Download Problem 7.15 The function υ(t) is the sum of two sinusoids, υ(t

Problem 7.15 The function υ (t) is the sum of two sinusoids,
υ (t) = 4 cos(ω t + 30◦ ) + 6 cos(ω t + 60◦ ) V.
(1)
(a) Apply the necessary trigonometric identities from Table 7-1 to show that
υ (t) = 9.67 cos(ω t + 48.1◦ ) V.
(2)
(b) Transform the expression given by Eq. (1) to the phasor domain, simplify it
into a single term, and then transform it back to the time domain to show that
the result is identical with the expression given by Eq. (2).
Solution:
(a)
υ (t) = 4 cos(ω t + 30◦ ) + 6 cos(ω t + 60◦ )
= 4[cos ω t cos 30◦ − sin ω t sin 30◦ ] + 6[cos ω t cos 60◦ − sin ω t sin 60◦ ]
= 6.46 cos ω t − 7.20 sin ω t.
If υ (t) = A cos(ω t + θ ) = A[cos ω t cos θ − sin ω t sin θ ], then
A cos θ = 6.46,
A sin θ = 7.20,
from which we deduce that
p
6.462 + 7.22 = 9.67,
7.2
−1
θ = tan
= 48.1◦ .
6.46
A=
Hence,
υ (t) = 9.67 cos(ω t + 48.1◦ )
(V).
(b)
◦
V = 4e j30 + 6e j60
◦
= 4 cos 30◦ + j4 sin 30◦ + 6 cos 60◦ + j6 sin 60◦
= 6.46 + j7.196
= 9.67e j48.1
◦
(V)
υ (t) = 9.67 cos(ω t + 48.1◦ )
(V).
c
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