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impedance value being attained at 8000 rad/s instead of 8 rad/s. Any other value of the impedance is likewise attained, after scaling, at a frequency 1000 times that at which it was attained before scaling. Consequently, the two graphs of jZ ! j di er only in the horizontal scale see Fig. 8-29. (The same would be true of the two graphs of Z ! .)

A (8, 1), Br(3

The s-domain transfer function for a system can be written in the form T s Kb z1 s 1 z2 s 1 sn p1 s 1 p2 s 1 1

8.22 In the RLC circuit of Fig. 8-30, the capacitor is initially charged to V0 200 V. after the switch is closed at t 0. Ans: 2e 1000t sin 1000t A Find the current transient

Br(8, 4), and C (5

where n may be positive, negative, or zero. Show that the Bode plot (for Mdb only) may be generated as a composite of individual Bode plots for three basic types of terms.

, L 0:1 H, and C 100 mF, has a voltage source of 200 V applied at t 0. Find the current transient, assuming zero initial charge on the capacitor. Ans: 1:055 e 52t e 1948t A What value of capacitance, in place of the 100 mF in Problem 8.23, results in the critically damped case Ans: 10 mF

(c) Ar(5 # 3, 5 # 1)

CHAP. 8]

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The frequency transfer function corresponding to (1) is T j! From de nition 3 of Section 8.2, Mdb 20 log jT j! j 20 log which may be written as Mdb 20 log Kb 20 log j1 j! z1 j 20 log j1 j! z2 j 20n log j j!j 20 log j1 j! p1 j 20 log j1 j! p2 j Kb j1 j! z1 jj1 j! z2 j j j! n jj1 j! p1 jj1 j! p2 j 3 Kb 1 j! z1 1 j! z2 j! n 1 j! p1 1 j! p2 2

Ar(15, 5), Br(5 # 3, 5 # 4)

Fig. 8-30

It is apparent from (4) that the Bode plot of T j! can be formed by point-by-point addition of the plots of three types of terms: 1. A frequency-invariant or gain-constant term Kb whose Bode plot is a horizontal line at Mdb 20 log Kb .

Br(15, 20), and Cr(5 # 5, 5 # 1)

Fig. 8-31

Cr(25, 5)

2. 3.

, L 0:1 H, C 5 mF. Ans: 1000 rad/s A voltage of 10 V is applied at t 0 to a series RLC circuit with R 5

(d) Ar( 1, 3), Br( 4, 3), and Cr( 1, 5) (e) Ar(1, 5 3) Ar(1, 2), Br(4, 5 3) Br(4, 2), and Cr(1, 5 5) Cr(1, 0)

, L 0:1 H, C 500 mF. Find the transient voltage across the resistance. Ans: 3:60e 25t sin 139t V In the two-mesh circuit shown in Fig. 8-31, the switch is closed at t 0. Find i1 and i2 , for t > 0. Ans: i1 0:101e 100t 9:899e 9950t A ; i2 5:05e 100t 5:00 0:05e 9950t A A voltage has the s-domain representation 100 308 V. Express the time function for (a) s 2 Np/s, (b) s 1 j5 s 1 . Ans: a 86:6 e 2t V ; b 100 e t cos 5t 308 V Give the complex frequencies associated with the current i t 5:0 10e 3t cos 50t 908 Ans: 0; 3 j50 s 1 A .

Poles or zeros of multiplicity n, ( j! n , whose amplitude ratio is Mdb 20n log !, where the plus sign corresponds to zeros and the minus sign to poles of the transfer function. (See Example 8.3.) First-order lead and lag factors, 1 j! n , as discussed in Examples 8.1 and 8.2. They are usually approximated with asymptotic Bode plots; if greater accuracy is needed, the asymptotic plots are corrected using Table 8-1:

A transformation that slides figures without flipping or rotating them is called a translation. The translation that slides everything to the right a units and up b units is P(x, y) A Pr(x a, y b).

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