Primary Realization of Inductance and Capacitance Scales With a Fully Digital Bridge

This article describes an automated electronic fully digital bridge for the comparison of four-terminal-pair (4TP) impedance standards in the audio frequency range. The bridge relative accuracy, which is on the order of 10-6, makes it suitable as a reference bridge for the realization of primary scales of inductance and capacitance in metrology institutes and calibration laboratories. The performances of this bridge were validated by comparing the results of the calibrations of inductance and capacitance standards with those obtained from an existing analog reference system based on the three-voltage method. The article also reports the results of this validation.


I. INTRODUCTION
T RACEABLE measurements of electrical impedance require the calibration of the meters employed, which, nowadays, are typically LCR electronic impedance bridges. The calibration is performed with artifact standards of ac resistance, inductance, and capacitance, and these require calibrations traceable to realizations of the impedance units ohm ( ), henry ( H) and farad ( F) within the International System of Units (SI). These units are linked by the SI unit of time, the second (s), by the relations 1 = 1 H s −1 = 1 F −1 s. In the SI, the ohm can be realized from a quantized Hall resistance standard and, therefore, the capacitance and inductance units can be derived from the ohm and the second.
Traditionally, and as proposed in the SI brochure [1, Appendix 2], the farad is realized from the ohm with a quadrature bridge [2] at a suitable capacitance level in the nF range, and then scaled up and down with a transformer ratio bridge [3], [4]. Inductance is then realized from capacitance by either a resonance method [5] (in the mH range, and then scaled with a transformer bridge) or a Maxwell-Wien bridge [6]. These kinds of realizations achieve high accuracies but require complex implementations and considerable efforts from highly skilled operators. Manuscript  Digital impedance bridges, which are based on mixed-signal electronics, have evolved from their original concept [7], [8], [9], [10], [11] to recent implementations [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] providing accuracy and flexibility suitable for the direct realization of inductance and capacitance scales. In particular, electronic fully digital impedance bridges [15], [16], [17], [21], [22], [23], [25], [26] do not include calibrated ratio transformers in their implementation but rely on the accuracy of the electronic waveform synthesizers employed. These can be more easily manufactured and operated, thus allowing the developing metrology institutes and calibration centers to realize impedance scales. At the Istituto Nazionale di Ricerca Metrologica (INRIM), the first step toward the use of electronic bridges in the realization of impedance scales was achieved in the 1980s [27], and a second one was achieved about 20 years ago, with an implementation of the threevoltage method, [28], [29], [30] and Section III. This method relies on a high-accuracy ac voltmeter and it is assisted by digital electronics, and allowed to achieve an uncertainty of parts in 10 5 or less. However, the operating ranges for fre- quency and impedance magnitude of the three-voltage method are limited. Moreover, the implementation is expensive with a suboptimal level of automation. More recently, INRIM was involved in the development of fully digital bridges [16], [23], [25], [26] in the framework of a number of European joint research projects [31], [32], [33], [34], [35], [36]. In particular, the project VersICaL 1 had the specific objective of developing a new fully digital bridge to be used as a primary reference bridge for the realization of impedance units and scales in national metrology institutes and calibration centers, and for the training of electrical metrologists [37].
This article discusses this new fully digital bridge and its application to the realization of inductance and capacitance scales from the ac resistance scale in a metrology laboratory. This bridge allows the comparison of four-terminal-pair (4TP) impedances in extended magnitude and frequency ranges, with a high level of automation and usability, and is suitable for an industrial uptake toward adoption in calibration centers.
This article reports a comparison between the calibrations performed with the digital bridge and the three-voltage method. This comparison constitutes a validation of the digital bridge and the basis of a future overhaul and extension of the Italian national scales of inductance and capacitance.

A. Working Principle
A simplified, noncoaxial schematic of the fully digital bridge is shown in Fig. 1. The 4TP [38] impedance standards being compared are Z 1 , which we assume to be the reference standard, connected to the bridge ports LC1, LP1, HC1 and HP1, and Z 2 , which we assume to be the standard under calibration, connected to the bridge ports LC2, LP2, HC2 and HP2. The voltage sources E 1 , E 2 , E 3 , E 4 , and E L correspond to the outputs of a polyphase digital sinewave synthesizer operating at frequency f . The bridge operation is based on the comparison of the impedance ratio Z 1 /Z 2 to the main voltage ratio E 1 /E 2 . The auxiliary voltage sources E 3 , E 4 , and E L realize the 4TP impedance definition: E 3 and E 4 generate, through the resistances R S3 and R S4 , the currents I 1 and I 2 driving Z 1 and Z 2 ; E L compensates the voltage across the series impedance between the low potential ports LP1 and LP2 of Z 1 and Z 2 , respectively.
The bridge is balanced, and the 4TP definition of the impedances fulfilled, when V LP1 = 0, V LP1 − V LP2 = 0 and I 1 = I 2 = 0. The balance is achieved by adjusting the voltages E 2 , E 3 , E 4 , and E L in magnitude and phase. The voltage E 1 is kept constant to fix the measurement current through the reference standard Z 1 .
The balance conditions are checked by detector D, a lock-in amplifier with two inputs A and B, referenced at f . Input A can be connected in turn to the detection ports LP1, DHP1 and DHP2. The ports DHP1 and DHP2 are connected to the transformers CT 1 and CT 2 , which detect the residual currents I 1 and I 2 . The input B is permanently connected to LP2, and it is used to check the condition V LP1 − V LP2 = 0 by operating D in differential mode with input A connected to LP1.
When the bridge is balanced the impedance ratio is given by The bridge reading is given by the settings E read 1 and E read 2 of the voltage phasors E 1 and E 2 , computed from the samples that synthesize the two sinusoidal waveforms. To cancel the effect of the synthesizer's gain error, the reading is obtained from two successive balances: in the forward configuration, the bridge is balanced as in Fig. 1; in the reverse configuration, the bridge is balanced with the channels E 1 and E 2 exchanged at the ports HS1 and HS2. The bridge reading is then computed as [16], [24], [26] are the synthesizer's settings in the forward (F subscript) and reverse (R subscript) configurations. The complex square root in (2) should be determined with a positive real part.

B. Implementation
A coaxial schematic of the fully digital bridge is shown in Fig. 2, and a picture in Fig. 3.
1) Synthesizer: The voltage sources from E 1 to E 5 and E ref represent the output channels of a 7-channel polyphase sinewave synthesizer (DSS2A-INRIM) designed and manufactured by the University of Zielona Góra (UZG) [37], [39]. The synthesizer is composed of a digital waveform instrument (National Instruments PXI-6541), and a digital-to-analog converter (DAC) and analog output unit. Each analog channel includes two 18-bit DACs, for the waveform synthesis and the amplitude scaling, low-pass filters and buffers. The output frequency range of the synthesizer is 20 Hz to 20 kHz. At present, the operating frequency range of the fully digital bridge is narrower to some extent, depending on the target uncertainty. At low frequency, the actual limit is around a few tens of hertz, determined by the reduced efficacy of the current equalizers in ensuring the coaxiality of the bridge network; at high frequency, the limit is determined by the reduced balancing resolution (at 20 kHz, the synthesized waves are composed of just 25 samples).
There are four available voltage ranges, 1, 2.5, and 5 V, with a maximum rms output current of 100 mA.
The voltage E L is obtained from E 5 with an injection transformer VT.
The resistors R S3 and R S4 are of either 10 or 100 , depending on the magnitudes of Z 1 and Z 2 and the operating current. 2 2) Detector: The detector is a Stanford Research SR 830 lock-in amplifier, switched between the single-ended and the differential modes of operation. The reference channel is driven at frequency f by the channel E ref of the synthesizer.
3) Switches: There are two coaxial switches. The output switch is connected to E 1 and E 2 to implement the forward and reverse configurations.

5) Current Equalizers:
The coaxiality of the bridge network is ensured by six coaxial current equalizers [2], represented by solid black marks in Fig. 2.
6) Software: The bridge control software, described in detail in [37], is based on a client-server architecture. The source code of the client application (BClient) is available online [41]. The server controls the instrumentation (synthesizer, detector, and switches), and the client provides the user interface and implements the balancing algorithm [42]. Both applications are developed with the programming environment National Instruments LabWindows/CVI. The client and the server communicate via a network variable server, supervised by the NI Distributed System Manager, allowing the remote control of the bridge over the Internet.
Associated with the BClient application there is a MAT-LAB/Octave script (BClientUncertaintyTool) that determines the unknown impedance and the associated uncertainty from the raw data recorded by the BClient application and the reference standard data. This uncertainty tool is also available online [41]. 7) Balancing and Measurement Procedure: Starting from the forward configuration shown in Fig. 2 (2) and (3), and the unknown capacitance C FD 2 or the unknown inductance L FD 2 (Section V) can be determined from W and the reference impedance Z 1 .

C. Uncertainty
A detailed analysis of the bridge error sources is presented in [26]. These are as follows.
3) Loading. 4) Imperfect low balance. 5) Imperfect high balance. Each of these contributions is dependent on the impedance values, the operating current, the operating frequency, the chosen ranges for the synthesizer's channels, and the balance thresholds. Giving an expression of the uncertainty for the full operating range is impractical because it requires extensive and time-consuming characterization of the synthesizer over several different parameters, and it depends also on the balancing parameters chosen by the operator and their optimization. Partial characterization of the synthesizer for magnitude ratios and frequencies different from those used in this work is reported in [26]. Qualitatively speaking, the main uncertainty component, which is the synthesizer nonlinearity, increases significantly for impedance magnitude ratios greater than about 3:1 or less than 1:3, but is not significantly affected by the operating current because the voltage channels work unloaded. In the comparison of impedances having equal nominal values, the fully digital bridge achieves a relative accuracy of 10 −6 or less.
Indeed, in addition to the above listed bridge-specific uncertainty sources, the uncertainty of the reference standard Z 1 should also be considered.
Since all quantities involved in the measurement model described in [26] are complex, the uncertainty evaluation should be performed according to [43]. The MATLAB/Octave uncertainty tool [41] provides the uncertainty evaluation for the bridge by propagating the distributions of the complex quantities by means of a Monte Carlo method [43], [44]. Table I reports an example uncertainty budget for the measurement of a 10 mH inductance standard against a 100 resistance standard taken as reference. The Monte Carlo evaluation was performed with a sample size of 10 6 . The main sources of uncertainty are the reference standard, and the synthesizer nonlinearity. The synthesizer nonlinearity was estimated by generating reference voltage ratios with an inductive voltage divider. The type A uncertainty was evaluated with six repeated measurements (other longer measurements yielded similar results) and it includes the effect of imperfect low and high balances.
The combined relative uncertainty of the measured inductance L FD 2 is u R (L FD 2 ) = u(L FD 2 )/L FD 2 ≈ 4.8 µH/H.
The principle of the method is described in [28]. As in the fully digital bridge, the impedance standards Z 1 and Z 2 being compared are in series and driven by the same current I . The voltages across Z 1 and Z 2 are V 1 = Z 1 I and V 2 = −Z 2 I , respectively. The average voltage V M = (V 1 +V 2 )/2 of the phasors V 1 and V 2 is generated with a precision inductive voltage divider set at a fixed ratio of 0.5. The voltage magnitudes |V 1 |, |V 2 | and |V M | are measured with a precision rms voltmeter. The relevant phasors form closed triangles whose angles can be computed by trigonometry from their sides, that is, from the phasor magnitudes [2]. The resulting measurement model is nonlinear and the uncertainty expression involved [49].
The INRIM implementation is described in [28], [29], and [30]. The impedances Z 1 and Z 2 are driven by digital waveform generators, and custom electronics including both slave digital generators [50] and analog synchronous filters [51] define the standards as 4TP impedances. A single high-accuracy thermal voltmeter (Fluke 5790A) is switched between the three measurand voltages, such that only its linearity enters in the uncertainty budget [52].
The accuracy of the system was validated by international intercomparisons [29], [30].
A drawback of the three-voltage method is a degradation of its performances in doing measurements at the low voltages needed to measure low impedances. One issue is the voltage noise. In fact, the rms voltmeter is wideband and hence prone to interferences, in particular those at the power line frequency and its harmonics. The performances reported in Section V are achieved in a shielded room. Another issue is the poor linearity of the rms voltmeter in the low voltage ranges. The analog electronics employed for the impedance definition have a limited frequency bandwidth (120 Hz to a few kHz) and manual gain ranges. The implementation relies on expensive commercial instruments (mainly the top-class voltmeter and the inductive voltage divider).

IV. REALIZATION OF INDUCTANCE AND CAPACITANCE SCALES
INRIM is a participant of the International Committee for Weights and Measures (CIPM) mutual recognition arrangement [53] and maintains declarations on calibration and measurement capabilities (CMCs), registered in the key comparison database (KCDB) [54]. The CMCs of interest for this article belong to services 4.3 Inductance: self inductance and 4.2 Capacitance: dielectric capacitors.
The inductance and capacitance scales corresponding to these services are realized with traceability to ac resistance.
The specific artifact standards considered in this work are involved in these scales and are described below. The twoterminal (2T), three-terminal (3T) and two-terminal-pair (2TP) standards are either re-encased as 4TP standards or a 3T-to-4TP adapter is used during the measurements.

A. AC Resistance
The ac resistance scale considered in this work is maintained by Wilkins resistors [55], in particular Tinsley 5685 4T resistance standards with nominal values 10 , 100 , 1 k, and 10 k, encased in shielding boxes provided with  III   COMPARISON OF THE FULLY DIGITAL BRIDGE AND THE THREE-VOLTAGE METHOD IN THE MEASUREMENT OF CAPACITANCE STANDARDS internal adapters from the original four-terminal connections to 4TP British Post Office Multiple Unit Steerable Array (BPO-MUSA) connectors. The dc resistance is periodically calibrated by 1:1 comparisons against the national dc resistance standard [56]. The ac-dc coefficient and the time constant were evaluated during an international intercomparison [57].

B. Inductance
The inductance scale considered in this work is maintained with commercial air-core toroidal self-inductors, with nominal values 1 mH, 10 mH, 100 mH, 1 H, and 10 H, defined as 2T or 3T standards. All standards belong to the General Radio/IET Labs 1482 series. To perform the measurements with the fully digital bridge or the three-voltage setup, the original 2T or 3T definition is converted to the 4TP definition with a special adapter having minimal stray parameters, and which was proven to introduce negligible effect in the course of an international intercomparison [58].

D. Environment
The fully digital bridge, the three-voltage setup and the standards are operated and maintained in a shielded room, thermostated at (23.0 ± 5) • C.

V. COMPARISON
Validation of the fully digital bridge was performed by taking the three-voltage setup as a reference. Both systems were employed to perform calibrations of an inductance or a capacitance standard Z 2 against a resistance standard R 1 taken as reference.
The list of comparisons is shown in Table II for the inductance standards and in Table III for the capacitance standards. The nominal values and frequencies of the comparisons are limited to the nominal values and frequencies covered by the three-voltage method and for which CMCs have been declared in the KCDB.
The first sections of Tables II and III (columns 1-4) report the measurand nominal value (L 2 or C 2 , respectively), the reference impedance R 1 , the measurement frequency f and the rms current I . The resistor nominal value R 1 is chosen, from the available decadal values, to have an impedance magnitude ratio |Z 2 /Z 1 | as close as possible to 1. For each pair of standards, the measurement with the fully digital bridge (L FD 2 or C FD 2 ) and the one with the three-voltage method (L 3 V 2 or C 3 V 2 , taken as reference) are performed in close time succession, to minimize the effect of possible thermal or time drifts. 3 The second sections of the tables (columns 5 and 6) report the 1-σ relative uncertainties of the two measurements, denoted by the subscript R. For the fully digital bridge, u R (L FD 2 ) and u R (C FD 2 ) are evaluated as in Section II-C. For the three-voltage method, u R (L FD 2 ) and u R (C FD 2 ) are evaluated as in [28] and published in the KCDB [54]. These uncertainties include the contribution of the reference impedance Z 1 and thus give the performances of the two systems when employed in regular calibrations.
The third section of the table (columns 6 and 7) reports the result of the comparison δ, that is, the relative difference between the values of L 2 or C 2 measured by the two instruments, and its 2-σ expanded relative uncertainty U R (δ). This uncertainty is evaluated by considering the values of the reference impedance Z 1 as totally correlated between the two measurements. Hence, the corresponding uncertainty u(Z 1 ) is not included in this calculation.
As Tables II and III show the values obtained with the two systems are compatible within the expanded relative uncertainty. The measured series resistances of the inductance standards are also compatible with most measurements but not reported in Table II for brevity. The measured dissipation factors of the capacitance standards were not compared because highly unstable.

VI. CONCLUSION
The results given in Tables II and III mutually validate the performances of the two instruments in their common working magnitude and frequency range, which is limited by the threevoltage method. The fully digital bridge achieves lower uncertainty in most measurements and overcomes the performance limitations of the three-voltage method (Section III).
Further characterizations of the fully digital bridge are ongoing to determine its performances in the magnitude and frequency ranges not covered by the three-voltage method. 3 All measurements were performed in a shielded and thermostated room at (23.0 ± 5) • C. Since air-core inductors have a significant inductance temperature coefficient (30 × 10 −6 /K), the 1 mH and 10 mH standards were further placed in a thermostated chamber with temperature stability of 4 mK. The 100 mH standard has an independent thermostat [60]. Even though small corrections can be performed to the readings on the basis of the equivalent series resistance, as suggested by the manufacturer [61], none were deemed necessary.
The new fully digital bridge is fully automated, can be operated remotely, and was the basis of a training program of European metrologists [37]. A bridge based on this design has been implemented at the National Standards Authority of Ireland (NSAI), the Irish metrology institute. The commercial instruments employed for the bridge implementation (Section II-C) can be replaced with purposely designed electronics on the road toward an industrial product and to extend its use in calibration laboratories. The validation of the bridge reported in this article is the main step toward its formal adoption in the Italian national standard and scales of inductance and capacitance, and the improvement of the Italian calibration and measurement capabilities.
The accuracy of the inductance and capacitance scales realized with the new fully digital bridge is now limited by the accuracy of the available ac resistance scale which is limited by the knowledge of the frequency and phase angle dependence of the artifact resistance standards employed. A set of Gibbings [62] calculable resistors, having values of 100 , 1 k, 10 k, and 12 906.4 is already available. Their frequency dependence can be calculated with uncertainty in the 10 −8 range for the ac-dc deviation and of about 1 ns for the time constant at 1 kHz. They will be characterized and put into service to partially replace the existing resistor set, thus improving the uncertainty of all the three impedance scales of ac resistance, inductance, and capacitance.

ACKNOWLEDGMENT
The authors are indebted to Danilo Serazio, INRIM, who constructed several electromagnetic devices employed in the experiment, and Dario Pilori, together with the INRIM IT help desk, for the contribution to the networking of the fully digital bridge. The work has been realized within the Joint Research Project 17RPT04 VersICaL: A versatile impedance calibration laboratory based on digital impedance bridges.