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If one thing can be accomplished in n1 different ways and after this a second thing can be accomplished in n2 different ways, . . . , and finally a kth thing can be accomplished in nk different ways, then all k things can be accomplished in the specified order in n1n2 c nk different ways.

and i1 0:2 cos 1000t 36:98

A diagram, called a tree diagram because of its appearance (Fig. 1-4), is often used in connection with the above principle.

EXAMPLE 1.15 Letting the shirts be represented by S1, S2 and the ties by T1, T2, T3, T4, the various ways of choosing a shirt and then a tie are indicated in the tree diagram of Fig. 1-4.

13.10 Two two-port networks a and b, with open-circuit impedances Za and Zb , are connected in series (see Fig. 13-12). Derive the Z-parameters equations (31a).

Fig. 1-4

12. Draw a system diagram for the B&B system (see Problem 4.3). See Fig. 8-17.

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From network a we have V1a Z11;a I1a Z12;a I2a V2a Z21;a I1a Z22;a I2a From network b we have V1b Z11;b I1b Z12;b I2b V2b Z21;b I1b Z22;b I2b From connection between a and b we have I1 I1a I1b I2 I2a I2b Therefore, V1 Z11;a Z11;b I1 Z12;a Z12;b I2 V2 Z21;a Z21;b I1 Z22;a Z22;b I2 from which the Z-parameters (31a) are derived. V1 V1a V1b V2 V2a V2b

Suppose that we are given n distinct objects and wish to arrange r of these objects in a line. Since there are n ways of choosing the 1st object, and after this is done, n 1 ways of choosing the 2nd object, . . . , and finally n r 1 ways of choosing the rth object, it follows by the fundamental principle of counting that the number of different arrangements, or permutations as they are often called, is given by

(25)

13.11 Two two-port networks a and b, with short-circuit admittances Ya and Yb , are connected in parallel (see Fig. 13-13). Derive the Y-parameters equations (32a).

where it is noted that the product has r factors. We call n Pr the number of permutations of n objects taken r at a time. In the particular case where r n, (25) becomes

From network a we have I1a Y11;a V1a Y12;a V2a I2a Y21;a V1a Y22;a V2a and from network b we have I1b Y11;b V1b Y12;b V2b I2b Y21;b V1b Y22;b V2b From connection between a and b we have V1 V1a V1b V2 V2a V2b Therefore, I1 Y11;a Y11;b V1 Y12;a Y12;b V2 I2 Y21;a Y21;b V1 Y22;a Y22;b V2 from which the Y-parameters of (32a) result. I1 I1a I1b I2 I2a I2b

(26)

n! (n r)!

Fig. 8-17.

CHAP. 13]

(27)

Fig. 12-6

n, we see that (27) and (26) agree only if we have 0!

1, and we shall actually take this as the definition of 0!.

The magnitude and angle are plotted in Fig. 12-7. This transfer function approaches unity at high frequency, where the output voltage is the same as the input. Hence the description low-frequency rollo and the name high-pass.

EXAMPLE 1.16 The number of different arrangements, or permutations, consisting of 3 letters each that can be formed from the 7 letters A, B, C, D, E, F, G is

Fig. 12-7

7P 3

13. Draw a system diagram for the dental o ce system (see Problem 4.3). See Fig. 8-18.

7! 4!

A transfer impedance of the RL high-pass circuit under no-load is H1 ! V2 j!L2 I1 or H1 ! ! j R1 !x

7 6 5

The angle is constant at 908; the graph of magnitude versus ! is a straight line, similar to a reactance plot of !L versus !. See Fig. 12-8.

Suppose that a set consists of n objects of which n1 are of one type (i.e., indistinguishable from each other), n2 are of a second type, . . . , nk are of a kth type. Here, of course, n n1 n2 c nk. Then the number of different permutations of the objects is

n! n1!n2! c nk!

Fig. 12-8

(28)

Fig. 8-18.

Interchanging the positions of R and L results in a low-pass network with high-frequency roll-o (Fig. 12-9). For the open-circuit condition

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