Digital MOS Integrated Circuits. Chapter 2 Gate Delay and RC Circuits. Practice: Inside the CMOS Inverter. Virtually Parallel Capacitors.

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1 apr 2 Ga Dlay and ircuis Digial MOS Ingrad ircuis MIPS From ompuaion, you av sn a digial Ingrad ircuis ar buil wi MOS ransisors a work as swics Today w will sar our sudy of ir swicing spd 3 firs ordr 1 3 firs ordr 2 Pracic: Insid MOS Invrr Virually Paralll apaciors 2.3 Tis is sudid in ompuaion 3 firs ordr 3 i 2 v 1 Ts wo circuis ibi sam ransin bavior Sow a currn i is idnical in bo circuis d(v V i Lf ck: 1 2 d d dv Eampl 1.4 d d d ( 1 2 d i v 1 2 i ( 1 2 d onsan bias volag across capacior dos no influnc ransin bavior 3 firs ordr 4 Invrr Swicing Modls n pulldown V pullup 5 V n p p n p abov ambin old off Your ar sudying so ard a you forg o drink your coff Skc mpraur as a funcion of im A B anonical circuis subjc of is lcur (Modl will b plaind in ompuaion W ar inrsd in Wavform a capaciors afr swic closs im urv could bs rprsn an acual cooling scdul, sinc ra of cooling (slop of curv is lfasr wn mpraur diffrnc (coff vs ambin is igr 3 firs ordr 5 3 firs ordr 6 1

2 Hydraulic Analogy ommunicaing Vssls Hydraulic Analogy Prssur Diffrnc Ponial Diffrnc War Flow urrn Flow Vry larg war ank Small war ank, mpy wn < Prssur diffrnc proporional o ig diffrnc Iniial condiion TUE/EE 5 nwrk analys 2/3 NvdM 1 fundamnal 8 Valv opns a Skc war ig in small ank as funcion of im War ig im 3 firs ordr 7 3 firs ordr 8 ircui Bavior: Iniial ondiion v i v W ar inrsd in bavior of circui afr swic closs No inrsd for < Eampl: wa is i jus afr swic closs? v mmbr i and no v v V v Tus i S ( ( (: iniial condiion (bginwaard For circuis wi mmory, if w wan o compu bavior afr som im (no only a, w nd o spcify iniial condiion(s a 3 firs ordr 9 ircui Bavior: Final Valu v i v iniial valu Wa is of? Final valu mans c sady sa condiion d i c v mmbr: i d Tus c i v d sady sa Final (sady sa valu is Noaion: ( ( 3 firs ordr 1 ircui Bavior: Final Valu v i v iniial valu Final valu is ( Dos of dpnd on iniial condiion (? No! For linar circuis, dos no dpnd on iniial condiions bu acual ransin rspons dos dpnd on iniial condiions. ircui Bavior: Slop of v i v iniial valu is a funcion of im: ( Wa is slop (llingsok of? i d v V v d i S W can find slop of rspons for ac valu of oupu Sow a i( ( (/ follows dircly as a spcial cas 3 firs ordr 11 3 firs ordr 12 2

3 Summary v i v iniial valu W can compu i( and of, ( and slop of as a funcion of, bu can w find ou mor? mmbr: w wan o say soming abou compur swicing spd using abov circui as a modl Implis a w wan o sudy bavior as a funcion of im spons im: ow fas is ransin from ( o V S Mor gnrally, w wan o find (: ransin rspons (ovrgangs vrscijnsl ircui Bavior: spons Tim v i v iniial valu Wic ings drmin ow fas rspons (ransiion from iniial valu o will b? T valu of (larg will carg mor slowly T valu of (larg will ak longr o rac W will sorly find a in suc (linar circuis wi on, rspons im is proporional o produc No valu of (diffrnc bwn final and iniial valu dos no influnc rspons im, only currns involvd 3 firs ordr 13 3 firs ordr 14 ircui Equaion v i v >: i d V v i S v d Tis diffrnial quaion, ogr wi iniial condiion, fully spcifis bavior of circui afr swic closs Our n callng: larn ow o solv suc quaions Diffrnial Equaions v d circui dy Ky c prooypical d Formulas Aad Equaion no only involving variabl, bu also is drivaiv(s rucial in many filds nginring, biology, conomics, adioaciv dcay, or laws of naur wnvr cang is influncd by prsn sa linar D.E. (diffrnial quaion wn cang proporional o sa Tis quaion, and many ors, producs ponnial soluions 3 firs ordr 15 3 firs ordr 16 f ( a a n sin( 1 Typ of Soluion o Epc v c V s d d d ~ v df ( d a a n n1 cos( ln(1 1 a of cang of v is proporional o v W nd compaibl LHS and HS Equaions of cang proporional o sa produc ponnial soluions Drivaiv proporional o funcion Drivaiv Proporional o Funcion If y b Proof: n dy cb d dy y( y( b b lim lim d Bu: b (b (b Tus: dy b b b lim d No: c dpnds on b b 1 b lim c Tis limi valuas o a consan cb ln( firs ordr 17 3 firs ordr 18 3

4 If W Would Know Abou ln,, c. Sow a a 1 lim is a consan ! ln a. ( ln a. a 1 ln a.... 2! a 1 1 ln a. 1 lim ln a QED T numbr db cb d c dpnds on bas b of ponn b 1 c lim Tr is a b suc a c 1 Tis valu of b is Tis numbr is calld markably spcial numbr is prfrd bas for prssing ponnial soluions o our diffrnial quaions 3 firs ordr 19 3 firs ordr 2 If y b n Wy as bas? dy cb If y n d Tis is wy w us as our bas: simplr formulas, wiou arbirary consans v v Tis circui dy d c d Tis diffrnial quaion v V ( c s 1 Has is soluion 3 firs ordr 21 v v v.5 Soluion c d v V ( c s firs ordr 22 Proof Using Diffrnaion Sow a is diffrnial quaion as is soluion c v d c V c V s s idnical d c v c c V d s d qd 3 firs ordr Evalua ingrals, us 1 d ln K 1 ad a K 2 Soluion by Ingraion. Sar wi Diffrnial Eq. 1. Spara variabls 2. Ingra LHS and HS 4. Absorb K s ino K 3 K 2 K 1 c d c d c d planaion ln( K 1 K 2 ln( K firs ordr 24 4

5 4. from prvious sp 5. ponnia LHS and HS 6. rsul Soluion coninud ln( K 3 K3 v c v v Soluion c d v V ( c s 1 7. wri (dfin K K 3 K v 8. Drmin K from iniial valu: ( V s K K.5 K 9. Final Soluion: V ( s 1 3 firs ordr firs ordr 26 Iniial Valu v v Volag across capacior (v dpnds on isory Any prvious circui aciviy could influnc carg prsn a a Iniial valu (bginwaard mus b known, ir plicily or implicily (from circui con Mamaically rlad o fac a ani drivaivs ar only spcifid up o a consan v V ( c s Tim onsan im consan (ijdconsan, us symbol τ Basic im scal of swicing vn V V s uni: ΩF s A V V s 3 firs ordr 27 3 firs ordr 28 Tim onsan spons can b normalizd wi rspc o τ and V S v ( v V ( c s 1 2 Eampl: (1. 86 ( τ Eac τsp givs 63% V.86 of rmaining swing s τ usually good (1% nginring approimaion for compl ransiion 2τ 4τ 6τ 5%.69τ 8τ Sow a 5%.69τ 3 firs ordr 29 Soluion Procss (viw Mamaics will lar provid mor formal drivaion Sparaion of variabls dy Bu dy (by M M( far ( N no ( y d d c V v c d all diffrnial s c quaions ar d sparabl! Ingra LHS and HS sparaly add consan K o b drmind from iniial condiion (s prvious slids...d... K or us dfini ingraions (s book...d 3 firs ordr 3 5

6 Gnral Iniial ondiion Assum swic closs a (insad of Assum v ( v (insad of v ( Eq 2.14 K /τ / τ K V K s / τ / τ ( / τ / τ ( ( / τ ( Simpl imsif of soluion, and scald ampliud of ransin 3 firs ordr 31 D sady sa rspons oal rspons omponns of Soluion ( o / τ (o ransin rspons 2τ 4τ 6τ 8τ Minus Transin rspons (scaklvrscijnsl sign From ig ampliud o zro: lim > Iniially opposs sady sa rspons: 1 Transin rspons significan jus afr swicing, lar D sady sa rspons dominas 3 firs ordr Gnral Firs Ordr ircuis c d Tis (linar firs ordr ordinary diffrnial quaion is valid for any firs ordr circui wi D sourcs Quick Proof: 1. Firs ordr circui mans a r is only on nod in circui wi mmory (sa (osand 2. If i as only on capacior, proof follows from compuing Tévnin quivaln for rs of circui, bcaus rsul is simpl sris ck. 3. If r ar mulipl capaciors, y mus b virually paralll and can us b rducd ino on quivaln capacior spcial cas τ c ( d gnral rsul 3 firs ordr 33 Unknown variabl as a funcion of im 2.1 of variabl iniial valu of variabl Gnral Soluion c Valid for any firs ordr τ ( d circui wi D sourcs ( / τ ( ( ( sady sa (saionair of variabl ransin [ (im of swicing ] im consan 3 firs ordr 34 Soluion (almos by Inspcion ( / τ ( ( ( 1. Idnify sa variabl: capacior volag 2. Drmin iniial valu of sa variabl 3. alcula 4. alcula im consan Sp 3: alcula Final Valu Sady sa or of capacior volag: I lim c lim c c i d d 1 2 v I 1 2 i mans: plac capacior by opn circui v ( D sady sa quivaln modl of capacior is an opn circui Q: (? I 2 3 firs ordr 35 3 firs ordr 36 6

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