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It is clear that ampli ers can be biased for operation at any point along the dc load line. Table 3-4 shows the various classes of ampli ers, based on the percentage of the signal cycle over which they operate in the linear or active region.

Table 3-4 Class A AB B C Percentage of Active-Region Signal Excursion 100 between 50 and 100 50 less than 50

notation There are other notations traditionally used for the derivative: Dx f (x) and dy dx dy . We shall use whichever notation is most convenient dx

Two common uses of capacitors (sized to appear as short circuits to signal frequencies) are illustrated by the circuit of Fig. 3-10(a).

LL _

When a variable y represents f (x), the derivative is denoted by y , Dx y, or or customary in a given case.

Note!

Fig. 3-10

1. 2.

The derivative is so important in all parts of pure and applied mathematics that we must devote a great deal of effort to nding formulas for the derivatives of various kinds of functions. If the limit in the above de nition exists, the function f is said to be differentiable at x, and the process of calculating f is called differentiation of f . EXAMPLES

Coupling capacitors (CC con ne dc quantities to the transistor and its bias circuitry. Bypass capacitors CE e ectively remove the gain-reducing emitter resistor RE insofar as ac signals are concerned, while allowing RE to play its role in establishing -independent bias (Section 3.6).

The capacitors of Fig. 3-10(a) are shorted in the circuit as it appears to ac signals [Fig. 3-10(b)]. In Fig. 3-10(a), we note that the collector-circuit resistance seen by the dc bias current ICQ % IEQ is Rdc RC RE . However, from Fig. 3-10(b) it is apparent that the collector signal current ic sees a collector-circuit resistance Rac RC RL = RC RL . Since Rac 6 Rdc in general, the concept of an ac load line arises. By application of KVL to Fig. 3-10(b), the v-i characteristic of the external signal circuitry is found to be vce ic Rac Since ic iC ICQ and vce vCE VCEQ , (3.11) can be written analogously to (3.9) as iC vCE VCEQ ICQ Rac Rac 3:12 3:11

All excursions of the ac signals ic and vce are represented by points on the ac load line, (3.12). If the value iC ICQ is substituted into (3.12), we nd that vCE VCEQ ; thus, the ac load line intersects the dc load line at the Q point.

(a) Let f (x) = 3x + 5 for all x. Then, f (x + h) = 3(x + h) + 5 = 3x + 3h + 5 f (x + h) f (x) = (3x + 3h + 5) (3x + 5) = 3x + 3h + 5 3x 5 = 3h 3h f (x + h) f (x) = =3 h h Hence, f (x) = lim

If liabilities are to be assumed by the partnership, they are credited to the respective liability ac counts.

Example 3.9. Find the points at which the ac load line intersects the axes of the collector characteristic. The iC intercept iC max is found by setting vCE 0 in (3.12): iC max The vCE intercept is found by setting iC 0 in (3.12): vCE max VCEQ ICQ Rac 3:14 VCEQ ICQ Rac 3:13

CHAP. 3]

(b) Let us generalize to the case of the function f (x) = Ax + B, where A and B are constants. Then, f (x + h) f (x) [A(x + h) + B] (Ax + B) Ax + Ah + B Ax B Ah = = = =A h h h h f (x + h) f (x) = lim A = A f (x) = lim h h 0 h 0

The noninverting ampli er of Fig. 9-3 is realized by grounding R1 of Fig. 9-2 and applying the input signal at the noninverting op amp terminal. When v2 is positive, vo is positive and current i is positive. Voltage v1 iR1 then is applied to the inverting terminal as negative voltage feedback.