IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 7, JULY 2011
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RLC Bridge Based on an Automated Synchronous Sampling System Frédéric Overney, Member, IEEE, and Blaise Jeanneret
Abstract—An RLC bridge based on an automated synchronous sampling system has been developed using commercially available high-resolution analog-to-digital and digital-to-analog converters. This bridge allows the comparison of any kind of impedance standards in the four-terminal-pair configuration at frequencies between 50 Hz and 20 kHz within a range from 1 Ω to 100 kΩ. An automatic balance of the bridge is carried out using a downhill simplex algorithm. Consistency checks have been realized by comparing resistance, inductance, and capacitance standards at different frequencies. The consistency of the measured voltage ratio is better than 20 μV/V over the whole frequency range and even smaller than 5 μV/V around 1 kHz. Finally, the results of the calibration of a 10-nF capacitance standard have been compared to those obtained using a commercial high-accuracy capacitance bridge. The difference is smaller than the commercial bridge specifications over the whole frequency range. Index Terms—Bridge circuits, calibration, impedance, measurement techniques, metrology, sampling methods.
I. I NTRODUCTION
I
MPEDANCE calibrations at the highest level of accuracy are usually carried out using coaxial ac bridges [1], [2]: a quadrature bridge [3], [4] to link the resistance to the capacitance, a Maxwell–Wien or a resonance bridge [5], [6] to link the inductance to the capacitance, and finally ratio bridges [7], [8] to scale impedances of the same kind up and down. Such bridges provide the best accuracies: a few 10−6 for the calibration of inductance standards and below a few 10−8 for the calibration of capacitance or resistance standards. However, they are time consuming to set up, tedious to operate, and usually limited to a narrow operating frequency range (usually around 1 kHz). These methods are therefore not adapted for daily and cost-effective calibration services. On the other hand, the commercially available impedance analyzers are automated and cover a broad bandwidth extending from few hertz to tens of megahertz. They can have six or even seven digits of resolution and a short-term stability sometimes better than 10−5 . However, their specified relative uncertainties are rarely better than 0.05%, and therefore, such instruments are not adequate for the calibration of impedance standards which will, in turn, be used to calibrate impedance analyzers.
Manuscript received June 11, 2010; revised November 10, 2010; accepted December 4, 2010. Date of publication March 28, 2011; date of current version June 8, 2011. The Associate Editor coordinating the review process for this paper was Dr. Wan-Seop Kim. The authors are with the Federal Office of Metrology, 3003 Bern-Wabern, Switzerland (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TIM.2010.2100650
To fill up the gap between high-accuracy narrow-bandwidth manual coaxial ac bridges and low-accuracy broad-bandwidth commercial impedance analyzers, many laboratories developed digitally assisted bridges [9], [10]. In some of them [11]–[15], voltage transformers or/and inductive voltage dividers of a classical coaxial ac bridge are replaced by digital sine-wave generators. Such generators have the advantage of generating voltages with an arbitrary phase relation and, therefore, greatly improve the ability of coaxial ac bridges to compare impedances of different kinds. In another class of setups, the impedance ratio does not directly rely on the accurate generation of voltages but is rather dependent on the accurate measurement of voltages, like in the three-voltmeter method [16] or in sampling-based bridges [17], [18]. At the Federal Office of Metrology, the sampling approach was implemented in the development of a Josephson-based sine-wave synthesizer [19], [20] as well as in the realization of the inductance scale from 1 μH to 10 H at frequencies ranging from 50 Hz to 20 kHz [21]. In this paper, we extend the use of this sampling system to the comparison of impedances of any kind (L–R, C–R, L–C, R–R, C–C, and L–L). After a brief description of the setup, the strategy implemented for the automatic balance procedure is detailed, and the effect of the Wagner balance on the measured voltage ratio is discussed. The stability of this Wagner balance significantly contributes to the uncertainty budget of L–R comparisons [21]. In addition, different consistency checks of the whole sampling bridge have been carried out analyzing the measured voltage ratios of different L–R, L–C, and C–R comparisons. Finally, the results of the calibration of a 10-nF capacitor at frequencies ranging from 50 Hz to 20 kHz are presented. II. D ESCRIPTION OF THE S AMPLING B RIDGE AND THE M EASURED S IGNALS Fig. 1 shows the setup of the sampling system in the configuration used for the comparison of two impedance standards Zt and Zb in a four-terminal-pair configuration. The outer conductor of the coaxial cables as well as the current equalizers has been omitted for clarity. The synchronous sampling bridge is built around different commercial boards mounted in a PXI chassis and has been described in detail in [21] and [22]. The only homemade component is the two-channel multiplexer which has the particularity to present a constant load to the network under test. Indeed, the channel which is not in use is
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Fig. 1. Schematic of the sampling-based RLC bridge. The outer conductor of the coaxial cables is omitted for clarity.
connected to the ground through an admittance Yi that has been adjusted to be equal to the input admittance of the digitizer. This precaution is crucial to prevent a current redistribution within the network under test between the two phases of the measurement sequence. The measurement sequence, which has been described in [21], consists in the successive synchronous digitization of the top and bottom voltages VtH and VbH . Therefore, any current redistribution between these two measurements will, indeed, deteriorate the accuracy of the measured voltage ratio VbH /VtH . Simultaneously to the measurement previously described, two supplementary digitizers (not shown in Fig. 1) are monitoring the low voltages VtL and VbL that are close to zero when the bridge is correctly balanced. Fig. 2 shows the different voltages digitized during the measurement sequence. The top of the figure shows the signal sampled by the main digitizer and consists of the voltage VtH during the first half of the sequence and of the voltage VbH during its second half. The second and third digitizers are monitoring VtL and VbL , respectively, during the whole duration of the measurement sequence (middle and bottom parts of Fig. 2). All these voltages are synchronously acquired, and therefore, the amplitude ratio and phase offset of the different signals can be calculated using a discrete Fourier transform (DFT) on different subsets of each measured signal. In our case, different quantities are therefore defined as follows: DF T VbH f jφ (1) A + jB = r · e = DF T VtH f DF T VtL f DW = (2) DF T VtH f DF T VtL f − DF T VbL f DK = (3) DF T VtH f where DF T [V ]f is the DFT coefficient of the fundamental component of a sinusoidal signal V having a frequency f . The
Fig. 2. Different signals acquired during a measurement sequence when the bridge is out of balance (DW = 0 and DK = 0). (On the top) The main digitizer is sampling VtH during the first half of the sequence and VbH during the second half. Two other digitizers are used to monitor (in the middle) VtL and (at the bottom) VbL during the whole measuring sequence. The symbols represent the subsets of each measured signal used to calculate the DFT.
subsets of the measured signals used in the calculation of the DFT are represented by the symbols in Fig. 2. A fourth quantity DS = DW [after] − DW [before] is the variation of the relative Wagner balance measured before and after the switch transition of the multiplexer. This quantity is proportional to the variation of the current, generated by a change of the apparent load of the multiplexer, flowing through the impedances to be compared. Since the same digitizer is used to measure the main voltages VtH and VbH , its gain error has no effect on the measured voltage ratio r. Only its short-term stability and its nonlinearity have to be taken into account. For the other quantities DW , DK , and DS , different digitizers are used. As these quantities are basically zero when the bridge is balanced, the gain error of the different digitizers does not introduce a significant error. III. AUTOMATIC BALANCING P ROCEDURE To get the bridge balanced, two auxiliary adjustments have to be carried out. Referring to Fig. 1, the first balance consists in the adjustment of the phase ϕ and amplitude ratio Abot /Atop of the top and bottom sources to zero the Wagner signal DW . The second adjustment consists in zeroing the Kelvin signal DK by injecting a small voltage VK between the low-current ports of the standards. Once these two balances are simultaneously obtained, VtL and VbL are both zero, and the impedance ratio Zb /Zt is directly given by the measured voltage ratio r · ejφ (or A + jB) Zb VH DF T [VbH ]f = − bH ≈ − = −r · ejφ . Zt Vt DF T [VtH ]f
(4)
In a coaxial ac bridge, the different adjustments required to balance the bridge and to obtain the defining conditions are time-demanding operations. The different balances are not
OVERNEY AND JEANNERET: RLC BRIDGE BASED ON AN AUTOMATED SYNCHRONOUS SAMPLING SYSTEM
necessary independent from each other, and successive cycles have to be repeated until all auxiliary signals are simultaneously zero. Although the automatization of these balances is not straightforward, it has already been investigated and successfully implemented [23], [24]. In our specific case, the balance condition is reached in two steps. First, an initial adjustment of each balance is obtained using a downhill simplex algorithm [25], [26]. Second, the fine adjustment of the Wagner balance is repeated between the different measurements. The downhill simplex algorithm only requires measurements of DW or DK ; no calculation of derivatives is involved. The voltages VtH , VtL , and VbL are simultaneously acquired, and DW and DK are calculated. The amplitude and the phase of the bottom source and the Kelvin source are then adjusted until the balance is obtained. The two balances are quite independent, and the whole procedure is typically carried out in a minute. After this first adjustment, the Kelvin signal is sufficiently small (typically below 1 μV/V) and stable that no further adjustment is required. Due to the short-term instabilities of the top and bottom sources, the Wagner balance has to be regularly adjusted to keep DW as small as possible during the whole series of measurements. One measurement consists of 20 repetitions of the measurement sequence shown in Fig. 2. The mean value of the voltage ratio A + jB is then calculated as well as the mean value of DW , DK , and DS . As each measurement sequence requires typically 24 periods of the measured signal, the time needed for one measurement varies from 3 s for a 100-Hz measurement to less than 100 ms with a 10-kHz signal. The measurement is then repeated about 20 times, and between each measurement, the Wagner balance is adjusted according the value of DW obtained in the previous measurement. The phase and amplitude corrections applied to the bottom source signal are calculated from the real and imaginary parts of DW , as described in [24].
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Fig. 3. (Top) Time variation of the modulus of the relative Wagner balance DW . (Bottom) Time variation of the difference between the modulus r of the measured voltage ratio and its mean rmean . Values measured in the 100 mH to 653 Ω comparison at 1 kHz.
Fig. 4. Correlation between the modulus of the measured voltage ratio r and the relative Wagner balance DW . (Solid symbols) For the data set shown in Fig. 3. (Open circles) For the comparison of 100 mH to 1 kΩ at 1592 Hz. (Open diamonds) For the comparison of 100 mH to 100 nF at 1592 Hz. (Solid lines) Least square fit of |DW | = |s|−1 |r − rmean |. s is defined in the text.
IV. ACCURACY OF THE WAGNER BALANCES In [21], it has been observed that the type-A uncertainty on the measured voltage ratio was limited to about 10 μV/V mainly due to the 1/f noise of the sources. Fig. 3 shows the stability of the measured voltage ratio r and the magnitude of the Wagner balance |DW | for the comparison of a 100-mH inductance to a 645-Ω resistance at a frequency of 1 kHz. The measurement sequence has been repeated every 200 ms over a period of 3 min, and no further adjustment of the Wagner balance was carried out between each sequence. As shown in Fig. 4, these two quantities are, in fact, strongly correlated. When the Wagner balance is not perfectly achieved, it can be shown [21] that the balance equation (4) becomes Zb = −r · ejφ + DW [1 + Zb /Zt + Zb Yi ] . Zt
(5)
Therefore, the error in the measured voltage ratio r · ejφ is proportional to the relative Wagner signal DW with a proportionality factor s = [1 + Zb /Zt + Zb Yi ]. In fact, (5) perfectly explains the “V” shape observed in Fig. 4. For the comparison
of a 100-mH inductor to a 1-kΩ resistor, the modulus of the impedance ratio Zb /Zt is close to 1 at 1592 Hz, and Zb Yi is √ negligible; therefore, the theoretical slope |s|−1 is close to 1/ 2. The least square fit gives |s| = 1.43. For the comparison of a 100-mH inductance to a 100-nF capacitance, the impedance ratio Zb /Zt is close to −1 at 1592 Hz, giving a small proportionality factor s and a large slope |s|−1 . In that case, the least square fit leads to the value |s| = 0.087. Fig. 4 also shows the importance of the repeated adjustment of the Wagner balance between the different measurements (open symbols). In this case, the voltage ratios are symmetrically distributed around the minimum, and the mean value corresponds to the perfect Wagner balance condition DW = 0. The situation is different when the Wagner balance is not adjusted between successive measurements (solid symbols): The mean value differs from the value corresponding the perfect Wagner balance, and a significant error is made in the impedance ratio determined from (4).
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TABLE I LIST OF THE IMPEDANCE STANDARDS USED FOR THE CONSISTENCY CHECKS CARRIED OUT AT DIFFERENT FREQUENCIES. THE LAST COLUMN INDICATES THE MODULUSOF ΔIN μV/V. THE UNCERTAINTIES ARE THE TYPE-A UNCERTAINTIES ALONE
V. C ONSISTENCY C HECK The ability of the sampling system to compare any kind of impedances gives the possibility to make different consistency checks. For this purpose, three different standards are required: a resistance standard ZR , an inductance standard ZL , and a capacitance standard ZC . The comparison of these three standards two by two gives three voltage ratios (Ai + jBi , i = 1, 2, 3) which can be combined in the following way: Δ=
ZL A1 + jB1 ZL /ZR − = + (A3 + jB3 ). ZC /ZR ZC A2 + jB2
Fig. 5. Distribution, in the complex plan, of Δ measured for the comparisons listed in Table I. The uncertainty bars are the type-A uncertainties alone. The circles have radii of 20 and 5 μV/V, respectively.
(6)
The resulting complex number Δ should ideally be zero. The main advantage of this consistency check is that the reference value of each standard does not need to be known. The standards have only to be stable during the time required for the three comparisons (typically less than 20 min). Ten different consistency checks have been carried out at different frequencies. The frequencies and the nominal value of the different standards are listed in Table I. They have been chosen to give a measured voltage ratio close to one. Fig. 5 shows the distribution of Δ around the origin of the complex plane. The uncertainty bars correspond to the type-A uncertainties alone. Most of the values are within a circle of 20 μV/V in radius, and the consistency checks carried out at 1592 Hz are even within 5 μV/V. This clearly demonstrates the performances of the bridge in the comparison impedance standards covering a wide impedance range over a broad frequency range. As can be seen in the first line of the last column in Table I, the 1 kΩ-1 H-1 μF consistency check carried out at 159 Hz leads to the largest measured |Δ| of about 22 μV/V with a relatively small type-A uncertainty. This particular measurement has been repeated, and the same result (within the type-A uncertainty) has been obtained. In this case, there is therefore a systematic effect which is clearly above the noise level of the measurement and deserves further investigations. VI. A PPLICATION As a final test, a capacitance standard of 10 nF has been calibrated at frequencies from 50 Hz to 20 kHz using both
Fig. 6. Relative deviation of an air-dielectric 10-nF capacitance standard (Tettex, type 3320) from its nominal value as a function of the frequency. (Open symbols) Values obtained using the sampling system presented in this paper. The uncertainty bars correspond to the expanded uncertainty (k = 2). (Solid symbols) Values obtained using a commercial high-accuracy capacitance bridge AH2700A. The uncertainty bars correspond to the specification of the bridge.
the sampling system described in this paper and a highaccuracy commercial capacitance bridge (Andeen-Hagerling AH2700A). Fig. 6 shows the results of the measurements. The uncertainty bars of the AH2700A results correspond to the bridge specifications whereas they correspond to the expanded uncertainty (k = 2) for the sampling system. Since the impedance of the capacitance standard decreases when the frequency increases, different resistors have been used as reference to maintain the voltage ratio r between 0.1 and 3 over the whole frequency range. In the frequency range where the measurements were performed with two different reference resistors, the results overlap within the uncertainties. This clearly indicates that the nonlinearities of the digitizer are not limiting the accuracy of the comparison and do not introduce an error in the measurement when the voltage ratios are other than one.
OVERNEY AND JEANNERET: RLC BRIDGE BASED ON AN AUTOMATED SYNCHRONOUS SAMPLING SYSTEM
Up to a frequency of 3 kHz, the results obtained with the sampling system and the commercial bridge are in excellent agreement (within 6 μV/V). This agreement is particularly meaningful around 1 kHz where the commercial bridge gives the smallest uncertainties. At higher frequencies, the frequency dependence obtained with the two measuring systems differs. However, this difference is not significant because it is still within the uncertainty of the commercial bridge. VII. C ONCLUSION The new synchronous sampling system as described in [21] has been extended to full operation in order to compare fourterminal-pair impedance standards of any kind (resistance, inductance, or capacitance) over a frequency range from 50 Hz to 20 kHz. The system is mostly based on commercial components. The whole system is computer controlled, and the balance of the bridge has been automatized using a downhill simplex algorithm. The analysis of the strong correlation observed between the main balance and the Wagner balance leads to the reduction of the type-A uncertainty below a level of few μV/V. The analysis of the voltage ratios measured in the comparison of the impedance standard of different kinds allows the consistency check of the whole system independently from the absolute value of the standards used. Consistency checks carried out at 159 Hz, 1592 Hz, and 15.92 kHz demonstrate an accuracy of the bridge below 20 μV/V over the whole frequency range and even below 5 μV/V around 1 kHz. Finally, an excellent agreement (within 6 μV/V) has been obtained between the calibrations of a 10-nF capacitance standard carried out with the sampling system and with a commercial high-accuracy capacitance bridge between 50 Hz and 3 kHz. At audio frequencies, such sampling systems are therefore valuable candidates to fill up the gap between high-accuracy narrow-bandwidth manual coaxial ac bridges and low-accuracy broad-bandwidth commercial impedance bridges. Moreover, the performances of commercial analog-to-digital and digitalto-analog converters are continuously improving, and new products will certainly be available in the near future to further increase the frequency range of these sampling-based RLC bridges.
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[5] T. L. Zapf, “Calibration of inductance standard in the Maxwell–Wien bridge circuit,” J. Res. Natl. Bur. Stand., vol.65C, no.3, pp. 183–188, 1961. [6] G. Rayner, B. Hibble, and M. Swan, “On obtaining the Henry from the Farad,” NPL Rep., vol. DES, no. 63, pp. 1–8, 1980. [7] R. Cutkosky, “Techniques for comparing four-terminal-pair admittance standards,” J. Res. Natl. Bur. Stand., vol. 74C, no. 3/4, pp. 63–78, 1970. [8] G. Rayner, “The calibration of inductors at power and audio frequencies,” Proc. Inst. Elect. Eng.—Monographs, vol. 106, no. 9, pp. 38–46, Mar. 1959, 315 M. [9] W. Helbach, P. Marczinowski, and G. Trenkler, “High-precision automatic digital ac bridge,” IEEE Trans. Instrum. Meas., vol. 32, no. 1, pp. 159– 162, Mar. 1983. [10] A. Corney, “Digital generator assisted impedance bridge,” IEEE Trans. Instrum. Meas., vol. 52, no. 2, pp. 388–391, Apr. 2003. [11] H. Bachmair and R. Vollmert, “Comparison of admittances by means of a digital double-sinewave generator,” IEEE Trans. Instrum. Meas., vol. 29, no. 4, pp. 370–372, Dec. 1980. [12] G. Ramm, “Impedance measuring device based on an ac potentiometer,” IEEE Trans. Instrum. Meas., vol. 34, no. 2, pp. 341–344, Jun. 1985. [13] A. Muciek, “Digital impedance bridge based on a two-phase generator,” IEEE Trans. Instrum. Meas., vol. 46, no. 2, pp. 467–470, Apr. 1997. [14] F. Cabiati and G. C. Bosco, “LC comparison system based on a two-phase generator,” IEEE Trans. Electron Devices, vol. 34, no. 2, pp. 344–349, Jun. 1985. [15] L. Callegaro, V. D’Elia, and B. Trinchera, Realization of the Farad from the DC Quantum Hall Effect With Digitally-Assisted Impedance Bridges, 2010. [Online]. Available: http://arxiv.org/abs/1003.1582 [16] L. Callegaro and V. D’Elia, “Automated system for inductance realization traceable to ac resistance with a three-voltmeter method,” IEEE Trans. Instrum. Meas., vol. 50, no. 6, pp. 1630–1633, Dec. 2001. [17] B. Waltrip and N. Oldham, “Digital impedance bridge,” IEEE Trans. Instrum. Meas., vol. 44, no. 2, pp. 436–439, Apr. 1995. [18] G. Ramm and H. Moser, “New multifrequency method for the determination of the dissipation factor of capacitors and of the time constant of resistors,” IEEE Trans. Instrum. Meas., vol. 54, no. 2, pp. 521–524, Apr. 2005. [19] B. Jeanneret, F. Overney, L. Callegaro, A. Mortara, and A. Rüfenacht, “Josephson-voltage-standard-locked sine wave synthesizer: Margin evaluation and stability,” IEEE Trans. Instrum. Meas., vol. 58, no. 4, pp. 791– 796, Apr. 2009. [20] A. Rüfenacht, F. Overney, A. Mortara, and B. Jeanneret, “Thermal transfer standard validation of the Josephson-voltage-standard-locked sine wave synthesizer,” IEEE Trans. Instrum. Meas., 2011, accepted for publication. [21] F. Overney and B. Jeanneret, “Realization of an inductance scale traceable to the quantum Hall effect using an automated synchronous sampling system,” Metrologia, vol. 47, no. 6, pp. 690–698, Dec. 2010. [22] F. Overney and B. Jeanneret, “Sampling based RLC bridge,” in Proc. CPEM Dig., 2010, pp. 400–401. [23] L. Callegaro, “On strategies for automatic bridge balancing,” IEEE Trans. Instrum. Meas., vol. 54, no. 2, pp. 529–532, Apr. 2005. [24] S. Giblin and J. Williams, “Automation of a coaxial bridge for calibration of ac resistors,” IEEE Trans. Instrum. Meas., vol. 56, no. 2, pp. 373–377, Apr. 2007. [25] C.U. Press, Ed., Numerical Recipes: The Art of Scientific Computing, 3rd ed. Cambridge, U.K.: Cambridge Univ. Press, 2007. [26] L. Kneip, Labview Implementation of the Downhill Simplex Algorithm. [Online]. Available: http://www.laurentkneip.de/DS.html
ACKNOWLEDGMENT The authors would like to thank H. Bärtschi for his technical skills and B. Jeckelmann for his continuous support. R EFERENCES [1] B. Hague and T. Foord, Alternating Current Bridge Methods, 6th ed. London, U.K.: Pitman, 1971. [2] B. Kibble and G. Rayner, Coaxial ac bridges. Bristol, U.K.: Adam Hilger, 1984. [3] F. Delahaye and R. Goebel, “Evaluation of the frequency dependence of the resistance and capacitance standards in the BIPM quadrature bridge,” IEEE Trans. Instrum. Meas., vol. 54, no. 2, pp. 533–537, Apr. 2005. [4] Y. Nakamura, A. Fukushima, Y. Sakamoto, T. Endo, and G. Small, “A multifrequency quadrature bridge for realisation of the capacitance standard at ETL,” IEEE Trans. Instrum. Meas., vol. 48, no. 2, pp. 351–355, Apr. 1999.
Frédéric Overney (M’07) was born in Yverdon, Switzerland, in 1967. He received the E.T.S. degree in microtechnics from the School of Business and Engineering Vaud (HEIG-VD), Switzerland, in 1990 and the M.S. degree in physics from the Swiss Federal Institute of Technology, Lausanne, Switzerland, in 1995. He is currently with the Federal Office of Metrology, Bern-Wabern, Switzerland, where he extended the measurement capabilities in the field of ac/dc transfer up to 1000 V and participated in the development of a new time and frequency transfer method based on geodetic GPS receivers. His current research interests include precision measurements of impedance and the development of measuring systems for electrical quantities.
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Blaise Jeanneret was born in Fleurier, Switzerland, in 1959. He received the Ph.D. degree in experimental condensed matter physics from the University of Neuchâtel, Neuchâtel, Switzerland, in 1989. In 1990 and 1991, he was a Guest Scientist with the National Institute of Standards and Technology, Boulder, CO, where he worked on the so-called hightemperature superconductors. In 1992, he was with the University of Neuchâtel. In 1993, he was with the Institute for Micro- and Optoelectronics, Swiss Federal Institute of Technology, Lausanne, Switzerland, where he was involved in the research on the quantum Hall effect. Since 1996, he has been with the Federal Office of Metrology (METAS), Bern-Wabern, Switzerland, where he was mainly engaged in work on the quantum Hall effect. Since 1997, he has been the Head of the Quantum Metrology Laboratory with METAS. His present wok is focused on electrical quantum metrology.
