3. Experimental results

Ux ¼ ð Þþ ax þ Δ j bð Þ <sup>x</sup> þ Δ ; U0 ¼ ð Þþ a0 þ Δ j bð Þ <sup>0</sup> þ Δ ; U0v ¼ ð Þþ a0v þ Δ j bð Þ <sup>x</sup> þ Δ (20)

<sup>p</sup> <sup>δ</sup><sup>n</sup> and <sup>Δ</sup><sup>a</sup> <sup>≈</sup> ffiffiffi

<sup>p</sup> <sup>δ</sup><sup>n</sup> and <sup>Δ</sup><sup>a</sup> <sup>≈</sup> ffiffiffi

where δ<sup>m</sup> and Δ<sup>a</sup> are the multiplicative and additive uncertainties caused by the relative noise

Formulas (21) and (22) show that the additive uncertainty Δ<sup>a</sup> caused by the relative noise (δ<sup>n</sup> ¼ Δ=U0) in both cases is the same. But these formulas also show that due to the variational

Calculation of the uncertainty by the formula (16) has the truncation error δ<sup>t</sup> caused by inequality K » 1 ð Þ δ<sup>t</sup> ¼ Z0Yc=K . This error sharply increases when K on high frequencies is low, so that calibration practically does not work when K ! 1. If amplifier gain K is so low, we cannot consider value Yc/K as the sole source of the uncertainty. As a result, we have to provide two separate variations: multiplicative variation of the gain K and additive variation of the admittance Yc (using variational admittance Yv and switcher Sv). DVV measures sequentially

> Ux=U<sup>0</sup> ¼ Zx=Z0½ � 1 þ YcZ0=ð Þ 1 þ K Ux=U<sup>0</sup> ¼ Zx=Z0½ � 1 þ YcZ0=ð Þ 1 þ α1K Ux=U<sup>0</sup> ¼ Zx=Z0½ � 1 þ ð Þ Yc þ Y<sup>ν</sup> Z0=ð Þ 1 þ K

2

2

<sup>0</sup> after multiplicative variation of the gain K and additive variation of

ð Þ α1K þ 1 ð Þ K þ 1 =Kð Þ 1 � α<sup>1</sup>

ÞA00 � <sup>A</sup><sup>0</sup> , c ¼ ðA<sup>0</sup>

� <sup>1</sup>ÞðA<sup>00</sup>

� <sup>1</sup>Þ, b ¼ ðYvZo <sup>þ</sup> <sup>A</sup><sup>0</sup>

Solution of the Eqs. (24) and substitution of these results in (15) gets the accurate results of

<sup>p</sup> <sup>δ</sup><sup>n</sup> (21)

<sup>p</sup> <sup>δ</sup><sup>n</sup> (22)

(23)

(24)

� 1Þ,

Let us substitute formula (20) in (14) and (19). It gets the following formulas for two cases:

<sup>δ</sup><sup>m</sup> <sup>≈</sup> ffiffiffi 2

δ<sup>m</sup> ≈ ffiffiffi 5

calibration, the multiplicative random uncertainty δ<sup>m</sup> increases 1.6 times.

Without variational calibration:

With variational calibration:

δ<sup>n</sup> of the VV.

38 Metrology

voltages Ux, U<sup>0</sup> and U<sup>0</sup>

here: <sup>a</sup> ¼ ½ð<sup>1</sup> <sup>þ</sup> <sup>α</sup>1Þ � <sup>α</sup>1ðA<sup>0</sup>

<sup>¼</sup> <sup>U</sup><sup>0</sup> <sup>0</sup>=U<sup>0</sup>

0=U0, A<sup>00</sup>

<sup>A</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup><sup>0</sup>

the admittance Yc.

0, U<sup>00</sup>

System of three equations describes these four measurements:

Solution of the system (23) gets following two equations:

<sup>Y</sup>cZ<sup>0</sup> <sup>¼</sup> <sup>A</sup><sup>0</sup>

aK<sup>2</sup> <sup>þ</sup> bK <sup>þ</sup> <sup>c</sup> <sup>¼</sup> <sup>0</sup>

� <sup>1</sup>Þ�ðA<sup>00</sup>

measurement which absolutely does not depend on the values Yc and K.

� 1 � � The earlier described approach was used in digibridge MNS1200. This digibridge was developed for Siberian Institute of Metrology (Novosibirsk), to be used in working inductance standard. Its short specification is as follows.

MNS1200 operates in frequency range of DC to 1 MHz.

Frequency set discreteness 2 <sup>10</sup><sup>5</sup> .

Capacitance range of measurement (F) 10<sup>17</sup>–105 .

Resistance range of measurement (R) 10<sup>6</sup> –1014.

Inductance range of measurement (H) 10<sup>12</sup>–1010.

Dissipation factor tgδ (tgφ) 10<sup>6</sup> –1.0.

Main uncertainty (ppm) 10.

Sensitivity (ppm) 0.5

Inner standard instability (24 hours, ppm) 2.

Weight (kg) 4

MNS1200 appearance is shown in Figure 6.

Instability of the MNS1200 inner standard can achieve 10<sup>4</sup> in a long period of time. To get maximal accuracy, MNS1200 can be calibrated by arbitrary R,L,C outer standard. In this case,

Figure 6. Digibridge MNS1200.

and dissipation factor in a whole range of measurements and reciprocal transfer of any units.

Variational Calibration

41

http://dx.doi.org/10.5772/intechopen.74220

Early autotransformer bridges were described in [24, 25]. These bridges have been widely used up to now [15, 16]. To eliminate the influence of the cable impedance (yoke) on the results of measurement, double autotransformer bridges are used [3, 5]. The wide-range double autotransformer bridge contains two inductive dividers, simultaneously controlled for bridge balance. For accurate measurements, these inductive dividers usually are of a two-stage design at least. Every stage of these inductive dividers [26] consists of a lot of turns and appropriate complicate switchers. They have to have multidigit capacity (up to seven or eight digits). This

• to eliminate the Yoke (Zn) influence on the results of measurement without using the

• to decrease sharply the number of the autotransformer divider decades without loss in the

The simplified measuring circuit of the automatic variational bridge (PICS) [27], which solves

The bridge consists of the supply unit (the generator G connected to the voltage transformer TV), the main autotransformer AT and the variationally balanced 90 phase shifter [28], which is calibrated through calibration circuit CC. The vector voltmeter VV (through the preamplifier PA and switchers S1 and S2) measures the bridge (U1, U2) and the calibration circuit CC (Uc)

Balance and calibration of these bridges are based on the variational method.

Development of the variational bridge has to solve two problems:

3.1.1. Autotransformer bridge: description and analysis

quite complicates the bridge.

double autotransformer bridge;

accuracy of measurement.

these problems, is shown in Figure 8.

Figure 8. Circuit diagram of the autotransformer bridge.

Figure 7. Results of the 24-hour 1 Ohm standard measurements.

uncertainty of measurement depends on short-time stability of inner standards. Results of the 24-hour 1 Ohm standard measurements are shown in Figure 7.

#### 3.1. Application of the VM in transformer bridge

Accurate comparison and unit dissemination of the impedance parameters are provided using many different, very complicated manual bridges with numerous different standards. The main world-renowned laboratories (BIPM, NIST, NML, NPL, PTB, VNIIM, etc.) in developed countries have their own primary standards, based on the calculable capacitor [13, 14] and the appropriate transformer bridges [15, 16], on the quantum hall resistance [17] and the appropriate bridges [18, 19] and very accurate quadrature transformer bridges for comparison of different impedance parameters [20, 21], that have original constructions. All these bridges contain complicated set of devices and have long and intricate handle balancing processes. In addition, these bridges and standards are of different kinds and are located in various laboratories. The process of calibration and traceability is, therefore, complicated and very expensive. Uncertainty of the measurement of these bridges achieves 10<sup>8</sup> –10<sup>9</sup> . It makes them an excellent instrument for fundamental investigations.

For practical needs of the metrologic calibration, it is enough to provide measurements with uncertainty about 10<sup>6</sup> . In this case, the equipment have to be universal, to compare arbitrary standards, to have low cost and weight and to be transportable. The complex of bridges described later satisfies these demands. Complex consists of autotransformer and quadrature bridges. Both of them are based on the variational calibration. Autotransformer bridge provides unit transfers in the whole range of the impedance of the C,L,R standards. Quadrature bridge provides cross transfers of the units. Last bridge is described in [22, 23].

This chapter describes the part of the results of this project, covering the development of the transformer bridge-comparators which transfer units of the resistance, inductance, capacitance and dissipation factor in a whole range of measurements and reciprocal transfer of any units. Balance and calibration of these bridges are based on the variational method.

### 3.1.1. Autotransformer bridge: description and analysis

uncertainty of measurement depends on short-time stability of inner standards. Results of the

Accurate comparison and unit dissemination of the impedance parameters are provided using many different, very complicated manual bridges with numerous different standards. The main world-renowned laboratories (BIPM, NIST, NML, NPL, PTB, VNIIM, etc.) in developed countries have their own primary standards, based on the calculable capacitor [13, 14] and the appropriate transformer bridges [15, 16], on the quantum hall resistance [17] and the appropriate bridges [18, 19] and very accurate quadrature transformer bridges for comparison of different impedance parameters [20, 21], that have original constructions. All these bridges contain complicated set of devices and have long and intricate handle balancing processes. In addition, these bridges and standards are of different kinds and are located in various laboratories. The process of calibration and traceability is, therefore, complicated and very expensive.

For practical needs of the metrologic calibration, it is enough to provide measurements with

standards, to have low cost and weight and to be transportable. The complex of bridges described later satisfies these demands. Complex consists of autotransformer and quadrature bridges. Both of them are based on the variational calibration. Autotransformer bridge provides unit transfers in the whole range of the impedance of the C,L,R standards. Quadrature

This chapter describes the part of the results of this project, covering the development of the transformer bridge-comparators which transfer units of the resistance, inductance, capacitance

bridge provides cross transfers of the units. Last bridge is described in [22, 23].

–10<sup>9</sup>

. In this case, the equipment have to be universal, to compare arbitrary

. It makes them an excel-

24-hour 1 Ohm standard measurements are shown in Figure 7.

Uncertainty of the measurement of these bridges achieves 10<sup>8</sup>

lent instrument for fundamental investigations.

uncertainty about 10<sup>6</sup>

40 Metrology

3.1. Application of the VM in transformer bridge

Figure 7. Results of the 24-hour 1 Ohm standard measurements.

Early autotransformer bridges were described in [24, 25]. These bridges have been widely used up to now [15, 16]. To eliminate the influence of the cable impedance (yoke) on the results of measurement, double autotransformer bridges are used [3, 5]. The wide-range double autotransformer bridge contains two inductive dividers, simultaneously controlled for bridge balance. For accurate measurements, these inductive dividers usually are of a two-stage design at least. Every stage of these inductive dividers [26] consists of a lot of turns and appropriate complicate switchers. They have to have multidigit capacity (up to seven or eight digits). This quite complicates the bridge.

Development of the variational bridge has to solve two problems:


The simplified measuring circuit of the automatic variational bridge (PICS) [27], which solves these problems, is shown in Figure 8.

The bridge consists of the supply unit (the generator G connected to the voltage transformer TV), the main autotransformer AT and the variationally balanced 90 phase shifter [28], which is calibrated through calibration circuit CC. The vector voltmeter VV (through the preamplifier PA and switchers S1 and S2) measures the bridge (U1, U2) and the calibration circuit CC (Uc)

Figure 8. Circuit diagram of the autotransformer bridge.

unbalances the signals. The differential voltage follower 1:1 compensates the voltage drop Un on the cable impedance Zn. The microcontroller μC transfers the results of the VV measurements to the personal computer PC and controls the bridge balance and calibration of the phase shifter 90�. The autotransformer AT Carries on its core windings m2, m1c and m1k. These windings are used to balance the bridge by the main (m1c) and secondary (m1k) parameters. The standards to be compared Z1 and Z2 are connected serially by the cable (yoke) and by their high potential ports, to voltage transformer TV and to the windings m1c and m2 of the autotransformer AT.

C ¼ ð Þ U<sup>2</sup> þ U<sup>1</sup> =ð Þ U2<sup>v</sup> � U<sup>2</sup> ; D ¼ ð Þ U<sup>2</sup> � U<sup>1</sup> =ð Þ U2<sup>v</sup> � U<sup>2</sup> ; δ<sup>v</sup> ¼ δm=ð Þ 1 þ δm ; δm ¼ Δmv=ð Þ m<sup>1</sup> þ m<sup>2</sup>

μC uses the results of the calculation of the bridge unbalance δZc by described algorithm in

• in the first stage, μC makes quick, automatic balance of the bridge on the four high-order

• in the second stage, μC increases the sensitivity of the voltmeter VV on 10<sup>4</sup> and decreases the value of the variation Δmv of the m1 turns in the same ratio. Then, μC repeats the measurements by described algorithm. Results of these measurements and calculations by formula (26) determine the balance point coordinates and find the

> Z1 Z2

<sup>¼</sup> <sup>m</sup><sup>1</sup> m<sup>2</sup>

The bridge balance and data processing by described variational algorithm reduce the number of the autotransformer dividers to only one and sharply (twice) reduce the number of the digits

The 90� phase shifter and the calibration circuit CC do not contain accurate internal standards of capacitance or resistance. To get good accuracy, we use the special phase shifter calibration procedure based on the variational method. Simplified structure of this phase shifter is shown

Phase shifter consists of serially connected inverter I and proper phase shifter PS. Firstly, calibrating circuit (resistors R1 and R2) and switchers S1 and S2 are used to calibrate inverter I. Secondly, calibrating circuit (resistor R1 and capacitor C1) and switchers S3 and S4 are used to calibrate the phase shifter PS. Vector voltmeter VV, through switcher S5 measures unbalance signals of the first or second calibration circuits and translates the results of measurements to microcontroller μC. Finally, one controls all calibration procedure and calculates PS

� δZ (27)

Variational Calibration

43

http://dx.doi.org/10.5772/intechopen.74220

two stages:

decades (balance stage);

impedance ratio:

of this divider.

in Figure 9.

real transfer coefficient.

Figure 9. Structure of the phase shifter.

Calibration procedure consists of two stages.

The final result is given in 8.5 digits.

The output of the 90� phase shifter is connected in series with the winding m1c to create the balance winding m<sup>1</sup> ¼ m1<sup>c</sup> þ jm1k.

The drop of the voltage Un acts on the impedance Zn of the cable which connects Z1 and Z2. This voltage is applied to the input of the differential voltage follower 1:1.

The two-channel VV has two digital synchronous demodulators, proper LF digital filters and Σ-Δ ADC. It simultaneously measures two orthogonal components of the bridge unbalance signals. This voltmeter has high selectivity (equivalent Q-factor is higher than 105 ). Its integral nonlinearity is better than 10�<sup>4</sup> and relative sensitivity is better than 10�<sup>5</sup> . The VV is calibrated automatically and periodically by variational algorithm, described in [29].

On the low impedance ranges, the drop Un of the voltage on the cable impedance increases. This increases the uncertainty of the bridge unbalance measurement. To decrease this effect, the voltage follower 1:1 is used. This follower places the named drop of the voltage between low potential pins of the windings m1 and m2. It decreases the effective cable impedance from Zn to the equivalent value Zne = Znδ, where δ is the uncertainty of the transfer coefficient of the voltage follower.

To decrease the number of the decades of the autotransformer divider and eliminate the influence of the Zn on the results of measurement, the bridge operates in a non-fully balance mode and use twice variational balance [27].

In compliance with developed variational algorithm, VV measures sequentially the bridge unbalance signals U1 and U2. After that, μC varies the turns of the winding m1 on Δmv and VV measures the variational signal U2v.

The system of Eqs. (30) describes these three measurements:

$$\begin{aligned} \mathbf{U}\_{0} \mathbf{U}\_{0} (\mathbf{Z}\_{1}/\mathbf{Z}\_{c}) - \mathbf{U}\_{0} [1 - \mathbf{Z}\_{n} (1 + \delta)/\mathbf{Z}\_{c}] m\_{1} / (m\_{1} + m\_{2}) - \mathbf{U}\_{1} &= \mathbf{0} \\ -\mathbf{U}\_{0} [1 - \mathbf{Z}\_{n} (1 + \delta)/\mathbf{Z}\_{c}] m\_{2} / (m\_{1} + m\_{2}) + \mathbf{U}\_{0} \mathbf{Z}\_{2} / \mathbf{Z}\_{c} + \mathbf{U}\_{2} &= \mathbf{0} \\ -\mathbf{U}\_{0} [1 - \mathbf{Z}\_{n} (1 + \delta)/\mathbf{Z}\_{c}] m\_{2} / (m\_{1} + m\_{2} + \Delta m\_{v}) + \mathbf{U}\_{0} \mathbf{Z}\_{2} / \mathbf{Z}\_{c} + \mathbf{U}\_{2v} &= \mathbf{0} \end{aligned} \tag{25}$$

where Zc = Z1 + Z2 + Zn, and δ is the uncertainty of the voltage follower 1:1, U<sup>0</sup> is the supply voltage.

The formula (26) gives the solution of the system (25):

$$\delta Z = -\frac{\delta\_v}{2} \frac{m\_1 + m\_2}{m\_2} \left( \mathcal{C} + \frac{m\_1 - m\_2}{m\_1 + m\_2} D \right) / \left[ 1 + (\mathcal{C} + D) \delta\_v \right] \tag{26}$$

where

$$\mathcal{C} = (\mathcal{U}\_2 + \mathcal{U}\_1) / (\mathcal{U}\_{2v} - \mathcal{U}\_2);\\\mathcal{D} = (\mathcal{U}\_2 - \mathcal{U}\_1) / (\mathcal{U}\_{2v} - \mathcal{U}\_2);\\\delta\_v = \delta m / (1 + \delta m);\\\delta m = \Delta m\_v / (m\_1 + m\_2)$$

μC uses the results of the calculation of the bridge unbalance δZc by described algorithm in two stages:


$$\frac{Z\_1}{Z\_2} = \frac{m\_1}{m\_2} - \delta Z \tag{27}$$

The final result is given in 8.5 digits.

unbalances the signals. The differential voltage follower 1:1 compensates the voltage drop Un on the cable impedance Zn. The microcontroller μC transfers the results of the VV measurements to the personal computer PC and controls the bridge balance and calibration of the phase shifter 90�. The autotransformer AT Carries on its core windings m2, m1c and m1k. These windings are used to balance the bridge by the main (m1c) and secondary (m1k) parameters. The standards to be compared Z1 and Z2 are connected serially by the cable (yoke) and by their high potential ports, to voltage transformer TV and to the windings m1c and m2 of the autotransformer AT.

The output of the 90� phase shifter is connected in series with the winding m1c to create the

The drop of the voltage Un acts on the impedance Zn of the cable which connects Z1 and Z2.

The two-channel VV has two digital synchronous demodulators, proper LF digital filters and Σ-Δ ADC. It simultaneously measures two orthogonal components of the bridge unbalance

On the low impedance ranges, the drop Un of the voltage on the cable impedance increases. This increases the uncertainty of the bridge unbalance measurement. To decrease this effect, the voltage follower 1:1 is used. This follower places the named drop of the voltage between low potential pins of the windings m1 and m2. It decreases the effective cable impedance from Zn to the equivalent value Zne = Znδ, where δ is the uncertainty of the transfer coefficient of the voltage follower.

To decrease the number of the decades of the autotransformer divider and eliminate the influence of the Zn on the results of measurement, the bridge operates in a non-fully balance

In compliance with developed variational algorithm, VV measures sequentially the bridge unbalance signals U1 and U2. After that, μC varies the turns of the winding m1 on Δmv and

�U<sup>0</sup> 1 � Znð Þ 1 þ δ =Ζ<sup>c</sup> ½ �m2=ðm<sup>1</sup> þ m<sup>2</sup> þ ΔmvÞ þ U0Z2=Zc þ U2<sup>v</sup> ¼ 0

where Zc = Z1 + Z2 + Zn, and δ is the uncertainty of the voltage follower 1:1, U<sup>0</sup> is the supply

m<sup>1</sup> � m<sup>2</sup> m<sup>1</sup> þ m<sup>2</sup>

D

=½ � 1 þ ð Þ C þ D δ<sup>v</sup> (26)

U0ð Þ� Z1=Zc U<sup>0</sup> 1 � Znð Þ 1 þ δ =Zc ½ �m1=ð Þ� m<sup>1</sup> þ m<sup>2</sup> U<sup>1</sup> ¼ 0 �U<sup>0</sup> 1 � Znð Þ 1 þ δ =Ζ<sup>c</sup> ½ �m2=ð Þþ m<sup>1</sup> þ m<sup>2</sup> U0Z2=Zc þ U<sup>2</sup> ¼ 0

C þ

). Its integral

(25)

. The VV is calibrated

This voltage is applied to the input of the differential voltage follower 1:1.

nonlinearity is better than 10�<sup>4</sup> and relative sensitivity is better than 10�<sup>5</sup>

automatically and periodically by variational algorithm, described in [29].

signals. This voltmeter has high selectivity (equivalent Q-factor is higher than 105

balance winding m<sup>1</sup> ¼ m1<sup>c</sup> þ jm1k.

42 Metrology

mode and use twice variational balance [27].

The system of Eqs. (30) describes these three measurements:

The formula (26) gives the solution of the system (25):

m<sup>1</sup> þ m<sup>2</sup> m<sup>2</sup>

<sup>δ</sup><sup>Z</sup> ¼ � <sup>δ</sup><sup>v</sup> 2

VV measures the variational signal U2v.

voltage.

where

The bridge balance and data processing by described variational algorithm reduce the number of the autotransformer dividers to only one and sharply (twice) reduce the number of the digits of this divider.

The 90� phase shifter and the calibration circuit CC do not contain accurate internal standards of capacitance or resistance. To get good accuracy, we use the special phase shifter calibration procedure based on the variational method. Simplified structure of this phase shifter is shown in Figure 9.

Phase shifter consists of serially connected inverter I and proper phase shifter PS. Firstly, calibrating circuit (resistors R1 and R2) and switchers S1 and S2 are used to calibrate inverter I. Secondly, calibrating circuit (resistor R1 and capacitor C1) and switchers S3 and S4 are used to calibrate the phase shifter PS. Vector voltmeter VV, through switcher S5 measures unbalance signals of the first or second calibration circuits and translates the results of measurements to microcontroller μC. Finally, one controls all calibration procedure and calculates PS real transfer coefficient.

Calibration procedure consists of two stages.

Figure 9. Structure of the phase shifter.
