Byrdsfan1948 My confusion is over exactly what you may mean by A.sin(3T/P).
All square integrable periodic function can be represented as a Fourier Series.
https://en.wikipedia.org/wiki/Fourier_series
I.e., if f(t) is periodic, you can write,
f(t) = A1.sin(t/P+ø1) + A2.sin(2t/P+ø2) + A3.sin(3t/P+ø3) + A4.sin(4t/P+ø4) + ...
The Ai are relative amplitudes of each harmonic component, and øi are phase terms.
the derivative f'(t) = d/dt( f(t) ) = d/dt( A1.sin(t/P+ø1) ) + d/dt( A2.sin(2t/P+ø1) ) + ... through the distributive property of the derivative (what engineers would call "linearity"). I.e., d/dt( a+b ) = d/dt(a) + d/dt( b).
Now, the derivative of each component of the sum series, d/dt( a.sin( nt/P + ø ) by chain rule is simply a.cos( nt/P + ø ) . d/dt(nt/P + ø ). I.e.,the derivative of a composite function like f(g(x)) is f'(g(x)). g'(x).
ø is constant, so d/dt( ø) = 0, and d/dt( nt/P ) = n/P, since n and P are constants.
So, d/dt( a.sin( nt/P + ø ) ) = a.(n/P ).cos( nt/P + ø)
Notice the "n" that now magnifies the amplitude of the cosine? For third harmonic, n = 3. So the effect of a third harmonic of a periodic function is three times greater than the fundamental component of the series. Intuitively, the third harmonic is 3 times more "compressed" in time, so the rate of change per unit time is 3 times greater than the fundamental.
So, for our periodic error curve, you want the relative amplitudes of all the higher harmonic terms to be as small as possible -- i.e., the ideal periodic error curve is a pure sinusoid.
See the n/P term? That means also that if P is large, the derivative is small (easy to guide). Which is part of the "rule of thumb" about auto-guiding -- i.e., long periods are easier to guide. The second rule of thumb is "smooth", which means a sinudoid with no harmonics -- i.e., no bumps.
The above analysis just shows the layman's "rule of thumb." But it also shows why there is a problem, and how one might try to counteract the problem (the engineer's task). P and n/P tells the story.
With visual work, you don't care about the derivatives, so as long as the amplitude of f(t) is small, you are good to go (Jupiter will move around by less than its own diameter, and your eye/brain can easily accommodate it.)
However, if you want to auto-guide (and you will need to autoguide unless you have a $8K premium mount, or use camera lenses), you want each guide exposure to be able to freeze the centroid of a guide star. You don't want the star centroid to move by more than a tenth or two of an arc second while you are taking that guide plate. Thus you want the rate of movement of the mount to be small. The amplitude of the periodic error itself is not important, as long as the derivative of the periodic error curve is small.
A mount can have a small periodic error, but if the third and fifth harmonic terms are large (i.e., lots of bumps and inflections in the curve) the mount is not going to guide well.
When I bought my RST-135, I started looking into why strain wave mounts are hard to guide, and came to the above conclusion. The solution is right there -- you need to use very short exposure times. Back then, PHD2 had multi-star centroid guiding, so they could use short exposures. ASIAIR did not, and short exposures had too much problem with atmosheric turbulence. And even after ASIAIR implemented multi-star centroiding, the device is so slow, although it allows short exposures, the frames per second never got better than 1 FPS (that is a different problem with auto-guiding, and has to do with stability of feedback loops). It was only after ASIAIR allowed Bin2 guiding that the FPS finally became 2 FPS for 0.5 sec exposures. I have not had any guiding problems since with my RST-135 (especially being able to limit the bounds of the max RA and declination pulses - I use 150 ms with my setup).
Chen
