ME 322: Instrumentation
Lecture 20
March 4, 2016
Professor Miles Greiner
myDAQ A/D converter, Temperature uncertainty,
First-order centered numerical differentiation and
random errors
Announcements/Reminders
• Will accept HW 7 Monday before class
– DW South lab in DeLaMare Library has DAQmx
– Marissa and Joseph will be in PE 2 tonight from 59 if students show up (will close at 7:30 if not)
– Joseph will be in the lab tomorrow from 12-2
– Have you found LabVIEW/DAQmx other places?
• HW 8 Due next Friday
• Science Olympiad tomorrow
– If you signed up, please show up (or lose 1%)
• Please complete the Lab Preparation Problems
and fully participate in each lab
A/D Converter Characteristics
• Full-scale range VRL ≤ V ≤ VRU
– FS = VRU - VRL
– For myDAQ the user can chose between two ranges
• ±10 V, ±2 V (FS = 4 or 20 V)
• Number of Bits N
– Resolves full-scale range into 2N sub-ranges
– Smallest voltage change a conditioner can detect:
• DV = FS/2N
– For myDAQ, N = 16, 216 = 65,536
• ±10 V scale: DV = 0.000,310 V = 0.305 mV = 305 mV
• ±2 V scale: DV = 0.000,061 V = 0.061 mV = 61 mV
• Sampling Rate fS = 1/TS
– For myDAQ, (fS)MAX = 200,000 Hz, TS = 5 msec
Input Resolution Error, IRE
• The reported voltage is the center of the digitization subrange in which the measured voltage is found to reside.
– So the maximum error due to the Digitization Process is half
the sub-range size, and random.
• Inside the voltage range
– 𝐼𝑅𝐸 =
1 𝐹𝑆
2 2𝑁
=
𝑉𝑅𝑈 −𝑉𝑅𝐿
2𝑁+1
• Fraction of FS error: 𝑉
– For myDAQ, N = 16
𝐼𝑅𝐸
𝑅𝑈 −𝑉𝑅𝐿
1
= 2𝑁+1
• ±10 V scale:
𝐼𝑅𝐸 = 0.000,153 V = 0.153 mV;
• ±2 V scale:
𝐼𝑅𝐸 = 0.000,031 V = 0.031 mV;
• For both Fraction of FS error = 0.000,8%
• At edge or outside of range
– 𝐼𝑅𝐸 → ∞
– To avoid this, before conducting an experiment estimate the
voltage range that must be measured, and choose appropriate A/D
converter and/or signal conditioners.
myDAQ Absolute Voltage Uncertainty
(0.11%FS)
(0.19%FS)
(0.12%FS)
(0.22%FS)
• More information myDAQ
– user guide, page 36-38
– http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/Labs/Lab%200
7%20Boiling%20Water%20Temperature/Lab7%20Index.htm
• What “should” the reading be if the input is shorted?
– Demonstration: Short myDAQ Analog Input 1 and observe signal
– In my office:
• ±10 V range: V ~ -0.0008 to -0.0026 V = -1.7 ± 0.9 mV (0.009% FS)
• ±2 V range: V ~ -0.0003 to -0.0009 V = -0.6 ± 0.3 mV (0.02% FS)
– What is it other at other voltages?
– Is it the same in the lecture hall?
Summary of myDAQ Uncertainties
Scale
±10 Volts
±2 Volts
Absolute Absolute
Accurcacy Accurcacy
23°C
18-28°C
22.8 mV
4.9 mV
38.9 mV
8.6 mV
0.1% FS
0.2% FS
Measurd
Shorted
Voltage Error
Input
Resolution
Error (IRE)
2.4 mV
0.9 mv
0.15 mV
0.03 mV
0.01 -0.02% FS 0.0008% FS
• What are these?
– AA: Maximum error of the voltage measurement reported
by the manufacturer for all voltage levels
• At different temperatures
– MSVE: Maximum error measured at V = 0V for one device
– IRE: Random error due to digitization process
• Which best characterizes voltage uncertainty?
Lab 7 Boiling Water Temperature in Reno
AA
wTC
• Boiling water temperature uncertainty
• Standard TC wire uncertainty
– Larger of 2.2°C or 0.75% of measurement
• Note: 0.0075 x 293°C = 2.2°C
• So for T < 293°C: wTC = 2.2°C; For T > 293°C: wTC = 0.0075*T (Which one for boiling?)
• For ±10 Volts, absolute accuracy at 23°C is AA= 0.0228V
– For TC signal conditioner SSC = 0.025 V/°C
– wTsc = AA/SSC = 0.0228V/(0.025 V/°C) = 0.916°C
• 𝑤𝑇 =
𝑤𝑇𝐶 2 + 𝑤𝑆𝐶 2 = 2.22 +.922 =2.4°C (~ 2.2°C)
A/D Converters are used to measure long
series of very rapidly-changing voltages
• Great for measuring a voltage signal
– How a voltage or measurand changes with time
– Would be very difficult using a regular voltmeter
• Allows determination of
– Rates of Change and
– Spectral (Frequency) Content
• The voltage and time associated with each
measurement has some error
– It is associated with the centers of the voltage sub-range and
sampling time.
– Additional systematic and random errors as well
• What can go wrong?
Example
Ti
TB
T(t)
• A small thermocouple at initial temperature Ti is
placed in boiling water at temperature TB
• Its measured temperature versus time T(t) is
shown
• What caused the temperature to change?
– What do you expect the time-dependent heattransfer rate to the thermocouple 𝑄 [joules/sec =
watts] to look like qualitatively?
– How can we determine it quantitatively?
t [sec]
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
T [oC]
20.599
20.387
20.646
20.316
20.905
20.528
20.716
20.858
20.693
20.905
20.669
20.811
20.811
20.716
20.246
20.646
20.387
20.387
20.693
20.222
1st Law of Thermodynamics
• 𝑄−𝑊 =
𝑑𝑈
𝑑𝑡
=
𝑑
−
𝑑
𝑑𝑇
𝑚𝑐𝑇 = 𝑚𝑐
𝑑𝑡
𝑑𝑇
time-derivative (𝑡)
𝑑𝑡
• How to estimate a
table of T versus t data?
from a
– ∆𝑡𝑆 is the sampling time step [sec] (TS)
• First-order numerical differentiation
– Centered differencing
–
𝑑𝑉
𝑑𝑡
𝑡
𝑉 𝑡+∆𝑡𝐷 −𝑉 𝑡−∆𝑡𝐷
= lim
∆𝑡𝐷 →0 𝑡+∆𝑡𝐷 − 𝑡−∆𝑡𝐷
𝑉 𝑡+∆𝑡𝐷 −𝑉 𝑡−∆𝑡𝐷
= lim
2∆𝑡𝐷
∆𝑡𝐷 →0
– ∆𝑡𝐷 is the differentiation time step [sec]
• ∆𝑡𝐷 = 𝑚∆𝑡𝑆 , m = integer (1, 2, or ?)
• Will we get the same result for different values of m?
– What is the best value for m? (1, 10, 20, ?)
U
t [sec]
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
0.015
0.016
0.017
0.018
0.019
𝑄
T [oC]
20.599
20.387
20.646
20.316
20.905
20.528
20.716
20.858
20.693
20.905
20.669
20.811
20.811
20.716
20.246
20.646
20.387
20.387
20.693
20.222
Sample Data
• Lab 9 Transient Thermocouple Measurements
– Download sample data
– http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/L
abs/Lab%2009%20TransientTCResponse/LabIndex.htm
• Plot T vs t for t < 2 sec
• Show how to evaluate and plot first-order centered
derivatives with different differentiation time steps
– Plot dT/dt vs t for m = 1, 10, 50
• Slow T vs t
– for 0.95s < t < 1.05s and 25°C < T < 45°C
– How do random errors affect “local” and “timeaveraged” slopes?
Effect of Random Noise on Differentiation
• Measured voltage has Real and Noise components
– VM = VR+VN
–
𝑑𝑉𝑀
𝑑𝑡
=
𝑉𝑅 −𝑉𝑁 𝑡+ − 𝑉𝑅 −𝑉𝑁 𝑡−
2∆𝑡𝐷
• ∆𝑉𝑅 = 𝑉𝑅𝑡+ − 𝑉𝑅𝑡− ≈
=
∆𝑉𝑅 +∆𝑉𝑁
2∆𝑡𝐷
𝑑𝑉𝑅
2∆𝑡𝐷
𝑑𝑡
• ∆𝑉𝑁 = 𝑉𝑁𝑡+ − 𝑉𝑁𝑡− ≈ 𝑤𝑉 ~ 𝐼𝑅𝐸 =
•
•
•
𝑑𝑉𝑀
𝑑𝑡
𝑑𝑉𝑅
𝑑𝑡
𝑊𝑉
=
+
2∆𝑡𝐷
𝑊𝑉
For small ∆𝑡𝐷 ,
2∆𝑡𝐷
𝑑𝑉𝑅
𝑊𝑉
Want
≫
𝑑𝑡
2∆𝑡𝐷
1 𝐹𝑆
2 2𝑁
RF, IRE, other random
errors, does not increase
with ∆𝑡𝐷
is large and random
– Want ∆𝑡𝐷 to be large enough to avoid random error but small
enough to capture real events
– If wV is mostly IRE, then decreases as FS gets smaller and N
increases
Common Temperature Measurement Errors
• Even for steady temperatures
• Lead wires act like a fin, cooling a hot the surface
compared to the case when the sensor is not there
• The temperature of a sensor on a post will be
between the fluid and duct surface temperature
High Temperature (combustion) Gas Measurements
Sensor
h, TS, A, e
Tgas
QConv=Ah(Tgas– TS)
TW
TS
QRad=Ase(TS4 -TW4)
• Radiation heat transfer is important and can cause errors
• Convection heat transfer to the sensor equals radiation
heat transfer from the sensor
– Q = Ah(Tgas – TS) = Ase(TS4 -TW4)
• s = Stefan-Boltzmann constant = 5.67x10-8W/m2K4
• e = Sensor emissivity (surface property ≤ 1)
• T[K] = T[C] + 273.15
• Measurement Error = Tgas – TS = (se/h)(TS4 -TW4)
Problem 9.39 (p. 335)
• Calculate the actual temperature of exhaust gas from
a diesel engine in a pipe, if the measuring
thermocouple reads 500°C and the exhaust pipe is
350°C. The emissivity of the thermocouple is 0.7
and the convection heat-transfer coefficient of the
flow over the thermocouple is 200W/m2-C.
• ID: Steady or Unsteady?
• What if there is uncertainty in emissivity?
Conduction through Support (Fin Configuration)
TS
T∞
h
x
L
A, P, k
T0
•
Sensor temperature TS will be between those of the fluid T∞ and duct surface T0
–
–
•
•
•
Support: cross sectional area A, parameter length P, conductivity k
Convection heat transfer coefficient between gas and support h
Fin Temperature Profile (from conduction heat transfer analysis):
–
𝑇(𝑥)−𝑇∞
–
cosh 𝑎 =
𝑇0 −𝑇∞
=
cosh[𝑚𝐿 1−𝑥/𝐿 ]
cosh 𝑚𝐿
𝑒 𝑎 +𝑒 −𝑎
𝑚𝐿 =
2
ℎ𝑃
𝑘𝐴
𝐿 (dimensionless length)
Sensor temperature at tip, 𝑇𝑆 = 𝑇(𝑥 = 𝐿)
Dimensionless Tip Temperature Error from conduction
𝑇𝑆 −𝑇∞
1
–
𝐸=
–
Decreases as 𝑚𝐿 =
•
•
𝑇0 −𝑇∞
=
cosh 𝑚𝐿
, (want this to be small)
L, h and P increase
k and A decrease
ℎ𝑃
𝑘𝐴
𝐿 decreases
Example
• A 1-cm-long, 1-mm-diameter stainless steel support
(k = 20 W/mK) is mounted inside a pipe whose
temperature is 200°C. The heat transfer coefficient
between gas in the pipe and the support is 100
W/m2K, and a sensor at the end of the support reads
350°C. What is the gas temperature? Assume esensor
=0
• Steady or unsteady
• Radiation or Conduction errors
Solution
• Sensor temperature:
•
𝑚𝐿 =
𝑇𝑆 −𝑇∞
𝑇0 −𝑇∞
=
1
cosh 𝑚𝐿
ℎ𝑃
𝐿
𝑘𝐴
• What is given and what must be found?
• What if esensor = 0.2?
T
TB
Ti
t=0
t
Example
A/D
N= 2
±10V
Interpret: 𝐼𝑜𝑢𝑡 → 𝑉𝑜𝑢𝑡,𝐷 = 𝐼𝑂𝑢𝑡 +
1
2
𝑉𝑟𝑢 −𝑉𝑟𝑙
2𝑁
+ 𝑉𝑐
Input
Range (v)
Iout
Vout,D
Max
Error (V)
-∞ to -5
-5 to 0
0 to 5
5 to ∞
0
1
2
3
-7.5
-2.5
2.5
7.5
∞
± 2.5V
± 2.5 V
∞
Transfer Function (Type-J-TC/DRE–TC-J TC)
Transfer
Function
10
Reading
VSC [V]
• For 0℃ < 𝑇𝑆 < 400℃
– 𝑉𝑆𝐶 = 10𝑉
𝑇𝑆
400℃
= 0.025
𝑉
? Out of
range
𝑆𝑆𝐶 =
0
𝑉
℃
0
Measurand, T [°C]
𝜕𝑉𝑆𝐶
𝑉
= 0.025
𝜕𝑇
°𝐶
400
𝑇𝑆 (linear)
𝑉
𝑆
• 𝑆𝑆𝐶 = 0.025 °𝐶 ; 𝑆𝑇𝐶,𝐽 = 0.00005 °𝐶; G𝑎𝑖𝑛 = 𝑆 𝑆𝐶 = 500
𝑇𝐶,𝐽
• For boiling water 𝑉𝑆𝐶 = 10𝑉
100℃
400℃
= 2.5 𝑉, For ice 𝑉𝑆𝐶 = ?
– To use: Inverted transfer function: TS = (40°C/V)*VSC
• Conditioner Provides
–
–
–
–
–
Reference Junction Compensation (𝑉𝑆𝐶 not sensitive to TT)
Amplification (Allows normal DVM or computer acquisition to be used)
Low Pass Filtration (Rejects high frequency RF noise)
Linearization (Easy to convert voltage to temperature)
Galvanic Isolation (TC can be used in water environments)