Answer:
a. r(t) = 6.40 cos (ωt + 38.66°) units
b. r(t) = 6.40 cos (ωt - 38.66°) units
c. r(t) = 6.40 cos (ωt - 38.66°) units
d. r(t) = 6.40 cos (ωt + 38.66°) units
Explanation:
To find the time-domain sinusoid for a phasor, given as a + bj, we follow the following steps:
(i) Convert the phasor to polar form. The polar form is written as;
r∠Ф
Where;
r = magnitude of the phasor = [tex]\sqrt{a^2 + b^2}[/tex]
Ф = direction = tan⁻¹ ([tex]\frac{b}{a}[/tex])
(ii) Use the magnitude (r) and direction (Φ) from the polar form to get the general form of the time-domain sinusoid (r(t)) as follows:
r(t) = r cos (ωt + Φ)
Where;
ω = angular frequency of the sinusoid
Φ = phase angle of the sinusoid
(a) 5 + j4
(i) convert to polar form
r = [tex]\sqrt{5^2 + 4^2}[/tex]
r = [tex]\sqrt{25 + 16}[/tex]
r = [tex]\sqrt{41}[/tex]
r = 6.40
Φ = tan⁻¹ ([tex]\frac{4}{5}[/tex])
Φ = tan⁻¹ (0.8)
Φ = 38.66°
5 + j4 = 6.40∠38.66°
(ii) Use the magnitude (r) and direction (Φ) from the polar form to get the general form of the time-domain sinusoid
r(t) = 6.40 cos (ωt + 38.66°)
(b) 5 - j4
(i) convert to polar form
r = [tex]\sqrt{5^2 + (-4)^2}[/tex]
r = [tex]\sqrt{25 + 16}[/tex]
r = [tex]\sqrt{41}[/tex]
r = 6.40
Φ = tan⁻¹ ([tex]\frac{-4}{5}[/tex])
Φ = tan⁻¹ (-0.8)
Φ = -38.66°
5 - j4 = 6.40∠-38.66°
(ii) Use the magnitude (r) and direction (Φ) from the polar form to get the general form of the time-domain sinusoid
r(t) = 6.40 cos (ωt - 38.66°)
(c) -5 + j4
(i) convert to polar form
r = [tex]\sqrt{(-5)^2 + 4^2}[/tex]
r = [tex]\sqrt{25 + 16}[/tex]
r = [tex]\sqrt{41}[/tex]
r = 6.40
Φ = tan⁻¹ ([tex]\frac{4}{-5}[/tex])
Φ = tan⁻¹ (-0.8)
Φ = -38.66°
-5 + j4 = 6.40∠-38.66°
(ii) Use the magnitude (r) and direction (Φ) from the polar form to get the general form of the time-domain sinusoid
r(t) = 6.40 cos (ωt - 38.66°)
(d) -5 - j4
(i) convert to polar form
r = [tex]\sqrt{(-5)^2 + (-4)^2}[/tex]
r = [tex]\sqrt{25 + 16}[/tex]
r = [tex]\sqrt{41}[/tex]
r = 6.40
Φ = tan⁻¹ ([tex]\frac{-4}{-5}[/tex])
Φ = tan⁻¹ (0.8)
Φ = 38.66°
-5 - j4 = 6.40∠38.66°
(ii) Use the magnitude (r) and direction (Φ) from the polar form to get the general form of the time-domain sinusoid
r(t) = 6.40 cos (ωt + 38.66°)
