Partial Element Equivalent Circuit Research Articles

Adaptive Multipoint Model Order Reduction Scheme for Large-Scale Inductive PEEC Circuits

The model order reduction techniques based on the multipoint projection Krylov methods have become the methods of choice to generate large macromodels of multiport RLC circuits. A well-known difficulty with such methods lies in the need for clever point selection to attain model compactness and accuracy. In this paper, we present an automatic methodology for optimizing sample point selection. This method is general, suitable for circuits obtained by the partial element equivalent circuit method coupled with an adaptive multilevel fast multipole method. Our algorithm has been validated on an industrial example to demonstrate the accuracy and robustness of the approach.

Influence of the Turn-to-Turn Contact Electrical Resistance on the Thermal Stability in Meter-Class No-Insulation REBCO Pancake Coils During a Local Normal-State Transition

In magnetic resonance imaging and nuclear magnetic resonance applications, it is important that the high-temperature superconducting coils provide a high magnetic field in a miniaturized form factor. Our research group considers these requirements to be compatible with no-insulation (NI) coils in which the thickness of the Cu stabilization layer and the turn-to-turn contact electrical resistance are key parameters that enable both high current densities and high thermal stabilities. When the Cu stabilization layer is thin, the current density is impaired, but the thermal stability is improved. If the turn-to-turn contact electrical resistance is very large, then the characteristics of the NI REBCO coil approach the characteristics of conventional insulated coils, and the thermal stability may be degraded. However, when the turn-to-turn contact electrical resistance is large, the excitation delay time is reduced. In this study, we used a combination of the partial element equivalent circuit method and the finite element method to perform a current distribution and thermal analysis of the behavior of NI REBCO coils when a local normal-state transition occurs. We also considered the appropriate turn-to-turn contact electrical resistance that could shorten the excitation delay time as much as possible while still suppressing the formation of hotspots, which is a disadvantage of the NI REBCO coil.

Current Behavior Simulation in Stacked NI REBCO Pancake Coils During Local Normal-State Transition

This paper presents the current behavior and axial Lorenz force of stacked no-insulation (NI) REBa2Cu3O7-x (REBCO, RE = Rare Earth) double-pancake coils during local normal-state transition. The NI winding technique enhances the thermal stability and increases the engineering current density in REBCO pancake coils. The high thermal stability of NI REBCO coil has been verified in experiments of overcurrent test. In addition, the numerical simulations of NI REBCO single-pancake coils have been performed to confirm the high stability. However, REBCO magnets usually consist of some stacked double-pancake coils to generate a high magnetic field for practical applications. When a normal-state transition occurs in one of the stacked pancake coils, it may affect the other pancake coils. For the more reliable verification, the current and electromagnetic force behavior of stacked NI REBCO pancake coils during a normal-state transition should be simulated. To examine the stacked NI REBCO pancake coils, we extend the previously proposed partial element equivalent circuit (PEEC) model to deal with a few double-pancake coils. In this paper, an NI REBCO double-pancake coil and their three stacked coils are simulated by using the extended PEEC model.

Detection Method for a Local Normal-State Transition in a No-Insulation REBCO Pancake Coil

In this paper, we have proposed a detection method for detecting and locating a local normal-state transition in a meter-class no-insulation (NI) coil that is deployed in next-generation high-field magnetic resonance imaging scanners (MRIs), using a copper pickup coil. There are two schemes for detecting and locating the normal zone: one is a ring pickup coil assembled on the midplane of a central bore and the other consists of several sector pickup coils deployed on the coil winding, namely “scanning pickup coil.” To verify the effectiveness of these detection methods, we performed a numerical simulation coupling consisting of a partial element equivalent circuit, a thermal analysis, and a magnetic field analysis. Based on the results of analyses, we reproduced the distributions of the temporal variations of the magnetic flux in the central bore and on the coil winding of the meter-class NI coil, and discussed the effectiveness of these pickup coil schemes for detecting and locating a local normal-state transition in a meter-class NI coil used in next-generation high-field MRIs.

Electrothermal formulation of the partial element equivalent circuit method

AbstractThis paper presents a combined electrothermal formulation of the partial element equivalent circuit (PEEC) method. It is shown that the distributed thermal problem can be formulated as an equivalent thermal network that shares the same topological and geometrical properties of the PEEC equivalent model. Temperature‐dependent thermal heat conduction and convection models are adopted thus leading to a time‐varying thermal network, which is studied by using pertinent techniques along with the electrical equivalent circuit. The proposed electrothermal PEEC solver is used to analyze 2 problems of interest and verified through comparison with a commercial software.

Quasi‐static partial element equivalent circuit models of magneto‐dielectric materials

This study aims to present the formulation of the partial element equivalent circuit (PEEC) for dispersive and lossy linear magneto-dielectric materials under the quasi-static hypothesis. The magnetic permeability and dielectric permittivity are usually available as tabulated data at different frequency samples. They are incorporated in the time-domain PEEC formulation by resorting to a rational approximation of their frequency-dependent constitutive relations. Thus, a set of ordinary differential equations is obtained besides those obtained by enforcing Kirchoff voltage and current laws to the PEEC equivalent circuit. The resulting augmented model preserves the circuit structure of the usual PEEC model and, hence, can be run in Spice-like circuit solvers. The robustness and accuracy of the proposed time-domain approach is tested by comparison with its frequency-domain counterpart despite the larger number of unknowns.

Digital Wave Simulation of Quasi-Static Partial Element Equivalent Circuit Method

The partial element equivalent circuit (PEEC) method is a well-established technique for obtaining a circuit equivalent for an electromagnetic problem. The time domain solution of such models is usually performed using nodal voltages and branch currents, or sometimes charge and currents. The present paper describes a possible alternative approach, which can be obtained expressing and solving the problem in the waves domain. The digital wave theory is used to find an equivalent representation of the PEEC circuit in the wave domain. Through a pertinent continuous to discrete time transformation, the constitutive relations for partial inductances, capacitances, and resistances are translated in an explicit form. The combination of such equations with Kirchhoff laws allows to achieve a semiexplicit resolution scheme. Three different physical configurations are analyzed and their extracted digital wave PEEC models are simulated at growing sizes using the general-purpose digital wave simulator. The results are compared to those obtained by using standard SPICE simulators in both linear and nonlinear cases. When the size of the model is manageable by SPICE, an excellent accuracy and a speed-up factor of up to three orders of magnitude are observed with much lower memory requirements. A comparative analysis of results including the effect of parameters like the simulation time step choice is also presented.

Nonorthogonal 2.5-D PEEC for Power Integrity Analysis of Package-Board Geometries

Design of the power ground layout of a multilayered printed circuit board (PCB) is crucial for low noise and stable power supply. 2.5-D tools are better suited for early stage power distribution network (PDN) analysis over 3-D full-wave electromagnetic solvers due to faster simulation times. For example, the multilayered finite difference method (MFDM), which is based on a 2.5-D formulation on an orthogonal mesh grid, can accurately model and analyze power planes. However, this method loses its advantage while analyzing planes with irregular shapes and holes, which require unnecessarily fine discretization at boundaries for a suitable staircase approximation in an orthogonal grid. In this paper, a nonorthogonal 2.5-D partial element equivalent circuit (PEEC) formulation is proposed, employing quadrilateral mesh elements for efficient simulation of the PDN. The individual stamps for resistance, inductance, capacitance, and conductance elements for a unit quadrilateral cell are derived. Further, the methodology is enhanced to capture coplanar coupling through the introduction of mutual inductance and capacitive terms between neighboring PEEC cell pairs. The numerical results demonstrate good accuracy compared with a 3-D full-wave commercial tool for layered PCB geometries. The efficiency of the proposed method is benchmarked against an orthogonal MFDM implementation and a commercial 2.5-D tool.

Partial Element Equivalent Circuit Models of Three-Dimensional Geometries Including Anisotropic Dielectrics

During recent years anisotropic materials have received an increasing interest and found important applications in the field of shielding and antennas. The anisotropy may be due to intrinsic properties, or as a consequence of mixing. Intentionally or not, the anisotropy impacts the electromagnetic (EM) behavior of a system. Therefore, it is desirable to be able to incorporate the anisotropic effects in an EM model, to allow design tasks and analysis. In this paper, the partial element equivalent circuit (PEEC) formulation is extended to handle nondispersive linear anisotropic dielectrics. The anisotropic dielectric PEEC cell is derived and the resulting PEEC equations are developed into a descriptor system form, which is well suited for implementation in SPICE-like solvers, and for reduction by model-order reduction techniques. A verification of the model is given by a numerical example of a patch antenna situated on an anisotropic substrate and the results are in good agreement with a finite-difference time-domain implementation. The proposed PEEC model is of interest for further work, i.e., in the modeling of setups involving mixtures of materials, with an orientational alignment, and engineered materials, encountered in different EM compatibility applications.

Fast and Accurate Calculation of System-Level ESD Noise Coupling to a Signal Trace by PEEC Model Decomposition

A fast and accurate calculation method of system-level electrostatic discharge (ESD) noise coupling based on model decomposition of the partial element equivalent circuit is proposed. The proposed method can significantly reduce the calculation time without loss of accuracy by separating small victim signal traces from large aggressor structures such as ground planes and ESD gun strap. Using the method, the ESD noise coupling at the terminations of a victim signal trace is rigorously calculated and validated with measurements and full-wave simulations in both frequency and time domains.

A Rigorous Proof on the Radiation Resistance in Generalized PEEC Model

This paper develops a theory on the proof of the radiation resistance in frequency-domain partial element equivalent circuit (PEEC) model. Specifically, the total power dissipated on the imaginary part of the complex inductance and capacitance derived from the PEEC model is found to be the total radiated or scattered power when the structure is internally excited by independent sources, or externally by incident waves, respectively. Accordingly, the imaginary part of the complex inductance and capacitance represents a radiation resistance network of the structure. Numerical simulations based on PEEC and commercial electromagnetic simulation tool are presented for comparison, and some physical implications are discussed.

Development of Simple Physics-Based Circuit Macromodel From PEEC

The evaluation of electromagnetic parasitics is important for electromagnetic interference within power electronics systems. Simple yet accurate physics-based lumped-circuit models of these parasitics can facilitate fast analysis and better understanding of the system performance. Methods exist to reduce complex electromagnetic models to much simpler representations but, while they speed up calculation, they do not preserve physicality. In this paper, a methodology is developed for obtaining a simplified SPICE circuit from a large-scale partial element equivalent circuit (PEEC) model, where there is a clear correlation between geometry and parasitic circuit elements. To preserve correlation between equivalent circuits and physical geometry, reduction is done first for inductance and resistance; then, reduction is performed for capacitance using nodes from the reduced LR model. The reduced model gives a better insight into the causes of and possible solutions to electromagnetic problems and can be used to improve the design. The method was validated on real power electronics geometries using frequency-dependent behavior of port impedances and admittances. The initial PEEC model for this example contained approximately 10 645 unknowns and was reduced to 195 elements. The calculated and measured impedances agree within 0.1 dB up to 200 MHz.

Time-Domain Partial Element Equivalent Circuit Solver Including Non-Linear Magnetic Materials

This paper presents a novel partial element equivalent circuit formulation to handle 3-D non-linear problems. Since the displacement currents are taken into account, the proposed formulation is not restricted to low-frequency applications, although the propagation delays are neglected. The non-linear problem is solved by means of the Newton–Raphson algorithm. Special care is taken in order to analytically compute the Jacobian of the problem. Furthermore, an adaptive setting of the time step is adopted in the transient analysis allowing to maintain the size of the time step as large as possible while preserving the accuracy. Computed results for a closed magnetic circuit and the testing electromagnetic analysis methods workshop problem 10 are reported.

Calculation of Power Losses in Litz Wire Systems by Coupling FEM and PEEC Method

The frequency-dependent resistance of inductive components with high-frequency litz wires is essential for the design of power electronic systems. A novel simulation approach is demonstrated, which combines the specific benefits of two numerical methods: The magnetic field distribution is calculated by standard finite-element method tools for complex 3-D geometries based on a solid conductor. The resulting magnetic field on cut sections of the conductor is used as boundary condition for the partial element equivalent circuit method. Based on the coupling of these methods, power losses on litz wire level have been calculated taking different bundle structures and pitch lengths into account. The work flow has been automatized and enables simulations of litz wire systems with hundreds of strands. For different simulation setups, the simulations have been verified by comparison to measurements up to 500 kHz. The average deviation is less than 5% in the relevant frequency range.

Rigorous DC Solution of Partial Element Equivalent Circuit Models

This paper presents a rigorous dc solution of 3-D Partial Element Equivalent Circuit models. Despite the decoupling of magnetic and electric fields at dc , the correct solution is not trivial. The proposed one is based on a two-step process where first the magnetostatic problem is solved; then, the electrostatic problem is faced by using the solution of the magnetostatic problem as boundary conditions and enforcing the neutrality of disconnected objects. A systematic partition computation is presented which is based on graph algorithms. Applications to the dc analysis of multiple connected and/or disconnected conductors demonstrate the accuracy of the proposed method when compared to the steady state analysis under step excitation.

Direct Mesh-Based Model Order Reduction of PEEC Model for Quasi-Static Circuit Problems

In this paper, a new concept of matrix-inversion-less derived physically expressive circuit method of model order reduction for large-scale high-speed/microwave circuit problems is proposed. It is the first model that is derived based on the mesh information of partial element equivalent circuit modeling. This new proposed method absorbs insignificant nodes and parallel branches while retaining the physical meaning of all the inductive and capacitive elements. This method does not involve any matrix inversions or decompositions and its computational overhead is dominated by outer products that can be simply accelerated by the massive GPU acceleration technique. The resultant order-reduced circuit is very useful for not only speeding up the analysis of a large-scale high-speed/microwave circuit but also interpreting the physical insight of the circuit layout from the circuit domain point of view. Three numerical examples are given, demonstrating the versatility, scalability, accuracy, and simplicity of the method.

A hybrid method for the calculation of the inductances of coils with and without deformed turns

Purpose – The purpose of this paper is to present an accurate and efficient hybrid method for the calculation of the inductance of a coil and its inductance change due to deformed turns using numerical methods. Design/methodology/approach – The paper opted for finite element method coupled with analytical method (FCA) to accurately calculate the inductance of a coil, which is used as reference value. An algorithm with a power function is presented to approximate the partial inductance matrix with the purpose of obtaining the percentage change of the inductance due to deformed turns by using the partial element equivalent circuit (PEEC) with an approximated model and an optimization process. The presented method is successfully validated by the numerical results. Findings – The paper provides a systematic investigation of the inductance of an arbitrary shaped coil and shows how to accurately and efficiently evaluate the inductance change of a coil due to its deformed turns. It suggests that the inductance of a coil can be accurately calculated by using FCA and its percentage change due to deformed turns can be efficiently calculated by using the PEEC_Approximation. Research limitations/implications – As this research is for the magnetostatics, the skin- and proximity-effects have not been taken into account. Practical implications – The paper includes implication for the worst-case analysis of the coil’s inductance due to mechanical damage or manufacturing tolerance. Originality/value – This paper fulfills an identified need to study how the inductance change of a coil can be obtained accurately and efficiently.

Partial Element Equivalent Circuit-Based Transient Analysis of Graphene-Based Interconnects

This paper presents a three-dimensional time-domain formulation of the partial element equivalent circuit (PEEC) method to model graphene-based interconnects. The frequency-dependent electrical conductivity is incorporated into the time-domain solver by resorting to a fitting process. It is proved that if only the intraband contribution to the electrical conductivity is considered, no additional unknowns are required with respect to a standard PEEC solver. If also the interband is to be modeled, auxiliaries unknowns are introduced to reproduce the dynamic of the material. Numerical examples are presented to validate and demonstrate the capabilities and advantages of the proposed approach. The comparison with a finite element method implementation and also with the frequency-domain PEEC solver shows a satisfactory agreement and a good speedup with respect to the frequency-domain counterpart.

Experiment and Simulation of Impregnated No-Insulation REBCO Pancake Coil

It is important to investigate the stability and behavior of an epoxy-resin-impregnated no-insulation (NI) REBCO pancake coil to implement high-field applications, such as ultra-high-field magnetic resonance imaging. We have performed sudden discharging and overcurrent tests for the impregnated NI REBCO pancake coil. From the discharging test, the contact resistivity is estimated, and it changes depending on the initial current. From the overcurrent test, the high thermal stability of the impregnated NI REBCO pancake coil is confirmed. The REBCO pancake coil is charged up to 67 A though the critical current is only 46 A, and no degradation has been found. To investigate in detail the electromagnetic behavior of an impregnated NI REBCO pancake coil, the simulation is performed by means of the partial element equivalent circuit (PEEC) model. In the sudden discharging test, the simulation results for the case of lower initial current are in good agreement with the experimental data. As can be inferred from the simulation results, the current drastically decreases from the inside of the impregnated NI REBCO pancake coil. The result of the overcurrent simulation is almost identical to the experimental one. However, since the contact resistivity is presumed to be constant in the simulation, the difference is observed in the high current region.

Transient Behaviors of No-Insulation REBCO Pancake Coil During Local Normal-State Transition

Recently, no-insulation (NI) winding techniques have been proposed for achieving high current density and high thermal stability. To verify the excellent characteristics achieved using these techniques, we investigated the transient behaviors of NI REBCO pancake coils. In this paper, we evaluate transient electromagnetic and thermal behaviors during a local normal-state transition, based on a partial element equivalent circuit and thermal coupled numerical analysis. Moreover, we investigate temporal and spatial current distributions in the NI coil winding after a local normal transition occurs. We also clarify the reason for the suppression of temperature increase caused by the local normal-state transition in the NI winding. Subsequently, we evaluate the influence of the thickness of a Cu stabilizer on the thermal stability of the NI and conventional insulation windings. Finally, we discuss the mechanism of high thermal stability attained in the NI coil winding technique.