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The coe cient L, in H, is called the self-inductance of the coil. Two conductors from di erent circuits in close proximity to each other are magnetically coupled to a degree that depends upon the physical arrangement and the rates of change of the currents. This coupling is increased when one coil is wound over another. If, in addition, a soft-iron core provides a path for the magnetic ux, the coupling is maximized. (However, the presence of iron can introduce nonlinearity.) To nd the voltage-current relation at the terminals of the two coupled coils shown in Fig. 14-2, we observe that the total magnetic ux linkage in each coil is produced by currents i1 and i2 and the mutual linkage e ect between the two coils is symmetrical. 1 L1 i1 Mi2 2 Mi1 L2 i2 334

4.53. Verify the expressions given for (a) the mean, (b) the variance, of the Weibull distribution, page 118.

E(X)

CHAP. 14]

Fig. 9-5.

b 1 e axb dx (1>b)

Fig. 14-2

E(X2)

where M is the mutual inductance (in H). The terminal voltages are time derivatives of the ux linkages. v1 t d 1 di L1 1 M dt dt d 2 di1 M L2 v2 t dt dt di2 dt di2 dt

30 u

2 a1

The coupled coils constitute a special case of a two-port network discussed in 13. The terminal characteristics (2) may also be expressed in the frequency domain or in the s-domain as follows. Frequency Domain V1 j!L1 I1 j!MI2 V2 j!MI1 j!L2 I2 3 s-Domain V1 L1 sI1 MsI2 V2 MsI1 L2 sI2 4

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Miscellaneous problems 4.54. The probability that an entering college student will graduate is 0.4. Determine the probability that out of 5 students (a) none, (b) 1, (c) at least 1, will graduate.

The values of variables in a program depend on the values of other variables. There are two basic dependencies: data dependencies, where the value of x a ects the value of y through de nition and use pairs, and control dependencies, where the value of x determines whether code that contains de nitions of y executes.

The coupling coe cient M is discussed in Section 14.2. The frequency domain equations (3) deal with the sinusoidal steady state. The s-domain equations (4) assume exponential sources with complex frequency s.

(a) P(none will graduate) (b) P(l will graduate)

0 5 5C0(0.4) (0.6)

EXAMPLE 14.1 Given L1 0:1 H, L2 0:5 H, and i1 t i2 t sin !t in the coupled coils of Fig. 14-2. v1 t and v2 t for (a) M 0:01 H, b M 0:2 H, and (c) M 0:2 H. a b v1 t 0:1 ! cos !t 0:01 ! cos !t 0:11 ! cos !t v2 t 0:01 ! cos !t 0:5 ! cos !t 0:51 ! cos !t V V Find

1 4 5C1(0.4) (0.6)

v1 t 0:1 ! cos !t 0:2 ! cos !t 0:3 ! cos !t V v2 t 0:2 ! cos !t 0:5 ! cos !t 0:7 ! cos !t V v1 t 0:1 ! cos !t 0:2 ! cos !t 0:1 ! cos !t v2 t 0:2 ! cos !t 0:5 ! cos !t 0:3 ! cos !t V V

(c) P(at least 1 will graduate)

P(none will graduate)

Fig. 2-25

4.55. What is the probability of getting a total of 9 (a) twice, (b) at least twice in 6 tosses of a pair of dice

The following code calculates the product of x and y . The output variable z has a data dependency on the variable x , since x is added to z. The output z has a control dependency on the variable y , since y controls how many times x is added to z.

Fig. 2-26

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