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EXAMPLE 10.16 A generator, with Vg 100 V(rms) and Zg 1 j, feeds a load Z1 2 (Fig. 10-12). (a) Find the average power PZ1 (absorbed by Z1 ), the power Pg (dissipated in Zg ) and PT (provided by the generator). (b) Compute the value of a second load Z2 such that, when in parallel with Z1 , the equivalent impedance is Z Z1 kZ2 Z g . (c) Connect in parallel Z2 found in (b) with Z1 and then nd the powers PZ ; PZ1 ; PZ2 (absorbed by Z, Z1 , and Z2 , respectively), Pg (dissipated in Zg ) and PT (provided by the generator). p p (a) jZ1 Zg j j2 1 jj 10. Thus I Vg = Z1 Zg 100= 2 1 j and jIj 10 10 A. The required powers are p PZ1 Re Z1 jIj2 2 10 10 2 2000 W p Pg Re Zg jIj2 1 10 10 2 1000 W PT PZ1 Pg 2000 1000 3000 W

(a) M(t) E(etX )

Fig. 10-12

1 2,

2 2 t

Then,

1 t2 2!

4. Rework The producer reviews the report and corrects the product. 5. Follow-up The moderator reviews the report and the correction. If it satis es the exit criteria, the inspection is completed. If not, the moderator can either have the producer rework the product or reinspection can be scheduled.

Z1 Z2 2 a jb 1 j Z1 Z2 2 a jb from which a b 2 0 and a b 2 0. Solving these simultaneous equations, a 0 and b 2; substituting into the equation above, Z2 j2: (b) Z Z1 kZ2 1 j and Z Zg 1 j 1 j 2. Then, I Vg = Z Zg 100= 1 j 1 j 100=2 50 A, and so PZ Re Z I2 1 502 2500 W Pg Re Zg I2 1 502 2500 W

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1 2,

t2 4 t3 3!

To p PZ1 and PZ2 , we rst nd VZ across Z: VZ IZ 50 1 j . Then IZ1 VZ =Z1 50 1 j =2 nd 25 2 458, and p PZ2 0 W PT Pg PZ1 5000 W PZ1 Re Z1 jIZ1 j2 2 25 2 2 2500 W Alternatively, we can state that PZ2 0 and PZ1 PZ 2500 W

3 4,

Consider a network containing two AC sources with two di erent frequencies, !1 and !2 . If a common period T may be found for the sources (i.e., !1 m!, !2 n!, where ! 2 =T and m 6 n), then superposition of individual powers applies (i.e., P P1 P2 ), where P1 and P2 are average powers due to application of each source. The preceding result may be generalized to the case of any n number of sinusoidal sources operating simultaneously on a network. If the n sinusoids form harmonics of a fundamental frequency, then superposition of powers applies. P

t3 8 mr 4 t4 4!

t4 16

EXAMPLE 10.17 A practical coil is placed between two voltage sources v1 5 cos !1 t and v2 10 cos !2 t 608 , which share the same common reference terminal (see Fig. 9-54). The coil is modeled by a 5-mH inductor in series with a 10-

A checklist is a list of items that should be checked during the review. Sometimes the items are expressed as questions to be answered. The value of a checklist is that it focuses the attention of the reviewer on potential problems. Every fault that is found should be analyzed to see if it warrants a checklist item to focus on that problem. (I recall a long debugging process caused by a semicolon directly after the condition in an IF statement in C . Now any checklist for C programs I write includes checking for semicolons at the end of decision conditions.) Checklist items that are not e ective in nding faults during inspections should be considered for removal. Too many checklist items will lessen the e ectiveness of the inspection.

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resistor. Find the average power in the coil for (a) !2 2!1 4000; b !1 !2 2000; p c !1 2000 and !2 1000 2, all in rad/s. Let v1 by itself produce i1 . Likewise, let v2 produce i2 . Then i i1 i2 . The instantaneous power p and the average power P are

3.19. Find the first four moments (a) about the origin, (b) about the mean, for a random variable X having density function f (x)

E(X)

2 2 p Ri2 R i1 i2 2 Ri1 Ri2 2Ri1 i2 2 2 P hpi Rhi1 i Rhi2 i 2Rhi1 i2 i P1 P2 2Rhi1 i2 i

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