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A transformer is a device for introducing mutual coupling between two or more electric circuits. The term iron-core transformer identi es the coupled coils which are wound on a magnetic core of laminated specialty steel to con ne the ux and maximize the coupling. Air-core transformers are found in electronic and communications applications. A third group consists of coils wound over one another on a nonmetallic form, with a movable slug of magnetic material within the center for varying the coupling. Attention here is directed to iron-core transformers where the permeability of the iron is assumed to be constant over the operating range of voltage and current. The development is restricted to twowinding transformers, although three and more windings on the same core are not uncommon. In Fig. 14-10, the primary winding, of N1 turns, is connected to the source voltage V1 , and the secondary winding, of N2 turns, is connected to the load impedance ZL . The coil resistances are shown by lumped parameters R1 and R2 . Natural current I2 produces ux 2 21 22 , while I1 produces 1 12 11 . In terms of the coupling coe cient k, 11 1 k 1 22 1 k 2

4.75. Find the area under the normal curve (a) to the left of z 1.78, (b) to the left of z 0.56, (c) to the right of z 1.45, (d) corresponding to z 2.16, (e) corresponding to 0.80 z 1.53, (f ) to the left of z 2.52 and to the right of z 1.83. 4.76. If Z is normally distributed with mean 0 and variance 1, find: (a) P(Z (c) P( Z Z Z 1). 1.64), (b) P( 1.96 Z 1.96),

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4.77. Find the values of z such that (a) the area to the right of z is 0.2266, (b) the area to the left of z is 0.0314, (c) the area between 0.23 and z is 0.5722, (d) the area between 1.15 and z is 0.0730, (e) the area between z and z is 0.9000. 4.78. Find z1 if P(Z z1) 0.84, where z is normally distributed with mean 0 and variance 1. 8).

Fig. 14-10

4.80. If the heights of 300 students are normally distributed with mean 68.0 inches and standard deviation 3.0 inches, how many students have heights (a) greater than 72 inches, (b) less than or equal to 64 inches, (c) between 65 and 71 inches inclusive, (d) equal to 68 inches Assume the measurements to be recorded to the nearest inch. 4.81. If the diameters of ball bearings are normally distributed with mean 0.6140 inches and standard deviation 0.0025 inches, determine the percentage of ball bearings with diameters (a) between 0.610 and 0.618 inches inclusive, (b) greater than 0.617 inches, (c) less than 0.608 inches, (d) equal to 0.615 inches.

From these ux relationships, leakage inductances can be related to the self-inductances: L11 1 k L1 The corresponding leakage reactances are: X11 1 k X1 X22 1 k X2 Hence, for two coils L22 1 k L2

4.82. The mean grade on a final examination was 72, and the standard deviation was 9. The top 10% of the students are to receive A s. What is the minimum grade a student must get in order to receive an A 4.83. If a set of measurements are normally distributed, what percentage of these differ from the mean by (a) more than half the standard deviation, (b) less than three quarters of the standard deviation 4.84. If is the mean and is the standard deviation of a set of measurements that are normally distributed, what percentage of the measurements are (a) within the range 2 (b) outside the range 1.2 (c) greater than 1.5 4.85. In Problem 4.84 find the constant a such that the percentage of the cases (a) within the range (b) less than a is 22%. a is 75%,

Fig. 2-9.

4.86. Find the probability that 200 tosses of a coin will result in (a) between 80 and 120 heads inclusive, (b) less than 90 heads, (c) less than 85 or more than 115 heads, (d) exactly 100 heads. 4.87. Find the probability that a student can guess correctly the answers to (a) 12 or more out of 20, (b) 24 or more out of 40, questions on a true-false examination. 4.88. A machine produces bolts which are 10% defective. Find the probability that in a random sample of 400 bolts produced by this machine, (a) at most 30, (b) between 30 and 50, (c) between 35 and 45, (d) 65 or more, of the bolts will be defective. 4.89. Find the probability of getting more than 25 sevens in 100 tosses of a pair of fair dice.

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