Transcript
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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