Appenix ore on gneiclly C Couple Coils Iel Trnsformers C. Equivlen Circuis for gneiclly Couple Coils A imes, i is convenien o moel mgneiclly couple coils wih n equivlen circui h oes no involve mgneic coupling. Consier he wo mgneiclly couple coils shown in Fig. C.. The resisnces R R represen he wining resisnce of ech coil. The gol is o replce he mgneiclly couple coils insie he she re wih se of inucors h re no mgneiclly couple. Before eriving he equivlen circuis, we mus poin ou n imporn resricion: The volge eween erminls mus e zero. In oher wors, if erminls cn e shore ogeher wihou isuring he volges currens in he originl circui, he equivlen circuis erive in he meril h follows cn e use o moel he coils. This resricion is impose ecuse, while he equivlen circuis we evelop oh hve four erminls, wo of hose four erminls re shore ogeher. Thus, he sme requiremen is plce on he originl circuis. We egin eveloping he circui moels y wriing he wo equions h rele he erminl volges o he erminl currens. For he given references polriy os, R L c R Figure C. The circui use o evelop n equivlen circui for mgneiclly couple coils. L + (C.) +. (C.) The T-Equivlen Circui To rrive n equivlen circui for hese wo mgneiclly couple coils, we seek n rrngemen of inucors h cn e escrie y se of equions equivlen o Eqs. C. C.. The key o fining he rrngemen is o regr Eqs. C. C. s mesh-curren equions wih s he mesh vriles. Then we nee one mesh wih ol inucnce of L H secon mesh wih ol inucnce of H. Furhermore, he wo meshes mus hve common inucnce of H. The T-rrngemen of coils shown in Fig. C. sisfies hese requiremens. R L c R Figure C. The T-equivlen circui for he mgneiclly couple coils of Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. 787 This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
788 ore on gneiclly Couple Coils Iel Trnsformers You shoul verify h he equions reling o reuce o Eqs. C. C.. Noe he sence of mgneic coupling eween he inucors he zero volge eween. The p-equivlen Circui We cn erive p-equivlen circui for he mgneiclly couple coils shown in Fig. C.. This erivion is se on solving Eqs. C. C. for he erivives > > hen regring he resuling expressions s pir of noe-volge equions. Using Crmer s meho for solving simulneous equions, we oin expressions for > >: L L - - L - ; (C.3) L L - - L - + L L -. (C.4) Now we solve for y muliplying oh sies of Eqs. C.3 C.4 y hen inegring: (0) + L - - L0 L - L0 (C.5) (0) - L - + L0 L L -. L0 (C.6) If we regr s noe volges, Eqs. C.5 C.6 escrie circui of he form shown in Fig. C.3. L B c (0) L A L C (0) Figure C.3 The circui use o erive he p-equivlen circui for mgneiclly couple coils. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
C. Equivlen Circuis for gneiclly Couple Coils 789 All h remins o e one in eriving he p-equivlen circui is o fin L A, L B, L C s funcions of L,,. We esily o so y wriing he equions for in Fig. C.3 hen compring hem wih Eqs. C.5 C.6. Thus (0) + v L + (v A L0 L - ) B L0 (0) + + v L A L - B L 0 L B L0 (C.7) (0) + v + (v C L0 - ) B L0 (0) + L B L0 + + L B L C L 0. (C.8) Then L B L -, (C.9) L A - L -, (C.0) L C L - L -. (C.) When we incorpore Eqs. C.9 C. ino he circui shown in Fig. C.3, he p-equivlen circui for he mgneiclly couple coils shown in Fig. C. is s shown in Fig. C.4. R (0) L L L L (0) c R Figure C.4 The p-equivlen circui for he mgneiclly couple coils of Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
790 ore on gneiclly Couple Coils Iel Trnsformers L R c R L L L Figure C.5 The p-equivlen circui use for sinusoil sey-se nlysis. Noe h he iniil vlues of re explici in he p-equivlen circui u implici in he T-equivlen circui. We re focusing on he sinusoil sey-se ehvior of circuis conining muul inucnce, so we cn ssume h he iniil vlues of re zero. We cn hus elimine he curren sources in he p-equivlen circui, he circui shown in Fig. C.4 simplifies o he one shown in Fig. C.5. The muul inucnce crries is own lgeric sign in he T- p-equivlen circuis. In oher wors, if he mgneic polriy of he couple coils is reverse from h given in Fig. C., he lgeric sign of reverses. A reversl in mgneic polriy requires moving one polriy o wihou chnging he reference polriies of he erminl currens volges. Exmple C. illusres he pplicion of he T-equivlen circui. Exmple C. ) Use he T-equivlen circui for he mgneiclly couple coils shown in Fig. C.6 o fin he phsor currens I I. The source frequency is 400 r>s. ) Repe (), u wih he polriy o on he seconry wining move o he lower erminl. Soluion ) For he polriy os shown in Fig. C.6, crries vlue of +3 H in he T-equivlen circui. Therefore he hree inucnces in he equivlen circui re L - 9-3 6 H ; - 4-3 H ; 3 H. Figure C.7 shows he T-equivlen circui, Fig. C.8 shows he frequency-omin equivlen circui frequency of 400 r>s. Figure C.9 shows he frequency-omin circui for he originl sysem. 500 j00 00 I V j00 00 300 0 V j3600 j600 j500 V 800 I 6 H H Figure C.6 The frequency-omin equivlen circui for Exmple C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge 3 H Figure C.7 The T-equivlen circui for he mgneiclly couple coils in Exmple C.. j400 j400 j00 Figure C.8 The frequency-omin moel of he equivlen circui. 400 r>s 500 j 00 00 j 400 j400 00 300 0 V I I 800 j00 j500 Figure C.9 The circui of Fig. C.6, wih he mgneiclly couple coils replce y heir T-equivlen circui.
Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge C. Equivlen Circuis for gneiclly Couple Coils 79 I Here he mgneiclly couple coils re moele y he circui shown in Fig. C.8. To fin he phsor currens I I, we firs fin he noe volge cross he 00 Æ inucive recnce. If we use he lower noe s he reference, he single noe-volge equion is V - 300 700 + j500 + V j00 + Solving for V yiels V 36 - j8 36.4 l -3.37 V (rms). Then I 300 - (36 - j8) 700 + j500 V 900 - j00 0. 63.5 l -7.57 ma (rms) 36 - j8 900 - j00 59.63 l63.43 ma (rms). ) When he polriy o is move o he lower erminl of he seconry coil, crries vlue of -3 H in he T-equivlen circui. Before crrying ou he soluion wih he new T-equivlen circui, we noe h reversing he lgeric sign of hs no effec on he soluion for I shifs I y 80. Therefore we nicipe h As efore, we firs fin he noe volge cross he cener rnch, which in his cse is cpciive recnce of -j00 Æ. If we use he lower noe s reference, he noe-volge equion is V - 300 700 + j4900 + V -j00 + V 900 + j300 0. Solving for V gives Then V -8 - j56 I I 56.57 l -98.3 V (rms). 300 - (-8 - j56) 700 + j4900 63.5 l -7.57 ma (rms) -8 - j56 900 + j300 59.63 l -6.57 ma (rms). I 63.5 l -7.57 ma (rms) j4800 j 800 I 59.63 l -6.57 ma (rms). We now procee o fin hese soluions y using he new T-equivlen circui. Wih -3 H, he hree inucnces in he equivlen circui re j00 Figure C.0 The frequency-omin equivlen circui for -3 H v 400 r>s. L - 9 - (-3) H ; - 4 - (-3) 7 H ; -3 H. A n opering frequency of 400 r>s, he frequency-omin equivlen circui requires wo inucors cpcior, s shown in Fig. C.0. The resuling frequency-omin circui for he originl sysem ppers in Fig. C.. 500 j00 00 j4800 j800 00 300 0 V I I 800 j00 j500 Figure C. The frequency-omin equivlen circui for Exmple C.().
79 ore on gneiclly Couple Coils Iel Trnsformers C. The Nee for Iel Trnsformers in he Equivlen Circuis The inucors in he T- p-equivlen circuis of mgneiclly couple coils cn hve negive vlues. For exmple, if L 3 mh, mh, 5 mh, he T-equivlen circui requires n inucor of - mh, he p-equivlen circui requires n inucor of -5.5 mh. These negive inucnce vlues re no roulesome when you re using he equivlen circuis in compuions. However, if you re o uil he equivlen circuis wih circui componens, he negive inucors cn e ohersome. The reson is h whenever he frequency of he sinusoil source chnges, you mus chnge he cpcior use o simule he negive recnce. For exmple, frequency of 50 kr>s, - mh inucor hs n impence of -j00 Æ. This impence cn e moele wih cpcior hving cpcince of 0. mf. If he frequency chnges o 5 kr>s, he - mh inucor impence chnges o -j50 Æ. A 5 kr>s, his requires cpcior wih cpcince of 0.8 mf. Oviously, in siuion where he frequency is vrie coninuously, he use of cpcior o simule negive inucnce is prciclly worhless. You cn circumven he prolem of eling wih negive inucnces y inroucing n iel rnsformer ino he equivlen circui. This oesn compleely solve he moeling prolem, ecuse iel rnsformers cn only e pproxime. However, in some siuions he pproximion is goo enough o wrrn iscussion of using n iel rnsformer in he T- p-equivlen circuis of mgneiclly couple coils. L L Iel Iel () () L L L L Iel Iel (L ) (L ) (L ) L (c) () Figure C. The four wys of using n iel rnsformer in he T- p-equivlen circui for mgneiclly couple coils. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
C. The Nee for Iel Trnsformers in he Equivlen Circuis 793 An iel rnsformer cn e use in wo ifferen wys in eiher he T-equivlen or he p-equivlen circui. Figure C. shows he wo rrngemens for ech ype of equivlen circui. Verifying ny of he equivlen circuis in Fig. C. requires showing only h, for ny circui, he equions reling o > > re ienicl o Eqs. C. C.. Here, we vlie he circui shown in Fig. C.(); we leve i o you o verify he circuis in Figs. C.(), (c), (). To i he iscussion, we rerew he circui shown in Fig. C.() s Fig. C.3, ing he vriles i 0 v 0. From his circui, L i 0 N N v 0 Iel () Figure C.3 The circui of Fig. C.() wih efine. i 0 v 0 L - + ( + i 0 ) (C.) v 0 - i 0 +. (C.3) (i 0 + ) The iel rnsformer imposes consrins on : v 0 i 0 v 0 ; i 0. (C.4) (C.5) Susiuing Eqs. C.4 C.5 ino Eqs. C. C.3 gives L + () (C.6) () +. (C.7) From Eqs. C.6 C.7, L + (C.8) +. (C.9) Equions C.8 C.9 re ienicl o Eqs. C. C.; hus, insofr s erminl ehvior is concerne, he circui shown in Fig. C.3 is equivlen o he mgneiclly couple coils shown insie he ox in Fig. C.. Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge
Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge 794 ore on gneiclly Couple Coils Iel Trnsformers i L k In showing h he circui in Fig. C.3 is equivlen o he mgneiclly couple coils in Fig. C., we plce no resricions on he urns rio. Therefore, n infinie numer of equivlen circuis re possile. Furhermore, we cn lwys fin urns rio o mke ll he inucnces posiive. Three vlues of re of priculr ineres: L, (C.0) L () Iel, (C.) ( k ) Iel () k Figure C.4 Two equivlen circuis when >L. L ( k ) k L () Iel i Iel () k Figure C.5 Two equivlen circuis when >. (C.) The vlue of given y Eq. C.0 elimines he inucnces L - > L - from he T-equivlen circuis he inucnces (L (L - )>( - )>( L - ) L - ) from he p-equivlen circuis. The vlue of given y Eq. C. elimines he inucnces ( > ) - (>) - from he T-equivlen circuis he inucnces (L - )>( - ) (L - )>( - ) from he p-equivlen circuis. Also noe h when >L, he circuis in Figs. C.() (c) ecome ienicl, when >, he circuis in Figs. C.() () ecome ienicl. Figures C.4 C.5 summrize hese oservions. In eriving he expressions for he inucnces here, we use he relionship kl. Expressing he inucnces s funcions of he self-inucnces L he coefficien of coupling k llows he vlues of given y Eqs. C.0 C. no only o reuce he numer of inucnces neee in he equivlen circui, u lso o gurnee h ll he inucnces will e posiive. We leve o you o invesige he consequences of choosing he vlue of given y Eq. C.. The vlues of given y Eqs. C.0 C. cn e eermine experimenlly. The rio >L is oine y riving he coil esigne s hving N urns y sinusoil volge source. The source frequency is se high enough h vl W R, he N coil is lef open. Figure C.6 shows his rrngemen. Wih he coil open, N C L. I V jvl jv N N V V jvi. Now, s jvl W R, he curren I is (C.3) Figure C.6 Experimenl eerminion of he rio >L. I V. jvl (C.4)
Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge C. The Nee for Iel Trnsformers in he Equivlen Circuis 795 Susiuing Eq. C.4 ino Eq. C.3 yiels V V I 0 L, (C.5) in which he rio >L is he erminl volge rio corresponing o coil eing open; h is, I 0. We oin he rio > y reversing he proceure; h is, coil is energize coil is lef open. Then V. V I 0 (C.6) Finlly, we oserve h he vlue of given y Eq. C. is he geomeric men of hese wo volge rios; hus C V V V I 0 V I 0 C L. CL (C.7) For coils woun on nonmgneic cores, he volge rio is no he sme s he urns rio, s i very nerly is for coils woun on ferromgneic cores. Becuse he self-inucnces vry s he squre of he numer of urns, Eq. C.7 revels h he urns rio is pproximely equl o he geomeric men of he wo volge rios, or N V V. CL N C V I 0 V I 0 (C.8)
Elecric Circuis, Eighh Eiion, y Jmes A. Nilsson Susn A. Rieel. This meril is proece y Copyrigh wrien permission shoul e oine from he pulisher prior o ny prohiie reproucion, sorge