Enginring 162: High Frquncy Efcts in BJT ircuits an Introduction Espcially for th Friday bfor Spring Brak I hav prpard ths nots bcaus on th day bfor a major vacation brak som popl find it ncssary to lav arly for travl connctions. As this matrial is only partially covrd in Sdra and Smith, I lt it worthwhil to summariz it for you. I hop it will also hlp you undrstand th calculations for Lab 5, a cascod amplifir for vido signals. Ovr arlir classs in th wk, I dvlopd a numbr of formula first for th capacitancs in th junctions of a transistor and thn for thir fct on th gain, input impdanc and output impdanc of a gnralizd common mittr amplifir. Lt m bgin by rcounting th capacitancs of a transistor. (I gav you a handout with th full drivation of ths rsults and hav addd that as an Appndix to this st of nots.) Th small-signal modl in its simplst form has two capacitancs: th bas-mittr capacitanc is commonly calld whil th collctor-bas capacitanc is variously OB or. (Th first nomnclatur is common on data shts whil th lattr is mor common in paprs. Th AD/SPIE usag is cmu. Thr is a slight but difficult to dtrmin difrnc btwn th two whn thr is parasitic sris rsistanc in th bas. W do not obsrv this distinction.) OB b OB is th capacitanc of a rvrs biasd c junction and oftn has th dpndnc on voltag of an abrupt-junction. For at last on gv m b valu of collctor-bas voltag, it is straightforwardly shown on any datasht. Th r transconductanc and th dynamic rsistanc of th bas-mittr junction ar ndd first bfor on can tas th valu of out of th datasht. qi Th Q-point dtrmins th rmaining modl paramtrs in th usual way: gm and kt 1 h kt r 1h r whr I am using h intrchangably with as th D qie currnt gain of th transistor. In class w showd that bcaus of th bas-mittr capacitanc, th currnt gain bcoms frquncy dpndnt. h For a groundd-collctor circuit, th gain bcoms i h c s. This is a singl pol ib 1 sr in db function that looks lik this sktch whn is graphd on a log-log Bod plot. Hr w hav dfind two frquncis: f at which th gain is down db f f T
3dB from its D valu and f at which th magnitud of th gain is unity. If T h is larg, ft thn f. Th datasht givs f T as a function of th quiscnt currnt bcaus 1 h that frquncy is not snsitiv to th D currnt gain. (It also hlps in advrtising th proprtis of a transistor to advrtis th largr numbr!) Th dpndnc of f T on currnt coms from r (or quivalntly r ) bing invrsly proportional to currnt. Part of th capacitanc is du to th dpltion layr and is roughly indpndnt of currnt. Th othr part of is du to charg in transit from mittr to collctor and whn that is dominant, r, th transit tim. Th typical curv for a 2N2222A dvic is shown blow nxt to a gnralizd common mittr circuit. V Z Q1 v bg v out Z E z tr Th capacitanc from collctor to bas complicats th simpl calculation of gain and input impdanc. To simplify th problm, w tak advantag of th Millr fct (namd aftr John W. Millr who publishd it in 192). Millr s thorm points out th quivalnc for input impdanc, output impdanc and gain of th two block diagrams blow as long as th gain of th amplifir is known with th dback capacitor in plac. (Th proof of th thorm simply quats th currnt in th dback capacitor to th two currnts through th capacitors of th scond configuration.) Bcaus th dback capacitor conncts btwn output and input and th output voltag is oftn biggr than th input, th currnt in th capacitor is gnrally gratr than it would b if th capacitor wr across th input to F ground. Th thorm points out that this is quivalnt to a largr, possibly frquncy-dpndnt ca- -A pacitor across th input and a marginally biggr on across th output. What oftn maks this thorm usful is that th low frquncy gain is known and that gain is constant nough to us ovr most of th usful frquncy rang of th amplifir. 1 1/ AF - -A W also dvlopd a st of formulas for th input impdanc and gain of th gnralizd common 1 A F 2
mittr amplifir shown at th top right. Th ida was to us Millr s thorm to mov th collctor bas capacitanc to two placs: th (1+1/A) componnt simply bcam part of Z and contributs to calculating th gain. Th (1+A) OB lmnt movd to th lft and ndd in paralll with th input. This transform lft only to complicat li. Th input impdanc with only that parasitic capacitanc was: 1h r ZE sr ZE ztr 1sr 1sr Th first trm is our old rsult for input impdanc at low frquncy but now thr is a dcras in impdanc with incrasing frquncy from an xtra singl pol at 1 ft f, a nic but not surprising rsult. Th scond trm assurs that vn 2 r 1 h whn th bas capacitanc shorts th bas-mittr junction, thr will still b Z E lft as part of z tr. This trm is not important until roughly f T and on would not usually try to us a dvic clos to its maximum frquncy limit. Notic that th first trm has th form r of a rsistor R 1h r ZE] in paralll with a capacitor of valu. r ZE Th rsistanc valu is th low-frquncy rsult w drivd a coupl of wks ago. Th capacitanc is proportional to but gnrally smallr than. W will us this rsult in an analysis xampl. Th gain formula has a similar form: G Z r ZE 1 srz E / r ZE Z in paralll with Millr s scond capacitor 1 1/ A 1 whr Z is. [I am playing fast and loos with xact rsults hr. Actually as A bcoms frquncy dpndnt on has to b carful to includ th fct of that chang on th input capacitanc. W will s this mor clarly in MOS circuits latr.] Th first factor is th low frquncy gain and th scond is a nw pol gnrally somwhat abov f. In othr words, T dos not hav a big fct on th Gain xcpt through a dcras in input impdanc that causs loading of th input signal sourc. Now lt us look at an xampl of ths fcts. Th circuit on th lft blow is on w usd as a low frquncy xampl som tim ago. It has a quiscnt point around 6 ma, a currnt gain about h = 12 typical, and thrfor r = 4.2 ohms and r 5 ohms. From th graph abov, ft 25 MHz. Th datasht valu of OB is 7 pf. From this, 1 1 12 OB 7 1 146pf. 8 2 f r 6.28*4.2*2.51 T 3
V 1K V VBIAS Q2 21K 1K 21K IN Q1 2N222A IN Q1 2N2222A v in 2.7K 75 Ω 25 Ω 16 μ Th mid ban v in d gain is.9911 3 34 BYPASS 2.7K 4.3 25. By mploying Millr s thorm, w can draw th small signal modl as: 75 Ω 25 Ω 16 μ IN 21 K r π = 5 ohm π = 146 pf g m v b v in 2.7 K μ (1+34) = 245 pf R E 25 ohm 1 K μ (1+1/34) = 7.2 pf v out R BB = 2.5 K z tr Z 4
For Z in w hav (nglcting IN bcaus it provids only a low-pass cutoff): 2.7 K 21 K 245 pf 121*29.2 = 3.5 K 1464.3 21 25 4.3 pf R BB = 2.5 K z tr Th rsulting systm is 1.42 K in paralll with 266 pf. Th input impdanc is dominatd by th Millr capacitanc (245 pf of 266 pf total) and its magnitud will start to dcras with singl pol bhavior at 421 KHz. By contrast th fct of on th gain is to introduc a high-frquncy low-pass pol at about 23 MHz from th 1K rsistor and 7 pf capacitanc in paralll. [If thr is apprciabl sourc loading at this frquncy, it is vn possibl that this scond pol will b vn highr in frquncy but that is a latr topic.] An amplifir that has constant gain to som high frquncy but has so low an input impdanc as to load th signal sourc wll blow its gain cutoff frquncy is a poor dsign bcaus on cannot us th gain for th full rang of th amplifir s potntial usfulnss. Th circuit on th right abov is calld a cascod amplifir and it attmpts to solv this problm with a scond transistor. Th tandm arrangmnt of a common mittr stag, Q1, with a common bas stag, Q2, is calld a cascod connction. (And no, this is not a splling rror.) Th voltag gain of th common mittr Q1 is vry low, fractional in this cas, bcaus Z for that stag is th input impdanc of th common bas stag, Q2. (That input impdanc is zin2 r 2 2. Th capacitanc of Q2 is not important at a w r.99 4.2 mgahrtz so th voltag gain of Q1 is G.15) Thr is no r ZE 4.2 25 longr a dirct capacitanc btwn input and output. Th output load no longr afcts th input impdanc. For that rason, cascod circuits ar somtims said to b unilatral. This tim th input impdanc is th sam 1.42 K rsistiv part but th capacitor is only 21+7 = 28 pf and th capacitiv part bcoms a factor in loading th input only abov 4.5 MHz. That is a full ordr of magnitud improvmnt in pol placmnt. 5
Appndix: Th Efct of π on Input Impdanc and Gain of a BJT E ircuit b c i b v b π r π g m v b v bg Z v out i Z E v g z tr Basic Equations: v KL at th mittr trminal, : ib gmvb Z ri b Ohm's law across bas-mittr: vb 1 s r KVL from input across bas-mittr and mittr to ground: vbg vb vbg vbg Dfinition of z tr : ztr ib onnction btwn hybrid-pi transconductanc modl and currnt controlld h- modl: gmr whr is th low-frquncy currnt gain. Stps to solv: g E vg gm r 1 s r ib1 ib ZE 1 sr 1 sr 1 s r v Z i g E b 1 sr and v b ri b 1 s r 6
v bg r Z 1 s r 1 sr E i b (Usd: r ) 1 r Input impdanc is: z tr 1 r Z s r Z E E 1 s r z tr 1 r Z srz 1s r 1s r E E Gain is th ratio: vout Zi Z gmvb Z gmrib Z gmrib Z Hs v v v v 1s r v 1s r z 1s r bg bg bg bg bg tr H s Summary: Z Z E E E E E 1 r Z s r Z 1 r Z 1 s rz / r Z Th input impdanc is lowrd by bginning at th bta cutoff frquncy. It is also asymptotic to Z E for frquncis abov f T. In this quation th first trm is th low frquncy impdanc with a nw pol at th bta cutoff frquncy. Th scond trm taks car of th bhavior that maks Z E th impdanc at vry high frquncy. 1 z tr 1 r Z srz 1s r 1s r E E Notic that th first trm has th form of a rsistanc in paralll with a capacitanc. As you did in th first lab, w can manipulat that into th form of a rsistanc with an quivalnt capacitanc in paralll by multiplying and dividing th frquncy in th dnominator by th rsistanc from th numrator. z tr 1 r Z 1 sr E if r and R 1 r Z r Z E E 7
Th gain is littl afctd xcpt thr is a nw pol at som frquncy abov f T. Th first factor is th low frquncy gain and th scond is a nw high frquncy pol. H s Z 1 r Z 1 s rz / r Z E E E 1 8