## Arm Geometry |

ARM GEOMETRYby Noel Keywood, (updated from Hi-Fi Answers, October 1979)
This article is a generally applicable technique that I have developed for optimal alignment of any commercial pickup arm. Since it is not based on a particular design method it is more appropriate to adjustment of an arm than adaptations of Baerwald's or Stevenson's methods, which are of most use to a designer. Records are cut on a lathe and the cutting stylus is driven across the disc's surface (this is a special 'acetate' disc or 'lacquer' that is subsequently processed) in a straight line, since it is driven on a lathe carriage. Because it is difficult to emulate this system for replaying discs, we use a simpler pivoted arm that swings across the surface of a disc in an arc. As you can see in Fig 1 this causes the cartridge stylus to travel along a different path from the cutting stylus and its this difference that leads to distortion. The cutter is always at a tangent to the groove and, for zero tracking error distortion the cartridge must maintain this geometry. In the Fig 1 example, however, where a straight arm is depicted, there is angular error - tracking error - right across the disc surface. While tracking error is the basic parameter that causes distortion, actual levels are also dependent upon groove speed relative to the cartridge. For a given amount of tracking error, distortion due to the error increases as groove speed decreases. Or we can say that if tracking error of an arm remains constant, the distortion it causes at the outside of a disc is much less than that at the inside. Therefore distortion increases as the cartridge travels inward and is greatest on inner grooves. This is very important since it means we can tolerate far less tracking error on inner grooves if distortion levels are to stay constant and it is in fact the basis of Stevenson's arm design technique to ensure that peak distortion levels remain constant as an arm tracks inward. Designers used to consider tracking error only, rather than the distortion it generates. Designing for minimum tracking error is a different ball-game and does not provide ideal results. Straight arms of the sort I have so far talked about don't exist. In fact they all have an angled headshell or arm tube - it amounts to the same thing where the cartridge is set at an angle to the arm tube, or its axis of movement. This is part of a geometric ruse, and a very clever one, where tracking error is reduced by offsetting the headshell and then placing the cartridge stylus arc of movement ahead of the turntable spindle. Figure 2 shows this idea and where we get the term 'overhang' with such a geometry. Note that headshell offset angle is relative to a notional line drawn from the arm's vertical pivot to the stylus tip, not the angle of the headshell to the arm tube. Overhang or the distance that the stylus swings past the platter pivot centre, is measured on the extension of a line from the arm pivot, through and past the turntable centre. The earliest mathematical solution to the problem of minimising tracking error in this fashion was presented by Percy Wilson in The Gramophone 1924. It carried the title 'Needle Track Alignment' (remember, they were steel needles in those days!) and described how the technique of 'offset and overlap' could reduce tracking error to less than 2° and distortion to less than 2%. The geometric technique Wilson described became common in Britain but was taken up in the USA later with the best known mathematical analysis presented by Baerwald in 1941, although he was in fact preceded by Bauer. Baerwald's equations were approximations, but a later analysis in Britain by J K Stevenson that I have already mentioned wasn't, and it provides a design technique that allows the best values of offset and overhang to be found for any particular length of arm. But you don't have to use the theoretical values and I am concerned here with making the best out of any arm, be it properly designed or not.
There are two ways in which the tracking error of an arm as it traverses a disc can be assessed - by calculation and my measurement. Both are reasonably simple and I will cover calculation first. Here we have one universal equation that contains no error through approximation and gives us the angle of offset a headshell must have for zero tracking error. If you subtract from this figure the actual offset of the headshell, obtained either from manufacturers figures or by measurement, then you end up with tracking error. For those who don't like the equation, and it's awkward to use without a calculator, there is a slightly less exact simplified form. Bauer developed this approximation for engineers to use in the days when calculators didn't exist. The first and most accurate equation, giving tracking angle needed for zero error at any particular distance from the disc centre, r, is - The notation sin -1 (arc sin) denotes 'the angle whose sine is' and you will need either mathematical tables that have Natural Sines, where the angle is found against the figure this equation provides (which is considerably less than 1) or, alternatively, a calculator that can handle the conversion, as many can. If all this seems nasty, try the following -
This is Bauer's approximation and as you can see it is much easier to operate, whilst also eliminating the need to use sines. Having got this far, the next problem is to find those values of l (effective length of the arm), d (overhang) and r (distance from disc centre) from a real-life situation so that they can be substituted into one or other of the equations. We start with the most fundamental property of arm effective length, designated l. Many manufacturers now specify this and their information can be relied upon to be accurate. Otherwise, you need to measure this length in millimetres with good accuracy (within 1mm), remembering that it is the distance, as shown in Fig 2, between stylus and the arm's vertical bearing centre. A majority of modern arms are adjusted for overhang by sliding the cartridge backward or forward in the headshell. This does of course alter effective length and you should set the arm up for overhang according to the manufacturer's instructions first. While effective length is often quoted, overhang is nearly always quoted. If for some reason you cannot obtain the figure however, there are a number of ways of finding it. For arms that will swing over the patter centre spindle it is usually easiest to mark up a small scale of distance in millimetres from the spindle centre on a 45rpm disc adaptor. The scale should be marked out on a straight line, perhaps on white paper glued to the surface, that runs through the centre of the adaptor's spindle hole. It is important to measure overhang with extreme accuracy, less than 0.5mm error. If the arm will not swing over the spindle, then you must measure armpivot to platter-centre distance (D in Fig 2) and subtract it from arm effective length, l. Experience shows that this is the least accurate way of finding overhang and it should be carried out with great care to achieve the degree of accuracy mentioned. Finally, there is r, or the distance from disc centre of the pickup arm. There is agreement that the outer groove of a disc is always 146mm from the centre and it is currently assumed that the inner groove lies at the IEC specified distance of 60mm from the centre, although a number of discs have an actual inner groove figure of 58mm; the graph of tracking error against distance from disc centre is plotted between limits of 60mm and 146mm. Here is an example and I hope this will lend hope to the mathematically faint-hearted who have eyed the equations with deep suspicion. Bauer's approximation is not beyond the capabilities of anyone who can add, subtract and multiply, although a calculator will take the tedium out of this process, especially if it is programmable and a spreadsheet will be even faster and more informative if it also plots the graph. Taking a typical arm with overhang, d = 15mm effective length, l = 220mm and r = 58mm As you can see, the second equation is simple, quick and adequately accurate. This angle of 22.4° is the amount of offset you need at the headshell of our example for zero tracking error on the inner groove of a disc. If you know or measure the actual designed offset of this arm, the difference between the two is the amount of tracking error that exists at this point. Few manufacturers specify headshell offset and invariably you will have to measure it. I place a steel rule along the top of the arm, from the centre of the pivots to the equivalent stylus position on top of the headshell. Remember that offset angle is that between the headshell and a line from the stylus to the vertical pivot centre of the arm. It is not the angle between the headshell and the arm tube, unless the arm is a design where the stylus does lie right beneath such a line. Place a protractor on top of the rule and measure 'headshell offset angle. Again, accuracy is needed and this depends entirely upon how careful you are in your technique. I should note that most headshells are offset at 23 degrees (0) or so and a common overhang figure is 15mm to 18mm. Having found actual (designed) offset of the arm - tracking error = ideal offset - actual offset Here 'ideal offset' is the calculated figure from the equation and 'actual offset' the angle of the headshell on the arm in question. When 'actual offset' is larger in value than 'ideal offset' you get a 'negative' angle (we call it negative by convention), as shown in Fig 1. To obtain this you do in fact subtract 'ideal' from 'actual' of course in this situation, and put a negative sign in front. Measuring in our example a headshell offset of 230 we can then find tracking error from the above equation, as follows - tracking error = 22.4 - 23 = - 0.6 degrees And so we now know that on inner grooves, or to be more precise at a distance of 58mm from the disc centre, our pickup arm example exhibits a tracking error of - 0.60
Using these calculations you can find the tracking error of an arm at various points across the surface of a disc, from outer to inner grooves. Using the radii that I quoted earlier of 58mm, 70mm etc you will have to run through this set of calculations ten times, or use a programmable calculator into which you first enter the arm dimensions and then the consecutive radii. It gives an immediate readout of tracking error and distortion for each point. Having found tracking error across the disc the values are then transferred onto a graph with a scale as shown for a 9in and 12in arm (below). From tracking error and distance, r, we can calculate distortion from one further simple calculation, attributable to Baerwald. It is - Obviously, both r and (Ø) come from our tracking error calculations. With negative tracking error you will end up with negative distortion of course and since this is impossible (the negative sign being a convention only), just forget the sign. Distortion values can also be plotted onto the graph shown using the left hand scale for distortion as well as tracking error. What you must remember here on joining up the points is that when tracking error is zero, which occurs as the tracking error line passes through the horizontal x-axis (showing distance, r), distortion is also zero and so the distortion curve always drops down to zero at this point. The actual distortion value is that produced under a particular set of circumstances, and these are that the replay stylus is conical (spherical) and it traces a signal of 1Ocms/sec rms level (modulation velocity) on a disc (LP) revolving at 33rpm.
The distortion itself is dominant second harmonic component produced by tracking error, corrected for level to account for attenuation experienced by RIAA correction in a disc pre-amplifier. Matters are slightly different for all elliptical stylus but distortion remains proportional to tracking error and inversely proportional to distance from the disc centre in the same way, so the distortion curve is still valid. However, as an elliptical stylus is rotated around its vertical axis by tracking error it sinks downwards slightly and one contact face becomes advanced on the groove wall relative to the other. For tracking error of 2 degrees the time difference on inner grooves of a 33rpm disc is 5µS (five millionths of a second), equivalent to one wavelength at 200kHz. The stylus will drop 0.4 x 10-6 in (0.4 millionths of an inch) which, even in the terms of minuscule groove modulations, is small. Having plotted the graph of tracking error and distortion due to the error you now have a complete picture of your arm's behaviour with regard to lateral tracking error. Also from the equations it is simple to calculate performance changes that result from adjustments of overhang and offset. Next I will explain how overall distortion performance can be improved by working backwards through the equations to find an ideal overhang figure.
It is easy to construct a graph of tracking error and distortion of any arm as it traverses a disc from outer to inner grooves. By inspection of such a graph, and by working 'backwards' through these relatively simple equations it is an easy matter to calculate ideal overhang such that distortion is reduced and performance generally improved for most arms available. If their fundamental design is suboptimal, as it is likely to be, you cannot easily correct this but re-adjustment of overhang will nearly always improve matters, often by an enormous degree, and result in much better sound quality. Since it is necessary to understand the tracking error graph, whether you construct it by calculation as explained earlier, or by measurement using a gauge (which I will go over later in this piece) the following discussion is important to both techniques. It was once quite usual to find tracking error approaching a minimum at the lEC minimum groove radius of 60mm, most alignment protractors being drawn up to accord with this geometry. Such geometry can produce a broad and significant distortion peak and there is only one distortion zero around which levels are low. Today's protractors place zeros at 62mm and 120mm approximately where the arm should be at a perfect tangent to the groove and you can see why by looking at the tracking error graph.
Our aim is to raise the tracking error curve so that it crosses the horizontal x-axis twice, providing two distortion zeros. In this way distortion right across the disc will be reduced, except on outer grooves. The increase in distortion on outer grooves is always less marked for changing tracking error though, since groove speed is higher. The trade-off is beneficial because distortion will be lower for a greater period of listening time. By Stevenson's design technique the arm's geometry is sub-optimal. since for an effective length of 220mm headshell offset should be 24.6°, while it is in fact 23°. This tells us that we cannot achieve perfect results, but it is obvious from inspection of the tracking error graph for this design that matters can be improved. When deciding how far to raise the curve (some will need lowering) the ideal should be to arrange zero-crossing just before minimum grove radius, or between 58mm and 70mm from the disc centre. We can work 'backwards' through the (transposed) equations provided earlier to find required overhang. From - ('ideal' offset = tracking error - actual offset) we know our arm has an actual offset of 23° and we want a tracking error at 80mm of -1°. Inserting these figures into the equation gives us--- 'Ideal' offset = -1°+23° = +22° Also, from the tracking error equation (transposed) we can find overhang - where 0 = ideal offset, r = distance from disc centre in mm, l = effective length of arm in mm We have just calculated an ideal offset figure of +22° to give us the required -1° tracking error at r = 80mm, for an arm with an effective length of 220mm. Substituting these figures into the equation we have - The manufacturer's original overhang figure was 15mm and we now have a revised figure just 1.2mm greater. This reflects the order of change one can expect in such re-alignments and it also emphasises the need for extreme accuracy in setting up. This fine increase in overhang will alone insert a second zero in the distortion and tracking error curves and completely change the overall characteristic. The tracking error curve has been raised as predicted to provide a maximum negative error at 80mm of -1° and this gives us two zero tracking error points, at 63mm and 11Omm. This reduces distortion across most of the disc with the peak level at 75mm for instance more than halved from 1 per cent to 0.45 per cent. Distortion on outer grooves has risen, but only by a small amount - in this instance 0.1 per cent. The most important feature to note however is that distortion levels are now low for a much greater period of listening time. If we take 0.5 per cent as a nominal upper distortion limit, the manufacturers recommended geometry gave distortion levels exceeding this amount for 54 per cent of playing time, while the revised geometry reduces this to 18 per cent of playing time. One way and another, that extra 1.2mm overhang makes a lot of difference!
Now let us finally consider what results optimal design by Stevenson's technique would have provided. After choosing outer and inner groove radius limits, this method first tells you where zero's will occur. Using 146mm and 58mm respectively, it puts zero·s at 63.6mm and 119.5mm. It then arranges the three distortion peaks to be equal in level, thus providing balanced performance. The equations give overhang and offset figures for any chosen effective length of arm and they are as follows- The second overhang equation is used when overhang adjustment is carried out at the headshell; the first when at the arm base. In the example we find that for an effective length of 220mm, headshell offset should be 24.6° and overhang 17.76mm. Our tracking error graph shows performance of a 9in and 12in arm designed using these equations. The 12in arm exhibits least distortion as expected, and the zero crossing points differ too. Constructing a graph of tracking error and distortion for a pickup arm and then carrying out adjustments through vertical repositioning of the tracking error curve as I have described, you not only get a clear picture of performance but can also select any wanted characteristic. For instance, some records sound best when inner-groove tracking error distortion is low, and you can adjust for this characteristic.
Pickup Arm Design, J.K. Stevenson, Wireless World, May & June 1966 |