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(a) The appropriate small-signal equivalent circuit is given in Fig. 6-12. By KVL around the B; C loop, with rm rc (from Problem 6.3), vB rb ib rm ib 1 rc ib ie rb rc ib 1 rc ie

1). What is the slope of the altitude to (a) AB; (b) BC; (12.14)

CHAP. 12]

12.37. Which of the following sets of points are collinear: (a) (2, 1), (4, 4), (8, 10); (b) ( 1, 1), (2, 4), (6, 8); (c) (1, 1), (3, 4), (5, 8) (12.15)

B rb ib + iS + LB _ rm ib RB _ (1 _ ) rc C ic RE RL ie re E iL + LL _

Fig. 12-36

12.38. What values of k will make the following trios of points collinear (a) A(0, 1), B(2, 7), C(6, k); (b) D( 1, 5), E(3, k), F(5, 11); (c) G(0, k), H(1, 1), I(3, 1) (12.15)

Fig. 6-12 Application of KVL around the C; E loop, again with rm rc , gives RE RL RE RL ie rc ib re 1 rc i RE RL RE RL e

2 shows that Pmax Veff =R, achieved at ! !0 , and that P 1 Pmax when jZin j2 2R2 ; that is, when 2

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12.39. State the equation of the line or pair of lines which is the locus of points (a) Whose abscissa is (b) Whose ordinate is 31 2 5 (f) 3 units from the line x (g) 6 units above the line y 2 2

!2

By Cramer s rule applied to the system consisting of (1) and (2), ie 2 = , where RE RL RE RL rc re rb re 1 rc RE RL RE RL 2 rc v B

(12.16)

Corresponding to the upper sign, there is a single real positive root: s 2 R R 1 2338:3 rad=s or !h 2L 2L LC and corresponding to the lower sign, the single real positive root s 2 R R 1 2138:3 rad=s or !l 2L 2L LC

12.40. State the equation of the locus of the center of a circle that (a) Is tangent to the x-axis at (6, 0) (b) Is tangent to the y-axis at (0, 5) (c) Is tangent to the lines x 4 and x 8 (d) Passes through the origin and (10, 0) (e) Passes through (3, 7) and (9, 7) (f) Passes through (3, 2) and (3, 8)

(b) The input impedance can be found as Zin RB k vB =ib . Now, in the system consisting of (1) and (2), by Cramer s rule, ib 1 = , where RE RL v 1 re 1 rc RE RL B RE RL RE RL RB rb re 1 rc R B rc re RE RL RE RL Hence, Zin RB k v RE RL RE RL 1 B RB rb re 1 rc rc re RE RL RE RL

(a) In the time domain, the instantaneous stored energy in the circuit is given by 1 q2 Ws Li2 2 2C For a maximum, dWs di q dq di q i L i vL vC 0 Li dt C dt dt C dt Thus, the maximum stored energy is Ws at i 0 or Ws at vL vC 0, whichever is the larger. Now the capacitor voltage, and therefore the charge, lags the current by 908; hence, i 0 implies q Qmax and 2 Q2 1 1 I I2 2 Ws ji 0 max CVC max C max max2 2 2 2C !C 2C! On the other hand, vL vC 0 implies vL vC 0 and i Imax (see the phasor diagram, Fig. 1237), so that

(12.18)

It follows that Ws max 8 2 < Imax =2C!2 :

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