4. Introduction and Chapter Objectives

Transcription

1 Rel Anlog Crcuts 1 Chpter 4: Systems nd Network Theorems 4. Introducton nd Chpter Ojectes In preous chpters, numer of pproches he een presented for nlyzng electrcl crcuts. In these nlyss pproches, we he een proded wth crcut consstng of numer of elements (resstors, power supples, etc.) nd determned some crcut rle of nterest ( oltge or current, for exmple). In the process of determnng ths rle, we he wrtten equtons whch llow us to determne ny nd ll rles n the system. For complex crcut, wth mny elements, ths pproch cn result n ery lrge numer of equtons nd correspondngly lrge mount of effort expended n the soluton of these equtons. Unfortuntely, much of the physcl nsght out the oerll operton of the crcut my e lost n the detled descrpton of ll of the nddul crcut elements. Ths lmtton ecomes prtculrly serous when we ttempt to desgn crcut to perform some tsk. In ths chpter, we ntroduce the concept of systems leel pproch to crcut nlyss. In ths type of pproch, we represent the crcut s system wth some nputs nd outputs. We then chrcterze the system y the mthemtcl reltonshp etween the system nputs nd the system outputs. Ths reltonshp s clled the nputoutput relton for the system. Ths representton of system leds to seerl network theorems whose use cn smplfy the nlyss of these systems. The network theorems essentlly llow us to model porton of complcted crcut s much smpler (ut equlent) crcut. Ths smplfed model cn then e used to fcltte the desgn or nlyss of the remnder of the crcut. The oe pproch for representng crcuts s prtculrly useful n crcut desgn; successful desgn pproches for lrge crcuts typclly use topdown strtegy. In ths desgn pproch, the oerll system s roken down nto numer of nterconnected susystems, ech of whch performs some specfc tsk. The nputoutput reltonshps for these nddul susystems cn e determned sed on the tsk to e performed. The susystems cn then e desgned to mplement the desred nputoutput relton. An udo compct dsc plyer, for exmple, wll nclude susystems to perform flterng, dgtltonlog conerson, nd mplfcton processes. It s sgnfcntly eser to desgn the susystems sed on ther nddul requrements thn to ttempt to desgn the entre system ll t once. We wll thus egn to thnk of the crcuts we nlyze s systems whch perform some oerll tsk, rther thn s collectons of nddul crcut elements Anlog Deces nd Dglent, Inc. 1

2 Chpter 4: Systems nd Network Theorems After completng ths chpter, you should e le to: Defne sgnls nd systems Represent systems n lock dgrm form Identfy system nputs nd outputs Wrte nputoutput equtons for systems Stte the defnng propertes of lner systems Determne whether system s lner Stte condtons under whch superposton cn e ppled to crcut nlyss Anlyze electrcl crcuts usng the prncple of superposton Defne the chrcterstc for crcut Represent resste crcut n terms of ts chrcterstc Represent resste crcut s twotermnl network Determne Théenn nd Norton equlent crcuts for crcuts contnng power sources nd resstors Relte Théenn nd Norton equlent crcuts to chrcterstcs of twotermnl networks Determne lod resstnce whch wll mxmze the power trnsfer from crcut 2012 Anlog Deces nd Dglent, Inc. 2

3 Chpter 4.1: Sgnls nd Systems 4.1: Sgnls nd Systems In ths secton, we ntroduce sc concepts relte to systemsleel descrptons of generl physcl systems. Lter sectons wll ddress pplcton of these concepts specfclly to electrcl crcuts. A system s commonly represented s shown n the lock dgrm of Fgure 4.1. The system hs some nput, u(t), nd some output, y(t). In generl, oth the nput nd output cn e functons of tme; the cse of constnt lues s specl cse of tmeryng functon. The output wll e represented s some rtrry functon of the nput: y( t) f { u( t)} (4.1) Equton (4.1) s sd to e the nputoutput equton goernng the system. The oe reltonshp hs only one nput nd one nput the system s sd to e snglenputsngleoutput (SISO) system. Systems cn he multple nputs nd multple outputs, n these cses there wll e n nputoutput equton for ech system outputs nd ech of these equtons my e functon of seerl nputs. We wll concern ourseles only wth SISO systems for now. Input, u(t) System Output, y(t) = f{x(t)} Fgure 4.1. Block dgrm representton of system. One mportnt spect of the systemsleel pproch represented y equton (4.1) nd Fgure (4.1) s tht we re representng our system s lck ox. We relly he no de wht the system tself s, eyond mthemtcl dependence of the output rle on the nput rle. The physcl system tself could e mechncl, therml, electrcl, or fludc. In fct, t s frly common to represent mechncl system s n equlent electrcl system (or ceers), f dong so ncreses the physcl nsght nto the system s operton. The crcuts we nlyze cn now e thought of s systems whch perform some oerll tsk, rther thn s collectons of nddul crcut elements. We wll lso thnk of the nputs nd outputs of the system s sgnls, rther thn specfc crcut prmeters such s oltges or currents. Ths pproch s somewht more strct thn we re perhps used to, so we wll prode some ddtonl dscusson of wht we men y these terms. Generlly, most people thnk of system s group of nterrelted elements whch perform some tsk. Ths ewpont, though ntutely correct, s not specfc enough to e useful from n engneerng stndpont. In these chpters, we wll defne system s collecton of elements whch store nd dsspte energy. The system trnsfers the energy n the system nputs to the system outputs; the process of energy trnsfer s represented y the nputoutput equton for the system. Exmples of the energy trnsfer cn nclude mechncl systems (the knetc energy resultng from usng force to ccelerte mss, or the potentl energy resultng from usng force to compressng sprng), therml systems (pplyng het to chnge mss s temperture), nd electrcl systems (dssptng electrcl power wth the flment n lght ul to produce lght). The tsk to e performed y the system of Fgure 4.1 s thus the trnsformton of some nput sgnl u(t) nto n output sgnl y(t). Sgnls, for us, wll e ny weform whch cn ry s functon of tme. Ths s n extremely rod defnton exmples of sgnls nclude: the force ppled to mss, the elocty of the mss s t ccelertes n response to the ppled force, the current ppled to crcut y power supply, 2012 Anlog Deces nd Dglent, Inc. 3

4 Chpter 4.1: Sgnls nd Systems the oltge dfference cross resstor whch s sujected to some current flow, the electrcl power suppled to hetng element, the temperture of mss whch s eng heted y n electrc col The trnsformton of the nput sgnl to the output sgnl s performed y the nputoutput relton goernng the system. The nputoutput relton cn e comnton of lgerc, dfferentl, nd ntegrl equtons. To prode some concrete exmples of the oe concepts, seerl exmples of systemsleel representtons of common processes re proded elow. Exmple 4.1: Mss sujected to n externl force Consder the mssdmper system shown n the fgure elow. The ppled force F(t) pushes the mss to the rght. The mss s elocty resultng from the ppled force s (t). The mss sldes on surfce wth sldng coeffcent of frcton, whch nduces force F = (t) whch opposes the mss s moton. The mss s ntlly t rest nd the ppled force s zero for tme efore tme t=0. Mss s elocty, (t) Externlly ppled force, F(t) Mss, m Surfce wth sldng coeffcent of frcton, The goernng equton for the system (otned y drwng free ody dgrm of the mss nd pplyng F m ) s d( t) m ( t) F( t) dt The goernng equton for the system s frst order dfferentl equton. Knowledge of the externlly ppled force F(t) nd the ntl elocty of the mss llows us to determne the elocty of the mss t ll susequent tmes. Thus, we cn model the system s hng n nput sgnl F(t) whch s known nd n output elocty (t) whch cn e determned from the nput sgnl nd the propertes of the system (the mss, m, nd coeffcent of frcton, ). The system cn then e represented y the lock dgrm elow: Input, F(t) d( t) m ( t) F( t) dt Output, (t) System s represented y nputoutput equton (It s rther unusul to plce the system goernng equton drectly n lock dgrm; we do t here to llustrte pont.) Exmple 4.2: Electrcl crcut For the electrcl crcut elow, wrte the equtons goernng the nputoutput reltonshp for the crcut. The ppled nput to the crcut s the oltge source V n nd the output s the oltge V cross the resstor Anlog Deces nd Dglent, Inc. 4

5 Chpter 4.1: Sgnls nd Systems 1 V 3V x 3 4 V n V X 5 We preously wrote mesh equtons for ths crcut (for specfc lue of V n ) n Chpter 3.2. We repet these mesh equtons here, long wth our defntons of the mesh currents: ( 3V X ) Vn 0 1 V n 3( 3V X ) V X 5 2 The output oltge V s relted to the mesh currents y: V 2( 1 3V x ) The oe four equtons prode n nputoutput descrpton of the crcut. If desred, they cn e cn e comned to elmnte ll rles except V n nd V nd rewrtten n the form V f V } { n per equton (1). Note tht ll nformton out the orgnl system, except the reltonshp etween the nput nd output sgnls, s lost once we do ths. The systemleel lock dgrm for the crcut mght then e drwn s: 4 1 V n 3V X 3 2 V X 5 3V x V n (t) Crcut V(t) 2012 Anlog Deces nd Dglent, Inc. 5

6 Chpter 4.1: Sgnls nd Systems Exmple 4.3: Temperture control system Our fnl exmple s of temperture control system. Ths exmple llustrtes the representton of complex system s set of nterctng susystems. A typcl temperture control system for uldng wll he thermostt whch llows the occupnts to set desred temperture, furnce (or r condtoner) whch prodes mens of djustng the uldng s temperture, some wy of mesurng the ctul uldng temperture, nd controller whch decdes whether to turn the furnce or r condtoner on or off, sed on the dfference etween the desred nd ctul tempertures. The lock dgrm elow prodes one possle pproch towrd nterconnectng these susystems nto n oerll temperture control system. Ths lock dgrm cn e used to dentfy nddul susystems, nd prode specfctons for the susystems, whch cn llow the desgn to proceed effcently. For exmple: 1. The temperture mesurement system mght e requred to produce oltge, whch s functon of the temperture n the uldng. The thermstorsed temperture mesurement systems we he desgned nd constructed n the l re good exmples of ths type of system. 2. The controller mght operte y comprng the desred temperture (generlly represented y oltge leel) wth the oltge ndctng the ctul temperture. For hetng system, f the ctul temperture s lower thn desred y some mnmum mount, the controller wll mke decson to swtch the furnce on. Desgn decsons mght e mde to determne wht mnmum temperture dfference s requred to turn the furnce on, nd whether to se the decson to turn on the furnce strctly upon temperture dfference or on rte of chnge n temperture dfference. 3. When the furnce turns on t wll pply het to the uldng, cusng the uldng s temperture to ncrese. Once the uldng temperture s hgh enough, the controller wll then typclly turn the furnce ck off. The furnce must e desgned to prode pproprte het nput to the uldng, sed on the uldng sze nd the ntcpted het losses to the uldng s surroundngs. (For exmple, lrger uldng or uldng n colder clmte wll requre lrger furnce.) 4. A model of the uldng s het losses wll generlly e necessry n order to sze the furnce correctly nd choose n pproprte control scheme. Desgn choces for the uldng tself my nclude nsulton requrements necessry to stsfy desred hetng costs. Desgns for the oe susystems cn now proceed somewht ndependently, wth proper coordnton etween the desgn cttes. Het nput from ment surroundngs Desred Temperture Controller On/Off Decson Furnce or r condtoner Het nput Buldng Buldng temperture Actul Temperture Temperture Mesurement System Secton Summry: Systems re set of components whch work together to perform some tsk. Systems re typclly consdered to he one or more nputs (whch re proded to the system from the externl enronment) nd one or more outputs (whch the system prodes to the enronment) Anlog Deces nd Dglent, Inc. 6

7 Chpter 4.1: Sgnls nd Systems Generclly, the nputs nd outputs of systems re sgnls. Sgnls re smply tmeryng functons. They cn e oltges, currents, eloctes, pressures, etc. Systems re often chrcterzed y ther nputoutput equtons. The nputoutput equton for system smply prodes mthemtcl reltonshp etween the nput to the system nd the output from the system. Once the nput s defned s prtculr numer or functon of tme, tht lue or functon cn e susttuted nto the nputoutput equton to determne the system s response to tht nput. Exercses: 1. The nput to the crcut elow s the current, U. The output s the current through the 10Ω resstor, I. Determne n nputoutput equton for the crcut. U 20 30V I The nput to the crcut s the oltge U. The output s the oltge V1. Determne n nputoutput relton for the crcut. 4 1A V 1 4 U 2012 Anlog Deces nd Dglent, Inc. 7

8 Chpter 4.2: Lner Systems 4.2: Lner Systems We he so fr ntroduced numer of pproches for nlyzng electrcl crcuts, ncludng: Krchoff s current lw, Krchoff s oltge lw, crcut reducton technques, nodl nlyss, nd mesh nlyss. When we he ppled the oe nlyss methods, we he generlly ssumed tht ny crcut elements operte lnerly. For exmple, we he used Ohm s lw to model the oltgecurrent reltonshps for resstors. Ohm s lw s pplcle only for lner resstors tht s, for resstors whose oltgecurrent reltonshp s strght lne descred y the equton = R. Nonlner resstors he een mentoned refly; n l ssgnment 1, for exmple, we forced resstor to dsspte n excesse mount of power, therey cusng the resstor to urn out nd dsply nonlner opertng chrcterstcs. All crcut elements wll dsply some degree of nonlnerty, t lest under extreme opertng condtons. Unfortuntely, the nlyss of nonlner crcuts s consderly more complcted thn nlyss of lner crcuts. Addtonlly, n susequent chpters we wll ntroduce numer of nlyss methods whch re pplcle only to lner crcuts. The nlyss of lner crcuts s thus ery perse for exmple, desgnng lner crcuts s much smpler thn the desgn of nonlner crcuts. For ths reson, mny nonlner crcuts re ssumed to operte lnerly for desgn purposes; nonlner effects re ccounted for susequently durng desgn ldton nd testng phses. The concept of tretng n electrcl crcut s system ws ntroduced n secton 4.1. In systemsleel nlyss of crcuts, we re prmrly nterested n the reltonshp etween the system s nput nd output sgnls. Crcuts goerned y nonlner equtons re consdered to e nonlner systems; crcuts whose goernng nputoutput reltonshp s lner re lner systems. In ths chpter, we formlly ntroduce the concept of lner systems. The nlyss of lner systems s extremely common, for the resons mentoned oe: structurl systems, flud dynmc systems, nd therml systems re often nlyzed s lner systems, een though the underlyng processes re often nherently nonlner. Lner crcuts re specl cse of lner systems, n whch the system conssts only of nterconnected electrcl crcut elements whose oltgecurrent reltonshps re lner. Lner systems re descred y lner reltons etween dependent rles. For exmple, the oltgecurrent chrcterstc of lner resstor s proded y Ohm s lw: R where s the oltge drop cross the resstor, s the current through the resstor, nd R s the resstnce of the resstor. Thus, the dependent rles current nd oltge re lnerly relted. Lkewse, the equtons we he used to descre dependent sources (proded n secton 1.2): Voltge controlled oltge source: s 1 Voltge controlled current source: s g1 Current controlled oltge source: s r1 Current controlled current source: s 1 ll descre lner reltonshps etween the controlled nd controllng rles. All of the oe reltonshps re of the form y( t) Kx( t) (4.2) 2012 Anlog Deces nd Dglent, Inc. 8

9 Chpter 4.2: Lner Systems where x(t) nd y(t) re oltges or currents n the oe exmples. More generlly, x(t) nd y(t) cn e consdered to e the nput nd output sgnls, respectely, of lner system. Equton 4.2 s often represented n lock dgrm form s shown n Fgure 4.2. Input, x(t) K Output, y(t) = Kx(t) Fgure 4.2. Lner system lock dgrm. The output s sometmes clled the response of the system to the nput. The multplcte fctor K reltng the nput nd output s often clled the system s gn. Elements whch re chrcterzed y reltonshps of the form of equton 4.2 re sometmes clled lner elements. The equton reltng the system s nput nd output rles s clled the nputoutput reltonshp of the system. Asde: Mny types of systems cn e descred y the reltonshp of equton (1). For exmple, Hooke s lw, whch reltes the force ppled to sprng to the sprng s dsplcement, s F k x where k s the sprng constnt, F s the ppled force, nd x s the resultng dsplcement s shown elow. In ths exmple, F s the nput to the system nd x s the system output. x F Notce tht we he llowed the nput nd output of our system to ry s functons of tme. Constnt lues re specl cses of tmeryng functons. We wll ssume tht the system gn s not tmeryng quntty. For our purposes, we wll defne lnerty n somewht more rod terms thn equton (4.2). Specfclly, we wll defne system s lner f t stsfes the followng requrements: 2012 Anlog Deces nd Dglent, Inc. 9

10 Chpter 4.2: Lner Systems Lnerty: 1. If the response of system to some nput x 1 (t) s y 1 (t) then the response of the system to some nput x 1 (t) s y 1 (t), where s some constnt. Ths property s clled homogenety. 2. If the response of the sme system to n nput x 2 (t) s y 2 (t), then the response of the system to n nput x 1 (t)x 2 (t) s y 1 (t)y 2 (t). Ths s clled the ddte property. The oe two propertes defnng lner system cn e comned nto sngle sttement, s follows: f the response of system to n nput x 1 (t) s y 1 (t) nd the system s response to n nput x 2 (t) s y 2 (t), then the response of the system to n nput x 1 (t) x 2 (t) s y 1 (t) y 2 (t). Ths property s llustrted y the lock dgrm of Fgure 4.3. The S symol n Fgure 4.3 denotes sgnl summton; the sgns on the nputs to the summton lock ndcte the sgns to e ppled to the nddul sgnls. x 1 (t) Lner System y 1 (t) x 1 (t) S Lner System y 1 (t) y 2 (t) x 2 (t) Lner System y 2 t) x 2 (t) Fgure 4.3. Block dgrm representton of propertes defnng lner system. The oe defnton of lnerty s more generl thn the expresson of equton (4.2). For exmple, the processes of dfferentton nd ntegrton re lner processes ccordng to the oe defnton. Thus, systems wth the nputoutput reltons such s: y xdt nd y dx dt re lner systems. We wll use crcut elements whch perform ntegrtons nd dfferenttons lter when we dscuss energy storge elements such s cpctors nd nductors. Dependent Vrles nd Lnerty: Lnerty s sed on the reltonshps etween dependent rles, such s oltge nd current. In order for system to e lner, reltonshps etween dependent rles must e lner plots of one dependent rle gnst nother re strght lnes. Ths cuses confuson mong some students when we egn to tlk out tme ryng sgnls. Tme s not dependent rle, nd plots of oltges or currents s functon of tme for lner system my not to e strght lnes. Although the oe defntons of lner systems re fundmentl, we wll not often use them drectly. Krchoff s oltge lw nd Krchoff s current lw rely upon summng multples of oltges or currents. As long s the oltgecurrent reltons for nddul crcut elements re lner, pplcton of KVL nd KCL to the crcut wll result n lner equtons for the system. Therefore, rther thn drect pplcton of the oe defntons of lner systems, we wll smply clm tht n electrcl crcut contnng only lner crcut elements wll e lner nd wll he lner nputoutput reltonshps. All crcuts we he nlyzed so fr he een lner Anlog Deces nd Dglent, Inc. 10

11 Chpter 4.2: Lner Systems Lnerty: If ll elements n crcut he lner oltgecurrent reltonshps, the oerll crcut wll e lner. Importnt note out power: A crcut s power s not lner property, een f the oltgecurrent reltons for ll crcut elements re lner. 2 2 Resstors whch oey Ohm s lw dsspte power ccordng to P R. Thus, the power dsspton R of lner resstor s not lner comnton of oltges or currents the reltonshp etween oltge or current nd power s qudrtc. Thus, f power s consdered drectly n the nlyss of lner crcut, the resultng system s nonlner. Secton Summry: Lner systems re chrcterzed y lner reltonshps etween dependent rles n the system. For electrcl system, ths typclly mens tht the reltonshp etween oltge nd current for ny crcut component s lner n electrcl crcuts, for exmple, ths mens tht plot of oltge s. current for eery element n the system s strght lne. Ohm s lw, for exmple, descres lner oltgecurrent reltonshp. Lner systems he ery mportnt property: the ddte prncple pples to them. Superposton essentlly mens tht the response of system to some comnton of nputs x 1 x 2 wll e the sme s the sum of the response to the nddul nputs x 1 nd x Anlog Deces nd Dglent, Inc. 11

12 Chpter 4.2: Lner Systems Excercses: 1. The 20Ω resstor elow oeys Ohm s lw, so tht V=20I. We wll consder the nput to e the current through the resstor nd the output to e the oltge drop cross the resstor. Determne:. The output V f the nput I=2A.. The output V f the nput I = 3A. c. The output V s the nput I = 2A 3A = 5A. Do your nswers oe ndcte tht the ddte property holds for ths resstor? Why? I 20 V 2. A lner electrcl crcut hs n nput oltge V1 nd prodes n output oltge V2, s ndcted n the lock dgrm elow. If n nput oltge V 1 = 3V s ppled to the crcut, the mesured output oltge V 2 = 2V. Wht s the output oltge f n nput oltge V 1 = 6V s ppled to the crcut? Lner V 1 electrcl V 2 crcut 2012 Anlog Deces nd Dglent, Inc. 12

13 Chpter 4.3: Superposton 4.3: Superposton In secton 4.2, we stted tht, y defnton, the nputoutput reltons for lner systems he n ddte property. The ddte property of lner systems sttes tht: If the response of system to n nput x 1 (t) s y 1 (t) nd the response of the system to n nput x 2 (t) s y 2 (t), then the response of the system to n nput x 1 (t) x 2 (t) s y 1 (t) 2 (t). Thus, f system hs multple nputs, we cn nlyze the system s response to ech nput nddully nd then otn the oerll response y summng the nddul contrutons. Ths property cn e useful n the nlyss of crcuts whch he multple sources. If we consder the sources n crcut to e the nputs, lner crcuts wth multple ndependent sources cn e nlyzed y determnng the crcut s response to ech source nddully, nd then summng, or supermposng, the contrutons from ech source to otn the oerll response of the crcut to ll sources. In generl, the pproch s to nlyze complcted crcut wth multple sources y determnng the responses of numer of smpler crcuts ech of whch contns only sngle source. We llustrte the oerll pproch grphclly y the lock dgrm of Fgure 4.4 (whch s relly just reersed form of the lock dgrm of Fgure 4.3). x 1 (t) x 2 (t) Lner System y 1 (t) y 2 (t) x 1 (t) x 2 (t) Lner System Lner System y 1 (t) y 2 t) Fgure 4.4. Addte property of lner systems. In Fgure 4.4, we he lner system wth two nput sgnls whch re ppled y sources n the crcut. We cn nlyze ths crcut y notng tht ech nput sgnl corresponds to n ndependent source n the crcut. Thus, f the crcut s oerll response to source x 1 (t) s y 1 (t) nd the crcut s response to source x 2 (t) s y 2 (t), then the totl crcut response wll e the sum of the two nddul responses, y 1 (t) y 2 (t). Thus, f we wsh to determne the response of the crcut to oth sources, x 1 (t) nd x 2 (t), we cn determne the nddul responses of the crcut, y 1 (t) nd y 2 (t) nd then sum (or supermpose) the responses to otn the crcut s oerll response to oth nputs. Ths nlyss method s clled superposton. In order to determne crcut s response to sngle source, ll other ndependent sources must e turned off (or, n more colorful termnology, klled, or mde ded). To turn off current source, we must mke the nput current zero, whch corresponds to n open crcut. To turn off oltge source, we must mke the nput oltge zero, whch corresponds to short crcut. Kllng Sources: To kll oltge source, replce t wth short crcut To kll current source, replce t wth n open crcut. To pply the superposton method, then, the crcut s response to ech source n the crcut s determned, wth ll other sources n the crcut ded. The nddul responses re then lgerclly summed to determne the totl response to ll nputs. To llustrte the method, we consder the exmples elow Anlog Deces nd Dglent, Inc. 13

14 Chpter 4.3: Superposton Exmple 4.4: Determne the oltge V n the crcut elow, usng superposton. 1 12V V 3A The crcut oe cn e consder to e the superposton of the two crcuts shown elow, ech wth sngle source (the other source, n oth cses, hs een klled). 1 1 V 1 3A 12V V 2 1 The oltge V 1 oe cn e determned to e the result of current dson: V1 3A 2V 1 2. V 2 cn e determned to e the result of oltge dson: V1 12V 8V. Thus, the oltge 1 V V V 10V Anlog Deces nd Dglent, Inc. 14

15 Chpter 4.3: Superposton Exmple 4.5: Determne the oltge V n the crcut elow, usng superposton. 1 V 2A 6V 6 We egn y determnng the response V 1 to the 6V source y kllng the 2A source, s shown n the fgure elow. 1 V 1 6V 6 1 The oltge V 1 s smply the result of oltge dson: V1 6V 2V. The response V 2 to the 2A source 3 cn e determned y kllng the 6V source, resultng n the crcut elow: 1 V 2 2A 6 Kllng the 6V source plces short crcut n prllel wth the 2A source, so no oltge s nduced n ny of the resstors y the 2A source. Thus, V 2 = 0V. The oltge V s the sum of the two nddul oltges: V = V 1 V 2 = 2V 0V = 2V Anlog Deces nd Dglent, Inc. 15

16 Chpter 4.3: Superposton Notes on Superposton: 1. Superposton cnnot e used drectly to determne power. Preously, we noted tht power s not goerned y lner reltonshp. Thus, you cnnot determne the power dsspted y resstor y determnng the power dsspton due to ech source nd then summng the results. You cn, howeer, use superposton to determne the totl oltge or current for the resstor nd then clculte the power from the oltge nd/or current. 2. When usng superposton to nlyze crcuts wth dependent sources, do not kll the dependent sources. You must nclude the effects of the dependent sources n response to ech ndependent source. 3. Superposton s powerful crcut nlyss tool, ut ts pplcton cn result n ddtonl work. Before pplyng superposton, exmne the crcut crefully to ensure tht n lternte nlyss pproch s not more effcent. Crcuts wth dependent sources, n prtculr, tend to e dffcult to nlyze usng superposton. Secton Summry: Superposton s defnng property of lner systems. It essentlly mens tht, for lner systems, we cn decompose ny nput to the system nto numer of components, determne the system output resultng from ech component of the nput, nd otn the oerll output y summng up these nddul components of the output. Superposton cn e used drectly to nlyze crcuts whch contn multple ndependent sources. The responses of the crcut to ech source (kllng ll other sources) re determned nddully. The oerll response of the crcut due to ll sources s then otned y summng (supermposng) these nddul contrutons. The prncple of superposton s fundmentl property of lner systems nd hs ery rodrngng consequences. We wll e nokng t throughout the remnder of ths textook, often wthout oertly sttng tht superposton s eng used. The fct tht superposton pples to lner crcuts s the sc reson why engneers mke eery possle ttempt to use lner models when nlyzng nd desgnng systems. Exercses: 1. Use superposton to determne the oltge V 1 n the crcut elow. 4 1A V 1 4 8V 2012 Anlog Deces nd Dglent, Inc. 16

17 Chpter 4.4: Twotermnl Networks 4.4: Twotermnl Networks As noted n secton 4.1, t s often desrle, especlly durng the desgn process, to solte dfferent portons of complex system nd tret them s nddul susystems. These solted susystems cn then e desgned or nlyzed somewht ndependently of one nother nd susequently ntegrted nto the oerll system n topdown desgn pproch. In systems composed of electrcl crcuts, the susystems cn often e represented s twotermnl networks. As the nme mples, twotermnl networks consst of pr of termnls; the oltge potentl cross the termnls nd the current flow nto the termnls chrcterzes the network. Ths pproch s consstent wth our systemsleel pproch; we cn chrcterze the ehor of wht my e n extremely complex crcut y reltely smple nputoutput reltonshp. We lredy he some experence wth twotermnl networks; when we determned equlent resstnces for seres nd prllel resstor comntons, we treted the resste network s twotermnl network. For nlyss purposes, the network ws then replced wth sngle equlent resstnce whch ws ndstngushle from the orgnl crcut y ny externl crcutry ttched to the network termnls. In ths chpter, we wll formlze some twotermnl concepts nd generlze our pproch to nclude networks whch contn oth sources nd resstors. We wll ssume tht the electrcl crcut of nterest cn e sudded nto two sucrcuts, nterconnected t two termnls, s shown n Fgure 4.5. Our gol s to replce crcut A n our oerll system wth smpler crcut whch s ndstngushle y crcut B from the orgnl crcut. Tht s, f we dsconnect crcut A nd crcut B t the termnls nd replce crcut A wth ts equlent crcut, the oltge nd the current seen t the termnls of the crcuts wll e unchnged nd crcut B s operton wll e unffected. In order to mke ths susttuton, we wll need to use the prncple of superposton n our nlyss of crcut A thus, crcut A must e lner crcut. We re not chngng crcut B n ny wy crcut B cn e ether lner or nonlner. It should e emphszed tht crcut A s not eng physclly chnged. We re mkng the chnge conceptully n order to smplfy our nlyss of the oerll system. For exmple, the desgn of crcut B cn now proceed wth smplfed model of crcut A s operton, perhps efore the detled desgn of crcut A s een fnlzed. When the desgns of the two crcuts re complete, they cn e ntegrted nd the oerll system hs hgh prolty of functonng s expected. Crcut A (Lner) Crcut B Fgure 4.5. Crcut composed of two, twotermnl sucrcuts. In order to perform the oe nlyss, we wll dsconnect the two sucrcuts n Fgure 4.5 t the termnls nd determne the currentoltge reltonshp t the termnls of crcut A. We wll generlly refer to crcut A s currentoltge reltonshp s ts chrcterstc. Our pproch, therefore, s to look t crcut A lone, s shown n Fgure 4.6, nd determne the functonl reltonshp etween oltge ppled to the termnls nd the resultng current. (Equlently, we could consder tht current s ppled t the termnls nd look t the resultng oltge.) Fgure 4.6 s t frst glnce somewht msledng the termnls should not e consdered to e opencrcuted, s cursory look t the fgure mght ndcte; we re determnng the reltonshp etween oltge dfference ppled to the crcut nd the resultng current flow. (Fgure 4.6 ndctes current I flowng nto the crcut, whch wll, n generl, not e zero.) 2012 Anlog Deces nd Dglent, Inc. 17

18 Chpter 4.4: Twotermnl Networks Systemsleel nterpretton: When we determne the chrcterstc for the crcut, we re determnng the nputoutput reltonshp for system. Ether the oltge or the current t the termnls cn e ewed s the nput to system; the other prmeter s the output. The chrcterstc then prodes the output of the system s functon of the nput. Crcut A Fgure 4.6. Twotermnl representton of crcut. Resste Networks: We he lredy (somewht nformlly) treted purely resste crcuts s twotermnl networks when we determned equlent resstnces for seres nd prllel resstors. We wll refly reew these concepts here n systems context n terms of smple exmple Anlog Deces nd Dglent, Inc. 18

19 Chpter 4.4: Twotermnl Networks Exmple 4.6: Determne the chrcterstc for the crcut elow. c 3 6 Preously, we would use crcut reducton technques to sole ths prolem. The equlent resstnce s (3 ) (6 ) R eq 4. Snce Req, the crcut s chrcterstc s We would now, howeer, lke to pproch ths prolem n slghtly more generl wy nd usng systemsleel ew to the prolem. Therefore, we wll choose the termnl oltge,, to e ewed s the nput to the crcut. By defult, ths mens the current wll e our crcut s output. (We could, just s esly defne the current s the nput, n whch cse the oltge would ecome our output.) Thus, our crcut conceptully looks lke system s shown n the lock dgrm elow. Input, Crcut Output, Applyng KCL to node c n the oe crcut results n c c. Ohm s lw, ppled to the resstor, 6 3 results n c 2. Elmntng c from the oe two equtons results n 4, whch s the sme result we otned usng crcut reducton. The chrcterstc for the oe crcut s shown grphclly elow; the slope of the lne s smply the equlent resstnce of the network. 1 R eq In the oe exmple, ewng the crcut s generl twotermnl network nd usng more generl systemsleel pproch to the prolem results n ddtonl work relte to usng our preous crcut reducton pproch. Vewng the crcut s more generl twotermnl network s, howeer, ery proftle f crcut reducton technques re not pplcle or f we llow the crcut to contn oltge or current sources. The ltter topc s ddressed n the followng susecton Anlog Deces nd Dglent, Inc. 19

20 Chpter 4.4: Twotermnl Networks Twotermnl networks wth sources: When the network conssts of resste elements nd ndependent sources, the crcut s chrcterstc cn e represented s sngle equlent resstnce nd sngle sourcelke term. In generl, howeer, we cnnot determne ths drectly y usng crcut reducton technques. The oerll pproch nd typcl results re llustrted n the followng exmples Anlog Deces nd Dglent, Inc. 20

21 Chpter 4.4: Twotermnl Networks Exmple 4.7: Determne the chrcterstc of the crcut elow. R 1 R 2 V S Although t s frly pprent, y pplyng Ohm s lw cross the seres comnton of resstors, tht ( R1 R2 ) V S, we wll (for prctce) use superposton to pproch ths prolem. The oltge source V s wll, of course, e one source n the crcut. We wll use the oltge cross the termnls s second source n the crcut. Kllng the oltge source V s results n the crcut to the left elow; the resultng current s the source results n the crcut to the rght elow; the resultng current s VS s, therefore, or ( R R2 ) VS R R R R V R 1 S R 1 2 R 1 R 2. Kllng. The totl current R 1 R 2 1 R 1 R 2 2 V S Plottng the oe chrcterstc results n the fgure elow. R1R2 1 VS 2012 Anlog Deces nd Dglent, Inc. 21

22 Chpter 4.4: Twotermnl Networks Exmple 4.8: Determne the chrcterstc of the crcut elow. R 2 I S R 1 Although not the most effcent pproch for ths prolem, we wll gn use superposton to pproch the prolem. One source wll, of course, e the current source I S. We wll ssume tht our second source s the current t node. Kllng the current source I S results n the crcut to the left elow; from ths fgure the oltge 1 cn e seen to e 1 ( R1 R2 ). Kllng the current source results n the fgure to the rght elow; from ths fgure the oltge 2 s seen to e R I (the ded current source results n n open crcut, so no current 2 1 flows through the resstor R 2 ). The totl oltge cross the termnls s, therefore, S 1 2 ) 1 ( R R R I. S R 2 R 2 =0 R 1 I S R 1 The chrcterstc for the crcut s, therefore, s shown n the fgure elow. 1 R 1 R 2 R 1 I S 2012 Anlog Deces nd Dglent, Inc. 22

23 Chpter 4.4: Twotermnl Networks Notes on lner crcut chrcterstcs: 1. All twotermnl networks whch contn only sources nd resstors wll he reltonshps of the form shown n exmples 1, 2, nd 3. Tht s, they wll e strght lnes of the form m. The y ntercept term,, s due to sources n the network; f there re no sources n the network, = 0 nd the chrcterstc wll pss through the orgn. 2. Due to the form of the chrcterstc proded n note 1 oe, ny twotermnl network cn e represented s sngle source nd sngle resstor. 3. The form of the soluton for exmples 2 nd 3 re the sme. Thus, the crcut of exmple 2 s V ndstngushle from smlr crcut wth current source S n prllel wth the resstor R 1. Lkewse, the crcut of exmple 3 s ndstngushle from smlr crcut wth oltge source I S R1 n seres wth the resstor R 1. The equlent crcuts re shown elow. R 1 R 1 R 2 R 2 V S V S R 1 R 1 R 2 R 1 R 2 I S R 1 I S R Anlog Deces nd Dglent, Inc. 23

24 Chpter 4.4: Twotermnl Networks Secton Summry: Electrcl crcuts, sucrcuts, nd components re often modeled y the reltonshp etween oltge nd current t ther termnls. For exmple, we re fmlr wth modelng resstors y Ohm s lw, whch smply reltes the oltge to the current t the resstor termnls. In Chpter 2, we used crcut reducton methods to extend ths concept y replcng resste networks wth equlent resstnces whch proded the sme oltgecurrent reltons cross ther termnls. In ths secton, we contnue to extend ths concept to crcuts whch nclude sources. For lner crcuts, the oltgecurrent reltonshp cross two termnls of the crcut cn lwys e represented s strght lne of the form m. If we plot ths reltonshp wth oltge on the ertcl xs nd current on the horzontl xs, the slope of the lne corresponds to n equlent resstnce seen cross the termnls, whle the yntercept of the lne s the oltge cross the termnls, f the termnls re opencrcuted. We wll formlze ths mportnt result n secton 4.5. Exercses: 1. Determne the IV chrcterstcs of the crcut elow, s seen t the termnls. I 3V 4 V 2012 Anlog Deces nd Dglent, Inc. 24

25 Chpter 4.5: Théenn's nd Norton's Theorems 4.5: Théenn s nd Norton s Theorems In secton 4.4, we sw tht t s possle to chrcterze crcut consstng of sources nd resstors y the oltgecurrent (or ) chrcterstc seen t pr of termnls of the crcut. When we do ths, we he essentlly smplfed our descrpton of the crcut from detled model of the nternl crcut prmeters to smpler model whch descres the oerll ehor of the crcut s seen t the termnls of the crcut. Ths smpler model cn then e used to smplfy the nlyss nd/or desgn of the oerll system. In ths secton, we wll formlze the oe result s Théenn s nd Norton s theorems. Usng these theorems, we wll e le to represent ny lner crcut wth n equlent crcut consstng of sngle resstor nd source. Théenn s theorem replces the lner crcut wth oltge source n seres wth resstor, whle Norton s theorem replces the lner crcut wth current source n prllel wth resstor. In ths secton, we wll pply Théenn s nd Norton s theorems only to purely resste networks. Howeer, these theorems cn e used to represent ny crcut mde up of lner elements. Consder the two nterconnected crcuts shown n Fgure 4.7 elow. The crcuts re nterconnected t the two termnls nd, s shown. Our gol s to replce crcut A n the system of Fgure 4.7 wth smpler crcut whch hs the sme currentoltge chrcterstc s crcut A. Tht s, f we replce crcut A wth ts smpler equlent crcut, the operton of crcut B wll e unffected. We wll mke the followng ssumptons out the oerll system: Crcut A s lner Crcut A hs no dependent sources whch re controlled y prmeters wthn crcut B Crcut B hs no dependent sources whch re controlled y prmeters wthn crcut A Crcut A (Lner) Crcut B Fgure 4.7. Interconnected twotermnl crcuts. In secton 4.3, we determned chrcterstcs for seerl exmple twotermnl crcuts, usng the superposton prncple. We wll follow the sme sc pproch here, except for generl lner twotermnl crcut, n order to deelop Théenn s nd Norton s theorems. Théenn s Theorem: Frst, we wll kll ll sources n crcut A nd determne the oltge resultng from n ppled current, s shown n Fgure 4.8 elow. Wth the sources klled, crcut A wll look strctly lke n equlent resstnce to ny externl crcutry. Ths equlent resstnce s desgnted s R TH n Fgure 4.8. The oltge resultng from n ppled current, wth crcut A ded s: R 1 TH (4.3) 2012 Anlog Deces nd Dglent, Inc. 25

26 Chpter 4.5: Théenn's nd Norton's Theorems Crcut A (Sources Klled) R TH 1 1 Fgure 4.8. Crcut schemtc wth ded crcut. Now we wll determne the oltge resultng from recttng crcut A s sources nd opencrcutng termnls nd. We opencrcut the termnls here snce we presented equton (4.3) s resultng from current source, rther thn oltge source. The crcut eng exmned s s shown n Fgure 4.9. The oltge OC s the opencrcut oltge. =0 Crcut A OC Supermposng the two oltges oe results n: or OC Fgure 4.9. Opencrcut response. 1 (4.4) R (4.5) TH OC Equton (4.5) s Théenn s theorem. It ndctes tht the oltgecurrent chrcterstc of ny lner crcut (wth the excepton noted elow) cn e duplcted y n ndependent oltge source n seres wth resstnce R TH, known s the Théenn resstnce. The oltge source hs the mgntude OC nd the resstnce s R TH, where OC s the oltge seen cross the crcut s termnls f the termnls re opencrcuted nd R TH s the equlent resstnce of the crcut seen from the two termnls, wth ll ndependent sources n the crcut klled. The equlent Théenn crcut s shown n Fgure 4.10 R TH V OC Equlent Crcut 2012 Anlog Deces nd Dglent, Inc. 26

27 Chpter 4.5: Théenn's nd Norton's Theorems Procedure for determnng Théenn equlent crcut: 1. Identfy the crcut nd termnls for whch the Théenn equlent crcut s desred. 2. Kll the ndependent sources (do nothng to ny dependent sources) n crcut nd determne the equlent resstnce R TH of the crcut. If there re no dependent sources, R TH s smply the equlent resstnce of the resultng resste network. Otherwse, one cn pply n ndependent current source t the termnls nd determne the resultng oltge cross the termnls; the oltgetocurrent rto s R TH. 3. Rectte the sources nd determne the opencrcut oltge V OC cross the crcut termnls. Use ny nlyss pproch you choose to determne the opencrcut oltge Anlog Deces nd Dglent, Inc. 27

28 Chpter 4.5: Théenn's nd Norton's Theorems Exmple 4.9: Determne the Théenn equlent of the crcut elow, s seen y the lod, R L. 6V 2A 6 3 R L We wnt to crete Théenn equlent crcut of the crcut to the left of the termnls. The lod resstor, R L, tkes the plce of crcut B n Fgure 1. The crcut hs no dependent sources, so we kll the ndependent sources nd determne the equlent resstnce seen y the lod. The resultng crcut s shown elow. 6 3 From the oe fgure, t cn e seen tht the Théenn resstnce R TH s prllel comnton of 3 resstor (6 )(3 ) nd 6 resstor, n seres wth resstor. Thus, R TH The opencrcut oltge OC s determned from the crcut elow. We (rtrrly) choose nodl nlyss to determne the opencrcut oltge. There s one ndependent oltge n the crcut; t s leled s 0 n the crcut elow. Snce there s no current through the resstor, OC = 0. R TH 6V 0 =0 2A 6 3 OC 0 6V 0 Applyng KCL t 0, we otn: 2A 0 0 OC 6V. Thus, the Théenn equlent 6 3 crcut s on the left elow. Rentroducng the lod resstnce, s shown on the rght elow, llows us to esly nlyze the oerll crcut V 6V R L Norton s Theorem: 2012 Anlog Deces nd Dglent, Inc. 28

29 Chpter 4.5: Théenn's nd Norton's Theorems The pproch towrd genertng Norton s theorem s lmost dentcl to the deelopment of Théenn s theorem, except tht we pply superposton slghtly dfferently. In Théenn s theorem, we looked t the oltge response to n nput current; to deelop Norton s theorem, we look t the current response to n ppled oltge. The procedure s proded elow. Once gn, we kll ll sources n crcut A, s shown n Fgure 4.8 oe ut ths tme we determne the current resultng from n ppled oltge. Wth the sources klled, crcut A stll looks lke n equlent resstnce to ny externl crcutry. Ths equlent resstnce s desgnted s R TH n Fgure 4.8. The current resultng from n ppled oltge, wth crcut A ded s: 1 (4.6) R TH Notce tht equton (4.6) cn e otned y rerrngng equton (4.3) Now we wll determne the current resultng from recttng crcut A s sources nd shortcrcutng termnls nd. We shortcrcut the termnls here snce we presented equton (4.4) s resultng from oltge source. The crcut eng exmned s s shown n Fgure The current SC s the shortcrcut current. It s typcl to ssume tht under shortcrcut condtons the shortcrcut current enters the node t ; ths s consstent wth n ssumpton tht crcut A s genertng power under shortcrcut condtons. Crcut A SC V=0 Fgure Shortcrcut response. Employng superposton, the current nto the crcut s (notce the negte sgn on the shortcrcut current, resultng from the defnton of the drecton of the shortcrcut current opposte to the drecton of the current ) so 1 (4.7) TH SC SC (4.8) R Equton (4.8) s Norton s theorem. It ndctes tht the oltgecurrent chrcterstc of ny lner crcut (wth the excepton noted elow) cn e duplcted y n ndependent current source n prllel wth resstnce. The current source hs the mgntude SC nd the resstnce s R TH, where SC s the current seen t the crcut s termnls f the termnls re shortcrcuted nd R TH s the equlent resstnce of the crcut seen from the two termnls, wth ll ndependent sources n the crcut klled. The equlent Norton crcut s shown n Fgure Anlog Deces nd Dglent, Inc. 29

30 Chpter 4.5: Théenn's nd Norton's Theorems SC R TH Equlent Crcut Fgure Norton equlent crcut. Procedure for determnng Norton equlent crcut: 1. Identfy the crcut nd termnls for whch the Norton equlent crcut s desred. 2. Determne the equlent resstnce R TH of the crcut. The pproch for determnng R TH s the sme for Norton crcuts s Théenn crcuts. 3. Rectte the sources nd determne the shortcrcut current SC cross the crcut termnls. Use ny nlyss pproch you choose to determne the shortcrcut current Anlog Deces nd Dglent, Inc. 30

31 Chpter 4.5: Théenn's nd Norton's Theorems Exmple 4.10: Determne the Norton equlent of the crcut seen y the lod, R L, n the crcut elow. 6V 2A 6 3 R L Ths s the sme crcut s our preous exmple. The Théenn resstnce, R TH, s thus the sme s clculted preously: R TH = 4. Remong the lod resstnce nd plcng shortcrcut etween the nodes nd, s shown elow, llows us to clculte the shortcrcut current, SC. 6V 0 SC 2A 6 3 =0 Performng KCL t the node 0, results n: 0 0 6V 6 0 2A 3 so 0 3V Ohms lw cn then e used to determne SC : 3V SC 1. 5A nd the Norton equlent crcut s shown on the left elow. Replcng the lod resstnce results n the equlent oerll crcut shown to the rght elow. 1.5A 4 1.5A 4 R L 2012 Anlog Deces nd Dglent, Inc. 31

32 Chpter 4.5: Théenn's nd Norton's Theorems Exceptons: Not ll crcuts he Théenn nd Norton equlent crcuts. Exceptons re: 1. An del current source does not he Théenn equlent crcut. (It cnnot e represented s oltge source n seres wth resstnce.) It s, howeer, ts own Norton equlent crcut. 2. An del oltge source does not he Norton equlent crcut. (It cnnot e represented s current source n prllel wth resstnce.) It s, howeer, ts own Théenn equlent crcut. Source Trnsformtons: Crcut nlyss cn sometmes e smplfed y the use of source trnsformtons. Source trnsformtons re performed y notng tht Théenn s nd Norton s theorems prode two dfferent crcuts whch prode essentlly the sme termnl chrcterstcs. Thus, we cn wrte oltge source whch s n seres wth resstnce s current source n prllel wth the sme resstnce, nd ceers. Ths s done s follows. Equtons (4.5) nd (4.8) re oth representtons of the chrcterstc of the sme crcut. Rerrngng equton (4.5) to sole for the current results n: R OC (4.9) TH R TH Equtng equtons (4.8) nd (4.9) leds to the concluson tht OC SC (4.10) RTH Lkewse, rerrngng equton (4.8) to otn n expresson for ges: R R (4.11) TH Equtng equtons (4.11) nd (4.5) results n: SC TH OC R (4.12) SC TH whch s the sme result s equton (4.10). Equtons (4.10) nd (4.12) led us to the concluson tht ny crcut consstng of oltge source n seres wth resstor cn e trnsformed nto current source n prllel wth the sme resstnce. Lkewse, current source n prllel wth resstnce cn e trnsformed nto oltge source n seres wth the sme resstnce. The lues of the trnsformed sources must e scled y the resstnce lue ccordng to equtons (4.10) nd (4.12). The trnsformtons re depcted n Fgure Anlog Deces nd Dglent, Inc. 32

33 Chpter 4.5: Théenn's nd Norton's Theorems R VS V S R R R I S R I S R Fgure Source trnsformtons. Source trnsformtons cn smplfy the nlyss of some crcuts sgnfcntly, especlly crcuts whch consst of seres nd prllel comntons of resstors nd ndependent sources. An exmple s proded elow Anlog Deces nd Dglent, Inc. 33

34 Chpter 4.5: Théenn's nd Norton's Theorems Exmple 4.11: Determne the current n the crcut shown elow V 6 2A We cn use source trnsformton to replce the 9V source nd 3 resstor seres comnton wth 3A source n prllel wth 3 resstor. Lkewse, the 2A source nd resstor prllel comnton cn e replced wth 4V source n seres wth resstor. After these trnsformtons he een mde, the prllel resstors cn e comned s shown n the fgure elow A 3 6 4V 3A 4V The 3A source nd resstor prllel comnton cn e comned to 6V source n seres wth resstor, s shown elow. 4 6V 4V The current cn now e determned y drect pplcton of Ohm s lw to the three seres resstors, so tht 6V 4V 0. 25A. 4 Voltge Current Chrcterstcs of Théenn nd Norton Crcuts: Preously, n secton 4.4, we noted tht the chrcterstcs of lner twotermnl networks contnng only sources nd resstors re strght lnes. We now look t the oltgecurrent chrcterstcs n terms of Théenn nd Norton equlent crcuts Anlog Deces nd Dglent, Inc. 34

35 Chpter 4.5: Théenn's nd Norton's Theorems Equtons (4.5) nd (4.8) oth prode lner oltgecurrent chrcterstc s shown n Fgure When the current nto the crcut s zero (opencrcuted condtons), the oltge cross the termnls s the opencrcut oltge, OC. Ths s consstent wth equton (4.5), eluted t = 0: R R 0. TH OC OC TH OC OC Lkewse, under shortcrcuted condtons, the oltge dfferentl cross the termnls s zero nd equton (4.8) redly prodes: R SC TH sc 0 R TH sc sc whch s consstent wth Fgure R TH OC 1 SC Fgure Voltgecurrent chrcterstc for Théenn nd Norton equlent crcuts. Fgure 4.14 s lso consstent wth equtons (4.10) nd (4.12) oe, snce grphclly the slope of the lne s OC oously RTH. SC Fgure 4.14 lso ndctes tht there re three smple wys to crete Théenn nd Norton equlent crcuts: 1. Determne R TH nd OC. Ths prodes the slope nd yntercept of the chrcterstc. Ths pproch s outlned oe s the method for cretng Théenn equlent crcut. 2. Determne R TH nd SC. Ths prodes the slope nd xntercept of the chrcterstc. Ths pproch s outlned oe s the method for cretng Norton equlent crcut. OC 3. Determne OC nd SC. The equlent resstnce R TH cn then e clculted from RTH to SC determne the slope of the chrcterstc. Ether Théenn or Norton equlent crcut cn then e creted. Ths pproch s not commonly used, snce determnng R TH the equlent resstnce of the crcut s usully eser thn determnng ether OC or SC Anlog Deces nd Dglent, Inc. 35

36 Chpter 4.5: Théenn's nd Norton's Theorems Note: It should e emphszed tht the Théenn nd Norton crcuts re not ndependent enttes. One cn lwys e determned from the other source trnsformton. Théenn nd Norton crcuts re smply two dfferent wys of expressng the sme oltgecurrent chrcterstc. Secton Summry: Théenn s theorem llows us to replce ny lner porton of crcut wth equlent crcut consstng of oltge source n seres wth resstnce. Ths crcut s clled the Théenn equlent, nd prodes the sme oltgecurrent reltonshp t the termnls s the orgnl crcut. The oltge source n the equlent crcut s the sme s the oltge whch would e mesured cross the termnls of the orgnl crcut, f those termnls were opencrcuted. The resstnce n the equlent crcut s clled the Théenn resstnce, t s the resstnce tht would e seen cross the termnls of the orgnl crcut, f ll sources n the crcut were klled. Norton s theorem llows us to replce ny lner porton of crcut wth equlent crcut consstng of current source n prllel wth resstnce. Ths crcut s clled the Norton equlent, nd prodes the sme oltgecurrent reltonshp t the termnls s the orgnl crcut. The current source n the equlent crcut s the sme s the current whch would e mesured cross the termnls of the orgnl crcut, f those termnls were shortcrcuted. The resstnce n the equlent crcut s the resstnce tht would e seen cross the termnls of the orgnl crcut, f ll sources n the crcut were klled; t s the sme s the Théenn resstnce. Theenn nd Norton s theorems llow us to perform source trnsformtons when nlyzng crcuts. Ths pproch smply llows us to replce ny oltge source whch s n seres wth resstnce wth current source n prllel wth the sme resstnce, nd ceers. The reltonshp etween the oltge nd current sources used n these trnsformtons re proded n equtons (4.10) nd (4.12) Anlog Deces nd Dglent, Inc. 36