The Compliance of a Cartridge and the Resonant Frequency of the Tonearm/Cartridge Combo

17.05.2012..

The symbol “C” refers to the compliance of a cartridge. The lower the compliance, the higher the value of C. Practically, there will be more elasticity if the value of C is lower. When a constant or frequent force is applied to the cantilever, its moving at the height of the stylus is indicated in cu (one millionth of a centimetre). This force is measured in Dyne.

Therefore, when we say that an X cartridge has a compliance of 10 cu/dyne at 10 hertz, we affirm that its cantilever will move over 10 millionth of a centimetre.

Consequently, the compliance should be as constant as possible to every direction. This outcome depends on many factors, in particular on how the combo suspension is devised (stylus, cantilever, rubber dampers, coils/magnets) and in which grade these elements can affect the compliance. Besides, for the stereo stylus, the movement of the cantilever inside the groove, which is recorded on both channels, must be aligned as much “circular” as possible.

The compliance can be measured both statically (zero Hz) and dynamically (10, 100 and very rarely 1.000 Hz).

When the compliance is measured in a static way, the mean value is approximately the double of the value we would measure at 10 Hz and the quadruple of the value at 100 Hz.

Tied to those elements that influence the elasticity (better called Elastic Constant or EC), are several expedients to better calculate these correspondences (EC and cu). On average, the value of the compliance at 100 Hz (usually referred to Japanese cartridges) must be simply doubled to find its correspondence at10 Hz. Instead, the static compliance (usually on US cartridges) must be halved in order to find its value at 10 Hz.

 

Knowing the compliance at 10 Hz is very important. In fact, to calculate the Resonant Frequency (RF) of the arm/cartridge combination there is a formula that considers the insertion of the compliance data rated at 10 Hz. This rate refers to the European cartridges, since their compliance is always at 10 Hz.

The RF is the frequency at which the arm/cartridge combination resonates and it is quite evident that is a negative element. A cartridge that resonates does not touch perfectly the groove and, by losing contact with both sides of the groove, it distorts. On the other hand, a cartridge must be kept perfectly stable and “stuck” to both sides of the groove. This result is obtained through the tonearm which has to be “solid as a rock”, to enable the stylus having the highest precision and dynamics in the instantaneous tracking and, at the same time, “light as a feather”, in order to guide the tip towards the centre of the LP all along the recorded spiral.

The RF is very important because it has to fall down into a precise interval of frequencies: some say between 8 -12 Hz, but reasonably it would be better between 9 -11 Hz. On one hand, if the RF is too high, the cartridge/arm combination will not be perfect because, in the record, the musical message is on those same frequencies and the stylus, when is going to track them, exhibits some resonance and influences negatively the sound signal. On the other hand, if the RF is too low, the stylus will resonate on the frequencies where often the records are affected by inaudible spurious notes that are inevitable in the recording phase. Therefore, if the RF is rated between 8 and 12 Hz, where there is no kind of signal, the stylus will not resonate at all.

This calculation takes roughly into account the fact that also the tonearm has its RF, which depends on many factors like the arm pivot, the materials, the shape of the rod and the “height” of the pivot with respect to the stylus. To avoid further complications, let us say that the following calculation offers a good approximation.

 

 

The formula for calculating the RF of an arm/cartridge combo is:

 

RF = A ÷ √ M × C

 

Where:

 

A = 1.000 ÷ 2 π = 159, 23 (you can also use the fixed value of 159)

M = sum of all masses (arm, screws and cartridge)

C = compliance (at 10 Hz)

 

Example. I have an arm of 12 g and a Japanese cartridge weighting 10 g, with a compliance of 5cu/dyne/100 Hz. I want to know the RF of the tonearm/cartridge combo. The first thing I have to do is to transform the compliance from 100 Hz to 10 Hz by doubling the value of 5, which will become 10.

Hence, I will add up the different masses: 10 g (stylus) + 12 g (arm) +1 g (screws) + other things (like gauges, in this case let us say they are zero) = total 23 g.

Then I multiply 10 x 23 = 230

The square root is around 15,17.

Finally: 159 ÷ 15,17 = 10,48 Hz, a value that places itself in the desired interval of 9 -11 Hz.

Finally: 159 ÷ 15,17 = 10,48 Hz; a value that places itself in the desired interval of 9 -11 Hz.

 

In the photo gallery below you can find some arms and cartridges corresponding to this example of RF calculation.

 

All that demonstrates how the compliance factor and the masses at stake are determinant in the choice of the cartridge that has to be matched with the arm or vice versa.

Currently, it is easy to find cartridges with low-compliance that are quite heavy and cartridges with high-compliance that are light because they have been conceived to match with medium mass tonearms. It follows that the problem of the wrong RF goes away also for the beginners, who maybe will not have an optimal but a satisfactory RF.

The big mistake of many users, is insisting in using arms, often aged but of good quality, with an extreme mass.

 

 

Koetsu Urushi Gold, exactly 10g with 5cu/Dyne at 100 Hz.
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by Massimo Riserbo
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