Class Note for ECE 449 at UA
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Date Created: 02/06/15
A I we 449549 ontinnons System mutualitth Electrical Circuits I 0 This lecture discusses the mathematical modeling of simple electrical linear circuits 0 When modeling a circuit one ends up with a set of implicitly fonnulated algebraic and differential equations DAEs Which in the process of horizontal and vertical sorting are converted to a set of explicitly fonnulated algebraic and differential equations 0 By eliminating the algebraic variables it is possible to convert these DAEs to a statespace representation September 3 2003 start of Presentation EASE I we 449549 Glontinnnns System mutualitth Table of Contents 0 Components and their models 0 The circuit topology and its equations An example 0 Horizontal sorting Vertical sorting Statespace representation 0 Transformation to statespace form September 3 2003 start of Presentation A I we 449549 untinnons System mmml Linear Circuit Components R Res1stors mi t vb u vu v u R i u i C u vu v o Capac1tors VaH H a Vb 1114 u t W 39 L M vu v o Inductors vac1m vb di quot u u L E September 3 2003 Start of Presentation I we 449549 Glontinnnns System mutualingl Linear Circuit Components 11 U 0 Voltage sources Va e vb U v Va U a ft U0 390 M v v Current sources v VILQ vb I fnt a du 0 Ground I oTel V 0 v September 3 2003 Start of Presentation A I we 449549 ontinnons System mutualitth Circuit Topology in ii v vv o Nodes Va vb quot ic In In tE 0 vc quota1 v11 vb 0 Meshes v umunum0 u be September 3 2003 start of Presentation EASE I we 449549 Glontinnnns System mutualitth An Example I ma my 3 11 Mggl i Gmund September 3 2003 start of Presentation A I we 449549 untinnons System mutualitth Rules for Systems of Equations I 0 The component and topology equations contain a certain degree of redundancy 0 For example it is possible to eliminate all potential variables vi without problems 0 The current node equation for the ground node is redundant and is not used 0 The mesh equations are only used if the potential variables are being eliminated If this is not the case they are redundant September 3 2003 start of Presentation I we 449549 Glontinnnns System mutualitth Rules for Systems of Equations 11 0 If the potential variables are eliminated every circuit component de nes two variables the current 139 through the element and die Voltage u across the element 0 Consequently we need two equations to compute values for these two variables 0 One of the equations is the constituent equation of the element itself the other comes from the topology September 3 2003 start of Presentation A I we 449549 ontinnons System mutualitth An Example II Component equations U0 ft is 039 due1t u R i ul L diLdt uz Rz 139 Node equations i0i1il i1izic The circuit contains 5 components 3 We require Mesh equations 10 equations in U u u u u u 10 unknowns 0 1 C L I z u u September 3 2003 start of Presentation I we 449549 Glontinnnns System mutualitth Rules for Horizontal Sorting I 0 The time t may be assumed as known 0 The state variables variables Which appear in differentiated form may be assumed as known U0ft i1iL U0ft IR139i1 i2ic MIR1i1 z Rz39 i2 U0 1 141 gt 142 Rz i2 iCCdqut ucuz iCCdugdt 0 uLLdiLdt ulu1u2 uLLdiLdt ulu1u2 September 3 2003 start of Presentation A I we 449549 ontinnons System mutualitth for it U0 ft I RI39 1391 142 R2 139 is C duedt ul L diLdt i0i1il i1izic U0u1uc 0 MLMIM2 The solved variables are now known Rules for Horizontal Sorting II 0 Equations that contain only one unknown must be solved U0 ft 1R139i1 uzRz i2 iCC dugdt ul L diLdt September 3 2003 start of Presentation A I we 449549 Glontinnnns System mutualitth Rules for Horizontal Sorting III 0 Variables that show up in only one equation must be solved for using that equation U0 ft I RI39 1391 142 R2 139 is C duedt ul L diLdt U0 ft 1 R13911 is C dugdt 14L L diLdt September 3 2003 start of Presentation A I we 449549 hummus System manslingl Rules for Horizontal Sorting IV 0 A11 rules may be used recursively U0ft i0i1il U0ft I R139 1391 i1 i2 ic I R139 1391 2 Rz39 1392 U0 I c gt z Rz39 l392 icC39 duedt 0 iCC dugdt uL L diLdt 14L 14 142 uL L diLdt September 3 2003 Start of Presentation we 449549 Glontinnnns System mutualingl U0ft i0i1il U0ft 1 RI39 1391 I I R139 1391 142 R2 12 gt 142 Rz i2 iCC duedt iCC duadt 6 uLLdiLdt ulLdiLdt uLu1uz U0 ft The algorithm is applied until I R1 1 every equation de nes exactly one variable that is solves for 2 R239 12 iCC dugdt ucuz uLLdiLdt uLu1uz September 3 2003 Start of Presentation A I we 449549 ontinnons System mutualitth Rules for Horizontal Sorting V 0 The horizontal sorting can now be performed using symbolic formula manipulation techniques Uoft i0i1il Uoft lI0iziz 1R139i1 i1lzic l391 1R1 ici1 39iz 2Rz39iz U0 1 c gt lIz zRz IU039 c iCC duedt c z duddticC 2 C MLLdiLdt ulu1uz diLdtuLL L 1 2 September 3 2003 3 start of Presentation I we 449549 Glontinnnns System mutualitth Rules for Vertical Sorting By now the equations have become assignment statements They can be sorted vertically such that no variable is being used before it has been defined Uoft U0ft lIz zRz i1 1R1 iCi1i2 1U039 c ici139iz izuzRz M1U0MC gt i1u1R1 ulu1uz dugdtiCC uz uc i0 i1 il dugdtiCC diLdtuLL MLM1M2 uzuc diLdtuLL September 3 2003 start of Presentation A I we 449549 untinnons System mutualitth potentials and Voltages equations of the topology ignored before can be ignored Rules for Systems of Equations III 0 Alternatively it is possible to work with both 0 In that case additional equations for the node potentials must be found These are the potential equations of the components and the potential Those had been 0 The mesh equations are in this case redundant and September 3 2003 start of Presentation A I we 449549 Glontinnnns System mutualitth An Example III Component equations U 0 ft u1R111 142 R2 12 is C dugdt ML L diLdt 0 The circuit contains v 5 components and 3 quot0 deS Node equations 3 We require i0 i1 iL 13 equations in 13 unknowns U0v1v0 1v1vz cvzv0 Lv1v0 lllzlc September 3 2003 start of Presentation A we 449549 untinnons System mutualitth Sorting The sorting algorithms are applied just like before 0 The sorting algorithm has already been reduced to a purely mathematical informational structure Without any remaining knowledge of electrical circuit theory 0 Therefore the overall modeling task can be reduced to two subproblems 1 Mapping of the physical topology to a system of implicitly formulated DAEs 2 Conversion of the DAB system into an executable program structure September 3 2003 start of Presentation lt I we 449549 Glontinnnns System mutualitth Statespace Representation A e an x n 0 Linear systems x 6 9V B 6 mm m AXBu xzox0 6mm cempxn Z yCxDu yegtp Dempxm x State vector 0 N onlmear systems In t t II pu V60 0139 mum x00 x0 y Output vector y gxut n Number of state variables m Number of inputs p Number of outputs September 3 2003 start of Presentation 10 A I was 449549 ontinnons System mutualitth Conversion to Statespace Form I duocit iCC U t 139 14 R 0 f z z z i1i2C u U 14 i i i I quot C C I z iICizC 11 1R1 L 1 z u1R139Q MzR239C i0i1il dugdtzCC U039 0R139Cv ucR239C u2uc dzLdtuLL diLdt uLL 1 zL u1LuzL U0ucL uCL U0L September 3 2003 start of Presentation I was 449549 Glontinnnns System mutualitth Conversion to Statespace Form 11 Welet x1uc I2 1 j x 1 u xziL 1 RI C RZ C 1 RI C39 u Ua 1 y quotC x2 I yx1 September 3 2003 start of Presentation 11 I lm 449 549 hummus 5mm mammal An Example IV gem UDBB kA I 59E R mu R2 zu L uuu15 c 175 mc JURan m sthCn all iuuc chl Aa1DDD h R1c 11 1 u 1 um s 5er 1a an t 1E752E73 u nmsmgmuum XE r 1 2 1 x laursucxm pmucm grld an Ecuxn 7 Q eueMEL39pFl Ready jam gt4 4 September 3 2003 Start of Presentation midi 449549 ontinunux system mnhelingl References Cellier FE 1991 Continuous SVstem Modeling SpringerVerlag New York Chapter 3 Cellier FE 2001 Matlab code of circuit example September 3 2003 Start of Presentation 12
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