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)(,)CBE CEifv v=Large signal(,)C C BE BE CE CEIifV vV v+∆ = +∆ +∆small signalDC (bias)(,)CcBEbeCEceIifV vV v+= + +small signal(less messy!)cbe ceBE CEQQffivvvv∂∂≈+∂∂transconductanceOutput conductance(,)BE CEQVV=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:2,0 011()( )22DqVkTndeppppQqAbhqAWx ne n=⋅ =⋅ − −00,()DqVnkTDpppdeppqADInenWx=−−()2,nDnpdeppDIQWx=−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,pdeppnTDnWxQIDτ−==nTDQIτ=ndTdTQICgVVττ∂∂== =∂∂Diffusion CoefficientDistance acrossP-type base()2,pdeppTnWxqkTτµ−=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 thvVCS CEAiIe vV=+• Differentiating and evaluating at Q = (VBE, VCE )/(1 )BEqV kTCSCE ABEQiqIe V VvkT∂=+∂CCmBEQiqIgvkT∂==∂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 kTBCF SF CEAII I e VVββ== +To find the change in base current due to changein base-emitter voltage:1BBCmBE C BE FQQQiiigvivβ∂∂∂==∂∂∂BbbeBEQiivv∂=∂mbbeFgivβ=Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadSmall Signal Current Gain0CFBiiββ∆==∆Since currents are linearly related, the derivative is a constant (small signal = large signal)0CBiiβ∆= ∆0cbiiβ=Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadInput Resistance rπ()11CmBBE F BE FQQigirvvπββ−∂∂== =∂∂In practice, the DC current gain βFand the small-signal current gain βoare both highly variable (+/- 25%)Typical bias point: DC collector current = 100 µAFmrgπβ=25mV100 25k.1mArπ==ΩiR =∞ΩMOSFETDepartment 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(/ACEVvSCVveIithBE+=Base (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 Modelbbeirvπ=1cmbe ceoigv 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,,01.4jBE jBECC≈bFmCgτ=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 non-linear! MOS gate & overlap caps are linearmgvπBaseCollectorEmitterReverse biased junctionReverse biased junction &Diffusion CapacitanceFictional Resistance(no noise)Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadComplete Small-Signal ModelReverse biased junctions“core” BJTExternal ParasiticsReal Resistance(has noise)Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A. NiknejadCircuits!When the inventors of the bipolar transistor first got a working device, the first thing they did was to build an audio amplifier to prove that the transistor was actually working!Department of EECS University of California, BerkeleyEECS 105 Fall 2003, Lecture 15 Prof. A.
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