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07-1 Temperature 07*
*-1
Definition: The translational (e.g., wiggling around) energy : : :
: : :of particles in a system.
No practical way to measure : : :
: : :velocity of every particle : : :
: : :in most systems of interest.
Instead, temperature scales are defined.
There are two types:
- The thermodynamic temperature scale.
"Really" measures temperature.
- Practical temperature scales.
Approximations of thermodynamic scale.
Much easier to measure temperature on a practical scale.
For temperatures of interest, differences are very small.
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07-2 07*
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Kelvin's Thermodynamic_________ Temperature Scale
Due to William Thomson, a.k.a., Lord Kelvin (1824-1907).
Start with a precise temperature that can easily be reproduced.
The triple point of water, Ttr, is used.
Ttr is temperature at which : : :
: : :water can simultaneously be in : : :
: : :the solid, liquid, and gas states: 0:01 ffiC.
- Confine an ideal gas in a container of fixed volume, S.
- Bring the gas to temperature Ttr.
Call the pressure of this gas Ptr.
By_definition___ (of the Kelvin scale) this temperature is Ttr , 273:16 K.
The ideal gas law: P S = n
07-3 07*
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Practical Temperature Scales_
Designed to be easy (relatively) to measure.
Scales are revised every few decades.
Latest revision in 1990, called ITS-90.
(International Temperature Scale.)
Older scale (1968), IPTS-68.
(International Practical Temperature Scale)
Difference between ITS-90 and IPTS-68 : : :
: : :is as large as 0:4 ffiCat 800 ffiC.
At human-tolerable temperatures, : : :
: : :difference is in hundreths of a degree.
All practical scales are identical at the triple point of water.
How a practical temperature scale is defined:
A set of fixed points is established, : : :
: : :for example the triple point of water.
A temperature is assigned to each fixed point, : : :
: : :based on the thermodynamic scale.
Accurate thermometers (transducers) are chosen.
Functions are defined mapping : : :
: : :the thermometers' output to temperature : : :
: : :so that they pass through the fixed points.
) Temperatures defined in terms of fixed points and special transducers.
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07-4 07*
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For ITS-90:
- (:65 K; 5:0 K)
Vapor-pressure relation between two isotopes of helium.
- (3:0 K; 24:5561 K)
Helium fixed-volume thermometer.
(Like thermometer used in thermodynamic scale, : : :
: : :except helium replaces the ideal gas.)
- (13:8033 K; 1234:93 K)
Resistance of platinum.
- > 1234:93 K:
Based on radiated light.
07-4 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07. *
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07-5 Temperature Transducers 07*
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Basic Types
- Thermistor.
Block of semiconductor material.
Resistance is a function of temperature.
- Resistance Temperature Device (RTD)
Strip of metal.
Resistance is a function of temperature.
- Thermocouple.
Potential across two metals is a function of temperature.
- Diode.
Forward-bias voltage is a function of temperature. (Not covered.)
Integrated Temperature Sensors
Transducer and factory-calibrated conditioning circuit : : :
: : :combined in a single package.
Usually available as current or voltage sources.
Current or voltage is a convenient, linear function of temperature.
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07-6 Thermistor 07*
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Name: Therm____al resistor__.
Symbols: (Both are used.)
Temperature range: about 100 ffiCto 200 ffiC. (Relatively narrow.)
Construction:
block of semiconductor material (without junction).
Principle of Operation
As with all semiconductors,: : :
: : :electron energy levels divided into two bands,: : :
: : :valence and conduction.
Electrons in conduction band participate in current flow.
Electrons in valence band do not.1
The number of electrons in conduction band: : :
: : :increases with temperature, : : :
: : :reducing resistance.
Resistance is determined by the density of conduction electrons.
____________________________
1 Actually they do, but that's a hole other story.
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07-7 07*
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Desirable Characteristics
- Sensitive.
(Small change in temperature yields an easily readable change in
resistance.)
- Can be made very small.
(Small devices react to temperature changes quickly.)
- High resistance.
(Easier to design conditioning circuit.)
Undesirable Characteristics
- Delicate.
Can be damaged (de-calibrated) by excessive heat.
- There are many non-standard types.
Transducer Model Functions
All functions will be approximations.
Very good, the Steinhart-Hart Equation:
1
1
Ht11 (y) = __________________________________3 ;
A + B ln y + C ln y
where A, B, and C, are experimentally determined constants.
fi_
Good: Ht2 (x) = R0 e x .
Later, a linear function will be derived.
07-7 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07. *
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07-8 Thermistor Sample Problem 07*
*-8
Convert process variable x 2 [10 ffiC; 50 ffiC], the temperature in
1
room 102 EE Building into H(x) = x ___, a floating-point number.
K
The number should have a precision of 0:05. Use a thermistor and
fi_
the function Ht2 (x) = R0 e x with fi = 3000 K and R0 = 0:059 .
Solution Plan:
- Choose ADC.
- Based on ADC input voltage, design conditioning circuit.
- Based on ADC precision (bits), write interface routine.
ADC Choice
Use ADC with function HADC(5 V;b) (y) : : :
: : :value of b chosen later.
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07-9 07*
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Conditioning Circuit
Input is a resistance (from thermistor), output is voltage.
Input range to ADC is 0 to 5 V, therefore:
0 Hc(Ht2(x)) 5 V for 10 ffiC x 50 ffiC
Choose conditioning circuit based on this constraint.
Conditioning circuit will not_ linearize x.
(This would be very difficult using analog circuits.)
Thermistor y = Ht2(x) is monotonic with temperature.
In this case, when x increases y always decreases.
Therefore, conditioning circuit must convert either:
Ht2(10 ffiC) = 0 V and Ht2(50 ffiC) = 5 V
or
Ht2(10 ffiC) = 5 V and Ht2(50 ffiC) = 0 V.
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07-10 *
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Will use gain/offset circuit.
Let Rmax = Ht2(263:15 K) = 5272 and
Rmin = Ht2(323:15 K) = 634:9 .
Hc(Ht2(50 ffiC)) = Hc(Rmin ) = A5(Rmin O5) = 0 V
Hc(Ht2(10 ffiC)) = Hc(Rmax ) = A5(Rmax O5) = 5 V
Therefore, O5 = Rmin and A5 = ______5_V_______R.
max Rmin
Recall O5 = vC_RD_RA___vand A5 = _RB_vB__.
B RC RD RA
Suppose, the following are convenient values:
RA = 1 k, RD = 5 k, vB = vC = 10 V.
Then choose RC = 7876 and RB = 539:2 .
Then A5 = 1:078 mA and O5 = 634:9 . So:
Hc(y) = 1:078 mA (y 634:9 )
07-10 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-11 ADC Output, ADC Precision Choice *
*07-11
Problem specified that H(x) should have a 0:05 precision.
ADC Output:
HADC(vADC ;b) (Hc(Ht2(x))) = z = ___1___v(2b 1)A5(R0efi=x O5) :
ADC
To determine precision evaluate at x1 = 323:10 K and x2 = 323:15 K.
Difference should be no less than one.
HADC(vADC ;b) (Hc(Ht2(x1))) HADC(vADC ;b) (Hc(Ht2(x2))) 1
__1____(2b 1)A (R efi=x1 R efi=x2) 1
vADC 5 0 0
Solving for b yields:
ss
b log2 ___________vADC_____________ + 1 = 13:
A5(R0efi=x1 R0efi=x2)
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07-12 *
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Interface Routine
HADC(vADC ;b) (Hc(Ht2(x))) = z = ___1___v(2b 1)A5(R0efi=x O5) :
ADC
Solving for x yields
1
x = fi ln O5__R+ ___1____zvADC____b
0 A5R0 (2 1)
Hf(HADC(vADC ;b) (Hc(Ht2(x)))) = H(x) = _x_K:
1
Hf(z) = H(x) = fi ln O5__R+ ___1____zvADC____b 1__:
0 A5R0 (2 1) K
Substituting values:
tee = 3000.0 / ( log( 10760.3 + 9.5949 * raw ));
where raw is the value read from the ADC output.
07-12 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-13 Linear Thermistor Model *
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Linear transducer functions are preferred.
Especially useful when there is no computer processing.
Thermistor response is close to linear over small temperature ranges.
(But non-linear over wide temperature ranges.)
A linear thermistor function will be derived.
Plan:
Call TM the "middle" temperature.
(Center of range of temperatures to measure.)
Goal: derive function in form Ht4(x) = RM (1 + ffx) : : :
: : :where RM and ff are constants to be determined : : :
: : :and x = x TM .
Temporarily set Ht4(x) = mx + b, the equation of a straight line.
fifix=TM
Let m = _d_dxHt2(x) fifi .
Solve for b in mTM + b = Ht2(TM ).
Transform mx + b into RM (1 + ffx). Then:
fi_ fi
RM = Ht2(TM ) = R0e TM and ff = ____T 2
M
Note: derivation can also be done using a more accurate model than Ht2(x).
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07-14 *
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Assuming the linear thermistor model, : : :
: : :a gain/offset amplifier could: : :
: : :convert temperature linearly into voltage.
i x j
For example, H(x) = ___K 300 K V, for x 2 [300; 301].
However, if a wide temperature range were used, : : :
: : :model error would be unacceptably high.
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07-15 Passive Conditioning Circuit *
*07-15
Idea:
Place thermistor in a resistor network to achieve some linearity.
A simple but effective example appears below.
For convenience, combination will be treated as a transducer.
i ff j
Transfer function: Ht3(x) = RM___2 1 + __2x ;
where RM is resistance at center of range,
x = x TM ,
TM is the temperature at the center of range,
fifix=TM
and ff = __1__Rd_Ht(x) fifi .
dMx
How much more accurate is this?
07-15 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-16 Thermistor Linearization Sample Problem *
*07-16
Compute the model error of thermistor functions Ht4 and Ht3 at
temperatures 250 K, 270 K, and 290 K for a thermistor and a ther-
mistor with a shunt resistor (the passive conditioning circuit just
presented) designed for temperature range [250 K; 290 K]. Base the
error on the following measurements:
Rt(250 K) = 9603 ,
Rt(270 K) = 3948 ,
Rt(290 K) = 1835 ,
where Rt(T ) is the measured resistance of the thermistor at temperature
T .
Thermistor model functions.
i ff j
Ht4(x) = RM (1 + ff(x TM )) and Ht3(x) = RM___2 1 + __2(x TM ) ,
where TM = 270 K, ff = _fi_T=20:04115_____ and RM = 3948 .
M K
The inverse of functions are:
H1t4(Rt4) = _1_ffRt4__R 1 + TM and H1t3(Rt3) = _2_ 2 Rt3__ 1 + TM :
M ff RM
Ideal Actual Actual
x=_K______Rt4=_______Rt3=________H1t4(Rt4)=_K_______Pct.__Err._____H1t3(Rt3)=_K________Pct.*
*__Err.____
250 9603 2798 235.2 5.91% 249.7 0.10%
270 3948 1974 270.0 0 270.0 0
290 1835 1253 283.0 2.42% 287.7 0.78%
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07-17 *
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Plotted below are the actual temperature and the temperature com-
puted using the two thermistor-model functions.
There are more elaborate networks which can be used to linearize
thermistor response. These will not be covered.
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07-18 Resistance Temperature Device (RTD) *
*07-18
Symbol:
Temperature range: about 220 ffiCto 750 ffiC(Platinum).
Construction:
Transducer is a metal, usually platinum.
Metal is wound into a long coil : : :
: : :or printed on a ceramic substrate in a serpentine pattern.
Inventor: C. H. Meyers in 1932.
Resistance change of metals discovered in 1821 by Sir Humphrey Davy.
Use of platinum for temperature measurement : : :
: : :suggested by Sir William Siemens in 1871.
Principle of Operation
In normal operation a current is flowing through RTD.
As the temperature increases, : : :
: : :electrons collide more frequently with metal atoms : : :
: : :increasing resistance.
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07-19 *
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Desirable Characteristics
- Accurate. (An RTD used to define part of ITS-90.)
- Stable.
- Wide temperature range.
- Reasonably linear.
- Available in standard types.
Undesirable Characteristics
- Low sensitivity.
(Small resistance change with temperature.)
- Expensive.
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07-20 *
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RTD Model Function
For a properly constructed transducer : : :
: : :the following function is exact for ITS-90:
9 !
X x= K 754:15 i
Ht0(x) = R0 C0 + Ci _________________ ;
i=1 481
for x 2 (0 ffiC; 961:78 ffiC), where R0 = Ht0(273:16 K) and the Ci are
constants defined in the ITS-90 standard.
This function is accurate (but not exact) and easier to use:
Ht1(x) = R0(1 + ff1x + ff2x2).
Platinum RTDs are usually made so that R0 = 100 .
For platinum, ff1 = 0:00398= ffiCand ff2 = 5:84 107 = ffiC2.
07-20 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-21 *
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Three-Wire Configuration
Problem with RTDs: lead resistance is significant.
Lead resistance may change with temperature.
Not necessarily the temperature being measured!.
Leads can be shortened or lengthened, changing resistance.
Solution: three-wire RTD.
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07-22 *
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Three-Wire RTD Use
When used in a Wheatstone bridge, : : :
: : :unwanted resistances have little effect on output.
Rw is resistance of wire. Output, vo, should not be a function of this.
Re-write RTD function as Ht(x) = R0 + R0H0t(x),
where H0t(x) = ff1x + ff2x2.
1_ 0
R0 Ht(x) AvE 0
vo = AvE ________2_____________________0 ss ______Ht(x).
R0 Ht(x) + 2Rw + 2R0 4
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07-23 Thermocouples *
*07-23
Symbol:
Temperature range: about [270 ffiC; 1820 ffiC].
Construction:
- Two metals joined (e.g., welded together).
- Transducer is junction (weld).
- Leads, connecting to metals, specially constructed.
Principle of Operation
When dissimilar metals joined, an EMF develops across the junction. (As
in semiconductor PN junctions.)
Strength of EMF is a function of metals used and junction temperature.
Temperature is determined by measuring voltage.
Measuring this voltage is not as easy as one might think: : :
History
Effect of EMF discovered in 1821 by Thomas Seebeck.
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07-24 *
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Desirable Characteristics
Very wide temperature range: freezer to furnace.
Rugged.
Accurate.
Highly standardized.
- Connectors and color of cables are all part of the standard.
- Tables of thermocouple voltages published by : : :
: : :National Institute of Standards and Technology (NIST).
No self-heating.
(Conditioning circuit does not warm transducer.)
Undesirable Characteristics
Difficult to measure voltage.
Non-linear.
Transducer Model Functions
Best: Ht1(x), thermocouple tables published by NIST.
Over a narrow temperature range:
Ht3(x) = vM (1 + ff(x TM )) : : :
: : :where ff is a constant called : : :
: : :the Seebeck Coefficient.
In most cases a lookup table, Ht1(x), would be used.
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07-25 Seebeck Voltage *
*07-25
Consider two joined metals:
Potential developed, vAB (x), a function of metals and temperature.
Inserted Metals
Consider two junctions in series, junction AX at x1 and junction XB at
x2:
v = vAX (x1) + vXB (x2).
_______________________________________
_ v = vAX (x) + vXB (x) = vAB (x) _
If x1 = x2 = x then _______________________________________._
This is referred to as the law of inserted metals.
Used to show, among other things, : : :
: : :that thermocouple junctions can be welded.
07-25 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-26 *
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Measurement of Seebeck Voltage
Consider the following setup:
Note that both voltmeter connections are at same temperature.
Schematically,
vM = vCA (Tr) + vAB (x) + vBC (Tr)
= vAB (x) + vBA (Tr)
= vAB (x) vAB (Tr)
To determine x must know Tr.
Note that vM is not a function of metal C, making life easier for us.
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07-27 *
*07-27
Isothermal Block
Block upon which connections to thermocouple leads are made.
All parts of block are kept at the same temperature.
Called the reference temperature, and denoted Tr.
Either : : :
: : :block is maintained at a known temperature.
(E.g., placed in an ice bucket.): : :
: : :or block also includes a temperature sensor.
Either way, Tr is treated as part of the conditioning circuit.
Using isothermal block, vAB (T ) can be determined.
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07-28 *
*07-28
Standard Thermocouple Tables
Published by NIST.
Some standard thermocouples:
- Type J: Iron vs. Copper-Nickel.
Temperature range: [210 ffiC; 760 ffiC].
- Type K: Nickel-Chromium vs. Nickel-Aluminum.
Temperature range: [270 ffiC; 1372 ffiC].
- Type R: Platinum- 13% Rhodium vs. Platinum.
Temperature range: [0 ffiC; 1767 ffiC].
Thermocouple Table Entries
Standard thermocouple tables give :
v = HXY (x) = vAB (x) vAB (0 ffiC)
and
x = H1XY (v), where XY is the type of thermocouple.
Temperatures in tables used for class are on the IPTS-68 scale.
Example:
A voltage of 6:86 mV is measured at an isothermal block connected to a
Type-R thermocouple. The block is at 0 ffiC. What is the thermocouple
temperature?
According to the table, HTypeR (710 ffiC) = 6:860 mV .
So, temperature is 710 ffiC.
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07-29 *
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When Isothermal Block is not at 0 ffiC:
Recall, HXY (x) = vXY (x) vXY (0 ffiC).
Consider a measurement where Tr 6= 0 ffiC.
Then we need: vXY (x) vXY (Tr).
This is equal to HXY (x) HXY (Tr).
Example:
A voltage of 6:860 mV is measured at an isothermal block connected to a
Type-R thermocouple. The block is at 23 ffiC. What is the thermocouple
temperature?
By the Type-R thermocouple table : : :
: : :HTypeR (710 ffiC) = 6:860 mV and HTypeR (23 ffiC) = 0:129 mV .
Measured voltage is vTypeR (x) vTypeR (Tr) = 6:860 mV .
Subtract vTypeR (0 ffiC) from both sides and solve for vTypeR (x)vTypeR (0 ffiC).
Substituting values, vTypeR (x) vTypeR (0 ffiC) = 6:989 mV .
Based on table, x = 721 ffiC.
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07-30 Ice-Bath Circuits *
*07-30
Compensate for temperature of isothermal block.
Other Names:
Electronic ice point.
Hardware compensation.
Details
Consider an isothermal block : : :
: : :with a built-in temperature transducer.
A circuit which converts the voltage at the thermocouple leads : : :
: : :from vXY (x) vXY (Tr), : : :
: : :to vXY (x) vXY (0 ffiC) : : :
: : :is called an electronic ice bath circuit.
These can be built from passive components or active devices.
An example of an ice-bath circuit : : :
: : :will follow integrated temperature sensors.
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07-31 Integrated Temperatures Sensors *
*07-31
Symbols: (current source type) (volt. source type).
Temperature range: about 100 ffiCto 200 ffiC. (Relatively narrow.)
Construction:
Transducer (usually diode) mounted : : :
: : :in same package as conditioning circuit.
Principle of Operation
Temperature is sensed by some transducer.
Conditioning circuit converts temperature to : : :
: : :a voltage or current : : :
: : :(depending on type).
Voltage or current output is in user (engineer)-friendly form.
Desirable Characteristic
- Linear, human-oriented output.
(E.g., current in microamps is temperature in Kelvins.)
Undesirable Characteristics
- Narrow temperature range.
- Slow response to temperature changes.
- Fragile.
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07-32 *
*07-32
Typical Functions
Voltage type: Ht1(x) = x 10_mV___K.
Current type: Ht1(x) = x A____K.
Use
Current type must have at least several volts bias.
Current type best : : :
: : :when resistance of leads may be significant, : : :
: : :as when long leads are used.
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07-33 Example: Thermocouple and Integrated Temperature Sensor *
*07-33
Design a circuit to convert a temperature, x 2 [100 ffiC; 1760 ffiC], to
a floating-point number stored in variable tee, where tee = H(x) =
1
x _____ffi. Use the following:
C
- A Type-R thermocouple.
- A 2100-entry Type-R thermocouple table : : :
: : :stored as an array in the computer's memory.
- An isothermal block : : :
: : :with an integrated temperature sensor : : :
A
: : :having response Hits(x) = x _____.
K
The isothermal block will be exposed : : :
: : :to temperatures in the range [5 ffiC; 50 ffiC].
Solution Plan:
- Design circuit.
- Choose component values.
- Write code to compute answer.
07-33 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-34 *
*07-34
Circuit
Call response of thermocouple HTypeR (x).
Use two ADCs, : : :
: : :one for thermocouple : : :
: : :and one for integrated temperature sensor.
ADC function: HADC(vADC ;b) (x) = HADC(10 V;16) (x).
Use instrumentation amplifier with gain A : : :
: : :to condition thermocouple output for ADC input.
Call response of this circuit Hc1.
Use resistor, RA , and voltage source : : :
: : :to condition integrated temperature sensor for ADC input.
Call response of this circuit Hc2.
07-34 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-35 *
*07-35
Instrumentation Amplifier Gain
ADC input cannot exceed 10 V.
Note: maximum occurs when x is at its maximum and Tr is at its mini-
mum.
Value for HTypeR (x) is found in Type-R thermocouple table.
0 V Hc1(HTypeR (x) HTypeR (Tr)) vADC
0 V A(HTypeR (x) HTypeR (Tr)) 10 V
Minimum voltage: HTypeR (100 ffiC) HTypeR (50 ffiC) = 0:351 mV .
Maximum voltage: HTypeR (1760 ffiC) HTypeR (5 ffiC) = 20:979 mV .
______________________
Therefore, A < 477 must be satisfied. __Choose_A_=_450._____ _
Makes use of more than 92% of ADC's dynamic range. (Good.)
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07-36 *
*07-36
Resistance Of RA
Constraint: 0 V Hc2(Hits(Tr)) vADC
Current-to-voltage circuit: Hc2(y) = yRA .
0 V A Tr_KRA 10 V.
RA < ____10_V_K_____323:15=K3A0:945 k
____________________________
_ Choose: R = 20 k. _
____________A________________
Makes use of < 10% of ADC's dynamic range. (Wasteful.)
Possible test or homework question: : : :
: : :"How can the circuit be modified : : :
: : : to make greater use of the ADC's dynamic range?"
07-36 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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07-37 *
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Interface Routine
Call the value read from the thermocouple input r1 : : :
: : :and call value from integrated temperature sensor r2.
r1 = HADC(10 V;16) (Hc1(HTypeR (x) HTypeR (Tr)))
Need to satisfy:
Hf(HADC(10 V;16) (Hc1(HTypeR (x) HTypeR (Tr)))) = H(x) = _x__ffiC
Let z = HADC(10 V;16) (Hc1(HTypeR (x) HTypeR (Tr))) and solve for x.
x = H1TypeR z vADC____2b1_1A+ HTypeR (Tr) .
Hf(z) = H(x) = _x_K 273:15
Next find Tr.
r2 = HADC (Hc2(Hits(Tr))). Solving, Tr = ___r2_KvADC_______(2b. 1) AR
A
Let function hTyR(T) return the thermocouple voltage : : :
: : :at temperature T with reference temperature 0 ffiC.
Let function hTyRi(v) return the thermocouple temperature : : :
: : :when the measured voltage is v with reference temperature 0 ffiC.
Then:
double t_ref = r2 * 7.6293E-9; /* = r2 _1__RvADC___b*/
2A 1
double tee = hTyRi( r1 * 3.390E-7 + hTyR( t_ref ) ) - 273.15;
07-37 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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Lookup Function
Store Type-R thermocouple table (from NIST) in a 2100-entry array.
Function hTyR(T) returns voltage if there is an entry for T.
Otherwise, it looks up two closest values in table.
A voltage is interpolated and returned.
Function hTyRi(T) works in a similar fashion.
07-38 EE 4770 Lecture Transparency. Formatted 8:22, 27 January 1999 from lsli07*
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| David M. Koppelman - koppel@ee.lsu.edu | Modified 27 Jan 1999 12:57 (18:57 UTC) |