### AN1001

```AN1001
IC Temperature Sensor Accuracy Compensation
with a PIC® Microcontroller
Author:
SOLUTION APPROACH
Ezana Haile
Microchip Technology Inc.
INTRODUCTION
Microchip Technology Inc. provides a number of analog
and serial output Integrated Circuit (IC) temperature
sensors. Typically, these sensors are accurate at room
temperature within one degree Celsius (±1°C). However, at hot or cold temperature extremes, the accuracy
decreases nonlinearly. Normally, that nonlinearity has a
parabolic shape.
This application note derives an equation to describe
the typical nonlinear characteristics of a sensor, which
is used to determine compensation for the sensor's
accuracy error over a specified range of operating temperatures. A PIC® microcontroller unit (MCU) can
compute the equation and provide a temperature reading with higher accuracy.
This application note is based on MCP9700 and
MCP9701 analog-output temperature sensors and
MCP9800 serial-output temperature sensors.
Silicon characterization data is used to determine the
nonlinear sensor characteristics. From this data, an
equation is derived that describes the typical performance of a sensor. When the corresponding coefficients for the equation are determined, the coefficients
are used to compute the compensation for the typical
sensor’s nonlinearity.
The error distribution is provided using an average and
±1 standard deviation (± before and after compensation. A total of 100 devices were used as representative
for the MCP9700 and MCP9701, while 160 devices
were used for the MCP9800.
Figure 1 shows the typical sensor accuracy before and
after compensation. It illustrates that the compensation
provides an accurate and linear temperature reading
over the sensor operating temperature range.
A PIC MCU is used to compute the equation and
compensate the sensor output to provide a linear
Typical Accuracy (°C)
3.0
2.0
Sensor Accuracy
1.0
0.0
-1.0
Compensated Sensor Accuracy
-2.0
-3.0
-55 -35 -15
FIGURE 1:
5
25 45 65
Temperature (°C)
85
105 125
Typical Sensor Accuracy Before and After Compensation.
 2010-2015 Microchip Technology Inc.
DS00001001C-page 1
AN1001
SENSOR ACCURACY
The typical sensor accuracy over the operating temperature range has an accuracy error curve. At hot and
cold temperatures, the magnitude of the error increases
exponentially, resulting in a parabolic-shaped error
curve. The following figures show the average and ±1°C
standard deviation of the sensor accuracy curve for the
MCP9800, MCP9700 and MCP9701 sensors.
The accuracy specification limits for these sensors are
published in the corresponding data sheets as plotted
in Figure 2, Figure 3 and Figure 4. Note that due to the
sensor nonlinearity at temperature extremes, the accuracy specification limits are widened. The reduced
accuracy at temperature extremes can be
compensated to improve sensor accuracy over the
range of operating temperatures.
3.0
Accuracy (°C)
2.0
1.0
Spec. Limit
0.0
-1.0
+ V
Average
- V
-2.0
-3.0
-55 -35 -15
5
25 45 65
Temperature (°C)
FIGURE 2:
(160 parts).
85
105 125
MCP9800 Accuracy
6.0
Accuracy (°C)
4.0
Spec. Limit
2.0
0.0
-2.0
+ V
Average
- V
-4.0
-55 -35 -15
5
25 45 65
Temperature (°C)
FIGURE 3:
(100 parts).
85
105 125
MCP9700 Accuracy
6.0
Accuracy (°C)
4.0
Spec. Limit
2.0
0.0
-2.0
-4.0
+ V
Average
- V
-15
5
FIGURE 4:
(100 parts).
DS00001001C-page 2
25
45
65
85
Temperature (°C)
105
125
MCP9701 Accuracy
 2010-2015 Microchip Technology Inc.
AN1001
SENSOR THEORY
Temperature sensors use a fully turned-on PNP transistor to sense the ambient temperature. The voltage
drop across the base-emitter junction has the characteristics of a diode. The junction drop is temperature
dependent, which is used to measure the ambient temperature. Equation 1 shows a simplified equation that
describes the diode forward voltage.
EQUATION 1:
DIODE FORWARD
VOLTAGE
VPTAT provides a linear voltage change with a slope of
(86 µV/°C)*ln(N)|N=10 = 200 µV/°C. The voltage is either
amplified for analog output sensors or is interfaced to an
Analog-to-Digital Converter (ADC) for digital sensors.
The accuracy of VPTAT over the specified temperature
range depends on the matching of both forward current
(IF) and saturation current (IS) of the two sensors [1].
Any mismatch in these variables creates inaccuracy in
the temperature measurement. The mismatch contributes to the temperature error or nonlinearity. The nonlinearity is described using a 2nd order polynomial
equation.
kT A I F
VF = --------- ln  -----, IF » I S
q
IS
Where:
k = Boltzmann’s Constant (1.3807 x 10-23 J/K)
q = Electron Charge (1.602 x 10-19 coulombs)
TA = Ambient Temperature
IF = Forward Current
IS = Saturation Current
IS is a constant variable defined by the transistor size.
A constant forward current (IF) is used to bias the
diode, which makes the temperature TA the only
changing variable in the equation. However, IS varies
significantly over process and temperature. The variation makes it impossible to reliably measure the
ambient temperature using a single transistor.
To minimize IS dependency, a two-diode solution is
used. If both diodes are biased with constant forward
currents of IF1 and IF2, and the currents have a ratio of
N (IF2/IF1 = N), the difference between the forward voltages (VF) has no dependency on the saturation currents of the two diodes, as shown in Equation 2. VF is
also called Voltage Proportional to Absolute
Temperature (VPTAT).
EQUATION 2:
VPTAT
 V F = V F1 – VF2
I F1
------kT A
IS
 V F = ---------  ln ---------------q
N  I F1
----------------IS
kT A
 VF = ---------  ln  N 
q
V F = VPTAT
Where:
VF = Forward Voltages
IF = Forward Currents
VPTAT = Voltage Proportional to Absolute
Temperature
 2010-2015 Microchip Technology Inc.
DS00001001C-page 3
AN1001
FITTING POLYNOMIALS TO THE
ERRORS
The accuracy characterization data is used to derive a
2nd order equation that describes the sensor error. The
equation is used to improve the typical sensor accuracy
by compensating for the sensor error.
1ST ORDER ERROR
Error T_1 = EC 1  T A – T cold  + Error T_cold
Where:
ErrorT_1 = 1st order temperature error
Linear Fit Derivation
To capture the parabolic-shaped accuracy error
between the temperature extremes (Figure 5), a 2nd
order term and the corresponding coefficient must be
computed.
3.0
2.0
Accuracy (°C)
EQUATION 4:
1.0
0.0
-1.0
MCP9800
MCP9700
-2.0
-3.0
-55 -35 -15
5
25 45 65
Temperature (°C)
FIGURE 5:
85
105 125
Typical Accuracy Plot.
Figure 5 shows a typical accuracy curve which indicates that the accuracy error magnitudes are not the
same at hot and cold temperatures. There is a 1st order
error slope, or temperature error coefficient (EC1), from
-55° to +125°C. The error coefficient is calculated using
an end-point-fit method:
EQUATION 3:
ERROR SLOPE
T A = T hot – T cold
Error
EC 1 = ------------------T A
Where:
Thot = Highest Operating Temperature
Tcold = Lowest Operating Temperature
ErrorT_hot = Error at Highest Operating
Temperature
ErrorT_cold = Error at Lowest Operating
Temperature
EC1 = 1st Order Error Coefficient
Once the error slope is calculated, the corresponding
offset is determined at cold by adjusting the error at
cold temperature as shown in Equation 4.
DS00001001C-page 4
Equation 5 shows that the 2nd order temperature error
coefficient, EC2, is solved by specifying a temperature
TA where the calculated 2nd order error, ErrorT_2, is
equal to the known error at TA. For example, if TA is
+25°C and ErrorT_2 is equal to the temperature error at
+25°C, then Equation 5 is rearranged to solve for EC2
as shown in Equation 6.
EQUATION 5:
2ND ORDER ERROR
Error T_2 = EC2  T hot – T A    T A – T cold  + Error T_1
Where:
ErrorT_2 = 2nd order temperature error
EC2 = 2nd order error coefficient
Equation 5 shows that when TA is equal to Thot or Tcold ,
the 2nd order term is forced to zero with no error added
to the 1st order error term. This is because the error at
the Thot and Tcold temperature extremes is included in
the 1st order error (ErrorT_1).
EQUATION 6:
2ND ORDER ERROR
COEFFICIENT
 Error T_2 – Error T_1 
EC 2 = ------------------------------------------------------------- T hot – T A    T A – T cold 
Equation 7 shows the complete 2nd order polynomial
equation that is used to compensate the sensor error.
EQUATION 7:
2ND ORDER POLYNOMIAL
EQUATION
Error T_2 = EC 2  T hot – T A    T A – T cold 
+EC 1  T A – T cold  + Error T_cold
 2010-2015 Microchip Technology Inc.
AN1001
ACCURACY COMPENSATION
Typical Results
nd
Equations 8, 9 and 10 show the 2 order error equation of the tested parts for the MCP9800, MCP9700 and
MCP9701, respectively. Since these devices have
functional differences, the operating temperature range
and temperature error coefficients differ.
To achieve higher accuracy in a temperature monitoring application, using Equations 8, 9 and 10 can compensate for the sensor error as shown in Equation 11.
EQUATION 11:
MCP9800 2ND ORDER
EQUATION
EQUATION 8:
Error T_2 = EC 2  125  C – T A    T A – – 55  C 
T compensated = T sensor – Error T_2
Tsensor = Sensor Output
Tcompensated = Compensated Sensor Output
Where:
EC1 = 7 x 10-3 °C/°C
Error-55 = -1.5°C
For example, if the MCP9800 temperature output
Tsensor = +65°C, the compensated temperature
Tcompensated is 64.6°C as shown below.
T compensated = 65  C – Error T_2
MCP9700 2ND ORDER
EQUATION
EQUATION 9:
+EC 1  T A – – 55  C  + Error -55
T compensated = 64.6  C
+EC 1  T A – – 40  C  + Error -40
EC2 = -244 x 10-6°C/°C2
EC1 = 2 x 10-12°C/°C  0 °C/°C
Error-40 = 2°C
Figures 6, 7 and 8 show the average sensor accuracy
with the 2nd order error compensation for all tested
devices. The figures indicate that, on average, the sensor accuracy over the operating temperature can be
improved to ±0.2°C for the MCP9800, and ±0.05°C for
the MCP9700 and MCP9701.
MCP9701 2ND ORDER
EQUATION
Error T_2 = EC 2  125  C – T A    T A – – 15  C 
+EC 1  T A – – 15  C  + Error -15
Where:
EC2 = -200 x 10-6 °C/°C2
EC1 = 1 x 10-3 °C/°C
Error-15 = 1.5°C
The preceding equations describe the typical device
temperature error characteristics.
 2010-2015 Microchip Technology Inc.
0.3
0.2
Accuracy (°C)
EQUATION 10:
T A = 65  C
= 65  C + EC2  125  C – 65  C   65  C – – 55  C 
Error T_2 = EC 2  125  C – T A    T A – – 40  C 
Where:
T A = T sensor
Where:
+EC 1  T A – – 55  C  + Error -55
EC2 = 150 x 10-6 °C/°C2
TEMPERATURE
COMPENSATION
0.1
0.0
-0.1
-0.2
Average
-0.3
-55 -35 -15
5
25 45 65
Temperature (°C)
85
105 125
FIGURE 6:
MCP9800 Average
Accuracy After Compensation (160 parts).
DS00001001C-page 5
0.3
6.0
0.2
4.0
Accuracy (°C)
Accuracy (°C)
AN1001
0.1
0.0
-0.1
2.0
0.0
+ V
Average
- V
-2.0
-0.2
Average
-0.3
-4.0
-55 -35 -15
5
25 45 65
Temperature (°C)
85
105 125
-55 -35 -15
6.0
0.2
4.0
Accuracy (°C)
0.3
0.1
0.0
-0.1
85
105 125
Spec. Limit
2.0
0.0
+ V
Average
- V
-2.0
-0.2
5
25 45 65
Temperature (°C)
FIGURE 10:
MCP9700 Accuracy After
Compensation (100 parts).
FIGURE 7:
MCP9700 Average
Accuracy After Compensation (100 parts).
Accuracy (°C)
Spec. Limit
Average
-4.0
-0.3
-15
5
25
45
65
85
Temperature (°C)
105
125
-15
5
25
45
65
85
Temperature (°C)
105
125
FIGURE 8:
MCP9701 Average
Accuracy After Compensation (100 parts).
FIGURE 11:
MCP9701 Accuracy After
Compensation (100 parts).
Figures 9, 10 and 11 show an average and ±1 standard
deviation of sensor accuracy for the tested parts with
the 2nd order error compensation.
When comparing the compensated accuracy from
Figures 9, 10 and 11 with the uncompensated accuracy
from Figures 2, 3 and 4, the accuracy error distribution
is shifted towards 0°C accuracy, providing a linear
3.0
Accuracy (°C)
2.0
1.0
Spec. Limit
0.0
-1.0
+ V
Average
- V
-2.0
-3.0
-55 -35 -15
5
25 45 65
Temperature (°C)
85
105 125
FIGURE 9:
MCP9800 Accuracy After
Compensation (160 parts).
DS00001001C-page 6
 2010-2015 Microchip Technology Inc.
AN1001
The 2nd Order Temperature Coefficient
nd
Among the compensations, the 2 order temperature
coefficient variable EC2 was evaluated at +25°C. For
most applications, the compensation characteristics at
this temperature are adequate. However, changing the
temperature at which EC2 is evaluated provides relatively higher accuracy at narrower temperature ranges.
For example, Figure 12 shows the MCP9700 EC2
evaluated at 0°, +25° and +90°C.
CALIBRATION
Calibration of individual IC sensors at a single temperature provides superior accuracy for high-performance,
embedded-system applications. Figure 13 shows that
if the MCP9700 is calibrated at +25°C and the 2nd order
error compensation is implemented, the typical sensor
accuracy becomes ±0.5°C over the operating
temperature range.
0.05
0.04
0.03
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
4.0
MCP9700
Accuracy (°C)
Accuracy (°C)
6.0
2.0
Spec. Limits
0.0
+ V
Average
- V
-2.0
EC2 @ 90°C
EC2 @ 25°C
EC2 @ 0°C
-55 -35 -15
-4.0
-50
0
25
50
75
100
125
Temperature (°C)
5 25 45 65 85 105 125
Temperature (°C)
FIGURE 12:
MCP9700 Average
Accuracy with Varying EC2 .
-25
FIGURE 13:
Accuracy.
MCP9700 Calibrated Sensor
When comparing EC2 at 0° and +25°C, accuracy is
higher at cold rather than hot temperatures. However,
for EC2 evaluated at temperatures higher than +25°C,
accuracy is higher at hot rather than cold temperatures.
However, the magnitude of accuracy error difference
among the various EC2 values is not significant.
Therefore, EC2 evaluated at +25°C provides practical
results.
 2010-2015 Microchip Technology Inc.
DS00001001C-page 7
AN1001
COMPENSATION USING
PIC® MICROCONTROLLERS
A PIC MCU can implement the 2nd order accuracy error
compensation for embedded temperature-monitoring
systems. The equation is relatively easy to implement
in a 16-bit core MCU since built-in math functions are
readily available. However, 12 and 14-bit cores require
firmware implementation of some math functions, such
as 16-bit add, subtract, multiply and divide. This application note includes firmware that can compute and
implement the compensation variables.
The file AN1001 Source Code.zip includes the
MCP9700 and MCP9800 compensation firmware versions. These firmware versions are intended to be
included in an existing embedded system firmware that
uses a PIC MCU. All registers required to execute this
routine are listed within the firmware. Once the temperature data from the device is retrieved using a serial
interface or ADC input, the binary data must be loaded
to the Bargb0 and Bargb1 registers. Detailed
instructions are included in the firmware files.
Figure 14 shows the firmware flowchart.
Determine 2nd Order Error
Determine 1st Order Error
Add 1st and 2nd Order Error to ErrorT_cold
Subtract Total Error from TA
Compensated
TA
FIGURE 14:
Firmware Flowchart.
TEST RESULTS
The MCP9800 and MCP9700 demo boards
(MCP9800DM-PCTL and MCP9700DM-PCTL, respectively) were used to evaluate the compensation firmware. A constant temperature air stream was applied
directly to the temperature sensors. A thermocouple
was used to accurately measure the air stream
temperature and compare the sensor outputs.
TABLE 1:
MEASUREMENT ACCURACY
TEST RESULTS
Temperature Error
Temperature
MCP9700
MCP9800
W/O
W
W/O
W
-40°C
0.9
0.2
-1.0
0.1
-25°C
0.6
0.2
-0.4
0.2
0°C
0.4
0.4
0.2
0.1
+25°C
0.3
0.6
0.1
0.1
+40°C
0.4
0.7
0.1
0.2
+90°C
1.2
0.8
0.3
0.3
+110°C
1.8
0.7
0.6
0.3
+125°C
2.3
0.6
0.9
0.1
Note 1:
The “W/O” and “W” columns indicate
accuracy without and with compensation.
The test result in Table 1 shows the accuracy improvement achieved using compensation firmware routines.
At hot and cold temperatures, accuracy is improved by
approximately 1° to 2°C, respectively.
CONCLUSION
The nonlinear accuracy characteristics of a temperature sensor is compensated for higher-accuracy
embedded systems. The nonlinear accuracy curve has
a parabolic shape that is described using a 2nd order
polynomial equation. Once the equation is determined,
it is used to compensate the sensor output. On average, the accuracy improvement using compensation is
±2°C (for all tested devices) over the operating temperature range. The compensation also improves the
wide temperature accuracy specification limits at hot
and cold temperature extremes. A PIC MCU can
compute the equation and compensate the sensor
output using the attached firmware.
WORK CITED
[1]. Bakker, A., and Huijsing, J. (2000). High-Accuracy CMOS Smart Temperature Sensors.
Boston: Kluwer Academic Publishing.
DS00001001C-page 8
 2010-2015 Microchip Technology Inc.
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ISBN: 978-1-63277-739-3
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DS00001001C-page 9
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Tel: 852-2943-5100
Fax: 852-2401-3431
Korea - Seoul
Tel: 82-2-554-7200
Fax: 82-2-558-5932 or
82-2-558-5934
China - Nanjing
Tel: 86-25-8473-2460
Fax: 86-25-8473-2470
Malaysia - Kuala Lumpur
Tel: 60-3-6201-9857
Fax: 60-3-6201-9859
China - Qingdao
Tel: 86-532-8502-7355
Fax: 86-532-8502-7205
Malaysia - Penang
Tel: 60-4-227-8870
Fax: 60-4-227-4068
China - Shanghai
Tel: 86-21-5407-5533
Fax: 86-21-5407-5066
Philippines - Manila
Tel: 63-2-634-9065
Fax: 63-2-634-9069
China - Shenyang
Tel: 86-24-2334-2829
Fax: 86-24-2334-2393
Singapore
Tel: 65-6334-8870
Fax: 65-6334-8850
China - Shenzhen
Tel: 86-755-8864-2200
Fax: 86-755-8203-1760
Taiwan - Hsin Chu
Tel: 886-3-5778-366
Fax: 886-3-5770-955
China - Wuhan
Tel: 86-27-5980-5300
Fax: 86-27-5980-5118
Taiwan - Kaohsiung
Tel: 886-7-213-7828
China - Xian
Tel: 86-29-8833-7252
Fax: 86-29-8833-7256
Germany - Karlsruhe
Tel: 49-721-625370
Germany - Munich
Tel: 49-89-627-144-0
Fax: 49-89-627-144-44
Italy - Milan
Tel: 39-0331-742611
Fax: 39-0331-466781
Italy - Venice
Tel: 39-049-7625286
Netherlands - Drunen
Tel: 31-416-690399
Fax: 31-416-690340
Poland - Warsaw
Tel: 48-22-3325737
Tel: 34-91-708-08-90
Fax: 34-91-708-08-91
Sweden - Stockholm
Tel: 46-8-5090-4654
UK - Wokingham
Tel: 44-118-921-5800
Fax: 44-118-921-5820
Taiwan - Taipei
Tel: 886-2-2508-8600
Fax: 886-2-2508-0102
Thailand - Bangkok
Tel: 66-2-694-1351
Fax: 66-2-694-1350
07/14/15
DS00001001C-page 10
 2015 Microchip Technology Inc.
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