AudioCircle
Other Stuff => Archived Manufacturer Circles => Hagerman Technology => Topic started by: acmn on 17 Oct 2014, 08:53 am

Dear Sir, Jim,
I'd like to comment on your article "Cartridge Loading" which is found here:
http://www.hagtech.com/loading.html
If the goal is to achieve critically damped frequency response, I think that there is an error
in the optimizing formula:
R_{opt} = sqrt(L/C).
Using this value results to underdamped system having slight peak in the gain curve and
ringing in the time response for the step input. That's because the damping factor is 0.5,
not 1 as it should be. So the formula should read like this:
R_{opt} = sqrt(L/C)/2
or, if the internal resistance of the cartridge coil (R_{C}) is taken into account, like this:
R_{opt} = L/(2sqrt(LC) + R_{C}C).
The deduction of the latter formula is attached below. The language used is finnish, but
the document should still be readable because mathematical language is international...
(vaimennuskerroin (finnish) = damping factor = 1/Q/2, Q = quality factor)
...regards Mika Korpela.
(http://www.audiocircle.com/image.php?id=107109)

Indeed, for critical damping one would use 0.71, which leads to zero peaking in the time domain. For frequency domain 0.50 will get you even more bandwidth, which is what cartridge manufacturers were trying to achieve. Basically, there is more than one way to optimize! You could also go for most linear phase, which I believe is something like 0.86.
My online article was merely attempting to demonstrate the general effects using a simplified model  aimed at audiophiles  not physicists.
jh

Thank you for your reply, Jim.
The fact is that there is no peak in the frequency domain (gain curve) until the damping factor
is reduced below 0.707 (= 1/sqrt(2)). But, by definition, the system is then already under
damped with tendency to ringing. The frequency of this ringing is given by formula:
f = f_{0}sqrt(1  z^{2}); z = damping factor, f_{0} = nominal corner frequency of the system.
(f_{0} = sqrt((1 + R_{C}/R_{L})/(LC))/(2π) = 1/sqrt(LC)/(2π) in practice.)
Here's some data showing what happens when damping factor is reduced below 1.0 by increasing R_{L}:
damping factor (z) gain at f_{0} ringing frequency peaking frequency (= f_{0}sqrt(1  2z^{2}))
1.0 6dB no ringing no peaking ; R_{L} = L/(2sqrt(LC) + R_{C}C)
0.707 3dB 0.707f_{0} no peaking
0.5 0dB 0.866f_{0} 0.707f_{0}
0.25 +6dB 0.968f_{0} 0.935f_{0}
Peaking frequency is always less than f_{0}. So f_{0} can be regarded as upper frequency limit in all
cases with z ≤ 1. It depends on R_{L}, but only slightly, and in fact increasing R_{L} makes it smaller.
So, here's the lesson to be learned: increasing R_{L} in order to decrease damping factor below
1.0 will not extend the gain curve, it only puts emphasis on the high frequencies...and causes
high frequency ringing in the time domain!
Now, in the light of all this, what would be the optimal choice for damping factor?
I would opt for z = 1.0, and I think, so would our ears too...
P.S. Optimal R_{L} for my Pickering cartridge is calculated below...for 3 values of C.
L = 930mH, R_{C }= 1.3kΩ
C/pF R_{L}/kΩ f_{0}/kHz
200 33.8 12.1
150 39.0 13.9
100 47.9 17.0

In a way you could say that there is more bandwidth when damping factor is reduced below 1.0
because the 3B point is then higher. But it will always stay below 1.55f_{0}. It is 0.64f_{0} for z = 1
and 1.27f_{0} for z = 0.5. So there is something gained but at the cost of ringing and peaking
in the frequency domain.
Critical damping being the best choice when tampering with R_{L} is not a new idea.
It was brought up by Mr. Van Alstine already in 80's, in March 1982 issue of Audio Basics.
The article is found here (starting from page 6):
http://www.avahifi.com/images/avahifi/root/audio_basics/ab_pdf/ab1982.pdf
The formula for optimal R_{L} is shown in page 7. It can be simplified to the form that
I gave in my first post.