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Sybolical Index Calculation For Linear Circuits

Sybolical Index Calculation For Linear Circuits Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-205$16.00 2001, Vol. 7, No. 2, pp. 205±214 Swets & Zeitlinger Symbolical Index Calculation for Linear Circuits * * * * C. CLAUû , P. SCHWARZ , B. STRAUBE and W. VERMEIREN ABSTRACT The condition for the DAE index to be higher than one is calculated symbolically. Using Analog Insydes a function for the computer algebra system Mathmatica is written. It calculates the index condition if the sparse tableau analysis method, or the modi®ed nodal analysis method is applied. 1 INTRODUCTION One of the central questions in the ®eld of differential-algebraic equations (DAE's) [1, 2, 5] or descriptor systems [11] is the determination of the index. Dif®culties in the numerical treatment [12, 16] of the DAE are expected if the index exceeds one (higher-index case). Therefore, a higher-index check is quite useful. The known index concepts coincide in the case of linear DAE's, in which the index calculation possibilities are well understood [3, 13]. Linear DAE's are generated if the most often used modi®ed nodal analysis (MNA) [8], or sparse tableau analysis (STA)[6] methods are applied to linear circuits [4]. Based on the matrices of the linear DAE's the higher-index condition can be checked. If the analysis approaches are realized symbolically, which is possible using a symbolical circuit analysis tool (e.g., Analog Insydes [7]) the symbol- ical matrices can be the basis for the calculation of a symbolical higher-index Fraunhofer Institute for Integrated Circuits, Design Automation Department (EAS) Dresden, Zeunerstraûe 38, D-01069 Dresden, Germany. Tel.: ‡49-351-4640 737; E-mail: (clauss, schwarz, straube, vermeire)@eas.iis.fhg.de 206 C. CLAUû ET AL. condition. Such a symbolical higher-index condition can be used for the index check by replacing the symbolical parameters by numerical ones. The advantage over the pure numerical calculation is the reduction of round-off errors. Furthermore, the symbolical higher-index condition allows to ®nd out how circuit element parameters have to be chosen to get / avoid a higher index. In this paper a symbolical calculation method for the higher-index condition is presented which uses Analog Insydes, and Mathematica. It is restricted to linear circuits in which only capacitors and inductors lead to derivatives of the variables with respect to the time. The method is applied to some simple examples. 2 THE HIGHER-INDEX CONDITION A linear DAE is considered with constant coef®cient matrices (e.g., a descriptor form, A can be singular) Ax …t†‡ Bx…t†‡ f…t†ˆ 0 …1† m mxm 1 (x…t†; f…t† : T ! R ; A; B 2 R ; T  R time interval, m > 0,' symbolizes the derivative with respect to the time). According to the tractability index de®nition [9] the following matrix chains have to be calculated: A ˆ A; B ˆ B …2:1† 0 0 A ˆ A ‡ B Q ; B ˆ B P …2:2† i‡1 i i i i‡1 i i with P ˆ I ÿ Q ; and Q is a projector on the kernel ker A ; i ˆ 0; 1; .. . The i i i i tractability index is the number  with A regular and A singular for all j < . Consequently, the index is higher than one if and only if A is singular. Therefore, the condition det…A‡ BQ †ˆ 0 …3† has to be evaluated. To get this condition symbolical circuit analysis is used. 3 SYMBOLICAL CIRCUIT ANALYSIS Analog Insydes [7] is a program for the symbolical analysing and sizing of analog electronic circuits. It is based on the computer algebra system Mathe- matica [19] utilizing its symbolical calculation capabilities. For handling SYMBOLICAL INDEX CALCULATION 207 circuit analysis a set of Mathematica functions is de®ned. We use Analog Insydes to derive equation (1) from a circuit description. The further calculation steps use Mathematica functions only. The higher-index condition (3) is derived by the following steps: (a) Describe the circuit in an element oriented way There is an input format which uses the Mathematica list capabilities. A netlist converter for reading SPICE netlists is available. (b) Set up the circuit equations It is carried out by the `CircuitEquations' analysis function. By default the MNA approach is used. If the `SparseTableau' option is set the STA equations are generated. (c) Extract the A and B matrices This is done by elementary Mathematica functions. (d) Construct the projector Q on ker A Considering the capacitors and inductors which are the only circuit components that cause derivatives with respect to the time the projector can be constructed by determining the zero rows of the matrix A. (e) Calculate the determinant, and factorize it The matrix A ˆ A‡ BQ is calculated. Symbolical determinant calcula- 1 0 tion and factorization are functions of the Mathematica system. The symbolical analysis steps are combined to a user de®ned Mathematica function. The steps (a), (b) are done by Analog Insydes functions, and the steps (c), and (e) are simple matrix calculations. The construction of the projector in step (d) is described in the next section. 4 PROJECTOR CONSTRUCTION Since the allowed circuit elements are restricted to only capacitors, and inductors, which produce time derivatives of circuit variables, the matrix A has a special structure which depends on the circuit analysis approach. The speci®c structure of A allows it to construct the projector Q easily. 4.1 Sparse Tableau Analysis (STA) Both all branch voltages and all branch currents become variables of the DAE. Only capacitors and inductors produce nonzero entries in the A matrix. Since 0 0 the equations Cv…t†ˆ i…t† of capacitors, and Li…t†ˆ v…t† of inductors are the 208 C. CLAUû ET AL. only equations which use time derivatives, the nonzero entries in A are all C- values, and all L-values. Moreover, each entry is the only entry both in its line and column. Due to this structure, Q can be easily given by a matrix with a 1 entry on the main diagonal if and only if the corresponding column of A is a zero column. All other elements of Q are zero, e.g.: 2 3 2 3 000 0 0 00 0 1 0 0 000 00 6 7 6 7 000 0 0 00 0 0 0 0 000 00 6 7 6 7 6 7 6 7 0 C 00 0 0 00 0 0 1 000 00 6 7 6 7 6 7 6 7 000 0 0 00 0 0 0 0 100 00 6 7 6 7 A ˆ Q ˆ 6 7 6 7 000 0 0 C 00 0 0 0 000 00 6 7 6 7 6 7 6 7 000 0 0 00 0 0 0 0 000 00 6 7 6 7 4 5 4 5 000 0 0 00 0 0 0 0 000 10 000 0 L 00 0 0 0 0 000 01 4.2 Modi®ed Nodal Analysis (MNA) All node voltages become variables of the DAE, and furthermore, all branch currents which cannot be eliminated, e.g. the currents of inductors or of independent voltage sources. Inductors and single grounded capacitors are treated like in the STA approach. A single ¯oating capacitor which is not grounded leads to the following submatrix of A which is singular. The corresponding submatrix q of the projector Q is added: 0 0 Connecting subgraphs of capacitors with n nodes, which do not have a capacitor to the ground lead to n-dimensional submatrices a of A, which are singular. The corresponding submatrices q of Q are n-dimensional matrices 0 0 with the entries 1/n each, e.g.: SYMBOLICAL INDEX CALCULATION 209 The submatrix is regular, if a connecting subgraph of capacitors is grounded by at least one capacitor. Consequently, the submatrix q of the projector Q is 0 0 zero. By subdividing the Matrix A into submatrices, the projector Q can be calculated using the above mentioned submatrices q . For the following sample matrix A with several submatrices due to capacitor subgraphs the projector Q is given: 2 3 0 0 0 0 0 000 0 0 6 7 0 C ‡ C ÿC 0 0 000 0 0 1 2 1 6 7 6 7 0 ÿC C 0 0 000 0 0 1 1 6 7 6 7 00 0 C ‡ C ÿC 0 ÿC 00 0 3 4 3 4 6 7 6 7 00 0 ÿC C 000 0 0 3 3 6 7 A ˆ 6 7 00 0 0 0 C 0 ÿC 00 5 5 6 7 6 7 00 0 ÿC 00 C 00 0 4 4 6 7 6 7 00 0 0 0 ÿC 0 C 00 5 5 6 7 4 5 0 0 0 0 0 000 C 0 0 0 0 0 0 000 0 L 2 3 10 0 0 0 0 0 0 00 6 7 00 0 0 0 0 0 0 00 6 7 6 7 0 001=31=30 1=30 0 0 6 7 6 7 0 001=31=3 000 0 0 6 7 6 7 00 0 0 0 1=20 1=20 0 6 7 Q ˆ 6 7 0 001=30 01=30 0 0 6 7 6 7 00 0 0 0 1=20 1=20 0 6 7 6 7 00 0 0 0 0 0 0 00 6 7 4 5 00 0 0 0 0 0 0 00 00 0 0 0 0 0 0 00 5 CIRCUIT EXAMPLES The following example gives a small impression of the calculation of the higher-index condition. It is a very simple example for demonstrating only the 210 C. CLAUû ET AL. calculation steps. The Analog Insydes input netlist of this RC circuit is shown on the left. The generated STA matrices A and B are: 2 3 2 3 0 0 000 000 ÿ11 11 00 0 0 6 7 6 7 0 0 000 000 000 0 0 0 0 0 6 7 6 7 6 7 6 7 0 0 000 000 000 0 0 ÿ11 0 6 7 6 7 6 7 6 7 0 0 000 000 000 0 0 0 ÿ11 6 7 6 7 A ˆ ; B ˆ 6 7 6 7 0 0 000 000 100 0 0 0 0 0 6 7 6 7 6 7 6 7 0 0 000 000 0 ÿ10 0 0 R 00 6 7 6 7 4 5 4 5 0 0 000 000 00 ÿ100 0 R 0 00 0 C 0 000 000 0 0 0 0 ÿ1 …4† The projector on the kernel of A is Q ˆ diag…1; 1; 1; 0; 1; 1; 1; 1†: …5† The calculated higher-index condition achieved by Mathematica is C…R ‡ R †ˆ 0: …6† 1 2 If this condition is ful®lled the DAE index is higher than 1. There are two factors in (6), therefore: C ˆ 0: This is not possible, because the construction of the projector presumed C not to be zero. R ˆÿR : It leads to a higher-index DAE indeed. 1 2 In this example a formula was calculated symbolically which allows to determine the circuit element parameters in such a way that the STA equation system is of higher index. Sometimes the determinant of the matrix A is zero independently from speci®c parameter values. Then the index is higher due to the DAE structure [18]. For example this can be observed if a mesh of capacitor branches occurs as in the following example. SYMBOLICAL INDEX CALCULATION 211 Without considering the intermediate steps the symbolically calculated higher-index condition is f0g. That means, the index is higher than 1 in any case of parameters R ; R ; C ; C ; C . Therefore, the symbolical higher-index 1 2 1 2 3 condition gives insight into the structural higher-index property. If the MNA is applied to this circuit, the index is equal to 1. This demon- strates that the index depends on the structure of the circuit, its numerical values, and the applied analysis method. 6 EXAMPLES FOR IDENTIFYING INDEX PROBLEMS IN TEST SIGNAL EVALUATION Analogue fault simulation [15] is applied to evaluate test signals for an analogue network for the purpose of fault detecting. A fault simulator creates a faulty network by fault injection. The electric behavior of a fault is usually modelled by network elements and their interconnections. Commonly, two- poles for shorts and opens or n-poles, e.g., for defective transistors, are injected into the network. Caused by such a fault injection the electric behavior of the now faulty network changes. As a ®rst example the RC-network [16, 17] with an operational ampli®er depicted in the following picture is considered. If the operational ampli®er is modelled ideally by a nullator-norator-pair, the fault free network can be simulated without any problems. In the case of the open fault no simulation was possible. If otherwise the operational ampli®er is modeled using a voltage controlled voltage source (ampli®cation V ˆ 15000) the fault-free network can be simulated too. Its behavior (node 7) is shown in the result plot (upper window). If the open fault is injected at G the simulator has calculated the behavior shown in the lower window of the result plot. Note the different 212 C. CLAUû ET AL. scales of the voltage ranges. An analogue fault simulator would report this fault as to be detected. However, there is an higher-index problem. The faulty network has the index ˆ 6. Thus, the high index is the reason for the above mentioned behavior. The application of the higher-index condition formula method allows to detect a higher-index already after fault injection but without the fault simulation run; to give a warning if in the case of a higher index a fault is reported as to be detected. On the other hand, the conditions for an index increase can be calculated for different description levels of the operational ampli®er. In this example, assumed an ideal operational ampli®er, the condition for a higher index is: C  C  C  C  C  G  R  R  R  R  R  R  R ˆ 0 …7† 1 2 3 4 5 7 0 1 2 3 4 5 6 That means the index increases if G ˆ 0 which is the open fault, or if R ˆ 0…i ˆ 0; .. . 6† which leads to meshes of capacitors. Assumed the operational ampli®er is represented by a voltage controlled voltage source the higher-index condition changes into C  .. . C  R  .. . R …R ‡ R ‡ G  R  R ÿ G  R  R  V†ˆ 0 …8† 1 5 2 6 0 1 7 0 1 7 0 1 The index increases if: G ˆ…R ‡ R †=…R  R …Vÿ 1†† …9† 7 0 1 0 1 SYMBOLICAL INDEX CALCULATION 213 In the case of the open fault the network can be simulated due to the fact that the index is not high. However, the result is doubtful since G is in the critical region according to (9) if G is closer to zero. Consequently, the symbolical higher-index condition gives a useful insight. The second example is a high-pass ®lter. If an ideal operational ampli®er is used the application of the above outlined method calculates the following symbolical higher-index condition: C  C  C  C …R  R ‡ R  R ‡ R  R † R  R ˆ 0 …10† 1 2 3 4 0 1 0 2 1 2 3 4 That means a higher index occurs if e.g. R is a short circuit. This is already known from fault simulation for a short fault injected at R . In such case the simulation runs into wrong results. 7 SUMMARY A method for the symbolical calculation of the higher-index condition of linear circuits is presented. It is quite useful both for the recognition and for the search of higher index DAE's. Critical circuit element parameters can be found which are responsible for an index increase. Due to the symbolical calculation, the method seems to be more robust than pure numerical calculations [10]. The performance of the method is demonstrated. ACKNOWLEDGEMENTS This research has been funded by the Deutsche Forschungsgemeinschaft (DFG) within the SFB 358 Automated System Design. 214 C. CLAUû ET AL. REFERENCES 1. Arnold, M.: Zur Theorie und zur numerischen Lo È sung von Anfangswertprolemen fu È r Differentiell-Algebraische Systeme von ho È herem Index. Fortschr.-Ber. VDI Reihe 20 Nr. 264, Du È sseldorf, VDI Verlag 1998. 2. Brenan, K.E., Campbell, S.L. and Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, New York, 1989. 3. Gantmacher, F.R.: Matrizentheorie II. Deutscher Verlag der Wissenschaften, Berlin, 1959. 4. Gu È nther, M., Hoschek, M. and Rentrop, P.: Differential-Algebraic Equations in Electric Circuit Simulation. Int. J. Electron. Commun. (AEU) 54 (2) (2000), pp. 101±107. 5. Hairer, E. and Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991. 6. Hachtel, G.D., Brayton, R.K., Gustavson, F. G.: The Sparse Tableau Approach to Network Analysis and Design. IEEE Trans. Circuit Theory, CT-18 (1971), pp. 101±113. 7. Hennig, E. and Halfmann, T.: Analog Insydes Tutorial. ITWM, Kaiserslautern, 1998. 8. Ho, C.W., Ruehli, A.E. and Brennan, P.A.: The Modi®ed Nodal Approach to Network Analysis. IEEE Transactions on Circuits Systems CAS 22 (1975), pp. 504±509. 9. Ma Èrz, R.: Numerical Methods for Differential-Algebraic Equations. Acta Numerica, 1991, pp. 141±198. 10. Matz, K. and Clauû, C.: Simulation Support by Index Computation. 15th IMACS World Congress, Berlin, Aug. 1997 Vol. 1, pp. 203±208. 11. Mu È ller, P.C.: Verallgemeinerte Luenberger-Beobachter fu È r lineare Deskriptorsysteme. ZAMM 79 (1999), S1 pp. 9±12. 12. Petzold, L.: Differential / algebraic equations are not ODEs. SIAM J. Sci. Stat. Comput. 3 (1982), pp. 367±384. 13. Ro È benack, K. and Reinschke, K.: Graph-Theoretically Determined Jordan Block Size Structure of Regular Matrix Pencils. Linear Algebra Appl. 263 (1997), pp. 333±348. 14. Ro È benack, K.: Polstellenordnung und Index bei singula È ren Deskriptorsystemen.4. Workshop Deskriptorsysteme, Liborianum Paderborn, 13.±17.7. 1998, pp. 221±233. 15. Straube, B., Mu È ller, B., Vermeiren, W., Hoffmann, C. and Sattler, S.: Analogue Fault Simulation by aFSIM. DATE'00, User Forum, Paris, March 27±30, 2000, pp. 205±210. 16. Straube, B., Reinschke, K., Vermeiren, W., Ro È benack, K., Mu È ller, B. and Clauû, C.: On the Fault-Injection-Caused Increase of the DAE-index in Analogue Fault Simulation. IEEE European Test Workshop ETW'99, Constance, Germany, May 25±28, 1999, pp. 118±122. 17. Straube, B., Reinschke, K., Vermeiren, W., Ro È benack, K., Mu È ller, B. and Clauû, C.: DAE- Index Increase in Analogue Fault Simulation. Workshop on System Design Automation SDA 2000, Dresden, March 13±14, 2000, pp. 99±104. 18. Tischendorf, C.: Topological Index Calculation of DAEs in Circuit Simulation. Surv. Math. Ind. 8 (3/4) 1999, pp. 187±199. 19. Wolfram, S.: Mathematica ± Ein System fu È r Mathematik auf dem Computer. Addison- Wesley Publishing Company, 1992. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

Sybolical Index Calculation For Linear Circuits

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Taylor & Francis
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1744-5051
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1387-3954
DOI
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Abstract

Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-205$16.00 2001, Vol. 7, No. 2, pp. 205±214 Swets & Zeitlinger Symbolical Index Calculation for Linear Circuits * * * * C. CLAUû , P. SCHWARZ , B. STRAUBE and W. VERMEIREN ABSTRACT The condition for the DAE index to be higher than one is calculated symbolically. Using Analog Insydes a function for the computer algebra system Mathmatica is written. It calculates the index condition if the sparse tableau analysis method, or the modi®ed nodal analysis method is applied. 1 INTRODUCTION One of the central questions in the ®eld of differential-algebraic equations (DAE's) [1, 2, 5] or descriptor systems [11] is the determination of the index. Dif®culties in the numerical treatment [12, 16] of the DAE are expected if the index exceeds one (higher-index case). Therefore, a higher-index check is quite useful. The known index concepts coincide in the case of linear DAE's, in which the index calculation possibilities are well understood [3, 13]. Linear DAE's are generated if the most often used modi®ed nodal analysis (MNA) [8], or sparse tableau analysis (STA)[6] methods are applied to linear circuits [4]. Based on the matrices of the linear DAE's the higher-index condition can be checked. If the analysis approaches are realized symbolically, which is possible using a symbolical circuit analysis tool (e.g., Analog Insydes [7]) the symbol- ical matrices can be the basis for the calculation of a symbolical higher-index Fraunhofer Institute for Integrated Circuits, Design Automation Department (EAS) Dresden, Zeunerstraûe 38, D-01069 Dresden, Germany. Tel.: ‡49-351-4640 737; E-mail: (clauss, schwarz, straube, vermeire)@eas.iis.fhg.de 206 C. CLAUû ET AL. condition. Such a symbolical higher-index condition can be used for the index check by replacing the symbolical parameters by numerical ones. The advantage over the pure numerical calculation is the reduction of round-off errors. Furthermore, the symbolical higher-index condition allows to ®nd out how circuit element parameters have to be chosen to get / avoid a higher index. In this paper a symbolical calculation method for the higher-index condition is presented which uses Analog Insydes, and Mathematica. It is restricted to linear circuits in which only capacitors and inductors lead to derivatives of the variables with respect to the time. The method is applied to some simple examples. 2 THE HIGHER-INDEX CONDITION A linear DAE is considered with constant coef®cient matrices (e.g., a descriptor form, A can be singular) Ax …t†‡ Bx…t†‡ f…t†ˆ 0 …1† m mxm 1 (x…t†; f…t† : T ! R ; A; B 2 R ; T  R time interval, m > 0,' symbolizes the derivative with respect to the time). According to the tractability index de®nition [9] the following matrix chains have to be calculated: A ˆ A; B ˆ B …2:1† 0 0 A ˆ A ‡ B Q ; B ˆ B P …2:2† i‡1 i i i i‡1 i i with P ˆ I ÿ Q ; and Q is a projector on the kernel ker A ; i ˆ 0; 1; .. . The i i i i tractability index is the number  with A regular and A singular for all j < . Consequently, the index is higher than one if and only if A is singular. Therefore, the condition det…A‡ BQ †ˆ 0 …3† has to be evaluated. To get this condition symbolical circuit analysis is used. 3 SYMBOLICAL CIRCUIT ANALYSIS Analog Insydes [7] is a program for the symbolical analysing and sizing of analog electronic circuits. It is based on the computer algebra system Mathe- matica [19] utilizing its symbolical calculation capabilities. For handling SYMBOLICAL INDEX CALCULATION 207 circuit analysis a set of Mathematica functions is de®ned. We use Analog Insydes to derive equation (1) from a circuit description. The further calculation steps use Mathematica functions only. The higher-index condition (3) is derived by the following steps: (a) Describe the circuit in an element oriented way There is an input format which uses the Mathematica list capabilities. A netlist converter for reading SPICE netlists is available. (b) Set up the circuit equations It is carried out by the `CircuitEquations' analysis function. By default the MNA approach is used. If the `SparseTableau' option is set the STA equations are generated. (c) Extract the A and B matrices This is done by elementary Mathematica functions. (d) Construct the projector Q on ker A Considering the capacitors and inductors which are the only circuit components that cause derivatives with respect to the time the projector can be constructed by determining the zero rows of the matrix A. (e) Calculate the determinant, and factorize it The matrix A ˆ A‡ BQ is calculated. Symbolical determinant calcula- 1 0 tion and factorization are functions of the Mathematica system. The symbolical analysis steps are combined to a user de®ned Mathematica function. The steps (a), (b) are done by Analog Insydes functions, and the steps (c), and (e) are simple matrix calculations. The construction of the projector in step (d) is described in the next section. 4 PROJECTOR CONSTRUCTION Since the allowed circuit elements are restricted to only capacitors, and inductors, which produce time derivatives of circuit variables, the matrix A has a special structure which depends on the circuit analysis approach. The speci®c structure of A allows it to construct the projector Q easily. 4.1 Sparse Tableau Analysis (STA) Both all branch voltages and all branch currents become variables of the DAE. Only capacitors and inductors produce nonzero entries in the A matrix. Since 0 0 the equations Cv…t†ˆ i…t† of capacitors, and Li…t†ˆ v…t† of inductors are the 208 C. CLAUû ET AL. only equations which use time derivatives, the nonzero entries in A are all C- values, and all L-values. Moreover, each entry is the only entry both in its line and column. Due to this structure, Q can be easily given by a matrix with a 1 entry on the main diagonal if and only if the corresponding column of A is a zero column. All other elements of Q are zero, e.g.: 2 3 2 3 000 0 0 00 0 1 0 0 000 00 6 7 6 7 000 0 0 00 0 0 0 0 000 00 6 7 6 7 6 7 6 7 0 C 00 0 0 00 0 0 1 000 00 6 7 6 7 6 7 6 7 000 0 0 00 0 0 0 0 100 00 6 7 6 7 A ˆ Q ˆ 6 7 6 7 000 0 0 C 00 0 0 0 000 00 6 7 6 7 6 7 6 7 000 0 0 00 0 0 0 0 000 00 6 7 6 7 4 5 4 5 000 0 0 00 0 0 0 0 000 10 000 0 L 00 0 0 0 0 000 01 4.2 Modi®ed Nodal Analysis (MNA) All node voltages become variables of the DAE, and furthermore, all branch currents which cannot be eliminated, e.g. the currents of inductors or of independent voltage sources. Inductors and single grounded capacitors are treated like in the STA approach. A single ¯oating capacitor which is not grounded leads to the following submatrix of A which is singular. The corresponding submatrix q of the projector Q is added: 0 0 Connecting subgraphs of capacitors with n nodes, which do not have a capacitor to the ground lead to n-dimensional submatrices a of A, which are singular. The corresponding submatrices q of Q are n-dimensional matrices 0 0 with the entries 1/n each, e.g.: SYMBOLICAL INDEX CALCULATION 209 The submatrix is regular, if a connecting subgraph of capacitors is grounded by at least one capacitor. Consequently, the submatrix q of the projector Q is 0 0 zero. By subdividing the Matrix A into submatrices, the projector Q can be calculated using the above mentioned submatrices q . For the following sample matrix A with several submatrices due to capacitor subgraphs the projector Q is given: 2 3 0 0 0 0 0 000 0 0 6 7 0 C ‡ C ÿC 0 0 000 0 0 1 2 1 6 7 6 7 0 ÿC C 0 0 000 0 0 1 1 6 7 6 7 00 0 C ‡ C ÿC 0 ÿC 00 0 3 4 3 4 6 7 6 7 00 0 ÿC C 000 0 0 3 3 6 7 A ˆ 6 7 00 0 0 0 C 0 ÿC 00 5 5 6 7 6 7 00 0 ÿC 00 C 00 0 4 4 6 7 6 7 00 0 0 0 ÿC 0 C 00 5 5 6 7 4 5 0 0 0 0 0 000 C 0 0 0 0 0 0 000 0 L 2 3 10 0 0 0 0 0 0 00 6 7 00 0 0 0 0 0 0 00 6 7 6 7 0 001=31=30 1=30 0 0 6 7 6 7 0 001=31=3 000 0 0 6 7 6 7 00 0 0 0 1=20 1=20 0 6 7 Q ˆ 6 7 0 001=30 01=30 0 0 6 7 6 7 00 0 0 0 1=20 1=20 0 6 7 6 7 00 0 0 0 0 0 0 00 6 7 4 5 00 0 0 0 0 0 0 00 00 0 0 0 0 0 0 00 5 CIRCUIT EXAMPLES The following example gives a small impression of the calculation of the higher-index condition. It is a very simple example for demonstrating only the 210 C. CLAUû ET AL. calculation steps. The Analog Insydes input netlist of this RC circuit is shown on the left. The generated STA matrices A and B are: 2 3 2 3 0 0 000 000 ÿ11 11 00 0 0 6 7 6 7 0 0 000 000 000 0 0 0 0 0 6 7 6 7 6 7 6 7 0 0 000 000 000 0 0 ÿ11 0 6 7 6 7 6 7 6 7 0 0 000 000 000 0 0 0 ÿ11 6 7 6 7 A ˆ ; B ˆ 6 7 6 7 0 0 000 000 100 0 0 0 0 0 6 7 6 7 6 7 6 7 0 0 000 000 0 ÿ10 0 0 R 00 6 7 6 7 4 5 4 5 0 0 000 000 00 ÿ100 0 R 0 00 0 C 0 000 000 0 0 0 0 ÿ1 …4† The projector on the kernel of A is Q ˆ diag…1; 1; 1; 0; 1; 1; 1; 1†: …5† The calculated higher-index condition achieved by Mathematica is C…R ‡ R †ˆ 0: …6† 1 2 If this condition is ful®lled the DAE index is higher than 1. There are two factors in (6), therefore: C ˆ 0: This is not possible, because the construction of the projector presumed C not to be zero. R ˆÿR : It leads to a higher-index DAE indeed. 1 2 In this example a formula was calculated symbolically which allows to determine the circuit element parameters in such a way that the STA equation system is of higher index. Sometimes the determinant of the matrix A is zero independently from speci®c parameter values. Then the index is higher due to the DAE structure [18]. For example this can be observed if a mesh of capacitor branches occurs as in the following example. SYMBOLICAL INDEX CALCULATION 211 Without considering the intermediate steps the symbolically calculated higher-index condition is f0g. That means, the index is higher than 1 in any case of parameters R ; R ; C ; C ; C . Therefore, the symbolical higher-index 1 2 1 2 3 condition gives insight into the structural higher-index property. If the MNA is applied to this circuit, the index is equal to 1. This demon- strates that the index depends on the structure of the circuit, its numerical values, and the applied analysis method. 6 EXAMPLES FOR IDENTIFYING INDEX PROBLEMS IN TEST SIGNAL EVALUATION Analogue fault simulation [15] is applied to evaluate test signals for an analogue network for the purpose of fault detecting. A fault simulator creates a faulty network by fault injection. The electric behavior of a fault is usually modelled by network elements and their interconnections. Commonly, two- poles for shorts and opens or n-poles, e.g., for defective transistors, are injected into the network. Caused by such a fault injection the electric behavior of the now faulty network changes. As a ®rst example the RC-network [16, 17] with an operational ampli®er depicted in the following picture is considered. If the operational ampli®er is modelled ideally by a nullator-norator-pair, the fault free network can be simulated without any problems. In the case of the open fault no simulation was possible. If otherwise the operational ampli®er is modeled using a voltage controlled voltage source (ampli®cation V ˆ 15000) the fault-free network can be simulated too. Its behavior (node 7) is shown in the result plot (upper window). If the open fault is injected at G the simulator has calculated the behavior shown in the lower window of the result plot. Note the different 212 C. CLAUû ET AL. scales of the voltage ranges. An analogue fault simulator would report this fault as to be detected. However, there is an higher-index problem. The faulty network has the index ˆ 6. Thus, the high index is the reason for the above mentioned behavior. The application of the higher-index condition formula method allows to detect a higher-index already after fault injection but without the fault simulation run; to give a warning if in the case of a higher index a fault is reported as to be detected. On the other hand, the conditions for an index increase can be calculated for different description levels of the operational ampli®er. In this example, assumed an ideal operational ampli®er, the condition for a higher index is: C  C  C  C  C  G  R  R  R  R  R  R  R ˆ 0 …7† 1 2 3 4 5 7 0 1 2 3 4 5 6 That means the index increases if G ˆ 0 which is the open fault, or if R ˆ 0…i ˆ 0; .. . 6† which leads to meshes of capacitors. Assumed the operational ampli®er is represented by a voltage controlled voltage source the higher-index condition changes into C  .. . C  R  .. . R …R ‡ R ‡ G  R  R ÿ G  R  R  V†ˆ 0 …8† 1 5 2 6 0 1 7 0 1 7 0 1 The index increases if: G ˆ…R ‡ R †=…R  R …Vÿ 1†† …9† 7 0 1 0 1 SYMBOLICAL INDEX CALCULATION 213 In the case of the open fault the network can be simulated due to the fact that the index is not high. However, the result is doubtful since G is in the critical region according to (9) if G is closer to zero. Consequently, the symbolical higher-index condition gives a useful insight. The second example is a high-pass ®lter. If an ideal operational ampli®er is used the application of the above outlined method calculates the following symbolical higher-index condition: C  C  C  C …R  R ‡ R  R ‡ R  R † R  R ˆ 0 …10† 1 2 3 4 0 1 0 2 1 2 3 4 That means a higher index occurs if e.g. R is a short circuit. This is already known from fault simulation for a short fault injected at R . In such case the simulation runs into wrong results. 7 SUMMARY A method for the symbolical calculation of the higher-index condition of linear circuits is presented. It is quite useful both for the recognition and for the search of higher index DAE's. Critical circuit element parameters can be found which are responsible for an index increase. Due to the symbolical calculation, the method seems to be more robust than pure numerical calculations [10]. The performance of the method is demonstrated. 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