1. Signal Conditioning and Linearization of RTD Sensors
9/24/11
This presentation will step through an overview on RTD sensors and their non-linearity, RTD measurement circuits, Analog Linearization and output methods, and Digital acquisition and lineariziation.

2. Contents RTD Overview RTD Nonlinearity Analog Linearization
Digital Acquisition and Linearization

3. What is an RTD? Resistive Temperature Detector
Sensor with a predictable resistance vs. temperature
Measure the resistance and calculate temperature based on the Resistance vs. Temperature characteristics of the RTD material
PT100
α =
Sometimes called a PRT for Platinum Resistive Thermometer. RTD is resistance Temperature Detector, a device that has a predictable resistance value vs. temperature. Measuring resistance allows for a back-calculation of the temperature.
Inject a current, measure the voltage, R= V/I
Temperature can then be calculated either directly, estimated using linear interpolations, or rounded to the closest look-up table value.
Over the full temperature range, the RTD resistance varies in a non-linear fashion from 35Ohms – 370Ohms

4. How does an RTD work? L = Wire Length A = Wire Area
e = Electron Charge (1.6e-19 Coulombs)
n = Electron Density
u = Electron Mobility
The product n*u decreases over temperature, therefore resistance increases over temperature (PTC)
Most will recognize the R=pl/A equation defining the resistance of a material with a defined length, area, and resistivity.
Less recognized is the equation that comprises the Resistivity (p) of the material. Resistivity is defined as 1/(e*n*u). The electron charge (e) is a constant value, but the electron density (n) and the electron mobility (u) change over temperature. For almost all conductors, the product of n*u decreases over temperature which results in an increase in resistance over temperature. As the electrons in a metal conductor are heated, they move faster, collide more and their ability to move decreases compared with the same conductor at a lower temperature. The result is a higher resistance as temperature increases.
Linear Model of Conductor Resistivity Change vs. Temperature

5. What is an RTD made of? Platinum (pt) Nickel (Ni) Copper (Cu)
Metal
Resistivity (Ohm/CMF)
Gold (Au) 13
Silver (Ag)
8.8
Copper (Cu)
9.26
Platinum (Pt) 59
Tungsten (W) 30
Nickel (Ni) 36
Platinum (pt)
Nickel (Ni)
Copper (Cu)
Have relatively linear change in resistance over temp
Have high resistivity allowing for smaller dimensions
Either Thin-Film or Wire-Wound
RTDs are commonly made of Pt, Ni, and Cu. This is mainly because over certain temperature ranges they experience fairly linear changes in resistance vs. temperature. Also, their properties allow for convenient resistance values to be created out of convenient sized sensors.
Platinum is most commonly used because it has a very high resistivity compared to the other common metals allowing for the creation of small thin-film elements.
Ohm/CMF = Ohms per “Circular Mil Foot”
*Images from RDF Corp

6. How Accurate is an RTD?
Absolute accuracy is “Class” dependant - defined by DIN-IEC Allows for easy interchangeability of field sensors
Repeatability usually very good, allows for individual sensor calibration
Long-Term Drift usually <0.1C/year, can get as low as C/year
Tolerance Class (DIN-IEC 60751)
**Temperature Range of Validity
Tolerance Values (C)
Resistance at 0C (Ohms)
Error at 100C (C)
Error over Wire-Wound Range (C)
Wire-Wound
Thin-Film
*AAA (1/10 DIN)
+/-( *t)
100 +/
0.08
AA (1/3DIN)
+/-( *t)
100 +/- 0.04
0.27
0.525 A
+/-( *t)
100 +/- 0.06
0.35
1.05 B
+/-( *t)
100 +/- 0.12
0.8
3.3 C
+/-( *t)
100 +/- 0.24
1.6
6.6
*AAA (1/10DIN) is not included in the DIN-IEC spec but is an industry accepted tolerance class for high-performance measurements
**Manufacturers may choose to guarantee operation over a wider temperature range than the DIN-IEC60751 provides
- The accuracy of an RTD depends on the Class of RTD that it falls into. DIN-IEC60751 defines 4 classes of RTDs: AA (1/3 DIN), A, B, and C. A fifth class, AAA (1/10 DIN) is industry accepted for high performance requiring applications. The class also dictates the usable temperature range of the RTD.
- Defining the classes allows for easy interchangeability of field sensors because it ensures the new sensor can only deviate from the previous sensor by a certain amount.
The repeatability of a single sensor is usually quite good allowing for individual RTD calibration for improved results.
Long-term shift on the sensor resistance typically equates to between 0.1C and C of temperature error per year.

7. Table Comparing Advantages and Disadvantages of Temp Sensors
Why use an RTD?
Table Comparing Advantages and Disadvantages of Temp Sensors
RTDs offer good linearity, temperature range, long-term stability, repeaability. The RTD is the measurement device used in establishing the International Temperature Scale of 1990 (ITS-90) between 14 and 962K.
RTD downsides are their cost (wire-wound are much more expensive than thin-film), the requirement for an excitation source, and fine resolution. A 1C change in temperature correlates to a change in resistance, demanding accurate acquisition circuitry.

8. How to Measure an RTD Resistance?
Use a…….
Current Source or
Wheatstone Bridge
The most straight-forward way to measure an RTD is to create a current source that drives through the RTD creating a voltage. By knowing the current and the measured voltage, the resistance can be calculated. The easiest method to implement is a wheatstone bridge, requiring only 3 external resistors, but in the long run the current source method provides less complexity in the linearization circuitry.

9. Note on Non-Linear Output of Bridge
It should be noted that although the bridge method seems like a simple option in comparison to building a current source, the inherent non-linearity of the bridge will have to be corrected for along with the non-linearity of the sensor itself adding an extra layer of complexity.
The plot shows that even for a linear change in sensor resistance (x-axis) the output is non-linear due to the (Ra+RTD) term in the denominator.

10. Simple Current Source / Sink Circuits
REF200
So instead of messing with a bridge, make a current source!!!
It’s really easy to make one with modern IC building blocks or purchase a solution like the REF200 which features dual 100uA current sources.

11. RTD Types and Their Parasitic Lead Resistances
2-Wire
3-Wire
2-Wire with Compensating Loop
4-Wire
Let’s compare the different types of RTDs showing the parasitic line resistances associated with their designs.

12. 2-Wire Measurements
In a 2-wire RTD the lead resistances of the wire connecting the RTD to the measurement circuitry appear in series with the RTD element and therefore appear as errors in the measurement. There is no way to cancel the lead resistances in a 2-wire RTD.
With a current source as the excitation, the error is simply 2*Rl * Isource

13. 3-Wire Measurements
With a 3-wire RTD an additional wire is connected to one end of the RTD element allowing for the cancellation of the lead resistances by placing one lead resistance in series with both the positive and negative measurement connections. When the differential measurement is made, the two lead resistances cancel in the subtraction.
Using current sources as the excitation source, the lead resistances can completely cancel as long as the two current sources are matched and the lead resistances are equal.
The 3-wire RTD provides improvements from the 2-wire solution by placing one lead resistance in series with both the top and bottom leg of the bridge. However the lead resistances unbalance the bridge resulting in a remaining error.
Error = 0 as long as Isource1 = Isource2 and RL are equal

14. 4-Wire Measurements
With a 4-wire RTD all lead resistance errors are removed from the measurement circuit by providing a Kelvin connection to the RTD. Two wires carry the current and the other two wires connect to the high impedance measurement circuitry. In this method the measurement circuitry does not include any lead resistance.
In a bride a “2-wire with compensating loop” can be used to provide further improvement by actually using the compensating loop to balance the bridge. ***Picture not correct needs updating!
System Errors reduced to measurement circuit accuracy

15. Self-Heating Errors of RTD
Typically 2.5mW/C – 60mW/C
Set excitation level so self-heating error is <10% of the total error budget
One of the negative aspects of RTDs is the self-heating of the element when large excitation sources are used to excite the sensor. The larger the excitation source the larger the output will be making the output easier to measure and increasing resolution. However with RTD sensors the larger the excitation current the larger the self-heating effects will be which will add to the measurement error cancelling out improvements in resolution.
Therefore the correct excitation current should be chosen so that the self-heating effects will not affect the accuracy of the circuit. As a simple rule choose the self-heating factor to account for a MAX of 1/10th of the total error in the system giving the other 90% of error tolerance band to the measurement circuit and RTD itself. Reducing the self-heating effects further may be required to work with the measurement circuitry to meet the required accuracy.
DIN/IEC requires self-heating to account for <25% of tolerance value when excited with max current (1mA /100Ω, 0.7mA/500 Ω, 0.3mA/1000 Ω)

16. RTD Resistance vs Temperature
Callendar-Van Dusen Equations
Equation Constants for
IEC PT-100 RTD (α = )
Now that we’ve successfully measured the RTD resistance we need to determine the temperature based on the resistance. The Callendar Van-Dusen equation defines the resistance vs. temperature characteristics of a platinum RTD. This is a non-linear model that is a fourth order polynomial for T < 0ºC and quadratic for T > 0ºC. Primary RTD standards have been created that specify the RTD resistance at 0C, the temperature coefficient, Alpha (α ), and the Callendar Van-Dusen constants.

17. RTD Nonlinearity
Linear fit between the two end-points shows the Full-Scale nonlinearity
Nonlinearity = 4.5%
Temperature Error > 45C
By creating a linear fit between the two endpoints of the RTD’s temperature the non-linear nature can be seen. Since the RTD’s curve is mostly second order (parabolic) the highest errors occur in the middle of the tempreature range and reduce near the end-points. Over the full -200C – 850C RTD temperature range an RTD has a little less than 4.5% FS nonlinearity resulting in a temperature error of 47degrees.

18. RTD Nonlinearity
B and C terms are negative so 2nd and 3rd order effects decrease the sensor output over the sensor span.
Because the second and third order terms in the CVD equation are negative, the resistance change per degree will decrease as the temperature delta away from 0C increases. This can be seen when comparing the actual RTD resistance to the resistance of a 1st order (linear) model for the RTD. The linear model only includes the “A” CVD coefficient.

19. Measurement Nonlinearity
When we apply an excitation source and measure the RTD voltage the non-linearity shows up as a measurement error unless we intend to correct it digitally. Shown in red is the resulting voltage out of the RTD when a 100uA current source is used to excite it over a temperature range of 0-800C. The result is a measurement offset of 10mV and span of 28mV.
The ideal transfer function is shown in blue. If we can obtain this transfer function then the math to calculate the temperature is only a linear y=mx+b model.

20. Correcting for Non-Linearity
Sensor output decreases over span? Compensate by increasing excitation over span! 
In an effort to create the linear transfer function, suppose we create a system where an additional voltage-controlled-current-source, “Icorrection.” is summed into the RTD to increase the excitation over the span of the RTD measurement? If it was done perfectly we could create a completely linear RTD output over the desired measurement span.
This type of analog compensation will only work for sensors whose output is a function of the excitation source. Since the measured output of the RTD is always in relation to the excitation source, the RTD is a candidate for this type of linearization.

21. Correcting for Non-linearity
Assuming a 500uA excitation source for the RTD, the uncorrected result can be seen in red. By performing a simple subtraction between the ideal (linear) sensor output and the actual sensor output and adding it back to the original source, we can create the appropriate correction current to linearize the RTD output. Using the correction current and the linear RTD model produces the blue curve showing an increasing sensor output over the span that is equal-to-but-opposite of the actual RTD sensor output. By exciting the actual RTD with the corrected current source a linear output is produced, shown in green.

22. Analog Linearization Circuits
We will now explore analog circuits that employ the linearization method previously covered.

23. Analog Linearization Circuits
Two-Wire Single Op-Amp
Example Amplifiers:
Low-Voltage:
OPA333 OPA376
High Voltage: OPA188 OPA277
This particular circuit was designed for a 0-5V output for a 0-200C temperature span. Components R2, R3, R4, and R5 are adjusted to change the temperature measurement range (not described in this presentation).
In this circuit the 4.99k resistor provides the ~1mA bias to the RTD. When correctly designed the I_Correction current is 0mA at the minimum sensor temperature and the RTD is only excited from the I_Bias current. As the RTD resistance increases, the voltage on the positive terminal of the op-amp increases, causing the output to increase. As the output increases, a small amount of the output is positively fed-back into the RTD through R5 providing an increase in RTD current. This creates the desired VCCS effect that was described previously.
This circuit is designed for a 0-5V output for a 0-200C temperature span. Components R2, R3, R4, and R5 are adjusted to change the desired measurement temperature span and output.

24. Analog Linearization Circuits
Two-Wire Single Op-Amp
Non-linear increase in excitation current over temperature span will help correct non-linearity of RTD measurement
Showing the non-linear increase in excitation current over the span of the measurement that will help to linearize the RTD. The correction current ranges from 0uA - 45uA causing the total RTD excitation current to vary from 982uA – 1010uA over the course of the measurement span. It should be noted that the initial bias current (i_bias) from the 4.99k resistor decreases over the span of the measurement as the RTD resistance increases which is why the final total RTD current does not equal 982uA +45uA = 1027uA.

25. Analog Linearization Circuits
Two-Wire Single Op-Amp
This type of linearization typically provides a 20X - 40X improvement in linearity
The results of the circuit are shown here. It can be seen by looking at where the curves pass through the graph grid lines that the corrected output is much more linear than the uncorrected model. Depending accuracy of the amplifier, the chosen resistor tolerance, and the proximity of the chosen resistors to the values calculated in the design of the circuit, this circuit will typically achieve a 20-40X improvement in non-linearity from the original uncorrected response.

26. Analog Linearization Circuits
Three-Wire Single INA
Example Amplifiers:
Low-Voltage:
INA333 INA114
High Voltage INA826 INA114
The previous circuit suffers from the issues mentioned earlier with all 2-wire connections. The previous circuit could be placed at the remote site with the RTD, but the voltage output would have loss over the cabling.
A 3-wire connection is shown here with the INA826. The RTD is placed into a bridge where a “Rzero” 100Ohm resistor is placed to match the zero-span resistance of the RTD (100Ohms) for this measurement. Similar to the previous circuit, as the RTD resistance increases, the voltage on the positive terminal increases causing the output to increase. As the output increases it causes a larger voltage drop to form across Rlin causing an increasing excitation current to flow through the RTD over the span of the measurement.
This method improves the errors from the lead resistances, but since the sensor is driven with a bridge instead of a current source the lead resistances are not completely cancelled and the error will be what was shown on the “3-wire RTD” slide.
This circuit is designed for a 0-5V output for a 0-200C temperature span. Components Rz, Rg, and Rlin are adjusted to change the desired measurement temperature span and output.

27. Analog Linearization Circuits
Three-Wire Single INA
This type of linearization typically provides a 20X - 40X improvement in linearity and some lead resistance cancellation
The results of the 3-wire circuit are shown here. Again, it can be seen that the corrected output is much more linear than the uncorrected model.
Compared to the single op-amp two-wire circuit, this circuit only depends on the matching of 4 resistors instead of 5 to produce the correct results and again, depending accuracy of the amplifier, the chosen resistor tolerance, and the proximity of the chosen resistors to the values calculated in the design of the circuit, this circuit will typically achieve a 20-40X improvement in non-linearity from the original uncorrected response.

28. Analog Linearization Circuits
XTR mA Current Loop Output
Shown here is an XTR105 circuit configured for operation with a 2-wire RTD. A solution like the XTR105 had advantages in the built-in current sources, which removes the non-linearities of the resistor biasing circuits shown previously. The XTR105 also integrates a 4-20mA driver allowing for long distances between the RTD+XTR circuit and the main measurement center.
The XTR105 is easier to understand when looking at a 2-wire system such as the one shown here, but it is easily configurable for a 3-wire circuit as well. Since the XTR105 integrates the current sources, the lead resistances are completely cancelled as good as the matching is of the current sources in the 3-wire approach.

29. Analog Linearization Circuits
XTR mA Current Loop Output

30. Analog + Digital Linearization Circuits
XTR mA Current Loop Output
An example of a mix of an analog solution with a digital solution is the XTR108. The XTR108 uses an EEPROM based look-up table to create the proper analog correction current to linearize the RTD output. Since the XTR108 does not rely on resistor tolerances and fixed values to determine the correction currents it can achieve accuracy much greater than the previous solutions and can reduce the errors to under 0.1%

31. Digital Acquisition Circuits and Linearization Methods
Move on to Digital Acquisition Circuits and Digital Linearization Methods

32. Digital Acquisition Circuits
ADS bit Delta-Sigma 2-Wire Measurement with Half-Bridge
The circuit shown here is a very simple 16-bit acquisition circuit with a very simple and easy to use converter. The RTD is excited in a bridge fashion for a 2-wire measurement. The 4-channels on the ADS1118 allow for up to 2 2-wire RTDs. This circuit would suffer the standard non-linearities associated with the bridge, as well as the resistor tolerance of the bias resistors, errors from the reference, and the inability to cancel the lead resistances.
For a limited temperature range and small lead resistances a circuit like this may function well.

33. Digital Acquisition Circuits
ADS bit Delta-Sigma Two 3-wire RTDs
The next level in performance would be to move to a solution like the ADS The ADS1220 integrates the current sources removing the bridge non-linearity. The integration of the current sources also allows for a ratiometric measurement to be made by creating the reference from the current sources and a reference resistor. Any noise on the excitation source will appear on the inputs which will also appear on the reference. Therefore when the differential measurement between the inputs and reference is made inside the converter, the noise is cancelled out.
Having two current sources also allows for three-wire RTDs which can cancel the lead resistances. Another advantage of the two current sources is the ability to create a differential input to the converter by placing an “Rcomp” compensation resistor in series with the RTD wire going back to the input. If the Rcomp value is chosen as the value of the mid-span RTD resistance This allows for the gain of the internal PGA to be increased drastically increasing resolution. The downside is the temperature drift and tolerance of the resistor which will need to be calibrated out to achieve high level accuracy.
The ADS1220 can support two 3-wire RTDs
3-wire + Rcomp shown for AIN2/AIN3

34. Digital Acquisition Circuits
ADS bit Delta-Sigma One 4-Wire RTD
The ADS1220 also allows for a 4-wire RTD to remove the lead resistance errors completely. Only one 4-wire RTD can be used with the ADS1220

35. Digital Acquisition Circuits
ADS bit Delta-Sigma Three-Wire + Rcomp
The highest level of accuracy comes from a solution like the ADS1247/ADS These are 24-bit converters like the ADS1220 but have better specifications on the inputs and current sources. Shown here is the ADS1247 with a 3-wire RTD and compensation resistor.

36. Digital Acquisition Circuits
ADS bit Delta-Sigma Four-Wire
Shown here is a 4-wire RTD with the ADS1247.

37. Digital Linearization Methods
Three main options
Linear-Fit
Piece-wise Linear Approximations
Direct Computations
So now we have directly acquired the RTD signal with the non-linearity and we must determine the temperature based on the non-linear voltage. We have 3 main options to do this:
1.) Linear-Fit
2.) Piece-wise Linear Fit
3.) Direct Computations

38. Digital Linearization Methods
Linear Fit
Pro’s:
Easiest to implement
Very Fast Processing Time
Fairly accurate over small temp span
Con’s:
Least Accurate
End-point Fit
Best-Fit
The least accurate but easiest to implement method is the linear fit method. In this method a single linear (y=m*x+b) equation is fit to the RTD resistance vs. temperature curve. Either an endpoint or best-fit linear method can be used with the best-fit providing higher accuracy.
This method takes almost no effort from the processor and requires no code overhead for coefficients or advanced math libraries. Over small temperature spans this method can be fairly accurate and may be attractive due to the low-overhead.
The downside is that this method is by far the least accurate. With no way to cancel or correct for the non-linearities of the RTD curve, the final answer is only as accurate as the non-linearity of the RTD over the designed span.

39. Digital Linearization Methods
Piece-wise Linear Fit
Pro’s:
Easy to implement
Fast Processing Time
Programmable accuracy
Con’s:
Code size required for coefficients
The next and most commonly used method is to use a piece-wise linear function to determine the temperature from the RTD resistance. In this method a series of linear fit equations are combined together with different slopes to fit the non-linear curve of the RTD.
Again this method takes very little effort from the processor to actually do the math for the calculations, and is comprised of a series of additions/subtractions with one multiplication and one divide per calculation so there is still not a requirement to load any math libraries. Also the user has control over the accuracy of the system by being able to choose the number of coefficients that will be stored in memory.
The main processing overhead with this method is due to the actual number of calibration coefficients used to do the linearization. The greater the number of points the greater the accuracy, but also the greater the code-size becomes to store the coefficients in memory. To achieve 0.001C accuracy around 900 coefficients are required.

40. Digital Linearization Methods
Direct Computation
Pro’s:
Almost Exact Answer, Least Error
With 32-Bit Math Accuracy to +/ C
Con’s:
Processor intensive
Requires Math Libraries
Negative Calculation Requires simplification or bi-sectional solving
Positive Temperature Direct Calculation
The most accurate method is to perform direct calculations on the measured RTD resistance based on a solution to the Callendar Van-Dusen equations. This method is rarely used in the field due to the processing requirements but can be successfully implemented on a more powerful processor or PC which has higher processing capabilities.
The main benefit of this method is the removal of any calculation errors in the RTD resistance to temperature conversion. With a fully capable 32-bit processor the rounding errors account for less than C which is well below the accuracy of the measurement system.
The downside to this method is the processing requirements. Math libraries will need to be loaded to successfully perform the calculations which will take up code space and memory. The time required to do the calculation is also non-trivial and can take several 10’s of milli-seconds.
In the positive direction the CVD equations have two solutions that can be found by solving the quadratic equation that defines the solution for all 2nd order systems.
Since the CVD equation is a 4-th order polynomial for temperatures less than 0C there are four solutions and the math to find them is simply not practical. Therefore one of two other methods is used to directly calculate the temperature in the negative direction. The first shown on this slide is to apply an inverse transfer function to the CVD equations which will yield a multi-order approximation to the solution. Shown here is a 4-th order approximation.
Negative Temperature Simplified Approximation

41. Digital Linearization Methods
Direct Computation
Bi-Section Method for Negative Temperatures
The second method for direct calculations in the negative direction is to apply a bi-sectional search for the correct answer by solving the CVD equation in the forward direction and comparing the results to the measured resistance. The binary search will eventually find the nearest solution based on the programmed accuracy.
In the example above the programmed accuracy is C and the loop runs through until the calculated resistance “Rcal” is within C of the measured resistance that is entered into the function called “Res”

42. Questions/Comments? Thank you!! Special Thanks to: Omega Sensors
RDF Corp