Quasi-stationary AC Analysis by Coupling Transient and Phasor Description Olaf Enge-Rosenblatt, Christoph Clauß, Peter Schneider, Peter Schwarz Fraunhofer Institut Integrierte Schaltungen, Institutsteil Entwurfsautomatisierung, Zeunerstraße 38, 01069 Dresden olaf.enge@eas.iis.fraunhofer.de Abstract Investigation of heterogeneous systems (“multi-nature systems” or “multi-domain systems”) by simulating long time periods requires effective methods. If only si- nusoidal quantities with fixed frequencies occur in one of the involved domains then phasor analysis can be used to describe steady-state behaviour of this part of the heterogeneous system. But an analysis of only steady states is seldom of interest. Rather, it would be advantageous to investigate different steady states in form of a sequence and their interaction with the other parts of the complete system. In these cases, a coupling of phasor analysis and transient analysis is desirable. In the present paper, modelling of electrical AC sys- tems using the well-known phasor method is shortly repeated. The main focus of the paper is the coupling of such phasor domain-based models with transient mod- els of other domains (e.g. mechanical). Some simulation results of electric machines considered as electromechanical systems (i.e. typical instances of a multi-domain system) are given. 1 Introduction Heterogeneous systems are characterized by a coupling of at least two different physical domains. Electrome- chanical systems are widely used in industry and households, but also electro-hydraulic or hydro-me- chanical components appear very often. Investigation of such systems by means of dynamic simulations can advantageously be carried out using a modern multi- domain language like Modelica (see e.g. [3], [13]). Oc- casionally, the time constants of dynamic processes within distinct domains are completely different. In these cases, special modelling methods become impor- tant to make dynamic simulations effective. Linear electrical circuits using alternating current are often called AC circuits. Mostly, such circuits operate with one fixed (nominal) frequency and nearly ideal si- nusoidal electrical quantities. A dynamic investigation of such systems can be performed using behavioural models (differential-algebraic equations). But studying steady-state analysis of such systems for a long time period (hours, days, years), this method is inefficient. In these cases, the phasor method – i.e. modelling with- in phasor domain – can be utilised. The system’s behaviour is only described by algebraic equations. Modelling of AC circuits using complex numbers (or phasors) was introduced by Steinmetz in the late 19 th century ([10], [11]). Nowadays it is well-known and can be found in many elementary textbooks ([1], [4], [5], [6], [7], [8], [9]). When combining a phasor domain-based model with a transient model, the steady-state analysis is only of lit- tle interest. The behaviour of the complete system can only be examined when the AC subsystem is in a so- called quasi-stationary mode. Quasi-stationary analysis of AC systems shall be understood as an investigation of a sequence of steady states on the following condi- tion: parameters (which would be constant at steady- state analysis) may vary extraordinary slowly com- pared to both the nominal frequency and the dominating time constant of the AC system. In this paper, Steinmetz‘ method for modelling of elec- trical AC systems – the well-known phasor description – is repeated. After some fundamentals, well-known laws in DC analysis (Ohm‘s Law and Kirchhoff‘s Laws) are formulated using phasors. Hereafter, this de- scription is applied to determine the quasi-stationary behaviour of an AC system. An appropriate method to couple such a phasor-based submodel with a transient submodel is shown. Finally, simulation results are giv- en for two electromechanical systems as typical samples of a multi-domain or multi-nature system. 2 AC systems in phasor domain 2.1 Fundamentals A sinusoidal signal can be described for instance in the following form: , (1) where stands for the amplitude, denotes the angu- lar frequency, and is the phase. Such a signal can be xt () x ˆ ωt ϕ + ( ) sin = x ˆ ω ϕ