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Describe the circuit which corresponds to each locus in Fig. 12-63 if there is only one variable element in each circuit. Ans: (a) A two-branch parallel circuit. Branch 1: xed R and XC ; branch 2: xed R and variable XC . (b) A three-branch parallel circuit. Branch 1: xed R and XC ; branch 2: xed XC ; branch 3: xed R and variable XL . (c) A two-branch parallel circuit. Branch 1: xed R and XC ; branch 2: xed XL and variable R.

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2.77. g(x)

Fig. 12-63 12.38 In the circuit of Fig. 12-64, L 1 mH. Determine R1 , R2 , and C such that the impedance between the two terminals of the circuit is 100 at all frequencies. Ans: C 100 nF, R1 R2 100 Given V2 =V1 10s= s2 2s 81 and v1 t cos !t , determine ! such that the amplitude of v2 t attains a maximum. Find that maximum. Ans: ! 9 rad/s, V2 5 V Given H s s= s2 as b determine a and b such that the magnitude of the frequency response jH ! j has a maximum at 100 Hz with a half-power bandwidth of 5 Hz. Then nd the quality factor Q. Ans: a 31:416; b 394784; Q 20

x 3e x/6 0

2.74. g(u)

Fig. 12-64

2.78. 1 > 4

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Given H s s 1 = s2 2s p determine where jH ! j is at a maximum, its half-power bandwidth and 82 , quality factor. Ans: !0 82 % 9 rad/s, ! 2 rad/s, Q 4:53 In a parallel RLC circuit R 10 k and L 20 mH. (a) Find C so that the circuit resonates at 1 MHz. Find the quality factor Q and the bandwidth in kHz. (b) Find the terminal voltage of the circuit if an AC current source of I 1 mA is applied to it at: (i) 1 MHz, (ii) 1.01 MHz, (iii) 1.006 MHz Ans: a C 1:267 nF; Q 79:6; f 12:56 kHz; (b) V2 10 V at 1 MHz, 5.34 V at 1.01 MHz, and 7.24 V at 1.006 MHz A coil is modeled as a 50-mH inductor in series with a 5- resistor. Specify the value of a capacitor to be placed in series with the coil so that the circuit would resonate at 600 kHz. Find the quality factor Q and bandwidth f in kHz. Ans: C 1:4 nF; Q 37:7; f 15:9 kHz The coil of Problem 12.43 placed in parallel with a capacitor C resonates at 600 kHz. Find C, quality factor Q, and bandwidth f in kHz. Hint: Find the equivalent parallel RLC circuit. Ans: C 1:4 nF; Q 37:7; f 15:9 kHz The circuit in Fig. 12-65(a) is a third-order Butterworth low-pass lter. Find the network function, the magnitude of the frequency response, and its half-power cuto frequency !0 . Ans: H s 1= s3 2s2 2s 1 ; jH ! j2 1= 1 !6 ; !0 1 rad/s

2.75. g(u)

1 = 14 2 = 5 = 19 2* = 3 N2 = 33 N = 84 N1 = 51 Est N = 14 log2 14 + 5 log2 5 = 14 38 + 5 23 = 647 + 115 = 762 V = 84 log2 19 = 84 425 = 357 V = 5 log2 5 = 5 23 = 115 L = 115 / 357 = 0032 8.

2.81. (a) 2 (b) F(x)

Fig. 12-65

(b) 27 > 64 (c) f1(x)

CHAP. 12]

(d) 0

A very important concept in probability and statistics is that of the mathematical expectation, expected value, or briefly the expectation, of a random variable. For a discrete random variable X having the possible values x1, c , xn, the expectation of X is defined as E(X) or equivalently, if P(X xj) E(X) x1P(X f (xj), x1 f (x1 ) c

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