Modelling Of Switched Mode Power Converters : A Bond Graph Approach
Umarikar, Amod Chandrashekhar
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Modelling and simulation are essential ingredients of the analysis and design process in power electronics. It helps a design engineer gain an increased understanding of circuit operation. Accordingly, for a set of specifications given, the designer will choose a particular topology, select component types and values, estimate circuit performance etc. Typically hierarchical modelling, analysis and simulation rather than full detailed simulation of the system provides a crucial insight and understanding. The combination of these insights with hardware prototyping and experiments constitutes a powerful and effective approach to design. Obtaining the mathematical model of the power electronic systems is a major task before any analysis or synthesis or simulation can be performed. There are circuit oriented simulators which uses inbuilt mathematical models for components. Simulation with equation solver needs mathematical models for simulation which are trimmed according to user requirement. There are various methods in the literature to obtain these mathematical models. However, the issues of multi-domain system modelling and causality of the energy variables are not sufficiently addressed. Further, specifically to power converter systems, the issue of switching power models with fixed causality is not addressed. Therefore, our research focuses on obtaining solutions to the above using relatively untouched bond graph method to obtain models for power electronic systems. The power electronic system chosen for the present work is Switched Mode Power Converters (SMPC’s) and in particular PWM DC-DC converters. Bond graph is a labelled and directed graphical representation of physical systems. The basis of bond graph modelling is energy/power flow in a system. As energy or power flow is the underlying principle for bond graph modelling, there is seamless integration across multiple domains. As a consequence, different domains (such as electrical, mechanical, thermal, fluid, magnetic etc.) can be represented in a unified way. The power or the energy flow is represented by a half arrow, which is called the power bond or the energy bond. The causality for each bond is a significant issue that is inherently addressed in bond graph modelling. As every bond involves two power variables, the decision of setting the cause variable and the effect variable is by natural laws. This has a significant bearing in the resulting state equations of the system. Proper assignment of power direction resolves the sign-placing problem when connecting sub-model structures. The causality will dictate whether a specific power variable is a cause or the effect. Using causal bars on either ends of the power bond, graphically indicate the causality for every bond. Once the causality gets assigned, bond graph displays the structure of state space equations explicitly. The first problem we have encountered in modelling power electronic systems with bond graph is power switching. The essential part of any switched power electronic system is a switch. Switching in the power electronic circuits causes change in the structure of the system. This results in change in dynamic equations of the circuit according to position of the switch. We have proposed the switched power junctions (SPJ) to represent switching phenomena in power electronic systems. The switched power junctions are a generalization of the already existing 0-junction and 1-junction concepts of the bond graph element set. The SPJ’s models ideal switching. These elements maintain causality invariance for the whole system for any operational mode of the system. This means that the state vector of the resulting state equation of the system does not change for any operating mode. As SPJs models ideal power switching, the problem of stiff systems and associated numerical stability problems while simulating the system is eliminated. Further, it maintains one to one correspondence with the physical system displaying all the feasible modes of operation at the same time on the same graph. Using these elements, the switched mode power converters (SMPC's) are modelled in bond graph. Bond graph of the converter is the large signal model of the converter. A graphical procedure is proposed that gives the averaged large signal, steady state and small signal ac models. The procedure is suitable for the converters operating in both Continuous Conduction Mode (CCM) and in Discontinuous Conduction Mode (DCM). For modelling in DCM, the concept of virtual switch is used to model the converter using bond graph. Using the proposed method, converters of any complexity can be modelled incorporating all the advantages of bond graph modelling. Magnetic components are essential part of the power electronic systems. Most common parts are the inductor, transformer and coupled inductors which contain both the electric and magnetic domains. Gyrator-Permeance approach is used to model the magnetic components. Gyrator acts as an interface between electric and magnetic domain and capacitor model the permeance of the magnetic circuits. Components like inductor, tapped inductor, transformer, and tapped transformer are modelled. Interleaved converters with coupled inductor, zero ripple phenomena in coupled inductor converters as well as integrated magnetic Cuk converter are also modelled. Modelling of integrated magnetic converters like integrated magnetic forward converter, integrated magnetic boost converter are also explored. To carry out all the simulations of proposed bond graph models, bond graph toolbox is developed using MATLAB/SIMULINK. The MATLAB/SIMULINK is chosen since it is general simulation platform widely available. Therefore all the analysis and simulation can be carried out using facilities available in MATLAB/SIMULINK. Symbolic equation extraction toolbox is also developed which extracts state equations from bond graph model in SIMULINK in symbolic form.
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