1.4 Small-signal models of BJT

Transcription

1 1.4 Small-sgnal models of J Analog crcuts often operate wth sgnal levels that are small compared to the bas currents and voltages n the crcut. Under ths condton, ncremental or small-sgnal models can be derved that allow calculaton of crcut gan and termnal mpedances wthout the necessty of ncludng the bas quanttes. = + c c v n = + b b p n n E = + e E e arrer concentraton - depleton regon + v E E n ΔQ h ΔQ e + c A A E W 22

2 = + c c v + n - = + b b (p) (n) E(n) E = + e E e arrer concentraton - depleton regon + v E E n ΔQ h ΔQ e + c A x A E W = quescent current = quescent current v = small-sgnal /p voltage otal current, b = + b otal current, c = + c Q e = change n the mnorty carrer charge Q h = change n the majorty carrer charge 23

3 1.4.1 ransconductance (pg. 27) g m = d / d E / E = d / d E g m = / E = c / v (1.90) g m = d d d E E S E g m = Sexp = exp = de g q m =,.e. g k m depends lnearly on. f = 1 ma and at 25, () 1 10 (A) -23 g m = = (J/K) 298(K) 8.91 ma/ rrespectve of the type of the J (npn or pnp), sze and made of materal (S, Ge, GaAs). 24

4 +v v E E c =Sexp =Sexp exp Snce E = exp, S then v =exp c (1.94) f v <, equaton (1.94) can be expanded n a power seres, 2 3 v 1 v 1 v c = c = c v v v c = f v <<, as (1.90). v g v = c = m whch s the same he small-sgnal analyss s vald. 25

5 he crteron for use of small-sgnal analyss s v = E << 26 m at 25. n practce, f E < 10 m, the small-sgnal analyss s accurate wthn 10% ase-chargng capactance + v E E arrer concentraton n ΔQ e ΔQ h + c - depleton regon A A E W he charge n E voltage, E = v, has caused a change n the mnorty-carrer charge n the,.e. Q e = q e. y charge neutralty requrements, there s an equal change, Q h = q h, n the majorty carrer charge n the. Snce majorty carrers are appled by the lead, the applcaton of voltage v requres the supply of charge q h to the. Due to ths, the devce has an nput capactance: b = q h / v (1.98) 26

6 Mnorty-carrer charge n the, Q= 1 e n(0)wqa p 2 (1.39) n(0) =qad p (1.33) W n Qe 2 1 W = D 2 n = τ (1.99) F τ F = transt tme n the forward drecton = average tme per carrer spent n crossng the ypcally, τ F From (1.99), = ps for npn = 1 40 ns for pnp Q= τ e F ΔQ= ΔQ=q= τ Δ = τf c (1.102) e n n F 27

7 From equatons (1.98).e. b = q h / v, and (1.102),.e. q = τ, n F c b = τ / v. F c From equaton (1.90),.e. g m = c / v, c = g m τ F c / b b = τ g m = τ = τ F F F herefore, b. q k nput resstance, r π (pg. 29) = / β F / = d /d Δ d = d β F Δ d d β = Small-sgnal current gan, β Δ o = = Δ c b = d d β F 1 28 F

8 f β F s constant, β o = β F. ypcally, β o β F. A sngle value of β s often assumed for a transstor and used for both ac and dc calculatons. Small-sgnal nput resstance, r π = v / b = v β o / c. Snce g m = c / v, r π = β o / g m = β o / Hence, r π 1/ Output resstance, r o (pg. 29) Small changes E n E produce correspondng changes n, where Δ = E Δ E Δ Δ E = E 29

9 From equaton (1.55), dw =- W d E E From equaton (1.57), E = A d E A =-W dw Δ Hence, E = A = r Δ A = Early voltage = r o = small-sgnal output resstance For = 1 ma, r o = kω. r o 1/ and g m = / o A q r= = A = 1 = 1 g gmk k g mη g m m o q A Slope A Δ 1 r = = ΔE o Δ E E3 E2 Δ E1 E 30

10 1.4.5 asc small-sgnal model of the J Hybrd-π model: r π b + - v gmv r ο E ald for both npn and pnp n the forward actve regon. ase-chargng capactance, b = τ F g m nput resstance, r π = β o /g m Output resstance, r o = 1/ηg m ransconductance, g m = / 31

11 resstance arrer concentraton ncreased depleton regon ntal depleton regon E +Δ A x=0 x A E ΔW Reduced wdth ntal wdth W 2 W 1 depleton regon wdens due to Δ E E - depleton-layer wdth ase wdth (W ) otal mnorty-carrer charge n. 1 (as 1 represents recombnaton of electron and holes n ) 32

12 Snce an ncrease E n E causes a decrease n, ths effect can be modeled by the ncluson of a resstor r µ from to. f E s assumed constant, Δ Δ Δ Δ r μ = = =r Δ Δ Δ Δ f = 1, E E ο Δ r μ =r ο =r c=r ο ο β Δ ο b hs s a lower lmt for r µ as 1 under ths condton s at ts maxmum value. n practce, 1 = 10% (as 2 s the one that domnates). Snce 1 s very small, the change 1 n 1 for a gven E and s also very small. r μ =r c ο =10βοr 0.1 ο b 33

13 1.4.7 Parastc elements n the smallsgnal model echnologcal lmtatons n the fabrcaton of transstors gve rse to a number of parastc elements that must be added to the equvalent crcut. ross-secton of a typcal npn transstor: E p cs n+ r c3 n r b je μ p n+ rex je r c1 njected moton p μ cs r c2 bured layer p-substrate cs n+ All pn junctons have a voltage-dependence capactance assocated wth the depleton regon. je = depleton regon capactance at E juncton µ = juncton capactance cs = -substrate juncton capactance 34