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Phasors: Numerical Experiment (Interference Patterns)

Module by: Louis Scharf. E-mail the author

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This module is part of the collection, A First Course in Electrical and Computer Engineering. The LaTeX source files for this collection were created using an optical character recognition technology, and because of this process there may be more errors than usual. Please contact us if you discover any errors.

Let's add two sinusoids whose amplitudes and frequencies are identical and whose phases are different:

x ( t ) = A cos ( ω t + φ ) + A cos ( ω t + φ + ψ ) . x ( t ) = A cos ( ω t + φ ) + A cos ( ω t + φ + ψ ) .
(1)

Show analytically that this sum has the phasor representation

X = 2 A c o s ( ψ 2 ) e j [ φ + ( ψ / 2 ) ] . X = 2 A c o s ( ψ 2 ) e j [ φ + ( ψ / 2 ) ] .
(2)

Interpret this finding. Then write a MATLAB program that computes and plots complex X on the complex plane as ψ varies from 0 to 2π2π and that plots magnitude, |X||X|, and phase, arg X argX, versus the phase angle ψ ψ. (You will have to choose ψ=n2πN,n=0,1,...,N-1ψ=n2πN,n=0,1,...,N-1, for a suitable N.N.) When do you get constructive interfelence and when do you get destructive interference? Now compute and plot x(t)x(t) versus t t (you will need to discretize t t) for several interesting values of ψ ψ. Explain your interference results in terms of the amplitude and phase of x(t)x(t) and the magnitude and phase of X X. Use the subplots discussed in "An Introduction to MATLAB" to plot all of your results together.

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