Analysis of Low-Frequency Electromagnetic Devices using Finite Elements

Transcription

1 Analysis of Low-Frequency Electromagnetic Devices using Finite Elements Ingeniería Energética y Electromagnética Escuela de Verano de Potencia 2014 Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

2 Overview The nodal method and its limitations. Modified nodal analysis (MNA): Implementation basics. Low-frequency electromagnetic equations. Circuit-field coupled problem. Example: Induction machine operating at steady state. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

3 Nodal Method Algorithm The system of circuit equations is easily constructed through building blocks of admittance elements. It is widely used in power network calculations. Forcing currents injected at specified circuit nodes. Building blocks are obtained from: Network-independent voltage-current relations (VCR) of an element. The numbering of the element terminals within an arbitrary network. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

4 Nodal Method Frequency Domain Consider only time-harmonic computations for the sake of simplicity. VCR of an admittance: y a v a = i a If the terminals are connected to nodes p and q of an arbitrary network of m non-trivial nodes, the branch voltage can be expressed as v a = v p v q The current contribution to nodes p and q is given by { } { } ip 1 = i 1 a i q Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

5 Nodal Method Frequency Domain Thus the VCR can be transformed into a building block: [ ] { } ya y a vp = y a y a v q { ip The final system of equations can be simply written as Yv = i i q } Y is known as the system admittance matrix. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

6 Nodal Method Attributes and Limitations Advantages Construction by building blocks gives an algorithm easily programmable. Mutually coupled admittances are readily incorporated. Source transformation may allow incorporation of non-ideal voltage sources. Current controlled current sources can also be accommodated. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

7 Nodal Method Attributes and Limitations Limitations Non-natural elements such as ideal voltage sources, dependent sources, transformers and auto-transformers cannot be directly considered in the analysis. Special post- and pre- processing are required with non-natural elements. Loss of information with the use of transformations (such as Norton s theorem). Topologically connected nodes are reduced in number. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

8 The Modified Nodal Analysis MNA Incorporation of VCR relations as additional equations. Injection of the element currents to nodes. Total elimination of the nodal method limitations with the advantage of keeping one of its main attributes: Building Block Construction. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

9 The Modified Nodal Analysis MNA The MNA always gives: [ ] { } { } Y A v i = B D Y is constructed in the usual way. Non-natural elements are erased from the Y building process. i e Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38 f v

10 The Modified Nodal Analysis MNA D, A and B are matrices whose element entries are zeros, ones, element resistances and dependent source gains. Zero diagonal may exist but this does not seem to be a limitation due to efficient sparse matrix solvers and reordering techniques. f v is a forcing vector that contains the voltage values of independent voltage sources and zeros for other non-natural elements. i e is the vector of unknown branch currents of non-natural elements. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

11 The Modified Nodal Analysis MNA A = B in the absence of equivalent circuit dependent sources. The MNA technique is widely used in most modern circuit simulation packages. It has a strong foundation on circuit theory. The procedure for constructing the building blocks is always the same: addition of VCR and incorporation of the element current into the nodal equations. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

12 MNA Building blocks Table : Independent Active Elements Circuit Element Stiffness Contribution Forcing Contribution p q s 1 p q 1 1 z s s { } 00 pq e s s { is is } pq Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

13 MNA Building blocks Table : Passive Elements Circuit Element Stiffness Contribution p q [ ] ya ya p ya ya q p q k l t 1 p q N k N l 1 1 N N 0 t Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

14 MNA Building blocks Table : Cont d Circuit Element Stiffness Contribution p q k a p (N + 1) q k a N 1 (N + 1) N 0 p q k l y a y a y m y m p y a y a y m y m q y m y m y b y b k ym ym y b y b l Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

15 MNA Building blocks Table : Dependent Sources Circuit Element Stiffness Contribution p q k l f 1 p 1 0 q 4 4 D F k D F l f p q k l g 1 p 1 0 q k 0 l D G g Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

16 MNA Building blocks Table : Cont d Circuit Element Stiffness Contribution p q k l e 1 p 1 0 q k 0 l D E D E e p q k l h i 1 0 p q k 0 1 l h G H i Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

17 2D Low-Frequency Electromagnetic Equations Let a 2D domain be constituted by pure non-conducting regions as well as by an arbitrary number of filamentary and solid conducting sub-domains. Neglect displacement current and free charge. Maxwell s equations lead to the following diffusion type equation ν A = J f J s = J f + σ( V + A t ) Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

18 Conductors Solid and Filamentary Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

19 2D Low-Frequency Electromagnetic Equations Domains: ν A = 0 : ν A = J f 2D ν A = J s = σ( V + A t ) Ω sol in Ω nc in Ω fil in Ω sol Solid conductor current equation: σ i s = v s dω σadω = v s σ AdΩ d s t Ω sol r s t Ω sol Solid conductor voltage equation: v s = r s i s + d s t Ω sol AdΩ S s Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

20 2D Low-Frequency Electromagnetic Equations Filamentary conductor voltage equation: Ω AdΩ v f = r f i f + d f n fil f t S f The field system can be set up as ν A = 0 : ν A = n f i f S f 2D ν A σ A t + σ t AdΩ S s = is S s in Ω nc in Ω fil in Ω sol Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

21 Finite-element system Finite-element discretization gives K a s (C s C s) da dt + f N f i f + s N s i s = 0 K = ν ζ T ζ dω e Ω e N f = n f ζ dω S f C s = σs s d s N s N T s = d s R s N s N T s C s = σζ T ζ dω e Ω e N s = 1 ζ dω S s Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

22 Connecting the FE model to the exterior world Figure : MVP controlled voltage sources and their interaction with the FE Model: (A) Filamentary and (B) Massive. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

23 Magnetic vector potential controlled voltage sources n f t t Ω fil AdΩ S f Ω sol AdΩ S s v f =r f i f + d f Nf T da dt v s =r s i s + d s Ns T da dt = n f ζ T dω da S f Ω fil dt = 1 S s Ω sol ζ T dω da dt = N T f = N T s Frequency domain counterparts are obtained by substituting t by jω. da dt da dt Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

24 Circuit-field uncoupled problem K 0 0 a f i 0 Y A v = i 0 B D K = K + jωc eq and f i = N f i f N s i s K a 0 0 N f 0 Y E 0 v i = i i 0 F D 0 e i e i c f v f v Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

25 Circuit-field coupled problem The VCR of solid and filamentary conductors can be written in the frequency domain as v p v q = ri + j d ωn T a subscripts f and s have been dropped out since filamentary and solid conductors can be treated in a unified way This VCR can be added to as an additional equation, leading to K 0 0 N a f i v i = i e f v 0 0 Y E 0 0 F D 0 j d ωn T l T 0 r i c Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

26 Building Block for Filamentary and Solid Conductors The coupling is completed adding i c to equation p of the nodal system and subtracting it to equation q: K 0 0 N a f i v i = i e f v 0 0 Y E l 0 F D 0 j d ωn T l T 0 r A building block can now be determined as i c N 0 l 0 j d ωn T l T 0 r Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

27 Coupled system Inclusion of all massive and solid conductors gives the final system K 0 0 N a f i 0 Y E L v i 0 F D 0 = i e f v j d ωn T L T 0 R 0 which can be conveniently rewritten as (κ 1 i c j d ω ) K 0 0 N a f i 0 κy κe κl T v κi 0 κf κd 0 = i e κf v N T κl 0 κr 0 i c Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

28 Coupled system The coupled field-circuit system is symmetric if E = F. This condition is achieved with circuits that do not contain voltage or current controlled sources. Notice that MVP-controlled sources do not lead to asymmetry. The coupled problem will not have zeros in the main diagonal for systems that consider non-ideal voltage sources and inductances, capacitances and resistances. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

29 Coupled system The incorporation of building blocks can be arbitrarily performed in any order. This means that the building blocks of solid and filamentary conductors are actually members of the family of non-natural circuit elements: K 0 N a f i 0 κy κe v N T = κi κf κd i e κf v Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

30 Induction motor coupled to circuits Quasi 3D FE model The test case involves the time-harmonic non-linear operation of a squirrel cage induction motor. It is a two-pole, 7.5 kw, 380 V, 50 Hz, three-phase star connected motor. Figure : 7235 nodes and 3464 second order elements (triangular and quadrilateral). Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

31 Circuits Stator and rotor Figure : Stator connections. Figure : Rotor connections. Antiperiodic boundary conditions are considered. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

32 Result Flux Plot: Rated Operating Condition. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

33 Result Table : FE Calculated Currents: Filamentary i a i b i c N.B. i a, i b and i c are the RMS stator phase currents. Current values in [A]. Table : FE Calculated Currents: Solid Bar 1 Bar 5 Bar N.B. RMS current values in [A]. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

34 Rotating magnetic field Show Movie Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

35 Conclusions The MNA scope has been broadened out with the addition of one new non-natural element (a MVP-controlled voltage source) The FE model has benefited from a well-posed formulation of unknown currents in conductor regions. A building block has been presented for systematic inclusion of both solid and filamentary conductors of FE regions within arbitrary topologies of circuit systems. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

36 Conclusions The circuit system does not see differences between these classes of conductors, it only sees controlled voltage sources. The proposed MNA-FE coupling technique is based on sound circuit theory known by all electrical and electronic engineers. Concepts of of topology theory are not needed. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

37 Conclusions The methodology can also be easily mounted on existing FE codes due to the building block approach that is also conventionally used in the FE method. Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38

38 Gracias! Rafael Escarela (UAM-A) Coupling Circuit and Field Systems August / 38