Abstract
Coupling of electrical circuits with 2D and 3D computational domains is very important for practical applications. To this aim, the notions of “electrical current” and “voltage” must be defined precisely and linked with local quantities (i.e., fields and potentials) in the computational domain. Apart from the static case, the definition of voltage is more complex than it may appear at a first glance, and it is usually tainted by unspoken and/or not justified assumptions. The purpose of this work is twofold: on one hand, to shed light on the definition and on the physical meaning of voltage in the case of time varying quasi-static fields and, on the other hand, to show how to establish coupling equations between lumped parameters circuit model and 2D/3D computational domains. It is demonstrated that a precise physical significance can be given to the voltage in terms of power balance only (the notion of potential is unnecessary). A couple of original operators which allow to express voltages and currents are introduced. Based on a critical analysis of the research literature, it is shown that existing coupling formulas can all be rewritten as particular cases of these two operators. The developed analysis is independent from any computational method and can be used to devise new coupling formulas.
Highlights
Coupling of external electrical circuits with Finite Element (FE) computational domains is very important for practical applications, and it has been investigated for a very long time
The notion of voltage is analysed and a general physical interpretation is given based on the electrical power balance of the computational domain
The classical definition of voltage (2) is obtained as a particular case of (34), by taking as test magnetic field h0γ, the Biôt–Savart field generated by a unit current which flows along the path γ: ˆ
Summary
Coupling of external electrical circuits with Finite Element (FE) computational domains is very important for practical applications, and it has been investigated for a very long time (see [1] for a historical perspective in 1993). Currents and voltages are global quantities that are applied to the computational domain through “ports”, that is, interfaces between internal regions, or surfaces on the external boundary of the domain. To this aim, the notions of “current” and “voltage” must be precisely defined and linked with the electric field e and current density j in the computational domain. The definition of the electrical current I which flows across a given surface Σ is expressed and it depends uniquely on the current density: ˆ
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