6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonRLThere are times when the most important metric of an amplifier is its low-frequency gain. Based onthe model we have given you so far:We would have you believe that the achievable gain for a single stage is unbounded. That is, for agiven gain, we just choose RLto be as high as necessary. As you might suspect, this is only possible upto a point.CLASS EXERCISEConsider the common emitter amplifier:Assume that with the bias voltage VB, we get perfect control over the collector current.(1) Suppose that the lowest allowable quiescent voltage at V0 is zero (ground). Express themaximum ICwe can have in terms of VCCand RL.(2) What is the maximum gain we can get out of this stage? What limits us?gmVπ↓rπVπ−+−+gm=ICVT, av= gmRL↓ICV0RLVCCVB−+VS6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonPage 2(Workspace)It is not uncommon, though, for an op-amp to achieve a gain of 106 in only two stages. If we allot 103of gain for each stage, that implies a power supply ≥ 1000 ⋅ VT= 25V. Since we routinely buy op-amps that operate on much smaller power supplies, it is clear that some other tricks are being played.One such trick looks like this:We need a more complete model of the transistor to understand this circuit, though. Let’s jump rightin…We return to device physics. Specifically, the excess minority carrier change in the base.The width of the base is decreased asBC junction is reverse biased.−+VIIC↓VCWBWB'0n 0( )CBE6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonPage 3When the width of the base decreases, the slope of the carrier distribution in the base increases. Thismeans the diffusion current must also increase. Recall,IC=n(0)WB× Dn× AE× q=ni2NBeqVBEkTWB× Dn× AE× qNow what we’re interested in is ∂IC∂VCE. We don’t see an explicit dependence on VCE, but we knowthat WBdepends on VCE…∂IC∂VCE= −ni2NBeqVBEkT⋅ −1WB2⎛⎝⎜⎞⎠⎟DnAEq ⋅∂WB∂VCERecognizing ni2NBeqVBEkTDnAEq = WBIC, we can simplify:∂IC∂VCE= −1WB∂WB∂VCEICNow we’ve gotten as far as we can without getting into more detailed device considerations. It turnsout that ∂WB∂VCE is well approximated by a constant. From a unit standpoint, we observe that1WB∂WB∂VCE must have units of V−1.We thus write∂IC∂VCE= kIC=ICVAslopeDiffusionconstantEmitterAreaElementaryCharge6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonPage 4Where we call VAthe “early voltage.”What does all of this mean? For one, it means that we will modify our large signal model of thebipolar transistor in the following way:IC= ISeqVBEkT1 +VCEVA⎛⎝⎜⎞⎠⎟Check :∂IC∂VCE=1VAISeqVBEkT=ICVA⎡⎣⎢⎢⎤⎦⎥⎥Second, it means that we have a new small-signal model:Wherer0=∂IC∂VCE⎛⎝⎜⎞⎠⎟−1=VAICTypical values of early voltage are 25 to 250 volts. For a collector current of 1mA, this gives andbetween 25kΩ and 250kΩ.So how does this behavior show up in the laboratory? Most obviously, it shows up on an instrumentcalled a curve tracer.Cr0EgmVπ↓rπVπ−+B6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonPage 5A curve tracer automates the graphing of ICvs VCEparameterized by different values of VBE.If you extrapolate according to the dotted lines back to where IC= 0, all of the lines meet at −VA.We can see this mathematically:ICVCE= −VA= ISeqVBEkT1 −VAVA⎛⎝⎜⎞⎠⎟= 0Looking at the graph, we can see that higher early voltages imply “flatter” ICvs. VCE curves. And theflatter the curve, the higher r0.There’s one more effect to consider:−VAVCEVBE1VBE 2VBE 3ICWBWB'n(0)CBE6.301 Solid State CircuitsRecitation 12: Base-Width ModulationProf. Joel L. DawsonPage 6Notice that when we increase VCEand therefore decrease WB, we decrease the amount of charge inthe base that is available for recombination. This means that the base current decreases, and we cancapture this behavior with an added resistance between the base and collector:It turns out that rµis typically on the order of βr0, and usually greater than 10βr0. We will almostalways ignore this