Analysis of Small-signal Transistor Amplifiers

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1 Analyss of Small-sgnal Transstor Amplfers On completon of ths chapter you should be able to predct the behaour of gen transstor amplfer crcuts by usng equatons and/or equalent crcuts that represent the transstor s a.c. parameters. easons for Adoptng ths Technque The gans of an amplfer crcut may be obtaned by drawng the load lnes on the plotted output characterstcs. Howeer, for a number of reasons, ths s not a truly practcal method. (a) Manufacturers do not prode graphs or data to enable the characterstcs to be plotted. (b) Een f such data were aalable, the process would be ery tme consumng. (c) Obtanng results from plotted graphs s not always ery accurate much depends upon the skll and nterpretaton of the nddual concerned. For these reasons an alternate method, whch noles the use of equatons and/or smple network analyss, s preferred. Ths method noles the use of the transstor parameters, the data for whch s proded by manufacturers. Ths nformaton s most commonly obtaned from component catalogues produced by supplers such as ado Spares and Mapln Electroncs. BJT Parameters You should already be famlar wth the d.c. parameters such as nput resstance ( IN ), output resstance ( OUT ), and current gan ( h FE ), and ther relatonshp to the transstor s output characterstcs. In addton, an a.c. amplfer crcut may be redrawn n terms of the appearance of the crcut to a.c. sgnals. Ths s llustrated n Fg.. 67

2 68 Analyss of Small-sgnal Transstor Amplfers V CC C C b B C S V ce C S b B V be V ce V S V S V be (a) crcut (b) a.c. equalent Fg. The a.c. equalent crcut of Fg. (b) s useful n that the current flow paths of the a.c. sgnal and the effecte a.c. load can be apprecated, but n order to analyse the complete amplfer crcut the load lnes would stll need to be drawn on the characterstcs. What s requred s a smple network representaton of the transstor tself, whch can then be nserted nto Fg. l(b) n place of the transstor symbol. There are a arety of transstor parameters that may be used n ths way. Amongst these are Z-parameters, Y-parameters, hybrd π parameters, and h-parameters. For the analyss of small-sgnal audo frequency amplfers the use of h-parameters s the most conenent, and wll be the method adopted here. Proded that the transstor s correctly based and the nput sgnal s suffcently small so as to cause excursons of currents and oltages that reman wthn the lnear portons of the characterstcs, then the transstor tself may be consdered as a smple four-termnal network as shown n Fg.. ν near ν network Fg. The relatonshps between the four quanttes of a lnear network can be expressed by a number of equatons, two of whch are: ν AB ν...[] C D ν...[]

3 Analyss of Small-sgnal Transstor Amplfers 69 Examnaton of the unts noled n these two equatons reeals that A must be an mpedance (ohm), B and C are dmensonless (ratos), and D must be an admttance (semen). Snce there s a mxture or hybrd of unts noled, they are known as the hybrd or h-parameters, hang the followng symbols: A h ohm; B h ; C h ; D h semen r f o If the transstor s conected n common emtter confguraton the two equatons would be wrtten as follows ν hehreν h h ν fe oe If the transstor s connected n common base confguraton then the parameters would be h b, h rb, h fb and h ob respectely. The h-parameters are defned as follows: h : s the nput mpedance wth the output short-crcuted to a.c. Thus, h ν ohm h r : s the reerse oltage feedback rato wth the nput open-crcuted to a.c. Thus, h r ν ν h o : s the output admttance wth the nput open-crcuted to a.c. Thus, h 0 semen h f : s the forward current gan wth the output short-crcuted to a.c. Notes: Thus, h f In modern transstors h r s ery small ( 0 4 ) so ths parameter wll be gnored. Just as conductance G / semen, so admttance, Y /Z semen. 3 The h-parameters wll ary wth temperature, ageng and frequency. For the analyss at ths leel we shall consder that they reman constant. 4 Snce the transstor s a current-operated dece t s conenent to represent ts collector crcut as a current generator wth ts nternal mpedance (/h o ) n parallel.

4 70 Analyss of Small-sgnal Transstor Amplfers Consderng the amplfer crcut of Fg., the complete h-parameter equalent crcut would be as shown n Fg. 3. b c S h e h fe B ν /h oe ν ν S e e Fg. 3 For practcal purposes t may be assumed that the h-parameters wll hae the same numercal alues as ther d.c. counterparts.e. h ; h h ; / h IN f F o OUT 3 h-parameter Equatons Ignorng h r the orgnal two equatons may be wrtten as: ν h... [] h h ν [ ] f o and usng these equatons the followng results can be obtaned. Amplfer current gan, A Amplfer oltage gan, A hf h ν o h f h ( h ) o A h () () Thus, knowng the alues for a transstor s h-parameters, the predcton of amplfer gans can smply be obtaned by ether usng the aboe equatons or by smple network analyss usng the h-parameter equalent crcut. Worked Example Q For the amplfer crcut of Fg. 4, (a) sketch the h-parameter equalent crcut and, (b) determne the amplfer current and oltage gans usng () network analyss, and () h-parameter equatons. The h-parameters are h e.5 k Ω ; h fe 90; h oe 50 S

5 Analyss of Small-sgnal Transstor Amplfers 7 V CC V B 68 kω C. kω S V S 600 Ω 0 kω 00 mv rms Fg. 4 A h e. 5 k Ω ; h fe 90; h oe S (a) The h-parameter crcut wll be as shown n Fg. 5. S 90 S 600 Ω B h e /h oe 68 kω.5 kω 0 kω ν ν ν S 0. V rms C. kω 0 kω n Fg. 5 6 () /h oe k Ω ; C C. 0 ohm k Ω. 8 k Ω. 0 Input crcut: n heb h Usng potental dder technque: e B k Ω n. 47 s olt 0. V s 7 mv amp 47.3 A h e Output crcut: ma Usng current dder technque: n / hoe h amp. / 0. 8 oe 39. ma

6 7 Analyss of Small-sgnal Transstor Amplfers () olt V A Ans A Ans hfe 90 A hoe ( ) 90 A 8. 6 Ans. 09 A A he. 5 A 99 Ans Thus, allowng for the cumulaton of roundng errors n part (), the results from the equatons agree wth those from the network analyss. The actual current that wll flow n the load of the preous example wll not n fact be, but only a fracton of that, and s shown n Fg. 5 as. Thus the power delered to the external load wll be less than the maxmum possble. Ths problem may be mnmsed by the use of a matchng transformer connected between the load and the amplfer crcut output termnals. Worked Example Q The transstor used n the crcut of Fg. 6 has the followng h-parameters h e k Ω ; h oe 6 0 S; h fe 00. Calculate (a) the amplfer current gan, (b) the actual power delered to the external load, and (c) the turns rato requred for a matchng transformer n order to maxmse the power delered to the load. V CC 0 kω C 4.7 kω V S 0. Vp-p S 600 Ω 0 kω E kω 5 kω Fg. 6 A h e kω ; h oe S; h fe 00

7 Analyss of Small-sgnal Transstor Amplfers 73 (a) The h-parameter equalent crcut s shown n Fg. 7. S s 600 Ω h e 00 /h oe C ν 0 kω 0 kω kω 6.7 kω 4.7 kω ν 5kΩ ν S 0. Vp-p n Fg. 7 Input crcut: h n n e semen ms 0 0 ms 0 0 so, n. 79 kω n. 79 s 00 mv pk-pk s n 50 mv pk-pk 0. 5 amp 75 µ A pk-pk h e 000 (b) Output crcut: C kω. C. 4kΩ ma pk-pk / hoe h.. ma pk-pk / oe 655. ma pk-pk A 75 0 A Ans 3 h Check: A fe h ( ) oe C ma pk-pk 97. C ma pk-pk P I watt, where I s the r.m.s. alue so, I 3. 7 ma. ma

8 74 Analyss of Small-sgnal Transstor Amplfers (c) P (. 03) 5000 P 63. mw Ans For maxmum power transfer, must match the parallel combnaton of / h oe and c call ths p. so, N N p s p k Ω 367. kω N p p N ohm N N p s s p : Ans 4 FET Parameters and Equalent Crcuts Snce a FET has an extremely hgh nput mpedance then ts nput crcut may be represented smply as an open crcut. Also, beng a oltage operated dece t s conenent to represent the output crcut as a oltage source wth the nternal resstance ( r ds ) n seres wth t. The small-sgnal equalent crcut wll therefore be as shown n Fg. 8. The FET parameters r ds and g m should already be famlar to you. From Fg. 8 : V o r ds g r V m ds olt V V o A ν g r m ds r ds (3) but n practce, r ds, so V V o gmrds r ds and, Aν g m (4) G d D r ds V g m r ds V V 0 S S Fg. 8

9 Analyss of Small-sgnal Transstor Amplfers 75 Worked Example 3 Q The FET used n the amplfer crcut of Fg. 9 has parameter alues of r ds 80 k Ω and g m 4 ms. Calculate (a) the amplfer oltage gan, and (b) the effecte nput resstance of the amplfer crcut. V DD 56 kω G MΩ D kω V 0 V S kω 4.6 kω 3kΩ Fg. 9 A r ds Ω ; g m S; 3 k Ω (a) For ths crcut, the effecte a.c. load, D 3 ohm kω 5 D. kω and snce r ds, then equaton (4) may be used so, A gm A 48. Ans 3 3 In order to check the aldty of usng the approxmaton of equaton (4), we can also calculate the gan usng equaton (3) and compare the two answers. g r mds Thus, A rds A 473., whch confrms the aldty of equaton (4) Note that a FET amplfer prodes ery much less oltage gan than a comparable BJT amplfer. (b) ookng n at the nput termnals, for a.c. sgnals, the gate resstor G s n seres wth the parallel combnaton of and, as shown n Fg. 0. n n G ohm M Ω Ans (say M Ω)

10 76 Analyss of Small-sgnal Transstor Amplfers G G n Fg. 0 S Thus, the nherently hgh nput resstance of the FET s presered n the amplfer crcut by the ncluson of G. 5 Practcal Implcatons It should be borne n mnd that when desgnng an amplfer crcut, the results of the equatons as shown n ths chapter ge only theoretcal answers. If an amplfer crcut thus analysed s then constructed and tested, the actual gan fgures acheed may well be dfferent to those predcted. There are a number of reasons for ths: the resstors wll hae actual alues dependng upon how close to tolerance they are, and the transstor parameters cannot be guaranteed to be exactly those quoted by the manufacturer. Indeed, manufacturers recognse ths by quotng mnmum, maxmum and typcal alues for such parameters as h f. In calculatons the typcal alue s normally used. Thus the mathematcal analyss should be consdered as only the frst step n the desgn process, and component alues wll then need to be adjusted n the lght of practcal tests. Summary of Equatons BJT amplfer: Current gan, A Voltage gan, A Power gan, A hf h A h A A p o FET amplfer: Approx. oltage gan, A g m or, more accurately, A g r m ds r ds

11 Analyss of Small-sgnal Transstor Amplfers 77 Assgnment Questons The h-parameters for the transstor used n the crcut of Fg. are h fe 50, h e 5 k Ω, and h oe 4 0 S. (a) sketch the h-parameter equalent crcut and hence, or otherwse, (b) calculate the amplfer current, oltage and power gans. 56 kω 3.9 kω V CC V CC 0. V pk-pk V 4.7 kω 500 Ω 7.5 kω 0 kω 4.7 kω 0.5 V pk-pk V Fg. The crcut of Fg. s now reconnected so that the transstor s connected n common base confguraton. If the common base parameters h b and h ob are 00 Ω and 0 S respectely, (a) sketch the equalent crcut, and (b) calculate the amplfer current, oltage and power gans. 3 Fgure shows a smply based common source FET amplfer, where the transstor parameters are g m 3 ms, and r ds 7 5 k Ω. Calculate the amplfer oltage gan. 5 kω V DD V 0 5 The parameters for the FET n Fg. 4 are r ds 8 5 kω and g m 4. ms. (a) calculate the amplfer oltage gan, and (b) the power dsspated n the external 5 k Ω load. 3V pk-pk V Fg kω 0 kω. MΩ 4.7 kω 4.7 kω V DD 30 V 5 kω Fg. 4 6 For the two equalent crcuts shown n Fgs. 5(a) and (b), sketch the amplfer crcuts that they represent, showng component alues, and also dentfy the alues for the transstor parameters n each case. 0 V 90 kω 5 kω MΩ S 0 kω V. MΩ 6.8 kω (a) Fg. 4 The transstor of the amplfer crcut shown n Fg. 3 has the followng parameters: h e.5 kω, h fe 0, and h oe 00 S. Sketch the equalent crcut and determne the amplfer current and oltage gans, and the power dsspated n the external 7.5 k Ω load. 600 Ω V 50 8 kω 0 kω 3kΩ 5 kω 4.7 kω 0 kω (b) Fg. 5

12 78 Analyss of Small-sgnal Transstor Amplfers Supplementary Worked Example Q Calculate the mnmum alue of h fe requred for the transstor n Fg. 6 n order that a power of 3.5 mw s dsspated n the 0 k Ω load resstor. The alues for h e and h oe are 4 k Ω and 50 µ S respectely. V CC 00 kω 4.7 kω 0.5 V pk-pk 600 Ω 0 kω Fg.6 A The h-parameter equalent crcut s shown n Fg. 7. Snce B h e then the shuntng effect of B wll be neglgble, and t has therefore been omtted from the calculaton. h e 4000 Ω ; h oe S; V 0.5 V pk-pk; P o W h fe 0.5 V pk-pk V S V 00 kω 4kΩ 0 kω V 4.7 kω 0 kω B h e /h oe C P V 0 Fg. 7 V watt so V P0 olt V he V h e V 0.7V pk-pk s 0.7 so, V 76.8mV r.m.s. 4 Vs olt pk-pk Voltage gan requred, A A V 5.96 V

13 Analyss of Small-sgnal Transstor Amplfers 79 A so, h fe h fe hoe / 47 where 3. k Ω h ( h ) / h 4.7 e oe A { h ( h )} 77{4( e oe )} 3. h e Ans oe Supplementary Worked Example Q The FET n the crcut of Fg. 8 has r ds 50 k Ω and g m 5 ms. Determne the alue of the output oltage, V, and the power deeloped n the 5 k Ω load. A r ds Ω ; g m S V DD 40 V 00 kω 39 kω V 4V pk-pk MΩ. kω V 5 kω Fg. 8 D ohm 5. kω D Now, snce r ds s NOT, then the approxmate equaton for oltage gan should not be used, hence g r mds A rds 65. A 7.48 Thus, V V pk-pk 70 V pk-pk so, V 4.75 V Ans V P watt mw Ans P 0 Note that had the approxmate equaton A g m been used n ths case an error of about % would hae resulted n the alue for A. Ths would be an unacceptably large error. The approxmate form of the equaton should be used only when r ds s at least 0 tmes larger than.

14 80 Analyss of Small-sgnal Transstor Amplfers Answers to Assgnment Questons (b) A 0; A 97; A p (b) A 0.9; A 4.8; A p A 95.5; A 98.; P o 6.5 mw 5 A 4.6; P o 45.4 mw

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