## Efficient Simulation of Coupled Circuit-Field Problems: Generalized Falk Method

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### BibTeX

@MISC{Vu-quoc_efficientsimulation,

author = {Loc Vu-quoc and Yuhu Zhai and Khai D. T. Ngo and Senior Member},

title = {Efficient Simulation of Coupled Circuit-Field Problems: Generalized Falk Method},

year = {}

}

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### Abstract

Abstract—In this paper, we present an efficient method to solve the coupled circuit-field problem, by first transforming the partial differential equations (PDEs) governing the field problem into a simple one–dimensional (1-D) equivalent circuit system, which is then combined with the circuit part of the overall coupled problem. This transformation relies on the generalized Falk algorithm, which transforms the coordinates in any complex system of linear first-order ordinary differential equations (ODEs) or second-order undamped ODEs, resulting from the discretization of field PDEs, into guaranteed stable-and-passive 1-D equivalent circuit system. The generalized Falk algorithm, having a faster transformation time compared with the traditional Lanczos-type methods, transforms a general finite-element system represented by possibly a system of full matrices—capacitance and conductance matrices in heat problems, or mass and stiffness matrices in structural dynamics and electromagnetics—into an identity capacitance (mass) matrix and a tridiagonal conductance (stiffness) matrix. We also discuss issues related to the stability and the loss of orthogonality of the proposed algorithm. In circuit simulation, the generalized Falk algorithm does not produce unstable positive poles, and is thus more stable than the widely used Lanczos-type methods. The stability and passivity of the resulting 1-D equivalent circuit network are guaranteed since all transformed matrices remain positive definite. The resulting 1-D equivalent circuit system contains only resistors, capacitors, inductors, and current sources. The generalized Falk algorithm offers an extremely simple and convenient way to incorporate field problems into circuit simulators to efficiently solve coupled circuit-field problems. Numerical examples show a significant reduction of simulation time compared to the solution without using the proposed transformation.

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Citation Context ...rthogonalization procedure to ensure the orthogonality among the generated trial vectors. The generalized Falk (GF) method does not produce any positive poles, while some other methods do (e.g., [7], =-=[8]-=-, etc.). The GF method is faster compared to traditional Lanczos-type methods, and generates a guaranteed stable-and-passive 1-D simple circuit system that is equivalent to the original complex system... |

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30 |
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Citation Context ...tworks and the thermal field problem to examine potential self-heating problems in power semiconductor devices requires an accurate electrothermal simulation of complete power electronic systems [1], =-=[2]-=-. The dense three-dimensional packaging used in compact electronic systems often produces magnetic interactions that interfere with system performance, and thus requires accurate simulation of this co... |

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Citation Context ... model-reduction methods, on the other hand, encounter the problem of having positive poles destabilizing the reduced systems [21], and thus require additional special treatment to avoid this problem =-=[22]-=-. Our method does not require any additional treatment, and is more efficient than the Lanczos method. In addition, transient behavior, which is important in power electronics, can be clearly predicte... |

8 |
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Citation Context ...ly used in fluid flow and transport problems [15]–[17]. Careful design of mass lumping techniques is important to obtain accurate results, compared to those from the consistent mass [13], [14], [18], =-=[19]-=-. The use of pentadiagonal mass matrix has also been suggested to improve the accuracy, compared to row-sum lumped mass [18]; even in this case, the proposed generalized Falk method still remains effe... |

2 |
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Citation Context ... one. Mass lumping techniques have been extensively developed to increase the efficiency of wave propagation analysis in electromagnetics (e.g., [11], [12]) or in solid and structural mechanics [13], =-=[14]-=-. Mass lumping has also been extensively used in fluid flow and transport problems [15]–[17]. Careful design of mass lumping techniques is important to obtain accurate results, compared to those from ... |

2 |
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Citation Context ...y of wave propagation analysis in electromagnetics (e.g., [11], [12]) or in solid and structural mechanics [13], [14]. Mass lumping has also been extensively used in fluid flow and transport problems =-=[15]-=-–[17]. Careful design of mass lumping techniques is important to obtain accurate results, compared to those from the consistent mass [13], [14], [18], [19]. The use of pentadiagonal mass matrix has al... |

2 |
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Citation Context ... simple one-dimensional mass-spring system via a class of trial vectors. Due to its simplicity and generality, it has been used for exact model reduction in structural dynamics for mechanical systems =-=[28]-=-. The idea is to generate a transformation matrix in such a way that the transformed system matrices will be diagonal and tridiagonal. Then, another scaling process will be performed to further transf... |

2 |
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Citation Context ...hrough moment matching is not preserved. The generalized Falk method generates a reduced-order model directly through the WYD process with a full orthogonalization and a selective reorthogonalization =-=[30]-=-. Time-domain transient analysis is performed with the reduced-order model without solving any eigenvalue problems or system transfer functions. III. STABILITY AND REORTHOGONALIZATION A. Lanczos Metho... |

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Citation Context ... is highly suited to high-power and high-voltage applications. A full-bridge transformer-isolated buck converter with four IGBT devices, shown in Fig. 8, is simulated with the circuit simulator SABER =-=[3]-=-. Fig. 9 shows the driving voltages for the full-bridge converter, with the on time s, the dead time-period s when all switches are off, and the switching time s. During the first subinterval , transi... |

1 |
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Citation Context ...o develop circuit networks equivalent to discretized field equations have been proposed [2], [3]. Even for certain field problems alone, such as structural dynamics, it has been demonstrated recently =-=[4]-=- that it is definitely quite advantageous to transform the discrete equation into equivalent circuit networks that are subsequently fed into circuit simulators with fast circuit solvers for startlingl... |

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Citation Context ...and tridiagonal conductance matrix, then the circuit network equivalent to the field problem can be constructed in a much easier fashion. In this paper, we first review the Falk transformation method =-=[5]-=- in structural dynamics, and establish a formal mathematical framework for this method. Unfortunately, the original Falk method displays a detrimental loss of orthogonality among the generated trial v... |

1 |
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Citation Context ...with the row-sum technique being a popularly used one. Mass lumping techniques have been extensively developed to increase the efficiency of wave propagation analysis in electromagnetics (e.g., [11], =-=[12]-=-) or in solid and structural mechanics [13], [14]. Mass lumping has also been extensively used in fluid flow and transport problems [15]–[17]. Careful design of mass lumping techniques is important to... |

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Citation Context ...y used one. Mass lumping techniques have been extensively developed to increase the efficiency of wave propagation analysis in electromagnetics (e.g., [11], [12]) or in solid and structural mechanics =-=[13]-=-, [14]. Mass lumping has also been extensively used in fluid flow and transport problems [15]–[17]. Careful design of mass lumping techniques is important to obtain accurate results, compared to those... |

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Citation Context ...wave propagation analysis in electromagnetics (e.g., [11], [12]) or in solid and structural mechanics [13], [14]. Mass lumping has also been extensively used in fluid flow and transport problems [15]–=-=[17]-=-. Careful design of mass lumping techniques is important to obtain accurate results, compared to those from the consistent mass [13], [14], [18], [19]. The use of pentadiagonal mass matrix has also be... |

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design using simulator coupling
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Citation Context ...rforming the computation independently from each other, and with the results from each simulator passed on to the other simulator to adjust input data before the next computational cycle begins [24], =-=[25]-=-. The circuit part can be simulated with a circuit simulator, while a mechanical sensor or thermal effects can be analyzed by employing the finite element method (FEM) program. Generally, the modeling... |

1 |
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Citation Context ...be modified by replacing the system matrices (capacitance and conductance matrices after finite element discretization) by the corresponding FE discretized system matrices. The reader should refer to =-=[27]-=- for the application to the Maxwell equation in electromagnetics model reduction. In [2], thermal networks equivalent to a discretization of the heat equation by the FEM were presented. The proposed m... |

1 |
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Citation Context ...as close eigenvalues. It is well known that, theoretically, the Lanczos method cannot treat multiple eigenvalues; however, in practice, the multiple eigenvalues are determined by round-off errors. In =-=[35]-=-, a modified version of the WYD method is used in which the method starts with block initial vectors, in a similar format to the subspace iteration method, but carries on iterations one by one. This w... |