15.3 Variation, Deviation, Wind Triangle, and E6B

ACS PA.VI.A.K5 wants heading, speed, course, wind correction angle, time, and true airspeed. PA.I.D.K3a is the same math on the planning task. Compass turning and acceleration errors (ANDS / UNOS) were Chapter 12; this section is the preflight conversion from a true line on paper to a number you can fly.

True, magnetic, compass — TVMDC

The sectional is drawn to true north. The magnetic compass points at magnetic north, then the airplane’s own iron and radios tug it a few more degrees (deviation). The memory stack is TVMDC:

True → Variation → Magnetic → Deviation → Compass.

Going down the stack (true toward the compass you actually steer), east is least (subtract) and west is best (add). Going back up the stack, reverse the signs.

• Variation is the angle between true north and magnetic north. It is a location error. You read it from the isogonic line nearest the route (or interpolate between two lines). It is the same for every airplane over that spot.
• Deviation is the angle between magnetic north and what this compass reads. It is an airplane error. You read it from the compass correction card in that cockpit, usually tabulated every 30°. It changes with heading and with which radios are on. There is no isogonic line for deviation.

Worked conversion. True heading 100°. Nearest isogonic line 7°E. Compass card on a 090° heading says 2°W (use the nearest tabulated heading).

• Magnetic heading = 100 − 7 = 093° (east is least).
• Compass heading = 093 + 2 = 095° (west is best).

If the variation had been 7°W, magnetic heading would have been 100 + 7 = 107°, then 107 + 2 = 109° compass. Students who apply variation to the course and then forget to apply it again after wind, or who apply the compass card to true heading, invent a 10-degree miss that the checkpoint will reveal 15 minutes later.

The heading indicator is slaved to nothing in a typical trainer. Set it to the compass in straight, unaccelerated flight (the only time Chapter 12 says the compass is honest), then fly the heading indicator, and reset it about every 15 minutes.

Wind triangle — a full numeric example

If there is no wind, true heading equals true course and groundspeed equals TAS. There is almost always wind. PHAK’s wind triangle is three vectors: the airplane through the air (TAS along the heading), the air over the ground (wind), and the airplane over the ground (GS along the course).

Given: true course 090°, wind 180° / 20 knots (from the south at 20), TAS 120 knots.

The wind is 90° to the course — a pure right crosswind if you fly east. You crab into the wind, to the right (south).

Crosswind component = 20 × sin(90°) = 20 knots.

sin(WCA) = 20 / 120 = 1/6 ≈ 0.1667.

WCA = arcsin(0.1667) ≈ 9.6°, which an E6B reads as 10°.

True heading = 090 + 10 = 100°.

Because the wind is exactly abeam the course, none of it is a headwind along the course, but the crab reduces the along-track component of TAS:

GS = √(TAS² − wind²) = √(120² − 20²) = √(14,400 − 400) = √14,000 ≈ 118.3 knots, which the E6B calls 118 knots.

Check: 120 × cos(9.6°) ≈ 118.3. Same number. A student who subtracts the entire 20 knots as a headwind (GS 100) or who adds it as a tailwind (GS 140) has not drawn the triangle. A student who crabs left to 080° will be blown north of course.

Apply the TVMDC numbers from above and the thing you actually steer is about 095° compass on this example, not 090, and not 100.

On the wind face of a mechanical E6B you place the wind dot upwind of the grommet by 20 knots, rotate the true course under the true index, and slide the card until the TAS 120 arc sits under the grommet. The dot’s left/right offset is the WCA; the number under the grommet is GS. An electronic E6B is the same triangle. If you enter magnetic wind into a true-course problem, or true wind into a magnetic-course problem, the box will still print a confident wrong heading. Winds aloft are true; METARs are magnetic. Know which one you typed.

Time, speed, distance, and fuel flow

Once GS exists, the rest is the 60-to-1 arithmetic the E6B was built for:

• Time (hours) = distance (NM) ÷ GS (knots)
• Distance = GS × time
• GS = distance ÷ time

177 NM at 118 knots is 177 / 118 = 1.50 hours = 1 hour 30 minutes. At 8.0 gallons per hour that cruise burns 12.0 gallons before reserve. Section 15.4 adds 91.151 and climb fuel; do not stop at 12.0 and call the airplane legal.

TAS itself is not the airspeed indicator. Calibrated airspeed plus pressure altitude and temperature (the E6B true-airspeed window, or the POH cruise chart) produce TAS. Density altitude that raises TAS also raises true fuel flow on a normally aspirated engine for the same indicated power setting — PA.VI.A.K5d and PA.VI.A.K6 are why you open the POH instead of inventing “we always burn 8 gallons.”

PAR scenario: TC 090°, wind 180/20, TAS 120, variation 7°E, deviation 2°W. You should be able to write, without a personality, WCA 10° right, TH 100°, GS 118, MH 093°, CH 095°, and 177 NM in 1+30. If any one of those numbers is a guess, the checkpoint in 15.2 will be in the wrong place, and the 91.151 fuel number in 15.4 will be a guess too.