## Tuesday, January 13, 2015

### Aircraft Simulation Mathematical Models

Measurements and performances are tightly coupled. From the basic telemetry applications to the most complex mission planning a mathematical model is needed to reach an usable result.

Georacer from BasicAirData started to write a document about the topic, in particular focused on airplane and sensor models.

To get the latest updates and docs please got to the BasicAirData forum.

## Tuesday, January 6, 2015

### Automatic Pitot-Static Calibrator part 2

Figure 1. BasicAirData Air Data Computer Prototype and flanged 8 mm Pitot-Static tube.
Ready to be tested with the new Pitot-Static Calibrator.

A precedent article highlighted requirements for the static pressure port of the instrument. In this article we will focus on the total pressure port requirements.

Airspeed calibration regards the indicated airspeed. The difference between total pressure $$p_t$$ and static pressure $$p_s$$ is denoted as $$q=p_t-p_s$$.

Indicated Airspeed is calculated with the following formula
$$IAS=\sqrt{\frac{2q}{\rho_{base}}}$$;$$\rho_{base}=1.225kgm^{-3}$$.

To constrain the total pressure range means to limit the maximum and the minimum speed that can be tested with the Pitot-Static Calibrator.

Good sense and experience with models at the airfield is useful to individuate a range of air speeds that covers the majority of flying envelopes, however we will follow a different path.
We will consider the USA Academy of Model Aeronautics documentation; this association publish safety documentation on a regular basis.

at point 4 states “For RC fixed wing aircraft: The maximum velocity will be 200 mph”
at point 4/c “Model aircraft flown FPV are limited to a weight (including fuel, batteries, and
\onboard FPV equipment) of 15lbs. and a speed of 70mph.”

We can observe that maximum airspeeds are really different between different types of models. As our test equipment should be able to handle calibrations of probes for every kind or common aircraft we chose the greater airspeed value as our requirement maximum value.

Is a common wish to start to measure from 0 m/s IAS. That seems an obvious and easy to accomplish requirement, but indeed it is not. In fact with the aircraft at rest on the ground, many commercially available, RC grade, airspeed loggers will measure an airspeed really different from zero.
Let's pretend we want to log air speeds starting from 2 mph (3.2 km/h, 0.9 m/s); the probe should handle the following speeds and diffirentials.

Minimum speed 2 mph 3.2 km/h 0.9 m/s, q=0.49 Pa
Maximum speed for FPV 70 mph 112 km/h 31,3 m/s, q=615 Pa
Maximum speed for turbines 200 mph 322 km/h 89,4 m/s, q=4895 Pa
Table 1. Differential pressure at different air speeds.

Assume that our differential pressure sensor have a range of (0, 5000) Pa and 12 bit of effective resolution; 12 bit resolution means achieve $$1.22Pa=5000/2^{n=12}$$ resolution. You can note that resolution value is greater than the minimum needed differential pressure of 0.49Pa. We can conclude that this pressure sensor cannot handle such a measurement. A feasible, and conservative, operating span for such a sensor should be (5000/2^{n-1}3,5000), substituting (7.3 , 5000)Pa, (12.3 km/h, 360) km/h or (3.4 , 90)m/s.

Recalling that the Pitot calibration should be feasible within the entire range of altitudes, the span of pressure at the total pressure port is then (61640, 103751+5000)Pa (61640, 108751)Pa.
Now that we've individuated the airspeed span it is necessary to define the accuracy requirements for the differential pressure sensor. We're not interested in exact values, a worst case study will be sufficient.

The differential pressure sensor is digital and we consider, for the sake of simplicity, two uncertainty terms, one derived by the digital resolution and the other from every other cause. The bounded digital resolution introduce an uncertainty $$u_r=Span/2^n=5000/4096=1.22Pa$$. The other term of uncertainty is correlated to the measured value of differential pressure $$q_m$$, and is denoted as $$u_o=f(q_m)$$ in many cases it is available a figure of uncertainty that have the following form $$f(q_m)=k(q_m)q_m$$. Let's pretend that our differential pressure sensor, after an accurate calibration, have an uncertainty term $$u_o=1/100q_m$$.
The total pressure measurement uncertainty is $$u_q(q_m)=u_r+u_o=1.3Pa+0.01q_mPa$$

You can note that for low pressure values the bigger contribution to uncertainty comes from the resolution  uncertainty term, at the contrary at higher pressure values is the term $$u_o$$ to give the greater contribution.

To proceed further a relation between $$u_q$$ and the IAS uncertainty $$u_{IAS}$$ is needed to be know. Using the IAS equation and the independent uncertainty linear propagation formula
$$u_{IAS}={(\frac{2q}{rho_{base}})}^{-0.5}/rho_{base}u_q$$
Let's check uncertainty impact at minimum and maximum speed.
@7.3 Pa, Minimum airspeed $$u_{IAS}= 0.15u_q=0.23\cdot1.373=0.32m/s$$
@5000 Pa, Maximum airspeed $$u_{IAS}= u_q=0.009\cdot51.23=0.46m/s$$
Note in the two equations above that the first multiplication operator, sensitivity, is not dependent in any way from the pressure sensor. The average of $$m$$ measurements of airspeed will have an uncertainty $$u_{\overline{IAS}}=u_{IAS}/\sqrt{m}$$.
Given $$m=10$$ we get
@7.3 Pa, Minimum airspeed $$u_{\overline{IAS}}= 0.10m/s$$
@5000 Pa, Maximum airspeed $$u_{\overline{IAS}}= 0.15m/s$$
During the use of the Pitot-Static Calibrator we compare the reading of an internal differential pressure sensor with the probe under test. Let's pretend it's necessary to calibrate the probe with an uncertainty of 0.5 m/s. To achieve such a result the reference sensor should soundly exceed that performance; without enter in sensitive, and by far trivial, topics let's say that our reference sensor should be five times, of course is a conservative requirement, more accurate than the calibrated probes. In our case the reference sensor should have an accuracy of 0.1m/s.

The sensor studied until now doesn't comply with that requirement because the uncertainty is too high in the maximum speed case, 0.15m/s.
We use the reverse equation to get the required uncertainty in a IAS single measurement.
$$u_{IASr}=u_{\overline{IASr}}\sqrt{m}=0.10\cdot3.16=0.32m/s$$
And we proceed backwards to get the required uncertainty $$u_{qr}$$
$$u_qr =u_{IASr}/{(\frac{2q}{rho_{base}})}^{-0.5}rho_{base}=35.41Pa$$
So to achieve the required performance we need a main sensor that can warrant an accuracy of 35.41Pa at 5000Pa.

Next article will expose the requirements for the test of the vertical speed indicator.