University of Warwick
School of Engineering
Laboratory Report
Strain Gauge
1222631
Report Reference Code: ES185 A3
Affiliation: Year 1 Engineering
th
Report of laboratory performed on: 9 January 2014
1
1. Summary
The objective of this laboratory is to provide the experience in the use of operational amplifier based circuits. The charact eristics of
three operational amplifiers – non-inverting amplifier, inverting amplifier, differential amplifier – will be looked at through its
amplification of DC signals. A transducer in the form of strain gauges will also be looked at.
The non-inverting amplifier is found to have a directly proportional relationship for its input
inpu t and output voltages. While th
the
e inverting
amplifier also the same relationship between its input and output voltages, the relationship is a negative one. The usage of strain
gauges allows the translation of gravitational potential energy to electrical energy. The change in energy output can also be adjusted
to the designer’s liking with the use of different amplifiers.
2
Contents
1.
Summary ........................................................................................................................................................................................................................................................... 2
2.
Introduction ...................................................................................................................................................................................................................................................... 4
3.
Theory ................................................................................................................................................................................................................................................................ 4
4.
3.1
Non-inverting Amplifier ....................................................................................................................................................................................................................... 4
3.2
Inverting Amplifier ................................................................................................................................................................................................................................ 4
3.3
Differential Amplifier ............................................................................................................................................................................................................................ 4
Apparatus and Methods ................................................................................................................................................................................................................................... 5
4.1
4.2
5.
Apparatus .............................................................................................................................................................................................................................................. 5
Method ................................................................................................................................................................................................................................................. 6
4.2.1
Exercise 1: Non-inverting Amplifier ............................................................................................................................................................................................... 6
4.2.2
Exercise 2: Inverting Amplifier ....................................................................................................................................................................................................... 6
4.2.3
Exercise 3: Strain Gauges and Differential Amplifier .................................................................................................................................................................... 6
Observations and Results ................................................................................................................................................................................................................................. 7
5.1
Exercise 1: Non-inverting Amplifier ..................................................................................................................................................................................................... 7
5.1.1
Theoretical Gain .............................................................................................................................................................................................................................. 7
5.1.2
Experimental Gain .......................................................................................................................................................................................................................... 7
5.1.3
Minimum and Maximum Output ................................................................................................................................................................................................... 7
5.1.4
Transfer Characteristics .................................................................................................................................................................................................................. 7
5.2
Exercise 2: Inverting Amplifier ............................................................................................................................................................................................................. 7
5.2.1
Theoretical Gain .............................................................................................................................................................................................................................. 7
5.2.2
Experimental Gain .......................................................................................................................................................................................................................... 8
5.2.3
Minimum and Maximum Output ................................................................................................................................................................................................... 8
5.2.4
Transfer Characteristics .................................................................................................................................................................................................................. 8
5.3
Exercise 3: Strain Gauges and Differential Amplifier to Non-inverting Amplifier ............................................................................................................................. 8
6.
Graphs ............................................................................................................................................................................................................................................................... 9
7.
Analysis and Discussion of Results ................................................................................................................................................................................................................. 11
7.1
7.1.1
7.2
Exercise 1 and Exercise 2: Non-inverting Amplifier and Inverting Amplifier.................................. ......................................... ...................................... .................. 11
Error Analysis ................................................................................................................................................................................................................................ 11
Exercise 3: Strain Gauges and Differential Amplifier ........................................................................................................................................................................ 12
8.
References and Bibliography .......................................................................................................................................................................................................................... 13
9.
Appendix .......................................................................................................................................................................................................................................................... 14
9.1
Briefing Sheet ..................................................................................................................................................................................................................................... 14
9.2
Deriving Gain and Output .................................................................................................................................................................................................................. 22
9.2.1
Non-inverting Amplifier ................................................................................................................................................................................................................ 22
9.2.2
Inverting Amplifier ........................................................................................................................................................................................................................ 22
9.2.3
Differential Amplifier (Subtractor) ............................................................................................................................................................................................... 22
3
2. Introduction
The purpose of this lab report is to look into the characteristics of 3 different amplifiers
amplifi ers – a non-inverting amplifier, an inverting
amplifier and a differential amplifier. These amplifiers will be investigated through its amplification of a DC signal. The characteristics
of the non-inverting amplifier and the inverting amplifier will be learned through its amplification and gain of the DC signal. The
relationship between the input, output and gain of these two amplifiers will be determined. In addition, the characteristics of resistive
strain gauges will be investigated using a differential amplifier. The experiment will also look at how transducers and amplifiers behave
to achieve an ideal output voltage.
The experiment will utilize OP177 as the controlled operational amplifier (op-amp) throughout the experiment. By using different
arrangements of resistors, the same op-amp will be able to act as the three different amplifiers.
3. Theory
3.1 Non-inverting Amplifier
The non-inverting amplifier as shown in Figure 1, has a given gain1 of,
   
=
=


(1)
By rearranging the equation, we can find that   =  ,  ∝  and that the
output,  , is directly proportional to the input,  . This suggest that when the input
voltage is plotted against the output voltage, a linear line will appear with the gain
value,  , as the slope.
Figure1:Non-invertingamplif
Figure1:Non-inv
ertingamplifierconfigur
ierconfiguration
ation
3.2
Inverting Amplifier
The inverting amplifier, as its name suggests, inverts the input signal’s polarity.
Hence, it will have a negative gain. The equation of the inverting amplifier is given
as2,
=


=


(2)
Figure2:Invertingamplifier
Figure2:Inve
rtingamplifierconfiguration
configuration
The above equation again suggests that the input will be directly proportional as
 ∝  . However, its difference with the non-inverting amplifier is that the gain is
negative.
3.3
Differential Amplifier
For the differential amplifier, the gain is unable to be found due to the presence of
two different input sources. However, the output value can be found from deriving
the following equation3,
 = (   )


(3)
Figure3:Differentialamplifie
rentialamplifierconfiguratio
rconfiguration
n
Figure3:Diffe
1
2
Calculations are derived in the Appendix under 10.2.1
Calculations are derived in the Appendix under 10.2.2
3
Calculations are derived in the Appendix under 10.2.3
4
The deflection of the transducer will then change the resistivity of the strain gauges on the
t he top and bottom, this would give two
different input voltages to the differential amplifier to subtract, and thus, generate an output.
4.
Apparatus and Methods
4.1 Apparatus
pparatus
Quantity
Analogue Experimental Transducer
Digital Volt Meter (with attached Red and Black
cables)
1
Power Supply
Pre-fabricated PCB
(contains 1 non-inverting amplifier using OP177,
1 inverting amplifier using OP177, 1 differential
d ifferential
amplifier using OP177)
Resistors
1
Variable Potentiometer
(University of Warwick)
M4 Washers
Wire Cutter
Wire Stripper
Wires
2
1
1 each
(10kΩ, 12kΩ, 120kΩ, 180kΩ,
1M2Ω, 2M7Ω)
1
20
1
1
1 roll each
(Black, Red, Blue, Yellow)
Table1:ApparatusList
Photograph1:AnalogueExper
Photograph1:Ana
logueExperimentalTransd
imentalTransducer
ucer
Photograph2:PrefabricatedPCB
containingthe3amplifiers
Photograph3:DigitalVolt
Meter
Photograph4:PowerSupply
Photograph5:Variable
Potentiometer
5
4.2 Method
4.2.1
1.
2.
3.
4.
5.
6.
Exercise 1: Non-inverting Amplifier
Insert the 120kΩ resistor into the  position of the prefabricated PCB.
Insert the 10kΩ resistor into the  position of the prefabricated PCB.
Connect the Power Supply to the prefabricated PCB.
Connect the variable potentiometer to the + , − and 0V lines to CON2 on the prefabricated PCB.
Wire the output of the potentiometer to IN1, making sure that PIN1 of the potentiometer goes to PIN1 of IN1.
Connect a Digital Volt Meter (DVM)
( DVM) to the prefabricated PCB, where RED goes into PIN1 of IN1 and BLACK goes into PIN2 of
IN1. This will measure the input voltage ( ) of the non-inverting amplifier.
7.
Connect the second DVM to the
t he prefabricated PCB, where RED goes into PIN1 of OUT1 and BLACK goes into PIN2 of OUT1.
This will measure the output voltage ( ) of the non-inverting amplifier.
8. Switch the Power Supply on.
9. Use the potentiometer to adjust  to 0.5V, indicated by the first DVM.
10. Measure  of the circuit on the second DVM.
11. Calculate the Gain (  =


) of the non-inverting amplifier.
12. Determine the maximum positive and negative  of the amplifier by producing a maximum positive and negative  .
13. Find out the transfer characteristic of the non-inverting
non -inverting amplifier by measuring the  using different values of  . Fix the
values of  at around 0.2V intervals.
14. Obtain 11 different results.
15. Switch the Power Supply off.
4.2.2
Exercise 2: Inverting Amplifier
Insert the 180kΩ resistor into the  position of the prefabricated PCB.
Insert the 2M7Ω resistor into the  position of the prefabricated PCB.
Connect the Power Supply to the prefabricated PCB.
Connect the variable potentiometer to the + , − and 0V lines to CON2 on the prefabricated PCB.
Wire the output of the potentiometer to IN2, making sure that PIN1 of the potentiometer goes to PIN1 of IN2.
Connect a Digital Volt Meter (DVM)
( DVM) to the prefabricated PCB, where RED goes into PIN1 of IN2 and BLACK goes into PIN2 of
IN2. This will measure the input voltage ( ) of the non-inverting amplifier.
7. Connect the second DVM to the prefabricated PCB, where RED goes into PIN1 of OUT2 and BLACK goes into PIN
PIN2
2 of OUT2.
This will measure the output voltage ( ) of the non-inverting amplifier.
8. Switch the Power Supply on.
9. Use the potentiometer to adjust  to 0.5V, indicated by the first DVM.
10. Measure  of the circuit on the second DVM.
1.
2.
3.
4.
5.
6.
11. Calculate the Gain (  =


) of the inverting amplifier.
12. Determine the maximum positive and negative  of the amplifier by producing a maximum positive and negative  .
13. Find out the transfer characteristic of the non-inverting
non -inverting amplifier by measuring the  using different values of  . Fix the
values of  at around 0.2V intervals.
14. Obtain 11 different results.
15. Switch the Power Supply off.
4.2.3
Exercise 3: Strain Gauges and Differential Amplifier
Part1
1.
2.
3.
4.
5.
6.
7.
Wire the Analogue Experimental Transducer to CON3 on the prefabricated PCB.
Connect the output of the Analogue Experimental Transducer to the IN3 of the prefabricated PCB.
Ensure the white/brown wire goes to PIN1 and the yellow wire goes to PIN2 of IN3.
Connect a DVM to the prefabricated
prefabr icated PCB, where RED goes into PIN1 of OUT3 and BLACK goes into PIN2 of OUT3. This will
measure the output voltage (  ) of the differential amplifier.
Switch on the Power Supply.
Add M4 washers to the nylon screw of the Analogue Experimental Transducer.
Tabulate the  from the differential amplifier against the number of washers.
6
8.
Record results for up to 20 M3 washers, using 2 M4 washers at a time for 10 readings.
Part2
9. Connect the output of the differential amplifier to the input
inpu t of the non-inverting amplifier. (OUT3 to IN1)
10. Calculate the gain required from the non-inverting amplifier such that
t hat one washer gives and  of 1V.
11. Switch the Power Supply off.
5. Observations and Results
5.1 Exercise 1: Non-inverting Amplifier
5.1.1
Theoretical Gain
=
  

=
120
120Ω
Ω  10Ω
10Ω
10Ω
 = 13
5.1.2
Experimental Gain
=

=
6.46
 0.50
 = 12.92
It is noted that the experimental value found for the gain differs from the theoretical value of the gain.
5.1.3
Minimum and Maximum Output
Min. Input
-0.960 V
Min. Output
-12.32 V
Max. Input
0.995 V
Max. Output
12.84 V
Table2:Mina
Table2:MinandMaxOutputof
ndMaxOutputofNon-inverting
Non-invertingAmplifier
Amplifier
5.1.4
Transfer Characteristics
Input Voltage
[V]
Output Voltage of Noninverting Amplifier [V]
-0.898
-11.58
-0.750
-9.68
-0.500
-6.45
-0.251
-3.23
-0.100
-1.28
-0.002
-0.02
0.101
1.31
0.251
3.25
0.500
6.46
0.750
9.68
0.904
11.68
Table3:ObservationsandRe
rvationsandResultsofNon-inve
sultsofNon-invertingAmplifie
rtingAmplifier
r
Table3:Obse
5.2 Exercise 2: Inverting Amplifier
5.2.1
Theoretical Gain
 =

=
27Ω
180Ω

 = 15
7
5.2.2
Experimental Gain
=

=
7.65
0.499

 = 15.33
It is noted that the experimental value found for the gain differs from the theoretical value of the gain.
5.2.3
Minimum and Maximum Output
Min. Input
0.855 V
Min. Output
-12.97 V
Max. Input
-1.006 V
Max. Output
12.98 V
Table4:MinandMaxOutputof
andMaxOutputofInvertingAmp
InvertingAmplifier
lifier
Table4:Min
5.2.4
Transfer Characteristics
Input Voltage
[V]
Output Voltage of Inverting
Amplifier [V]
-0.804
12.36
-0.751
11.58
-0.503
7.74
-0.248
3.82
-0.101
1.56
0.002
-0.02
0.102
-1.56
0.250
0.501
-3.82
-7.67
0.750
-11.50
0.799
-12.25
Table5:ObservationsandResultsofInvertin
Table5:ObservationsandRe
sultsofInvertingAmplifier
gAmplifier
5.3 Exercise 3: Strain Gauges and Differential Amplifier to Non-inverting Amplifier
Number of Washers
Output
Voltage
[V]
Output Voltage
without
Systematic Error
[V]
0
0.011
0.000
2
4
0.018
0.025
0.007
0.014
6
0.031
0.020
8
0.037
0.026
10
0.043
0.032
12
0.050
0.039
14
0.056
0.045
16
0.062
0.051
18
0.068
0.057
20
0.074
0.063
Table6:ObservationsandRe
rvationsandResultsofDiffe
sultsofDifferentialAmplifie
rentialAmplifier
r
Table6:Obse
8
6. Graphs
RELATIONSHIP BETWEEN INPUT VOLTAGE AND OUTPUT VOLTAGE
14.00
12.00
10.00

8.00
= 12.908 + 0.0069
6.00
4.00
]
2.00


[
E
G
A
T
L
O-1.000
V
T
U
P
T
U
O
0.00
-0.900 -0.800 -0.700 -0.600 -0.500 -0.400 -0.300 -0.200 -0.100 0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
-2.00
-4.00
-6.00
=
-15.36 + 0.0176
-8.00
-10.00
-12.00
-14.00
INPUT VOLTAGE [ ]
Outputt Voltage
Outpu
Voltage of Inve
Inverting
rting Ampl
Amplifier
ifier [V]
Outputt Voltage
Outpu
Voltage of NonNon-inve
inverting
rting Ampl
Amplifie
ifierr [V]
[V]
Graph1:ResultsofNon-inver
Graph1:Resul
tsofNon-invertingAmplifie
tingAmplifierandInverti
randInvertingAmplifier
ngAmplifier
9
RELATIONSHIP
RELA
TIONSHIP BETWEEN NUMBER OF WASHERS AND OUTPUT
VOLTAGE
Outp
Ou
tput
ut Volt
Voltag
age
e [V]
[V]
Outpu
Ou
tputt Volt
Voltag
age
e with
withou
outt Syste
Systema
mati
ticc Error
Error [V]
[V]
0.080
0.070
Vo = 0.0031x + 0.0119
0.060
y = 0.0031x
] 0.050
V
[
E
G
A
T
L
O0.040
V
T
U
P
T
U
O
0.030
0.020
1.000
0.010
0.000
0
2
4
6
8
10
12
14
16
18
20
22
NUMBER OF WASHERS
Graph2:Results
Graph2:ResultsofOutputVoltage
ofOutputVoltageandWashers
andWashers
10
7. Analysis and Discussion of Results
7.1 Exercise 1 and Exercise 2: Non-inverting Amplifier and Inverting Amplifier
From Graph 1, the non-inverting amplifier is shown to have a linear transfer characteristic. As the theory suggests, the input voltage
and the output voltage has a directly proportional relationship. In this particular
p articular OP177, the non-inverting amplifier is calculated to
have a gain of 13, however, the experiment yielded a gain of 12.92. The difference
d ifference of these two values is minimal, u
up
p to 0.06%.
The values are different because the theoretical value ignores the resistance of the wires in the circuit. The prefabricated PCB is also
set up so that there are several extra components between the power supply and the input terminal of the OP177. These components
are there to protect the operational amplifier, however, at the same time, they may alter the input voltage after the readings on the
DVM. These components consists of various diodes, capacitors and resistors of unknown value, thus, a fu
further
rther investigation to how
these components affect the input voltage is void. Another explanation of the experimental yield is that the OP177 is not an ideal
amplifier and has internal resistances
r esistances (Covington, 2007). All three causes will, according to Ohm’s Law,  = , lower the output
voltage..
Another observation to the exercise is that the resistors used in the experiment are gold band resistors. Gold band resistors have a
tolerance of 5% (Storr, 2014), meaning that the values of the resistors may differ by 5% of its value. If so, the theoretical value of this
amplifier’s gain will then range from 11.7 to 14.34. The value of the experimental gain sits insides this range. From Graph1, the noninverting amplifier has a gain of 12.908, lower than the theoretical value and experimental value at an input voltage of 0.5V. The
above suggestions can also be used to explain the discrepancies found for the gain value.
Yet, in this particular OP177, the amplifier is designed to saturate maximum and minimum output voltage at +12V and -12V
respectively. However, the minimum and maximum output voltages exceed these values, as shown in Table 2. However, each
operational amplifier is made differently (Storey, 2009) , other circuits in this arrangement is fault. The inverting amplifier, has the
same OP177 as its operational amplifier, but its output values are within the +VCC and –VCC as shown below.
Graph 1 also shows that the experiment had a systematic error. When the input is 0V, the amplifier is shown to have a positive output
voltage. However,
However, this does not change the value of the gain as the input and output are directly proportional. The systematic error
may have come from the calibration of the DVM or the imperfect makeup of the circuit.
Moving on to the inverting amplifier, Graph 1 shows that it has a negative linear transfer
tran sfer characteristic. As the theory suggests, the
input voltage and the output voltage has a negative directly proportional relationship. In this particular
partic ular OP177, the non-inverting
amplifier is calculated to have a gain of -15, however, the ex
experiment
periment yielded a gain of -15.33. The difference of these two values is
2.2%. Graph 1 then indicates that the inverting amplifier has a gain of -15.36.
The explanation of the discrepancies in the inverting amplifier is the same as the non-inverting amplifier. The circuit has extra
components, resistors with tolerance and internal resistance within the operational amplifier, similar to the non-inverting
non -inverting amplifier.
Graph 1 also shows that the experiment contains a systematic error, being in the same situation as the non-inverting amplifier.
However, the minimum and maximum output values for this particular OP177 is within the range of -12V to 12V, fulfilling the
description of the amplifier. This shows that typical operational amplifiers differs from each other and do
d o not give identical results
everytime.
7.1.1
Error Analysis
The equation of the gain for the non-inverting amplifier is given as,
  
=

As gold band resistors, each value of the resistors have a 5% tolerance, hence,
Δ
Δ(   )
 =
Δ
 
  
(4)
4
Error Calculations in 7.1.1
11
Δ

=5
5%
%  5% = 10%
10%
(5)
The theoretical value of the gain will have a relative uncertainty of 10%.
The DVM also have uncertainties to 0.01V, using, the first experimental value as an example,
6.46
6 ± 0.0
0.01
1
 =  = 6.4
0.50  0.0
0.01
1
 0.50
Δ

=
∆


∆

=
0.01
6.46

0.01
0.50
(6)
Δ

= 2.15%
This gives the uncertainty of the result yielded for the experimental gain.
The equation of the gain for the inverting amplifier is given as,
 =


As gold band resistors, each value of the resistors have a 5% tolerance, hence,
Δ

=
Δ


Δ

(7)
Δ

=5
5%
%  5% = 10%
10%
The theoretical value of the gain will have a relative uncertainty of 10%.
The DVM also have uncertainties to 0.01 V or 0.001V, using, the first experimental value as an example,
=

=
7.65
7.6
5 ± 0.0
0.01
1
0.499 ± 0.001
0.001
∆ 0.01 0.001
Δ

=

=
7.65 0.499




∆
Δ

= 0.33%
As stated above, there will also be unknown losses through the circuit and components. The several extra components increases the
uncertainty of the experiment. The uncertainty cannot be calculated unless the components were disclosed.
7.2 Exercise 3: Strain Gauges and Differential Amplifier
Exercise 3 experiments the change of gravitational potential energy to electrical energy by the Analogue Experimental Transducer. As
expected from section 2, the theory, the strain gauges give the differential amplifier two different input voltages, where the
differential amplifier amplifies. The change of energy form is created from the strain gauges as the compression of it increases
resistivity of the connected device.
From the results, as the number of washers increase, the output voltage from the differential amplifier increases as well. However,
from Table 6, it is found that the original data had a systematic error. Due to the lack of time, the Analogue Experimental Transducer
was not calibrated with the potentiometer. To remedy this, a column is inserted next to it in Table 6, indicating the expected output
voltage without the systematic error.
12
The results are then displayed graphically in Graph 2. Graph 2 indicates that each M4 washer’s weight translates to 0.0031V through
the differential amplifier. To fulfill the requirement of the briefing
br iefing sheet, the weight of each washer is ideally transformed into 1V. To
do this, a non-inverting amplifier will need to be designed.
The gain of this non-inverting amplifier will be,
=
=


1
0.0031
 = 323
The final results then look like the following,
RELATIONSHIP
RELA
TIONSHIP BETWEE N NUMBER OF WASHERS
AND FINAL OUTPUT VOLTAGE
10
9
8
]
V
[
E
G
A
T
L
O
V
T
U
P
T
U
O
L
A
N
I
F
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
NUMBER OF WASHERS
Graph3:IdealResult
This shows that with the correct combination of resistors, many different forms of energy can be translated into electrical energy by
transducers. The role of the operational amplifier allows the designer to freely design and manipulate the desired output from
fro m the
desired input.
8. References and Bibliography
Covington, J. A. (2007). StrainGaugeLaboratoryBriefingSheet. School of Engineering.
Storey, N. (2009). Operational Amplifiers. In B. Crofts, & K. Godfrey, CircuitsandSystems (p. 101). Essex: Pearson Education Limited.
Storr, W. (2014, March 5). ResistorColorCodes. Retrieved March 11, 2014, from Basic Electronics Tutorials Site:
http://www.allaboutcircuits.com/vol_5/chpt_2/1.html
13
UltraprecisionOperationalAmplifierDataS
UltraprecisionOp
erationalAmplifierDataSheetOP177.
heetOP177. (2012). Retrieved March 11, 2014, from Analog Devices:
http://www.analog.com/static/imported-files/data_sheets/OP177.pdf
9. Appendix
9.1 Briefing Sheet
14
15
16
17
18
19
20
21
9.2 Deriving Gain and Output
9.2.1
Non-inverting Amplifier
 = 

  

  
= =

9.2.2

Inverting Amplifier
− = + = 0
  −
 =

 =
  −

=
=
  0

  0

=
=




 = 


=


=
9.2.3


=


Differential Amplifier (Subtractor)
+ = −


  
=   (   )

  
  =  (   )  (    )
  =  (   )  (    )
  =           
  =     
 = (   )


22