chap09

# chap09 - Chapter 9 Solution 1(a(b angular frequency...

This preview shows pages 1–6. Sign up to view the full content.

Chapter 9, Solution 1. (a) angular frequency ω = 10 3 rad/s (b) frequency f = π ω 2 = 159.2 Hz (c) period T = f = 1 6.283 ms (d) Since sin(A) = cos(A – 90 ° ), v s = 12 sin(10 3 t + 24 ° ) = 12 cos(10 3 t + 24 ° – 90 ° ) v s in cosine form is v s = 12 cos(10 3 t – 66 ° ) V (e) v s (2.5 ms) = 12 ) 24 ) 10 5 . 2 )( 10 sin(( 3 - 3 ° + × = 12 sin(2.5 + 24 ° ) = 12 sin(143.24 ° + 24 ° ) = 2.65 V Chapter 9, Solution 2. (a) amplitude = 8 A (b) ω = 500 π = 1570.8 rad/s (c) f = π ω 2 = 250 Hz (d) I s = 8 -25 ° A I s (2 ms) = ) 25 ) 10 2 )( 500 cos(( 8 3 - ° × π = 8 cos( π 25 ° ) = 8 cos(155 ° ) = -7.25 A Chapter 9, Solution 3. (a) 4 sin( ω t – 30 ° ) = 4 cos( ω t – 30 ° – 90 ° ) = 4 cos( ω t – 120 ° ) (b) -2 sin(6t) = 2 cos(6t + 90 ° ) (c) -10 sin( ω t + 20 ° ) = 10 cos( ω t + 20 ° + 90 ° ) = 10 cos( ω t + 110 ° )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Chapter 9, Solution 4. (a) v = 8 cos(7t + 15 ° ) = 8 sin(7t + 15 ° + 90 ° ) = 8 sin(7t + 105 ° ) (b) i = -10 sin(3t – 85 ° ) = 10 cos(3t – 85 ° + 90 ° ) = 10 cos(3t + 5 ° ) Chapter 9, Solution 5. v 1 = 20 sin( ω t + 60 ° ) = 20 cos( ω t + 60 ° 90 ° ) = 20 cos( ω t 30 ° ) v 2 = 60 cos( ω t 10 ° ) This indicates that the phase angle between the two signals is 20 ° and that v 1 lags v 2 . Chapter 9, Solution 6. (a) v(t) = 10 cos(4t – 60 ° ) i(t) = 4 sin(4t + 50 ° ) = 4 cos(4t + 50 ° – 90 ° ) = 4 cos(4t – 40 ° ) Thus, i(t) leads v(t) by 20 ° . (b) v 1 (t) = 4 cos(377t + 10 ° ) v 2 (t) = -20 cos(377t) = 20 cos(377t + 180 ° ) Thus, v 2 (t) leads v 1 (t) by 170 ° . (c) x(t) = 13 cos(2t) + 5 sin(2t) = 13 cos(2t) + 5 cos(2t – 90 ° ) X = 13 0 ° + 5 -90 ° = 13 – j5 = 13.928 -21.04 ° x(t) = 13.928 cos(2t – 21.04 ° ) y(t) = 15 cos(2t – 11.8 ° ) phase difference = -11.8 ° + 21.04 ° = 9.24 ° Thus, y(t) leads x(t) by 9.24 ° . Chapter 9, Solution 7. If f( φ ) = cos φ + j sin φ , ) ( f j ) sin j (cos j cos j -sin d df φ = φ + φ = φ + φ = φ φ = d j f df
Integrating both sides ln f = j φ + ln A f = Ae j φ = cos φ + j sin φ f(0) = A = 1 i.e. f( φ ) = e j φ = cos φ + j sin φ Chapter 9, Solution 8. (a) 4 j 3 45 15 ° + j2 = ° ° 53.13 - 5 45 15 + j2 = 3 98.13 ° + j2 = -0.4245 + j2.97 + j2 = -0.4243 + j4.97 (b) (2 + j)(3 – j4) = 6 – j8 + j3 + 4 = 10 – j5 = 11.18 -26.57 ° j4) - j)(3 (2 20 - 8 + ° + j12 5 - 10 + = ° ° 26.57 - 11.18 20 - 8 + 144 25 ) 10 )( 12 j 5 - ( + = 0.7156 6.57 ° 0.2958 j0.71 = 0.7109 + j0.08188 0.2958 j0.71 = 0.4151 j0.6281 (c) 10 + (8 50 ° )(13 -68.38 ° ) = 10+104 -17.38 ° = 109.25 – j31.07 Chapter 9, Solution 9. (a) 2 + 8 j 5 4 j 3 + = 2 + 64 25 ) 8 j 5 )( 4 j 3 ( + + + = 2 + 89 32 20 j 24 j 15 + + = 1.809 + j0.4944 (b) 4 -10 ° + ° 6 3 2 j 1 = 4 -10 ° + ° ° 6 3 63.43 - 236 . 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
= 4 -10 ° + 0.7453 -69.43 ° = 3.939 – j0.6946 + 0.2619 – j0.6978 = 4.201 – j1.392 (c) ° ° ° + ° 50 4 80 9 20 - 6 10 8 = 064 . 3 j 571 . 2 863 . 8 j 5628 . 1 052 . 2 j 638 . 5 3892 . 1 j 879 . 7 + + + = 799 . 5 j 0083 . 1 6629 . 0 j 517 . 13 + = ° ° 86 . 99 886 . 5 81 . 2 - 533 . 13 = 2.299 -102.67 ° = -0.5043 – j2.243 Chapter 9, Solution 10. (a) z 9282 . 6 4 z and , 5 66 . 8 z , 8 6 3 2 1 j j j = = = 93 . 19 66 . 10 3 2 1 j z z z = + + (b) 499 . 7 999 . 9 3 2 1 j z z z + = Chapter 9, Solution 11. (a) = (-3 + j4)(12 + j5) 2 1 z z = -36 – j15 + j48 – 20 = -56 + j33 (b) 2 1 z z = 5 j 12 4 j 3 - + = 25 144 ) 5 j 12 )( 4 j 3 (- + + + = -0.3314 + j0.1953 (c) = (-3 + j4) + (12 + j5) = 9 + j9 2 1 z z + 2 1 z z = (-3 + j4) – (12 + j5) = -15 – j 2 1 2 1 z z z z + = ) j 15 ( - ) j 1 ( 9 + + = 2 2 1 15 j) - 15 )( j 1 ( 9 - + = 226 ) 14 j 16 ( 9 - + = -0.6372 – j0.5575
Chapter 9, Solution 12. (a) = (-3 + j4)(12 + j5) 2 1 z z = -36 – j15 + j48 – 20 = -56 + j33 (b) 2 1 z z = 5 j 12 4 j 3 - + = 25 144 ) 5 j 12 )( 4 j 3 (- + + + = -0.3314 + j0.1953 (c) = (-3 + j4) + (12 + j5) = 9 + j9 2 1 z z + 2 1 z z = (-3 + j4) – (12 + j5) = -15 – j 2 1 2 1 z z z z + = ) j 15 ( - ) j 1 ( 9 + + = 2 2 1 15 j) - 15 )( j 1 ( 9 - + = 226 ) 14 j 16 ( 9 - + = -0.6372 – j0.5575 Chapter 9, Solution 13.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern