of the bridges with a remotelycontrolled coaxial switch. Since the digital
assistance of primary impedance bridges has proved as a successful approach, the
realization of a digitallyassisted 10:1 ratio bridge for scaling 1000pF to maintained
10pF standards is under consideration.
Acknowledgments
The authors are indebted with their colleagues F. Francone and D. Serazio for the
physical construction of the electromagnetic devices; and to C. Cassiago for the
calibration of RQ.
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Realization of the farad from the dc quantum Hall effect with digitallyassisted impedance bridges14
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