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Chapter3Single-Transistor and Multiple-Transistor Amplifiers---Single-Transistor Amplifiers---Two-Transistor BJT Amplifier---Two-Transistor MOSFET Amplifier---Differential Amplifiers (Mismatch)DEVICE MISMATCH IN DIFEERENTIALAMPLIFIERSOutline---The general approach to analyzing mismatches ---Input voltage and current offset of BJTdifferential amplifiers---Input voltage offsets of MOS differential amplifiers---Small-Signal Characteristics of Unbalanced Diff.Amp.General MethodSuppose performance parameters p 1and p 2can be written as),,,(),,,(22221111L L z y x f p andz y x f p ==Ideally, x 1=x 2, y 1=y 2, z 1=z 2…,→p 1=p 2. But, in practice, they are not and an error exists:),,,,,,,(),(22211121L L z y x z y x f p p e Error ==where x1,y1,z1,…and x2,y2,z2,…can be expressed in terms of their difference and average values, For example,22121x x x andx x x +=−=∆MISMATCH ANALYSIS METHODS22121x x x andx x x +=−=∆xx x andxx x ∆−=∆+=⇒5.05.021thus,);5.0,5.0;5.0,5.0(),(21L y y y y x x x x f p p e ∆−∆+∆−∆+=Generally:x x <<∆, the following approximations canεεεε−≈++≈−111111orneglecting high power values of ).,.(2εεe i be concluded:INPUT VOLTAGE AND CURRENT OFFSET OF THE BJT DIFF. AMPLIFIERModel for Input Offset Voltage and CurrentBase width, base & collector doping level, effective emitter area, collector load resistance.•V OS :If v in =0)()(OS C TCOS C m C V R V I V R g v ∆=∆=∆If an offset exists in:21&C C R R )/(/)/()(_C C T m C C R OS C C C R R V g R v V R i v C ∆=∆=→∆=∆→If an offset exists in area or doping level, in I S)/(/)/(_/S S T m C I OS S C S V V S C I I V g i V I I I eI i S TBE ∆=∆=→∆=∆=∆→mOS C C C g V i i i =−=∆21•I OS :If an offset exists in i C ,→β/C OS i I ∆=)/)(/(/)()/)(/(/)/(__ββββββββC B OS C C C C C C R OS i i I i R R R R i I C ∆=∆=∆=∆=⇒RMS-sum (polarity of error is not important):2_2_2_2_&βOS R OS OS I OS R OS OS III VVV CSC+=+=DC performance:⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛=⇒12212211ln ln ln S S C C T S C T S C T IDI I I I V I I V I I V V 22222221121121)()()()(A V Q D qn A V W N D qn I A V Q D qn A V W N D qn I CB B n i CB B A n i S CB B niCB B A n iS ====where W B (V CB )is the base width as a function of V CB , N A is theacceptor density in the base and A is the emitter area.122122110C C C C C C C C OD R R I I R I R I V =⇒=⇐=⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=⇒)()(ln 211212CB B CB B C C T OSV Q V Q A A R R V V 021=+−BE BE ID V V VNow, define:222&212121CC C C C C C C C C C C R R R and R R R R R R R R R ∆−=∆+=⇒+=−=∆222&212121AA A and A A A A A A A A A ∆−=∆+=⇒+=−=∆222&212121BB B B B B B B B B B B Q Q Q and Q Q Q Q Q Q Q Q Q ∆−=∆+=⇒+=−=∆Substituting these values into the expression for V OS gives⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛∆−⎟⎠⎞⎜⎝⎛∆−⎟⎟⎠⎞⎜⎜⎝⎛∆−≈⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∆+∆−⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∆+∆−⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∆+∆−=B B C C T BB BB C C C C T OSQ Q A A R R V Q Q Q Q A A A A R R R R V V 111ln 222222ln BB C C Q Q and A A R R <<∆<<∆<<∆,Expanding the logarithm and neglecting higher order terms gives⎟⎟⎠⎞⎜⎜⎝⎛∆−∆−=⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−∆−≈S S C C T B B C C T OSI I R R V Q Q A A R R V V whereBBS S Q Q A A I I ∆−∆=∆Example:05.0&01.0//==∆∆SSI IR R σσmV mV I I R R V V S S C C T OS 5.1)05.001.0)(26(−=+=⎟⎟⎠⎞⎜⎜⎝⎛∆−∆−≈mVmV V V SS CC I I R R T OS 3.1)05.0()01.0()26(22/2/2=+=+=∆∆σσCalculation of dV OS /dT:Assuming V OS (270C)=2mV,K V KmVT T V T V or I I R R T V T V OS OS S S C C T OS /67.63002)(µ=≈=∂∂⎟⎟⎠⎞⎜⎜⎝⎛∆+∆=∂∂Offset voltage can be cancelled using external circuitrybut to cancel the temperature drift, the external circuitry must have the temperature dependence.Temperature Dependence of the Input Offset VoltageThe input offset current of the BJT Differential Amplifier can be written as :112212215.0&5.0F C F C B B OS OS B B OS B B I I I I I I I I I I I ββ−=−=⇒−=+=Define:222&222&212112212112FF F F F F F F F F F F CC C C C C C C C C C C and I I I and I I I I I I I I I ββββββββββββ∆+=∆−=⇒+=−=∆∆+=∆−=⇒+=−=∆Input Offset Current of the BJT Differential AmplifierqCombining the previous expressions into the function for I OS gives:FF C C F F CC F C F FC C F F C C OS I I when I I I I I I I I βββββββββ<<∆<<∆⎟⎟⎠⎞⎜⎜⎝⎛∆−∆≈⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛∆−∆−−∆+∆+=,2222, then,CCC C CC CCC C C C R R I I R R R R I I I I ∆=∆−⇒∆+∆−=∆+∆−21212121122122110C C C C C C C C OD R R I I R I R I V =⇒=⇐=Therefore,⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−=FFC C F C OSR R I I βββINPUT VOLTAGE OFFSET OF THE MOSFETDIFF. AMPLIFIERModel for Input Offset Voltage•V OS : If v in =0mOS D D D g V i i i =−=∆21If an offset exists in:2/))(/(2/)/(_t GS D D D D D R OS V V R R I R v V D −∆=∆=→β21&D D R R LW k '=βDD D D D D R v i orR i v /)(∆=∆∆=∆→If),,('k L W or ∆∆∆∆β2/))(/(2/)/(2/)/(/_t GS D D D m D OS V V I I I g i V −∆=∆=∆=∆=βββββββββ),()(/)]}(2[2{/22_t tt tt GS t GS t t m D V OS V Vt small is V if V V V V V V V g i V t ∆<<∆∆∆≈−−∆+∆=∆=ββIf exists:•I OS =0tV ∆2'21'1212'221'1121)/(2)/(2)/(2)/(20L W k I L W k I V V L W k I V L W k I V V V V V D D t t D t D t ID GS GS ID −+−=−−+=⇒=+−122122110D D D D D D D D OD R R I I R I R I V =⇒=⇐=DC performance:Define:222&212121DD D D D D D D D D D D R R R and R R R R R R R R R ∆−=∆+=⇒+=−=∆222&212121ββββββββββββ∆−=∆+=⇒+=−=∆and 222&212121DD D D D D D D D D D D I I I and I I I I I I I I I ∆−=∆+=⇒+=−=∆222&212121tt t t t t t t t t t t V V V and V V V V V V V V V ∆−=∆+=⇒+=−=∆Substituting these values into the expression for V OS givestD D D t D D D D D t D D D D D tt t t D D D D OSV I I I V I I I I I V I I I I I V V V V I I I I V ∆+⎥⎦⎤⎢⎣⎡∆−∆=∆+⎥⎦⎤⎢⎣⎡∆+∆−−∆−∆+=∆+⎥⎦⎤⎢⎣⎡∆+∆−−∆−∆+=∆−−∆++⎟⎟⎠⎞⎜⎜⎝⎛∆−∆−−⎟⎟⎠⎞⎜⎜⎝⎛∆+∆+=βββββββββββββββββ222)221()221(2)21)(21()21)(21(2)2()2(5.05.025.05.02where,tt D D D D V V and I I R R <<∆<<∆<<∆<<∆,,,ββWhen the variation in 3 parameters are uncorrelated:mV V V R R V V V V t GS D D t GS t OS1072)(2)(22222=⎟⎟⎠⎞⎜⎜⎝⎛∆−+⎟⎟⎠⎞⎜⎜⎝⎛∆−+∆=ββVV V mV V R R t GS t D D 5.1&100%,5/%,1/=−=∆=∆=∆ββFor the correlated case:mV R R V V V V D D t GS t OS5522)(=⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−−∆=ββ⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−∆≈⇒∆−=∆βββ222D D D t OS D D D D R R I V V R R I I tGS DV V I −=β2⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−−∆=⇒ββ22)(D D t GS t OSR R V V V VTemperature Dependence of the MOS InputOffset Voltage⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−∆=βββ222D D D t OS R R I V V TT T T T T dT d T T T L WT k T )(5.1)/)((5.1)()()(5.1005.100'βββββ−=−=⇒⎟⎟⎠⎞⎜⎜⎝⎛==−−⎟⎟⎠⎞⎜⎜⎝⎛+∂∂⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−=⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂⎟⎟⎠⎞⎜⎜⎝⎛∆+∆−=2/12/342322122 22122122βββββββββT I T I I R R T I T I I R R dT dV DD D D D D D D D D OSepredictabl T V T ∝epredictabl so not T T I D ∂∂∂∂β&R D , ,V tR C , (A, Q B ) →I SV OSV OS , I OS MOSFETBJT ↓⇒↑OS dm V A SUMMARYβ22221121221121R i R i v v v R i R i v v v c d +=+=−=−=2/)(,&,2/)(,21212121R R R R R R i i i i i i c d +=−=∆+=−=21R R ≠if:Definite:4)(2)2)(2()2)(2()()2)(2()2)(2(R i R i R R i i RR i i v R i R i RR i i RR i i v d cd c dc c cd d c dc d ∆+=∆−−+∆++=∆+=∆−−−∆++=Half circuitSMALL-SIGNAL CHARACTERISTICS ofUNBALANCED DIFF. AMP.2/)(,&,2/)(,21212121m m m m m m c d g g g g g g v v v v v v +=−=∆+=−=42)2)(2()2)(2( 22 )2)(2()2)(2( 221121221121d m c m d c mm d c m m m m c cm d m dc m md c m m m m d v g v g v v g g v v g g v g v g i i i v g v g v v g g v v g g v g v g i i i ∆+=−∆−++∆+=+=+=∆+=−∆−−+∆+=−=−=⇒21m m g g ≠Definite:if:22&0222=+∆+=∆++Rc idm m mid m Rd i v g v g v g v g i tail Rc ic tail ic r i v v v v 2+=−=)212()212(22tailm m tail m m ic tail m mtailm m id Rd r g g r g g v r g g r g g v i +∆+∆−++∆∆+−=tailm idm ic m Rc r g v g v g i 2122+∆+−=⇒⎟⎟⎠⎞⎜⎜⎝⎛+∆+∆−==+∆−∆∆+−====tail m m m v icoddmcm tailm m m tailm m v idod dmr g R g R g v v A r g Rg R g r g R g v v A id ic 21212220_0R i R i v RdRc od 222+∆=icdm cm id dm od v A v A v _+=Ri R i v Rc Rd oc +∆=22iccm id cm dm oc v A v A v +=_⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+∆∆+−==⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛∆∆−∆+∆−====tail m m m v ic oc cm tail m m m tail m m m m v id oc cm dm r g R g R g v v A r g g g r g R g R g R g v v A id ic2122212241020_SUMMARY•Differential Mode and Common Mode •Characteristics of Differential Amplifier •CMRR•ICMR•Offset Voltage•Mismatch。
