Re: [Phys-L] Math List Serves

[David Bowman <David_Bowman@georgetowncollege.edu>]

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