c Copyright 2010. W. Marshall Leach, Jr., Professor, Georgia Institute of Technology, School ofElectrical and Computer Engineering.The BJTNotationThe notations used here for voltages and currents correspond to the following conventions: Dc biasvalues are indicated by an upper case letter with upper case subscripts, e.g. VDS, IC. Instantaneousvalues of small-signal variables are indicated by a lower-case letter with lower-case subscripts, e.g.vs, ic. Total values are indicated by a lower-case letter with upper-case subscripts, e.g. vBE, iD.Circuit symbols for independent sources are circular and symbols for controlled sources have adiamond shape. Voltage sources have a ± sign within the symbol and current sources have anarrow.Device EquationsFigure 1 shows the circuit symbols for the npn and pnp BJTs. In the active mode, the collector-basejunction is reverse biased and the base-emitter junction is forward biased. For the npn device, theactive-mode collector and base currents are given byiC= ISexpvBEVTiB=iCβ(1)where VTis the thermal voltage, ISis the saturation current, and β is the base-to-collector currentgain. These are given byVT=kTq= 0.025 V for T = 290 K = 25.86 mV for T = 300 K (2)IS= IS01 +vCEVA(3)β = β01 +vCEVA(4)where VAis the Early voltage and IS0and β0, respectively, are the zero bias values of ISand β.Because IS/β = IS0/β0, it follows that iBis not a function of vCE. The equations apply to thepnp device if the subscripts BE and CE are reversed.The emitter-to-collector current gain α is defined as the ratio iC/iE. To solve for this, we canwriteiE= iB+ iC=1β+ 1iC=1 + ββiC(5)It follows thatα =iCiE=β1 + ββ =iCiB=α1 − α(6)Thus the currents are related by the equationsiC= βiB= αiE(7)1Figure 1: BJT circuit symbols.Transfer CharacteristicsThe transfer characteristics are a plot of the collector current iCas a function of the base-to-emittervoltage vBEwith the collector-to-emitter voltage vCEheld constant. From Eqs. 1 and 3, we canwriteiC= IS01 +vCEVAexpvBEVT(8)It follows that iCvaries exponentially with vBE. A plot of this variation is given in Fig. 2. Itcan be seen from the plot that the collector current is essentially zero until the base-to-emittervoltage reaches a threshold value. Above this value, the collector current increases rapidly. Thethreshold value is typically in the range of 0.5 to 0.6 V. For high current transistors, it is usuallysmaller. The plot shows a single curve. If vCEis increased, the current for a given vBEis larger.However, the displacement between the curves is so small that it can be difficult to distinguishbetween them. The small-signal transconductance gmdefined below is the slope of the transfercharacteristics curve evaluated at the quiescent or dc operating point.Figure 2: BJT transfer characteristics.2Output CharacteristicsThe output characteristics are a plot of the collector current iCas a function of the collector-to-emitter voltage vCEwith the base current iBheld constant. From Eqs. 1 and 4, we can writeiC= β01 +vCEVAiB(9)It follows that iCvaries linearly with vCE. A plot of this variation is given in Fig. 3. For smallvCEsuch that 0 ≤ vCE< vBE, Eq. (9) does not hold. This is the region on the left in Fig. 3. Inthis region, the BJT is saturated. The small-signal collector-to-emitter resistance r0defined belowis the reciprocal of the slope of the transfer characteristics curve evaluated at the quiescent or dcoperating point to the right of the saturation region in Fig. 3.Figure 3: BJT output characteristics.Hybrid-π ModelLet each current and voltage be written as the sum of a dc component and a small-signal accomponent as follows:iC= IC+ iciB= IB+ ib(10)vBE= VBE+ vbevCE= VCE+ vce(11)If the ac components are sufficiently small, we can writeic=∂IC∂VBEvbe+∂IC∂VCEvceib=∂IB∂VBEvbe(12)where the derivatives are evaluated at the dc bias values. The transconductance gm, the collector-to-emitter resistance r0, and the base-to-emitter resistance rπare defined as follows:gm=∂IC∂VBE=ISVTexpVBEVT=ICVT(13)3r0=∂IC∂VCE−1= IS0VAexpVBEVT−1=VA+ VCEIC(14)rπ=∂IB∂VBE−1= IS0β0VTexpVBEVT−1=VTIB(15)It is convenient to define the current icas follows:ic= gmvπwhere vπ= vbe(16)It follows that the collector and base currents can thus be writtenic= ic+vcer0ib=vπrπ(17)The small-signal circuit which models these equations is given in Fig. 4(a). This is called thehybrid-π model. The resistor rxin series with the base is called the base spreading resistance. Thisresistor arises from the resistance of the base connection. There is no equation for it for it must bemeasured. It is often neglected in small-signal analyses.Figure 4: Rev(a) Hybrid-π model. (b) T model.The small-signal base-to-collector ac current gain β is defined as the ratio ic/ib. It is given byβ =icib=gmvπib= gmrπ=ICVT×VTIB=ICIB(18)Note that icdiffers from icby the current through r0. Therefore, ic/ib= β unless r0= ∞.T ModelThe T model replaces the resistor rπin series with the base with a resistor rein series with theemitter. This resistor is called the emitter intrinsic resistance. The current iecan be writtenie= ib+ ic=1β+ 1ic=1 + ββic=icα(19)where α is the small-signal emitter-to-collector ac current gain given byα =β1 + β(20)4Thus the current iccan be writtenic= αie(21)The voltage vπcan be related to ieas follows:vπ= ibrπ=icβrπ=αieβrπ= ieαrπβ= ierπ1 + β= iere(22)The above equation defines the intrinsic emitter resistance regiven byre=vπie=rπ1 + β=VT(1 + β) IB=VTIE(23)The T model of the BJT is shown in Fig. 4(b). The currents in both models are related by theequationsic= gmvπ= βib= αie(24)The Collector Equivalent CircuitIf the BJT output is taken from the collector, the input can be either applied to the base or to theemitter. If it is applied to the base, the circuit is called a common-emitter amplifier. If it is appliedto the emitter, the circuit is called a common-base amplifier. In some cases, separate inputs canbe applied to both the base and the emitter. In any of these cases, the collector output can besolved for by first making a small-signal Thévenin or Norton equivalent circuit seen looking intothe collector. We solve for the Norton equivalent circuit here. We assume that the circuits externalto the base and the emitter can be