A full-wave rectifier turns a sinusoidal input of period into an output voltage

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

Mathematical Methods for Physics, Ch Fourier series and periodic decomposition

Problem

A full-wave rectifier turns a sinusoidal input of period LL into an output voltage v(x)v(x), which we expand in the harmonics cos⁡(2πnx/L)\cos(2\pi nx/L) and sin⁡(2πnx/L)\sin(2\pi nx/L) of the input period. The output satisfies v(−x)=v(x)v(-x) = v(x) for all xx (it is even), and also v(x+L/2)=v(x)v(x + L/2) = v(x) for all xx (it repeats after half of LL, so its own period is L/2L/2, and it is half-wave symmetric rather than half-wave antisymmetric). ((a)) Without evaluating any integral, state which of the coefficients bnb_n must vanish for every nn, and why. ((b)) Without evaluating any integral, state which of the coefficients ana_n (for n≥1n \ge 1) must vanish, and why. ((c)) Name the lowest three harmonics, other than a0a_0, that the two symmetries allow to be non-zero.

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