Equivalent circuit of the dynamic system of a phono cartridge

If we searched for an individual who transformed the phonograph from a Victorian novelty to a modern marvel an excellent candidate would be Frederick V. Hunt who, as a young professor of physics at Harvard in 1936, was given the task of making phonograph recordings of the university's tri-centenary festival.

In order to improve the quality of the recordings, Hunt made enormous headway in the understanding of the interaction of the stylus and groove9. The invention of the LP and stereo would have been unthinkable without Hunt's work and the subsequent work of a series of talented doctural students he tutored and encouraged.

Counted amongst the eager beneficiaries of the work at Harvard was the American company Shure Brothers, Inc. of Evanston, Illinois. Shure's development team, not only took time to understand Hunt's work on the dynamical system of the phono cartridge; especially in terms of its ability to track the groove modulation accurately at a downforce below that which would damage the PVC material, but also, they took the time and trouble to develop the work and transmit that understanding to the wider audio community1,2,6. Hunt's work was still the basis for the design of phono cartridges in America and Japan up to the end of the LP era.5

Shure's "analog"

Various methods of investigation are available to the engineer to investigate the dynamic system of the phono cartridge: Mathematical analysis, physical measurements, and mechanical equivalent-circuit techniques. Because a study of optimization requires a great many variations (and presumably because most of the engineering team were, at heart, electrical engineers), Shure chose an analogue equivalent circuit as their analytical tool.¹

To anyone who has a similar background, this analogue circuit model of the physical, dynamic system is enormously helpful in understanding the engineering compromises inherent in phono cartridge design (see Appendix 2).

We will start by considering a simplified model which eliminates the cantilever effects; essentially this element is treated as infinitely stiff. We will re-visit the effect of the cantilever below.

The simplified model is illustrated. The various mechanical elements are represented by the electrical components as follows:

C1, represents - Record tip compliance µF = µ(cm/dyne)
C2 represents - Bearing compliance µF = µ(cm/dyne)
L1 represents - Inertia of armature (tip, cantilever and magnet) mH = mg
R represents - Viscous damping of bearing ohms = dyne-sec/cm
Ii represents - Recorded velocity Amperes = cm/s
Io represents - Output velocity of stylus Amperes = cm/s
Vab represents - Tracking force volts = grams

Now the last of these quantities (Vab) is a little tricky to understand. Effectively, it is the force of reaction from the groove upon the stylus as it tracks the modulation. It is thereby more or less equivalant (by a small margin) with the force required to hold the stylus in the groove provided by the tracking weight. Vab is thereby a direct measure of the downforce (expressed as a weight) required by the cartridge at various frequencies. Generally, we want this to be a small as possible to avoid unnecessary wear to the record.

The quantity Io (the output velocity of the stylus magnet) is easiest to visualise as the voltage generated across the damping (R) expressed as a voltage (V). This gives us the frequency-response of the dynamic system (ignoring the electrical circuit of the motor part of the cartridge.) Generally we want this to be as flat as possible across the audio band.

Concentrating of the circuit itself, we can see that it simpy two resonant circuits formed by L1 resonating in series with C2 and in parallel with C1; each being damped by R. Because of the huge difference in the values of C1 and C2, the two resonance frequencies will be widely separated and, to all intents and purposes they have little effect upon one another. Translating this back to a mechanical model, the mass of the armature (stylus, cantilever and magnet = L1) resonates with the main bearing compliance (C2) at an mid-band audio frequency, in this case at,

   [2 × π × (L1 × C2)½ ]-1 = 1125Hz, ....... (1)

and again with the compliance of the record material (C1) at a high frequency, in this case at,

   [2 × π × (L1 × C1)½ ]-1 = 25kHz....... (2)

Plotting voltage Vab and altering the value of R we get the following set of curves.

Because one resonance is series and the other parallel, they work in opposite directions in terms of the tracking downforce required. The resonance due to the main bearing compliance and the armature mass tends to make the stylus easier to drive and thereby requires less downforce to keep the stylus engaged with the groove. On the other hand, the parallel resonance due to the armature and the vinyl compliance makes the cartridge much harder to drive and thereby demands more downforce to keep it in the groove.

We can see that the critical component here is the damping (R) because it is "shared" between the two resonances. Too little damping and the high-frequency peak becomes very significant. Too much and, although the HF peak is "tamed", the cartridge needs more tracking downforce in the mid-range.

This is the core of the designer's dilemma.

The role of each part

Now let's consider the effects of the various components (mechanical and electrical) across the frequency band.

R - If you consider the point of series resonance due to the armature (L1) and the compliance of the main bearing (C2), it's obvious that when their reactances are equal and opposite, they "disappear" electrically and the only load the generator "sees" is the damping (R). So R represents the limiting factor in the midrange. We must not over-damp or we will sacrifice mid-range tracking. In real life C2 and R are not separable, they are both part of the characteristic of an elastomer³.

C2 - The bearing compliance (C2) is what "decouples" R from the stylus at very low frequencies which is the equivalent of saying that the compliance of the bearing must be large enough to allow for the large amplitude groove excursions at low-frequencies.

In order to track a record with maximum modulation amplitude of about 0.002" (50µm) at a maximum tracking force of 1 g, the required compliance for C2 is 5µ(cm/dyne). Viscoelastic bearings decrease in compliance as frequency increases so a considerable safety margin needs to be provided at the lowest frequency of interest, say 20Hz. That is why this component is set at 25 × 10-6 cm/dyne (or 25µF) in the equivalent circuit. But neither must the compliance of C2 be so great that sub-audio movements (due to warps etc) cause the armature to move relative to the pole-pieces.

L1 - The resonance due to the effective mass of the armature (L1) and the record-surface compliance (C1) has a number of side-effects over and above its effect on tracking. The increased current in the circuit mesh including R indicates that the transfer-function of IoIi will peak at this point and this will lead to a frequency-response peak. And, in the stereo cartridge, the resonance also leads to interchannel crosstalk as the movement of the armature becomes uncontrolled.

In the 1960s, it was believed the best the cartridge designer could do was simply to postpone this resonance to as high a frequency as possible and then damp adequately to keep the tracking demand down without over-damping the midrange. It seemed that keeping this resonance in abeyance could only be achieved by reducing the effective mass of the armature (L1) because the cartridge designer had no control over the compliance of the record surface (C1). This was uncomfortable because, even with the selection of the lightest (and sometimes exotic) materials, this was inevitably a game of diminishing returns.

The birth of CD-4 quadraphonic in the 1970s placed excrutiating demands on the phono cartridges at the time. Shure's solution to the problems fostered by CD-4 quadraphonic was to add a parasitic compliance and mass to the moving armature tuned to a frequency close to the main L1/C1 resonance. In terms of the equivalent circuit, this involved adding a parallel resonant circuit in series with L1. Shure called this innovation the Dynamic Vibration Absorber² and it acted rather like the port in a bass-reflex loudspeaker, sucking energy out of the system and controlling the main resonance.

In Japan, Norio Shibata of JVC4 realised that if he could increase the contact area of the stylus in the groove, he could vary the effective springiness of the record surface and do the equivalent of altering the value of C1 in the model. The Shibata stylus has a contact area about 4× greater than an elliptical type stylus and thus this resonance was "stepped up" an entire octave; a very great accomplishment at the time. (Remember, resonance is related to the square-root of the product of L1 and C1, see equation 2).


In the mathematical model above (formalised as a electrical circuit analogy), the cantilever effects were eliminated. This element was treated as infinitely stiff. (The anatomy of a practical cartridge is illustrated here to clarify the cantilever element.)

The cantilever must obviously be lightweight in order to reduce the effective mass for an extended frequency response. So, what lightweight materials might approach this ideal for infinite stiffness? Here is a table of some materials suitable for cantilever construction5.

Table. 1 Cantilever material characteristics

Material Modulus of Elasticity (E) Specific density (ρ)
Ti Titanium 11000 (kgf/mm²) 4.54
Al2O3 Sapphire or Ruby 32000 (kgf/mm²) 4.00
Al Aluminium 7400 (kgf/mm²) 2.69
B Boron 45000 (kgf/mm²) 2.30
Be Beryllium 27000 (kgf/mm²) 1.84
C(s, diamond) Diamond 100500 (kgf/mm²) 3.52

The modulus of elasticity (or elastic modulus) is a quantity that measures a material's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. A stiffer material has a higher elastic modulus.

In this application, we want the highest elastic modulus we can get for the cantilever, and we want the lowest density too (to ensure the lowest mass). But, in the table we have density and modulus of elasticity jumbled-up. We want a single measure for the best material for highest stiffness for a given weight.

The answer is to divide the figure for elastic modulus by the figure for the relative density, that's to say, E/ρ. We say that the cantilever material should have a high modulus of elasticity ratio. The materials are ranked (best to worst) in this way in Table. 2.

Table. 2 Cantilever material ranked by modulus of elasticity ratio

Material Modulus of Elasticity Ratio (E/ρ)
C(s, diamond) Diamond 28551 (kg/mm²)
B Boron 19565 (kg/mm²)
Be Beryllium 14674 (kg/mm²)
Al2O3 Sapphire or Ruby 8000 (kg/mm²)
Al Aluminium 2750 (kg/mm²)
Ti Titanium 2422 (kg/mm²)

The data, organised in this way, reveal that diamond is the best material for a stiff cantilever. But diamond (apart from the obvious expense), is notoriously hard to work and is rarely used in this application. Only the very top of the top-end products have attempted to use diamond as a cantilever. An example is the Koetsu Bloodstone Platinum cartridge, at the price of a new, small car.

Boron (atomic number 5, see appendix) is a metallioid — a chemical element with properties intermediate between those of typical metals and nonmetals. It has the highest modulus of elasticity ratio of any of the known elements except carbon in the form of diamond. Like diamond, boron forms covalent bonds which make it different from the metals aluminium, titanium, and beryllium, all of which form metallic bonds. The main uses of boron in industry are in the production of high-strength glass and ceramics (e.g. Pyrex) and as a component of metal alloys (boron steels). The isotope boron-10 is also used to make the control rods in atomic power stations.

It is difficult to produce a dense body of boron by casting or rolling methods so, the production of a boron cantilever involves "growing" the part by chemical vapor deposition (CVD). This is an involved and expensive process. Boron is an excellent choice for a rigid cantilever material and many high-end cartridges employ boron cantilevers, although, price and scarcity of the element are making it a more infrequent choice.

Beryllium, lightest member of the alkaline-earth metals (atomic number 4, see appendix) would seem to be the next best choice for the cantilever material. Shure made high-end phono cartridge cantilevers from beryllium for a time and some manufacturers use beryllium as a component of an alloy. But this metal has serious problems in manufacture due to its toxidity. Breathing in beryllium in the workplace may cause berylliosis — a dangerous and persistent (and sometimes fatal) lung disorder which can also damage other organs, such as the heart. Beryllium is therefore nowadays rarely used for this application.

Emerald is a naturally occurring compound of beryllium.

The mineral corundum (Al2O3) in the form of ruby or sapphire are both suitable for rigid cantilever construction (especially as both are made industrially) and are the choice of material in a range of high-end phono cartridges. The photograph (right) illustrates a cantilever formed from a solid rod of industrial ruby. From a conspicuous-consumption viewpoint, nothing comes close on to ruby on aesthetic grounds. Even diamond doesn't look as good. Boron — for all its material credentials — is, by contrast, positively drab.

The transition metal titanium (Ti), despite its low density, and high strength and its extensive use in aerospace and medicine, actually scores worse than aluminium for this application. Its most important property of corrosion resistance is hardly an issue in such a coddled application.

Shape and form

Cantilever performance is not simply dependent on the choice of materials: form matters too.

The tapered cantilever shape offers a refinement over a pure cylindrical form. Effective mass (seen from the stylus' point of view) increases the further mass is from the cantilever fixing (bearing). So, if the mass of the cantilever can be minimised towards the stylus, performance will be optimised. And a hollow pipe rather than a solid structure lowers effective mass too compared with a solid rod5.

It is in these ways that metallic aluminium (and lightweight aluminium alloys such as Duralumin) despite their somewhat "dowdy" image next to the exotic materials of diamond, boron, rubies and sapphires may score as a designer's choice of cantilever material8. Aluminium remains the material used for the great majority of phono cartridge cantilevers. Whilst not possessing the stiffness of the other materials, it is light, plentiful, cheap and easy to work and shape. Along with another surprise.....

Compliant cantilever

It turns out that an infinitely stiff cantilever may not be what's required anyway! It is possible to modify the simple electrical circuit analogy above to include the effect of the cantilever¹. First, let's reconsider the simplified model which eliminated the cantilever effects.

The various mechanical elements are represented by the electrical components as follows:

C1, represents - Record tip compliance µF = µ(cm/dyne)
C2 represents - Bearing compliance µF = µ(cm/dyne)
L1 represents - Inertia of armature (tip, cantilever and magnet) mH = mg
R represents - Viscous damping of bearing ohms = dyne-sec/cm
Ii represents - Recorded velocity Amperes = cm/s
Io represents - Output velocity of stylus Amperes = cm/s
Vab represents - Tracking force volts = grams

Now, let's split L1 (which represents inertia of armature: tip, cantilever and magnet) into two parts: L2 and L3; the former representing the tip end of the stylus assembly, and the latter, the magnet end of the stylus assembly. And let's add a new compliance (C3) to represent the compliance of the cantilever (as illustrated below).

We perform the same analysis as we did before for the voltage Vab - which represent tracking force (volts = grams). But, this time, we include C3. We notice significant changes to the results obtained as we alter the value of this capacitor (representing a compliance).

The curves are as illustrated (right). Compare this with the earlier graph. The red curve in this illustration, is the same as the red curve in the earlier graph to make comparison easier. That's to say a dampling value of 40Ω and C3 = 0.

Note the effect as the value of C3 is increased.

Above a certain point, C3 > 0.1µF (where µF = µ(cm/dyne)) the performance arguably deteriorates. But there is very clearly a broad range of values, where the tracking force requred to keep the stylus in the groove is reduced and the tracking is improved over the last octave of the recorded range with little or no effect upon the midrange performance.

This improvement is due to the non-zero compliance of the cantilever.

And, it's not simply the tracking which is improved. If we solve for Io, the output velocity of the stylus (in Amperes = cm/s), it demonstrates that the frequency-response is flattened too 7.

A mechanical explanation

We can see a mechanical explanation for this counterintuitive result illustrated (left). The diagram shows the performance of the stylus, cantilever and magnet as overlaid images at successive instants of time, so that we see the positions of the components as if in consecutive frames of a movie film6.

The image illustrates that the bending of the cantilever partially decouples the movement of the stylus from the movement of the magnet. Whilst we might assume that we normally don't want this (on the basis of arm-waving, maximum information retrieval, maximum transmission of microvibrations type arguments), the model clearly demonstrates that, if the resonance of the cantilever is arranged to be broadly the same as the resonance of the tip end of the shank and the record material, the destructive effect of this latter resonance may be ameliorated by the former.

Which goes to prove that superficial arguments that a boron cantilever is automatically better than an aluminium cantilever seriously oversimplify the situation. The cantilever mass distribution (which may be modelled by altering the ratio of L2 and L3) and its compliance play an indispensable part in the overall performance of the cartridge.

".....superficial arguments that a boron cantilever is better than an aluminium cantilever seriously oversimplify the situation......... replacing the original manufacturer's cantilever with a .. component of more exotic material ought to be approached with caution."

Cantilever fever

Shure's Roger Anderson and his team¹ demonstrated that cantilever design was an integral part of a high-performance phono-cartridge. (It should be noted that the Shure team took great pains that this circuit analogy echoed the performance of real-world physical components.)

An infinitely stiff cantilever may not always offer the best solution and should certainly only be employed when an advanced stylus profile (like Shibata) ensures that the tip-plastic resonance is pushed to as high a frequency as possible. A rigid cantilever will offer no relief from this inevitable resonance on the tracking ability of the stylus. And tracking is king: an unyielding cantilever could introduce high-frequency mistracking with its insidious ability to damage records without betraying the damage it is doing.

Mathematical analysis (via circuit analogy) warns us that "upgrading" an existing cartridge by replacing the original manufacturer's cantilever with a "better" higher-performance component of more exotic material ought to be approached with caution.

Any cartridge (irrespective of its use of extravagant materials) needs to be carefully evaluated with extensive tracking tests before it is used for precision needle-drop recordings of valuable records.

Appendix 1 - Cantilever materials in the periodic table of the elements

It's fascinating to note how the finest cantilever materials are adjacent in the periodic table. Beryllium (Be), Boron (B) and Carbon (C) are all adjacent in the same period. Beryllium (Be) is suitable but for its toxidity. Aluminium (Al) is in the same group as Boron and aluminium is suitable either as a refined metal or as aluminium oxide, in the crystalline form of ruby and sapphire (Al2O3). Magnesium (Mg) is sometimes found in an alloy with aluminium. Carbon, in the crystalline form of diamond, is best of all.

Appendix 2 - Mechanical equivalents

There exist fundamental similarities between electrical and mechanical networks. Not only is this simply interesting, it means that there are novel ways of approaching old problems. For example, the design of loudspeakers or phonograph pickups which, as engineering problems are normally conceived in the mind’s eye as a combination of springs and weights and dampers, may be modelled as the design of AC electrical networks and analysed using electrical filter theory.

There is a delightful irony here. First lessons in electricity always liken voltage to force or pressure, resistance to friction, and so on. But electrical engineers (despite starting later) have overtaken mechanical engineers to such an extent that it is often worthwhile translating mechanical quantities into electrical to get the benefit of the highly developed state of electrical circuit analysis and measurement techniques.

In the, so called, direct analogy:

  • Force is represented by e.m.f. (voltage),
  • Velocity is represented by current, and thus because d(displacement)/dt = velocity and dQ/dt = I,
  • Displacement is represented by charge
  • Mass by inductance,
  • Compliance by capacitance (in mechanics, compliance is the inverse of stiffness)
  • Resistance, reactance, impedance, etc., are common to both.
  • Mechanical links, such as rods, are assumed massless, or their masses are represented separately in lumps and are largely treated in the same way as electrical connections.

This means that the electrical Ohm's Law has mechanical analogues. So that since we know,

   V= I × R or V = I Z (if we allow for impedance) , we can deduce,

   force = velocity × mechanical resistance (or impedance), and rearranging,

   mechanical resistance = force ⁄ velocity

This process reveals more, and often less obvious, relationships, such that, electrical power = V × I, so mechanical power = force × velocity. And that compliance which we normally think of in mechanics as displacement ⁄ force is modelled in the electrical equivalent so that capacitance is equal to charge ⁄ voltage.

There even exists an analogy between the voltage that appears on a coil as a result of a change of current (like an HT coil in a car) and the relationship we all learnt at school between force, mass and acceleration because the voltage generated across an inductor is equal to the rate of change of current within its turns. Mathematically we can express this,

   V = L dI/dt

By substituting the analogous quantities in the table above, we can see this is an analogue of,

   Force = mass . d(velocity)/dt or,

   F = mass . acceleration ....... or Newton's Second Law

Bringing in frequency

All the above has ignored frequency. Rather than thinking in terms of rates of change, and because we're thinking about sound reproduction, it helps enormously to know that just as the impedance of a capacitor is purely imaginary and is given by the well known equation,

   Z = 1 ⁄ ( jωC)

the analogous mechanical impedance of a mechanical compliance (Cm) is given by,

   Z = 1 ⁄ ( jωCm )

Similarly, the mechanical impedance of a mass (M) = jωM, just as the electrical impedance of an inductor (L) = jωL


1. Optimizing the Dynamic Characteristics of a Phonograph Pickup. Anderson, C.R., et al. Journal of the Audio Engineering Society April 1966, volume 14, number 2.

2. The Dynamic Vibration Absorber Principle Applied to a High-Quality Phonograph Pickup. GROH, A.R. Presented October 31, 1976, at the 55th Convention of the Audio Engineering Society, New York. Journal of the Audio Engineering Society June 1977, volume 25, number 6

3. Elastomers are not perfectly elastic and lose energy during the compression (or tension) and subsequent recovery. Unlike purely elastic substances, elastomers are viscoelastic which means that a substance has an elastic component and a viscous component. A viscoelastic element may be modelled as a perfect spring with a damper; or, in electrical terms, as a capacitance and a resistor.

4. Further Improvements in the Discrete Four-Channel Disc System CD-4. Owaki, I. et al. JAES June 1972, Vol. 20, No. 5.

5. A Hi-Fi Moving-Magnet Cartridge Using Recent Technology Obata, S. et. al. J.Audio Eng. Soc., Vol.32, No.3, 1984 March

6. Dynamic Modeling and Analysis of a Phonograph Stylus Happ, L.R. J. of the Audio Engineering Society, Volume 27, No. 1/2, 1979 January/February

7. Stylus velocity vs. variation in value of C3 (cantilever compliance).

8. What about high-performance plastics such as carbon fibre reinforced polymers ? Could these materials not be used for cantilevers? They have: Audio Technica make a low-end cartridge (the AT81CP) with a carbon reinforced ABS cantilever. With a modulus of elasticity ratio better than aluminium, it would seem that materials which play such a large rôle in contemporary engineering could be employed.

9. The Rational Design of Phonograph Pickups Hunt, F. V. JAES Vol. 10 No. 4, October 1962. This article sums up Hunt's work and that of his associates.


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