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Lecture 2: Introduction to electronic analog circuits 361-1-3661 1 2. Elementary Electronic Circuits with a BJT Transistor © Eugene Paperno, 2008 Our main aim in the two next lectures is to build all the possible practical circuits (amplifiers) by using a BJT transistor and resistor. (We use the resistor to translate the output current of the circuit into voltage; otherwise the circuit will not be able to provide a voltage gain.) We then analyze and compare the circuits' small-signals gains to understand for what applications they can be suitable. We are particularly interested in the applications where there is a need to amplify power and dc signals. In this lecture, we develop all the models for the transistors − as we did this for the diode − and then will build and analyze − with the help of these small-signal models − all the possible single-transistor amplifiers. C p n B B p n n E p E Injection Extraction W W' n++ p+ n * VBE / VT n Bo e E * iE iB 2.1. BJT transistor: symbol, physical structure, analytical model, and graphical characteristics The symbols of the npn and pnp BJT transistors and the physical structure of the npn transistor are given in Fig. 1. We will analyze in the lectures only npn transistors. The only difference between the npn and pnp transistors is in their static states: the static state of the pnp transistors is reverse to that of the npn ones because of their opposite structures. There will be no difference in the small-signal behavior and models. The circuits analyzed in home exercises, the lab, and the exam will comprise both npn and pnp transistors. In analog circuits, the operating point of transistors is usually defined in active (linear) region, where the emitter junction is forward biased and the collector junction is reverse biased. Thus, the emitter injects the electrons into the base, and the collector collects them. The amount of the injected electrons is controlled by the emitter-base voltage, vBE (or base-to-emitter current, iB). The collector collects almost all the electrons from the base if its potential is sufficiently high: is greater or equal to that of the base. The base is very thin and the electrons prefer entering the collector − even its potential equals that of the base − and not the base, because the resistance that they see looking into the base is much greater than that they see looking into the collector. To define the operating point of the transistor in active region, we ground the emitter and bias the transistor junctions with a current and voltage source as shown in Fig. 1. A single transistor circuit (with no other components, except independent sources) with grounded emitter is called the common-emitter configuration. Although we develop all the models of the transistor for the common-emitter C iC iR pCo nBo pEo B' B vCE VCE >VBE + 180 mV rB iB C e180/26 = 1015.44 V'CE >VCE VCE=VBE I'B= 0.5 IB; I'C= 0.5 IC * VCE = 0 Fig. 1. Symbol of the n-p-n and p-n-p BJT transistors and the physical structure of the npn transistor. Note that for a fixed iB, v'BE is also fixed. configuration, they can also be used (see the Appendix) for any transistor in a circuit, no matter which terminal of the transistor is grounded (if at all). Analytical model: transistor equations Let us first write the equations for the transistor current based on the concentrations of the minor charge carriers in Fig. 1: iC Dn q n Bo e v ' BE / VT ABE iR w iC i R . n Dn q Bo ABE e v ' BE / VT I CS e v ' BE / VT w I CS (1) Lecture 2: Introduction to electronic analog circuits 361-1-3661 iC VCE IC 2 iC Q iC VCE IC Q IB IC Q hoe gm hfe 0.5IB vCE 0.5 IC 0.5 hoe vCE I"B I'B IB iB VBE iB 1/ro vBE VA VCE 0.7 V I"B V'CE vCE vBE VCE 1/hie Q Q IB VBE IB hre vCE VBE iE vBE VCE V'CE vCE VCE iC Q IE iB 1/re vCE C B vCE iB vBE VBE iE E vBE Fig. 2. Common-emitter characteristics of an npn transistor. iB D p q ABE p Eo L pE i BS n Dn q Bo ABE e v ' BE / VT iC w F D p q ABE p Eo v ' BE / VT iB e L pE ( e v ' BE / VT 1) i B i BS . (4) (2) I BS e v ' BE / VT iE iC iB I CS e v ' BE / VT I BS e v ' BE / VT D p p Eo w (3) F I ES We now can define the static current gains 1 n Bo p Eo ; L p E W and . ( I CS I BS ) e v ' BE / VT I ES e v ' BE / VT Dn n Bo L pE iC iC i /i C B i E i B iC 1 iC / i B . F 1 F 1 F 1 (5) Lecture 2: Introduction to electronic analog circuits 361-1-3661 3 Note that according to (4), the transistor iC-iB characteristic should be a linear one (see Fig. 2), of course, provided that F is constant (in a real transistor, F depends on iB, but we will neglect this in our theory). It is also apparent from (1)-(3) that the iC-vBE, iB-vBE, and iE-vBE characteristics are exponential. Since according to (1), the collector current is a function of the base width, w, and w decreases with increasing vCE, the transistor output characteristics have a slope that is proportional to IC. (This is unlike the Ebers-Moll model, where the transistor output characteristics are horizontal.) Indeed, iC w Q n Dn q Bo ABE eV ' BE / VT w w Small-signal parameters Having all the needed transistor characteristics, we can define the small-signal gains as the slopes of the characteristic at their operating points. The small-signal current gains h fe f Q , v ce 0 F , (7) Q , vce 0 ic ib ic Q , v ce 0 ic / ib 1 ic / ib Q , v ce 0 . Q 1 n 1 Dn q Bo ABE eV ' BE / VT I C W W W iC ic ie ic ib w IC . W h fe 1 h fe F F 1 F The small-signal conductance and resistance of the emitter i 1 e re v ' be (6) Q , vce 0 I ES eV ' BE / VT I E VT VT . For w vCE : W VCE iC (8) vCE I C iC I C . VCE Due to the linear dependence of the slope of the output characteristics on IC, their extrapolations meet at the one and the same point on the vCE axis, so-called Early voltage, VA. When vCE increases, the base width w decreases, and the base resistance, rB, increases. Therefore, the static VBE voltage should increase for the same static bias current IB (see the iBvBE and vBE-vCE characteristics in Fig. 2). As a result, the iB-vBE, characteristic decreases a bit with increasing VCE. Since decreasing w causes much more substantial increase in iC and iE than in vBE, the iC-vBE and iE-vBE characteristics increase with increasing vCE. The effect associated with the change (modulation) of the base width by the collector voltage, vCE, and with the corresponding behavior of the transistor characteristics is called Early effect. re VT IE (9) 26 300 K, I E 1 mA The small-signal (mutual) conductance gain gm ic v ' be Q , vce 0 I CS eV 'BE /VT I C, VT VT . gm ic v ' be Q , vce 0 f ie v ' be Q , vce 0 (10) f re The small-signal input conductance and resistance 1 i b hie v 'be Q , v ce 0 ie (1 h fe )v 'be hie (1 h fe )re ic / h fe v 'be Q , v ce 0 Q , v ce 0 f ie / h fe v 'be Q , v ce 0 1 ; (1 h fe ) re 300 K, I E 1 mA, h fe 100 . (11) 2.6 k The small-signal output conductance and resistance ("r-out", not "r-zero") Lecture 2: Introduction to electronic analog circuits 361-1-3661 4 IC i'c C hfeib gmv'be ic IB IC i"c B IB rb B' i"C VCE ib ib re hrevce ro hrevce IB E i"C i'C iC hfeib gmv'be i"C ib E VBE+ v'be IE B Fig. 3. "Large"-signal equivalent circuit (model) for the transistor. Note that another VCE source is added to cancel the effect of the static collector-toemitter voltage, VCE, on the current through ro. Thus, only the small-signal collector-to-emitter voltage, vce, generates the small-signal current through ro, which is in accordance with the Early effect. Note also that alternating the polarity of the vs source causes the corresponding alternating the polarity of the hfeib source. 1 i c roe vce Q , ib 0 VA VCE IC rb B' ib C B (1+hfe) re v'BE hrevce IC VA VCE 1+hfe hfe hrevce vce re ro (b) E rb . (12) ro (a) vce i'c IB hoe ro re V ib hfeib gmv'be C B' v'BE IC ib iC vce VCE IB B rb i'C 100 k i'C B' ib v'BE C i"C hie I C 1 mA, V A 100 V VCE iC hfeib gmv'be hrevce vce ro (c) And finally, the small-signal reverse-voltage gain hre vbe vce . (13) E Q , ib 0 i'C B iC C "Large"-signal model for the transistor To develop a "large"-signal model (see Fig. 3) for the transistor, we first replace the base-emitter diode with the "large"-signal model of the diode, add the IB dependent source (this completes the static signal translation), and then add the hfeib, or what is the same gm v'be dependent source to represent the effect of v'be on ic, add ro together with an additional independent voltage source VCE to represent the effect of vce on ic, and finally add the hrevce source to represent the effect of vce on v'be. Note that we add another VCE source to cancel the effect of the static collector-to-emitter voltage, VCE, on the current through ro. Only the small-signal collector-toemitter voltage, vce, should generate the small-signal current through ro, which is in accordance with the Early effect. ib i"C vBE hfeib gmvbe hie ro vce (d) E Fig. 4. Small-signal equivalent circuits (models) for the transistor. (a) T smallsignal model of the BJT transistor, (b) separating the input and output loops of the T model by applying the Miller theorem, (c) hybrid- small-signal model, (d) simplified hybrid- model with the hrevce source and rb neglected. Lecture 2: Introduction to electronic analog circuits 361-1-3661 5 i ki (1+k) Z vin Electronic circuit V 1+k k i ki Z Z vo V vin vo V vCE Fig. A2. Miller's theorem (for voltages). iB Fig. A1. Transistor in an arbitrary electronic circuit connected to equivalent signal sources. According to the substitution theorem, a branch of the network that is not coupled to other branches can be replaced by an equivalent independent current or voltage source without affecting any other branch current or branch voltage. To apply the substitution theorem, the network has to have a unique solution for all its branch currents and branch voltages. The network does not have to be linear. source. We can omit these two components because they do not affect the hfeib source and, therefore, do not affect the model output voltage and current: vce and ic. Neglecting the hrevce source (the typical value of hre is very small, about 10-3), we obtained in Fig. 4(d) a simplified hybrid- model. We will use this model in all our further analysis. Either the T or models can be used in a small-signal analysis to replace a transistor in an electronic circuit. Naturally, all the small-signal parameters of the models should be found in advance as a function of the transistor operating point. APPENDIX Small-signal model for the transistor Note that the circuit in Fig. 3 is a linear one. Hence, to obtain a small-signal model for the transistor [see Fig. 4(a)], we simply suppress all the static sources in Fig. 3. The circuit in Fig 4(a) is called the T small-signal model of the BJT transistor. The T model can be simplified by separating its input and output loops [see Fig. 4(b)] by applying the Miller theorem for voltages (see the Appendix). Such a separation provides us with so-called hybrid- small-signal model shown in Fig 4(c). Note that in Fig. 4(b) we short-circuited the resistor and the voltage source that are connected in series with the hfeib Fig A1 illustrates that the effect of the electronic circuit on a transistor can be modeled with two independent sources. Fig. A2 illustrates the Miller theorem for voltages: the input and output loops of a T network can be separated without changing the states of the network ports if the values of the impedances Zin and Zo are increased to compensate for the reduction of the currents through them relative to the current in the impedance Z of the original T network. REFERENCES [1] [2] J. Millman and C. C. Halkias, Integrated electronics, McGraw-Hill. A. S. Sedra and K. C.Smith, Microelectronic circuits.