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The graph shown in Fig. 12-2 is formed by combining two number lines at right angles to each other so that their zero points coincide. The horizontal number line is called the x-axis, and the vertical number line is the y-axis. The point where the two lines cross each other is, again, called the origin. A point is located on a graph by its coordinates, which are its distances from the axes. The abscissa or x-coordinate of a point is its distance from the y-axis. The ordinate or y-coordinate of a point is its distance from the x-axis. When the coordinates of a point are stated, the x-coordinate precedes the y-coordinate. Thus, the coordinates of point P in Fig. 12-2 are written (4, 3); those for Q are ( 4, 3). Note the parentheses. The quadrants of a graph are the four parts cut off by the axes. These are numbered I, II, III, and IV in a counterclockwise direction, as shown in Fig. 12-2.

For the Darlington-pair emitter-follower of Fig. 5-10, nd ICQ1 as a function of the six temperature-sensitive variables ICBO1 ; ICBO2 ; VBEQ1 ; VBEQ2 ; 1 ; and 2 .

Addition and Subtraction of Matrices Two matrices of the same order are conformable for addition or subtraction; two matrices of di erent orders cannot be added or subtracted. The sum (di erence) of two m n matrices, A aij and B bij , is the m n matrix C of which each element is the sum (di erence) of the corresponding elements of A and B. Thus, A B aij bij .

12.1 Locating points on a graph Give the coordinates of the following points in Fig. 12-3:

APP. B]

4 3 P 2 1

Fig. 5-10 By KVL, VCC IBQ1 RF VBEQ1 VBEQ2 IEQ2 RE By KCL, IEQ2 IEQ1 ICQ2 1 (2)

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Heretofore, the frequency response of a network has been exhibited by plotting separately the magnitude and the angle of a suitable network function against frequency !. This same information can be presented in a single plot: one nds the curve (locus diagram) in the complex plane traced by the point representing the network function as ! varies from 0 to 1. In this section we shall discuss locus diagrams for the input impedance or the input admittance; in some cases the variable will not be !, but another parameter (such as resistance R). For the series RL circuit, Fig. 12-28(a) shows the Z-locus when !L is xed and R is variable; Fig. 1228(b) shows the Z-locus when R is xed and L or ! is variable; and Fig. 12-28(c) shows the Y-locus when R is xed and L or ! is variable. This last locus is obtained from Y 1 1 q tan 1 !L=R R j!L R2 !L 2

Fig. 12-3

Using the result of Problem 3.36 in (2) and then substituting IBQ2 IEQ1 , we obtain IEQ2 IEQ1 2 IBQ2 2 1 ICBO2 2 1 IEQ1 2 1 ICBO2 Assuming 1 ; 2 ) 1 and substituting for IEQ1 according to (5.5), we obtain IEQ2 % 2 1 ICQ1 2 1 ICBO2 ICBO1 Also, from (5.6), IBQ1 % ICB1 ICBO1 1 4 3

Fig. 12-28

(a) (2, 2) (b) ( 3, 3)

Note that for !L 0, Y 1=R 08; and for !L ! 1, Y ! 0 908. When !L R, 1 Y p 458 R 2 A few other points will con rm the semicircular locus, with the center at 1/2R and the radius 1/2R. Either Fig. 12-28(b) or 12-28(c) gives the frequency response of the circuit. A parallel RC circuit has the Y- and Z-loci shown in Fig. 12-29; these are derived from Y 1 j!C R and R Z q tan 1 !CR 1 !CR 2

Now we substitute (3) and (4) into (1) and rearrange to get ICQ1 VCC VBEQ1 VBEQ2 ICBO1 RF 2 RE ICBO2 2 RE RF = 1 2 RE 5

(c) (0, 0) (d) ( 4, 3)

Fig. 12-29

(a) Determine a rst-order approximation for the change in ICQ1 in the circuit of Fig. 5-10, in terms of the six variables ICBO1 ; ICBO2 ; VBEQ1 ; VBEQ2 ; 1 ; and 2 . (b) Use ICQ1 as found in Problem 5.17 to evaluate the sensitivity factors (that is, the coe cients) in the expression determined in part a.

(e) ( 11, 11) 2 2 (f) (11, 2 21) 2

CHAP. 12]

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