Bond Graph and Wave-Scattering Models

Transcription

1 Bond Graph and Wave-Scattering Models of Switched Power Conversion R.G. Longoria and J.A. Kypuros Department of Mechanical Engineering University of Texas at Austin Austin, Texas , USA H.M. Paynter Massachusetts Institute of Technology P.O. Box 568 Pittsford, VT 05763, USA ABSTRACT Insight is gained by taking a wave-scattering perspective in the bond graph representation of power conversion processes that occur via physical switching mechanisms. Examples are presented to compare and contrast the approach necessary when considering certain hydraulic, mechanical, and power electronic systems. 1. INTRODUCTION Switched power conversion is achieved by a class of systems designed to transfer energy efficiently by taking advantage of changes in power flow connections. Ftealization of this concept can be identified in many areas and disciplines, and significant attention has been given in recent years to electronic systems. In this area alone the focus can vary, requiring decisions to be made regarding how to best represent the switching mechanisms. This requires reconciling accurate componentlevel physical models of switch-based devices with efficient system-level analysis and design methodologies. In this respect, this area offers many challenges opportunities for advancing the state-of-the-art in bond graph modeling and system simulation. This paper demonstrates how bond graph and wavescattering models can be used to study switched power conversion. The boost converter as realized in both the hydraulic and electrical energy domains is used for this purpose. 2. MODELING SWITCHES WITH SCATTERING MATRICES Switching mechanisms can be either physically or functionally represented, depending on the application and end use of the model. There has been considerable discussion in the literature regarding switch models, although it would seem that the two prevailing schools of thought are applicable for their respective problems of interest. It has been well demonstrated, for exam- ple, that one would generally not choose what is regarded as an ideal switch model to account for thermal effects [lo], particularly since the ideal switch is non-dissipative. There also arise questions regarding causality, and those have been addressed in many different ways as well. Many of the challenges faced relate to discontinuities and problems can arise from the conventional use of causality to derive independent states of a system. In this respect, the problems with changes in causality due to switching can be resolved by building a single model as suggested by Karnopp [4, 51, although this seemingly preferred approach has not been universally accepted. A scattering approach as presented by Longoria [9] is reviewed here as a companion or alternative to conventional bond graph approaches, although the bases are fundamentally related [SI. A particular advantage is that the ideal and non-ideal (lossy) switch models can be formulated in a consistent manner, providing a way to evolve toward a more complex or simplified system model. Bond graphs can be generally thought of as powerflow graphs, and the conventional use employs quantifying power flow on a bond via effort-flow variables (e, f). A wave-scattering approach employs instead duplex signals to represent the transfer of energy and flow of power on bonds and through and within system elements [6,9]. On power bonds, wave variables are used (G7,Z) to quantify power, and at the port of a system or device scattering variables are employed (U, v). The utilization of a scattering formulation for system modeling is beginning to attract attention outside of the classical microwave applications, and, for example, has become popular as a basis for computer-aided analysis of photonic systems. Scattering operators relate inwave U and outwave 1) scattering variables (which can be directly related to most classically defined effort-flow variables). A useful 1997 IEEE 1522

2 formulation for system modeling is t,hrough the causal matrix relation, v = S(u)u, where U and v are inwave and outwave vectors for the multiport system. The matrix S is shown here as inwave modulated, and as such is general1.y nonlinear. Most classical uses of scattering matrices have primarily been concerned with linear systems. 3. PHYSICAL NONEXISTENCE OF THE IDEAL SWITCH An ideal switching model has the scattering matrix, with z = 1 for on, and z = 0 for off. The scattering matrix is modulated by 2 to satisfy switch functionality. As long as the dissipation is not an issue, this model may be applicable for those cases where only functionality is important. However, the dual circuits and bond graphs of Figure 1 suffice to show the hazards entailed in using such models in physical systems. ciates (see [7, 101 anld other related works). The topology of choice involves a resistive element appended to a modulated transformer (M I F, or here T). This T-R combination is fundamental, and was earlier examined by Paynter and Busch-Vishniac [6] from the standpoint of modeling nonlineair elements using scattering representations. As such, it is a fundamental model for representing many types of lossy mechanisms. To derive the scattering relation for the T-R combination, use the cascade-load relationship in the form PI, Sc = Cii + Ziz(E, - SiCzz)-lSiCzi, where the Cij represent components of the scattering matrix for a coupling system. For the T-R combination, these components are the scalar modulated scattering coefficients for an ideal transformer and the result is 191, STR = (a +?)/(I + a.r>, where y = (T - l)/(r + 1) is the scattering coefficient of the one-port resistive load, and T is the normalized resistance. This formulation can be used to derive the scattering matrices for the two basic switch function topologies denoted as Type 0 (shunt>) and 1 (series) in Table 1 [9]. The switching models are represented in Table 1 by a wave-scattering bond graph, and the scattering variables at the two ports are related through the scattering relation, For a lossy Type 10 switch, the scattering matrix is, Figure 1: (a) Two common and dual switching circuits. (b) Wave-scattering bond graphs witih a general switching subsystem, S,,,,. If the right-hand circuit is interpreted in a thermal context, we recall that it has long been used in thermodynamics (as the hot-brick problem) to establish that entropy is necessarily generated upon closing the switch unless the two energy states are at the same temperature (= effort). General second law considerations then require that corresponding conditions must hold for all interpretations of these bond graphs, generally requiring a dissipative structure for the switch model. 4. THE LOSSY SWITCH Dissipation can be incorporated by including resistive elements, as done by G. Dauphin-Tanguy and asso- where, and, a (1- a) SI1 = 622 :=- 3 + a + 7. (1 + 37) These scattering matrices depend on the inwavemodulated function, a = a(ul,uz), which takes on a value of &1 to represent ideal switching. For values -1 < a < 1, this scattering formulation can be used to model dissipative or lossy switching.

3 1: Basic Wave-Scattering Switch Models. TvDe 0 I TvDe 1 I R Y 5. CLASSIFICATION SCHEME FOR SWITCH MODELS One of the classical uses of scattering methods was in formulating fundamental theorems in network theory. The scattering formulation is based on conservation of energy and has an inherent causal structure. It turns out to be useful not only in interpreting switching functions and models, as in the previous section, but also in helping to classify switching models. A classification is here proposed, as one does not seem to exist in the literature, as a tool to guide system modelers in determining what type of switch model is applicable for a particular application. For example, in low-power (chopper) controllers for dc motors, it may be sufficient to treat the semiconductor switches using functionallymodulated ideal switches, although this may not be the case for high-power switching of ac drives where the flux of energy being controlled is significant, and thermal effects are critical. A stick-slip switch model, on the other hand, as might be used to model power flow through a clutch, is best described by both modulation (e.g., control of the force between plates via hydraulics) and the level of energy flowing through the system. These problems suggest the following classification: I. Functionally modulated switch (includes two classes: ideal and lossy). 11. Energy-fiux (or inwave) modulated switch Combination of Type I and Type 11. This classification is used below for modeling switched power conversion. 6. SWITCHED POWER CONVERSION In the modern arena of electronics, switched power conversion refers to the process of either increasing or decreasing an electrical state (typically voltage) in a controlled manner. While this form of power conversion may be most prevalent (owing to the preponderance of electrical energy transmission), there is much to be gained by broadening the scope of this concept. In fact, it is found that one form of switched power conversion dates back 200 years to the French balloonist J.M. Montgolfier, who combined two hydraulic valves and an air vessel supplied by a water drive pipe to build what may be the first boost-converter. Some time earlier, B. Franklin may have also experimented with boosting electrical states through experiments with seriescascaded leyden jars, a process that likely describes the genesis of the electronic design. However, the topology of Montgolfier s hydraulic power converter, which is now referred to as a hydraulic ram, offers a striking analogy. Indeed, this self-acting pump has much to offer in a parallel study with its electrical cousin. The boost converter analogy is shown in Fig. 2. Figures 3(a) and (b) illustrate wave-scattering bond graphs with switching elements indicated by SI and SII, to denote switches of Type I and 11, respectively. The scattering matrix used in each case should correspond to the switch models of Type 0 and 1 (see Table 1). Note that both transformers are modulated (MTF or T here), 1524

4 a) = dg)/(l and m = v'm/(l- -a). The classification scheme serves to identify the inherent difference in the operating principle governing each of these systems-indeed the analogy is found to have distinct differences. To begin with, the hydraulic ram model requires both the R and C elements at the inlet side in order to model the pumping action. The switches are also different types, as indicated in the bond graphs of Figure 3. In the electronic boost converter, the transformer modulus is controlled by an external circuit, while the downstream Type I1 switch (the diode) is modulated by the inwave variables. The hydraulic ram is designed to be self-acting-a mode achieved in this ram through the connection of two Type I1 switches. The self-acting nature implies this is an open-loop conversion process> whereas the electronic boost converter takes advantage of feedback. Indeed, this explains how an electronic converter can be much more efficient compared to an equivalent self-acting hydraulic ram. Additional insight can be gained by deriving a single, twwport (modulated) scattering matrix for the switching sub-system in the boost converter. The wave matrices of each component are multiplied to give one wave matrix, which can be used in analysis or converted to a scattering matrix (see [SI). These scattering matrices provide insight into the controlled action required for the electronic boost converter. For the case of the hydraulic ram, the scattering matrix will reveal how the system can be designed to achieve the required switching performance. This scattering matrix captures the critical switching losses, and is a useful tool for efficiency studies [Ill. 7. DISCUSSION AND CONCLUSIONS The use of an ideal switch is attractive in that it is conceptually simple; it certainly fits the Boolean nature of modern day computing. Abandoning the second law, however, is not advisable, and physical system modelers recognize the need to capture the lossy nature of switching. This paper has shown how bond graph models can be used to represent most basic switching topologies using scattering matrices. The benefit of scattering matrices is that they cm model both the ideal and lossy switching behavior in a uniform fashion. A classification scheme has been proposed in order to guide the development and/or selection of switch models for a given application. The use of two-port scattering models for switches permits a system modeler to incorporate the effect of switching mechanisms. To illustrate this, the bond graph and wave-scattering models for two analogous systems have been contrasted in this paper. The electronic boost converter and the U Deliverv Figure 2: A boost converter analogy: (a) hydraulic ram, (b) electronic boost converter. E- I I C E a C Control Figure 3: Bond graphs of the anaalogous systems: (a) hydraulic ram, (b) electronic boost converter. F

5 hydraulic ram are switched power conversion systems that require modeling of interactions between switching mechanisms and auxiliary components. Being able to represent the interaction of switches with other systems over a broad range of parameter changes offers many challenges. One issue of importance arises when studying the hydraulic ram, for instance. In the model presented in this paper, the drive pipe has been modeled with lumped resistive and inertive elements. The operation of the waste valve, however, requires that the pressure at the junction of the two switching valves undergo sequenced variations. This effect will arise from a compressibility effect that has been approximated in the lumped model of Fig. 2(a) by a lumped C. In order to best reflect back impedance, a more accurate representation would require at least a modal or transmission line model of the drive pipe wave dynamics. In this case, the scattering approach may be even more appropriate for representing the interaction of switches and distributed-parameter models. 8. ACKNOWLEDGMENTS We gratefully acknowledge support by the Dynamic Systems and Controls Division at the National Science Foundation through grant number CMS REFERENCES H.M. Paynter, Analysts and Design of Engineering Systems. MIT Press, Cambridge, Massachusetts, R.W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, New York, [7] J.P. Ducreux, G. Dauphin-Tanguy, and C. Rombaut, Bond Graph Modeling of Commutations in Power Electronics Circuits, Znternatnonal Conference on Bondgraph Modelang (ICBGM 93), Proceedings of the 1993 Western Simulation Multiconference, January 17-20, 1993, La Jolla, California, USA, pp [SI H.M. Paynter, An Epistemic Prehistory of Bond Graphs, Bond Graphs for Engzneers, ed. P.C. Breedveld and G. Dauphin-Tanguy, Elsevier Science Publishers B.V. (North-Holland), [9] R.G. Longoria, Wave-Scattering Formalisms for Multiport Energetic Systems, Journal of the Franklin Institute, Vol. 333(B), No. 4, pp , [lo] J. Garcia, G. Dauphin-Tanguy, and and C. Rombaut, A Bond Graph Approach for Modeling Switching Losses of Power Semiconductor Devices, International Conference on Bondgraph Modeling (ICBGM 93), Proceedings of the 1997 Western Simulation Multiconference, January 12-15, 1997, Tuscon, Arizona, USA, pp [ll] H.M. Paynter and R.G. Longoria, Two-Port Canonical Bond Graph Models of Lossy Power Transduction, IEEE/SMC 97, Paper No. 634 (this conference). Woodson, H.H. and W.F. Weldon, Energy Considerations in Switching Current from an Inductive Store into a Railgun, Proc. Fourth IEEE Pulsed Power Conference, Albuquerque, New Mexico, pp. 2225, Karnopp, D., Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems, Journal of Dynamic Systems Measurement and Control (ASME), Vol. 107, pp Karnopp, D., General Method for Including Rapidly Switched Devices in Dynamic System Simulation Models, Duns. of the Society of Computer Simulation, Vo. 2, No. 1, pp Paynter, H.M. and I. Busch-Vishniac. 1988, Wave-scattering Approaches to Conservation and Causality. ) Journal of the Franklin Institute 325( 3) ~