All right - all the experts have had a chance to have their say, now it's my turn...
I'll talk as if I'm talking to a student. You'll probably know all this Unsched Stops, but hopefully you'll see how
I try to explain this. Good diagrams are excellent - there are lots of places on the www to find them if you don't already have some.
There are two main sorts of errors: Acceleration Error & Turning Error. An explanation of either error requires an understanding of the E2 compass construction.
Construction: Consider the compass as an inverted bowl placed over a pivot, with the numbers around the outside and a bar magnet sitting on top so the bowl remains orientated to magnetic north. When an aircraft turns, the compass stays pointing to the pole and the aircraft turns around the compass.
Problem: The bowl will sit level on its pivot only where the earth's magnetic field is parallel with the earth's surface, which is the case only in a few regions near the equator. In fact, the earth's magnetic flux lines have a vertical (or "z") component called "dip" - this is the angle at which the flux lines enter the earth's surface. Where I am, in New Zealand, dip is approximately 60°, so our magnetic bowl shaped compass will tip over 60° towards the south in order to align the magnet with the local magnetic field.
Solution: A counterweight is placed at some distance from the pivot so that when the aircraft is level, gravity exerting a force on the weight will counter the magnetic influence of the vertical component of the earth's flux lines. Because dip always angles down towards the nearest pole and varies with latitude, the centre of gravity of the compass bowl is always on the equator side of the magnet, and a compass will have to be adjusted if it is to work at a different latitude from that for which it was originally calibrated (or swung).
Acceleration Error is caused by the inertia of the counterweight. Consider an aircraft pointing 270°M, and accelerating. The compass bowl will be accelerated by the pivot (the only point where it touches the aircraft), but a moment is set up because the pivot is some distance from the centre of gravity of the compass bowl. The counterweight will lag behind the pivot, causing the compass to indicate a more southerly value in the southern hemisphere (northerly in the N.H.) Deceleration gives the opposite effect.
To see for yourself, put a large pen (eg whiteboard marker) on a table, and give it a sudden push from the side but close to one end (so the centre of gravity of the pen is to one side of where you push). The pen will rotate slightly. This is the case when you accelerate an aircraft on heading E or W. Now give the pen a push on one end, so the centre of gravity of the pen is in line with the direction of your push - it simply moves, with no rotation. This is the case when an aircraft accelerates due N or S.
So we see that acceleration error is at a maximum when the counterweight is laterally furthest from the pivot - ie E or W - and is zero when the counterweight and pivot are in line with the direction of acceleration - ie N or S. The magnitude of this error depends upon magnitude of acceleration, aircraft heading, and the latitude for which the compass has been swung. It is not practically possible to correct for this error in flight, and the compass will stabilise quickly if the aircraft maintains constant speed. (US - I assume you are familiar with the mnemonics SAND or ANDS to determine whether acc or decel. will cause a swing to N or to S.)
Turning Error is caused by magnetic dip. During a balanced turn, when passing through magnetic north or south, the magnet will remain laterally level with respect to the aircraft (a design feature to make it easy to read in flight). However, remember that dip acts vertically downwards with respect to the earth: the Z component of the flux lines are no longer parallel with the aircraft's normal axis. For example, consider an aircraft at 90° angle of bank as if in an extreme turn (!) - Z component of flux lines will be parallel with aircraft's lateral axis. Because of this, and because the compass is always tending to align itself with the local flux lines, the compass rotates slightly so that it is more closely aligned with the local flux:
the compass counterweight does not prevent this because the aircraft's normal axis, or local "up", is not parallel with vertical component of flux, or earth "up". The concept of an offset compass c.g. is designed to minimise the effect of dip only when aircraft normal axis = earth vertical reference.
If the aircraft is in an unbalanced turn - an unbanked turn for example - then the aircraft's normal axis may very well be parallel with the vertical component of the earth's flux lines. There will be no turning error: this is why turning error is only predictable when the aircraft is in balance.
Turning errors are at a maximum on headings N and S, and zero on E and W. For example, take our compass bowl mentioned at the beginning of this explanation, and, with the bowl banked at 90° turn it through north. In the northern hemisphere, the north-seeking end of the magnet will dip down towards the surface of the earth; in the S.H., the south-seeking end will dip down. This will make the apparent heading of the compass change, as it rotates to dip downwards. (Have the student sit behind the bowl as it turns - this simulates the position of the pilot, and allows the student to see the numbers on the bowl moving as dip causes the bowl to rotate.)
Turning error is predictable, providing we keep the aircraft in balance and angle of bank is reasonably constant (which it is for most aircraft turning at Rate 1 or 25°AoB, whichever is the lesser). In New Zealand, turning error is 30° on N or S and 0° on E or W, but that will vary in other localities depending on the local magnitude of the vertical component of magnetic flux. One rule of thumb for turning error is that TE = 15° + half your latitude.
Now it remains only to teach the ONUS mnemonic for overturn on North, underturn on South - or whichever mnemonic you N.H. types use...
Hope this helps. If not, try
www.pilotsweb.com, or searching Google.com for "compass turning error".
Cheers,
O8