Magnetic Dip Compass Errors Explained

Disclaimer: I am only a paragliding hobbyist, not an aeronautical engineer nor a private or commercial pilot. No guarantee that the following is accurate.

I stumbled upon magnetic compass errors in my HAGAR exam, mandatory to obtain a P3 rating and above from HPAC. The nerd in me really wanted to understand the source of these errors, especially after reading conflicting explanations from different sources and having long arguments with my mechanical engineer friends, so I had to dig deeper. This was purely out of curiosity since magnetic compasses have long been reduced to the role of calibrating gyro compasses, which do not suffer from the errors below but from divergence and need to be reset periodically from straight and level flight. And I believe the latter are themselves only used as a fallback for GPS / electronic compasses.

Background

Magnetic dip

First you have to understand that as you leave the equator towards the magnetic poles, the force from Earth's magnetic field gains a downward vertical component. See these drawings.

Compass construction

See design diagrams to help visualize the following description.

The magnetic dip is already quite strong at mid-latitudes. In order for the compass to not tip down farther than its operating range (usually around 18° by design) it uses an upside-down bowl resting on a pin, which brings its center of mass below the pivot (the pin). It may also be the only construction feasible. Note that this only reduces the magnetic dip effect in magnitude, symmetrically in all directions. Attached to the bowl are two magnetic bars that are influenced by the earth’s magnetic field and will rotate and tip the bowl. One source of confusion comes from the inscriptions on the bowl being rotated 180° such that the S end reads N, N end reads S, E → W and W → E so that on a given heading, the side of the compass (in front of you) that faces you reads the direction you’re going.

Note that except in the polar regions, the horizontal component of the magnetic force vector is stronger than its vertical component. This is why both errors below slowly become nil as the tipping of the compass bowl becomes longitudinal. In the absence of a lateral pull from the magnetic dip (relative to the low side of the bowl) the compass will point towards the magnetic north, without error.

Furthermore, I believe some compasses have their center of mass shifted back towards the S side to counter the N magnetic dip, either by moving the pivot point towards the N end, or adding weights on the S end. Note that this only works in the northern hemisphere and would require recalibration (or a different southern compass) if one were to cross the equator into the other hemisphere. The following explanations disregard this type of adjustment.

Latitudinal effect on magnitude

Both errors are caused by the vertical component of the magnetic field (aka the magnetic dip), nonexistent at the equator and progressively stronger closer to the poles. They are therefore proportional to latitude.

Reflection symmetry around the equator

On top of their own respective reflection symmetry (e.g. acceleration vs deceleration), both errors are further flipped (opposite) between the northern and southern hemispheres. This means that under the same parameters described below, an error to the N in the N hemisphere will be an error to the S in the S hemisphere and vice-versa.

Acceleration / deceleration error

Note the following 4 properties are formulated to be true independently of hemisphere.

Effect: compass error towards the nearest pole

Reflection: acceleration vs deceleration

Proportional to: heading angle from due N-S (0%) to due E-W (100%)

Irrespective of: latitudinal heading (E or W)

Mnemonic (N hemisphere): ANDS (Accelerate North, Decelerate South)

Mnemonic (S hemisphere): ASDN (Accelerate South, Decelerate North)

In straight and level flight on an easterly heading, the compass N end will point to the left side of the aircraft and will also dip down due to magnetic dip, matched by the weight of the other side (S end) that it is trying to lift. 

When accelerating, the center of mass being lower than the pivot point means that the rear (W end) of the bowl tips up. In that case, the N end of the compass still wants to tip down and it can now do so by rotating the compass, without movement/lift, only rotation. Therefore the N end will rotate towards the lower side of the bowl (front of aircraft), indicating a turn towards the north (no matter if you’re accelerating E or W).

When decelerating, the compass bowl’s forward section tips up, and the N end of the compass wants to turn aft, therefore indicating a turn to the S (despite no actual change of heading happening).

Accelerating or decelerating in a southerly or northerly direction, the magnetic dip and acceleration-induced tipping low points align so no lateral pull is felt by the bowl and the horizontal pull applies, with no error.

In the southern hemisphere, the exact same effect happens but with the S end being moved towards the front of the aircraft, therefore the errors are mirrored: Accelerate South, Decelerate North.

Turning error

Note the following 4 properties are formulated to be true independently of hemisphere.

Effect: compass leading or lagging the actual change in heading when heading (not turning) towards or away from the pole (respectively)

Reflection: polar vs equatorial heading (roughly S vs roughly N)

Proportional to: heading angle from due E-W (0%) to due N-S (100%)

Irrespective of: turn direction (left, right, towards or away from poles)

Mnemonic (N hemisphere): UNOS (the pilot should Undershoot N, Overshoot S) or S leads, N lags

Mnemonic (S hemisphere): USON (Undershoot S, Overshoot N)

A coordinated turn is one in which the bank angle of the aircraft shifts the angle of the pull of gravity on the aircraft enough that it matches the centrifugal force induced by the turn, so that the resulting force applied onto the aircraft is towards the bottom of the aircraft, i.e. a ball on the floor of the aircraft would not roll to either side but remain in place, and a passenger would not feel pulled in or out of the turn, only feel heavier.

The compass bowl being on a pivot means that it tips and banks separately from the aircraft and has its own bank angle dependent on the centrifugal force (and where the center of mass is relative to the pivot). In a roughly northerly heading, if the pilot initiates a banked turn in any direction, the compass bowl will also bank i.e. dip down on the inside of the turn. The vertical component of the magnetic field will then pull the N end of the compass towards the low side (inside) of the turn, turning the compass (inducing error).

In a roughly northerly heading, the compass N end normally wants to turn towards the outside of the turn (since you’re turning away from the starting heading and indication) but the magnetic dip pulls it towards the low side, thus the compass undershoots (lags) the turn.

Conversely, on a roughly southerly heading, both the change in direction and the magnetic dip want to pull the N end towards the low side, thus the compass overshoots (leads) the turn.

As a turn approaches a directly E or W heading, either the low side and N side align (no difference thus no error) or oppose (the magnetic dip applies no lateral force and the magnetic north wins the battle).

Again, in the southern hemisphere the exact same effect happens when going towards and away from the pole, except this time you are going S and N respectively, so the mnemonic is flipped.

Sources

• Wikipedia
• AvStop.com
  

The other web sources I found on the matter were either incomplete or incorrect.