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Old 19th Oct 2017, 3:10 pm   #318
Argus25
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Join Date: Oct 2016
Location: Maroochydore, Queensland, Australia.
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Default Re: Audiophoolery. 'Cable Break In' - I never knew that!

Quote:
Originally Posted by merlinmaxwell View Post
Quote:
you can't take the root of a negative number
Yes you can, it's is called X times i (or j) commonly and frequently used in mathematics. Used also to good (perfect) effect in engineering too.
No, this is not the situation I'm describing, no i or j operators to consider.

Consider the equation solution for the angular frequency of a resonant circuit involving L,C and R:

w = 2.pi.f = square root of ( 1/LC - R^2/4L^2 )

(In most electronic circuits R is small, we use the abbreviated formula).

But as you can see from the formula above if R is greater than 2 times the square root of L/C or over damping, the term in the bracket above is negative, and you cannot take the square root of a negative number (try it on your calculator). So the equation solution cannot provide any utility as a result of the increased damping exceeding a boundary value.

The three equation solutions for damped simple harmonic motion systems are presented elegantly in Erwin Kreyszig's book, Advanced Engineering Mathematics 7th Edn page 84 & 85. Wiley.

Basically for the three scenarios :

Overdamping, has two distinct real roots, critical damping has a real double root and underdamping has complex conjugate roots.

So the the damping conditions have to be tested to select the correct equation format to provide a solution. In other words the form of the mathematical solution is dependent on the damping.

So for one example, if you have a series circuit with L,C and R, and you want to know for arguments sake what the voltage is across the capacitor as a function of time after you apply a fixed voltage across the circuit, you have to select the correct equation solution first, depending on the damping, or the equation will fail to solve it. The same applies to damped resonant mechanical systems, say if you wanted to know the displacement or velocity of the mass with time.
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