Electrical Metrology

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QUIRDER ELECTRICAL METROLOGY THEORETICAL BASICS 1-

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F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g

1 . T h e o r e t i c a l B a s i c s

Preliminary it was already noted that a measurement is always defective. Therefore, it should be generally examined first, how to describe measuring errors, how they must be considered in the measuring result and how they can be explained by the type of measurement. In accordance with The German Institute for Standardization (DIN 1319-1) you do not speak any more from a measuring error but from a measuring derivation. Nevertheless let us talk about errors.

1.1 Measuring error

The absolute error fa is defined as the difference between the actual value xA which

bears the error, and the desired value xW which should be genuine.

W A

a x x

f = − (1.1.1)

The absolute error is a part of the measured value and therefore must be indicated as a measure with its dimension unit.

Also a relative error fr may be defined which represents an absolute error related to

the desired value. Therefore it is dimensionless.

W W A r x x x f = − (1.1.2)

While electrical engineers mostly use an indication error. In doing so the absolute error is related to the final value in the measuring range xMB and given in percent.

This error is called the indication error fAr.

% 100 ⋅ − = MB W A Ar x x x f (1.1.3)

Wherefrom the errors result, it was not mentioned yet. Following DIN 1319 there are two different kinds of errors, namely the systematic and the random ones.

Systematic errors are a type of errors whose sign and value either are already known or, however, are to be calculated anyhow, therefore a correction of the measured value is possible. Beneath others you will find most of the errors resulting of environmental effects, as for example a subject to the ambient air temperature.

Random or stochastically errors permit no statement about their sign and their value. Beneath this you find, e.g., errors which appear while reading the display. Therefore, they can be estimated only with the methods of the theory of probabilities.

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QUIRDER ELECTRICAL METROLOGY

THEORETICAL BASICS

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2 1.1.1 Average Value, Standard Deviation, Variance

A statistical error will be determined by calculating the average value x of n measurements under the same conditions and with the same equipment

(

x x xn

)

n x= ⋅ 1+ 2++ 1 (1.1.4)



= ⋅ = n i i x n x 1 1 (1.1.5)

furthermore the standard deviation s as the root mean square deviation

( ) (

)

(

)

[

2 2

]

2 2 1 1 1 x x x x x x n s ⋅ − + − + + _n− − ± =  (1.1.6)

( )



= − ⋅ − ± = n i i x x n s 1 2 1 1 (1.1.7)

or with an algorithm which is used by your pocket calculator,

              ⋅ − ⋅ − ± =



= = 2 1 1 2 1 1 1 n i i n i i x n x n s (1.1.8)

The transformation of Gl. 1.1.7 to the Gl. 1.1.8 you find in the appendix.

For a sufficiently large number n goes s to the limit σ , the standard deviation of the population. This also applies if for an actually insufficiently large number of values all values are included in the calculation.

The square of this limit is called variance of the population

( )

_    − ⋅ − = =



= ∞ → ∞ → n i i n n s _n x x 1 2 2 2 1 1 lim lim σ (1.1.9)

The so calculated average value is indicated frequently as the result of the

measuring. But this may not represent the true value of the measured parameter, however. Therefore, one defines the so-called confidence limit v within which the true value with the statistic confidence S is to be expected.

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3 1.1.2 Normal Distribution

For a better understanding of the confidence limit some terms of the theory of probabilities should be explained first.

In the metrology statistical, independent and in advance unknowable errors lead to random variables as measured values which are subject to a so-called normal or Gaussian distribution. Besides, it is a continuous distribution whose probability density can be calculated with the function f(x).

( ) ( ) 2 2 2 2 2 1 _σ σ π ⋅ − − ⋅ ⋅ ⋅ = x x x e f (1.1.10)

(

)

(

−∞< <+∞

)

=ϕ x;x;σ2 x

A measured value x, a random variable, where such a density function may be applied, is called normal-distributed. The expected value x, that means the average value, and the deviation σ2 _{are the parameters of this distribution. If these both}

parameters are known, the probability density is uniquely determined.

Fig. 1.1.1: Gaussian Bell-Shaped Curve

The bell-shaped curve of the normal distribution is symmetrically to the maximum, the average value. Where the curve shows its inflexion points, there the values of the standard deviation lie. Then the curve progression asymptotically approaches the abscissa. Therefore the more the curve is rampant, the less the deviation σ2_{will be.}

The distribution function F(x) of the normal distribution indicates the probability that

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F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g ( )=



₋_∞

(

)

⋅ x x z x dz F ϕ ; ;σ2 (1.1.11) ( ) ( )



−∞ ⋅ − − ⋅ ⋅ ⋅ ⋅ = x x z x e dz F 2 2 2 2 2 1 _σ σ π

(

2

)

; ;xσ x Φ =

If you enlarge the upper integration limit to x=+∞, the distribution function will enclose all values and the probability will be one.

Fig. 1.1.2: The Distribution Function

( )=



(

; ; 2

)

⋅ =1 +∞

∞

− z x dz

Fx ϕ σ (1.1.12)

The distribution makes use of the shape shown in figure 1.1.2.

For easier handling very often the function will be used as a normalized or standardized normal distribution. Doing so the function values may be given in tabular form. With x=0 and σ2 =1the functions will have the following form.

(

x

) ( )

ϕ x ϕ ;0;1 = (1.1.13)

(

−∞< <+∞

)

⋅ ⋅ = e− x x 2 2 2 1 π

as density of the standardized normal distribution and

(

x

)

=Φ

( )

x Φ ;0;1 (1.1.14) dz e x z ⋅ ⋅ ⋅ =



∞ − − 2 2 2 1 π

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as the distribution function of the standardized normal distribution. Then the associated curves change to the shape as shown in figure 1.1.3.

The function values are present in the relevant literature as tables mostly for x≥0. The existing symmetry causes the following:

( ) ( )

−x =ϕ +x

ϕ (1.1.15)

and

( )

−x = −Φ

( )

+x

Φ 1 (1.1.16)

Fig. 1.1.3: Curve of the Density- and the Distribution-Function of the Standardized Normal Distribution

If the table is calculated with the lower integration limit x=0, the distribution function follows as:

( )

+x = +Φ

( )

+x Φ 0 2 1 (1.1.17) and

( )

−x = −Φ

( )

+x Φ 0 2 1 (1.1.18)

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6 1.1.3 Confidence Region

The confidence region marks out the limits, within those and with the specification of the statistical confidence a measured value may belong to a certain normal distribution following the distribution function Φ

( )

x . To define the confidence region you may use tabular values as they were calculated for the t test, also student's test called, (s. Appendix). Then the confidence limit v becomes:

s n t

v= ⋅ (1.1.19)

The genuine value of the measurement lies with a statistical confidence S given in percent within the confidence region

v x x v x− ≤ ≤ + (1.1.20)

1.1.4 Uncertainty of Measurement

By adding the systematic error xF – estimated or calculated -, the measuring

uncertainty u will be determined:

      + ⋅ ± = s xF n t u (1.1.21)

Therefore after the correction of the registered systematic errors the measuring result follows from the average value and the measuring uncertainty u:

u x

x_e = ± (1.1.22)

1.2. Error Propagation

As far as measured values are linked together by a mathematic algorithm their errors have to be calculated too. Again you differentiate between the systematic and the random errors.

The measuring values with their systematic mistakes are examined, as follows: a) Addition:

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F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g b) Subtraction:

(

x1±x1F

) (

− x2 ±x2F

)

=x1−x2±

(

x1F +x2F

)

(1.2.2) c) Multiplication:

(

) (

)

_      ± ⋅ ⋅       ± ⋅ = ± ⋅ ± 2 2 2 1 1 1 2 2 1 1 1 1 x x x x x x x x x x F F F F (1.2.3)

with x1F <<x1 and x2F <<x2 it is valid:

(

) (

)

            + ± ⋅ ⋅ ≈ ± ⋅ ± 2 2 1 1 2 1 2 2 1 1 1 x x x x x x x x x x F F F F (1.2.4) d) Division:       ± ⋅       ± ⋅ = ± ± 2 2 2 1 1 1 2 2 1 1 1 1 x x x x x x x x x x F F F F (1.2.5)

If you consider, e.g., only the positive sign and further on you extend the denominator, so this applies:

      − ⋅       +       − ⋅       + ⋅ =       + ⋅       + ⋅ 2 2 2 2 2 2 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 x x x x x x x x x x x x x x x x F F F F F F (1.2.6) 2 2 2 2 2 1 1 2 2 1 1 2 1 1 1       − ⋅ − − + ⋅ = x x x x x x x x x x x x F F F F F (1.2.7) Then with 1 2 2 1 1 ⋅ << x x x x_F _F and 1 2 2 2 <<       x x _F

it applies to both signs of the relative errors:

            + ± ⋅ ≈ ± ± 2 2 1 1 2 1 2 2 1 1 1 x x x x x x x x x x _F _F F F (1.2.8)

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e) Raising to a Higher Power:

(

)

      ⋅ ± ⋅ = ± 1 1 1 1 1 1 x x m x x x m m F F (1.2.9)

Generally it applies to the linkage of systematic errors:

If you add or subtract measured values, you calculate the sum of the absolute errors. If you multiply or divide measured values, you calculate the sum of the relative errors.

If the measuring result is calculated by statistically processed measuring values, you get

(

x x xm

)

f

y= 1, ,, (1.2.10)

The m different values will be measured n-times, so you may calculate m standard deviations s1, s2 ... sm. Now you calculate the deviation dyi of a value yi due to the

small changes dx1, dx2..., dxm using the entire differential.

mi m i i i dx x y dx x y dx x y dy ⋅ ∂ ∂ + + ⋅ ∂ ∂ + ⋅ ∂ ∂ = 2  2 1 1 (1.2.11)

If you square this equation and sort it according to the purely square and mixed terms, you get:

( )



= ≠ =         ⋅ ⋅ ∂ ∂ ⋅ ∂ ∂ ⋅ +         ⋅ ∂ ∂ = m k j ki ji k j m j ji j i dx dx x y x y dx x y dy 1 2 1 2 2 (1.2.12)

Because of the statistically distributed signs in the second term, this can be neglected. For the sum of all square deviations you get:

( )



= =               ⋅ ∂ ∂ + +       ⋅ ∂ ∂ +       ⋅ ∂ ∂ = n i mi m i i n i i dx x y dx x y dx x y dy 1 2 2 2 2 2 1 1 1 2  (1.2.13)

And therefore the standard deviation will become:

2 2 2 2 2 2 2 1 2 1 m m y s x y s x y s x y s _ ⋅      ∂ ∂ + + ⋅       ∂ ∂ + ⋅       ∂ ∂ ± =  (1.2.14)

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THEORETICAL BASICS

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9 1.3. Dynamic Measuring Errors

Beside the static measuring errors which will be considered if the whole system to be measured is in steady state condition, there are additional errors which result from the measurement of time variant values, and then you speak of dynamic errors. Now the dependence between the measured values xa and the result xe can be described

with a differential equation.

( ) ( ) ( ) ++ ⋅ + = ⋅ + ⋅ + ⋅ ₋ − ₋ a e a n a n n n a n n x B x t d x d T t d x d T t d x d T 1 1 0 1 1 (1.3.1)

The dynamic behaviour of measuring systems is described like time dependent systems by their character functions and values in the time and frequency domain. Dynamic measuring errors are not treated for their complexity.

1.4. Reporting Measured Values

Measuring you may follow three different aims:

* Investigation of the dependence between two or several measured values as a function of an independently changeable parameter (e.g., current/voltage characteristic curve of an electric light bulb).

* Investigation of continuous stochastic measured values (e.g., voltage with overlaid noise voltage measured continuously).

* Investigation of stochastic measured values by single measurements (e.g., repeated measurement of a resistance).

1.4.1 Pairs of Measured Values

A typical report of pairs of measured values, which are a function of an independent changeable parameter, is a chart. The uncertainty u may be drawn as a vertical dash, so that you can draw a curves adaptation within the framework of the measuring uncertainty with estimation by sight. If you do it with a computer, you can arrange the curves adaptation mathematically by a so-called curves-fit, i.e. you calculate the coefficients of the equation so you get the minimum square deviation to all measured pairs.

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Fig. 1.4.1: Presentation of Measured Value Pairs

For n measured pairs of values xi and yi you may get a linear function x

a a

y= ₀ + ₁⋅ (1.4.1)

where the sum of the square deviation in each measuring pair becomes a minimum. This kind of the curves adaptation is called linear regression and the function is a regression straight line. The coefficients of the regression straight lines are:

(

)

( )



= = = = =       − ⋅ ⋅ − ⋅ ⋅ = n i n i i i n i n i i n i i i i x x n y x y x n a 1 2 1 2 1 1 1 1 (1.4.2)       ⋅ − ⋅ =



= = n i i n i i a x y n a 1 1 1 0 1 (1.4.3)

A further analysis determines the correlation coefficient r,

(

)

( )

              − ⋅ ⋅               − ⋅ ⋅ − ⋅ ⋅ =



= = = = = = = n i n i i i n i n i i i n i n i i n i i i i y y n x x n y x y x n r 1 2 1 2 1 2 1 2 1 1 1 (1.4.4)

It indicates the closeness of agreement between the linear function and every value pair, with r=±1 there an accurately linear relationship exists. If no relationship exists, r becomes zero, i.e. with the correlation coefficient you can check the result of the curves-fit on its utility. Logarithmic, exponential or power functions can also be examined by taking the logarithm from xi and/or yi.

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11 1.4.2 Continuous Stochastic Measured Values

The evaluation and graphic presentation of continuous, stochastic measuring parameters result in an average value in examined section.

Fig. 1.4.2: Continuous Stochastic Measured Values mathematically seen you apply an integration:



⋅ ⋅ = T x dt T x 0 1 (1.4.5)

Besides the correlation analysis finds its application on such measured values. You distinguish two kinds of correlation.

There we have the autocorrelation, which enables us to find periodic signals in a strongly jammed signal. By shifting a sequence of the signal as far as the product of it and the measured values becomes a maximum (mathematical: convolution) (radio astronomy). ( ) ( )



⋅ ⋅ ⋅ = Φ Tu t u t₋ dt T 0 1 1 ) ( 11 1 τ τ (1.4.6) Fig. 1.4.3: Block Diagram of a Correlator (Autocorrelator: Switch Position S1)

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The other type is the cross correlation which you apply to measured values to find out the interrelation between two stochastic signals by a convolution (controlled systems). ( ) ( )



⋅ ⋅ ⋅ = Φ T u t u t₋ dt T 0 1 2 ) ( 12 1 τ τ (1.4.7) Fig. 1.4.4: Example of an Autocorrelation

1.4.3 Stochastic Measured Values

For the presentation of stochastic measured values you arrange the statistic distribution of each measured values around a desired value, a histogram, a bell curve or a curve of the cumulative frequency.

Moreover you divide the area determined by the measured values into k equal classes and assign the measured values to the classes. Class width m should be chosen smaller than 1/3 of the standard deviation s (with 9 < k < 25).

In the cumulative frequency curve the cumulative frequency values ΣH will be inserted every time on the lower border of a class. They are formed by the addition the frequency values H of the separate classes, beginning with the highest class. Moreover, the frequency values are applied mostly as per cent values. This qualified presentation gives you the chance of a straight comparison of different measurements.

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The average value is located at 50% of the cumulative frequency curve. The value of the standard deviation you find as the difference between 50% and 84,13% or between 50% and 15,87% of the cumulative frequency.

For the presentation of the cumulative frequency curve you find special paper whose ordinate is divided in such a way so that you get a straight line by a normal distribution.

Fig. 1.4.5: Cumulative Frequency Curve on Special Paper

If one applies the class frequency H of the measured values in the middle of each class, one can draw a histogram or a bell curve too. The maximum of the curve represents the average value. The difference between the average value and the 60,6% value indicates the standard deviation.

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14 1.5. Measuring Parameters and Unities

Measured values consist, how already mentioned, of the unit and the absolute measure. Therefore the units have to be defined and as far as possibly to be internationalized. The units have been named as the so-called „international unity system“ (SI) in 1960 by the 11th_{General conference for measurement and weight of}

the international meter convention. Finally in 1971 it was extended to altogether seven basic unities and two supplement unities.

Base item Formular symbol Basic unit Short symbol

length l meter m 1)

mass m kilogram kg 2)

time t second s 3)

current I ampere A 4)

temperature T kelvin K 5)

light intensity IV candela cd 6)

quantity of material n mol mol 7)

plane angle α,β,γ radiant rad 8)

solid angle ω,Ω steradiant sr 9)

1)

1 meter are 1.650.763,73 vacuum wavelengths of the radiation which corresponds to the crossing between the levels 2p10 and 5d5 of the atom krypton 86.

2)

1 kilogram is the mass of the international kilogram prototype.

3)_{1 second is the duration of 9.192.631.770 periods of the radiation which corresponds to the crossing between both hyperfine}

structure levels of the initial state of the atom caesium 133.

4)_{1 ampere is the strength of the time invariant current through two straight-line, parallel, infinitely long conductors of the}

relative permeability 1 and from negligible cross section which have the distance 1 m and between those the force caused electro-dynamically by the current for each 1 m of the double line amounts 2·10-7_{N in free space.}

5)_{1 Kelvin is the 273,16th part of the (thermodynamic) temperature of the tripe point of water.} 6)

1 candela is the light intensity which a black body of the surface 1/600.000 m2_{radiates vertically to its surface at the}

solidification temperature of the platinum and the pressure of 101.325 Pa.

7)

1 Mol is the material amount of a system of a certain compounding which exists of the same number of particles as atoms are contained in 12/1000 kg of the atom carbon 12.

[ Avogadro constant: 12 g 12_{C contain (6,02252 ± 0,00028) · 10}23_atoms]

8)_{1 Radiant is the angle which is shaped by two rays going out of the centre of a circle with the radius 1 m and including a}

segment of the circle with the length 1 m. (1 rad = 57°17 ' 44,8 ")

9)_{1 Steradiant is the solid angle which is shaped by a band of rays going out of the centre of a ball with the radius 1 m and}

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In a lot of applications these unities are either too big or, however too small. Therefore, headers are internationally defined in order to form decimal multiple and parts.

SI-header abbreviaton factor SI-header abbreviation factor exa E 1018 _deci _d ₁₀-1 peta P 1015 _centi _c ₁₀-2 tera T 1012 _milli _m ₁₀-3 giga G 109 _micro _µ ₁₀-6 mega M 106 _nano _n ₁₀-9 kilo k 103 _Pico _p ₁₀-12 hecto h 102 _femto _f ₁₀-15 deka da 101 _atto _a ₁₀-18

1.6. Principle of Indication

Referring to the principle of indication one selected measuring instruments into analogue and digital ones, today the difference is not so easy to explain. Measuring instruments constructed in analogue technology may have a digital display and those in digital technology may have an analogue scale reading (wrist watch!).

1.7. Principle of Measuring

You may measure in two ways:

One method is to set as much units against the measuring parameter till the system is balanced. The advantage of this method is that you need no calibration of the measuring system, because only the equality is stated by the measuring parameter and the measure. The units required for the balance represent the measuring result (beam balance).

The other way is to use a pointer which will be deflected in relation to the measuring parameter. The result may be read out on a scale. Moreover the scale must be

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calibrated, i.e. it must have been compared with a normal of the measuring parameter previously and scaled.

1.8. Measuring Instruments in the Measuring Circuit

If you bring a measuring instrument in a system to measure it, the system is always affected and the measuring result is falsified generally.

In the electric metrology this particularly appears measuring current and voltage simultaneously, due to the fact that the distribution of the current and the voltage in the measuring circle are disturbed. Because this is, however, a systematic error, a correction is possible.

Measuring simultaneously the current and the voltage you may distinguish between two systematic errors, i.e. the "current-correct" or the "voltage-correct" circuit of the measuring instruments; one names both variants also "voltage-error" or "current-error".

In the "current-correct" circuit the current measuring instrument – ampere meter - measures the correct current I =IA . The voltage measuring instrument – voltmeter -

measures the total voltage minus the voltage drop across the ampere meter I

R U

U = _V − _ia⋅ .

Fig. 1.8.1: "current-correct" or "voltage-error" Measurement

In the "voltage-correct" circuit the voltmeter measures the correct voltage U =UV.

The ampere meter measures the total current minus the current needed in the voltmeter iv A R U I I = − .

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Fig. 1.8.2: "voltage-correct" or "current-error" Measurement

Therefore the application of the measuring circuit is corresponding to the value of the load resistor Rx:

voltage-error method: Rx ≥100*RiA

current-error method: Rx ≤0,01*RiV

With a measuring resistor in the range of the internal resistor Ria of the ampere meter

the voltage drop across the ampere meter can not be neglected, i.e. it must be measured voltage-correct. With a measuring resistor in the range of the internal resistance Riv of the voltmeter, however, the current flow across the voltmeter can not

be neglected, i.e. it must be measured current-correct.

1.9. Limits of the Measurability

As a result of the thermal agitation of the charge carriers stochastic fluctuations are superposed to the currents and the voltages. This limits the measurability and the measuring accuracy of small parameters. In addition the mechanically indicating measuring instruments are influenced by the Brownian motion of the mobile measuring element.

For a critically damped galvanometer, e.g., with the closing resistor R_gr = k4 Ω, the time of oscillation T₀ =10s , the temperature of the measuring circuit ϑ =290 K, the

Boltzmann constants 23 1 10 38 , 1 ⋅ − ⋅ − = J K

k and the efficiency factor of the

galvanometer η=0,2 the average fluctuation squares of the current 2 i

δ and the voltage 2

u

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F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g ϑ η δ η δ ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅k⋅ R T u T R i gr gr 2 1 0 2 0 2

Now we solve the equation for iδ :

0 2 1 T R k i gr⋅ ⋅ ⋅ ⋅ = η ϑ δ And we get: pA i=0,5 δ and δu=R_gr ⋅δi=2nV This is a theoretical value which can not be reached in practice.

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19 Appendix:

Calculation to form the algorithm of the standard deviation:

( )



= − ⋅ − ± = n i i x x n s 1 2 1 1 and               ⋅ − ⋅ − ± =



= = 2 1 1 2 1 1 1 n i i n i i x n x n s

should be same, or:

( )

2 1 1 2 1 2 1       ⋅ − = −



= = = n i i n i i n i i x n x x x

First the bracket on the left side of the equation is dissolved:

( )

(

)

(

)



= = = = − ⋅ ⋅ − = + ⋅ ⋅ − = − n i i n i i n i i i n i i x x x x x x x x x x 1 2 1 2 1 2 2 1 2 2 2

Now it is to be proved that this applies:

(

)

2 1 1 2 1 2       ⋅ = − ⋅ ⋅



= = n i i n i i x n x x x

If one forms the sum of all

(

2⋅x_i⋅x−x2

)

, the term x2 exists n-times in this sum, i.e. it is follows:

(

)

(

)

2 1 1 2 2 2 x x x x x n x n i i n i i⋅ − = ⋅ ⋅ − ⋅ ⋅



= =

If you now use the equation for the average value (see Gl.1.5), so you find:

(

)

. . . 1 2 1 1 1 2 2 2 1 2 1 1 1 2 1 1 1 2 1 d e q x n x x x n x n n x n x x n x x n i i n i i n i i n i i n i n i i n i i i n i i       ⋅ =               − ⋅ ⋅ ⋅ =       ⋅ ⋅ − ⋅ ⋅ ⋅ = ⋅ − ⋅ ⋅



= = = = = = = =

(20)

THEORETICAL BASICS

1-

20 Table 1: Distribution Function:

(

x

)

( )

x xe dz z ⋅ ⋅ ⋅ = Φ = Φ



∞ − − 2 2 2 1 1 ; 0 ; π x Φ

( )

x x Φ

( )

x x Φ

( )

x x Φ

( )

x 0,00 0,50000 0,05 0,51994 2,00 0,97725 2,05 0,97982 0,10 0,53983 0,15 0,55962 2,10 0,98214 2,15 0,98422 0,20 0,57926 0,25 0,59871 2,20 0,98610 2,25 0,98778 0,30 0,61791 0,35 0,63683 2,30 0,98928 2,35 0,99061 0,40 0,65542 0,45 0,67364 2,40 0,99180 2,45 0,99286 0,50 0,69146 0,55 0,70884 2,50 0,99379 2,55 0,99461 0,60 0,72575 0,65 0,74215 2,60 0,99534 2,65 0,99598 0,70 0,75804 0,75 0,77337 2,70 0,99653 2,75 0,99702 0,80 0,78814 0,85 0,80234 2,80 0,99744 2,85 0,99781 0,90 0,81594 0,95 0,82894 2,90 0,99813 2,95 0,99841 1,00 0,84134 1,05 0,85314 3,00 0,99865 1,10 0,86433 1,15 0,87493 3,10 0,99903 1,20 0,88493 1,25 0,89435 3,20 0,99931 1,30 0,90320 1,35 0,91149 3,30 0,99952 1,40 0,91924 1,45 0,92647 3,40 0,99966 1,50 0,93319 1,55 0,93943 3,50 0,99977 1,60 0,94520 1,65 0,95053 3,60 0,99984 1,70 0,95543 1,75 0,95994 3,70 0,99989 1,80 0,96407 1,85 0,96784 3,80 0,99993 1,90 0,97128 1,95 0,97441 3,90 0,99995

(21)

21 Table 2: t – Distribution

α - values in per cent

Bilateral problem m\α 50 25 10 5 2 1 0,2 0,1 1 1,00 2,41 6,31 12,7 31,82 63,7 318,3 637,0 2 .816 1,60 2,92 4,30 6,97 9,92 22,33 31,6 3 .765 1,42 2,35 3,18 4,54 5,84 10,22 12,9 4 .741 1,34 2,13 2,78 3,75 4,60 7,17 8,61 5 .727 1,30 2,01 2,57 3,37 4,03 5,89 6,86 6 .718 1,27 1,94 2,45 3,14 3,71 5,21 5,96 7 .711 1,25 1,89 2,36 3,00 3,50 4,79 5,40 8 .706 1,24 1,86 2,31 2,90 3,36 4,50 5,04 9 .703 1,23 1,83 2,26 2,82 3,25 4,30 4,78 10 .700 1,22 1,81 2,23 2,76 3,17 4,14 4,59 11 .697 1,21 1,80 2,20 2,72 3,11 4,03 4,44 12 .695 1,21 1,78 2,18 2,68 3,05 3,93 4,32 13 .694 1,20 1,77 2,16 2,65 3,01 3,85 4,22 14 .692 1,20 1,76 2,14 2,62 2,98 3,79 4,14 15 .691 1,20 1,75 2,13 2,60 2,95 3,73 4,07 16 .690 1,19 1,75 2,12 2,58 2,92 3,69 4,01 17 .689 1,19 1,74 2,11 2,57 2,90 3,65 3,96 18 .688 1,19 1,73 2,10 2,55 2,88 3,61 3,92 19 .688 1,19 1,73 2,09 2,54 2,86 3,58 3,88 20 .687 1,18 1,73 2,09 2,53 2,85 3,55 3,85 25 .684 1,18 1,71 2,06 2,49 2,79 3,45 3,72 30 .683 1,17 1,70 2,04 2,46 2,75 3,39 3,65 40 .681 1,17 1,68 2,02 2,42 2,70 3,31 3,55 60 .679 1,16 1,67 2,00 2,39 2,66 3,23 3,46 120 .677 1,16 1,66 1,98 2,36 2,62 3,17 3,37 ∞ .674 1,15 1,64 1,96 2,33 2,58 3,09 3,29 m_/α ₂₅ _12,5 ₅ _2,5 ₁ _0,5 _0,1 _0,05 Unilateral problem

(22)

22 Example 1.1

A resistor measurement under the same conditions shows the following 20 values of Ri in ohm:

680 684 684 672 664 692 685 681 676 668 693 681 676 696 689 672 688 682 677 680

Please calculate the average value, the standard deviation, the confidence region for a statistic safety S = 95% and the measuring result. The estimated systematic error amounts to 0,1%.

Example 1.2

the measuring result is the function of the measuring parameters: A) y=a⋅x1+b⋅x2−c⋅x3 B) 3 2 1 x x x y= ⋅

Calculate in each case formally the standard deviation.

Example 1.4.1

Between the current I and the voltage U of an electric light bulb exists a dependence which can be described with the power function:

a U U I k I _      ⋅ ⋅ = 0 0

With the measured values U and I

U in V 30 50 70 90 110 130 150 170 190 210 230 I in mA 340 421 496 563 623 677 726 772 817 860 903

(23)

23 Example 1.4.2

A resistor measurement under the same conditions shows the following 20 values of Ri in ohm:

680 684 684 672 664 692 685 681 676 668 693 681 676 696 689 672 688 682 677 680

(24)

ACCURACY OF MEASURING INSTRUMENTS

2 ‐

1

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n ‐ U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g

2 . A c c u r a c y o f M e a s u r i n g I n s t r u m e n t s

The indication error should not exceed given limits under nominal and test condi‐ tions within the measuring range. Analog and digital measuring instruments define the indication error in different ways. Nevertheless with this specification you will calculate the value of the rated systematic error of your measurement.

2 . 1 A c c u r a c y o f a n A n a l o g M e a s u r i n g I n s t r u m e n t

The graduation takes place in accuracy classes with the maximum error related to the full scale value of the measuring range. It is a percent specification. You differentiate between precision measuring instruments: Class: 0,05; 0,1; 0,2 or 0,5 and operational measuring instruments: Class: 1,0; 1,5; 2,5 or 5,0 with the corresponding percental indication errors. Measuring instruments without a mechanical zero or with a very non‐linear scale de‐ fine the class as the percental error of the usable scale length. You find the accuracy class with other specifications on the dial of your instrument. The indication error fAR is calculated by A MB AR x x k f   (2/1/1) with k   class in percent, xMB  full scale range and xA  measured value This is the rated systematic error of your measurement.

(25)

ACCURACY OF MEASURING INSTRUMENTS

2 ‐

2

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n ‐ U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g A more individual error is the not vertically directed look at the pointer, parallax named. To prevent or at least minimize this error the scale of precision instruments is built with a lamellar mirror next to the scale marks. As well, one holds the distance between pointer and scale as low as possible. This is a typical random error. Fig. 2.1.1: Parallax

2 . 2 A c c u r a c y o f a D i g i t a l M e a s u r i n g I n s t r u m e n t

With digital measuring instruments you find no specification into different accuracy classes. There is only given a limit of indication errors %) 100 ) % 100 / % (      a x blsd fAR A (2.2.1) with a indication error in percent, xA  measured value b number of digits and lsd least significant digit

(26)

QUIRDER ELECTRICAL METROLOGY SUPPORTING UNITS 3-

1 3 . S u p p o r t i n g U n i t s

To extend the application of measuring instruments, very often supporting units are added to the basic instruments.

3.1 Direct and Alternating Magnitudes

Especially for the measurement of alternating magnitudes you need various types of supporting units. Therefore, the definition of direct and alternating magnitudes fol-lows now:

3.1.1 Definition of Direct Magnitudes

Direct magnitude says that during the measuring interval a constant value exists, i.e. the magnitude during the measuring time does not change:

( )t f U ≠ (3.1.1) or ( )t f I ≠ (3.1.2)

The ratio of voltage and current is called the ohmic resistance R

I U

R= (3.1.3)

and its reciprocal value is the conductance G U

I R

G= 1 = (3.1.4)

The product of voltage and current results in the power P, the active power

I U

P= ⋅ (3.1.5)

3.1.2 Definition of Alternating Magnitudes

Alternating magnitude says that during the measuring interval no constant value must exist. The value is dependent on time. For sine wave-form alternating values you usually write:

( )t u

(

t u

)

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2

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g or ( )t i

(

t i

)

i = sinˆ⋅ ω +ϕ (3.1.7)

With û and î as the peak values of the voltage and the current, ϕu and ϕi the phase

displacements of the voltage and the current related to a reference phase angle and ω the angular frequency

(

ω=2⋅π⋅ f =2⋅π T

)

with T the time of oscillation.

The ratio of the voltage u (t) and the current i(t) is called the complex resistance1) or the

impedance Z. The reciprocal value is the complex conductance or the admittance Y. If you multiply a voltage by a current, you will get the instantaneous power of an al-ternating voltage circuit with a current flow i (t) and in the voltage u (t).

( )t u( ) ( )t it

p = ⋅ (3.1.8)

(

t u

)

i

(

t i

)

u⋅ ω +ϕ ⋅ ⋅ ω +ϕ

= ˆ sin ˆ sin

with the addition theorem:

(

)

(

)

[

α β α β

]

β α⋅ = ⋅ cos − −cos + 2 1 sin sin follows: ( )t

[

(

u i

)

(

t u i

)

]

i u p = ⋅ ⋅ cos ϕ −ϕ −cos 2⋅ω +ϕ +ϕ 2 ˆ ˆ (3.1.9)

Due to the fact that the reference phase angle can be freely chosen, you may select the phase angle ϕi as the reference phase angle, so the equation 3.1.9 simplifies to

( )t

[

u

(

t u

)

]

i u p = ⋅ ⋅ cosϕ −cos 2⋅ω +ϕ 2 ˆ ˆ (3.1.10)

If you use the notation of the root mean square value of a sine wave-form current

I I i eff = = 2 ˆ or u =U_eff =U 2 ˆ you find 1)_Definition:

Complex Resistance: Impedance (Z) = Resistance (R) + j Reactance (X) Complexer Conductance: Admittance (Y) = Conductance (G) + j Suszeptance (B)

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3

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g ( ) =U⋅I⋅

[

ϕ−

(

⋅ωt+ϕ

)

]

p_t cos cos 2 (3.1.11)

If you look at the power within a period of the oscillation frequency, so you find:

( )



⋅ ⋅ = T t dt p T P 0 1 (3.1.12)

(

)

[

]

(

)

(

)

[

(

)

(

)

]

_    _⋅ _⋅ ₊ ₋ ₊ ⋅ − ⋅ − ⋅ ⋅ ⋅ =     _⋅ _⋅ ₊ ⋅ − ⋅ ⋅ ⋅ ⋅ = ⋅ + ⋅ − ⋅ ⋅ =



ϕ ϕ ω ω ϕ ϕ ω ω ϕ ϕ ω ϕ 0 sin 2 sin 2 1 cos 0 1 2 sin 2 1 cos 1 2 cos cos 1 0 0 T T I U T t t I U T t d t I U T T T

finally you get:

ϕ cos ⋅ ⋅ =U I P (3.1.13)

The integration over one period shows that the average power value of the alterna-ting current corresponds to that of a direct current and that it is dependent on the phase angle between current and voltage.

The second term of the integral supplies no contribution to the result, because it is an alternating value with the amplitude uˆ i⋅ˆ 2 which oscillates with the doubled fre-quency around the zero baseline. Fig. 3.1.1 shows the run of the curve for three dif-ferent phase angles.

Fig. 3.1.1: alternating current power vs. ϕ [a) ϕ = 0; b) ϕ = 60 °; c) ϕ = 90 °]

This part of the alternating current power which spends no contribution to the active power is call the reactive power. If you choose the complex manner of writing, how-ever, it is obvious to disassemble the alternating current power in a real part and an imaginary part of the complex numbers plane.

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4 { }

S m I U Q= ⋅ ⋅sinϕ =ℑ (3.1.14)

and together you get the amount of the apparent power:

S Q P I U S = ⋅ = 2+ 2 = (3.1.15)

Writing with exponential terms you must remember the definition of the phase angle namely ϕu −ϕi =ϕ . So that the product of voltage and current must be build with the

current as a conjugate complex value.

Apparent power written in complex manner:

* I U S= ⋅ (3.1.16) with _U =_U⋅_ej(ω⋅t+ϕu) and _I =_I⋅_ej(ω⋅t+ϕi) or * j( t _i) e I I = ⋅ − ω⋅ +ϕ you get: ) ( ) ( t _u j t _i j e I e U S = ⋅ ω⋅ +ϕ ⋅ ⋅ − ω⋅ +ϕ (3.1.17) ( )

(

ϕ ϕ

)

ω ϕ ϕ sin cos j I U e I U e I U j j _u _i + ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = −

3.1.3 Multiphase System

If several coils which are dedicated to each other on a common shaft, turn in a mag-netic field, a voltage is induced in every coil which corresponds to the momentary position in the field. Also one of the voltages represents the spatial allocation related to the other induced voltages by the phase angles to these. If the coils of such a mul-tiphase system are identical and, moreover, are evenly distributed against each other, this system is called a symmetric multiphase system with a phase displacement

m

π

⋅

2 according to m coils. In a homogeneous magnetic field the instantaneous value of the voltage induced in the kth_{coil amounts to:}

( ) =u⋅  +

(

k−

)

⋅ _m⋅ + t

u_k_t ˆ cos ϕ_u 1 2 π ω (3.1.18)

With the peak value û of the voltage in each coil and ϕu the zero phase angle of the

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SUPPORTING UNITS

3-

5

Fig. 3.1.2: Three-Phase System

Limiting the implementation to a three-phase system the voltages can be formulated in accordance with Fig. 3.1.2:

( )t u

(

t u

)

u₁ = cosˆ⋅ ω +ϕ (3.1.19) ( ) = ˆ⋅cos_ + +1⋅2₃⋅ _ 2 π ϕ ω u t u t u (3.1.20) ( ) = ˆ⋅cos_ + +2⋅2₃⋅ _ 3 π ϕ ω u t u t u (3.1.21)       ₊ ₋ ⋅ ⋅ = 3 2 cos ˆ ωt ϕ_u π u

In the practice the coils will be linked, there are two basic kinds of wiring. On the one hand the phases will be connected as a delta with the sum of the voltages ΣU = 0 and on the other as a star with the sum of the currents ΣI = 0. Therefore a distinction is drawn between three-wire and four-wire systems.

For the power measurement in a three-wire system a few characteristics are to be considered.

Within a four-wire system, a star circuit with neutral line, you can measure the active power in every phase and sum it up.

3 2 1 P P P Pges = + + (3.1.22) 3 3 3 2 2 2 1 1

1⋅ ⋅cosϕ + ⋅ ⋅cosϕ + ⋅ ⋅cosϕ

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6

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g Fig. 3.1.3: Four-Wire System

If you are sure that the load is symmetrically, you may reduce the number of meas-urement to the active power measmeas-urement in one phase. Then the measured value is multiplied by three. with U₁=U₂ =U₃ =U I I I I1= 2 = 3 = and ϕ1 =ϕ2 =ϕ3 =ϕ

follows out of equation 3.1.22

3 3 3 2 2 2 1 1

1⋅ ⋅cosϕ + ⋅ ⋅cosϕ + ⋅ ⋅cosϕ

=U I U I U I

Pges (3.1.23)

=3⋅U⋅I⋅cosϕ

With a symmetrically loaded three-wire system which is based either on a star circuit without a neutral line or on a delta circuit, you may simulate an artificial neutral point by resistors as a voltage reference point for the three phases and measure the active power with only one wattmeter.

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7

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g Fig. 3.1.5: Aron-Circuit

If symmetry is not guaranteed, you apply a so-called Aron measuring circuit within a three-wire system. In such a system the sum of the phase currents must be zero, therefore you may determine the active power by two watt meters. The voltage is measured in each case against the phase in which no wattmeter is inserted. Therefore three measurement setups are possible:

32 12 P P Pges = + (3.1.24) 23 13 P P P_ges = + (3.1.25) 31 21 P P P_ges = + (3.1.26)

It should be mentioned that while the watt meters are installed in the same way one of the partial powers can become negative. Nevertheless the partial powers must be added with their correct signs.

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SUPPORTING UNITS

3-

8

If you consider the measuring circuit under the aspect of a symmetric load, P12 and

P32 show the partial powers:

      ₊ ⋅ ⋅ = π ϕ 6 cos 1 12 12 U I P (3.1.27)      ₋ ₊ ⋅ ⋅ = π ϕ 6 cos 3 32 32 U I P (3.1.28)

with U₁₂ =U₃₂ =U₁⋅ 3=U and I₁ =I₃ =I follows:

                  − − +       ₊ ⋅ ⋅ = π ϕ π ϕ 6 cos 6 cos I U P_ges (3.1.29)

With the symmetry of the cosine       − =             − − π ϕ π ϕ 6 cos 6 cos

And with the addition theorem

2 cos 2 cos 2 cos cosα+ β = ⋅ α+β ⋅ α−β with α =π +ϕ 6 and ϕ π β = −

6 and a symmetric load the active power calculates to:

ϕ ϕ π cos 866 , 0 2 cos 6 cos 2 ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = I U I U P_ges ϕ cos 3⋅ ⋅ ⋅ = U I (3.1.30)

In this case the reactive power may be calculated by the two partial power measure-ments too.

Now this formula applies:

                  − − −       ₊ ⋅ ⋅ = − π ϕ π ϕ 6 cos 6 cos 32 12 P U I P (3.1.31)

Again under the aspect of the cosine symmetry and with the addition theorem 2 sin 2 sin 2 cos cosα− β =− ⋅ α+β ⋅ α−β follows:

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9

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g ϕ π _sin 6 sin 2 32 12−P =−U⋅I⋅ ⋅ ⋅ P (3.1.32) ϕ sin 5 , 0 2 12 32−P =U⋅I⋅ ⋅ ⋅ P (3.1.33) 3 Q =

Or for the reactive power with symmetric load

(

32 12

)

3 P P

Q= ⋅ − (3.1.34)

For this special case you may also calculate the phase angle:

(

)

32 12 12 32 3 tan P P P P P Q + − ⋅ = = ϕ (3.1.35) 3.2 Rec

t

ifiers

Measuring circuits with rectifying diodes are called according to their way of opera-tion. You find half-wave rectifier and bridge rectifier to measure the mean value, the root mean square value and the peak value.

The most common electric rectifier circuits are: Half-wave rectifier

Only that part of the measuring magnitude, which lies in the pass band of the electric rectifier element (diode), is included in the result of the measurement.

Warning: Used only for the measurement of voltages, because every time one half of

the current is blocked.

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10

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g Graetz circuit

A bridge rectifier consists of four rectifier elements. It has the disadvantage of the se-ries connection of two rectifier elements with the DC-measuring circuit by which the sensitivity is decreased and the temperature dependence is raised.

Fig. 3.2.2: Graetz Circuit

Modified Graetz Circuit (Pfannenmüller)

You avoid the disadvantages of the Graetz circuit and, however, the current selectiv-ity decreases. The resistors are shunt circuit to the whole circuit and raise the power demand.

Fig. 3.2.3: Pfannenmüller Circuit

Greinacher Circuit

A circuit which doubles the peak value of an alternating voltage. It loads the capaci-tors lying parallel to the consumer in the same direction.

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SUPPORTING UNITS

3-

11

Fig. 3.2.4: Greinacher Circuit

Ring-Modulator

The ring-modulator also belongs to the rectifiers, namely it is a phase depending rec-tifier. The carrier frequency f1 closes and opens alternately the rectifiers Gl1 and Gl2 or

the diagonal rectifiers Gl3 and Gl4. The carrier voltage must be so high that the

meas-uring signal can not compensate it and change the conducting direction. To avoid that the carrier voltage appears in the output of the circuit, the transformer and the rectifiers must be well selected.

Fig. 3.2.5: Ring-Modulator

3.3 Phase Rotator

The circuits used to rotate the phase between two electric values by ϕ = 90° are summarized as bumblebee-circuits.

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12

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g Fig. 3.3.1: Bumblebee-Circuit

A disadvantage of these circuits is its frequency dependent; therefore, they only can be used for a certain frequency.

1 2 1 2 2 2 1 2 1 2 1 R L L R L R R L j I U ⋅ ⋅ − +       +       + ⋅ ⋅ = ω ω (3.3.1) with 1 2 2 2 1 R L L

R ⋅ =ω ⋅ ⋅ the real part becomes zero. Between the voltage U and the cur-rent I2 there is phase rotation of π , as demanded. 2

The phase shifter represents an interesting circuit in this field. If the voltage U2 re-mains unstressed, it can take all phase angles in the relation to the total voltage U1.

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SUPPORTING UNITS

3-

13

Under the condition

C L R R

R₁ = ₂ = = the input resistance becomes independent of the frequency, real and like R.

With the oscillating frequency

C L⋅

= 1

ω you get U1 = U2 and the phase between

both voltages becomes π . 2

3.4 Extension of the Measuring Range

Extensions of the measuring range are needed if either the current or the voltage to be measured overruns the final value of the measuring range.

For the current measurement you put a resistor (shunt) in parallel to the measuring element to divide the current flow in known ways.

Fig. 3.4.1: Extension of the Measuring Range by a Parallel Resistor

You find:

p M

ges i i

i = + (3.4.1)

With iges ≡ the total current to be measured

≡

M

i the current of the measuring element ≡

p

i the current of the parallel resistor, shunt furthermore must the condition:

p

A u

u = and

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14

F a c h h o c h s c h u l e F r a n k f u r t a m M a i n - U n i v e r s i t y o f A p p l i e d S c i e n c e s E l e c t r i c a l E n g i n e e r i n g p p iA M R i R i ⋅ = ⋅ be fulfilled.

For a voltage measurement uges a resistor Rv is assembled in series with the

measur-ing element to distribute the voltage so that above the internal resistor RiV of the

measuring element the maximal voltage may drop which is related to the permissible measuring current.

Fig. 3.4.2: Extension of the Measuring Range by a Resistor in Series

(

iV V

)

M

ges i R R

u = ⋅ + (3.4.2)

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PHENOMENA AND THEIR APPLICATION

4-

1 4 . P h e n o m e n a a n d T h e i r A p p l i c a t i o n

If you want to measure a physical parameter, you must realize which phenomenon is released by this parameter and how you can use this in a measuring device.

In the electric metrology the following parameters are measured:

current, voltage, power, resistance, capacity, inductance, phase angle between cur-rent and voltage, frequency, etc..

Now for these parameters the phenomena which can be used for their measurement are to be examined.

4.1 Thermal Energy

When a current flows in an electric conductor, i.e. charge carriers move under the ef-fect of electric field forces through the material, they transfer a part of their kinetic energy to the material particles as an undirected kinetic energy as thermal energy. Therefore the current which flows through a conductor with the resistance R pro-duces thermal energy, also called Joule energy or current thermal energy:



⋅ =



⋅ ⋅ =



⋅ ⋅ =



⋅ ⋅ = u d Q u i d t R i d t G u d t

W_J 2 2 (4.1.1)

Then the supplied electric power is:

2 2 u G i R i u t d W d P= J = ⋅ = ⋅ = ⋅ (4.1.2)

This has found its implementation in the following instruments:

4.1.1 Hot Wire Measuring Instrument

Thermal expansion 2

~ i

By the influence of the supplied current thermal energy extends the conductor. The amperage is shown by the changed slack span of the conductor.

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4-

2

Fig. 4.1.1: Hot Wire Instrument

The response time amounts to some seconds. It is also suitable for the radio fre-quency metrology, because no cut off frefre-quency is to be considered. In most cases thermal instruments are applied even to the demonstration of the current thermal energy.

Disadvantage: It is sensitive to overload

4.1.2 Thermal Converter

Thermo-electric voltage 2

~ i

Here a linear conductor is flown by a current to be measured too. The produced heat is measured with a thermocouple and the thermo-electric voltage proportional to the temperature of the conductor is advertised.

Fig. 4.1.2: Thermal Converter

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4-

3

The thermocouple can be welded with the hot wire (direct heating) or coupled elec-trically isolated (indirect heating).

These instruments find application in the current measurement, especially in the area of the radio frequency engineering. Higher amperages must be measured with cur-rent transformer. Because the interrelationship is square, power can be measured too, only the phasing of alternating parameters must be considered.

Fig. 4.1.3: Thermal Converter Measuring Power

Therefore, two thermal converters are used to measure the active power. Their hot wires are flown by two current, by the sum and the difference of the current through the load and of the voltage falling over the load. The thermocouples are arranged on the hot wires insulated and are switched again each other. Because the thermo-electric voltage uth depends on the square of the heating current, it applies:

2 1 th th th u u u = − (4.1.3)

(

) (

)

[

]

2 1 2 2 1 2 2 1 4 k i i i i i i k ⋅ ⋅ ⋅ = − − + ⋅ =

with i₁=k₁⋅uand i₂ =k₂⋅i follows: i u k k u_th = ₁⋅ ₂⋅ ⋅ (4.1.4) i u kth⋅ ⋅ =

and therefore for alternating parameters:

ϕ cos ⋅ ⋅ ⋅ =k U I u_th _th (4.1.5)