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EXAMPLE 11.5 A three-phase, 339.4-V, ABC system [Fig. 11-15(a)] has a -connected load, with ZAB 10 08 ZBC 10 308 ZCA 15 308

3 6 1 2.

The essential functions in this diagram are file manipulation (create, save, and open). Insert and modify are intended as higher-level phrases covering the typical text-editing functions. Note that some capabilities such as search, copy, and move may be neglected, since they are not mentioned explicitly.

Fig. 11-14 Obtain phase and line currents and draw the phasor diagram. IAB VAB 339:4 1208 33:94 1208 A ZAB 10 08

Also P(A > B)

1 3.

Similarly, IBC 33:94 308 A and ICA 22:63 2708 A. Then, IA IAB IAC 33:94 1208 22:63 2708 54:72 108:18 A Also, IB 65:56 458 A and IC 29:93 169:18 A. The voltage-current phasor diagram is shown in Fig. 11-15(b), with magnitudes and angles to scale.

P(A > B) P(A)

Fig. 11-15

Hence, the added knowledge that the toss results in an odd number raises the probability from 1 > 2 to 2 > 3.

Theorem 1-9 For any three events A1, A2, A3, we have P(A1 > A2 > A3) P(A1) P(A2 u A1) P(A3 u A1> A2) (19)

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A scenario is a sequence of actions that accomplishes a user task. Alternative sequences are only shown by having a separate scenario for each alternative. Scenarios are used to illustrate an important capability or proposed use of the system. In UML, an interaction diagram (see 2) is used to specify the scenarios. Scenarios can also be speci ed by listing the sequence of actions.

11.11 UNBALANCED WYE-CONNECTED LOAD Four-Wire The neutral conductor carries the unbalanced current of a wye-connected load and maintains the line-to-neutral voltage magnitude across each phase of the load. The line currents are unequal and the currents on the phasor diagram have no symmetry.

In words, the probability that A1 and A2 and A3 all occur is equal to the probability that A1 occurs times the probability that A2 occurs given that A1 has occurred times the probability that A3 occurs given that both A1 and A2 have occurred. The result is easily generalized to n events. Theorem 1-10 If an event A must result in one of the mutually exclusive events A1, A2, c, An, then P(A) P(A1) P(A u A1) P(A2 ) P(A u A2) c P(An ) P(A u An ) (20)

EXAMPLE 11.6 A three-phase, four-wire, 150-V, CBA system has a Y-connected load, with ZA 6 08 ZB 6 308 ZC 5 458

If P(B u A) P(B), i.e., the probability of B occurring is not affected by the occurrence or non-occurrence of A, then we say that A and B are independent events. This is equivalent to P(A > B) P(A)P(B) (21)

We say that three events A1, A2, A3 are independent if they are pairwise independent: P(Aj > Ak ) and P(Aj)P(Ak ) P(A1 > A2 > A3) j2k where j, k 1, 2, 3 (22) (23)

See Figure 11-16(a).

P(A1)P(A2 )P(A3 )

VAN 86:6 908 14:43 908 A ZA 6 08 V 86:6 308 14:43 08 A IB BN ZB 6 308 V 86:6 1508 17:32 1058 A IC CN ZC 5 458 IN 14:43 908 14:43 08 17:32 1058 10:21 167:08 A Figure 11-16(b) gives the phasor diagram.

Note that neither (22) nor (23) is by itself sufficient. Independence of more than three events is easily defined.

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