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The Korf Blog

The inside story: our research,
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2 April 2020
Tonearm and Cartridge Matching, Part IV
This is the fourth and final post in the series on low frequency interaction between the tonearm and the cartridge. Unfortunately, this series hasn't really gone as planned. I've hoped to quickly go through the experiments and then focus on the solution. But the current coronavirus epidemics threw a spanner into the works. We'll try to complete it anyway, but some of the most interesting bits would probably have to be left for later.

In our previous posts, we've formulated The Plan, and done the three bullet points: measured the Ortofon/Jelco and Denon/Jelco combos, plus we checked how the LF content of real LPs is handled.
Make sure our test rig is working fine and is picking up both high and low frequency resonances. We'll measure the low frequency set with it, and superimpose it over the usual 20Hz-20kHz sweep.
Change the cartridge to the one with different compliance, and see what the effect on the low frequency resonance would be.
See what the low frequency content of the usual LPs looks like. We'll use some nearly unplayable LPs from our collection to try and get the effects of warps and excentricity.
Do the analysis of the data and see if there are some recommendations to be made on matching tonearms and cartridges.
The data we have gathered so far completely disagrees with existing theoretical predictions. The observed motion of the tonearm and cartridge combination does not look like the resonant motion at all.

Now is the time for bullet point 4: what do we actually see then?

The Recap
We have used 3 cartridges with widely varying compliance (Denon DL-103, Ortofon SL-15E and Shure M97xE) to see how low frequency excitation causes tonearm to move.

The current, but very much dated, wisdom suggests there is a well defined low frequency elastic resonance formed by the cartridge's compliance and the tonearm's effective mass. Such resonance should cause defined peaks in tonearm acceleration. With the change of either the compliance or the effective mass, these peaks should shift in frequency. The illustration to the right shows what the measurement result for cartridges with compliance of 6 and 12\( \cdot 10^{-6}\) cm/dyne should look like.

And this is the typical picture of what is happening in reality. The differences in compliance result mainly in amplitude changes, while the frequency remains the same. It's determined by the stimulus and does not resemble the calculated tonearm/cartridge resonance.

But the amplitude of peaks does seem to inversely track the compliance. Is there a reasonable explanation to all of this?

The Theory
The original Carlson equation is purely in frequency domain. There's mass m and compliance c, the result is frequency f, but there is no motion to be seen:

$$f={1\over{2\pi\sqrt{m \cdot c}}}$$

But what if we try to solve it in the motion domain? What if, instead of just concerning ourselves with frequency, we'll try to model the amplitude instead?
Here's a rough mechanical representation. The mass M is excited by the motion of point A (the stylus) through the compliant spring C (cartridge suspension). The motion of B would then give us the movement of the tonearm at headshell.

There is one problem with this approach. If we assume C is perfect, we'll be back to square one—solving this in the motion domain would give us infinite amplitude at the natural frequency. This clearly isn't happening, so we must introduce losses in C. This is easiest to do with a Q factor that defines the relationship between the energy stored and the energy lost.
I am not a maths wizard, and I have asked a brilliant engineer Mr Alexander Valencia-Campo to help me with derivation. Here's what he came up with:

$$A_{b} = A_{a}w_{0}^{2}\dfrac{1}{\sqrt{ (w_{0}^{2} - w^{2})^{2} + w^{2}w_{0}^{2}/Q^{2} }}$$

\(A_{a}\) is amplitude with which the external force excites point A (the stylus motion, basically)
\(A_{b}\) is the resulting amplitude in point B (how far the tonearm deflects as measured at headshell)
\(Q\) is the damping factor
\(w\) is the frequency with which point A moves
\(w_{0}\) is the system's natural frequency \(\sqrt{c \over m}\). Exactly the same as in Carlson's formula, only written in terms of angular frequency instead of Hertz. It's worth noting that once we start to account for losses and introduce Q, it's no longer the same as the system's resonant frequency. Here's how they are related:

$$w_{res}=w_{0}\sqrt{1-\dfrac{ 1}{2Q^{2}}}$$

The derivation is available in a linked PDF file.
Quite a mouthful, isn't it? Now you see the appeal of thinking in pure frequency domain—it's simple. Simple, but wrong.

But, of course, complexity doesn't guarantee correctness. How does it fit our experimental data?
To do a test, we've assumed the tonearm effective mass to be 25 gram, and plotted the frequency dependence for 3 different compliances: 5, 15 and 25 \(\cdot 10^{-6}\) cm/dyne.
If we look just at the motion, it's pretty clear that we are dealing with a simple lowpass filter. The higher the compliance, the lower the cutoff frequency. Very high compliance results in steeper slope, but more "ringing". Insufficient compliance gives us a very shallow slope, with bass energy ending up in the tonearm instead of being picked up by the cartridge.

Magnetic cartridges register the rate of change, so recalculating the above in terms of acceleration might give us a better picture. Also, we've measured acceleration, so this would allow for direct comparison.
That's pretty cool, isn't it?

Compare it with the low frequency sweep measurements from two posts ago. That hump at 6 Hz—that's probably our "ringing". And if we smooth the curves and adjust for wildly different levels , I think we'll get a pretty good match.

The Summary
Just like with our previous travels in the land of azimuth adjustment, the accepted wisdom turned out to be completely wrong. It took us a while to find the right path, and we needed outside help to tame the maths. Here's what we have learned through our journey in the low frequency domain:
Carlson's formula of a low frequency resonance does not describe the measured low frequency behaviour of the cartridge/tonearm interaction
Modern cartridges (meaning all those built in the last 60 years or so) have too much suspension damping and non-linearity for the resonances to dominate
The frequency of the observed motion is determined largely by the frequency of the excitation
The cartridge/tonearm system acts as a lowpass filter for vibrations picked up by the stylus
Too low an effective mass for a given compliance (or too low a compliance for a given effective mass) results in low frequency attenuation and excessive tonearm motion.
Too high an effective mass for a given compliance (or too high a compliance for a given effective mass) results in "ringing"—a small resonant peak—that is largely benign and barely registers in the measurements
But how can we make this insight usable? Is there an easy way to tell, what effective mass is too low for a given compliance, and what is just right? We are developing an online calculator that would plot the motion and acceleration curves for any given compliance and effective mass. Hope to open it up to everyone in the next couple of weeks. Stay with us!
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