Lecture-10 BJT Switching Characteristics, Small Signal Model

Transcription

1 1 Lecure-1 BJT Swiching Characerisics, Small Signal Model BJT Swiching Characerisics: The circui in Fig.1(b) is a simple CE swich. The inpu volage waveform v s shown in he Fig.1(a) is used o conrol he sae of he swich (beween collecor and emier). For < T 1, v s = V 1 and he emierbase diode is reverse-biased. If we neglec he reverse-curren componens, since he collecor-base diode is also reverse-biased, he BJT is cu-off and no curren exiss anywhere in he circui. Consequenly, v o = V CC, and wih i C =, his is an open swich. Acually, i C I CO and v o = V CC I CO R L. However, wih I CO of he order of a nano-ampere and R L of he order of kilohms, v o differs from V CC by only a few microvols. Thus, for pracical purposes, v o = V CC. The inpu volage becomes V 2 for T 1 < < T 2. The value of V 2 is seleced o ensure ha he BJT is a leas a he edge of sauraion. From Table-1 in LN-7, v CE = v o = V CE(sa).3 V and i C = (V CC V CE(sa) )/R L ; hese values approximae he closed swich. Noe ha he curren in he closed swich is deermined by he exernal elemens V CC and R L. For V CC.3 V, i C = V CC /R L. A = T 2, he inpu wave form swiches back o V 1, evenually causing he BJT o reurn o cuoff. Skeches of boh v o and i C are depiced in Fig.1. The causes of he swiching ransiens are described laer in his secion. The naure of he swiching characerisics is readily discernible from he ransfer characerisic, a graph of v o versus v s, for he circui. v () s +V CC V 2 i B R L T 1 T 2 + v () s R B i C + v o () V 1 (a) (b) Figure 1: (a) Inpu Waveform applied o BJT (b) A simple BJT used as swich in CE configuraion BJT Swiching Speed: Our descripion of he circui in Fig. 1(a) and (b) a he beginning of his secion focused on he ON and OFF saes of he swich. We now consider he swiching ransiens indicaed in he waveforms in Fig.2. As seen in Fig.2, he curren does no immediaely respond o he inpu signal. Insead, here is a delay, and he ime ha elapses during his delay, ogeher wih he ime required for he curren o rise o 1 percen of i maximum (sauraion) value, is

2 2 Figure 2: The waveforms for v o and i c displaying he rise ime, fall ime, delay and sorage ime during swiching called he delay ime d. The curren waveform has a nonzero rise ime r, which is he ime required for he curren o rise hrough he acive region from 1 o 9 percen of I C(sa). The oal urn-on ime ON is he sum of he delay and rise ime, ON d + r. When he inpu signal reurns o is iniial sae a = T 2, he curren again fails o respond immediaely. The inerval which elapses beween he ransiion of he inpu waveform and he ime when i C has dropped o 9 percen of I CS or I C is called he sorage ime s. The sorage inerval is followed by he fall ime f, which is he ime required for i C o fall from 9 o 1 percen of I C(sa). The urn-off ime OF F is defined as he sum of he sorage and fall imes, OF F s + f. We shall consider now he physical reasons for he exisence of each of hese imes. The exac calculaions of hese imes are complex; Three facors conribue o he delay ime: (1) when he driving signal is applied o he ransisor inpu, a nonzero ime is required o charge up he emier-juncion ransiion capaciance so ha he ransisor may be brough from cuoff o he acive region; (2) even when he ransisor has been brough o he poin where minoriy carriers have begun o cross he emier juncion ino he base, a ime inerval is required before hese carriers can cross he base region o he collecor juncion and be recorded as collecor curren; and (3) some ime is required for he collecor curren o rise o 1 percen of is maximum.

3 3 The rise ime and he fall ime are due o he fac ha if a base-curren sep is used o saurae he ransisor or reurn i from sauraion o cuoff, he ransisor collecor curren mus raverse he acive region. The collecor curren increases or decreases along an exponenial curve whose ime consan T r. The failure of he ransisor o respond o he railing edge of he driving pulse for he ime inerval s resuls from he fac ha a ransisor in sauraion has excess minoriy carriers sored in he base. The ransisor canno respond unil his excess charge has been removed. Consider ha he ransisor is in is sauraion region and ha a = T 2 an inpu sep is used o urn he ransisor off, as in Fig.1. Since he urn-off process canno begin unil he abnormal carrier densiy has been removed, a relaively long sorage delay ime s may elapse before he ransisor responds o he urn-off signal a he inpu. In an exreme case his sorage-ime delay may be several imes he rise or fall ime hrough he acive region. I is clear ha when ransisor swiches are o be used in an applicaion where speed is a a premium, i is advanageous o reduce he sorage ime. A mehod for prevening a ransisor from sauraing, and hus eliminaing sorage ime, is he use of a Schoky diode in conjuncion wih he BJT. p (x) p () p (x) Q B x = W B x Figure 3: Disribuion of excess minoriy carriers in he base of a pnp ransisor Sored Charge and Delay ime d (or Transi Time r ): We have seen ha BJT does no respond insananeously o he fas changing signals. This is because BJT speed of response is limied mainly by he sorage or diffusion capaciance, which accompanies he sorage of minoriy carriers in he base. Le W B be he widh of he base diffusion lengh L p or L n of minoriy carriers so ha disribuion is linear in boh acive and sauraion modes. Le he linear disribuion of minoriy carriers is as shown in Fig.3, for he operaion of acive region. For pnp he hole curren is he majoriy curren, herefore excess hole curren is relaed as J C = qd p dp dx (1)

4 4 I C = AqD p dp dx (2) dp dx = p () W B (3) I C = qad pp () W B (4) The excess minoriy carrier charge sored in he base is given by Q B = concenraion volume = area under he curve shown in Fig.2 qa ( p ) ()W B = qa (5) 2 Now aking he raio of Eqns.(5) and (4) we ge, Q B I C = W 2 B 2D p τ B (6) Le us find he relaionship beween τ B and approximae ime i akes a hole o ravel from emier juncion o he collecor juncion. dx d = v(x) = velociy of he hole (7) Le d r be he ime for he hole o cross base of widh W B. Therefore, we have r = WB The curren due o holes is given by d = WB I p = qad p dp /dx dx/v(x) (8) = qad p p ()/W B (9) This curren can also be wrien in erms of velociy of he carriers. Thus we have, I p = qp (x)av(x) (1) Bu equaion for concenraion gradien as shown in he Fig.2 is given by ( p (x) = p () 1 x ) W B (11)

5 5 Subsiuing Eqn.(11) in (1) we ge, ( I p = qav(x)p () 1 x ) W B (12) Now equaing Eqn.(9) and (1) we ge, Therefore, r = v(x) = WB D p ( 1 x W B ) (13) ( 1 x D p W B ) dx = W 2 B 2D p (14) Thus r = τ B. Charge Conrol Relaions: The general expression for he excess hole densiy charge sored in he base of pnp ransisor, Q B is given by WB Q B = qa p (x)dx (15) where p (x) is he excess hole densiy a x. p (x) = p(x) p ha implies dp (x) = dp(x). From he coninuiy equaion for he holes we have, p p = p p 1 J p τ p q x = p (x) 1 J p τ p q x (16) Muliplying boh sides by qa and inegraing from o W B wih respec o x we ge WB WB p qap (x) WB (x)qa dx = dx qa J p (17) τ p The firs inegral in he above equaion represens he rae of change of he sored charge, Q B wih ime; he second inegral is he raion of he sored charge o he life-ime of holes; and he hird inegral becomes i C + i E = i B (). Thus equaion can be wrien as, dq B d = Q B() τ p i B () = Q B() τ p ( i B ()) + dq B() d (18)

6 6 Turn-ON Time: The ransisor swiches from cu off o sauraion by applicaion of a sep of base curren. Le i B () = I B for cerain duraion of ime. Then he Eqn.(18) becomes, I B = Q B() τ p + dq B() d Therefore solving his firs-order differenial equaion, and using he iniial condiion a =, Q B =. Q B () = I B τ p (1 e /τp ) (2) i C () = Q B() τ B (19) = I Bτ p τ B (1 e /τp ) (21) Q B () I B τ p Q SAT i () C I CSAT ON Figure 4: Skeches of Q B () and i C () As, implies Q B ( ) = I B τ p and i C ( ) = I Bτ p τ B. We define Q SAT is he value of Q B a i C = I CSAT. Thus we have, i C () = I Bτ p τ B (1 e /τp ) for Q B Q SAT (22) Therefore a = ON i C ( ON ) = I CSAT = Q SAT /τ B I CSAT = I Bτ p (1 e ON /τ p ) = V CC (23) τ B R L

7 7 Thus he expression for urn-on ime is given by 1 ON = τ p ln 1 ( ) ( ) V CC τb (24) I B R L τp Turn-OFF Time: To urn a ransisor OFF, he excess sored charge in he base mus be removed and he collecor curren mus be made almos zero (= I C or I CEO ). This is done by making i B () =. Thus Eqn.(18) becomes, dq B d = Q B τ p (25) The soluion of he above firs-order homogeneous differenial equaion is given by, Q B () = Q B ()e /τp for > (26) The urn off ime defined as he ime required o reduce he collecor curren o almos zero, is made up of wo incremens: he ime i akes Q B o reach Q SAT, is known as he sorage ime s and second, he ime f, i akes collecor curren o reach zero, or more pracically o a value of abou.1i CSAT. The decrease of sored charge and curren are shown in he Fig.5. A = s we have Q B ( s ) = Q SAT. For > s, he ransisor is in acive region, so ha i C is given by i C () = Q B ()/τ B = (Q SAT /τ B )e /τp = I CSAT e /τp (27) The sorage ime s is found from, Solving for s, we obain Q B ( s ) = Q SAT = Q B ()e s/τp (28) s = τ p ln ( ) QB () Q SAT The BJT Small-Signal Model: The circui shown in he Fig.6 is an elemenary CE amplifier sage. The capacior C B (called a blocking capacior) is used o isolae he dc bias from he signal source v s = V sm sin ω and is source resisance R s. This capacior acs as an open-circui under he quiescen condiions (no inpu signal) because he reacance of a capacior is infinie a zero frequency (dc). We assume ha a he angular frequency of he signal, he reacance of C B is sufficienly small compared wih R s ha he series combinaion of hese elemens is R s. Consequenly, he effec of he capacior on he signal ransmied from he source v s o he amplifier inpu is negligible. The ampliude V sm is chosen o provide he base curren i b = I bm sin ω. The oal insananeous base curren i B is he superposiion (29)

8 8 Q () B Q () B Q SAT i C () I CSAT.1I CSAT s f ON Figure 5: Skeches of Q B () and i C () of he dc bias level and signal curren. Hence, i B = I BQ + i b = I BQ + I bm sin ω A (3) As seen from he Fig.7, he effec of his signal causes boh i C and v CE o vary (approximaely) sinusoidally abou heir quiescen levels. These quaniies are expressible as, i C = I CQ + i b = I CQ + I cm sin ω A (31) v CE = V CEQ + v ce = V CEQ + V cem sin ω V (32) The small signal equivalen circui of he BJT is shown in he Fig.8(a). The elemens forming he equivalen circuis relae he changes in volages and currens abou he operaing poin. Each elemen in he model is a funcion of he quiescen volages and currens esablished by he bias. As he inpu signal causes he changes, he equivalen circui permis us o relae he oupu signal o he inpu signal. The hybrid-π equivalen circui for CE conneced BJT is shown he Fig.8(a). We can idenify he elemens in his model wih coupled-diode represenaion of he

9 9 + V CC R B R c C B R s v s = V sin ω s Figure 6: CE Amplifier ransisor. The forward-biased emier-base juncion is modeled by r π and C π where C π is essenially he diffusion capaciance and r π is he incremenal resisance of he emier-base diode.the capaciance C µ is he depleion capaciance of he reversebiased collecor-base juncion. The incremenal resisance r µ is shown by he dashed line. This resisance accouns for he feedback (base widh modulaion) beween he inpu and oupu due o Early effec. r µ is very large and usually negleced. Coupling beween he juncion is modeled by he conrolled curren source g m v π and is proporional o he inpu curren i b. The oupu resisance r o is he resul of Early effec and equals he reciprocal of he slope of he dashed lines in he Fig.3 of LN-7. The resisance r b is he base spreading resisance and accouns for he volage drop in he pah beween he base conac and he acive base region under emier. Because of he larger cross-secional area of he collecor region, he collecor-spreading resisance is in he order of 1 Ω and is usually negleced. The Low Frequency Model: From he LN-9 we find he expressions for C π (or C D ) and C µ (or C T ) are dependen on he operaing poin values of he BJT volages and currens. A ypical quiescen levels, for boh IC and low-power discree ransisors, values of C π are in he order of ens of picofarads o one or wo hindered picofarads. Values of C µ are generally a few picofarads (1 o 5 pf). A low signal frequencies, he reacance of boh capaciors are exremely high. A such frequencies he effecs of C π and C µ can be negleced and consequenly ac as open circui. This

10 1 leads o he low frequency model shown in he Fig.8(b). We observe ha, v π = r π i b (33) For v ce =, no currens exiss in r o and, i c = g m v π = g m r π i b (34) I is convenien o inroduce, i c i b = g m r π (35) Figure 7: CE oupu characerisics showing load line and sinusoidal signal componens β o = i C i B V CE = cons = V CEQ = i c i b v ce = (36)

11 11 The parameer β o is he incremenal (ac) common-emier forward shor-circui curren gain and is evaluaed a he operaing poin. The consan value of v CE is indicaive of incremenal change in he quaniy, and hus only v ce =. (The condiion v ce = and i c represens a shor circui beween collecor and emier relaive o he signal. I does no, however, indicae a physical shor-circui connecion beween hese erminals.) Thus we have, β o = g m r π (37) r µ b r b + C µ c r π C π v π g v m π r o e (a) e b r b r π + v π g v m π r o c e (b) e Figure 8: (a) Small Signal Hybrid-π equivalen circui of BJT. (b) The Low frequency Hybrid-π The parameer g m = i c /v π, called ransconducance, reflecs he incremenal changes in i C abou he operaing poin produced by he incremenal change in he emier-base volage. The volage drop i b r b is small so ha changes in he baseemier volage can be assumed o appear across he juncion. Quaniaively, g m

12 12 is expressible as, g m = i C v BE = i C v BE V CE = cons = V CEQ v ce = (38) We know ha i C = α F i E for eiher npn or pnp ransisor and Eqn.(38) becomes, i E g m = α F (39) v ce = v BE We wish o relae g m o he conducance of he emier-base diode. The incremenal conducance of he diode is given by, Eqn.(3) of LN-8, as g d = di D dv D (4) where i D and v D are he forward curren and volage of he diode. For an npn ransisor, v BE forward-biases he emier diode and v BE = v D ; however i E is in he i opposie direcion of i D (from n o p) so ha i E = i D. Therefore, E v BE = di D dv D and g m = α F g d (41) Eqn.(41) remains valid for pnp ransisor because forward-biasing he emier juncion makes i E = i D and v BE = v D. The emier-diode conducance g d is expressed in he Eqn.(5) of LN-8, wih η = 1. Hence g d = I EQ /V T for an npn ransisor and g d = +I EQ /V T for an pnp device. For npn (pnp) ransisor, I EQ is negaive (posiive); hus g d is posiive in boh he insances and can wrien as g d = I EQ /V T, hus we obain he following simple expression for he ransconducance: g m = α F I EQ V T = I CQ V T (42) Figure of Meri: A measure of he qualiy of a high frequency ransisor is is figure of meri f T. As we shall see f T is a measure of he raio of g m o he oal capaciance of he ransisor. By neglecing r µ, r b we will deermine an expression for he shor-circui gain β(jω), of he ransisor. I is defined as he raio of he curren I o in a shor-circui placed a he oupu, as shown he Fig.9. Because of he shor circui, C µ is in parallel wih C π. By neglecing he curren in C µ compared

13 13 o g m v π, I o is g m v π and v π is I i r π /[1 + jωr π (C π + C µ )]. The curren gain is given by I o I i = β o 1 + jωr π (C π + C µ ) (43) A high frequencies, he magniude of he imaginary par of he denominaor of Eqn.(43) is much greaer han uniy, so ha β(jω) = g m ω(c π + C µ ) (44) where g m = β o /r π. The symbol, f T is defined as he frequency a which he magniude of he shor-circui curren gain is uniy, so ha a f = f T, β(jω) = 1 and g m f T = (45) 2π(C π + C µ ) I i I o b + + C µ c v π r π C π v π g v m π r o e e Figure 9: Circui for calculaing f T of a ransisor