### cd00174666

```AN2648
Application note
Increasing the resolution of analog temperature sensors
Introduction
For a recent trade show, a demonstration board was required to display ST's new low-cost
temperature sensor, the STLM20. The idea was to connect it to a basic microcontroller with
on-board analog-to-digital converter (ADC) and display the temperature on seven-segment
LEDs. Any user would be able to affect the temperature by applying his finger directly to the
sensor and then watch the display change as a result.
The goal was to keep things simple and use the ADC converter on the micro thus avoiding
the expense and engineering effort of using an external ADC.
Using the STLM20 temperature sensor with low-cost microcontrollers
The design flow used for the demonstration board can be applied in many applications. This
document describes how any user can implement a temperature sensor design using the
LM20 and a low-cost microcontroller with integral A-D converter.
Figure 1.
STLM20 demonstration board
200mm / ~8
144mm / 5.7
34mm / 1.34
280mm / ~11
December 2007
Rev 1
1/15
www.st.com
STLM20 temperature-to-output transfer function
1
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STLM20 temperature-to-output transfer function
Upon examining the temperature sensor's output characteristics, several issues become
apparent. The STLM20 is a voltage output device with a 2nd order transfer function:
V0 = (−3.88 × 10 −6 × T 2 ) + (−1.15 ×10 −2 × T ) + 1.8639 V
This is rather unwieldy math for a simple 8-bit micro, but a linear approximation is available
which matches the curve very closely. As shown in Figure 2, the dashed blue line is a very
good fit to the pink parabola of the 2nd order equation above, and has the following equation:
Equation 1
V0 = (−0.01169 V/°C) × T + 1.8663 V
This is much more manageable for coding in assembly language without a math package. In
this figure, the reader will note that the curve deviates only slightly from the line at the
extreme ends. In the middle, they are almost indistinguishable.
Figure 2.
STLM20 transfer function
2.5
2.3
Output voltage, VO , (V)
2.1
1.9
2nd Order
Linear
1.7
1.5
1.3
1.1
0.9
0.7
0.5
-60
-40
-20
0
20
40
60
80
100
Temperature, T (˚C)
ai13986
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STLM20 temperature-to-output transfer function
The second issue arises with the output voltage range. The micro's ADC performs ratiometric conversion on the input using VCC and ground as the upper and lower references,
respectively. That means that when the input voltage equals VCC, the 8-bit ADC's output will
be 255 (FFHEX), and when the input is ground, the output will be 0 (00HEX). Any voltage in
the range VCC to ground is converted proportionally according to the following relationship:
ADC output (dec) = (VIN /VCC ) × 255
For this application, the nominal VCC is 5 V, so this equation becomes:
ADC output (dec) = (VIN /5) × 255 = VIN × 51
Referring to Figure 3, while the ADC can accept inputs over the full range 0 to 5 V, the
temperature sensor output will vary over a much smaller range, between 0.87 and 2.33 volts
across the temperature range –40° to +85°C.
STLM20 output voltage range
5
Output voltage, VO (V)
Figure 3.
4.5
4
3.5
3
2.5
STLM20 full range,
linear, ind temp
2
Available
voltage
range
1.5
STLM20
output voltage
range
1
0.5
0
-60
-40
-20
0
20
40
60
80
100
Temperature, T (˚ C)
ai13987
This amounts to using only 1.46 V of the available 5 V range. Furthermore, in the intended
demonstration board application, the necessary temperature range was approximately room
temperature –15°/+25°, or 10° to 50°C.
For this target temperature range, the voltage range is 1.28 to 1.75 V, an interval of less than
0.5 V, less than one tenth the available 5 V range. This is depicted in Figure 4.
3/15
STLM20 temperature-to-output transfer function
Figure 4.
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Application temperature and voltage ranges
5
4.5
4
Output voltage, VO (V)
3.5
3
2.5
Temperature range of interest
2
1.5
Used voltage range
of application
1
STLM20 full temperature range
0.5
Available voltage range
0
- 60
- 40
-20
0
20
40
60
80
100
Temperature, T (˚C)
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In Equation 1, the slope is –11.69 mV per °C. For the 8-bit ADC, the step size is 5 V/255
steps or 0.019608 V/step. Comparing this to the slope, we get:
0.019608 V/step
= − 1.677318 °C/step
− 0 .0 1169 V/ °C
This means that, with the temperature sensor connected directly to the ADC, the resolution
is only 1.68°C/step. The smallest temperature increment the ADC can resolve is
1.68°C. This is very coarse resolution.
4/15
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STLM20 temperature-to-output transfer function
Given that the expected voltage range is less than 0.5 V, it should be possible to amplify the
STLM20 output signal so that is uses more of the available voltage range. A 10x
amplification should be possible without exceeding the available 5 V range. Furthermore, a
positive slope is more intuitive to the user, so inverting it would be helpful, too. Hence, a
gain of –10 should be used. Multiplying equation 1 by –10, we get the curve as shown at the
bottom in Figure 5.
Figure 5.
Application transfer function after gain, inversion and offset
5.5
3.5
1.5
-50
-30
-0.5
-10
10
30
50
70
90
-2.5
Voltage (V)
-4.5
+17.5
x –10
-6.5
-8.5
-10.5
-12.5
-14.5
< 5V
-16.5
-18.5
Temperature, T (˚C)
ai13989
The result has positive slope and spans from –17.5 to –12.8, an interval of about 4.7 V,
much closer to the available 5 V range.
5/15
STLM20 temperature-to-output transfer function
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With an offset of approximately +17.5 V the curve is moved back into the range 0-5 V to give
the transfer function shown as the light blue line in Figure 5.
So how is this done? What circuit will perform this transformation without adding intolerable
cost to the system?
It turns out that a difference amplifier can be used to add gain and offset, and can do the
inversion as well. And it requires only one op-amp to implement.
An additional op-amp is used as a voltage follower out of the temperature sensor to prevent
output loading from affecting the STLM20's output level. The net amplifier requirement then
is a dual op-amp. A voltage reference is also required to provide the offset level going into
the difference amp. The overall circuit is shown in Figure 6.
The STLM20 output voltage, V0, appears identically at the first op-amp output. That then
drives the inverting input of the difference amp with a gain of –50k/5k = –10.
This translates the curve from a voltage range of 1.28 V to 1.75 V to a range of –12.8 V to
–17.5 V so an offset of about +17.5 V must be applied. The 1.7676 V reference sees a gain
of 10 (R4/R3 = ~50k/5k = 10) on the op-amp's non-inverting input. Thus, 17.676 volts is
With this gain and offset, the original curve, which had an interval of less than 0.5 V and a
negative slope, is transformed to one which spans close to 5 V and has a positive slope as
illustrated by the light blue line in Figure 5.
Figure 6.
STLM20 with gain and offset circuit
VCC
VCC
0.1uF
0.1uF
T
VO
= VO
0.1uF
STLM20
LMV358
VCC
R3
50k
V1
R1
5k
VOUT
V2
5k R2
LMV358
1.7676V
0.1uF
R4
50k
0.1uF
TS431
ai13990
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STLM20 temperature-to-output transfer function
Equation 2 is the basic difference amplifier equation.
Equation 2
VOUT =
V2 ( R3 + R1) R4 V1 ⋅ R3
−
(R4 + R2) R1
R1
If R1 = R2 and R3 = R4, then it reduces to:
VOUT = (V2 − V1 )
R3
R1
However, obtaining perfectly identical resistors is nearly impossible. Thus, the more
complex expression of Equation 2 is used. Starting with Equation 1 on page 2, for Figure 6,
we have:
VO = −0.01169 ⋅ T + 1.8663 V
In Figure 6, V1 is driven by V0, so Equation 2 becomes:
VOUT =
V2 ( R3 + R1) R4 VO ⋅ R3
−
(R4 + R2) R1
R1
Substituting the expression for V0 into this, we get:
VOUT = V2 ⋅
( R3 + R1) R4 ( − 0.01169 ⋅ T + 1.8663) ⋅ R3
−
(R4 + R2) R1
R1
VOUT = V2 ⋅
( R3 + R1) R4
R3
R3
+ 0.01169 ⋅
⋅ T − 1.8663 ⋅
(R4 + R2) R1
R1
R1
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STLM20 temperature-to-output transfer function
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Equation 3
VOUT = V2 ⋅
( R3 + R1) R4
R3
R3
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
(R4 + R2) R1
R1
R1
The ADC inside the microprocessor is shown in Figure 7.
Figure 7.
VOUT
VCC = 5V
A/D
8-bit
CPU
I/O
MEM
MICROCONTROLLER
ai13991
For the ADC output, we have the following, where INT means the integer portion of the
expression:
⎡ 255
⎤
VOUT ⎥
⎣ VCC
⎦
Substituting Equation 3, we get:
⎡ 255 ⎛
( R3 + R1) R4
R3
R3
⎜⎜ V2 ⋅
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
(R4 + R2) R1
R1
R1
⎣ VCC ⎝
Rearranging, and ignoring the integer notation for now, we get:
8/15
VCC
( R3 + R1) R4
R3
R3
= V2 ⋅
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
255
(R4 + R2) R1
R1
R1
⎞⎤
⎟⎟⎥
⎠⎦
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STLM20 temperature-to-output transfer function
To get the measured temperature, TMEASA, we must solve for T above:
VCC
( R3 + R1) R4
R3
R3
− V2 ⋅
+ 1.8663 ⋅
= 0.01169 ⋅
⋅T
255
(R4 + R2) R1
R1
R1
VCC R1
( R3 + R1) R4 R1
R3 R1
R3 R1
⋅
− V2 ⋅
⋅
+ 1.8663 ⋅
⋅
= 0.01169 ⋅
⋅
⋅T
255 R3
(R4 + R2)R1 R3
R1 R3
R1 R3
VCC R1
( R3 + R1) R4
⋅
− V2 ⋅
+ 1.8663 = 0.01169 ⋅ T
255 R3
(R4 + R2) R3
Equation 4
TMEASA = T = VADC ⋅
VCC R1
( R3 + R1) R4
1
1.8663
1
⋅
⋅
− V2 ⋅
⋅
+
255 R3 0.01169
(R4 + R2) R3 0.01169 0.01169
This has the form:
Equation 4b
TMEASA = VADC ⋅ K S + K O
But, rather than using this equation, it is better to filter the temperature, to average multiple
samples together for display. This has two benefits. It smoothes the changes resulting in a
displayed value which updates more gradually. And it increases the effective resolution of
the sensor. By adding two more bits, this essentially makes the A/D appear to have 10-bit
resolution.
So, we use a modified version of the previous math, but sum four consecutive samples
together:
⎡ 255
⎤
VOUT ⎥
⎣ VCC
⎦
⎡ 255 ⎛
( R3 + R1) R4
R3
R3
⎜⎜ V2 ⋅
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
(R4 + R2) R1
R1
R1
⎣ VCC ⎝
⎞⎤
⎟⎟⎥
⎠⎦
9/15
STLM20 temperature-to-output transfer function
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VSUM4 = 4 samples of VADC, range 0 to 1020.
⎡ 255 ⎛
( R3 + R1) R4
R3
R3
⎜⎜ V2 ⋅
VSUM4 = 4 ⋅ INT ⎢
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
(R4 + R2) R1
R1
R1
⎣ VCC ⎝
⎞⎤
⎟⎟⎥
⎠⎦
Rearranging and ignoring the integer operator, we get:
VSUM4 ⋅
VCC
( R3 + R1) R4
R3
R3
= V2 ⋅
− 1.8663 ⋅
+ 0.01169 ⋅
⋅T
1020
(R4 + R2) R1
R1
R1
VSUM4 ⋅
VCC
( R3 + R1) R4
R3
R3
− V2 ⋅
+ 1.8663 ⋅
= 0.01169 ⋅
⋅T
1020
(R4 + R2) R1
R1
R1
VSUM4 ⋅
VCC R1
R3 R1
( R3 + R1) R4 R1
R3 R1
⋅
− V2 ⋅
⋅
+ 1.8663 ⋅
⋅
= 0.01169 ⋅
⋅
⋅T
1020 R3
(R4 + R2) R1 R3
R1 R3
R1 R3
VSUM4 ⋅
VCC R1
( R3 + R1) R4
⋅
− V2 ⋅
+ 1.8663 = 0.01169 ⋅ T
(R4 + R2) R3
1020 R3
Equation 5
TMEAS4 = T = VSUM4 ⋅
VCC R1
( R3 + R1) R4
1
1.8663
1
⋅
⋅
− V2 ⋅
⋅
+
1020 R3 0.01169
(R4 + R2) R3 0.01169 0.01169
This is a linear equation and has the form:
Equation 5b
TMEAS4 = VSUM4 ⋅ K S /4 + K O
10/15
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STLM20 temperature-to-output transfer function
Notice that the upper limit for the sum of four samples is 1020 (4 x 255) and not 1023 as
might be inferred for a 10-bit number.
The actual values of KS/4 and KO depend on the four resistors in the difference amp, the
voltage reference output and VCC. In the case of the temperature sensor demonstration
board, we have these values:
R1 = 5107 Ω
R2 = 5105 Ω
R3 = 50850 Ω
R4 = 50850 Ω
VCC = 4.9058 V
V2 = VREF = 1.7676 V
For KS, we insert these values into Equation 4 and get this for a single sample:
TMEASA = T = VADC ⋅
VCC R1
1
( R3 + R1) R4
1
1.8663
⋅
⋅
− V2 ⋅
⋅
+
255 R3 0.01169
(R4 + R2) R3 0.01169 0.01169
TMEASA = VADC ⋅ 0.16528 + 8.43771
Thus, for Equation 4b, KS = 0.16528 and KO = 8.43771. Therefore, KS/4 = 0.04132.
Substituting this into Equation 5b, we get:
TMEAS4 = 0.04132 ⋅ VSUM4 + 8.43771
This gives the four-sample average of the temperature. However, more processing is still
required to get it into a form suitable for implementing in a simple processor. Since the
processor is an 8-bit, integer machine, multiplication by a real number - in this case a
fraction - is difficult and requires lots of math software.
Instead of doing such complicated math, it's easier to first multiply by 1000, then later divide
by 1000.
So, Equation 5b becomes:
1000 ⋅ TMEAS4 = 41⋅ VSUM4 + 8438
11/15
STLM20 temperature-to-output transfer function
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Since the numbers are strictly integers, some truncation error has occurred. Running error
analysis on the calculated results shows an average error of –0.21°C, so as a final
1000 ⋅ TMEAS4 = 41 ⋅ VSUM4 + 8438 + 200 = 41⋅ VSUM4 + 8638
The above equation is then implemented in the software. The final step is dividing the result
by 1000 and writing it to the demonstration board's display.
The overall solution looks like this (Figure 8):
Figure 8.
Overall solution
VCC
STLM20
VO = –11.69mV / ˚C × T + 1.8663V
slope is 0.01169 V/˚C
VOUT =
TEMPERATURE
TO
VOLTAGE
V2 ( R3 + R1) R4 VO •R3
(R4 + R2) R1
R1
= V2 •
( R3 + R1) R4
R3
R3
1.8663 •
+ 0.01169 • • T
(R4 + R2) R1
R1
R1
VOUT = 0.1164 • T – 0.9821
V1
R1
V2
R2
R3
VOUT
1.7676V
255
VOUT
VCC
VCC R1
( R3 + R1) R4
V2 •
+ 1.8663 = 0.01169• T
•
255 R3
(R4 + R2) R3
TMEASA = T = VADC •
VCC R1
1
( R3 + R1) R4
1
1.8663
•
•
•
+
V2 •
255 R3 0.01169
(R4 + R2)R3 0.01169 0.01169
TMEASA = VADC • 0.165284 + 8.437709
VREF
R4
VCC =
T
= VSUM4 • 0.041321 + 8.437709
4.9058V MEAS4
41
8438
+
1000
A/D
8-bit
CPU
1000
I/O
MEM
MICROCONTROLLER
ai13992
Thus, for the minor expense of a voltage reference and dual op-amp, the resolution has
been improved from 1.677°C per ADC step to better than 0.1°C steps on the display.
12/15
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2
Summary
Summary
The STLM20 is very accurate over a wide temperature range and has a nearly linear output
which spans the voltage range 2.33 down to 0.87 volts over the corresponding temperature
range –40° to +85°C. While it can be directly connected to an 8-bit A-D converter such as
those often found integrated into low-cost microcontrollers, the resolution may be too coarse
for some applications, amounting to –1.677°C per A-D step. For applications requiring
higher resolution, users can use an op-amp and a voltage reference to add gain and offset
to the STLM20 output. This has the effect of increasing the resolution over the temperature
range of interest while avoiding the need for a more expensive, higher resolution A-D
converter. Using averages of four samples also increases the effective resolution. The final
result is that users can achieve resolution as fine as 0.1°C with the STLM20.
13/15
Revision history
3
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Revision history
Table 1.
14/15
Document revision history
Date
Revision
20-Dec-2007
1
Changes
Initial release.
AN2648
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