Explain the determine red Example FTaking the z-transform of each term of the second-order equationYk+2 + a1Yk+1+a2Yk = R¢(4.313)gives(2?F(2) – 2²yo – zyı) + a1[zF(2) – zyo) + a2F(2) = G(z),(4.314)%3DorG(2) + yoz² + (a1Y0 + Y1)zF(2)(4.315)z2 + a1% + a2where G(2) = Z(Rx).Let a1 =0, a2 =4, and R= 0. This corresponds to the differenceequationYk+2 -4Yk = 0.(4.316)The z-transform is, from equation (4.315), given by the expressionF(z)zYo + Y12y0 – Y112y0 + Y11+(4.317)(z + 2)(2 – 2)4z + 24Z – 2From Table 4.1, we can obtain the inverse transform of F(z); doing this givesYk = /¼(2y0 – Y1)(-2)* + '/¼(2yo + y1)2*.(4.318)Since yo and yı are arbitrary, this gives the general solution of equation(4.316).The equationYk+2 + 4yk+1 + 3yk2k(4.319)corresponds to a1 = 4, a2 = 3, and R = 2k. Using the fact thatG(2) = Z(2*)(4.320)2 - 2we can rewrite equation (4.315), for this case, asF(2)(- E+1)(: + 3) * (z + 1)(z + 3)yo(z + 4)Y11(4.321)(z – 2)(2 + 1)(z + 3)*The three terms on the right-hand side of this equation have the followingpartial-fraction expansions:1111(4.322)(z + 1)(z + 3)z + 4(z + 1)(z + 3)2 z + 12 z + 33111(4.323)2 z + 12 z + 31111111(4.324)(z – 2)(z + 1)(z + 3)15 z – 26 z + 110 z + 3

Answered: Image /qna-images/answer/869f0c33-bb8b-4e73-8ed5-0b032ecc8534.jpg

we can rewrite equation (4.315), for this case, as F(2) yo(z +4) + Y1 1 (4.321) (z + 1)(2 + 3) + (z + 1)(z + 3) ' (z – 2)(z+ 1)(z+ 3)' The three terms on the right-hand side of this equation have the following partial-fraction expansions: 1 1 1 1 (4.322) (z + 1)(z + 3) z + 4 (z + 1)(z + 3) 2 z + 1 2 z + 3' 1 1 1 (4.323) 2 z + 1 2 z +3 1 1 1 1 1 1 1 (4.324) %3D (z – 2)(z+1)(z +3) 15 z - 2 6 z +1 10 z +3

we can rewrite equation (4.315), for this case, as F(2) yo(z +4) + Y1 1 (4.321) (z + 1)(2 + 3) + (z + 1)(z + 3) ' (z – 2)(z+ 1)(z+ 3)' The three terms on the right-hand side of this equation have the following partial-fraction expansions: 1 1 1 1 (4.322) (z + 1)(z + 3) z + 4 (z + 1)(z + 3) 2 z + 1 2 z + 3' 1 1 1 (4.323) 2 z + 1 2 z +3 1 1 1 1 1 1 1 (4.324) %3D (z – 2)(z+1)(z +3) 15 z - 2 6 z +1 10 z +3

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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