Vinyl Asylum Welcome Licorice Pizza (LP) lovers! Setup guides and Vinyl FAQ. 

A Treatise on Cartridge Alignment  Part IIContinued from Above

Antiskating force
The skating force, is a torque about the tonearm pivot that arises because a tangent to the record groove at the point of contact of the stylus is offset from the tonearm pivot. The line of action of the frictional force does not pass through the tonearm pivot, so a torque is generated. This torque tends to turn the tonearm inwards towards the centre of the record. If the perpendicular distance from a tangent to the record groove at the point of contact of the stylus and the tonearm pivot is defined as the groove offset, then the skating torque is proportional to the groove offset times the frictional force between the stylus and the record groove. The skating force is not constant across the record surface, it is largest at the inner and outer groove radii, and is minimum somewhere between the two null radii. To counteract the skating force, an antiskating force is applied.
Conveniently, when the stylus is at one of the null radii, the groove offset is the same as the linear offset so the following equation arises.
Skating torque at null radii = Friction * linear offset.(17)
Clearly, if the user selects an alternative linear offset to that preferred by the manufacturer, the required antiskating force will be different to that when the intended linear offset is used, in this case the antiskating force should be adjusted after alignment, using a test record instead of using the scale on the tonearm.

Equations of LÃ¶fgren and Baerwald.
The equations (10), (11) and (12) give the arm parameters in terms of the null radii and this is the most logical form. As stated above it is also possible to give the arm parameters in terms of the inner and outer groove radii if the Tchebichef method of minimising maximum distortion is used. The result is the form in which the arm parameters are usually presented, and is the form presented by LÃ¶fgren and Barewald.
In equations (4) and (5) the null radii are expressed in terms of the inner and outer groove radii, so rearranging gives
N1*N2 = 2*P/[1+A^2/P](18)
and
(N1+N2)/2 = 2*A/[1+A^2/P](19)
In this case
A = (R1+R2)/2
P = (R1*R2)These results can be substituted into equations (10) (11) and (12) to give the following expressions for mounting distance, overhang and offset angle.
Lm^2 = Le^2  {2*P/[1+A^2/P]}(20)
D = Le *(1sqrt{12*P/[Le^2*[1+(A^2)/P]]}(21)
Sin(theta) = {2*A/[1+A^2/P]}/Le(22)
However it can be seen clearly that these equations are rather more cumbersome than equations (10), (11) and (12) above. Note that Baerwald's final result  which is not included here  was an approximation for overhang, which in fact is quite an inaccurate approximation so should not be used..

Conclusions.
Setting tonearm alignment involves selecting three parameters. Mounting distance effective length and offset angle. The null radii are defined by this selection.
The difference between the square of the effective length and the square of the mounting distance is the product of the two null radii.
The linear offset is equal to the average of the two null radii.
The necessary and sufficient condition for the existence of two null radii is that the square of the linear offset be greater than or equal to the difference between the square of the effective length and the square of the mounting distance  appendix (2).
The radius of maximum distortion between the null radii is given by the equation (3), which is given again below.
R Max Distortion = 2*(N1*N2)/(N1+N2)(3)
It can be seen that this radius is identically equal to the product of the null radii divided by the average of the null radii. From the opening lines of this section, it can be seen that the radius of maximum distortion between the null radii is also given by the difference between the square of the effective length and the square of the mounting distance divided by the linear offset.
The optimum alignment system is based on Tchebichef's method and was proposed by LÃ¶fgren and Baerwald. This system minimises the maximum tracking distortion between a given inner groove radius and a given outer groove radius.
Using this system of alignment, the three maxima in the distortion curve have the same value (Peak Distortion Equivalence). The three distortion maxima occur at the inner groove radius, the outer groove radius, and at a radius given by equation (3) above.
For this method, a one to one relationship exists between the null radii and the groove radii. Correspondingly the tonearm parameters can be given in terms of groove radii instead of the null radii. (For example the equations of LÃ¶fgren and Barewald).
For this method, another expression for the radius of maximum distortion exists, and is given below in equation (23).
R Max Distortion = 2*(N1*N2)/(N1+N2) = 2*(R1*R2)/(R1+R2)(23)
Which is just the product of the null radii divided by the average of the null radii OR the product of the groove radii divided by the average of the groove radii.
For the IEC standard groove radii, the "Peak Distortion Equivalence" method gives null radii at 66.00mm and 120.89mm. For the DIN standard groove radii, this alignment method gives null radii at 63.10mm and 119.17mm.
The null radii that result from the "Minimum RMS Distortion" method can be calculated numerically. Applied to the IEC standard groove radii, and for typical tonearm dimensions, this method gives null radii of 70.15mm and 116.23mm. For the DIN standard groove radii, this method gives null radii of 67.29mm and 114.53mm.
LÃ¶fgrens "Minimum Distortion Method", which uses the same linear offset as the "Peak Distortion Equivalence Method" gives almost the same null radii as those above. 67.29mm and 114.53mm (IEC); 67.41mm and 114.86mm (DIN).
Stevenson's method, is essentially a modified form of "Peak Distortion Equivalence". Using Stevenson's method the inner null radius is placed at the inner groove radius, and the peak distortion between the null radii is equal to the distortion at the outer groove radius. Stevenson's method, applied to the IEC standard groove radii gives null radii at 60.325mm and 117.42mm. For the DIN standard groove radii, Stevenson's alignment method gives null radii of 57.5 and 115.526mm.

Appendix 1. Calculating the tracking error for a given set of arm parameters
For a given set of arm parameters (Effective Length, Le; Mounting distance, Lm; offset angle, theta), the tracking error, e varies with groove radius, R.
The relationship between tracking error and the arm parameters is given below.
e = Arcsin{[(Le^2Lm^2)/R +R]/(2*Le)}  theta
In terms of the null radii, and the tonearm effective length, the above expression is equivalent to
e = Arcsin{[(N1*N2)/R +R]/(2*Le)}  Arcsin{(N1+N2)/(2*Le)}

Appendix 2. Necessary conditions for the existence of two null radii.
The null radii were given in terms of the arm parameters in equations (15) and (16)
N1 = Le * Sin(theta)  Sqrt{[Le * Sin(theta)]^2[Le^2  Lm^2]}(15)
N2 = Le * Sin(theta) + Sqrt{[Le * Sin(theta)]^2[Le^2  Lm^2]}(16)The necessary and sufficient condition for the existence of at two null radii is
Le * Sin(theta)]^2 > [Le^2  Lm^2]
This is equivalent to the condition that the square of the linear offset must be greater than the difference between the squares of the effective length and the mounting distance.
It should be noted here that some tonearms (Dual CS5000 for example) have recommended parameters that fail the condition above. Adopting these parameters will result in an alignment that give zero distortion nowhere!

Appendix 3. LÃ¶fgren's Minimum RMS distortion Null Radii.
For minimum RMS distortion, LÃ¶fgren gave the following equation for the product of the two null radii in terms of the linear offset.
N1N2 = 3Rp(Lo*RsRp)/(Rs^2Rp)
Where
Lo is the linear offset,
Rp = R1*R2
Rs = R1+R2This can be rearranged to give an expression for the null radii in the same way as equations (15) and (16) were constructed, as follows
N1 = Lo  Sqrt[Lo^23Rp(Lo*RsRp)/(Rs^2Rp)]
N2 = Lo + Sqrt[Lo^23Rp(Lo*RsRp)/(Rs^2Rp)]As stated in the text above, these expressions for the null radii are not a full solution to the "Minimum RMS Distortion" method, as the result depends on the Linear Offset, Lo. That is, to determine the null radii, firstly a linear offset must be selected. According to Dennes, (an English translation of the LÃ¶fgren paper is not available) LÃ¶fgren argued that the linear offset which results form the "Peak Distortion Equivalence" method should also be used for the "Minimum RMS Distortion" method. This has a value of 93.445mm for the IEC standard groove radii, and this gives null radii of 70.29mm and 116.60mm.
For the DIN standard groove radii, the linear offset is 91.138mm, and this gives null radii of 67.41mm and 114.86mm.

References
[1] Baerwald : Analytic treatment of tracking error and notes on optimal pickup design, Journal of the Society of Motion Picture Engineers 1941, p.591
[2] LÃ¶fgren: Ãœber die nichtlineare Verzerung bei der Wiedergabe von Schallplatten infolge Winkelabweichungen des Abtastorgans (On the nonlinear distortion in the reproduction of phonograph records caused by angular deviation of the pickup arm) Akustische Zeitschrift, Nov.1938, p.350
[3] http://www.riaa.com
[4] Stevenson : Pickup arm design, Wireless World, May 1966, p.214, June 1966, p.314
[5] Graeme Dennes: "A comparison of six major papers on tracking distortion" Available directly from the author.
[6] http://www.enjoythemusic.com
[7[ http://www.AudioAsylum.com/audio/vinyl/messages/44594.html
This post is made possible by the generous support of people like you and our sponsors:
Topic  A Treatise on Cartridge Alignment  Part II  bkearns 04:42:34 04/20/01 (3)
 Re: Erratum  bkearns 20:09:10 04/22/01 (0)
 These Two Posts should be placed in FAQ! nt  Jeff Starr 10:32:02 04/21/01 (1)
 I AGREE !!! nt  Jim E 13:32:55 04/23/01 (0)