Time Constant

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Dr3d Sl3d
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Time Constant

Post by Dr3d Sl3d »

The 19th Greek letter, Tau (Image), appears in the formula used to derive the "time constant" of the corner (or, transitional) frequency of an audio filter, such as one of the pole or zero nodes of the RIAA emphasis equalization standard, used by LP and 45 rpm records' groove modulation signals. The Ortofon manual gives a simplified formula: 159 / F = Image. (F = the filter's center-, or, transitional, frequency, in kHz.)


The fact that audio filters are easily modeled by RC circuits results in the nodes of the filters often being referred to as their time constant value, rather than their frequency in Hertz, since capacitors (the "C," in "RCL" filter) not only behave electrically in a way that causes energy to be transferred with frequency-dependent sensitivity, but achieve this effect by requiring finite time to fill and empty their stored charge, based on package construction and materials. There may or may not be enough time for the cycle of a given frequency to be fully stored and retrieved from the metaphorical bucket that is the series (or parallel) capacitor in a communication line. The bucket size is its capacity and is relative to the capacitance of the chosen condenser in the analogy. So, the "time" needed to fill the specified bucket is "constant," but the duration of the flood of each electron (hole) tide (caused by the cyclical hang time of the excursions of the electrical analog of the audio waves into the positive or negative side of the ground reference), may vary... and may not be able to survive the electrical journey intact. Part of each "flood" will be caught by the bucket-like aspect of the capacitor and keep that part of the bandwidth out of the flood.

The formula often seen, and in fact given in the Ortofon cutting amplifier manual, is perhaps over-simplified in the following term: 159. For, as it appears in the formula, "159/f (in kHz)" = Image, there appears to be numbers significantly truncated to the right of the unwritten decimal place. If you used 159, as is, you'd think that 75 µs = 2,120 cycles per second. However, the origin of this constant, "159," is, I believe, the formula for deriving the center/transitional frequency of an RC filter. The solution is:

1/2πRC.


π = 3.14159265358979...

2π = 6.28318530717958...

1/6.28318530717958... = 0.1591549430919...

But, we want to use a frequency centered around 1 kHz and we want to have a relatively simple number for the time constant, so, we multiply 0.1591549430919.... by 1 k and get: 159.1549430919...


Now, if we plug that into the Ortofon formula, in order to derive the time constant, but this time, invert the equation so that we can find the true frequency of the standardized RIAA time constants, we see that 75 µs is not 2,120 cycles per second. It is just above 2,122 cycles per second, if you don't round off the numerator in the original formula.

It might seem as if the difference between 2120 cps and 2122 cps is insignificant, but if they were street numbers, you could be at the wrong house!

Also, it's, therefore, not really 500 cycles per second; it's 500.487... cycles per second (;

And same with 3,180 µs and 50.0487... cycles per second...
Whereas, 3,183 µs is much closer to 50.00 cycles per second.


Saùde,
Dr3d Sl3d
Cappin' the Rap in the M&M's...
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