Engineering 1620: High Frequency Effects in BJT Circuits an Introduction Especially for the Friday before Spring Break

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Amber Owen
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1 nginring 162: High Frquncy ffcts in BJT Circuits an Inoduction spcially 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 avl connctions. As this maial is only partially covrd in Razavi, I flt 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 rcnt classs, I dvlopd a numbr of formula first for th capacitancs in th junctions of a ansistor and thn for thir ffct on th gain, input impdanc and output impdanc of a gnralizd common mit amplifir. Lt m bgin by rcounting th capacitancs of a ansistor. (A full drivation of ths rsults is in an Appndix to this st of nots.) Th small-signal modl in its simplst form has two capacitancs: th bas-mit capacitanc is commonly calld C whil th collctor-bas capacitanc is variously C or C µ. (Th first nomnclatur is common on data shts whil th lat is mor common in paprs. Th CAD/SPIC usag is cmu. Thr is a slight but difficult to dmin diffrnc btwn th two whn thr is parasitic sris rsistanc in th bas. W do not obsrv this distinction.) OB b C r g v m b C OB c COB is th capacitanc of a rvrs biasd junction and oftn has th dpndnc on voltag of an abrupt-junction. For at last on valu of collctor-bas voltag, it is saightforwardly shown on any datasht. Oftn is is givn as a graph against VC. Th ansconductanc and th dynamic rsistanc of th bas-mit junction ar ndd first bfor on can tas th valu of C out of th datasht. qic Th Q-point dmins th rmaining modl params in th usual way: gm and kt ( hf ) kt r = ( hf ) r whr I am using h f inchangably with β as th DC qi h currnt gain of th ansistor. In class, w showd f that bcaus of th bas-mit capacitanc, th currnt gain bcoms frquncy dpndnt. For a β groundd-collctor circuit, th gain bcoms in db i h c f β ( s) = =. This is a singl pol i 1 + sr C + C db f β f T b µ function that looks lik this sktch whn β is graphd on a log-log Bod plot. Hr w hav d-

2 find two frquncis: f β at which th gain is down 3dB from its DC valu and f at T ft which th magnitud of th gain is unity. If h f is larg, thn fβ. Th datasht h givs f T as a function of th quiscnt currnt bcaus that frquncy is not snsitiv to th DC currnt gain. (It also hlps in advrtising th proprtis of a ansistor to advrtis th largr numbr!) Th dpndnc of f on currnt coms from r (or quivalntly T r ) bing invrsly proportional to currnt. Part of th C capacitanc is du to th dpltion layr and is roughly indpndnt of currnt. Th othr part of C is du to charg in ansit from mit to collctor and whn that is dominant, rc = τ, th ansit tim. Th typical curv for a 2N2222A dvic is shown blow nxt to a gnralizd common mit circuit. f VCC Z C Q1 v bg v out Z 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 ffct (namd af 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 fdback capacitor in plac. (Th proof of th thorm simply quats th currnt in th fdback capacitor to th two currnts through th capacitors of th scond configuration.) Bcaus th fdback capacitor conncts btwn output and input and th output voltag is oftn CF biggr than th input, th currnt in th capacitor is -A gnrally gra than it would b if th capacitor wr across th input to ground. Th thorm points out that this is quivalnt to a largr, possibly frquncy-dpndnt capacitor across th input and a marginally biggr on across th output. What oftn maks this thorm usful is that th low frquncy ( 1/ AC ) F - gain is known and that gain is constant nough to us -A ovr most of th usful frquncy rang of th amplifir. z ( + ) 1 ACF 2

3 W also dvlopd a st of formulas for th input impdanc and gain of th gnralizd common mit 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/A)C componnt simply bcam part of ZC and conibuts to calculating th gain. Th (A)COB lmnt movd to th lft and ndd in paralll with th input. This ansform lft only C to complicat lif. Th input impdanc with only that parasitic capacitanc was: ( hf )( r + Z ) src Z z = + sr C sr C Th first m is our old rsult for input impdanc at low frquncy but now thr is a dcras in impdanc with incrasing frquncy from an xa singl pol at 1 ft fβ = =, a nic but not surprising rsult. Th scond m assurs that vn 2 rc 1 + h f whn th bas capacitanc shorts th bas-mit junction, thr will still b Z lft as part of z. This m is not important until roughly f T and on would not usually y to us a dvic clos to its maximum frquncy limit. Notic that th first m has th form rc of a rsistor R= ( h f )( r + Z )] in paralll with a capacitor of valu C =. r + Z Th rsistanc valu is th low-frquncy rsult w drivd a coupl of wks ago. Th capacitanc is proportional to but gnrally smallr than C. W will us this rsult in an analysis xampl. α Z C 1 Th gain formula has a similar form: G = r + Z 1 + srz C / r + Z ( ) ( ) Z in paralll with Millr s scond capacitor C ( 1 1/ A) whr Z C is C µ +. [I am playing fast and loos with xact rsults hr. Actually as A bcoms frquncy dpndnt on has to b carful to includ th ffct of that chang on th input capacitanc. W will s this mor clarly in MOS circuits la.] Th first factor is th low frquncy gain and th scond is a nw pol gnrally somwhat abov f. In othr words, C T dos not hav a big ffct 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 ffcts. 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 f = 12 typical, and thrfor r = 4.2 ohms and r = 5 ohms. From th graph abov, ft 25 MHz. Th datasht valu of COB is 7 pf. From this, C = COB = 7 1 = 146pf f r 6.28*4.2*2.5 1 T 3

4 VCC 1K VCC VBIAS Q2 21K 1K 21K C IN Q1 2N222A C IN Q1 2N2222A v in 2.7K 75 Ω 25 Ω 16 μ Th mid ban d gain is 2.7K = By mploying Millr s thorm, w can draw th small signal modl as: v in C BYPASS 75 Ω 25 Ω 16 μ C IN 21 K r = 5 ohm C = 146 pf gm vb vin 2.7 K C μ (34) = 245 pf R 25 ohm 1 K C μ (1/34) = 7.2 pf vout R BB = 2.5 K z Z C 4

5 For Zin w hav (nglcting CIN bcaus it provids only a low-frquncy, high-pass cutoff): 2.7 K 21 K 245 pf 121*29.2 = 3.5 K = pf R BB = 2.5 K z 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 conast th ffct of C µ on th gain is to inoduc 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 la 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 ansistor. Th tandm arrangmnt of a common mit 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 mit Q1 is vry low, fractional in this cas, bcaus ZC for that stag is th input impdanc of th common bas stag, Q2. (That input impdanc is zin2 = r 2 C 2. Th capacitanc of Q2 is not important at a fw αr mgahrtz so th voltag gain of Q1 is G = =.15) Thr is no r + Z longr a dirct capacitanc btwn input and output. Th output load no longr affcts th input impdanc. For that rason, cascod circuits ar somtims said to b unilaal. This tim th input impdanc is th sam 1.42 K rsistiv part but th capacitor is only 27 = 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

6 Appndix: Th ffct of C on Input Impdanc and Gain of a BJT C Circuit b c i b v b C r g m v b v bg ZC v out i Z v g z Basic quations: v KCL at th mit minal, : = ib + gmvb Z ri b Ohm's law across bas-mit: vb = 1 + sc r KVL from input across bas-mit and mit to ground: vbg = vb + vbg vbg Dfinition of z : z ib Connction btwn hybrid-pi ansconductanc modl and currnt conolld h- modl: gmr = β whr β is th low-frquncy currnt gain. Stps to solv: g vg gm r β + sc r = ib = ib Z 1 + scr 1 + scr 6

7 β + sc r v = Z i g b scr and v b ri b = 1 + sc r v bg ( 1 β ) r + Z + + sc r i = b scr (Usd: r ( β ) = + ) 1 r Input impdanc is: z = ( 1 β ) + r + Z + sc r Z sc r z ( β )( + ) r Z sc r Z sc r sc r = + Gain is th ratio: vout ZCiC ZC gmvb ZC gmrib ZC gmrib ZCβ H( s) = = = = = = v v v v sc r v sc r z sc r bg bg bg bg bg H s Summary: Z β Z β C C = = ( β )( r + Z ) + sc r Z ( β )( r + Z ) 1 ++ sc rz /( r + Z ) Th input impdanc is lowrd by C bginning at th bta cutoff frquncy. It is also asymptotic to Z for frquncis abov ft. In this quation th first m is th low frquncy impdanc with a nw pol at th bta cutoff frquncy. Th scond m taks car of th bhavior that maks Z th impdanc at vry high frquncy. 1 z ( β )( + ) r Z sc r Z sc r sc r = + Notic that th first m 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. 7

8 z = ( β )( r + Z ) sc R if C r C = and R = r + Z ( r + Z ) ( β ) Th gain is littl affctd xcpt thr is a nw pol at som frquncy abov ft. Th first factor is th low frquncy gain and th scond is a nw high frquncy pol. H s Z β C = ( β )( r + Z ) 1 ++ sc rz /( r + Z ) 1 8

ECE 2210 / 00 Phasor Examples

ECE 2210 / 00 Phasor Examples EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain