For example, the UNI-DIN alignment that is available with the Smartractor/Uni-Protractor is not a "new geometry", as is often stated, but simply the Lofgren A formula with different recorded radii of around 58 and 134mm, giving nulls at around 63.5 &112.5mm. What Lofgren did was to find ways to minimise tracking distortion by "loading" the basic horizontal tracking error equations to allow for minimum distortion. In three cases this is relatively easy (which is why there are three common alignments). So, (for given inner and outer recorded radii, eg IEC or DIN), solving the Lofgren equations for three equal distortion maxima gives Lofgren A/Baerwald; minimising the average distortion at the expense of higher inner and outer maxima gives Lofgen B, and solving for a null (ie distortion at zero) at the inner recorded limit, gives Stevenson. With computers it is now easy to customise these distortion curves by varying the arm parameters to see how they affect where the distortions are maximised and minimised.
Registering and logging in to the Vinyl Engine to access their tools make this much easier to do than it was, (with calculators 30 years ago when I was doing it, or with paper and pencil and logarithms, when Lofgren was doing it). You can enter any arm and record parameters and get a read out of distortion for the standard alignments or for any arm you want. You can see that for the IEC standard radii, UNI-DIN has lower distortion towards the end of side and more at the beginning, as does VPI's own alignment, so it is more like an extreme DIN alignment. Make or obtain a protractor with nulls at around 64 and 112 and try it.
Any budding designer might wish to know more about the way those parameters are linked, as it gives an insight into the way some alignment protractors work.
So, working back from the nulls (for a null based approach rather than from arm parameters), given the two nulls (N1 & N2), that is, the radii at which tracking error (and distortion) is zero:
Linear offset (Lo) equals the mean of the sum of the nulls
Lo = (N1+N2)/2
The Dennesen and its clones, the Pro-ject and the expensive Soundtractor/Uni-Protractor, use the fact that Lo -N1 (the distance the slide is offset from the spindle) is a constant. This fixes a specific effective length (L) at N1 for any given mounting distance (D), as (referring to the drawing of the basic geometry rectangle below):
L^2 = D^2 + Lo^2 -(Lo-N1)^2
or, alternatively, for a given effective length,
D^2 = L^2 - Lo^2 + (Lo-N1)^2
The distance Lo - N1 is half of the difference of the nulls.
Therefore for any alignment, this distance can be found if the nulls are known.
From the basic geometry rectangle above:
Given that for any given alignment ( LofgrenA/Baerwald, Lofgren B, Stevenson, VPI, UNI-DIN, etc) together with given minimum and maximum recorded radii, the nulls are constant, then the Linear Offset is constant, and so Lo-N1 is constant. Therefore if D is fixed, then L is fixed at N1, at which point the cantilever should be at 90 degrees to the radius.
Also, the offset angle OA (note: not necessarily the arm headshell angle) is easily found, as the sine of the offset angle is: linear offset divided by effective length:
sin OA = Lo/L
So, knowing the nulls and one parameter, the others can be found.
And entered in the VE spreadsheet.
Or, more easily, vice versa...
