Generalized Thévenin/ Helmholtz and Norton/ Mayer Theorems of Electric Circuits With Variable Resistances

Transcription

1 WSES TRSTOS on RUTS and SSTEMS Penn lexandr eneralzed Thévenn/ Helholtz and orton/ Mayer Theores of Electrc rcuts Wth arable Resstances PE EXDR "D htu" nsttute of Electronc Engneerng and anotechnologes cadey of Scences of Moldova cadee str 3/3 MD-8 hsnau Republca Moldova bstract: - The generalzed equvalent crcuts whch develop the known theores are forulated t appears that the load straght lne at varous values of a changeable eleent (resstor of an actve twopole passes nto a bunch of these lnes The bunch centre coordnates do not depend on ths changeable eleent t s proposed to use as the paraeters of the generalzed equvalent generator such a load current and voltage whch proved the current across ths eleent equal to zero The applcaton of projectve coordnates nstead of resstance values allows obtanng sutable forulas of the recalculaton of the load current to defne the scales for the load and varable eleent Key-Words: - equvalent crcut actve two-pole load straght lne projectve geoetry geoetrc crcut theory ntroducton n the theory of the electrc crcuts n case of varable paraeter of eleents one of the analyss probles s the establshent of the dependence of the rege paraeter changes on the respectve change of the eleent paraeter n practce t can be D power supply systes wth a varable load To splfy the calculaton of such networks Thévenn/ Helholtz and orton/ Mayer theores are used [ - 4] ccordng to these theores the fxed part of a crcut concernng the ternals of the dedcated load s replaced by an equvalent crcut or equvalent generator The open crcut voltage nternal resstance or short crcut current s the paraeters of ths equvalent generator onsderng portance of deas of the equvalent generator the attenton s gven to the respectve theores n educaton [5 6] lso these theores attract the attenton of researchers [7 8 9] The paraeters of the equvalent generator can be used as scales for noralzed values of the load paraeters or reges Such a defnton of reges (herenafter referred as relatve reges allows coparng or settng the reges of the dfferent systes However ths known equvalent generator does not copletely dsclose the property of a crcut For exaple power supply systes wth the basc (prorty load and the varable auxlary (buffer load or voltage regulator n ths case the change of such an eleent leads to change of the open crcut voltage and short crcut current as the paraeters of the equvalent generator Thus the proble of calculaton of ths crcut and fndng of the equvalent generator paraeters arses agan n a nuber of prevous papers of the author the generalzed equvalent generator whch develops Thévenn/ Helholtz and orton/ Mayer theores s proposed t appears that the load straght lne at varous values of a changeable eleent passes nto a bunch of these lnes Snce the bunch centre coordnates do not depend on ths changeable eleent they can be accepted as the paraeters of the generalzed equvalent generator [ ] lso the approach based on projectve geoetry for nterpretaton of changes (kneatcs of reges s developed [3] t allows revealng the nvarant propertes of a crcut e such expressons whch turn out dentcal to the load and eleent changes Such nvarant expressons allow obtanng convenent forulas of recalculaton of the load current The ethodcally spler and reasonable stateent of the basc obtaned results s offered further E-SS: 4-66X 4 olue 3 4

2 WSES TRSTOS on RUTS and SSTEMS Penn lexandr Equvalent enerator of an ctve Two-Pole wth arable Eleents et us consder an electrc crcut wth a conductvty of the basc load and a conductvty of the auxlary load (varable eleent n Fg The short crcut S current of all the loads are S S y y y y S S y yσ Dsadvantages of the known equvalent generator Takng nto account the current Σ ( and syste of equaton ( we obtan the expresson of a load straght lne Fg ctve two-pole wth a load and a varable eleent (3 Ths crcut can be consdered as an actve two-port network relatvely to the specfed loads n Fg The operatng pont wth varable coordnate ( oves on the load straght lne wth the paraeter at the expense of change of the load conductvty as shown n Fg3 Fg ctve two-port network wth the specfed loads Takng nto account the specfed drectons of currents ths network s descrbed by the followng syste of the equatons + S S S S ( where the paraeters are y + y yσ y + y + y + y y Σ y y y Σ y + y y y Σ Fg3 Faly of load straght lnes wth the paraeters and load The voltage n the short crcut rege n ths case the short crcut current or coordnate of the ntersecton pont of load straght lne (3 wth the axs s S + (4 + E-SS: 4-66X 5 olue 3 4

3 WSES TRSTOS on RUTS and SSTEMS Penn lexandr Slarly the current n the open crcut rege Then the open crcut voltage s S O (5 where the value + (6 + + s the nternal conductvty of crcut relatvely to the load ; the s the deternate of the atrx paraeters Takng nto account the entered paraeters equaton (3 becoes as S (7 + Ths expresson we present as S (8 Thus we obtan the equaton of the straght lne S passng through the pont n turn the nternal conductvty defnes a slope angle of ths lne So the values S are the paraeters of orton/ Mayer equvalent generator y analogy to (8 and takng nto account (5 we obtan the equaton of the straght lne passng through the pont O O ( (9 O So the values are the paraeters of known Thévenn/ Helholtz equvalent generator We note that the paraeters S O depend on the conductvty of a changeable eleent eneralzed orton/ Mayer equvalent generator et us study features of load straght lne (3 Ths expresson (3 represents a bunch of straght lnes wth the paraeter To fnd the coordnates of the bunch centre of these lnes t s convenent to use the extree values of paraeters e These lnes are shown n Fg3 n ths case expresson (3 gves the followng syste of equatons ( ( + For the pont of ntersecton we have that ( ( ( The solvng of syste ( gves the values of voltage y ( y and current y + y + ( y The obtaned values and nternal conductvty allow to present equatons (3 or (8 n another for ( + (3 Thus we obtan the equaton of the straght lne passng through the pont The nternal conductvty defnes a slope angle of ths lne So the values and are the paraeters of the generalzed orton/ Mayer equvalent generator n Fg4 Fg4 eneralzed orton/ Mayer equvalent generator E-SS: 4-66X 6 olue 3 4

4 WSES TRSTOS on RUTS and SSTEMS Penn lexandr We note that besdes the basc energy source of one knd (a current source there s an addtonal energy source of another knd (a voltage source that t s possble to consder as a correspondng theore t s natural when the value we obtan the known orton/ Mayer equvalent generator n S ths case et us note that physcally the centre of the bunch corresponds to such a voltage current of the load and when the current of the eleent s equal to zero ccordng to ths condton fro ( t s also possble to fnd values ( ( of the paraeters the correspondng load conductvty + y Then + ( y (4 The load s a power source because of a negatve value of the conductvty 3 eneralzed Thévenn/ Helholtz equvalent generator n the above case the centre of the bunch s n the second quadrant of coordnate syste and < > so t s natural to consder a case that the centre s n the fourth quadrant as shown n Fg5 Fg5 entre of a bunch s n the fourth quadrant To do ths we consder an electrc crcut wth the basc load R and varable eleent (voltage regulator r n Fg6 Fg6 rcut wth a varable eleent r Slarly there s a bunch of the straght lnes wth the paraeter et us defne the centre of ths r bunch t once t s vsble that the current across the resstance r wll be equal to zero f the voltage Then the load voltage and current are r + r > (5 r (6 r The load resstance R ( r + r (7 n turn the nternal resstance r r R r + r + r (8 Thus the equaton of the straght lne passng through the pont has the for So the values R and + (9 R are the paraeters of Thévenn/ Helholtz equvalent the generalzed generator n Fg7 We note that besdes the basc energy source of one knd (a voltage source there s an addtonal energy source of another knd (a current source that t s possble to consder as the correspondng theore E-SS: 4-66X 7 olue 3 4

5 WSES TRSTOS on RUTS and SSTEMS Penn lexandr c + ( + y (3 + y Fg7 eneralzed Thévenn/ Helholtz equvalent generator t s natural when the value we obtan the known Thévenn/ Helholtz equvalent generator n ths case O whch corresponds to the bea and defnes a zero- order source when the current and voltage of the load are always equal to zero for all ts values 3 nalyss of Operatng Reges of the eneralzed Equvalent enerator The centre poston (the second or fourth quadrant s defned by the knd of an actve twopole as an energy source f the actve two-pole shows ore propertes of a current source the case of Fg 3 takes place f t shows ore propertes of a voltage source we have the case of Fg 7 et us deonstrate how the nternal conductvty and respectvely conductvty nfluence on the knd or type of the generalzed equvalent generator n Fg4 The correspondng faly of the load straght lnes s shown n Fg8 We use further the nverse expresson to (6 ( The conductvtes have the followng characterstc value: a ( Fg 8 Faly of the load straght lnes wth characterstc values of and The Fg9 presents ths case of generalzed equvalent generator that deonstrates the zeroorder generator The load voltage because whch defnes the generalzed equvalent generator as an deal current source; b ( whch defnes the generalzed equvalent generator as an deal voltage source; Fg9 Zero- order generator E-SS: 4-66X 8 olue 3 4

6 WSES TRSTOS on RUTS and SSTEMS Penn lexandr So the varable eleent can have these three specfed characterstc values These characterstc values are defned at a qualtatve level Ths brngs up the proble of deternaton n the relatve or noralzed for of conductance value regardng of these characterstc values n ths case t s possbly to defne the knd of an actve two-pole as an energy source Therefore as though obvous values are not characterstc ones concernng load et us vew possble load characterstc values The tradtonal values as and too wll be characterstc values accordng to Fg8 The physcal sense of these values s clear et us show how the nternal resstance R and respectvely the changeable eleent r nfluence on the type of the generalzed equvalent generator n Fg7 The correspondng faly of the load straght lnes s shown n Fg We use further the nverse expresson to (8 The resstances have the followng characterstc value: a R r r r (5 r + r whch defnes the generalzed equvalent generator as an deal voltage source; b R r r (6 whch defnes the generalzed equvalent generator as an deal current source; c R R r + r r (7 whch corresponds to the bea and defnes the zero- order source The Fg presents the generalzed equvalent generator that deonstrates the zero- order generator The load voltage because r r ( R r (4 r ( R r Fg Zero- order generator So the varable eleent and load can have these three specfed characterstc values 4 nvarant haracterstcs of Operatng Rege hanges Fg Faly of the load straght lnes wth characterstc values R and r et us consder a coon change of the load and varable eleent of a crcut n Fg The correspondng load straght lnes are shown n Fg et the ntal value of varable eleent be and subsequent value be Slarly the ntal value of load equal and subsequent one s et us consder the straght lne of the ntal load The three straght lnes (wth characterstc E-SS: 4-66X 9 olue 3 4

7 WSES TRSTOS on RUTS and SSTEMS Penn lexandr values of eleent as a paraeter of these lnes and the two lnes wth paraeters ntersect ths lne The ponts F are ponts of ths ntersecton n turn the pont F are ponts of ntersecton of the lne of ponts of straght lnes present transforatons of projectve geoetry s shown n a nuber of papers t s convenent to use projectve geoetry for the analyss of crcuts wth varable eleents [456] The projectve transforaton s also set by three pars of respectve ponts s pars of these ponts t s convenent to use the ponts correspondng to the characterstc values of a load and varable eleent The projectve transforatons preserve a cross rato of four ponts or otherwse the cross rato s an nvarant of the projectve transforatons Further we wll show applcaton of such nvarants 4 Defnton of relatve operatng rege at load change The cross rato of four ponts (three of these are the characterstc ones of the lne and the fourth s a pont of the ntal rege has the vew ( (8 The sae value of the cross rato wll be for the ponts of the lne Fg oon change of the load and varable eleent ( (9 Therefore a projectve ap (conforty of ponts of one lne nto ponts of other lne takes place Ths conforty s set by a projecton centre Slarly we consder the straght lne The three straght lnes (wth the characterstc values of load and two lnes wth paraeters ntersect ths lne The ponts are ponts of ths ntersecton n turn the pont are ponts of ntersecton of the lne Therefore the projectve ap nto ponts of takes place Ths conforty s set by (conforty of ponts of one lne other lne the projecton centre The presented confortes For fndng the value of ths cross rato t s necessary to defne coordnates of these ponts To do ths we ap the lne nto the axs of current There s also the projectve transforaton whch s set by an nfntely reote centre Then the cross rato s expressed by the coponents of current ( (3 The correspondng values of ths cross rato are shown n Fg3 The cross rato n geoetry underles the defnton of a dstance between the ponts concernng the extree or base E-SS: 4-66X olue 3 4

8 ponts n turn the pont s a scale or unt pont Fg3 Mutual corresponds of the load current conductvty and cross rato Slarly the cross rato of the subsequent rege ponts ( (3 The vew or structure of expressons (3 and (3 shows that t s possble to use the dvson of these cross ratos e ( (3 Ths cross rato s the dstance between the ponts of the ntal and subsequent reges of the lne The sae dstance wll be between the ponts of the ntal and subsequent reges of the lne ( (33 Physcally t s evdently because the ntal and subsequent rege s set by the change of the coon load The equalty of cross ratos (8 (9 s also explaned Then t s possble to express cross rato (3 or (3 by load conductvtes et us present equaton (3 of the generalzed equvalent generator by the followng vew + ( (34 Ths fractonally lnear expresson s a projectve transforaton whch aps the values of the load conductvty nto the values of the load current The correspondng ponts of these values are shown n Fg3 Therefore t s possble by foralzed ethod to express at once cross rato (3 by the load conductvtes ( ( (35 So we consder cross rato (35 as the projectve coordnate of the ntal or runnng rege ponts Ths coordnate s expressed by nvarant (dentcal anner by varous rege paraeters Too t s possble by foralzed ethod to express cross rato (3 by load conductvtes ( ( (36 So we consder the cross rato of type (3 (36 as the change of runnng rege Ths change s expressed by nvarant anner through varous rege paraeters Therefore usually used rege changes by ncreents (as foral are elnated n turn the values are the scales for noralzng of values of current and conductvty Then expressons (3 (36 present the relatve reges t perts to copare or set the rege of the dfferent crcuts wth varous paraeters WSES TRSTOS on RUTS and SSTEMS Penn lexandr E-SS: 4-66X olue 3 4

9 et us note the propertes of a cross rato f to nterchange the coponents of expresson (3 then we get / (37 lso the group property takes place (38 et us obtan the subsequent value of current fro expresson (3; we have ( + (39 The obtaned transforaton wth the paraeter allows realzng the drect recalculaton of the current at load change Ths expresson s especally convenent n case of group or set of load changes on account of perforance of group property (38 4 Defnton of relatve operatng rege at change of eleent The cross rato of the four pont (three of these are the characterstc ones F of lne and the fourth pont of ntal rege has the vew ( F F F (4 The ponts F are chosen as base ones That wll be explaned later The sae value of the cross rato wll be for the ponts F of the lne ( F F F (4 ross rato (4 s expressed by the coponents of current ( (4 So the cross rato s the dstance between the ponts concernng the base ponts n turn the pont s a unt pont Slarly the cross rato of the ponts of subsequent rege ( (43 The dstance between the ponts of ntal and subsequent reges of lne has the vew ( (44 The sae dstance s between the ponts of ntal and subsequent reges of lne ( (45 et us express cross rato (4 or (44 by the load conductvtes Expresson (34 ( ( for the gven load s the projectve transforaton whch aps the ponts nto the ponts of current Therefore t s possble by foralzed ethod to express at once cross rato (4 944 by the conductvtes ( ( ( (46 ( ( ( (47 WSES TRSTOS on RUTS and SSTEMS Penn lexandr E-SS: 4-66X olue 3 4

10 WSES TRSTOS on RUTS and SSTEMS Penn lexandr lso the group property takes place (48 et us obtan the subsequent value of current fro expresson (44 Then ( + (49 The obtaned transforaton wth the paraeter allows realzng the drect recalculaton of current 43 Defnton of relatve operatng rege at coon change of load and eleent et the coon or coposte change of rege be gven as Then the vew of expressons (45 and (3 shows that t s possble to use the ultplcaton of these cross ratos as a copound change of rege ( (5 n ths resultant expresson the nteredate coponents are reduced at the expense of the choce of dentcal basc ponts Therefore we obtan the resultant value of current for the pont 5 Exaple ( + (5 et the eleents of a crcut n Fg be gven as follows 5 5 y y 5 y 5 y 5 y 5 y The syste of equaton ( et us consder the conductvty 5 The paraeters of the ntal rege pont 5 79 The paraeters of the subsequent rege pont 74 Paraeters (4 (5 of the known equvalent generator ponts nternal conductvty (6 of the crcut ( Paraeters ( ( of the orton/ Mayer equvalent generator pont Equaton (3 of orton/mayer equvalent generator 5 (5 + 7 alue (4 of the load conductvty pont 5 et us now consder the conductvty 5 The paraeters of the ntal rege pont The paraeters of the subsequent rege pont 7649 The paraeters of the ponts nternal conductvty (6 of the crcut 68 The value densons are not ndcated E-SS: 4-66X 3 olue 3 4

11 WSES TRSTOS on RUTS and SSTEMS Penn lexandr ext we are fndng the characterstc values of nternal conductvty and varable eleent Expresson ( alue ( of the deal current source 57 alue ( of the deal voltage source 975 alues (3 of the zero-order source 5 5 The subsequent value of current ( ( Operatng rege at load change ross rato (3 of the ntal rege pont ( et us check the cross rato value of the pont ( We are checkng cross rato value (35 by the load conductvty ross rato (3 of the subsequent rege pont ( et us check ths value by the load conductvty Rege change (3 and ( Fg4 Exaple of utual corresponds of rege paraeters 3 et the load once agan be changed 5 Then rege change (36 or (38 n regard to the load orrespondng current value ( ( The coon rege change relatvely to the load We have obtaned the sae value (39 of the current ( The corresponded ponts of values are shown n Fg4 E-SS: 4-66X 4 olue 3 4

12 WSES TRSTOS on RUTS and SSTEMS Penn lexandr 5 Operatng rege at change of eleent Dstance (44 between the ntal pont and the subsequent pont et us check the sae value (45 of ponts and the sae value (47 of conductvtes Subsequence current value ( ( oon change of load and oon rege change ( The resultant current value pont ( Developent of the Obtaned Results The presented results are generalzed for actve twoport [7] and ultport [8] The applcaton of projectve coordnates gves capablty to obtan forulas of recalculaton of load currents of actve ultport for varous cases [9 ] 7 oncluson -The generalzed equvalent generator of actve twopole wth the varable paraeters splfes of crcut analyss gves ore profound dea about the nterrelaton of operatng reges and paraeters of eleents can be useful n the educaton purposes - The applcaton of the projectve coordnates nstead of conductvtes or resstance allows obtanng the sutable forulas of the recalculaton of load current to defne the scales for load and varable eleent -The obtaned results can be appled n partcular to lnear crcuts to crcuts wth dependent sources of voltage and current References: [] K harles O Matthew Sadku Fundaentals of Electrc rcuts 4-th Edton Mcraw-Hll 7 [] J D rwn and R M els asc engneerng crcut analyss -th edton Wley Publshng 8 [3] S T Karrs rcut nalyss Wth MT pplcatons Orchard publcatons 4 [4] D Johnson Orgns of the equvalent crcut concept: the voltage-source equvalent Proceedngs of the EEE 94 3 pp [5] J andewalle Shortcuts n crcuts and systes educaton wth a case study of the Thévenn/Helholtz and orton/mayer equvalents rcuts and Systes (SS EEE nternatonal Syposu on EEE [6] hatzaraks ew Method for Fndng the Thevenn and orton Equvalent rcuts Engneerng 5 p38 [7] E luskn and Patlakh n deal ndependent source as an equvalent -port arxv preprnt arxv:8459 [8] M Hosoya Dervaton of the Equvalent rcut of a Mult-Ternal etwork ven E-SS: 4-66X 5 olue 3 4

13 WSES TRSTOS on RUTS and SSTEMS Penn lexandr by eneralzaton of Helholtz-Thevenn's Theore UET-OEE OF SEE UERST OF THE RUKUS 84 7 [9] R Hashean Hybrd equvalent crcut and alternatve to Thévenn and orton equvalents ts propertes and applcaton Proc Mdwest Syp On rcuts and Systes MWSS 9 pp 8-83 [] Penn Utlzaton of the projectve coordnates n the lnear electrcal network wth the varable reges uletnul cadee de Stnte a Republc Moldova Fzca s tehnca o 99 pp 64-7 (Russan [] Penn haracterstcs of odfed equvalent generator of actve twoternal network wth varable resstor Electrchestvo o4 995 pp (Russan [] Penn near- fractonal relaton n the probles of analyss of resstve crcuts wth varable paraeters Electrchestvo o 999 pp 3-44 (Russan [3] Penn Deternaton of reges of the equvalent generator based on projectve geoetry: The generalzed equvalent generator nternatonal Journal of Electrcal and oputer Engneerng ol3 o5 8 pp [4] Penn Projectvely - affne propertes of resstve two-ports wth varable load Tekhncheskaa elektrodnaka 99 pp 38-4 (Russan [5] R E ryant J D Tygar and P Huang eoetrc characterzaton of seres-parallel varable resstor networks rcuts and Systes : Fundaental Theory and pplcatons EEE Transactons on pp [6] Penn Projectve eoetry Method n the Theory of Electrc rcuts wth arable Paraeters of Eleents nternatonal Journal of Electroncs ouncatons and Electrcal Engneerng ol3 o 3 pp essue/-3---d-pdf [7] Penn bout the defnton of paraeters and reges of actve two- port networks wth varable loads on the bass of projectve geoetry WSES Transactons on rcuts and Systes ol o5 pp lbrary/transactons/crcuts//53-346pdf [8] Penn Paraeters and haracterstcs of the Modfed Equvalent enerator n an ctve Multport etwork Electrchestvo o5 pp 3-39 (Russan [9] Penn Recalculaton of the oads urrent of ctve Mult-Port etworks on the ass of Projectve eoetry Journal of rcuts Systes and oputers ol o5 3 pages 3 c [] Penn Recalculatng the load currents of an actve ult-pole wth varable paraeters on the bass of projectve geoetry Electrchestvo o pp 3-39 (Russan [] Penn oralzed representaton of the equatons of actve ult- port networks on the bass of projectve geoetry Moldavan Journal of the Physcal Scences ol o3 E-SS: 4-66X 6 olue 3 4