Dorf, R.C., Wa, Z., Joho, D.E. Laplace Traform The Electrical Egieerig Hadbook Ed. Richard C. Dorf Boca Rato: CRC Pre LLC,
6 Laplace Traform Richard C. Dorf Uiverity of Califoria, Davi Zhe Wa Uiverity of Califoria, Davi David E. Joho Birmigham-Souther College 6. Defiitio ad Propertie Laplace Traform Itegral Regio of Abolute Covergece Propertie of Laplace Traform Time-Covolutio Property Time-Correlatio Property Ivere Laplace Traform 6. Applicatio Differetiatio Theorem Applicatio to Itegrodifferetial Equatio Applicatio to Electric Circuit The Traformed Circuit Thévei ad Norto Theorem Network Fuctio Step ad Impule Repoe Stability 6. Defiitio ad Propertie Richard C. Dorf ad Zhe Wa The Laplace traform i a ueful aalytical tool for covertig time-domai igal decriptio ito fuctio of a complex variable. Thi complex domai decriptio of a igal provide ew iight ito the aalyi of igal ad ytem. I additio, the Laplace traform method ofte implifie the calculatio ivolved i obtaiig ytem repoe igal. Laplace Traform Itegral The Laplace traform completely characterize the expoetial repoe of a time-ivariat liear fuctio. Thi traformatio i formally geerated through the proce of multiplyig the liear characteritic igal x(t) by the igal e t ad the itegratig that product over the time iterval (, + ). Thi ytematic procedure i more geerally kow a takig the Laplace traform of the igal x(t). Defiitio: The Laplace traform of the cotiuou-time igal x(t) i + X () xte () - The variable that appear i thi itegrad expoetial i geerally complex valued ad i therefore ofte expreed i term of it rectagular coordiate + j w where Re() ad w Im() are referred to a the real ad imagiary compoet of, repectively. The igal x(t) ad it aociated Laplace traform X() are aid to form a Laplace traform pair. Thi reflect a form of equivalecy betwee the two apparetly differet etitie x(t) ad X(). We may ymbolize thi iterrelatiohip i the followig uggetive maer: -t by CRC Pre LLC
X() [x(t)] where the operator otatio mea to multiply the igal x(t) beig operated upo by the complex expoetial e t ad the to itegrate that product over the time iterval (, + ). Regio of Abolute Covergece I evaluatig the Laplace traform itegral that correpod to a give igal, it i geerally foud that thi itegral will exit (that i, the itegral ha fiite magitude) for oly a retricted et of value. The defiitio of regio of abolute covergece i a follow. The et of complex umber for which the magitude of the Laplace traform itegral i fiite i aid to cotitute the regio of abolute covergece for that itegral traform. Thi regio of covergece i alway expreible a + < Re() < where + ad deote real parameter that are related to the caual ad aticaual compoet, repectively, of the igal whoe Laplace traform i beig ought. Laplace Traform Pair Table It i coveiet to diplay the Laplace traform of tadard igal i oe table. Table 6. diplay the time igal x(t) ad it correpodig Laplace traform ad regio of abolute covergece ad i ufficiet for our eed. Example. To fid the Laplace traform of the firt-order caual expoetial igal x (t) e at u(t) where the cotat a ca i geeral be a complex umber. The Laplace traform of thi geeral expoetial igal i determied upo evaluatig the aociated Laplace traform itegral X () e ute () e + + -at - t -( + at ) - -( + at ) + e - ( + a) (6.) I order for X () to exit, it mut follow that the real part of the expoetial argumet be poitive, that i, Re( + a) Re() + Re(a) > If thi were ot the cae, the evaluatio of expreio (6.) at the upper limit t + would either be ubouded if Re() + Re(a) < or udefied whe Re() + Re(a). O the other had, the upper limit evaluatio i zero whe Re() + Re(a) >, a i already apparet. The lower limit evaluatio at t i equal to /( + a) for all choice of the variable. The Laplace traform of expoetial igal e at u(t) ha therefore bee foud ad i give by -at L [ e ut ()] for Re( ) > -Re( a) + a by CRC Pre LLC
TABLE 6. Laplace Traform Pair Propertie of Laplace Traform Liearity Let u obtai the Laplace traform of a igal, x(t), that i compoed of a liear combiatio of two other igal, where a ad a are cotat. The liearity property idicate that Time Sigal Laplace Traform Regio of x(t) X() Abolute Covergece. e at u(t) Re() > Re(a) + a. t k e at u( t) k! Re() > Re(a) ( + a) k + 3. e at u( t) Re() < Re(a) ( + a) 4. ( t) k e at u( t) k! Re() < Re(a) ( + a) k + 5. u(t) Re() > 6. d(t) all 7. k d d( t) k all k 8. t k u(t) k! Re() > k + ì, t ³ 9. g t í Re() î, t < w. i w t u(t) Re() > + w. co w t u(t) Re() > +w. e at i w t u(t) w Re() > Re(a) ( + a) + w 3. e at co w t u(t) + a Re() > Re(a) ( + a) + w Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 33. With permiio. x(t) a x (t) + a x (t) [a x (t) + a x (t)] a X () + a X () ad the regio of abolute covergece i at leat a large a that give by the expreio by CRC Pre LLC
FIGURE 6. Equivalet operatio i the (a) time-domai operatio ad (b) Laplace traform-domai operatio. (Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 38. With permiio.) where the pair ( +; + )< Re() < mi( ; ) idetify the regio of covergece for the Laplace traform X () ad X (), repectively. Time-Domai Differetiatio + + - - max( ; ) < Re( ) < mi( ; ) The operatio of time-domai differetiatio ha the bee foud to correpod to a multiplicatio by i the Laplace variable domai. The Laplace traform of differetiated igal dx(t)/ i é dxt () ù ê ë ú û X() Furthermore, it i clear that the regio of abolute covergece of dx(t)/ i at leat a large a that of x(t). Thi property may be eviioed a how i Fig. 6.. Time Shift The igal x(t t ) i aid to be a verio of the igal x(t) right hifted (or delayed) by t ecod. Right hiftig (delayig) a igal by a t ecod duratio i the time domai i ee to correpod to a multiplicatio by e t i the Laplace traform domai. The deired Laplace traform relatiohip i [ xt ( - t )] e -t X() where X() deote the Laplace traform of the uhifted igal x(t). A a geeral rule, ay time a term of the form e t appear i X(), thi implie ome form of time hift i the time domai. Thi mot importat property i depicted i Fig. 6.. It hould be further oted that the regio of abolute covergece for the igal x(t) ad x(t t ) are idetical. FIGURE 6. Equivalet operatio i (a) the time domai ad (b) the Laplace traform domai. (Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 4. With permiio.) by CRC Pre LLC
FIGURE 6.3 Repreetatio of a time-ivariat liear operator i (a) the time domai ad (b) the -domai. (Source: J. A. Cadzow ad H. F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 44. With permiio.) Time-Covolutio Property The covolutio itegral igal y(t) ca be expreed a where x(t) deote the iput igal, the h(t) characteritic igal idetifyig the operatio proce. The Laplace traform of the repoe igal i imply give by where H() [h(t)] ad X() [x(t)]. Thu, the covolutio of two time-domai igal i ee to correpod to the multiplicatio of their repective Laplace traform i the -domai. Thi property may be eviioed a how i Fig. 6.3. Time-Correlatio Property yt () h( t) xt ( - t) d t - Y () HX () () The operatio of correlatig two igal x(t) ad y(t) i formally defied by the itegral relatiohip fxy ( t ) xtyt () ( + t) The Laplace traform property of the correlatio fuctio f xy (t) i - F xy () X( -Y ) () i which the regio of abolute covergece i give by max( - -, + ) < Re( ) < mi( - +, - ) x y x y by CRC Pre LLC
Autocorrelatio Fuctio The autocorrelatio fuctio of the igal x(t) i formally defied by fxx( t ) xtxt () ( + t) The Laplace traform of the autocorrelatio fuctio i - F xx () X( -X ) () ad the correpodig regio of abolute covergece i Other Propertie max( -, ) < Re( ) < mi( -, ) - + + - x y x A umber of propertie that characterize the Laplace traform are lited i Table 6.. Applicatio of thee propertie ofte eable oe to efficietly determie the Laplace traform of eemigly complex time fuctio. y TABLE 6. Laplace Traform Propertie Sigal x(t) Laplace Traform Regio of Covergece of X() Property Time Domai X() Domai + < Re() < Liearity a x (t) + a x (t) a X () + a X () At leat the iterectio of the regio of covergece of X () ad X () Time differetiatio X() At leat + < Re() ad X () Time hift x(t t ) e t X() + < Re() < Time covolutio Time calig x(at) H()X() At leat the iterectio of the regio of covergece of H() ad X() Frequecy hift e at x(t) X( + a) + Re(a) < Re() < Re(a) Multiplicatio (frequecy covolutio) x (t)x (t) t Time itegratio x( t) At leat + < Re() < X() for X ( ) Frequecy differetiatio Time correlatio Autocorrelatio fuctio dx() t h( t) x( t - t ) - - ( t) k x(t) + x() t yt ( + z ) - + x() t x( t + z ) - æ X ö æ ö ç < Re ç < a è a ø è a ø c + j X( ux ) ( - ud ) pj c - j k d X() k d + - + < Re( ) < + () ( ) () ( ) + + - - + < c < + () ( ) () ( ) + + - - At leat + < Re() < X( )Y() max( x, y+ ) < Re() < mi( x+, y ) X( )X() max( x, x+ ) < Re() < mi( x+, x ) Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985. With permiio. by CRC Pre LLC
Ivere Laplace Traform Give a traform fuctio X() ad it regio of covergece, the procedure for fidig the igal x(t) that geerated that traform i called fidig the ivere Laplace traform ad i ymbolically deoted a xt () ± [ X()] The igal x(t) ca be recovered by mea of the relatiohip I thi itegral, the real umber c i to be elected o that the complex umber c + jw lie etirely withi the regio of covergece of X() for all value of the imagiary compoet w. For the importat cla of ratioal Laplace traform fuctio, there exit a effective alterate procedure that doe ot eceitate directly evaluatig thi itegral. Thi procedure i geerally kow a the partial-fractio expaio method. Partial Fractio Expaio Method c + j t xt () Xe () d pj c - j A jut idicated, the partial fractio expaio method provide a coveiet techique for reacquirig the igal that geerate a give ratioal Laplace traform. Recall that a traform fuctio i aid to be ratioal if it i expreible a a ratio of polyomial i, that i, X () B () A () m m m - m - - - b + b + + b + b + a + + a + a The partial fractio expaio method i baed o the appealig otio of equivaletly expreig thi ratioal traform a a um of elemetary traform whoe correpodig ivere Laplace traform (i.e., geeratig igal) are readily foud i tadard Laplace traform pair table. Thi method etail the imple five-tep proce a outlied i Table 6.3. A decriptio of each of thee tep ad their implemetatio i ow give. I. Proper Form for Ratioal Traform. Thi diviio proce yield a expreio i the proper form a give by B () X () A () R () Q () + A () TABLE 6.3 Partial Fractio Expaio Method for Determiig the Ivere Laplace Traform I. Put ratioal traform ito proper form whereby the degree of the umerator polyomial i le tha or equal to that of the deomiator polyomial. II. Factor the deomiator polyomial. III. Perform a partial fractio expaio. IV. Separate partial fractio expaio term ito caual ad aticaual compoet uig the aociated regio of abolute covergece for thi purpoe. V. Uig a Laplace traform pair table, obtai the ivere Laplace traform. Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 53. With permiio. by CRC Pre LLC
i which Q() ad R() are the quotiet ad remaider polyomial, repectively, with the diviio made o that the degree of R() i le tha or equal to that of A(). II. Factorizatio of Deomiator Polyomial. The ext tep of the partial fractio expaio method etail the factorizig of the th-order deomiator polyomial A() ito a product of firt-order factor. Thi factorizatio i alway poible ad reult i the equivalet repreetatio of A() a give by A( ) ( - p)( - p)...( - p ) The term p, p,..., p cotitutig thi factorizatio are called the root of polyomial A(), or the pole of X(). III. Partial Fractio Expaio. With thi factorizatio of the deomiator polyomial accomplihed, the ratioal Laplace traform X() ca be expreed a X () - B () b + b - + + b A () ( - p )( - p ) ( - p ) (6.) We hall ow equivaletly repreet thi traform fuctio a a liear combiatio of elemetary traform fuctio. Cae : A() Ha Ditict Root. X () a + a a a + + + - p - p - p where the a k are cotat that idetify the expaio ad mut be properly choe for a valid repreetatio. a k ( - p ) X( ) for k,,..., k p k ad a b The expreio for parameter a i obtaied by lettig become ubouded (i.e., + ) i expaio (6.). Cae : A() Ha Multiple Root. X () B () A () B () q - ( p ) A ( ) The appropriate partial fractio expaio of thi ratioal fuctio i the give by X() a + a a q + + ( - p ) ( - p ) q + ( - q) other elemetary term due to the root of A ( ) by CRC Pre LLC
The coefficiet a may be expedietly evaluated by lettig approach ifiity, whereby each term o the right ide goe to zero except a. Thu, a lim X () + The a q coefficiet i give by the coveiet expreio a q q ( - p ) X () Bp ( ) A ( p ) p (6.3) The remaiig coefficiet a, a,, a q aociated with the multiple root p may be evaluated by olvig Eq. (6.3) by ettig to a pecific value. IV. Caual ad Aticaual Compoet. I a partial fractio expaio of a ratioal Laplace traform X() whoe regio of abolute covergece i give by < Re( ) < it i poible to decompoe the expaio elemetary traform fuctio ito caual ad aticaual fuctio (ad poibly impule-geerated term). Ay elemetary fuctio i iterpreted a beig () caual if the real compoet of it pole i le tha or equal to + ad () aticaual if the real compoet of it pole i greater tha or equal to. The pole of the ratioal traform that lie to the left (right) of the aociated regio of abolute covergece correpod to the caual (aticaual) compoet of that traform. Figure 6.4 how the locatio of caual ad aticaual pole of ratioal traform. V. Table Look-Up of Ivere Laplace Traform. To complete the ivere Laplace traform procedure, oe eed imply refer to a tadard Laplace traform fuctio table to determie the time igal that geerate each of the elemetary traform fuctio. The required time igal i the equal to the ame liear combiatio of the ivere Laplace traform of thee elemetary traform fuctio. + - FIGURE 6.4 Locatio of caual ad aticaual pole of a ratioal traform. (Source: J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985, p. 6. With permiio.) by CRC Pre LLC
Defiig Term Laplace traform: A traformatio of a fuctio f(t) from the time domai ito the complex frequecy domai yieldig F(). F () f() te - where + jw. Regio of abolute covergece: The et of complex umber for which the magitude of the Laplace traform itegral i fiite. The regio ca be expreed a -t where + ad deote real parameter that are related to the caual ad aticaual compoet, repectively, of the igal whoe Laplace traform i beig ought. Related Topic 4. Itroductio Referece J.A. Cadzow ad H.F. Va Ladigham, Sigal, Sytem, ad Traform, Eglewood Cliff, N.J.: Pretice-Hall, 985. E. Kame, Itroductio to Sigal ad Sytem, d Ed., Eglewood Cliff, N.J.: Pretice-Hall, 99. B.P. Lathi, Sigal ad Sytem, Carmichael, Calif.: Berkeley-Cambridge Pre, 987. 6. Applicatio David E. Joho < Re( ) < I applicatio uch a electric circuit, we tart coutig time at t, o that a typical fuctio f(t) ha the property f(t), t <. It traform i give therefore by + - which i ometime called the oe-ided Laplace traform. Sice f(t) i like x(t)u(t) we may till ue Table 6. of the previou ectio to look up the traform, but for implicity we will omit the factor u(t), which i udertood to be preet. Differetiatio Theorem Time-Domai Differetiatio F () f() te If we replace f(t) i the oe-ided traform by it derivative f (t) ad itegrate by part, we have the traform of the derivative, -t Baed o D.E. Joho, J.R. Joho, ad J.L. Hilbur, Electric Circuit Aalyi, d ed., Eglewood Cliff, N.J.: Pretice- Hall, 99, chapter 9 ad. With permiio. by CRC Pre LLC
[ f ( t)] F( ) - f( ) (6.4) We may formally replace f by f to obtai [ f ( t)] [ f ( t)] - f ( ) or by (6.4), [ f ( t)] F( ) - f( ) - f ( ) (6.5) We may replace f by f agai i (6.5) to obtai [f (t)], ad o forth, obtaiig the geeral reult, ( ) - - ( -) [ f ( t)] F( ) - f( ) - f ( ) -L - f ( ) (6.6) where f () i the th derivative. The fuctio f, f,, f ( ) are aumed to be cotiuou o (, ), ad f () i cotiuou except poibly for a fiite umber of fiite dicotiuitie. Example 6.. A a example, let f(t) t, for a oegative iteger. The f () (t)! ad f() f () f ( ) (). Therefore, we have [!] [ t ] or! [ t ] [!] ;,,, ¼ + (6.7) Example 6.. A aother example, let u ivert the traform F () 3 8 ( + ) which ha the partial fractio expaio where A B C D F () + + + + 3 ad 3 A F() 4 D ( + ) F( ) - - by CRC Pre LLC
To obtai B ad C, we clear F() of fractio, reultig i 3 8 4( + ) + B( + ) + C ( + ) - Equatig coefficiet of 3 yield C, ad equatig thoe of yield B. The traform i therefore F ()!! - + - 3 + o that f() t t - t + - e - t Frequecy-Domai Differetiatio Frequecy-domai differetiatio formula may be obtaied by differetiatig the Laplace traform with repect to. That i, if F() [ f(t)], df() d d d f() t e -t Aumig that the operatio of differetiatio ad itegratio may be iterchaged, we have df() d d -t [ f( t) e ] d -t [-tf( t)] e From the lat itegral it follow by defiitio of the traform that [ tf( t)] - df() d (6.8) Example 6..3 A a example, if f(t) co kt, the F() /( + k ), ad we have [ t co kt] - d d æ ç è ö + k ø - k ( + k ) We may repeatedly differetiate the traform to obtai the geeral cae df () d -t [(-t) f( t)] e from which we coclude that by CRC Pre LLC
TABLE 6.4 Oe-Sided Laplace Traform Propertie f(t) F(). cf(t) cf(). f (t) + f (t) F () + F () 3. df() t F() f() 4. d f() t - - F () - f ( ) - f ( ) - - f ( ) - - f ( - L )( ) 5. t f( t) F () 6. e at f(t) F( + a) 7. f(t t)u(t t) e t F() t 8. f * g f gt - d F()G() ( t) ( t) t 9. f(ct), c > c F æ ö ç è c ø. t f(t),,,,... ( ) F () () () [ t f ()] t d F (- ) ;,,, ¼ d (6.9) Propertie of the Laplace traform obtaied i thi ad the previou ectio are lited i Table 6.4. Applicatio to Itegrodifferetial Equatio If we traform both member of a liear differetial equatio with cotat coefficiet, the reult will be a algebraic equatio i the traform of the ukow variable. Thi follow from Eq. (6.6), which alo how that the iitial coditio are automatically take ito accout. The traformed equatio may the be olved for the traform of the ukow ad iverted to obtai the time-domai awer. Thu, if the differetial equatio i the traformed equatio i a x ( ) a x ( - ) a x f t + + ¼ + ( ) - - - a X x x ( ) [ ( ) - ( ) - ¼ - ( )] - - - + a X( ) - x ( ) - ¼ - x ( ) ( ) - [ ] + ¼ + ax( ) F ( ) The traform X() may the be foud ad iverted to give x(t). by CRC Pre LLC
Example 6..4 A a example, let u fid the olutio x(t), for t >, of the ytem of equatio x + 4x + 3x e -t x( ), x ( ) Traformig, we have X () - - + 4 [ X() - ] + 3X () + from which X () + 8 + 3 ( + )( + )( + 3) The partial fractio expaio i X () 3 - - + + + 3 from which Itegratio Property xt () 3e - e - e Certai itegrodifferetial equatio may be traformed directly without firt differetiatig to remove the itegral. We eed oly traform the itegral by mea of -t -t -3t t é ù ê f( t) ë ú û F () FIGURE 6.5 A RLC circuit. Example 6..5 of equatio, A a example, the curret i(t) i Fig. 6.5, with o iitial tored eergy, atifie the ytem di t + i + 5 i ut () i( ) Traformig yield 5 I() I () I + + () by CRC Pre LLC
or I () é ù ê ú + + 5 ë( + ) + 4 û Therefore the curret i Applicatio to Electric Circuit A the foregoig example how, the Laplace traform method i a elegat procedure tha ca be ued for olvig electric circuit by traformig their decribig itegrodifferetial equatio ito algebraic equatio ad applyig the rule of algebra. If there i more tha oe loop or odal equatio, their traformed equatio are olved imultaeouly for the deired circuit curret or voltage traform, which are the iverted to obtai the time-domai awer. Superpoitio i ot eceary becaue the variou ource fuctio appearig i the equatio are imply traformed ito algebraic quatitie. The Traformed Circuit Itead of writig the decribig circuit equatio, traformig the reult, ad olvig for the traform of the circuit curret or voltage, we may go directly to a traformed circuit, which i the origial circuit with the curret, voltage, ource, ad paive elemet replaced by traformed equivalet. The curret or voltage traform are the foud uig ordiary circuit theory ad the reult iverted to the time-domai awer. Voltage Law Traformatio Firt, let u ote that if we traform Kirchhoff voltage law, we have -t it () 5. e i t A v () t + v () t + + v () t V () + V () + + V () where V i () i the traform of v i (t).the traformed voltage thu atify Kirchhoff voltage law. A imilar procedure will how that traformed curret atify Kirchhoff curret law, a well. Next, let u coider the paive elemet. For a reitace R, with curret i R ad voltage v R, for which the traformed equatio i v R Ri R V () RI () R R (6.) Thi reult may be repreeted by the traformed reitor elemet of Fig. 6.6(a). Iductor Traformatio For a iductace L, the voltage i v L L di L / by CRC Pre LLC
FIGURE 6.6 Traformed circuit elemet. Traformig, we have V () LI () - Li ( ) L L L (6.) which may be repreeted by a iductor with impedace L i erie with a ource, Li L (), with the proper polarity, a how i Fig. 6.6(b). The icluded voltage ource take ito accout the iitial coditio i L (). Capacitor Traformatio I the cae of a capacitace C we have which traform to v t i + v C C C C ( ) V C I () () + v ( ) C C C (6.) Thi i repreeted i Fig. 6.6(c) a a capacitor with impedace /C i erie with a ource, v C ()/, accoutig for the iitial coditio. We may olve Eq. (6.), (6.), ad (6.) for the traformed curret ad ue the reult to obtai alterate traformed elemet ueful for odal aalyi, a oppoed to thoe of Fig. 6.6, which are ideal for loop aalyi. The alterate elemet are how i Fig. 6.7. Source Traformatio Idepedet ource are imply labeled with their traform i the traformed circuit. Depedet ource are traformed i the ame way a paive elemet. For example, a cotrolled voltage ource defied by FIGURE 6.7 Traformed elemet ueful for odal aalyi. by CRC Pre LLC
FIGURE 6.8 (a) A circuit ad (b) it traformed couterpart. v (t) Kv (t) traform to V () KV () which i the traformed circuit i the traformed ource cotrolled by a traformed variable. Sice Kirchhoff law hold ad the rule for impedace hold, the traformed circuit may be aalyzed exactly a we would a ordiary reitive circuit. Example 6..6 To illutrate, let u fid i(t) i Fig. 6.8(a), give that i() 4 A ad v() 8 V. The traformed circuit i how i Fig. 6.8(b), from which we have I () [ / ( + 3)] + 4 - ( 8/ ) 3 + + ( / ) Thi may be writte I () - 3 + - + + 3 + 3 o that it () - 3 e + e - 3e -t -t -3t A Thévei ad Norto Theorem Sice the procedure uig traformed circuit i idetical to that uig the phaor equivalet circuit i the ac teady-tate cae, we may obtai traformed Thévei ad Norto equivalet circuit exactly a i the phaor cae. That i, the Thévei impedace will be Z th () ee at the termial of the traformed circuit with the ource made zero, ad the ope-circuit voltage ad the hort-circuit curret will be V oc () ad I c (), repectively, at the circuit termial. The procedure i exactly like that for reitive circuit, except that i the traformed circuit the quatitie ivolved are fuctio of. Alo, a i the reitor ad phaor cae, the opecircuit voltage ad hort-circuit curret are related by V () Z () I () oc th c (6.3) by CRC Pre LLC
FIGURE 6.9 (a) A RLC parallel circuit ad (b) it traformed circuit. Example 6..7 A a example, let u coider the circuit of Fig. 6.9(a) with the traformed circuit how i Fig. 6.9(b). The iitial coditio are i() A ad v() 4 V. Let u fid v(t) for t > by replacig everythig to the right of the 4-W reitor i Fig. 6.9(b) by it Thévei equivalet circuit. We may fid Z th () directly from Fig. 6.9(b) a the impedace to the right of the reitor with the two curret ource made zero (ope circuited). For illutrative purpoe we chooe, however, to fid the ope-circuit voltage ad hort-circuit curret how i Fig. 6.(a) ad (b), repectively, ad ue Eq. (6.3) to get the Thévei impedace. The odal equatio i Fig. 6.(a) i V oc () 3 + + Voc () 4 6 from which we have Voc () 4( - 6) + 8 From Fig. 6.(b) Ic () -6 6 The Thévei impedace i therefore Z th () Voc () I () c é4( - 6) ù ëê + 8 ûú é - 6 ù ëê 6 ûú 4 + 8 FIGURE 6. Circuit for obtaiig (a) V oc () ad (b) I c (). by CRC Pre LLC
FIGURE 6. Thévei equivalet circuit termiated i a reitor. ad the Thévei equivalet circuit, with the 4 W coected, i how i Fig. 6.. From thi circuit we fid the traform V () 4( - 6) - 6 + ( + )( + 4) + + 4 from which - t -4t vt () - 6 e + e V Network Fuctio A etwork fuctio or trafer fuctio i the ratio H() of the Laplace traform of the output fuctio, ay v o (t), to the Laplace traform of the iput, ay v i (t), aumig that there i oly oe iput. (If there are multiple iput, the trafer fuctio i baed o oe of them with the other made zero.) Suppoe that i the geeral cae the iput ad output are related by the differetial equatio a dv o - d v a a dv o o + - + L + + - av o o b d m v m - d v b b dv i i i m + m m - + L + + m - bv o i ad that the iitial coditio are all zero; that i, v ( ) o - m - dvo( ) d vo( ) dvi( ) d vi( ) v i( ) m L L - - The, traformig the differetial equatio reult i - ( a + a + L + a + a ) V () - from which the etwork fuctio, or trafer fuctio, i give by o m m - ( b + b + L + b + b ) V() m m - i H () Vo() V () i m m m - m - - - b + b + + b + b a + a + + a + a (6.4) by CRC Pre LLC
FIGURE 6. A RLC circuit. Example 6..8 A a example, let u fid the trafer fuctio for the traformed circuit of Fig. 6., where the trafer fuctio i V o ()/V i (). By voltage diviio we have H () Vo() 4 4 V () + 4 + ( 3/ ) ( + )( + 3 ) i (6.5) Step ad Impule Repoe I geeral, if Y() ad X() are the traformed output ad iput, repectively, the the etwork fuctio i H() Y()/X() ad the output i Y () HX () () (6.6) The tep repoe r(t) i the output of a circuit whe the iput i the uit tep fuctio u(t), with traform /. Therefore, the traform of the tep repoe R() i give by R () H () / (6.7) The impule repoe h(t) i the output whe the iput i the uit impule d(t). Sice [d(t)], we have from Eq. (6.6), ht () - [ H() ] - / [ H()] (6.8) Example 6..9 expaio, A a example, for the circuit of Fig. 6., H(), give i Eq. (6.5), ha the partial fractio H () - + + 6 + 3 o that - t - t ht () - e + 6e 3 V If we kow the impule repoe, we ca fid the trafer fuctio, H() [ ht ()] from which we ca fid the repoe to ay iput. I the cae of the tep ad impule repoe, it i udertood that there are o other iput except the tep or the impule. Otherwie, the trafer fuctio would ot be defied. by CRC Pre LLC
Stability A importat cocer i circuit theory i whether the output igal remai bouded or icreae idefiitely followig the applicatio of a iput igal. A ubouded output could damage or eve detroy the circuit, ad thu it i importat to kow before applyig the iput if the circuit ca accommodate the expected output. Thi quetio ca be awered by determiig the tability of the circuit. A circuit i defied to have bouded iput bouded output (BIBO) tability if ay bouded iput reult i a bouded output. The circuit i thi cae i aid to be abolutely table or ucoditioally table. BIBO tability ca be determied by examiig the pole of the etwork fuctio (6.4). If the deomiator of H() i Eq. (6.4) cotai a factor ( p), the p i aid to be a pole of H() of order. The output V o () would alo cotai thi factor, ad it partial fractio expaio would cotai the term K/( p). Thu, the ivere traform v o (t) i of the form v () t At e + A t e + + Ae + v () t o - pt - pt pt - (6.9) where v (t) reult from other pole of V o (). If p i a real poitive umber or a complex umber with a poitive real part, v o (t) i ubouded becaue e pt i a growig expoetial. Therefore, for abolute tability there ca be o pole of V o () that i poitive or ha a poitive real part. Thi i equivalet to ayig that V o () ha o pole i the right half of the -plae. Sice v i (t) i bouded, V i () ha o pole i the right half-plae. Therefore, ice the oly pole of V o () are thoe of H() ad V i (), o pole of H() for a abolutely table circuit ca be i the right-half of the -plae. From Eq. (6.9) we ee that v i (t) i bouded, a far a pole p i cocered, if p i a imple pole (of order ) ad i purely imagiary. That i, p jw, for which which ha a bouded magitude. Ule V i () cotribute a idetical pole jw, v o (t) i bouded. Thu, v o (t) i bouded o the coditio that ay jw pole of H() i imple. I ummary, a etwork i abolutely table if it etwork fuctio H() ha oly left half-plae pole. It i coditioally table if H() ha oly imple jw-axi pole ad poibly left half-plae pole. It i utable otherwie (right half-plae or multiple jw-axi pole). Example 6.. A a example, the circuit of Fig. 6. i abolutely table, ice from Eq. (6.5) the oly pole of it trafer fuctio are, 3, which are both i the left half-plae. There are coutle example of coditioally table circuit that are extremely ueful, for example, a etwork coitig of a igle capacitor with C F with iput curret I() ad output voltage V(). The trafer fuctio i H() Z() /C /, which ha the imple pole o the jw-axi. Figure 6.3 illutrate a circuit which i utable. The trafer fuctio i which ha the right half-plae pole. pt e co wt + j i wt H () I ()/ V () / ( - ) i FIGURE 6.3 Utable circuit. by CRC Pre LLC
Defiig Term Abolute tability: Whe the etwork fuctio H() ha oly left half-plae pole. Bouded iput bouded output tability: Whe ay bouded iput reult i a bouded output. Coditioal tability: Whe the etwork fuctio H() ha oly imple jw-axi pole ad poibly left halfplae pole. Impule repoe, h(t): The output whe the iput i the uit impule d(t). Network or trafer fuctio: The ratio H() of the Laplace traform of the output fuctio to the Laplace traform of the iput fuctio. Step repoe, r(t): The output of a circuit whe the iput i the uit tep fuctio u(t), with traform /. Traformed circuit: A origial circuit with the curret, voltage, ource, ad paive elemet replaced by traformed equivalet. Related Topic 3. Voltage ad Curret Law 3.3 Network Theorem. Itroductio Referece R.C. Dorf, Itroductio to Electric Circuit, d ed., New York: Joh Wiley, 993. J.D. Irwi, Baic Egieerig Circuit Aalyi, 3rd ed., New York: Macmilla, 989. D.E. Joho, J.R. Joho, J.L. Hilbur, ad P.D. Scott, Electric Circuit Aalyi, 3rd ed., Eglewood Cliff, N.J.: Pretice-Hall, 997. J.W. Nilo, Electric Circuit, 5th ed., Readig, Ma.: Addio-Weley, 996. by CRC Pre LLC