Initial NIST AC QHR Measurements

We demonstrate that dc quantized Hall resistance (dc QHR) guideline properties and dc and ac QHR values can be measured without changing sample probe lead connections at the QHR device, and report ac QHR values that converge to the dc QHR value when using four-terminal-pair ac QHR measurements. This was accomplished during one cooldown using single-series and quadruple-series connections outside the sample probe. The QHR was measured from 0 Hz to 5500 Hz in 1:1 ratio at 20 µA to ±1 part in 107 uncertainties with a poor-quality QHR device. A good device would allow an order of magnitude smaller uncertainties over this frequency range. We exchanged positions of the QHR device and reference resistor in the bridge and remeasured the resistance ratios to remove dominant ac bridge effects.


QHR Device, Header, and Sample Probe
The only ac QHR device available was a LEP 175 GaAs/AlGaAs heterostructure fabricated by the Laboratories d'Electronic Philips 1 in France that we labeled ac1. K. C. Lee of NIST mounted it on one of our custom-built headers using 100 µm diameter platinum wires to avoid vibrational effects of ac currents in a magnetic field. The header, constructed from gold-patterned 1.6 mm (1/16") thick printed circuit board, had a single ground plane over most of its back surface to minimize wire-to-wire capacitances at the device. (This matters because Eq. (55) of Ref. [16] predicts that the largest frequency-dependent correction term in quadruply-connected ac QHR measurements is -ω 2 C X′X′ C X′X′ R H R H , which involves the squares of: the angular frequency ω; the summed-total C X′X′ of those wire-to-wire capacitances that have the quantum Hall voltage across them; and the quantized Hall resistance R H .) A single ground plane assures symmetry on magnetic field reversal. Tests showed that this grounded back-plane reduced wire-to-wire capacitances, with only slight increases in capacitances-to-shield, whereas a ground plane on the front surface of the header significantly increased capacitances-to-shield for similar reductions in wire-to-wire capacitance.
We predipped the eight semi-rigid coaxial cables several times in liquid helium before sample probe assembly to minimize Teflon insulation flow during cooldown. (B. W. Ricketts of the National Measurement Laboratory in Australia cautioned us that differential compression when cooling the coaxial cables squeezes the Teflon, causing it to flow during the first few cooldowns and thus stress solder joints at the coaxial socket.) Cernox thermometers sensitive to a mK were located above and below the device, which could only be cooled to 1.5 K at maximum pumping rates (rather than the 1.3 K of our dc sample probes) because semi-rigid coaxial cables have more thermal conduction.

Device Properties
We pumped the sample probe extensively to avoid trapped gasses in the coaxial leads, and then cooled the device slowly over two days to eliminate electron-hole pair production in the device (which creates additional electrons in the two-dimensional fluid and a slow downward drift in the magnetic field positions of QHR plateaus as the pairs recombine). The device was then maintained at or below 4.2 K for 3 months. Figure 1 shows a magnetic field sweep for the central quantum Hall voltage probes V H (3,4) and the longitudinal voltage probes V x (2,6) at a I = 20.0 µA dc source-drain current and T = 1.59 K temperature. (The device source, drain, and potential probe contacts S′, D′, 1′ -6′ are identified in Fig. 7. Primes refer to connections at the device contact pads, while unprimed numbers and letters refer to connections outside the sample probe.) The device looks promising in this sweep. However, three-terminal contact resistance measurements at the approximate currents that will pass through each contact in the quadruple-series mode on the i = 2 (12 906.4 Ω) quantum Hall plateau at 8.3 T and 1.58 K tell a different story: S′ = 9 Ω, D′ = 0.04 Ω, 1′ = 115 Ω, 2′ = 238 Ω, 3′ = 1417 Ω, 4′ = 73 Ω, 5′ = 179 Ω, and 6′ = 7644 Ω after measuring the voltages with both current directions and subtracting lead resistances. Furthermore, the 7644 Ω potential contact 6′ exhibited Corbino-like behavior: its contact resistance increased with decreasing current at small currents due to isolated spikes into the two-dimensional electron fluid, rather than a uniform diffusion. Current circulated around the spikes, as well as along the device. That behavior (pointed-out by K. C. Lee of NIST) would remain undetected if its contact resistance had been measured at the 20.0 µA source-drain current, rather than the ≈ 0.01 µA quadruple-series current appropriate 392 Volume 109, Number 4, July-August 2004 Journal of Research of the National Institute of Standards and Technology 1 Certain commercial equipment, instruments, or materials are identified in this paper to foster understanding. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for this purpose. for that probe. It is important to measure contact resistances at the approximate multi-series connection currents, and for each current direction. Resistances of the 1′, 3′, and 5′ potential contacts were also currentdependent, but in a more typical way: their resistances increased with increasing current.
The Corbino-like behavior of potential contact 6′ created several problems: static voltages induced when reversing dc current directions or changing ac bridge cables sometimes required minutes or hours to decay (making measurements time-consuming), and the QHR values R H = V H /I often fluctuated by about 1 part in 10 7 (1 × 10 -7 R H ) within a day or between days. A device with properties this poor would normally be immediately discarded, but no others were available. The following measurements are intended only as indicators of what could be done with a good device.
The device was cooled from 4.2 K to either 1.6 K or 1.5 K in about 1 h each measurement day, but it required another 2 h to stabilize the QHR value. An example is shown in Fig. 2, where T is the temperature of the lower Cernox thermometer, and X M and Y M are the in-phase and 90° out-of-phase ac QHR bridge main signal components (defined in Sec. 6). A 6.5 V change in X M corresponds to a 1 part in 10 6 change in the QHR value at 1592 Hz; so the thermal stabilization problem is significant. We have no explanation for this puzzling and time-consuming feature. (It did not arise in previous temperature-dependence experiments using a dc sample probe, and was not due to thermal oscillations because the Cernox thermometers above and below the device remained synchronized and their temperatures decreased monotonically. We never filled the variable temperature insert with smaller heights of liquid helium before pumping to see if that reduced the stabilization time.)

Reference Resistors
All measurements were made in 1:1 ratio to avoid scaling ambiguities. The main reference standard was a 12 906.4 Ω wire-wound resistor labeled 12.9WW1. It was assembled from Tegam resistor components made of wire wound on mica cards and placed in a shielded container. It was designed for minimum capacitancesto-shield, trimmed to within about 6 parts in 10 6 of the dc QHR value, and maintained in an oil bath at (25.00 ± 0.04)°C. This resistor type is used in the ac part of the NIST calculable capacitor chain from the Farad to the Ohm [19]. They are very stable, except for small linear drift rates. The 12.9WW1 drift rate is not yet determined, but typically this type drifts about 1.5 parts in 10 7 per year.
This reference resistor was compared with the 12 906.4 Ω, i = 2 plateau of QHR device ac1, and also with a NL Engineering (Norman Lloyd) quadrifilar resistor (Gibbings [20] resistor) of 12 906.4 Ω nominal value that we labeled 12.9QF1. Quadrifilar resistor 12.9QF1 is maintained in a self-contained air bath and thermally-lagged with additional insulation. It lacks an independent sensor to monitor temperature stability, and looses control when the room temperature exceeds 24.5°C. Self-heating occurs at currents above 50 µA. Its resistance value differs from the dc QHR value by about 48 parts in 10 6 , and can unpredictably change a few parts in 10 7 over several hours, necessitating computational adjustment to a reference value (which we chose at 1592 Hz).

DC Measurements
It is important to understand the dc properties of the QHR device when making ac measurements; so we begin with those. Contact resistances were discussed in Sec. 3.   voltages of the three Hall probe sets probably arise from thermoelectric effects and are of no consequence since we find later in this section that the three V H values are comparable when including reverse-current measurements.
Large voltage shifts (about 1 part in 10 5 of V H ) appeared in all three Hall probe sets around 9.15 T in Fig. 4. They probably resulted from an hourly reading of the liquid helium level of the variable temperature insert into which the sample probe is placed since a reading occurred at that time. Liquid helium level readings were never made during subsequent measurements after daily cooldown from 4.2 K. Figure 5 shows magnetic field sweeps of the longitudinal voltage probe sets V x (2,4), V x (4,6), and V x (2,6) in the i = 2 plateau region at 1.59 K and 20.0 µA dc. V x minima involving the 7644 Ω contact resistance of probe 6 were much noisier and narrower, and shifted to smaller magnetic fields. The centroid of the V x (2,6) minima is about 8.3 T.
We measured the V x minima to 1 part in 10 9 type A, 1σ uncertainty at 24.0 µA dc, 1.58 K, and 8.3 T with an automated potentiometeric system POTSYS [21]. Longitudinal resistance r x = V x /I was negligible for r x (2,4), but r x (4,6) and r x (2,6) were both 0.4 mΩ. The 7644 Ω probe 6 contact resistance clearly has an effect on r x . (V x measurements would have also been made on the high-voltage side of the device if it had been standards quality.) The three single-series-connected R H = V H /I i = 2 quantized Hall resistances R H (1,2), R H (3,4), and R H (5,6) of QHR device ac1 were compared with the 12 906.4 Ω wire-wound resistor 12.9WW1 at I = 29.1 µA dc, T = 1.58 K, and B = 8.3 T using an automated measurement system that reverses currents, exchanges positions of two digital multimeters, and exchanges the positions of ac1 and 12.9WW1 in the bridge. The type A, 1σ, ac1/12.9WW1 ratio values for the three single-series measurements differed from unity by (-5.97 ± 0.02) ×10 -6 , (-5.97 ± 0.02)×10 -6 , and (-5.91 ± 0.02)×10 -6 . The device therefore seems homogeneous, with the possible exception of R H (5,6) which involves the 7644 Ω contact. We note, however, that none of these three measurement sets were repeated, and we sometimes observed shifts and decays after switching. Also, we later found in the ac measurements that the R H values were stable and reproducible only to within 1 part in 10 7 because of the Corbino effect. Therefore we assign uncertainties of ±0.1×10 -6 to these measurements, and assume homogeneity only within that uncertainty. (DC cryogenic current comparator measurements on a good device would allow several parts in 10 9 uncertainties.) Cage, Jeffery, and Matthews [16] predicted that devices with two sets of external quadruple-series connections allow dc guideline properties and dc and ac QHR values with small and predicable quadratic frequency dependences to be measured with all sample  (2,4), V x (4,6), and V x (2,6) in the i = 2 plateau region at 1.59 K and 20.0 µA dc. probe leads attached at the device. Figure 6 is a magnetic field sweep of the quadruple-series-connected i = 2 plateau V H (Y,Z) at 20.0 µA dc and 1.58 K, where room temperature locations Y and Z are defined in Fig.  7. The plateau appears flat in this 1 part in 10 6 resolution plot. However, the 1 part in 10 8 resolution plot of Fig. 12 that will be shown for ac currents suggests the dc "plateau" is likely an inverted "U", with no flat region. Furthermore, Fig. 12   We see from Eq. (55) of Ref. [16] that V H (Y,Z) is primarily the quantity V H (3,4) -V x (2,6) in homogeneous, quadruple-series-connected devices. That is of no consequence in good devices when cooled to temperatures where V x (2,6) is negligible over the V x minima region. But V x (2,6) is not negligible here because of poor contacts. That may be the source of the possible inverted "U" shape of V H (Y,Z).
The quadruple-series-connected resistance R H (Y,Z) was compared with the 12 906.4 Ω wire-wound reference resistor 12.9WW1 at I = 29.1 µA dc, T = 1.58 K, and B = 8.3 T using the automated double-multimeter measurement system. The ratio differed from unity by (-6.03 ± 0.02) × 10 -6 . That value is slightly smaller than the three single-series ratios because the quadruple-series mode measures R H -[r x (2,4) + r x (4,6)] [16] and the value of r x (4,6) is 3 × 10 -8 R H . The dc measurements would have been repeated at least once, and at different places along the plateau, but programmatic constraints interceded. As mentioned earlier, we found during subsequent ac measurements for this device that the QHR values were stable and reproducible only to within ±1 part in 10 7 because of the Corbino effect. Therefore we assign a dc value of [1-(6.0 ± 0.1) ×10 -6 ] to this 1:1 ratio. No effort was made to see how well the i = 2 dc quadruple-series QHR value of this poorquality device approximates the 12 906.403 5 Ω von Klitzing constant R K since we were only investigating convergence of ac and dc values in this experiment, and would in any case not use ac1 as a standard.

AC Bridge
The NIST multifrequency transformer bridge can measure 1:1, 2:1, and 10:1 ratios. We used interchanged 1:1 ratios here to minimize ambiguities, and the same set of bridge windings in every interchanged measurement. Figure 7 shows a simplified representation of the three-terminal-pair (3TP) mode in 1:1 ratio. ("Terminal-pair" is an accessible coaxial connection (port) consisting of an inner conductor and its shield. "Three" is the number of terminal-pair connections of a 4-port standard that meet the 4TP balance conditions [18].) A 20.0 µA rms drive current, generated by primary voltage source signal P and an auxiliary (drive) transformer, passes through: coaxial cable on the High (H) side of the bridge to drive port Dr(H) at external "star" connector Y; QHR device ac1 (connected in quadrupleseries between external stars Y and Z; coaxial cables between out ports Ot(H) and Ot(L) on the High and Low (L) sides of the bridge; reference resistor 12.9WW1 (shown here as resistor R R , with internal connection points A and B) to drive port Dr(L); and back to the auxiliary transformer. Passive coaxial chokes [22] (current equalizers) Ck1 and Ck2 (with 20 turns wound around a magnetic core) assure nearly equal and opposite currents in the inner and outer conductors of the coaxial drive cables.
The primary voltage signal P is also supplied to the main (potential) transformer. Main balance is achieved by adjusting the six-decade in-phase α dials and 90°o ut-of-phase β dials on the main (potential) transformer until the in-phase and out-of-phase impedance signal components X M and Y M are amplified at A and nulled in the main lock-in detector D. There is negligible current in the inner conductor to the main detector at balance. Coaxial choke Ck3 then assures negligible current in the outer conductor. The inner conductor of "star" connector G is at virtual ground at balance. Systematic errors may arise if the network is overchoked or under-choked. No effort has yet been made to vary the number or location of chokes. We chose the coaxial cable between contact 2′ and star connector Z to be unchoked to avoid over-choking the QHR "standard" itself because that cable is near ground potential and carries the smallest current. The outer shields of all bridge components are grounded via one point at the main (potential) transformer. (The effect of adding extra ground connections will be tested in the future.) This circuit satisfies the 3TP conditions at both the QHR "standard" and the reference resistor: (1) negligible current at the inner and outer conductors of potential ports Pt(H) and Pt(L), and therefore no inphase currents in the potential cables to external connection star Y of the QHR "standard" or to internal con-nection point A of the reference resistor; (2) negligible current and voltage in the inner and outer conductors to the main detector D; (3) the same current to/from the QHR "standard" and the reference resistor at connection star G (where the inner conductor is at virtual ground at balance); and (4) equal and opposite currents in the inner and outer conductors of cables between external connection star Z of the QHR "standard" and internal connection point B of the resistor. (In addition, detection ports Dt(H) and Dt(L) have no current since they are open-circuited. That meets part of the 4TP definition. However, ports Dt(H) and Dt(L) are not at ground potential because of cable loses to star G.) The 3TP circuit of Fig. 7 works perfectly well, but Fig. 8 shows modifications to minimize cabling changes between the 3TP and 4TP measurement modes. Voltage injectors VI(H) and VI(L) were added to the coaxial cables leaving the out ports Ot(H) and Ot(L), and the VI output windings were shorted. This only increased cable impedances from each out port Ot(H) or Ot(L) to star G by 0.5 mΩ, 4.4 pF, and 1 µH. (The purposes of IVDs Ot(H) Balance and Ot(L) Balance and their capped cables will be explained for the 4TP mode. They have no influence on 3TP measurements.) 3TP balances are made in an iterative process using amplifier A, lock-in detector D, and the bridge main adjustment dials α and β to null the X M and Y M impedance signals at star G. A second lock-in detector D moves between defining transformers DF(H) and DF(L) to assure negligible currents in the inner conductors at potential ports Pt(H) and Pt(L) by adjusting IVDs Dr(H) Balance and Dr(L) Balance. This second detector is always replaced with a shorted inner/outer connector at the DF(H) or DF(L) output winding, and is completely removed on final bridge balance.
We initially inserted a "combining network" like that of Fig. 6 in Cutkosky's paper [18]  for an earlier NIST bridge.) However, measurement errors arose that were sometimes as large as 18 parts in 10 6 under "proper" 4TP balance conditions using different internal resistance values of the combining network when measuring 10:1 resistance ratios. (Perhaps this resulted from using out-of-phase β signal injection into a ground potential location of the combining network, rather than into a location on the higher potential side. This will be checked in future. ) We therefore removed the combining network and made all 4TP measurements by "brute force" using five in-phase and five out-of-phase balances, as indicated in Fig. 9. Injectors VI(H) and VI(L) insert voltages into the coaxial cables between out ports Ot(H) and Ot(L) to make up for cable loses such that the in-phase and out-of-phase signals are nulled at detection ports Dt(H) and Dt(L), as well as at star G. This is accomplished with IVD adjustments Ot(H) Balance and Ot(L) Balance to null the in-phase and out-of-phase signals at ports Dt(H) and Dt(L) using amplifiers A and lock-in detectors D. Thus the inner conductors of ports Dt(H), Dt(L), and G are all at virtual ground at balance. Those two IVDs are connected to +1 turn and -1 turn taps on a separate winding of the main (potential) transformer. (It is possible that they slightly load the potential transformer. That will be tested in future by connecting them to existing +1 and -1 or +10 and -10 taps (not shown) of the drive transformer.) Cable lengths were carefully matched to minimize errors. Inner and outer contacts of every British Post Office (BPO) cable connector were individually tested and polished to avoid ratio measurement shifts of order 1 part in 10 7 . (All outer contacts of a recent BPO purchase were too loose, and the inner contacts very tight; so the connections "felt" good, but occasionally caused large shifts.) Each week we twisted every connector several times to maintain polish and avoid shifts of order 1 part in 10 8  It requires several iterations and some experience to simultaneously null all three sets of low voltage inphase and out-of-phase signals because the main, Ot(H), and Ot(L) balance adjustments interact. Although tedious to initially determine at each frequency, the bridge adjustment parameters are reproducible, and should provide a close approximation to ideal 4TP measurements.
The 4TP bridge, at balance, measures the ratio of the QHR "standard" (with its in-phase impedance component defined between external connection stars Y and Z) and the wire-wound resistor (defined between internal connection points A and B). (Figure 4 Figure 10 demonstrates the bridge sensitivity of the balanced and amplified in-phase main signal component X M for the ac1/12.9WW1 ratio with a -1 part in 10 7 change of the dial settings. The bridge has a 5 parts in 10 9 resolution at 1592 Hz and 20.0 µA. This is comparable with any bridge in the NIST calculable capacitor chain from the Farad to the Ohm [19]. Its inphase resolution increases at lower frequencies and decreases at higher frequencies. Figure 11 shows a magnetic field sweep over the quadruple-series-connected i = 2 plateau V H (Y,Z) at 1.51 K compared in 4TP mode with the 12.9 kΩ wirewound resistor 12.9WW1 at 20.0 µA rms and 1592 Hz. The 90° out-of-phase signal component Y M is offset for clarity from the in-phase main lock-in detector signal output X M . X M and Y M sweeps at 700 Hz, 3000 Hz, and 5000 Hz have similar shapes and magnetic field locations. Figure 12    flat region. (As mentioned before in Sec. 5, it appears just as flat as the dc quadruple-series plateau of Fig. 6 when plotted to that resolution.) The "peak" feature near 8.3 T was reproducible, and occurs at the same magnetic flux density as the V x (2,6) minimum. We suspect it is due to the [V H -V x (2,6)] aspect of quadrupleseries measurements, and would not appear on plateaus of good devices when cooled enough that V x (2,6) was negligible. This feature may contribute to the ±1 part in 10 7 fluctuations in the resistance ratio. AC bridge measurements were always made at the B = 8.3 T value used for dc measurements. 3TP and 4TP plots of α vs f in Fig. 13 at 20.0 µA rms, 8.3 T, and 1.51 K are the in-phase six-decade main dial readings of the resistance ratio ac1/12.9WW1 vs frequency when the device and reference resistor are in their "normal" positions, with ac1 on the High side of the bridge and 12.9WW1 on the Low side. The bridge is constructed such that: a perfect unity ratio, with no frequency dependences or bridge corrections, would be α = 555.555, where a change of +1.0 in the α dials reading represents a +1.0 part in 10 6 shift in the inphase ratio signal X M . (We deliberately plot this "raw data" in that format to indicate that no bridge or cable corrections are applied at this stage.) 3TP balances were achieved between 350 Hz and 5500 Hz; 4TP balances between 700 Hz and 5000 Hz. (A 5500 Hz 4TP datum point was not taken when it appeared for a while that the project was going to be terminated.)

AC QHR Measurements
Corbino-like behavior of the 7644 Ω potential contact meant that static voltages induced when changing bridge leads sometimes required minutes, or even hours, to decay. As a result, every measurement has a ±1 part in 10 7 uncertainty; which is frustrating when compared with the 5 parts in 10 9 resolution.
Second-order polynomial fits to the data in Fig. 13 assume a constant term M 0 , a linear frequency term M 1 f, and a quadratic frequency term M 2 f 2 . Coefficients M 0 , M 1 , and M 2 are listed in the figure. Large linear and quadratic frequency dependences occur for both 3TP and 4TP measurements. Figure 14 shows the 3TP and 4TP main dial "raw data" readings α e vs f with the device and reference resistor positions "exchanged" in the 1:1 ratio measurements: 12.9WW1 on the High side of the bridge and ac1 on the Low side. We tried two exchange methods:  The in-phase ac bridge contributions α 0 to the α and α e measurements of Figs. 13 and 14 are obtained by averaging the sum of the "normal" and "exchanged" "raw data" at each frequency to obtain interchanged bridge in-phase components α 0 : (1) Figure 15 plots these α 0 vs f results, which are the same for both 3TP and 4TP measurements. There is a significant offset from the ideal unity value: (555.725 -555.555) × 10 -6 = (+1.7 ± 1.0) × 10 -7 . Note also the large (-5.0 ± 0.1) × 10 -7 /kHz linear frequency dependence. A linear dependence term was unexpected, but definitely exists given the ±1 part in 10 7 data uncertainties. The (-1.207 ± 0.010) × 10 -6 /kHz 2 quadratic dependence is very large.
Interchanged in-phase deviations ∆ from the unity resistance ratio are obtained by averaging the difference between the "normal" and "exchanged" "raw data" for each frequency: (2) as shown in Fig. 16. Note that the 4TP interchanged ac QHR deviation measurements converge within the ±1 part in 10 7 relative standard deviation uncertainty to the dc value of (-6.0 ± 0.1) parts in 10 6 , whereas the interchanged 3TP measurements are in error by (-3.2 ± 0.1) parts in 10 6 on extrapolation to dc when using the "raw data" with no cable corrections. Cable corrections are not necessary in our 4TP bridge of Fig. 9 because voltage injectors VI(H) and VI(L) make up losses in the out cables, and Dt(H) and Dt(L) are both at zero voltage. (However, the ac QHR "standard" is defined between external stars Y and Z; it therefore includes effects due to coaxial cables from the QHR device to those stars, as shown in the equivalent circuit model of Ref. [16].) Differences in cable impedance between star Z to star G and point B to star G contribute to the -3.2 parts in 10 6 error in the 3TP mode from 4TP at dc. Only -2.4 of this -3.2 parts in 10 6 error is due to differences in the out cable resistances. Furthermore, it would be difficult to correct for these differences to high accuracy: a 0.13 mΩ variation of a connector contact resistance in the out cables would change the measured resistance ratio by 1 part in 10 8 . 3TP measurements are not adequate.
Unfortunately the device characteristics prevented measurements to parts in 10 8 . That would have rigorously tested the quality of fits and convergence to the dc value.

AC Quadrifilar Resistor Measurements
Similar 20.0 µA rms ac measurements and 29.1 µA dc measurements comparing the 12 906.4 Ω quadrifilar resistor 12.9QF1 with the wire-wound reference resistor 12.9WW1 had been made nine months earlier. They are shown in Figs. 17-20. The ratio value was remeasured each day at 1592 Hz and adjusted to account for shifts of resistor 12.9QF1. Each datum point has a ±5 parts in 10 8 1σ type A uncertainty. That could be reduced with more frequent measurements of the 12.9QF1 resistance shift, but this experiment was only intended as an initial mapping of bridge parameters.
"Raw data" 4TP interchanged resistance ratio deviation measurements ∆ again converge to the (42.0 ± 0.1) parts in 10 6 dc value in Fig. 20. The "raw data" 3TP convergence error is about 4 parts in 10 7 . That is much smaller than the -3.2 parts in 10 6 error in Fig. 16 for the ac1/12.9WW1 ratio because out cable lengths (points B to star G) between the two resistors are more closely-matched than the star Z to star G length to the ac QHR "standard".

Frequency Dependences
Quadratic frequency dependences will be discussed in a future paper [24], but we see here that the bridge quadratic dependence was at least ten times larger than for any resistance ratio. Large linear frequency dependences appeared not only in measurements involving the ac QHR "standard," but also in the ac bridge, and in comparisons of two different types of resistors. Because of programmatic constraints, and since the device was of such poor quality, we did not pursue sources of the linear frequency dependence of the bridge, the ac1/12.9WW1 resistance ratio, or the 12.9QF1/12.9WW1 ratio. (Such as: moving the unchoked cable from probe 2 to other probe positions; adding chokes to the network; removing chokes; and verifying that the currents are indeed equal and opposite in the inner and outer conductors of every choke to test Schurr and Melcher's observations [25] that linear frequency dependences can be induced by improper choking and imperfect current equalization.)

Conclusions
DC QHR guideline properties and the dc and ac QHR values can be determined during one cooldown using single-series and quadruple-series connections outside the sample probe. The ac QHR values converged to the dc QHR value to within ±1 part in 10 7 with a poor device when using external quadrupleseries connections and four-terminal-pair measurements. Convergence could be tested in the future to an order of magnitude smaller uncertainty with good devices. It was crucial to use 1:1 resistance ratios and make interchanged measurements to remove dominant ac bridge frequency-dependence effects. Fig. 19. AC bridge contributions α 0 to the α and α e measurements, obtained by averaging the sums of "normal" and "exchanged" "raw data" of Figs. 17 and 18. These α 0 vs f results are the same for both 3TP and 4TP modes, and are identical to those of Fig. 15 for interchanged ac1/12.9WW1 measurements. Fig. 20. Interchanged in-phase deviations ∆ from the unity resistance ratio 12.9QF1/12.9WW1, obtained by averaging differences between the "normal" and "exchanged" "raw data" of Figs. 17 and 18. The 4TP interchanged measurements converge within the ±1 part in 10 7 uncertainty to the +42.0 parts in 10 6 dc value.