energygraphwave
I don't understand how the top graph was used to approximate the bottom graph, could someone please explain to me what was going on?
I know that in regions 1, 3, 5, 7, there is no oscillation and therefore E<U, but I don't know what other information I can extract from the top graph.
That's a "log log" graph. You may notice that the major divisions go up by factors of ten.
The minor divisions in the example, say starting after 100, are 200, 300, 400, and so on up to 900.
log(100) is 2
log(1000) is 3
So halfway between is \$10^{2.5}\$ or about 316.
It is said that just about everything is approximately linear if you plot it on a log-log graph with a fat marker.
A motor like transformers has some excitation current that causes some <10% phase lead on current with respect to voltage (not given/shown.)
But what is shown is the current of each phase with a dominant 3rd harmonic current from the non-linear core ( partial saturation at peak voltage).
By hand drawing a similar sine wave with a 3rd harmonic and adjusting the phase and amplitude, I see that the 3rd harmonic is approximately 41% of the fundamental and the phase is 1.95 radians phase lead. Since reactance increases 300% with 3rd harmonic, we expect this for an inductor. But the core having partial saturation properties shows that the 3rd harmonic content is between 1/3 and 1/2 of the fundamental. THis is stored energy and not contributing to work by the motor. (VAR)
The pump load will also emphasize the current harmonics and sub-harmonics if a gear ratio is involved.(not observed or analyzed) as the load is not linear throughout the pump cycle.
The fundamental above is normalized to 1 @ 0 deg phase near 50Hz with the 3rd harmonic only shown in yellow with composite waveform in red.
You can see the website where to analyze FFT and settings that I used.
Best Answer
I figured it out. From the Schrodinger wave function, psi, we know the following:
$$-\frac{h^2}{2m_0}\frac{d^2}{dx^2}{\psi(x)}+U(x)~\psi(x)=E~\psi(x)$$
Where \$ E > U, ~~~ {\psi(x)}=Ae^{jkx}+Be^{-jkx}\$
Which is sinusoidal/oscillating.
Where \$ E < U, ~~~ {\psi(x)}=Ae^{kx}+Be^{-kx}\$
Which is not sinusoidal, it's exponential.
Now note in the first graph, regions 2, 4, and 6 have an oscillating wave and therefore E > U. On the contrary, regions 1, 3, 5, and 7 aren't oscillating and look exponential. So for each of these regions, they have a corresponding estimate whether U should be above or below \$E\$.