OK. The formula we commonly know is (29.92-Altimeter)x1000+Elevation.
However, the E6B calculator I have does not fit this formula at all. For example, if the airport elevation is 175 ft and the barometric pressure is 30.21 in.hg, the general formula gives -115 ft. But on the calculator it comes out as -91ft.
I know that the calculator is obviously more accurate. And recently, I heard that the formula that is commonly known came out for convenience. So, how do I get the real formula?
The approximation is used because the actual formula is too complicated to commit to memory, and too complicated to do by hand in flight. There’s an exponential aspect to the atmospheric pressure lapse rate that makes it a hard problem to solve on paper on your knee board. But most of our flying is done in conditions where pressure altitude will tend to be in a range of -1,000 to +1,500 feet of our indicated altitude, and we’re using pressure altitude to look up performance data that is likely listed in 1,000 or 2,0000 foot increments in the tables. So a rule of thumb that is accurate to within 100 feet was always good enough. But when you’re doing other calculations and you want to start with a more precise pressure altitude, then:
Pressure Altitude = Indicated Altitude + $145442.2*\left(1- \left( \frac{Alt Setting}{29.92126} \right)^.190261\right)$
AltSetting is the altimeter setting in the Kollsman in inches hg. Indicated Altitude and resulting Pressure Altitude are feet.
It’s non-linear:
In your example, 30.21 works out to be -265.99 feet, plus your elevation of 175 = -90.99 feet, almost exactly what you got.
If I wanted something more accurate than the old rule of thumb, but still as easy, I think I’d use (29.92-altimeter) x 940 or something like that. That’s accurate to within 40 feet or better across almost the entire normal range of altimeter settings.
