( E) 1 PE ( 1 ) will be the probability to have not an. electron in this state (on this energy level). For the energy level configuration, depicted in

Transcription

1 hatr. troducto to Sold Stat Physcs... Frm Drac Dstrbuto ad th Dsty of Ergy Stats a Sold Lt b P ( E ) th robablty to ha a lctro th stat charactrsd by th rgy E, th PE ( ) wll b th robablty to ha ot a lctro ths stat (o ths rgy ll). For th rgy ll cofgurato, dctd Fg.., th total robablty to ha such a stat (E ad E flld ad E 3 ad E 4 uflld) s g by th formula: P ( E ) P( E ) ( P( E )) ( P( )) 3 E4 Th robablty for th comlmtary stuato s: ( P( E )) ( P( E )) P( E ) P( ) 3 E 4 Both robablts must b qual th cas of thrmal qulbrum, thrfor w ca wrt th followg qualty: P E ( ) ( ) ( ) ( ) 3 P E4 P E P E... But th rcl of rgy cosrato rqurs that E E E3 E 4 ad ths cas oly th fucto Ax( β E) ca b dtfd wth P(E), whr β / kt ad A E kt x ( / ). F Th th robablty to ha a lctro th stat charactrsd by rgy E s: P(E) ad th robablty to ha a mty stat s : ( E) f... EE kt F

2 f..3. E E ( E) f ( E) F kt Th fucto f s kow as th Frm-Drac dstrbuto ad s rrstd fg... At T K, th sha of ths fucto s lk th sha of a st fucto (s dottd l) ; At T K, for E EF, th robablty to ha a lctro o ths stat s /. Th sha of Frm-Drac dstrbuto for ths tmratur s rrstd by a cotuous l. Th stat charactrsd by E EF s kow as Frm ll ad rrsts a rtual rgy ll charactrstc for ay sold stat matral. Ths ll s th ur lmt of rgy lls whch ca b flld wth lctros at T K. (Oly th cas of mtals xst such stuato. For solators ad smcoductors th ur lmt b lowr, as you wll s xt aragrah) Nxt roblm, sold stat hyscs, s to obta th formula for dsty of such rgy stats (th umbr of rgy lls th ut olum). ordr to accomlsh that, w must work th momtum sac. Th quatum mchacs assrts that a statoary stat, a lctro ca b dscrbd by a statoary wa fucto. That mas that a bulk matral hag th charactrstc dmso L, oly lctros that ha th assocatd walgths λ rfyg L λ ca xst, whr s a ost tgr. Ths formula must hold o all thr coordats (x, y, z). But th walgth s lkd to th momtum (or muls) through d Brogl formula λ h. osqutly w ha th followg rlatos btw momtum (o ach sac drcto) ad th dmso of th bulk matral:

3 x h h x y zh ; y ; z. For th ut cll th sac of momts L L L ( x y z ), wth th olum 3 h, w ca ha two stats (Paul s Prcl), L rrstd Fg..3 by two arrows (s quatum umbr ±. ) osqutly, th dsty of lctros stats th ut cll wll b: 3 L d h h L ths cas w ca calculat whch s th umbr of lctros whch ha th momtum lyg btw ad d, usg th Fg..4 whch rrsts oly th ost rgo of th sac of momtum, bcaus all comots of th lctro s momtum ar ost. 3 L d d d 4, d π d h 8 Startg from ths formula, w ca fd th dsty of lctroc stats whch ha th momtum btw ad d d 8π, d ( ) d But th ktc rgy for th quas-fr lctro ca b wrtt as Ek. Thrfor, w m ca rwrt th.4 formula as th dsty of lctroc stats whch ha th rgy btw E ad EdE * / ( ) ( m ) E k N Ek N 4π 3 / h whr m* s th ffct mass of th lctro. L h d 3

4 A smlar formula ca b foud for hols (th dsty of uflld lctroc stats that ha th rgy btw E ad EdE N * / ( E ) ( m ) E k k 4π 3 / h whr m* s th ffct mass of th hol... Th Dsty of harg arrrs a Pur Smcoductor. a ur smcoductor, as w mto th troducto, w ca rrst th rgy stats of a lctro or hol usg th modl of rgy bads. Lt s cosdr for xaml th cas of th Grmaum crystall lattc. As ca b s Fg..5, th boudd alc lctros ar th alc Bad, charactrsd by th ur rgy ll E, but ca xst too a xctd stat oducto Bad, charactrsd by th lowr rgy ll E. th oducto Bad th lctros ar ot boudd to th atom ad thy ca ha a mog through th crystall lattc from atom to atom. Th sam thg ca b do by th hol, whch rrsts th mty stat whch rma th alc Bad aftr th jum of th lctro from alc Bad to th oducto Bad by thrmal xctato. But for lctros E k E E ad for hols c E k E E. Now w ca comut th dsty of charg carrrs B or B, usg xt formulas: E f (E)N(E) de... or, for hols, E f (E)N (E) de... W wll comut th dsty of charg carrrs usg th..,..3,..6 ad..7 substtutd to.. ad..: 4

5 Ec 4π * ( m ) 3 / / ( E E ) de h 3 c EEF kt ad for hols: E 4π ( m * ) 3 / / ( E E) de h 3 EF E kt ordr to b abl to tgrat th abo quatos, w ha to mak crta aroxmatos, allowd by th tycal romt codtos. For stac, at room tmratur, T3 o K, th dx of xotal from th domator of Frm-Drac dstrbuto s ry hgh, ad ths cas w ca glct th factor from th domator. ths stuato th Frm-Drac dstrbuto bcoms for lctros: f E E F ( E) kt, whl for hols t bcoms f ( E) kt EEF kt Now w ca us th xt mathmatcal artfc EF E kt EF E EEF Ec E c kt Ec E F kt EE c kt, whr th frst trm s a costat for th smcoductor matral, that ca b mod out of th tgral. Fally th xrsso for dsty of lctros oducto Bad s : 4π E E E E ( m * ) 3 / c F c kt ( E E ) / kt de h 3 c..3. Ec ad, corrsodgly, th dsty of hols w wll b: 4π ( m* ) 3 / EF E E E E / kt ( E E) kt de h3..4. By makg a chag of arabl both tgrals, E Ec E E x or x, th xrsso for lctro dsty bcoms kt kt 5

6 4π ( m * 3 / Ec E ) F 3 / x kt ( kt) x dx 3 h corrsodgly, th xrsso for hols dsty bcoms: 4π * ( m ) 3 / EF E 3 / kt ( kt) x x dx 3 h Now both tgrals ca b comutd by th arts mthod, as show blow: x x dx x x x dx π..5. whr th last tgral s half of th Posso tgral x dx π Now w ha th fal xrsso for both dsts of charg carrrs f w troduc th..5. quato..3. ad..4. quatos: rsctly * E ( ) 3 / Ec EF c πmkt kt kt EF N h ( * E ) 3 / EF E F πmkt kt kt E N h Th costats N c ad N ar so calld dsty of rgy stats.b., rsctly.b. * N < N bcaus m < m * Th osto of Frm ll ur smcoductors. a ur smcoductor, th dsty of th two tys of charg carrrs s th sam bcaus ths carrrs ar gratd by thrmal xctato from alc Bad to oducto Bad, as show Fg..5. Thus, w ca wrt th followg qualty: 6

7 N E E kt N E E kt F F..8. Now, w ca trasform th quato..8. to: N N EF E Ec EF kt, to whch w ca aly th logarthm ad th, wth a sml orato w ca xtract th alu of Frm ll: Ec E kt N E F l..9. Nc Th quato shows that th Frm ll s th mddl of forbdd bad, at T K. f tmratur s crasg, th Frm ll shfts towards th oducto Bad (s fgur.6) Th most mortat quatos, ald ay smcoductor ar th law of charg cosrato. Basd o quato..8. w ca ro that th roduct of dsty charg carrrs s a costat of smcoductor matral, bcaus ths roduct dos ot dd o Frm ll. Ths roduct s th so calld ur dsty : N N Ec E kt... For th most commo smcoductor matrals, at room tmratur, th alus of ths costats ar: Grmaum: N. 4 cm ; N 6 cm ;.4 x 6 cm -6 Slco: N cm ; N. 4 cm ; cm

8 .3. Extrsc Smcoductors (Dod Smcoductors). dod Smcoductors. f a matral lk Slco or Grmaum w troduc atoms lk Al, Ga or, whch ar atoms from th rd grou of Mdl s Tabl, th osato ottal of ths atoms wll dramatcally dcras. Ths ffct s xlad by th ddc of osato ottal by th / ε r, whr ε r s th rlat dlctrc costat of th mdum whch ar ths atoms, rsctly th rlat dlctrc costat of Grmaum or Slco. For ths matrals th rlat dlctrc costat s ε r ad rsctly ε r 6. Th osato ottal for such atoms from grou of Mdl s Tabl, srtd G or S, s g Tabl. Tabl B Al Ga S G Ths rgs ar rrstd th S modl of bad rgs by th xstc of a acctor rgy ll, ry clos to alc Bad (dstac btw ths acctor rgy ll ad th ur rgy ll of alc Bad - E, s th osato rgy of murty atoms) as fgur.7. Arrows dcat trastos of lctros from th alc Bad to acctor ll or to oducto Bad. Bcaus th acctor ll s closr to alc Bad tha oducto Bad, th robablty to ha such a trasto acctor ll s hghr tha th robablty to ha such a trasto to oducto Bad. For that w wll ha mor lctros o acctor ll tha th oducto Bad, but all ths lctros ar boudd lctros, osg th G 8

9 acctor murts. Thy do ot artcat to coducto homa, but hols, gratd by such trastos ca artcat to coducto homa ad thy ar mor tha th lctros from oducto Bad. Th hols ar majorty charg carrrs. For ths raso w amd ths smcoductors P smcoductors. Th robablty to ha a lctro o acctor ll has th sam form lk Frm-Drac dstrbuto for lctros oducto Bad, f w dd ot tak to accout th dgracy factor A f (E) EA EF kt th th dsty of osd acctors wll b: NA EA E F N kt AfA(E) NA.3.. ad th dsty of hols obtad by th homa of such osato wll b: E F E N kt NA From ths qualty w ca fd th osto of Frm ll th P smcoductor N EF E kt A E A E F kt N.3.. thus, by usg th sam rocdur w ald for ur smcoductors, w fd : EA E kt N E F l.3.3. NA 9

10 Ths formula shows us tha at T K th Frm ll s at th mddl of th dstac btw acctor ll ad ur ll of alc Bad. f th tmratur s crasg th Frm ll shfts to th mddl of th Forbdd Bd (s Fgur.8), f NA<N, as th cas of tmratur hghr tha K. From Fg..8 w coclud that at room tmratur all murts ar osd. That mas that th dsty of majorty carrrs s N A At ths tmratur w ha morty charg carrrs too, gratd by bad to bad trastos of alc lctros (s Fg.7). Th dsty of ths carrrs ca b calculatd wth th hl of... quato Thrfor.3.4. NA Dod Smcoductors. Th sam homo of dcrasg of osato ottal s hag a ur smcoductor dod wth lmts from th grou of Mdl s Tabl, lk P, As, St. th Tabl you ca s th modfd osato ottals of such murts. Tabl P As Sb B S G

11 Th dagram of rgy bads of such smcoductor s show Fgur.9. ths cas a door ll rrsts th rgy of osato. ths cas th majorty carrrs ar lctros bcaus th robablty of a jum from door ll s hghr tha th robablty of a bad to bad jum. Th robablty of osato of a murty wll b smlar wth Frm-Drac dstrbuto, f w dd ot tak to accout th dgracy factor fd(e) EF E kt thrfor th dsty of osd doors s D N D DfD D EF E D kt N (E) N.3.5. ad th dsty of lctros obtad by th homa of such osato wll b: E c EF N kt ND.3.6. From ths qualty w ca fd th osto of Frm ll th P smcoductor N Ec EF kt N D EF E D kt th, usg th sam rocdur lk th cas of ur smcoductors w wll fd : EF ED Ec kt N l.3.7. ND Ths quato shows us tha at T K th Frm ll s at th mddl of th dstac btw door ll ad lowr ll of oducto Bad. f th tmratur s crasg ur to

12 K, th Frm ll shfts to th mddl of Forbdd Bad bcaus Nc bcoms hghr tha N D. (s Fgur.). By lookg at Fg.. w ca s that at room tmratur, ractcally all murts ar osd. That mas that th dsty of majorty carrrs s N But at ths tmratur w ha too, morty carrrs gratd by bad to bad jums of alc lctros (s Fg..9). Th dsty of ths carrrs ca b calculatd usg th... quato. Thrfor,.3.8. N D D.4. Physcal Phoma Smcoductors. oducto. Dffrt from mtals, smcoductors two dffrt kds of charg carrrs artcat to coducto homa: gat charg carrrs (lctros) ad ost charg carrrs (hols). th rsc of a lctrc fld both charg carrrs wll mo to th drcto of ths fld (lctros th oost way ad hols th sam way of th fld). Th, a smcoductor w wll ha two comots of th currt dsty: j E.4.. µ j E.4.. µ whr, E ad E, ar th drft locts of charg carrrs. Ths locts µ µ ar roortoal to th tsty of lctrc fld E, th costat of roortoalty rrstg th moblty of th charg carrr, µ.

13 Th th total currt dsty ca b wrtt as th sum of both comots, g by quatos.4. ad.4.: tot E E j µ j j µ.4.3. Usg quato.4.3. w ca fd th xrsso for lctrcal coductty of th smcoductor matral: j tot σe j σ.4.4 E tot th, ( µ µ ) tmratur ddc of th moblty of charg carrrs. A ur smcoductor has, thrfor th ddc of σ fucto of tmratur wll ha th sam sha as trsc dsty of charg carrrs fucto of tmratur, as t s show Fg.., f w glct th Th dottd l rrsts th coductty of mtals. Obously, ths fgur rrsts just a qualtat lot of th coductty. From ths lot w ca s th dffrc btw mtals ad smcoductors: smcoductors, th coductty crass xotally wth th tmratur, ad at room tmratur th coductty of smcoductors s lowr th th coductty of mtals. Ths rorty s usd a umbr of ass dcs, for stac thrmstors. th cas of xtrsc smcoductors th quato.4.3. rmas ald, but th dsty of charg carrrs must b ramd accordac wth th smcoductor ty. th cas of N ty smcoductors th total currt wll b: tot E E j j j µ µ 3

14 ad th coductty such smcoductors wll b rdomatly mdatd by lctros, sc >> N ; D. Thrfor: σ.4.5. N N Dµ orrsodgly, for th P ty smcoductors th total currt wll b: tot E E j j j µ µ ad th coductty such smcoductors wll b rdomatly mdatd by hols, sc >> ; NA, thrfor: σ P N Aµ.4.6. Dffuso. f thr s a dsty gradt of charg carrrs a smcoductor s rgo (s Fg..), th carrrs th dsly oulatd rgo wll td to mgrat towards th dltd aras. Thrfor, a carrr dffuso currt wll occur. At thrmal balac, th moto of charg carrrs ( our xaml lctros) s radom. Th, ddg o th dsty of lctros o ach sd of th scto through th smcoductor at x o, th umbr of lctros whch mo through th la at x o, th ma fr tm (th tm btw two collsos) wll b dffrt. Th umbr of lctros assg from rght to lft, through th la x s: N [ ( x l) ( x )] S R L l.4.7. whras th umbr of lctros that ass from th rght to th lft of th sam la s, 4

15 N l [ ( x l) ( x )] S L R.4.8. whr l s ma fr ath, whch t s assumd to b th sam for both carrrs, hc most of th collsos occur wth th lattc, ts dfcts or murts, ad thrfor s ddt of th carrr dsty. Factor s g by th qual robablty for th momt from rght to lft or from lft to rght. Th th total umbr of lctros, whch ass through ths la, s th dffrc btw quatos.4.8 ad.4.7: d NT N L R N R L 4 dx [ ( x l) ( x l) ] l S l S x.4.9. Ths momt of charg carrrs crats a currt that has th dsty j Q NT S τs τs l l d τ dx D D d dx.4.. whr th costat D s th so calld dffuso costat. Usg a smlar dmostrato w ca fd th currt dsty of hols: j D d D.4.. dx Th mus sg s dtrmd by th gradt of charg dsty, whch s gat ad, at th sam tm, by th charg of th hol, whch s tak as ost. Grato ad Rcombato of charg carrrs. Th dsty of charg carrrs ca ot buld u dftly tm, bcaus at th sam tm wth th grato homa thr ar th rcombato homa, whch scal wth th dsty of charg carrrs. At thrmal balac, th grato rat must qual th rcombato rat. 5

16 Th rcombato rat s roortoal to th roduct of th dsts of charg carrrs: R ost..4.. th cas of P ty smcoductor NA ;. Thrfor quato.4. bcoms: R ost. N A.4.3. τ whr τ s ma lf tm of morty carrrs gratd xcss. th cas of N ty smcoductor ND ;. Thrfor quato.4. bcoms: R ost. ND.4.4. τ whr τ s ma lftm of morty carrrs gratd xcss. Th, f w ha a xcss of morty carrrs, lt that b a P ty smcoductor, from ay rasos, w ca fd th tm oluto of ths xcss: [ G R] dt d but G ad τ R (t) th ths rlato ca b wrtt τ d dt (t) whch s a frst ordr dffrtal quato, whch has followg soluto: τ (t) τ [ () ].4.9. Th morty carrr xcss has thrfor a xotal dcay tm, as show fg..3. t 6

17 .5. Equato of cotuty (Law of charg cosrato). Lt thr b a lmtary olum a smcoductor (s Fg..4). sd ths lmtary olum w may ha grato homa, that taks lac at a rat g ad rcombato homa, at a rat r. ths olum trs th currt ad gos out th currt d. ths cas th balac quato for th tm arato of total charg sd th lmtary olum Sdx ca b wrtt as: Sdx t Sdx τ Sdxg d.5.. th statoary cas t ; d τ g, thus th quato.5.. bcoms : t d d µ E D τ dx.5.. dx whr w rlac th total currt by ts two comots (drft currt ad dffuso currt) jts µ E D d S dx Thrfor, th fal form of.5.. quato bcoms: t τ µ ( ) d E dx D d dx.5.6. Equato.5.6. rrsts th balac quato for morty carrrs a N ty smcoductor, bcaus th dsty of morty carrrs s sst to accdtal arato of 7

18 charg dsty. For that raso w ca rlac by. a smlar way w ca fd th quato for N ty smcoductors: ( E) d d µ D t τ dx dx.5.7. Th mus sg bfor th dffuso costat aars from th xrsso of dffuso currt for lctros. Partcular cass of cotuty quato. Lt thr b a smcoductor of P ty. Th frst artcular cas s basd o th followg smlfyg assumtos: ddc of dsty to dstac (x axs) ad ull lctrc fld. Accordgly, th quato.5.6 w ha: ; E x ad th quato bcoms: t τ whch has th kow soluto (t) [ () ] t τ, smlar wth.4.5. quato, lottd th Fg..3 Th scod artcular cas s: ddc of th carrr dsty tm ad ull lctrc fld: ; t E ad th quato.5.6. bcoms: d D dx τ whch has th soluto (x) A x L B x L whr w df L D τ whch rrsts th so calld dffuso lgth. 8

19 Th costat A must b zro (sc th carrr dsty caot cras towards fty wth crasg x), thrfor from boudary codtos w ca fd th alu of costat B: B () ad ow w ca wrt th fal form of ths soluto: (x) ( () ) x L that has th grahcal rrstato lottd fg..5. hatr. P - N JUNTON... Physcal Phoma P-N Jucto Th P - N jucto s formd a bulk smcoductor, whch s cosdrd to ha th sz largr th th dffuso lgth of charg carrrs. Two dffrt rgos of dog ar cratd th structur, o of P ty ad othr o of N ty. Th boudary btw ths two rgos rrsts th P - N Jucto. Bcaus ths structur has a hgh gradt of majorty charg carrrs from P ty smcoductor to N ty smcoductor, dffuso homa wll aar at th boudary btw ths two tys of smcoductors. Th majorty carrrs of P ty, wll dffus to N ty smcoductor whras th majorty carrrs of N ty wll dffus to P ty smcoductor. But N ty smcoductor th hols ar morty carrrs, thrfor a homo of rcombato btw hols ad lctros wll occur. Th sam homa wll occur P ty smcoductor btw lctros ad hols. 9

20 Followg dffuso ad rcombato, both sds of th jucto, a dlto layr wll occur du to mass rcombato. At th sam tm, thr s gog to b a t lctrcal chargg th rgo, bcaus ths rgos w wll ha oly th fxd chargs, th o chargs. ths rgo a tral lctrcal fld wll aar ad, of cours, a oltag gradt (s Fgur.). lot a) w lottd th charg dsty dlto layr; lot b) w lottd th tsty of tral lctrc fld fucto of dstac; lot c) w lottd th oltag gradt fucto of th dstac. As w ca s from lot (a), th charg cosrato law ca b wrtt as: NA LS NDL S... whch ca b furthr rducd to: N AL ND L... To fd th xrsso of lctrc fld th dlto layr of P ty smcoductor w must aly th Gauss law for th ay S surfac rdcular to th ost x axs. E N L A ( x)s ε ( x)s

21 rsultg : E ( x) N A (L ε x)..3a. th sam way w ca obta th quato for th lctrc fld th drcto of th gat x axs.: E ( x) N D (L ε x)..3b. From th last two quatos w ca obta th alu of maxmum lctrc fld: NAL NDL E max ε ε E ()..3c. Th alu of th barrr ottal ca b obtad smly by tgratg th lctrc fld or th lgth of th jucto: b L max (L L) E (x)dx..4. L E From quatos..3c. ad... w wll obta th fal formula for th barrr ottal: N L (L N L D A b..5. ε ε All ths formulas ar calculatd at thrmal balac. L ) (L L )

22 Rlato..5. s usd to dtrm th dffuso lgth of majorty charg carrrs whch dffus th rgo whr thy bcom morty charg carrrs : ε ε ) L L ( L N ) L L ( L N A D b..6. From rlatos..6. ad... w wll obta th fal formula for th dffuso lgth: b A D D N N N L ε..7a. rsctly, b D A A N N N L ε..7b. Th dffuso of charg carrrs wll cotu utl th lctrc fld cratd by ths charg dslacmt wll buld u to a alu that wll comltly sto th chargs o crossg th jucto. Oc ths qulbrum has b attad, th total currt of hols, or lctros, wll b zro (assumg also thrmal balac): dx d D j j j d c t µ E..8a dx d D j j j d c t µ E..8b. f w rlac th lctrcal fld wth th oltag gradt dx d b E w ca tgrat ths formulas. Lt tak as a xaml th formula..8a. :

23 db d D d µ, thrfor: µ d b dx dx D Th lattr dffrtal quato has th followg soluto: l µ b ost...9. D From th boudary codtos, w wll fd th alu of tgrato costat: at b ost. l ad at b b, th quato..9. bcoms: µ b D... But at room tmratur w ha th followg rlatos, rstd arlr ths chatr, for th dsty of charg carrrs: E E E E F F N kt kt ; N whr, at thrmal qulbrum, E F E F. Th, th rato btw majorty carrrs ad morty carrrs ca b wrtt as: E E kt... Ths rato ca b obtad from quato... too, but ordr to ha a qualty btw ths two ratos t s rqurd that: µ D b E kt E bo kt... From quato... w wll obta th followg rlatos btw carrr moblty, dffuso costat, ad tmratur: 3

24 D µ kt..3a. whl for lctros, a smlar way, D µ kt..3b. Rlatos..3. ar th so calld Est s rlatos for smcoductors. Th xt cocluso xtractd from quato... s E E, whch shows us that a such structur b ( PN jucto) th rgy bads of smcoductor ar brok or shftd at th ll of th jucto, ordr to ha th sam Frm ll o both sds of th jucto (s Fgur.) at thrmal balac. From quatos... ad... w ca obta th formula for th barrr ottal (whch matchs th maxmum of th oltag gradt): b kt kt kt N N b l l..4. A D Th rato kt T s th so calld thrmal ottal, ad at room tmratur has th alu: T.6 olt ( T 3 K). 4

25 .. Th urrt-oltag haractrstc for th P - N Jucto. Th dsty of charg carrrs must b a cotuos fucto or th whol lgth of th smcoductor (s fgur.3). W also dtrmd, chatr.., that th dffuso lgth of majorty charg carrrs, whch dffus through th jucto, dds o th squar root of th barrr ottal. Th, f w chag ths barrr ottal by alyg a xtral ottal / (oltag), w wll modfy ths lgths ad th hght of barrr ottal: L, Kb, whr th hght of barrr ottal s kt N N b l, at thrmal balac. But th hght of A D barrr ottal ca b wrtt as ottal s as follows: b b xt, whr th coto for xtral xt xt f ths s th so calld rrs basg ottal/oltag (th ost lctrod of xtral sourc s coctd to th N ty smcoductor); xt xt f ths s th so calld forward basg ottal/oltag (th ost lctrod of xtral sourc s coctd to th P ty smcoductor). 5

26 Th chags ducd by th xtral oltag to th lgth of dlto rgo, th hght of barrr ottal ad th currts whch flows through th jucto ar rstd Fg..7. Now, as you ca s fgur.4, w ca ot chag th ddc of dsty charg carrrs fucto o th x axs. Th, at forward olarsato, at th w dffuso lgth L thr wll b a jcto of morty charg carrrs th N ty smcoductor, rsultg a carrr dsty dffrt from th o at thrmal balac (a smlar rocss wll tak lac th P ty smcoductor). Now, f w tak th org of X axs o th boudary of dlto rgo, w wll b th codtos of th quato of cotuty artcular cas E ;, whr th soluto s: [ () ] x L (x)... Th th dffuso currt that wll b stablshd ca b wrtt as: x L d D [ () ] j(x) D.. dx L takg to accout th quato... t 6

27 A smlar xrsso wll rsult for th lctro comot of th currt. But th dsty of morty charg carrrs ca also b wrtt as: ( ) b b xt kt kt () kt xt..3. Thrfor th fal formula for th currt g by hols bg: j (x) x xt D kt L..4 L whl for lctros t wll b: j (x) xt kt x L D..5 L Th total currt s th sum of..4 ad..5. But ths currt dos ot dd of th X abscssa. Th w ca comut ths currt at x j t cost. j (x) j (x) x D L D L xt kt..6 f w multly th quato..6 wth th cross ara of th jucto, w wll obta th currt-oltag charactrstc or so calld olt-amr charactrstc of th dal dod: xt kt..7. 7

28 Th lot of rlato..7 s g Fg.5, whr th gat currt axs has b magfd sral ordrs of magtud wth rsct wth th ost axs. Th currt s th so calld th saturato currt or rrs currt through th jucto. Th alu of ths currt dds o th aramtrs of th crystall lattc ad tmratur. For Slco, alus of aoamrs ar commo, ad for Grmaum lattc, alus of mcroamrs ar commo for th saturato currt. Ths currt s mdatd by th morty carrrs, ad ts xrsso ca b dtrmd from quato.. : D D S..8. L L Th og ottal of th dod, γ, s dfd as th forward basg oltag for whch th currt s µa. ractcal alcatos ad crcut aalyss, th lot of currt fucto of forward basg oltag s aroxmatd by a lar ddc of th oltag o currt abo a thrshold, as you ca s Fg..6. 8

29 9

30 3.3. aactac Effcts th P-N Jucto. W dtrmd aragrah. that th dffuso lgth s roortoal wth th squar root of th barrr ottal, at thrmal balac, as show th followg formulas: b A D D N N N L ε ad b D A A N N N L ε f w basd th jucto wth a rrs ottal, th hght of barrr ottal wll cras to a alu b b xt ad th dffuso lgth wll cras: ( ) / xt b A D D N N N L ε quato whch ca b wrtt, f multld by / b b, a w form : / b xt L L (w ha cosdrd, th last two quatos, th xtral oltag as gat). W ca obta, a smlar way, a mathmatcal formula for dffuso lgth of lctros: / b xt L L. Now f w df th dyamc caactac of barrr ottal as d dq B.3..

31 w ca calculat ts alu th followg sts: B dq d dq dl dl, but Q NALS d, th NA dl S ad dq dl d N A ε N N A D L th: B / / εs εs εs xt xt B.3.. NA L L ( L L ) b b L N D Th formula.3. gs us th alu of barrr caactac of th P-N jucto, whch looks lk th formula of th caactac of a la caactor. Ths caactac s a charactrstc of ry dod at rrs olarsato. Ths rorty s usd a class of scal dcs, calld arca dods, whch ar usd lk caactors whos caactac s cotrolld by th ald rrs oltag. Th commo alu of ths caactac s lyg th rag 5- F. ARabl APactac 3

32 Now lt us s what wll ha f w forward bas th P-N jucto. ths stuato w wll ha a jcto of morty carrrs a rgo whch thy ar xcss (s fg..8 whch s smlar wth Fg..4) Th ara blow th cur, whch dscrbs th dsty of xcss morty carrrs utral rgo, wll rrst th amout of charg jctd ths rgos. From th quato of cotuty w wll obta: (x) x x L L ( () ) P (x) P ().3.3. Th, th amout of charg jctd utral rgo of N ty smcoductor s: Q x L SP (x)dx LSP () SP () L.3.4. Th dyamc caactac s dfd by th formula.3., thrfor th caactac ca b wrtt as: dq dp() D SL d d.3.5 Now, f w tak to accout oly th hol comot of th total currt whch flows through SDP () L th jucto,.. P (), w ca calculat th drat of P ( ) L SD fucto of oltag: dp() L d.3.6. d SD d Now f w rlac th xrsso.3.6 th quato.3.5, w wll fd th alu of th so calld "dffuso/storag caactac of hols" : D L d.3.7. D d 3

33 a smlar way w ca calculat th "dffuso/storag caactac of lctros": D L d.3.8. D d Th alu of ths caactac s hghr tha th barrr caactac. alus of F, or mor, ar commo for ths caactac..4. Dyamc rsstac of th dod. W ca df, at forward bas of P-N jucto, th dyamc rsstac by xt formula: r d g d.4.. d Th dal dod quato ca b aroxmatd, at forward bas oltag, by: xt xt kt kt T.4.. xt Th th quato.4. bcoms: g.4.3. T By lookg at Fgur.9, you ca gt a flg of what s rrstg ths "coductac" o th lot fucto of : Th slo of th tagt l th ot "" to th cur whch rrsts th currt through dod fucto of basg oltag, s th charactrstc "coductac" for th dod at currt "". Th "coductac" s dfd as th rs of "rsstac", as quato

34 .5. Th Zr Dod. th quato of dal dod xt T, th saturato currt s a mortat aramtr of th dod ad has th followg D D P xrsso: N S LP LN. As you ca s, ths currt s fucto of morty charg carrr dsty, whch s a costat at a g tmratur. Sc ad N, th xrsso of saturato currt bcoms: N D A G DP DN DP DN 3 T.5.. S ost T LPND LNN A LPND LNN A but th dffuso costats ar rsly roortoal wth th tmratur, th w ca rwrt.5. th fal form: G T.5.. K T Th, for costat tmratur w xct to ha costat currt. ractc, ths formula holds oly for modrat rrs oltags (s Fg..9). Whl crasg th rrs bas oltag ald to th dod, th currt s costat u to a thrshold oltag, calld th brakdow oltag. For oltag alus largr tha brakdow oltag, th saturato currt crass abrutly. Ths bhaour of th saturato currt may ha two dffrt orgs: 34

35 - th aalach multlcato of morty charg carrrs (classcal homa) whch ca occur at oltag hghr tha ; - th tullg of boudd charg, through th barrr ottal, from alc bad of P ty smcoductor, drctly to alc bad of N ty smcoductor (quatum homa), whch ca occur at oltag lowr tha. rrsct of th actually brakdow mchasm, th dods whch work ths rgm ar calld ZENER dods, ad ar grally usd oltag stablsato..6. Th Tul Dod. Th tul dod s a scal dc that works at ry hgh frqucs (mor tha 5Mhz). Ths dod s mad a form of a P- N jucto wth a hay dog of both smcoductors (N A ad N D hghr tha 9 cm -3 ). ths codtos th wdth of th barrr ottal s ry short, ad th Frm ll, at thrmal balac, ls th alc bad of P ty smcoductor ad, corrsodgly, coducto bad of N ty smcoductor (s Fg..) 35

36 Fgur cato: A) Thrmal balac; B) Forward basg oltag (tullg currt s crasg) ) Forward basg oltag, hghr tha th cas B (maxmum tullg currt) D) Forward basg oltag, hghr tha th cas (tullg currt s zro) Th currt-oltag charactrstc s show Fg... W ca s Fg.. four dffrt rgos of ths lot. Rgo : hr th rrs currt crass radly, bcaus all lctros whch ar alc bad of P ty smcoductor ar tullg through th barrr ottal bcaus thy s uflld stats th coducto bad of N ty smcoductor, ur tha th Frm ll. Rgo : At a small forward basg oltag bgs a momt of majorty carrrs or th barrr ottal, lk a ormal dod, but ths currt s cocurrc wth th gat currt gratd by th tullg ffct. Wh th tullg currt bcoms rdomat, th currt through th dod mos rgo 3 (cas B Fg..) Rgo 3: Th tullg currt s hghr tha th ormal currt, thrfor th currt through th dod dcrass wth th bas oltag cras, utl a mmum alu (at th d of rgo 3), at whch th tullg currt s maxmal (cas Fg..). ths rgo th dod s charactrsd by a gat dyamc rsstac (th slo of ths art of th charactrstc s gat, as you ca s). 36

37 Rgo 4: Wth th crasg of th oltag, th rgy bads of N ty smcoductor ar shftd mor to ur rgs, ad th rgo wth fr lctros from coducto bad of N ty smcoductor bgs to look at a rgo of forbdd bad, th th tullg ffct dcrass, utl ths ffct ashs (cas D Fg...) ad all th currt through th dod wll b a ormal forward basg currt. hatr 3. Th Bolar Jucto Trasstor (BJT). 3.. Phomologcal dscrto of Bolar Trasstor. 37

38 Th Bolar Jucto Trasstor (BJT) or smly Jucto Trasstor, has th structur show Fg.3.. As you ca s Fg. 3., th currts whch flow through th bolar trasstor, th codtos of forward basg of E-B jucto ad rrs basg of -B jucto, ar: E PE NE ; P ; B E W ca df th followg "trasstor's costats" : Th ffccy of mttr: PE γ (th dal alu of ths costat s ) E td by Shockly 38

39 * P Th carrr factor: β (th dal alu of ths costat s ) PE Th currt ga : α (th dal alu of ths costat s ) E From th last rlato w ca fd th "trasstor's quato" αe 3... ractc th currt ga costat has alus amog Th w ca us th aroxmat rlato : α E 3... Lk th cas of P-N jucto, such a dc must b bult th sam c of smcoductor matral, ordr to assur th cotuty of th crystall lattc. Ay dfct th lattc would gratly mar carrr moblty ad would dstort th rgy bads. Th madatory codtos for hag such rlatos btw currts, th to ha a trasstor bhaour, ar: - th dog of Emttr s hghr tha th dog of th Bas,.. N A(E) >> ND(B ) - th bas s th ough that th dffuso lgth of morty charg carrrs whch ar jctd Bas s hghr tha th wdth of utral rgo of th Bas,.. L > w Fg. 3. w rrstd th cas of a NPN trasstor, basd th act rgm (Emttr jucto forward basd 39

40 ad ollctor jucto rrs basd). ths cas all currts ha rs drctos comard to th cas of PNP trasstor. Now, f w tak to accout th quato 3..., w ca aroxmatly calculat th owr ga of ths dc. Through th jucto of mttr, charactrsd by dyamcally rsstac currt E. Th th ut owr, dssatd o ths jucto s P Erd r d < Ω flows th Th outut owr s th owr dssatd o collctor jucto, whch s rrs basd, th t s charactrsd by a hghr rsstac R > 4 Ω. Th th outut owr s aroxmatly Pout R O has to ot that ths alus ar ot th actual total owr dssatd o th mttr ad collctor, sc rd ad R ar ot th statc rsstacs, but th dyamc os, rlatd to th A sgal. Thrfor P ad P out wll b A sgal owrs at th ut ad outut. Th owr ga s gog to b: G out R αer 4 Erd Erd P P Now, from th last rlato w ca udrstad why ths dc was calld "tras-rsstor" or "trasstor". Ths dc maks ossbl th trasfr of a currt whch flows through a rgo wth low rsstac, a rgo wth hgh rsstac, wthout a sst modfcato of th currt. 3.. Th aalytcal quatos of trasstor s currts. Th mttr currt has two comots, as w saw last scto. Th lctro comot of ths currt must ha th sam xrsso lk th lctro comot of a dal dod, thrfor: 4

41 E AD T E L 3... ths quato w wll aotat th dffuso lgth of lctros utral rgo of mttr by L E ad th dsty of morty charg carrrs mttr by E. Wth ths w aotatos, formula 3.. bcoms: E AD E T E L 3... E To comut th hol comot of th mttr currt, w must tak to accout that ths currt s a dffuso currt a utral bas rgo, whr t s a morty carrr currt. Thrfor, th xrsso of ths currt s ry much alk th formula of a dffuso currt: j E D d dx But alrady w kow th xrsso of th dsty of morty charg carrrs jctd a utral rgo: x L (x) K Bcaus th trast of ths charg carrrs through th utral rgo of th bas s fast, du to th small thckss of th bas, ( w < ) L, w ca aroxmat th quato by: (x) K K x Th, by rlacg , w wll obta : j E D K W ca obta th alus of costats K ad K from th boudary codtos of quato 3..5: at x, w ha 4

42 4 K () ad at xw, w ha: w K K (w) But th dsty of th jctd carrrs at x, whch rrsts th boudary btw th sac charg rgo of mttr jucto ad utral rgo of th bas, s g by: T E () Now, f w rlac th w wll obta th alu of th K costat: K T E 3... th sam way, takg to accout that: T (w) w wll obta from quato th alu of th K costat: w K T E T 3... Now, f w rlac th alu of K costat, g by quato w wll obta th alu of hols currt whch flows through th mttr jucto: w A D T E T E 3... Fally, th total currt whch flows through th mttr ca b xrssd as:

43 43 w AD w AD L AD T T E E E E E E th sam way w ca obta th xrsso for th collctor currt, whch s g by: Th saturato currt s lk th lctroc comot of th currt of a dal dod: L AD T Now, f w glct th rcombato homo th utral rgo of th bas,.. E, th sum of ad 3... quatos wll g us th xrsso for collctor currt: w AD w AD L AD T E T Th quatos ad rrst th aalytc xrssos for currts whch flow through th trasstor. Ths rlatos ca b wrtt codsd forms such: a a a a T T E T T E E whr th coffcts a j ar: w AD L AD a E E ; w AD a

44 AD a w ; AD AD a L w Th rlatos ar so calld "Ebrs-Moll" rlatos Ebrs-Moll Modl of Bolar Trasstor. Takg to accout th quato 3... ad th gral quato for o, th rrs currt of th collctor towards th bas jucto, th gral quato 3.. ca b wrtt as: (x α NE o ) T whr α N s th currt ga udr ormal codtos of basg (mttr to bas jucto forward basd ad bas to collctor jucto rrs basd). Now, f w tak th trasstor lk a rrsbl dc ad rrsg th basg, w ca rwrt th as: E E α R Eo(x ) T whr αr s th currt ga rrs codtos of th basg, hag lowr alu tha αn bcaus th trasstor dos t work ormal rgm. Th 3.3. ad 3.3. rlatos ca b usd to dscrb a sml modl of bolar trasstor, amd Ebrs-Moll modl, show Fg.3.3. whr E s th forward bas of th mttr ad s th rrs bas of th collctor. Th th frst trms of 3.3. ad 3.3. rlatos ar rrstd th Ebrs-Moll modl as costat currt grators ad th scods trms of ths rlatos ar rrstd by th currts whch ar flowg through two qualt dods, frst o basd wth E, ad th scod basd wth. Now, f w tak to accout that th currt g by costat currt grator a R s lowr tha th currt g by th costat currt grator a NE, ad th rrs currt of th qualt dod of th collctor jucto s ry small comarso wth th currt g by forward 44

45 basg qualt dod of th mttr jucto, th Ebrs-Moll modl ca b smlfd as Fg.3.4. Usg th modl showd Fg.3.4, th dmostrato of th owr ga g at th bgg of ths chatr bcoms asr to udrstad Statc haractrstcs of Bolar Trasstor. Th most commo coctos for th bolar trasstor ar th "ommo Bas octo" ad th "ommo Emttr octo", amd ths way bcaus th Bas, rsctly th Emttr, ar coctd to th commo cocto btw ut ad outut, cocto whch s cotoally tak as groud. Fg. 3.5 ar show ths two basc coctos of bolar trasstor. Each o s charactrsd by two ut coctos ad two outut coctos. Th w ha four trmals, of whch two of thm coctd to th commo groud. For that raso, such a dc s amd a four-trmal twork. Th bhaour of such dc ca b charactrsd by th ut ad outut currts ad oltags. Usually w tak as ddt 45

46 arabls th currt of ut ad th oltag of outut. ths way w ca wrt th ddt arabl (oltag of ut ad currt of outut) fucto of th ddt arabl. f (, ); f (, ) BE E B E B wll b th rlatos for B four-trmal twork, ad f (, ); f (, BE E B E B wll b th rlatos for E fourtrmal twork. ) fgur 3.6 you ca s such charactrstcs for B crcuts, ad fgur 3.7 for E crcuts. 46

47 Th way w dfd th currt ga for B cocto, th rlato btw th ut currt ad th outut currt s rodd by th followg quato: α E Ths rlato s so calld "th dc quato" for th trasstor B cocto. th cas of E cocto th "dc quato" ca b foud by rlacg th mttr currt by :. ths cas th quato bcoms: E β B ( β ) B whr β α α s th currt ga th E cocto. A tycal alu for b s. Th basg crcuts for bolar trasstor. th cas of B cocto w may ha a basg crcut comrsg two d.c. sourcs, lk Fg. 3.8 th trasstor s act rgo, th jucto E-B s forward basd ad th jucto -B s rrs basd. Udr 47

48 th assumto EB>>kT/ T, w ca fr that th mttr currt s hghr tha th rrs collctor currt, th w ha a wd rag of alus for th E th rlato EB cost ad th quato bcoms α E E Th quatos ad rrst th "dc quatos". Now w wll wrt th "crcut quatos", whch wll b th basd o scod Krchhoff s law for th ut ad outut crcuts: EE EB ERE R B From w ca calculat th alu of E for a g crcut: R cost. E EE E assumg that EE >> EB. Th, f w tak to accout th rlato , w ca assrt that E cost. Th quato rrsts th so calld "load l quato", from whch w ca calculat th bas oltag of th -B jucto. B R

49 Th trscto btw th load l ad th outut statc charactrstc corrsodg to th mttr currt calculatd by quato rrsts th so calld "statc oratg ot" for th trasstor. Ths ot s markd fg.3.9 wth a Q lttr. ths ot th bas oltag for -B jucto s rodd by quato Equatos ad ro that th B cocto s th most stabl oratg cofgurato of bolar trasstor. Ths s rodd by th fact that w cotrol th outut currt wth a currt E, hghr tha th rsdual currt of th collctor ad th ga currt α s aroxmatly costat, hag alus th rag Th crcut for th E cocto, whch s by th way th most usual crcut, s show Fg. 3.. Th basg crcut s commoly calld "oltag ddr basg" or ursal basg crcut, bcaus th rsstors R B ad R B rod th basg of E-B ad -B juctos usg a sgl owr suly. ths cofgurato, th stuato s qut dffrt as comard to th B cocto bcaus th currt ga β may ha a wd dsrso or ddual trasstors. f α s.98, β s 49 whl f α s o rct hghr,.99, β s 99. For ths cofgurato th "dc quato" s g by q. 3.4., but usually for Slco trasstors s usd th smlfd formula β B Th oltag ddr basg crcut showd Fg.6 has a qualt d.c. crcut whch s usd to fd th crcut quatos. To obta ths qualt crcut w must us xt sts:. W wll assum all caactors hag "ft rsstac";. W wll aly th "Th's thorm" for th oltag ddr basg crcut; 49

50 ths cas th qualt crcut of Fg.3. s th crcut showd Fg.3.. Now f w wrt th scod Krchhoff s law for ut ad outut crcuts of Fg.3., w wll fd xt quatos: whr BB BRB BE ERE R E ERE R R R B B B ad BB RB as rsult from Th's RB RB RB RB thorm. Th quatos 3.4. ad 3.4. ar th "crcut quatos" for E cocto. f th rsstac RE, from rlatos 3.4. w wll fd that B BB BE RB ad usg rlato 3.4.9, rsults that: ( ) β BB BE RB But β may ha a wd dsrso, th for a g bas currt w ca ha a lot of outut currts. ths cas th "oratg ot" of th trasstor s ot stabl. To rt ths 5

51 stuato t s cssary to ha th codto R E. ths cas th quato bcoms: B BB BE RB RE ( β ) ( quato w tak to accout that EB B(b ), f w usd th rlato ) ths cas quato bcoms: ( ) β BB BE RB RE ( β ) Now, f w ha mt th crtro R E ( β ) >> RB, th outut currt bcoms ddt of b, th th "oratg ot" bcoms stabl, ad th quato.4.5. bcoms: ( ) β( ) β BB BE BB BE BB BE cost RB RE ( β ) RE ( β ) RE Bcaus th currt ga E s larg, w ca aroxmat b» b. 5

52 Th quato 3.4. rrsts th "load l quato" for E of bolar trasstor. Fg.3. you ca s th outut charactrstcs for E of th bolar trasstor ad th oratg ot, obtad th sam way lk th cas of B. 3.5.Th stablsato of workg codtos for bolar trasstor. Th outut currt of B or E crcuts s th collctor currt. Th alu of ths currt s fucto of tmratur by hs ddc o: saturato collctor currt, bas oltag of th E-B jucto BE ad th currt ga b. Th w ca wrt that (, ) BE,β whr ach arabl dds o tmratur followg a commo law: (T) (T ) a ( T T ) whr T 3 K; th alu of th costat a dds o th atur of th smcoductor, bg hghr for Grmaum whch has th ga rgy (wdth of th forbdd bad) lowr tha Slco. β (T) β T T ( T ) K whr th alu of costat K s for Grmaum ad 5 for Slco. BE.m/ T Th strogst ddc o tmratur s for th saturato currt, bcaus ths currt s rodd by morty carrrs, ad th thr coctrato dds xotally o tmratur. Now, f w tak th drat of xrsso wth rsct to tmratur, w wll obta th followg quato: 5

53 T T BE BE T β β T whr th coffcts of th tmratur drats ar th so calld sstty factors : S ; SU BE ; β β S Th sstty factor of currt S s th most mortat, ts mmsato ladg to th mmsato of all othr factors. th am to fd th xrsso of S, w must calculat th xt drat: [ β ( β ) ] B from whch w ca obta : S β B β Th alu of th drat B dds o th ty of crcut usd for basg th trasstor. Th smlst basg crcut s show fgur 3.3. Th bas currt s g by th xt rlato : B BE, whch ca b foud by wrtg th scod Krchhoff's law for th ut R B crcut. Bcaus >> BE w ca gor th alu of BE th xrsso of B, thrfor t rsults that th bas currt s costat, ad, as cosqutly, ts drat wth rsct to s zro. ths cas th quato bcoms S β 53

54 whch s a larg alu, thrfor th sstty wth th tmratur s ry hgh. Ths crcut has a bad stablty fucto of tmratur. A good stablty wth th tmratur has th crcut show Fg.3.4. Usg th scod Krchhoff law, w ca wrt th xt quato: ( B ) R BB BE R from whch, gorg th BE oltag bcaus t s much smallr tha, w wll fd: B Tha, f w rlac ths drat quato, w wll fd: R R R B B R R R B S R R β B whch has alus th rag 3 to, ddg o R R β R RB th alus of rsstors usd th crcut. Ths s a low sstty, rsultg a good stablty of th crcut wth rsct to th tmratur aratos. th artcular cas RB w gt th bst sstty alu, S, but ths cas th trasstor has th -B jucto shutd. Howr, such a crcut s usd to stabls th scod trasstor. Ths mthod of stablsato s calld "currt mrror stablsato". Th bst alu, that mas th lowst alu for S, s obtad th cas of oltag ddr basg crcut show Fg

55 55 From th d.c. qualt crcut, dscussd arlr ths chatr, w ca fd th alu of B as a fucto of th currt E B E B R R R. Th th xrsso bcoms: E B E B E E R R R R R R S β β whch may ha tycal alus th rag to 6.

56 hatr 4. Small Sgals Oratg Rgm. Notato otos for th Dyamc Rgm. th dyamc rgm w ha our crcut both currts ad oltags to look at. gral currts ad oltags ar dsgatd by catal lttrs or hag a subscrt whch rrsts th lttr charactrstc for th trasstor trmal (E for mttr, for collctor ad B for bas). Th a.c. comots ar dotd by talc small lttrs hag small lttrs as subscrts, dsgatg trasstor trmal, as th d.c. cas ( for mttr, c for collctor ad b for bas). Th, th d.c. collctor currt s otd by ad th a.c. collctor currt by c. Th sum of both comots s dotd by talc catal lttrs as you ca s th followg xaml: c c fgur 4. you ca s a grahcal aalyss of th commo mttr amlfr dyamc rgm. Th ut sgal s ald th bas of th trasstor, whch has th statc 56

57 oratg ot Q g by th trscto of load l wth th statc charactrstc for th bas currt 6 µa. Ths cas wll rsult a total outut currt/oltag sgal c ad c as a fucto of tm t that ca b s o lowr lft al of fg.4.. As you ca s Fg.4., a small ut sgal (currt or oltag sgal) s amlfd by th trasstor. Th usual ut for th ga of a amlfr s th dcbl, dfd as: Numbr of dcbls log Pout/P f th ut ad outut owr of a amlfr s masurd o th sam rsstor, th dfto for th dcbl bcoms: Numbr of dcbls log P out /P log ( out ) /( ) log out / But, st of ot bg tchcally corrct, t has bcom customary to df th dcbl oltag ga of a amlfr trms of th oltag ga, that th ut ad outut rsstacs ar ot quals. Thrfor, last formula bcoms: Dcbl oltag ga G log A 4. Th Small Sgals Modl for Bolar Trasstor. Th bhaour of four-trmal twork, as a gral class of crcuts, ca b charactrsd by lar quatos oly for small sgals. Ths affrmato ca b dmostratd usg th Ebrs-Moll quatos of th trasstor, wrtt for total currt (both comots): c a a E T E T a a T T 57

58 58 whr E ; E ad c c accordg to coto adotd th bgg of ths chatr. For small sgals, th amltud of a.c. comots s small ough to allow us to k oly th frst ordr trms from th Taylor (owr) srs whch ca b aroxmatd th xotal of a.c. comots. Lt s tak as a xaml th frst Ebrs- Moll quato:... T T E T E rsctly... T c T T c Groug th trms of d.c. ad a.c. comots w wll obta th followg quato: T c T E a a a a T T E T T E whr th frst two trms rrst just th d.c. comot of th mttr currt ad th xt two trms rrst th a.c. comot of th mttr currt. Now, by dffrtatg th abo quato, w obta: c y y 4... whr T a y T E ad T a y T Usg a smlar mthod, w ca obta th quato for small aratos of th collctor currt g by th scod Ebrs-Moll quato: c c y y 4... Th quatos 4... ad 4... df th so calld admttac aramtrs, dtrmd usg th xt quatos: c y, whch rrsts th ut admttac

59 y, whch rrsts th rrs trasfr admttac c y c, whch rrsts th forward trasfr admttac c y c, whch rrsts th outut admttac c Th qualt crcut dscrbd by rlatos 4.. ad 4.. s drawg Fg.4. Ths four-trmal twork rrst th qualt crcut for B of th bolar trasstor usg admttac aramtrs. As you ca s, th qualt ut crcut of th trasstor accordg to th quato 4... comrss th ut admttac y ad th costat currt grator y c whch rrsts th fluc (fdback) of th outut crcut to th ut crcut. Smlarly, th qualt outut crcut comrss th outut admttac y ad th costat currt grator y whch rrsts th fluc of th ut crcut to th outut crcut. quatos 4.. ad 4.. w ha tak as ddt arabls th ut oltag ad th outut oltag c, by wrtg th ut currt ad th outut currt c usg lar rlatos allowd by th small a.c. sgal aroxmato. 59

60 Aothr trasstor modl ca b foud by takg as ddt arabls th ut ad outut currts. ths cas th modllg aramtrs ar mdacs, dfd by th followg rlatos: z, whch rrsts th ut mdac c z, whch rrsts th rrs trasfr mdac c z c, whch rrsts th forward trasfr mdac c z c, whch rrsts th outut mdac c Th qualt crcut usg mdac aramtrs s show Fg.4.3. As th cas of admttac aramtrs modl, ths four-trmal twork whch modlld th B of bolar trasstor, has a ut crcut mad by th ut mdac z ad a costat oltag grator z c, ad a outut crcut mad by th outut mdac z ad a costat oltag grator z. Th, th quatos 4..3 ad 4..4 wll aroxmat th bhaour of th bolar trasstor: z z c c z z c

61 Now, f w tak as ddt arabls th ut currt ad th outut oltag, lk th cas of statc charactrstcs, th aratos of th ut oltag ad outut currt ca b wrtt as follows: h h o h h o o whr aramtrs hj, amd hbrd aramtrs, ar dfd by followg rlatos: h h ct. o ct. o rrsts th ut mdac. Usually s otd by h. rrsts th rrs trasfr factor. Usually s otd by h r. h h o ct. o o ct. o rrsts th forward trasfr factor. Usually s otd by h f. rrsts th outut admttac. Usually s otd by h o. All ths factors ha a scod dx to charactrs th trasstor s cocto; th lttr for E of th trasstor, b for B of th trasstor, rsctly c for of th trasstor. Equatos 4..5 ad 4..6 allow us to mag a w four trmal twork for th bolar trasstor, whch s most usd lctrocs. Ths crcut, amd hybrd aramtrs modl of th trasstor s show Fg.4.4 6

62 As you ca s, ths crcut s a mxtur btw mdac aramtrs ad admttac aramtrs crcuts, for that raso bg amd hybrd aramtrs modl. Tycal alus for hybrd aramtrs. Scod dx Paramtr (E) (B) b () c h. x 3 W - W -5 x 3 W h r -4-4 or lss h f (mdum).99 (mdum) (mdum) h o -5 W -7 W 3 W 4. Gral haractrstcs of a Amlfr. Ery amlfr s charactrsd by oltag amlfcato A, currt amlfcato A, ut mdac z ad outut admttac y o. ordr to b abl to calculat ths aramtrs t s cssary to trasform th actual crcut t s a.c. qualt. Hr ar two ruls to b followd: ry caactac s a short-crcut a.c. th d.c. basg sourc s a short-crcut to th groud a.c. 6

63 Lt s tak th most usual amlfr crcut usg trasstors ractcal alcatos, th ommo Emttr octo amlfr, showd Fg.4.5. Th qualt a.c. crcut, takg to accout th abo ruls, s show Fg.4.6. By dottd ls ar rrstd th crcut of Fg.4.5 th ut sgal sourc ( g ad R g th outut crcut th load rsstor (RL). Now, fg.4.6 w must rlac th trasstor wth ts qualt hybrd crcut, ths artcular cas wth th hybrd crcut for th E trasstor. t wll rsult th fal qualt crcut, showd fg.4.7. Th rsstor R B s th qualt rsstac of th oltag ddr bas crcut (rsstors R B ad RB aralll cocto). By dfto, th currt ga A s L A 4... For a smlfd calculato, fg.4.7 w wll tak to accout th load rsstor as th qualt rsstor R LR R L /(R R L ), thus th fg.4.7 bcoms fg

64 Th th currt ga s: A A c b h h f o h R L f b ' b h o o h f h o L R b L ' h f h o A R L ' 4... Th ut mdac s dfd by z th, from fg.4.8 w ca calculat ths mdac by alyg th scod Krchhoff s law for th ut crcut: z h h b r o h hrar L' b Now, w ca calculat asly th oltag ga, whch s dfd by: o A th, th sam way usd last dmostratos, w wll fd: R ' RL' A z o L L A z Th last aramtr whch w must kow s th outut admttac (mdac). Ths s dfd by th xt rlato: 64

65 y o o g Ths aramtr s dfd codto of ut sgal sourc shot-crcut ( g ). Th y o h h h o o f f ho o o g But from ut crcut alyg scod Krchoff s Law w ca wrt R ' g h h r o o g R h r ' g h ad f w rlac ths xrsso w wll fd y o h h f r ho ' Rg h 4.3. Th Smlfd Hybrd rcut for Bolar Trasstor. For a E amlfr hag a qualt crcut as th o dctd th fgur blow w foud arlr th followg quatos: urrt ga (4..) A h h f o R ' L ad ut mdac (4..4.) z h h A R ' r L ' th cas whr w ha satsfd th codto h R o L < ( artcular that mas a maxmum alu for th load rsstor of 4 ohms), th scod trm from th domator of A 65

66 ca b gord ad th currt ga of th amlfr ca b aroxmatd by th hybrd factor h f. That mas, from a ractcal stadot of w, that th hybrd modl of th trasstor, w ca glct th outut admttac h o. Now, f w tak to accout th actual tycal alus for th hybrd aramtrs th quato for th ut mdac (h r -4 ; A h f ; R L ' 4-3 ; h 3 ), w wll s that th scod trm th xrsso of ut mdac ca ha alus th rag -. ths cas w ca gor ths trm rsus th frst trm whch has a alu to tms hghr. That mas, from a ractcal stadot of w, that th hybrd modl of th trasstor, w ca glct th rrs trasfr factor h r. Takg to accout ths two aroxmatos, th hybrd modl of th trasstor bcoms mor sml, lk th crcut show Fg.4.9. th cas of th E amlfr, usg th smlfd qualt crcut Fg.4.9, th gral aramtrs of a commo mttr amlfr ar: A h f ; z h ; y o ; A u h f R L '/h 4.3. Th rror calculatg ths aramtrs, usg th smlfd hybrd modl, s aroud 4%. Ths rror s lss tha th dsrso th alus of commoly usd rsstors, whch maks t acctabl. 66

67 A scal cas of commo mttr amlfr s th commo mttr amlfr wth mttr rsstor. ths cas th E caactor dos t xst. Th th qualt a.c. crcut must tak to accout th rsc of ths rsstor ad Fgur 4.6 bcoms 4.6bs. Rlacg th trasstor ow wth hs smlfd hybrd crcut w wll obta fgur 4.7bs. Th currt ga rmas th sam lk for commo mttr amlfr, but th ut mdac ad oltag ga wll b dramatcally chagd. ( ) [ h R ( h )] h b R E b c b E ad usg th dfto formula for ut mdac w wll obta z h R E ( h ) f whch has a hghr alu tha th alu of commo mttr mdac. Now, usg th gral formula for oltag ga w wll obta th w alu of A. f A h ' hfrl hfrl RL RE( hf ) RE( hf ) RE ' ' As you ca s th oltag ga, ths cas, dos t dd o th trasstor rformacs, th amlfr hag a stabl oltag ga. 67

68 Th smlfd crcut show Fg.4.9 ca b usd too, for comutg th aramtrs of ay amlfr, ddt of th ty of th trasstor cocto. Lt s tst that rght ow ommo Bas Amlfr. Fg.4. s dctd th ommo Bas Amlfr wth oly o basg sourc. To obta th qualt c.a. crcut w must follow th sam ruls as th gral cas. Usg ths ruls w wll obta th qualt a.c. crcut show Fg.4.. f w rlac th trasstor wth ts smlfd crcut from Fg.4.9, w wll obta th fal qualt crcut, show Fg. 4., from whch w wll b abl to comut th gral charactrstcs of ths amlfr. Bcaus - -( b c ) th currt ga wll b: A L c b hfb hf ( h f) h f whos alu s a lttl bt smallr tha, whch s accordac wth th dfto of a factor. Th ut mdac s: z hb h b( hf ) hf whch has a alu aroud ohms, smallr tha th ut mdac of E amlfr. 68

69 Th oltag ga s smlar wth A of E amlfr, ad outut admttac s mor clos to zro tha th cas of E amlfr bcaus h ob -7 W -, a alu two ordrs of magtud lowr tha ho ommo ollctor Amlfr. Ths amlfr s show Fg.4.3. Th dffrcs btw ths amlfr ad E amlfr ar: th absc of a rsstor th collctor crcut ad th outut ot th mttr of th trasstor. Now, by alyg th ruls for a.c. qualt crcut ad rlacg th trasstor wth ts smlfd hybrd crcut, w wll fd th crcut drow Fg ths cas th currt ga has xt formula: L A b h f Th ut mdac ca b calculatd from two stadots of w, th ut mdac of trasstor zt ad th ut mdac of amlfr za. 69