Comparative Evaluation of the Two Current Source Supplied Strain Gauge Bridge

Metrological properties of a two-current-source bridge circuit were tested with the use of the method of measuring resistance increments of strain gauges. An unconventional system was investigated in comparison with the commonly used Wheatstone’s half-bridge, quarter-bridge and Anderson’s loop. Input-output characteristics of the systems tested with a current supply were examined experimentally. Error values of offset and gain of the characteristics in relation to the characteristics of reference were taken as the criterion of comparison. Moreover, standard uncertainties of y-intercept and slope coefficients (of the straight lines) were analysed. The coefficients with their uncertainties are presented in tables. Errors for three tested systems with two metal strain gauges or with one semiconductor are presented on graphs. Additionally, the errors change, resulted from the spread of initial resistances as the quantity influencing the uncertainties of offset and gain coefficients, was defined for the bridge circuits. DOI: http://dx.doi.org/10.5755/j01.eie.22.6.17220


I. INTRODUCTION
This article presents an attempt to compare metrological properties of selected direct current supplied systems, i.e. a two-current-source bridge [1], a Wheatstone bridge [2]- [4] and an Anderson's loop [5].It is widely known that the type of the system used in a device influences the linearity of the output voltage and the sensitivity of the system to the measured quantity change [3], [6], [7].It is described in articles where strain gauge deflection measurements are presented.Works [8], [9] show significant differences between parameters values of a regular voltage supplied quarter-bridge and a two-current-source supplied system.The author of [8], [9] analysed nonlinearity errors and sensitivity changes of the output voltage at a great range of metal strain gauge deflection for both systems.It is only briefly mentioned that the ratio of the output voltage to the supply voltage is two times greater in the Anderson's loop than in the Wheatstone's bridge (at equal power dissipation in its elements).In the Anderson's system it was also possible to obtain a greater ratio of signal to noise (6 dB) [10].
The aim of experiments presented in this paper is to examine the usefulness of a 2J+2R two-current-source bridge in indirect resistance measurements.Other measurements were conducted at the same time with the use of the same type sensors applied in commonly used systems: a Wheatstone's current supplied bridge and an Anderson's loop.Input-output characteristics of the tested systems were determined experimentally.The values of the obtained offset errors and gain errors towards the characteristics of reference were taken as the criterion of comparison.The authors consider these elements of the article to be original.

II. TESTED DC MEASUREMENT SYSTEMS AND THE REFERENCE SYSTEM
The following systems were tested: 2J+2R two-currentsupplied bridge (Fig. 1), Wheatstone's bridge (Fig. 2), Anderson's loop (Fig. 3).A Keithley 2000 multimeter is a reference system (Fig. 4).Two configurations of each system are analysed respectively.First, one with one semiconducting strain gauge R1 (the range of resistance change -1 Ω, resistance relative increment -|ε1| ≤ 0.01, ε2 = 0).And the second one with two metal strain gauges R1 and R2, (the range of the resistance modules mean average -0.1 Ω, resultant relative increment of sensors resistance -ε = 0.5(|ε1| + |ε2|) ≤ 0.001).The earlier analysis of a two-current-source circuit provided information about the range of linearity of output voltages in the function of resistance relative increments of ε1, ε2 sensors.The range of relative increment of sensors resistance is relevant to the linearity condition of output voltages of a two-current-source bridge, i.e. |ε1 + ε2| << 1. Laboratory measurements showed that relative errors of the measured increments differences and sums of two resistance variables are not greater than 9.7 %.
The tested systems were built of identical elements, which enabled reliable comparison of characteristics parameters.Additionally, the same sensor (or a set of identical sensors) was used.Strain gauges working conditions were also identical for each case, e.g.equal values of power emitted by a sensor (or sensors) and the same temperature of its activity.

III. THE WAY OF STRAIN GAUGES DISTRIBUTION ON A METAL BEAM AND THE BENDING MECHANISM
The strain gauges were stuck on thin, cuboidal beams made of tool steal.On the top surface of the first beam one semiconductor gauge AP 120-6-12 (OPS Gottwaldov) was placed.On the other case, one metal gauge (foil) TF-3/120 (Tenmex) was stuck on each side of the beam (top and bottom), at the same distance from the point of its attachment (Fig. 5).
The resistance increments of the strain gauges ε1, ε2 were imposed by a mechanism deflecting the beam with the use of a micrometer screw gauge and providing a good repetitiveness of the deflections (Fig. 6).Fig. 5.The way of metal strain gauges distribution on a one-sided attached cantilever beam undergoing deflection (X).Symbols: ε1 -positive relative increment of the top gauge resistance (the gauge is stretched), ε2 -negative relative increment of the bottom gauge (the gauge is compressed).Fig. 6.Laboratory stand with a mechanism enabling simple bending of the beam.The photo presents a cantilever beam being deflected (black dotted line color).The micrometer screw gauge is used to deflect the beam.

IV. AMPLIFIED RELATIVE RESISTANCE INCREMENT IN THE REFERENCE SYSTEM
The resistances of strain gauges in the reference system were measured directly with the use of a precise Keithley 2000 multimeter.The resistance relative increments were multiplied by a constant W. Its value equals the voltage amplification of amplifiers applied to the outputs of the systems shown in Fig. 1-Fig.3.
The resistance relative empirical increment of a semiconductor strain gauge was determined according to the following equation where W -constant (W = 100 was assumed), R1i -measured value of resistance for deflection Xi (where the number of measurements i = 1 to 100), R10 -initial resistance of a strain gauge (for deflection X1 = 0 mm).The resistance average relative empirical increment for a set of two foil strain gauges, however, was determined according to the following equation where R1i, R2i -measured value of resistance for deflection Xi (where the number of measurements i = 1 to 100), R10, R20 -initial resistances of strain gauges (for deflection X1 = 0 mm).

V. MEASUREMENT EQUATIONS OF DC TESTED CIRCUITS
The analysed circuit can work with one pair of resistance sensors and may be used to measure two increments, as well as the sum and difference of resistances, at the same time.
The following equations ( 3) and ( 4) can be used as the measurement equations for a two-current-source bridge circuit: It is assumed that J1 = J2 = J because an inequality of currents results in additional components of ( 3) and ( 4).Then the output voltages depend also on a difference of currents ΔJ.
As it can be observed, the voltage UAB' changes for subsequent beam deflections Xi, and the UDC' is close to zero.This derives from equations ( 3), ( 4) and from the way of strain gauges arrangement on the beam presented in Fig. 5 (during beam deflecting, the increment ε1 is always positive while ε2 is always negative, and the modules have the same values |ε1| = |ε2|).After transformations of equation ( 3), for a circuit with one strain gauge (ε2 = 0), equation ( 5) was obtained, and for the circuit with two strain gauges (ε1 > 0 ∩ ε2 < 0 => ε1 -ε2 = |ε1| + |ε2|) -equation ( 8), presented in Table II.
Additionally, circuits from Fig. 2 Fig. 3 were analysed, assuming that R10 = R20 = R30 = R40 = Rr = R0.As a result, measurement equations of other circuits were obtained.Those equations for different configurations are also included in Table I Anderson's loop ( '''-''') ''' Current J , existing in equations ( 3) and ( 4), is a mean average of sources 1 J and 2 J currents.It was measured through voltage decreases of low value.In the case of the Wheatstone's bridge circuit, the current of the supply source J was measured in the same way.Circuit with two strain gauges ( Anderson's loop

VI. CRITERION OF CIRCUITS COMPARISON AND DATA ACQUISITION
The values of gain and offset errors of appropriate processing characteristics were taken as the comparison criterion of the tested circuits.The fact of the eleven-fold beam deflection X (Fig. 5, Fig. 6) in each configuration was the starting point of the research.As a result, output voltages (Fig. 1-Fig.3) occurred.They were amplified one hundred times, measured and averaged (out of 200 samples) in a data acquisition system presented in Fig. 7 (LabJack UE-9Pro).This enabled calculating strain gauges resistance average relative increments W  in LabVIEW with the use of equations ( 5)- (10).Thereafter, measurement results were worked out with the weighted least squares regression method [11].Estimators of average relative increments ˆW  were determined in this way where au -characteristics gain coefficient of the tested circuit (ad, aw or ap), bu -characteristics offset coefficient of this circuit (bd, bw or bp), where subscript stand for d -twocurrent-source bridge, w -Wheatstone's bridge, p -Anderson's loop.Figure 8 presents a geometrical interpretation of absolute gain Δn and offset Δp errors.Likewise, relative errors of linear regression models of tested circuits were defined [12] in the following way: max 100% 100%, 100% 100%, where εWmax -measurement range (processing) εWmax = aKj Xmax, Xmax -maximum deflection of the beam, aK1reference characteristics gain coefficient of a semiconductor strain gauge (j = 1), aK2 -reference characteristics gain coefficient of two metal strain gauges (j = 2).As it can be observed, errors ( 12), ( 13) were determined by comparing linear regression models (determined for three circuits) with a reference model (regarded as close to ideal).Those errors should have possibly smallest values.Reference models were determined regarding the data obtained as a result of gradual, linear deflection of strain gauges and direct measurements of their resistance changes with a precise Keithley 2000 multimeter (Fig. 4).Regression lines were recognized as reference characteristics.Moreover, coefficients standard uncertainties au, bu of linear regression models were calculated [11].
The proposed comparisons of parameters let us evaluate metrological properties of a two-current-source bridge 2J+2R in collation with classic measurement systems.

VII. UNCERTAINTY ANALYSIS OF REGRESSION LINES COEFFICIENTS
The uncertainties of resistance relative increments were calculated assuming that the input values ', AB U J in equation (3) were correlated.According to the GUM guide [13], all standard uncertainties were denoted by small letters u.The combined uncertainty of the resistance relative increment uncertainties, was calculated with the use of equation [13, Annex H] (A and B type [13], from the measurements) and u(R0) (B type, from estimations), they were geometrically added, in compliance with the rule of uncertainty propagation [13].The approximated combined standard uncertainty value of the resistance relative increment measurement was obtained in this way In equation ( 11), the resistance relative increment is the dependent variable.Different measurement uncertainties were obtained for particular Wm  (m = 1, 2 to 11).
The uncertainties result from sources of both A and B types.Whereas deflection X is an independent variable.
As uncertainties have different values, the line coefficients (au and bu) were determined with the weighted least squares regression method [11].Expanded uncertainties U(au) and U(bu), however, were determined taking into account the coverage factor k = 2 and the confidence level p = 95 %.Additionally, average estimation error (square of residual variance) was calculated for each model ( 5)-( 10) where L -number of observation (L = 11), K -number of estimated parameters (K = 2).The relative average estimation error was related to the average increment module 1 100%.

VIII. MEASUREMENT RESULTS AND ANALYSIS
Estimated parameters au and bu of linear regression models, their expanded uncertainties U(au) and U(bu), as well as average relative estimation errors sew are given in Table III and Table IV.
Figure 9-Fig.12 was made on the basis of ( 12), ( 13) and Table II, Table III.The differences between particular tested circuits are visible.Except for Wheatstone's half-bridge, offset errors (Fig. 9 Low resistance increment measurement (up to 1 Ω) is one reason of this situation.It is also worth stressing that in both experiments gain errors for a two-current-source bridge appeared significantly smaller (Fig. 9(a), Fig. 10(a)).As it can be observed in Fig. 11 and Fig. 12, accepting resistance dispersion R0 of ± 0.5 % value affects the change in both gain and offset errors.Wheatstone's half-bridge with two metal strain gauges appeared to be the least sensitive to the circuits initial resistances dispersion (Fig. 11, Fig. 12).

IX. CONCLUSIONS
The following conclusions and remarks can be formed on the basis of the research results: For a two-current-source bridge 2J+2R with two metal sensors (Table III), the uncertainty of linear models parameters ˆW u u a X b    reach the greatest values.In the case of cooperation with semiconductor sensors (Table II), those parameters reach the greatest values for the Wheatstone's quarter-bridge (model with dispersion R0).
In the case of circuits with one semiconducting strain gauge (Fig. 9(a)), smaller values of gain error were obtained for the Anderson's loop and 2J+2R bridge than for the Wheatstone's quarter-bridge.The two-current-source bridge 2J+2R appeared to be less sensitive to resistance dispersion R0 than the classic quarter-bridge (Fig. 11).Moreover, a better adjustment (smaller average estimation error sew) of the linear model to empirical data from the 2J+2R bridge in relation to data from the Wheatstone's bridge (Table III) was obtained.
In the other experiment (with two metal strain gauges), smaller values of gain error were also obtained for the Anderson's loop and the two-current-source circuit, whereas greater values -for the Wheatstone's half-bridge (Fig. 10(a)).The Wheatstone's half-bridge appeared to be significantly more sensitive to resistance dispersion R0 than the 2J+2R bridge (Fig. 12).
Resistance R0 occurrence in measurement equations ( 5), ( 6), ( 8), ( 9) is a drawback of bridges in relation to Anderson's loop.If the R0 value is defined imprecisely in a two-current-source bridge, it affects both the gain and the offset error (Fig. 11, Fig. 12).
The interpretation of the results was done without considering the influence of parameters of operational amplifiers on the measurement uncertainty.Identical amplifiers were applied in all three tested circuits.It was accepted that they have the same influence on the circuits input-output characteristics.
The unconventional circuit 2J+2R allows to measure two parameters simultaneously.It can be utile in industry where there is a need to measure mechanical strain and the change of temperature of strain gauges in a specific localization.A disadvantage is that two current sources in the circuit should provide equal currents.
In the research presented above, the influence of one parameter (mechanical deflection) on the resistance increment of sensors was analysed.Further work will concern a two-current-source bridge application in simultaneous measurements of two parameters, e.g.deflection and temperature.
Manuscript received 22 November, 2016; accepted 8 June, 2016.The paper was prepared at Bialystok University of Technology within a framework of the S/WE/1/2013 project funded by Ministry of Science and Higher Education.

Fig. 4 .
Fig. 4. Direct measurement of resistance with the use of a multimeter (reference system).

Fig. 8 .
Fig. 8.The way of determining characteristics offset Δp and gain Δn errors of tested circuits towards the reference characteristics (Keithley).
During the following stage of calculations, an additional source of uncertainty, resulting from resistance dispersion R10 = R20 = R30 = R40 = Rr1 = Rr2 = R0 of the bridge, was taken into consideration.The resistance boundary error R0 was estimated with the total differential method, obtaining ± 0.5 %.Considering a different character of uncertainties 0.866201) and average relative estimation error ew s = 1.16 % for L = 100, K = 1.

Fig. 11 .
Fig. 11.Offset/gain error change for a two-current-source bridge 2J+2R and a Wheatstone's bridge (with one semiconductor strain gauge) after considering dispersion R0 as an input quantity affecting the coefficients uncertainty au and bu.

Fig. 12 .
Fig.12.Offset/gain error change for a two-current-source bridge 2J+2R and a Wheatstone's bridge (with two metal strain gauges) after considering dispersion R0 as an input quantity affecting the coefficients uncertainty au and bu.

TABLE I .
MEASUREMENT EQUATIONS OF TESTED CIRCUITS.
Circuit with one strain gauge (1

TABLE II .
MEASUREMENT EQUATIONS OF TESTED CIRCUITS.

TABLE III
Note: parameter of the reference (Keithley) model

TABLE IV .
COMPARISON OF COEFFICIENTS au AND bu OF THE DETERMINED STRAIGHT LINES AND THEIR EXPANDED UNCERTAINTIES (FOR k = 2 AND p = 95 %) AND AVERAGE RELATIVE ESTIMATION ERRORS FOR CIRCUITS WITH TWO METAL STRAIN GAUGES (THE RANGE OF DEFLECTION Xmax = 10 mm, εWmax = 0.083545).
Note: parameter of the reference (Keithley) model