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When x t Xest and y t Yest , an sn an 1 sn 1 a1 s a0 est bm sm bm 1 sm 1 b1 s b0 est Then, H s Y s a sn an 1 sn 1 a1 s a0 nm X s bm s bm 1 sm 1 b1 s b0

Obviously, the network phase angle is a constant 908. By (8.10), Mdb 0 when ! 1; further, Mdb increases by 20 db for each order-of-magnitude (decade) change in !. A graph of Mdb versus the logarithm of ! would thus have a slope of 20 db per decade of frequency. A complete Bode plot is shown in Fig. 8-3:

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33. (a) 8; (b) 6; (c) 262 3 34. (a) 42 ft; (b) 66ft 37. (a) 8:5; (b) 3:5; (c) halved (in each case) 38. (a) 15; (b) 60; (c) 25, 35, 40; (d) 4; (e) 6, 3 39. (a) 3:7; (b) 7:2; (c) quadrupled; (d) 7 43. (a) 5; (b) 14; (c) 6; (d) 5; (e) 12; ( f ) 13; (g) 48; (h) 2 44. 30, 18 45. (a) 8; (b) 6; (c) 12; (d) 5; (e) 7; ( f ) 12; (g) 30; (h) 71; (i) 5; ( j ) 8 2 46. (a) 8; (b) 13; (c) 21; (d) 6; (e) 9; ( f ) 14; (g) 3; (h) 8 47. (a) a (d) p 49. (a) b 4, h 18, h 16; (b) a 212 or 2 23; (b) c 2108 6 23 8; (d) b 2 23; (e) b 5 22; b 23 9, h 220 or 2 25; (c) q 4 and b 280 or 4 25;

In linear circuits made up of lumped elements, the network function H s is a rational function of s and can be written in the following general form H s k s z1 s z2 s z s p1 s p2 s p

Fig. 8-3

48. (a) 25; (b) 39; (c) 241; (d) 10; (e) 7 22 2 27; (c) a 50. (a) 9, 12; (b) 10, 24; (c) 80, 150; (d) 2 25, 4 25 51. (a) 41; (b) 5 25 52. (a) 12; (b) 10 22; (c) 5 25 53. All except (h) 54. (a) yes; (b) no, since (2x)2 (3x)2 (4x)2 55. (a) 8; (b) 6; (c) 219; (d) 5 23

where k is some real number. The complex constants zm m 1; 2; . . . ; , the zeros of H s , and the pn n 1; 2; . . . ; the poles of H s , assume particular importance when H s is interpreted as the ratio of the response (in one part of the s-domain network) to the excitation (in another part of the network). Thus, when s zm , the response will be zero, no matter how great the excitation; whereas, when s pn , the response will be in nite, no matter how small the excitation.

EXAMPLE 8.8 A passive network in the s-domain is shown in Fig. 8-13. current I s due to an input voltage V s . H s Obtain the network function for the

56. (a) 15; (b) 2 25; (c) 6 57. (a) 16; (b) 30; (c) 4 23; (d) 10 58. (a) 10; (b) 12; (c) 28; (d) 15 59. (a) 5; (b) 20; (c) 15; (d) 25 60. (a) 12; (b) 24 61. 12 62. 30 63. (a) 10 and 10 23; (b) 7 23 and 14; (c) 5 and 10 64. (a) 11 23; (b) a 23; (c) 48; (d) 16 23 65. (a) 25 and 25 23; (b) 35 and 35 23 66. (a) 28, 8 23; (b) 17, 14 23 67. (a) 17 22; (b) a 22; (c) 34 22; (d) 30 68. (a) 20 22; (b) 40 22 69. (a) 45, 13 22; (b) 11, 27 22; (c) 15 22, 55 70. 6 22, 5 22

The exact Bode plot of a network frequency transfer function is tedious to construct. Frequently, su ciently accurate information can be obtained from an asymptotic Bode plot (see Problem 8.1).

Fig. 8-13

8

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