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In the analyses of the two preceding chapters, we assumed operation in the midfrequency range, in which the reactances of all bypass and coupling capacitors can be considered to be zero while all inherent capacitive reactances associated with transistors are in nitely large. However, over a wide range of signal frequencies, the response of an ampli er is that of a band-pass lter: Low and high frequencies are attenuated, but signals over a band (or range) of frequencies between high and low are not attenuated. The typical frequency behavior of an RC-coupled ampli er is illustrated by Fig. 8-1(a). In practical ampli ers the midfrequency range spans several orders of magnitude, so that terms in the gain ratio expression which alter low-frequency gain are essentially constant over the high-frequency range. Conversely, terms that alter high-frequency gain are practically constant over the low-frequency range. Thus the high- and low-frequency analyses of ampli ers are treated as two independent problems.

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This is a combination of the following two principles: 1. Complements of the same angle are congruent. Thus, j a > j b in Fig. 2.23 and each is the complement of j x. 2. Complements of congruent angles are congruent. Thus, j c > j d in Fig. 2-24 and their complements are the cons gruent j x and y.

Fig. 8-6

Fig. 2-23

v A 1 es 1 t A 2 es 2 t 2 s q 1 1 2 1 2 !2 where s1 0 2RC 2RC LC s q 1 1 2 1 2 !2 s2 0 2RC 2RC LC p where 1=2RC and !0 1= LC . Note that , the damping factor of the transient, di ers from in the series RLC circuit.

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|A| Low-frequency range Amid Amid 2 Midfrequency range I1 (s) + V1 (s) _ T (s) = N (s) D (s) I2 (s) + V2 (s) _

Fig. 8-7

PRINCIPLE 4:

Fig. 8-1

The transient response is easiest to visualize by assuming an initial charge Q0 on the capacitor and a switch that closes at t 0. However, a step function voltage applied to the circuit will initiate the same transient response. Overdamped Case 2 > !2 0 In this case, the solution (2) applies.

Fig. 2-24

CHAP. 8]

, C 0:167 mF, and L 1:0 H, has an initial voltage V0 50:0 V on the capacitor. Obtain the voltage v t when the switch is closed at t 0. We have 1 2994 2RC 2 8:96 106 !2 0 1 5:99 106 LC

This is a combination of the following two principles: 1. Supplements of the same angle are congruent. Thus, j a > j b in Fig. 2-25 and each is the supplement of j x. 2. Supplements of congruent angles are congruent. Thus, j c > j d in Fig. 2-26 and their supplements are the congruent angles x and y.

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Since 2 > !2 , the circuit is overdamped and from (2) we have 0 q q s1 2 !2 1271 and s2 2 !2 4717 0 0 dv At t 0; and s A s2 A 2 V0 A1 A2 dt t 0 1 1 From the nodal equation (1), at t 0 and with no initial current in the inductance L, V0 dv dv V0 0 or C dt dt t 0 R RC Solving for A1 , A1 Substituting into (2) v 155:3e 1271t 105:3e 4717t See Fig. 8-8. V V0 s2 1=RC 155:3 s2 s1 and A1 V0 A1 50:0 155:3 105:3

Fig. 2-25

Any linear two-port electrical network that is free of independent sources (including a small-signal ampli er equivalent circuit) can be reduced to the form of Fig. 8-1(b), where T s N s =D s is the Laplace-domain transfer function (a ratio of port variables). Of particular interest in ampli er analysis are the current-gain ratio (transfer function) T s Ai s and voltage-gain ratio (transfer function) T s Av s . For a sinusoidal input voltage signal, the Laplace transform pair v1 t V1m sin !t $ V1 s is applicable, and the network response is given by V2 s Av s V1 s Av s V1m ! s2 ! 2 8:1 V1m ! s2 ! 2

Fig. 8-8

PRINCIPLE 5:

Without loss of generality, we may assume that the polynomial D s 0 has n distinct roots. Then the partial-fraction expansion of (8.1) yields V2 s k1 k2 k3 k4 k n 2 s j! s j! s p1 s p2 s pn 8:2

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