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1 16.5 ADDITIONAL EXAMPLES For reiew purposes, more examples of boh piecewise linear and incremenal analysis are gien in he following subsecions. No new maerial is presened, so readers who do no need addiional pracice can omi his secion wihou loss of coninuiy PIECEWISE LINEA EXAMPLE: CLIPPING CICUIT The oupu olage o in he diode clipper circui, shown in Figure 16.21a, will resemble he inpu olage i, excep ha he boom of he waeform will be clipped off. The circui has only one diode, so ha he Théenin soluion mehod discussed in Chaper can be used, bu here we will use he mehod of assumed diode saes. Assume ha he diode in Figure is ideal, hen draw he wo subcircuis, one wih he diode OFF, and he oher wih he diode ON, as shown in Figures 16.21b and 16.21c. By inspecion, he oupu olage wih he diode OFF is consan, because here is a fixed curren I O flowing up hrough. Thus o1 = I O. (16.35) The curren source and he olage source are in series, so he olage source has no effec on o1. Furhermore, when he diode is OFF, i = o1 D. (16.36) Because he diode is in he OFF sae, D mus be negaie. I follows ha in he OFF sae i mus always be more negaie han I O. Nex, when he diode is ON, he oupu is direcly conneced o he inpu: o2 = i. (16.37) In he ON sae, i mus be more posiie han I O. Hence he alid porions of he waeforms in he subcircuis are he darkened segmens, and he complee oupu waeform is as shown in Figure 16.21d. As promised, he circui has clipped off he boom of he inpu wae EXPONENTIATION CICUIT The circui shown in Figure produces an oupu olage OUT ha is proporional o he exponenial of he inpu olage IN for sufficienly large IN. To analyze his circui, assume ha he Op Amp is ideal, he sauraion curren of he diode is I s = 1 12 A, and he emperaure of he diode is approximaely 29 C so ha is hermal olage is TH = 26 m. Because 918f

2 I O i o (a) o1 i I O i i D = D alid for i < I O o1 (b) I O I O o2 i = o2 FIGUE Diode clipper. i D = o2 I O alid for i > I O (c) o o (d) I O i 918g

3 = 1 kω FIGUE A diodebased exponeniaion circui. IN i D OUT he Op Amp is ideal, and used in a sable negaiefeedback configuraion, he olage a he inering erminal of he Op Amp is zero. As a resul, ) ( ) i D = I s (e IN/ TH 1 = 1 12 A e IN/(26 m) 1. For sufficienly large IN, for example for IN 12 m, he exponenial erm dominaes, and his relaion simplifies o i D 1 12 A e IN/(26 m). Nex, because he olage a he inering erminal of he Op Amp is zero, and because he curren ino ha erminal is zero, he oupu olage is gien by OUT = i D 1 7 e IN/(26 m), which exhibis an exponenial dependence on IN. For example, for IN = 2 m, 3 m, and 4 m, OUT =.219 m, 1.3 m, and 48 m, respeciely PIECEWISE LINEA EXAMPLE: LIMITE The circui in Figure is useful for making square waes ou of sine waes, and for limiing he ampliude of an oupu waeform when he inpu waeform ampliude aries oer a wide range. To analyze he circui, we noe ha he Théenin approach is no helpful, and a graphical soluion migh be messy because of he wo diodes (no so, in fac, bu ha is no obious ye). So resor o analysis by assumed diode saes. The subcircuis for he four saes, assuming an idealdiode model, are shown in Figures 16.23b, 16.23c, 16.23d, and 16.23e 918h

4 i D 1 D 2 o (a) Limier circui o1 i i D1 OFF OFF D2 o1 (b) Boh diodes OFF o2 i OFF o2 FIGUE Diode limier. (c) D 1 ON o3 i OFF o3 (d) D 2 ON o i ON ON o4 i o (e) Boh diodes ON (f) Complee waeform 918i

5 along wih he appropriae subcircui oupu olages, obainable by inspecion. From Figure 16.23b, o1 = i (16.38) because here is no curren hrough. When eiher diode is ON, he oupu olage is independen of he source olage i. For D 1, ON, for example, o2 =. The fourh diode sae, Figure 16.23e, canno be reached wih his opology, (assuming is a posiie quaniy) because here is no alue of i ha will force boh diodes ON a he same ime. Now we mus idenify he alid segmens of hese waeforms. In Figure 16.23b, boh diodes are assumed OFF, so d1 and d2 mus boh be less han zero. Hence, using KL: i = D1 < (16.39) i < (16.4) i = D2 < (16.41) i >. (16.42) Thus i mus be beween and. Likewise o1, from Equaion This range of alidiy is indicaed by he darkened segmens of he waeform in Figure 16.23b. I follows ha he complee oupu wae mus be as shown in Figure 16.23f. If he peak ampliude of i is en or weny imes, hen o is a reasonable approximaion of a square wae EXAMPLE: FULLWAE DIODE BIDGE Figure shows one of he mos common recifier circuis found in elecronic equipmen, he fullwae diode bridge. We assume herefore ha i is a 6Herz sinusoid wih 1ol peak ampliude, and we wish o find he waeform o across he oupu resisor. A fullblown assaul using assumed diode saes would yield 16 subcircuis, bu i will urn ou ha only wo of hese are possible, suggesing a more insighful approach. Suppose ha i is a small posiie olage. Then curren mus flow down hrough he bridge. The only aailable pah is D 1, L, and D 4, because of he orienaion of D 3 and D 2. Similarly, for i negaie, curren mus flow up, and hus mus follow he pah D 3, L, D 2. The wo corresponding subcircuis, assuming ideal diodes, are shown in Figures 16.24b and 16.24c. Now, by inspecion, for i posiie, and for i negaie, o = i (16.43) o = i. (16.44) 918j

6 (a) i D 1 D 2 o L D 3 D 4 i D 3 OFF oa D 1 OFF i D 1 OFF ob D 4 OFF i D 3 OFF oc D 4 OFF FIGUE Fullwae diode bridge. (b) Subcircui for i posiie (c) Subcircui for i negaie (d) D 1 and D 2 ON i o (e) All oher subcircuis are degenerae. Consider, for example, he subcircui for boh D 1 and D 2 ON, as in Figure 16.24d. Clearly no curren can flow in L. Also, curren can flow down hrough D 3, or up hrough D 4, so all diode currens mus be zero in his sae. A similar argumen holds for all adjacen diode pairs, wheher ON or OFF. Hence he wo saes depiced in Figures 16.24b and 16.24c are he only ones we need o consider. Noe ha he curren always flows in he same direcion hrough L, regardless of he polariy of i. For sinusoidal inpu he waeforms appear as in Figure 16.24e. The circui is called a fullwae recifier because curren flows hrough L on boh hales of he inpu wae. Neglecing diode olage drops, he aerage alue of he oupu olage, ha is he DC olage, is.637 imes he peak of he inpu sinusoid. Furher, by symmery here is no frequency componen in he oupu waeform a he inpu frequency, 6 Herz in our example, or odd muliples hereof. Hence he circui has a much higher percenage of DC relaie o harmonics compared o he halfwae recifier discussed in Secion k

7 i i 5 m AC FIGUE Zenerdiode regulaor. 2 DC o (a) (b) INCEMENTAL EXAMPLE: ZENEDIODE EGULATO All semiconducor diodes will break down and conduc appreciable curren under reerse bias condiions if he reerse olage across he diode is large enough. This breakdown is nondesrucie if he curren is no excessie; he diode reurns o normal reersebias behaior if he olage is reduced. A Zener diode is a semiconducor diode in which his breakdown under reerse bias is carefully conrolled by he manufacuring process so ha he breakdown occurs a a specified olage, he socalled Zener olage of he diode. A ypical i cure is shown in Figure 16.25a. Because he breakdown olage of a Zener diode can be carefully conrolled by he manufacuring process, and he incremenal resisance in he breakdown region is quie small (around 1 o 5 ), Zener diodes are quie useful as olage regulaors. A simple example is shown in Figure 16.25b. Equaion 4.74 is clearly inappropriae for finding he incremenal resisance in breakdown, because his par of he characerisic is no an exponenial. Hence he alue mus be obained from he daa shee for he Zener diode in quesion. Inegraedcircui regulaors, wih ransisors, Zener diodes, and resisors all on a single chip, will cerainly ouperform eiher of he crude regulaor circuis discussed here INCEMENTAL EXAMPLE: DIODE ATTENUATO I should be clear from Secion 4.5, and paricularly from Equaions 4.63 and 4.74, ha for small incremens of olage or curren, he semiconducor diode looks like a linear resisor whose alue depends on he DC curren flowing hrough i. Thus i should be fairly easy o build an aenuaor wih an aenuaion consan ha can be changed by means of an exernal olage or curren. Figure shows a simple example. Here a diode is used in he shun branch of a olage diider on a smallsignal source i. The DC curren hrough he diode is conrolled by he DC olage C hrough he large 918l

8 C = 1 C (a) C i 1 OC = C 1 1 C (c).6 d C C FIGUE Diode aenuaor. (b) 1 C I D (d) i 1 r d o resisor C. If we assume ha i produces less han a5mchange in he diode olage, hen he incremenal analysis approach discussed in Secion 4.5 can be applied. Firs, draw he circui for he calculaion of he DC curren I D is o form he Théenin equialen of he linear par of he circui, as shown in Figure 16.26b. An easy way o sole for I D is o form he Théenin equialen of he linear par of he circui, as shown in Figure 16.26c. A he same ime we replace he diode by is piecewise linear model. Then I D = ( OC.6) d. (16.45) The incremenal resisance r d of he diode will hus be a funcion of C : r d = kt (16.46) qi D = kt ( 1 C ) d. (16.47) q 1 C.6 1 C Now draw he subcircui ha relaes he incremenal ariables. eplace he diode in Figure 16.26a by he incremenal resisance r d, and se all DC sources, in his case C, o zero, as indicaed in Figure 16.26d. By inspecion, C r o = d i 1 ( C r d ). (16.48) The aenuaion is clearly dependen on he DC olage C, as desired. 918m

9

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