Question

1. (40 points) Write down the phasor representation for each of the following signals a. (5 pts) x, () 12cos(21) b. pts) x(1)-20cos(3t+45) c. (5 pts)x,(t)-10sin(21 + 300) d. (5 pts) x,(t)-35cos(20+22 sinQ2 +45") e. (10 pts) )- d 135cos(21)+22 sin(2i+4')leas ao express the result in the time domain. f (10 pts) ,)-[B5cos(21)+22sin(2r +45°ydr. Please also express the result in the time domain. Notes: Phasors are ideal for steady-state analysis of sinusoidal signals at a fixed frequency, so we cannot do 35cos(21) +22 sin(3t +45) using phasors. The frequency does not appear explicitly in the phasor representation. . .To add sinusoidal signals using phasors, we determine the phasor representation for each sinusoid, add the phasors using complex arithmetic, convert the sum to polar form, and then write out the result as a time function

Answer 1

Let x (t) = Am cos [wt + phi] 1 ts phasor x (t) = Am < phi we can use Am or A = Am/squareroot 2 Here let us consider x (t)^rightarrow = Am < phi x_1 (t) = 12 cos [2t] x_1 (t) = 12 cos [2t + 0 degree] x_1 (t) = 12 < 0 degree x_2 [t] = 20 cos [3t + 4 pi degree] x2[t]^rightarrow = 20 angle 456 degree \r\n x_3 [t] = 10 sin [2t + 30 degree] sin theta = cos [theta - 90 degree] x_3 [t] = 10 cos [2t + 30 degree - 90 degree] = 10 cos [2t - 60 degree] x_3 [t] degree = 10 angle -60 degree x_a (t) = 35 cos 2t + 22 sin (2t + 4 n degree) 22 cos [2t + 415 degree - 90 degree] x_4 (t) = 35 cos 2t + 22 cos [2t - 45 degree] x_4 (t) = 35 angle 0 degree + 22 angle -45 degree phish to rectangular (complex) form r angle phi = r cos phi + jr sin phi