A pulse-train voltage of period is for and elsewhere in

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fourier analysis and integral transforms · Function spaces and operator theory

Mathematical Methods for Physics, Ch Fourier series and periodic decomposition

Problem

A pulse-train voltage of period

L

v(x) = h = 5.00\,\mathrm{V}

for

-L/8 < x < L/8

and

v(x) = 0

elsewhere in

-L/2 < x \le L/2

. Its Fourier coefficients are

a_0 = h/2

a_n = \dfrac{2h}{n\pi}\sin\dfrac{n\pi}{4}

for

n \ge 1

, and

b_n = 0

. ((a)) Compute the mean square of

v

over one period directly from its definition. ((b)) Use Parseval's theorem to find the power carried by the constant term, by the

n=1

harmonic and by the

n=2

harmonic, in

\mathrm{V^2}

. ((c)) Find the fraction of the mean square carried by all the harmonics with

n \ge 3

together, without summing any of them. ((d)) In the language of Chapter 16, say in one sentence why the separate powers add with no cross terms between harmonics.

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