MCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems

Transcription

1 MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a physical lctrical circuit, currnt lows rom highr to lowr potntial whn ollowing a circuit path. Within a voltag sourc (say, a battry), currnt lows rom (-) to (). S Fig. 1. Not that w us th convntional or nginring intrprtation o currnt low. A mor laborat xplanation involving movmnt o vacancis or hols is mployd in rigurous physics. 2 Voltag and Currnt Sign Convntions and Intrprtation Whn voltags (including sourcs) can chang polarity with tim, w ix a particular st o polaritis and assign voltag variabls as positiv or that coniguration. S Fig. 2. Nothing has bn assumd in th pictur on th lt. On th right, w hav assumd that point a has highr potntial than point b, which in turn has highr potntial than point c. This dos not man this will b th cas at all tims. W indicat our assumd coniguration with () signs at on nd o ach sourc or componnt and associat a voltag variabl which is dind positiv whn masurd rom () to (-). This applis to all componnts xcpt voltag sourcs, or which th sourc voltag is positiv whn dirctd rom (-) to () within th sourc. Latr, i som solution mthod givs, or xampl 1 > 0 and 2 < 0, w intrprt this in th physical systm as point a having highr potntial than point b (matching th assumd coniguration), but point b having lowr potntial than point c. For currnts, th natural choic is to introduc variabls which ar positiv whn dirctd rom highr to lowr potntial. Howvr, on may b intrstd in solving or th ngativ variabl. rring to Fig. 3 (lt), w s that i a numrical solvr givs positiv valus or i 1, i 2 and i 3, th currnts will b dirctd as shown. Th solvr should giv i 1 = i 2 in any cas (sam sign). For high potntial within batt: rom - to obsrvd sns o circulation: rom to - low potntial Figur 1: Convntional currnt circulation 1

2 1 a b 2 Nothing assumd Potntials assumd c Figur 2: Voltag convntions i i 1 i 3 i 2 i i 1 i 2 i 3 Natural choic (thru-powr) Arbitrary choic Figur 3: Currnt convntions th choic on th right, i th solvr givs i 1 < 0, thn th actual currnt is lowing rom point a to point b. Th solvr should always giv i 1 and i 2 with opposit signs undr this sign convntion. 3 Ect o th Convntions on th Equations Suppos that th convntion shown in Fig. 3 (lt) is usd. For nod a w writ i = i 1 i 3 whil or nod b w hav For voltags w writ Finally, or th componnts w writ i 1 = i = = i 3 1, V 1 = i 1 2, i 2 = Cė 2 Now suppos th convntion shown on th right is usd. For nod a w writ i = i 1 i 3 whil or nod b w hav i 1 = i 2 2

3 = = (t) 1 = = (t) 1 = = (t) = = (t) Figur 4: Bond Graph sign convntions For voltags w writ and or componnts w hav 1 2 = V a V c = = i 3 1, V a V b = 1 = i 1 2, Cė 2 = i 2 4 Sign Convntions in Bond Graphs and Physical Intrprtation Examin Fig. 4. I a 1-port (xcpt sourcs) has th hal-arrow dirctd towards it, w writ th constitutiv quations without ngativ signs. I th hal arrow is dirctd away rom th 1-port, w nd to introduc a minus sign in th constitutiv quation. For xampl, considr th bottomright cas. I th solvr givs > 0 and < 0 at som instant, w intrprt this physically in th circuit as th top bing at highr potntial than th bottom and thror th currnt circulating clockwis. Th bond graph should conirm ths orintations. In act, th product is ngativ, thror powr lows rom th sourc to th lmnt. 5 Ect o Bond Graph Sign Convntions on th Equations Figur 4 shows th quations that nd to b writtn or ach sign convntion. Th sam applis to C and I lmnts. Not that, or, th quations and powr intrprtation o th hal-arrow orbid th situation o powr lowing rom into th sourc. In th cas o 2-ports (TF and GY), th basic principl is that th 2-port passs powr thru with 100% icincy. garding sign convntion, i powr ntrs a port, thn it must lav th othr. For transormrs, rr to Fig. 5. In th top igur, w s rom th quations that i 1 and ar positiv, thn so must b 2 and 2. Thus, powr lows rom lt to right. I ithr port variabl changs sign, thn th quations dictat that th corrsponding variabl in th opposit port must do so. I, or instanc, 1 < 0 and > 0, thn 2 < 0 and 2 > 0 and powr lows rom right to lt. I w assum th sam or th bottom igur, howvr, w gt 2 < 0 and 2 < 0, but powr still ntrs rom th right and lavs to th lt. Work out th maning o convntions or GY yoursl. 3

4 TF = m 2 2 = m TF = m 2 2 = m Figur 5: Convntions or Transormrs V o 6 Exampl Figur 6: Exampl circuit Obtain a bondgraph or th circuit o Fig. 6. Assign arbitrary powr dirctions and obtain a valid st o quations. Thn intrprt in th physical diagram. Th bondgraph with arbitrary convntions is shown in Fig. 7. Th ollowing is th list o corrct quations: 1 = (t) = 2 = = 0 2 = TF 1 0 SF : i = 0 Figur 7: Bondgraph or xampl 4

5 = V o = Figur 8: Intrprtation o sign convntions 3 = m 4 4 = 5 = 6 5 = = 0 6 = 7 = = 0 7 = = 0 Th physical intrprtation o our signd bondgraph is shown in Fig. 8. I (t) is givn and th solvr givs, or xampl, 7 < 0 at som instant, w anticipat that th mtr will hav a positiv rading at that instant. 7 Laboratory Dmonstration A simpl circuit will b brought to class. A bondgraph will b obtaind and signd. Thn th quations will b solvd by computr using som (t) input. Th sign convntion will b intrprtd with th physical schmatic and vriid with th xprimntal stup. 8 Conclusions Sign convntions in a bond graph cannot b wrong until th wrong quations ar writtn. A consistnt bond graph-quations combination has to b intrprt proprly in a physical situation. Th ntir lin o rasoning can b applid to thrmal, mchanical and hydraulic systms, i th abstract notions o ort and low ar rtaind. Th powr signs ar arbitrary, but th thru-powr convntion is mor intuitiv. 5