EE306-BJT Transport Model

1Bipolar Junction Transistor Transport ModelDavid Braun, Senior Member, IEEE TABLE ISUPPLEMENT OUTLINEThis supplement presents the bipolar junction transistorTransport model, explains how it predicts DC BJT behavior, andshows how to simplify the model for four common modes of BJToperation: forward active, saturation, reverse active, and cut-off.I. INTRODUCTION AND LEARNING OBJECTIVESII. ASSEMBLING THE TRANSPORT MODELA. NPN BJTB. PNP BJTIII. SIMPLIFYING THE TRANSPORT MODELA. Forward Active ModeB. Reverse Active ModeC. Cut-off ModeD. Saturation ModeE. Edge of Conduction and Edge of SaturationAPPENDIX—THE EBERS-MOLL MODELREFERENCESIndex Terms—Bipolar transistors, bipolar integrated circuits,integrated circuit modeling, semiconductor device modeling,semiconductor devices.I. INTRODUCTION AND LEARNING OBJECTIVESTHE Ebers-Moll[1] and Gummel-Poon[2] models predictdetailed large- and small-signal BJT behavior to supporthand analysis or computer simulations. The Transport modeloffers a slightly simpler approach while preserving intuitionand accuracy. The Transport model circuit topology derivesdeductively from the Ebers-Moll model via circuit theory.Alternately, superposing forward active and reverse activemode equivalent circuits permits assembling the Transportmodel equivalent circuit inductively [3]. The latter strategy,conveniently proceeds from the EE 306 text treatment [4].This supplement informs the following learning objectives:(a) an ability to draw the Transport model equivalent circuit(b) an ability to use the Transport model to predict BJTterminal voltages and currents(c) an ability to draw simplified Transport model equivalentcircuits for the forward active, saturation, reverse active, andcut-off operation modes.Table I outlines the supplement organization. Section IIpresents the Transport model equivalent circuits and circuitequations for the NPN and PNP model versions. Section IIIsimplifies the Transport model for the four BJT operationmodes. An appendix describes the Ebers-Moll model. [exp exp] [exp 1] (1C) [exp exp] [exp 1] [exp 1] [exp 1](1E)(1B)II. ASSEMBLING THE TRANSPORT MODELA. NPN BJTWe begin with an overview. Equations 1 relate the BJTterminal currents (iC, iE, and iB) to the Base-Emitter (VBE) andBase-Collector (VBC) voltage drops at a given thermal voltage(VT). The saturation current (IS), the forward current gain (βF),and the reverse current gain (βR) describe a given transistor.Figure 1 contains the Transport model equivalent circuit for anNPN-BJT. Equations 2 define the transport current (iT),forward current (iF), and reverse current (iR).Manuscript prepared August 28, 2015. The author is with CaliforniaPolytechnic State University, San Luis Obispo, CA 93407 USA (e-mail:dbraun@calpoly.edu). 2015Fig. 1. NPN–BJT Transport model equivalent circuit juxtaposed on theNPN-BJT cross section. Per convention, positive base current (iB) andcollector current (iC) flow into the device. Positive emitter current (iE) leavesthe device.

2 [exp 1](2F) [exp 1](2R)(2T)Applying Kirchhoff’s Current Law at the Collector, Emitter,and Base terminals permits translation between Equations 1and 2. The NPN transistor formulation assumes positivecurrents flow into the base and collector terminals, whilepositive current flows out the emitter terminal. See Figure 2A.B. PNP BJTThe PNP transistor formulation assumes positive currentsopposite to those in the NPN BJT. Positive current flows inthe emitter terminal, while positive currents flow out the baseand collector terminals. Figure 2B shows the currentconventions.Equations 3 relate the BJT terminal currents (iE, iC, and iB)to the Emitter-Base (VEB) and Collector-Base (VCB) voltagedrops at a given thermal voltage (VT). The saturation current(IS), the forward current gain (βF), and the reverse current gain(βR) describe a given transistor. Figure 3 contains theTransport model equivalent circuit for a PNP-BJT. Equations4 define the transport current (iT), forward current (iF), andreverse current (iR). [exp exp] [exp 1] (3C) [exp exp] [exp 1] 1] [exp 1](3E) [exp [exp 1](4F) [exp 1](4R)(3B)(4T)Fig. 3. PNP–BJT Transport model equivalent circuit juxtaposed on thePNP-BJT cross section. Per convention, positive emitter current (iE) flowsinto the device. Positive base current (iB) and collector current (iC) leave thedevice.III. SIMPLIFYING THE TRANSPORT MODELFor general BJT operation, applying Equations 1 or 3permits calculating unknown BJT terminal voltages orcurrents. Many practical circuit applications bias BJTs so thatthe equations simplify into one of the four modes of operationshown in Table II. This section covers how to simplify theTransport model for these four operation modes. The strategytakes advantage of the relative magnitudes of the exponentialterms in Equations 1. Following the convention that a Siliconpn junction “turns on” when its voltage drop reaches 0.5 V to0.9 V means at room temperature that the exponential termexp(VBE/VT) or exp(VBC/VT) exceeds 1 by somewhere between8 and 15 orders of magnitude. Such a great difference inmagnitudes justifies ignoring small or negligible terms.Simplifying the Transport model equations leads tocorrespondingly simplified equivalentcircuits anddramatically simplified circuit analysis.This section only present results for the NPN BJT, since thePNP version proceeds so similarly.TABLE IIISIMPLIFIED BJT OPERATION MODESA) NPN BJTB) PNP BJTFig. 2. BJT circuit symbols showing positive current and voltage namingconventions. For the NPN BJT, positive base current (iB) and collectorcurrent (iC) flow into the device, while positive emitter current (iE) leavesthe device. For the PNP BJT, positive emitter current (iE) flows into thedevice, while positive base current (iB) and collector current (iC) leave thedevice. Note the voltage naming conventions use appropriate subscripts toindicate the voltage polarities. NPN and PNP devices use reversedsubscripts.BC Junction OffBC Junction OnCut-offReverse ActiveIC IE IB 0IE – R IBForward ActiveSaturatedIC F IBIC F IBBE Junction OffBE Junction On

3A. Forward Active ModeIn Forward Active Mode (FA), the Base-emitter junction(BE junction) turns on, and the Base-collector junction (BCjunction) turns off. The large BE junction exponential termsgreatly exceed the negligible BC junction terms and –1 terms,so Equations 1 simplify as follows for FA: [exp exp [exp] [exp exp [exp](] [exp 1](5C)] [exp 1]) [exp 1] [exp](5E)[exp 1](5B)Equations 5 support the equivalent circuit simplificationshown in Figure 4. As required, Kirchhoff’s Current Lawapplies in FA at the emitter terminal: ( 1)(6)B. Reverse Active ModeRA operation occurs less frequently in practice than theother operation modes, primarily due to the low reversecurrent gain (βR) compared to forward current gain (βF). βRusually has a value less than 10 and often less than 1, whereasβF usually exceeds 20 and often exceeds 100.In Reverse Active Mode (RA), the Base-emitter junction(BE junction) turns off, and the Base-collector junction (BCjunction) turns on. The large BC junction exponential termsgreatly exceed the negligible BE junction terms and –1 terms,so Equations 1 simplify into Equations 7 as follows: [exp exp [exp [expFig. 4. Simplified Transport model for Forward Active mode. With the BEjunction on and the BC junction off, the reverse current term, iR, becomesnegligible, and the Transport model on the left simplifies to the equivalentcircuit on the right. The equivalent circuit on the right corresponds toEquations 5.]( exp [exp] [exp 1] [exp]] [exppositive emitter current flowing into the emitter terminal andpositive collector current flowing out the collector terminal.This convention, shown in Figure 5, does not agree with theconvention used for forward active mode shown in Figure 4.Rewriting Equations 7 using the new convention yieldsEquations 8 for RA. [exp]( [exp] [exp])(8C)(8E)(8B)Equations 8 support the equivalent circuit simplificationshown in Figure 5. As required, Kirchhoff’s Current Law 1])(7C)] [exp 1](7E)[exp 1](7B)Because the negative signs on the collector and emitterterminals can make circuit analysis more confusing thannecessary, it often proves more convenient to redefine thedirections of positive collector current (iC) and positive emittercurrent (iE). Therefore, for reverse active mode, we defineFig. 5. Simplified Transport model for Reverse Active mode. With the BCjunction on and the BE junction off, the forward current term, iF, becomesnegligible, and the Transport model on the left simplifies to the equivalentcircuit on the right. The equivalent circuit on the right corresponds toEquations 8. It uses a convention for positive collector and emitter currentopposite to that used for the other three BJT modes.

4applies in RA at the collector terminal: ( 1)(9)C. Cut-off ModeIn Cut-off Mode (CO), both junctions turn off. With bothjunctions reversed biased, the small exponential terms becomenegligible compared to the –1 terms, so Equations 1 simplifyas follows for CO, leaving only leakage currents: [exp exp] [exp 1] (10C)[exp exp] [exp 1] (10E)[exp 1] [expFig. 6. Simplified Transport model for Cut-off mode. With both junctionsoff, the transport current term, iT, becomes negligible, leaving the Transportmodel on the left. Ignoring leakage currents produces the equivalent circuiton the right. The open circuit equivalent circuit on the right corresponds toEquation 11. 1]Applying Kirchhoff’s Voltage Law around the BJT gives (10B)(Equations 10 support the equivalent circuit simplificationshown in the left pane of Figure 6. Typically, handcalculations ignore the negligible leakage currents and use thefollowing approximation: 0(11)In CO, the BJT behaves as an open circuit. Figure 6 showsthe Equation 11 open circuit approximation in its right pane.D. Saturation ModeIn Saturation Mode (SAT), both junctions turn on, so the –1terms in Equations 1 become negligible compared to theexponential terms. Equations 1 simplify as follows for SAT: [exp exp] [exp](12C) [exp exp] [exp](12E) [exp] [exp] )() ().(15)Inserting Equations 13 and 14 into Equation 15 produces( )ln().(16)The preceding formulations use the common-emitter currentgain, β. Equivalent versions based on the common-basecurrent gain, α, also exist. To translate from one to the other,use (17)or .(18)(12B)In Saturation Mode, , so hand analysis proceedsmore conveniently by considering terminal voltages.Combining Equations 12C and 12B produces() ln) ln()(13)and(.(14)Fig. 7. Simplified Transport model for Saturation mode. With both junctionson, the Transport model on the left simplifies to the equivalent circuit on theright. The equivalent circuit on the right corresponds to Equations 14 – 16.