MM cartridges and capacitance

Al,
Thanks.
With regard to the fraction... My math is OK. I misspoke.
I see frequency as being a number X greater than one. I see a higher frequency as being a number greater than X which is greater than 1. If I take one cycle, it's frequency (stated as a fraction) is a number Y less than one and greater frequency is an even lower number.

I expect it is the case that I do not understand where the decimal place is and how sqrt(faradsxhenrys) changes to frequency.

T-bone, if we use units of henries for L, and units of farads for C (which results in F being in units of hertz), then both L and C will be far less than 1. In fact C will be on the order of 0.0000000001 farads (or 100 pf). Therefore we will end up with a number in the denominator that is much smaller than 1, with the numerator being equal to 1. Therefore F, in units of Hertz, will be a number that is much larger than 1.

As for Farads x Henries resulting in frequency, after application of the square root and reciprocal functions and some constants, yes that is not intuitively obvious. But the derivation is as follows:

Resonance occurs at the frequency at which the magnitude of the impedance of L becomes equal to the magnitude of the impedance of C. Since the impedances of L and C have opposite polarities in the complex plane (don't ask!), the equal and opposite magnitudes will cancel each other out at that frequency, resulting in a net impedance of zero (apart from the resistance that is present).

The magnitude of the impedance of an inductor, aka its inductive reactance, is measured in ohms and is equal to 2 x pi x F x L.

The magnitude of the impedance of a capacitor, aka its capacitive reactance, is also measured in ohms and is equal to 1/(2 x pi x F x C).

So to find the resonant frequency we set those two formulas equal to each other, and solve for F.

2 x pi x F x L = 1/(2 x pi x F x C).

Re-arranging that equation:

1 = 2 x pi x F x L x 2 x pi x F x C

Therefore F x F = 1/(2 x pi x L x 2 x pi x C)

Therefore F = 1/(2 x pi x (sqrtLC))

Voila!

Best regards,
-- Al

Al,
You are a scholar and a gentleman.
Thank you very much for that explanation.

Al, Thanks so much. This is the first clear explanation for the apparently paradoxical effect of added capacitance on the treble response that I have come across. Makes perfect sense, now. Your post above should be archived somewhere.

Perhaps a naive question:

If "higher than standard 47k" capacitance was known to optimize frequency response by an MM cart's designers, wouldn't they have included that higher capacitance value in the spec sheet for that cart?

I use MMs almost exclusively and find these discussions most informative, though I've no simple and reliable means to increase the capacitance loading via my MM phono stage.