Question 5.8: (a) Find the period and the cyclic and radian frequencies fo......
(a) Find the period and the cyclic and radian frequencies for each of the following sinusoids:
v1(t) = 17 cos(2000t – 30°) V
v2(t) = 12 cos(2000t + 30°) V
(b) Find the waveform of v3(t) = v1(t) + v2(t) V:
(a) The two sinusoids have the same frequency ω0 = 2000 rad/s since a term 2000t appears in the arguments of v1(t) and v2(t). Therefore, f0 = ω0/2π = 318.3 Hz and T0 = 1/f0 = 3.14 ms.
(b) We use the additive property, since the two sinusoids have the same frequency. Beyond this checkpoint, the frequency plays no further role in the calculation. The two sinusoids must be converted to the Fourier coefficient form using Eq. (5–23)
a = V_{A}\cos \phi
b = -V_{A} \sin \phi (5–23)
a1 = 17 cos(-30°)= +14.7 V
b1 = -17 sin(-30°)= +8.50 V
a2 = 12 cos(30°) = +10.4 V
b2 = -12 sin(30°)= -6.00 V
The Fourier coefficients of the signal v3 = v1 + v2 are found as
a3 = a1 + a2 = 25.1 V
b3 = b1 + b2 = 2.50 V
The amplitude and phase angle of v3(t) are found using Eqs. (5–24) and (5–25):
V_{A} = \sqrt{a^{2} + b^{2}} (5–24)
\phi = \tan ^{-1} \frac{-b}{a} (5–25)
V_{A} = \sqrt{a^{2}_{3} + b^{2}_{3} } = 25.2 V
\phi = \tan^{-1} (-2.5/25.1) = -5.69°
Two equivalent representations of v3(t) are
v3(t) = 25.1 cos(2000t) + 2.5 sin(2000t) V
and
v3(t) = 25.2 cos(2000t – 5.69°) V