Bond Graph Model Of A Generalised Multiphase Electromagnetic Device With Magnetic Nonidealities
Abstract
The electromagnetic machines like the dc, induction, synchronous motor/generator and the transformer have an energy flow framework that is similar. All these machines deal with electrical energy in the electrical domain that is interfaced with the magnetic domain. Except for the transformer, the other machines also have one more energy interface i.e. with the mechanical domain. In all these machines, the magnetic domain acts as the silent energy manager. The electrical and the mechanical domain energies will have to pass through the magnetic domain and appropriately get routed. In recognition of the commonality of this pattern of energy flow, this thesis proposes a generalised model of a multiphase electromagnetic device wherein the dc machine, induction machine, synchronous machine and the transformers are special cases of the proposed generalised model. This is derived using bond graphs that is based on the underlining principle of Energy Flow rooted in the concept of Conservation of Energy.
A model is a set of mathematical equations representing a physical system. A model is as good as a modeller understanding of the physical system and the underlying approximation he makes while writing down the equations describing the models behaviour to the stimulus. A modelling language tool, which can cut down the approximations made by using the power of identified analogous characteristics across the physical domain, can help make a model more close to real life situation. Bond Graph is such a modelling language which is powerful enough to model the nonlinear, multidisciplinary, hybrid continuousdiscrete phenomena encountered in a real life physical system. Bond graphs as a modelling tool was introduced by Professor H.Paynter at Massachusetts Institute of Technology in 1959. The Bond Graph methodology is based on consideration of energy flows between the ports of the components of an engineering system. Bond Graph methodology enables one to develop a graphical model that is consistent with the first principle of energy conservation without having the need to start with establishing and reformulating equations. The derivation of a mathematical model from the graphical description is automated by software tools. As a consequence, a modeller using this methodology can focus on modelling of the physical system. In the graphical representation of bond graph the vertices of a bond graph denote subsystems, system components or elements, while the edges, called power bonds, represent energy flows between them. The nodes of a bond graph have power ports where energy can enter or exit. Bond graph models are developed in a hierarchical topdown or bottomup approach by using component models or elements from model libraries.
An electromagnetic machine is a black box having an assemblage of windings in iron resulting in a combination of input/output ports on shaft and electrical terminals. Abstraction of an machine model by a modeller matching the vision of the observer above is an ideal goal. Bond graph methodology is an appropriate tool for trying to reach this goal as it is based on object oriented modelling techniques. There have been few attempts to model electric machine in bond graph earlier. A well established DC motor bond graph has been widely used in all bond graph literature. But AC rotating machine being a higher order nonlinear system poses a tougher challenge. Here too, there have been few attempts in modelling AC machines. It is observed that majority of AC machine bond graph models have been built up from their mathematical models. But as the bond graph modelling technique is based on the unifying theory of energy exchange, better insight into the system is achievable if the model is conceptualised from its physical structure. This thesis starts from the basic theory of energy port to conceptualise the generalised model from physical correspondence.
In this thesis a Rotating Electrical Machine is studied as a physical system. The energy ports inside this physical system is identified. When a physical system receives the energy through its energy port in one energy cycle, it processes this energy in one of the three ways. The received energy is converted into useful work or it is dissipated or stored. The storage can further be classified into two ways, either as kinetic energy or as potential energy. For a rotating electric machine the inputoutput port for energy exchange are either in electrical or mechanical domain depending on the class of the machine. The magnetic domain across all class of electromagnetic device acts as the energy manager.
In order to capture the features of the energy jumping across the air gap in a rotating electrical machine, wherein the magnetic fields from spatially distributed windings of the stator and rotor interplay, an Axis Rotator (AR) element a mathematical commutator, is introduced in this thesis as a new bond graph element. In a multiphase device, the energy from the various phases and spatial axes are transferred through the axis rotator element. The Axis Rotator is a critical element which helps distinguish between the various classes of electromagnetic devices. The defining features of the Axis Rotator helps in deriving the various special electromagnetic devices (such as the dc machine, induction machine, synchronous machine and the transformer) from the generalised model. The Axis Rotator exists in the magnetic domain. It naturally inherits the characteristics of the magnetic domain.
The Axis Rotator as a bond graph element is complex. In a specific case of 3φ Induction Motor an alternative bond graph model with all integral elements is developed. By one to one correspondence with the AR bond graph model, the inner component of ’AR’ can be identified. Another advantage of using this model is that saturable and nonsaturable magnetic permeance can be separated out, a useful feature in the nonlinear model discussed next.
One of the most distinguishing features of the magnetic domain is the existence of Magnetic Hysteresis. Magnetic Hysteresis is a well understood and studied subject. But this physical process is wilfully ignored by the modelling community at large. The main reason for this is the difficulty of modelling a nonlinear phenomena. The bond graph modelling naturally allows the inclusion of such nonidealities within its framework. This thesis proposes the generalised model along with the inclusion of magnetic nonlinearities and nonidealities into the model of the system. This inherent strength of bond graph model flows from the fact that the models in bond graph are developed from the first principles of energy conversation and the mathematical equations are derived later from the evolved graph.
The tools that are available for bond graph simulation are not adequate for power electronics systems. The existing tools do not address space vectors and frame transformations. As a consequence it is difficult to simulate the electromagnetic device models developed in this thesis. The need for a bond graph tool to address vectors and frame transformations, a common occurrence in electric machines dynamic model study was acutely felt. This necessitated a support for handling complex data class from the underlying mathematical engine of the software. MATLAB/Simulink is the commonly available mathematical tool which has a support for complex variables. Therefore during the course of this research work a new software tool box was developed which meets the need of electromagnetic machines in particular and other engineering domains in general. For developing the new bond graph simulation software, the language extender approach was chosen, as it combines the capabilities of existing popular mathematical engine with its tested graphical frontend and the flexibility of combining different modelling technique like bond graph, block diagram, equations etc. It also ensures portability as they are compiled by interpreted language compiler of the mathematical engine and are thus independent of the computer operating system. CMEX Sfunction methodology was used to develop the software as it has access to lower level functions and methods of the underlying mathematical engine. This helps in speeding up the software execution time alongwith the flexibility in defining new complex elements like the Nonlinear Axis Rotator.
In conclusion, this thesis makes the following contributions: (i) The Axis rotator concept to handle space vectors and frame transformations, (ii) generalised model of the electromagnetic device, (iii) introduction of the saturation and hysteresis nonlinearity in the magnetic domain, (iv) development of the bond graph toolbox to handle vector and frame transformations.
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