... My gut tells me the top one with less pincushion.
For the sake of completeness; you mean less barrel distortion instead of less
pincushion.
I don't even know why the word pin is used at all to describe the pincushion
effect. The so-called pincushion effect for TV has nothing to do with a
classic pincushion, i.e. a cushion where one puts some pins on. The pins have
no meaning in describing the effect. It basically should only be called
cushion distortion (like barrel distortion) which is also the term used in
non-English languages like German or Russian, i.e. Kisseneffekt or
Kissenverzerrung in German. The associations with a cushion basically stems
form the fact that the corners of squares, when projected by a TV (under
certain conditions), form elongated points on the screen.
The real classic (pin-)cushion effect in TV stems from trying to project an
orthogonal raster on a flat screen under magnetic deflection. If one has a
uniform magnetic field for deflecting the beam horizontally and vertically,
then one will come to the conclusion that the horizontal deflection depends on
the vertical one, and wise-versa, in a non-linear fashion. So basically, to
project a linear raster on a flat screen under uniform magnetic deflection
produces a non-linear dependency of the horizontal and vertical deflection
(which is quit interesting on its own). Hence, using linear deflection
currents to drive the coils won't work in producing a linear raster on the
screen. This discrepancy was a core issue in TV design, because to get a
uniform raster again on the screen under uniform magnetic deflection requires
a rather complex non-linear current which is very difficult to produce with
analog circuits. This issue is basically the reason for all the additional
permanent magnets placed around the neck of the tube.
Well, how does the current needs to look like? I "needs" to look like this;
Code:
i(x) = x*arctan(sqrt(x²+y²))/sqrt(x²+y²) // horizontal deflection current
i(y) = y*arctan(sqrt(x²+y²))/sqrt(x²+y²) // vertical deflection current
Some assumption were being made here, i.e. the distance from the screen to the
center of magnetic deflection equals 1 (normalized TV), sensibility of the CRT
equals 1 in x and y, coils produce a uniform magnetic field, and we do not
consider any flyback.
So for i(x) we see the non-linear dependency on y, similar for i

. Producing
such a current shape is very difficult. So what was done to counter this
probem? Roughly speaking, the ideal current was approximated and the remaining
distortions were countered by distorting the magnetic field by shaping the
coils and by using some additional magnets around the neck of the tube. Hence,
the better the shape of the current, the less corrections one needs to make.
But since no magnetic field is linear, the non-linearity of the magnetic field
produced by the coils on the far end of the deflection system and all the
additional magnets around the neck of the tube were factored in to produce a
less demanding non-linear current.
Guess we could have the best possible current shape, i.e. linear one;
Linear currents can be produced to a good degree. However, now we would need
to shape the magnetic field rather drastically (doesn't really work, will
produced lots of distortions which can't be countered solely by rigging the
magnetic field on certain points). So a compromise was found by balancing the
shape for the current with respect to the shape of the magnetic field in
balancing the non-linearity on both, i.e. for the current and the magnetic
field. That's why we see the interestingly shaped coils and all the magnets
around the neck of a common TV tube - they match (as good as possible, for the
better TVs) the slightly non-linear current driving the deflection coils.
There is one more interesting thing. The arctan behaves linear around zero,
hence, if the deflection isn't large, i.e. a small TV, we can use an almost
linear current and only need a minor if any correction of the magnetic field.
For example;
Code:
i(x) = x*arctan(sqrt(x²+y²))/sqrt(x²+y²)
arctan(sqrt(x²+y²)) ~= sqrt(x²+y²) for sqrt(x²+y²) < epsilon
[arctan(x) = x - x^3/3 + x^5/5+ O(x^6) (Taylor series)]
Hence,
Code:
i(x) = x*sqrt(x²+y²)/sqrt(x²+y²) = x.
So in general, smaller TVs produce less geometric distortions than larger
ones. As larger a TV becomes, as more difficult it becomes to counter the
geometric distortions making a TV more expensive when required to retain
an almost distortion free image.
Edit:
Let me express the stuff written above with modern terms. The modification of
the magnetic field due to the placement of permanent magnets around the neck
of the tube or by geometrically adjusting the coils is called static
convergence. Modifying the shape of the deflection currents (to counter the
distortions towards the edges of the screen) is called dynamic convergence. The
text of mine tells you why the separation exists and how they depend on each
other.