# Comparative evaluation of the two current source supplied strain gauge bridge.

I. IntroductionThis 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-current-supplied 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 [R.sub.1] (the range of resistance change - 1 [ohm], resistance relative increment - [absolute value of [[epsilon].sub.1]] [less than or equal to] 0.01, [[epsilon].sub.2] = 0). And the second one with two metal strain gauges [R.sub.1] and [R.sub.2], (the range of the resistance modules mean average 0.1 [ohm], resultant relative increment of sensors resistance - [epsilon] = 0.5([absolute value of [[epsilon].sub.1]] + [absolute value of [[epsilon].sub.2]]) [less than or equal to] 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 [[epsilon].sub.1], [[epsilon].sub.2] 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. [absolute value of [[epsilon].sub.1] + [[epsilon].sub.2]] << 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 [[epsilon].sub.1], [[epsilon].sub.2] 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).

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

[[epsilon].sub.Wi] = W[[epsilon].sub.1i] = W [R.sub.1i] - [R.sub.10]/[R.sub.10], (1)

where W--constant (W = 100 was assumed), [R.sub.1i]--measured value of resistance for deflection [X.sub.i] (where the number of measurements i = 1 to 100), [R.sub.10] - initial resistance of a strain gauge (for deflection [X.sub.1] = 0 mm).

The resistance average relative empirical increment for a set of two foil strain gauges, however, was determined according to the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [R.sub.1i], [R.sub.2i]--measured value of resistance for deflection [X.sub.i] (where the number of measurements i = 1 to 100), [R.sub.10], [R.sub.20]--initial resistances of strain gauges (for deflection [X.sub.1] = 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:

[U.sub.AB'] [congruent to] [V.sub.A] - [V.sub.B] = J [R.sub.0]/4 ([[epsilon].sub.1] - [[epsilon].sub.2]), (3)

[U.sub.DC'] [congruent to] [V.sub.D] - [V.sub.C] = J [R.sub.0]/6 ([[epsilon].sub.1] + [[epsilon].sub.2]), (4)

It is assumed that [J.sub.1] = [J.sub.2] = 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 [DELTA]J.

As it can be observed, the voltage [U.sub.AB] changes for subsequent beam deflections [X.sub.i], and the [U.sub.DC]' 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 [[epsilon].sub.1] is always positive while [[epsilon].sub.2] is always negative, and the modules have the same values [absolute value of [[epsilon].sub.1]] = [absolute value of [[epsilon].sub.2]]). After transformations of equation (3), for a circuit with one strain gauge ([[epsilon].sub.2] = 0), equation (5) was obtained, and for the circuit with two strain gauges ([[epsilon].sub.1] > 0 [intersection] [[epsilon].sub.2] < 0 => [[epsilon].sub.1] - [[epsilon].sub.2] = [absolute value of [[epsilon].sub.1]] + [absolute value of [[epsilon].sub.2]])--equation (8), presented in Table II.

Additionally, circuits from Fig. 2 Fig. 3 were analysed, assuming that [R.sub.10] = [R.sub.20] = [R.sub.30] = [R.sub.40] = [R.sub.r] = [R.sub.0]. As a result, measurement equations of other circuits were obtained. Those equations for different configurations are also included in Table I and Table II.

Current [bar.J], existing in equations (3) and (4), is a mean average of sources [[bar.J].sub.1] and [[bar.J].sub.2] currents. It was measured through voltage decreases [[bar.U].sub.RJ1] and [[bar.U].sub.RJ2] on additional resistors [R.sub.J1] = [R.sub.J2] = [R.sub.J] of low value. In the case of the Wheatstone's bridge circuit, the current of the supply source [bar.J] was measured in the same way.

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 [[bar.[epsilon]].sub.W] in Lab VIEW 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 [[??].sub.W] were determined in this way

[[??].sub.W] = [a.sub.u] X + [b.sub.u], (11)

where [a.sub.u]--characteristics gain coefficient of the tested circuit ([a.sub.d], [a.sub.w] or [a.sub.p]),[b.sub.u]--characteristics offset coefficient of this circuit ([b.sub.d], [b.sub.w] or [b.sub.p]), where subscript stand for d--two-current-source bridge, w--Wheatstone's bridge, p--Anderson's loop.

Figure 8 presents a geometrical interpretation of absolute gain [[DELTA].sub.n] and offset [[DELTA].sub.p] errors. Likewise, relative errors of linear regression models of tested circuits were defined [12] in the following way:

[[delta].sub.n] = [[DELTA].sub.n]/[[epsilon].sub.Wmax] x 100% = [a.sub.Kj] - [a.sub.u]/[a.sub.Kj] x 100%, (12)

[[delta].sub.p] = [[DELTA].sub.p]/[[epsilon].sub.Wmax] x 100% = [b.sub.u]/[a.sub.Kj] [X.sub.max] x 100%, (13)

where [[epsilon].sub.Wmax]--measurement range (processing) [[epsilon].sub.Wmax] = [a.sub.Kj] [X.sub.max], [X.sub.max] - maximum deflection of the beam, [a.sub.K1]--reference characteristics gain coefficient of a semiconductor strain gauge (j = 1), [a.sub.K2] - 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 [[??].sub.W] = [a.sub.K1] X and [[??].sub.W] = [a.sub.K2] X were recognized as reference characteristics. Moreover, coefficients standard uncertainties [a.sub.u], [b.sub.u] of linear regression models (11) 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 [[bar.U].sub.AB]', [bar.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 [u.sub.c] ([[bar.[epsilon]].sub.W]'), considering only u ([[bar.U].sub.AB]'), u ([bar.J]) uncertainties, was calculated with the use of equation [13, Annex H]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where r ([[bar.U].sub.AB]', [bar.J]) - coefficient of correlation between input values [[bar.U].sub.AB]' and [bar.J].

During the following stage of calculations, an additional source of uncertainty, resulting from resistance dispersion [R.sub.10] = [R.sub.20] = [R.sub.30] = [R.sub.40] = [R.sub.r1] = [R.sub.r2] = [R.sub.0] of the bridge, was taken into consideration. The resistance boundary error [R.sub.0] was estimated with the total differential method, obtaining [+ or -] 0.5 %. Considering a different character of uncertainties [u.sub.c] ([[bar.[epsilon]].sub.W]') (A and B type [13], from the measurements) and u([R.sub.0]) (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

[u.sub.c] ([[bar.[epsilon]].sub.W]) [approximately equal to] [square root of ([u.sub.c.sup.2] ([[bar.[epsilon]].sub.W]') + [([delta][[bar.[epsilon]].sub.W]/[delta][R.sub.0]).sup.2] [u.sup.2] ([R.sub.0]))]. (15)

In equation (11), the resistance relative increment is the dependent variable. Different measurement uncertainties [u.sub.c] ([[bar.[epsilon]].sub.W]) were obtained for particular [[bar.[epsilon]].sub.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 [u.sub.c] ([[bar.[epsilon]].sub.W]) have different values, the line coefficients ([a.sub.u] and [b.sub.u]) were determined with the weighted least squares regression method [11]. Expanded uncertainties U([a.sub.u]) and U([b.sub.u]), 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)

[s.sub.e] = [square root of ([L.summation over (m=1)] [([[??].sub.Wm] - [[bar.[epsilon]].sub.Wm]).sup.2])]/L - K], (16)

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

[[bar.[epsilon]].sub.W] = [[SIGMA].sup.L.sub.m=1] [[bar.[epsilon]].sub.Wm]/L [right arrow] [s.sub.ew] = [s.sub.e]/[absolute value of [[bar.[epsilon]].sub.W] x 100%. (17)

VIII. MEASUREMENT RESULTS AND ANALYSIS

Estimated parameters [a.sub.u] and [b.sub.u] of linear regression models, their expanded uncertainties U([a.sub.u]) and U([b.sub.u]), as well as average relative estimation errors [s.sub.ew] 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(b) and Fig. 10(b)) for models without dispersion [R.sub.0] are quite big (over 10 %).

Low resistance increment measurement (up to 1 [ohm]) 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 [R.sub.0] of [+ or -] 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 [[??].sub.W] = [a.sub.u] X + [b.sub.u] 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 [R.sub.0]).

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 [R.sub.0] than the classic quarter-bridge (Fig. 11). Moreover, a better adjustment (smaller average estimation error [s.sub.ew]) 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 Ro than the 2J+2R bridge (Fig. 12).

Resistance Ro occurrence in measurement equations (5), (6), (8), (9) is a drawback of bridges in relation to Anderson's loop. If the [R.sub.0] 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.

http://dx.doi.org/10.5755/j01.eie.22.6.17220

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Wojciech Walendziuk (1), Jerzy Golebiowski (1), Adam Idzkowski (1)

(1) Department of Electrical Engineering, Bialystok University of Technology, Wiejska 45D St., 15-351 Bialystok, Poland w.walendziuk@pb.edu.pl

TABLE I. MEASUREMENT EQUATIONS OF TESTED CIRCUITS. Circuit with one strain gauge ([[bar.[epsilon].sub.W] = W [[epsilon].sub.1]) Two-current- [[bar.[epsilon]].sub.W] = (5) source bridge 4W[[bar.U].sub.AB]'/[bar.J][R.sub.0] (2J+2R) Wheatstone's [[bar.[epsilon]].sub.W] = 4W[[bar.U].sub.DC]'' (6) bridge /[bar.J][R.sub.0]-[[bar.U].sub.DC"] Anderson's [[bar.[epsilon]].sub.W] = (7) loop W([[bar.U].sub.AB]"' - [[bar.U].sub.C]"')/ [[bar.U].sub.C]"' TABLE II. MEASUREMENT EQUATIONS OF TESTED CIRCUITS. Circuit with one strain gauge ([[bar.[epsilon].sub.W] = 0.5W ([absolute value of [[epsilon].sub.1]] + [absolute value of [[epsilon].sub.2]]) Two-current- [[bar.[epsilon]].sub.W] = (8) source bridge 2W[[bar.U].sub.AB]'/[bar.J][R.sub.0] (2J+2R) Wheatstone's [[bar.[epsilon]].sub.W] = 2W[[bar.U].sub.DC]" (9) bridge [bar.J][R.sub.0] Anderson's [[bar.[epsilon]].sub.W] = (10) loop W([[bar.U].sub.AB]"' - [[bar.U].sub.BC]"')/2[[bar.U].sub.C]"' TABLE III. COMPARISON OF PARAMETERS [a.sub.u] AND [b.sub.u] OF THE DETERMINED MODELS, THEIR EXPANDED UNCERTAINTIES U([a.sub.u]), U([b.sub.u]), (FOR k = 2 AND p = 95%) AND AVERAGE RELATIVE ESTIMATION ERRORS FOR CIRCUITS WITH ONE SEMICONDUCTOR STRAIN GAUGE (THE RANGE OF DEFLECTION [X.sub.max] = 1 mm, [[epsilon].sub.Wmax] = 0.866201). Two-current-source [a.sub.d] [b.sub.d] U([a.sub.d]) bridge 2J+2R 0.5 % dispersion [R.sub.0] 0.915619 0.152809 0.000075 without dispersion [R.sub.0] 0.875020 0.174259 0.000017 Wheatstone's bridge [a.sub.w] [b.sub.w] U([a.sub.w]) 0.5 % dispersion [R.sub.0] 1.017133 0.090555 0.000133 without dispersion [R.sub.0] 0.935297 0.124349 0.000032 Anderson's loop [a.sub.p] [b.sub.p] U([a.sub.p]) 0.874489 0.578160 0.000060 Two-current-source U([b.sub.d]) [s.sub.ew] [%] bridge 2J+2R 0.5 % dispersion [R.sub.0] 0.000023 4.12 without dispersion [R.sub.0] 0.000009 3.13 Wheatstone's bridge U([b.sub.w]) [s.sub.ew] [%] 0.5 % dispersion [R.sub.0] 0.000041 7.59 without dispersion [R.sub.0] 0.000020 5.00 Anderson's loop U([b.sub.p]) [s.sub.ew] [%] 0.000032 1.71 Note: parameter of the reference (Keithley) model [[??].sub.W] = [a.sub.K1]X ([a.sub.K1] = 0.866201) and average relative estimation error [s.sub.ew] = 1.16 % for L = 100, K = 1.A55 TABLE IV. COMPARISON OF COEFFICIENTS [a.sub.u] AND [b.sub.u] 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 [X.sub.max] = 10 mm, [[epsilon].sub.Wmax] = 0.083545). Two-current-source [a.sub.d] [b.sub.d] U([a.sub.d]) bridge 2J+2R 0.5 % dispersion 0.00840671 -0.10963994 0.00000073 [R.sub.0] without dispersion 0.00831815 -0.10960964 0.00000012 [R.sub.0] Wheatstone's bridge [a.sub.w] [b.sub.w] U([a.sub.w]) 0.5 % dispersion 0.00882729 0.00260378 0.00000012 [R.sub.0] without dispersion 0.00882690 0.00260405 0.00000011 [R.sub.0] Anderson's loop [a.sub.p] [b.sub.p] U([a.sub.p]) 0.00837743 -0,01357607 0.00000006 Two-current-source U([b.sub.d]) [s.sub.ew] bridge 2J+2R 0.5 % dispersion 0.00000441 0.12 [R.sub.0] without dispersion 0.00000065 0.12 [R.sub.0] Wheatstone's bridge U([b.sub.w]) [S.sub.ew] 0.5 % dispersion 0.00000040 0.92 [R.sub.0] without dispersion 0.00000040 0.92 [R.sub.0] Anderson's loop U([b.sub.p]) [S.sub.ew] 0.00000038 0.26 Note: parameter of the reference (Keithley) model [[??].sub.W] = [a.sub.K2] X ([a.sub.K2] = 0.00835456) and average relative estimation error [S.sub.ew] = 3.13 % for L = 100, K = 1. Fig. 9. Gain errors (a) and offset errors (b) for three tested circuits with one semiconductor strain gauge (models without dispersion [R.sub.0]). a) two-current bridge 2J+2R -1.0 Wheatstone quarter bridge -8 0 Anderson loop -1.0 b) two-current bridge 2J+2R 20.1 Wheatstone quarter bridge 14.3 Anderson loop 66.7 Note: Table made from bar graph. Fig. 10. Gain errors (a) and offset errors (b) for three tested circuits with two metal strain gauges (models without dispersion [R.sub.0]). a) two-current bridge 2J+2R 0.4 Wheatstone quarter bridge -5.7 Anderson loop -0.3 b) two-current bridge 2J+2R -131.2 Wheatstone quarter bridge 3.1 Anderson loop -16.2 Note: Table made from bar graph. 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 [R.sub.0] as an input quantity affecting the coefficients uncertainty [a.sub.u] and [b.sub.u]. difference in difference in [[delta].sub.p][%] [[delta].sub.n][%] two-current bridge 2J+2R 3.37 4.69 Wheatstone half bridge 3.90 9.45 Note: Table made from bar graph. 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 [R.sub.0] as an input quantity affecting the coefficients uncertainty [a.sub.u] and [b.sub.u]. difference in difference in [[delta].sub.p][%] [[delta].sub.n][%] two-current bridge 2J+2R 0.04 1.06 Wheatstone half Bridge 0.01 0.01 Note: Table made from bar graph.

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Author: | Walendziuk, Wojciech; Golebiowski, Jerzy; Idzkowski, Adam |
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Publication: | Elektronika ir Elektrotechnika |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Dec 1, 2016 |

Words: | 4230 |

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