# AudioCircle

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

Post by: acmn on 17 Oct 2014, 08:53 am
Dear Sir, Jim,

If the goal is to achieve critically damped frequency response, I think that there is an error
in the optimizing formula:

Ropt = sqrt(L/C).

Using this value results to under-damped 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:

Ropt = sqrt(L/C)/2

or, if the internal resistance of the cartridge coil (RC) is taken into account, like this:

Ropt = L/(2sqrt(LC) + RCC).

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)

Post by: hagtech on 22 Oct 2014, 02:10 am
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
Post by: acmn on 27 Oct 2014, 11:01 am

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 = f0sqrt(1 - z2); z = damping factor, f0 = nominal corner frequency of the system.
(f0 = sqrt((1 + RC/RL)/(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 RL:

damping factor (z)        gain at f0      ringing frequency    peaking frequency (= f0sqrt(1 - 2z2))
1.0                                -6dB             no ringing                no peaking         ; RL = L/(2sqrt(LC) + RCC)
0.707                            -3dB             0.707f0                    no peaking
0.5                                 0dB             0.866f0                    0.707f0
0.25                              +6dB            0.968f0                    0.935f0

Peaking frequency is always less than f0. So f0 can be regarded as upper frequency limit in all
cases with z ≤ 1. It depends on RL, but only slightly, and in fact increasing RL makes it smaller.
So, here's the lesson to be learned: increasing RL 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 RL for my Pickering cartridge is calculated below...for 3 values of C.

L = 930mH, RC = 1.3kΩ

C/pF         RL/kΩ      f0/kHz
200          33.8       12.1
150          39.0       13.9
100          47.9       17.0