These notes discuss the various pitot static system errors and describe techniques to reduce most of them, and determine the residual error. An Excel spreadsheet, (ssec.zip for Excel 7, ssec4.zip for Excel 4) is available to do the required calculations. The ZIP files include these notes.
The spreadsheet uses a method which is accurate at any subsonic speed, up to 36,089 ft altitude (see Technical Notes for more detail).
The spreadsheet makes use of a method to determine TAS from three legs on any track method developed by Doug Gray (Measuring True Airspeed using GPS).
The pitot tube will give accurate readings as long as it is:
The pitot system is subject to leaks and it should be leak checked.
The airspeed indicator (ASI) is subject to instrument error and it should be calibrated.
In general, a leak-free pitot system with a calibrated airspeed indicator will be error free, except at high angles of attack (near the stall), where there may be a large angle between the local flow and the pitot tube. Any other errors are most likely due to the static system.
Static system - The static system is much more prone to errors than the pitot system. The purpose of the static system is to measure the free stream ambient pressure. This is a bit of a trick because the aircraft causes the airflow to accelerate and decelerate as it passes by. Bernoulii's Law tells us that the pressure will change as the speed of the air changes, so it is very difficult to find a place where the local pressure is the same as the free stream ambient pressure.
The static system is subject to Position Error, which is caused by the fact that the pressure at the static port differs from the free stream ambient pressure.
The static system is also subject to leaks and instrument error.
TAS Determination - It is recommended to use data from a GPS to determine TAS. Formula are available to take data from three runs at headings 90 degrees apart, or three runs at tracks 90 degrees apart, or three runs at any track. The spreadsheet provided with these notes uses the three runs at any track method developed by Doug Gray (Measuring True Airspeed using GPS). This method simplifies the flying, and does not rely on an accurate compass swing. Doug Gray's Excel spreadsheet is also available.
Fly three legs at a constant heading, altitude and airspeed. The run directions should be about 90 to 120 degrees apart. All runs must be at the same airspeed and altitude. For each run, record:
The more accurately you can fly the test points the more accurate a result you will obtain. You can fly more accurately if you don't have to record the data your self. It is useful to have a second person in the aircraft to record the data and watch for other traffic.
The spreadsheet will calculate the TAS for each set of three legs. Compare the calculated winds for all runs that were done at the same altitude. They should be very similar. If any run shows a wind that differs from the others, something is wrong with the data and that run should be discarded.
If you use another method to calculate TAS, you can directly input the TAS into column AH. You will have to unprotect the worksheet (Tools menu in Excel).
Enter the recovery factor for your OAT gauge. Use a value of 0.8 for a general aviation type temperature probe if the manufacturer does not provide another value (see discussion in the Technical Notes section).
Enter the units for your airspeed indicator (kt, mph or km/h), GPS groundspeed (kt, mph or km/h) and OAT gauge (C or F) for degrees Celsius or Fahrenheit.
Enter a reference altitude that you want to correct your data to (this could be sea level, or maybe the altitude of your home airport).
Enter the GPS data from the three tracks, indicated airspeed, altitude and OAT. If you know the instrument errors for the airspeed indicator and altimeter, enter them. Airspeed indicator instrument error goes in column Z. (corrected indicated airspeed = indicated airspeed + instrument error. i.e. the instrument error is positive if the corrected value is greater than the indicated value). Similarly, the correction (if known) for altimeter instrument error goes in column AD (corrected altimeter reading = indicated altitude + instrument error. i.e. the instrument error is positive if the actual pressure altitude is greater than the indicated value).
The Calibrated Airspeed for each test point is calculated and displayed in column AS. The CAS error, at the test condition, is in column AT. Note that this value does not include the instrument error, it is only the error in CAS caused by the static source instrument error. For example, if you had an IAS of 150 kt, with a +3 kt instrument error, your corrected IAS would be 153. If the CAS is calculated as 155, you have a +2 kt error due to position error.
If you did an airspeed indicator calibration to determine the instrument error, and put the corrections in the spreadsheet, we can assume that all remaining errors are due to position error. The amount of error in indicated altitude, caused by this position error, is provided in column AU, and the resulting error in the altitude indication is provided in column AU.
Important Note - the calculated altitude errors are only valid if you have entered airspeed indicator instrument error. Otherwise there is no way to know how much of the error in IAS is due to static source position error, and how much is due to instrument error.
The position error may be affected by flap and landing gear position and gross weight. You may want to gather data at light and heavy weights. The position error is certainly affected by airspeed, so you need to gather data over the full airspeed range of your aircraft. This technique will only work for speeds that can be attained in level flight, but you may be able to plot airspeed error versus airspeed and extrapolate to VNE.
Method of Calculation - The TAS is determined by flying three legs at the same airspeed. The GPS ground speed and tracks are used to calculate the TAS using formulae developed by Doug Gray (Measuring True Airspeed using GPS - documentation, spreadsheet & both docs & spreadsheet in a ZIP file).
Once the TAS is known, the rest of the method is as described in the US Air Force Test Pilot School (USAF TPS) course notes, except where specifically noted.
The indicated air temperature is corrected for the ram temperature rise by applying the probe recovery factor, to calculate the outside air temperature.
The speed of sound is calculated given the outside air temperature (speed of sound in knots = 38.968 * SQRT(OAT + 273.15), with the OAT in degrees C). The Mach number is calculated given the TAS and speed of sound.
Then, a rather complex bunch of mathemagic is performed to compare the IAS (corrected for instrument error) against what the CAS should be for that mach number and altitude.
Assuming that all the remaining error in airspeed is due to the static source position error (see above), the amount of static source position error can then be calculated. The formulae from the USAF TPS notes calculate the position error as a ratio of pressure error to the measured differential pressure (pitot pressure - static pressure). This parametre (dPp/qcic) is in one of the hidden columns (column AQ, the parameter symbols are in hidden row 18). The static pressure error is converted to an altitude error at the test altitude, and at the desired reference altitude.
The USAF TPS notes also convert dPp/qcic to an error in CAS, and this is presented as CAS error at the reference altitude. The result at the test altitude had a bit more error than I liked. The result was that for a given test condition (TAS, altitude and temperature), the calculated CAS (calculated by adding the USAF TPS notes calculated CAS error to the IAS corrected for instrument error) varied if the IAS varied. The error was small, but was not completely satisfying. So, I jumped out on a limb, and calculated the CAS at the test condition given the Mach and pressure altitude, and then calculated the CAS error as the difference between the CAS and the IAS corrected for instrument error. [CAS = speed of sound at sea level * SQRT(5*(((((1-0.00000687558*Hc)^5.2559)*((1+0.2*M^2)^3.5-1)+1)^(2/7))-1)), with Hc = pressure altitude and M = Mach].
There are other documented methods of calculation, but many of the older ones use simplified, approximate formulae, and are only accurate at low speeds (up to about 130 kt TAS in some cases). This was necessary in the days when slide rules were used to do this sort of calculation, but today, more accurate methods of calculation can be used. You will still be limited to the accuracy of your instruments, and the accuracy of your data points, but there is no need to introduce additional errors in the calculation methods given the power of modern computers.
Recovery Factor - We need to measure the actual static temperature of the free stream air (Outside Air Temperature - OAT). However, the air is moving relative to the aircraft, and its kinetic energy will get converted to heat as the air stops against the temperature probe (I am describing this as if the aircraft was stationary and the air moving - the same things happen if the air is stationary and the aircraft is moving). So, our temperature probe does not measure the OAT, but actually sees some temperature warmer than that. There are many different names for this increased temperature measured by the probe, depending on how technical we want to get. I will call it the Indicated Air Temperature (IAT). In an ideal world, the IAT would be equal to the theoretical value.
|1.0||31 Dec 98||Original release. The method of calculation used the formulae relating pitot and static pressures to CAS, pressure altitude and Mach. This method was developed by myself because all published methods that I could find made assumptions that limited the accuracy above 130 kt TAS.|
|1.0.1||01 Jan 99||Editorial change to clarify section on ground testing following flight test. Corrected bad link to ZIP files.|
|1.0.2||23 Jan 99||Added Technical Notes section, with new material, plus some material moved from other sections. The Excel spreadsheet, version 1.0.2, now allows a selection of units for GPS ground speed, and also allows km/h as units for IAS and GPS. The method of calculation was completely reworked to follow that described in the USAF TPS course notes (from the Society of Flight Test Engineers Symposium, September 1998). The results are similar to the original method of calculation, but the pedigree is better. The new method also allows correction of the results to other altitudes.|
|1.0.3||14 Apr 00||Changed links to Doug Gray's spreadsheet and documentation to reflect the new location.|
|1.0.4||31 Aug 00||Error correction - one of the spreadsheets included with ver 1.0.2 was actually still the 1.0.1 version. This is now fixed.|
|1.0.5||17 Dec 00||Fixed a major spelling error. No technical changes were made.|
|1.0.6||02 Nov 01||Fixed bad link to Doug Gray's information on Measuring True Airspeed Using GPS. No technical changes were made.|
|1.0.7||19 Nov 03||Fixed bad links to the other info on my web site. No technical changes were made.|
|1.0.8||08 Feb 04||Corrected a bad link in the spreadsheet, and raised it to v1.0.5. Added a link to a new spreadsheet for ASI calibration. No technical changes were made.|
|1.0.9||08 Mar 05||Changed URLs to point to my new domain name. No technical changes were made.|
|1.1||01 Aug 05||Revised reference to asi2.zip spreadsheet. No technical changes were made.|