Wednesday, January 15, 2014

Linear sensors measurement primer part 2

Link to precedent linear sensor measurement primer article

Preliminary estimation of sensor uncertainty

A reliable, digital, compensated and non ratiometric sensor is available from Sensortechnics, with part number HCLA12X5B. The datasheet can be found online at this link.
This sensor measures differential pressure input and represents it with a hexadecimal value. Its nominal range is from -1250 Pa to 1250 Pa.

Let's estimate the out-of-the-box expected uncertainty of measurement. Of course, that is the uncertainty we attain if we use the nominal values reported into the datasheet or implicitly assume a linear model for our sensor. In this preliminary analysis, the statistical distribution of error will not be taken into account; instead, a worst case approach is used. On those assumptions, the concept of uncertainty is almost identical to the error.

The wiring scheme of the HCLA sensor and some basic firmware can be freely downloaded from here. We assume that all uncertainty contributions are statistically independent with respect to the others. Refer to page four of datasheet to retrieve all the sensor performance-related numerical data.

Datasheet reports an expected zero pressure offset between \((37A0)_{16}\) or \((14090)_{10}\) and \((3C28)_{16}\) or \((15400)_{10}\), with a central value of \((3999)_{16}\) or \((14745)_{10}\).

The nominal output span is (-12,5mBar,12,5mBar) or \(((0666)_{16},6CCC_{16})\) \((1638)_{10},(27852)_{10})\)

For the sake of simplicity, from now on we will use the decimal notation.  In the given output span, every decimal count corresponds to a \(\frac {25e-3} {26214} \)Bar change or 0,0953 Pa. Let's examine the offset related uncertainty terms.

The initial offset value can have a variation range of ±655 counts over a nominal full span variation of 26214 count, otherwise stated 2.5% full scale span. At the same time "Thermal effects (-25 to 85°C)" can cause a change to the offset of 1%FSS. To further reduce the performance, the term of "Offset warm-up shift" must be considered. The maximum rated value for this parameter is 33 counts or 0.13%.FSS.

In the worst case scenario, without warm up offset correction, the sensor will have an uncertainty of 3.83 %FSS. For our sensor this corresponds to 96 Pa. Using IAS definition this value leads to a reading of about 12.5 m/s or 45 km/h. So if the sensor is used with a Pitot probe, we can expect an erratic speed indication even when the Pitot is at rest on the table.
Now that the offset is evaluated, we must also add the uncertainty caused by span variations.

Page four of the datasheet reports that full scale span is in the interval \((63D6_{16},68F5_{16})\) or \(((25558)_{10},(26869)_{10})\) and the center value is \((6666)_{16}\) or \((26214)_{10}\).

Using equation ILS.4

$$m=\frac{2500Pa}{FSS count}$$

\(m\in(m_{max}=0,0978 ; m_{min}=0.0929)\)


In this case the uncertainty value is proportional to the magnitude of the measured \(q_{counts} \) pressure count.

$$u_m^o=( m_{max}- m_{nominal})q_{counts_{nominal}}$$

Or in a relative form

$$u_m^o=2,6=\frac{abs(m_{max}- m_{nominal})q_{counts_{nominal}}} {FSS}100$$

The uncertainty due to hysteresis and non-linearity \(u_{NL}\) is considered to have a maximum value of 0.25% FSS. This should be further expanded upon, but I choose not to make this discussion for now.

Until now we have a total uncertainty of \(u=6.68\%FSS=u_{offset}+ u_m^o +u_{NL}=3,83\%+2.6\%+0.25\%\)

If we consider the bare value of 6.68%FSS, the sensor seems to have poor performance, but it is not the case. First, we should note that this uncertainty estimation corresponds to a full swing of the output. Even if this is formally correct, it is unrealistic to use that value as a single pressure measurement uncertainty indication. Typically, we can get a much better measurement because we usually operate in the central region of the output range.

Is now quite clear, that the use of the sensor which is treated as linear, using nominal values, leads to an unacceptable level of uncertainty. So what we need to do in order to get a better measurement?

A viable procedure is to reduce the offset and sensitivity uncertainty term By removing those terms, we get \(u=0.25\%FSS\). To achieve a result of this grade, it is necessary to compare our sensor readings against a reference manometer. To compensate for temperature-related deviation of the bias over the operating range requires collecting a hefty amount of experimental data. Details on the procedure will be presented on the following sections.

Statistical behavior of the sensor

Usually datasheets do not report statistical information. To reach top notch measurement levels it is necessary to collect statistical data for the sensor. This may seem a bit cumbersome but it can be done with a reasonable amount of effort.
Information about statistical distribution allows us to have a better indication on the most probable measurement value. There are a few statistical distributions that can be employed to model sensor data. In this text, our approach is to use experimental data to construct the probability density function of our sensor.
In this link you can find 16870 pressure samples collected from a HCLA12X5B pressure sensor at a sample frequency of 10Hz. The two pressure ports have been left unconnected to have a zero reading. As room temperature is constant and the input is fixed to zero we expect to have a data set without any trend. By inspection of the plot of the sample data in figure ILS.5 it seems this is a correct assumption.
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