Electrical Measurement Techniques by Strain Gauges - Lab Report Example

Different Electrical Measurement Techniques for Strain Measurement Using Strain Gauges Hasmit 16 May 2006 Summary Experiments have been carried out to measure strain by means of electrical signals using strain gauges. Three variant of the electrical signal measurements have been explored – 1. Basic current and voltage measurement method, 2. Wheatstone bridge measurement and 3. Wheatstone bridge and amplifier method. It has been observed that strain resolution of the techniques has improved from first method i.e the basic current and voltage measurement method towards the third method i.e Wheatstone and amplifier method. 1. Introduction Strain measurement is a very important activity in many critical applications to assess and monitor the loading conditions on different component. Strain gauge is very commonly used for this purpose. The basic principle of operation of strain gauge is variation of resistance of a metallic conductor when load is applied onto it. The applied load is limited to elastic deformation regime and the change in resistance of the strain gauge is very small in the range of 0.1% of the resistance of the strain gauge. Therefore, direct measurements of the resistance change by current and voltage measurements is not expected to give accurate result and strain resolution of such a method is also not expected to be very good. Therefore, the change in resistance is generally measured by means of imbalance in Wheatstone bridge and the imbalanced voltage signal is magnified as well by using an amplifier. In these experiments the three electrical techniques i.e. direct measurement of current and voltage across the strain gauge resistor, direct measurement of the imbalance in Wheatstone bridge arrangement and Wheatstone bridge and amplifier techniques have been explored. The objectives of this experiment are: to demonstrate various strain measurement techniques; to show the limitations of measurement techniques and how these may be overcome; to show the problems that may occur due to electrical noise and instrument resolution; to analyse the error inherent in measurements. 2. Theory The experimental rig, shown in Figures 1 and 2, has a cantilever beam with a pair of 120  strain gauges, attached one on the upper and one on the lower beam surfaces. The strain-setting device is a bolt with a metric thread. The strain produced at the surface of a deflected cantilever beam varies along its length. Close to the gripping point (where the strain gauges are mounted) it has a maximum value of: . .......[1] Where:  - deflection of the end of the beam (max of 45 mm), d - beam thickness (0.8mm in the rig) and L - beam length (250 mm in the rig). The strain in the cantilever beam gets transmitted to the strain gauges mounted on it and causes corresponding change in the resistance of these gauges. The strain gauge mounted on the upper side is under tensile strain and its resistance increases while the one mounted on the lower side gets compressed and its resistance decreases. An analysis to calculate strain theoretically from the measured voltage signal is presented below: Measuring strain with strain gauges Let Ro be resistance of the strain gauge with zero strain. Then Ro is given by Ro = lo/Ao .........[2] Where: r is receptivity of the strain gauge material, lo is length of the strain gauge and Ao is the area of cross-section at zero strain When strain is applied on the cantilever beam, the strain is transmitted to the strain gauge and the strain gauge resistance changes. Let us consider the upper strain gauge. This strain gauge undergoes tensile strain i.e. its length increases and the area decreases accordingly. Let us assume that the length become l, cross-sectional area becomes A and the resistance becomes R. Then R = l/A ..........[3] Combining, [2] and [3] R = Ro(l/lo)(Ao/A) ..........[4] Applying constancy of volume for the strain gauge resistor Aolo = Al or, Ao/A = l/lo ..........[5] By [5] in [4] R = Ro (l/lo)2 ...........[6] Now, l/lo = (lo+l)/lo = 1 + l/lo = 1+  Where  = l/lo is strain of the strain gauge ...........[7] By [7] in [6] R = Ro (1+)2 = Ro(1+2+2) = Ro(1+2) Because e is small, therefore, e2 can be neglected .........[8] Now applying ohm’s law i.e. V = iR following relationship between the voltage and strain can be derived V = Vo(1+2) ...........[9] Same analysis can be applied for the lower strain gauge and following relationship is derived V = Vo(1-2) ...........[10] Equations [9] and [10] can be clubbed together as Modulus (V)/2Vo =  ............[11] This formula can be used to theoretically calculate strain from the voltage signal. To improve accuracy one can average out the differential voltage signal and then calculate strain by using equation [11]. Another more practical approach can be plotting a linear calibration curve between the applied strain and the electrical signals i.e. voltage across the strain gauge resistor or imbalance in the Wheatstone bridge circuit. Strain value can be directly read from this chart. Figure 1: The experimental rig. Figure 2: The strain gauge mounted on the top of the beam, with the connection wires 3. Apparatus a. Experimental rig (figures 1 and 2) comprising of a cantilever beam, strain setting arrangement and strain gauges mounted onto the cantilever beam. b. +/- 15 V power supply c. Digital voltmeter d. Amplifier AD 620 from Analog Devices 4. Procedure Basic measurement circuit (figure 3) was set up. Supply voltage (VS) and voltage across strain gauge (VSG) was measured by the digital voltmeter under zero strain condition and recorded. Subsequently, different magnitude of strain was set by the strain setting arrangement and corresponding VSG was also measured and recorded for both the strain gauges i.e. the one on the upper side and another on the lower side of the of the cantilever beam. The Supply voltage (VS) was also measured in between the experiments. Figure 3: Basic measurement circuit Wheatstone bridge circuit was set up using the upper strain gauge and three inactive strain gauges (figure 4). The Wheatstone bridge circuit was balanced using a 67.3 kresistance (figure 5). Again, different magnitude of strain was set by the strain setting arrangement and corresponding imbalance voltage was measured across the Wheatstone bridge and the values were recorded. The same experiment was repeated with having both the upper and the lower strain gauges active and remaining two inactive gauges in the Wheatstone bridge. Figure 4: Wheatstone bridge Figure 5: Balancing of the Wheatstone bridge Amplifier circuit was constructed using 750 kresistance. The amplifier was connected to the Wheatstone bridge with two active gauges and keeping the voltmeter connected across the bridge. The bridge was balanced again and the output of the amplifier was noted for zero strain. This corresponds to noise. Again, some stain was applied and the input and output voltages of the amplifier was noted and the gain was verified. Different magnitude of strain was applied and corresponding amplifier output voltage was recorded. Using oscilloscope amplitude of noise in the input as well as output signal was measured. 5. Results and Discussion Strain value measured by equation [1] is listed in table 1 Table 1: Strain as calculated from equation [1] Deflection (mm) 5 10 15 20 25 30 35 Strain (x10-6) 96 192 288 384 480 576 672 Error analysis of equation [1] is presented below / = /+d/d+2*L/L However as d and L were provided to us and not measured in this experiment it was assumed to be highly accurate and the error in measurement of d and L were not considered. As  was measured with a scale of least count of 1 mm, therefore, 0.5 mm was taken as Therefore, / = / and the value of  comes out to be +/- 9.6 microstrain. One can notice here that error in strain measurement is associated with error in deflection measurement and therefore, to minimize error in strain measurement one has to minimize the same in deflection measurement. For this purpose one alternative can be to count number of threads of the strain setting device to measure the deflection. Suppose, pitch of the strain setting thread is 0.5 mm then in one revolution the amount of deflection will be 0.5 mm and the error can be reduced to 50%. To further reduce the error one can mark a vertical line on the thread and total angle of revolution i.e. 360o can be subdivided into 12 marking of 30o at the base of the plate from where the strain setting screw goes inside. This way location of the verical line can be read with an error of just 15o. One should note that 360 revolution of the strain setting screw corresponds to 0.5 mm of the linear deflection. Therefore, 15o will correspond to just 0.5/24 mm which is ~0.02 mm. Thus, the error in strain measurement will reduce by approximately 25 times i.e. the error will be just 0.4 microstrains in the mechanical strain measurement. This is fairly high degree of accuracy. Basic electrical measurement results Measured voltages : VS = . . 15.33V. . . . . . . .(VS - VSG) = . . . . 14.185V . . . VSG = . . .1.145V. Voltage resolution : Vmin = . . . 2000mV. . . . . . . Calculate the resistance of the strain gauge. Calculated RSG : . . . .242.16  Variation of strain gauge resistance and the measured strain Vs. predicted strain is plotted in figure 6 below. The linear interpolation line is also imposed on the scatter plot. In resistance variation plot there is lot of scatter, this is because of the fact that the least count of the voltmeter which is 2 mV is almost comparable with the measured voltage differences of interest. As, the voltage differentials caused by change in resistance of the strain gauge is in 1-3 mV range therefore, measurements with a voltmeter of 2mV least count has to be error prone. Similarly, the error has been transmitted to the measured strain value as well but this has been minimized by averaging the voltage differentials from both the upper as well as the lower strain gauge. Strain measurement from current and voltage measurement as per equations [9], [10] and [11] as derived in the theory section has been presented in table 2. Table 1: Strain as calculated from equation [1] Deflection (mm) 5 10 15 20 25 30 35 Strain (x10-6) (2c) Upper strain gauge 437 437 437 873 873 873 1310 Strain (x10-6) (2d) Lower strain gauge 0 437 437 437 437 873 873 Strain (x10-6) Average voltage signal (2c & 2d) 218 437 437 655 655 873 1091 Error Analysis of electrical measurement technique: = (Modulus of V)/(2*Vo) Therefore, V)/V + Vo/Vo and V = V) = 0.1 mV Therefore, 0.05 to 0.1 or 5% to 10%. This is reasonably high error and is almost comparable with the error encountered while using equation 1. However, this is very much on expected line given the fact that least count of the digital voltmeter is 0.1 mV. Strain sensitivity of this methods is about 40 microstrains. Wheatstone bridge Method With upper active and three inactive strain gauges Measured: VS : . . . 15.34 V . . . VB1 : . . . . . .2.31 V . . . . VBimbalance : . . . 0.002 . . . Noted or measured : R3 : . . . . . 67.3 k . . . . . . . With two active (upper and lower) and two inactive strain gauges: Measured: VS : . .15.35V . . . VB1 : . . . . . 2.31V. . . . . VBimbalance : . . . . 0.0013. . . . Noted or measured balancing resistance R4 : . . . . 12.8 k . . . . . . . . . . . . With one active strain gauge and three inactive strain gauges following formula for strain measurement was derived using basic electrical circuit laws  = 2*Vo/Vs where Vo is the imbalance produced when strain is applied and Vs is the supply voltage .......[12] Similarly with two active and two inactive strain gauges the formula becomes  = Vo/Vs ........[13] Strain was calculated from the imbalance voltage values obtained from the Wheatstone bridge arrangement method with one and two active strain gauges using the formulae [12] and [13]. The plot is linear with very high R2 values indicating this method is very sensitive and therefore, can be used for measuring very small strains. Sensitivity of this method is close to 6 microstrains. 6. Conclusion While the direct current and voltage method appears to be most inaccurate, the accuracy and sensitivity of the electrical methods is considerably improved while using the Wheatstone bridge method and Wheatstone bridge with an amplifier and this can be used for measuring very small strain with reasonable accuracy. Appendix: Pre lab Calculation: 1.a) Predict the maximum strain (in microstrains) to expect during this exercise. * Answer – max = 864 microstrains Strain is equal to where:  - deflection of the end of the beam (max of 45 mm), d - beam thickness (0.8mm in the rig) and L - beam length (250 mm in the rig). 2.a) Choose whether to use 15V or ±15V supply, provided within the breadboard unit used in Lab1. Calculate a suitable value for R1 to ensure that the current through the 120  strain gauge is below the recommended maximum current of 10mA, but not too low. * Answer – chosen VS = 15V R1 = 3 k. Let us choose the resistor value slightly higher using the nearest standard value. Therefore R1 = 3000  = 3 k. 3.a) Calculate a suitable value for resistor R2. * Answer – R2 : 3 k. Calculations are similar to the previous problem: Again let us choose the resistor value slightly higher using the nearest standard value. Therefore R2 = 3000  = 3 k. 3.b) If the bridge resistors were identical, no voltage would be detected by the meter. In practice a small voltage (up to 2mV) appears as the resistors are not identical. Propose a method for balancing the bridge (zeroing the voltage when no strain is applied). Equations (voltages are measured between the points 0 and 1,2,3,4): Exact solution of these equations is rather complicated. But we are interested in some qualitive result. Let us assume Ra,Rb,Rc,Rd we can write According to this formula we can choose appropriate value of R3 for the given disbalance (difference between V1/2 and real V2). 4.a) Design a suitable amplifier circuit. Calculate the gain that you expect from the circuit. The amplifier needs to be able to amplify differential signals of up to about 3mV and produce an output in the range of about 1V to 5V. It should have a high Common Mode Rejection Ratio (CMRR) and a (reasonably) high input impedance. Answer The best way to obtain high CMRR is to use the so called instrumentation amplifier. The internal structure of the instrumentation amplifier is shown in the figure (above). It is connected to the bridge similar to the ordinary differential amplifier using the inverting (“-“) and the non-inverting (“+”) inputs. It can be build from 3 op amps and the appropriate resistors, but it is more convenient to use the integrated circuit with all needed components. One of the most popular (and cheap) instrumentation amplifiers is AD620 from Analog Devices. Some of the parameters are given below (from the datasheet): EXCELLENT DC PERFORMANCE (“B GRADE”) 50 mV max, Input Offset Voltage 0.6 mV/8C max, Input Offset Drift 1.0 nA max, Input Bias Current 100 dB min Common-Mode Rejection Ratio (G = 10) LOW NOISE 9 nV/Hz, @ 1 kHz, Input Voltage Noise 0.28 mV p-p Noise (0.1 Hz to 10 Hz) EXCELLENT AC SPECIFICATIONS 120 kHz Bandwidth (G = 100) 15 ms Settling Time to 0.01% Required gain is defined by one external resistor. Its value is given by For 3mV input and 3V output gain is 1000 and Rex = 49.4 Read More