5. Measuring amplifiers
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5. MEASURING AMPLIFIERS
5.1. Tasks of the measurement
1.
Measure the voltage of the given thermocouple using a DVM for one position of the
thermostat switch.
2.
Using operational amplifier OP7 propose the circuit diagram
1) of an inverting voltage amplifier with voltage gain 100 and input resistance 1 k
2) of a noninverting amplifier with voltage gain 100 and input resistance 100 k
3.
Use the inverting amplifier from point 5.1.2 to amplify the thermocouple voltage.
The amplifier output voltage should be measured by the same DVM that was used in
point 5.1.1 and for the same position of the thermostat switch. Make correction of the
error of the method caused by finite input resistance of the amplifier.
4.
Find the expanded uncertainty of the measurement of the thermocouple voltage
(coverage factor k = 2) both for the direct measurement of the thermocouple voltage
and for the thermocouple voltage amplified by the inverting amplifier according to
point 5.1.2. In the latter case take into account not only error of the DVM and
tolerances of the resistors used, but also the maximum input offset voltage of the
operational amplifier (disregard the influence of input bias currents of the operational
amplifier). The values that you need for computation are given below.
5.
Find the temperature measured by the thermocouple according to points 5.1.1 a
5.1.3, if the thermocouple constant is K = 54 V/°C. Suppose that the temperature of
the reference end of the thermocouple is 20 °C (temperature of the laboratory).
6.
Verify that the actual input offset voltage of the used operational amplifier is lower
than the maximum (or even typical) value of input offset voltage from the amplifier
data sheet.
Note to symbols of operational amplifiers used in schematic diagrams
In this textbook we use OA symbols showing neither power supplies nor symbolic grounds.
Hints to the measurement
1.
Parasitic thermoelectric voltages at the connecting points (both terminals and soldered
connections) of wires used in the measuring circuit can cause comparatively large
relative error of measurement. Therefore start measuring after sufficient time
necessary for the temperature balance of the circuit - wait till the voltmeter reading
does not change monotonically (allow for possible changes caused by noise).
2.
Tolerances of the resistances are written on resistors. Thermocouple resistance is
given on the thermocouple box.
3.
The used operational amplifier allows for the input voltage offset compensation. This
compensation is seldom used in practice and we do not use it in this laboratory
exercise either.
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5.2. Schematic diagram
TC
U1
DV
1
0
Fig. 5.1 Direct measurement of thermocouple voltage by digital voltmeter
R2
RT
I1
R2
R1
+
UT
UX
1k
UX
+
U2
100k
U2
R1
Fig. 5.3 Noninverting amplifier with input
resistance of 100 k
Fig. 5.2 Inverting amplifier for amplification of
the thermocouple voltage
5.3. List of the equipment used
DV - digital voltmeter, model number: ..., accuracy: ± ... % of reading ± ... % of range,
voltage range: ...;
TC - thermocouple in thermostat;
OA - operational amplifier type OP07;
DC - power supply +15 V, -15 V
Tab. 5.1 Basic parameters of selected operational amplifiers
OA property
OA ICL 7650
741
LT 1097
OP 07
LM 155
Voltage offset typ./max.
(µV)
0.7
1500/5000
10/60
60/150
1000
Voltage offset temperature
drift (µV/°C)
0.02
10
0,3
0,5
5
5
50000
350
1800/7000
50
Input bias current typ. / max.
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(pA)
CMRR (dB)
120
90
130
110
100
Slew rate (V/µs)
2.5
0.5
0.2
0.3
5
Notes:
ICL 7650
automatically nulled OA
741
low cost obsolete bipolar OA
LT 1097
high accuracy OA
OP 07
high quality OA, the given parameters correspond to the lowcost version of the OA (industry standard)
LM 155
low-cost BIFET OA (using FE transistors in input stage)
5.4. Theoretical background and notes to the measurement
Output voltage of a thermocouple is in a limited temperature range directly proportional to the
temperature difference between the temperature of the cold (or reference) terminal of the
thermocouple and the temperature of the hot (or heated) end of the thermocouple ( 1 -  0 ).
Temperature of the cold terminal is supposed to be 20 °C. Thermocouple voltage is in the
range of mV; special care must be therefore paid to the method of its measurement.
Temperature of the hot end  1 can be found as  1 = U 1 /K +  0 , where  0 is the ambient
temperature (+20 °C) and K = 54×10-6 V/°C is the thermocouple constant.
The thermocouple voltage will be measured directly by a digital voltmeter, and by the same
digital voltmeter after amplification using an amplifier.
If the thermocouple voltage U 1 is measured directly by a digital voltmeter (see Fig. 5.1), the
standard uncertainty of measured voltage can be found as
uU 1 
1

U1  2 U R
100
100
3
(5.1)
where  1 is error of the DV in % of the measured voltage and  2 is error of the DV in % of
the measurement range (of full scale) U R . Relative standard uncertainty of measurement of
voltage using digital voltmeter is
uU 1,r 
uU 1
 100 %
U1
(5.2)
The same relations are valid for measured voltage U 2 at the amplifier output (according to
Fig. 5.2) when changing voltage U 1 for U 2 in relations (5.1) a (5.2).
Thermocouple voltage is measured in this task. This voltage is so small that using common
digital voltmeters the voltmeter reading is closed to the beginning of the measurement range.
If we measure voltage after amplification with amplifier gain equal to -100 using the same
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measurement range of the voltmeter as by direct measurement, the uncertainty according to
(5.1) is higher. However (for voltmeters with error in percent of reading not much higher then
error in percent of range) this change as compared to the corresponding component of
uncertainty of direct thermocouple voltage measurement is not very distinct. Contrary to that
relative uncertainty according to (5.2) is lower so many times, how many times the voltage
was amplified, in our case 100-times. That is why it is useful to amplify the thermocouple
voltage before its measurement. This effect would however be not very distinct if it is
necessary to change the voltmeter range when measuring amplified voltage and if a lowquality operational amplifier having large input voltage offset is used for measurement.
Resistance of the thermocouple R T is given on the thermocouple case and it is several Ohm,
typically 5 . Voltage drop on this resistor due to voltmeter input current results in decreasing
the measured voltage as compared to the thermocouple voltage. The measured voltage in this
case is in fact the output voltage of the voltage divider. The divider input voltage is the (open
circuit) thermocouple voltage U T , and the resistors of the divider are thermocouple resistance
R T and the voltmeter resistance R V . The systematic error caused by this divider (error of the
method, methodical error) can be corrected. The input resistance of digital voltmeters is
usually 10 M (and much higher at the lowest voltage range, when the voltmeter input
voltage divider is not used). By measuring the amplified thermocouple voltage, we measure
instead of voltage U T the voltage U X given by the relation
RV
UT
(5.3)
UX 
RV  RT
For the above given numerical values the divider division ratio (found using (5.3)) 0.9999995
and the error of the method is  m =U X - U T = 0.5 V. This value is negligible as compared to
both the measured value and the measurement uncertainty. It is therefore not necessary to
make the correction of methodical error by direct thermocouple voltage measurement.
If we measure the amplified voltage, then by estimating the measurement uncertainty also
components of type B uncertainty caused by tolerances of resistors of the feedback loop of the
amplifier, input bias currents of the amplifier and the input voltage offset of the amplifier
have to be taken into account. The above mentioned methodical error caused by thermocouple
resistance might not be negligible here, since amplifier input resistance in this laboratory task
is much lower than the digital voltmeter input resistance
Schematic diagram of the inverting voltage amplifier is shown in Fig. 5.2. For ideal
operational amplifier there is:
UX  
R1
U2
R2
(5.4)
where U 2 is output voltage of amplifier and U X is the measured voltage of thermocouple.
Resistance of the resistor R 1 is equal to the input resistance of the amplifier and therefore it
should be according to point 5.1.2.1 of Tasks of the measurement equal to 1 k. Resistor R 2
should have resistance 100 k to reach the prescribed gain -100.
Measured voltage of the thermocouple is in this case influenced also by the above-mentioned
error of the method, caused by loading effect of the input resistor of the measuring device.
Here this device consists of inverting amplifier with digital voltmeter at its output, so
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thermocouple output is loaded by resistance R 1 . Resistor R 1 creates together with
thermocouple resistor a resistive divider similar to the divider with division ratio according to
equation (5.3), where resistance R V of the voltmeter should be replaced by resistance R 1 .
Systematic error (error of the method) caused by this divider can be corrected from the result
of measurement by multiplication of the measured value U X by correction factor K F
according to relation (5.5).
UT  UXKF  UX
 R 
R1  RT
 U X 1  T 
R1
R1 

(5.5)
If R T were 5 , then for R 1 = 1 k is K F = 1.005. Since there is R T / R 1 <<1, it is possible to
disregard the uncertainty of the correction factor in estimation of the resulting measurement
uncertainty.
The component of the type B standard uncertainty of the measured voltage U X caused by
resistance tolerances and error of the digital voltmeter is for the case of ideal operational
amplifier given by the relation
2
uUx (id )
2
2

  U
  U
 U
  X u R1    X uU 2    X u R 2  

  R2
  U 2
 R1
2
2

 U 2
 R
   U 2 R1

 
u R1    1 uU 2   
u
2
R
2

R
R
R
 2
  2
 
2

2
(5.6)
where
u R (1, 2 )  R(1,2)
R (1,2)
uU 2
3
 R(1,2)
100 3
R(1, 2 )
are standard uncertainties of
resistances R 1 or R 2 ,
are tolerances of resistance R 1
or R 2 in %,
is standard uncertainty of
measurement of the output
voltage of the amplifier, found
using (5.1) and (5.2).
R2
R1
UDO
+
UX
I1N
U2
I1P
Obr. 5.4 Equivalent circuit of the inverting
amplifier with a real OA
Besides the component of the standard uncertainty found by (5.6) the type B uncertainty is
influenced also by properties of the real operational amplifier (non-zero input bias currents I 1P
and I 1N and non-zero input voltage offset U D0 ). The equivalent circuit of the inverting
amplifier respecting input bias currents and input voltage offset is shown in Fig. 5.4.
The expression for the output voltage of the circuit from the Fig. 5.4 including contributions
of all sources in the circuit can be found by using principle of superposition. Using this
expression for finding value of voltage U X we get
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UX  

R1
R 
U 2  I1N R1  U DO 1  1 
R2
R2 

(5.7)
(The current source I 1P plays no role here, since it is in the circuit from the Fig. 5.4 shortcircuited by the zero resistance of the voltage source U DO . The input bias current I 1N should
not be disregarded, if resistance R 1 were so high that the voltage drop across resistance R 1
caused by current I 1N were comparable with voltage U x . In this measurement task this
component should be disregarded. The value of U DO can be found from the Tab. 5.1. As we
suppose that the probability distribution of the current of this source is uniform around the
zero value in the band U DO , corresponding component of the type B standard uncertainty can
be found as U DO /3.
The component of the standard uncertainty U X from the (5.5) corresponding to the input
voltage offset U DO is therefore
u OA(UDO ) 
U DO 
R 
1  1   87 V
3  R2 
since the input offset voltage in the circuit in Fig. 5.4 is amplified by a noninverting voltage
amplifier.
The total (type B) uncertainty of the measured voltage U X by using a real (non ideal)
operational amplifier is therefore
uUx ( OA )  u
2
Ux ( id )
u
2
OA (U DO )
 u
2
Ux ( id )
 U 1  R1 R2  
  DO

3


2
(5.8)
Note to calculation of the total measurement uncertainties:
Since geometrical sum is used in finding total uncertainty consisting of several components,
contributions of individual uncertainty components lower than one tenth of the largest
component can be disregarded
Finding the temperature
Thermocouple hot end temperature can be found using the approximate formula from the
thermocouple output voltage U 1 as
1 
U1
 0
K
(5.9)
where K = 54×10-6 V/C. We suppose that the temperature of the laboratory (equal to
temperature of the cold end of the thermocouple) is  0 = 20 °C.
Measurement of the input voltage offset
Input voltage offset of the inverting amplifier can be found by measurement of the amplifier
output voltage when the amplifier input is short-circuited, and by division of the measured
voltage by the amplifier gain for the input offset voltage. This gain is in our case 101 (there is
R 1 = 1 k and R 2 = 100 k, and input offset voltage of the inverting amplifier is amplified by
noninverting amplifier, see Fig. 5.4).
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