A fully digital bridge towards the realization of the farad from the quantum Hall effect

In the International System of Units (SI), the units of electrical impedance ohm, henry and farad are linked by the relations $1\ \Omega = 1\ \textrm{Hs}^{-1} = 1\,\,{\textrm{F}}^{-1}\textrm{s}$. Since the 2018 revision of the SI,

The farad F can be realized [...] by comparing the impedance of a known resistance obtained using the quantum Hall effect and the value of the von Klitzing constant [...], including a quantized Hall resistance itself, to the impedance of an unknown capacitance using, for example, a quadrature bridge; [...] [1, appendix 2]

The quadrature bridge mentioned in the SI brochure is a double product impedance bridge [2, 3], which compares two resistors R₁ and R₂, and two capacitors C₁ and C₂, at a properly chosen fixed angular frequency ω. The standard implementation of the quadrature bridge principle [4–8] results in a very complex electrical network with many electromagnetic components, including several multi-decade inductive voltage dividers. These variable dividers must be skilfully operated in order to achieve both the two main balances and, simultaneously, several auxiliary ones. The latter are in fact required to match the proper impedance definition of each of the four standards being compared. Automating the quadrature bridge, although possible, is cumbersome [9, 10]. Digitally-assisted implementations [11, 12] can reduce the network complexity of the quadrature bridge to some extent, and allow a partial automation. A quadrature bridge calibrates the product $C_1 C_2$; an additional measurement with a ratio bridge is required to determine the individual values of C₁ and C₂.

The quoted SI brochure entails that R₁ and R₂ are either resistance standards having an AC value traceable to a quantized Hall resistance (QHR) standard, or that they are QHR standards themselves, measured in the AC regime (ACQHR standards). In the first case [12–14], special resistors having a calculable frequency dependence [15, 16] are calibrated in DC versus the QHR, and their values are numerically corrected. In the second case [6, 17], the effect of unwanted frequency dependencies caused by parasitic effects in the Hall device must be carefully considered [18, 19]. To date, only one laboratory worldwide operates a permanent quadrature bridge with two quantum Hall effect devices in a single cryostat [17], a massive and complex experiment filling an entire laboratory.

Fully-digital bridges [20–25] can compare a single resistor R and a single capacitor C. The fully-digital approach allows the design and implementation of much simpler bridges suitable to be industrially manufactured. Together with the increasing availability of quantum Hall graphene devices, which allow the realization of QHR standards in simpler experimental environments [26], the fully-digital approach underpins an affordable quantum realization of the unit farad by metrology institutes and calibration centres [27].

In the following, we present an electronic ⁴ fully-digital bridge suitable for the realization of the farad ⁵ . The bridge calibrates a capacitance standard of nominal value 8 nF versus a resistor R of nominal value $R_{\textrm{H}} = R_{\textrm{K}}/2 \approx 12\,906 \Omega$ at the frequency ⁶ f ≈ 1541 Hz.

The bridge network is very simple and is based on a multi-channel polyphase electronic digital waveform generator [33]. The bridge balance is semi-automated, and the duration of an individual measurement is around 15 min. The transfer relative expanded uncertainty is about 3 × 10⁻⁷ in a single measurement (15 min) and down to 2 × 10⁻⁷ in about 5 measurements (75 min).

This work reports the principle of operation of the bridge (section 2), its implementation (section 3), a thorough analysis of the uncertainty sources (section 4), and a validation of the bridge performance, by comparison with the traceability chain of the Italian national standard of capacitance [12] (section 5).

The four-terminal-pair voltage-ratio fully-digital impedance bridge described herewith is based on the principle schematic of figure 1, where the coaxial shield and terminal pairs, and a number of auxiliary components have been omitted for simplicity (the complete schematic is reported in section 3).

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The driving voltages and currents E₁, E₂, E₀, E_L, I₁ and I₂ are synthesized by a polyphase digital sinusoidal waveform generator operating at frequency f and can be individually adjusted in magnitude and phase ⁷ . Each voltage or current corresponds to an output channel of the generator. E₁ and E₂ are the bridge main voltages; E_L, I₁ and I₂ are auxiliary voltage and currents needed to realize the impedance definition; and E₀ is an auxiliary injection voltage which, together with the injection impedance Z₀, allows to fine-tune the bridge balance.

The impedances under comparison are Z₁, with terminals HC1, HP1, LC1 and LP1, and Z₂, with terminals HC2, HP2, LC2 and LP2. In this specific application, Z₁ is the reference impedance that can be either an ACQHR standard with resistance R_H, connected with a multiple-connection scheme (see section 6), or a calibrated standard resistor with nominal resistance R_H, such that $Z_1\approx R_{\textrm{H}}$; Z₂ is a capacitor with capacitance C, such that $Z_2\approx 1/(\mathrm{j} 2\pi f C)$; and f is chosen so that 2π f R_H C ≈ 1 (that is, $Z_1/Z_2\approx \mathrm{j}$).

The bridge is balanced when ${V_{{LP1}}} = {V_{{LP2}}} = 0$ and ${I_{{HP1}}} = {I_{{HP2}}} = 0$, where the voltages V_LP1 and V_LP2 are measured at the impedance terminals LP1 and LP2, and the currents I_HP1 and I_HP2 are measured at the detection terminals DHP1 and DHP2 through the current transformers CT₁ and CT₂. The balance can be checked by cycling the synchronous detector D, referenced at the frequency f, through the detection terminals LP1, LP2, DHP1 and DHP2, and can be attained by adjusting E₂ (or E₁), E₀, E_L, I₁ and I₂ (the details of the balance procedure are discussed in section 3).

When the bridge is balanced, the impedance ratio $Z_1/Z_2$ is directly compared with the voltage ratio $E_1/E_2$, and the following balance equation holds:

$$\begin{equation} W = \frac{Z_1}{Z_2} = -\frac{E_1}{E_2}\left(1+\frac{E_0/Z_0}{E_1/Z_1}\right). \end{equation} \tag{ 1 }$$

Alternatively, by considering the admittances $Y_1 = 1/Z_1$, $Y_2 = 1/Z_2$ and $Y_0 = 1/Z_0$,

$$\begin{equation} W = \frac{Y_2}{Y_1} = -\frac{E_1}{E_2}\left(1+\frac{E_0Y_0}{E_1Y_1}\right). \end{equation} \tag{ 2 }$$

As can be seen from the above equations, by choosing $|Z_0|\gg|Z_1|$, the voltage E₀ can be used to fine-tune the bridge balance around the ratio $-E_1/E_2$. This is instrumental to achieve the highest measurement accuracy, as described below.

The voltage phasors E₁ and E₂ can be computed from the samples used to synthesize the two waveforms. We take the readings $E_1^{\,\textrm{read}}$ and $E_2^{\,\textrm{read}}$ from the fundamental components at frequency f of the Fourier series representing the waveform samples. Due to the generator non-idealities, the actual voltages will differ from the readings and we can write $E_k = [1+g_k(E_k^{\,\textrm{read}})]E_k^{\,\textrm{read}}$, k = 1, 2, with $g_k(E_k^{\,\textrm{read}})$ representing a possibly voltage-dependent complex gain error (for magnitude and phase errors), thus considering a possible generator non-linearity. This is actually the main source of error in fully-digital bridges [23, 34].

The error described above can be perfectly cancelled by performing two measurements: one with the impedances connected as in figure 1 (forward configuration) and one with the two impedances exchanged (reverse configuration), and by imposing that the samples used to generate E₁ and E₂ are exactly the same in the two configurations, at most shifted in time, as shown on the left of figure 1 (how this condition can be imposed is described in section 3). This is feasible because with the conditions introduced at the beginning of this section, the magnitude of the impedance ratio is about 1 in both the forward and reverse configurations, $|Z_2/Z_1|\approx |Z_1/Z_2|\approx 1$, and its phase changes by about 180° because $\arg (Z_2/Z_1) = -\arg (Z_1/Z_2)\approx 90^\circ$. Therefore, to balance the reverse configuration, we just need to change the sign of all the samples of either E₁ or E₂ (equivalent to a 180° shift), and re-adjust the injection voltage E₀.

For the forward (F subscript) and reverse (R subscript) configurations, the balance equations are

$$\begin{equation} W = W_{\textrm{F}} = -\frac{E_{1{\textrm{F}}}}{E_{2{\textrm{F}}}}\left(1+\frac{E_{0{\textrm{F}}}Y_0}{E_{1\textrm{F}}Y_1}\right), \end{equation} \tag{ 3 }$$

$$\begin{equation} W = W_{\textrm{R}} = -\frac{E_{2{\textrm{R}}}}{E_{1{\textrm{R}}}}\left(1+\frac{E_{0{\textrm{R}}}Y_0}{E_{2\textrm{R}}Y_1}\right), \end{equation} \tag{ 4 }$$

from which

$$\begin{equation} W = \sqrt{W_{\textrm{F}}W_{\textrm{R}}}, \end{equation} \tag{ 5 }$$

where the complex square root in (5) should be determined with a positive imaginary part because $W\approx\mathrm{j}$.

According to the foregoing argument, $E_{1{\textrm{R}}}/E_{2{\textrm{R}}} = -E_{1{\textrm{F}}}/E_{2{\textrm{F}}}$, exactly, by construction of the waveform samples, such that, from (3)–(5),

$$\begin{equation} W = \mathrm{j}\sqrt{\left(1+\frac{E_{0{\textrm{F}}}Y_0}{E_{1{\textrm{F}}}Y_1}\right)\left(1+\frac{E_{0\textrm{R}}Y_0}{E_{2{\textrm{R}}}Y_1}\right)}, \end{equation} \tag{ 6 }$$

thus cancelling any major error due to $g_k(E_k^{\,\textrm{read}})$ (minor errors are discussed in section 4).

From W and the reference impedance Z₁, the capacitance C can be finally determined as

$$\begin{equation} C = \frac{1}{2\pi f}\mathrm{\textrm{Im}}\frac{W}{Z_1} = \frac{1}{2\pi f}\mathrm{\textrm{Im}}(W Y_1), \end{equation} \tag{ 7 }$$

where $\mathrm{\textrm{Im}}$ denotes the imaginary part of its argument.

Figures 2 and 3 show the coaxial schematic and a photograph of the bridge.

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This section describes only the implementation of the bridge network; the impedances Z₁ and Z₂ employed in the validation are described in section 5.

At the core of the bridge there is the adjustable polyphase digital sinusoidal waveform generator described in [35] ⁸ . This digital signal source is based on 18 bit digital-to-analogue converters with adjustable range (1 V, 2.5 V, 5 V and 10 V) and isolated precision filter/buffer output stages [37]. The relative amplitude and phase stability of the source are about 10⁻⁷h⁻¹. A National Instruments NI-DAQ 6541 board generates the digital codes of the waveform samples, which are then transmitted to the source. The waveform is composed of a fixed integer number of samples per period ⁹ and the frequency is adjusted through the sample rate and the number of samples. This yields the possibility to control the output samples and obtain a simple and computable output spectrum. A channel reading is then obtained from the latter. The clock of the digital source, which defines the sample rate, is locked to a 10 MHz signal from the Italian national time scale.

The digital source provides seven independent output channels, necessary for the complete implementation of the bridge. Two channels generate the voltages E₁ and E₂, which provide the voltage ratio reference against which the impedance ratio is compared. The $R_{\textrm{S1}} = R_{\textrm{S2}} = 10\ \Omega$ resistors (Vishay S102 series), in series to E₁ and E₂, isolate the channel outputs from the capacitive load, avoiding possible self-oscillations of the output buffers.

Three channels generate the voltages E₃, E₄ and E_L which drive the auxiliary circuits realizing the four terminal-pair impedance definition. The currents I₁ and I₂ are generated by E₃ and E₄ through the $R_{\textrm{S3}} = R_{\textrm{S4}} = 100\,\,\Omega$ resistors (Vishay S102 series). Two dummy loads C_L ≈ C and ${R_{L}} \approx {R_{H}}$ are added at the HC ports of the impedances to symmetrize the source loading in both forward and reverse configurations.

One channel generates the voltage E_L through a 200 : 1 feedthrough injection voltage transformer. One channel generates the voltage E₀ which provides the auxiliary injection through the C₀ = 1 pF capacitor, which corresponds to Z₀ in figure 1. One further channel generates the reference signal for the phase-sensitive detector.

The phase-sensitive detector is a Stanford Research SR830 lock-in amplifier and is used to detect the bridge balance on the fundamental component of the signal.

The detector is manually switched across the positions V_DHP1 and V_DHP2, measuring the currents I_HP1 and I_HP2 through the 1:200 feedthrough transformers CT₁ and CT₂, and the positions V_LP1 and V_LP2.

Both the digital source and the detector are controlled by a software purposely coded in the National Instruments LabWindows/CVI environment and based on the balancing algorithm presented in [39]. The bridge balance procedure is thus semi-automated and a single complete measurement (forward/reverse) requires about 15 min. According to what was introduced in section 2, when switching between the forward and reverse configurations, we can change the phase of E₁ by 180° and keep E₂ fixed. The digital source employed in this implementation allows to read the generated samples verifying that they are exactly matched with a 180° phase shift.

Before beginning a series of measurements, it is convenient to preset the main voltages E₁ and E₂ to minimize the auxiliary injection E₀. Starting from the set-up shown in figure 2, the preliminary procedure can be done in the following way:

• P1.  
  Set E₀ to 0 V, and E₁ to the operating value of interest.
• P2.  
  Short circuit both V_LP1 and V_LP2.
• P3.  
  Adjust E₃ to null the detector connected to V_DHP1.
• P4.  
  Adjust E₄ to null the detector connected to V_DHP2.
• P5.  
  Remove the short circuit at V_LP1.
• P6.  
  Adjust E₅ to null the detector connected to V_LP1 (this actually nulls the differential voltage ${V_{{LP1}}} - {V_{{LP2}}}$).
• P7.  
  Remove the remaining short circuit at V_LP2.
• P8.  
  Adjust E₂ to null the detector connected to V_LP1.
• P9.  
  Adjust E₄ to null the detector connected to V_DHP2.
• P10.  
  Repeat steps P8 and P9 until the detector is nulled simultaneously within the chosen thresholds for both detection ports.
• P11.  
  Record the voltage reading $E_{2{\textrm{F}}}^{\,\textrm{read}}$ for the forward configuration.
• P12.  
  Change set-up from the forward configuration to the reverse one by swapping the cables between $E_1\leftrightarrow E_2$ and $E_3\leftrightarrow E_4$, and by shifting the phase of E₁ by 180°.
• P13.  
  Repeat steps P1 through P10 and record the voltage reading $E_{2{\textrm{R}}}^{\,\textrm{read}}$ for the reverse configuration.
• P14.  
  Set E₂ in both configurations to the geometric mean $E_2 = \sqrt{E_{2{\textrm{F}}}^{\,\textrm{read}} E_{2\textrm{R}}^{\,\textrm{read}}}$.

The measurement procedure is then as follows:

• M1.  
  In the forward configuration, execute the steps P2 through P7 of the preliminary procedure and finally adjust E₀ to null the detector connected to V_LP1. Record the readings $E_{0{\textrm{F}}}^{\,\textrm{read}}$, $E_{1{\textrm{F}}}^{\,\textrm{read}}$ and $E_{2{\textrm{F}}}^{\,\textrm{read}}$.
• M2.  
  In the reverse configuration, execute the steps P2 through P7 of the preliminary procedure and finally adjust E₀ to null the detector connected to V_LP1. Record the readings $E_{0{\textrm{R}}}^{\,\textrm{read}}$, $E_{1{\textrm{R}}}^{\,\textrm{read}}$ and $E_{2{\textrm{R}}}^{\,\textrm{read}}$.
• M3.  
  Calculate W from (19), section 4.5.
• M4.  
  The capacitance C is determined, from W and the impedance Z₁, as in (7).

We note the following: i) even though the bridge balance procedure can be completely automated with an external switch, we preferred the semi-automated approach to avoid the risk of worsening the crosstalk between the source channels (see section 4.2); and ii) the steps P5 and P6 of the preliminary procedure can be substituted with a measurement with the detector set in the differential input configuration.

This section presents an evaluation of the uncertainty for the ratio $W = Z_1/Z_2$ as defined by (5). The uncertainty sources are first described separately and then the relevant ones are combined in a measurement model which is used in reporting the uncertainty budget (for a review on the error sources in digital impedance bridges see also [34]).

In the following, according to the notation introduced in section 2, the subscripts 'F' and 'R' designate quantity values obtained in the forward and reverse configurations, respectively; the superscript 'read' designates quantity values read off the generator settings, as described in section 1. From (3)–(5) we also define

$$\begin{equation} W_{\textrm{{F}}}^{\,\textrm{read}} = -\frac{E_{1{\textrm{F}}}^{\,\textrm{read}}}{E_{2{\textrm{F}}}^{\,\textrm{read}}}\left(1+\frac{E_{0\textrm{F}}^{\,\textrm{read}}Y_0}{E_{1{\textrm{F}}}^{\,\textrm{read}}Y_1}\right), \end{equation} \tag{ 8 }$$

$$\begin{equation} W_{\textrm{R}}^{\,\textrm{read}} = -\frac{E_{2{\textrm{R}}}^{\,\textrm{read}}}{E_{1{\textrm{R}}}^{\,\textrm{read}}}\left(1+\frac{E_{0\textrm{R}}^{\,\textrm{read}}Y_0}{E_{2{\textrm{R}}}^{\,\textrm{read}}Y_1}\right) \end{equation} \tag{ 9 }$$

and

$$\begin{equation} W^{\,\textrm{read}} = \sqrt{W_{\textrm{F}}^{\,\textrm{read}}W_{\textrm{R}}^{\,\textrm{read}}}. \end{equation} \tag{ 10 }$$

4.1. Type A uncertainty

4.2. Crosstalk

Crosstalk is the phenomenon by which the voltage at one channel of the polyphase generator is coupled to that of another channel. Considering just the channels that may induce significant variations when switching between the forward and reverse configurations, and that $E_3\approx E_1$ and $E_4\approx E_2$, we can therefore write

$$\begin{eqnarray} E_1 & = E_1^{\,\textrm{read}} + E_{10} + a_{12}E_2^{\,\textrm{read}} + a_{10}E_0^{\,\textrm{read}}, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} E_2 & = E_2^{\,\textrm{read}} + E_{20} + a_{21}E_1^{\,\textrm{read}} + a_{20}E_0^{\,\textrm{read}}. \end{eqnarray} \tag{ 12 }$$

In the above equations, E₁₀ and E₂₀ represent possible residual voltages at the source outputs, independent of $E_0^{\,\textrm{read}},E_1^{\,\textrm{read}}$ and $E_2^{\,\textrm{read}}$, and which can be due to the crosstalk from $E_{\,\textrm{ref}}$ or a clock feedthrough; and a_ij is the (complex) coupling coefficient from channel j to channel i.

Due to crosstalk, the equality $E_{1{\textrm{R}}}/E_{2{\textrm{R}}} = -E_{1{\textrm{F}}}/E_{2{\textrm{F}}}$, obtained in section 1 from the construction of the samples, no longer holds exactly. As a consequence, an error arises. By combining (11) and (12) with (3)–(5), and simplifying, we obtain at first order in the coupling coefficients

$$\begin{equation} W = W^{\,\textrm{read}}-\Delta W^{\textrm{CT}} \end{equation} \tag{ 13 }$$

with the error ΔW^CT given by

$$\begin{align} \frac{\Delta W^{\textrm{CT}}}{W^{\,\textrm{read}}} & \approx -\frac{1}{2}\bigg[E_{10}\left(\frac{1}{E_{1\textrm{F}}^{\,\textrm{read}}}-\frac{1}{E_{1{\textrm{R}}}^{\,\textrm{read}}}\right) \nonumber \\ & \quad -E_{20}\left(\frac{1}{E_{2{\textrm{F}}}^{\,\textrm{read}}}-\frac{1}{E_{2{\textrm{R}}}^{\,\textrm{read}}}\right) + a_{12}\left(\frac{E_{2{\textrm{F}}}^{\,\textrm{read}}}{E_{1{\textrm{F}}}^{\,\textrm{read}}}-\frac{E_{2{\textrm{R}}}^{\,\textrm{read}}}{E_{1\textrm{R}}^{\,\textrm{read}}}\right) \nonumber \\ & \quad - a_{21}\left(\frac{E_{1{\textrm{F}}}^{\,\textrm{read}}}{E_{2{\textrm{F}}}^{\,\textrm{read}}}-\frac{E_{1{\textrm{R}}}^{\,\textrm{read}}}{E_{2\textrm{R}}^{\,\textrm{read}}}\right) +a_{10}\left(\frac{E_{0{\textrm{F}}}^{\,\textrm{read}}}{E_{1{\textrm{F}}}^{\,\textrm{read}}}-\frac{E_{0\textrm{R}}^{\,\textrm{read}}}{E_{1{\textrm{R}}}^{\,\textrm{read}}}\right) \nonumber \\ & \quad -a_{20}\left(\frac{E_{0{\textrm{F}}}^{\,\textrm{read}}}{E_{2{\textrm{F}}}^{\,\textrm{read}}}-\frac{E_{0{\textrm{R}}}^{\,\textrm{read}}}{E_{2\textrm{R}}^{\,\textrm{read}}}\right)\bigg]. \end{align} \tag{ 14 }$$

The measured average magnitudes of E₁₀ and E₂₀ are of the order 30 nV, with a high phase uncertainty. For a reasonably conservative evaluation of the uncertainty, for E₁₀ and E₂₀, we assumed uncorrelated real and imaginary parts centred around 0 with a standard uncertainty of 30 nV for each part, corresponding to a square coverage region. The measured magnitudes of the coupling coefficients between the different channels of the polyphase generator are less than 2 × 10⁻⁸, with a high phase uncertainty. For these coefficients, we assumed uncorrelated real and imaginary parts centred around 0 with a standard uncertainty of 2 × 10⁻⁸ for each part.

4.3. Injection uncertainty

The injection terms in (3) and (4) depend on the voltage ratios $E_{0{\textrm{F}}}/E_{1\textrm{F}}$, $E_{0{\textrm{R}}}/E_{2{\textrm{R}}}$ and on the admittances Y₀ and Y₁. The voltage ratios can be written as

$$\begin{equation} \frac{E_{0{\textrm{F}}}}{E_{1{\textrm{F}}}} = (1+g_{01})\frac{E_{0{\textrm{F}}}^{\,\textrm{read}}}{E_{1{\textrm{F}}}^{\,\textrm{read}}} \end{equation} \tag{ 15 }$$

and

$$\begin{equation} \frac{E_{0{\textrm{R}}}}{E_{2{\textrm{R}}}} = (1+g_{02})\frac{E_{0{\textrm{R}}}^{\,\textrm{read}}}{E_{2{\textrm{R}}}^{\,\textrm{read}}} \end{equation} \tag{ 16 }$$

where the complex error terms g₀₁ and g₀₂ account for the generator non-linearity as described in section 1.

A characterization of the source yielded for 1 + g₀₁ and 1 + g₀₂ a magnitude uncertainty of about 10⁻⁴ and an angle uncertainty of about 10⁻⁴ rad. The uncertainty contributions of Y₀ and Y₁ are typically negligible.

4.4. Source loading

When the bridge is balanced, the currents crossing CT1 and CT2 are approximately zero, but E₁ and E₂ are still loaded by the stray capacitances C_H1 and C_H2 of the cables connecting the channels to the current transformers.

When reversing the channels at the source ports, as is done in this experiment, this causes an error such that [23, 34]

$$\begin{eqnarray} W&\approx W^{\,\textrm{read}}\left[1-\frac{1}{2}(R_{\textrm{s1}}+R_{\textrm{s2}})(Y_{\textrm{H1}}-Y_{\textrm{H2}})\right] \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} &\approx W^{\,\textrm{read}}\left[1-\mathrm{j}\pi f (R_{\textrm{s1}}+R_{\textrm{s2}})(C_{\textrm{H1}}-C_{\textrm{H2}})\right], \end{eqnarray} \tag{ 18 }$$

where Y_H1 and Y_H2 are the admittances of C_H1 and C_H2, respectively. If the bridge construction is symmetric, the difference $C_{\textrm{H1}}-C_{\textrm{H2}}$ is small. Since $W^{\,\textrm{read}}\approx\mathrm{j}$, the expression in the brackets changes at first order only the real part of $W^{\,\textrm{read}}$ and not its imaginary part. Therefore, from (7), the resulting error on C is negligible at the target uncertainty level of this experiment and will not be considered in the measurement model of the next section.

4.5. Measurement model and uncertainty budget

By combining the above results with (6), we obtain the measurement model

$$\begin{eqnarray} &&W = \mathrm{j}\sqrt{\left[1+(1+g_{01})\frac{E_{0{\textrm{F}}}^{\,\textrm{read}}Y_0}{E_{1\textrm{F}}^{\,\textrm{read}}Y_1}\right]\left[1+(1+g_{02})\frac{E_{0{\textrm{R}}}^{\,\textrm{read}}Y_0}{E_{2{\textrm{R}}}^{\,\textrm{read}}Y_1}\right]} \nonumber \\ &&\qquad\qquad -\Delta W^{\textrm{CT}}. \end{eqnarray} \tag{ 19 }$$

Table 1 reports the uncertainty budget of an example measurement for the quantity $\Delta = \mathrm{\textrm{Im}} W-1$ (because $W\approx\mathrm{j}$). All the uncertainties are standard uncertainties (k = 1). The measurement was performed on 20 February 2020 and it is also discussed in section 5.1. The driving rms voltages $|E_{1,2}|\approx 0.25~\textrm{V}$ are chosen so that the driving current I ≈ 20 µA flowing in the standards is suitable for the QHR. We performed the propagation of uncertainty according to [40] with the help of Metas.UncLib [41].

Table 1. Example uncertainty budget for Δ = ImW − 1. All the uncertainties are standard uncertainties (k = 1).

i	Quantity	Type	$u_i(\Delta)\times 10^{6}$
1	Bridge reading (n = 13)	A	0.022
2	Crosstalk (ΔWCT)	B	0.089
5	Injection	B	0.01
		RSS	0.092

The measurement consisted in n = 13 repetitions of forward and reverse measurements, for an overall measurement time of about 200 min and a type A uncertainty component of 2.2 × 10⁻⁸, in reasonable agreement with the value obtainable from the projection of section 4.1 ($\approx \times 10^{-7}/\sqrt{13}$). The most significant uncertainty component is that associated to ΔW^CT, which was evaluated from (14) and the measurements of the coupling coefficients.

The combined uncertainty is $u_{\textrm{c}}(\Delta) \approx 9.2 \times 10^{-8}$, which is competitive for the primary realization of the unit of capacitance. Since the type A uncertainty is much smaller than the type B, 5 measurements are enough to reach a combined uncertainty of 2 × 10⁻⁷.

The validation of the bridge performance was obtained by comparing the 8 nF capacitor calibration performed with the four-terminal-pair fully-digital bridge with that performed with the traceability chain of the Italian national standard of capacitance [12]. The latter is typically employed to calibrate a 1 nF capacitance standard against a calibrated 12 906 Ω quadrifilar resistance standard, but it is here adapted to calibrate the 8 nF capacitance standard.

Figure 4 shows the measurement chain adopted in the validation process.

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On one side of the chain, two resistance standards R₁ and R₂ of nominal value $R_{1,2} \approx 8 R_\mathrm{H} \approx 103.251\ {\textrm{k}} \Omega$ are calibrated with a 8:1 transformer-ratio resistance bridge against a resistance standard of nominal value $R_{\textrm{H}} = R_{\textrm{K}}/2 \approx 12\,906\ \Omega$. These resistance standards are then employed in a transformer-ratio quadrature bridge to calibrate the product $C_1 C_2$ of two 1 nF capacitance standards. By means of a double calibration of the 8 nF capacitance standard against C₁ and C₂ with a 1:8 transformer-ratio bridge, $C^{\,\textrm{ref}}$ is then obtained from the product $C_1 C_2$.

On the other side, the calibration C^fd of the 8 nF is performed with the four-terminal-pair fully-digital bridge against the same 12 906 Ω resistance standard employed before, as described in section 3.

The two calibrations are performed at the same operating frequency of 1541.4326 978 Hz and current of 20 µA. This frequency is chosen to minimize the injections of both the fully-digital bridge and the quadrature bridge. For the fully-digital bridge, the residual deviation from the optimum condition $W = \mathrm{j}$ is about 3.5 × 10⁻⁵.

During the measurement procedure, the 8 nF capacitance standard is kept in a temperature-controlled chamber (Kambič TK-190 US) with a temperature stability better than 4 mK [42]. Since the temperature coefficient of the capacitance standard is $4\pm 6 \times 10^{-6}\ ^\circ$C⁻¹ [43], the maximum effect due to the temperature dependence is about 4 × 10⁻⁸ and can be considered virtually negligible with respect to the other uncertainty components. However, even though hermetically sealed in dry nitrogen, the 8 nF capacitance standard shows a significant dependence on the atmospheric pressure [44], evaluated in section 5.1.

5.1. Results

The result of the comparison is the relative difference

$$\begin{equation}\delta = \frac{{{C^{{fd}}} - {C^{{ref}}}}}{{{C^{{ref}}}}}.\end{equation} \tag{ 20 }$$

Figure 5 reports the results of the comparisons performed on 17 and 20 February 2020. Each point in the plot represents the result of the comparison of the average of two successive measurements of C^fd with one measurement of $C^{\,{\textrm{ref}}}$.

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For the setup of 17 February, the thresholds were set to 50 nV for the balances at LP1 and LP2 and to 500 nV for the balances at DHP1 and DHP2. For the setup of 20 February, the thresholds were set to 30 nV for the balances at LP1 and LP2 and to 200 nV for the balances at DHP1 and DHP2. The latter set of parameters lead to an improved repeatability and it will be therefore considered an optimal trade-off between repeatability and balancing time.

The dependence on the atmospheric pressure of the 8 nF capacitance standard, introduced in section 5, is checked at regular time intervals with an ultra-precision capacitance bridge (Andeen-Hagerling 2700A). The instability component of the capacitance standard is thus considered for each measurement in the uncertainty of δ, which is calculated by combining the uncertainties of C^fd and $C^{\,\textrm{ref}}$. The expanded uncertainty of $C^{\,\textrm{ref}}$ is $U(C^{\,\textrm{ref}})\approx 2 \times 10^{-7}$, higher than the one stated in [12] to take into account the adaptation of the measuring chain.

In figure 5, the uncertainty bars represent the expanded uncertainties with coverage factor k  =  2.

For the measurements of 20 February, the mean value is δ ≈ 3.0  ×  10⁻⁷ with an expanded uncertainty U(δ)  ≈  2.9 × 10⁻⁷.

We presented the design, the implementation and the uncertainty evaluation of a four-terminal-pair fully-digital impedance bridge, optimized for RC comparisons with 1:1 magnitude ratio. Here the bridge was applied to compare a resistance standard with nominal value of R_H with an 8 nF capacitance standard at 1541 Hz (a 10 nF standard capacitor can be for instance measured at 1233 Hz). We reported the uncertainty budget of an example measurement consisted in 13 repetitions of forward and reverse measurements (overall measurement time of about 200 min). The type A uncertainty component is of 2.2 × 10⁻⁸, while the most significant uncertainty component is that associated to the crosstalk among the channels of the generator, which is about 8.9 × 10⁻⁸. The combined uncertainty is 9.2 × 10⁻⁸, suitable for the primary realization of the unit of capacitance from a QHR standard. It should be noted that, since the type A uncertainty component is much less than the type B component, 5 measurements (overall measurement time of about 75 min) are enough to reach a combined expanded uncertainty of 2 × 10⁻⁷. The bridge was validated against a suitably modified version of the traceability chain of the Italian national standard of capacitance.

This bridge is intended to be a much simpler, easy-to-operate and affordable alternative to traditional bridges, suitable to be industrially manufactured. Together with the increasing availability of graphene quantum Hall resistance standards, this can provide an affordable quantum realization of the unit farad for metrology institutes and calibration centres. The accuracy can be further improved by means of internal shielding and filtering of the generator to reduce the crosstalk among the channels.

To employ this bridge with a QHR standard it is necessary to adopt a multiple-connection scheme [45] and possibly a double-shield technique [46]. The integration of the bridge with a QHR standard will be the subject of a future work.

This project (18SIB07 GIQS) received funding from the European Metrology Programme for Innovation and Research (EMPIR) co-financed by the Participating States and from the European Unions' Horizon 2020 research and innovation programme.