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In particular, for two parallel impedances, Zeq Z1 Z2 = Z1 Z2 . Impedance Diagram In an impedance diagram, an impedance Z is represented by a point in the right half of the complex plane. Figure 9-8 shows two impedances; Z1 , in the rst quadrant, exhibits inductive reactance, while Z2 , in the fourth quadrant, exhibits capacitive reactance. Their series equivalent, Z1 Z2 , is obtained by vector addition, as shown. Note that the vectors are shown without arrowheads, in order to distinguish these complex numbers from phasors.

Fig. 2-23

Fig. 2-24

Fig. 9-8

g(u)

Combinations of Admittances Replacing Z by 1/Y in the formulas above gives admittances in series admittances in parallel 1 1 1 Yeq Y1 Y2 Yeq Y1 Y2

The same result is obtained for u Fig. 2-24.

2.36. A floor has parallel lines on it at equal distances l from each other. A needle of length a l is dropped at random onto the floor. Find the probability that the needle will intersect a line. (This problem is known as Buffon s needle problem.)

Thus, series circuits are easiest treated in terms of impedance; parallel circuits, in terms of admittance.

b 81

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Let X be a random variable that gives the distance of the midpoint of the needle to the nearest line (Fig. 2-24). Let be a random variable that gives the acute angle between the needle (or its extension) and the line. We denote by x and u any particular values of X and . It is seen that X can take on any value between 0 and l > 2, so that 0 x l > 2. Also can take on any value between 0 and p > 2. It follows that P(x i.e., the density functions of X and X x dx) 2 dx l P(u 2 > l, f2(u) 2 p du 1 du) 2 p du

Admittance Diagram Figure 9-9, an admittance diagram, is analogous to Fig. 9-8 for impedance. Shown are an admittance Y1 having capacitive susceptance and an admittance Y2 having inductive susceptance, together with their vector sum, Y1 Y2 , which is the admittance of a parallel combination of Y1 and Y2 .

are given by f1(x)

From Fig. 2-24 it is seen that the needle actually hits a line when X event is given by 4 p>2 (a>2) sin u dx du lp 3u 0 3x 0 2a lp

Fig. 9-9

When the above expression is equated to the frequency of hits observed in actual experiments, accurate values of p are obtained. This indicates that the probability model described above is appropriate.

In view of the analogy between impedance in the frequency domain and resistance in the time domain, Sections 3.6 and 3.7 imply the following results. (1) Impedances in series divide the total voltage in the ratio of the impedances: Vr Zr Vs Zs See Fig. 9-10. or Vr Zr V Zeq T

2.37. Two people agree to meet between 2:00 P.M. and 3:00 P.M., with the understanding that each will wait no longer than 15 minutes for the other. What is the probability that they will meet

c 16

Let X and Y be random variables representing the times of arrival, measured in fractions of an hour after 2:00 P.M., of the two people. Assuming that equal intervals of time have equal probabilities of arrival, the density functions of X and Y are given respectively by f1(x) f2( y) e e 1 0 1 0 0 x 1 otherwise 0 y 1 otherwise

Fig. 9-10

Then, since X and Y are independent, the joint density function is (1) Since 15 minutes (2)

Fig. 9-11

f (x, y)

f1(x) f2(y)

(2) Impedances in parallel (admittances in series) divide the total current in the inverse ratio of the impedances (direct ratio of the admittances): Ir Zs Yr Is Zr Ys See Fig. 9-11. or Ir Zeq Y IT r IT Zr Yeq

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