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In the context of complex frequency of RLC circuits, the real part is called neper frequency, according to units it's understandable that it has 1/s as the unit which is same as frequency but what is repeating at this frequency as I nowhere see any repetition? Why is it a frequency? I only understand that it decreases or increases the amplitude of sinusoidal function as time increases.

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2 Answers 2

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the neper represents attenuation, or energy loss. the rate of energy loss will then be so many nepers in such an amount of time, or so many nepers per second. As such it "looks" like a frequency (cycles per second) even though it is not oscillating.

Bear in mind though that a more useful and convenient measure of energy loss in oscillating systems is the damping coefficient

zeta = (nepers per second)/(cycles per second at resonance)

which is less confusing to work with.

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  • $\begingroup$ thank you for the answer but could please elaborate on " As such it "looks" like frequency although not oscillating" because I think oscillating is what defines frequency. $\endgroup$ Jul 5, 2019 at 15:17
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Here's one way.

Expand the exponential function into its MacLaurin series:

$e^{x} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \tag*{}$

We know that in physics, equations must be dimensionally consistent. In other words, both sides of an equation must have the same dimensions (units if you will). Let's substitute $x = \sigma t$ in the previous series:

$e^{\sigma t} = 1 + \sigma t + \dfrac{(\sigma t)^2}{2!} + \dfrac{(\sigma t)^3}{3!} + \cdots \tag*{}$

Notice that in order for the previous equation to be dimensionally consistent, because the first term of the right-hand side is dimensionless, all other terms as well must also be dimensionless. This means that the second term, $\sigma t$, must be dimensionless. We know $t$ has dimension of time (SI units: seconds), thus $\sigma$ must have dimension of inverse of time (SI units: 1/seconds), i.e. dimension of frequency. $\sigma$ is the Neper frequency, measured in Nepers per second (Np/s), which is really just 1/s, similar to how the angular frequency is really measured in 1/s instead of rad/s.

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