Question

Convert the following pairs of voltage and current waveforms to phasor form. Each pair of waveforms corresponds to an unknown element. Determine whether each element is a resistor, a capacitor, or an induotor, and compute the value of the corresponding parameter R, C, or L. 2.

a. v(t) = 20 cos(400t +30), i (t) = 4 sin(400t – 120)
b. b. (t) = 9 cos(900t - 60), i (t) = 4 sin(900t + 280)
c. c.vt) - 13 cos(250+ + 60), 1 (t) = 7 sin(250t + 240)

Answer 1

a) V = 20 ∠30⁰ , I = 4 ∠-210⁰ Z inductive L = 0,0125 H

b) V = 9∠-60⁰ , I = 4 ∠ 190⁰ Z capacitive C = 4,94 *10⁻⁴ F

c) V = 13 ∠240⁰ , I = 7 ∠ 150⁰ Z Inductive L = 0,0074 H

Step-by-step explanation:

a) v(t) = 20 cos (400*t + 30 )

Phasor form V = 20 ∠30⁰

i(t) = 4 sin (400*t - 120)

First we need to transform 4sin( 400t - 120 ) as function cosine

we know that sin ( x + 90 ) = cos x

Then sin ( 400*t -120 ) = cos ( 400*t - 120 -90 ) = cos ( 400t - 120 - 90)

Phasor form I = 4 ∠-210⁰

To have the impedance nature we compute

Z = V / I ⇒ Z = 20 ∠30⁰ / 4 ∠-210⁰ Z = 5 ∠-180⁰

We notice that voltage advances the current then we are in presence of an inductive impedance

5 = wl ⇒ 5 = 400 *L ⇒ L = 0,0125 H

b) v(t) = 9 cos ( 900t - 60 )

V = 9∠-60⁰

i(t) = 4 sin ( 900t + 280 ) ⇒ i(t) = 4 cos ( 900t + 280 - 90)

i(t) = 4 cos (900t + 190 ) ⇒ I = 4 ∠ 190⁰

Z = V/I ⇒ Z = 9∠-60⁰ / 4 ∠ 190⁰ Z = 2,25 ∠-250

In this case the current advances the voltage. Impedance capacitive

1/wc = 1/ 900*C 1/wc = Z ⇒ 2,25 = 1/ 900*C

2,25*900 = 1/C ⇒ 2025 =1/C ⇒ C = 4,94 *10⁻⁴ F

c) v(t) = - 13 cos ( 250t + 60 )

v(t) = 13 cos ( 250t + 60 +180 ) ⇒ v(t) = 13 cos ( 250t +240)

Phasor Form

V = 13 ∠240⁰

i(t) = 7 sin (250t + 240 - 90) ⇒ i(t) = 7 sin (250t + 150)

Phasor Form I = 7 ∠150⁰

Z = 13∠240⁰ / 7 ∠150⁰ ⇒ Z = 1,86 ∠ 90⁰

Voltage advances the current then the impedance is inductive

wl = 250L 250 L = 1,86 L = 1,86/250 L = 0,0074 H