Small-Signal Midfrequency JT Amplifiers 6.. INTRODUTION For sufficiently small emitter-collector voltage and current excursions about the quiescent point (small signals), the JT is considered linear; it may then be replaced with any of several two-port networks of impedances and controlled sources (called small-signal equivalent-circuit models), to which standard network analysis methods are applicable. Moreover, there is a range of signal frequencies which are large enough so that coupling or bypass capacitors (see Section 3.7) can be considered short circuits, yet low enough so that inherent capacitive reactances associated with JTs can be considered open circuits. In this chapter, all JT voltage and current signals are assumed to be in this midfrequency range. In practice, the design of small-signal amplifiers is divided into two parts: () setting the dc bias or Q point (hapters 3 and 5), and (2) determining voltage- or current-gain ratios and impedance values at signal frequencies. 6.2. HYRID-PARAMTR MODLS General hybrid-parameter analysis of two-port networks was introduced in Section.7. Actually, different sets of h parameters are defined, depending on which element of the transistor (,, or) shares a common point with the amplifier input and output terminals. ommon-mitter Transistor onnection From Fig. 3-3(b) and(c), we see that if i and v are taken as dependent variables in the transistor configuration, then v ¼ f ði ; v ð6: i ¼ f 2 ði ; v ð6:2 63 opyright 2002, 988 by The McGraw-Hill ompanies, Inc. lick Here for Terms of Use.
64 SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS [HAP. 6 If the total emitter-to-base voltage v goes through only small excursions (ac signals) about the Q point, then v ¼ v be ; i ¼ i c,and so on. Therefore, after applying the chain rule to (6.) and (6.2), we have, respectively, v be ¼ v dv ¼ @v @i i b þ @v Q @v v ce ð6:3 Q i c ¼ i di ¼ @i @i i b þ @i Q @v v ce ð6:4 Q The four partial derivatives, evaluated at the Q point, that occur in (6.3) and (6.4) are called hybrid parameters and are denoted as follows: Input resistance h ie @v @i v Q i ð6:5 Q Reverse voltage ratio h re @v @v v Q v ð6:6 Q Forward current gain h fe @i @i i Q i ð6:7 Q Output admittance h oe @i @i ð6:8 Q @v v Q The equivalent circuit for (6.3) and (6.4) isshown in Fig. 6-(a). The circuit is valid for use with signals whose excursion about the Q point is sufficiently small so that the h parameters may be treated as constants. h ie i b i c L be h re L ce h fe i b h oe (S) L ce (a) small-signal equivalent circuit h ib i e L eb h rb L cb h fb i e (S) (b) small-signal equivalent circuit Fig. 6- i c L cb ommon-ase Transistor onnection If v and i are taken as the dependent variables for the transistor characteristics of Fig. 3-2(b) and (c), then, as in the case, equations can be found specifically for small excursions about the Q point. The results are v eb ¼ h ib i e þ h rb v cb ð6:9 i c ¼ h fb i e þ v cb ð6:0
HAP. 6] SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS 65 The partial-derivative definitions of the h-parameters are: Input resistance h ib @v @i v Q i Reverse voltage ratio h rb @v @v v Q Forward current gain h fb @i @i i Q i Q Output admittance @i i Q @v Q v Q v Q ð6: ð6:2 ð6:3 ð6:4 A small-signal, h-parameter equivalent circuit satisfying (6.9) and(6.0) isshown in Fig. 6-(b) ommon-ollector Amplifier The common-collector () or emitter-follower (F) amplifier, with the universal bias circuitry of Fig. 6-2(a), can be modeled for small-signal ac analysis by replacing the -connected transistor with its h-parameter model, Fig. 6-(a). Assuming, for simplicity, that h re ¼ h oe ¼ 0, we obtain the equivalent circuit of Fig. 6-2(b). V i b a h ie b i e R 2 i c i i L R R 2 R R 2 h fe i b R L e R R L a b (a) (b) i b h ie R R 2 R R 2 (h fe )R L e (c) Fig. 6-2 amplifier An even simpler model can be obtained by finding a The venin equivalent for the circuit to the right of a; a in Fig. 6-2(b). Application of KVL around the outer loop gives v ¼ i b h ie þ i e R þ i b h ie þðh fe þ i b R The The venin impedance is the driving-point impedance: ð6:5 R Th ¼ v i b ¼ h ie þðh fe þ R ð6:6
66 SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS [HAP. 6 The The venin voltage is zero (computed with terminals a; a open); thus, the equivalent circuit consists only of R Th. This is shown, in a base-current frame of reference, in Fig. 6-2(c). (See Problem 6.3 for a development of the h-parameter model.) 6.3. T-QUIVALNT IRUIT The tee-equivalent circuit or r-parameter model is a circuit realization based on the z parameters of hapter. Applying the z-parameter definitions of (.0) to(.3) tothe small-signal equivalent circuit of Fig. 6-(b) leads to (See Problem 6.7.) If we now define z ¼ h ib h rbh fb z 2 ¼ h rb z 2 ¼ h fb z 22 ¼ Substitution of these z parameters into (.8) and (.9) yields v eb ¼ h ib h rbh fb i h e þ h rb ð i ob h c ob v cb ¼ h fb i h e þ ð i ob h c ob ð6:7 ð6:8 ð6:9 ð6:20 ð6:2 ð6:22 r b ¼ h rb r e ¼ h ib h rb ð þ h fb r c ¼ h rb ð6:23 ð6:24 ð6:25 0 ¼ h fb þ h rb h rb ð6:26 then (6.2) and (6.22) can be written v eb ¼ðr e þ r b i e r b i c ð6:27 and v cb ¼ð 0 r c þ r b i e ðr b þ r c i c (6.28) Typically, 0:9 > h fb > and 0 h rb. Letting h rb 0in(6.26), comparing (6.3) with (3.) while neglecting thermally generated leakage currents, and assuming that h F ¼ h fb (which is a valid assumption except near the boundary of active-region operation) result in 0 h fb ¼ Then the tee-equivalent circuit or r-parameter model for operation is that shown in Fig. 6-3. Problems 6.3 and 6.5 for r-parameter models for the and configurations, respectively.) ð6:29 (See
HAP. 6] SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS 67 =i e i e r e r c i c L eb r b L cb Fig. 6-3 6.4. ONVRSION OF PARAMTRS Transistor manufacturers typically specify h F ð h fe and a set of input characteristics and collector characteristics for either or connection. Thus the necessity arises for conversion of h parameters among the,, and configurations or for calculation of r parameters from h parameters. Formulas can be developed to allow ready conversion from a known parameter set to a desired parameter set. xample 6.. Apply KVL and KL to Fig. 6-(a) toobtain v eb ¼ g ði e ; v cb and i c ¼ g 2 ði e ; v eb. ompare these equations with (6.9) and (6.0) tofind the h parameters in terms of the h parameters. Use the typically reasonable approximations h re and h fe þ h ie h oe to simplify the computations and results. KVL around the ; loop of Fig. 6-(a) (with assumed current directions reversed) yields ut KL at node requires that v eb ¼ h ie i b h re v ce i b ¼ i e i c ¼ i e h fe i b h oe v ce or i b ¼ h fe þ i e þ h oe h fe þ v ce (6.3) In addition, KVL requires that v ce ¼ v cb v eb Substituting (6.3) and (6.32) into(6.30) and rearranging give ð h re ðh fe þ þh ie h oe v h fe þ eb ¼ h ie h fe þ i e þ h ieh oe h fe þ h re v cb ð6:30 ð6:32 ð6:33 Use of the given approximations reduces the coefficient of v eb in (6.33) tounity, so that v eb h ie h fe þ i e þ h ieh oe h fe þ h re v cb Now KL at node of Fig. 6-(a) (again with assumed current directions reversed) yields i c ¼ h fe i b þ h oe v ce Substituting (6.3), (6.32), and (6.34) into(6.35) and solving for i c give " i c ¼ h # " # fe h fe þ þ h oeh ie h ðh fe þ 2 i e h ie h oe oe ðh fe þ 2 h re þ h fe þ v cb ð6:34 ð6:35 ð6:36 Use of the given approximations then leads to h fe h oe i c h fe þ i e þ h fe þ v cb ð6:37
68 SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS [HAP. 6 omparing (6.34) with (6.9) and (6.37) with (6.0), we see that h ie h ib ¼ h fe þ h rb ¼ h ieh oe h fe þ h re h fe h fb ¼ h fe þ h oe ¼ h fe þ ð6:38 ð6:39 ð6:40 ð6:4 6.5. MASURS OF AMPLIFIR GOODNSS Amplifiers are usually designed to emphasize one or more of the following interrelated performance characteristics, whose quantitative measures of goodness are defined in terms of the quantities of Fig. 6-4:. urrent amplification, measured by the current-gain ratio A i ¼ i o =i in. 2. Voltage amplification, measured by the voltage-gain ratio A v ¼ v o =v in. 3. Power amplification, measured by the ratio A p ¼ A v A i ¼ v o i o =i o i in. 4. Phase shift of signals, measured by the phase angle of the frequency-domain ratio A v ð j! or A i ð j!). 5. Impedance match or change, measured by the input impedance Z in (the driving-point impedance looking into the input port). 6. Power transfer ability, measured by the output impedance Z o (the driving-point impedance looking into the output port with the load removed). If Z o ¼ Z L, the maximum power transfer occurs. Z s i in i o L s L in Amplifying circuit L o Z L Z in Z o Fig. 6-4 6.6. AMPLIFIR ANALYSIS A simplified (bias network omitted) amplifier is shown in Fig. 6-5(a), and the associated smallsignal equivalent circuit in Fig. 6-5(b). xample 6.2. In the amplifier of Fig. 6-5(b), let h ie ¼ k; h re ¼ 0 4 ; h fe ¼ 00; h oe ¼ 2 S, and R L ¼ 2k. (These are typical amplifier values.) Find expressions for the (a) current-gain ratio A i, (b) voltage-gain ratio A v, (c) input impedance Z in, and (d) output impedance Z o. (e) valuate this typical amplifier. (a) ycurrent division at node, i L ¼ =h oe ð h =h oe þ R fe i b ð6:42 L
HAP. 6] SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS 69 L s R L i b i i c L h ie i L L L L s L be h re L ce h fe i b h oe L ce R L Z in Z o L L (a) (b) Fig. 6-5 and h fe A i ¼ i L 00 ¼ ¼ i b þ h oe R L þð2 0 6 ð2 0 3 ¼ 97:7 ð6:43 Note that A i h fe, where the minus sign indicates a 808 phase shift between input and output currents. (b) y KVL around ; mesh, v s ¼ v be ¼ h ie i b þ h re v ce ð6:44 Ohm s law applied to the output network requires that v ce ¼ h fe i b kr h L ¼ h fer L i b ð6:45 oe þ h oe R L Solving (6.45) fori b,substituting the result into (6.44), and rearranging yield A v ¼ v s h fe R L ¼ v ce h ie þ R L ðh ie h oe h fe h re ð00ð2 0 3 ¼ 0 3 þð20 3 ½ð 0 3 ð2 0 6 ð00ð 0 4 ¼ 99:2 ð6:46 Š (c) Observe that A v h fe R L =h ie, where the minus sign indicates a 808 phase shift between input and output voltages. Substituting (6.45) into(6.44) and rearranging yield Z in ¼ v s ¼ h i ie h reh fe R L ¼ 0 3 ð 0 4 ð00ð2 0 3 b þ h oe R L þð2 0 6 ð2 0 3 ¼ 980:5 ð6:47 (d) Note that for typical amplifier values, Z in h ie. Wedeactivate (short) v s and replace R L with a driving-point source so that v dp ¼ v ce. Then, for the input mesh, Ohm s law requires that i b ¼ h re h ie v dp ð6:48 However, at node (with, now, i c ¼ i dp ), KL yields i c ¼ i dp ¼ h fe i b þ h oe v dp ð6:49 Using (6.48) in(6.49) and rearranging then yield Z o ¼ v dp ¼ ¼ i dp h oe h fe h re =h ie 2 0 6 ð00ð 0 4 =ð 0 3 ¼ 500 k ð6:50 The output impedance is increased by feedback due to the presence of the controlled source h re v ce.
70 SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS [HAP. 6 (e) ased on the typical values of this example, the characteristics of the amplifier can be summarized as follows:. Large current gain 2. Large voltage gain 3. Large power gain ða i A v ) 4. urrent and voltage phase shifts of 808 5. Moderate input impedance 6. Moderate output impedance 6.7. AMPLIFIR ANALYSIS A simplified (bias network omitted) amplifier is shown in Fig. 6-6(a), and the associated small-signal equivalent circuit in Fig. 6-6(b). L s i L R L L L L s L eb Z in i e i c h ib i L h rb L cb h fb i e L cb R L Z o L L (a) (b) Fig. 6-6 amplifier xample 6.3. In the amplifier of Fig. 6-6(b), let h ib ¼ 30 ; h rb ¼ 4 0 6 ; h fb ¼ 0:99; ¼ 8 0 7 S, and R L ¼ 20 k. (These are typical amplifier values.) Find expressions for the (a) current-gain ratio A i, (b) voltage-gain ratio A v, (c) input impedance Z in, and (d) output impedance Z o. (e) valuate this typical amplifier. (a) ydirect analogy with Fig. 6-5(b) and (6.43) h fb 0:99 A i ¼ ¼ þ R L þð80 7 ð20 0 3 ¼ 0:974 ð6:5 Note that A i h fb <, and that the input and output currents are in phase because h fb < 0. (b) ydirect analogy with Fig. 6-5(b) and (6.46), h fb R L A v ¼ h ib þ R L ðh ib h oc h fb h rb ¼ ð 0:99ð20 0 3 30 þð20 0 3 ½ð30ð8 0 7 ð 0:99ð4 0 6 ¼ 647:9 ð6:52 Š Observe that A v h fb R L =h ib, and the output and input voltages are in phase because h fb < 0. (c) ydirect analogy with Fig. 6-5(b) and (6.47) Z in ¼ h ib h rbh fb R L ¼ 30 ð4 0 6 ð 0:99ð20 0 3 þ R L þð80 7 ð20 0 3 ¼ 30:08 ð6:53 It is apparent that Z in h ib. (d) y analogy with Fig. 6-5(b) and (6.50), Z o ¼ ¼ h fb h rb =h ib 8 0 7 ð 0:99ð4 0 6 ¼ :07 M ð6:54 =30 Note that Z o is decreased because of the feedback from the output mesh to the input mesh through h rb v cb.
HAP. 6] SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS 7 (e) ased on the typical values of this example, the characteristics of the amplifier can be summarized as follows:. urrent gain of less than 2. High voltage gain 3. Power gain approximately equal to voltage gain 4. No phase shift for current or voltage 5. Small input impedance 6. Large output impedance 6.8. AMPLIFIR ANALYSIS Figure 6-7(a) shows a amplifier with the bias network omitted. circuit is drawn in Fig. 6-7(b). The small-signal equivalent L s h i b i ic e i L i L R L L L s L bc h rc L ec h fc i b h oc L R L ec L Z in Z o L L (a) (b) Fig. 6-7 amplifier xample 6.4. In the amplifier of Fig. 6-7(b), let h ic ¼ k; h rc ¼ ; h fc ¼ 0; h oc ¼ 2 S, and R L ¼ 2k. Drawing direct analogies with the amplifier of xample 6.2, find expressions for the (a) current-gain ratio A i, (b) voltage-gain ratio A v, (c) input impedance Z in, and (d) output impedance Z o. (e) valuate this typical amplifier. (a) In parallel with (6.43), h fc 0 A i ¼ ¼ þ h oc R L þð2 0 6 ð2 0 3 ¼ 98:6 ð6:55 Note that A i h fc, and that the input and output currents are in phase because h fc < 0. (b) In parallel with (6.46), h fc R L A v ¼ h ic þ R L ðh ic h oc h fc h rc ¼ ð 0ð2 0 3 0 3 þð20 3 ½ð 0 3 ð2 0 6 ¼ 0:995 ð6:56 ð 0ðŠ Observe that A v =ð h ic h oc =h fc. Since the gain is approximately and the output voltage is in phase with the input voltage, this amplifier is commonly called a unity follower. (c) In parallel with (6.47), Z in ¼ h ic h rch fc R L ¼ 0 3 ðð 0ð2 03 þ h oc R L þð2 0 6 ð2 0 3 ¼ 8:4 M ð6:57 Note that Z in h fc =h oc. (d) In parallel with (6.50), Z o ¼ ¼ h oc h fc h rc =h ic 2 0 6 ð 0ð=ð 0 3 ¼ 9:9 Note that Z o h ic =h fc.
72 SMALL-SIGNAL MIDFRQUNY JT AMPLIFIRS [HAP. 6 (e) ased on the typical values of this example, the characteristics of the amplifier can be summarized as follows:. High current gain 2. Voltage gain of approximately unity 3. Power gain approximately equal to current gain 4. No current or voltage phase shift 5. Large input impedance 6. Small output impedance 6.9. JT AMPLIFIR ANALYSIS WITH SPI Since SPI models of the JT (see hapter 3) provide the device terminal characteristics, a transistor amplifier can be properly biased and a time-varying input signal can be directly applied to the completely modeled amplifier circuit. Any desired signal that results can be measured directly in the time domain to form signal ratios that yield current and voltage gains. With such modeling, any signal distortion that results from nonlinear operation of the JT is readily apparent from inspection of signaltime plots. Such an analysis approach is the analytical equivalent of laboratory operation of the amplifier where the time plot of signals is analogous to oscilloscope observation of the amplifier circuit signals. SPI capabilities also lend themselves to JT amplifier analysis using the small-signal equivalent circuits. In such case, the voltage-controlled voltage source (VVS) and the current-controlled current source (S) introduced in Section.3 find obvious application in the small-signal equivalent circuits of the type shown in Fig. 6-. ither time-varying analysis (.TRAN command statement) or sinusoidal steady-state analysis (.A command statement) can be performed on the small-signal equivalent circuit. xample 6.5. For the amplifier of Fig. 3-0(a), let v i ¼ 0:25 sinð2000t V; V ¼ 5 V; ¼ 2 ¼ ¼ 00 F; R ¼ 6k; R 2 ¼ 50 k; R ¼ R L ¼ k; and R i ¼ R ¼ 00. The transistor is characterized by the model of Problem 5.4. Use SPI methods to determine the hybrid parameters of (6.5) through (6.8) for this transistor at the point of operation. The netlist code below describes the circuit. X65.IR vi 0 SIN(0V 250mV 0kHz) Ri 2 00ohm 2 3 000uF 2 4 7 000uF R 3 0 6kohm R2 3 6 50kohm R 6 4 kohm R 5 0 00ohm RL 7 0 kohm V 6 0 5V Q4 35QNPNG.MODL QNPNG NPN(Is=0fA Ikf=50mA Isc=0fA f=50 r=3 Rb=ohm Rc=ohm Va=75V jc=0pf je=5pf).tran us 0.ms.PRO.ND After executing hx65.iri, the plots of Fig. 6-8 can be generated by use of the Probe feature of PSpice. resulting h-parameter value is indicated on each of the four plots of Fig. 6-8. The