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Concept# Equivalent circuit

Résumé

In electrical engineering, an equivalent circuit refers to a theoretical circuit that retains all of the electrical characteristics of a given circuit. Often, an equivalent circuit is sought that simplifies calculation, and more broadly, that is a simplest form of a more complex circuit in order to aid analysis. In its most common form, an equivalent circuit is made up of linear, passive elements. However, more complex equivalent circuits are used that approximate the nonlinear behavior of the original circuit as well. These more complex circuits often are called macromodels of the original circuit. An example of a macromodel is the Boyle circuit for the 741 operational amplifier.
Examples
Thévenin and Norton equivalents
One of linear circuit theory's most surprising properties relates to the ability to treat any two-terminal circuit no matter how complex as behaving as only a source and an impedance, which have either of two simple equivalent circuit forms:

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For the simulation of electrical power systems (adjustable speed drives, wind farms, complete grids, etc.) the Kirchhoff's model is used. Each of the components of this model (transmission line, circuit breaker, electrical machines, etc.) is represented by an equivalent circuit. These equivalent circuit models are unable to take precisely into account the non-linearities of the electrical machines. These non-linearities (eddy currents, magnetic saturation of the materials, skin effect) are however accurately predicted by the finite element method. The goal of this thesis is to add a finite element model of an electrical machine, the hydro generator, to a grid solver. The nature of the link between the grid solver and the finite element model is first investigated. Then, a finite element program used solely to the simulation of the hydro generator and to its link with a grid solver is designed. The features required for such a program are mandated by the physic of the device modelled: dealing with non-linear materials, eddy currents and taking the movement of the rotor into account. Furthermore, it is possible to use the symmetries of the studied device to reduce both the calculating time and the necessary memory. All these features were validated individually, before being used together in the simulation of a hydro generator.

Modeling the interaction of ionizing radiation, either light or ions, in integrated circuits is essential for the development and optimization of optoelectronic devices and of radiation-tolerant circuits. Whereas for optical sensors photogenerated carriers play an essential role, high energy ionizing particles can be a severe issue for circuits, as they create high density of excess carriers in ICs substrate, causing parasitic signals. In particular, recent advances in CMOS scaling have made circuits more sensitive to errors and dysfunctions caused by radiation-induced currents, even at the ground level. TCAD simulations of excess carriers generated by light or radiation are not dedicated to large scale circuit simulations since only few devices can be simulated at a time and computation times are too long. Conversely, SPICE simulations are faster, but their accuracy is strictly dependent on the correctness of the compact models used to describe the devices, especially when dealing with photocurrents and parasitic radiation-induced currents.
The objective of this thesis is to develop a novel modeling approach for SPICE-compatible simulations of electron-hole pairs generated by light and by high energy particles. The approach proposed in this work is based on the Generalized Lumped Devices, previously developed to simulate parasitic signals in High Voltage MOSFET ICs. Here, the model is extended to include excess carriers generation. The developed approach allows physics-based simulations of semiconductor structures, hit by light or radiation, that can be run in standard circuit simulators without the need for any empirical parameter, only relying on the technological and geometrical parameters of the structure, and without any predefined compact model. The model is based on a coarse mesh of the device to obtain an equivalent network of Generalized Lumped Devices. The latter predicts generation of excess carriers and their propagation, recombination and collection at circuit nodes through the definition of equivalent voltages, proportional to the excess carrier concentrations, and equivalent currents, proportional to the excess carrier gradients. The model is validated with Sentaurus TCAD numerical simulations for different scenarios. Regarding light effects, the proposed strategy is applied to simulate various optoelectronic devices. Complete DC I-V characteristics of a solar cell and transient response of a photodiode are studied. Next, phototransistors are considered. After, a full pixel of a 3T-APS CMOS image sensor is analyzed. The photosensing device, described with Generalized Devices, is co-simulated with the in-pixel circuit, described with compact models. The impact of semiconductor parameters on pixel output and on crosstalk between adjacent pixels is predicted. Finally, radiation-induced soft errors in ICs are examined. Alpha particles at different energies hitting the substrate are simulated. Parasitic currents collected at contacts are studied as a function of particles position and energy. Funneling effect, which is a phenomenon specific to high injection, is also included in the model.
This work shows that the Generalized Lumped Devices approach can be successfully used for SPICE simulations of optoelectronic devices and for prediction of radiation- induced parasitic currents in ICs substrate. This thesis is a first step towards a complete and flexible tool for excess carriers modeling in standard circuit simulators.

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The paper proposes a novel method for the optimal parameter selection of the discrete-time switch model used in circuit solvers that adopt the Fixed Admittance Matrix Nodal Method (FAMNM) approach. As known, FAMNM-based circuit solvers allow to reach efficient computation times since they do not need the inversion of the circuit nodal admittance matrix. However, these solvers need to optimally tune the so-called discrete switch conductance, since this parameter might largely affect the simulations accuracy. Within this context, the method proposed in the paper minimizes the distance between the eigenvalues of the original circuit’s nodal admittance matrix with those associated with the presence of the discrete-time switches. The method is proven to provide values of the discrete-time switch conductance that maximize the simulation accuracy and minimize the losses on this artificial parameter. The performances of the proposed method are finally validated by making reference to two test cases: (i) a circuit composed of RLC elements, (ii) a network model that includes a single-phase transmission line.