Question

4.) Find the amplitudes of each of the following harmonic synchronous oscillations and the phase angle between the oscillations. (20 pts) xi(t) = 3cos(20t) - 4sin(20) x:(t)= 1.5sin(20t - 1/6) - 2cos(20t) 5.) Represent the two harmonics from problem #4 in the time domain, frequency domain, using phasors and complex numbers. (20 pts) 6.) Calculate the amplitude and the phase of the oscillation that results from the composition of the two harmonic synchronous oscillations in problem #4. (10 pts)

Answer 1

given data: X, (t) = 3 os (20t) - de fin (dot) 2,64): 1.5 Sin (2017) - 2 w(dot) Solution: Take X, (t) = 3 s (2st) - 48h (206) x.() - JIH? 3 es(206) h. 24 Sih laot 2² +22 Sin aut 5 5 . + knits 2.(4.5 Casa wa 2012 2. (t) = 5 (cas & cos aot - gin of Sin 20t] 2 (te con sotto where O= cost (2) 53.18 Resultant R = 5 Take no(t) = 1.5. Sin (Qut- ) - 2as (oot) kg (t) = 1,5 (Sin 2st was % - Bor 2st &n 7/6) - 2 eus 20k X, 6 ( Sh 80t - los cos 20t] got х,0) = (sin(ast) (1.299) - (0.15) cws (20t)] - 2 cos 20t : (1.219) In 20t - 2.15 cas (201) Mg(t) JC: 29.9797 6.15) Sin (Qot) dots soslaot) 1.2992 28762 ~ 1.299, (275) Ka (6) Sin 200- Sin a los 20t Peix (204-6) = 1.5 2 con 1.299 3.0h1 (caso 12 (1) = 3.041

9, = cosi (1.289 3.0L) = 64.91 (sultant R = 3.04) units Dhase angle between di(k) & Kalt) = x.(t) = 5 cos (2017¢ ) ) 5 8h (206+8, +1) Yo lo 3. Ohl Bin (206-80) =) 206+ 0 got + 4 + 1 h = 53.15't 62.71 + a 0 +82 + 1 / (90) . 207.8 a + 207.87 Positive sign indicates xi(t) dogs beg [207.84 oms