( ) B. Application of Phasors to Electrical Networks In an electrical network, let the instantaneous voltage and the instantaneous current be

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1 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 Analyss of Electrcal Networks Usng Phasors: A Bond Graph Approach Israel Núñez-Hernández Peter. Breedeld Paul B.. Weustnk Glberto Gonzalez-A Abstract hs paper proposes a phasor representaton of electrcal networks by usng bond graph methodology. A so-called phasor bond graph s bult up by means of two-dmensonal bonds whch represent the complex plane. Impedances or admttances are used nstead of the standard bond graph elements. A procedure to obtan the steady-state alues from a phasor bond graph model s presented. Besdes the presentaton of a phasor bond graph lbrary n SIDOPS code also an applcaton example s dscussed. Keywords Bond graphs phasor theory steady-state complex power electrcal networks. W I. INODUION HEN an electrc system s operatng n steady-state dfferental are not requred to descrbe ts behaor snce all arables are ether constants or n the A case snusodal aratons n tme wth constant frequency. In the latter case a phasor representaton [] s approprate. he use of phasor notaton not only brngs a sgnfcant mathematcal smplfcaton but also reduces the capacty requrements for computatonal processng. At the other hand bond graph methodology [] results n a concse graphcal representaton of energy storage dsspaton and exchange n a system. he oerall purpose of ths methodology s the doman-ndependent representaton of any engneerng system whch s noled n dfferent domans. he paper conssts of eght sectons. In Secton II the phasor representaton and ts usage n electrcal networks are presented. In Secton III we ge a bref ntroducton to bond graph methodology. Secton IV contans the descrpton of the phasor bond graph elements. In secton V the methodology proposed to obtan the steady-state alues s descrbed and llustrated by means of an example. Secton VI shows the phasor bond graph elements mplementaton n SIDOPS code. Secton VII llustrates the smulaton results n sm of an electrcal network by usng phasor bond graphs. Fnally the conclusons are stated n Secton VIII. Israel Núñez-Hernández and Peter. Breedeld are wth the obotcs and Mechatroncs Group Unersty of wente P.O. Box 7 75 AE Enschede he Netherlands (e-mal:.nunezhernandez@utwente.nl P..Breedeld@utwente.nl). Paul B.. Weustnk s wth the ontrollab Products B.V. Hengelosestraat 5 75 AN Enschede he Netherlands (e-mal: paul.weustnk@controllab.nl) Glberto Gonzalez-A s wth the Faculty of Electrcal Engneerng Unersty of Mchoacan Morela Mchoacán Mexco (e-mal: glmchga@yahoo.com). II. PHASO EPESENAION A. Mathematcal Descrpton he snusodal analyss by means of phasors s an elegant way to analyze electrcal crcuts wth snusodal nputs and responses wth a gen constant frequency.e. when the system s n steady-state wthout the need to sole dfferental. A phasor represents the temporal behaor of an electrcal sgnal relate to a fxed reference. Smlar to a ector a phasor has magntude and phase. It can be represented as an nstant n tme of a rotary ector. In general a snusodal functon wth  ampltude θ phase angle and frequency; can be descrbed by f ( t) = Âcos( t + θ) () where  may be expressed n root-mean-square (MS) alue A. For snusodal waes  = A. It s possble rewrte () n complex notaton by usng Euler s formula and substtutng ts ampltude alue by ts MS alue ( + ) ( ) j t θ jθ j t F( t) = f t = { Ae } = { Ae e } () where {} s the real operator. he part that does not depend on tme Ae jθ n () s known as a phasor [3]. A phasor F may also be wrtten as j F = Ae θ = A θ = A(cosθ + j sn θ ) he tme ntegral of F( t ) s F ( t) dt = Ae e dt = j F e j j t j t whch mples that the ntegral of the phasor s lagged by π/ radans and scaled by /. Smlarly the tme derate of F( t) s d dt F t Ae e j Fe j d j t j t ( ) = θ = Hence the derate of a phasor s leaded by π/ radans and multpled by. hs means that n phasor notaton the ntegraton and dfferentaton operatons can be performed by scalng and phase shftng. dt (3) (4) (5) 945

2 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 B. Applcaton of Phasors to Electrcal Networks In an electrcal network let the nstantaneous oltage and the nstantaneous current be ( t) = cos( t + θ ); ( t) = cos( t + θ ) (6) he phasor representaton of (6) may be obtaned by usng () and (3) thus jθ jθ V = V e = V θ ; I = I e = I θ he mpedance Z s the relatonshp between the oltage and current. Snce ths relatonshp s between two phasors t wll be a phasor too. he mpedance may be expressed as Z = V I = Z θ = + j X X ( ) where θ = θ - θ s called the mpedance angle. he real part s gen by the resste elements and the magnary or reacte part s gen by the nducte and capacte reactances n the system respectely X and X. able I shows a lst of the three basc elements n an electrcal network and ther mpedances. ABE I IMPEDANES me Phasor Impedance ( t) = ( t) V = I d ( t) = ( t) V = jx I jx = j t dt ( ) ( ) = t dt V = jx I jx = j Obously the mpedance of capacte or nducte elements s a functon of the constant frequency. In power engneerng oltages and currents are often represented n a phasor dagram. A phasor dagram s lke a pcture at any nstant of these rotary ectors whch we can determne the angular dfference between them at that tme.. omplex Power he nstantaneous power consumed by the network may be wrtten as wth (7) (8) p( t) = ( t) ( t) = cos( t + θ )cos( t + θ ) (9) After applyng some trgonometrc denttes [4] we hae p( t) = P( + cos( t + θ )) + Qsn ( t + θ ) () P = V I cos θ ; Q = V I snθ () where P s called real or acte power defned n watts (W). It represents the absorbed power by the resste elements n the load. At the other hand Q s referred as reacte power defned n olt-ampere reacte (ar) and ths power supples the stored energy n reacte elements. Snce cosθ plays an mportant role n the amount of real power n the system t s called power factor [4] [5]. eal and reacte power are represented together as a complex or apparent power S ts unt s olt-ampere (VA) [6]. he apparent power may be represented as S = P + jq = V I = Z I = I + jx I () where I s the conjugate current. hese three powers are normally descrbed n a so-called power trangle. III. BOND GAPH MODEING he bond graph methodology s a graphcal notaton of a port-based descrpton for modelng dynamcal systems. hs graphcal technque s based on representng power transfer by means of bonds. Gen that energy s a doman-ndependent quantty t forms the bass for ths doman ndependent approach. he labelled nodes of a bond graph descrbe a fundamental behaor wth respect to energy lke storage transducton etc. In each physcal doman the power s the product of two arables effort and flow. hese par of arables s called power arables. hese two arables are represented as pared sgnals flowng n opposte drecton. herefore just one of them may be an nput. Hence the bond also represents a blateral sgnal flow. In a bond graph the way n whch these sgnal are specfed as nput and output s by means of the causal stroke. It s a perpendcular lne put at one end of a bond ndcatng the drecton of the effort sgnal or called also causalty. Momentum and dsplacement are consered arables or energy arables. hey are obtaned by ntegraton of one of the power arables and represent the energy accumulated n an deal energy storage element. here are some basc types of elements necessary to represent energy behaor n a doman ndependent way. he -port elements whch dsspate power (resstor ) store energy (nerta I capactor ) and supply power (sources S e S f ). In addton t s necessary nterconnect two or more elements n a power conserate way n order to create structure. At one hand we hae the transducers elements (- port): transformers (F) and gyrators (GY). At the other hand the nterconnecton elements (multport): the -juncton s a node where all the efforts are equal and represents n an electrcal crcut a parallel connecton; the -juncton descrbes a seres connecton n an electrcal crcut t means that all the flows are equal. Note howeer that seres and parallel are specal cases whle the junctons are more general n the sense that they apply to each common flow (-juncton) and common effort (-juncton) stuaton. It s possble to obtan the state-space of the system usng a bond graph model. onsder Fg. scheme of a multport lnear tme-narant system whch ncludes the key ectors [7]. 946

3 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 In Fg. the state ector Fg. Bond graph key ectors n x s composed of energy p n arables; u denotes the system nput; z the cor r energy ector; fnally Dn and Dout represent the energy exchanges between the dsspaton feld and the juncton structure. he storage and dsspaton feld relatonshps are z = F x D = D (3) ; out n he juncton structure relatonshps are defned by x ɺ S S S 3 z D = S S S D n 3 out y S 3 S 3 S 33 u q where the ector y s the plant output. he entres of the S matrx take alues nsde the set { ± ±m ± n} where m and n are transformer and gyrator modules; S and S are square skew-symmetrc matrces; S = S and S 4 = S. 4 onenently representng the stated as xɺ = Ax + Bu ; y = x + Du (5) where A = ( S + S MS ) F = ( S + S MS ) F 3 3 B = S + S MS D = S + S MS beng M = ( I S ) wth I as an dentty matrx. IV. PHASO BOND GAPHS EEMENS he aplace transform can also be appled to the bond graph models [8] [9]. Wth ths transformaton the -port elements become mpedances or admttances. If a -port has effort-out out causalty s modelled as mpedance whle a -port wth flow-out out causalty s modelled as admttance. We must assume that the consttute relatonshps of the components are lneal thus Fg. -port elements a) ) mpedances and b) admttances (4) (6) Note that the preferred ntegral causalty of an I-element s an admttance. We are now able to buld up mpedance bond graphs by usng these elements and followng the same procedure that n the case of dynamc models. If we substtute s = j Fourer operator the phasor bond graph elements are obtaned. We propose to express the mpedances n matrx form. In ths way we may represent esent the phasor elements usng D multbonds [] []. Where one bond wll represent the real part of the phasor and the second bond wll represent the magnary part of the phasor. From able I we can rewrte the mpedances n matrx form see able II. ABE II -PO IMPEDANES IN PHASO BOND GAPH FOM Phasor D multbond { V} { I} esste = { V} I I { I} { V} { I} Inducte = X { V} I I { I} { V} { I} apacte = X I{ V} I{ I} I{} s the magnary operator Note that n the phasor model we wll keep the same preferred causaltes as n a dynamc model effort-out for capactors flow-out out for nductors and ndfferent for resstors. he -port elements are modelled n the same way as n case of D multbonds [] see able III. ABE III -PO EEMENS IN PHASO BOND GAPH FOM Phasor D multbond V I m ransformer = ; = I V m V G I n Gyrator = ; G = V G I n m and n are the scalar modulus of the transformer and gyrator respectely We can change (7) nto ts matrx form usng (3) so we can obtan the D multbond sources. See able IV. ABE IV SOUES IN PHASO BOND GAPH FOM Source Phasor D multbond Voltage V = V [cosθ sn ] θ urrent I = I [cosθ sn θ ] V and I are the MS alue of oltage and current respectely V. PHASO BOND GAPH ANAYSIS In ths secton we wll show how to obtan the system steady-state state alues by usng the phasor bond graphs. In Fg. 3 the feld structure for a phasor bond graph s depcted 947

4 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 he key ectors are xɺ = f e4 { f } { f } { e4} { e4} 3 = I I 3 3 z = e f 4 = { e } I{ e } { f } I{ f } 4 4 () Fg. 3 Phasor bond graph feld structure he reactance and dsspaton fled contan the power demandng elements of the system and may be defned as z = F x ɺ D = D ; out n where F s flled wth the mpedance reacte elements. he matrx or admttance of the contans the mpedance or admttance of the resste elements. he juncton structure relatonshps may be changed thus x ɺ S S S = 3 z Dn S S S 3 D out (8) y S 3 S3 S33 u Equaton (8) keeps the same characterstcs as (4). After some algebrac manpulatons the system output s gen by y = ( ( F A ) B + D ) u (7) (9) where A = S + S M S = S 3 + S 3 M S () B = S 3 + S M S 3 D = S 33 + S 3 M S 3 beng M = ( I S ). In order to clarfy ths procedure we wll show an example. onsder the crcut shown n Fg. 4. D n = f = { f } { f } ; Dout e { e} { e} = = u = e = { e } { e } ; y = f = { f } { f } We can construct the consttute relatons of the -port bond graph nodes by applyng (8) (9) and () thus F = dag X ; = X (3) and the juncton structure s gen by I S S S = ; I = = I S S S S S S 3 = 3 = I ; = 3 = 3 = 33 = Fnally we substtute () (3) and (4) nto (9) and to obtan the steady-state state alue of the output: y = + ( X ) X VI. IMPEMENAION ON -SIM Normally the electrcal networks contan a lot of elements and sources. Due to ths fact t would be conenent to automate phasor analyss wth bond graphs by usng a software tool. For ths purpose we wll use -sm [] software. In able V we show the SIDOPS code [9] for the effort and flow. ABE V SOUES PHASO BOND GAPH SIDOPS ODE SIDOPS code real V =. {V}; real ang =. {deg}; X X u X X () (4) (5) Fg. 4 crcut We can change the dynamc model to the phasor model by usng the D multbonds Fg. 5. Fg. 5 Phasor model of the crcut Effort source Flow source arables real flow[]; p.u = V * [cos(ang); sn(ang)]; flow= p.; real I =. {A}; real ang =. {deg}; arables real effort[]; p. = I * [cos(ang); sn(ang)]; effort= p.u; 948

5 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 he SIDOPS code for the passe phasor elements s shown n able VI. ABE VI -PO EEMENS PHASO BOND GAPH SIDOPS ODE SIDOPS code real r =. {ohm}; he transformer and gyrator are both already ncluded n the -sm lbrary. VII. SIMUAION OF AN EEIA NEWOK In ths secton we wll compare the response of a phasor bond graph model wth a dynamcal bond graph model. onsder the electrcal network shown n Fg. 7. esstor arables real []; Inducte reactance apacte reactance = r * [ ; ]; p.u = * p.; real xl =. {ohm}; arables real X[]; X = xl * [ -; ]; p.u = X * p.; real xc =. {ohm}; arables real X[]; X = xc * [ ; - ]; p.u = X * p.; Due to the mportance of complex power n the analyss of electrcal networks t s necessary defne a bond graph sensor able to determne the complex power durng smulaton. We can rewrte () nto ts matrx form thus S = V I = [ { V} + ji{ V} ] ji { I} { I} = { V } { I} + I{ V } I { I} + j I{ V } { I} { V } I{ I} (6) P Q he power sensor s depcted n Fg. 6 and ts SIDOPS representaton s shown n able VII. Fg. 6 omplex power sensor n bond graph ABE VII POWE SENSO PHASO BOND GAPH SIDOPS ODE SIDOPS code arables real P {W} Q {VA} S {VA} PF; real V_mag {V} I_mag {A}; real V_ang {deg} I_ang {deg} th {deg}; Fg. 7 Electrcal network where f = 5Hz = π f = Ω = Ω 3 = 4Ω 4 = 5Ω = 63.66mH = 3.83mH = µ F ( t) = 3cos t. he crcut s conerted nto a bond graph by usng the standard methodology for electrcal systems [9] n Fg. 8. Fg. 8 Bond graph of the electrcal network We added the MS blocks from the -sm lbrary n order to obtan the real power. Fg. 9 shows the nstantaneous oltage source total current and total power; together wth ther MS alues. We can obsere from Fg. 9 that the system contans a transent and the MS blocks are calculatng the MS alue drectly from the nstantaneous sgnals. Power sensor p.u = p.u; p. = p.; P = p.u[] * p.[] + p.u[] * p.[]; Q = p.u[] * p.[] - p.u[] * p.[]; S = sqrt(p^ + Q^); th = atan(qp); PF = cos(atan(qp)); 949

6 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 (t) {V} V rms Q = {VA} tme {s} (t) {A} I rms p(t) {W} P.4.5 Fg. 9 Voltage current and real power n the electrc network tme {s} PF =.956 theta = 7. {deg} In order to construct the phasor bond graph we substtute all the elements by ther equalent D representaton. hus Fg. shows the phasor bond graph ncludng a complex power sensor. Fg. Phasor bond graph model It s mportant remark that n phasor bond graph model all our sgnals are MS alues. omparng the responses from the phasor bond graph model wth the MS alue gen by the blocks n the regular bond graph n Fg. 9 we obtan Fg tme {s} (t) {V} V rms V = 3 {V} (t) {A} I rms I =. {A} p(t) {W} P P = 5 {W}.4.5 Fg. omparson between the nstantaneous responses and phasor alues Obously the sgnals from the phasor model are constants whle the MS alues of the dynamc model conerge to these alues once the steady-state state has been reached by the system. One adantage of a phasor model s that the computatonal tme has been reduced drastcally. Besdes we are now able to show the reacte e power the mpedance angle and thus the power factor n a more drect manner. Fg. shows these last three quanttes taken from the complex power sensor. Fg. eacte power power factor and mpedance angle VIII. ONUSION We hae gen a bref descrpton of the phasor theory wdely used n the study of electrcal systems. he bond graph methodology was ntroduced together wth the standard methodology for obtanng the steady-state from a phasor bond graph model. We saw that the bond graph elements could be replaced by ther equalent mpedances. In order to automate the phasor analyss by usng bond graphs the Fourer transform was substtuted nto the mpedance bond graphs. In ths way we were able to create a D multbond bond graph by splttng the real and magnary part of the mpedances ths model was called phasor bond graph. he phasor bond graph model allows knowng the angle and magntude of mportant arables as oltage currents and complex power. he phasor modellng of the electrcal networks ge us the steady-state state behaor of the system. hs nformaton may be useful for fault studes stablty and wdely used n the descrpton of power conerson of electrcal machnes. AKNOWEDGMEN he authors thank the support gen by ontrollab crew deelopers of -sm by ther help n smulaton matters. Specals thank to ONAY (the Mexcan Natonal ouncl of Scence and echnology) and to SEP (the Mexcan Secretary of Educaton) by the fundng of ths research. EFEENES [] has. Proteus Stenmetz omplex Quanttes and ther use n Electrcal Engneerng Proceedngs of the Internatonal Electrcal ongress AIEE Proceedngs 893 pp [] Paynter H. M. Analyss and Desgn of Engneerng Systems MI press 96. [3] A. Veltman D.W.J. Pulle and.w. De Doncker Fundamentals of Electrcal Dres Sprnger 7. [4] Had Saadat Power System Analyss McGraw-Hll New York 999. [5] Mohamed E. El-Hawary Electrcal Power Systems John Wley & Sons New York 995. [6] Stephen J. hapman Electrc Machnery Fundamentals 4 th edton McGraw-Hll New York 5. [7]. Sueur G. Dauphn-anguy Structural controllablty/obserablty of lnear systems represented esented by bond graphs Journal of the Frankln Insttute 36(6)

7 World Academy of Scence Engneerng and echnology Internatonal Journal of Electrcal obotcs Electroncs and ommuncatons Engneerng Vol:8 No:7 4 [8] Kypuros Jaer A. System Dynamcs and ontrol wth Bond Graph Modelng aylor & Francs Group 3 USA. [9] Wolfgang Borutzky Bond Graphs: A Methodology for Modellng Multdscplnary Dynamc Systems Sprnger ondon. []. S. Bonderson Vector Bond Graphs Appled to One-Dmensonal Dstrbuted Systems 975 J. Dyn. Sys. Meas. ontrol 97() [] P. Breedeld Multbond graph elements n physcal systems theory Journal of the Frankln Insttute (/): 36. [] ontrollab Products B.V. -sm 95