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EXAMPLE 11.2 A three-phase, three-wire, ABC system, with an e ective line voltage of 120 V, has three impedances of 5:0 458 in a -connection. Determine the line currents and draw the voltage-current phasor diagram. p The maximum line voltage is 120 2 169:7 V. Referring to Fig. 11-7(a), the voltages are: VAB 169:7 1208 V VBC 169:7 08 V VCA 169:7 2408 V

A graph of the density function (7), sometimes called the standard normal curve, is shown in Fig. 4-1. In this graph we have indicated the areas within 1, 2, and 3 standard deviations of the mean (i.e., between z 1 and 1, z 2 and 2, z 3 and 3) as equal, respectively, to 68.27%, 95.45% and 99.73% of the total area, which is one. This means that P( 1 Z 1) 0.6827, P( 2 Z 2) 0.9545, P( 3 Z 3) 0.9973 (10)

Draw a DFD for the simple Unix, vi-like editor. See Fig. 8-3.

Double subscripts give the phase-current directions; for example, IAB passes through the impedance from line A to line B. All current directions are shown in Fig. 11-8. Then the phase currents are IAB IBC ICA VAB 169:7 1208 33:9 758 Z 5 458 A

Fig. 4-1

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A table giving the areas under this curve bounded by the ordinates at z 0 and any positive value of z is given in Appendix C. From this table the areas between any two ordinates can be found by using the symmetry of the curve about z 0.

Fig. 11-8

Table 4-2

CHAP. 11]

Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function M(t) f(v)

3 4 2

By KCL, line current IA is given by IA IAB IAC 33:9 758 33:9 1958 58:7 458 A Similarly, IB 58:7 758 A and IC 58:7 1658 A. The line-to-line voltages and all currents are shown on the phasor diagram, Fig. 11-9. Note particularly the balanced currents. After one phase current has been computed, all other currents may be obtained through the p p symmetry of the phasor diagram. Note also that 33:9 3 58:7; that is, IL 3IPh for a balanced delta load.

Fig. 8-3.

(s2t2>2) (s2v2>2)

Fig. 11-9

If n is large and if neither p nor q is too close to zero, the binomial distribution can be closely approximated by a normal distribution with standardized random variable given by Z X np !npq (11)

Here X is the random variable giving the number of successes in n Bernoulli trials and p is the probability of success. The approximation becomes better with increasing n and is exact in the limiting case. (See Problem 4.17.) In practice, the approximation is very good if both np and nq are greater than 5. The fact that the binomial distribution approaches the normal distribution can be described by writing lim P aa S

Three identical impedances connected as shown in Fig. 11-10 make up a balanced Y-connected load. the currents in the impedances are also the line currents; so the directions are chosen from the source to the load, as before.

(12)

EXAMPLE 11.3 A three-phase, four-wire, CBA system, with an e ective line voltage of 120 V, has three impedances of 20 308 in a Y-connection (Fig. 11-10). Determine the line currents and draw the voltage-current phasor diagram.

Let X be a discrete random variable that can take on the values 0, 1, 2, . . . such that the probability function of X is given by f (x) P(X x) lxe x!

Behavioral modeling refers to the behavior of the system, usually from the user point of view. These diagrams are used to specify aspects of the proposed system. It is important that the diagrams capture the essential aspects of the system and are able to communicate those aspects both to the developer and to the user for con rmation that this is the system that he or she wants.

(13)

Fig. 11-10 p The maximum line voltage is 169.7 V, and the line-to-neutral magnitude, 169:7= 3 98:0 V. 11-7(b), VAN 98:0 908 V Then IA VBN 98:0 308 V VCN 98:0 1508 V From Fig.

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