Sensors - March 2002 - Resistive-Element Sensor Temperature Compensation

For optimum versatility, accuracy, and speed, try a conditioning system that uses temperature-driven look-up tables in its calibration schemes.

Resistive-element sensors, most notably the Wheatstone bridge–configured piezoresistive devices, have dominated the low- to medium-accuracy pressure-sensing industry since the early 1980s. The principal source of measurement error with these devices is the change in sensitivity and output offset brought about by fluctuations in temperature. The quest for ever-increasing accuracy and lower cost has led to numerous solutions to the requirement for signal conditioning and calibration. One of the most effective solutions is the analog path conditioning architecture, which uses four D/A converters (DACs) to provide the necessary temperature corrections.

A four-DAC system can give you good accuracy and speed in many applications. With such a system, you can easily implement your own first-order calibrations once you understand the basics of dealing with sensor sensitivity and offset behavior.

Dealing with Sensor Sensitivity
Defining Sensitivity. The temperature sensitivity of gain for piezoresistive sensors originates from two primary sources: the thermal coefficient of sensitivity (TCS) and the thermal coefficient of resistance (TCR). TCS effects arise from dimensional and stiffness changes in the sensor over temperature. TCS is always negative (i.e., sensitivity reducing with increasing temperature). TCR describes the change in sensor bridge resistance with temperature, and it is normally positive.

Defining Offset. The offset behavior of a piezoresistive sensor is similarly affected by dimensional changes within the sensor as temperature varies. This effect produces a dominantly first-order temperature sensitivity of sensor offset that can be either positive or negative in sign. Of less significance are local thermal variations within the sensor bridge, which can lead to the generation of second-order (or higher-order) behavior over temperature.

Designing the Sensor. Most resistive-element sensors are designed to make best use of the opposing signs of these two thermal coefficients. The aim is to produce a sensor with TCS slightly lower in magnitude than TCR. When driven from a constant current source, such a sensor exhibits a much-reduced total temperature sensitivity and allows external temperature compensation to be easily applied.

The resulting condition is portrayed in Figure 1, which contains normalized temperature responses for bridge resistance (R_b) and pressure sensitivity. The slopes of the two responses represent the TCR and TCS characteristics of the sensor. The third curve in Figure 1 represents the ideal response of sensor bridge voltage (V_b) required to balance the sensitivity curve and produce a sensor with a null temperature coefficient of gain.

Figure 1. This graph shows normalized responses of piezoresistive pressure sensor bridge resistance (R_b) and pressure sensitivity versus temperature. The ideal V_b characteristic shown is the bridge voltage response that would perfectly compensate changes in pressure sensitivity with temperature.

Meeting Conditioning Requirements. The first task of a conditioning circuit, which acts on the bridge drive current, is to produce the ideal V_b curve (see Figure 1).

The second task is to produce a signal that balances the sensor’s output offset behavior. Resistive bridge sensors typically exhibit output offset behavior that can be described in terms of a fixed component plus a temperature-dependent component. Both of these components can be corrected by summing offsets into the signal path. One of these offset components should have a fixed value. The second should have a temperature characteristic opposing that of the sensor offset temperature-dependent component.

Providing Compensation. An analog path, four-DAC system that satisfies these requirements is shown in Figure 2.

Figure 2. The illustration shows a system of D/A converters (DACs) for temperature compensation of resistive element sensors, as found in the MAX1452 and similar devices. The system uses the temperature dependence of the sensor bridge voltage to control the temperature-related correction elements. Two of the DACs provide fixed compensation components for sensitivity and offset; the other two provide temperature compensation.

In this system, the Span DAC and SpanTC DAC combine to provide the necessary modification to the bridge voltage V_b. The Span DAC takes its reference from the positive supply rail V_dd and provides an output that is ratiometric to V_dd and independent of temperature. The SpanTC DAC is referenced to the bridge voltage V_b and has an output that is temperature dependent (because V_b is temperature dependent). The resulting bridge voltage response characteristic is then of the form given in Equation 1:

Span	= Span DAC value, range 0–1
SpanTC	= SpanTC DAC value, range 0–1
K₁	= arbitrary constant
K₃	= arbitrary constant
R_b	= sensor bridge resistance
V_dd	= conditioner IC supply voltage

The Offset DAC and OTC DAC provide compensation for the sensor output offset characteristic. As with the Span and SpanTC DACs, the offset correction DACs are arranged to provide both fixed (referenced to V_dd) and temperature-dependent (referenced to V_b) outputs. The Offset and OTC DACs provide outputs that are a positive or negative fraction of their reference voltages. The modification to the output of the system of Figure 2 provided by these offset-correcting DACs is then of the form shown in Equation 2:

Offset	= offset DAC value, range 0 to ±1
OTC	= offsetTC DAC value, range 0 to ±1
K₂	= arbitrary constant
K₄	= arbitrary constant
V_b	= sensor bridge voltage
V_dd	= conditioner IC supply voltage

Implementing a First-Order Calibration
By using the four-DAC architecture shown in Figure 2, you can calibrate virtually any piezoresistive bridge pressure sensor to a first-order fit, over the entire operating temperature range. The objective of a first-order, fixed-value calibration is to find unique calibration values for each of the four DACs in the compensation system. These are the values required to produce the desired bridge voltage response of Figure 1 and to correct for the sensor offset response. You can do this with a simple two-temperature, two-pressure calibration.

Span Compensation. The first step in the calibration is achieving span compensation of the sensor by modifying the sensor bridge voltage response. The method used requires that you determine an ideal value of bridge voltage at each temperature. Then find pairs of values for the Span and SpanTC DACs and the corresponding value for bridge voltage at each temperature. The bridge voltage values that are generated should be close to the ideal bridge voltage value at the current temperature. You should avoid matching values for V_b at each temperature. The mathematical solution to the resulting set of equations is in matrix form and will generate a singular matrix if you use identical values for V_b.

Begin the process at the first calibration temperature (T₁) by loading the Span and SpanTC DAC registers with nominal values and measuring the bridge voltage and conditioner IC output span. You can calculate the value of bridge voltage required at this temperature by using the expression given in Equation 3.

Next, choose three values for the SpanTC DAC (ß₁₁, ß₁₂, ß₁₃) and load them, in turn, into the SpanTC DAC register. For each of these SpanTC DAC settings, determine a corresponding value for Span DAC (

) that produces a bridge voltage approximately equal to the required V_b value.

After completing the measurements at T₁, change the test chamber temperature to the second calibration temperature (T₂) and repeat the process. Figure 3 illustrates this process and the measurements required at each temperature.

Figure 3. Shown here are D/A converter (DAC) settings (

,ß) and measurements (V_b) required during a first-order, fixed-value span calibration. Once the required value for Vb at the test temperature is determined, three measurements are required to uniquely determine the required settings for the two Span DACs. To complete the calibration, this process is repeated at a second test temperature.

Finally, complete the Span calibration process by calculating the required values for Span (

) and SpanTC (ß) and loading them into the respective signal conditioner registers. The calculations required are:

Offset Compensation. Figure 4 shows the full, first-order calibration measurement system using four DACs.

Figure 4. Shown here are the D/A converters settings and measurements required for a full first-order, fixed-value, calibration. The full calibration consists of the measurements performed for span calibration, as shown in Figure 3, plus three additional measurements for offset calibration.

The measurements in brackets are those required to perform the offset compensation for the sensor. While at temperature T₁ and with minimum pressure applied to the sensor, adjust the Offset DAC to produce an output voltage approximately equal to the desired zero output. Normally, you set the OTC DAC to zero during this phase of the calibration. After adjusting Offset DAC, record the values of Offset DAC (

₁), OTC DAC (

₁), output voltage (V_out1), and bridge voltage (V_b1) for later use.

When you’ve completed the span calibration, while at temperature T₂ and with minimum pressure applied to the sensor, load the Offset and OTC DAC registers with the values

₁ and

₁₁ recorded at T₁B Next, measure and record the resulting value of output voltage (V_out2) and the current value for bridge voltage (V_b2). Then adjust the OTC DAC to produce the output voltage value given by Equation 4.

Finally, complete the offset calibration by adjusting the Offset DAC to produce an output equal to the desired zero output level.

Simplifying Calibration—The Look-Up Table
Including a temperature-driven look-up table in the signal conditioning architecture adds flexibility to the system and can simplify the calibration process. You need only two DACs for a basic system driven by a look-up table—one for span and one for offset (see Figure 5). One advantage of this type of system is calibration speed.

Figure 5. This system includes two D/A converters with a look-up table, as typified by such devices as Maxim's MAX1452 and MAX1455. The inclusion of a temperature-driven look-up table in the signal conditioning architecture adds flexibility to the system and can simplify the calibration process.

For a first-order temperature compensation, the calibration process used with a system based on look-up tables is similar to that described for the four-DAC, fixed-value calibration. As with the fixed-value system, you need two calibration temperatures. At each temperature, determine the ideal value for sensor bridge voltage by using Equation 3. Then find a value for Span DAC (

) that produces the required bridge voltage value. The Span DAC value thus determined is the required look-up table value at that temperature. The Offset DAC look-up table value (

) required is that which, with minimum pressure applied to the sensor, produces the desired output zero reading. Figure 6 illustrates this process and the measurements required at each temperature.

Figure 6. This illustration shows D/A converter (DAC) settings and measurements required during a full, first-order calibration driven by a look-up table. Once the required value for V_b at the test temperature is determined, only two measurements are needed to uniquely determine the required settings for the Span and Offset DACs. A minimum number of two test temperatures are required to complete a first-order calibration.

You complete the first-order calibration by loading the look-up tables for Span DAC and Offset DAC with values obtained from a linear interpolation between the values recorded for each DAC at each temperature.

Comparing Calibration Schemes
So far, we’ve discussed two calibration schemes: fixed-value and look-up-table derived. Their strengths and weaknesses can be analyzed in terms of system error conditions and calibration efficiency. Both schemes provide first-order compensation for temperature-related errors. The accuracy of the compensation scheme depends in part on the accuracy of the compensation temperature measurement. In the fixed-value calibration, we derived all of the temperature information from the sensor bridge resistance. The system should minimize any errors produced by transient thermal conditions.

With the look-up table system calibration, the temperature information is typically derived from an onchip temperature sensor used to drive the look-up table pointer. Any temperature difference between the sensor and the conditioning IC will therefore give rise to errors in the sensor compensation by pointing to adjacent locations in the look-up table.

An additional source of error for a first-order calibration is any second-order (or higher) term in the sensor temperature response. The fixed-value calibration scheme can be applied only as a first-order compensation and cannot correct for higher-order effects in the temperature response curve. The look-up-table compensation, on the other hand, can be applied at as many temperature intervals as required, accommodating virtually any order of temperature curve.

A further consideration is the time required to complete a sensor calibration. Calibration throughput is largely a function of temperature and pressure settling times and the time taken to determine and program the compensation coefficients for each sensor. With small-volume production, the settling times tend to dominate the calibration time. With high-volume production, the time required for the measurements on each sensor becomes more significant. It may take only one or two seconds to perform a single measurement set at one temperature. The fixed-value calibration scheme requires nine such measurement sets for a full, first-order calibration, whereas the look-up table system requires only four.

A Modified Look-Up-Table Calibration
Using a combination of aspects from the fixed-value and look-up-table calibrations, you can tailor the calibration scheme to provide an optimum compromise between sensor accuracy under transient temperature conditions and calibration speed. One such scheme uses the offset correction method from the fixed-value calibration, together with a generic value for the SpanTC DAC, while retaining most of the speed of the calibration based on the look-up table.

In the look-up-table calibration scheme previously described, the SpanTC DAC was not considered part of the calibration system and was set to a nominal value. For any sensor, a unique value for SpanTC will perfectly compensate the sensor’s thermal gain response at any two temperature points. This is the value that would have been determined by the application of the fixed-value calibration method. Had the SpanTC register been set to this value at the start of a look-up-table calibration, the values determined for Span DAC would have been the same at both measurement temperatures. Under this condition, all of the span-related temperature information would be derived from the sensor bridge resistance.

Similarly, had the SpanTC register been loaded with a generic value close to the actual value required, the Span DAC values would have differed only slightly at the two temperatures. The result would be a system in which most of the temperature information would be bridge-resistance derived, a system that would be relatively insensitive to thermal transients.

A statistical analysis of sensor calibration data will yield an average SpanTC DAC value that you can use as the generic value for a particular sensor type. Ideally, the data used for this should be the actual SpanTC DAC values recorded from a suitable number of sensor calibrations.

Further, having calculated the distribution of values for SpanTC, you can predict the error distribution of the calibrated sensors as a function of any temperature difference that may exist between the sensor and the conditioning IC. The precise formula for this error prediction will depend on the conditioning architecture you use. The expression in Equation 5 has been determined for the MAX1452 signal conditioning IC. (The MAX1452 signal conditioning IC from Maxim Integrated Products, a good example of the current generation of conditioning products, features a four-DAC architecture with a fine-pitched, temperature-driven look-up table. The data sheet for MAX1452 is available on the Maxim Web site: www.maxim-ic.com.) This expression can be used to calculate the errors produced by the use of a nonideal (generic) SpanTC value as a function of the temperature difference between the sensor and the conditioning IC.

T	= temperature difference between sensor and conditioning IC
K_R	= temperature coefficient of bridge resistance (/°C)
ß₀	= required SpanTC DAC value, range 0–1
ß₁	= actual (generic) SpanTC DAC value, range 0–1
R_b	= sensor bridge resistance
R_stc	= setting resistor value associated with the SpanTC DAC
N	= current multiplier gain (value internally set in conditioner IC)

Sensor offset behavior does not lend itself to the programming of generic DAC values because the offset could be either positive or negative. The preferred solution for offset compensation is to use the system provided by the fixed-value calibration method.

The resulting calibration technique, which uses a look-up table for Span DAC together with a generic value for SpanTC and a fixed-value calibration method for Offset and OTC, is illustrated in Figure 7.

Figure 7. These are the D/A converter settings and measurements required for a first-order, look-up table compensation, using the generic SpanTC value and fixed-value offset determination method. This method generally provides the best compromise between sensor accuracy performance and calibration simplicity for first-order compensation systems.

You complete the first-order calibration by loading the Span DAC look-up table with interpolated values and the Offset (table) and OTC (register) with figures calculated by using the fixed-value calibration offset technique. The resulting solution retains much of the calibration speed advantage of the system based on look-up tables but exhibits substantially reduced susceptibility to thermal transient errors.

Augmenting Higher-Order Look-Up-Table Schemes
In addition to the obvious advantages in calibration throughput, the look-up-table architecture can accommodate virtually any order of temperature response, most easily by providing additional temperature points in the look-up-table calibration previously described. If thermal transient errors are of concern, you can use an extension of the generic SpanTC value method or, for ultimate accuracy, a true combination of the fixed-value and look-up-table calibrations.

For example, you could perform a second-order temperature compensation by applying a first-order, fixed-value calibration at two temperatures and then use the normal look-up-table method for the third temperature. You would accommodate the higher-order temperature calibrations by simply using the look-up-table method to add further temperature points. The fixed-value calibration at the first two temperatures determines the values for the SpanTC and OTC DACs. Typical temperatures for a three-point calibration would be minimum, ambient, and maximum.

You’ll usually get the best results if the first two temperatures are the minimum and maximum. For convenience, however, using ambient as the first temperature is often desirable. Figure 8 illustrates this method for temperature measurements in the sequence of ambient temperature, minimum temperature, and maximum temperature.

Figure 8. These are the D/A converter settings and measurements required for a combination calibration that includes a first-order, fixed-value and a second-order, look-up-table compensation. The system would be used in applications in which higher accuracy is required. The measurements will accommodate a second-order error response in both sensor span and offset. The method can be expanded to accommodate higher order responses.

Following measurements at the first two test temperatures (T₁, T₂), calculate and program the required SpanTC and OTC DAC values. Similarly, find and record the values of Span DAC and Offset DAC. These values are valid at both T₁ and T₂. At the third (and any subsequent) temperature, you need to determine only the values for Span DAC and Offset DAC. Complete the calibration by applying a suitable polynomial fit to the Span and Offset data and subsequently loading the look-up tables.

A Versatile Sensor Signal Conditioning Solution
A four-DAC, analog path conditioning system based on look-up tables is undeniably versatile. With a variety of sensor calibration schemes supported, an optimum mix of accuracy and speed is readily available for most applications.

Four such schemes have been described in this article, each addressing particular calibration throughput and temperature performance requirements. You can easily find variations of these schemes and optimize them to meet particular requirements.