OLYMPUS DIGITAL CAMERA In the national airspace system, altitudes are barometric, despite the fact that we can now determine altitude within about five feet with a WAAS GPS. Those GPS altitudes are used for terrain clearance using a TAWS system (terrain awareness and warning system). But otherwise, we rely on altimetry for flight altitudes and instrument approaches. From the static pressure in our aircraft we deduce our altitude.
So how does an altimeter determine your altitude, and how accurate (or inaccurate) is baro-altitude? And why are there errors, and what is their source? To explore these questions and to explain how an altimeter works, we need to understand how pressure in the atmosphere changes as we go up in altitude. We’ll find that the pressure drops roughly exponentially with height above MSL.
In an altimeter, static pressure is introduced into the chamber and a bellows or wafers moves up and down as the pressure goes down and up. (See a mechanical schematic in my website article.) The thin wafer discs may either be evacuated or pressurized to 29.92 inches of mercury. In either case the stack expands when the chamber pressure drops.
Since a bellows behaves like a spring, you expect the displacement up or down should be proportional to the force on it. The force is the pressure differential (inside and outside the wafers) times the wafer area, so you expect linear changes in pressure will give a linear change in displacement. Clearly, looking at this mechanical arrangement, linear changes in the length of the wafers will produce a linear increase in angle of the shaft driving the gears, and then a linear rotation of the needle, through the gearing arrangement. Unfortunately, the atmosphere does not behave this way, so an altimeter is much more complex than this simple arrangement, and the non-linearity of pressure vs altitude must be incorporated into the design.
We’ve been taught that a pressure drop of 0.1 inches corresponds to an altitude change of about 100 feet. This is one-tenth of a full rotation of the needle, or 36 degrees. So the next 0.1 inch drop would change the altitude another 100 feet, rotating the needle 36 degrees more. All would be well if these linear changes in altitude with pressure continued to all altitudes, but unfortunately the atmosphere isn’t linear, but is exponential. We’ll show later that this 0.1 inch drop gives increasing heights as you go up in altitude.
One final point on the altimeter is that you can mimic changing the pressure in the chamber by rotating the adjustment knob and setting the local correction in the Kollsman Window. When calibrated, the rotation angle matches the pressure change from 29.92 inches to the altimeter setting (baro-correction). The range of that correction is 28 inches to 31 inches, but if you get altitude from an air data computer there is no limit. The PFD on which these data are displayed can make a baro correction of an arbitrary amount.
Pressure changes with altitude
So what is the correct relationship between pressure and altitude in the atmosphere?
The so-called barometric law states that when you change altitude (height h) by a small amount ∆h, there will be a small pressure drop ∆p given by H ∆p = – p ∆h. Here, p is the pressure and H is a height scale factor. The factor H is proportional to temperature, greatly complicating the solution, but if H were constant the solution is a simple exponential in the pressure ratio R (pressure at sea level divided by pressure at the aircraft). Then you could easily find your altitude h if you measure the pressure at your aircraft and know the pressure at sea level; that solution is h = LH0, where L is the natural logarithm of R and H0 is the value of H at sea level, 27,500’.
But H is not a constant at all altitudes because it is proportional to temperature, which typically drops with altitude. So to compute the altitude from solutions to the barometric law you have to assign a temperature for each altitude above sea level up to your aircraft. Here, the International Standard Atmosphere (ISA) model of the Troposphere comes into play.
That model states that the pressure at sea level is 29.92 inches, the temperature there is 15 degrees C (288 degrees Kelvin) and decreases 1.98 degrees C per thousand feet of altitude (the lapse rate). So the equation that relates temperature and height is T = 288° – 1.98° (h/1000) = 288°(1 – 2ah). This defines “a,” and it gives 1/a = 290,909. An approximate solution is now h ≈ LH0 (1 – a LH0). An exact solution for pressure, and also density altitude, is given in my website article.
Clearly, the ISA model is only an approximation to the actual conditions when you’re flying, but altimeters are designed assuming that model. Herein lies the largest error in your baro-altitude. The temperature profile below you is unknown and on a given day is surely different from that of the ISA model. Consequently, errors grow with altitude.
In Figure 1 these two solutions, exponential pressure and the ISA model pressure, are plotted vs altitude in thousands of feet. You can see that the temperature corrections have a modest but important effect. For example, using the exponential law the pressure is half of 29.92 inches at 19,061’. The above approximate solution using the ISA model gives 17,812’, while the exact solution is 17,966’.
Figure 1. Pressure vs altitude, with and without temperature effects. The ISA model gives results close to those from an exponential model (constant air temperature).
So why have I pressed all this mathematics on you? Because to understand how an altimeter works you need to know how pressure relates to altitude in the atmosphere and in the altimeter chamber, which tries to mimic it. We solved the barometric equation for pressure vs height, but we can also solve it for the change in height ∆h vs change in pressure -∆p. We have ∆h = (-∆p)(H/p) = (-∆p)(H0/p)(1 -2ah). This equation tells you how the altitude changes when you increment pressure, either in the atmosphere or altimeter chamber.
Let’s plot that height increment at different altitudes with a pressure decrease of 0.1 inch at each height. Those increments are shown in Figure 2, in which the change is 92’ at sea level. They are clearly not constant with altitude. What happened to the old rule of thumb that this increase is 100 feet? It’s only roughly true at low altitudes, say between 0 and 5,000 feet. The fact that the height increase gets bigger as altitude increases is not due to the temperature profile but is basically due to the increase with dropping pressure. It’s the product of p∆h (approximately Ho∆p) in the above equation that’s roughly constant (except for the small temperature correction). Decrease p and ∆h increases. So at 18,000 feet where the pressure is halved, the increment ∆h is 155 feet, not quite doubled because of the temperature correction. (Figure 2.)
Figure 2. Altitude changes ∆h for a pressure change ∆p, as a function of altitude
If we plot that height change versus the drop in pressure from 29.92 we get the profile in Figure 3. Our 92-foot change at sea level rises to 225 feet when the pressure has dropped 20 inches. This curve determines how the altimeter needle should change (∆h) when there are pressure increments (-∆p) in the chamber, plotted for different chamber pressures. It’s a non-linear curve, so the linkages between wafers and the needle must reflect this non-linear motion. A basic design directly linking a bellows, wafers, or diaphragm would clearly be missing such elements, since the wafers alone would give a linear response. (Figure 3.)
Our purpose is not to explain design solutions, but first to elucidate the challenges for a designer to come up with the appropriate non-linear responses to changes in the bellows pressure to match the atmospheric model. Second, it is to explain the atmospheric behavior based on the ISA model.
Figure 3. Change in height ∆h when the pressure drops by ∆p, vs the pressure drop from 29.92 inches.
Air Data Computers
There is an alternative to finding altitude from these mechanical devices that use data from solid-state pressure sensors to compute altitude, and other parameters. Computations are done by an ADC (air data computer), which uses static and dynamic pressures (from the static and pitot ports) and temperature (from an OAT probe).
An ADC is often part of a Primary Flight Display (PFD) such as an Aspen, the Garmin G5, G500, or G950 integrated flight deck, and others. It can compute pressure altitude, vertical speed, calibrated and true airspeed, and density altitude. The altitude calculation comes from the static pressure and the ISA model equations described here, solved in exact form as in my website article.
The ADC is often combined with an AHRS system (Attitude and Heading Reference System) in a PFD and used by both PFD and GPS. This allows the PFD to show altitude and speed tapes, including VSI, and compass headings and to enter the baro-correction. You can also set a HDG bug there. On a GPS you can track heading legs and calculate wind speed and direction. With the altitude input a GPS can automatically sequence the three flight legs (VA, CA, HA) that end at an altitude.
Summary
The pressure in the atmosphere would drop exponentially with altitude if the temperature were constant everywhere, but temperature changes as incorporated in the ISA model make a modest change to the exponential. The ISA model does not compensate for actual temperature profiles, which are not known, and is the reason altimeter readings get less and less accurate as you get higher above a ground reporting station. But if you set the altimeter to the value at a nearby ground station, the error goes to zero as you get to that altitude. This is comforting when you shoot an approach to minimums. So, trust your altimeter operating at low altitude with local baro-corrections applied. To avoid rocks in the sky, trust your GPS altitude, which can differ from baro-altitude by as much as 1000 feet, and has very small errors.
The roughly exponential behavior of pressure with altitude also prescribes how the pressure in the chamber of the altimeter behaves with altitude. It complicates design because a simple linear mechanism to translate wafer motion to uniform altitude changes will not work. Instead, non-linear mechanical designs are required.
If you no longer have an old fashioned altimeter in your panel, and instead have a newer glass instrument, it will show your altitude based on a pressure sensor feeding an Air Data Computer. The computer can translate the pressure reading to altitude using the barometric equation, including the correction for temperature change with altitude in the ISA model. But that still does not eliminate altitude errors because the ISA model cannot represent the atmosphere on a given day.
Dr. Thomassen has a PhD from Stanford and had a career in teaching (MIT, Stanford, UC Berkeley) and research in fusion energy (National Labs at Los Alamos and Livermore). He has been flying for 62 years, has the Wright Brothers Master Pilot Award, and is a current CFII. See his website (www.avionicswest.com) for all his manuals, and his “Aviating and Aviation in the Modern World”, describing the increased GPS capabilities from a series of new technologies that have been introduced in recent years; AHRS, Air Data Computers, Magnetometers, Digital Autopilots, and more.
