Validation of combinatorial evaluation of strain-gauge amplifier linearity


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 205 

ACTA IMEKO 
ISSN: 2221-870X 
December 2020, Volume 9, Number 5, 205 - 209 

 
 

VALIDATION OF COMBINATORIAL EVALUATION OF STRAIN-GAUGE 

AMPLIFIER LINEARITY 
 

M. Hiti1 

 
1 Slovenian National Building and Civil Engineering Institute (ZAG), Ljubljana, Slovenia, miha.hiti@zag.si  

 

Abstract: 

This paper describes a validation of a 

combinatorial calibration technique based 

calibration procedure for strain-gauge amplifier 

calibration. The deviation between two strain-gauge 

amplifiers at calibration using the combinatorial 

technique is compared to the deviation between the 

same two amplifiers at calibration on a force 

calibration machine as a transducer-amplifier chain 

under unchanged conditions, serving as an amplifier 

linearity comparator. This enables the validation to 

be confirmed with a suitable expanded 

measurement uncertainty of 3 nV/V. 

Keywords: strain-gauge amplifier; calibration; 

combinatorial technique; validation 

1. INTRODUCTION 

The calibration of strain-gauge amplifiers 

(bridge amplifiers) used in measuring chains for 

force, torque and pressure measurements is 

potentially subjected to large measurement 

uncertainties when calibrated with traditional 

voltage ratio (mV/V) standards – bridge standards. 

In previous publication [1] we presented an 

alternative calibration procedure which reduces the 

calibration uncertainty to a more acceptable level by 

calibrating the bridge amplifier at a single reference 

point with a traditional bridge standard and 

additionally performing linearity evaluation 

between the relative 0 mV/V and the calibrated 

reference point using the ZAG LinCheck 

combinatorial system [2]. This procedure has 

already been offered to customers as a calibration 

service within ISO/IEC 17025 accredited scope 

with an expanded relative measurement uncertainty 

of 3 nV/V or 0.003 % (whichever is higher) in the 

range from 0.05 mV/V to 2.5 mV/V for positive and 

negative voltage ratios, covering the typically 

required range, Figure 1. 

One of the requirements at assessment of a 

calibration laboratory’s competence regarding 

accredited services is the validation of calibration 

procedures. While the combinatorial procedure has 

been reproduced also by other laboratories to verify 

the linearity of amplifiers with comparable results 

[3], the validation of the procedure by comparison 

with a typical best-available calibrated voltage ratio 

standards is not sufficient. The measurement 

uncertainty of the latter exceeds the potentially 

achievable measurement uncertainty of the 

combinatorial calibration procedure even after the 

best available calibration procedure for bridge 

standard calibration has been improved, reducing 

the expanded uncertainty for bridge standard 

calibration from 10 nV/V to 5 nV/V (for k = 2) [4]. 

Lower uncertainties for linearity evaluation were 

not possible until recently, when a cascaded 

inductive voltage divider setup was presented for 

bridge amplifier linearity evaluation, achieving 

expanded uncertainty of 2 nV/V [5] and offering 

107 steps within the ±5 mV/V range. 

 

Figure 1: Expanded measurement uncertainty for bridge amplifier calibration with traditional bridge standards calibration 

and combinatorial procedure (left) and expressed as relative expanded uncertainty (right) 

http://www.imeko.org/
mailto:miha.hiti@zag.si


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 206 

A direct comparison of the combinatorial 

procedure with the newly presented unique 

cascaded inductive voltage divider setup could 

provide the reference needed for the validation, but 

in this paper we present a different approach: to use 

a force calibration machine as a relative standard for 

voltage ratio linearity comparison. Force calibration 

machines are widely available at NMIs and 

calibration laboratories and offer expanded 

measurement uncertainties of generated force down 

to 0.001 %. When calibration of a single force 

transducer is repeated with different bridge 

amplifiers under unchanged conditions, the 

amplifier linearity could be compared by analysing 

the calibration results, and thus acting as a 

comparator for relative bridge linearity evaluation. 

2. DESCRIPTION OF THE WORK  

Previous investigation of calibration of the same 

transducer with different amplifiers has shown that 

the agreement between comparable amplifiers of 

the same type and supply voltage can be within 

0.003 % [6], Figure 2. 

 
Figure 2: Results of deviation of transducer calibration 

with different amplifiers compared to an HBM DMP41 

(amplifier sensitivity errors corrected) 

In these investigations, the same force transducer 

(1 kN) was calibrated with different amplifiers: 

225 Hz carrier frequency type amplifiers HBM 

DMP41, HBM ML38B and HBM DK38, 4.8 k Hz 

carrier frequency type amplifier HBM ML55B and 

a DC excitation type amplifier HBM ML10B. Each 

of these amplifiers was sequentially connected to 

the same transducer and measurements were 

performed in the same 1 kN dead-weight force 

calibration machine (FCM) with < 0.002 % 

reproducibility (the accredited CMC of the force 

calibration machine is 0.005 %, but this value 

includes many uncertainty contributions which are 

irrelevant in the case of comparative measurements). 

The transducer and the force calibration machine 

remained in unchanged configuration, as the aim of 

this investigation was to compare the effect of 

different amplifiers on the calibration result. 

Measurements with one ML38B amplifier were 

done at the beginning of the investigation, at the end 

of the investigation, and between measurements 

with other amplifiers to verify the stability and 

reproducibility of the measurement setup and 

measurement procedure itself. Before each 

measurement series, each amplifier was calibrated 

by a calibrated bridge standard HBM K3608 with 

expanded measurement uncertainty of 20 nV/V for 

the 225 Hz carrier frequency, 330 nV/V for 4.8 kHz 

carrier frequency, and 200 nV/V for DC excitation 

voltage. 

The deviation of resulting characteristics in 

Figure 2 shows the indication deviation of each 
amplifier from a reference characteristic - in this 

case the characteristic of DMP41 amplifier. But 

these measurements offered also the possibility to 

compare the linearity of the amplifiers, as the 

relevant information can be extracted from the 

results. Results suggested that the amplifier linearity 

could be very good, as can typically also be 

observed by calibration with voltage ratio standards, 

albeit with a high measurement uncertainty 

(> 20 nV/V), and the much lower relative 

uncertainty of the force measurement setup could 

help to prove these assumption. 

 
Figure 3: Measurement setup with 1 kN force transducer 

positioned in 1 kN force calibration machine with HBM 

DMP41 amplifier and HBM MGCplus system with 

ML38B amplifier. A HBM K3608 bridge standard and a 

ZAG LinCheck system for combinatorial calibration are 

also shown. 

Based on these results the measurements with 

the 1 kN HBM Z30A-TOP transducer in the 1 kN 

dead-weight force calibration machine were 

repeated for the HBM DMP41 and HBM ML38B 

amplifiers under more stringent conditions with the 

aim to investigate the comparability of linearity of 

these amplifiers, Figure 3. 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 207 

     
Figure 4: Deviation of two different amplifiers from 

linear characteristics (0 mV/V – 2 mV/V) for the same 

force transducer in the same FCM – results normalized to 

2 mV/V 

For the best possible reproducibility of the force 

steps, the force transducer was positioned in a single 

rotational position, its position was not changed 

during the whole process, and the loading frame of 

the force calibration machine (the first 10 N weight) 

was never unloaded to keep the force introduction 

as repeatable as possible. This way most uncertainty 

sources arising from the force calibration machine 

and the force transducer have been significantly 

reduced or eliminated, such as the effects of 

machine and transducer alignment, reproducibility 

of force transducer in different transducer rotations 

and the effect of change in load application. By 

eliminating most uncertainty sources, the 

uncertainty of force results mainly from the 

repeatability of the relative force steps. The 

contributions of mass and gravitational field only 

influence the process as relative terms during the 

short measurement period and their exact absolute 

values do not even have to be known. To reduce the 

effect of force transducer creep, the force calibration 

machine was controlled in automatic mode with 

constant time intervals between each measurement 

and the same pre-defined loading profile for all 

measurements. Both amplifiers were constantly 

powered on and a dummy load was used to simulate 

the transducer load when the actual 1 kN force 

transducer was occupied by the other amplifier, to 

keep the operating conditions of the amplifiers as 

constant as possible. Amplifier parameters were set 

to the same values on both amplifiers: 5 V 

excitation voltage, 0.1 Hz Bessel filter settings. The 

selected time interval between measurements was 

60 s. The final measurements were done 

sequentially within two hours. Measurements were 

performed in the range from 10 N to 990 N in an A-

B-B-A sequence (DMP41-ML38B-ML38B-

DMP41). The same amplifiers were later calibrated 

also with the ZAG LinCheck system for 

combinatorial calibration. 

3. RESULTS  

The results of calibration of the 1 kN transducer 

with the DMP41 and the ML38B are shown in 

Figure 4, where the characteristic sensitivities are 

normalized to 2 mV/V. The data obtained from the 

calibration of the amplifiers in the FCM includes 

also the transducer nonlinearity contribution, which 

is in this case unknown, so the nonlinearity of the 

amplifier cannot be simply extracted from the 

results. However, the difference in nonlinearity 

results can be used as a reference for the comparison 

as it should depend mainly on the amplifier, if other 

system parameters are kept constant. 

It can be seen in Figure 4 that both measured 

DMP41 series and both ML38B series are in good 

agreement, with the largest deviation between all 

series being 3 nV/V (at 1.2 mV/V output ratio). The 

deviation between average ML38B and average 

DMP41 characteristics is shown in Figure 5 as 

absolute and relative deviation.  

 

Figure 5: Deviation between average ML38B and average DMP41 as absolute deviation (left) and relative deviation 

(right). Standard uncertainty of the deviation is also shown with error bars (contributions of repeatability and resolution). 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 208 

    
Figure 6: Linearity evaluation of deviation of two 

different amplifiers from linear characteristics (0 mV/V 

to 2 mV/V) with the ZAG LinCheck system. Results 

adjusted at 2 mV/V. 

The result suggests that the linearity of the 

amplifiers agrees for the most part, with deviations 

of less than 2 × 10-5. For the uncertainty attributed 

to the measured deviation, contributions of the 

repeatability of two series and contributions of 

amplifier resolution were taken into account. It is 

within 2 nV/V or up to 5 × 10-5 for the lowest ratios. 

Immediately after the linearity measurements on the 

force calibration machine, both amplifiers were also 

evaluated with the ZAG LinCheck system 

employing the combinatorial method. The amplifier 

parameters were the same as during measurements 

on the force calibration machine, but the evaluated 

range was in this case up to 2.5 mV/V, Fehler! 

Verweisquelle konnte nicht gefunden werden.. 

The resulting characteristics are shown with 

standard uncertainty of nonlinearity determination 

of 0.9 nV/V for DMP41 and 1.2 nV/V for ML38B. 

The nonlinearity results of both amplifiers have 

been adjusted to a 2 mV/V reference point.  

For comparison with linearity results obtained 

by evaluation of the amplifiers on the force 

calibration machine, the difference between 

ML38B and DMP41 results was calculated, shown 

in Figure 7 as absolute and relative difference. The 

determined nonlinearity difference of the two 

amplifiers using combinatorial method up to 

2 mV/V is within 2 nV/V, below 2 × 10-6 if 

expressed as relative deviation, with standard 

uncertainty up to 7.5 × 10-5. 

     

Figure 7: Deviation between ML38B and DMP41 as absolute deviation (left) and relative deviation (right) measured by 

employing combinatorial method (ZAG LinCheck). Standard uncertainty of the deviation is also shown with error bars. 

   

Figure 8: Comparison of nonlinearity deviation between ML38B and DMP41 determined in a FCM (FCM) and by 

combinatorial method (comb). Standard uncertainty of the deviation is shown with error bars. 

http://www.imeko.org/


ACTA IMEKO | www.imeko.org December 2020 | Volume 9 | Number 5 | 209 

The comparison of both methods of linearity 

evaluation is shown in Figure 8. The figure shows 

the deviation in nonlinearity difference of both 

amplifiers, once obtained from calibration in the 

FCM and once obtained by combinatorial 

calibration. By comparing the results from both 

methods, it can be seen that the two methods agree 

to within their respective standard uncertainties. 

4. DISCUSSION 

The evaluation of the linearity of amplifiers with 

different methods produced comparable results. As 

the combinatorial method has already been 

demonstrated to produce comparable results to 

traditional linearity evaluation with bridge 

standards, with large uncertainty of the latter, the 

additional linearity comparison via force calibration 

machine confirms the suitability of the 

combinatorial calibration procedure for amplifier 

calibration. The expanded measurement uncertainty 

of the comparison was 3 nV/V. This confirmation 

suggests, that it is possible to realize the linearity 

evaluation of 225 Hz carrier frequency amplifiers 

with the ZAG LinCheck system to within 2 nV/V 

expanded measurement uncertainty. 

At the same time it also confirms the 

comparability of linearity determination by 

transducer calibration in a FCM if an amplifier with 

known linearity is available. In combination with 

force calibration machines offering low expanded 

uncertainties of generated force, the aim is to further 

improve the amplifier calibration to be able to 

subtract the amplifier from the calibration result of 

the force transducer under calibration, specifying 

only the transducer characteristics. 

5. SUMMARY 

This paper presents an example of a procedure to 

validate the combinatorial evaluation procedure for 

strain-gauge amplifier linearity evaluation. By 

using a single force calibration machine and a single 

force transducer with various bridge amplifiers, the 

force calibration system acts as a comparator to 

compare the linearity of amplifiers. The lower 

uncertainty of generated force enables better 

measurement uncertainties than offered by 

traditional bridge calibrators and is in line with the 

requirements for the combinatorial system (3 nV/V). 

The proposed procedure is suitable for the 

confirmation of performance of the combinatorial 

calibration procedure and should enable the 

validation and application of the procedure in 

ISO/IEC 17025 accredited calibration laboratories. 

6. REFERENCES 

[1] M. Hiti, “Reducing the uncertainty of strain gauge 
amplifier calibration”, ACTA IMEKO, vol. 6, no. 4, 

pp. 69-74, 2017. Online [accessed 20200821]: 

https://acta.imeko.org/index.php/acta-

imeko/article/view/IMEKO-ACTA-

06%20%282017%29-04-11/pdf 

[2] M. Hiti, “Resistor network for linearity check of 
voltage ratio meters by combinatorial technique”, 

Meas. Sci Technol., vol. 26, no. 5, 2015. 

[3] M. M. Schäck: “Long Term Proven and Optimized 
High-Precision 225 Hz Carrier Frequency 

Technology in a Modern and Universal Data 

Acquisition System”, Proc. IMEKO 23rd TC3, 13th 

TC5 and 4th TC22 International Conference, 

Helsinki, Finland, 30 May to 1 June 2017. 

[4] M. F. Beug, A. Kölling, H. Moser, “A new 
calibration transformer and measurement setup for 

bridge standard calibrations up to 5 kHz”, IEEE 

Transactions on Instrumentation and Measurement, 

vol. 66, no. 6, pp. 1531-1538, June 2017. 

[5] M. F. Beug, H. Moser, A. Kölling: “Bridge 
Amplifier Linearity Investigation with a Cascaded 

Inductive Voltage Divider Setup”, J. Phys.: Conf. 

Ser. 1065 042010, 2018. 

[6] M. Hiti: “Analysis of exchanging of measuring 
amplifiers on force transducer calibration results”, 

ERK 2017, Portorož, Slovenia, 

25 - 26 September 2017. 

 

 

http://www.imeko.org/
https://acta.imeko.org/index.php/acta-imeko/article/view/IMEKO-ACTA-06%20%282017%29-04-11/pdf
https://acta.imeko.org/index.php/acta-imeko/article/view/IMEKO-ACTA-06%20%282017%29-04-11/pdf
https://acta.imeko.org/index.php/acta-imeko/article/view/IMEKO-ACTA-06%20%282017%29-04-11/pdf