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These expressions are not of much use in a problem where calculations can be carried out with the numerical values as in the following example.

(a) If AD > BC, then nI (b) If /1 /4, then nI nII by SSS. nII by SAS. nII by ASA. nII by ASA. nII by SAS.

Fig. 8-20 (a) The small-signal equivalent circuit is given in Fig. 8-20. ZE RE k Then, by KCL, Ie Ib hfe Ib hfe 1 Ib KVL around the input mesh requires that Vi Ri hie Ib ZE Ie Substituting (2) into (3) and solving for Ib yields Ib Vi Ri hie hfe 1 ZE 4 3 2 We rst de ne 1

EXAMPLE 9.5 The phasor voltage across the terminals of a network such as that shown in Fig. 9-7(b) is 100:0 458 V and the resulting current is 5:0 158 A. Find the equivalent impedance and admittance. Z Y V 100:0 458 20:0 I 5:0 158 I 1 0:05 V Z 308 17:32 j10:0

Selecting corresponding parts of congruent triangles In each part of Fig. 3-10, the equal parts needed to prove nI parts that are congruent.

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Current division at the collector node gives IL and Ohm s law and (5) yield V L R L IL RL RC h I RC RL 1=sCC fe b 6 RC h I RC RL 1=sCC fe b 5

Combinations of Impedances The relation V IZ (in the frequency domain) is formally identical to Ohm s law, v iR, for a resistive network (in the time domain). Therefore, impedances combine exactly like resistances: impedances in series impedances in parallel Zeq Z1 Z2 1 1 1 Zeq Z1 Z2

Fig. 3-10

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.

Substituting (4) and (1) into (6) and rearranging now lead to the desired voltage-gain ratio: V Av s L Vi hfe RL RC CC s sRE CE 1 hfe 1 RE hie Ri CE RE Ri hie sCC RC RL 1 s 1 hfe 1 RE Ri hie

Congruent corresponding sides lie opposite congruent angles. Congruent corresponding angles lie opposite congruent sides. (a) Opposite 45 , AC > DE, Opposite 80 , BC > DF. Opposite the side of length 12; /C (b) Opposite AB and CD, /3 /A /C. /3. Opposite BE and EC, /1 /4. Opposite /5 and /6, AB > CD. (c) Opposite AE and ED, /2 /2. Opposite BC and AD, /1 /D.

Fig. 9-8

/4. Opposite common side BD,

(b) The Laplace-domain transfer function (7) is of the form T s Kb s 2 s 1 1 s 1 3 s 1

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

Fig. 3-11

1 1 1 25 rad=s 1 CC RC RL 10 10 6 4000 1 1 1 50 rad=s !2 2 RE CE 100 200 10 6 1 hfe 1 RE Ri hie 101 100 100 1000 !3 509:1 rad=s CE RE Ri hie 3 200 10 6 100 1100 !1

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

(a) Since nI nII, by SSS, corresponding angles are congruent. Hence, 2x or y 20. (b) Since nI and y 5 24 or x 20 12, and 3y 26 or x 6, 60

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