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Coil ends and slip rings are not shown; however, it is evident that counterclockwise rotation results in the coil sides A, B, and C passing under the pole pieces in the order . . . A-B-C-A-B-C . . . Voltage polarities reverse for each change of pole. Assuming that the pole shape and corresponding magnetic ux density are such that the induced voltages are sinusoidal, the result for the three coils is as shown in Fig. 11-3. Voltage B is 120 electrical degrees later than A, and C is 2408 later. This is referred to as the ABC sequence. Changing the direction of rotation would result in . . . A-C-B-A-C-B . . . ; which is called the CBA sequence.

Suppose that we have an experiment such as tossing a coin or die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a red marble. In some cases this probability will not change from one trial to the next (as in tossing a coin or die). Such trials are then said to be independent and are often called Bernoulli trials after James Bernoulli who investigated them at the end of the seventeenth century. Let p be the probability that an event will happen in any single Bernoulli trial (called the probability of success). Then q 1 p is the probability that the event will fail to happen in any single trial (called the probability of failure). The probability that the event will happen exactly x times in n trials (i.e., successes and n x failures will occur) is given by the probability function

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n! x!(n x)!

Fig. 8-1.

Fig. 11-3

2 6 2

The voltages of a balanced ABC sequence in the time and phasor domains are given in (5) and (6), respectively. The phasor diagram for the voltage is shown in Fig. 11-4.

6! 1 1 a b a b 2!4! 2 2

15 64

Fig. 11-4

The discrete probability function (1) is often called the binomial distribution since for x corresponds to successive terms in the binomial expansion

CHAP. 11]

Some of the important properties of the binomial distribution are listed in Table 4-1.

Draw an object model for simplified vi-like editor. See Fig. 8-2.

Table 4-1 Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function s a3

5 6

a4 M(t) f(v)

The ends of the coils can be connected in wye (also designated Y; see Section 11.8), with ends A 0 , B 0 , and C 0 joined at a common point designated the neutral, N; and with ends A, B, and C brought out to become the lines A, B, and C of the three-phase system. If the neutral point is carried along with the lines, it is a three-phase, four-wire system. In Fig. 11-5, the lines are designated by lowercase a, b, and c at the supply, which could either be a transformer bank or a three-phase alternator, and by uppercase A, B, and C at the load. If line impedances must be considered, then the current direction through, for example, line aA would be IaA , and the phasor line voltage drop VaA .

EXAMPLE 4.2 In 100 tosses of a fair coin, the expected or mean number of heads is m 2(100) A 1 B A 1 B 5. standard deviation is s 2 2

The law of large numbers, page 83, has an interesting interpretation in the case of Bernoulli trials and is presented in the following theorem. Theorem 4-1 (Law of Large Numbers for Bernoulli Trials): Let X be the random variable giving the number of successes in n Bernoulli trials, so that X>n is the proportion of successes. Then if p is the probability of success and is any positive number, X lim Pa 2 n p2 Pb 0 (3)

Fig. 11-5

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