Recall from our discussion on pressure sensors that changes in ambient temperature affected both the zeros setting of the sensor and the span. This is a common occurrence for all types of sensors. As one example, let's illustrate how manufacturers perform temperature compensation on piezoresistive pressure sensors. We showed this slide when discussing a thermal errors on pressure sensors. The sensor output at the reference temperature of 25 degrees C is represented by the red line going through the zero point. In this case, we are ignoring the zero error and the span error at 25 degrees C. The yellow lines represent the change in the slope of the line due to the thermal span error. There are two yellow lines because the span error could be positive as shown at 50 degrees C or negative as shown at five degrees C. The blue lines represent the change in the zero offset of the line due to the thermal zero error. There are two blue lines because the error could be positive as shown at 50 degrees C or negative as shown at five degrees C. The thermal span and the thermal zero errors are additive, so the sensor will operate somewhere between the green lines. The right side of this slide shows a two digital to analog converters, DACs providing a partial correction for the thermal span and zero errors. This change in span over temperature results from dimensional and stiffness changes in the sensor and is always negative. This means that the sensor becomes less sensitive as the temperature increases. The change in zero over temperature results from the increase in resistance of the four sensors in the bridge as the temperature increases. As such, the resistive elements sensors are designed to make best use of the opposing signs of the thermal coefficients. The effects of temperature on the span and zero are shown in the graph on the left side of the slide. The slopes of the responses represent the change in zero and span as a function of temperature. With the ideal of bridge response, these will be the effective sum of the two. The DACs labeled span and offset modify the bridge voltage V_b. For span and zero compensation, values for the DACs and the bridge voltage are found at the four extremes of maximum and minimum ambient pressure and temperature plus some combination of other temperatures and pressures between the extremes. The temperatures and pressures are simulated by placing the pressure sensor in a temperature controlled chamber and then piping pressure through airlines to the sensor from a calibrated pressure source. Look-up tables for span DAC and offset DAC are loaded into the sensors microprocessor. Minor adjustments to the bridge output V_b are made per the temperature reading of the independent temperature sensor. We don't have an environmental chamber nor do we have access to the internal firmware and circuits running a pressure sensor. So to illustrate mathematically how temperature compensation works in a pressure sensor, we have the following example for you. This is fully documented in one of your references. The slope of a curve of bridge voltage versus pressure is the sensitivity of the pressure sensor. It depends not only on the zero output of the device, but on the sensitivity to actual pressure S_t. This sensitivity to pressure S_t in turn depends on the temperature coefficient of zero offset TCZ. If we can define this variable, we can partially compensate, but not fully eliminate our thermal zero shift. I say partially compensate because we're using a linear model and the full causes of a thermal zero shift include non-linear effects too. S_t also depends on the temperature coefficient of span TCS. In a similar manner, we can partially compensate for it, but not fully eliminate it. Our job will be to find compensation data at the four extremes of operating pressure and temperature so that we can define TCC and TCS. Let's assume our pressure sensor is rated from 0-100 PSI gauge with an operating range of minus 125 C to plus 125 C and in nominal calibration temperature of 25 degrees C. We're going to measure values of the bridge output at 25 degrees C for 0-100 PSI and at 125 degrees C for 0 and 100 PSI. This gives us a compensation methods solely for running the pressure sensor at temperatures above nominal. To fully characterize the sensor, you would also need to take data at minus 25 degrees C for 0-100 PSI. Your full characterization of your pressure sensor might include a continuous function for all temperatures between minus 25 degrees C and plus 125 degrees C or it might have two separate functions. One for temperatures below 25 degrees C and one for above it. This would depend on the data you got in your experiment. I made up data for bridge output involves assuming a nominal peak bridge output of 4.5 volts. It does not represent any real sensor only in approximation of data you might get for typical pressure sensors. You would place your pressure sensor in an environmental chamber such as the one shown on the right. You would first run the chamber at 25 degrees C with the sensor first on pressurize, in other words, zero-gauge pressure and then pressurize with 100 PSI. For these two data points, you need to measure the bridge output of your circuit very precisely so that you get the three decimal accuracy shown in the chart. Voltage measuring instruments with this capability will cost you $1,000 or more. You'll also need a very precise pressure calibrator for the 100 PSI measurement. Those instruments will cost you $5,000 or more. Then you increase the chamber temperature to 125 degrees C waiting an hour or two for the sensor inside to reach thermal equilibrium. After that you measure the bridge outputs at 0-100 gauge pressure again. With the raw data available, you first calculate the sensitivity of the sensor at the 25 degrees C ambient temperature and at the 125 degrees C maximum temperature. These numbers are shown as 0.0451 and 0.450 volts per PSI respectively. With the sensitivities calculated, you can calculate the temperature coefficient of span TCS. Sometimes TCS is quoted as a percentage of full-scale per degree C. In this example, we show TCS as a decimal number per degrees C. TCS will be a negative number for a pressure sensor. Next, you calculate the temperature coefficient of zero offset, TCZ. This number is also often quoted as a percentage of full-scale per degrees C. In this example, we show TCZ as a decimal number per degree C. TCZ will be a positive number for a pressure sensor with an absolute value higher than the absolute value of TCS. With TCS and TCZ calculated, you can take any nominal reading from your pressure sensor and adjusts its value for pressures measured at temperatures other than 25 degrees C. Use the formula shown on the right side. In this example, suppose your sensor bridge output measured 2.269 volts at 60 degrees C, you would use your newly calculated TCS and TCC factors and adjust your reading to 50.33 PSI then your sensor would output that value. Here's a screenshot of this spreadsheet I use to calculate all the data. Overall, you have a four-step process together, the raw bridge output voltages that you need for the calculations. Then it takes three more steps to calculate your coefficients TCS and TCC. The last step labeled step 8 involves taking a simple pressure data point and adjusting its value for elevated temperature. Remember that you're reading still won't be totally precise because we have used only a linear model to compensate the sensor for operating at a temperature other than 25 degrees C ambient. The second-order effects will still exist which will be quoted as a thermal zero error and thermal span error. Let's recap. You've learned the first two steps that manufacturers take to process their sensors, linearization, and temperature compensation. The third major step is calibration, which we will review next.