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The use of a magnetic model to analyze the load dependent behaviour of a power supply Lind, Magnus G. J. 2006

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T H E U S E OF A M A G N E T I C M O D E L TO A N A L Y Z E T H E L O A D D E P E N D E N T B E H A V I O U R OF A P O W E R S U P P L Y by M A G N U S G. J. LESTD BSEET, Western Washington University, Bellingham, W A , USA, 2002 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES (Electrical and Computer Engineering) T H E UNIVERSITY OF BRITISH C O L U M B I A February 2006 © Magnus G. J. Lind, 2006 Abstract It has been known since the first half of the 1990s that power supplies based on the ferroresonant transformer technology are incompatible with certain more recent types of loads such as actively power factor corrected devices. This is an issue that still remains today. The objective of this thesis is to analyze whether a specific ferroresonant uninterruptible power supply, operating in line mode, can be controlled by a linear time-invariant controller. This work shows that whether the ferroresonant-based circuit can be compatible with the aforementioned type of load is dependent on the characteristics of the same. The study includes a comprehensive nonlinear model of the controlled ferroresonant transformer also known as the controlled constant voltage transformer. Saturation is a normal mode of operation for this device. The level of detail includes winding resistances, core resistances, continuously nonlinear magnetizing inductances, tapped windings, and leakage inductances. A complete nonlinear state-space model of the power supply where the constant voltage transformer constitutes the major part is derived. The investigation encompasses stability evaluation of the device with several types of loads using a generalized approach. For the analysis, the nonlinear system is converted into a linearized periodically time-varying system, which in turn is sampled by means of Poincare mapping to provide a discrete-time representation of the model. An analytical approach explains, in the discrete-time domain, why certain loads may be impossible to control by a linear time-invariant controller. The study emphasizes this type of controller since in general, with unknown loads, the assumption is that the controller should be relatively simple and less dependent on specific load parameters or specific type of loads. To complement the theory, this work also proposes an approximate procedure that leads to an implementation of a practical feedback controller that stabilizes a specific actively power factor corrected load. Laboratory experiments show that, for line mode operation, the power supply can indeed be compatible with the aforementioned load. i i Table of Contents Abstract....: 1 Table of Contents i i i List of Tables vii List of Figures viii List of Symbols xiii List of Abbreviations xviii Acknowledgement xx Chapter 1 Introduction 1 1.1 Literature Overview 1 1.1.1 Historical Development of the Ferroresonant Transformer 2 1.1.2 The Ferroresonant UPS in Recent Times 4 1.1.3 The Ferroresonant UPS and Instability 5 1.1.4 The Future of the Ferroresonant UPS 8 1.1.5 Thesis Motivation and Objective 8 1.2 Thesis Contribution 9 1.3 Thesis Overview 10 Chapter 2 Problem Statement 12 2.1 The Topology and Operation of a Specific UPS 12 2.2 Characterization of the Existing Control Loop 14 2.3 Measurements of Circuit Variables 15 2.3.1 Variables of the Tank Circuit 16 2.3.2 Instability in the Output Voltage 18 ii i Table of Contents iv 2.3.3 S ystem B ehaviour in Open-Loop Control 21 2.4 Stabilizing Attempts 23 2.5 Summary 24 Chapter 3 Modeling of the Ferroresonant Transformer 25 3.1 The Traditional Two-Winding Transformer 25 3.2 The Ferroresonant Transformer 28 3.2.1 The Equivalent Circuit of the Ferroresonant Transformer 29 3.2.2 Determination of Circuit Parameters 39 3.2.2.1 Parameters Related to the Magnetizing Flux 40 3.2.2.2 Parameters Related to the Leakage Paths 45 3.2.2.3 Parameters Related to the Physical Structure 47 3.3 Measurement Techniques ; 49 3.4 Experimental Results.... 51 3.5 Summary 60 Chapter 4 State-Space Representation of the Model 62 4.1 The Model Including a Generic Load 62 4.1.1 Circuit Equations of the Comprehensive Model 64 4.1.2 Circuit Equations of the Reduced Model 69 4.2 Equivalent Switched Inductance 73 4.3 Some Notes on Simulation Models 79 4.4 Experimental Verification 81 4.5 Summary 85 Chapter 5 Power Factor Corrected Loads 87 5.1 Typical Actively Power Factor Corrected Loads 87 5.2 Measurements of the Input Impedance 91 Chapter 6 Stability Analysis of the System 94 6.1 The Poincare Maps 95 6.2 The Differential Form of the Model 96 Table of Contents v 6.3 Circuit Equations of the Model 97 6.4 Equations for the Stability Analysis 98 6.4.1 Switching Regions 99 6.4.2 Linearization of the System 102 6.4.3 Typical Loads 108 6.5 Results of the Stability Analysis 110 6.5.1 Resistive, Inductive, and Capacitive Loads 110 6.5.2 PFCLoad 114 6.6 An Example of Controller Design 119 6.6.1 Load Identification 120 6.6.2 Sample Considerations 121 6.6.3 Transformation to the Continuous-Time System 123 6.6.4 Control Loop 126 6.6.5 Experimental Results 130 6.6.6 Comments on the Experimental Results 134 6.7 Summary 135 Chapter 7 Conclusion and Future Work 136 7.1 Conclusion 136 7.2 Future Work 138 Bibliography 140 Appendix A Theory and Derivations of Equations 149 A. 1 The State-Space Equation 149 A.2 Poincare Maps 150 A.3 Floquet Theory 156 A.4 Derivation of Equation (6.6) 159 A. 5 Derivation of Equations (6.16) and (6.17) 162 Appendix B Hardware Data 164 B. 1 Hardware Data 164 Table of Contents vi B.2 Data of Laboratory Equipment 164 List of Tables Table 3.1 Results from the polynomial fit of the hysteresis loops 57 Table 3.2 Comparison of the winding resistances 57 Table 3.3 Comparison of the leakage inductances 58 Table 6.1 Continuous-time poles and zeros 124 vi i List of Figures Fig. 1.1 A schematic of the C V T regulator 3 Fig. 1.2 A rudimentary equivalent circuit of the C V T 3 Fig. 1.3 A general ferroresonant UPS arrangement 5 Fig. 1.4 Approximated possible circuit of the ferroresonant UPS and PFC PS 7 Fig. 2.1 The essential parts of the transformer circuit 13 Fig. 2.2 A rudimentary equivalent circuit of the power circuit of the ferroresonant UPS 13 Fig. 2.3 Overview of the schematic of the transformer part of the ferroresonant UPS 14 Fig. 2.4 The control loop of the ferroresonant UPS 15 Fig. 2.5 Plots of measured input and output variables. The UPS was loaded at 56.0 % of the rated load with the PFC PSs 17 Fig. 2.6 Plots of measured tank circuit variables. The UPS was loaded at 56.0 % of the rated load with the PFC PSs 17 Fig. 2.7 Plots of the UPS output voltage. The UPS was loaded at 56.0 % of the rated load with the PFC PSs 18 Fig. 2.8 Plots of the UPS output voltage. The UPS was loaded at 80.1 % of the rated load with the PFC PSs 19 Fig. 2.9 Poincare map of the UPS output variables. The UPS was loaded at 56.0 % of the rated load with the PFC PSs 20 Fig. 2.10 Poincare map of the UPS output variables. The UPS was loaded at 80.1 % of the rated load with the PFC PSs 21 Fig. 2.11 Measured normalized output variables of the UPS. The UPS operated in open-loop control, and the firing angle was adjusted to vary the output voltage 22 Fig. 2.12 Measured normalized output variables of the UPS. The UPS operated in closed-loop control, and the purely resistive load was adjusted to vary the output current... 23 Fig. 3.1 An elementary two-winding transformer: (a) the physical structure, (b) the Steinmetz equivalent circuit, and (c) a more detailed equivalent circuit 27 viii List of Figures ix Fig. 3.2 A traditional model of a multi-winding transformer 28 Fig. 3.3 A general and simplified circuit of the uncontrolled C V T 29 Fig. 3.4 The physical structure of the ferroresonant transformer 31 Fig. 3.5 The equivalent magnetic circuit of the C V T 33 Fig. 3.6 The dual equivalent magnetic circuit of the C V T 34 Fig. 3.7 The reduced dual equivalent magnetic circuit of the C V T 34 Fig. 3.8 The reduced dual equivalent magnetic circuit of the ferroresonant transformer referenced to the turns ratio N2 a D 35 Fig. 3.9 The reduced dual equivalent magnetic circuit of the C V T referred to the (N2ab) side and to the time derivative of the flux sources and the current variables 35 Fig. 3.10 The electrical equivalent circuit of the C V T 36 Fig. 3.11 The electrical equivalent circuit of the ferroresonant transformer including the ideal transformers for the bifilar windings for the inverter, charger, and reference... 37 Fig. 3.12 The reduced electrical equivalent circuit of the C V T 38 Fig. 3.13 The electrical equivalent circuit of the ferroresonant transformer based on the simplified magnetic model 39 Fig. 3.14 Flow chart for the routine that produced the single-valued magnetization curve 44 Fig. 3.15 Configuration for the measurements of the hysteresis loops 50 Fig. 3.16 The voltage across the series combination of the auxiliary windings 52 Fig. 3.17 The normalized hysteresis loop of the primary magnetizing inductance referred to the N2ab side. The applied voltage was 115 % of the rated 54 Fig. 3.18 The normalized hysteresis loop of the secondary magnetizing inductance. The applied voltage was 115 % of the rated 54 Fig. 3.19 The zoomed-in normalized hysteresis loops of the primary magnetizing inductance for three different levels of the applied primary and secondary voltages 55 Fig. 3.20 The zoomed-in normalized hysteresis loops of the secondary magnetizing inductance for three different levels of the applied primary and secondary voltages 56 Fig. 3.21 Complete equivalent circuit of the part of the ferroresonant transformer that served the purpose of this work including component values 58 List of Figures x Fig. 3.22 Step response - the load resistance was step-changed between 49.9 % and 111 %... 59 Fig. 3.23 Step response - the load resistance was step-changed between 49.9 % and 111 %... 60 Fig. 4.1 The equivalent circuit of the controlled C V T including a generic load 64 Fig. 4.2 The equivalent reduced circuit of the controlled C V T including a generic load 70 Fig. 4.3 Quantitative analysis of the equivalent reduced circuit of the C V T including a resistive load (a) phasor diagram and (b) the characteristic of X L m 2 72 Fig. 4.4 Voltage and current waveforms across and through L 73 Fig. 4.5 Plots of the normalized equivalent inductance and the polynomial fit 78 Fig. 4.6 Plots of the normalized harmonic currents through the equivalent inductances 78 Fig. 4.7 Plots of the normalized rms currents through the equivalence inductance of the SCR-switched inductor 79 Fig. 4.8 An example of implementation of a nonlinear inductance that was used for circuit-based software analysis 80 Fig. 4.9 Simulink® model of two anti-parallel SCRs 80 Fig. 4.10 Plots of input and output voltage variables. The device operated in open-loop control with a resistive load of 101 % 82 Fig. 4.11 Plots of input and output current variables. The device operated in open-loop control with a resistive load of 101 % 83 Fig. 4.12 Plots of output voltage and current variables when the resistive load was step changed from 101 % to 0 %. With constant rated source voltage, without harmonic content, the device operated in open-loop control 84 Fig. 4.13 Plots of output voltage and current variables for the SCR-switched L and the continuous L e q . With constant rated source voltage, including harmonic content, the device operated in open-loop control with a resistive load of 101 % 85 Fig. 5.1 A rudimentary constant power load circuit 88 Fig. 5.2 Frequency response for the second-order transfer function of the input admittance.. 90 Fig. 5.3 Input voltage and current to the PFC PS when sourced from the power grid. The load was 80.9 % of that of the rated 91 Fig. 5.4 Magnitude and phase plot of the input admittance of the PFC PS 92 Fig. 6.1 Orbit of a three-dimensional state vector 96 Fig. 6.2 Waveforms of the capacitor voltage and switched inductor current 99 List of Figures xi Fig. 6.3 Trajectory of a general state variable 100 Fig. 6.4 Simulation diagram of the linearized UPS and load combination 106 Fig. 6.5 Pole plot for the model with R load I l l Fig. 6.6 Zero plot for the model with R load I l l Fig. 6.7 Pole plot for the model with R L load 112 Fig. 6.8 Zero plot for the model with R L load 112 Fig. 6.9 Pole plot for the model with RC load 113 Fig. 6.10 Zero plot for the model with R C load 113 Fig. 6.11 Experimental results for v2ab(t) at 80.1 % PFC PS load 115 Fig. 6.12 Pole plot for the UPS with PFC loads Set I 116 Fig. 6.13 Zero plot for the UPS with PFC loads Set 1 116 Fig. 6.14 Zoomed-in pole and zero plot for the UPS with PFC loads Set 1 117 Fig. 6.15 Phase plot of the transfer function of the plant 119 Fig. 6.16 Pole plot of the families of the five poles for the system with PFC load Set F 120 Fig. 6.17 Zero plot of the families of the four zeros for the system with PFC load Set F 121 Fig. 6.18 Transformation between the s and z-planes 122 Fig. 6.19 Family of Bode plots of the plant, the power supply and PFC load Set F, for Rn 0rm = {0.1, 0.2, 1} with the matched pole-zero method 125 Fig. 6.20 Control loop 128 Fig. 6.21 Bode plot of the controller response 129 Fig. 6.22 Bode plot of the loop response of the system 129 Fig. 6.23 Experimental results of the response to a step change in Rn 0rm from 62.5 to 100 % with PFC load 130 Fig. 6.24 Experimental results of the response to a step change in R n 0rm from 62.5 to 100 % with PFC load 131 Fig. 6.25 Experimental results of the response to a step change in Rn 0rm from 100 to 62.5 % with PFC load 132 ) Fig. 6.26 Experimental results of the response to a step change in Rnorm from 0 to 100 % with R load 132 Fig. 6.27 Experimental results of the response to a step change in Rn o rm from 100 to 0 % with R load 133 List of Figures xii Fig. 6.28 Experimental results of the response to a step change in Rn o rm from 0 to 100 % with PFC load 133 Fig. 6.29 Experimental results of the response to a step change in Rn0rm from 100 to 0 % with PFC load 134 Fig. A . l Orbit of a three-dimensional state vector 153 Fig. A.2 Periodic orbit of a three-dimensional state vector 154 List of Symbols Symbols used only one time or rarely are not tabulated below. A symbol consisting of a listed character with an added argument, subscript, superscript, or prime is not tabulated. A symbol may denote more than one property. For this case, all representations are explained below and the context determines the meaning. Generally, lower-case letters represent instantaneous quantities whereas upper-case letters indicate steady-state quantities. Lower-case boldface letters symbolize vector quantities whereas upper-case boldface letters represent matrices. S c a l a r s A cross sectional area (m2) a length (m) B Bernoulli number B magnetic flux density (T) b length (m) c complex exponential form of the Fourier Series coefficient C capacitance (F) G transfer function H gain H magnetic field intensity (A-turns/m) H scaling factor I steady-state current (A) i instantaneous current (A) i sqrt(-l) j sqrt(-l) 1 length (m) xiii List of Symbols xiv k positive integer k general constant L inductance (H) M modulation function m modulation function m number of samples N turns ratio n positive integer n slope of a Bode plot trace (20 dB/decade) P power (W) P transfer function of a plant p pole R resistance (Q) K reluctance (A-turns/Wb) S hypersurface s s steady state t continuous time (s) T period of a waveform (s) T transfer function u forcing function V steady-state voltage (V) v instantaneous voltage (V) v load dependent variable X reactance (Q.) V admittance (S) V reciprocal of a parallel combination consisting of a resistance, representing the core loss, and a magnetizing inductance of a particular magnetic path (S) Z impedance (O.) z zero List of Symbols xv Scalars - Greek Letters a firing angle for the SCR (rad) P turn-off angle for the SCR (rad) P coefficients for regions of the magnetizing inductance (1/H) A small signal deviation y reciprocal of the inductance for the primary magnetizing branch (1/H) 8 correction factor for air gaps c damping factor reciprocal of the inductance for the secondary magnetizing branch (1/H) 0 phase angle (rad) I eigenvalue I magnetic flux linkage (Wb-turns) P- permeability (Wb/(A-turns-m)) Po 47il0"7; permeability of free space (Wb/(A-turns-m)) a conduction angle for the SCR (rad) I state space X dummy variable of integration Y subset of state space <P magnetic flux (Wb) 9 phase angle (rad) CO angular frequency (rad/s) Phasors I current (A) V voltage (V) X reactance (Q) Vectors b input vector d feed forward vector f load dependent vector List of Symbols xvi f general purpose vector g general purpose vector h load dependent vector u forcing function vector w vector for nonlinear elements X state vector y observation vector y output vector Vectors - Greek Letters P parameter vector E residual vector Matrices A system matrix B input matrix C output matrix D feedforward matrix E coefficient matrix for nonlinear elements G coefficient matrix for nonlinear elements I identity matrix J Jacobian matrix P change of basis matrix U fundamental solution matrix X design matrix Matrices - Greek Letters <D state-transition matrix Mathematical Symbols arg(-) phase angle for complex numbers in polar form conj(-) complex conjugate of a complex number List of Symbols xvii D first order time derivative det(-) determinant Im(-) the imaginary part of a complex number mod modulus function Re() the real part of a complex number sqrt(-) square root taylor(-) Taylor Series expansion AHB the intersection of set A and set B A c B every element in set A is contained in set B; A is a subset of B a e A a is an element of the set A a => b a implies b; equivalently "not a" implies "not b" V for all f: A —> B f maps the domain A into the codomain B K field of real numbers Rn n-dimensional vector space Z field of integers ~ approximately equal to = identically equal to := defined as | | absolute value for scalars | • | magnitude for phasors ||||2 Euclidean norm * dot product d partial derivative T transpose * transpose and complex conjugate * approximately referred to a particular winding of the transformer * denotation of a more involved expression ' referred to a particular winding of the transformer 0 degrees List of Abbreviations Acronyms ac alternating current C C M continuous conduction mode CPL constant power load CVT constant voltage transformer dc direct current D C M discontinuous conduction mode EMI electromagnetic interference ESR equivalent series resistance FFT Discrete Fourier Transform LHP left-half plane LTI linear time invariant LTP linear time periodic L T V linear time variant mp minimum phase nmp nonminimum phase norm normalized PFC power factor corrected PUD proportional-plus-integral-plus-derivative PS power supply PSIM® Powersim PSpice Personal Computer Simulation Program with Integrated Circuit Emphasis RHP right-half plane rms root mean square xviii List of Abbreviations xix SCR silicon controlled rectifier SEPIC single-ended primary inductance converter SMPS switched mode power supply UPS uninterruptible power supply ZOH zero order hold Acknowledgement I wish to express my deepest gratitude to my research supervisor Prof. Guy A. Dumont for the assistance and guidance throughout this process. The effort of my supervisor Prof. William G. Dunford and committee member Prof. Juri V . Jatskevich is also thankfully acknowledged. Numerous interactions with my colleges Kenneth Wicks and Weidong Xiao have as well inspired me throughout my graduate studies. In particular, I would also like to express my thanks to Dr. Khosro Kabiri who introduced me to the Poincare mapping technique. I wish also to show appreciation to Donovan C. Davidson for letting me use part of his laboratory facilities. Finally, I am expressing my sincerest gratitude to my parents and my wife for their financial support during my studies. Vancouver, BC, Canada, December, 2005 xx Magnus Lind Chapter 1 Introduction The uninterruptible power supply (UPS) has become a necessity in our society. Whenever the ordinary ac source fails, the UPS can temporarily deliver power to its load, examples of which are computer servers and cable television networks. One category of the UPS is the technology based on the ferroresonant transformer. The ferroresonant UPS featured excellent characteristics as a source of power until the mid 1990s, when governmental and regulatory bodies demanded power supplies to be power factor corrected (PFC) [1], [2]. At that time, it became apparent to a number of users that the voltage regulation of the ferroresonant UPS featured instability for many actively PFC loads. This problem persists today. This thesis analyzes this matter based on a commercially produced ferroresonant UPS. 1.1 Literature Overview The ferroresonant UPS, in its present state, was developed during the 1970s. During this period and in the 1980s, this was the predominant UPS topology. As modern power electronics have been introduced to form new UPS topologies, the usage of the ferroresonant power supply has declined. However, due to some of its advantageous characteristics, it is today often advertised as a heavy duty UPS for severe and critical conditions of operation. For example, it suppresses line transients extremely well and is therefore capable of protecting upstream and downstream equipment [3]-[5]. 1 Chapter 1 Introduction 2 1.1.1 Historical Development of the Ferroresonant Transformer Another name for the main element of the ferroresonant UPS, the ferroresonant transformer, is the constant voltage transformer (CVT). The C V T first appeared in the 1930s, and in fact, some of the first magnetic voltage stabilizers materialized in 1910. In the late 1950s, the importance of regulated voltage became evident when the utilization of electronically based control systems and early computers increased. At that time, various voltage regulators could handle a slowly changing source voltage, but they lacked the ability to regulate transients down to a few cycles of duration. Variations of the C V T concept solved this problem. Around 1970, the dc power supply (PS) industry brought further attention to the CVT. The implementation of silicon-controlled rectifiers (SCRs) and an inductor in the output branch of the C V T emulated saturation of the secondary core in order to produce an output voltage that had the shape of a quasi square-wave. The addition of a diode bridge with a capacitor connected across its output terminals completed the regulated dc power supply [6]-[10]. The core of the primary winding of a typical C V T normally operates in the linear region, and as Fig. 1.1 indicates, due to the air gap not all of the primary flux links the secondary winding. The magnetic shuntdiverts some of the magnetizing flux through the air gap. Likewise, not all of the secondary flux links the primary winding. In a traditional transformer with only one core flux path, the primary current will vary to counterbalance the influence of the secondary flux. For the unloaded arrangement in Fig. 1.1, the capacitor current introduces a secondary flux that is in phase with the magnetizing flux. Not all of the former flux links the primary winding, due to the magnetic shunt, and thus, the primary current will not fully compensate for the effect of the secondary flux. Consequently, there will be more flux in the magnetic path to the right of the shunt in the figure than to the left. With proper design, the secondary core must have saturation capabilities. Thus, operating the core of the primary winding below the knee of the B - H curve and the core of the secondary winding above the knee of the same curve facilitates regulation of the output voltage. In general, the simplified explanation above holds, but a practical application also includes one or more of the following: compensation windings, resonant filters, additional transformers, further magnetic shunts, inductors, and semiconductor switches [6]-[8], [11]-[17]. Chapter 1 Introduction 3 i(t) p r imary v(t) p r imary Q i(t) secondary 2 i c (0 v(t) secondary Fig . 1.1 A schematic of the C V T regulator. The schematic in F i g . 1.2 presents a basic equivalent circuit of the C V T in Fig . 1.1 with the same capacitor C i connected across the output terminals. A voltage divider consisting of a linear inductor Lo in series with the nonlinear inductor L i approximates the C V T . v 0 ( t ) Fig. 1.2 A rudimentary equivalent circuit of the C V T . A general instability predicament that can arise at light loads with the C V T is the ferroresonance jump phenomenon. This can occur when X L O > X c i , but not when X L O < X c i for the circuit parameters in F ig . 1.2. The following explains the phenomenon. Assuming that the circuit is slowly energized, and that the input voltage Vi(t) ramps up towards its rated value, then the output voltage v„(t) should similarly ramp up. The ferroresonance jump phenomenon occurs if v0(t) suddenly jumps up, at some point in its ramp-up period, close to its rated value, during the ramp period of Vj(t). Likewise, suppose that the circuit is slowly de-energized. When v;(t) ramps down and reaches a sufficiently low value, v0(t) is supposed to begin to ramp down smoothly as well. The ferroresonance jump phenomenon transpires i f v0(t) remains at approximately its rated value, and then suddenly jumps down to a value from which it begins its ramp down period; the aforementioned materializes in the middle of the ramp down period of Chapter 1 Introduction 4 v;(t). The above occurs in a fashion that features hysteresis. Thus, if Vj(t) were to change moderately, vQ(t) would exhibit relatively large and sudden changes of its voltage level [6], [9], [18]. 1.1.2 The Ferroresonant UPS in Recent Times The controlled ferroresonant UPS is a fairly complicated device consisting of a nonlinear transformer with multiple windings, the nonlinear tank circuit, and typically an unknown load, which may or may not be actively PFC. Experience has shown that this type of UPS is very rugged due to its simple design. It has several other good qualities as well. For instance, irregularities of the ac mains that may occur during lightning do not easily damage the UPS or its load. The C V T of the ferroresonant UPS inherently also features limitation of the output current. An example of this limitation is 150 % of the rated secondary current [19]. The high input power factor characteristic of this UPS has also contributed to its extensive use. This type of UPS is useful in switching applications since it provides harmonic filtering between the power source and the load. Moreover, the C V T typically has the capability of storing about 20 ms worth of energy at 50/60 Hz unless the ac mains is shorted, which consequently quickly drains this energy. The energy storage capability facilitates a smooth transition between the ac mains and battery operation. The C V T also provides galvanic isolation [11], [19]-[23]. Prior to the first half of the 1990s, the fundamental design of the switched mode power supply (SMPS) had remained untouched for two to three decades. Practically all computers at that time used this design, which featured low power factor. When the number of computers increased, as did power supplies for other applications as well, regulatory laws forced manufacturers to implement PFC power supplies in the mid 1990s. Redesigned SMPS for powers below 300 V A often involved passive harmonic filters, while larger ones incorporated actively PFC circuits. The latter used high frequency switching schemes to shape the waveform of the input current in order to lessen the harmonic content [2], [24], [25]. It unfolded that several actively PFC circuits exhibited instability problems with power protection equipment and especially when connected to ferroresonant UPSs. This issue still remains, and to make matters worse, in recent times, the passive filter technique to accomplish power factor correction has become less popular due to its bulk and cost [2], [24]-[26]. The Chapter 1 Introduction 5 possibility of unstable behaviour has led to difficulties in marketing the ferroresonant UPS. For example, Cisco Systems recommends not using ferroresonant based UPSs, regardless of brand, to power their Integrated Communications System 7750, due to instability concerns. 1.1.3 The Ferroresonant UPS and Instability A ferroresonant UPS may supply several PFC power supplies, each of which might power one or more loads. The arrangement in Fig. 1.3 shows a fundamental connection. During instability conditions, when a ferroresonant UPS powers an actively PFC power supply, the output voltage and current of the former often becomes oscillatory. This occurs in a fashion such that these waveforms not only feature the frequency of the line voltage but also unwanted and significant harmonics including subharmonics. The oscillations may also become unbounded and as such incapacitate the system. For example, a UPS with a rated 120 V at 60 Hz secondary may, instead of a single frequency sine wave, produce an output in a quasi amplitude-modulated fashion. This output could have an envelope that varies between 90 V m , . , and 150 V r , ™ before over-voltage protection circuitry disables the UPS. If a subsequent enable is automatic, as frequently is the case, the pattern repeats indefinitely, or until the stresses on individual components, of the ferroresonant UPS or the PFC power supply, result in incapacitated equipment [24], [25]. Unstable voltage regulation might also depend on the output configuration of the UPS. Reference [25] reported that for one of the tested ferroresonant UPSs, instability problems occurred for 230 V but not for 120 V output configuration. This suggests that the tapped secondary transformer winding, or perhaps rather the impedance transformation, may play a significant role. AC Source ac Ferroresonant ac PFC dc Load (Mains or Generator) ^ UPS Power Supply Fig. 1.3 A general ferroresonant UPS arrangement. Chapter 1 Introduction 6 Instability due to the actively PFC power supply occurs in general when the load in Fig. 1.3 is moderate or large. For example, if the system operates under stable conditions with a moderate to large load, the addition of a small power consumer to the load such as a disc drive may trigger instability, bounded or not, whereas a further increase of the load could cause the system to oscillate without bounds [25]. Other types of unstable behaviours of the output regulation of the ferroresonant UPS can arise at light loads or at abrupt variations of the load [20], [27]; however, these do not specifically originate due to the PFC load. The general concept of instability regarding power supplies is not new. In the 1970s, [28], [29] presented some of the first comprehensive analyses on the subject [1], [24]. The investigated instability issue arose because of the implementation of filters, often a passive L C filter, to the front end of the dc-to-dc converter to reduce the ripple of the input current. These references also discuss the concept of negative resistance. The analysis of the characteristics of the tank circuit of the ferroresonant UPS and the actively PFC power supply at a system level is not trivial, but the elementary concept of [28], [29] give some insight to this task. The input average power to the PFC power supply is nearly constant if its losses are small and if its load is constant. Thus, to maintain approximate unity power factor, the power supply decreases the input current when the input voltage increases and vice versa. For a typical actively PFC power supply design, this can cause the real part of its input envelope impedance to be incrementally negative. The incremental input resistance can be negative only at frequencies below the frequency of the ac source, usually 50/60 Hz. For a stiff source with low output impedance such as the ac mains, the negative input resistance of the PFC power supply is not influential enough to affect the ac source. However, if the ac source features noteworthy dynamic output envelope impedance below its output frequency, as the ferroresonant UPS does for example, the output voltage regulation of the same may become unstable. To make matters worse, this category of UPS often has a relatively high output impedance at some frequency, for example, as large as 50 Q,. This envelope impedance, in general, peaks at a frequency that is below the rated output frequency of the UPS. This further promotes instability [1], [2], [24], [25], [28]. Another source that may exhibit instability dilemmas when it feeds active PFC loads is the generator due to its generally highly inductive output impedance [26]. Several other types of non-ferroresonant based UPSs have also demonstrated instability in combination with actively C h a p t e r 1 I n t r o d u c t i o n 7 P F C p o w e r s u p p l i e s . S o m e e x a m p l e s o f s u c h s o u r c e s are t h e s t e p p e d s i n e w a v e U P S a n d t h e h i g h f r e q u e n c y s w i t c h i n g s i n e w a v e U P S [24]. T h e i n t r o d u c t i o n o f a d m i t t a n c e s i n s t e a d o f i m p e d a n c e s , as t h e s c h e m a t i c i n F i g . 1.4 e x e m p l i f i e s , l e a d s t o a n e l e m e n t a r y p o s s i b l e c i r c u i t f o r t h e p a r t i c u l a r c a s e w h e r e Ri s y m b o l i z e s the e f f e c t i v e d a m p i n g r e s i s t a n c e o f t h e t a n k c i r c u i t , a n d t h e l o a d R2 r e p r e s e n t s t h e a c t i v e l y P F C p o w e r s u p p l y . T r a n s f o r m e r ! Lo 1 F r o m ac Source C l = R i : R 2 : i Q 2 Ferroresonant U P S L o a d ( P F C P o w e r Supply) F i g . 1.4 A p p r o x i m a t e d p o s s i b l e c i r c u i t o f the f e r r o r e s o n a n t U P S a n d P F C P S . F o r negative v a l u e s o f R2 a n d i f t h e m a g n i t u d e o f R2 i s s m a l l e r t h a n t h e r e s i s t a n c e o f Ri, (1.1) s h o w s that t h e t a n k c i r c u i t w i l l f e a t u r e n e g a t i v e d a m p i n g . T h i s p r o m o t e s o s c i l l a t i o n s . A n u n e c o n o m i c a l a n d t h e r e f o r e u n r e a l i z a b l e s o l u t i o n t o t h e i n s t a b i l i t y p r o b l e m w o u l d b e t o i n t r o d u c e d a m p i n g i n t h e f o r m o f a r e s i s t a n c e i n t h e o u t p u t c i r c u i t r y o f t h e f e r r o r e s o n a n t U P S . A n o v e r s i z e d f e r r o r e s o n a n t U P S w o u l d a l s o i m p r o v e the i n s t a b i l i t y m a t t e r , b u t t h i s i s , as w e l l , a n i m p r a c t i c a l s o l u t i o n [1], [24], [25]. (R 1>0)n(R 2<0)n(|R 2|<R 1 ) ^ ^ i^-<0 (1.1) Chapter 1 Introduction 8 1.1.4 The Future of the Ferroresonant UPS In addition to the instability problem, the ferroresonant UPS is also inherently noisy, bulky, and heavy due to its transformer and inductor; however, this type of UPS also produces a very high quality power output, matched only by the highest performance high frequency switching based UPSs. The ferroresonant UPS is rugged, inexpensive, and the load is electrically decoupled from the source. Therefore, the realization of a stable output regulation, and above all, the verification that the implementation would function for all typical cases of operations, would reintroduce the ferroresonant UPS as an attractive source of power for certain applications. 1.1.5 Thesis Motivation and Objective The ferroresonant based power supply is an established topology with several advantageous characteristics as described in the previous sections. The introduction of actively PFC devices revealed a weakness of the aforementioned technology. Since the ferroresonant based power supply is commonly used in numerous applications, it would be exceptionally desirable to make it compatible with actively PFC loads as well as with other load types. The question also arises whether the specific ferroresonant topology can be controlled by a LTI controller or not, and if not, under which circumstances is the topology not controllable from a practical and robust standpoint. Thus, a model that can predict stability or instability through computer simulations is highly desirable to develop. Apart from the papers being a product of this work, to the best of the knowledge of the author, publications describing such models are not available. Moreover, if stability can be predicted, a natural development is a practical implementation of a controller than can stabilize a ferroresonant power supply sourcing an actively PFC load. The objective of this thesis is to investigate why and under what circumstances actively PFC loads may be incompatible with ferroresonant power supplies. To fulfill this objective, the study creates a mathematical simulation model of the physical device including a generic load. Chapter 1 Introduction 9 Since instability has been reported to occur at heavy loads, we limit the model to be accurate only for this case. The model is in turn analyzed from a stability prediction perspective. It is shown whether an LTI controller can control the aforementioned combination or not. Higher performance controllers are left for future work and loosely discussed in Chapter 7. In case a specific system is controllable, a practical LTI controller is implemented and proven adequate through laboratory experiments. To limit the scope of the analysis, but without loss of generality, only the line mode operation of a particular UPS is investigated. 1.2 Thesis Contribution The contribution of this work comprises of the development of an analysis describing source and load interaction between a ferroresonant UPS, operating in line mode, and typical loads including actively PFC ones. The strength of the proposed technique is its usefulness in that it probably applies to a majority of the ferroresonant class of UPSs sourcing general loads. The details of the contribution unfold as follows. > The development of a comprehensive equivalent circuit of the ferroresonant transformer where the circuit elements, linear as well as nonlinear, relate closely to the physical structure of the device. > The realization of practical and accurate measurement techniques that is required for the extraction of the parameters for the equivalent circuit. > The derivation of a state-space representation of the equivalent circuit, which includes a universal load. This provides for an exhaustive stability analysis using a general approach that can accommodate generic loads in state-space format. > Analytical substantiation that explains why the ferroresonant UPS may be incompatible with certain loads and specifically with those that are actively power factor corrected. Chapter 1 Introduction 10 1.3 Thesis Overview The current chapter first describes a literature review and an introduction to the ferroresonant topology for power supplies. This is followed by the goals and contribution of the thesis. The remaining chapters are organized as follows. Chapter 2 introduces the problem statement and the topology and operation of a specific ferroresonant power supply, but deemed general, that is currently manufactured and marketed. An experimental investigation describes the waveforms of typical circuit variables when the ferroresonant source is connected to an actively PFC load. It also includes the instability phenomena that occur under certain load conditions. A stabilizing attempt is also considered. Chapter 3 presents a detailed modeling of a constant voltage transformer, which constitutes the major part of a typical ferroresonant power supply. This model includes winding resistances, core resistances, continuously nonlinear magnetizing inductances, tapped windings, and leakage inductances. Methods to extract the circuit parameters from the physical device are developed as well and experimental results verify the validity of the derived circuit. Chapter 4 implements a comprehensive as well as a reduced nonlinear state-space model of the ferroresonant power supply. The model includes a generic load to facilitate analyses of the system with different loads in state-space representation. This chapter also discusses the equivalent switched inductance and a few notes on simulation models. Chapter 5 gives a short introduction to modeling of the input impedance of a device or in this case an actively PFC power supply. Some measurement techniques are discussed and the envelope input admittance of a specific actively PFC power supply is established. Chapter 6 presents a stability analysis of the source and load combination described in the previous chapters. For the analysis, the periodic nonlinear system operating in a steady-state orbit is converted into a linearized periodically time-varying system. This system is in turn sampled by means of Poincare mapping to provide a linear discrete-time representation of the model. The perturbations from the periodic orbit form the basis of the stability prediction. For typical loads, an analytical explanation is proposed, in the discrete-time domain, why certain Chapter 1 Introduction 11 loads may be impossible to control by an LTI controller. The study emphasizes this type of controller since typically, with unknown loads, the postulation is that the controller should be comparatively simple and less dependent on detailed load parameters. Two cases of PFC loads are investigated; one case, which is impossible to control by an LTI controller and a second case that indeed can be controlled by such a controller. An approximate method is proposed that can produce a stabilizing feedback controller. For a specific case, the effectiveness of this controller is demonstrated on the real physical system. Finally, Chapter 7 summarizes the key results from this study and future research directions are contemplated as well. Chapter 2 Problem Statement 2.1 The Topology and Operation of a Specific UPS The study used a particular ferroresonant U P S to exemplify the analysis and to produce insight to the instability problem. The single-phase 1670 W / 2500 V A U P S featured tapped input and output transformer windings that covered several voltages from 120 V to 240 V at 50/60 Hz. Comprehensive data is available in Appendix B . l . The schematic in F ig . 2.1 shows the transformer circuit of the particular device. The transformer has analogous characteristics as previously explained for the C V T in Fig . 1.1. This particular U P S provides galvanic isolation not only between the source and the load, but also among both of the aforementioned, the battery, and the feedback signal to the controller. This may be required in certain telecommunication applications. One advantage of this type of ferroresonant U P S is its simple design as the schematic in F ig . 2.1 indicates. To improve the voltage regulation, the inductor in the secondary branch emulates saturation of the secondary core as needed [6], [10], [19]. Previous works have presented various equivalent circuits of the ferroresonant power source. The schematic in F ig . 2.2 displays one of the most simple ones [6], [9], [10], [18], [20], [30]-[33]. The inductor Lo represents the transformer and the primary and secondary voltages are labelled Vj(t) and v 0(t) respectively. The tank circuit connected to the secondary of the transformer consists of the capacitor C\, the inductor L j , and two anti-parallel SCRs . For further details, [34] illustrates some more elaborate equivalent circuits, and [35] investigates in detail transformers in which air gaps and saturation are present. 12 Chapter 2 Problem Statement 13 AC Source 120/208/240 V 60 Hz 230 V 50 Hz To Controller Feedback From Inverter (Battery) To Battery Charger Outputs 230/240 V 208 V 120 V Fig. 2.1 The essential parts of the transformer circuit. Fig. 2.2 A rudimentary equivalent circuit of the power circuit of the ferroresonant UPS. Chapter 2 Problem Statement 14 2.2 Characterization of the Existing Control Loop The control circuitry of the particular ferroresonant UPS is based on discrete components and small ICs such as transistors and operational amplifiers. The generality pertaining to the particular design, at least regarding the power circuitry, is probably common for comparable products. Although the schematic in Fig. 2.3 appears simple, the actual implementation involves a fair number of discrete components not disclosed here. The block diagram in the figure also omits several features, for example, over voltage protection and soft start. The ac source and the load can be connected in various fashions to accommodate commonly used voltage levels. The transformer portion of the schematic is the equivalent of the physical layout presented in Fig. 2.1. Fig. 2.3 Overview of the schematic of the transformer part of the ferroresonant UPS. The control circuit functions as follows. When the value of the ramp of the saw tooth generator exceeds that of the controller, the comparator outputs a latched rectangular pulse to the Chapter 2 Problem Statement 15 firing circuitry of the SCRs. Referenced to a sine wave output voltage at the relevant transformer tap, which feeds the inductor L i , the theoretical range of the firing angle is 90° < a < 180°. Thus, the inductor always operates in discontinuous conduction mode. This is also a necessity for meaningful operation, as the circuit arrangement of the SCRs in the figure indicates. The capacitor d connects to the transformer tap, which outputs the highest voltage of the secondary winding. The reason for this is to facilitate the largest possible energy storage capability, which relates quadratically to the voltage but not so to the cost of a capacitor with a higher voltage rating. A block diagram of the control loop is shown in Fig. 2.4 including the controller, saw tooth generator, and feedback gain H . The label alpha represents the firing angle of the SCRs. The output voltage of the tank circuit is a function of several variables, as Fig. 2.4 indicates. One main disadvantage of the ferroresonant UPS is the transformer and tank circuit combination with its characteristically high output impedance. The incremental negative input resistance of an actively PFC power supply amplifies this weakness. AC Source Transformer Controller ^ Sawtooth alpha Lj i out (not available in existing design) Tank Circuit H PFC Power Supply M Load Fig. 2.4 The control loop of the ferroresonant UPS. 2.3 Measurements of Circuit Variables Extensive experimental procedures facilitated the study of the existing design of the UPS and resulted in further insight into the instability problem. Unless otherwise specified, all the measurements performed in this chapter employed the system configuration in Fig. 1.3. The Chapter 2 Problem Statement 16 source was the 120 V and 60 Hz ac mains, and the UPS output was configured for 208 V and 60 Hz. The UPS had the nameplate output power rating 1670 W / 2500 V A . Since all experiments featured loads with approximately unity power factor, rated conditions refer to a UPS output of 1670 W at 208 V. To confine the investigation to the case with the ac mains supplying the power for the output of the UPS, the measurements presented in this document do not involve inverter mode operation. However, the study found that the output of the ferroresonant UPS featured similar instability characteristics when it operated in inverter mode, compared to the ac mains mode of operation. The load of the UPS consisted of two PFC power supplies that each had two 56.2 V d c outputs and individually capable of delivering 650 W. Comprehensive data is available in Appendix B . l . Four separately adjustable resistor banks loaded the PFC PSs. The equipment used for all the experimental results in this thesis is listed in Appendix B.2. 2.3.1 Variables of the Tank Circuit The plots in Fig. 2.5 and Fig. 2.6 display some tank circuit variables of interest. The trace of the voltage of the ac mains vi i n e is present in all three figures as a reference. Each cycle of the variables consists of 167 data points. The UPS output variables refer to the relevant transformer taps of the secondary winding in Fig. 2.3 whereas the line variables refer to the ac source in the same figure. The definitions of positive current directions for the nonzero voltage taps of the transformer are as follows. The input current iane enters the primary winding, and the output current iu p s,output leaves the secondary winding. The plus and circle markers in Fig. 2.3 are regarded as positive polarity for the voltages across the capacitor vci and the inductor v u respectively. The currents through the aforementioned elements enter the respective element at the marker for positive direction. The UPS output voltage featured stable behaviour up to a load of 56.0 % of the rated load. The plots in Fig. 2.5 and Fig. 2.6 represent the case when the UPS operated in steady state at 56.0 % PFC load. Although it may be difficult to discern in this particular graph, the plot of V H n e in Fig. 2.5 illustrates how distorted the sine wave of the ac mains can be due to semiconductor-based loads. The ferroresonant UPS partially offsets this distortion as the plot of v u p S j 0 U t p U t shows. Chapter 2 Problem Statement 17 0.0042 0.0083 0.0125 0.0167 0.0208 Time (s) 0.025 0.0292 0.0333 Fig. 2.5 Plots of measured input and output variables. The U P S was loaded at 56.0 % of the rated load with the P F C PSs. Time (s) Fig. 2.6 Plots of measured tank circuit variables. The U P S was loaded at 56.0 % of the rated load with the P F C PSs. Chapter 2 Problem Statement 18 2.3.2 Instability in the Output Voltage The plots of the time and frequency domain of the output voltage of the ferroresonant UPS in Fig. 2.7 and Fig. 2.8 show the behaviour of this variable for two different loads. Each cycle of the output voltage consists of 333 data points. The magnitude spectrum is normalized to the rated output voltage, sqrt(2)*208 V . The time window of the sampled output voltage of the magnitude spectrum was such that frequencies down to 1 Hz could be detected in the magnitude plot. The magnitude of the fundamental was normalized to unity. The particular choice of the vertical scale of the magnitude plot facilitates readings of the frequency components that are of small magnitude. For the plots in both figures, the UPS was loaded with the two PFC PSs. 3001 1 1 1 1 1 1 1 1 1 Il I I I I I. I . 1 1.1 0 50 100 150 200 250 300 Frequency (Hz) Fig. 2.7 Plots of the UPS output voltage. The UPS was loaded at 56.0 % of the rated load with the PFC PSs. Chapter 2 Problem Statement 19 0.1 0.08 0.06 0.04 0.02 0.04 0.06 0.08 0.1 0.12 Time (s) 0.14 0.16 0.18 0.2 50 100 150 200 Frequency (Hz) 250 300 Fig. 2.8 Plots of the UPS output voltage. The UPS was loaded at 80.1 % of the rated load with the PFC PSs. As mentioned before, the output of the UPS featured stable behaviour up to a load of 56.0 % of the rated one. The plot in Fig. 2.7 shows this case. The oscillations of the system became increasingly larger with growing load from this level. The time-domain trace in Fig. 2.8 shows the instance when the load was 80.1 %. At slightly larger loads, the oscillations began to increase without bounds. Decreasing the load within a short period of time, less than a second, from the aforementioned condition, transitioned the system back to steady-state oscillations. During stable operation, the only significant harmonic was the third, as the magnitude plot in Fig. 2.7 shows. This harmonic is due to the effect of the switched inductor of the tank circuit. The relative magnitude of the third harmonic did seemingly not change when the system operated in the unstable region at 80.1 % load; however, as the frequency spectrum in Fig. 2.8 illustrates several other harmonics including sub-harmonics emerged. For intermediate loads in the range from 56.0 Chapter 2 Problem Statement 20 % to 80.1 % of the rated load, the relative harmonic spectrum was similar to that in Fig. 2.8; however, each individual harmonic became larger in relative magnitude as the load increased. Measurements for the same operating conditions that produced the plots in Fig. 2.7 and Fig. 2.8 were also undertaken and plotted on Poincare maps. The output variables of the UPS were sampled at 60 Hz at the same fixed instant in time relative to the grid voltage. The plots in Fig. 2.9 and Fig. 2.10 show these results for 56.0 % and 80.1 % PFC PS loads respectively. The output variables of the UPS are shown as iups.output versus vups.output representing Ch 2 and Ch 1 respectively. The particular scaling is 0.5 A/division for iups.output and 20 V/division for vups,output-For steady state and stable output variables, the oscilloscope readings should consist of overlapping sample points. This is approximately the case for the samples in Fig. 2.9. The oscilloscope readings in Fig. 2.10 show the operation with 80.1 % load. Since the sample points do not coincide here, we conclude that this system does not operate as desired at a single frequency sinusoid. Appendix A.2 addresses the fundamentals of Poincare maps at length. TekaCUB 500MS /S 312 Acqs I { T ] 0 [flffl 200mv Ch2 50mV M 10ms Line J D 5ns 0 V 9 Aug 2003 10:47:56 Fig. 2.9 Poincare map of the UPS output variables. The UPS was loaded at 56.0 % of the rated load with the PFC PSs. Chapter 2 Problem Statement 21 TekSERB 500MS/S 1024 Acqs HUH 200mv Ch2 somv rvi 10ms Line J - o V D 5ns 9 Aug 2003 10:39:53 Fig. 2.10 Poincare map of the UPS output variables. The UPS was loaded at 80.1 % of the rated load with the PFC PSs. 2.3.3 System Behaviour in Open-Loop Control The system was also evaluated when it operated in open-loop control. The system featured stable behaviour over the full resistive load range, from 0 % to 100 % of the rated UPS load, when the ac mains as well as the battery via the inverter supplied power to the output of the UPS. Measured data taken at operation with the aforementioned open-loop arrangement, with the ac mains being the sole source supplying the UPS, and with the UPS supplying a purely resistive load instead of feeding the two PFC PSs, produced the plots in Fig. 2.11. The value of the fixed resistive load represented 50 % of the rated load at rated output voltage. Chapter 2 Problem Statement 22 A P.ave ups_out Linear (P.ave ups_out) V.rms ups_out I • Linear (V.rms ups_out) I l,rms ups_out — - Linear (l.rms ups_out) 101.0 103.0 105.0 107.0 109.0 111.0 113.0 115.0 117.0 119.0 121.0 123.( Rring Angle (deg) Fig. 2.11 Measured normalized output variables of the UPS. The UPS operated in open-loop control, and the firing angle was adjusted to vary the output voltage. The plots indicate that the dc input signal to the comparator allowed the firing angle to vary, such that the output voltage ranged from about 75 % to 110 % of its rated value. The output current changed accordingly. The data points of the respective plotted normalized variables, the UPS output voltage V r m S i U p s _ 0 U t , current Irms,uPs_out, and power PaVe,ups_out featured close linear relationships with the firing angle of the SCRs as each linear regression line in the figure indicates. The respective equations and R 2 values of the regression lines are also present in the figure. A second experiment, this time with closed-loop control as indicated in Fig. 2.3, with the ac mains supplying the UPS and the UPS powering a purely resistive load, produced the plots in Fig. 2.12. The variation of the value of the load resistor produced different firing angles, output currents, and powers. As indicated in the figure, the voltage remained practically constant, independent of the size of the load. The respective data points of the plotted normalized variables featured an approximately linear relationship with the firing angle of the SCRs. As can be determined from the R values for the respective current in Fig. 2.11 and Fig. 2.12, the data points in the latter figure deviated more from the linear regression line than was the case for the Chapter 2 Problem Statement 23 test displayed in Fig. 2.11. The respective data points and regression lines of the measured power in Fig. 2.11 and Fig. 2.12 also illustrate this difference. It is also interesting to observe that the large load range of approximately 85 % produced the rather small firing interval from about 116° to 122°. | a> (0 A P.ave ups_out Linear (P,ave ups_out) 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 1 V.rms ups_out • - Linear (V.rms ups_out) I I, rms ups_out — - Linear (I,rms ups_out) y =-0.0011x + 1.1581 R 2 = 0.9318 A y = 0.135x - 15.593 R 2 = 0.9605 A;, - •' y = 0.1326x- 15.307 A - ' ' " R 2 = 0.9615 * — ~~ -m- I I I i i i 2.0 1.8 1.6 1.4 1.2 (A/A) 1.0 c CD 0.8 o 0.6 0.4 0.2 0.0 16.0 116.5 117.0 117.5 118.0 118.5 119.0 119.5 120.0 120.5 121.0 121.5 122.0 122.1 Firing Angle (deg) Fig. 2.12 Measured normalized output variables of the UPS. The UPS operated in closed-loop control, and the purely resistive load was adjusted to vary the output current. 2.4 Stabilizing Attempts Considerable effort was carried out in an experimentally oriented fashion to stabilize the system. The sentences below describe only the successful such attempt. When the UPS sourced the PFC load and instability transpired, laboratory experiments established the size of a resistor, to introduce damping, that prevented the occurrence of the oscillations. The resistor was connected in parallel with the output of the UPS. For this purpose and at rated power of the UPS output, it required a resistance constituting of about seven percent of the rated load. The overall system losses due to this added resistance would be unacceptable for most practical purposes. Chapter 2 Problem Statement 24 2.5 Summary Overall, instability may occur in many types of power electronic circuits. References [28], [29] provide good insight into filter instabilities. A solution to instability problems involving the CVT at light loads is available in [20], [27]. A relatively new area of research of power electronic circuits involves nonlinear concepts such as instability, sub-harmonics, bifurcation, and chaos [36]-[44]. The study performed extensive empirical research to obtain knowledge regarding the instability issue. The objective of this chapter was to model the existing design quantitatively in a reverse engineering fashion. We now regard the instability issue to be well defined and its characteristics to be understood in an approximate nature. This chapter also complemented the literature review in Chapter 1 in that numerous references were given to similar systems and phenomena as those discussed in the sections above. The experimental results presented in this chapter were a motivation factor that initiated the modeling approach and stability analysis presented in the subsequent chapters. Chapter 3 Modeling of the Ferroresonant Transformer 'After a brief introduction to transformer modeling, this chapter presents a comprehensive electrical equivalent circuit of the CVT. The level of detail includes winding resistances, core resistances, continuously nonlinear magnetizing elements, tapped windings, and leakage inductances. The equivalent circuit relates to the physical structure of a typical transformer design in the sense that it is economical to manufacture. The following sections describe methods how to extract the circuit parameters from a physical ferroresonant transformer including the hysteresis loops and how to fit the same into single-valued functions. At the end of the chapter, computer simulation results of the model are compared with those obtained analytically and experimentally. The subsequent chapter makes further use of the equivalent circuit and includes the load connected to the secondary side. This affords a complete state-space representation of the system that facilitates computer simulations. It is the hope that this work will aid designers and analysts of the C V T specifically in the area of computer simulations and subsequent hardware prototyping. 3.1 The Traditional Two-Winding Transformer The physical structure in Fig. 3.1 (a) represents a rudimentary two-winding transformer. Although a transformer would normally not feature this arrangement, it serves the purpose of this 1 A version of this chapter has been publicized. M. G. J. Lind, W. Xiao, and W. G. Dunford, "Modeling of a Constant Voltage Transformer," IEEE Trans, on Circuits and Systems I: Regular Papers, Vol. 5, No. 2, pp. 409-418, Feb. 2006. A version of this chapter has been accepted for publication. M. G. J. Lind, G. A. Dumont, and W. G. Dunford, "Analysis of a Circuit Exhibiting Ferroresonance," IEEE International Symposium on Circuits and Systems ISCAS 2006, Kos, Greece, May 2006. Accepted for Publication. 25 Chapter 3 Modeling of the Ferroresonant Transformer 26 discussion. Individual placement, one around each vertical leg, of the primary and secondary winding characterizes this structure. The figure displays three flux paths, the primary and secondary leakage paths and the magnetizing path. The character x indicates that current is flowing into the paper and a dot specifies current flow of opposite direction. The points a, b, c, and d indicate the mean length path of the magnetizing flux. The general flux pattern in Fig. 3.1 (a) occurs when the magnetomotive force of winding one is greater than that of winding two. A thorough presentation of flux paths for a traditional two-winding transformer given the instantaneous magnitudes and directions of the winding currents is presented in [45]. Several degrees of desired detail or representation of this transformer is possible. The most common model is the Steinmetz equivalent circuit also called the T-model that Fig. 3.1 (b) presents. This model consists of one linear magnetizing branch, two leakage inductances, and the resistances of the primary and secondary windings. The figure also shows two ideal transformers to emphasize that the circuit elements in the figure are not referred to a particular side. This representation considers the permeability of the paths of the non-airborne leakage flux to equal infinity, an assumption, which may be unsuitable for the inclusion of nonlinear phenomena. However, saturation effects due to the leakage flux are absent in this model. The circuit in Fig. 3.1 (c) exemplifies a more detailed description of the circuit. This model is a variation of the n-equivalent circuit, which characteristic is a leakage inductance flanked by a magnetizing inductance on either side. The reluctance method was used to obtain this magnetic circuit where the topological principle of duality or the Duality Theorem provided the transition from the magnetic equivalent circuit to that of the electrical. The subsequent sections describe in detail how such a derivation is performed. The lettered subscripts of the designators of the circuit elements correspond to the respective distance in the physical structure. The added lines, resembling a Z, to individual resistors and inductors signify nonlinear elements. Thus, this circuit consists of four nonlinear magnetizing inductances that separately characterize each side of the rectangular magnetic path of the iron. Once the magnetic circuit and its electrical equivalent is established, (3.1) and (3.2) produce the lumped reluctance, or energy storage element ft of the magnetic path and the corresponding inductance L such that K = — pA (3.1) and Chapter 3 Modeling of the Ferroresonant Transformer 27 The parameter 1 corresponds to the mean length of the section, is the absolute permeability of the particular material, A represents the net cross-sectional area of the magnetic path, and N is the turns ratio of the winding to which L is referred. * / / \ \ 4 / — ^ T -r<-\ t \ t \ / / 1 • \ 1 / \ ' / i ' i X \ 1 \ 1 \ 1 x f t \ i • i i • i i • i 1 1 X i I f 1 \ \ \ \ \ 1 / t 1 * / I \ 1 \ 1 / \ 1 \ i \ t \ / \ 2 (a) «1 N ^ N -AAA N : N 2 r 2 AAA-(b) N : N 2 R 2 A A A -+ v2 (C) Fig. 3.1 A n elementary two-winding transformer: (a) the physical structure, (b) the Steinmetz equivalent circuit, and (c) a more detailed equivalent circuit. A third equivalent circuit of the standard two-winding transformer based on coupled coils may also be used. However, this model seems to be less popular in the literature, probably because the mutual inductance does not explicitly appear on the schematic. Chapter 3 Modeling of the Ferroresonant Transformer 28 For multi-winding transformers, the modeling becomes more complex. The schematic in Fig. 3.2 shows a conventional model of a multi-winding transformer where the magnetizing flux is represented by a single inductance, which is placed on the primary side. The core and winding resistances are omitted in this figure. A leakage inductance is placed in series with each winding. However, the leakage inductances that model the reluctances of the leakage paths may not be directly related to the physical composition of the transformer in this configuration. This model also inaccurately predicts the values of the leakage inductances to be larger than the actual ones [46]. Fig. 3.2 A traditional model of a multi-winding transformer. 3.2 The Ferroresonant Transformer The operation of the C V T differs from that of a traditional transformer in that the former purposely features saturation by design. Little literature covering detailed equivalent circuits of the CVT exists. One of the very early publications of the ^-equivalent circuit for the traditional two-winding transformer is [47]. Further treatment on the topic exists in [45], [48]. The latter also covers nonlinear circuit elements. The ^-equivalent circuit also applies to the CVT. One of the earliest extensive attempts to model the C V T appears in [35]. This model does not cover hysteresis loops, winding resistances, or tapped windings. Furthermore, this publication does not directly derive the equivalent circuit from a particular physical structure of the transformer. A few other publications discuss the equivalent circuit of the C V T extensively. A thorough flux derivation based on a physical model appears in [12], but the treatment excludes continuously nonlinear magnetizing elements and tapped windings. Similar facts apply to [31], which, however, includes winding resistances. Chapter 3 Modeling of the Ferroresonant Transformer 29 Previous studies, examples of which are [6], [10], [18], [32] have well documented the various behaviours of the CVT. However, most works have based the equivalent circuit of the CVT on the circuit in Fig. 3.3 without questioning its origin. Besides, the nonlinear inductance, labelled L i , has generally been considered a piecewise linear inductance taking only two values. Fig. 3.3 A general and simplified circuit of the uncontrolled CVT. Publications on the SCR-controlled C V T are uncommon. This study found a limited number of references [27], [32], [49] that treat the topic. Each of these works eliminates the nonlinear secondary magnetizing inductance of the C V T from the equivalent circuit. The literature review found one reluctance-based derivation of the C V T in [34], but this treatment is based on the superposition principle and as such is not valid for nonlinear systems. It is the opinion of this author that the lack of publications of a comprehensive equivalent circuit of the controlled CVT, apart from [50], is due to historical reasons. The circuit in Fig. 3.3 used to consist of individual circuit elements with Lo being an external inductor representing leakage inductance, and L i representing a transformer or an inductor that featured saturation. This circuit configuration has remained since. It may fairly well represent the essentials of the static behaviour of the CVT, but one purpose of this study is to point out that it less accurately describes the dynamic behaviour of the circuit. 3.2.1 The Equivalent Circuit of the Ferroresonant Transformer This section presents the equivalent circuit of the ferroresonant transformer that the study deemed best to describe the physical device. The research considered several other models as well, but the one presented below was the most detailed model achievable where experimental Chapter 3 Modeling of the Ferroresonant Transformer 30 results followed by calculations could reliably generate the circuit parameters. The conventional transformer model described in Section 3.1 does not directly relate to the physical structure of the CVT. Thus, a physically based model of the ferroresonant transformer has the potential of producing a more accurate electrical circuit. The structure in Fig. 3.4 shows a typical controlled C V T used for ferroresonant UPS applications. This E-type core with concentric windings is economical to manufacture. The primary side or upper part of the core contains the windings for the line and inverter voltages. This side also feeds the battery charger. The secondary side or lower part of the core houses the reference winding for the control system, the tapped winding for the load, the SCR-controlled inductor, and the capacitor. The maximum energy storage capacity of this capacitor relates as V2CV 2. Thus, a large operating voltage reduces the size and consequently the cost of the capacitor for the same capacity of energy storage. The air gaps, one at each end of the horizontal shunts, and their interaction with the external capacitor, provide for different flux densities in the primary and secondary portions of the center leg. The hatched areas in the figure represent the iron core and the arrows indicate the various flux paths. As the arrows implicitly indicate, leakage flux within the windings themselves is not modeled. The study used the Electrical-Magnetic Duality Theorem and the reluctance method to derive the electrical circuit analytically from its corresponding magnetic one. Originally, the authors in [48], [51] developed these techniques. This approach was not widely accepted at the beginning, the non-existence of computers probably being the main reason, and it has been uncommonly used since. However, present day researchers have recently reactivated these modeling techniques since they produce circuits that are topologically more accurate [52], [53]. The advantage of this approach is that the electrical circuit is related to the physical structure of the transformer, and the derived parameters have a one-to-one relationship with equivalent magnetic and physical quantities. Thus, this methodology produces representations that describe the consequences of leakage effects and saturation in each individual section of the core. The Duality Theorem states that the dual of the topology of a circuit occurs when meshes are replaced by nodes or vice versa. Likewise, voltage sources replace current sources of the same polarity and reciprocals of impedances replace the latter. Resistances become conductances and inductances transform into capacitances. Open circuits translates to short circuits. The Chapter 3 Modeling of the Ferroresonant Transformer 31 polarity signs of the circuit elements should be rotated in the same direction in the transformation from the original circuit to that of the dual. The dual of reluctance is permeance, which is the inductance of one turn of an electrical conductor. Certain networks, however, such as nonplanar ones do not possess duals. Fig. 3.4 The physical structure of the ferroresonant transformer. Although magnetic and electric circuits are in different realms, their duals are truly equivalent. The application of the Duality Theorem to convert a magnetic circuit to that of the electric unfolds as follows. > > Construct the magnetic equivalent circuit from the physical structure of the device in terms of reluctance elements and magnetomotive sources. Place a mesh label at the center of each mesh and a reference label outside all meshes. Chapter 3 Modeling of the Ferroresonant Transformer 32 > Interconnect all mesh labels and the reference label with lines that go through each and every circuit element once and only once. > Draw a new circuit where the mesh labels now become nodes and where the dual of the circuit elements of the mesh-based circuit constitutes the node-based circuit in that the dual elements now connect the nodes. > Scale the circuit elements according to the turns ratio of the winding to which side the final circuit is referred and use ideal transformers wherever appropriate. The equivalent magnetic circuit in Fig. 3.5 corresponds to the physical structure in Fig. 3.4. The study assumed symmetry about the center leg and considered only the right side of the structure. Consequently, the values of the elements subjected to this assumption were halved. The lettered subscripts of the components of the circuit correspond to the distance of the mean magnetic path between the same letters in Fig. 3.4. Furthermore, the positive direction of the flux is established from the first letter of the subscript to the last. A comparison of Fig. 3.4 and Fig. 3.5 reveals that the circuit accounts for all mean-length flux paths external to the windings of the right half of the physical structure. The study assumed that the leakage fluxes that are established in-between the windings are parallel to the center leg. This is a common supposition when the windings have the same height [46]. The reluctances Kae and K n e correspond to the parallel combination of the magnetic path of the center leg and that between the center leg and the innermost primary or secondary winding. The encircled letters label each mesh in Fig. 3.5, and the arched lines interconnect the same through the circuit elements. The lines that are broken outside the perimeter of the circuit are understood to return to the reference point J. Each source of the magnetomotive force in the figure labelled Nyiy corresponds to the respective winding in Fig. 3.4. The lines resembling a Z, added to individual elements, signify nonlinear elements. The figure reveals that the model considers the permeability of the magnetic paths of the air-born leakage fluxes to be constant as opposed to the variable permeability of the iron paths. The usage of the Duality Theorem converted the mesh labels in Fig. 3.5 to node labels as the circuit in Fig. 3.6 demonstrates. The reciprocals of the reluctances appear in the latter figure and each magnetomotive source has become a flux source. Each boxed element labelled Yu Chapter 3 Modeling of the Ferroresonant Transformer 33 symbolizes the reciprocal of a nonlinear parallel combination consisting of a resistance, representing the core loss, and a magnetizing inductance of the particular section. Fig. 3.5 The equivalent magnetic circuit of the CVT. Since the circuit in Fig. 3.6 is too complex to produce a useful practical electrical representation of the CVT, the study made two assumptions that generated the circuit in Fig. 3.7. First, that the elements 2Yef, 2V fg, 2Ygn, 2Yhi, 2fy 2Yjk, 2Yef, 2Ycb, 2Ydc, 2Ymd, 2Ypo, 2Yqp, 2Ygr, and lYmr in Fig. 3.6 are larger than the remaining magnetizing inductances. Second, that the shunts never saturate and that the reluctances of the air gaps are much larger than those of the shunts. This made the elimination of 2Yim possible and resulted in only one element +l/^ kr = VX representing the leakage path between the primary and secondary groups of windings. C h a p t e r 3 M o d e l i n g o f t h e F e r r o r e s o n a n t T r a n s f o r m e r 34 0 1 a b N 1ab ' lab | H 2 K ZY, id (t) 'ae 2 R Q b B AAA/— 0 „ 4ab N4ab'4ab| 2* ba flhf CJ) 2V, 'c c ->vW— 0, 5ab 5ab'5ab 2V, rcb J—2—lf2jk|-1 Rw 2 R k d D ->*1 - > x 2 ->Xg - » x 4 r-AAAH 2Vr dc 2Vi Im 2Vr md x i>n x 2 > -x 5>-0 3ab N 3abi3ab x 6 > -ffi 2 R fo F •AAA— •0. 2ab N 2ab '2ab 2 K hp G •AAA— 0. 2c N2c'2c 2K £ 0 (t) H •AAA/— 02d N 2 d i 2 d 2V, © 2 R k r •AAAr-4 2V, l4 2V„ F i g . 3.6 T h e d u a l e q u i v a l e n t m a g n e t i c c i r c u i t o f t h e C V T . 0 1 a b N 1ab ' l ab | F i g . 3.7 T h e r e d u c e d d u a l e q u i v a l e n t m a g n e t i c c i r c u i t o f t h e C V T . S i n c e t h e l o a d i s a t t a c h e d t o t h e N2ab s i d e o f the t r a n s f o r m e r , i t i s m o s t c o n v e n i e n t to r e f e r a l l c i r c u i t e l e m e n t s t o t h i s s i d e . I n t h i s p r o c e s s , t h e s c h e m a t i c s i n F i g . 3.8 a n d F i g . 3.9 s h o w that Chapter 3 Modeling of the Ferroresonant Transformer 35 each reluctance and current variable was multiplied and divided respectively by N2ab- A second multiplication by the same turns ratio with the reluctances and the flux sources was also performed. The flux sources and their related respective current variables were differentiated to accommodate the fact that the voltage is the derivative of the flux linkage. The following parameters were also relabelled, to facilitate conformation to the final circuit: 1Zsb = 7^^, K =ft«,. K =n^, KP =^ab, nf0 =^ab, v a e//2V b a =y e n I , and v n e//2V n o =vcm2. 2 N 2 a b 2 N 2 a b 2 N 2 a b 2 N 2 a b 2 N 2 a b 2 N 2 a b 0 1 a b N 1ab '1ab N 2 a b R 1 a b B R 4 a b C -WV—< W v -04ab(J) 05ab(J) 02d(J) 02c(J) 02ab(J) '2ab N 2ab y cm1 J N 4 a b ' 4 a b N 2 a b 0 5 a b N 5 a b ' 5 a b N 2 a b -VAr I H K 2 c G 0 2 d i N2d<2d N2ab -AAAr-R 2 a b F R 3 a b E 0 2 c N 2 c '2c N 2 a b -VW- -VvV-03ab(J) N 3 a b ' 3 a b N2ab NZab^crrg Fig. 3.8 The reduced dual equivalent magnetic circuit of the ferroresonant transformer referenced to the turns ratio N2ab-2 ( N 2 a b ) 2 2 ( N 2 a b ) 2 2 ( N 2 a b ) 2 2 ( N 2 a b ) 2 2 ( N 2 a b ) 2 2 ( N 2 a b ) 2 R 1 a b B R 4 a b c R l . (J) 3 1ab N 1ab'1ab N 2ab 0 1ab N 2 a b •AAAr-(D ib N 4ab '4ab N 2 a b 0 4 a  ,N2ab N2abycm1 I -AAAr N 0 vL' N 5ab '5ab N2ab - A A V N 2 a b 0 2 d N 2 d i 2 d N2ab K 2 c -AAV-2ab F K 3 a b E N 2 a b 0 2 c T N 2c '2c l N 2 a b 0 2 a b N 2ab CP '2ab -AAAr-N 2 a b 0 3 a b N 3ab '3ab N 2ab N2abvcm2| T Fig. 3.9 The reduced dual equivalent magnetic circuit of the C V T referred to the (N 2 a b ) 2 side and to the time derivative of the flux sources and the current variables. The equivalent magnetic circuit became the electrical of the same as Fig. 3.10 presents. The transformers in this figure are ideal and winding resistances are present. This circuit intuitively seems to model the transformer correctly. In the hypothetical case that power is applied to the reference winding N 3 a b , this model accurately predicts the secondary magnetizing Chapter 3 Modeling of the Ferroresonant Transformer 36 inductance to be closest to the aforementioned winding rather than to any other winding. The flux linkages produced by N 3 a b links N2 ab the most compared to the other windings and N 2 ab is also the winding physically closest to N 3 ab. In this model, the leakage in-between the individual windings of the primary and secondary groups of windings respectively, build up stepwise with increasing distance from the center leg. The geometry of the windings is therefore considered, and experimental procedures can quite accurately extract the variables necessary for the calculations of the leakage parameters. Fig. 3.10 The electrical equivalent circuit of the CVT. The schematic in Fig. 3.11 shows further modifications of the model. The primary circuit variables and parameters were referred, as (3.3) and (3.4) indicate below, to the N 2 ab side. The introduction of the prime signs indicates a referred quantity. The windings N3ab, N4ab, and Nsab that were previously considered being single windings are now also truly implemented according to their physical structure, namely bifilar center-tapped windings. N V lab - V l a b 2ab N N 1 lab ^ a b lab lab N R ' l a b - R l a b 2ab N \2 2ab V N l a b 7 (3.3) N \ 2 2ab V ^ l a b J L' m l ' m l N 2ab V N l a b J (3.4) Chapter 3 Modeling of the Ferroresonant Transformer 37 Fig. 3.11 The electrical equivalent circuit of the ferroresonant transformer including the ideal transformers for the bifilar windings for the inverter, charger, and reference. For the purpose of this work, without loss of generality, the charger and inverter windings were removed from the circuit. The schematic in Fig. 3.12 shows the final electrical equivalent circuit. The primary circuit variables and parameters are referred, as (3.3) and (3.4) specify, to the N 2 a b side. The prime signs indicate a referred quantity. The N 3 a b winding that was previously considered being a single winding is implemented according to its physical structure, a bifilar center-tapped winding. The C V T has excellent transient suppression. It can be deduced from Section 3.4 that the natural frequencies of the circuit in Fig. 3.12 based on the extreme values of the nonlinearities are typically below 120 Hz. Concerning impedances, observe that X L i is substantial and that Xc* and X L * where * indicate approximate referral to the v 2 a b terminals, are smaller than the load. Due to the relatively large value of X u at 60 Hz, the load is decoupled from the source. The 60 Hz impedance of L ^ , in the extreme nonlinear region, is roughly 130 % of that of the rated load. Thus, transients are quite small and frequency dependent or nonlinear eddy current effects [54] Chapter 3 Modeling of the Ferroresonant Transformer 38 are of less concern. The primary and secondary core losses R ' c i and RC2 are therefore approximated as being linear. The nonlinear inductances L ' m i and Lm2 correspond to the magnetizing fluxes of the primary and secondary sections of the core respectively. The leakage inductance Li represents the air gaps of the shunts. As was postulated earlier, to keep this inductance relatively constant throughout normal operating conditions, the cross sectional area of the shunts should be sufficiently large to maintain the leakage fluxes without the possibility of saturation. Three ideal transformers are present in the circuit. Two of these are due to the particular connection of the bottom wire of the secondary windings of N2ab : N2 C and N2 3b : N2d- This model accurately describes the fact that the effective turns ratio of the respective windings deviate more from that of the ideal the further the windings are located from the center leg. The further a winding is located from the center leg, the larger is the number of leakage elements in the current path feeding that particular winding. Fig. 3.12 The reduced electrical equivalent circuit of the CVT. An interesting aside is the derivation of the equivalent circuit of the C V T based on the same methods that was used to obtain the Steinmetz equivalent circuit in Fig. 3.1 (b). This simplified method produced a more straightforward circuit than that of the more detailed circuit in Fig. 3.1 (c) that the comprehensive approach created. The physical structure is the same as previously presented. However, the airborne fluxes in-between the windings are here not Chapter 3 Modeling of the Ferroresonant Transformer 39 considered to cause any saturation effects in any of the iron paths e to m, m to a, a to e, m to n, or n to e and similarly in the paths of the left half of the transformer. The application of the Duality Theorem and the reluctance method led to the electrical equivalent circuit in Fig. 3.13. The most noteworthy aspect of this model is that the order of the circuit is now increased by one in form of an additional leakage inductance. Furthermore, this model positions the secondary magnetizing branch next to the leakage inductance Lj. This less accurate representation of the model may incorrectly define the values of the elements constituting the secondary magnetizing branch as far as the execution of the parameter extraction from the physical model is concerned. Fig. 3.13 The electrical equivalent circuit of the ferroresonant transformer based on the simplified magnetic model. Thus, an interesting perspective is that a simpler application of the reluctance method produced a final model that not only is less accurate than that obtained from the more comprehensive treatment, but it also produces a more complex equivalent electrical circuit. 3.2.2 Determination of Circuit Parameters Once the study had established the equivalent circuit of the controlled CVT, analytical relationships describing the parameters of the model were derived. This section describes the analytical relationships of circuit variables and parameters. This is an important step in the Chapter 3 Modeling of the Ferroresonant Transformer 40 modeling process. Although previous works have developed generic design equations [12], [33], however, mainly for the uncontrolled C V T , the design process still typically involves iterations of several manufactured prototypes. The goal of this section is to establish reliable methods to define the values of the circuit parameters. 3.2.2.1 Parameters Related to the Magnetizing Flux In this subsection, a method is proposed to model the hysteresis loops suitable for the CVT. For the traditional two-winding transformer, factory test data frequently inadequately describe the parameters pertaining to the core. These parameters are therefore best determined by experimental tests [52], [55], [56]. This is also the preferable method regarding the magnetizing branch of the C V T . The approach is based on the Taylor Series expansion of the tangent function and curve fitting techniques to minimize the least square error. The schematic in Fig. 3.12 reveals that the flux linkages of L ' m i and Lm2 are V m i ( t ) = jv ' l a b ( t ) -R ' l a b i ' l a b ( t )d t (3.5) ^ L m 2 ( t )= j v 2 a b ( t ) - R 2 a b i 2 a b ( t ) d t (3.6) respectively where the voltage drop across Lo ab has been ignored due to the small impedance of this element compared to that of the adjoining elements at the rated frequency. This approximation is substantiated at the end of this chapter where the impedances of the various circuit elements are established. The term involving the resistor in either equation can be eliminated, but to a loss of accuracy of probably one percent. In the discrete time domain, with the introduction of the cumulative trapezoidal integration technique (3.5) and (3.6) become ^ L - O , , W = X L M (k 0 i l )+X L . m I (k-l)+i-[v ' I l b (k)+v'lab (k-l)-R\ab ( i ' l a b (k)+i'lab (k-1))] (3.7) T ^ ( k ^ A ^ k J + A ^ k - l H ^ ^ ^ (3.8) where the sampling time is defined as T s = tk - tk-i and Xj(ko,i) is the initial flux linkage. These equations will be used below in the software routine that calculates the flux linkages. Chapter 3 Modeling of the Ferroresonant Transformer 41 For the description of magnetization curves, the Langevin function [57] has been frequently used in the past. For the particular case such as for A.(i(t)), this function is expressed as Langevin(i(t)) = coth(i(t)) - . (3.9) However, for the model of the hysteresis loop, the current through the inductance can be considered a function of the flux linkage instead of the contrary. This suggests that the tangent function can be used to represent a single-valued magnetization curve. The Taylor Series expansion of this function is taylor(tan(X)) = ± B " ^ | ' 4 ' W , \X\ < £ (3.10) where B2 n is the Bernoulli number. However, this representation is less useful since it is not valid for all A. such as for a heavily saturated inductance. The electrical definition of inductance is such that di(t) (3.11) Since (3.10) consists of odd and positive powers of X, it suggests that a one-term polynomial with a power greater than unity can be used. However, for the C V T , a one-term polynomial would possess either too much curvature or a too step slope about the origin. The latter can also be considered as a quasi-linear region of L . The addition of a linear term gives the equation of a two-term polynomial in X such that i(t) = P,X(t) + Pn^(t)n, n =3,5,7,... (3.12) where the coefficients represent the linear and nonlinear regions respectively as ft = 77— a n d v°=r^—• (313) linear nonlinear Based on (3.12) and (3.13), the designer could approximate the worst-case inrush current of the primary side provided the retentivity is known. The equivalent circuit suitable for this analysis would only consist of the series combination of v ' i a b, R ' i a b , and L ' m i . The reminder of the circuit branches can be neglected since their respective currents are very small compared to Chapter 3 Modeling of the Ferroresonant Transformer 42 the worst-case peak current through the primary magnetizing inductance. The latter current can easily exceed ten times the rated input current to the UPS. Although R ' i a b is very small, it is necessary to include this resistance in the analysis since, for a substantial inrush current, it will contribute to a considerable voltage drop. The peak value of the flux linkage occurs when v'i ab is of the form sin(cot). Prospective residual flux must also be considered. The equation for the inrush current is then (3.12) combined with (3.5), where v ' l a b (t)= V ' l a b m a x sin(d)t), and (3.13). This first order nonlinear differential equation is most conveniently analyzed numerically. It must also be noted that the model of the magnetization curve in (3.12) may be less accurate at larger values of the current since the experimental procedure of this part of the modeling require a source that can provide a very large current with significant di/dt at various phase angles. Such a source might not be readily available. Additional polynomial terms can be implemented in (3.12) should a higher degree of accuracy be desired. To fit an experimentally obtained magnetization curve, the procedure of which the subsequent section describes, the study used the least-squares approximate solution for the over-determined system. This curve fitting technique is based on a multi-term polynomial fit such that y = xp + s (3.14) where the objective is to minimize the length of e. The quantities, y, X , p, and E, represent the observation vector, the design matrix, the parameter vector, and the residual vector respectively [58]. In expanded form, (3.14) is described by " I X ? X ? • "p." y 2 = I 2 3 X 2 X 2 - X 2 p 2 + y m _ I 2 3 X m X m • • • (3.15) where the x-elements represents all positive powers of the polynomial. However, such a solution would be unwieldy. The expression in (3.10) suggests that only odd powers are needed and based on (3.12), a two-term polynomial might suffice. This results in a reduced version of X defined as X r that consists of elements of only the first and the k t h power such that Chapter 3 Modeling of the Ferroresonant Transformer 43 x (3.16) x m The least-squares solution for the over-determined system of (3.14) is f»=(x;xr)_,x;y. (3.17) A measure of deviation, the root mean square of the Euclidean norm of the residual vector, is where m equals the number of samples. To facilitate a minimized-error solution of (3.14), the study designed a software routine that converted the bi-valued function of the hysteresis loop into a single-valued function. A representative cycle of a sine wave based on the average of ten cycles of and i(t) where X,(i(t)) represented the hysteresis loop was used for this purpose. Linear interpolation provided the conversion of each set of similar dual data points into that of a single. The linear interpolation was based on the integrated quantity X,(t), which inherently possessed a smooth waveform, and this aided the linearization process. The routine calculated the power of the terms of the second column of X that minimized ||e||2. The flowchart in Fig. 3.14 describes the main scope of this routine. To simplify the analysis, the core losses are considered to obey a linear relationship that is best obtained through experimental procedures. For this approximation, the winding resistances Riab and R2ab are neglected since their values are very small compared to that of the resistances of the magnetizing branches. A second assumption is that no currents flow through the windings N 2 c , N 2d, N 3 a , and N 3 a . A third supposition states that the current that flows through Li , which represents the air gap, is essentially zero. Thus, for an unloaded transformer, the primary core loss referred to the N2ab side is calculated as (3.18) Chapter 3 Modeling of the Ferroresonant Transformer 44 R'c = N \ 2 2ab v N i a b y Re (3.19) where 0i ab is the phase angle between the rms values of the voltage and current of the primary side. Likewise, the secondary core loss referred to the N2ab side is obtained from the relationship R c 2 = • ^ 2 a b - ^ ^ 2 a b ^ Re v V 2 a b Z 0 ° (3.20) How to obtain the rms variables for the conditions described above and in particular, the relationship i u ~ 0 is explained below in Section 3.3. For one complete cycle of averaged data of a sine wave, for all j , input i(j) and Find j for the half-cycle zero-crossing, z, of Let z = j . * For the first half cycle, increment j from j = 1. For each j , decrement k from k = z and find k that minimizes \x(j)-X{k) and let \{h)= X $ + ^ and i(h) = i^±iW for each j and the related k. Perform the operations above for the second half cycle. Reorganize the vectors i and k. For all odd n, from n = 3 to n - n m a x , calculate the least square regression for i(k)=bt>.(k)+bnA.(k)°, n = 3,5,7,... Also, calculate E = ||i-xp||. For the n that produced the smallest norm of E , output n, i, X, p\ and e. Fig. 3.14 Flow chart for the routine that produced the single-valued magnetization curve. Chapter 3 Modeling of the Ferroresonant Transformer 45 3.2.2.2 Parameters Related to the Leakage Paths The analysis in this section is analogous to that of the short-circuit test of the standard two-winding transformer. However, for the CVT, there is not a generic procedure, but each transformer topology must be considered according to its equivalent circuit. The analysis of the circuit in Fig. 3.12 affords the basis of the experimental extraction of the values of the leakage inductances and winding resistances. To source certain terminals and to short others is the preferred method. One commonly used technique is to short all windings but one and to source the remaining winding with a voltage level that will produce significant currents in the shorted windings. Although such techniques regarding multi-winding transformers are well documented in [45], [47], [59], they are unsuitable for the CVT. For example, an applied voltage of a thirty percent of the rated value may cause saturation of various levels and parts of the core, interfering with the mathematical treatment of the measured variables. In addition, a winding such as the reference winding is designed to carry only a small current. In order to facilitate the calculation of the leakage inductance L i and the winding resistances Riab and R2d in Fig. 3.12, viab can be sourced whereas V 2 a b c and V2abcd are grounded or the circuit can be sourced between V2abcd and V 2 a b c whereas vi a b is grounded. Although it seems as if either relationship derived in (3.21) or (3.22) below may be used to produce these parameters; this choice is not arbitrary. The study found that the magnetizing branch that has the lowest impedance should be shorted. The rationale for this is that if the impedance of one magnetizing branch is significantly lower than that of the other, it will also draw a significant current. This in turn would cause an excessive voltage drop across the adjacent winding resistance and possibly also introduce a small phase shift of the voltage across L i compared to the applied voltage. This would result in a too large extracted winding resistance and in a slightly incorrect value of Lj. It must also be observed whether the applied voltage causes either magnetizing branch to operate far outside its linear region, in practice the secondary one. The study observed that the applied voltage needed usually was less than one third of the rated one. A condition under which only limited saturation occurred. The calculation of the two resistances in (3.21) and (3.22) is based on the ratio of their respective dc resistances, and the prime signs in (3.21) through (3.24) indicate that these elements are referred from the Ni ab and N2d sides to the N2ab side. Chapter 3 Modeling of the Ferroresonant Transformer 46 - V J^I -f-J^' +iX V ' 2abcd 2abc lab 2d J LI ) N 2 2ab I N N i l a b 1 > l a b 1 ^ 2d R lab R' lab R 2 d - R > 2 d v _ v l a b ^ 2ab I N N A 2 a b c d i > l a b i > i > lab 2 V N 2 a b > f N } i y 2d 2 N V 2ab J (3.21) (3.22) (3.23) (3-24) For the particular CVT, (3.21) better represented the model than (3.22) did. It should also be pointed out that for the short-circuit test described above, the source voltage and the short circuit current are used as (3.21) indicates. This is in opposition to the traditional short-circuit test, which typically utilizes the applied voltage and current. The relationships in (3.25)-(3.27) below describe the calculation of the leakage inductance Li2ab and the winding resistances Pv2ab and R.2C. For this parameter evaluation, the circuit is energized at V2abc and V2ab is connected to ground. The equation assumes that no current flows through Li2c or L [3 a b since the impedances of the magnetizing branches are large compared to those of the elements under investigation. It should be observed that since one of the terminals of both the N2C and N2d winding is now connected to ground, the calculations below make use of the full turns ratio N23bc = N2ab + N2C. The prime signs in (3.25)-(3.27) indicate that these elements are referred from the N2ab side to the N2C side. The calculation of the two resistances in (3.25) is based on the ratio of their respective dc resistance. R2c+R '2ab+JX' v. 2abc L12ab R 2ab ~ R 2ab N 2^abc \ 2 2abc V N 2 a b J Y — Y ' "^L12ab — L12ab N ^ ^ 2abc V ^2ab J (3.25) (3.26) (3.27) The expressions derived in (3.28) and (3.29) below produce the values of the leakage inductance L i 2 c and the winding resistance R 2 c i . The circuit is energized at v2abcd whereas v 2 a b and Chapter 3 Modeling of the Ferroresonant Transformer 47 V2abc are connected to ground. The equation assumes that no current flows through L i or L i 3 a b since the impedances of the magnetizing branches are large compared to those of the elements under investigation. Since either end of the N 2 c and N2d windings is connected to ground, the calculations below make use of the full turns ratios N2abc = N23b + N2C and N2abcd = N2ab + N 2 c + N2d- The prime signs in (3.28) and (3.29) indicate that these particular elements are referred from the N2d side to the N 2 c side. R ' 2 d + J X N 2ab L12c I N A 2 a b c d 1 > 2abcd V - I R v 2 a b c d A 2 a b c d i > ' 2 d N R 2 d - R ' 2 d N 2abcd V - ^ 2 a b c R 2 c N 2 a b c J 2abcd V N 2 a b J (3.28) (3.29) Finally, the relationships in (3.30) and (3.31) illustrate the computation of the leakage inductance L i 3 a b and the winding resistance R 3 ab = R 3 a + R3b- The circuit is energized at v2ab whereas the v 3 a and v 3 b terminals are connected. This equation assumes that no current flows through Li2c, RC2, or Lm2 since the impedances of the magnetizing branches are large compared to those of the elements under investigation. The prime signs in (3.30) and (3.31) indicate that these elements are referred from the side of N 3 ab = N 3 a + N3b to the N2ab side. R 2 a b + R 3 a b + J X U 3 a b V _ v 2 a b l 2 a b R 3 a b - R ' 3 a b 3ab V N 2 a b 7 (3.30) (3.31) 3.2.2.3 Parameters Related to the Physical Structure It is an essential part of the work of a designer to establish the transformer parameters from the dimensions and properties of the materials. Although the purpose of this thesis is not to describe and develop design methods for the CVT, some facts are noteworthy to mention and the following paragraphs discuss some details. The designer can predict winding resistances based on the cross-sectional area and length of the conductor. A correction factor that takes the stray losses and the skin effect into account Chapter 3 Modeling of the Ferroresonant Transformer 48 should also be part of the equation. The nonlinear magnetizing inductance and the core loss resistance are more difficult to predict. The magnetizing curve for the particular ferromagnetic material curves is an asset in this calculation. However, small alterations in chemical composition, by heat treatment, and by mechanical work may cause the properties of different specimens of the transformer steel to deviate significantly from those predicted by the aforementioned curves [45]. Two commonly used schemes to calculate the inductance representing the leakage in-between windings are the reluctance and energy storage methods. The references [45], [46] thoroughly describe this for standard transformers. Although the energy storage technique may produce a better result, it is less suitable for the C V T since this method assumes that the airborne leakage fluxes enter iron paths of infinite permeability. Since saturation is a normal mode of operation of the CVT, the aforementioned condition does not hold. This study used the reluctance technique to calculate the leakage inductances, which amounts to N 2 L = - (3.32) and ft = — - (3.33) H 0 A where U is the lumped reluctance, 1 corresponds to the height of the winding, u 0 is the permeability of free space, A represents is the cross-sectional area of the volume between two windings, and N is the turns ratio of the winding to which the leakage inductance L is referred. It should be noted that the leakage flux within the windings themselves is neglected in the calculations in (3.32) and (3.33). For the leakage calculation of the air gaps of the shunts, the cross-sectional area is obtained from A = (a + 6 l g ) ( b + 6 l g ) (3.34) where A equals the equivalent cross-sectional area of the air gap, a and b constitute the sides of the cross section, 5 is a correction factor, which default value is unity, and l g represents the length of the air gap. When one of the surfaces facing the gap is significantly larger than that of Chapter 3 Modeling of the Ferroresonant Transformer 49 the other, experience has shown that 28 is more appropriate as long as the correction does not exceed one fifth of the dimension of the cross-sectional area [45]. 3.3 Measurement Techniques To provide a detailed model of the transformer, the study proposed measurement techniques to extract the parameters of the CVT. A l l the measured variables in the plots below indicate experimental results that were obtained from a C V T that was designed according to the structure in Fig. 3.4. The methods below resemble, but are also quite different from, the open-circuit and short-circuit tests of the standard two-winding transformer. An inspection of the equivalent circuit in Fig. 3.12 reveals that to source the primary side of the transformer and to leave the secondary side open, or vice versa, would not produce individual information of the magnetizing branches. However, if the primary and secondary sides are sourced simultaneously, each with a phase and magnitude that would produce zero current through L i , a separate excitation of these branches is possible. This study derived such a method. It also describes how to extract the leakage inductances and winding resistances. The schematic in Fig. 3.15 shows a diagram of the configuration for the measurements of the magnetizing branches. An inspection of the equivalent circuit provides the information on which windings to source. As Fig. 3.12 indicates, the primary magnetizing branch is physically closest to the v i a D terminals, making it an obvious choice for the connection of source one. The secondary magnetizing branch is closest to the reference winding. However, in general, the wire gauge of this winding would not support the magnetizing current. Therefore, the second best choice is to connect the second source to the V 2 a D terminals. Only little accuracy will be lost due to this approximation. The introduction of two RC circuits, one on each side of the transformer, provides for the integration of the applied voltages and consequently produces the flux linkages. Although this is a well-known method for the standard transformer, a further refinement required for the C V T was implemented. Most importantly, the first part of the explanation below serves only the purpose of being a visual aid during the measurements. Especially designed software routines provided the actual retrieval of the flux linkages. Chapter 3 Modeling of the Ferroresonant Transformer 50 Fig. 3.15 Configuration for the measurements of the hysteresis loops. The series network of Rj and Ci in Fig. 3.15 integrates the respective applied voltage. At the operating frequency, the series impedance of the network should be very large and that of the capacitor must be very small as (3.35) indicates. R ; » 0)C; The voltage across either capacitor is v B l(t) = -v S l (t)-v c , ( t)-R; (3.35) (3.36) and letting v S i ( t)-v C i ( t ) « v S i (t) (3.37) gives (3.38) Letting R i d equal unity and assuming that the condition in (3.35) is fulfilled, provides for a unity time constant of the integration. This results in vei(0= J v > 1 ( t ) d t = X 1 ( t ) . (3.39) To perform the measurements of the flux linkages, both sources must excite the circuit. The sources should be in phase and their respective voltage levels should be ramped up simultaneously from zero to their rated values to prevent excessive currents. The relationship Chapter 3 Modeling of the Ferroresonant Transformer 51 above facilitates that Xi(t) versus i,(t), the primary and secondary hysteresis loops, can be displayed on the oscilloscope. For a standard transformer, this procedure is unproblematic since only one source is involved. However, the study found that for the C V T , a change in voltage level of either source of as little as 0.5 % ruined the shapes of the loops. Furthermore, the visual interpretation of the shapes of the loops is inadequate and may lead to subjective observations. In addition, due to transformer temperature considerations, measurements may only be executed after the circuit has been energized for several hours. To ensure that the current through L | is zero on an instantaneously basis, to achieve faithfully-shaped hysteresis loops, is virtually impossible due to unlike phase shift of the respective current though the winding resistances and potential harmonic content in the applied voltages. This study initially considered power balance and minimum power supplied by the two sources. However, this work defined a much more adequate method. The introduction of auxiliary windings in the outer leg of the transformer between m and d and r and m in Fig. 3.4 provide for an indirect measurement of the flux through the air gap of the shunt. An equal turns ratio of these windings results in a flux linkage in the air gap such that MO^M+MO- (3.40) To be able to detect that the fluxes in the primary and secondary parts of the core are as equal as possible on a sinusoidal cycle basis, the auxiliary windings should be connected in series and of opposing polarities. Thus, the output of the series combination should ideally equal zero at all times. 3.4 Experimental Results The plot in Fig. 3.16 illustrates the waveform of DA,g(t) normalized to the average of the peaks of those of the individual windings. The plot shows the condition when each source was set close to the rated voltage of the respective winding and then fine-tuned to minimize the flux in the shunt. The scale of the vertical axis indicates a very small difference between the voltages of the auxiliary windings. To make the relationship A ^ t ) = Km(t) as true as possible, to achieve waveform symmetry about the quarter and three-quarter mark respectively of the period is the main objective. The adjusted levels of the applied voltages provided the aforementioned. Chapter 3 Modeling of the Ferroresonant Transformer 52 For an objective numerical evaluation, the waveform in Fig. 3.16 is integrated in real time, to acquire the flux linkage, and the source voltages are adjusted to obtain the relationships among the areas in the figure to minimize | A 1 - A 2 | + | A 3 - A 4 | + | ( A 1 + A 2 ) - ( A 3 + A 4 ) | and in terms of voltages, this is 71 2(0 jv a l ( t )dt+ }v a 2(t)dt 2 M + 3n 2ra 2 n |v a l ( t )dt+ jv a 2 ( t )dt 2H 2co + In (i) | v a l ( t ) - v a 2 ( t ) d t - J v a l ( t ) - v a 2 ( t ) d t 0) (3.41) (3.42) where vai(t) and va 2(t) are the voltages across the auxiliary windings. The minimization of the expression above ensures that on an averaged basis, the respective source winding magnetizes the primary and secondary sections of the core separately. Once flux balance has been achieved, the voltage and current variables entering the transformer terminals are recorded. These variables Chapter 3 Modeling of the Ferroresonant Transformer 53 now made available adequately shaped dynamic hysteresis loops. Noteworthy is that, in general, the saturation flux density decreases when the temperature of the core material increases. The resistivity of the core increases and, consequently, the eddy currents decreases. This results in a narrower loop. A software routine based on the cumulative trapezoidal integration defined in (3.7) and (3.8) provided the flux linkages needed for the reconstruction of the hysteresis loops. This was a two-step process. As previously described, the establishment of the impedances of the magnetizing branches must be present before the measurements of the winding resistances can occur. The first step used the dc values of the resistances in (3.7) and (3.8) to provide an estimate of the magnetizing parameters. Based on this, the true ac resistances of the windings were extracted using the equations as described in Section 3.2.2.2. A second application of the software routine using the aforementioned resistances produced the data required for the curve-fitting routine of the magnetization curves. The curve fitting routine described in Section 3.2.2.1 produced the plots in Fig. 3.17. The figure shows the hysteresis loop of Lmi referred to the N2ab side. The applied voltage was 115 % of the rated one. Thus, the plot indicates indirectly, that for rated voltage, the primary side operated largely in the linear region. The horizontal axis represents the current normalized to that of the rated of i'iab(t). The single-valued fit of the hysteresis loop is located in the center of the same and the best-fit curve of the two-term polynomial quite accurately models the true loop. The plots in Fig. 3.18 describe the hysteresis loop of L ^ - The same procedure as described above produced this plot and the applied voltage was 115 % of the rated one. The current was normalized to that of the rated of i2ab(t). The shape of this loop is remarkably different from a typical hysteresis loop of a standard two-winding transformer. The linear region, some engineers might conclude that there is none at all, is very limited. The experimental results, based on the two figures below, provided the following relationships for the magnetizing inductances (t) 1 1 L ' m l 2.114 0.04233 (3.43) (t) 1 ^Lm2(0 + 1 ^ m 2 ( t ) -0.7830 0.1101 (3.44) Chapter 3 Modeling of the Ferroresonant Transformer 54 0.8 0.6 E ? 0.2 5 1 [ 1 1 1 1 1 1 1 1 ] 1 A// y ] 1 1 i i 1 " ( I J! J I T i 1 r 7 / : * l 1 CJ..1 / " / ! i 7 7 fa j t 1 ->~/~ i-i / _ L 1 1 r r - /- 71 1 - Z -1 True Hysteresis — • Single-Valued Hysteresis Two-Term Polynomial Fit . U 1 1 1 1 i i I I i I I -0.125 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05 0.075 0.1 0.125 Normajized Current (A/A) Fig. 3.17 The normalized hysteresis loop of the primary magnetizing inductance referred to the N 2 ab side. The applied voltage was 115 % of the rated. Fig. 3.18 The normalized hysteresis loop of the secondary magnetizing inductance. The applied voltage was 115 % of the rated. Chapter 3 Modeling of the Ferroresonant Transformer 55 The plots in the next two figures, Fig. 3.19 and Fig. 3.20, show the normalized primary and secondary hysteresis loops respectively for three different levels of the applied voltages vi a b(t) and V2ab(t). Only the first quadrant is shown in Fig. 3.20 for better readability. The three voltage levels are as indicated in the plots 85 %, 100 %, and 115 % of the rated ones. The respective polynomial fit that (3.43) and (3.44) produced is also present in the corresponding figure. For both figures, the hysteresis loops are in good correspondence with each other as well as with the fitted curve. It should be noted that the fits were based on 1.15viab(t) and 1.15v2ab(t) respectively. c * _Q CD CC X 3 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 L — — ^ r r ^ n _1 Jr ' ''yJj r yj / yy / / JJ j / J J / ^ / w ' J 1 /A * (J J // 1 / // ( / J i / JJ yrf A J/J )y r Jy f ) / Jr ^ y /7Tt J J ) r / J * J J s 1 J J J /1 J J J i * fj *' \ Js / I / / * J J J \ r r ~\ / j f J ' 1 l 1 - ^ " c ^ - < f - J l ^ ! ' True Hysteresis 85,100,115% Two-Term Polynomial Fit 1 ' 1 i i i [ I I -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 Normalized Current (A/A) Fig. 3.19 The zoomed-in normalized hysteresis loops of the primary magnetizing inductance for three different levels of the applied primary and secondary voltages. Chapter 3 Modeling of the Ferroresonant Transformer 56 0 0.05 0.1 0.15 0.2 0.25 0.3 Normalized Current (A/A) Fig. 3.20 The zoomed-in normalized hysteresis loops of the secondary magnetizing inductance for three different levels of the applied primary and secondary voltages. The entries in Table 3.1 show the rms error of the Euclidian norm ||s||2/sqrt(m) for selected orders of the second term of the polynomials including those indicated in (3.43) and (3.44) for the primary and secondary magnetization curves respectively. The data reveal that the rms error is very small for the respective powers in the equations above. The error reflects the deviation in the current variable. The entries in the table indicate that for the primary curve, the errors for n = 15 and n = 17 are very similar. Thus, in practice, either one of these powers could have been used to model the magnetization curve. Calculations based on the measured rms voltages and currents of the circuit configuration in Fig. 3.15 and the usage of (3.19) and (3.20) produced the resistances R ' c i and RC2 of the magnetizing branches. The calculations were a straightforward procedure analog to that of the Chapter 3 Modeling of the Ferroresonant Transformer 57 open-circuit test of a standard two-winding transformer. The calculations produced the core loss resistances to the following values at 60 Hz: R ' c i = 3500 Q. and R C2 = 1561 d . As a comparison, the values of the external elements approximately referred to the V2ab terminals were X L * = 7.09 Q and Xc* = 12.4 CI. The rated load was Ri o ad, rated = 25.9 CI. Table 3.1 Results from the polynomial fit of the hysteresis loops. Primary Hysteresis Loop Secondary Hysteresis Loop Order of the Second Polynomial Term RMS Error Order of the Second Polynomial Term RMS Error 11 0.03334 3 0.04835 13 0.02543 5 0.02240 15 0.02091 7 0.01464 17 0.02123 9 0.02297 19 0.02347 11 0.03304 The measurements and the related calculations to obtain the values of the leakage inductances and winding resistance were performed as Section 3.2.2.2 describes. Table 3.2 provides the data of the resistances whereas Table 3.3 presents the data of the leakage inductances from the calculations based on the physical structure and those based on the measurements respectively. The former table indicates that the ac resistances were all larger than their dc counterparts were. Table 3.2 Comparison of the winding resistances. Parameter Directly Measured DC Value (mfi) Measured and Calculated AC Value (mQ.) Percentage Difference (%) R'lab 186 246 -24.4 R2ab 155 166 -6.68 R2c 35.0 37.5 -6.67 275 363 -24.3 R3ab 630 720 -12.5 Chapter 3 Modeling of the Ferroresonant Transformer 58 Table 3.3 Comparison of the leakage inductances. Parameter Calculated Based on the Physical Structure Measured Percentage Difference L~ 18.6 mH 21.5 mH -13.4 % L l 2 a b 95.5 uH 81.5 uH 17.2% L 1 2 c 146 uH 168 uH -13.0% L , 3 a b 64.6 uH 70.1 uH -7.81% The schematic in Fig. 3.21 shows the complete equivalent circuit including the values of the parameters that formed the basis for the remainder of this work. The magnetizing inductances in the figure are defined as given in (3.43) and (3.44). The inductances of the aforementioned, for the purely linear and purely nonlinear regions respectively, are indicated in the schematic to give a comparative insight to their values. In addition, as the numbers reveal, the core loss of the secondary branch was about twice of that of the primary. 2abcd Fig. 3.21 Complete equivalent circuit of the part of the ferroresonant transformer that served the purpose of this work including component values. The study simulated the circuit in Fig. 3.21 in PSIM®, where the nonlinear elements were implemented using mathematical function blocks. The simulation results were compared with those obtained from experiments. The assessment was performed for the controlled CVT. Additional circuitry was added to the C V T to facilitate this. A series combination of an inductor Chapter 3 Modeling of the Ferroresonant Transformer 59 L and two anti-parallel SCRs was connected between V2at>c and ground whereas a capacitor C was connected between v 2 a bcd and ground. The controlled C V T operated in open-loop control -the firing angle of the SCRs was kept constant at a typical value. For constant rated source voltage, the load resistance, connected between V2ab and ground, was step-changed between 49.9 % and 111 %. These particular values were selected because they corresponded to a ±15 % change of the rated output voltage V2 ab,rated and the rated output current l2 a b , r a ted was obtained at the lower level of this voltage. The plots in Fig. 3.22 and Fig. 3.23 show the response of the normalized input and output voltages and currents respectively to a step-up and step-down change, in that order, of the resistive load. The labels in the figures correspond to those in Fig. 3.21. The arrows indicate the instant in time when the step changes occurred. The study implemented the circuit in PSIM® with a source that had the same harmonic content, up to the first five odd harmonics, as the line voltage to facilitate a true comparison. The plots in both figures below show that the measured variables compared extremely well with those obtained from the computer simulations. The extraordinary results were considered due to the combination of the detailed modeling technique of the transformer and the methods that were derived to extract accurate transformer parameters. Time (s) Fig. 3.22 Step response - the load resistance was step-changed between 49.9 % and 111%. Chapter 3 Modeling of the Ferroresonant Transformer 60 Time (s) Fig. 3.23 Step response - the load resistance was step-changed between 49.9 % and 111 %. 3.5 Summary The chapter showed that the values of the leakage elements and the winding resistances of the C V T could be approximated from the physical structure of the CVT. This knowledge combined with the experimental results that produced the hysteresis loops can generate a comprehensive equivalent circuit of a C V T and perhaps even for a certain family of CVTs. This is an important aid for the designer. Studies based on computer simulations can analyze the behaviour of the system for different control schemes and for several values of the elements L and C external to the transformer. The characteristics of the load may also be varied to observe how this affects the behaviour of the overall system. The designer can also evaluate particular phenomena such as worst-case inrush current and parameter dependent behaviour for the control system design. Since the equivalent circuit described in this chapter relates physically to the CVT, prototypes consisting of the actual individual physical circuit elements can also be built. Chapter 3 Modeling of the Ferroresonant Transformer 61 Such a circuit can also be used for experimental verification of the system before the actual transformer is being built. Chapter 4 State-Space Representation of the Model Accurate modeling followed by computer simulations can aid the prototyping process in that it reduces costly delays that a large number of prototypes or even trial and error attempts bring about. A crucial step in modeling is to implement the circuit in appropriate software. Furthermore, for a detailed mathematical analysis of the model, accurate equations must be available and introduced in computerized form as well. This chapter presents a state-space representation of the equivalent circuit of the controlled ferroresonant transformer including the switched inductance, capacitance, and generic load. Since the mathematical representation describes a typical C V T topology, the proposed equations probably apply to a vast majority of the ferroresonant UPS class. A discussion of a reduced model is included as well. An equivalent inductance of the switched inductor is also addressed. Experimental findings verify and demonstrate the behaviour of the model in accordance to the computer simulations in PSTM®, which is circuit-element based, and the equation-based simulations in Simulink®. 4.1 The Model Including a Generic Load A normal mode of operation of the C V T is in the saturated region. This operation is also called ferroresonance. This is quite different from the operation of a traditional transformer. The expression resonance when used in association with nonlinear resonant circuits is not proposed to signify that a nonlinear inductor and a linear capacitor resonate at co = l/sqrt(LC) when 1 A version of this chapter has been submitted for publication. M. G. J. Lind, W. Xiao, and W. G. Dunford, "State-Space Formulation of a Controlled Constant Voltage Transformer," IEEE Trans, on Circuits & Systems II: Express Briefs. 62 Chapter 4 State-Space Representation of the Model 63 connected to a sinusoidal source of this angular frequency. Linear elements resonate according to this relationship. Nonlinear circuits are called resonant when they react to changes in voltages or currents, of constant frequency, somewhat analogous to how linear circuits respond to change in frequency of the aforementioned variables [60]. Few publications describing comprehensive equivalent circuits of the controlled C V T exist, and [35] is probably one of the earliest and most extensive proposed models of the uncontrolled CVT. This model does not cover tapped windings, hysteresis loops, or winding resistances. A more recent study [50] proposes a detailed equivalent circuit of the CVT. A few other works have presented state-space realizations of magnetic stabilizers. Reference [12] describes an uncontrolled C V T with a piecewise linear secondary magnetizing inductance taking two values. Primary magnetizing elements, winding resistances, and tapped windings are excluded in this model. In [31], primary and secondary magnetizing branches are included, but this model describes only the uncontrolled C V T as well. The focus in [31] is on a computer algorithm to solve for the circuit variables. Adequate simulation software was probably nonexistent at the time of the study. Publications on the SCR-controlled C V T are scarce. The literature review revealed a limited number of references [27], [32], [34], [49] that treat the topic. None of these references, however, includes a state-space representation of the magnetic stabilizer. The former three works also eliminates the nonlinear secondary magnetizing inductance of the C V T from the equivalent circuit. A comprehensive state-space representation as the one derived in this chapter is necessary for the analysis of the compatibility of the controlled C V T with sources such as generators [26] and loads such as PFC power supplies [25]. The sections below describe the structure of the model of the magnetic stabilizer that was derived in the Chapter 3 and three equivalent circuits are used in the description of the device. The first circuit is the one derived in the previous chapter in Fig. 3.21. This model includes several leakage elements and is therefore less suitable for state-space representation due the increased complexity that comes with a large number of energy storage elements. The second and third circuits named the comprehensive and reduced model respectively are presented in state-space format below. Chapter 4 State-Space Representation of the Model 64 4.1.1 Circuit Equations of the Comprehensive Model The schematic in Fig. 4.1 presents the controlled C V T including the generic load. The proposed schematic reveals some obvious approximations. The reference winding in Fig. 3.21 is absent since it is of little interests for the analysis in this chapter. Furthermore, the leakage elements Li2at>, Li2C, and L i 3 a b are eliminated from the previous circuit. The rational for this is that their relatively small values compared to those of the adjoining elements would only marginally affect the operation of the circuit [50]. Thus, fewer energy storage elements, three in this case, greatly simplify the analysis. The elements external to the C V T , the inductor L , the anti-parallel SCRs, and the capacitor C, are also included in this circuit. The resistances R L and Rc represent the winding resistance of L and the equivalent series resistance (ESR) of C respectively. Fig. 4.1 The equivalent circuit of the controlled C V T including a generic load. The continuous-time state space equation that describes the nonlinear model can be represented by x(t)= Ax(t)+Ew(x(t))+fv(t)+bu(t) (4.1) that corresponds to, term for term, in expanded form Chapter 4 State-Space Representation of the Model 65 *U»2(0 A L ( t ) _vc(0 . l21 l31 l 41 l12 L22 l 32 l 42 0 0 " 0 a 24 0 a 3 4 MO 0 a 4 4 . . v c ( t ) + 0 0 0 0 0 '22 '32 '42 '23 '33 '43 w,(XL.m l(t)) (*u*(t)) ,(M0) w Linear Part Nonlinear Part + "0" V f 2 i 2 a b ( t ) + 0 f 3 0 . f 4 . 0 (4.2) lab (.) Load Dependent Part Input Part where ( A , m l ( t ) ) W 3 ( A L ( t ) ) W w l L ' m l l L m 2 (KM) i L ( A L ( t ) ) Y , *L™(t)+Y„A . °» i ( t ) ^ l A ^ (37i-a2)/a> J v L ( t ) d t + J v L ( t ) d t a,/o) (7t+a2)/d) ((2t-a, )/o> 71 < a , , a 2 <7t (4.3) Four distinct terms constitute the right hand side of the relationship in (4.2). The linear and nonlinear parts intrinsic to the system, the forcing function, and the load dependent part. The latter is dependent on the load current, which in general is unknown due to the unknown load. Four state variables describe the system. These are, in row order, the primary magnetizing flux linkage referred to the N2ab side, the secondary magnetizing flux linkage, the flux linkage of the switched inductor, and the capacitor voltage. Each vector element of the nonlinear part consists of a function of flux linkage and (4.3) further defines these functions. The first two entries in this column vector correspond to (3.12) and (3.13) that express the values of iL-mi(t) and iLm2(t) respectively. The last entry describes the flux linkage of the switched inductance where cii and 012 symbolize the firing angles of the first and second half cycles of the applied voltage across the anti-parallel combination of the SCRs, in that order. The parameter co denotes the angular frequency. The voltage drop across the SCRs is ignored in (4.3). Although the firing angles may be considered as inputs, they are not represented Chapter 4 State-Space Representation of the Model 66 in the input part in the equation above. The prime signs in (4.2) and (4.3) indicate a referred quantity. Extensive derivations of the state variables according to the notations of the circuit in Fig. 4.1 produced the individual matrix and vector elements of A , E , f, and b. To simplify the expressions for the subsequent matrix and vector elements, a few auxiliary constants were defined as k, - NL(R 2 a b + R 2 c R 2 d + R c 2)+ 2 N 2 a b ( N 2 c + N 2 d ) R c 2 + ( N 2 c + N 2 d ) 2 R c 2 (4.4) k 2 = N 2 a b [N 2 o b R 2 d + N 2 c R 2 d - N 2 d (R 2 a b + R 2 c )]R c 2 (4.5) k 3 = R 2 a b + R 2 c + R 2 d + R c 2 . (4.6) The following equations where the parameters are related to the circuit in Fig. 4.1 demonstrate the results of the derivations and the values of ay, bj, e^ , and fj. The elements of A amount to a = _ a 1 2 = - J - R ' ' ° b R ' " (4.7) = 7 ^ - N 2 a b R c 2 ( R 2 a o + R 2 c + R 2 d ) a 2 i =~a22 T - ^ 2 a b R c 2 ( R 2 a b + c + R 2 ) (4.8) L,k , *24 = 7 - N 2 a b R c 2 ( N 2 a b + N 2 c + N 2 d ) k, (4.9) 1 , a 31 = ^ 3 2 = T - r - k 2 (4.10) L . k i 34 = 7 - T - N 2 a b R c 2 L 2 N 2 c R 2 d - N 2 d ( R 2 a b + R 2 c - R 2 d +R c 2 ) ] k , k 3 + r T r ( N ^ + N 2 d ) R c 2 [ N 2 c R 2 d - N 2 d ( R 2 a b + R 2 c +R e 2 )]—-2-. + 1 (4.11) = 7 4 r - N 2 a b R c 2 ( N 2 a b + N 2 c + N 2 d ) (4.12) a 4 4 = - ^ - N 2 a b . (4.13) The vector b contains only one nonzero element, Chapter 4 State-Space Representation of the Model 67 " • - u c s r - ( 4 1 4 > lab ' A V cl The elements of E for the nonlinear part of the equation were derived as " " ' T ^ - ( 4 1 5 ) K l a b + K cl e 2 2 = - T ^ N 2 a b R c 2 ( R 2 a b + R 2 c + R 2 d ) (4.16) e 2 3 = - 7 ^ N 2 a b R c 2 [ N 2 a b R 2 d + N 2 c R 2 d - N 2 a b ( R 2 a b +R 2 c ) ] (4.17) k 2 en = —r- (4.18) K i (4.19) e 3 3 = - R L + 7 ^ [ N 2 a b R 2 d + N 2 c R 2 d - N 2 d ( R 2 a b + R 2 c ) N 2 d R c 2 ] k, + 7 M N 2 a b R 2 d + N 2 a b N 2 d R c 2 + (N 2 c + N 2 d ) N 2 d R c 2 - k J ( R 2 d + R C ) ^ - ( N 2 a b + N 2 c + N 2 d ) N 2 a b R c 2 (4.20) 6 4 2 " Ck, 1 e 4 3 = " Ck, [ N 2 a b ( R 2 a b + R 2 c + R c 2 ) + N 2 a b ( 2 N 2 c + N 2 d ) R c 2 + N 2 c ( N 2 c + N 2 d ) R di- (4.21) Finally, the elements of f were derived as f 2 = - ^ - N 2 a b R c 2 [ N 2 a b ( R 2 c + R 2 d ) - ( N 2 c + N 2 d ) R 2 a b ] (4.22) k, U = 7 L [ N 2 a b ( R 2 c + R 2 d ) " ( N 2 c + N 2 d ) R 2 a b ] N 2 d R c (4.23) k «- 4.au \ t \ m / a^u J 4 U 1"c2 i + • k - L N L ( R 2 c + R 2 d ) + N 2 a b ( N 2 c + N 2 d ) R c 2 + ( N 2 c + N 2 d ) 2 R c 2 - k 1 ] ( R 2 d + R c ) f 4 = - ^ [ N L ( R 2 a b + R c 2 ) + ( N 2 c + N 2 d ) R c 2 ] . (4.24) Chapter 4 State-Space Representation of the Model 68 The equations above indicate that the acquirement of the elements of the matrices and vectors were quite involved. This was due to the particular configuration of the secondary side of the transformer and its tapped winding. The output equation of the system comes in the form of y(t) = Cx(t)+Gw(x(t))+hv(t)+du(t) (4.25) that in expanded form, term for term, corresponds to ' 2ab lab (t) W "21 '12 -22 0 0 Linear Part + 0 g 2 l § 1 2 0 § 1 3 0 *L'ml(0 M2(0 MO . v c ( t ) Wi(A L . m l ( t)) W 2 (A L m 2 ( t)) w3(M0) v \ " 0" + l 2 a b U 0+ 0 . d 2 . lab Nonlinear Part Load Dependent Part Input Part Analogue to the state space equation, the output equation contains four parts as well. The input and output current variables were defined as i'lab W i2ab(0 R ' l a b + R ' c l (4.26) (4.27) V2ab(0 Load(t) where Load(t) represents an unknown quantity. It is not indicated in (4.27), but for computer simulations of the system, in the discrete time domain, i2at>(t) would lag one time step behind that of the other circuit variables and in particular that of v2ab(t). Thus, the system was broken up into two parts, the C V T and the load. The output voltage of the C V T v 2 ab(0 was defined as the independent variable and the output current i2ab(t) constituted the dependent variable. The derivations of the state variables of the circuit produced the individual elements of C, G, h, and d for the output equation. The following equations where the parameters are related to the circuit in Fig. 4.1 demonstrate the results of the derivations and the values of Cj, di, gj, and hi. Chapter 4 State-Space Representation of the Model 69 '12 = T \ ~ tN 2ab (2R 2 a b + R 2 c + R 2 c ) + ( N 2 c + N 2 d )R 2 a b ] R c 2 N L , k 2ab • . 4 = - T L [ N 2 a b ( R 2 a b - R 2 c ) - ( N 2 c + N 2 d ) R c 2 ] N k, 2ab C21 — C22 d 2 = 1 1 L l R ' lab+R ' c l R ' lab+R ' c . g . 2 = - 7 ^ [ N 2 a b ( 2 R 2 a b + R 2 c + R 2 d ) + ( N 2 c + N 2 d ) R 2 a b R c 2 N 2 a b ] K l g.3 =T7[N2ab(R2ab ^ ^ d - N 2 a b [ N 2 c R 2 d " N 2 d ( 2 R 2 a b + R j ] R + ( N 2 c + N 2 d ) N 2dR2abR C 2 J R'„, g 21 D -R ' lab+R ' c l N 2 a b ( R 2 a b - R c 2 ) ( R 2 c + R 2 d ) + 2 N 2 a b ( N 2 c + N 2 d ) R 2 a b R 2 c + ( N 2 c + N 2 d ) 2 R 2 a b R c2 (4.28) (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) 4.1.2 Circuit Equations of the Reduced Model Since the model in the previous section resulted in expressions that analytically are difficult to investigate, a reduced circuit was developed as well. This simplified model is shown in Fig. 4.2, and it approximates the circuit in Fig. 4.1. The former circuit has in the past, with the exclusion of the SCR-controlled L ' and continuously nonlinear 1^2, been the basis for analyses of the uncontrolled C V T , examples of which are [6], [10], [18], [32]. These references have documented various behaviours of the uncontrolled CVT. Only the elements that significantly were deemed to contribute to the operation of the circuit are present in this schematic. The primary magnetizing branch that undoubtedly affects the input section of the circuit, and specifically the input current, is eliminated. The resistances are also absent in the model. Since all the series resistances are very small and the parallel Chapter 4 State-Space Representation of the Model 70 resistances are relatively large, this approximation may be justifiable. However, the resistors definitely provide some damping in the circuit, however small, that now except for the load and the SCRs only consists of energy storage elements. Fig. 4.2 The equivalent reduced circuit of the controlled C V T including a generic load. When the model is reduced and the resistors and the primary magnetizing inductance are excluded, the following relationship can model the system x(t) = Ax(t) + Ew(x(t)) + fv(t) + bu(t). (4.36) In expanded form this corresponds to, term for term, " i L 1 ( t ) " 0 0 0 - 1 / L , " " iu(0 " MW 0 0 0 1 *Lm2vO MO 0 0 0 1 MO _v(0. _ l / C * 0 0 0 .v(0. + 0 0 0 - l / C Linear Part 0 0 0 - l /C* w .Mo) 0 "1/L." 0 / \ 0 + 0 l2ab(0 + 0 - l / C 0 lab Nonlinear Part Load Dependent Part Input Part (4.37) where L and C are referred to the N2ab side as indicated by the asterisk notation. It should be noted that these referrals are approximations. The number of state variables remains the same compared to the non-reduced system. However, they are not all identical. They are, in row order, the current though the leakage inductance related to the air gaps of the shunts, the secondary Chapter 4 State-Space Representation of the Model 71 magnetizing flux linkage, the flux linkage of the switched inductor, and the capacitor voltage. Similar to the derivation of the comprehensive circuit, the elements of the column vector of the nonlinear part are described in more detail as i f _w,(X L . ( t ) )_ . v-M<)). -a,)/a> (37i-a2)/(0 J v c . ( t ) d t + J v c . ( t ) d t i,/o) (7i+a 2)/(o <a 1 ;a 2 <n X L . ( t ) (4.38) where the voltage drop across the SCRs is ignored. The referred parameters from the circuit in Fig. 4.1, the inductance and capacitance, to the N 2 ab side of the circuit in Fig. 4.2 are approximately defined as L = L N 2ab V N 2 a b + N 2 c y and C * = C ' N 2 a b + N , „ + N , ' 2 c 1 x , 2 d N 2ab respectively. The output equation for the reduced model is defined as y(t) = Cx(t) and in corresponding expanded form V2ab(0' i ' l ab W. 0 0 0 1 1 0 0 0 Linear Part ' iu(0 " MO vr.(0 which merely amounts to substitution of variables. The output current is computed as V2ab(0 i 2 a b W " L o a d ( t ) -(4.39) (4.40) (4.41) (4.42) (4.43) Chapter 4 State-Space Representation of the Model 72 As the equations in this section reveal, the constant elements of the matrices and vectors in the state-space and output equations are much simpler than those derived for the comprehensive system in the previous section. Furthermore, most entries in the matrices and vectors equal zero. If the reduced model system would approximate the static and dynamic behaviour of the physical system, much would be gained in the analysis of the behaviour of the transformer model. Although necessary for a stability analysis, the state-space representation does not provide for an intuitive understanding of the circuit behaviour. However, a phasor diagram can afford such an insight. The regulatory behaviour of Lm2 with the exclusion of L * of the reduced circuit in Fig. 4.2 is quantitatively presented in the frequency domain by the phasor diagram and magnitude plot in Fig. 4.3. For the primary side, a C V T is typically designed with leading phase angle at minimum input voltage, that is, Vi a b ;min lags Iiab.min and with the power factor approaching, but not reaching, unity as the input voltage increases toward its maximum value [61], [62]. 1 L r a 2 , max t (a) (b) Fig. 4.3 Quantitative analysis of the equivalent reduced circuit of the C V T including a resistive load (a) phasor diagram and (b) the characteristic of XLm2-The phasor diagram in Fig. 4.3 (a) show the relationships among of the circuit phasors at the minimum and maximum values of Vi a b denoted "min" and "max" respectively. The characteristic of XLm2 in Fig. 4.3 (b) indicate that with purely resistive load and for a small Chapter 4 State-Space Representation of the Model 73 increase in V2ab, that probably was caused by a small increase in Vi a b, there is a large increase in l L m 2 - This corresponds to significant saturation in the time domain. A large current through X L m 2 or equivalently a small value of X L m 2 results in a magnitude and phase variation in the current through X T J . This consequently affects the voltage drop across X L i and the output voltage is accordingly reduced. The slope of V 2 a b versus l L m 2 determines the effectiveness of the regulation. The smaller the slope, the better is the regulation. 4.2 Equivalent Switched Inductance To gain further insight, the switched inductance L may be expressed in terms of n continuous inductances connected in parallel that produces the equivalent rms current through the combination. For the SCR-controlled inductor L, the plot in Fig. 4.4 shows the shape of the voltage and current variables for the firing angles oti and a2. The aforementioned are relative to the zero crossings of the applied voltage. 1 ' \ s ' \ X 1 1 a 2 / i A f \ i i 1 / 3 \ / ' / f V L 90 180 270 360 450 tot (deg) 540 Fig. 4.4 Voltage and current waveforms across and through L. For an arbitrary phase angle 0, the fundamental component of the voltage Vi , L across a linear inductor produces a fundamental current such that Chapter 4 State-Space Representation of the Model 74 I 1 L Z 0 - 9 O ° = v . ,L^e° X L Z90° ' (4.44) The separation of the SCR-switched inductor into rms-equivalent inductances produces an inductance for each harmonic component. For an applied voltage free of harmonic content, a nonlinear inductance produces current harmonics. This can be described for uncorrected waveforms as i = Y i 2 = rms \\£.u n.rms n=0 ( n=0 V„ Y n, L V . X (4.45) eq where X n ,L is the reactance that produces the n t h harmonic component of the current and X e q = coiL is the rms-equivalent inductance. For the particular application coi = 60 Hz. The one-sided spectrum of the voltage across the switched inductor vL(t) relative to cosine is V,L(0 = | c 0 , n = 0 |2|c n|cos(ntot-r-Zc n), n>0 (4.46) where co is the angular frequency and the Fourier Series coefficients are c . = ^ - J v L ( t ) ^ , d t (4.47) L l T, with Ti and coi being the period and angular frequency of the fundamental respectively. Relating the components of (4.46) to sine and combining with (4.45) gives after some derivations, the equivalent continuous inductance L e q in terms of the Fourier Series coefficients such that | n=3,odds 2c. V2nV, (4.48) rms J where L is the physical SCR-switched inductance. The Fourier Series coefficients for the waveform of the voltage in Fig. 4.4 are Chapter 4 State-Space Representation of the Model 75 ( 71-0 c -L , max 271 - a , 27t-a, 2TI Jsin(cot) e" j nMl t dcot + Jsin(a)t) e^"10'' dtot + Jsin(tot) e" j n t0 l t dcot (4.49) where VL,maxSin(cot) is the applied voltage to the branch of the switched inductor. Combining the components of (4.49) with (4.48) gives after some tedious derivations, the equivalent continuous inductance L e q in terms of the firing angle a = ai = a 2 and the harmonic component n such that Leq(a,n) = i ( s in(2a) -2a + 27i) L 7t , n = l .2J n7i ' sin [(n + l) a] sin[(n-l)a]N ^ n+l n -1 j , n = 3,5,7,. (4.50) The rms summation of all the harmonic components gives the equivalent continuous inductance L e q in terms of the firing angle such that L (4.51) L(a): -eq ' 1 n 2a + 2Tt) 2 + z - - ( n=3, odds n + l n - 1 where n represents the cumulative number of the frequency components. The derivation above is applicable for the full cycle of the fundamental and rms equivalence. This means that balanced or equal firing angle for both SCRs is assumed. Thus, only the odd orders are represented in the equations above. Each normalized harmonic component of the current expressed in rms and as a function of a and n can now be calculated in terms of the equivalent inductance. This leads to Innrm „ (Ct,n) = -norm, rms \ ' / norm, rms CO Leq(a,n) (4.52) 1, norm where coi>n0rm is the normalized angular frequency of the fundamental waveform and V norm,rms IS the applied voltage across the combination of the SCRs and the inductor. The normalized inductance can be expressed as Ln0rm(a,n) = L/L e q (a,n) where 0 < Lnormfa.n) < 1. Chapter 4 State-Space Representation of the Model 76 The conduction angle, the interval in time, during which either SCR is turned on is related to the firing angle as a = n — , 0 < a<7i. 2 (4.53) Thus, an increase in the firing angle leads to two major phenomena. First, the dissipation of real power in the inductor decreases. Secondly, the waveform of the current through the inductance becomes more distorted, that is, the ratio of the harmonic content to that of the fundamental increases. Since the expression in (4.51) is quite complex to integrate into the state space equations, the study performed a polynomial fit of L e q . This curve fitting technique is based on the multi-term polynomial fit such that y = xp + s (4.54) " y i " " i y 2 i V + Pn. _ym_ I _ £ m_ where the objective is to minimize the length of e. The notations, y, X , p, and £, represent the observation vector, the design matrix, the parameter vector, and the residual vector respectively [58]. In expanded form, using only two polynomial terms, (4.54) becomes (4.55) As previously described in Chapter 3, the least-squares solution for the over-determined system is 0 = (x*x)_ ,X*y. (4.56) A measure of deviation, the root mean square of the Euclidean norm of the residual vector, is HL |y-xPlL Vm Vm (4.57) where m equals the number of samples. The general form of the fitted curve for the equivalent inductance using two polynomial terms is Chapter 4 State-Space Representation of the Model 77 = p \+P>, n - (4.58) L A circuit model based on L e q would use this equation for the third state in (4.2) and (4.3) or (4.37) and (4.38) for the complete and reduced model respectively where ii/t) would replace A,L(t) as the state variable. The plots in Fig. 4.5 show the results of the curve fitting. To maximize the performance of the fit, a typical firing range was assumed to be in the interval 100° < a; < 160°. The first 999 terms of the Fourier Series were used to plot L e q / L . An increase in the number of terms beyond this value did virtually not change L e q / L . As the plots indicate, the fitted two-term polynomial curve is in good agreement with the true curve. The smallest rms error of ||e||2 occurred for n = 16. The figure also shows the equivalent inductance using only the fundamental component. This curve adequately approximates L e q / L up to the medium range of the firing angle. The equation of the best fit in Fig. 4.5 is with ||£Jj2/sqrt(m) = 0.5031. The plots in Fig. 4.6 display the individual first nine odd harmonics of the inductor current versus the firing angle for the switched inductor. The waveforms were normalized to that of the fundamental at a = 90° and steady-state firing angle was assumed. It is notable that the maximums of each harmonic do not occur at the same firing angle. The third harmonic has a strong presence. Calculations revealed that it peaks with 0.138 (A/A) at 120°. What is more noteworthy is that for a steady-state firing angle of 120°, the fundamental is 0.391 (A/A). Thus, the ratio of the third harmonic to the fundament is substantial at this particular value of a. The traces in Fig. 4.7 present the rms values of three different currents versus the firing angle. These currents are the fundamental, all components but the fundamental, and all of the first 999 odd components. The plots show that although the harmonic content is small in absolute values, it increases in relative values with increasing firing angle. For small and up to medium firing angles, the ratios of lL,i,rms/a and lL,rms,aii/a are approximately linear relationships. For a low harmonic content, L should be sized so that a is small in steady state. This also promotes an = 2.064 + 2.224 -10"6 a 16 (4.59) L Chapter 4 State-Space Representation of the Model 78 approximately linear relationship between the firing angle and the reactance of the controlled inductance. - - 1 s t 1s tthrough 999 t h in// - - - Pol. fit - ; /» i 4 J T 7 if L tl - JA r ,f if i 7 A _ , r " ° 9 0 100 110 120 130 140 150 160 a (deg) Fig. 4.5 Plots of the normalized equivalent inductance and the polynomial fit. a (deg) Fig. 4.6 Plots of the normalized harmonic currents through the equivalent inductances. Chapter 4 State-Space Representation of the Model 7 9 4.3 Some Notes on Simulation Models This section discusses some remarks on the simulation models that were developed. Although many basic versions of circuit-based simulation packages such as PSpice® and PSIM® do not feature nonlinear magnetizing inductances, such elements can be created from the standard mathematical blocks that usually are available. This work implemented the nonlinear inductor model in Fig. 4.8 based on (3.12). This model contains an integrator, two gain blocks, a power function, a summer, and a voltage controlled current source of unity gain. The control side of the latter could also have been shorted directly to ground, thus eliminating this block. However, most simulation packages would not allow such a connection unless there is a small resistance in the path. The schematic also indicates the variables pertaining to the inductor. Chapter 4 State-Space Representation of the Model 80 v_L lambda_L i_L Fig. 4.8 An example of implementation of a nonlinear inductance that was used for circuit-based software analysis. Standard versions of equation-based software packages may generally not contain circuit elements. This study implemented the model in Fig. 4.9 that represents two anti-parallel SCRs switching an inductor. The labels v L_ i n, fire SCRs, i L , and VSCRS symbolize the applied voltage across the series combination of L and the SCRs, the firing command, the consequential current, and the voltage across the SCRs respectively. (reverse current o f f level) SCRs Fig. 4.9 Simulink® model of two anti-parallel SCRs. The simulation time for this model was quite low, an intentional design. The block labelled "SCRs allowed to turn on" controls the minimum forward biased voltage at which either SCR is Chapter 4 State-Space Representation of the Model 81 allowed to turn on. The model also indirectly features the carrier recovery time at turn-off in terms of the reverse current level. This block is labelled "turn off SCRs" in the figure. The model does not consider the forward biased voltage drop since it would have only very little influence on the operation of the circuit. 4.4 Experimental Verification Based on the derivations of the comprehensive and reduced equivalent circuits of the loaded ferroresonant transformer, the study performed computer simulations to compare the behaviour of the mathematical model to that of the physical device. The variables and their respective suffixes A , B, and C of the plot labels in the figures below, correspond to the circuit configurations in Fig. 3.21, Fig. 4.1, and Fig. 4.2 respectively. The subscript "meas." attached to the variables in the plots indicates experimental results that were obtained from a C V T that was designed accordingly to the structure in Fig. 3.4. Both the comprehensive and reduced models were implemented in PSIM® and Simulink®. The plots in Fig. 4.10 and Fig. 4.11 of the normalized voltage and current variables show the results of the simulations. The source was implemented with the same harmonic content, up to the first five odd harmonics, as the line voltage, at 50 % resistive load, to facilitate a true comparison. The actual line voltage dropped a little bit from this value when the load was 101 %, which was the load used in this study. The respective traces of the variables regarding the measured values and the values obtained from the simulations of the circuit configurations A and B indicate very good agreement. These respective traces in the plots essentially overlap. The small disagreement between the traces corresponding to the circuit configurations A and B justifies the utilization of the latter for the analysis in the subsequent chapters. The study simulated the circuit in open-loop control with a resistive load of 101 %. Although this value of the load provided for the largest damping in the system, the results obtained from the simulations of the reduced PSIM® model were quite discouraging. Especially noteworthy is the plot of i2ab(t) in Fig. 4.11. The negative peak of this current is about 275 % larger than the true value obtained experimentally. For smaller values of the load, the same Chapter 4 State-Space Representation of the Model 82 results deteriorated substantially. However, the result for the reduced model is not surprising since the source did not consist of a single-frequency cosine signal that would have provided for flux balance in the magnetizing inductance. Moreover, the reduced model did not include internal damping elements. A conclusion from the simulations is that in case the source voltage suddenly changes its phase, shape, or magnitude, the reduced model inadequately models these phenomena. Apart from the results of the reduced PSIM® model, the plots of the computer simulations compared extremely well with those obtained experimentally. 1 > 0.5 CD CD 1 o > E -0.5 o •1 \ r v. . larjmeas. V, u „ . A 1ab Psim v „ . C 1ab Psim & V 1abSimu.^ i 0.41 0.42 0.43 0.44 0.45 I CD CJ) CO o > E o 1 0.5 0 -0.5 -1 J L r ><fi—x I ^ i.l/-'_>_ A 77 ; A if ' V V 2ab meas. V A 2ab Psim ^ V C 2ab Psim v B 2ab Simu. i 1 / vV 1 / \\ ,/ A T V V-_ \ \ . . . . / . : V v / 1 \ , _ _ _ _ ^ r r : J _ _ ; _ _ _ _ _ _ A - T 0.4 0.41 0.42 0.43 Time (s) 0.44 0.45 Fig. 4.10 Plots of input and output voltage variables. The device operated in open-loop control with a resistive load of 101 %. Chapter 4 State-Space Representation of the Model 83 I •i—• c o I t_ O E o ^ - ; ^ - - \ V - - i \ \ 2ab meas. j / \ 2ab Psim j Q 2ab Psim j g 2ab Simu. i \ / 0.41 0.42 0.43 Time (s) 0.44 0.45 Fig. 4.11 Plots of input and output current variables. The device operated in open-loop control with a resistive load of 101 %. The plots in Fig. 4.12 show the step response of the normalized output variables for the Simulink® simulations of the comprehensive and reduced circuits respectively. The load was stepped down from 101% to 0%. As the respective traces in the figure indicate, the reduced model in Fig. 4.2 was in very good agreement with the complete model in Fig. 4.1. Perhaps surprisingly, the C V T was able to maintain a quite acceptable magnitude of the output voltage despite the drastic change in the load. The plot shows that the overshoot of the voltage was about 135 %. A 60 Hz cosine signal sourced the circuits, a requirement for adequate performance of the reduced model. Chapter 4 State-Space Representation of the Model 84 Fig. 4.12 Plots of output voltage and current variables when the resistive load was step changed from 101 % to 0 %. With constant rated source voltage, without harmonic content, the device operated in open-loop control. The traces in Fig. 4.13 show the response of the normalized output variables for the Simulink® simulations of the circuit configuration in Fig. 4.1, with and without the equivalent inductance. The source voltage was identical to that of Fig. 4.10. As the plots indicate, L e q quite well modeled the true inductance of the SCR-switched L. This agreement would be sufficient to validate the usage of L e q for the simplification process of the state-space equations - especially for steady-state operation. The open-loop control was set to produce the same rated output rms voltage and current for the two cases. The traces corresponding to L also reveals one of the Chapter 4 State-Space Representation of the Model 85 inherent disadvantages of the controlled CVT. The output variables have a harmonic content due to the switched inductance. 1.91 | c CD k_ i _ O E o 1 0.5 0 -0.5 -1 1.91 1.915 1.92 1.925 1.93 1.935 1.94 1.945 L h SCRs-L B 2ab - - - - 'o K L B 2ab eq K // ft \ \ i v \ 1 J . If r x V if 1 1 1.915 1.92 1.925 1.93 Time (s) 1.935 1.94 1.945 Fig. 4.13 Plots of output voltage and current variables for the SCR-switched L and the continuous L e q . With constant rated source voltage, including harmonic content, the device operated in open-loop control with a resistive load of 101 %. 4.5 Summary This chapter derived two state-space representations of a typical magnetic stabilizer. The reduced model was deemed for elementary purposes suitable for computer simulations. However, the comprehensive model is required when it comes to analyses regarding the primary current including inrush of the same and non-ideal sources or sudden changes of the Chapter 4 State-Space Representation of the Model 86 characteristics of the same. The reduced model is less reliable at very small loads due to the absence of damping in the circuit. The comprehensive model is a valuable tool in the analyses of certain sources such as generators and particular loads such as PFC switchmode PSs; in particular, PSs that feature incremental negative input resistance. This is the subject of the next two chapters. The state-space equations presented in this chapter make the Simulink® implementation of the circuit a straightforward task. The study also compared simulations using Simulink® and PSIM®. Chapter 5 Power Factor Corrected Loads 'This chapter presents a condensed introduction and generalized analysis of a typical actively PFC load. Specifically, a generic model of the input admittance of a possible but deemed typical PFC PS is introduced. Since numerous topologies can be employed for PFC devices each having combinations of several different control schemes and input filters, a single model covering a large portion of PFC PSs is unlikely to exist. The sections below also contain notes on measurement techniques regarding the extraction of the input admittance and experimental results obtained from of a particular PFC PS are also discussed. 5.1 Typical Actively Power Factor Corrected Loads The boost converter is the classical converter topology used for actively PFC devices. Since it is normally desired to maintain power factor correction continuously throughout each individual cycle of the sinusoidal input voltage of a device, a converter that increases the voltage level is of interest. Consequently, this also implies that the level of the output voltage of the boost stage must be higher than the maximum of that of the input side. Other topologies such as the Cuk and single-ended primary inductance converter (SEPIC) may be used as well. They have inherent over-current and short-circuit protection. These converters also feature both step-down and step-up capabilities [63]. Various types of electromagnetic interference (EMI) filters that are A version of this chapter has been submitted for publication. M. G. J. Lind, G. A. Dumont, and W. G. Dunford, "Stability Analysis of a Ferroresonant Transformer with Generalized Loads," IEEE Trans, on Industrial Electronics. 87 Chapter 5 Power Factor Corrected Loads 88 implemented in particular to attenuate high frequency components may also precede the converter that performs the power factor correction [63], [64]. In the mid 1980s, power factor correction for a device implied a sinusoidal input current that is in phase with the input voltage where the current features a very low harmonic distortion. For lower power levels, say below 300 W, passive filters is an inexpensive alternative when regulations permit. For active power factor correction, there exist three popular topologies. A typical topology for this application is a control system with two control loops - the inner loop that shapes the input current to be sinusoidal and the outer loop that keeps the output voltage constant. The dynamics of the output voltage is thus inherently slow. Some single-stage converters inherently emulate resistive behaviour in discontinuous conduction mode (DCM). This solution is inexpensive, but since the current level for a given power level is higher than for a continuous conduction mode (CCM) converter, the trade off is lower efficiency. The D C M converter type is used for lower power levels as well. A third category is a cascade combination of a boost and buck converter. The first stage can be designed for optimum power factor correction whereas the dynamics of the second converter can be made rapid. Galvanic isolation is also possible with this configuration. Since this implementation is rather expensive, it typically applies to power levels above 300 W [63], [65], [66]. The schematic in Fig. 5.1 shows an elementary circuit consisting of a source, a full wave rectifier, and a constant power load (CPL). This combination emulates a rudimentary PFC converter [67]. Fig. 5.1 A rudimentary constant power load circuit. We assume here that vs(t) is a sinusoid of single frequency and that is(t) is free of distortion. The latter is untrue for a full-wave diode bridge, but in terms of real average power, Chapter 5 Power Factor Corrected Loads 89 the harmonics of is(t) are not a contributing factor. The steady-state fundamentals of the source variables of the circuit in Fig. 5.1 are defined as v:10 = V^ x s in(co t ) (5.1) i: s(t) = Iss:maxsin(cot + (p) (5.2) where the phase angle, tp = 0, for id > 0, or for C C M . The input variables can be divided into steady state and deviations thereof such that v i(t) = vr(t) + Av,(t) (5.3) i s(tHr(t) + Ai s ( t) (5.4) where the deviated variables are sinusoids analog to those in (5.1) and (5.2). The corresponding average power for the period is P = ^ rjvi(t)ii(t)dt T T (5-5) V I s.max s.max where it is assumed that the average power due to the deviations equals zero in steady state over a period. For average constant power consumed by the load the expression can be written as P =P S S +AP V i (vs +AV )(r +AI ) ( 5 -°) s,max s,max \ s.max s,max/\ s.max s.max / 2 ~ 2 and a first order approximation in terms of perturbations and average power leads to V s s A l +AV Vs s, max s,max s, max s, max Q ^ _ _ _ That in terms of the load resistance can be written as V s s A V ^ = L = _ ^ = « = > A R R ( 5 8 ) I Al s, max s, max Chapter 5 Power Factor Corrected Loads 90 Thus, the small signal model features negative incremental input resistance. This is consistent with the corresponding dc case where the aforementioned is described as A R d c dV, dc dl d c dl d c _ d c V J d c J - P d c - r = - T ^ = - R c i c - (5-9) * d c A d c The derivation above excludes the dynamical behaviour of the PFC device. Although the dynamical response often is topology and in particular control system specific, there exist some common characteristics. Based on previous research work regarding specific topologies [25], [63], [64], [66], it can be concluded that a generalized second order model describing the input admittance of a PFC converter may be written as where the numerator contains one left-half plane (LHP) and one right-half plane (RHP) zero and the denominator consists of a L H P complex pole pair. The constant parameter R describes the real power dissipation whereas C, and co represent the damping factor and natural angular frequency respectively. The magnitude and phase plots in Fig. 5.2 depict the frequency response of Yin(s) for four different values of £ and co = 188 rad/s. Chapter 5 Power Factor Corrected Loads 91 5.2 Measurements of the Input Impedance To exemplify that the input current of a particular actively PFC converter may not exhibit a desirable behaviour, the input variables of the aforementioned was measured when it was sourced from the stiff power grid. It was hereby assumed that the output impedance of the source could be neglected. The plots in Fig. 5.3 show the input voltage and current when the PFC PS was 80.9 % loaded. An FFT analysis reveals that the current leads the voltage by 13.7° with respect to the fundamental. It is also noticeable that the rate of change of the current about the zero crossings of the voltage is approximately constant. Reference [68] investigated a similar behaviour. The phenomenon was there classified as being caused by limited control bandwidth of the current loop. However, the objective here is not to analyze a specific behaviour of PFC converters. —. 200 CD CD 0 as o > -200 t— • • . . . . I I I 1 1 1 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 10 7 r 7 — ^ 1 5 7 / 1 \ 1 0 1 1\_ ---T---;---W--10 ; ; \s* ""'I ! ! ! -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 Time (s) Fig. 5.3 Input voltage and current to the PFC PS when sourced from the power grid. The load was 80.9 % of that of the rated. To measure the envelope input impedance of the PFC load, a power amplifier, whose output voltage was modulated, sourced the PFC PS with the following signal Chapter 5 Power Factor Corrected Loads 92 v t a(t) = (V c + V m cos(comt))Ac cos(coct) (5.11) where the subscripts c and m denotes carrier and modulating variables respectively and A c is the gain constant of the carrier. The voltage and current at the input of the PFC PS were measured and spectrum analyzed, using the Hanning window for good frequency resolution, to obtain the input admittance of the envelope at each frequency jco as However, the modulated signal should ideally be multiplied into the actual UPS with its load configuration, without affecting the same, and as such, the procedure is hardware limited. The plots in Fig. 5.4 show the measured and curve fitted magnitude and phase plots of the input admittance of the commercial PFC PS when it was sourced from the signal generator. The plots show the normalized magnitude only since the shape of the measured magnitude and phase plots were approximately independent of the value of ijn(t) for the following normalized values {0.263,0.513,0.813, 1.01} A / A . Fig. 5.4 Magnitude and phase plot of the input admittance of the PFC PS. Chapter 5 Power Factor Corrected Loads 93 The experimental procedure to obtain the data points in Fig. 5.4 was conducted as follows. The time window for the Fourier Series analysis was set to one second to accommodate low frequencies as well as to average cycle-to-cycle variations of the carrier. The input variables were sampled at 50 kHz for superior frequency resolution. The amplitudes and frequencies of the carrier and modulation signal were kept constant during the time window of each data recording procedure. It should be pointed out that the forcing function in (5.11) produces a carrier and two sidebands. The amplitude of V C A C was kept at the rated value of the input voltage of the PFC PS whereas the amplitude of the modulation was set to V m A c = 0.1V CA C . During the experiments, it was observed that for the frequency of the carrier and for each frequency of the modulation signal only the respective currents corresponding to these frequencies featured significant magnitude. This is an important aspect, which showed that the system operated in the linear region about a steady-state operating point. Furthermore, each data point in Fig. 5.4 was produced individually, following a separate experiment, using a software routine written in Lab view. Due to hardware limitations, the data points for frequencies below 30 Hz are less reliable, and data points for frequencies below 20 Hz are not shown in the figure. As frequently is typical for the particular linearization process, the non-linearity of the measured input admittance was too severe to allow for a fit of the entire spectrum. The part of the spectrum, below 60 Hz, that is of interest for our study was curve fitted according to (5.10) with C, = 0.195 and co = 188 rad/s. It is known that in the aforementioned frequency range, an actively PFC PS may feature negative incremental input impedance, which in turn can cause instability problems for the ferroresonant UPS [25], [67]. Chapter 6 Stability Analysis of the System 'This chapter presents a stability analysis of a controlled magnetic stabilizer where the ferroresonant transformer constitutes the major part. The investigation encompasses stability evaluation of the device with several types of loads using a generalized approach. The nonlinear system is converted to a linearized time-varying system, which in turn is sampled to provide a discrete-time representation. It has previously been reported that certain loads such as power factor corrected power supplies often cause instability for the aforementioned system. This chapter proposes an analytical explanation for this situation. Many electronic loads such as power supplies feature active power factor correction of their input variables. Power factor correction is often a requirement by regulatory bodies to preserve power quality. However, this correction also poses stability problems for certain source and load combinations. An example of such a source is the ferroresonant UPS, which has a large output impedance. Historically, the input impedance of various loads has featured positive incremental characteristics. This is contrary to several semiconductor loads; especially, actively power factor corrected ones, whose input impedance may possess negative incremental characteristics [63], [67]. As has also been described in Chapter 2, it has been experimentally verified that UPSs based on the ferroresonant transformer demonstrate stability problems when they source actively power factor corrected loads [25]. Versions of this chapter have been submitted for publication and patent application. M. G. J. Lind, G. A. Dumont, and W. G. Dunford, "Load Interaction Analysis of a Ferroresonant UPS with Generalized Loads," IEEE Trans, on Industrial Electronics. M. G. J. Lind, G. A. Dumont, and W. G. Dunford, "Practical Design of a Stabilizing Controller for a Ferroresonant Power Supply," IEEE International Symposium on Industrial Electronics ISIE 2006." M. G. J. Lind, G. A. Dumont, and W. G. Dunford. Patent Application: Invention Disclosure No. 06-077: Improved Stability of Ferroresonant Power Supplies. Filed with the UBC University-Industry Liaison Office (UILO), Sep. 2005. 94 Chapter 6 Stability Analysis of the System 95 The controlled C V T for UPS applications is not often described in the literature. The references [10], [27], [32], [49] provide analyses of the simplified circuit of the C V T with a purely resistive load. More recently, [50], [69] have developed a comprehensive circuit of the controlled CVT, and a stability analysis can be found in [70]. The aforementioned circuit is general in nature and probably represents a majority of the ferroresonant class of UPSs. The purpose of the stability analysis is to comparatively and analytically explain the behaviour of the output variables of the ferroresonant UPS when it is connected to resistive, inductive, capacitive, and actively power factor corrected loads. The derived model separates the state-space representation of the load from that of the source. This is a convenient arrangement since any load described by state-space equations can effortlessly be integrated with the system for a stability analysis. The intent is that the material presented in this chapter will assist research of the C V T specifically in the area of source and load interaction analysis. 6.1 The Poincare Maps The model of the controlled C V T represents a nonlinear and periodically time-varying system where the exact knowledge of the load is often not available. Such a system imposes severe restrictions on the method of analysis. The approach we propose in this chapter is linearization of the system about a steady-state operating point. This system is in turn uniformly sampled, and the Poincare mapping technique is used for the analysis. For a closed orbit under small periodic forcing [71], the state equation can be expressed as x = f(x,t)+eg(x,t) (6.1) where f (x,T + t) = f(x,t) and the periodic forcing term is g(x,T + t) = g(x,t). This system has a periodic orbit of period T for a sufficiently small nonzero s. The stability characteristic of the closed orbit is that of the sampled steady-state trajectory xss(t). The system is stable when the linearization of the Poincare map, Df(x), has all its eigenvalues inside the unit circle. The trajectory in Fig. 6.1 shows on example of a three-dimension state vector and the condition when steady state is not established. For the Poincare maps, we sample the state vector periodically. The figure shows that this is analogous to imagining having a hypersurface S of dimension n-1. The hypersurface need not be planar as indicated in the figure, but it must be chosen such that Chapter 6 Stability Analysis of the System 96 the orbit of the state vector is everywhere transverse to S. This translates to (f(x,t) + g(x,t)) • n(x) 4- 0 where n(x) is the unit normal to S at the point x. We define the Poincare map to be on the surface of S. The Poincare analysis method is valid as long as the sampled values of the state variables are in the neighbourhood of their respective steady-state values. Appendix A.2 discusses the fundamentals of Poincare maps whereas Appendix A.3 describes the essentials of the related topic, the theory of Floquet, both of which form the basis of the analysis in this chapter. Fig. 6.1 Orbit of a three-dimensional state vector. 6.2 The Differential Form of the Model The differential form of a forced time-varying periodic system operating in steady state can be written in integral form as t x(x(t 0 ) , t 0 , t ) = x(t 0)+ |f(x,x)dx (6.2) where the external forcing function u(t) is implicit in f(x,t). This representation can be viewed as when x(x(t0),to,t,u(t)), with Au(t) = 0, traces out the steady-state orbit in state space, the trajectory remains undisturbed. For the system under investigation, the forcing function is the grid voltage. In the analysis, we assume that any potential cycle-to-cycle changes of this voltage can be neglected. The partial derivatives of (6.2) are dx _ Vdf dx Chapter 6 Stability Analysis of the System 97 dx „/ / \ \ \df dx , — = - f (x ( t 0 ) , t 0 ) + j ^ - y — d x (6.4) dt 0 c „ a x a t o f = f(x , t ) . (6.5) It was noted in [72] that by successive substitution, (6.3) and (6.4) can be written as dx dx 3t„ dx(t 0) - f (x( t„) , t 0 ) . (6.6) The above relationships will be used in the derivation of the discrete-time model of the system. A complete derivation of the above equations is presented in Appendix A.4. 6.3 Circuit Equations of the Model In Chapter 4, we derived the state-space equations that describe the nonlinear model of the ferroresonant power supply. These equations are now slightly modified to accommodate the analysis in this section. This leads to the state-space model of the circuit in Fig. 4.1 such that x(t) = Ax(t)+Ew(x(t))+fv(t)+bu(t) (6.7) where the output equation is y(t) = cx(t) + gw(x(t)) + h, v(t). (6.8) In expanded form when an SCR is turned on, (6.7) is described by "M(0" M(0 M O _ v c ( t ) . l21 l31 l41 0 0 0 '12 '22 '32 l 42 0 0 0 0 Linear Part 0 0 '22 '32 '42 '23 '33 '43 . 0 " 'KM' a 2 4 KM a 3 4 M O a 4 4 . VC(0. + ™AKM) MM2W) M O / L + L2ab (0 + b, 0 0 0 lab (t) ( 6 . 9 ) Nonlinear Part Load Dependent Part Input Part Chapter 6 Stability Analysis of the System 98 and (6.8) is written as v 2ab W = [c„ c12 0 c14] Linear Part + [0 gn g i 3 ] w 2 ( X L m 2 ( t ) ) +h 1 i 2 a b (t) L M O / L J w,(^ L.m l(t))' (6.10) where _ w 2 ( A ^ 2 ( t ) ) J h^^W+ri^^W. (6.11) The third state variable in (6.9) and (6.10) is merely eliminated when an SCR is turned off. Similarly to what has been previously described, four distinct terms constitute the right hand sides of the relationships in (6.9) and (6.10). The linear and nonlinear parts intrinsic to the system, the forcing function, and the load dependent part. The latter is dependent on the load current. Four state variables describe the system in (6.9). These are, in row order, the primary magnetizing flux linkage referred to the N2ab side, the secondary magnetizing flux linkage, the flux linkage of the switched inductor, and the capacitor voltage. Each of the first two vector elements of the nonlinear part consists of a function of the flux linkage and (6.11) further defines these functions. The voltage drop across the SCRs is ignored in this model. The prime signs in the equations indicate a referred quantity. In the subsequent few sections, we address the switching regions, which are inherent characteristics of circuits involving SCRs, and the linearization of the system. The linearization of the system is a two-step process. First, we linearize the system about the switching instants of the SCR. This is described in Section 6.4.1. Second, we linearize the system about a periodic sinusoidal trajectory in each of the regions of the different circuit configurations with an SCR being turned on or off. This is addressed in the first part of Section 6.4.2. At the end of Section 6.4.2, we put the two linearizations together and complete the discrete-time linearized system. 6.4 Equations for the Stability Analysis Chapter 6 Stability Analysis of the System 9 9 Finally, to exemplify the generic derivation of the linearized system, the stability of the system is evaluated for a few typical types of loads. 6.4.1 Switching Regions For the system in Fig. 4.1, the variables pertaining to the switched inductor are of interest in the selection of the sampling instants. The plots in Fig. 6.2 show typical shapes of these variables vc(t) and iL(t). Although the circuit in Fig. 4.1 indicates that vc(t) is not directly across the SCRs and L combination, apart from the amplitude, it can be thought of as approximately being so. Four time instants to, a, p, and ti are indicated in the figure below. These instances represent the boundaries of the three regions of different circuit topology: an SCR turned off in to < t < a; an SCR turned on in a < t < P; and an SCR turned off in p < t < ti. These intervals are closed to facilitate the analysis below. The regions sufficiently amount to a full sampling cycle of the discrete-time system. Due to symmetry, the sampling period constitutes half of that of the forcing voltage. T(sT Fig. 6.2 Waveforms of the capacitor voltage and switched inductor current. In addition to the nonlinear magnetizing elements, the circuit features a change in topology due to the SCRs. There are a few techniques to circumvent this problem. One such approach is to Chapter 6 Stability Analysis of the System 100 let each time region in Fig. 6.2 have its own state-space representation. The regions are further described in Fig. 6.3, which shows a waveform of a general state variable xj(t). The steady-state trajectory of this variable is denoted with the superscript "ss," and the subscripts "o f f and "on" signify whether an SCR is turned off or on respectively. Observe that the trajectory of XjSS(t) flows exactly through, or rather defines, the center of each square that divides the regions. The perturbation about XjSS(t) is designated Xi(t). The plots in Fig. 6.3 demonstrate that the perturbed signal does in general not intersect with the center of each square. 2 A X xf s ( t )+A X i ( t ) 2A alpha Region 2 Region 1 • — • Region 3 • o o n o n • * SCR 0 f f x ,>\ SCR 0 f f Xoff(t) X o f f ( t ) Fig. 6.3 Trajectory of a general state variable. For a half cycle of an orbit, given the initial value of vc(to), the control objective is to predict the deviation Ax(t0 from the steady-state value x s s(ti) of the state at time ti. Referenced to the traces in Fig. 6.3, the deviation of the state, from steady-state conditions, at a 0ff depends on the state at to, the instant of the sampling time about to, and the firing angle a0ff. This amounts to Ax o f f(cc) = d*Sr(t?) at: da (6.12) Similarly, the dependence of the state at p on on deviations at a o n and p on is A*„„(P) = On \ / *"\ CB ' x r k « ' d<n(ass) da* (6.13) Chapter 6 Stability Analysis of the System 101 Finally, at tj we have A X o f f ( t , ) : dx0sff(Pss) ap off (6.14) To transition from one region to the next, we use the change of basis matrix P [73]. For our system, the transition from <x0ff to a o n amounts to the pre-multiplication of x0ff(a) by P = 1 0 0 0 1 0 0 0 0 0 0 1 (6.15) Consequently, the change of the dimension of the state space at pon to that at p0ff necessitates pre-multiplication of xon(P) by P T . For the system Fig. 4.1, the state variables consist of continuous functions that describe variables pertaining to the circuit elements of energy storage. This validates the pre-multiplications by P and its transpose since the sufficient condition for a unique solution is that all elements of the system matrix are continuous. Each of the first terms, the partial derivative, of the right-hand side of (6.12)-(6.14) is actually the state-transition matrix that takes the state from the initial time to the final time of the respective region in the absence of the input. For the transition of the state vector from x(to) to x(ti), we obtain the Ad-matrix of the sampled discrete-time system as A d = (tr.esr te. <c (<c, t?). (6.i6> It was explained in [72] and in its references that the coefficients involving the turn-off time of a thyristor, for a different application but where the switching circuit is similar to the one for the CVT, are zero. The combination of (6.5), (6.6), and (6.12)-(6.14) leads to the following Bd-matrix Bd=*i(tr,Pi)pT*ite.oc) (6.17) where q = [0 0 -1 0] T and the first and second columns of B d are related to Aa and At 0 respectively. The particular construction of (6.16) and (6.17) is further explained in Appendix A.5. Chapter 6 Stability Analysis of the System 102 6.4.2 Linearization of the System The particular selection of the state variables allow for a feasible steady-state linearization of the same. Although some natural choices of the state variables are currents or flux linkages of the inductors and voltage or charge of the capacitor, the selection of the state variables was not made arbitrarily. This is further addressed below. To linearize the system, the time-varying Jacobians of the terms of the right hand sides of (6.7) and (6.8) are calculated. The general Jacobian evaluated at steady state, regardless of the region of operation, for the nonlinear parts and steady-state trajectories is J L , E w ( t ) = - £ ( A x ( t ) + Ew(x(t))) (6.18) «-(0 for the state equation whereas for the output equation we have (6.19) J^(0 = ^ Mt) + gw(x(t))) The state variables are divided into steady-state quantities and deviations thereof and defined as ^ , m l ( t ) = ^ m , ( t ) + A X , m l ( t ) ^(0 = ^ (0 + ^ ( 0 (6.20) X L ( t ) = X t ( t ) + A A L ( t ) v c ( t ) = v - ( t ) + A v c ( t ) . The steady-state variables, which are approximated as being sinusoids, free of harmonic content, are defined as Chapter 6 Stability Analysis of the System 103 K M = K m , ^ sin(cot + esLs.ml) ^2(0 = ^ 2 . ™ sin(«t + eL 2) (6.21) V c W ^ v - ^ sin(cot + escs) where the phase angles are chosen to be relative to v c(t). Thus, 9c s s = 0. The deviated variables are analogous to the variables in (6.21) also approximated as being sinusoids. Since it cannot be arbitrarily assumed that the variables in (6.20) and (6.21) are well defined sinusoids operating in steady state, further explanation is needed. An inspection of the circuit in Fig. 4.1 shows that X,L'mi(t) is closely coupled to v'i ab(t), which in turn is a very stable sinusoid. In fact, we regard this variable to be periodically constant although we allow v'iab(t) to have a harmonic content. In the simulations in the result section below, we consider the third through the eleventh harmonic. The variable ^Lm2(t) is closely coupled to V2ab(t), which in turn we attempt to keep as a periodically constant sinusoid. The flux linkage in L and the voltage across C are also in shape and form quite closely coupled to V2ab(t). The amplitudes of the aforementioned state variables are essentially independent of the magnitude of the load. The firing angle, however, is load dependent but its range is surprisingly small. It was shown in Section 2.3.3 that the range of a is about 10°. Furthermore, in Section 4.2, we analyzed the equivalent inductance as a function of a. The small signal state and output equations are Ax(t) = J s A s X ( E w(t)Ax(t)+fAv(t)+bAu(t) (6.22) and Ay( t ) = J " B i r ( t ) A x ( t ) + h I A v ( t ) (6.23) respectively where it turns out that Av(t) will cancel out for the combined system of the UPS and load that is derived below. For the regions where an SCR is turned on and off, the coefficients in (6.22) are Chapter 6 Stability Analysis of the System 104 J Ax,Ew,on (0 l21 '41 42 0 ,1 e 3 2 / L e 3 2 / L "12 a 22 e 2 2 A " l 24 l 34 l 44 "o" V f 2 0 , f = z , b = U 0 0 (6.24) and * an a 1 2 0 " "o" V J Ax,Ew,off (0~ a 2 1 * a 2 2 a 2 4 , f = U , b = 0 a 4 1 * a 4 2 a 4 4 .f4. 0 (6.25) respectively where a n _ a u + e n a 22 _ a 22 + e 22 a 3 2 - a 3 2 + e 3 2 a 4 2 _ a 4 2 + e 4 2 Yi+Y m m(X£. m l (t))" ^ i + ^ „ n ( ^ 2 ( t ) ) n _ 1 ] ^ l + ^ l n ^ S L 2 ( 0 ) n ni+n n n (> iL2 ( t ) ) n (6.26) The Jacobians of the output equation for the regions where an SCR is turned on and off are Jrx,gw,o„(0 = [ c u c; 2 g 1 3 / L c j (6.27) and Jcx,gw,off (0— [ C l l C 1 2 C 1 4 ] (6.28) respectively where C12 — C 1 2 + § 1 2 ^ l + ^ „ n ( ^ m 2 ( t ) ) n ~ (6.29) The truncated Binomial Series expansion was used in the derivations of the Jacobians above. The small signal state equation for the system of the UPS and load combination in Fig. 6.4 below is Chapter 6 Stability Analysis of the System 105 Ax(t) = Ax 2(t) Jxu(t) j s ; 2 1 ( t ) Jx2 2 ( t ) Ax,(t) Ax 2(t) + A U l (t) (6.30) where xi(t) and X2(t) pertain to the UPS and the load respectively and the Jacobians of the partitioned A(t)-matrix are J x U W = J Ax,Ew (0 + f [ i " JSjh, J 2 D J r x > g w W JxS12(t) = f[l-J2Dh 1]" 1J 2 Sc(t) Jx2 1(0=J2B(t)[l-h 1J 2 S D]" 1Jrx > gw(t) Jx22(t) = J2A ( t ) + J 2 B ( t ) [ l - h 1 J 2 S D ] " 1 h 1 J 2 S c ( t ) . (6.31) It is assumed that the above inverses are nonsingular. This is always the case when the feed-forward matrix equals zero. The output equation for the small signal system of the UPS and load combination is Ay(t) = j;sn(0 J s y s 2 1 ( t) \ J ^ ( t ) where the Jacobians are Ay,(t) Ay2(0 j ; sn(t)=[i-h 1j 2 D]" Ijrx, gw(t) J",2W=[l-h,JS,f ,h1JiBW(t) j ; 2 2 ( t ) = [ i - j 2 D h 1 ] " 1 j s 2 c ( t ) . Ax,(t) Ax 2(t) (6.32) (6.33) The actual variable to be controlled, the output voltage of the UPS, is yi(t), in (6.32). It should be noted that the Jacobian of the D-matrix of the load is made time invariant. This is necessary to validate (6.40) below. The simulation diagram in Fig. 6.4 shows the small signal combined system of the UPS and a general load. The advantage of this representation is that any load in state-space representation can be integrated with the system. Chapter 6 Stability Analysis of the System 106 A"i(t)| UPS 'Ax, Ew (t) • hi Load J 2 A ( t ) Ay2(t)i Fig. 6.4 Simulation diagram of the linearized UPS and load combination. The schematic also provides for a physical insight. For example, the output Ay2(t) of the system is fed back to the input at the point where the load disturbance normally would enter for a general case. This signal is normally not random here and its strength is unfortunately dependent on the load conditions. In addition, there is no controller in the input path in the usual context. The control of this system is de facto a change of the dimension of the state space corresponding to the switched-on or switched-off SCRs. In order to be able to solve (6.30), we need to find the state-transition matrices in (6.16) that advance the state in time from to to ti. Recall that any matrix U(t) satisfying DU(t) = A(t)U(t) provided det(U(to)) ^ 0 is called the fundamental solution matrix. There exists a unique fundamental matrix 0(t,x), called the state-transition matrix, of the aforementioned equation such that 0(x,x) = I. The state-transition matrix ®(t,x) = U(t)U~'(x), in absence of any input u(t) and given the state at x(x), maps the state at time t to x(t). A convenient choice, but not a necessary restriction, is to select the initial condition to equal the identity matrix or U(to) = I. This leads to the following relationship between the state-transition matrix and system matrix such that 6 ( t , t 0 ) = A(t)o(t,t0) , O ( t 0 , t 0 ) = I . (6.34) Chapter 6 Stability Analysis of the System 107 The solution of the above equation is used to advance the state from the beginning to the end of a particular time interval such that The Jacobians in (6.30) are periodically linear time variant (LTV). A closed form of O(t,to) can only be obtained when the commutative property [74] M(t,t0)A(t) = A(t)M(t,t0) is met for This occurs when A is constant and for time-varying systems only for some special cases such as when A(t) is diagonal or triangular. For the aforementioned systems, the state-transition matrix is computed as Otherwise, apart from successive approximation, O(t,to) has to be evaluated numerically. Unfortunately, this is the case for our model. To find the respective state-transition matrices for each of the three regions, the Jacobians in (6.30) are utilized and (6.34) is computed in integral form as Ax(t + t 0 ) = O ( t , t 0 )Ax(t 0 ) . (6.35) (6.36) O S S (ass tss) * - ( P - . c C ) (6.38) a; 'on Poff where JxjSS(t) represents the concatenation of the elements of the partitioned 2-by-2 matrix in (6.30) for the respective region i corresponding to an SCR being turned on or off. Care must be taken in the evaluation of (6.38) since, in general, there is no guarantee that the columns of a Chapter 6 Stability Analysis of the System 108 state-transition matrix turn out to be linearly independent when numerical integration is involved [75]. The combination of (6.16), (6.17), and (6.38), leads to the following state equation for the discrete-time system Ax(n + l) = AdAx(n)+BdAu(n) ,Au(n) = [Aa AtJ (6.39) where due to symmetry of the state variables, n represents a sample frequency of 120 Hz. The discrete-time output equation can be determined from (6.32) and (6.33) as Ay,(n) = Cd(n)Ax(n) (6.40) where for control purposes, we are typically interested in either the magnitude or the rms value Of V2ab(t). 6.4.3 Typical Loads In the special case when the load is purely resistive, denoted by R, Dx2(t) = X2(t), J S S2A(0 = 0, J s s 2 B(t) = 0, J s s 2 C ( t ) = 1/R, and J S S 2 D = 0, (6.30) reduces to AX l(t) Ax2'(t)' ! fJ?c(0 J ^ g j t ) j h . J ^ t j AX l(t) Ax2(t) + Au(t) and the output equation (6.32) becomes Ay(t) = Ay.W Ay2(t) 0 i J"c(0 Tss ft) " cx,gw \ / ! h , J 2 s c ( t ) Ax,(t) Ax2(t) This results in the following system representation A x ( t ) = [ j ^ E J t ) + ^ Ay(t) = [ i + h , J 2 S C ( t ) [ l - h , J S 2 S C ( t ) ] j ; s x > g w (t)] Ax(t). (6.41) (6.42) (6.43) In the general case when the load contains states, P in (6.15) needs to be augmented such that Chapter 6 Stability Analysis of the System 109 P 0 P * = M 4 x n (6.44) _ " n x 3 * n where n equals the number of states of the load. For the simple cases with R L and RC loads, we also need to address series versus parallel loads. Since the output section of the circuit including the external capacitor and switched inductor in Fig. 4.1 can be thought of as essentially consisting of admittances, the load may likewise be expressed as such. However, this is disadvantageous from a simulation perspective since the voltage across a linear loss-less inductance must be of the form cos(cot) to maintain the flux balance. The desired parallel R L and RC load is therefore converted to its series equivalent in a normalized fashion such that R ' + i X s = R p R p N X p R p X p X p N ~ J Y IT. ^ \ 2 . ~ \ 2 _ J / ~ „ \ 2 U „ \ 2 ipA5) R SN x s N ( R P X P N ) 2 + ( R P N X p ) 2 ( R P X p N ) 2 + ( R P N X P ) : where the subscripts, p and s, denote parallel and series maxima allowed or rated parameters respectively. Additionally, pN and sN indicate parallel and series normalization factors living in the interval [0,1] correspondingly. The reactance of either L or C is expressed as X and the positive and negative signs on both sides of the equation indicate inductive and capacitive reactance in that order. Although the circuit conversion in (6.45) is true for the frequency domain, the corresponding series and parallel combinations generally have different transient responses. However, the stability analysis is carried out for steady-state conditions. For the simple cases, we have the following load equations. The state-space representation of the series R S L S load is Ax 2 (t) = AJ L , l o a d (t) = [- R , / L , ] Ai u l o a d (t) + [1/L, ] Au 2 (t) Ay 2 ( t) = A i L l o a d ( t ) = [i]Ai L i l o a d (t) . For the series R S C S load, the state-space representation is Ax 2 ( t ) = A v C i l o a d (t) = [-1/(R SC S)] A v c , l o a d (t)+ [ l / (R s C s )] Au 2 (t) A y 2 ( t ) = Ai c , l o a d ( t ) = [ - l /R s ]Av c > l o a d ( t )+[ l /R s ]Au 2 ( t ) . (6.46) (6.47) For the transfer function of the input admittance of a typical PFC load, which typically is a boost converter, we use (5.10) from Chapter 5 repeated here for convenience such that Chapter 6 Stability Analysis of the System 110 1 s 2 +2£a)s-u) 2 R s 2 +2Cws + co2 (6.48) where the direct state-space realization in observable canonical form is Ax 2(t)= 1 Ax 2(t) + [_ - or 0 J Ay 2(t) = [l 0]Ax2(t) + R- 1Au 2(t). 0 -R- ' to 2 ( l + 2Co)) Au 2(t) (6.49) The parameters are selected such that Yj„(s) possesses real zeros, one stable and one unstable, whereas a stable complex pair constitutes the poles. A Simulink14' model based on the circuit in Fig. 4.1 was implemented to obtain the steady-state conditions necessary for the calculations of the state-transition matrices. These conditions were also verified with those of the real physical UPS, the parameters of which, the circuit in the Fig. 4.1 is based on. The circuit variables were sampled at 120 Hz at -90° relative to the zero crossings of the state variable v c(t). The ideal sampling deviation Ato = 0 was used. The Tustin approximation, z ~ (l+T ss/2)/(l-T ss/2), was used in the mapping from the discrete-time domain to that of the continuous time for the pole and zero plots. Several types of loads were investigated regarding the presence of unstable poles and zeros. In particular, the relative locations of the aforementioned were studied as to determine whether one controller can robustly control the system with commonly, but in general unknown, encountered loads. 6.5.1 Resistive, Inductive, and Capacitive Loads The plots in the next six figures show the continuous-time pole and zero locations for R, RL, and RC loads, where the latter two were implemented as described by (6.46) and (6.47) respectively. For the first purely resistive case, the normalized value of R was swept in the interval 0.001 < Rnorm < L For the remaining two cases, the normalized parameters were inter-6.5 Results of the Stability Analysis Chapter 6 Stability Analysis of the System 111 swept in the ranges 0.15 < R p N < 1, and 0.03 < X p N > L , X p N > c < 1. The value of Xj was not allowed to exceed that of the corresponding R to keep the power factor, pf > 0.707, lagging or leading. It was assumed that a typical UPS load would obey this relationship. r r T i [ [ i tx, 1 1 \ i i / i i i r ' • Ix i I I i r T F T 1 ~l 1 1 1 / 1 1 1 1— Increasing R - J - - Increasing R '< i a norm / 3 norm , -x-xx-x-x-x- • X - X - - - X - ->§s K 1 1 1 X 1 ; ; ; , ^ _ _ X 1 r 1 — — — — — i i -§00 -250 -200 -150 -100 -50 0 Real Axis Fig. 6.5 Pole plot for the model with R load. I n n r o a e i n n R * norm i 1 i r\ ; 1 1 1 .11 1 I I I I I I -150 -100 -50 0 50 100 150 200 250 Real Axis Fig. 6.6 Zero plot for the model with R load. Chapter 6 Stability Analysis of the System 112 200 150 100 •R 50 « o "En CO E -50 -100 -150 i i i i i i 1 i • j ^ ^ x -x^mssi T 1 mm.- - - j xi 00 -250 -200 -150 -100 Real Axis -50 Fig. 6.7 Pole plot for the model with R L load. CO c CO E 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 I J. i I I I I i r i I I I I i i i BD-(|) i ;--(E>---I ; i i i i i i i i 300 -200 -100 0 1 00 Real Axis 200 300 Fig. 6.8 Zero plot for the model with R L load. Chapter 6 Stability Analysis of the System 113 15 10 co 'x < ra c cn ca E -10 i i X 1 . - - X _ J C J X 1 X i >< - - - x - • ) * C - - X ^ mom; I x i x r x" X ' i X i 1 ?00 -350 -300 -250 -200 -150 -100 -50 0 Real Axis Fig. 6.9 Pole plot for the model with R C load. C O "x < CO c "CO CO E i <p ; mmmammmmmtifrmi. ^ + i i i i I i i i i i i ; i i ! <b ; -300 -200 -100 0 100 Real Axis 200 300 Fig. 6.10 Zero plot for the model with R C load. Chapter 6 Stability Analysis of the System 114 As the plots reveal, the systems with R, RL, and RC loads are all well behaved. Apart from the sampling zeros, there are no unstable poles or zeros. This fact significantly simplifies the controller design. The sampling zero at 240 Np/s is here of less importance since the theoretical bandwidth of the controller is limited to one-half of the sampling frequency. The delta-domain sampling zero is computed as z = (exp(-Ts)-l)/(Ts+exp(-Ts)-l). 6.5.2 PFC Load The input characteristics of the specific PFC load were determined as described in Section 5.2. As discussed there, the modulated signal should ideally be multiplied into the actual UPS-load configuration. This also implies that the system should operate in the stable region or that at least unbounded instability cannot occur. Due to hardware limitations of such multiplication, we begin with an estimation of the input envelope impedance of the load based on the characteristics of the output voltage of the UPS when it operates in the bounded unstable region. The purpose of the aforementioned estimation is not only due to practicalities, but as it is shown below and most importantly, we also demonstrate a case, or a certain load set of characteristic for which the system cannot be controlled by an LTI controller. Unless proven otherwise, it cannot be ruled out that such a load exists. We will therefore come to the important realization that there are true limitations to what the system can accomplish. In the subsequent sections, we contrast the aforementioned load with a case that indeed can be controlled by an LTI controller. To exemplify the discussion above, the parameters of (6.48) were estimated based on the experimentally obtained results of V2ab(t) during unstable but bounded conditions. The UPS corresponding to the equivalent circuit in Fig. 4.1 was connected to the commercial PFC PSs that in turn were loaded with resistor banks. The region of bounded oscillations of the output voltage, V2ab(t) was found to occur in the interval 56.0 % to 80.1 % of rated load. The plot in Fig. 6.11, which is a detailed plot of Fig. 2.8, shows v2ab(t) for the latter case. Above 80.1 %, the oscillations became too severe for proper operation of the system. It should be noted that instability for this system is defined as when the waveform of V2ab(t) deviates sufficiently enough from its desired periodical shape and rated amplitude. Chapter 6 Stability Analysis of the System 115 Based on the envelope in Fig. 6.11, the load was modeled with the following parameters for (6.48): R = 10 Cl, C, = 0.044, and to = 251 rad/s. The damping factor was subjectively chosen such that very similar, but not long-term steady state sustainable, oscillations to those in Fig. 6.11 were obtained in the computer simulations of the large signal model in Simulink®. Due to the linear approximation of the load, long-term steady state sustainable oscillations may not be expected considering the nonlinearities involved of the real physical UPS and load. Therefore, the results of these simulations are not shown here. Fig. 6.11 Experimental results for v 2 a b(t) at 80.1 % PFC PS load. When the system was studied with the PFC load, some interesting results were obtained. The normalized resistance of R was swept such that 0.1 < R n 0 r m < 1 in increments of 0.1. The plots in Fig. 6.12 through Fig. 6.14 show the locations of the poles and zeros of the UPS loaded with Set I, which denotes the transfer function in (6.48) with the aforementioned parameters. The plots show that not only do two complex pairs of poles and zeros respectively cross the j co-axis, this is zoomed-in on in Fig. 6.14, but there is also a closeness of the real parts of the unstable poles and zeros. Unstable poles and zeros impose a lower and upper limit respectively, on the control bandwidth. Ideally, unstable zeros should be much faster, at least five times, than unstable poles to achieve reasonable control design. The plots in Fig. 6.14 indicate that this is not Chapter 6 Stability Analysis of the System 116 the case. This indicates that the system is extremely difficult to control by a linear time-invariant (LTI) controller. 500 400 300 200 £ 100 CO c 'cn co E 0 •100 -200 -300 -400 -50 T 1 X r T --XX>OO<>0<X^ - - -Increasing R a n o r m 1 i n c r e a s i n g n -xkxxx-x-i i i n o r m y « - - 3 8K- h _x > 1 1 i i i i i l 1 r " X X J O O C ^ ^ - - n T 250 -200 -150 -100 -50 Real Axis 50 Fig. 6.12 Pole plot for the UPS with PFC loads Set I. to 'x < CO c 'cn co E 500r 400-300 200 100 0 -100 -200--300 -400 -50( Increasing R Increasing R -G>: o o -50 50 100 150 Real Axis 200 250 Fig. 6.13 Zero plot for the UPS with PFC loads Set I. Chapter 6 Stability Analysis of the System 117 430 425 420 •8 4 1 5 % 410 to •§> 405 CO E 400 395 390 385 R =0.1 norm R = 0.5 norm - X -R = 1 norm o R =0.1 norm -40 X X Increasing R •---^-•---x X R =0.5 norm x-X o -x-R = 1 norm -30 -20 -10 0 Real Axis 10 20 30 Fig. 6.14 Zoomed-in pole and zero plot for the UPS with PFC loads Set I. When the loop gain of a system possesses unstable poles or zeros, time delays, or uncertainties in the transfer function of the plant, there are limitations in the design of a linear controller [76]. The phase angle of the non-minimum phase part of the transfer function of the plant can be calculated using the unstable poles pu,i and zeros zU j; as ^Tarctan —— +^arctan V z « . i J r u,i > Z u , i ' P u , i >0 (6.50) where cogc is the crossover frequency of the loop gain. For the system to be stable in closed loop, the inequality arg(PnmpKc))>-^  + 9 m - n g c | > - 2 < n g c < 0 (6.51) must be satisfied where (pm and n g c denote the desired phase margin and the desired slope of the minimum phase part of the transfer function of the loop gain in units of 20 dB/decade respectively. For the sampled system with the load in (6.48), the corresponding continuous-time Chapter 6 Stability Analysis of the System 118 transfer function of the plant divided into the minimum and non-minimum phase parts turns out to be of the form r M - K(s + Z l )(s + z 2 ) ( s 2 - 2 ^ 8 + 0 ) , ) z > Q (s + P l ) ( s + p 2)(s + P 3 ) (s 2-2C 2s + ( 0 2 ) ' P " ' (6-52) where K is the gain. To make |Pnmp(s)| = 1 and to possess a negative phase without affecting P(s), we change the signs of the unstable factors and augment the right hand terms of (6.52) with these and obtain / \ _ K(s + Z l )(s + z 2 ) ( s 2 + 2 ^ 5 + 0 3 . ) (s + p1)(s + p 2)(s + p 3 ) (s2 +2C2s + ( 0 2 ) P M _ ( s 2 - 2 C , s + Q),) (s2+2C2s + co2) [s +2C)1s + bol) (s -2C 2 s + co2j The plots in Fig. 6.15 show the phase of Pm p(s) and Pnmp(s) for the UPS with the PFC load, Set I, with Rnorm = 0.1, Rnorm = 1, 9m = 7t/4, and n g c = -1/2. The minimum phase allowed, P n m p , nun, of PnmP(s) is also plotted. The plots are becoming more inexact with increasing frequency due to the Tustin approximation and the relative closeness to the sample frequency of 120 Hz. However, this is of less significance regarding the qualitative interpretation of the plots at large loads. The two extreme values of Rnorm were chosen here to exemplify, and in particular to verify, why the system featuring the particular relative pole and zero location in Fig. 6.14 is essentially impossible to control. In between theses two values, at about 60 to 70 % load, we lose the capability to control the system robustly with an LTI controller. The two traces in Fig. 6.15 for light load, Rnorm = 0.1, for Pm p(s) and Pnmp(s), indicate that we have sufficient phase margin for Pnmp(s) throughout the particular frequency spectrum. However, for maximum load, Rnorm = 1, there is no phase margin for Pnmp(s), not even at very low frequencies. This system is essentially impossible to control with an LTI controller. The type of controller is also limited for this system. Since the load in general is unknown, state feedback is not quite applicable. A controller of the proportional-plus-integral-plus-derivative (PUD) type is better suited for this system. Chapter 6 Stability Analysis of the System 119 6.6 An Example of Controller Design Despite the fact that the stability analysis in this chapter focuses on open-loop stability and, due to its nature, as such is less suitable for controller design, we propose that the particular technique can be extended to encompass controller design albeit in an approximate fashion. We also address sampling issues. In control system problems regarding the theory of Floquet or Poincare maps, a continuous time-varying system is converted to a discrete-time equivalent. Examples of excellent studies regarding such applications are [77], [78]. The Floquet theory is often used for state-space systems with feedback and pole-placement design. Although pole-placement techniques generate stability, robustness may not be guaranteed [76]. Often, the literature does not seem to address the subtleties regarding the accuracy of the transformation provided by the Floquet theory. However, this is of particular interest when subparts of a time-varying system are time invariant. We would like to verify the preservation of the modes belonging to the subparts throughout the discretization process. This may not always be possible; for example, when the Chapter 6 Stability Analysis of the System 120 aforementioned are close to other modes that result from the discretization process. The problem is nontrivial since the eigenvalues of a continuous periodically time-varying system are undefined. In this study, we transform the discrete-time system back to the continuous-time domain for eigenvalue validation. The design discussed in the subsequent sections is partially based on [79]. 6.6.1 Load Identification The experimentally obtained model of the PFC load described by (5.10) with R n 0 rm = {0.1, 0.2, 1}, 4 = 0.195, and co = 188 rad/s was incorporated in (6.7) and (6.8), and the system was discretized to generate the form of (6.39) and (6.40). The plots in Fig. 6.16 and Fig. 6.17 show the resulting locations of the poles and zeros respectively. For unknown loads and for the particular system, open-loop stability is advantageous regarding the LTI controller design. The pole and zero complex conjugate pairs that tend towards the unit circle with increasing R n 0 rm are pertinent to the PFC load. Should these cross the unit circle, we would have Neimark bifurcation in R n 0 rm. Fig. 6.16 Pole plot of the families of the five poles for the system with PFC load Set F. Chapter 6 Stability Analysis of the System 121 Fig. 6.17 Zero plot of the families of the four zeros for the system with PFC load Set F. 6.6.2 Sample Considerations In this section, we address sampling issues not considered in the previous sections since they did not focus on controller design. We first discuss sampling fundamentals. The s and z planes in Fig. 6.18 show the relationship between the continuous and discrete time domains [80]. When the poles or zeros have imaginary parts, folding is inevitable for frequencies larger than cos/2. However, the real parts of the poles are unaffected by the folding phenomenon and continuous-time poles with negative real parts are mapped inside the unit circle of the z domain by z = e s T s = e 0 7 ' e j l o T s . Furthermore, a movement from -jcos/2 to jcos/2 in the s-plane corresponds to -7i/Ts to 7i/Ts in the z-plane and a movement from jcos/2 to j3cos/2 in the s-plane corresponds to 7t/Ts to 3%/Ts in the z-plane. Consider a sampled signal at co = oo2 in the frequency spectra coi with the sample frequency cfls < 2coi and © 2 < coi. The magnitude of the signal at co = 0 0 2 does not only appear as that of CO2. Due to aliasing, co also contains additional and indistinguishable frequency magnitude Chapter 6 Stability Analysis of the System 122 components, if present, from ncos ± CO2 where n is an integer. Thus, we obtain magnitudes at co, such that co = C02 + (ncos ± © 2 ) where we customarily only consider positive frequencies. jco Complementary Strip Complementary Strip Primary Strip Complementary Strip Complementary Strip j5cos/2 j3cos/2 jcos/2 -j(os/2 -j3cos/2 -j5oos/2 s-plane o co72 = JI/T -G>J2 = -71/T. z-plane Fig. 6.18 Transformation between the s and z-planes. In practice, some degree of folding is always present. For the zero-order-hold (ZOH), we have the well-known transfer function G Z O H (jw) = T s sin(coTs 12) < D T . / 2 (6.55) which magnitude is zero at ncos. The state-transition matrices in (6.38) are obtained by integration of the continuous periodically time-varying and small-signal equivalent system of (6.9) and (6.10) over the first half of the period. This can be considered as a discretization of a continuous time-varying state-space system where the input Au(n) in (6.39) is constant between any two consecutive sampling instants. The discrete-time state equation that is obtained by means of a general integration process of the continuous-time counterpart is known as the zero-order-hold equivalent [80]. This integration is not an approximation since it involves the solution equation of the continuous time-varying state equation where the input is constant within each region of integration. However, the zero-order-hold process may produce aliasing errors, albeit the method inherently provides for moderate high-frequency attenuation. It may seem obscure to deal with the system in (6.39) from a sampling standpoint. However, for a detailed analysis of the nonlinear system in (6.9) the available methods are Chapter 6 Stability Analysis of the System 123 limited. Normally, we would resort to a higher sampling rate. However, this is impossible in this case due to the particular periodicity of the system and the analysis application using Poincare mapping. Therefore, we propose to make the best use of the sampled information at hand. Although not pursued here, but for better frequency accuracy, a signal may be completely defined up to frequencies of the sampling frequency, i.e., twice the Nyquist frequency. This requires that both the amplitude as well as the derivative of the signal is available at the sampling instants [81], [82]. This is actually the case for (6.39). It is important to realize that the Poincare mapping technique described in the previous sections may fold frequencies above the Nyquist frequency into the region below the same. This is due to the principle of analysis and aggravates the interpretation of the same. However, it is by no means necessarily related to possible aliasing occurring at the input of the controller. Actually, for an analog controller implantation, the sampling rate can be assumed to be extremely fast, i.e., continuous sampling rate. The Poincare mapping is inherently limited to the period of the system, or to subintervals, that features symmetry. This fact is often not considered. Therefore, we consider only loads with a damped natural frequency less than the Nyquist frequency. In particular, it should be observed that the inter-sample response of the system in (6.39) is due to the step response, with initial conditions, occurring at the instant of the control action, i.e., the system is running in open loop in-between control actions. Inherently, such as system is easier to control if it is open loop stable. Furthermore, the time interval of open-loop operation is substantial or half a cycle of the periodic system. 6.6.3 Transformation to the Continuous-Time System For typical parameters of the circuit in Fig. 4.1, the matrices in (6.39) and (6.40) are generally ill-conditioned. To enhance the numerics, the balanced realization of the system with the similarity transformation T and the added subscript b may be written as To be able to study the system at the periodic level, we may also apply time-lifting [83]. Ax b (n +l) = TA d T" 1 Ax b (n )+TB d Au(n) , Au(n) = Aa(n Ay(n) = C d T ' 1 A x b ( n ) ) (6.56) (6.57) Chapter 6 Stability Analysis of the System 124 To transform the discretized system back to the continuous time domain, we use the matched pole-zero mapping method and consider the numerator and denominator of the transfer function separately. We use the relationship z = e s T and adjust the gain, for a low-pass system, such that the gain at z = 1 equals that of s = 0. However, for the zeros z ~ esT, i.e., an approximation, and it is generally impossible to give a simple formula for this relationship [84], [85]. The frequency response of the discrete-time system in 0 < co < cos/2 corresponds to that of the continuous-time counterpart in 0 < co < oo. Thus, aliasing does not occur, and the matched pole-zero method preserves the general characteristics of the frequency response [82]. To exemplify the accuracy of the entire process of analysis from the continuous-time system to the small-signal discrete system in (6.39) and (6.40) and back to the continuous time domain using the matched pole-zero method, Table 6.1 presents a representative case with R n 0 rm = 0.5. The table lists the continuous-time poles and zeros of the time-invariant PFC load for different values of co before (6.48) was inserted into the model of the system. The table also lists the poles and zeros of the whole system after it was converted back to the small-signal continuous-time domain. Table 6.1 Continuous-time poles and zeros. Load Before Discretization System After Discretization (rad/s) / Poles Zeros Poles Zeros (Hz) (Modes) (Modes) (Modes) (Modes) 4,5 3 4 1 2 3 4,5 1 2 3,4 6.28 /1 -1.23 ± 6.16i -7.63 5.18 -218 -94.6 -2.34 -1.21 ±6.16i -50.2 -0.356 -1.21 ± 6.16i 75.4/12 -14.7 ± 74.0i -91.5 62.1 -216 -96.5 -2.43 -13.7±74.1i -52.0 -0.351 -13.5 ± 74.2i 188/30 -36.8 ± 185i -229 155 -211 -92.9 -2.34 -28.9 ± 185i -52.8 -0.250 -29.6 ± 186i 314/50 -61.3 ± 308i -381 259 -129 -80.5 -2.51 -39.9 ± 305i -47.6 0.390 -47.2 ± 311i 346 / 55 -67.4 ± 339i -419 205 -74.3±27.1i -2.60 -44.3 ± 335i -40.5 1.60 -56.9 ± 344i 440/70 -85.8 ± 431i -534 362 -90.9 ± 6.07i -2.20 -45.8 + 340i -47.5 1.05 -75.9 + 330i Although the low sampling rate degrades the accuracy of the analysis, the complex poles of the load before discretization versus after are quite similar as are the complex zeros. The latter Chapter 6 Stability Analysis of the System 125 are addressed below. Closer to the Nyquist frequency, the accuracy deteriorates. However, since actively PFC loads may feature incremental negative input resistance, i.e., the damping factor, for a second order approximation, is small or negative, it is less feasible to consider such loads having a natural frequency at or in the neighbourhood of the rated supply frequency since that could cause severe oscillations or beating respectively. A characteristic of this system due to the construction of (6.48) is that the complex zeros after discretization are similar to those of the poles before discretization. As the entries in Table 6.1 indicate, this is true for low values of co. For co > 50 rad/s, the analysis produces an unstable zero. The last row in the table exemplifies an under-sampled case for which folding occurs. The plots in Fig. 6.19 present the response of the transfer function, after discretization, of the system in (6.9) with the load described by (6.48) for Rn0rm = {0.1, 0.2, 1}. The response does not change significantly with the variation in Rnorm- This is favourable since a general controller that can cope with all load cases may be designed. Although not shown here, results similar to those in Fig. 6.19 were obtained when (6.48) was replaced with loads such as R, RL , and RC loads. CO CD TJ. CD CO CO 10 10 Frequency (Hz) 10 Fig. 6.19 Family of Bode plots of the plant, the power supply and PFC load Set F, for R n {0.1, 0.2, 1} with the matched pole-zero method. Chapter 6 Stability Analysis of the System 126 6.6.4 Control Loop To make the periodically time-varying signal of the output voltage useful for the controller, we use a rectifier in the feedback path. We consider the controller possessing low-pass characteristics so that the harmonics generated by the rectifier are filtered out. This is analogous to the describing function method. Recently, there has been a renewed interest in implementations of rectifiers in the control loop [86]. Should harmonics pass though the controller, it is only of little disadvantage due to the location of the comparator in the control loop and the ratio in (6.58) shown below. The rms value of the output voltage is the controlled variable. Therefore, we consider the ratio of the average to the rms value. Moreover, the particular topology inherently produces harmonics in V2ab(t). We consider here only the significant one, the third harmonic. Also, see the plots in Fig. 2.7 and Fig. 2.8 regarding integer multiples of 60 Hz. As shown in Section 2.3.3, the range of the firing angle of the SCRs is surprisingly small. Thus, the harmonics generated by the switched inductor can be approximated as being of fixed amplitude. The average to the rms ratio of the rectifier output voltage v d is such that where V2 a b, max is the amplitude of the fundamental of the output voltage, T is the period of the fundamental, coi = 377 rad/s, and © 3 = 3cpi. Note, that although integration is used to obtain the average value of V2ab(t), the switches of the rectifier cannot store energy and an additional state is not considered in the control loop here. Using (6.58), the gain of the rectifier can be described by T d,rms,1,3 d,ave,l,3 62V202 3037t (6.58) Chapter 6 Stability Analysis of the System 127 f T 2 2 d,ave - { V 2 a b > n a x sin(a>,t)dt d,rms,1,3 d,ave,1,3 3V202 62 V. 2ab,max ' (6.59) and we denote this transfer function H 2 . Due to the nonlinear plant and, in general, the unknown load, we chose a simple lead controller combined with a integrator to minimize the steady-state error such that C ( S ) = K y m - + i > , » . < « , , s ( s / (O p +l ) (6.60) This transfer function increases the closed-loop bandwidth, which is desirable since the plant itself is sluggish. The lead stage makes the system less sensitive to parameter variations. We also consider any high frequency noise at the control output to be insignificant due to the low frequencies involved. To produce the firing angle, we modulate the control output by the modulation function m(t) such that m(t) = M m i n + ( M m a x - M m i n ) f — modi vTs (6.61) J where T s is the switching period and Mmjn and M m a x are the boundaries of the amplitude and the transfer function of m(t) is M = -1 M m a x - M m i „ (6.62) Normally, in SCR applications, we consider an average delay time td,ave [87] between the change of command to the change in the firing angle such that V d , a v e =k( t - t d a v J (6.63) where k is the gain of the SCR circuitry. The Laplace transform, of this function approximated to the first order is Chapter 6 Stability Analysis of the System 128 (6.64) However, for the circuit in Fig. 4.1 and the control loop shown below, the actual firing angle may be approximated as occurring simultaneously with the commanded firing angle. The diagram in Fig. 6.20 shows the control loop where G(s) represents the ferroresonant power supply and its load, Hi and H 3 are scaling factors, and a is the firing angle. We consider the comparator and the SCR drive circuitry to have insignificant influence on the dynamics of the control loop. a XH c(s) m(t) a, SCR Drive G(s) V2ab Comparator H 3 H 2 H i ^ Fig. 6.20 Control loop. The small signal transfer function can now be written as C(s)MG(s) T(s) = 1 + C ( S ) M G ( S ) H 1 H 2 H 3 (6.65) Based on the worst-case information from Fig. 6.19 and (6.65), the controller was designed with ICjc, gain = 0.2, coz = 94.0 rad/s, © p = 50.0 krad/s, infinite gain margin, and a rather low phase margin of 31.3° at 13.2 kHz. However, this high frequency would never occur in practice. The controller was implemented with operational amplifiers. Due to the approximate nature of the analysis, the idea behind the controller design is to keep the magnitude constant and minimize its phase shift in the typical frequency range of the dynamics. See also the representative spectrum plots in Fig. 2.7 and Fig. 2.8. The plots in Fig. 6.21 show the controller response whereas the traces in Fig. 6.22 display the loop response of the system. The exact locations of the poles and zeros are also indicated as "x" and "o" respectively in these two figures. Chapter 6 Stability Analysis of the System 130 6.6.5 Experimental Results To verify the analysis, rated line voltage of rate frequency sourced the ferroresonant power supply, which was loaded with the actively PFC loads that in turn sourced the resistor banks. The PFC power supply possessed the characteristics that were discussed in Section 6.6.4 and the resistance of resistor bank corresponded to Rnorm in (6.48). The experimentally obtained plots in Fig. 6.23 portray the normalized response in V2at>(t) and i2at>(t) to a step change in Rn0rm from 62.5 % to 100 % at t = 5.92 s. This response is rather sluggish for all practical purposes. Time (s) Fig. 6.23 Experimental results of the response to a step change in R n 0 rm from 62.5 to 100 % with PFC load. Realizing not only the approximate nature of the design but also that the dynamics of the original nonlinear system were evaluated at steady state, practical tuning of the controller may be Chapter 6 Stability Analysis of the System 131 necessary. We found that the system performed better with 10K<iCigain. It produced an adequate response and hints of unstable behaviour were not detected. Factors contributing to the disagreement in K d C i g a i n are: (a) the low sampling rate of the discretization process generally results in lower magnitude response and more phase lag for G(s) than would have been the case if higher sampling rate would have been possible (b) K d C i g a i n is a worst-case design (c) the plots in Fig. 5.4 is a steady-state response from which a transient response is likely to deviate (d) the model of the system was analyzed operating in steady-state orbit. The experimentally obtained plots in Fig. 6.24 and Fig. 6.25 show the response for a step-up and step-down change in Rn 0rm between 62.5 % and 100 % with 10 Kd C , g a i n . The arrows indicate the instant of the step. A disadvantage with this type of power supply is the sluggish response to step changes in the load due to the low bandwidth of the firing of the SCRs. Thus, it is less suitable for loads with large and rapid step changes. Finally, the experimentally obtained plots in Fig. 6.26 through Fig. 6.29 show the response for a step-up and step-down change in Rnorm between 0 % and 100 % for resistive as well as PFC load. The arrows indicate the instant of the step. Fig. 6.24 Experimental results of the response to a step change in R n 0rm from 62.5 to 100 % with PFC load. Chapter 6 Stability Analysis of the System 132 CD -I O ) 1 CO | 0.5 O S U | -0.5 I "I Kit rt-ti - U - i rrn 3.1 3.15 3.2 3.25 3.3 c 1 (13 Jj) .5 | l ° g -0.5 o 2 -1 3.1 3.15 3.2 Time (s) Jul r rA-fl-f j2ab -vC \J. \J AJ \J Li I J y i 1 3.25 3.3 Fig. 6.25 Experimental results of the response to a step change in R n o r m from 100 to 62.5 % with PFC load. I I L _ 1 1 I I I I 8.43 8.44 8.45 8.46 8.47 8.48 8.49 8.5 8.51 Time (s) Fig. 6.26 Experimental results of the response to a step change in R n 0 r m from 0 to 100 % with R load. Chapter 6 Stability Analysis of the System 133 Fig. 6.27 Experimental results of the response to a step change in R n 0 r m from 100 to 0 % with R load. 5.7 5.8 5.9 I F 2ab 6.1 5.6 5.7 5.8 5.9 6 6.1 Time (s) Fig. 6.28 Experimental results of the response to a step change in R n 0 r m from 0 to 100 % with PFC load. Chapter 6 Stability Analysis of the System 134 CD CO O > ^ -a > N > ca o 1 0.5-0--0.5 -1 7T i f "If TT 2ab 4.05 4.1 4.15 4.2 4.25 4.3 Time (s) Fig. 6.29 Experimental results of the response to a step change in R n 0 rm from 100 to 0 % with PFC load. Regarding the traces in Fig. 6.23 through Fig. 6.29, it must be noted that the response is somewhat dependent on where in the cycle of the waveform the step change occurs. The cases shown were selected to be representative for a general response of the system. 6.6.6 Comments on the Experimental Results It may seem unsatisfactory that the design method of the controller contains substantial approximations due to the low sampling rate. However, considering the particular system consisting of nonlinear elements and in essence an unknown load, we consider that the proposed method is of significant value and an approach that can be used with reasonable accuracy. We note the instrumental discovery that a ferroresonant power supply can indeed be compatible with certain PFC loads. Note that the system is inherently very sluggish, i.e., the control action is limited to twice the period of the output frequency. This is generally several orders of magnitude slower than the dynamics of the unknown load. Chapter 6 Stability Analysis of the System 135 The particular transfer function of the PFC load in this chapter did not produce discrete-time poles with negative real parts. Such poles may be dealt with using specific algorithms discussed in [88]. 6.7 Summary This chapter presented a comprehensive stability analysis of the ferroresonant power supply using Poincare maps and the theory of Floquet. These techniques are suitable for periodic systems operating in steady state. It was shown that the system could be controlled with passive R, RL, and RC loads. However, for a specific, but deemed general, actively PFC load, an LTI controller can essentially not control all possible characteristics of the particular topology. It must be emphasized that the equation representing the PFC load was an approximation and a more accurate transfer function of the load may lead to different conclusions. However, the purpose of this chapter is the generality of the analysis. The results of this chapter are instrumental in that the reason for the instability or stability is analytically demonstrated. The derived model has the significant advantage that a general load in state-space representation can be studied, including various PFC topologies, without having to repeat the cumbersome circuit analysis of the system. Based on the pole and zero plots, a worst-case or typical-case controller may conveniently be designed provided that the system can be robustly controlled. We also demonstrated a particular case where an LTI controller was able to stabilize a specific actively PFC load. We note that the controller design was approximate in nature, but deemed an adequate approach considering the complexity of the system. It should also be observed that the actively PFC load that was shown not to be controllable by an LTI controller might actually be controllable by other types of controllers. A particular adequate controller for the system in general is a periodically time varying controller. This area of research is left for future work. Finally, although not specifically addressed in this chapter, controllability and observability were also verified for the system with the various types of loads. Chapter 7 Conclusion and Future Work 7.1 Conclusion This study has focused on the stability analysis of a system constituting of a ferroresonant power supply and actively P F C loads. Instability for this type of system has been known to exist since the beginning of 1990s. However, prior to this work, to the best of the knowledge of this author, an analytical explanation for the phenomenon has previously not been presented. The main objective of this study was to derive a model for the system and to determine whether it can be stabilized by a L T I controller in line-mode operation and with heavy loads. The study focused on the line-mode operation for a particular source and load combination, since without proper operation in this mode, the system cannot perform its intended task. The thesis consists of the following major subjects: ferroresonant transformer modeling, a generic state-space representation of the complete system, and stability analysis. In Chapter 2, we empirically established and verified the instability problem. It was thoroughly addressed that the use of Poincare mapping is necessary to determine, with certainty, unstable operating conditions. A quantitative model of the system was also discussed. The next chapter addressed the main component of the modeling process, the modeling of the C V T . A comprehensive equivalent circuit of this type of transformer was produced. Unlike a traditional two-winding transformer, the ferroresonant transformer is designed to operate in the saturated region. This complicates the modeling process. Ferroresonance is in itself also a subtle phenomenon, which may be uncomplicated to model on a cycle-to-cycle basis in an rms fashion, 136 Chapter 7 Conclusion and Future Work 137 but cumbersome to describe in the instantaneous time frame. To better facilitate the analysis in the subsequent chapters, the derived circuit was also reduced to minimize the number of states. In Chapter 4, the equivalent circuit of the ferroresonant power supply was presented in a nonlinear state-space format where the load was integrated in a generic fashion. Both a reduced and a comprehensive model were derived. Earlier works have relied on mathematical models similar to the reduced model. Generally, these works have considered the secondary magnetizing inductance as being linear or piecewise linear taking only two values. In this study, we regarded the aforementioned inductance as a continuously nonlinear element. However, even so, we concluded that the comprehensive model is superior; especially, when transients are to be analyzed. Thus, for computer simulations, this model is essentially necessary for meaningful studies. How to characterize the envelope of the input impedance approximately of a device was addressed in Chapter 5. It was discussed how with small means such an investigation may be performed. However, the difficulty lies in the interpretation of the results. In the case the system is significantly nonlinear, the alternatives are to model it as such or to use parts of the empirical results to create a linear model that is deemed sufficiently adequate. This was also the approach taken in this work. A substantial stability analysis was undertaken using Poincare maps and the theory of Floquet. The large signal nonlinear source and load combination was linearized into an LTP system. This system was in turn sampled on a half-periodic basis. The obtained discrete-time system was then analyzed regarding its stability characteristics. We also addressed a sampling rate issue, which does not seem to be discussed in the literature, regarding the theory of Floquet. Two load cases were presented. A load that could not be robustly controlled by an LTI controller and one that could. Thus, the conclusion is that the particular system under investigation can be made compatible with some actively PFC loads but probably not with all. The difficulty with this postulation is that a comprehensive characterization of all actively PFC loads is unlikely to exist. As is further discussed in the subsequent section, a more elaborate controller may be able to control a large number of actively PFC loads. We also exemplified an approximate procedure of a controller design that could stabilize the system in line-mode operation. Chapter 7 Conclusion and Future Work 138 7.2 Future Work Due to the complexity of the particular system treated in this thesis, a wealth of options of future research work exists. One consideration for such work is to implement nonlinear core resistances or hysteresis, especially for the secondary magnetizing branch. The range of the instantaneous power VLm(t)iLm(t) for the core branches, both referred to the N2ab side, and at 100 % excitation are [-34, 56] V A and [-133, 174] V A respectively. The maximum energy stored in C* = 214 /xF is 1/2C*V22ab,max = 9.26 J. Assuming the energy function is sinusoidal with co = 2*377 rad/s and since power is the time derivative of the energy, this gives the amplitude of the instantaneous power as (9.26/2)*co = 3.49 kVA. Similar calculations give for Li and L* o n 5.34 k V A and 5.80 k V A respectively. The model investigated in this study is less accurate for light loads since the instantaneous V A due to hysteresis is not considered. As a matter of fact, the model with linear core resistances provides damping to the system at light loads at all instances in time during any given period. However, the main purpose of the study was the stability analysis at large loads. Besides, due to overall efficiency concerns, the ferroresonant type of UPS is not well suited to operate with small loads due to the large circulating currents in the tank circuit. An additional contemplation is the investigation of having less saturation in the secondary magnetizing branch and instead increasing the size of the external L that consequently in turn would need to be controlled more frequently than twice the period. This may improve the stability of the system due to the increase of the control bandwidth. It would also significantly decrease the harmonic content of the output voltage of the UPS. However, it must be observed that the excellent transient characteristics of the C V T are maintained in such an alteration process. 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Appendix A Theory and Derivations of Equations A.l The State-Space Equation For a nonlinear system with the state-space representation x(t) = f(x(t),t,u(t)) , x(t 0) = x 0 y(t) = g(x(t),t,u(t)) (A.l) the right hand side, for periodically time-varying state variables, can be expressed as f (x(t + kT), t + kT, u(t))=f (x(t), t, u(t)) g(x(t + kT), t + kT, u(t))= g(x(t), t, u(t)). (A.2) The linearization of (A.l) about steady-state trajectories in the n-dimensional state space yields the time-varying steady-state Jacobians « ( 0 = £ < B M ) « ( 0 - £ ( c ( t ) ) • JSW = ^ (B(,)) . J ; ( 0 - £ ( D ( 0 ) (A.3) that similar to (A.2) also are periodic such that 149 Appendix A Theory and Derivations of Equations 150 j;(t+kr)=j;(t) , js(t+kT)=j;(t) (A 4) j ^ s ( t + k T ) = j - ( t ) , j ; ( t+kT )= j ; ( t ) . The steady-state linearized small signal and periodically time-varying system now takes the form of Ax s s(t) = JsAs(t)Axss(t) + J^s(t)Auss(t) , Ax s s s ( t 0 ) = Ax*s Ay s s (t) = J^ s (t)Axss (t)+J£ (t)Auss (t) (A.5) having the solution equation t Ax s s(t) = O s s ( t , t 0 )Ax s s ( t 0 )+ ja>ss(t,T)j^(T)Auss(x)dT (A.6) where O(t,to) represents the state-transition matrix and Au(t) is the input. The output equation is Ay s s (t) = J^ s (t)0>ss (t, t 0) Ax s s (t0)+ Jj£ (t)O s s (t, x) J^ s (x)Auss (x)dx +J^s (t)Auss (t). (A.7) to In the periodic case, when O(t+T,to+T) = O(t,to) where T is the period, the state-transition matrix O(t+T,to+T) is called the monodromy matrix. A.2 Poincare Maps This chapter is adopted from [89], and it introduces the fundamentals of Poincare maps. Before the introduction of the fundamentals of Poincare maps near closed or periodic orbits, we first state a few motivating propositions without proofs. Proposition A . 1. Let xss be a hyperbolic equilibrium or fixed point, that is, none of the eigenvalues of the linearized Poincare map under steady state conditions are on the unit circle, of the unforced nonlinear system [71] x = f(x). (A.8) Then, for a closed orbit under small periodic forcing, the state equation can be expressed as Appendix A Theory and Derivations of Equations 151 x = f(x,t)+eg(x,t), x e l " , e e l ' (A.9) where f (x ,T + t) = f(x, t) and the periodic forcing term is g(x,T + t) = g(x, t) . For a specific initial condition, this system has a periodic solution curve, also known as orbit or trajectory. The periodic solution is such that x(T + t) = x(t). The vector field, which is the right hand side of (A.8), generates a flow cp(x,t) of period 0 < T < co for a sufficiently small nonzero e. The families of the solution curves define the global behaviour of the flow. The stability characteristic of the closed orbit is that of the periodically sampled steady-state trajectory x s s(t). The system is locally stable when the linearization of the Poincare map, Df(x s s(t)), has all its eigenvalues inside the unit circle. • Proposition A . 2. The above also implies that a continuous flow produces a discrete map. A discrete dynamical linear system and its flow cp(x,t) for t fixed, can be described by the difference equation x k + 1 =<D(k + l , k ) x k (A. 10) where 0 ( k +1, k) is the state-transition matrix. Analogous, for the discrete nonlinear dynamical system, we have the nonlinear map x k + 1 = G ( x k ) (A.11) where the nonlinear map G(x k ) = (p(x,tflxed) is a nonlinear vector-valued function. The corresponding small signal system is described by A x k + i = 3 G dx A x k . (A. 12) If there is a cycle of k distinct points p; = G'(po), i = l , 2 , . . . , k - l , and G k(p 0) = po where Gk(po) denotes k iterations G(G(G(po))) • • •, there exists an orbit of period k. The local stability of such an orbit is determined by the linearization DGk(po). The chain rule gives DGk(po) = DG k(p 0)DG k" 1(po)...DG 2(po)DG(p 0). • Proposition A . 3. The corresponding Poincare map to the system in (A.8) is a discrete-time reduced order system described by Appendix A Theory and Derivations of Equations 152 x k+l (A.13) where x k e l ° 1 is given by the sampling instant on E e M n at time k. In steady state, x k s + 1 = x s k s and the map p ( x k ) has a fixed point at the periodically fixed sample instant denoted x k (t l e ) A fixed point is said to be locally stable if a solution x(t) intersects with the neighbourhood of the fixed point at the sample instant for all t. Fixed points of the map correspond to periodic orbits of the system. The Poincare map analysis method is valid as long as the sampled values of Poincare maps near closed or periodic orbits are defined as follows. Let y be a periodic orbit of period T of some flow cp(x,t) in state space due to a nonlinear vector field f(x,t). The zeros or equilibria of f(x,t) are called the fixed points of (p(x,t). The trajectory in Fig. A . l shows an example of the flow of a three-dimensional state vector in R 3 and the condition when steady state is not established. For the local Poincare map, we sample the state vector periodically. The figure shows that this is analogous to imagining having a hypersurface E e M ° of dimension n -1 inserted in the path of tp(x,t). We call this hypersurface the local cross section. The local hypersurface E need not be planar as depicted, but it must be chosen such that the orbit of cp(x,t) is everywhere transverse to E. This translates to f(x,t) • n(x) ^ 0 where n(x) is the unit normal to E at the sampled point p of cp(x,t) in the neighbourhood Y c E . When y has multiple intersections with £ , E is to be shrunken until only one intersection remains. Two points are indicated in Fig. A . l , po and pi . We define the local Poincare map to be on the surface of E. The first return or Poincare map P: Y —> E is defined for p 0 , P ; e Y by pi = P(Po) = <p(Po>t) for the initial point p 0 at time to and the first return point pi at time ti. Observe, that generally, (p(p 1,T + t )^cp(p 0 , t ) , but in the limit, pi —> p 0 , (p(pj,T + t) = tp(p 0 ,T). We now say that p is a fixed point in the discrete time-domain of the map P: Y —*• E and that the local stability of y for the flow (p(x,t) in the neighbourhood Y can be predicted by the analysis of P(p). For a locally stable system, we require all y in Y to approach p . The periodic orbit is locally stable if and only if all the eigenvalues of the linearized Poincare map DP(p) are of modulus less than one. the state variables are in the neighbourhood of their respective steady-state values. • Appendix A Theory and Derivations of Equations 153 dy Fig. A . 1 Orbit of a three-dimensional state vector. The plot in Fig. A.2 shows the two-dimensional local map for the three-dimensional state vector of a nonlinear system x = f (x),xe I " ,x(o) = x0 with a fixed unstable hyperbolic point p of type saddle. For the periodic orbit y, p has a stable local manifold W , ^ . ( Y ) = W s(p)e IR1 and an unstable local manifold W 1 O \ , ( Y ) = W u(p)e Rl. Unless the solution stays exactly on the former, the orbit is unstable. The local manifolds are nonlinear analogies of the eigenspaces of the linear system. Using the local manifolds as initial conditions generate the corresponding global manifolds W s(y)e R2 and W u (y)e R 2 , indicated in the figure, by letting Ws(p) and W u (p) flow forwards and backwards in time respectively along the periodic orbit. The global manifolds describe the vector field about the orbit in state space. For W ^ Y ) , (p(x,t) approaches y as t —• co. Note, that the local manifolds are present wherever the orbit is sampled at some fixed point, and unless all such fixed points are stable, the periodic orbit will not be stable in steady state. Existence and uniqueness of the solution of a nonlinear system ensure that two W l o c (y) or two W , ^ . ( Y ) of distinct fixed points cannot intersect. Likewise, W s (xss) or W u (xss) cannot intersect itself. For forced periodic oscillations, and for the system augmented with a state variable that expresses time explicitly, we define the Poincare map on the global cross section or hypersurface Appendix A Theory and Derivations of Equations 154 £ as P: E —» E and the map is described by P(x(t0)) = x(x(t 0), t + T) . It is noteworthy that the time of flight, the period T, is the same for all points x e E . Fig. A.2 Periodic orbit of a three-dimensional state vector. The Poincare map is found by (a) converting the system of differential equations to an n+1 order autonomous differential equation where time is expressed explicitly in the augmented state variable n+1 (b) finding the solution of the system (c) finding the period T for steady-state conditions, thus obtaining the Poincare map P(-) from the solution with t = T. The local stability can be evaluated by (a) evaluating the solution at steady state, t = T (b) differentiating P(-) with respect to the state variable (c) evaluating DP(p) at the steady state orbit. The local stability of y is now determined from the eigenvalues of DP(p). Appendix A Theory and Derivations of Equations 155 Generally, it is difficult if not impossible to compute P(p) since the system of differential equations must first be solved involving integration, and second, the steady-state vector xss(t) must be found at the sample instant typically involving the solution of a transcendental equation. However, using Floquet theory, the system can be linearized about y. A periodically time-varying nonlinear system can be written as x(t) = f(x,t), x(t 0) = x 0 , t > t 0 (A. 14) where x(T +1) = x(t) and f(T + t) = f(t). The corresponding small signal system evaluated at steady state is such that Ax(t): df dx (xss(t))Ux, J A ( t ) : = ^ ( x " ( t ) ) e R ' » (A.15) dx where J A ( T + t) = J A ( t ) . As detailed in the next section, the state-transition matrix of (A.15) may be written as O(t , t 0 ) = L ( t , t 0 ) e F ( t - t ° ) (A.16) where <D(T +1,T +1 0) = 0(t, t 0 ) is nonsingular V t since it is a state-transition matrix of a linear time-varying system, the nonsingular matrix L(T + t ) : = L ( t ) e K ™ with L(0) = I, and the constant matrix F = (l/T)log(o(T + t 0 , t 0 ) ) . The state-transition matrix of (A.15) may be found from ^ ( t , t 0 ) = J A ( t )o ( t , t 0 ) , O ( t 0 , t 0 ) = I (A.17) where a closed form of O(t , t 0 ) can only be found for special cases [74]. Thus, numerical integration is generally required. We can now write (A.15) as the mapping Ax(t) = O( t , t 0 )Ax( t 0 ) . (A.18) Thus, the behaviour of the orbit y on £ is predicted by the eigenvalues A, d i of O( t , t 0 ) . These eigenvalues are called the Floquet or characteristic multipliers whereas the eigenvalues X c ; of F are called the Floquet or characteristic exponents of y. The eigenvalues are related as A, c i =e?"diT, i = l,2,... ,n. For a locally stable periodic system, all | A , d i | < 1 except for one, call it Appendix A Theory and Derivations of Equations 156 A,d n , which is unity. This eigenvalue is associated with the manifold that is tangent to the orbit y at the fixed point xps e £ . Unless A,d n = 1, a periodic orbit would not be sustained. Thus, the Floquet multipliers Xdl, X d 2 , Xd3 A , d n . , equals the eigenvalues of the linearized Poincare map, which is the Jacobian of the Poincare map evaluated at the fixed point DP(xps). The Xdi of DP(xps) determine the amount of contraction and expansion in the neighbourhood of the periodic solution during one cycle. Moreover, the n-1 eigenvectors of O(t , t 0 ) corresponding to the n-1 Floquet multipliers form a basis of the Poincare map. A.3 Floquet Theory It is well known that analysis methods for LTI system generally do not apply to L T V systems. However, a special case is linear time-periodic (LTP) systems, which behave in a fashion similar to LTI systems and consequently the methods of analysis are also analogous. The Floquet theory is such an example [78], [90]. Using a transformation that preserves stability in the sense of Lyapunov, the Floquet theory maps an LTP system into an LTI system. Consider the LTP system x(t)=A(t)x(t) + B(t)u(t), t> t 0 , x(t0) = x0 (A 19) y(t) = C(t)x(t)+D(t)u(t) where A(t + T)=A(t) , T > 0 . (A.20) The unforced part of (A. 19) is written as x(t) = A(t)x(t), t > t 0 , x(t 0) = x 0. (A.21) The state-transition matrix of such a system is usually not periodic but nevertheless it is such that O(t + T , t 0 + T ) = O(t , t 0 ) . (A.22) By transitivity, we also have Appendix A Theory and Derivations of Equations 157 O(t + k T , t 0 ) = O ( t , t 0 ) Q k ( T , t 0 ) , k e Z . (A.23) The state-transition matrix of the system in (A.21) can be factored as O(t , t 0 ) = L ( t , t 0 ) e F ( t - t ° ) (A.24) where L(t) is periodic, nonsingular for all t, continuous with piecewise continuous derivative, possibly complex, and L(0) = I. The constant matrix F may also be complex. Differentiation of (A.24) gives 6(t, t 0 ) = (L(t, t 0 )+L(t, t 0 )F) e F ( t- t o ) . (A.25) This implies that and A( t , t 0 ) = L( t , t 0 )+L( t , t 0 )F (A.26) 0(T + t 0 , t 0 ) = e F ( T + t o ) . (A.27) Consider the transformation *(0 = L(t)x e (t) , (A.28) which gives the continuous LTI system such that [91] *e(0 = F*e(t), x e ( t 0 ) = x 0 (A.29) and for the system in (A. 19) we have x e(t) = Fx e(t)+L" ,B(t)u(t) y(t) = C(t)L(t)x e(t)+D(t)u(t). (A.30) Due to the characteristics of L(t), this matrix is a Lyapunov transformation. It follows that (A.21) and (A.29) are equivalent in the sense of Lyapunov as are (A. 19) and (A.30). Thus, if the transformed system is stable in the sense of Lyapunov, so is also the original system where A(t + T) = A(t). This is also referred to as the theory of Floquet. Generally, however, a transformation does not preserve stability. Appendix A Theory and Derivations of Equations 158 The state-transition matrix is called the monodromy matrix, which we denote O ( t 0 + T , t 0 ) , if O(t + T , t 0 ) = O( t , t 0 ) . The eigenvalues X6i of O ( t 0 + T , t 0 ) are called the Floquet multipliers, a.k.a. the characteristic multipliers or Poincare multipliers. The eigenvalues Xc i of F are called the Floquet exponents or the characteristic exponents. For an LTP system, these exponents are not unique whereas the Floquet multipliers are. Since the Floquet multipliers relate to a discrete-time system, the system is stable if and only if all the eigenvalues of 0>(t0 + T, t 0 ) are inside the unit circle. This corresponds to the eigenvalues of F having negative real parts. From (A.27), it follows that since A(t) is also periodic for the period 2T. Consequently, for a complex F, we must have that F = ± l n ( * ( t 0 + T , t 0 ) ) (A.31) or if F is complex F = ^ln(o(t 0+T,t 0)o(t 0+T,t 0)) = ^ l n ( * ( t 0 + 2 T , t 0 ) ) = ^ln[(O>(t 0 +2T,t 0 )) 2 ] (A.32) L(t + 2T) = L(t), L(0) = I . (A.33) From (A.31) it also follows that the eigenvalues are related as (A.34) Considering the polar representation of A, d i such that 7 t < 0 d i <n (A.35) Then the continuous and discrete time eigenvalues are related as Appendix A Theory and Derivations of Equations 159 (A.36) where the real part is R e ^ J ^ h f , , 2T 1 ln[X d iconj(^ d i)] (A.37) and the imaginary part for k = 0 is ( I r 4 d , i ) Im(A,c j )= arctan (A.38) V J where - 7t/T < Im(XC i l) < n/T . For the periodic system of period T, 7i/T corresponds to the Nyquist frequency. Although eigenvalues are undefined for continuous time-varying systems, we give here an example of folding. Assume that the time-varying system (A. 19) is a result of a partitioned system with at least one LTI subpart. We note then that the modes with the damped natural frequency - °° < to d a m p c < o o of the original continuous-time periodic system are transformed into the discrete-time system according to the particular period, which may be considered being the sampling frequency. When transformed back to the continuous-time domain, the resulting damped natural frequency is - 7i/T < co d a m p c d c <7i/T. Finally, we note that the possible LTI modes described above may not be nicely preserved under the discretization process. These modes may interact with other modes resulting from the discretization process and consequently become altered. The equations below show the derivations of the partial derivatives of (6.2), which is repeated below for convenience. The subsequent derivations follow those of [92]. However, after private discussions with the author of the aforementioned reference, it was concluded that a few derivations needed minor modifications. Consequently, these alterations are implemented below. A.4 Derivation of Equation (6.6) Appendix A Theory and Derivations of Equations 160 x(x(t 0 ) , t 0 , t ) = x(t 0)+ Jf(x,x)dx. to The partial derivative of (6.2) with respect to x(to) is (6.2) 3x( t 0 ) 3 x(x(t 0 ) , t 0 , t ) 8 3x(t 0 ) d*(0 i x(t0)+jf(x(x(t0),t0,x))dx Jf(x(x(t0),t0,x))dX (A.39) = I + k 7—^dx where the last row leads to (6.3). The substitution of (A.39) into the last row of the right hand side of the same yields df 3x(t0) -x(x(t 0 ), t 0 , t) = I+ I — 1+ J df dx •dx, dx. ' df T2=t df T'=t df 3x = 1+ f—dx+ f — f :——dx.dx 1AV J 3 t;ax t2ftoaxTiitoaxax(t0) l u " 2 (A.40) A further substitution of the last row of (A.39) into (A.40) gives 3x(t0) x(x(t 0 ) , t 0 , t ) = I+J—dx + J — J — 1+ J , — d x , dxr ±. dx. x ,= ,„ a x 9x ( t 0 ) dx 2dx 3 = 1+ I—dx+ I — I — dx,dx2 , ox , : , d x T : , dx + J T ~ J T ~ J / \ dx,dx 2dx 3 J t 0^x i t oa x itoaxaxt0 (A.41) The partial derivative of (6.2) with respect to to is Appendix A Theory and Derivations of Equations 161 3t 3 3 { ' ^ x(x(t0),t0,t) = — x(t0)+ Jf(x(x(t0),t0,x))dx :0--^-jf(x(x(t 0 ) , t 0 ,x))dx 0 t 3 'f. = -f(x(x(t0),t0,t0))-— ff(x(x(t0),t0,x))dx 3 t 0 t J = -f(x(x(t0),t0))+j|I|fdx (A.42) where the last row leads to (6.4). The substitution of (A.42) into the last row of the right hand side of the same gives 3t 3 x(x(t 0 ) , t 0 , t ) = -f(x(x(t 0),t 0)) + { |1 ' V3f ' •f(x(x(t0)>t0))+TlJ2^^dx •dx, V t ^ X / V 'o J T,=t - V - T,=T. 3f '72 3f 3x 1+J—dx f(x(x(t 0),t 0))+ { — { ^ - i ^ d x 1 d x 2 . t i = t o 3 x t i J t o 3 x 3 t 0 (A.43) A further substitution of the last row of (A.42) into (A.43) gives a at, -x(x(t0),t0,t) = - f(x(x(t 0),t 0)) ( + J J >^  -'M*(0.to))+ J — — ^ T 3 _ T 0 T 2 _ T 0 |dx2dx3 ' T V3f, T3?' af T 27'3f, , A V 'o a x, i, a x T 3 _ T O T 2 _ T 0 f(x(x(t 0),t 0)) J r af T 273 af T i7 2 3f ax J J + J T ~ J T " J T ~ ^ T d T i d T 2 d T 3 T ax 3xT_j ax at0 (A.44) The respective last term of the right hand side of (A.41) and (A.44) are identical and further substitution leads to the same pattern. We assume that the aforementioned term converges Appendix A Theory and Derivations of Equations 162 to zero since otherwise the system may be unstable. A comparison of (A.41) and (A.44) with the omission of the respective last term thus gives — x(x(t0),t0,t) = — * — f(x(x(t0),t0)) = -o(t,t0)f(x(x(t0),t0)) (A.45) d t 0 3x(t 0 ) where we also show the state-transition matrix O(t,to) for emphasis. A.5 Derivation of Equations (6.16) and (6.17) Replacing the appropriate terms of (6.12), (6.13), and (6.14) with the respective state-transition matrix gives the following state vector at time ti Ax o f f (t,)=os0sff (tr, p0sff ) P T « C (p- , < )po:sff (< f f, t?) A x o f f ( t l ) + ®3r(tr.P3r)PT®-(P-.a-) q ^ ( « o „ ) 3*5r (0 3t* A a and • • i ( . r . P i ) p > : t e . « : ) i ^ dtn At f (A.46) + A S S / fss oss ) p T 3Xo n (t) ^ o f f VL1 'Pott lr ™ss +-dxl (t) ap SS off AP B d = [b M b 2 i d] Au = [Aa A t 0 ] T (A.47) (A.48) Using (A.45), the last term of the last row of (A.46) can be expressed as Appendix A Theory and Derivations of Equations 163 a*sosff(t)l =-*£ (tr, P i ) f te )- P i )) =-*:sff frr. p:sf )?Tf te k « te \ P: S„ )). (A.49) which results in the AP term in (A.46) to equal zero. Appendix B Hardware Data B.l Hardware Data This work used the following ferroresonant UPS for the modeling and analysis. Manufacture: Alpha Technology. Input: 120 V , 60 Hz, 17.6 A . Output: 120 V , 60 Hz, 1670 W, 2500 V A . Model: 017-073-54-001. S/N: 50553-0994. S.P.: CFR 2500 M E D . Drawing #: 017-071-05. The following power supply was used as actively PFC load. Manufacture: Lucent Technologies, M99DJ10, 98DJ12327380, Date: 981218, CS774B, Engineering Sample. Series: 1:0 G. Input: 170-264 V , 50/60 Hz. Output 1: 56.2 V D C . Output 2: 56.2 V D C . Output total: 1500 W, 26.7 A. B.2 Data of Laboratory Equipment The following equipment was used to obtain the experimental results. Oscilloscope: Tektronix TDS 340A, Serial # B014407. Voltage isolators: U B C E E No 2 and U B C E E No 3. Current transformers: U B C E E No 4 and U B C E E No 6. Function generator: Philips P M 5132, No 9445 051 3200B. Signal generator: Hewlett Packard 4204A. Oscillator: California Instruments 3001TCA. Data acquisition: Labview 7.0, PCI-6025 E, 12 bits. Due to inaccuracy in the laboratory equipment, most waveforms in this thesis were signal processed before they were 164 Appendix B Hardware Data 165 plotted. This procedure was not deemed to have altered or manipulated the results in a fashion to present more idealized outcomes. 

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