2. The variational calibration

#### 2.1. Theoretical basis of the variational calibration

Every measuring circuit (MC) has the input value, which has to be measured and generates measured output value. In an ideal case, the results of measurement depend on the input value and the transfer function k of the MC only.

Formula (1) describes the result of measurement of ideal MC:

$$\mathbf{Z}\_{\mathbf{x}} = \mathbf{k} \mathbf{Z}\_{0} \tag{1}$$

The much more complicated mathematic model (3) of the real MC now describes the results of

Zx<sup>0</sup> ¼ γ Zx; δ1…δi…δj; δ0; δ<sup>s</sup>

In the simplest case, every disturbing factor z1::zi::zj creates appropriate uncertainty components δ1…δi…δj. In more complicated cases, some disturbing factors z1…zi can influence some complex Zi…Ziþ<sup>m</sup> of the results of measurement. But we know functions δ<sup>i</sup> ¼

ð Þ z1…zi…zn and do not know just the constant coefficients, which enters into these depen-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

. Last ones influence the uncertainty sources z1…zj and change

. It creates the output of the proper results of MC measurement Zx1…Zxj.

Formula (4) describes the standard uncertainty δ<sup>r</sup> of the measurement of the real MC:

q

δ2 <sup>0</sup> <sup>þ</sup> <sup>Σ</sup><sup>j</sup> 1δ2 <sup>i</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> s

To eliminate the influence of the uncertainties δ1…δi…δ<sup>j</sup> on the results of measurement, the variation method was developed (VM) [9]. Figure 2 illustrates this method. Here, MC contains

Variators cannot change the uncertainties δ<sup>0</sup> and δs. These uncertainties are supposed to be

2. Then, MC consequently varies the influence of the disturbing factor zi on the well-known

δ<sup>r</sup> ¼

Usually the model (3) is well known from preliminary investigations of the MC.

� � (3)

(4)

Variational Calibration

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measurement:

f i

dences.

n additional variators V1…Vj

VM consists of the following steps:

Figure 2. Variational measuring circuit.

known or equal to zero during the VM calibration.

1. First, MC measures initial value Zx0 of the input value Zx.

the uncertainty δ1…δ<sup>j</sup>

value αi.

Formula (2) describes the standard uncertainty δid of such measurement:

$$
\delta\_{\rm id} = \sqrt{\delta\_0^2 + \delta\_s^2} \tag{2}
$$

Here δ<sup>0</sup> and δ<sup>s</sup> are the uncertainties, of the standard Z0 and the uncertainty, caused by the sensitivity of the MC.

In the real MC, the results of measurement Ζx0 also depend on the complex of the disturbing factors z1…zi…zj as well (for simplicity of the description, these factors on the Figure 1 are shown being out of MC). These factors create proper complex of the uncertainties of measurements δ1…δ<sup>i</sup> …δ<sup>j</sup> and shift the appropriate result Zx<sup>0</sup> of measurement from its ideal value Zx.

Figure 1. Real measuring circuit.

The much more complicated mathematic model (3) of the real MC now describes the results of measurement:

$$Z\_{x0} = \gamma \left( Z\_x; \delta\_1...\delta\_i...\delta\_j; \delta\_0; \delta\_s \right) \tag{3}$$

Usually the model (3) is well known from preliminary investigations of the MC.

In the simplest case, every disturbing factor z1::zi::zj creates appropriate uncertainty components δ1…δi…δj. In more complicated cases, some disturbing factors z1…zi can influence some complex Zi…Ziþ<sup>m</sup> of the results of measurement. But we know functions δ<sup>i</sup> ¼ f i ð Þ z1…zi…zn and do not know just the constant coefficients, which enters into these dependences.

Formula (4) describes the standard uncertainty δ<sup>r</sup> of the measurement of the real MC:

$$
\delta\_r = \sqrt{\delta\_0^2 + \Sigma\_1^j \delta\_i^2 + \delta\_s^2} \tag{4}
$$

To eliminate the influence of the uncertainties δ1…δi…δ<sup>j</sup> on the results of measurement, the variation method was developed (VM) [9]. Figure 2 illustrates this method. Here, MC contains n additional variators V1…Vj . Last ones influence the uncertainty sources z1…zj and change the uncertainty δ1…δ<sup>j</sup> . It creates the output of the proper results of MC measurement Zx1…Zxj.

Variators cannot change the uncertainties δ<sup>0</sup> and δs. These uncertainties are supposed to be known or equal to zero during the VM calibration.

VM consists of the following steps:

results of measurement. System of equations describes these results. Solution of this system eliminates influences of the disturbing factors and gets the accurate results of measurement. This method significantly simplifies the accurate devices, reducing their weight, dimension

Every measuring circuit (MC) has the input value, which has to be measured and generates measured output value. In an ideal case, the results of measurement depend on the input value

Zx ¼ kZ<sup>0</sup> (1)

(2)

and cost, but increases the time of measurement.

2.1. Theoretical basis of the variational calibration

Formula (1) describes the result of measurement of ideal MC:

Formula (2) describes the standard uncertainty δid of such measurement:

δid ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ2 <sup>0</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> s

q

Here δ<sup>0</sup> and δ<sup>s</sup> are the uncertainties, of the standard Z0 and the uncertainty, caused by the

In the real MC, the results of measurement Ζx0 also depend on the complex of the disturbing factors z1…zi…zj as well (for simplicity of the description, these factors on the Figure 1 are shown being out of MC). These factors create proper complex of the uncertainties of measurements δ1…δ<sup>i</sup> …δ<sup>j</sup> and shift the appropriate result Zx<sup>0</sup> of measurement from its ideal value Zx.

and the transfer function k of the MC only.

sensitivity of the MC.

Figure 1. Real measuring circuit.

2. The variational calibration

30 Metrology


Figure 2. Variational measuring circuit.

Variations could be provided in any order. To simplify the system of equations, it is preferable to perform variations sequentially and to switch ON the variation α<sup>i</sup> when all other variations are switched OFF.

Variations could have any law. To simplify the system of equation, it is preferable to provide the multiplicative variation (when we multiply the appropriate uncertainty component δ<sup>i</sup> on well-known ratio α<sup>i</sup> (δiv = αiδi)) or additive variation (when we add the appropriate well-known uncertainty Δ<sup>v</sup> to the uncertainty component Δim (Δiv = Δim + Δv)).


$$\begin{aligned} \mathbf{Z}\_{x0} &= \gamma \left( \mathbf{Z}\_x, \delta\_1 \dots \delta\_i \dots \delta\_j, \delta\_0, \delta\_s \right) \\ \mathbf{Z}\_{x1} &= \gamma \left( \mathbf{Z}\_x, \delta\_1, \alpha\_1 \dots \delta\_i \dots \delta\_j, \delta\_0, \delta\_s \right) \\ \mathbf{Z}\_{xj} &= \gamma \left( \mathbf{Z}\_x, \delta\_1 \dots \delta\_i \dots \delta\_j, a\_j, \delta\_0, \delta\_s \right) \end{aligned} \tag{5}$$

tc <sup>¼</sup> <sup>Σ</sup><sup>j</sup>

Let us suppose that δα<sup>i</sup> = 0 and δ<sup>0</sup> = 0. In this case, formula (9) describes the standard uncertainty

• Variation method needs n+1 measurement instead one only. It sufficiently increases the

• Variation method increases the contribution of measurement sensitivity δsi in the common

We can overcome these two disadvantages of the variation method in different ways. Here, we

Usually, different uncertainty sources have different typical speeds of drift. We can divide the thesaurus of j uncertainty sources into clusters, which have congruous time of drift. Figure 3 illustrates this approach. In Figure 3, thesaurus of the j uncertainty components is divided into

The first cluster (T1) joins m of the most stable uncertainty sources. It could be instability of the internal standards or arms ratios in transformer bridges, and so on. MC provides their calibration very seldom, for example, one time per year. For this calibration, MC performs sequential

ffiffiffiffiffiffiffiffiffiffiffiffi Σj <sup>0</sup> <sup>δ</sup><sup>2</sup> si <sup>q</sup>

δ<sup>c</sup> ¼

Formulas (8) and (9) show that the variation method has two disadvantages:

shortly describe time and space clustering of the thesaurus of the uncertainty sources.

of measurement caused by sensitivity of measurements only:

time of measurement.

2.1.1. Time clustering

uncertainty of measurement.

three clusters T1, T2 and T3 (j = m + n + k).

Figure 3. Variation calibration with time clustering.

<sup>0</sup>ti (8)

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Variational Calibration

33

: (9)

The system (5) contains j + 1 unknown quantities: Zx and uncertainties of measurement δ1…δ<sup>j</sup> , and j + 1 results of measurement Zx0…Zxj. Solution (6) of this system gets the true value of the results of measurement Zx and the values of the uncertainties δ1…δ<sup>j</sup> of the measurement:

$$\begin{aligned} Z\_{\mathbf{x}} &= \rho\_0 \left[ \left( \mathbf{Z}\_{\mathbf{x}0} - \mathbf{Z}\_{\mathbf{x}j} \right), \left( \alpha\_1 - \alpha\_j \right), \delta\_0, \delta\_s \right] \\ \delta\_1 &= \rho\_1 \left[ \left( \mathbf{Z}\_{\mathbf{x}0} - \mathbf{Z}\_{\mathbf{x}j} \right), \left( \alpha\_1 - \alpha\_j \right), \delta\_0, \delta\_s \right] \\ \delta\_j &= \rho\_j \left[ \left( \mathbf{Z}\_{\mathbf{x}0} - \mathbf{Z}\_{\mathbf{x}j} \right), \left( \alpha\_1 - \alpha\_j \right), \delta\_0, \delta\_s \right] \end{aligned} \tag{6}$$

Periodical variation calibration lets us to observe the behavior of every disturbing factor, to determine their stability, to monitor measuring circuit and to ensure precision of the period of the variational calibration.

Let the uncertainty caused by the finite sensitivity of the i-measurement be δsi and the uncertainty of the variation α<sup>i</sup> be δαi. In this case, formula (7) describes the resulting standard uncertainty δ<sup>c</sup> of the measurement with variation calibration:

$$\delta\_c = \sqrt{\delta\_0^2 + \Sigma\_0^{\dagger} \left(\delta\_i^2 \delta \alpha\_i^2 + \delta\_{si}^2\right)}\tag{7}$$

Eq. (7) shows that the VM sharply decreases influence of the uncertainty components δ<sup>i</sup> on the common uncertainty of measurement (on the 1/δα<sup>i</sup> times).

Let us suppose uncertainty source zi creates uncertainty δ<sup>i</sup> = 10�<sup>3</sup> and we need to decrease it to the value 10�<sup>6</sup> . It means that we have to provide appropriate variation with uncertainty better than 10�<sup>3</sup> only. It is a very important result of the VM. This effect is restricted only by the stability of the uncertainties δ1…δ<sup>j</sup> during the time of measurement.

Let us suppose that time of every measurement is ti. It means that the common time tc of measurement increases to the value:

$$t\_c = \Sigma\_0^j t\_i \tag{8}$$

Let us suppose that δα<sup>i</sup> = 0 and δ<sup>0</sup> = 0. In this case, formula (9) describes the standard uncertainty of measurement caused by sensitivity of measurements only:

$$
\delta\_c = \sqrt{\Sigma\_0^j \, \delta\_{si}^2}. \tag{9}
$$

Formulas (8) and (9) show that the variation method has two disadvantages:


We can overcome these two disadvantages of the variation method in different ways. Here, we shortly describe time and space clustering of the thesaurus of the uncertainty sources.

#### 2.1.1. Time clustering

(5)

(6)

Variations could be provided in any order. To simplify the system of equations, it is preferable to perform variations sequentially and to switch ON the variation α<sup>i</sup> when all other

Variations could have any law. To simplify the system of equation, it is preferable to provide the multiplicative variation (when we multiply the appropriate uncertainty component δ<sup>i</sup> on well-known ratio α<sup>i</sup> (δiv = αiδi)) or additive variation (when we add the appropriate well-known uncertainty Δ<sup>v</sup> to the uncertainty component Δim (Δiv = Δim + Δv)).

3. After every variation, MC measures the results of the measurement Zx1…Zxi…Zxj.

Zx<sup>0</sup> ¼ γ Zx; δ1…δi…δj; δ0; δ<sup>s</sup> � �

Zx<sup>1</sup> ¼ γ Zx; δ1; α1…δi…δj; δ0; δ<sup>s</sup> � �

Zxj ¼ γ Zx; δ1…δi…δj; aj; δ0; δ<sup>s</sup> � �

The system (5) contains j + 1 unknown quantities: Zx and uncertainties of measurement

� �; <sup>α</sup><sup>1</sup> � <sup>α</sup><sup>j</sup>

� �; <sup>α</sup><sup>1</sup> � <sup>α</sup><sup>j</sup>

Periodical variation calibration lets us to observe the behavior of every disturbing factor, to determine their stability, to monitor measuring circuit and to ensure precision of the period of

Let the uncertainty caused by the finite sensitivity of the i-measurement be δsi and the uncertainty of the variation α<sup>i</sup> be δαi. In this case, formula (7) describes the resulting standard

Eq. (7) shows that the VM sharply decreases influence of the uncertainty components δ<sup>i</sup> on the

Let us suppose uncertainty source zi creates uncertainty δ<sup>i</sup> = 10�<sup>3</sup> and we need to decrease it to

than 10�<sup>3</sup> only. It is a very important result of the VM. This effect is restricted only by the

Let us suppose that time of every measurement is ti. It means that the common time tc of

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

. It means that we have to provide appropriate variation with uncertainty better

<sup>q</sup> � � (7)

� �; <sup>α</sup><sup>1</sup> � <sup>α</sup><sup>j</sup>

Zx ¼ r<sup>0</sup> Ζx<sup>0</sup> � Zxj

δ<sup>1</sup> ¼ r<sup>1</sup> Ζx<sup>0</sup> � Zxj

δ<sup>j</sup> ¼ r<sup>j</sup> Ζx<sup>0</sup> � Zxj

uncertainty δ<sup>c</sup> of the measurement with variation calibration:

common uncertainty of measurement (on the 1/δα<sup>i</sup> times).

δ<sup>c</sup> ¼

stability of the uncertainties δ1…δ<sup>j</sup> during the time of measurement.

δ2 <sup>0</sup> <sup>þ</sup> <sup>Σ</sup><sup>j</sup> <sup>0</sup> <sup>δ</sup><sup>2</sup> <sup>i</sup> δα<sup>2</sup> <sup>i</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup> si

, and j + 1 results of measurement Zx0…Zxj. Solution (6) of this system gets the true value of the results of measurement Zx and the values of the uncertainties δ1…δ<sup>j</sup> of the

� �

� �

� �

� �; δ0; δ<sup>s</sup>

� �; δ0; δ<sup>s</sup>

� �; δ0; δ<sup>s</sup>

4. The system of Eqs. (5) describes these measurements:

variations are switched OFF.

δ1…δ<sup>j</sup>

32 Metrology

measurement:

the value 10�<sup>6</sup>

measurement increases to the value:

the variational calibration.

Usually, different uncertainty sources have different typical speeds of drift. We can divide the thesaurus of j uncertainty sources into clusters, which have congruous time of drift. Figure 3 illustrates this approach. In Figure 3, thesaurus of the j uncertainty components is divided into three clusters T1, T2 and T3 (j = m + n + k).

The first cluster (T1) joins m of the most stable uncertainty sources. It could be instability of the internal standards or arms ratios in transformer bridges, and so on. MC provides their calibration very seldom, for example, one time per year. For this calibration, MC performs sequential

Figure 3. Variation calibration with time clustering.

variation of all sources of uncertainty and provides m+n+k+1 measurements. The system (5) of equations describes the results of these measurements. Solution (6) of this system gets us values of the m uncertainties of the first cluster.

The second cluster (T2) joins the n less stable sources of the uncertainty. It could be the temperature dependences of the operational amplifiers parameters, and so on. Calibration of these sources is provided more frequently, for example, one time per hour. During this calibration, we suppose that the m uncertainties of the first clusters are stable. Values of these uncertainties enter in the system (5) as constants. To find values of the n uncertainties of the second cluster, MC varies sequentially the uncertainty sources n+k, provides proper measurements and solves the system (5). It needs n+k+1 measurements.

The third cluster (T3) joins the k uncertainty sources which change most quickly. This cluster mostly includes the sources, which directly depends on the parameters of the object to be measured. This calibration is aimed to find the true results of measurement and values of the last k uncertainties. During this calibration, we suppose that uncertainties of the first and second clusters are stable. Their appropriate values are entered in system (5) as constants. Calibration now consists of sequential variation of the k uncertainties of third cluster and appropriate measurements. Solution of the system (5) gets us the true results of measurement Zx and last k uncertainties. This calibration needs k+1 measurements only.

Let us suppose that any measurement needs time ti. Formula (10) describes the weighted average tc of the measurement with variation calibration:

$$t\_c = \Sigma\_1^k t\_i \left( 1 + \frac{n+k}{m+n+k} \frac{T\_k}{T\_m} + \frac{k}{m+n+k} \frac{T\_k}{T\_m} \right) \tag{10}$$

where Zx and Zx0 are the MC input and output values, respectively, ΔΚ1… ΔΚi… ΔΚ<sup>n</sup> are the

The following formula expresses the dependence of the measurement uncertainty δ<sup>r</sup> on the

Let us provide n well-known variations ν1…νi…ν<sup>n</sup> of the quadripole transfer coefficients K1…Ki…Kn. MC provides the new measurements Zx0…Zxi…Zxn of the unknown value Zx after

> Zx<sup>0</sup> ¼ f Zx;K1;ΔK1…Ki ð Þ ;ΔKi…Kn;ΔKn Zx<sup>1</sup> ¼ f Zx;K1;ΔK1; v1…Ki; ΔKi ð Þ ;…Kn; ΔKn Zxn ¼ f Zx;K1;ΔK1…Ki ð Þ ;ΔKi…Kn;ΔKn; vn

Solution of the system (13) of equations gets accurate results of measurement together with all

Formulas (8) and (9) describe the uncertainty and time of measurement when using the space clustering as well. However, the number of measurements in case of space clustering is much

We can decompose the measuring circuit in different ways. Optimal decomposition depends on the structure of the measuring circuit. Here, it is impossible to analyze all these possibilities.

less. Error accumulation and common time of measurement are much less as well.

In most cases, we are forced to use time and space clustering together.

δ2 <sup>0</sup> <sup>þ</sup> <sup>δ</sup><sup>2</sup>

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>s</sup> <sup>þ</sup> <sup>Σ</sup><sup>n</sup> 1ΔK<sup>2</sup> i

(12)

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35

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(13)

uncertainties of the quadripole transfer coefficients K1…Ki…Kn.

where δ<sup>s</sup> is the uncertainty caused by the finite MC sensitivity.

every variation. The system of Eq. (13) describes these measurements:

δ<sup>r</sup> ¼

components of the decomposed MC:

Figure 4. Variation correction with space clustering.

uncertainties of the quadripoles.

where Σ<sup>k</sup> <sup>1</sup>ti is the time of the k cluster calibration and measurement, Tn=Tm is the ratio of the periods of the second Tn and first Tm clusters calibrations and Tk=Tm is the ratio of the periods of the third Tk and first Tm cluster calibrations.

Formula (10) shows that the time of measurement decreases only slightly during the time of calibration of the third cluster. It means sufficient diminution of the time of measurement.

#### 2.1.2. Space clustering

Sometimes, we do not need to separately study every component of the measurement uncertainty. In this case, we use space clustering. During the space clustering, MC is represented as a complex of the n quadripoles and standards to be compared. Figure 4 shows such decomposition of the measurement circuit.

In Figure 4, K1 … Kn are the quadripoles of the MC and the V1…Vn are the variators used to vary the transfer coefficient of the proper quadripole.

The following formula describes the decomposed MC:

$$Z\_{x0} = f(Z\_x, K\_1, \Delta K\_1 \dots K\_i, \Delta K\_i \dots K\_n, \Delta K\_n) \tag{11}$$

Figure 4. Variation correction with space clustering.

variation of all sources of uncertainty and provides m+n+k+1 measurements. The system (5) of equations describes the results of these measurements. Solution (6) of this system gets us

The second cluster (T2) joins the n less stable sources of the uncertainty. It could be the temperature dependences of the operational amplifiers parameters, and so on. Calibration of these sources is provided more frequently, for example, one time per hour. During this calibration, we suppose that the m uncertainties of the first clusters are stable. Values of these uncertainties enter in the system (5) as constants. To find values of the n uncertainties of the second cluster, MC varies sequentially the uncertainty sources n+k, provides proper mea-

The third cluster (T3) joins the k uncertainty sources which change most quickly. This cluster mostly includes the sources, which directly depends on the parameters of the object to be measured. This calibration is aimed to find the true results of measurement and values of the last k uncertainties. During this calibration, we suppose that uncertainties of the first and second clusters are stable. Their appropriate values are entered in system (5) as constants. Calibration now consists of sequential variation of the k uncertainties of third cluster and appropriate measurements. Solution of the system (5) gets us the true results of measurement

Let us suppose that any measurement needs time ti. Formula (10) describes the weighted

periods of the second Tn and first Tm clusters calibrations and Tk=Tm is the ratio of the periods

Formula (10) shows that the time of measurement decreases only slightly during the time of calibration of the third cluster. It means sufficient diminution of the time of measurement.

Sometimes, we do not need to separately study every component of the measurement uncertainty. In this case, we use space clustering. During the space clustering, MC is represented as a complex of the n quadripoles and standards to be compared. Figure 4 shows such decompo-

In Figure 4, K1 … Kn are the quadripoles of the MC and the V1…Vn are the variators used to

Tk Tm þ

<sup>1</sup>ti is the time of the k cluster calibration and measurement, Tn=Tm is the ratio of the

k m þ n þ k

Zx<sup>0</sup> ¼ f Zx;K1;ΔK1…Ki ð Þ ;ΔKi…Kn;ΔKn (11)

Tk Tm

(10)

n þ k m þ n þ k

surements and solves the system (5). It needs n+k+1 measurements.

Zx and last k uncertainties. This calibration needs k+1 measurements only.

<sup>1</sup>ti 1 þ

average tc of the measurement with variation calibration:

tc <sup>¼</sup> <sup>Σ</sup><sup>k</sup>

of the third Tk and first Tm cluster calibrations.

vary the transfer coefficient of the proper quadripole. The following formula describes the decomposed MC:

where Σ<sup>k</sup>

34 Metrology

2.1.2. Space clustering

sition of the measurement circuit.

values of the m uncertainties of the first cluster.

where Zx and Zx0 are the MC input and output values, respectively, ΔΚ1… ΔΚi… ΔΚ<sup>n</sup> are the uncertainties of the quadripole transfer coefficients K1…Ki…Kn.

The following formula expresses the dependence of the measurement uncertainty δ<sup>r</sup> on the components of the decomposed MC:

$$
\delta\_r = \sqrt{\delta\_0^2 + \delta\_s^2 + \Sigma\_1'' \Delta K\_i^2} \tag{12}
$$

where δ<sup>s</sup> is the uncertainty caused by the finite MC sensitivity.

Let us provide n well-known variations ν1…νi…ν<sup>n</sup> of the quadripole transfer coefficients K1…Ki…Kn. MC provides the new measurements Zx0…Zxi…Zxn of the unknown value Zx after every variation. The system of Eq. (13) describes these measurements:

$$\begin{aligned} Z\_{\mathbf{x}0} &= f(Z\_{\mathbf{x}}, K\_1, \Delta K\_1 \dots K\_i, \Delta K\_i \dots K\_n, \Delta K\_n) \\ Z\_{\mathbf{x}1} &= f(Z\_{\mathbf{x}}, K\_1, \Delta K\_1, \upsilon\_1 \dots K\_i, \Delta K\_i, \dots K\_n, \Delta K\_n) \\ Z\_{\mathbf{x}n} &= f(Z\_{\mathbf{x}}, K\_1, \Delta K\_1 \dots K\_i, \Delta K\_i \dots K\_n, \Delta K\_n, \upsilon\_n) \end{aligned} \tag{13}$$

Solution of the system (13) of equations gets accurate results of measurement together with all uncertainties of the quadripoles.

Formulas (8) and (9) describe the uncertainty and time of measurement when using the space clustering as well. However, the number of measurements in case of space clustering is much less. Error accumulation and common time of measurement are much less as well.

We can decompose the measuring circuit in different ways. Optimal decomposition depends on the structure of the measuring circuit. Here, it is impossible to analyze all these possibilities. In most cases, we are forced to use time and space clustering together.

It should be noted that variation method was used earlier in some measurements (e.g., elimination of the uncertainty caused by self-heating of the resistive thermometer in temperature measurements). Here, we consider generalization and dissemination of this method in different areas, first in impedance measurements.

results of measurement to microcontroller μC. μC controls the operation of the ΜC, processes results of the voltages measurements and calculates the ratio of two impedances Zx and Z0.

The amplifier A protects measuring circuit and decreases the influence of the parasitic admit-

Let gain K be finite. In this case, admittance Yc between the amplifier inputs cause one of the biggest sources of the measurement uncertainty. This uncertainty (δΖ) strongly limits the measurements of the high impedances on high frequencies. δΖ is described by the

Here, the values Yc and K are the disturbing factors. The quotient of the Yc and K can be considered as the sole source of the uncertainty. Let us provide the multiplicative variation of the gain K of the amplifier A. To vary K on ratio α1, the divider Dv with transfer coefficient 1 or

α<sup>1</sup> (Figure 5) is used. After this variation, MC measures the additional voltage U0v.

Solution of this system gets the following formula (18):

The system of three equations describes the measurements of the voltages Ux, Uo and Uov.

Analysis of the formula (18) shows that the uncertainty of the variation calibration has minimal

Formula (19) shows that the ratio Zx/Z0 does not depend on the quotient of the Yc and K.

But here increases component of the uncertainty, caused by the increased number of measurements. VV measures quadrature components a and b of three voltages: Ux, Uo and U0v. Let us suppose that effective input noise of the VV in all these measurement has the same value Δ and the results of measurement are not correlated. In this case, the following

Ux ¼ IxZx; U0ð Þ¼ 1 � YcZ0=K IxZ0; U0ð Þ¼ 1 � YcZ0=K IxZ<sup>0</sup> (17)

Zx=Z<sup>0</sup> ¼ Ux½ � 1 � δU � α1=ð Þ 1 � α<sup>1</sup> =U<sup>0</sup> (18)

Zx=Z<sup>0</sup> ¼ Uxð Þ 1 � δU =U<sup>0</sup> (19)

Zx=Z<sup>0</sup> ¼ Ux=U<sup>0</sup> (14)

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δΖ ¼ YcZ0=ð Þ 1 þ K (15)

δZ ffi YcZ0=K (16)

tance Yc between the amplifier inputs on the results of measurement.

In case if gain K is infinite, Eq. (14) describes the process of measurement:

Display D shows results of measurements.

equation:

If K » 1, we can write:

where δU ¼ 1 � U0=U0<sup>v</sup>

formulas are justified:

if α1=0.5. Then:

#### 2.2. Experimental developments of the VM

VM was used in several developments. It is too complicated to analyze all these possible applications. Here, we consider only some applications of this method in very important cases of widely used digibridges and in accurate transformer bridges.

#### 2.2.1. Application of the VM in digibridges

Development of the integral operational amplifiers and microprocessors resulted in the new class of measuring devices—digibridges [10–12]. Nowadays, digibridges cover most part of the specific market of the impedance meters. Now many companies manufacture digibridges (HP, Agilent, TeGam, IetLab, Wine Kerr, etc).

#### 2.2.1.1. Operation and analysis

A usual digibridge consists of two serially coupled impedances Zx and Z<sup>0</sup> (see Figure 5) These impedances are connected between outputs of the generator G and the protecting amplifier A. Negative input of this amplifier is connected to the common point of the impedances Zx and Z0. Amplifier A creates in this point the potential, close to zero (virtual ground). The same current Ix flows through both impedances Zx and Z<sup>0</sup> and creates voltages Ux and U0. Differential vector voltmeter DVV, through switcher S0, measures these voltages and transfers the

Figure 5. Structure of the digibridge with variational calibration.

results of measurement to microcontroller μC. μC controls the operation of the ΜC, processes results of the voltages measurements and calculates the ratio of two impedances Zx and Z0. Display D shows results of measurements.

The amplifier A protects measuring circuit and decreases the influence of the parasitic admittance Yc between the amplifier inputs on the results of measurement.

In case if gain K is infinite, Eq. (14) describes the process of measurement:

$$\mathbf{Z}\_{\mathbf{x}}/\mathbf{Z}\_{0} = \mathbf{U}\_{\mathbf{x}}/\mathbf{U}\_{0} \tag{14}$$

Let gain K be finite. In this case, admittance Yc between the amplifier inputs cause one of the biggest sources of the measurement uncertainty. This uncertainty (δΖ) strongly limits the measurements of the high impedances on high frequencies. δΖ is described by the equation:

$$
\delta \mathcal{Z} = \Upsilon\_{\mathcal{C}} Z\_0 / (1 + \mathcal{K}) \tag{15}
$$

If K » 1, we can write:

It should be noted that variation method was used earlier in some measurements (e.g., elimination of the uncertainty caused by self-heating of the resistive thermometer in temperature measurements). Here, we consider generalization and dissemination of this method in different areas, first in imped-

VM was used in several developments. It is too complicated to analyze all these possible applications. Here, we consider only some applications of this method in very important cases

Development of the integral operational amplifiers and microprocessors resulted in the new class of measuring devices—digibridges [10–12]. Nowadays, digibridges cover most part of the specific market of the impedance meters. Now many companies manufacture digibridges

A usual digibridge consists of two serially coupled impedances Zx and Z<sup>0</sup> (see Figure 5) These impedances are connected between outputs of the generator G and the protecting amplifier A. Negative input of this amplifier is connected to the common point of the impedances Zx and Z0. Amplifier A creates in this point the potential, close to zero (virtual ground). The same current Ix flows through both impedances Zx and Z<sup>0</sup> and creates voltages Ux and U0. Differential vector voltmeter DVV, through switcher S0, measures these voltages and transfers the

ance measurements.

36 Metrology

2.2. Experimental developments of the VM

2.2.1. Application of the VM in digibridges

2.2.1.1. Operation and analysis

(HP, Agilent, TeGam, IetLab, Wine Kerr, etc).

Figure 5. Structure of the digibridge with variational calibration.

of widely used digibridges and in accurate transformer bridges.

$$
\delta \Omega \cong \Upsilon\_c \mathbb{Z}\_0 / \mathbb{K} \tag{16}
$$

Here, the values Yc and K are the disturbing factors. The quotient of the Yc and K can be considered as the sole source of the uncertainty. Let us provide the multiplicative variation of the gain K of the amplifier A. To vary K on ratio α1, the divider Dv with transfer coefficient 1 or α<sup>1</sup> (Figure 5) is used. After this variation, MC measures the additional voltage U0v.

The system of three equations describes the measurements of the voltages Ux, Uo and Uov.

$$\mathbf{U}\_{\mathbf{x}} = \mathbf{I}\_{\mathbf{x}} \mathbf{Z}\_{\mathbf{x}}; \quad \mathbf{U}\_{0} (\mathbf{1} - \mathbf{Y}\_{c} \mathbf{Z}\_{0}/\mathbf{K}) = \mathbf{I}\_{\mathbf{x}} \mathbf{Z}\_{0}; \quad \mathbf{U}\_{0} (\mathbf{1} - \mathbf{Y}\_{c} \mathbf{Z}\_{0}/\mathbf{K}) = \mathbf{I}\_{\mathbf{x}} \mathbf{Z}\_{0} \tag{17}$$

Solution of this system gets the following formula (18):

$$\mathbf{Z}\_{\mathbf{x}}/\mathbf{Z}\_{0} = \mathbf{U}\_{\mathbf{x}}[1 - \delta \mathbf{U} \cdot \boldsymbol{\alpha}\_{1}/(1 - \boldsymbol{\alpha}\_{1})]/\mathbf{U}\_{0} \tag{18}$$

where δU ¼ 1 � U0=U0<sup>v</sup>

Analysis of the formula (18) shows that the uncertainty of the variation calibration has minimal if α1=0.5. Then:

$$\mathbf{Z}\_{\mathbf{x}}/\mathbf{Z}\_{0} = \mathsf{U}\_{\mathbf{x}}(\mathbf{1} - \delta \mathsf{U})/\mathsf{U}\_{0} \tag{19}$$

Formula (19) shows that the ratio Zx/Z0 does not depend on the quotient of the Yc and K.

But here increases component of the uncertainty, caused by the increased number of measurements. VV measures quadrature components a and b of three voltages: Ux, Uo and U0v. Let us suppose that effective input noise of the VV in all these measurement has the same value Δ and the results of measurement are not correlated. In this case, the following formulas are justified:

$$\mathbf{U}\_{\mathbf{x}} = (\mathbf{a}\_{\mathbf{x}} + \boldsymbol{\Delta}) + \mathbf{j}(\mathbf{b}\_{\mathbf{x}} + \boldsymbol{\Delta});\\\mathbf{U}\_{0} = (\mathbf{a}\_{0} + \boldsymbol{\Delta}) + \mathbf{j}(\mathbf{b}\_{0} + \boldsymbol{\Delta});\\\mathbf{U}\_{0\mathbf{v}} = (\mathbf{a}\_{0\mathbf{v}} + \boldsymbol{\Delta}) + \mathbf{j}(\mathbf{b}\_{\mathbf{x}} + \boldsymbol{\Delta})\tag{20}$$

Let us substitute formula (20) in (14) and (19). It gets the following formulas for two cases: Without variational calibration:

$$
\delta\_m \approx \sqrt{2}\delta\_n \text{ and } \Delta\_d \approx \sqrt{2}\delta\_n \tag{21}
$$

The described approach could be used for the accurate calibration of any amplifier with positive or negative

Variational Calibration

39

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

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

.

–1014.

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,

gain, followers, gyrators, and so on. It could be used for calibration of any control system as well.

3. Experimental results

standard. Its short specification is as follows.

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

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

Inner standard instability (24 hours, ppm) 2.

MNS1200 appearance is shown in Figure 6.

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

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

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

Main uncertainty (ppm) 10.

Sensitivity (ppm) 0.5

Figure 6. Digibridge MNS1200.

Weight (kg) 4

MNS1200 operates in frequency range of DC to 1 MHz.

.

–1.0.

With variational calibration:

$$
\Delta \delta\_m \approx \sqrt{5} \delta\_n \text{ and } \Delta\_a \approx \sqrt{2} \delta\_n \tag{22}
$$

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

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

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 voltages Ux, U<sup>0</sup> and U<sup>0</sup> 0, U<sup>00</sup> <sup>0</sup> after multiplicative variation of the gain K and additive variation of the admittance Yc.

System of three equations describes these four measurements:

$$\begin{aligned} \text{LI}\_{\text{x}}/\text{LI}\_{0} &= \text{Z}\_{\text{x}}/\text{Z}\_{0}[1 + \text{Y}\_{\text{c}}\text{Z}\_{0}/(1 + \text{K})] \\ \text{LI}\_{\text{x}}/\text{LI}\_{0} &= \text{Z}\_{\text{x}}/\text{Z}\_{0}[1 + \text{Y}\_{\text{c}}\text{Z}\_{0}/(1 + \alpha\_{1}\text{K})] \\ \text{LI}\_{\text{x}}/\text{LI}\_{0} &= \text{Z}\_{\text{x}}/\text{Z}\_{0}[1 + (\text{Y}\_{\text{c}} + \text{Y}\_{\text{v}})\text{Z}\_{0}/(1 + \text{K})] \end{aligned} \tag{23}$$

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

$$\begin{aligned} \mathbf{Y}\_{\mathcal{C}} \mathbf{Z}\_{0} &= \left(\mathbf{A}^{'} - \mathbf{1}\right) (\alpha\_{1} \mathbf{K} + \mathbf{1}) (\mathbf{K} + \mathbf{1}) / \mathbf{K} (1 - \alpha\_{1}) \\ \mathbf{a} \mathbf{K}^{2} + b \mathbf{K} + \mathbf{c} &= \mathbf{0} \end{aligned} \tag{24}$$

here: <sup>a</sup> ¼ ½ð<sup>1</sup> <sup>þ</sup> <sup>α</sup>1Þ � <sup>α</sup>1ðA<sup>0</sup> � <sup>1</sup>Þ�ðA<sup>00</sup> � <sup>1</sup>Þ, b ¼ ðYvZo <sup>þ</sup> <sup>A</sup><sup>0</sup> ÞA00 � <sup>A</sup><sup>0</sup> , c ¼ ðA<sup>0</sup> � <sup>1</sup>ÞðA<sup>00</sup> � 1Þ, <sup>A</sup><sup>0</sup> <sup>¼</sup> <sup>U</sup><sup>0</sup> 0=U0, A<sup>00</sup> <sup>¼</sup> <sup>U</sup><sup>0</sup> <sup>0</sup>=U<sup>0</sup>

Solution of the Eqs. (24) and substitution of these results in (15) gets the accurate results of measurement which absolutely does not depend on the values Yc and K.

The described approach could be used for the accurate calibration of any amplifier with positive or negative gain, followers, gyrators, and so on. It could be used for calibration of any control system as well.
