Lecture 15: Small Signal ModelingLecture OutlineNotation ReviewDiffusion RevisitedDiffusion CapacitanceDiffusion Capacitance (cont)BJT Transconductance gmTransconductance (cont)BJT Base CurrentsSmall Signal Current GainInput Resistance rπOutput Resistance roGraphical Interpretation of roBJT Small-Signal ModelBJT CapacitorsBJT Cross SectionCore BJT ModelComplete Small-Signal ModelCircuits!Modern ICsA Simple Circuit: An MOS AmplifierSelecting the Output Bias PointFinding the Input Bias VoltageApplying the Small-Signal VoltageSolving for the Output Voltage vOSmall-Signal CaseLinearized Output VoltagePlot of Output Waveform (Gain!)There is a Better Way!Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15Lecture 15: Small Signal ModelingProf. NiknejadDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadLecture OutlineReview: Diffusion RevisitedBJT Small-Signal ModelCircuits!!!Small Signal ModelingExample: Simple MOS AmplifierDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadNotation ReviewSince we’re introducing a new (confusing) subject, let’s adopt some consistent notation Please point out any mistakes (that I will surely make!)Once you get a feel for small-signal analysis, we can drop the notation and things will be clear by context (yeah right! … good excuse)( , )C BE CEi f v v=Large signal( , )C C BE BE CE CEI i f V v V v+D = +D +Dsmall signalDC (bias)( , )C c BE be CE ceI i f V v V v+ = + +small signal(less messy!)c be ceBE CEQQf fi v vv v� �� +� �transconductanceOutput conductance( , )BE CEQ V V=Quiescent Point(bias)Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadDiffusion RevisitedWhy is minority current profile a linear function?Recall that the path through the Si crystal is a zig-zag series of acceleration and deceleration (due to collisions)Note that diffusion current density is controlled by width of region (base width for BJT):Decreasing width increases current!Density here fixed by potential (injection of carriers)Physical interpretation: How many electrons (holes) have enough energy to cross barrier? Boltzmann distribution givethis number.WpDensity fixed by metal contactHalf go left,half go rightDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadDiffusion CapacitanceThe total minority carrier charge for a one-sided junction is (area of triangle)For a one-sided junction, the current is dominated by these minority carriers:2 , 0 01 1( )( )2 2DqVkTn dep p p pQ qA bh qA W x n e n= � = � - -0 0,( )DqVnkTD p pp dep pqADI n e nW x= --( )2,nDnp dep pDIQW x=-Constant!Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadDiffusion Capacitance (cont)The proportionality constant has units of timeThe physical interpretation is that this is the transit time for the minority carriers to cross the p-type region. Since the capacitance is related to charge:( )2,p dep pnTD nW xQI Dt-= =n T DQ It=nd T d TQIC gV Vt t��= = =� �Diffusion CoefficientDistance acrossP-type base( )2,p dep pTnW xqkTtm-= MobilityTemperatureDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadBJT Transconductance gmThe transconductance is analogous to diode conductanceDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadTransconductance (cont)Forward-active large-signal current:/(1 )BE thv VC S CE Ai I e v V= +•Differentiating and evaluating at Q = (VBE, VCE )/(1 )BEqV kTCS CE ABEQiqI e V Vv kT�= +�C CmBEQi qIgv kT�= =�Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadBJT Base CurrentsUnlike MOSFET, there is a DC current into thebase terminal of a bipolar transistor:( )/(1 )BEqV kTB C F S F CE AI I I e V Vb b= = +To find the change in base current due to change in base-emitter voltage:1B B CmBE C BE FQ QQi i igv i v b� � �= =� � �Bb beBEQii vv�=�mb beFgi vb=Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadSmall Signal Current Gain0CFBiib bD= =DSince currents are linearly related, the derivative is a constant (small signal = large signal)0C Bi ibD = D0c bi ib=Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadInput Resistance rπ( )11C mBBE F BE FQ Qi girv vpb b-��= = =� �In practice, the DC current gain F and the small-signal current gain o are both highly variable (+/- 25%)Typical bias point: DC collector current = 100 AFmrgpb=25mV100 25k.1mArp= = WiR =�WMOSFETDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadOutput Resistance roWhy does current increase slightly with increasing vCE?Answer: Base width modulation (similar to CLM for MOS)Model: Math is a mess, so introduce the Early voltage)1(/ACEVvSCVveIithBEBase (p)Emitter (n+)Collector (n)BWDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadGraphical Interpretation of roslope~1/roslopeDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadBJT Small-Signal Modelb bei r vp=1c m be ceoi g v vr= +Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadBJT CapacitorsEmitter-base is a forward biased junction depletion capacitance:Collector-base is a reverse biased junction depletion capacitanceDue to minority charge injection into base, we have to account for the diffusion capacitance as well, , 01.4j BE j BEC C�b F mC gt=Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadBJT Cross SectionCore transistor is the vertical region under the emitter contactEverything else is “parasitic” or unwantedLateral BJT structure is also possibleCore TransistorExternal ParasiticDepartment of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadCore BJT ModelGiven an ideal BJT structure, we can model most of the action with the above circuitFor low frequencies, we can forget the capacitors Capacitors are
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