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-----Original Message-----
From: Phys-l <phys-l-bounces@mail.phys-l.org> On Behalf Of David Bowman
Sent: Thursday, September 22, 2022 6:49 AM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Math List Serves
Regarding Don P's response:
... Yes, I would like some more of your challenge questions. I very
much enjoyed working on (and finally solving) your cosine of multiple
values of pi/5 challenge question. ...
OK, since you asked for more, here are three more, this time from my Fourier
Series series. Their level is probably more like *at least* the upper
undergraduate level than the 1st two years, but that should mean that they
will give your brain cells a better workout than only a typical 1st/2nd year type
problem. Of these, the 1st two are related to AC/DC/AC power conversions,
and the 3rd one is just pure math.
1) Consider a wye-connected 3-phase AC ideal sine wave voltage source, (say
from a power company or an alternator) Suppose, relative to the central
neutral, each arm of the 3-phase system has an RMS voltage, V. The voltage
from all 3 arms is full-wave rectified (using a half-dozen ideal diode rectifiers)
and superimposed to make a DC voltage source (such as for the charging
system for a car's lead-acid battery & electrical system, or for a high power DC
supercharging system for an e-vehicle). This means the DC output voltage is
the instantaneous maximum of the 3 absolute values of the 3 arms' individual
AC voltages. A consequence of this is that the resulting "DC" voltage has a
relatively small AC ripple on it whose fundamental frequency is 6 times that of
the original AC source. The DC output is unfiltered.
A) What is the mean DC voltage in terms of V?
B) What is the AC ripple factor for this DC voltage source? The ripple factor is
the quotient (often expressed as a percentage) of the RMS AC ripple voltage
difference between the instantaneous voltage and its DC background mean
value, divided by the mean voltage value.
C) Extra credit: What is the Fourier series for the AC ripple in terms of V and
the original 3-phase AC source? And how much of the AC power is in the
ripple's fundamental frequency mode?
2) Consider a solid-state inverter that converts a DC voltage source to an AC
one. Suppose the inverter's circuitry can't efficiently convert the DC power to
an actual sine wave, but it can efficiently make piecewise constant and
constant-sloped wave forms. So the AC output is designed to be a piecewise
approximation to a sine wave consisting of locally constant parts
approximating the peaks and troughs of the sine wave, and constant-sloped
parts approximating the parts of the sine wave near the zero crossings. The
whole approximated wave is continuous at the splice points, and each cycle of
the wave has 4 splice points, 2 of them connecting a peak's constant piece to a
constant-sloped rising and to a constant-sloped falling piece, and 2 more splice
points connecting the constant-sloped pieces to a corresponding constant
trough's piece. Thus, each cycle has a rising-sloped part, a flat top, a falling-
sloped part, and a flat bottom part, all continuously connected to each other.
(I think this wave form might be called a truncated triangle wave, but I'm not
sure about that.) Suppose the sine wave to be approximated has an RMS
amplitude of V and a period of T.
A) Find the location of the splice points and the height/depth of the
peak/trough values in terms of V & T that minimizes the total harmonic
distortion, THD of the piecewise approximation to the sine wave being
approximated. BTW, the THD is the quotient of the mean squared deviation of
the approximation from the ideal sine wave divided by the sine wave's mean
squared value, i.e. V^2, over a full cycle.
B) What is that minimized THD?
C) What fraction of the total distortion power is in the lowest nonzero
harmonic distortion frequency above the sine wave fundamental? IOW, what
is the quotient of the 1st nonzero distortion harmonic squared amplitude
divided by the sum over all squared harmonic distortion amplitudes. How fast
does the Fourier distortion series converge?
Somewhat simplifying hint: There are no even-order harmonics present.
Why?
3) Consider a periodic function f(x) defined as a definite double integral over
other coordinates y & z:
f(x) ≡ ∫∫dydz(sin(x)*sin(y)/√(1-(sin(x)*sin(y)*sin(z))^2)) where the lower limit
for both y & z is zero, and the upper limit for both of them is π/2.
A) What shape periodic function is f(x)? How would you describe it?
B) What is the Fourier Series for f(x)?
Somewhat simplifying hint: Note f(x) has the same symmetry and
fundamental frequency as sin(x).
These problems should keep you busy and off the streets for a good long
while. BTW, when these were up on the whiteboard last year there were no
takers for them.
Maybe next time I'll give out a couple of easier ones that are doable with much
less work & involve lower level mathematics with no numerical analysis.
David Bowman
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