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 Appendix 1 to plot all of your results together.
