This article describes the calibration and compensation of piezoresistive pressure sensors. Basic circuit properties of external circuits are calculated as an example of bare C32 pressure dies and prehoused ASB1200E dies and the advantages and disadvantages of external circuits compared to integrated circuits are discussed, especially considering the restrictions of modern production lines.
The Uncompensated Device
A piezoresistive MEMS pressure sensor consists of a bridge circuit which is implanted into a silicon MEMS structure carrying a membrane. Figure one shows the typical geometry and the layout of the C32 family pressure sensor chips. The readout of such an uncompensated device like a bare C32 pressure sensor die or housed ASB1200E, seen in figure two, can be considered as
(Equation 1.1) V_{OUT} = V_{DD} • vo • p • s • V_{DD} + O(p^{2})
V_{OUT} = V_{DD} • v_{o} • p • s • V_{DD} + O(p^{2})
V_{OUT} = Output Voltage
V_{DD}: Input Voltage
v_{o}: Offset voltage (as ratio of V_{DD})
p: Pressure
s: Sensitivity
O(p2): The quadratic contribution in pressure refers to the nonlinearity error of the reading which is typically in the range of 0.3% fullscale output.
The offset voltage V0 is part of this equation as ratiometric contribution of the driving voltage:
v_{o} = V_{DD} • v_{o}
Due to the given tolerances of the offset, the sensitivity, and the bridge resistance, the readout of a device deviates from the nominal values. The accuracy of a compensated device is then only limited by the nonlinearity and drift, if not for the fact that important parameters of Equation 1 significantly change with temperature. In fact all piezoresistive pressure sensors show the following inherent characteristics:

The decrease in sensitivity with increasing temperature, which is referred to as the negative temperature coefficients of span TCS.
(Equation 1.2) S(T)=S_{N} • (1 + α_{S}(T  T_{N}) + β_{S}(T  T_{N})^{2})

An increase with bridge resistance with temperature
(Equation 1.3) R_{S}(T)=R_{SN} • (1 + α_{RS}(T  T_{N}) + β_{RS}(T  T_{N})^{2})
With T_{N} being the nominal temperature (typically 25°C), S_{N} and R_{SN} the sensitivity and bridge resistance at T_{N} and α_{S} , β_{S} , α_{RS}_{}and β_{Rs} are the coefficients for the first and second order of the quadratic approximations.
The temperature dependence of the offset voltage is due to thermomechanical stress on the device. This stress is mainly caused by the mounting situation of the chip and is therefore not referred to as an intrinsic temperature dependency. It is often given in the form of the two secants slopes TCV_{0}+ and TCV_{0}. From these secants the quadratic coefficients can be calculated. A first indication of the temperature impact on the offset can be calculated by the linear approximation i.e. for elevated temperatures:
V_{0} (T_{Max}) = V_{DD} • (V_{0}(T_{N}) + TCV_{0}• (T_{Max}  T_{N}))
The change of the output voltage (1.1) with temperature is given by:
(Equation 1.4)
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With , in case of constant driving voltage and , in case of constant current excitation where R_{S} is the total bridge resistance and I_{DD} is the input current.
In case of a compensated device one may consider V_{0} = 0. Just taking into account the linear contributions in temperature one finds:
for constant voltage
for constant current
Comparing the magnitude of these contributions to the temperature behavior of an ASB1200E at 75°C one finds temperature errors of sensitivity and bridge resistance are typically 10% of the full scale readings and the temperature shift of the offset is one order of magnitude smaller than the contributions of α_{S}, and α_{RS}. Additionally, the contributions of as, and aRs have different signs and compensate each other in case of constant current excitation: TCVO>>(α_{RS} + α_{S}) • s • p
Compensation And Calibration Via External Circuits
The Compensation of Offset and Sensitivity
With open (half) bridge devices the compensation of the offset can be done adding a fixed resistor into the bridges circuit. As this procedure is most often the origin of additional temperature shifts closed (full) bridge devices are used together with external circuits. An example for such a circuit is shown in figure three. This circuit consists of a typical instrumentation amplifier with the two operational amps A and B. The resistor R gain in the resistor network is used to set the gain and compensate the deviations in sensitivity:
The offset is compensated by the voltage divider R offset 1 and R offset 2.
To determine the settings for the offset one can start to measure the output at a given pressure p_{1} at 25°C,
with v0^{N} = 0 and s_{N} being the nominal values of the ratiometric offset and sensitivity. The actual reading now differs form the nominal expected reading:
Given only this one measurement point (p1, V_{OUT} 1), it cannot be distinguished how much of the deviation is due to the offset and how much is due to the applied pressure (sensitivity). A second measuring point (p2, V_{OUT} 2) is needed to adjust offset and sensitivity:
One finds the shift of the offset for a specific device:
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Using these values one can now adjust R offset 1 and R offset 2 until the signal is shifted by δv_{0} • V_{DD}, than the gain can be adjusted to rotate the curve i.e. by varying R gain at pressure p1 until the desired reading V_{OUT1}^{N} is reached. The following instruction should be carefully respected for each calibration procedure:
1. Allow the circuit to heat up and stabilize temperature.
2. Make sure that the sensor temperature is well controlled at 25°C
3. Applied pressures shall be applied with the highest possible accuracy. The use of calibrated references is recommended.
4. The adjustment of the offset and gain should be done at the pressure point where highest accuracy is required.
5. After adjusting the gain the compensation of the offset should be rechecked.
6. The procedure needs to be performed on every single device
The accuracy that can be reached by this method mainly depends on the accuracy of the reference pressures and the stability and robustness of the circuit. If the pressures can only be adjusted with a limited accuracy of Δ_{p} also the compensation will show limitations.
Temperature Compensation With External Circuits
The calibration of a device includes the compensation of offset and sensitivity as well as elimination of the temperature influences. During the design of the circuit one not only needs to take measures to calibrate the inherent temperature behavior of the sensor, but also needs to consider additional temperature effects added by the external circuit.
In a circuit such as figure three, the temperature impact is not only limited to α_{s}, β_{s}, a_{Rs},β_{Rs}and TCV0 as discussed earlier, but also has contributions of the temperature dependence of R_offset1, R_offset2, R_gain and from the other components in the circuit.
One also needs to consider that the temperature of the pressure die is not necessarily the same as the temperature of the circuit. In extreme cases a separate determination and calibration of the circuits and the chips temperatures need to be applied.
The most important temperature effect of the pressure die is the decrease of sensitivity at elevated temperatures. As we have seen before, this effect is typically one order of magnitude higher than the temperature dependence of the offset TCV0 or the quadratic contributions β_{s}. Now we simplify and use the following equations:
(Equation 2.5a for constant voltage)
(Equation 2.5b for constant current)
To deal with the decreasing sensitivity two basic methods can be applied:

Using thermistors or other temperature dependent devices in combination with a constant voltage supply.

Using internal chip characteristics – like the temperature behavior of the bridge resistance – in combination with constant current source.
Calibration With A Constant Voltage Supply
Equation 2.5a is a ratiometric dependency in VDD. The idea behind constant voltage supply calibration with thermistor networks is to make V_{DD} temperature dependent. Figure four sketches this basic idea. According to equation 1.1 with compensated offset and neglecting all second order effects:
With VN being the voltage drop over the thermistor network RN. With the condition of proper temperature compensation one finds:
(Equation 2.7)
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Using the linear approximations for the temperature behavior of s and Rs equation (2.7) can be rearranged to a condition for the temperature coefficient of RN:
(Equation 2.8)
With RNN and RSN as resistance values at the reference temperature, the given characteristic has a negative temperature coefficient. The bridge supply voltage VDD rises when RN decreases with temperature, the drop in sensitivity is then compensated by the increase of the bridge output.
Temperature Compensation With A Thermistor
Already indicated in figure four, the resistor network can be realized with a thermistor RT and a fixed parallel resistor RP. The rate of rise of this linearized characteristic is …
(Equation 2.9)
… with B the BValue of the thermistor, TM the middle temperature in the measuring range, and R_{T} the thermistors resistance at T_{M}. . Setting 2.8 equal to 2.9, one finds an equation for the parallel resistor. The easiest way to determine R_{P} is to start with given thermistor values R_{T} and B and than solve the equation in R_{NN} first:
(Equation 2.10)
In this example, it is assumed that the application temperature range is between 0°C to +80°C with the bridge resistance of ASB1200E being 3.5 k? at +40°C (T_{M} = 313 K). The compensation is via an NTC B57881S212F, which has the following parameters at 40°C: B = 3560 K and R_{T} = 1.2 k?
Using 2.10 one finds as solution…
= 0.5 k?
… and therefore R_{P} = 0.86 k?.
With the parameters given it is possible to reduce the temperature impact of sensitivity from 2.2 •10^{3}K^{1} by a factor 5 to 0.4•10^{3}K^{1}.
Calibration With A Constant Current Source
Figure five shows the configuration used in case of constant current source configuration. Similar to the previous analysis we start again with Equation 1.1, assuming again the offset to be compensated and neglecting all second order effects:
(Equation 2.11)
With the condition of proper temperature compensation, one finds after the same procedure as above as compensation condition:
(Equation 2.12)
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On Temperature Compensation
The response curve of piezoresistive pressure sensors show linear and quadratic dependencies in pressure and temperature. A useful sensor needs to be compensated and calibrated. The simple compensation and calibration techniques previously described can reduce the linear impacts of temperature and improve the typical accuracy within the temperature range to 3% to 5% full scale or reading whatever the higher value is. Further more sophisticated circuit ideas are shown in figures six and eight.
Very often users of pressure dies however see certain drawbacks:
1. The calibration procedure needs to be done for each single device and the sensors need to be matched with selected resistors individually.
2. For the calibration procedure measurements at two pressure points and at two temperatures are needed as minimum requirement. The integration of the measurement into the production flow is difficult.
3. The sensor chip’s technical parameters are altered by the assembly and mounting of the chip. The use of premeasured chips is not possible.
4. A/D conversion or amplification of the signal needs to be done separately.
Therefore external circuits are more and more replaced by integrated signal processors, i.e., DSPs, ASSPs, or ASICs, especially for new designs and higher volume applications. By using ICs it is possible to completely separate the mechanical assembly of a sensor device from the calibration and compensation step. Ready packaged sensors can be calibrated by software measures.
Calibration Using ICs
Figure seven shows a typical circuit and architecture of a calibration circuit with Signal Processor. As indicated by the ASIC's architecture the IC provides power management, a communication interface, EEPROM storage block for calibration parameters, signal transmission with multiplexing and ADC/DAC, amplification in case of analog outputs, an option DSP, optional temperature reference or temperature signal input, and an optional V_DD regulator.
Simple low cost ASICs are designed for the use with external DSPs and come without temperature reference, signal processor, and V_DD regulator. They primarily offer three functions: storage of calibration parameters, conversion and transmission of sensor parameters, and the communication with an external signal processing or control unit. If the ASIC does not provide V_DD regulation, this has to be provided by an external circuit.
It is not recommended to use the internal temperature reference of an ASIC for the calibration and compensation of a pressure sensor. The chip’s temperature is more important and can deviate from the ASIC’s temperature.
Calibration Procedure
Whether the signal processing is done using an internal or an external signal processor does not matter for the general calibration procedure. The five free parameters s_{N}, α_{s}, β_{s}, r_{off}(T_{N}), and TCV0 in equation 1.1 to1.4 need to be implemented in a model. The calibration is based on the following general equation which describes a quadratic dependency of temperature and pressure…
(Equation 3.1)
… with a, b, c, d, and e being the five calibration parameters and VOUT the raw output of the resistance bridge. This equation is implemented in the signal processing unit and can also be given in different forms, for example…
(Equation 3.2a)
(Equation 3.2a)
… with F, G, H, J, and K being another set of five calibration parameters.
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One can easily show that equation 3.1 and 3.2 are similar if all contributions of higher, nonquadratic orders are ignored. The parameters g and g’ are fixed gain factors to map the desired pressure range to a specific output signal range. To determine all five calibration parameters of equation 1.11.4 or 3.1 or 3.2 one needs to measure five points of the transfer characteristic: (p_{i},T_{i},V_{outi})with i = 1.5
The number of measuring points can be reduced if the temperature dependency in the application is small or the quadratic impact of output voltage shall be ignored. In this case one my consider to preselect some parameters as 0 or 1. i.e. d=e=0 in equation 3.1 leads to a similar calibration problem as G=1 and K=0 in equation 3.2 with 3 free parameters each.
The actual calculation of the calibration parameters and the storage of the parameters in the EEPROM of the ASIC is an easy software task and does not require any physical changes of the device itself. Therefore the calibration can be completely separated from the rest of the assembly line which allows a more efficient set up and usage of the production equipment.
Accuracy And Resolution
The terminology of accuracy and resolution of a device often causes confusion, both refer to completely different properties of the transmitter. Some more refined definitions might be useful for a deeper understanding, such as accuracy refers to the reading of a sensor compared to a well known reference pressure, resolution is the smallest short term pressure change that can be detected by the sensitive device, and resolution is not a property of the sensitive chip or calibration but of the signal processor.
As an example the ASB1200V is a fully calibrated device using a mixed signal ASIC with a minimum output resolution of 11 bits. The remaining nonlinearity L is in the order of magnitude of the temperature and pressure hysteresis HT and Hp and equals 0.1% full scale. The remaining temperature dependencies of sensitivity and offset TCVFS and TCVAO equal 0.015% FS/K.
The accuracy therefore can be estimated as per:
The resolution however can be estimated as , which equals 0.6 h Pa.
Conclusion
Pressure sensitive elements, i.e., silicon MEMS chips, show variations of offset and sensitivity as well as cross sensitivities to temperature and thermomechanical stress. The aim of compensation and calibration is to eliminate the sensitivity variations as well as reduce the cross sensitivities.
Compensation and calibration can be done by either external passive circuits or ICs, especially ICs as they’ve become more popular in recent years as the availability of standard ICs for sensor purposes increases. Packaging is an important issue related to calibration for two reasons: a major part of the thermomechanical stresses originate from the packaging of the sensitive elements and pressure and temperature profiles need to be applied during calibration. Pressure ports must be installed on the device.
As a consequence, the calibration needs to be done after the packaging is complete. The use of prepackaged devices like the ASB1200E can be very useful as the prepackage already protects the sensitive element in production. However, the decision of which calibration concept to use requires a detailed analysis of more factors such as the application, design, and production.
In terms of the application, temperature and pressure measuring range and accuracy requirements of the sensitive device itself need attention. A number of factors influence the design including the electrical environment (availability of a stabilized power supply and an external signal processor), necessary output (analog current, analog voltage, digital interface, etc.), packaging concept and accessibility of variable components, and the availability of temperature signals.
Production entails the availability of pressure measuring equipment, integration of calibration procedures in the production line, and inhouse or external calibration. Especially for high volume productions, it is most often desirable to separate the assembly of a sensor from its calibration and testing. In this case, integrated signal processing units or an ASIC seem to be the best choices.
Very often the calibration equipment does not fit into a highly optimized pick and place production. In such cases, the use of ready calibrated devices like ASB1200V (see fig. 8) may be the way to go.
Dr. Bernhard Ostrick is head of marketing of temperature and pressure sensors systems for the Sensors Business Group of TDK EPC. He joined the company, then EPCOS, in 2002. Dr. Ostrick can be contacted via email at [email protected].