Question

Find the time-domain sinusoid for the following phasors:_________

(a) 5+j4
(b) 5 -j4
(c)-5+j4
(d)-5-j4

Answer 1

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

Step-by-step 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 =
`√(a^2 + b^2)`

Ф = direction = tan⁻¹ (
`(b)/(a)`)

(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 =
`√(5^2 + 4^2)`

r =
`√(25 + 16)`

r =
`√(41)`

r = 6.40

Φ = tan⁻¹ (
`(4)/(5)`)

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 =
`√(5^2 + (-4)^2)`

r =
`√(25 + 16)`

r =
`√(41)`

r = 6.40

Φ = tan⁻¹ (
`(-4)/(5)`)

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 =
`√((-5)^2 + 4^2)`

r =
`√(25 + 16)`

r =
`√(41)`

r = 6.40

Φ = tan⁻¹ (
`(4)/(-5)`)

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 =
`√((-5)^2 + (-4)^2)`

r =
`√(25 + 16)`

r =
`√(41)`

r = 6.40

Φ = tan⁻¹ (
`(-4)/(-5)`)

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°)