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Tone arm geometry and tracking distortion (longish)

212.206.88.5

Posted by Klaus on December 4, 2000 at 08:26:10

1. Baerwald's analysis :

Distortion due to lateral tracking error affects only the laterally cut modulation of the V-shaped groove, not the vertical signal (which are affected by vertical tracking error). The distortion figures as obtained by Baerwald's equation (1) and (1*) will have to be divided by approximately
two [3]. Distortion values with optimum arm design are below 1 per cent.

In his paper [1] Baerwald analyses the distortion induced by a pivoting arm tracking grooves cut by a radial arm for laterally cut records. The arm's pivotal axis and the groove tangent do not coincide, the difference in angle being called tracking error eta (expressed in radians or degrees).

When looking on top onto a lateral modulation (taking the centerline of the groove as reference), the instantaneous position of the stylus is somewhat advancing (for a modulation sloping towards the record's centre) the recorded signal (its abscissa being displaced by delta s). The effect constitutes an alternating advance and delay of the reproduced signal with
respect to the recorded on or a frequency modulation of the signal itself, which results harmonic distortion. Harmonic distortion of a given signal increases with decreasing groove velocity, i.e. it is greater in the inner grooves than in the outer grooves.

The subsequent analysis is based on a sinusoidal signal resulting in the equation for distortion epsilon :

epsilon = A x omega / (r x OMEGA x tan eta)(1)

where

A = recorded amplitude
omega = recorded angular frequency in 1/s
r = groove radius
OMEGA = angular disk speed = pi /30 x speed in rpm

or

epsilon = (2 pi A / lambda ) x tan eta

where

A = recorded amplitude
lambda = recorded wavelength = 2 pi r OMEGA /omega

or

epsilon = tan alpha x tan eta
where alpha is the inclination angle of the sinusoidal modulation.

tan alpha is expressed as

tan alpha = 2 pi A / lambda = omega x A / (OMEGA x r) = V / (OMEGA x r)

From this it can be seen that distortion increases with recorded amplitude, velocity and frequency. Knowing that according to RIAA records are cut with constant amplitude below 500 Hz and above 2120 Hz and with constant velocity between those two frequencies, calculation of precise distortion figures appears to be rather complicated. Values of lower than 1 % are given by Löfgren.

For complex signals, the 2nd-order cross-modulation products are the prevalent distortion components (whatever that may be).

Baerwald considers 2 per cent distortion at 8 cm/sec due to tracking error as upper limit, using this figure as basis for his following approach for optimum arm design. "Optimum design should minimize tracking distortion over the entire playing range of the record, which, as analysis shows, is by no means synonymous with minimizing the tracking error itself".

Weighted tracking error eta' = rm/ (r x eta)(2)

where (dimensions in inch)

rm = sqrt (r1r2 )= mean groove radius
r = radius of an arbitrary groove
r1, r2 = inner, outer recorded groove radius

Maximum distortion combining equations 1 and 2 :

epsilonmax (in per cent) (17 to 22) x eta' (in radians) (1*)

The factor of proportionality is 17 for transcription recordings with 8 cm/sec and 22 for commercial recording with 16 cm/sec.

Another formula for calculating distortion comes from Stevenson [4] :

epsilon (in per cent) = 9.28 Vrms x phi / Ls (1**)
2nd harmonic distortion

where

Vrms = mean recorded velocity = Vo / sqrt 2 (Vo = peak recorded velocity) in cm/sec
phi = 360 eta/ 2 pi (eta = tracking error in radians) in degrees
L = effective length in inch
s = turntable speed in rpm

Third and higher harmonic distortion amount to less than 10% of the value of 2nd harmonic distortion and considered negligible.

The following geometrical analysis results in equations for optimum arm design, for overhang and offset angle (for given effective lengths of the arm). For the offset arm, the tracking error passes twice through zero and the distortion reaches maximum value three times, i.e. for r1 and r2 and
an intermediate radius r0.

For single-purpose offset arms the equations are (all dimensions in inches)

Optimal offset angle :

sin alpha = a / L (1 + b*2/c) (3)

a = r1 + r2
b = (r1 + r2)/2
c = r1 x r2
L = effective arm length = distance pivot axis - stylus tip

Optimal overhang :

-delta = a / L (1 + b*2/c)(4)

where

a = r1 x r2
b = (r1 + r2)/2
c = r1 x r2
L = effective arm length = distance pivot axis - stylus tip

Maximal weighted tracking error :

eta' = (a*2 / (8L sqrt b)) x 1 / (sqrt [(1+c*2/d)*2] - e*2) (5)

where

a = r2 - r1
b = r1 x r2
c =( r2 + r1)/2
d = r2 x r1
e = (r2 + r1)/L

Null radii :

r = 2 x r2 x r1 / [ (1 -+ 1/sqrt2) x r2 + (1± 1 /sqrt2) x r1] (6)

For an inner recorded radius of 60.325 mm and an outer radius of 146.05 mm, the null radii are 66.04 and 120.9 mm.

For determining design parameters of an arm, the only parameter to select is inner recorded groove radius r1.

I have measured that radius on some 230 records of my collection and found the average radius to be 66.8 mm. This would mean that with an arm having Baerwald geometry, the inner radius is close to the inner null radius (66.04 mm) and that consequently inner groove distortion is close to zero.

The diagram of weighted tracking error versus groove radius starts shows three peaks of equal magnitude, at outer and inner grooves and (being negative) between the two null points.

2. Löfgren's analysis

According to Dennes Löfgren's approach results in equations identical to those of Baerwald. A second solution (Löfgren B), however, is proposed. The reason Löfgren indicates is that the three peaks of weighted tracking error are not of equal importance. The middle (negative) peak should receive
greater consideration because 1. weighted tracking error slopes slowly in that peak's vicinity, and quickly at the inner and outer peaks and 2. the outer and inner radii (as selected for computation) are not fully used for all records. This would lead to allowing greater values of weighted
tracking error at the inner and outer peaks than at the middle peak, which could be achieved by a slightly smaller offset angle.

Another approach for considering the greater importance of the middle peak is the method of the least squares. The integral (between inner and outer radii r1 and r2 ) of the squared weighted tracking error eta' (see equation 2) has to be minimized when changing overhang and linear offset p (linear
offset p = effective length L x sin offset angle).

For doing so, a further parameter, the effective distortion factor K is introduced. A diagram is presented showing curves of identical K in a overhang vs linear offset system. A straight line indicates the optimum conditions. The equation describing that line is further developed. It allows to calculate overhang for a given linear offset (read offset angle) that has been calculated using the equations (equation 3) for optimum arm design.

Overhang delta = L - sqrt (L*2 - a*2) (7)

where L = effective length in cm

a*2 = ( 3 x r1 x r2 (p (r1 + r2 ) - r1 x r2 ) / ( r1*2 + r1 x r2 + r2*2 ) (8)

r1, r2 = inner and outer radii in cm
p = linear offset = L x sin (offset angle) in cm (9)

Interestingly this second solution is mentioned in Baerwald's paper without going into further detail.

As explained above, offset arms with Baerwald geometry have weighted tracking error reaching maximum value at three points, the outermost groove, the innermost groove and between the null points. With optimum design the weighted error is the same for each peak, the value being minimized. A lot of arms, however, have a geometry with the inner null radius at or close to the inner groove (Stevenson approach). Other arms
have near zero error at the beginning and end of the record, creating a larger error in the middle [2].

Effective length, overhang and mounting distance are related parameters and, with a design for optimum performance, unique for every tone arm. They fit together like a jigsaw puzzle - a wrong dimension will not fit. The ideal arm would have infinite length (and hence zero overall tracking
error), infinite stiffness and zero mass. From a design standpoint it is desirable to have the longest effective length practical. The size of the turntable base sets limits as regards length. There is one arm, the Garrard Zero 100, which has been designed such that maximum tracking error is 0.022 degrees (as compared to 2.279 for the SME 3009 III)

In order to achieve optimum geometry, the mounting distance, which is related to the arm's effective length, must be correct, i.e. effective length minus optimum overhang.

Effective arm length, just like offset angle, cannot be measured, it is a design parameter and has to be known from the arm's specifications. In the case that the length for a given arm is not known, one can obtain the value by placing the cartridge somewhere in the headshell slots, measure the
length, calculate corresponding optimum overhang using equation (4) above (or look it up in a table such as of US 4,326,283 which can be viewed at http://www.uspto.gov/patft/index.html), move the cartridge to the the corresponding position in the slot, measure the new length, calculate the new overhang and so forth. The arm should be mounted for this exercise on a "dummy" turntable having marking for the platter axis and a hole for the
arm base allowing the mounting distance to be adjusted in correspondence with the different effective length - overhang values. The use of the Dennesen soundtractor, which is based on Baerwald geometry, makes the determination of the correct mounting distance easier in that it avoids the steps of measuring the unknown effective length and the subsequent
calculation. Once a position (read mounting distance) on your dummy table is found where the stylus is on the grid marking and the vertical reference pin on the pivot centre, you just measure the distance pivot centre and platter axis.

Offset angle is the angle between the line drawn from the pivot centre through the stylus tip and a line parallel to the cartridge body (read cantilever) through the stylus tip. Again, this angle is a result of design specification and not a measurement after the fact of assembly. Use equation (3) for calculation.

Another parameter, independent of and yet linked to arm geometry, is the orientation of the vertical arm bearings (those that allow up-and-down movement of the arm). Many arms allow for adjustment of VTA. A lot of records are warped so that the headshell, just like when adjusting VTA, will not remain parallel to the record's surface.
If the angle (read axis) of the vertical bearings is perpendicular to the line through the offset angle (read cantilever), there will be only one angular change, that of the VTA. If the angle is not perpendicular, the cartridge moves not only in the vertical but also in the horizontal plane, such that both VTA and azimuth are affected. When visualizing this effect, if the arm could be rotated in its bearing until it was straight up, the arm whose bearings are perpendicular to the cantilever line would have the
front (line) of the cartridge still parallel to the record surface, whereas the arm not so designed would have one front edge of the cartridge higher than the other front edge.

It seems in consequence advisable to use a protractor that is designed for being used with the arm or, if such protractor is not available, a protractor that is based on the same geometrical design as is the arm (including input parameter inner recorded groove radius).

In my personal case, the use of a Dennesen type protractor on my SME 309 would compromise tracking of warped records by inducing azimuth changes, since the 309's offset angle does not correspond to the optimum angle as obtained by Baerwald's equations based on 60.325 mm inner recorded radius, which optimum angle is used for the Dennesen tool. The 309 design is
actually based on an inner recorded groove radius of 58 mm (reply from SME to a corresponding e-mail).

From a recent follow-up on the Vinyl Asylum it appears, however, that such azimuth changes are far below the tolerance with which azimuth can hoped to be set, hence of negligible magnitude.

Arms using standard Baerwald geometry (including the above mentioned inner and outer radii) are, as far as I have obtained arm data :
SME IV, V
Graham
Wheaton Triplanar
Linn
Rega (the arm only, the protractor included has its inner null point at the inner radius)
Immedia
Dual CS 5000

Protractors also using standard Baerwald geometry (including inner and outer radii) are :
Dennesen Geometric Soundtractor (US 4,295,277)
Geodisc (US 4,326,283)

Protractors also using Baerwald geometry (inner and outer radii used not known) are
DB systems DBT-10 phono alignment protractor
JML universal tonearm alignment protractor
Wallytractor

Mounting errors : small errors in offset angle and overhang do affect the geometrical setup and hence tracking error and distortion. Longer arms are more critical in that the same absolute error has a greater effect on the location of the null radii than for shorter arms (errors of over two degrees in angle may out the null radii somewhere off the record: adding 1 mm to optimum overhang shifts the null radii from 66.04 and 120.9 mm to 73.5 and 113.3 mm for the shorter, and to 79.9 and 106.9 mm for the longer arm (200 and 30 mm effective length)). Errors of 2.5 mm in overhang or 2°
on offset angle will more than double the distortion [4].

A mathematical comparison of the classical papers on tone arm geometry (1, 4, 5, 6, 7) has been made by Graeme Dennes. The results of this comparison are published in 1. Audio magazine, May 1983, p.48 and 2. Wireless World, Feb.1983, p.60.

Dennes' results :

Löfgren, Baerwald, Seagrave and Stevenson produced mathematically identical, and exact, design equations for optimum offset angle and overhang.
Löfgren and Baerwald also produced identical, but approximate, equations for overhang.

Bauer's offset angle equation is an approximation, and his overhang equation is identical to the Löfgren/Baerwald approximation. Seagrave also produced an approximation for overhang, actually more accurate than the Löfgren/Baerwald/Bauer approximations.

There is a single exact solution from four different sources plus three different kinds of approximation.

Two particular approaches :

Löfgren aims at minimizing tracking distortion by minimizing the weighted tracking error (solution A) with null radii at 66.04 and 120.9 mm.
He considered also an alternative approach, taking into account what he called the annoyance factor (Löfgren actually does not employ this or a similar term in his work; what he says is that the objection one could express against his standard optimum geometry that the three peaks of the
weighted tracking error curve are not of equal importance) : he altered overhang in order to lower the maximum distortion between the null radii, thereby accepting a higher distortion at the beginning and end of the record (considering that beginning and end of the record are of shorter duration than the time between the two null radii) : solution B. Solution B
thus gives a different overhang and different null radii (116.6 and 70.29 mm). Offset angle is identical for both solutions.

Stevenson considered tracking and tracing distortion in the neighbourhood of the inner groove to be annoying, so that he placed his inner null radius at the inner groove, which results in an increase in overall tracking distortion across the record. This approach is of little value for modern
styli, since the inner grooves are tracked better by these styli than a spherical stylus did the outer grooves.

In short : there are basically two approaches to tone arm geometry :
1.Löfgren A and Baerwald
2. Löfgren B.

Any arm not being designed according to one of these two approaches, produces higher tracking distortion than necessary and should be disregarded.

A simple way to check for Baerwald standard geometry is to look at the linear offset, which is given in equation (9) above : for inner and outer radii of 60,325 and 146,05 mm the linear offset for optimum design has a value of 93,47.

Values different from 93,47 do not necessarily mean that the arm's geometry is incorrect since the simple fact of selecting a different inner radius gives a different offset angle and hence a different linear offset, the design still obtaining optimum geometry for that particular selected recorded record surface. Such is the case for the SME 309 : the selected
inner radius is 58 mm which results in a linear offset of 91,54.

[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] Kessler et al.: Tonearm geometry and setup demystified, Audio Jan.1980, p.76

[3] Randhawa : Pickup-arm design techniques, Wireless World, March 1978, p.73, April 1978, p.63

[4] Stevenson : Pickup arm design, Wireless World, May 1966, p.214, June 1966, p.314

[5] Löfgren: Über die nichtlineare Verzerung bei der Wiedergabe von Schallplatten infolge Winkelabweichungen des Abtastorgans (On the non-linear distortion in the reproduction of phonograph records caused by angular deviation of the pickup arm) Akustische Zeitschrift, Nov.1938, p.350

[6] Bauer (Shure)
Tracking angle Electronics, 1945, March, p.110

[7] Seagrave
Minimizing pickup tracking error
Audiocraft Magazine, Dec.1956, p.19 : Jan.1957, p.25 : Aug.1957, p.22

There are two earlier papers by Wilson, published in The Gramophone in 1924, 1925.

Klaus

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