# Why is neper frequency called a frequency?

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.

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.

• 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. Commented Jul 5, 2019 at 15:17

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.

It is just another naming convention, as far as I observe.

The explanation through comparing units is actually misleading. The unit of stochastic radionuclide events per second (also called "activity") Bequerel(Bq) also has units of inverse time, but does not describe a "repetition" as one may imagine (in fact, the "activity" decreases exponentially with time, let alone repeating in any sense).

The "neper frequency" does not only appear at the "real part" of the complex frequency, as I will demonstrate below. The equation describing a transient response of the series RLC circuit is:

$$\frac{d^2I}{dt^2}+2\alpha\frac{dI}{dt}+\omega_0^2I=0$$

where $$2\alpha=R/L$$ and $$\omega_0^2=1/(LC)$$. For now, think of $$\omega_0$$ and $$\alpha$$ as just symbols to make the equation more concise, not having any sort of physical meaning. The equation above has characteristic equation:

$$\lambda^2+2\lambda\alpha+\omega_0^2=0$$

Which has two roots $$\lambda_1$$ and $$\lambda_2$$ so that the general solution of the original differential equation is:

$$I(t)=Ae^{\lambda_1t}+Be^{\lambda_2t}$$

These $$\lambda$$ values are referred to as "complex frequencies" because when they are purely imaginary, they describe periodic oscillatory behavior. However, that is not the general case: solving the characteristic equation, we compute these values as:

$$\lambda_1=-\alpha+\sqrt{\alpha^2-\omega_0^2}$$ $$\lambda_2=-\alpha-\sqrt{\alpha^2-\omega_0^2}$$

So not only the neper frequency $$\alpha$$ is "the real part of the complex frequencies," but it is also involved in the imaginary part when $$\omega_0^2 > \alpha^2$$ (underdamped) and not at all equal to the real part when $$\omega_0^2 \leq \alpha^2$$ (overdamped, critical when equality holds).

Let us properly assign "physical meanings" to $$\alpha$$ and $$\omega_0^2$$. In the case $$\alpha=0$$ (no damping), you may prove easily that the $$\lambda$$ values will become purely imaginary and the system will periodically oscillate with angular frequency equal to $$\omega_0$$. You yourself seem to understand the physical meaning of $$\alpha$$ as the exponential decay rate, but I demonstrate it again: the general solution factors as:

$$I(t)=e^{-\alpha t}(Ae^{\sqrt{\alpha^2-\omega_0^2}t}+Be^{-\sqrt{\alpha^2-\omega_0^2}t})$$

So in the case when the square root expressions are imaginary, the factor $$e^{-\alpha t}$$ "modulates" (as an exponential decay) the amplitude of the oscillations supplied by the expressions in the parentheses. So why not call it a "decay rate?"

However, looking back at $$\lambda_1$$ and $$\lambda_2$$, we see that a squared angular frequency $$\omega_0^2$$ is subtracted from $$\alpha^2$$, implying (if calculations are correct) that they have identical units, despite having different "physical meanings." Nevertheless, in order not to appear as "subtracting apples (frequency) from oranges (decay rate)," this quantitiy $$\alpha$$ is also named a "frequency" (unlike other units (like Bq) with units of inverse time and exempt from similar problems.)

In analogy to the angular resonance frequency, the neper frequency is more aptly called the exponential transience rate. It is typically a factor in the arguement of an exponential (i.e., $$\exp(-\alpha\,t)$$). Note that the measure of frequency and the measure of time rate have the same dimension (e.g., $$s^{-1}$$).