Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
1. Introduction
This document is intended to explain how the Thermal Compensation of the MLX90251 works and
how it can be used in order to evaluate the TC of a complete system.
In the first section of this document, a theoretical approach to the thermal behaviour of the
MLX90251 will be given. This section will give a general overview of the Linear Hall Sensor Transfer
Function vs. Temperature and explain the item “TC – Thermal Compensation”.
The second section will explain how the thermal compensation works inside the MLX90251.
The third section will show how to evaluate the TC of a system by using the MLX90251.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Index
1. Introduction........................................................................................................................... 1
2. Theory .................................................................................................................................... 3
2.1. Linear Hall Effect Sensor Output Transfer Function vs Temperature.............................. 3
2.1.1. Second order term neglected in Equation 5.............................................................. 4
2.1.1.1. Ideal case ............................................................................................................ 5
2.1.1.2. Non ideal case ..................................................................................................... 5
2.1.2. Second order term not neglected in Equation 5........................................................ 5
2.1.3. Summary ................................................................................................................... 7
General Output Transfer Characteristic ........................................................................ 7
IC Temperature Coefficient........................................................................................... 7
Magnet Temperature Coefficient .................................................................................. 7
2.2. Approximations ............................................................................................................... 7
2.2.1 Summary .................................................................................................................... 9
1st approximation ............................................................................................................. 9
2nd approximation ............................................................................................................ 9
3. MLX90251 – TC Table .......................................................................................................... 10
4. TC Determination................................................................................................................. 13
4.1.
Characterize TC of the MLX90251 ........................................................................ 13
4.2.
Determine TC of the system ................................................................................. 14
Appendix I................................................................................................................................ 16
Appendix II .............................................................................................................................. 18
Appendix III ............................................................................................................................ 19
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
2. Theory
This section is intended to give a theoretical description of the Temperature Coefficient, TC.
2.1. Linear Hall Effect Sensor Output Transfer Function vs Temperature
In a general way the output transfer characteristic of a Linear Hall Effect Sensor can be expressed by
equation 1.
Vout ( B, T ) = Voq(T ) + σ (T ) × B(T ),
Equation 1
where:
Vout(B,T) = Output Voltage of the IC at temperature T and when a magnetic field of
strength B is applied,
Voq(T) = Vout(0,T) = Quiescent Output Voltage (Zero Gauss Output Voltage) at
temperature T,
σ(T) = Magnetic Sensitivity of the IC at temperature T,1
B(T) = Applied Magnetic Flux Density at temperature T,
T = Ambient Temperature.
In equation 1, each parameter can be split into two parts2:
one temperature independent part (value of the dedicated parameter at 25°C)
and one temperature dependant part (proportional to [T-25°C])
By developing equation 1 in this way, equation 2 can be obtained.
Vout ( B, T ) = Voq ( 25°C ) + ∆Voq (T ) + σ ( 25°C ) × [1 + TCσ × (T − 25°C )] × B ( 25°C ) × [1 + TC B × (T − 25°C )]
Equation 2
where:
1
2
3
σ (T ) =
∆Voq(T) = Voq(T) – Voq(25°C) = Thermal Voq Drift = Thermal Offset Drift,
TCσ = Sensitivity Temperature Coefficient of the IC3,
TCB = Magnetic Flux Density Temperature Coefficient (this TC is in general only due to
the magnet, but sometimes the module can also have some influence).
Vout ( P 2, T ) − Vout ( P1, T )
, where P1 and P2 are two different magnetic field positions,
B( P 2, T ) − B( P1, T )
25°C is taken as the reference temperature.
For all temperature coefficients the unit “ppm/°C” will be used.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Vout ( B , T ) = Voq(25°C ) + σ (25°C ) × B (25°C ) + ∆Voq (T )
+ σ ( 25°C ) × B ( 25°C ) × [TCσ × (T − 25°C ) + TC B × (T − 25°C ) + TCσ × TC B × (T − 25°C ) 2 ]
Equation 3
Since Voq(25°C) + σ(25°C) x B(25°C) = Vout(B,25°C), equation 3 becomes
Vout ( B, T ) = Vout ( B,25°C ) + ∆Voq (T )
+ σ ( 25°C ) × B ( 25°C ) × [TCσ × (T − 25°C ) + TC B × (T − 25°C ) + TCσ × TC B × (T − 25°C ) 2 ]
Equation 4
which can also be written like:
Vout ( B, T ) = Vout ( B,25°C ) + ∆Voq(T ) + σ (25°C ) × B(25°C ) × ∆TC × (T − 25°C )
+ σ ( 25°C ) × B ( 25°C ) × [TCσ × TC B × (T − 25°C ) 2 ] ,
Equation 5
where ∆TC = TCσ + TCB.
Equation 5 is the general expression of the Output Transfer Function of a Linear Hall Effect Sensor
versus Temperature.
2.1.1. Second order term neglected
If the second order term in equation 5 can be neglected
(i.e. valid if TCσ X TCB< 0.25ppm/°C2, i.e. if TCB < 500ppm/°C), a simplified equation can be
obtained.
Vout ( B , T ) ≈ Vout ( B,25°C ) + ∆Voq (T ) + σ ( 25°C ) × B ( 25°C ) × ∆TC × (T − 25°C )
Equation 6
Summary:
General Equation:
Vout ( B, T ) = Vout ( B,25°C ) + ∆Voq(T ) + σ (25°C ) × B(25°C ) × ∆TC × (T − 25°C )
+ σ ( 25°C ) × B ( 25°C ) × [TCσ × TC B × (T − 25°C ) 2 ]
Simplified Equation (i.e. TCB < 500ppm/°C):
Vout ( B, T ) ≈ Vout ( B ,25°C ) + ∆Voq (T ) + σ ( 25°C ) × B ( 25°C ) × ∆TC × (T − 25°C )
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
2.1.1.1.
Ideal case
!
No Thermal Offset Drift
=> ∆Voq(T) = 0
!
Perfect matching between the Temperature Coefficient of the Sensitivity (TCσ) and
the magnet (TCB)
=> TCσ = -TCB => ∆TC = 0
In this case equation 6 becomes:
Vout ( B, T ) ≈ Vout ( B,25°C )
Equation 7
which means that the output voltage of the IC would only be dependant of the applied
magnetic flux density.
2.1.1.2.
Non ideal case
From equation 6, we can define the Thermal Error as follows:
Thermal Error = ∆Voq(T ) + ∆TC × σ (25°C ) × B(25°C ) × (T − 25°C )
Equation 8
If we focus on the TC (i.e. suppose that ∆Voq = 0), it can be observed that for this simplified
situation it is sufficient to chose TCσ = -TCB in order to compensate the temperature
coefficient of the magnet (TCB).
For a numerical and graphical example, please refer to Appendix I.
2.1.2. Second order term not neglected
If the second order term in equation 5 cannot be neglected
(i.e. valid if TCσ X TCB> 0.25ppm/°C2, i.e. if TCB > 500ppm/°C), Vout(B,T) follows equation
5.
Vout ( B, T ) = Vout ( B,25°C ) + ∆Voq (T ) + σ ( 25°C ) × B( 25°C ) × ∆TC × (T − 25°C )
+ σ ( 25°C ) × B ( 25°C ) × [TCσ × TC B × (T − 25°C ) 2 ] .
Even for ∆Voq(T) = 0, there is a non negligible amount of drift coming from the sensitivity. In
this case, the whole term
σ (25°C ) × B(25°C ) × [(TCσ + TC B ) × (T − 25°C ) + TCσ × TC B × (T − 25°C ) 2 ]
has to be minimized over the whole temperature range.
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Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
In order to get the optimal temperature compensation, the Sensitivity Temperature Coefficient
TCσ has to be defined by:
TCσ =
− TC B
1 + TC B × (T − 25°C )
Equation 9
From equation 9, it can be seen that the Sensitivity Temperature Coefficient TCσ has to vary with
temperature (it can be assumed that TCB is constant over the temperature range).
This behavior is more or less implemented in the MLX90251. That is why the “TC vs TC Code”
specification can be written in such a way that the TC codes refer to the “to-be-compensated”
temperature coefficient instead of the “pure IC sensitivity” temperature coefficient.
However, we have (see equation 2):
TCσ =
σ (T ) − σ (25°C )
σ (25°C ) × (T − 25°C )
Equation 10
and
TC B =
B(T ) − B(25°C )
B(25°C ) × (T − 25°C )
Equation 11
Based on equation 9, we have also:
TC B =
− TCσ
1 + TCσ × (T − 25°C )
Equation 12
If TCσ is replaced in equation 12 by equation 10, TCB can be expressed as:4
1
1
)−(
)
σ (T )
σ (25°C )
TC B =
1
(
) × (T − 25°C )
σ (25°C )
(
Equation 13
4
To get the complete development of this formula, please refer to Appendix II.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
If σ1(T) = 1/σ(T), equation 13 can also be written as follows:
TC B =
σ 1 (T ) − σ 1 (25°C )
σ 1 (25°C ) × (T − 25°C )
,
Equation 14
which is similar to equation 10. The only difference is, that TCB is a function of 1/σ whereas TCσ
is a function of σ.
2.1.3. Summary
General Output Transfer Characteristic:
Vout ( B, T ) = Vout ( B,25°C ) + ∆Voq(T ) + σ (25°C ) × B(25°C ) × ∆TC × (T − 25°C )
+ σ (25°C ) × B(25°C ) × [TCσ × TC B × (T − 25°C ) 2 ] ,
IC Temperature Coefficient:
TCσ =
TCσ =
Magnet Temperature
Coefficient):
− TC B
1 + TC B × (T − 25°C )
σ (T ) − σ (25°C )
σ (25°C ) × (T − 25°C )
Coefficient
TC B =
(or
Magnet
+
Module
Temperature
σ 1 (T ) − σ 1 (25°C )
σ 1 (25°C ) × (T − 25°C )
with σ1(T) = 1/σ(T).
2.2. Approximations
Some simplified equations exist in order to determine the TC of a magnet/system.
Those simplified equations are based on equation 10:
TCσ =
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σ (T ) − σ (25°C )
σ (25°C ) × (T − 25°C )
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Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Also remember that the sensitivity σ is defined as follows:
σ (T ) =
Vout ( P 2, T ) − Vout ( P1, T )
.
B( P 2, T ) − B( P1, T )
Therefore, TCσ can also be expressed like in equation 15.
Vout ( P 2, T ) − Vout ( P1, T ) Vout ( P 2,25°C ) − Vout ( P1,25°C )
−
B( P 2, T ) − B( P1, T )
B( P 2,25°C ) − B( P1,25°C )
TCσ =
,
Vout ( P2,25°C ) − Vout ( P1,25°C )
× (T − 25°C )
B( P2,25°C ) − B( P1,25°C )
Equation 15
where P1 and P2 are two different positions with different magnetic flux densities.
If it is assumed that
B(P2,T) - B(P1,T) = B(P2,25°C) – B(P1,25°C),
i.e. the span of the magnetic flux density is constant over temperature, equation 15 becomes:
TCσ =
[Vout ( P2, T ) − Vout ( P 2,25°C )] − [Vout ( P1, T ) − Vout ( P1,25°C )]
.
[Vout ( P2,25°C ) − Vout ( P1,25°C )] × (T − 25°C )
Equation 16
where [Vout(P2,25°C) – Vout(P1,25°C)] is the output voltage span at 25°C.
If Vout(P1,T) = Voq(T) (= Offset voltage position), equation 16 becomes:
TCσ =
[Vout ( P 2, T ) − Vout ( P 2,25°C )] − [Voq (T ) − Voq (25°C )]
[Vout ( P2,25°C ) − Voq (25°C )] × (T − 25°C )
Equation 17
Furthermore, if it could be assumed that the Thermal Offset Drift (thermal Voq drift) is compensated
(Voq(T) – Voq(25°C) = 0), equation 17 can still be simplified. In this case we have:
TCσ =
Vout ( P 2, T ) − Vout ( P2,25°C )
[Vout ( P 2,25°C ) − Voq(25°C )] × (T − 25°C )
Equation 18
Equation 18 gives the simplest expression of the IC Temperature Coefficient, TCσ.
However, it should be emphasized that this expression is only an approximation.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
2.2.1. Summary
Following simplified equations can be used in order to determine the Temperature coefficient of a
system:
1st approximation: magnetic field span is constant vs temperature
TCσ =
[Vout ( P 2, T ) − Vout ( P2,25°C )] − [Vout ( P1, T ) − Vout ( P1,25°C )]
[Vout ( P 2,25°C ) − Vout ( P1,25°C )] × (T − 25°C )
2nd approximation: ∆Voq(T) = Voq(T) – Voq(25°C) = 0
TCσ =
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Vout ( P 2, T ) − Vout ( P 2,25°C )
[Vout ( P 2,25°C ) − Voq (25°C )] × (T − 25°C )
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Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
3. MLX90251 – TC Table
This section is intended to give a description of the TC Table implemented into the MLX90251.
In fact, the MLX90251 is a Programmable Linear Hall Effect Sensor, whose Temperature Coefficient, TCσ,
is also programmable. This allows the user of this IC to program a TC into the chip so that he can
compensate - at least partially – the thermal drift of the applied magnetic flux density. This former drift
can be due either only to the magnet which is used in the application or even to the interaction of magnet
and module.
The TCσ (TC of the IC) is defined through three programmable parameters: TCW (3 bits), TC (5 bits) and
TC2ND (6 bits).
All three parameters can be programmed in the final application and are stored in the EEprom of the IC.
In order to determine which triplet [TCW, TC, TC2ND] has to be programmed into the IC to compensate
a pre-defined TCB (see equation 9 to get the relationship between TCB and TCσ) a kind of look-up table is
used. This table is what we call the TC Table.
The following paragraph gives more details about the implementation of this TC Table.
The TC Table of the MXL90251 is based on two equations, equation 19 and equation 20.
TChot = ( K 2 + A1 × TC + A2 × TCW ) × TC 2 ND + ( K 3 + A3 × TC + A4 × TC 2 ) × TCW + K 4 × TC + A5 ×
TC 2
+ K5
2
Equation 19
TCdiff = K1 × TC 2 ND + ( K 7 + A6 × TC ) × TCW + A7 ×
TCW 2
TC 2
+ K 8 × TC + A8 ×
+ K9
2
2
Equation 20
where:
TChot = TC between 25°C and 125°C
TCdiff = TChot – TCcold, with TCcold = TC between -40°C and 25°C.
Both equations have been evaluated through a statistical analysis performed on the MLX90251.
It should also be emphasized to the fact, that TChot equals TC target, i.e. the “to-becompensated” temperature coefficient (the TC of the magnet).
Normally, the concept of this TC Table would require the coding of all the information needed to calculate
the TChot at any point of the space defined by the triplet [TCW, TC, TC2ND]. Therefore, the 16
parameters which define the previous 2 expressions should be stored into the EEprom. However, this
storage would require too much memory-space.
This is why a simplified method for calculating the value of TChot as a function of (TCW, TC) only has
been evaluated. In a similar way, a simplified equation for calculating the value of TC2ND as a function of
(TCW, TC) only has been defined. This former one is defined in such a way to minimize TCdiff, i.e. to
compensate the difference between TChot and TCcold. Therefore, this parameter is called TC2ND; it
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
compensates the second order effect, which normally would imply a little difference between the TChot
and the TCcold of the IC.
Finally, the TCσ is based on two TC Tables: one table, which gives TChot as a function of TCW and TC
and a second one which gives TC2ND as a function of TCW and TC, where TC2ND is chosen in order
to minimize TCdiff.
By using this approach, only six TChot values and six TC2ND values must be stored inside the EEprom in
order to be able to restore the whole TC characteristic of the IC. All intermediate values can be calculated
either via a linear interpolation or via a quadratic interpolation.
Those 12 values, which have to be stored, are determined individually for each IC during the three
temperature Final Test at Melexis.
The following figure illustrates the interpolation scheme.5
TChot
TC
TCW
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
0
TChot(0,0)
1
2
3
4
5
6
7
TChot(7,0)
TChot(0,16)
TChot(7,16)
TChot(0,31)
TChot(7,31)
5
TChot(x,y) represents the value of TChot stored in the EEprom for TCW = x and TC = y.
TC2ND(x,y) represents the value of CT2ND stored in the EEprom for TCW = x and TC = y.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
TC2ND
TC
TCW
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
0
TC2ND(0,0)
TC2ND(0,31)
1
2
3
TC2ND(3,0)
4
TC2ND(3,31)
5
6
7
TC2ND(7,0)
TC2ND(7,31)
The yellow colored cases are recorded TChot or TC2ND values.
The solid arrows are representative for a linear interpolation (along the TCW axis for TChot and along the
TC axis for TC2ND), whereas the dashed arrows are representative for a quadratic interpolation (along
the TC axis for TChot and along the TCW axis for TC2ND).
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
4. TC Determination
This section is intended to describe how the Temperature Coefficient of a system can be determined by
using a chip like the MLX90251.
First of all it should be specified what the denomination “system” does mean. In the following section,
when speaking about the “system” we refer either to a simple magnet or to a magnet implemented in a
module (w/o IC, if not otherwise specified).
Each part of the system can have a certain temperature coefficient. Therefore, we have the general
equation:
TC measured = TC IC + TC system = TC IC + TC M + TC mod ule
Equation 21
,
where:
TCIC represents the temperature coefficient of the IC
TCM represents the temperature coefficient of the magnet
TCmodule represents the temperature coefficient of the module (w/o magnet and w/o
IC)
TCsystem+IC represents the temperature coefficient of the complete system (module +
magnet + IC).
TCmeasured represents the measured temperature coefficient, which will be measured
via the MLX90251.
All our measurement results will be based on measurements made by using the MLX90251. In order to
determine the TC of the system we should know/characterize the TC of the MLX90251. To do so, the TC
of the MLX90251 should be programmed to 0ppm/°C. In this way equation 21 becomes:
TC measured = TC IC + TC system = 0 + TC M + TC mod ule = TC M +TC mod ule
Equation 22
This means that we can directly evaluate the TC of the system (magnet + module). However, in order to
get accurate estimations of this TC, we need first to characterize the TC of the IC. To do so, the following
procedure should be applied.
4.1. Characterize TC of the MLX90251
As mentioned before, the TC of the MLX90251 should be programmed to 0ppm/°C. However, due to
the limited accuracy of the chip, the perfect 0ppm/°C behavior will never been achieved. Therefore,
the TC of the IC has to be characterized. This should be done by performing the following procedure:
1. Program the TC of the IC to 0ppm/°C via the PTC-03/PTC-04
2. Evaluate the real thermal behavior of the IC
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
In order to evaluate the TC of the IC, the output voltage should be measured at at least three
different temperatures (Cold, Ambient, Hot) and per temperature for two different positions, i,e, for
two different magnetic fields.
Concerning the two different positions, the best approach would be to measure Vout for a positive
magnetic field and for the corresponding negative magnetic field, i.e. Vout(+B Gauss, T) and Vout(-B
Gauss, T).
To calculate the TC of the IC equation 15 should be used.6
Vout ( P 2, T ) − Vout ( P1, T ) Vout ( P 2,25°C ) − Vout ( P1,25°C )
−
B( P 2, T ) − B( P1, T )
B( P 2,25°C ) − B( P1,25°C )
TCσ =
Vout ( P2,25°C ) − Vout ( P1,25°C )
× (T − 25°C )
B( P2,25°C ) − B( P1,25°C )
If however, the magnetic field strength is not known one of the approximating equations can be used
(see equation 16 and equation 18).
In order to represent the TC of the IC (and even the TC of a system) a nice method is to use the
normalized sensitivity, i.e. Sensitivity(25°C)/Sensitivity(T °C). This approach will be explained in
Appendix III.
4.2. Determine TC of the system
Once the thermal behavior of your IC is fully characterized you can use it to characterize your system
(magnet or magnet + module).
To do so you only need to implement the characterized IC into your system and to perform the same
measurements as explained in the previous section, i.e. measure the output voltage at at least three
different temperatures (Cold, Ambient, Hot) and per temperature for two different positions.
Once those measurements have been performed you can evaluate the TC of your system by using
equation 13.
1
1
)−(
)
σ (T )
σ (25°C )
.
=
1
(
) × (T − 25°C )
σ (25°C )
(
TC system
In order to compensate this thermal TC of your system you should program the corresponding value
into your IC.
6
If the strength of the magnetic field B is not known you can also use one of the approximating formulas (see
section 2.2.1.). However, in this case you will loose on accuracy.
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Example:
Assume that TCsystem = -450ppm/°C.
Your IC should compensate the TC of your system. Therefore, the IC should have a TC of
+450ppm/°C. However, it should be emphasized that by using the User Interface of the PTC-03/PTC04 you have to define the “To-Be-Compensated TC”, TCsystem, and not the TC that your IC should
have.
This means that in this case you should define as Target TC -450ppm/°C. The software of the PTC03/PTC-04 will then automatically calculate the corresponding TC, i.e. +450ppm/°C, and program the
corresponding TC-triplet [TCW, TC, TC2ND] into the IC.
To get a nice representation of the thermal TC behavior of your system it is also recommended to use
the normalized sensitivity approach (see Appendix III).
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Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Appendix I
Remember that the Thermal Error was defined as:
Thermal Error = ∆Voq (T ) + ∆TC × σ (25°C ) × B (25°C ) × (T − 25°C )
We will take the following assumptions:
∆Voq(T) = ±20mV i.e. 0.4%Vdd if Vdd = 5V (tolerance on the Thermal Offset Drift)
∆TC = ±100ppm/°C (tolerance on the mismatching between the calibrated [Melexis]
Sensitivity TC and the whole magnetic and mechanical circuit TC)
Output Span from 10%Vdd up to 90%Vdd
0.8%Vdd/mT i.e. 4mV/G if Vdd = 5V
The Thermal Voq Drift Error, ∆Voq(T), can be represented graphically as follows:
Figure 1: Error induced by the Thermal Voq Drift i.e. ∆Voq(T)
Whereas the error induced by the TC mismatch, i.e. ∆TC, can be represented like:
Figure 2: Error induced by the TC mismatch i.e. ∆TC
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MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Finally, the total Thermal Error will have the following shape:
Figure 3: Total Thermal Error
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Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Appendix II
σ(T) = σ(25°C) x [1 + TCσ x ∆T]
B(T) = B(25°C) x [1 + TCB x ∆T]
with ∆T = T – 25°C.
!
!
TCσ = [σ(T) - σ(25°C)] / [σ(25°C) x ∆T]
TCB = [B(T) – B(25°C)] / [B(25°C) x ∆T]
[a]
[b]
Furthermore we have:
TCσ = -TCB / [1 + TCB x ∆T]
TCB = -TCσ / [1 + TCσ x ∆T]
[c]
[d]
If we put [a] in [d]:
TCB = - {[σ(T) - σ(25°C)] / [σ(25°C) x ∆T]} / {1 + [σ(T) - σ(25°C)] / [σ(25°C) x ∆T]] x ∆T}
= - {[σ(T) - σ(25°C)] / [σ(25°C) x ∆T]} / {[σ(25°C) + σ(T) - σ(25°C)] / σ(25°C)}
= - {[σ(T) - σ(25°C)] / [σ(25°C) x ∆T]} / {σ(T) / σ(25°C)}
= - {σ(T) - σ(25°C)} / {σ(T) x ∆T}
by multiplying nominator and denominator with {1 / [σ(25°C) x σ(T)]}
= - {[σ(T) - σ(25°C)] / [σ(25°C) x σ(T)]} / {[σ(T) x ∆T] / [σ(25°C) x σ(T)]}
= - {[σ(T) - σ(25°C)] / [σ(25°C) x σ(T)]} / {∆T / σ(25°C)}
= - {[σ(T) - σ(25°C)] / [σ(25°C) x σ(T)]} / {∆T x [1 / σ(25°C)]}
= - {[1 / σ(25°C)] – [1 / σ(T)]} / {∆T x [1 / σ(25°C)]}
if σ1(T) = 1 / σ(T):
= {σ1(T) - σ1(25°C)} / {σ1(25°C) x ∆T}.
390119025101
Rev 001
Author: PHI
Page 18 of 22
Mar-2006
Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Appendix III
The normalized Sensitivity Approach
This section will give a nice way to represent graphically the TC of a system or of an IC.
This approach is called the normalized sensitivity approach because it represents the normalized sensitivity versus
the normalized temperature, where
-
normalized sensitivity means:
1
Sensitivity (T )
NormalizedSensitivity =
1
Sensitivity (25°C )
-
normalized temperature means:
NormalizedTemperature = T − 25[°C ] .
Example:
Assume that we should have an IC with a target TC of -350ppm/°C, and that the spec is +/-150ppm/°C.
Assume the following measurements results:
Chip ID Temperature [Deg.C.] Magnetic Filed [Gauss] Output Voltage [V]
52259
-40
-500
3.259074
52259
-40
0
2.486386
52259
-40
500
1.712807
52259
25
-500
3.287673
52259
25
0
2.497092
52259
25
500
1.70554
52259
85
-500
3.31161
52259
85
0
2.503244
52259
85
500
1.693924
52259
125
-500
3.319199
52259
125
0
2.504777
52259
125
500
1.689406
52259
150
-500
3.319829
52259
150
0
2.504135
52259
150
500
1.687745
Table 1
390119025101
Rev 001
Author: PHI
Page 19 of 22
Mar-2006
Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Based on those measurements, the normalized Sensitivity and the normalized Temperature can be
calculated. The results are summarized by Table 2.
Normalized Temperature [Deg.C.]
-65
0
60
100
125
Normalized Sensitivity
1.023195218
1
0.978022311
0.970757023
0.969394345
Table 2
Normalized Sensitivity
TC Evaluation
Normalized Sensitivity Approach
-65
-50
-35
-20
1.04
1.035
1.03
1.025
1.02
1.015
1.01
1.005
1
-350ppm/Deg.C. (Ideal behavior)
0.995
0.99
0.985
0.98
0.975
0.97
0.965
0.96
0.955
-350ppm/Deg.C. +/-150ppm/Deg.C.
0.95
0.945
0.94
0.935
0.93
-5
10
25
40
55
70
85
100
115
52259
Ideal behavior
Limit 1
Limit 2
Normalized Temperature [Deg.C.]
How to determine the Normalized Sensitivity characteristic versus the normalized temperature?
390119025101
Rev 001
Author: PHI
Page 20 of 22
Mar-2006
Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
This approach is based on equation 13:
1
1
)−(
)
σ (T )
σ (25°C )
TC B =
.
1
(
) × (T − 25°C )
σ (25°C )
(
This equation can be transformed into:
(
TC B =
σ (25°C )
σ (25°C )
)−(
)
σ (T )
σ (25°C )
(T − 25°C )
(
=> TC B =
σ (25°C )
) −1
σ (T )
(T − 25°C )
.
This equation represents the slope of the ideal behaviour line in our Normalized Sensitivity Approach.
In fact, we have:
NormalizedSensitivity (T ) =
Sensitivity(25°C )
Sensitivity(T °C )
= [TC B ∗ (T − 25°C )] + 1
If T = 25°C, the Normalized Sensitivity equals to 1. Therefore, at the Normalized Temperature of 0°C our Normalized
Sensitivity function crosses the Y-axis at point 1.
390119025101
Rev 001
Author: PHI
Page 21 of 22
Mar-2006
Application Note
MLX90251
Programmable Linear Hall Effect Sensor
TC – Temperature Compensation
Numerical Example:
How to determine the Ideal behavior characteristic?
Suppose that TC target = -350ppm/°C.
At T = 125°C, we have:
(
− 350 ppm / °C =
=>
σ (25°C )
) −1
σ (T )
(125 − 25°C )
σ (25°C )
= ([−350 ppm / °C ] * [100°C ]) + 1
σ (T )
= −0.035 + 1
= 0.965
Furthermore, the Normalized Sensitivity equals to σ(25°C)/ σ(T). So we have:
Normalized Sensitivity = 0.965, at T = 125°C (Normalized Temperature = 100°C).
To calculate the specification limit lines, same procedure has to be used for -500ppm/°C
(=-350ppm/°C – 150ppm/°C ) and for -200ppm/°C (=-350ppm/°C + 150ppm/°C).
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390119025101
Rev 001
Author: PHI
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