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Statistical analysis of the characteristics of some basic mass-produced passive electrical circuits used in measurements
Measurement 44 (2011) 1713–1722
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Statistical analysis of the characteristics of some basic mass-produced passive electrical circuits used in measurements Koviljka Stankovic´ a, Miloš Vujisic´ a, Dragan Kovacˇevic´ b, Predrag Osmokrovic´ a,⇑ a b
Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, P.O. Box 3554, 11000 Belgrade, Serbia Institute of Electrical Engineering ‘‘Nikola Tesla’’, Koste Glavinic´a 8a, 11000 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 21 February 2010 Received in revised form 8 April 2011 Accepted 6 July 2011 Available online 22 July 2011 Keywords: Measurement uncertainty Non-linearity Probability density function Voltage divider Wheatstone bridge Monte Carlo method
a b s t r a c t Statistical properties of some basic mass-produced passive electrical circuits used in measurements are analyzed in this paper, using methods for expressing measurement uncertainty of indirectly measured quantities. The focus of this paper is on electrical circuits whose defining parameters are obtained as non-linear functions of component parameters. Variants of voltage divider and Wheatstone bridge circuits significant for measurement practice are investigated in detail. Even if distributions with symmetrical probability density functions (PDFs), such as uniform or normal, are adopted for the parameters of the components comprising a circuit, non-linearity of the circuit function gives rise to asymmetry in the PDF of the circuit’s parameter. The asymmetry of the PDF causes the mean and the nominal value of the circuit parameter to differ. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Equality of electrical circuit parameters is always a goal in mass production. However, the manufacture of circuits utilizing discrete components (i.e. circuit elements, such as resistors, capacitors, or inductors) can never achieve ideal equality of circuit parameters, due to the finite tolerances of individual discrete components. One way to attempt to attain this equality is to preselect the components. Selection from a population of electrical components (resistors, capacitors, etc.) can be performed either directly by using an appropriate digital instrument (multimeter), or by means of an automatic testing system based on a digital sensor. Either way, the tightest tolerance of component characteristics over a series of circuits is obtained by picking out the components with identical readings of the parameter value (resistance, capacitance, etc.), while keeping the conditions of the measurements unchanged. Within the framework of measurement uncertainty, identical readings of components’ parameters, ⇑ Corresponding author. Tel.: +381 11 370 186; fax: +381 11 3370 187. E-mail address: [email protected] (P. Osmokrovic´). 0263-2241/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.07.007
observed on the display of a digital instrument, imply that the parameter value ascribed to the population of selected components represents a random quantity with a uniform distribution. This uniform distribution is centered on the indicated value, which is taken as the nominal value for the population of components. Assuming a digital instrument with ideal characteristics is used for selecting the components, the half-width of the uniform distribution is equal to one half of the smallest significant digit on the instrument’s indicating device [1,2]. When low cost is the principal requirement, components acquired directly from a manufacturer are used, without further selection. Tolerance (i.e. relative uncertainty) of the parameter value for such components typically ranges from 0.1% to 1%. In this case, the value of the component parameter, regarded as a random quantity, is usually assumed to follow a normal distribution. It is possible to assume some other distribution, if details of the production process or the type of component selection conducted by the manufacturer are known [3,4]. A statistical analysis of the characteristics of some basic passive electrical circuits is carried out in this paper, using methods originally intended for evaluating uncertainty in
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measurement. Statistical characteristics of complex circuits whose defining parameters are expressed as linear functions of component parameters, such as the serial connection of resistors or the parallel connection of capacitors, have been investigated in detail [5–7]. The focus of this paper is on the electrical circuits whose defining parameters are obtained as non-linear functions of component parameters. Even if distributions with symmetrical probability density functions (PDFs), such as uniform or normal, are adopted for the parameters of the components comprising the circuit, non-linearity of the circuit function gives rise to asymmetry in the PDF of the circuit’s parameter. It is pointed out herein that the asymmetry of the PDF causes the mean and the nominal value of the circuit parameter to differ. 2. General considerations The following circuits with non-linear circuit functions are investigated: 1. Voltage divider with two resistors, in its three variants: 1.1. Attenuator with two fixed-value resistors produced by the same technological process. In this case, relative uncertainties of the resistances are typically equal DR1/R01 = DR2/R02. 1.2. Half-bridge consisting of a resistive sensor and a fixed-value resistor. Since the technologies of these two kinds of resistors generally differ, relative uncertainties of their resistances also differ DR1/ R01 – DR2/R02. This holds both for discrete component dividers and for dividers produced in MEMS technology. Tolerances of sensor resistors are usually larger than those of fixed-value resistors. 1.3. Voltage dividers with two differential piezoresistive sensors, used for measuring mechanical quantities (force, pressure, acceleration, etc.). Sensors are produced by the same technological process, but are installed on elastic consoles in such a way that they manifest opposite changes in resistance when the measured mechanical quantity is applied. The analysis of dividers with resistive sensors pertains to unloaded circuits, i.e. to the initial condition when the measured quantity is not applied. 2. Wheatstone bridge in the following forms: 2.1 Wheatstone bridge with only one resistive sensor, such as the bridge for measuring temperature with a platinum thermistor, or the bridge with a hot wire mass airflow sensor. 2.2 Wheatstone bridge with two identical differential piezoresistive sensors (i.e. two strain gauges) and two fixed-value resistors. 2.3 Wheatstone bridge with four active arms, containing sensors with equal characteristics, such as microelectronic sensors for pressure and acceleration, or strain gauges. Bridges are analyzed in terms of the PDF of the output voltage when the measured quantity is not applied,
characterizing the imbalance of an unloaded bridge. Knowledge of this PDF provides an opportunity for designing an optimal circuit for balancing (nulling) the bridge. In addition to the aforementioned resistive circuits, capacitive dividers and bridges are widely used, especially in high-voltage engineering. Basic electrical circuit theory demonstrates that circuit functions of capacitive dividers and bridges are analogous in form to those of the corresponding resistive circuits. Analysis of capacitive circuits is therefore omitted, since it can easily be carried out on the basis of the results for resistive circuits. 2.1. The nominal and the mean value of a circuit parameter Due to the uncertainties inherent in electrical component production and selection, parameters characterizing individual electrical components in a circuit are treated as random quantities X, Y, Z, . . . Components used in a complex circuit each come from a respective population with a known nominal value and tolerance, obtained through preselection or acquired from a manufacturer. Nominal values of component parameters can be designated x0, y0, z0, . . . An investigated complex circuit parameter F is generally expressed as a non-linear function F = f(X, Y, Z, . . .), for which component parameters serve as input quantities. Complex circuit parameter F is evidently also a random quantity with a PDF of its own. The nominal value of the complex circuit parameter is obtained as F0 = f(x0, y0, z0, . . .). According to the recommendations laid out in the Guide to the Expression of Uncertainty in Measurement (GUM) [1], it is customary to adopt symmetrical distributions for input quantities. Considerations in this paper refer to the two basic cases of uniformly and normally distributed component parameters (input quantities). Each of the input quantities can be represented as consisting of a deterministic part x0, equal to its mean, i.e. nominal value, and a stochastic part dX, that for symmetrical PDFs has a zero mean value ðdX ¼ 0Þ. The random value of an input parameter is then expressed as the sum X = x0 + dX. In the case of a multivariate circuit function f(X, Y, Z, . . .), where X, Y, Z, . . . are continuous random variables with a joint PDF p(x, y, z, . . .), the circuit parameter mean is determined as the expectation value:
F¼
Z
f ðx; y; z; . . .Þ pðx; y; z; . . .Þ dx dy dz
ð1Þ
where integration is performed over the whole ranges of all input quantities. For a circuit function with two input quantities X and Y, the mean value of the complex circuit parameter F = f(X, Y) is obtained as:
F¼
Z
þ1
1
Z
þ1
f ðx; yÞ pðx; yÞ dx dy
ð2Þ
1
where p(x, y) is the joint PDF of the input quantities. This paper makes use of the special case of two independent input random variables with uniform distributions X Unif(x0 Dx, x0 + Dx) and Y Unif(y0 Dy, y0 + Dy), applied to the investigated electrical circuits. The circuit parameter mean value then becomes:
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
F¼
1 4 DxDy
Z
x0 þ D x
Z
x0 Dx
y0 þDy
f ðx; yÞ dx dy
ð3Þ
y0 Dy
An alternative way of determining the parameter mean is by developing the function f(X, Y) into a Taylor series. Odd-powered terms in the series expansion have zero mean values, and only the means of the even-powered terms contribute. In most cases, an estimate of the mean value which is accurate enough is obtained by neglecting the terms beyond the square term. If the circuit function is linear, as in the case of a sum or an arithmetic mean of several input quantities, the nominal and the mean value of the circuit parameter are equal, F ¼ F 0 . It is pointed out in this paper that if the circuit function is non-linear, the mean value F generally differs from the nominal value F–F 0 . 2.2. The shape of the circuit parameter’s PDF In addition to providing an estimate of the mean value, the Taylor series expansion of the circuit function can also yield information on the shape of the circuit parameter’s PDF, once the component parameters are assigned specific PDFs. For the simplest case of a multivariate circuit function with only two input variables F = f(X, Y), the Taylor series expansion is:
F ¼ F0 þ
@f @f dX þ dY @X @Y
1 @2f @2f @2f 2 þ dXdY þ ðdXÞ þ 2 ðdYÞ2 2 @X 2 @X@Y @Y 2
! þ
ð4Þ
where dX = X x0, dY = Y y0, and all partial and mixed derivatives are determined at the point F0 = f(x0, y0), wherefore they are constant values. Digital instruments used for component preselection give rise to relative measurement uncertainties (i.e. tolerances of component parameters) represented by uniform distributions with half-widths equal to the halves of the respective instrument’s smallest digits. Using the notation introduced in Section 2.1., the relative uncertainties of the two input quantities X and Y can be represented as Dx/x0 and Dy/y0, where Dx and Dy are the half-widths of the parameters’ uniform distributions, and x0 and y0 are their mean or nominal values. For modern instruments, the uncertainties Dx/x0 and Dy/y0 are of the order of 103 or smaller. General conclusions about the shape of the circuit parameter’s PDF can therefore be drawn by using only the first (linear) term of the expansion (4), since higher order terms are negligible. Deviation of the circuit parameter dF = F F0 thus becomes:
dF
@f @f dX þ dY @X @Y
ð5Þ
Seeing that dF is a linear combination of two random quantities, the shape of the resulting PDF that describes the random nature of the circuit parameter F is determined by the convolution of component parameters’ PDFs [8]. In the special case when maximum values of the addends in @f @f Eq. (5) are equal @X Dx ¼ @Y Dy, convolution of the two input uniform distributions yields a triangular resultant PDF. If
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higher order terms in the Taylor series expansion of Eq. (4) were not neglected, this would correspond to an approximately triangular resultant PDF. For the general case @f @f of @X Dx– @Y Dy, an approximately trapezoidal shape PDF of the circuit parameter ensues. Departure from ideal triangular and trapezoidal shapes, resulting from the influence of higher order terms in the Taylor series, is more pro@f Dx @f Dy nounced when the relative deviations @X and @Y are F0 F0 larger. In another case of practical importance, when component parameters are normally distributed, on the basis of the approximate Eq. (5), convolution of input PDFs yields a resultant normal distribution. On account of the higher order terms in the Taylor series, the true PDF of the circuit parameter is bell-shaped, but only approximately normal. This means that the circuit parameter value corresponding to the maximum of its PDF, i.e. the most probable value, can differ from its mean value. For a circuit function with three input quantities, the approximate deviation of the circuit parameter obtained from the linear term of the Taylor series is:
dF
@f @f @f dX þ dY þ dZ @X @Y @Z
ð6Þ
Discussion regarding the shape of the resulting PDF is significantly more complex than for a two-variable function. In the special case of all three addends in Eq. (6) being equal and all three component parameters X, Y, and Z uniformly distributed, convolution yields a bell-shaped symmetrical PDF of the circuit parameter, consisting of several parabolic segments [9,10]. However, due to the presence of higher order terms in the Taylor series, the actual PDF is asymmetrical, causing the most probable and the mean value to differ. Aside from this case, circuit parameter’s PDF is generally shaped as a curvilinear quadrilateral and can be determined by a Monte Carlo method (MCM). A three-component circuit with normally distributed component parameters can be discussed the same way as the two-component case. The actual resultant PDF is generally bell-shaped and asymmetrical, and can be determined by the MCM. For circuits with four and more components, the circuit parameter’s PDF becomes increasingly similar to that of the normal distribution, as expected from the central limit theorem (CLT) [11–13]. 3. Voltage divider Voltage divider is used in many electrical and electronic circuits. Output voltage Vo of a divider is smaller than the input voltage Vi and can have a value in the range 0 < Vo < Vi. Voltage divider consists of two (or more) resistors and can be applied to either dc or ac voltage. Parameter characterizing a voltage divider circuit is the divider’s voltage ratio. It is defined as output-to-input voltage ratio a = Vo/Vi. If the input voltage is assumed to be obtained from an ideal voltage source, with negligible internal resistance and an electromotive force E, voltage ratios for a resistive and a capacitive divider (Fig. 1a and b) are:
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a¼
Vo R1 ¼ E R1 þ R2
aC ¼
ð7Þ
Vo C2 ¼ E C1 þ C2
ð8Þ
Seeing that expressions for a and aC are formally very similar, the rest of this paper considers only the resistive dividers. Eq. (7) is the non-linear circuit function of the voltage divider, dependent on two input quantities, namely the two resistances. It is valid only when the divider is not loaded by other circuits connected to its output terminals. This is practically achieved by making the output resistance of the divider Ro = R1R2/(R1 + R2) significantly lower than the input resistance of the next stage circuit. Voltage divider can be used to achieve attenuation of an input voltage by a desired ratio, in which case the divider is called an attenuator. Voltage ratio of an attenuator can have any value in the range 0 < a < 1. Attenuator can be constructed as a potentiometer or with two fixed-value resistors. Attenuators are commonly used as input circuits in voltmeters and other measuring instruments. Another use of the voltage divider circuit is as a halfbridge, when one of the two resistors is a sensor, while the other is a fixed-value resistor. Resistors in a half-bridge generally differ in both nominal value and tolerance. Dividers with two sensing resistors arranged in a differential connection are also widely in use. In this case, resistors usually have equal resistances and tolerances. They can be two equal strain gauges, piezoresistors produced by MEMS technology, thermoresistive sensors, or photoresistors. The equality of the resistances provides maximum measuring sensitivity in both static and dynamic operating modes. Nominal voltage ratio of a half-bridge with equal nominal resistances of the two resistors R01 = R02, is a0 = 0.5. 3.1. Mean value of the voltage ratio If resistances are assigned uniform distributions centered on their respective nominal values (R1 Unif(R01 DR1, R01 + DR1) and R2 Unif(R02 DR2, R02 + DR2)), the mean value of the voltage ratio is obtained by substituting expression (7) for the circuit function f(x, y) in Eq. (3). For brevity, the limits of the distributions are marked a = R01 DR1, b = R01 + DR1, c = R02 DR2, and d = R02 +
(a)
DR2. This yields the following expression for the mean voltage ratio: 1 aþc ðd cÞðb aÞ þ a2 ln a ¼ 2ðb aÞðd cÞ aþd bþd cþb dþa 2 2 ð9Þ þ c2 ln þ d ln þb ln bþc cþa dþb Considering the complexity of Eq. (9), it is more convenient to use its approximation based on the Taylor series expansion. Keeping only the constant and the quadratic terms of the series (the linear term is zero), and going back to the half-width notation, the expression for the mean voltage ratio becomes:
"
a a0 1 þ
!# 1 R02 DR 2 2 DR1 2 R R 02 01 3 ðR01 þ R02 Þ2 R02 R01 ð10Þ
where a0 = R01R02/(R01 + R02) is the nominal value of the voltage ratio. If both the nominal values and the relative expanded uncertainties of the two resistances are equal, i.e. R01 = R02 and DR1/R01 = DR2/R02, the quadratic term in Eq. (10) is zero, and the mean value of the voltage ratio is ¼ a0 . If any of these two conequal to its nominal value, a ditions is not fulfilled, two distinctive cases are possible: > a0 , and R02 ðDR02 =R02 Þ2 > R01 ðDR01 =R01 Þ2 , in which case a < a0 . For example, R02 ðDR02 =R02 Þ2 < R01 ðDR01 =R01 Þ2 , when a ¼ 1:01 and R01 = R02, DR1/R01 = 0.1, DR2/R02 = 0.2, yields a a0 = 0.505. For R01 = R02, DR1/R01 = 0.2, DR2/R02 = 0.1, the ¼ 0:99 and a0 = 0.495. mean and the nominal value are a 3.2. The shape of the voltage ratio’s PDF The shape of the voltage ratio’s PDF can be deduced from an approximate expression for its deviation da relative to the nominal value a0. According to Eqs. (5) and (7), this deviation can be expressed as:
da ¼ a a0
dR1 dR2 ðR01 þ R02 Þ R01 R02 R01 R02
2
where dR1 = R1 R01 and dR2 = R2 R02. If distributions assigned to the resistances are uniform (R1 Unif(R01 DR1, R01 + DR1) and R2 Unif(R02 DR2, R02 + DR2), i.e. dR1 Unif(DR1, +DR1) and dR2 Unif
(b) C2
R2 U
E R1
Vo = E
R1
ð11Þ
C1
R1 + R2
Fig. 1. Voltage divider: (a) resistive and (b) capacitive.
Vo = U
C2 C1 + C2
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
(DR2, +DR2)), the convolution rule, valid for the linear combination of random quantities, implies that for DR1/ R01 = DR2/R02 the PDF of the voltage ratio is approximately triangular, while for DR1/R01 – DR2/R02 it is approximately trapezoidal.
3.2.1. Some characteristic examples of the voltage ratio PDFs To make the effect of the circuit function’s non-linearity on the shape of the voltage ratio’s PDF visible, it is assumed that the relative expanded uncertainties of the resistors in the divider (DR1/R01 and DR2/R02) are much larger than typical resistor tolerances in metrological practice. The large expanded uncertainties of the resistances, combined with the non-linearity of the function defining the voltage ratio, make higher terms of the Taylor series expansion relevant and Eq. (11) inadequate for assessing the shape of the resulting PDF. A more detailed insight into the shape of the voltage ratio’s PDF can be obtained by using MCM,
α = α 0 = 0.5
Probability density
(a)
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as described in [14]. Labeling of resistances in this section conforms to that of Fig. 1a. Fig. 2 is the result of a MCM calculation for a divider with uniformly distributed resistances, with equal nominal values and relative expanded uncertainties (R01 = R02 = 100 X and DR1/R01 = DR2/R02 = 50%). This example relates to a divider with two sensing resistors arranged in a differential connection. The histogram in Fig. 2a, with the interval width of 0.001 X, was produced on the basis of 107 trials, which means that the MCM code calculated the value of the voltage ratio for 107 pairs of values of R1 and R2, sampled independently from the corresponding uniform distributions. The envelope of the histogram is the PDF of the divider’s voltage ratio. It is symmetrical, but deviates from a triangle insofar as its sides are curvilinear with inflection points. The mean and the nominal value of ¼ a0 ¼ 0:5, represented by the the voltage ratio are equal a solid vertical line. Plot in Fig. 2b is the cumulative distribution function (CDF) which corresponds to the PDF of Fig. 1a, determined from the same MCM calculation. The symmetrical coverage interval for the voltage ratio with 95% level of confidence was found to be a95% e (0.301, 0.699). Bounds of this coverage interval are marked in Fig. 2b. For an attenuator with R01 – R02 and DR1/R01 = DR2/R02 (R01 = 60 X, R02 = 100 X, DR1/R01 = DR2/R02 = 50%), MCM results are presented in Fig. 3. Both resistances are assigned uniform distributions (R1 Unif(30 X, 90 X) and R2 Unif(50 X, 100 X)). Voltage ratio’s PDF (the envelope of the histogram) is a highly asymmetrical, skewed, curvilinear triangle. Due to the asymmetry of the PDF, the mean > a0 Þ. and the nominal value of the voltage ratio differ ða In this and all subsequent figures, the nominal and the mean value of the circuit parameter calculated by MCM are indicated by the solid and the dotted line, respectively. The calculated values, stated next to the corresponding ¼ 0:3802. vertical lines in the graph, are a0 = 0.375 and a
Voltage ratio α
(b) α 0 = 0.375
α = 0.3802
Probability
Probability density
Δα r = + 1.39 %
Voltage ratio α Fig. 2. (a) Histogram of voltage ratio values, determined by MCM with 107 trials, for a divider with equal uniformly distributed resistances (R01 = R02 = 100 X, DR1/R01 = DR2/R02 = 50%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean ¼ a0 ¼ 0:5, and the nominal value of the voltage ratio are equal a represented by the solid vertical line. (b) Voltage ratio’s CDF. Bounds of the symmetrical coverage interval with 95% level of confidence are marked at 0.301 and 0.699.
Voltage ratio α Fig. 3. Histogram of voltage ratio values, determined by MCM with 107 trials, for a divider with different nominal resistances that have equal relative expanded uncertainties (R1 Unif(30 X, 90 X) and R2 Unif(50 X, 100 X), i.e. R01 = 60 X, R02 = 100 X, DR1/R01 = DR2/ R02 = 50%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively.
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Relative deviation of the mean from the nominal value is r ¼ ða a0 Þ=a0 ¼ þ1:39%. It can also be noticed that, Da in spite of the asymmetry, the nominal and the most probable value of the voltage ratio coincide. Results for the general case when R01 – R02 and DR1/ R01 – DR2/R02, corresponding to the conditions in a halfbridge, are presented in Fig. 4. Example uniform distributions assigned to the resistances are R1 Unif(5 X, 15 X) and R2 Unif(15 X, 25 X), which equates to R01 = 10 X, R02 = 20 X, DR1/R01 = 50%, DR2/R02 = 25%. Voltage ratio’s PDF is shaped as an irregular curvilinear quadrilateral, with marked asymmetry. Again, the asymmetry of the PDF gives rise to a difference between the mean and the nominal va < a0 Þ. The calculated values of lue of the voltage ratio (a the nominal and the mean voltage ratio, round up to four ¼ 0:3301, which decimal places, are a0 = 0.3333 and a r ¼ 0:96%. gives Da It can be noted that the conclusions about the relation and a0 drawn from the approximate Eq. (11) in Section of a > a0 if R02 ðDR02 =R02 Þ2 > R01 ðDR01 =R01 Þ2 and 3.1, i.e. that a a < a0 if R02 ðDR02 =R02 Þ2 < R01 ðDR01 =R01 Þ2 , are confirmed by the MCM calculations for the previous two investigated dividers. The last investigated example of a voltage divider has one sensing resistor with large relative expanded uncertainty, and another high quality fixed-value resistor with low tolerance that can be modeled by zero uncertainty. MCM calculation results for two possible configurations of such a divider are presented in Fig. 5. Fig. 5a refers to the case of DR1/R01 – 0 and DR2/R02 0, while Fig. 5b refers to DR1/R01 0 and DR2/R02 – 0. Both the asymmetry of the PDF and its departure from the trapezoidal shape are particularly manifested in these cases. This is attributed to the fact that partial compensation of the circuit function’s non-linearity, achieved through the differential connection of resistors in the divider, is removed when only one of them is treated as having a distributed resis and tance, while the other is fixed to a constant value. a a0 differ as expected, with the relative deviation r j ¼ 2:16% larger than for the previously analyzed jDa dividers. In addition to the histograms, CDFs of the voltage
4. Equal-resistance Wheatstone bridge A Wheatstone bridge is one of the basic electrical circuits for measuring physical quantities with resistive sensors. This is especially true of the measurements where small changes of resistance occur, such as with strain gauges or piezoresistive semiconductor sensors of pressure and acceleration. Its structure is that of two voltage dividers connected in parallel, as presented by the circuit diagram of Fig. 6. Output signal of the bridge is the potential difference between the outputs of the two dividers. The ratio of the output voltage to the electromotive force of a source connected to the bridge is the principal circuit parameter. Assuming that the resistance of a voltmeter used to measure the bridge output voltage is much higher than the output resistance of the bridge, the bridge ratio ab can be expressed as a non-linear function of the four resistances:
ab ¼
Vo R1 R3 ¼ E R1 þ R2 R3 þ R4
ð12Þ
If stochastic deviations of the four resistances from their respective nominal values are designated dR1 = R1 R01, dR2 = R2 R02, dR3 = R3 R03, and dR4 = R4 R04, deviation of the bridge ratio from the nominal value ab0 = (R01/ (R01 + R02)) (R03/(R03 + R04)) can be approximated by the first term of the Taylor series expansion, as expounded in Section 2.2:
dab ¼ ab ab0
1 dR1 dR4 dR2 dR3 þ 4 R01 R04 R02 R03
ð13Þ
Expression (13) can be used for assessing the shape of the bridge ratio’s PDF. Since expression (13) is a linear combination of four random quantities, the resulting PDF is obtained by convoluting four uniform PDFs assigned to the resistances. This yields a symmetric bell-shaped curve,
α 0 = 0.3333 Δα r = − 0.96 %
Probability density
α = 0.3301
ratio for the two cases are shown in Fig. 5. There is a marked difference between these plots, one being concave and the other convex.
Voltage ratio α 7
Fig. 4. Histogram of voltage ratio values, determined by MCM with 10 trials, for a divider with uniformly distributed resistances that differ in both their nominal values and their relative expanded uncertainties (R1 Unif(5 X, 15 X) and R2 Unif(15 X, 25 X), i.e. R01 = 10 X, R02 = 20 X, DR1/R01 = 15%, DR2/ R02 = 25%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively.
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
(a) α = 0.4892
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α 0 = 0.5
Probability
Probability density
Δα r = − 2.16 %
Voltage ratio α
Voltage ratio α
(b)
Δα r = + 2.16 %
Voltage ratio α
Probability
α = 0.5108
Probability density
α 0 = 0.5
Voltage ratio α
Fig. 5. Histogram of voltage ratio values and the corresponding CDF, determined by MCM with 107 trials, for a divider consisting of one resistor with uniformly distributed resistance, and another having zero uncertainty. The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively. (a) R1 Unif(50 X, 150 X) and R2 = 100 X, i.e. DR1/R01 – 0 and DR2/R02 = 0. (b) R1 = 100 X and R2 Unif(50 X, 150 X), i.e. DR1/R01 = 0 and DR2/R02 – 0.
R2
R4
Vo = V1 − V2 E
V1
V2
R1
R3
Fig. 6. Wheatstone bridge circuit diagram.
made up of third-power polynomial segments. The true PDF of the bridge ratio is in effect asymmetrical, since the circuit function of the bridge (Eq. (12)) is non-linear, which can be observed by the MCM calculations, as for the voltage dividers in Section 3.2.1.
A large number of different Wheatstone bridge configurations and modifications is encountered in practice. For reasons of maximum sensitivity and temperature dependence compensation, mass-produced measuring devices typically use equal-resistance Wheatstone bridges, in which all four resistances have equal nominal values R01 = R02 = R03 = R04 = R0. The nominal value of the bridge ratio for an unloaded equal-resistance Wheatstone bridge (with no measured quantity applied to any of the sensors) equals zero, ab0 = 0. One of the following three configurations is possible: (a) bridge with one active arm (containing a resistive sensor) and fixed-value resistors in other three arms, (b) bridge with two differentially connected sensors (R1 and R3 in Fig. 6) and two fixed-value resistors (R2 and R4), or (c) bridge with all four active arms. Non-sensing fixed-value resistors are commonly produced with low tolerances, i.e. approximately zero uncertainties. In the following, only the sensing resistors are treated as having distributed resistances. A Wheatstone bridge with one active arm is similar to a voltage divider with one sensor and one zero uncertainty
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
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fixed-value resistor. If R1 is the sensing resistor in such a bridge, and if all four nominal resistances are equal, Eq. (12) reduces to:
R1 1 R1 þ R0 2
ð14Þ
This means that the PDF of the bridge ratio is obtained by shifting the PDF of a corresponding voltage divider (Fig. 5a) to the left by 1/2. For a bridge in which only R1 and R3 are sensors, Eq. (13) becomes:
dab
1 dR1 dR3 4 R01 R03
If resistances in the previously analyzed circuits are assigned normal distributions, PDFs of the circuit parameters are bell-shaped and only approximately normal. PDFs obtained by the MCM for these circuits are presented in Figs. 8–11. Only in the case of a symmetrical voltage divider (R01 = R02 = R0) for which the resistors have the same normal distributions (R N(R0, (rR)2), where notation N(l, r2) designates a normal distribution with mathematical
α = α 0 = 0.5
ð15Þ
which is analogous to Eq. (11) for voltage dividers. It follows that PDF of the bridge ratio for this type of Wheatstone bridge has one of the shapes presented in Figs. 2, 3 or 4, depending on the specific nominal values R01 and R03, as well as on the relative expanded uncertainties DR1/R01 and DR3/R03. For an equal-resistance Wheatstone bridge of this type (R01 = R03, DR1/R01 = DR3/R03) the symmetric histogram of Fig. 2a applies. Circuit function of a Wheatstone bridge with four active arms depends on four input random variables. Due to the differential connection of the resistors in the bridge, the asymmetry of the bridge ratio’s PDF is slight, even when relative expanded uncertainties of the resistances are very large. This is illustrated by an example equal-resistance bridge in which all four resistances are uniformly distributed over the interval (50 X, 150 X). The bridge ratio’s PDF (Fig. 7) resembles that of a normal distribution, and the mean value of the bridge ratio is nearly equal to its b ab0 ¼ 0. nominal value, a
Probability density
ab ¼
5. Examples with normally distributed resistances
Voltage ratio α Fig. 8. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider with equal normally distributed resistances (R01 = R02 = 10 X, rR1 = rR1 = 2 X, i.e. rR1/R01 = rR2/R02 = 20%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The ¼ a0 ¼ 0:5, mean and the nominal value of the voltage ratio are equal a represented by the solid vertical line.
α 0 = 0.25
α = 0.25675 α m = 0.238 Δα r = + 2.7 %
Probability density
Probability density
α b ≈ α b0.= 0
Voltage ratio α Bridge ratio α b 7
Fig. 7. Histogram of voltage ratio values, determined by MCM with 10 trials, for an equal-resistance Wheatstone bridge with all four resistances uniformly distributed (R Unif(50 X, 150 X) i.e. R0 = 100 X, DR/ R0 = 50%). The interval width is 0.0001 X. Bridge ratio’s PDF is the envelope of the histogram. Two vertical lines marking the mean and the nominal value are irresolvable.
Fig. 9. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider with different nominal resistances, normally distributed with equal relative standard deviations (R1 N(1 X, (0.25 X)2) and R2 N(3 X, (0.75 X)2), i.e. R01 = 1 X, R02 = 3 X, rR1/R01 = rR2/R02 = 25%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The nominal (a0), the mean ), and the most probable (am) value of the voltage ratio are indicated by (a the solid, the dotted, and the broken line.
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722 α 0 = 0.25
α = 0. 25994 α m = 0. 234
Probability density
Δα r = + 3.98 %
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examples and their standard deviations (uncertainties) are provided in figure captions. For a Wheatstone bridge with four active arms containing resistors with equal normal distributions, the resulting PDF is symmetrical and bell-shaped (Fig. 11). Accordance with an equivalent normal distribution is even more visible here than in the case of uniformly distributed resistances (Fig. 7). 6. Conclusions
Voltage ratio α Fig. 10. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider consisting of one resistor with normally distributed resistance, and another having zero uncertainty (R1 = 1 X and R2 N(3 X, (0.75 X)2). The interval width is 0.001 X. The nominal (a0), ), and the most probable (am) value of the voltage ratio are the mean (a indicated by the solid, the dotted, and the broken line.
Probability density
α b ≈ α b0.= 0
Bridge ratio α b Fig. 11. Histogram of voltage ratio values, determined by MCM with 107 trials, for an equal-resistance Wheatstone bridge with all four resistances uniformly distributed (R N(10 X, (2 X)2)). The interval width is 0.0001 X. Bridge ratio’s PDF is the envelope of the histogram. Two vertical lines marking the mean and the nominal value are irresolvable.
expectation l and standard deviation r), does the PDF of the voltage ratio have a symmetrical shape seen in Fig. 8. For a divider with different nominal resistances (R01 – R02), normally distributed with equal relative standard deviations with relation to their nominal values, voltage ratio’s PDF is an asymmetrical bell-shaped curve (Fig. 9), which means that the most probable, the mean and the nominal value of the voltage ratio differ. PDF of an asymmetrical divider in which R1 is a fixed-value resistor with negligible uncertainty and R2 is normally distributed is shown in ), Fig. 10. In Figs. 9 and 10 the nominal (a0), the mean (a and the most probable (am) value of the voltage ratio are indicated by the solid, the dotted, and the broken line, respectively. Specific nominal values of resistances in these
We have shown in this paper that the non-linearity of a circuit function gives rise to an asymmetry of the circuit parameter’s PDF, and that consequently the mean and the nominal value of the circuit parameter differ. This is of particular concern in mass-produced passive electrical circuits used in measurements, in which statistical variations inherent in their production may result in a discrepancy between the two values that affects measurement readings. Some of the basic circuits of this kind, such as the voltage divider and the Wheatstone bridge circuit, have been analyzed using methods originally intended for evaluating uncertainty in measurement. The presented analysis refers either to the case of preselecting circuit components on the basis of a digital instrument indication, or relying on a specified tolerance of the components acquired directly from a manufacturer. Within the framework of measurement uncertainty, identical readings of components’ parameters, observed on the display of a digital instrument, imply that the parameter value ascribed to the population of selected components represents a random quantity with a uniform distribution, centered on the indicated value, and with a half-width equal to one half of the smallest significant digit on the instrument’s indicating device. In the other case, of using components with a tolerance specified by the manufacturer, the value of the component parameter, regarded as a random quantity, is assumed to follow a normal distribution, with a relative standard deviation proportional to the tolerance. General conclusions about the shape of circuit parameters’ PDFs have been drawn based on a one-term Taylor series expansion of the circuit function. A more detailed insight into the shape of an output quantity’s PDF has been obtained by a Monte Carlo method, as an envelope to the histogram of circuit parameter’s values. MCM calculations also provide results for the mean and the nominal value of the circuit parameter. The degree of asymmetry of the resultant PDF and the deviation between the mean and the nominal value of the circuit parameter depend on the specific nominal values of component parameters, as well as on their relative expanded uncertainties. Relative deviation of the mean from the nominal value can generally be either positive or negative. Acknowledgment The Ministry of Science and Technological Development of the Republic of Serbia supported this work under Contract 171007.
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References [1] BIPM, IEC, IFCC, ISO, IUPAC, IUPAP and OIML, Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland, 1995. [2] M. Vujisic´, K. Stankovic´, P. Osmokrovic´, A statistical analysis of measurement results obtained from nonlinear physical laws, Applied Mathematical Modelling 35 (2011) 3128–3135. [3] K.Dj. Stankovic´, M.Lj. Vujisic´, Lj.D. Delic´, Influence of tube volume on measurement uncerainty of GM counters, Nuclear Technology and Radiation Protection 25–1 (2010) 46–50. [4] A. Kovacˇevic´, D. Brkic´, P. Osmokrovic´, Evaluation of measurement uncertainty using mixed distribution for conducted emission measurements, Measurement 44 (2011) 692–701. [5] European Association of National Metrology Institutes, Calibration of Thermocouples, EURAMET/cg-08/v.01, 2007. [6] European Association of National Metrology Institutes, Guidelines on the Calibration of Temperature Indicators and Simulators by Electrical Simulation and Measurement, EURAMET/cg-11/v.01, 2007. [7] European Association of National Metrology Institutes, Calibration of Temperature Block Calibrators, EURAMET/cg-13/v.01, 2007.
[8] L. Kirkup, R.B. Frenkel, An Introduction to Uncertainty in Measurement: Using the GUM, Cambridge University Press, 2006. [9] M. Grabe, Measurement Uncertainties in Science and Technology, Springer, 2005. [10] B.N. Taylor, C.E. Kuyatt, NIST Technical Note 1297: Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Physics Laboratory, National Institute of Standards and Technology, Gaithersburg, MD, 1994. [11] American National Standards Institute, US Guide to the Expression of Uncertainty in Measurement, ANSI/NCSL Z540-2-1997, 1997. [12] K. Stankovic´, M. Vujisic´, Influence of radiation energy and angle of incidence on the uncertainty in measurements by GM counters, Nuclear Technology and Radiation Protection 23–1 (2008) 41–42. [13] G. Wübbeler, M. Krystek, C. Elster, Evaluation of measurement uncertainty and its numerical calculation by a Monte Carlo method, Measurement Science and Technology 19 (2007) 1–4. [14] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, Evaluation of Measurement Data – Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’ Propagation of Distributions Using a Monte Carlo Method, Joint Committee for Guides in Metrology, 2007.
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Measurement journal homepage: www.elsevier.com/locate/measurement
Statistical analysis of the characteristics of some basic mass-produced passive electrical circuits used in measurements Koviljka Stankovic´ a, Miloš Vujisic´ a, Dragan Kovacˇevic´ b, Predrag Osmokrovic´ a,⇑ a b
Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, P.O. Box 3554, 11000 Belgrade, Serbia Institute of Electrical Engineering ‘‘Nikola Tesla’’, Koste Glavinic´a 8a, 11000 Belgrade, Serbia
a r t i c l e
i n f o
Article history: Received 21 February 2010 Received in revised form 8 April 2011 Accepted 6 July 2011 Available online 22 July 2011 Keywords: Measurement uncertainty Non-linearity Probability density function Voltage divider Wheatstone bridge Monte Carlo method
a b s t r a c t Statistical properties of some basic mass-produced passive electrical circuits used in measurements are analyzed in this paper, using methods for expressing measurement uncertainty of indirectly measured quantities. The focus of this paper is on electrical circuits whose defining parameters are obtained as non-linear functions of component parameters. Variants of voltage divider and Wheatstone bridge circuits significant for measurement practice are investigated in detail. Even if distributions with symmetrical probability density functions (PDFs), such as uniform or normal, are adopted for the parameters of the components comprising a circuit, non-linearity of the circuit function gives rise to asymmetry in the PDF of the circuit’s parameter. The asymmetry of the PDF causes the mean and the nominal value of the circuit parameter to differ. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Equality of electrical circuit parameters is always a goal in mass production. However, the manufacture of circuits utilizing discrete components (i.e. circuit elements, such as resistors, capacitors, or inductors) can never achieve ideal equality of circuit parameters, due to the finite tolerances of individual discrete components. One way to attempt to attain this equality is to preselect the components. Selection from a population of electrical components (resistors, capacitors, etc.) can be performed either directly by using an appropriate digital instrument (multimeter), or by means of an automatic testing system based on a digital sensor. Either way, the tightest tolerance of component characteristics over a series of circuits is obtained by picking out the components with identical readings of the parameter value (resistance, capacitance, etc.), while keeping the conditions of the measurements unchanged. Within the framework of measurement uncertainty, identical readings of components’ parameters, ⇑ Corresponding author. Tel.: +381 11 370 186; fax: +381 11 3370 187. E-mail address: [email protected] (P. Osmokrovic´). 0263-2241/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2011.07.007
observed on the display of a digital instrument, imply that the parameter value ascribed to the population of selected components represents a random quantity with a uniform distribution. This uniform distribution is centered on the indicated value, which is taken as the nominal value for the population of components. Assuming a digital instrument with ideal characteristics is used for selecting the components, the half-width of the uniform distribution is equal to one half of the smallest significant digit on the instrument’s indicating device [1,2]. When low cost is the principal requirement, components acquired directly from a manufacturer are used, without further selection. Tolerance (i.e. relative uncertainty) of the parameter value for such components typically ranges from 0.1% to 1%. In this case, the value of the component parameter, regarded as a random quantity, is usually assumed to follow a normal distribution. It is possible to assume some other distribution, if details of the production process or the type of component selection conducted by the manufacturer are known [3,4]. A statistical analysis of the characteristics of some basic passive electrical circuits is carried out in this paper, using methods originally intended for evaluating uncertainty in
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measurement. Statistical characteristics of complex circuits whose defining parameters are expressed as linear functions of component parameters, such as the serial connection of resistors or the parallel connection of capacitors, have been investigated in detail [5–7]. The focus of this paper is on the electrical circuits whose defining parameters are obtained as non-linear functions of component parameters. Even if distributions with symmetrical probability density functions (PDFs), such as uniform or normal, are adopted for the parameters of the components comprising the circuit, non-linearity of the circuit function gives rise to asymmetry in the PDF of the circuit’s parameter. It is pointed out herein that the asymmetry of the PDF causes the mean and the nominal value of the circuit parameter to differ. 2. General considerations The following circuits with non-linear circuit functions are investigated: 1. Voltage divider with two resistors, in its three variants: 1.1. Attenuator with two fixed-value resistors produced by the same technological process. In this case, relative uncertainties of the resistances are typically equal DR1/R01 = DR2/R02. 1.2. Half-bridge consisting of a resistive sensor and a fixed-value resistor. Since the technologies of these two kinds of resistors generally differ, relative uncertainties of their resistances also differ DR1/ R01 – DR2/R02. This holds both for discrete component dividers and for dividers produced in MEMS technology. Tolerances of sensor resistors are usually larger than those of fixed-value resistors. 1.3. Voltage dividers with two differential piezoresistive sensors, used for measuring mechanical quantities (force, pressure, acceleration, etc.). Sensors are produced by the same technological process, but are installed on elastic consoles in such a way that they manifest opposite changes in resistance when the measured mechanical quantity is applied. The analysis of dividers with resistive sensors pertains to unloaded circuits, i.e. to the initial condition when the measured quantity is not applied. 2. Wheatstone bridge in the following forms: 2.1 Wheatstone bridge with only one resistive sensor, such as the bridge for measuring temperature with a platinum thermistor, or the bridge with a hot wire mass airflow sensor. 2.2 Wheatstone bridge with two identical differential piezoresistive sensors (i.e. two strain gauges) and two fixed-value resistors. 2.3 Wheatstone bridge with four active arms, containing sensors with equal characteristics, such as microelectronic sensors for pressure and acceleration, or strain gauges. Bridges are analyzed in terms of the PDF of the output voltage when the measured quantity is not applied,
characterizing the imbalance of an unloaded bridge. Knowledge of this PDF provides an opportunity for designing an optimal circuit for balancing (nulling) the bridge. In addition to the aforementioned resistive circuits, capacitive dividers and bridges are widely used, especially in high-voltage engineering. Basic electrical circuit theory demonstrates that circuit functions of capacitive dividers and bridges are analogous in form to those of the corresponding resistive circuits. Analysis of capacitive circuits is therefore omitted, since it can easily be carried out on the basis of the results for resistive circuits. 2.1. The nominal and the mean value of a circuit parameter Due to the uncertainties inherent in electrical component production and selection, parameters characterizing individual electrical components in a circuit are treated as random quantities X, Y, Z, . . . Components used in a complex circuit each come from a respective population with a known nominal value and tolerance, obtained through preselection or acquired from a manufacturer. Nominal values of component parameters can be designated x0, y0, z0, . . . An investigated complex circuit parameter F is generally expressed as a non-linear function F = f(X, Y, Z, . . .), for which component parameters serve as input quantities. Complex circuit parameter F is evidently also a random quantity with a PDF of its own. The nominal value of the complex circuit parameter is obtained as F0 = f(x0, y0, z0, . . .). According to the recommendations laid out in the Guide to the Expression of Uncertainty in Measurement (GUM) [1], it is customary to adopt symmetrical distributions for input quantities. Considerations in this paper refer to the two basic cases of uniformly and normally distributed component parameters (input quantities). Each of the input quantities can be represented as consisting of a deterministic part x0, equal to its mean, i.e. nominal value, and a stochastic part dX, that for symmetrical PDFs has a zero mean value ðdX ¼ 0Þ. The random value of an input parameter is then expressed as the sum X = x0 + dX. In the case of a multivariate circuit function f(X, Y, Z, . . .), where X, Y, Z, . . . are continuous random variables with a joint PDF p(x, y, z, . . .), the circuit parameter mean is determined as the expectation value:
F¼
Z
f ðx; y; z; . . .Þ pðx; y; z; . . .Þ dx dy dz
ð1Þ
where integration is performed over the whole ranges of all input quantities. For a circuit function with two input quantities X and Y, the mean value of the complex circuit parameter F = f(X, Y) is obtained as:
F¼
Z
þ1
1
Z
þ1
f ðx; yÞ pðx; yÞ dx dy
ð2Þ
1
where p(x, y) is the joint PDF of the input quantities. This paper makes use of the special case of two independent input random variables with uniform distributions X Unif(x0 Dx, x0 + Dx) and Y Unif(y0 Dy, y0 + Dy), applied to the investigated electrical circuits. The circuit parameter mean value then becomes:
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
F¼
1 4 DxDy
Z
x0 þ D x
Z
x0 Dx
y0 þDy
f ðx; yÞ dx dy
ð3Þ
y0 Dy
An alternative way of determining the parameter mean is by developing the function f(X, Y) into a Taylor series. Odd-powered terms in the series expansion have zero mean values, and only the means of the even-powered terms contribute. In most cases, an estimate of the mean value which is accurate enough is obtained by neglecting the terms beyond the square term. If the circuit function is linear, as in the case of a sum or an arithmetic mean of several input quantities, the nominal and the mean value of the circuit parameter are equal, F ¼ F 0 . It is pointed out in this paper that if the circuit function is non-linear, the mean value F generally differs from the nominal value F–F 0 . 2.2. The shape of the circuit parameter’s PDF In addition to providing an estimate of the mean value, the Taylor series expansion of the circuit function can also yield information on the shape of the circuit parameter’s PDF, once the component parameters are assigned specific PDFs. For the simplest case of a multivariate circuit function with only two input variables F = f(X, Y), the Taylor series expansion is:
F ¼ F0 þ
@f @f dX þ dY @X @Y
1 @2f @2f @2f 2 þ dXdY þ ðdXÞ þ 2 ðdYÞ2 2 @X 2 @X@Y @Y 2
! þ
ð4Þ
where dX = X x0, dY = Y y0, and all partial and mixed derivatives are determined at the point F0 = f(x0, y0), wherefore they are constant values. Digital instruments used for component preselection give rise to relative measurement uncertainties (i.e. tolerances of component parameters) represented by uniform distributions with half-widths equal to the halves of the respective instrument’s smallest digits. Using the notation introduced in Section 2.1., the relative uncertainties of the two input quantities X and Y can be represented as Dx/x0 and Dy/y0, where Dx and Dy are the half-widths of the parameters’ uniform distributions, and x0 and y0 are their mean or nominal values. For modern instruments, the uncertainties Dx/x0 and Dy/y0 are of the order of 103 or smaller. General conclusions about the shape of the circuit parameter’s PDF can therefore be drawn by using only the first (linear) term of the expansion (4), since higher order terms are negligible. Deviation of the circuit parameter dF = F F0 thus becomes:
dF
@f @f dX þ dY @X @Y
ð5Þ
Seeing that dF is a linear combination of two random quantities, the shape of the resulting PDF that describes the random nature of the circuit parameter F is determined by the convolution of component parameters’ PDFs [8]. In the special case when maximum values of the addends in @f @f Eq. (5) are equal @X Dx ¼ @Y Dy, convolution of the two input uniform distributions yields a triangular resultant PDF. If
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higher order terms in the Taylor series expansion of Eq. (4) were not neglected, this would correspond to an approximately triangular resultant PDF. For the general case @f @f of @X Dx– @Y Dy, an approximately trapezoidal shape PDF of the circuit parameter ensues. Departure from ideal triangular and trapezoidal shapes, resulting from the influence of higher order terms in the Taylor series, is more pro@f Dx @f Dy nounced when the relative deviations @X and @Y are F0 F0 larger. In another case of practical importance, when component parameters are normally distributed, on the basis of the approximate Eq. (5), convolution of input PDFs yields a resultant normal distribution. On account of the higher order terms in the Taylor series, the true PDF of the circuit parameter is bell-shaped, but only approximately normal. This means that the circuit parameter value corresponding to the maximum of its PDF, i.e. the most probable value, can differ from its mean value. For a circuit function with three input quantities, the approximate deviation of the circuit parameter obtained from the linear term of the Taylor series is:
dF
@f @f @f dX þ dY þ dZ @X @Y @Z
ð6Þ
Discussion regarding the shape of the resulting PDF is significantly more complex than for a two-variable function. In the special case of all three addends in Eq. (6) being equal and all three component parameters X, Y, and Z uniformly distributed, convolution yields a bell-shaped symmetrical PDF of the circuit parameter, consisting of several parabolic segments [9,10]. However, due to the presence of higher order terms in the Taylor series, the actual PDF is asymmetrical, causing the most probable and the mean value to differ. Aside from this case, circuit parameter’s PDF is generally shaped as a curvilinear quadrilateral and can be determined by a Monte Carlo method (MCM). A three-component circuit with normally distributed component parameters can be discussed the same way as the two-component case. The actual resultant PDF is generally bell-shaped and asymmetrical, and can be determined by the MCM. For circuits with four and more components, the circuit parameter’s PDF becomes increasingly similar to that of the normal distribution, as expected from the central limit theorem (CLT) [11–13]. 3. Voltage divider Voltage divider is used in many electrical and electronic circuits. Output voltage Vo of a divider is smaller than the input voltage Vi and can have a value in the range 0 < Vo < Vi. Voltage divider consists of two (or more) resistors and can be applied to either dc or ac voltage. Parameter characterizing a voltage divider circuit is the divider’s voltage ratio. It is defined as output-to-input voltage ratio a = Vo/Vi. If the input voltage is assumed to be obtained from an ideal voltage source, with negligible internal resistance and an electromotive force E, voltage ratios for a resistive and a capacitive divider (Fig. 1a and b) are:
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
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a¼
Vo R1 ¼ E R1 þ R2
aC ¼
ð7Þ
Vo C2 ¼ E C1 þ C2
ð8Þ
Seeing that expressions for a and aC are formally very similar, the rest of this paper considers only the resistive dividers. Eq. (7) is the non-linear circuit function of the voltage divider, dependent on two input quantities, namely the two resistances. It is valid only when the divider is not loaded by other circuits connected to its output terminals. This is practically achieved by making the output resistance of the divider Ro = R1R2/(R1 + R2) significantly lower than the input resistance of the next stage circuit. Voltage divider can be used to achieve attenuation of an input voltage by a desired ratio, in which case the divider is called an attenuator. Voltage ratio of an attenuator can have any value in the range 0 < a < 1. Attenuator can be constructed as a potentiometer or with two fixed-value resistors. Attenuators are commonly used as input circuits in voltmeters and other measuring instruments. Another use of the voltage divider circuit is as a halfbridge, when one of the two resistors is a sensor, while the other is a fixed-value resistor. Resistors in a half-bridge generally differ in both nominal value and tolerance. Dividers with two sensing resistors arranged in a differential connection are also widely in use. In this case, resistors usually have equal resistances and tolerances. They can be two equal strain gauges, piezoresistors produced by MEMS technology, thermoresistive sensors, or photoresistors. The equality of the resistances provides maximum measuring sensitivity in both static and dynamic operating modes. Nominal voltage ratio of a half-bridge with equal nominal resistances of the two resistors R01 = R02, is a0 = 0.5. 3.1. Mean value of the voltage ratio If resistances are assigned uniform distributions centered on their respective nominal values (R1 Unif(R01 DR1, R01 + DR1) and R2 Unif(R02 DR2, R02 + DR2)), the mean value of the voltage ratio is obtained by substituting expression (7) for the circuit function f(x, y) in Eq. (3). For brevity, the limits of the distributions are marked a = R01 DR1, b = R01 + DR1, c = R02 DR2, and d = R02 +
(a)
DR2. This yields the following expression for the mean voltage ratio: 1 aþc ðd cÞðb aÞ þ a2 ln a ¼ 2ðb aÞðd cÞ aþd bþd cþb dþa 2 2 ð9Þ þ c2 ln þ d ln þb ln bþc cþa dþb Considering the complexity of Eq. (9), it is more convenient to use its approximation based on the Taylor series expansion. Keeping only the constant and the quadratic terms of the series (the linear term is zero), and going back to the half-width notation, the expression for the mean voltage ratio becomes:
"
a a0 1 þ
!# 1 R02 DR 2 2 DR1 2 R R 02 01 3 ðR01 þ R02 Þ2 R02 R01 ð10Þ
where a0 = R01R02/(R01 + R02) is the nominal value of the voltage ratio. If both the nominal values and the relative expanded uncertainties of the two resistances are equal, i.e. R01 = R02 and DR1/R01 = DR2/R02, the quadratic term in Eq. (10) is zero, and the mean value of the voltage ratio is ¼ a0 . If any of these two conequal to its nominal value, a ditions is not fulfilled, two distinctive cases are possible: > a0 , and R02 ðDR02 =R02 Þ2 > R01 ðDR01 =R01 Þ2 , in which case a < a0 . For example, R02 ðDR02 =R02 Þ2 < R01 ðDR01 =R01 Þ2 , when a ¼ 1:01 and R01 = R02, DR1/R01 = 0.1, DR2/R02 = 0.2, yields a a0 = 0.505. For R01 = R02, DR1/R01 = 0.2, DR2/R02 = 0.1, the ¼ 0:99 and a0 = 0.495. mean and the nominal value are a 3.2. The shape of the voltage ratio’s PDF The shape of the voltage ratio’s PDF can be deduced from an approximate expression for its deviation da relative to the nominal value a0. According to Eqs. (5) and (7), this deviation can be expressed as:
da ¼ a a0
dR1 dR2 ðR01 þ R02 Þ R01 R02 R01 R02
2
where dR1 = R1 R01 and dR2 = R2 R02. If distributions assigned to the resistances are uniform (R1 Unif(R01 DR1, R01 + DR1) and R2 Unif(R02 DR2, R02 + DR2), i.e. dR1 Unif(DR1, +DR1) and dR2 Unif
(b) C2
R2 U
E R1
Vo = E
R1
ð11Þ
C1
R1 + R2
Fig. 1. Voltage divider: (a) resistive and (b) capacitive.
Vo = U
C2 C1 + C2
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
(DR2, +DR2)), the convolution rule, valid for the linear combination of random quantities, implies that for DR1/ R01 = DR2/R02 the PDF of the voltage ratio is approximately triangular, while for DR1/R01 – DR2/R02 it is approximately trapezoidal.
3.2.1. Some characteristic examples of the voltage ratio PDFs To make the effect of the circuit function’s non-linearity on the shape of the voltage ratio’s PDF visible, it is assumed that the relative expanded uncertainties of the resistors in the divider (DR1/R01 and DR2/R02) are much larger than typical resistor tolerances in metrological practice. The large expanded uncertainties of the resistances, combined with the non-linearity of the function defining the voltage ratio, make higher terms of the Taylor series expansion relevant and Eq. (11) inadequate for assessing the shape of the resulting PDF. A more detailed insight into the shape of the voltage ratio’s PDF can be obtained by using MCM,
α = α 0 = 0.5
Probability density
(a)
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as described in [14]. Labeling of resistances in this section conforms to that of Fig. 1a. Fig. 2 is the result of a MCM calculation for a divider with uniformly distributed resistances, with equal nominal values and relative expanded uncertainties (R01 = R02 = 100 X and DR1/R01 = DR2/R02 = 50%). This example relates to a divider with two sensing resistors arranged in a differential connection. The histogram in Fig. 2a, with the interval width of 0.001 X, was produced on the basis of 107 trials, which means that the MCM code calculated the value of the voltage ratio for 107 pairs of values of R1 and R2, sampled independently from the corresponding uniform distributions. The envelope of the histogram is the PDF of the divider’s voltage ratio. It is symmetrical, but deviates from a triangle insofar as its sides are curvilinear with inflection points. The mean and the nominal value of ¼ a0 ¼ 0:5, represented by the the voltage ratio are equal a solid vertical line. Plot in Fig. 2b is the cumulative distribution function (CDF) which corresponds to the PDF of Fig. 1a, determined from the same MCM calculation. The symmetrical coverage interval for the voltage ratio with 95% level of confidence was found to be a95% e (0.301, 0.699). Bounds of this coverage interval are marked in Fig. 2b. For an attenuator with R01 – R02 and DR1/R01 = DR2/R02 (R01 = 60 X, R02 = 100 X, DR1/R01 = DR2/R02 = 50%), MCM results are presented in Fig. 3. Both resistances are assigned uniform distributions (R1 Unif(30 X, 90 X) and R2 Unif(50 X, 100 X)). Voltage ratio’s PDF (the envelope of the histogram) is a highly asymmetrical, skewed, curvilinear triangle. Due to the asymmetry of the PDF, the mean > a0 Þ. and the nominal value of the voltage ratio differ ða In this and all subsequent figures, the nominal and the mean value of the circuit parameter calculated by MCM are indicated by the solid and the dotted line, respectively. The calculated values, stated next to the corresponding ¼ 0:3802. vertical lines in the graph, are a0 = 0.375 and a
Voltage ratio α
(b) α 0 = 0.375
α = 0.3802
Probability
Probability density
Δα r = + 1.39 %
Voltage ratio α Fig. 2. (a) Histogram of voltage ratio values, determined by MCM with 107 trials, for a divider with equal uniformly distributed resistances (R01 = R02 = 100 X, DR1/R01 = DR2/R02 = 50%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean ¼ a0 ¼ 0:5, and the nominal value of the voltage ratio are equal a represented by the solid vertical line. (b) Voltage ratio’s CDF. Bounds of the symmetrical coverage interval with 95% level of confidence are marked at 0.301 and 0.699.
Voltage ratio α Fig. 3. Histogram of voltage ratio values, determined by MCM with 107 trials, for a divider with different nominal resistances that have equal relative expanded uncertainties (R1 Unif(30 X, 90 X) and R2 Unif(50 X, 100 X), i.e. R01 = 60 X, R02 = 100 X, DR1/R01 = DR2/ R02 = 50%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively.
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
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Relative deviation of the mean from the nominal value is r ¼ ða a0 Þ=a0 ¼ þ1:39%. It can also be noticed that, Da in spite of the asymmetry, the nominal and the most probable value of the voltage ratio coincide. Results for the general case when R01 – R02 and DR1/ R01 – DR2/R02, corresponding to the conditions in a halfbridge, are presented in Fig. 4. Example uniform distributions assigned to the resistances are R1 Unif(5 X, 15 X) and R2 Unif(15 X, 25 X), which equates to R01 = 10 X, R02 = 20 X, DR1/R01 = 50%, DR2/R02 = 25%. Voltage ratio’s PDF is shaped as an irregular curvilinear quadrilateral, with marked asymmetry. Again, the asymmetry of the PDF gives rise to a difference between the mean and the nominal va < a0 Þ. The calculated values of lue of the voltage ratio (a the nominal and the mean voltage ratio, round up to four ¼ 0:3301, which decimal places, are a0 = 0.3333 and a r ¼ 0:96%. gives Da It can be noted that the conclusions about the relation and a0 drawn from the approximate Eq. (11) in Section of a > a0 if R02 ðDR02 =R02 Þ2 > R01 ðDR01 =R01 Þ2 and 3.1, i.e. that a a < a0 if R02 ðDR02 =R02 Þ2 < R01 ðDR01 =R01 Þ2 , are confirmed by the MCM calculations for the previous two investigated dividers. The last investigated example of a voltage divider has one sensing resistor with large relative expanded uncertainty, and another high quality fixed-value resistor with low tolerance that can be modeled by zero uncertainty. MCM calculation results for two possible configurations of such a divider are presented in Fig. 5. Fig. 5a refers to the case of DR1/R01 – 0 and DR2/R02 0, while Fig. 5b refers to DR1/R01 0 and DR2/R02 – 0. Both the asymmetry of the PDF and its departure from the trapezoidal shape are particularly manifested in these cases. This is attributed to the fact that partial compensation of the circuit function’s non-linearity, achieved through the differential connection of resistors in the divider, is removed when only one of them is treated as having a distributed resis and tance, while the other is fixed to a constant value. a a0 differ as expected, with the relative deviation r j ¼ 2:16% larger than for the previously analyzed jDa dividers. In addition to the histograms, CDFs of the voltage
4. Equal-resistance Wheatstone bridge A Wheatstone bridge is one of the basic electrical circuits for measuring physical quantities with resistive sensors. This is especially true of the measurements where small changes of resistance occur, such as with strain gauges or piezoresistive semiconductor sensors of pressure and acceleration. Its structure is that of two voltage dividers connected in parallel, as presented by the circuit diagram of Fig. 6. Output signal of the bridge is the potential difference between the outputs of the two dividers. The ratio of the output voltage to the electromotive force of a source connected to the bridge is the principal circuit parameter. Assuming that the resistance of a voltmeter used to measure the bridge output voltage is much higher than the output resistance of the bridge, the bridge ratio ab can be expressed as a non-linear function of the four resistances:
ab ¼
Vo R1 R3 ¼ E R1 þ R2 R3 þ R4
ð12Þ
If stochastic deviations of the four resistances from their respective nominal values are designated dR1 = R1 R01, dR2 = R2 R02, dR3 = R3 R03, and dR4 = R4 R04, deviation of the bridge ratio from the nominal value ab0 = (R01/ (R01 + R02)) (R03/(R03 + R04)) can be approximated by the first term of the Taylor series expansion, as expounded in Section 2.2:
dab ¼ ab ab0
1 dR1 dR4 dR2 dR3 þ 4 R01 R04 R02 R03
ð13Þ
Expression (13) can be used for assessing the shape of the bridge ratio’s PDF. Since expression (13) is a linear combination of four random quantities, the resulting PDF is obtained by convoluting four uniform PDFs assigned to the resistances. This yields a symmetric bell-shaped curve,
α 0 = 0.3333 Δα r = − 0.96 %
Probability density
α = 0.3301
ratio for the two cases are shown in Fig. 5. There is a marked difference between these plots, one being concave and the other convex.
Voltage ratio α 7
Fig. 4. Histogram of voltage ratio values, determined by MCM with 10 trials, for a divider with uniformly distributed resistances that differ in both their nominal values and their relative expanded uncertainties (R1 Unif(5 X, 15 X) and R2 Unif(15 X, 25 X), i.e. R01 = 10 X, R02 = 20 X, DR1/R01 = 15%, DR2/ R02 = 25%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively.
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722
(a) α = 0.4892
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α 0 = 0.5
Probability
Probability density
Δα r = − 2.16 %
Voltage ratio α
Voltage ratio α
(b)
Δα r = + 2.16 %
Voltage ratio α
Probability
α = 0.5108
Probability density
α 0 = 0.5
Voltage ratio α
Fig. 5. Histogram of voltage ratio values and the corresponding CDF, determined by MCM with 107 trials, for a divider consisting of one resistor with uniformly distributed resistance, and another having zero uncertainty. The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The mean and the nominal value of the voltage ratio are indicated by the solid and the dotted line, respectively. (a) R1 Unif(50 X, 150 X) and R2 = 100 X, i.e. DR1/R01 – 0 and DR2/R02 = 0. (b) R1 = 100 X and R2 Unif(50 X, 150 X), i.e. DR1/R01 = 0 and DR2/R02 – 0.
R2
R4
Vo = V1 − V2 E
V1
V2
R1
R3
Fig. 6. Wheatstone bridge circuit diagram.
made up of third-power polynomial segments. The true PDF of the bridge ratio is in effect asymmetrical, since the circuit function of the bridge (Eq. (12)) is non-linear, which can be observed by the MCM calculations, as for the voltage dividers in Section 3.2.1.
A large number of different Wheatstone bridge configurations and modifications is encountered in practice. For reasons of maximum sensitivity and temperature dependence compensation, mass-produced measuring devices typically use equal-resistance Wheatstone bridges, in which all four resistances have equal nominal values R01 = R02 = R03 = R04 = R0. The nominal value of the bridge ratio for an unloaded equal-resistance Wheatstone bridge (with no measured quantity applied to any of the sensors) equals zero, ab0 = 0. One of the following three configurations is possible: (a) bridge with one active arm (containing a resistive sensor) and fixed-value resistors in other three arms, (b) bridge with two differentially connected sensors (R1 and R3 in Fig. 6) and two fixed-value resistors (R2 and R4), or (c) bridge with all four active arms. Non-sensing fixed-value resistors are commonly produced with low tolerances, i.e. approximately zero uncertainties. In the following, only the sensing resistors are treated as having distributed resistances. A Wheatstone bridge with one active arm is similar to a voltage divider with one sensor and one zero uncertainty
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fixed-value resistor. If R1 is the sensing resistor in such a bridge, and if all four nominal resistances are equal, Eq. (12) reduces to:
R1 1 R1 þ R0 2
ð14Þ
This means that the PDF of the bridge ratio is obtained by shifting the PDF of a corresponding voltage divider (Fig. 5a) to the left by 1/2. For a bridge in which only R1 and R3 are sensors, Eq. (13) becomes:
dab
1 dR1 dR3 4 R01 R03
If resistances in the previously analyzed circuits are assigned normal distributions, PDFs of the circuit parameters are bell-shaped and only approximately normal. PDFs obtained by the MCM for these circuits are presented in Figs. 8–11. Only in the case of a symmetrical voltage divider (R01 = R02 = R0) for which the resistors have the same normal distributions (R N(R0, (rR)2), where notation N(l, r2) designates a normal distribution with mathematical
α = α 0 = 0.5
ð15Þ
which is analogous to Eq. (11) for voltage dividers. It follows that PDF of the bridge ratio for this type of Wheatstone bridge has one of the shapes presented in Figs. 2, 3 or 4, depending on the specific nominal values R01 and R03, as well as on the relative expanded uncertainties DR1/R01 and DR3/R03. For an equal-resistance Wheatstone bridge of this type (R01 = R03, DR1/R01 = DR3/R03) the symmetric histogram of Fig. 2a applies. Circuit function of a Wheatstone bridge with four active arms depends on four input random variables. Due to the differential connection of the resistors in the bridge, the asymmetry of the bridge ratio’s PDF is slight, even when relative expanded uncertainties of the resistances are very large. This is illustrated by an example equal-resistance bridge in which all four resistances are uniformly distributed over the interval (50 X, 150 X). The bridge ratio’s PDF (Fig. 7) resembles that of a normal distribution, and the mean value of the bridge ratio is nearly equal to its b ab0 ¼ 0. nominal value, a
Probability density
ab ¼
5. Examples with normally distributed resistances
Voltage ratio α Fig. 8. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider with equal normally distributed resistances (R01 = R02 = 10 X, rR1 = rR1 = 2 X, i.e. rR1/R01 = rR2/R02 = 20%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The ¼ a0 ¼ 0:5, mean and the nominal value of the voltage ratio are equal a represented by the solid vertical line.
α 0 = 0.25
α = 0.25675 α m = 0.238 Δα r = + 2.7 %
Probability density
Probability density
α b ≈ α b0.= 0
Voltage ratio α Bridge ratio α b 7
Fig. 7. Histogram of voltage ratio values, determined by MCM with 10 trials, for an equal-resistance Wheatstone bridge with all four resistances uniformly distributed (R Unif(50 X, 150 X) i.e. R0 = 100 X, DR/ R0 = 50%). The interval width is 0.0001 X. Bridge ratio’s PDF is the envelope of the histogram. Two vertical lines marking the mean and the nominal value are irresolvable.
Fig. 9. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider with different nominal resistances, normally distributed with equal relative standard deviations (R1 N(1 X, (0.25 X)2) and R2 N(3 X, (0.75 X)2), i.e. R01 = 1 X, R02 = 3 X, rR1/R01 = rR2/R02 = 25%). The interval width is 0.001 X. Voltage ratio’s PDF is the envelope of the histogram. The nominal (a0), the mean ), and the most probable (am) value of the voltage ratio are indicated by (a the solid, the dotted, and the broken line.
K. Stankovic´ et al. / Measurement 44 (2011) 1713–1722 α 0 = 0.25
α = 0. 25994 α m = 0. 234
Probability density
Δα r = + 3.98 %
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examples and their standard deviations (uncertainties) are provided in figure captions. For a Wheatstone bridge with four active arms containing resistors with equal normal distributions, the resulting PDF is symmetrical and bell-shaped (Fig. 11). Accordance with an equivalent normal distribution is even more visible here than in the case of uniformly distributed resistances (Fig. 7). 6. Conclusions
Voltage ratio α Fig. 10. Histogram of voltage ratio values, determined by MCM with 3 107 trials, for a divider consisting of one resistor with normally distributed resistance, and another having zero uncertainty (R1 = 1 X and R2 N(3 X, (0.75 X)2). The interval width is 0.001 X. The nominal (a0), ), and the most probable (am) value of the voltage ratio are the mean (a indicated by the solid, the dotted, and the broken line.
Probability density
α b ≈ α b0.= 0
Bridge ratio α b Fig. 11. Histogram of voltage ratio values, determined by MCM with 107 trials, for an equal-resistance Wheatstone bridge with all four resistances uniformly distributed (R N(10 X, (2 X)2)). The interval width is 0.0001 X. Bridge ratio’s PDF is the envelope of the histogram. Two vertical lines marking the mean and the nominal value are irresolvable.
expectation l and standard deviation r), does the PDF of the voltage ratio have a symmetrical shape seen in Fig. 8. For a divider with different nominal resistances (R01 – R02), normally distributed with equal relative standard deviations with relation to their nominal values, voltage ratio’s PDF is an asymmetrical bell-shaped curve (Fig. 9), which means that the most probable, the mean and the nominal value of the voltage ratio differ. PDF of an asymmetrical divider in which R1 is a fixed-value resistor with negligible uncertainty and R2 is normally distributed is shown in ), Fig. 10. In Figs. 9 and 10 the nominal (a0), the mean (a and the most probable (am) value of the voltage ratio are indicated by the solid, the dotted, and the broken line, respectively. Specific nominal values of resistances in these
We have shown in this paper that the non-linearity of a circuit function gives rise to an asymmetry of the circuit parameter’s PDF, and that consequently the mean and the nominal value of the circuit parameter differ. This is of particular concern in mass-produced passive electrical circuits used in measurements, in which statistical variations inherent in their production may result in a discrepancy between the two values that affects measurement readings. Some of the basic circuits of this kind, such as the voltage divider and the Wheatstone bridge circuit, have been analyzed using methods originally intended for evaluating uncertainty in measurement. The presented analysis refers either to the case of preselecting circuit components on the basis of a digital instrument indication, or relying on a specified tolerance of the components acquired directly from a manufacturer. Within the framework of measurement uncertainty, identical readings of components’ parameters, observed on the display of a digital instrument, imply that the parameter value ascribed to the population of selected components represents a random quantity with a uniform distribution, centered on the indicated value, and with a half-width equal to one half of the smallest significant digit on the instrument’s indicating device. In the other case, of using components with a tolerance specified by the manufacturer, the value of the component parameter, regarded as a random quantity, is assumed to follow a normal distribution, with a relative standard deviation proportional to the tolerance. General conclusions about the shape of circuit parameters’ PDFs have been drawn based on a one-term Taylor series expansion of the circuit function. A more detailed insight into the shape of an output quantity’s PDF has been obtained by a Monte Carlo method, as an envelope to the histogram of circuit parameter’s values. MCM calculations also provide results for the mean and the nominal value of the circuit parameter. The degree of asymmetry of the resultant PDF and the deviation between the mean and the nominal value of the circuit parameter depend on the specific nominal values of component parameters, as well as on their relative expanded uncertainties. Relative deviation of the mean from the nominal value can generally be either positive or negative. Acknowledgment The Ministry of Science and Technological Development of the Republic of Serbia supported this work under Contract 171007.
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