MULTI-WINDING MODEL WITH DIRECT EXTRACTION OF PARAMETERS FROM VOLTAGE MEASUREMENTS AJAY GANGUPOMU

Transcription

1 MULTI-WINDING MODEL WITH DIRECT EXTRACTION OF PARAMETERS FROM VOLTAGE MEASUREMENTS By AJAY GANGUPOMU A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003

2 Copyright 2003 by Ajay Gangupomu

3 I dedicate this document to my mom, dad, and brother; and to my teacher.

4 ACKNOWLEDGMENTS I thank Dr. Khai D.T. Ngo for his continuous guidance, encouragement, and support throughout the course of this study. I would also like to thank Dr. Alexander Domijan and Dr. William R. Eisenstadt for their suggestions which have helped me a lot in completing this work. iv

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS... iv LIST OF TABLES... vii LIST OF FIGURES... viii ABSTRACT... xi CHAPTER 1 INTRODUCTION Background Integrated Magnetics Transformer Equivalent Circuits And Models Multi-Winding Models And High-Frequency Effects Motivation For Multiwinding Transformer Models MULTI-WINDING TRANSFORMER MODELING METHODS Introduction General Multi-Winding Transformer Equivalent Circuit Model Description Equivalent Circuit Cross-Coupled-Secondaries Model Model Description Parameter Extraction Frequency Dependent Fully Distributed Model Model Description Parameter Extraction Extended Cantilever Model Model Description Parameter Extraction Ladder Model Model Description Parameter Extraction...18 v

6 2.7 Leakage Impedance Model Model Description Parameter Extraction Synopsis MESH MODEL Introduction Model Description Parameter Extraction Procedure Extraction Of Open-Circuit Parameters Z jk, J K Extraction Of Short-Circuit Parameters N k, Z Comparison Of Mesh Model With ECM Parameter Extraction Extraction Of ECM Parameters From Mesh Model Synopsis EXPERIMENTAL VERIFICATION OF THE MESH MODEL Introduction Model And Measurements Flyback Transformer Example Synopsis CONCLUSION Summary Future Work Prospects LIST OF REFERENCES...62 BIOGRAPHICAL SKETCH...64 vi

7 LIST OF TABLES Table page 3-1 Comparison of number of measurements for Mesh model and ECM Results of Load test on Mesh model at 50 KHz Results of Load test on Mesh model at 25 KHz Results of Load test on ECM at 50 KHz Results of Load test on ECM at 25 KHz...55 vii

8 LIST OF FIGURES Figure page 1-1 Micro-transformers fabricated on silicon The T Model of a two-winding transformer The Pi Model of a two-winding transformer Lumped parameter Model with all elements referred to primary winding Frequency Dependent Fully distributed model for a two winding transformer Extended Cantilever model for a Four-Winding Transformer Equivalent circuit of a four winding transformer Equivalent circuit of a four winding transformer neglecting the excitation currents Cross coupled secondaries winding model of 4-winding transformer Flow Chart for generating the fully distributed model Frequency dependent fully distributed model of a 3-winding transformer Extended cantilever model of an N-winding transformer Broadband Extended Cantilever Model Ladder Model of a multiwinding transformer Leakage inductance measurement setup Leakage Impedance Model for a three winding Transformer Leakage Impedance model of a four winding transformer Block Diagram of a Mesh Model for the n windings of a transformer or inductor Mutual Coupling Based Mesh Model for a three-winding transformer viii

9 3-3 Mesh Model for the N windings of a transformer or inductor Extraction of Z 1k for the n winding mesh model of a transformer or inductor Extraction of Z 12 and Z 13 for a three-winding transformer Extraction of Z jk for an n-winding mesh model Extraction of Z 23 for a three-winding mesh model Extraction of Z 11 for the n winding mesh model of a transformer or inductor Extraction of Z 11 for a three-winding mesh model Extraction of Z kks for the n winding mesh model Extraction of Z 22s for a three-winding mesh model Extraction of Z 1ks for the n winding mesh model of a transformer or inductor Extraction of Z 12s for a three-winding mesh model Saber Schematic of Mesh Model derived for the test specimen Magnitude variation of Z 11 over frequency Phase variation of Z 11 over frequency Magnitude variation of Z' 12 over frequency Phase variation of Z' 12 over frequency Magnitude variation of Z' 13 over frequency Phase variation of Z' 13 over frequency Magnitude variation of Z' 23 over frequency Phase variation of Z' 23 over frequency Magnitude variation of Z 22S over frequency Phase variation of Z 22S over frequency Magnitude variation of Z 12S over frequency Phase variation of Z 12S over frequency Magnitude variation of Z 33S over frequency...43 ix

10 4-14 Phase variation of Z 33S over frequency Magnitude variation of Z 13S over frequency Phase variation of Z 13S over frequency Saber Schematic of Mesh Model derived for the test specimen Simulation of Z 11 variation over frequency Simulation of Z' 12 variation over frequency Simulation of Z' 13 variation over frequency Simulation of Z' 23 variation over frequency Simulation of Z 22s variation over frequency Simulation of Z 12S variation over frequency Simulation of Z 33S variation over frequency Simulation of Z 13S variation over frequency Mesh Model schematic of 4-winding flyback transformer Mesh Model schematic of 4-winding flyback transformer Mesh Model schematic of 4-winding flyback transformer Flyback converter example circuit Waveforms from Flyback converter circuit simulation Waveforms from Flyback converter circuit measurement...59 x

11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science MULTI-WINDING MODEL WITH DIRECT EXTRACTION OF PARAMETERS FROM VOLTAGE MEASUREMENTS By Ajay Gangupomu December 2003 Chair: Khai D.T. Ngo Major Department: Electrical and Computer Engineering Recent advances in the field of power electronics (such as multiple output switched mode power supplies) have resulted in multiple winding magnetic devices operating at high frequencies being increasingly used in these circuits. Along with the development of technologies such as integrated magnetics and matrix transformers and inductors this has created a need to model multiple magnetic windings accurately. This task is complicated by the interaction between windings leading to phenomenon such as cross-regulation among the windings. A good model should be able to take into account the cross-coupling between windings; and should predict transformer winding behavior such as winding current ripples and output voltage cross-regulation under various conduction modes and excitation conditions. At the same time the model should be general enough to be extended to any number of windings; and should have a simple procedure to extract its parameters accurately. A significant problem in most existing transformer models is that xi

12 to measure the model parameters short-circuit current sensing is required, which introduces an error in the extracted model. We propose a model for predicting the winding behavior of a multi-winding transformer or inductor based on the Mesh topology. The Mesh model is a general winding model with directly measurable parameters. It accurately models the mutual coupling among the windings. We outline a procedure to extract the parameters of the Mesh model without the need for any short-circuit current measurements. Thus we avoid errors in model parameters due to non-ideal short-circuit measurements. A three winding transformer was characterized; and its equivalent Mesh model was obtained using the parameter-extraction procedure. The generated model was simulated in a circuit simulator and the measurements were verified. The model was also used to predict transformer behavior under various loading conditions; and its performance was compared to the actual transformer behavior and was found to be satisfactory. The validity of the model was thus proved. xii

13 CHAPTER 1 INTRODUCTION 1.1 Background Power electronic equipment and circuits have developed rapidly in recent years; and find applications in a diverse range of fields from DC power supplies, uninterruptible power supplies, and inverter systems for non-conventional energy generation to high-voltage DC transmission and motor drive applications in speed control, motion control and electric vehicles. Inductors and transformers are important components of most power electronic circuits. Inductors are used for filtering switched waveforms, for limiting the rate of change of current, and for transient current limiting. Transformers are used for transforming energy, for voltage and current sensing, for providing electrical isolation, for transforming impedances, and for phase shifting. Recent developments like the matrix transformers inherently have a large number of distributed windings. Switching frequencies of converter circuits for telecommunications and computer applications are entering the 100 MHz range. The design of high-power high-frequency magnetic devices with high efficiency and compact size and weight is essential in most of these applications. At such high frequencies the stray effects (such as the skin and proximity effects) come into play and introduce parasitics. Thus low-frequency transformer models become obsolete at high frequencies. These frequency-dependent stray effects confound the modeling of transformers for high-frequency applications. 1

14 2 1.2 Integrated Magnetics With miniaturization of semiconductor components for electronic circuits progressing at a rapid pace, a corresponding reduction in size of the magnetic components is needed to facilitate the integration of magnetic components on to ICs. The integrated magnetics concept [Blo98, Cuk83] merges the inductor and transformers into a single magnetic circuit with multiple windings (Figure1-1). Depending on the fulfillment of certain prerequisites, the inductors and transformers in an electronic circuit can be interconnected magnetically, resulting in a single magnetic structure with multiple windings. Figure 1-1. Micro-transformers fabricated on silicon Such technologies created a need for accurate modeling of multiple-winding magnetic components. The modeling is important in the analysis of voltage distribution, winding integrity, transient analysis, and performance evaluation of applications such as multiple-output power supplies.

15 3 1.3 Transformer Equivalent Circuits And Models Models of magnetic components such as transformers are being widely used for circuit simulation and system optimization. Transformer behavior can be described by various modeling methods. In one method, the geometric description of the transformer is used to calculate reactances and capacitances of the model [Hei91, Hei93, Oky99]. But the derivation of these models involves complex mathematical calculations and requires detailed geometric description. Other models requiring finite element analysis of the transformers also involve complex mathematic formulations [Ark92, Che94]. Figure 1-2. The T Model of a two-winding transformer The simplest of the transformer models are the equivalent circuit models (such as the T-model and the Pi- model which are widely used for modeling low frequency operation of transformers). The basic T-model as shown in Figure 1-2 accounts for the magnetizing and leakage inductances of the transformer windings. The magnetizing inductance models the magnetization of the core material. The primary winding acts as an inductor, with its inductance equal to the magnetizing inductance when all the secondary windings are disconnected.

16 4 The leakage inductance models the phenomenon of flux leakage in the transformer windings. In a physical transformer, each of the windings has some flux that links with no other winding. The Pi-model is an equivalent for the T-model with the equivalent inductances arranged as shown in Figure 1-3. Both the T and Pi models are found wanting for modeling of high-frequency transformer operation and for multiwinding transformer modeling, as they do not address issues such as cross regulation and high-frequency effects. Figure 1-3. The Pi Model of a two-winding transformer For modeling high-frequency operation of transformers, the models should include first-order effects (like magnetizing inductances, stray inductances, and inductive coupling) and second-order effects (such as the hysteresis and eddy current losses, magnetic saturation, capacitive coupling, and parasitic effects such as skin and proximity effects). The models should be useful for both frequency and transient analysis and have a meaningful circuit topology. 1.4 Multi-Winding Models And High-Frequency Effects Multi-winding transformers can be represented by lumped elements. Models that include the stray inductances and other parasitic effects give good characterization. The

17 5 simplest model for a multi-winding transformer is the lumped parameter model (Figure 1-4). Figure 1-4. Lumped parameter Model with all elements referred to primary winding The lumped parameter model is based on some very unrealistic assumptions (such as perfect coupling between windings, and all the parasitic effects being concentrated in a single winding). The simple distributed model is a better model, because it considers the parasitic effects to be distributed among the different windings. However, the simple distributed model cannot be used at high frequencies, because it does not consider frequency-dependent parasitic effects. The frequency-dependent fully distributed model proposed by Asensi et al [Ase99] performs better, because it considers frequency-dependent parasitic effects; and hence can be used for modeling high-frequency transformer operation. The frequency-dependent fully distributed model also has the disadvantage that it involves short circuit measurements. Figure 1-5. Frequency Dependent Fully distributed model for a two winding transformer

18 6 The Extended cantilever model proposed by Erickson et al [Eri98] is an extension of the cantilever model or T-model to the multi-winding case. Figure 1-6 shows a simple model with easily extractable parameters which quite accurately predicts the cross regulation among the different windings of the multi-winding transformers. Figure 1-6. Extended Cantilever model for a Four-Winding Transformer A broadband version of the ECM was proposed by Shah et al [Sha00], but it has the disadvantage that the extraction of short circuit parameters requires short circuit current measurement which is a difficult process at high frequencies due to the unavailability of an ideal short at high frequencies. Other methods such as the one proposed by Ngo et al [Ngo01] involve complex mathematical formulation. 1.5 Motivation For Multiwinding Transformer Models With the advent of switched mode power supplies that are widely used in portable battery powered equipment, a large number of multi-output power converter topologies which utilize multi-winding magnetic components have come into use. Most of these circuits such as the quasi-resonant flyback converter operate at mega-hertz frequency range. This has created a need for accurate models of multi-winding transformers which

19 7 can predict their behavior at high frequencies and can be used in design and analysis of these power electronic circuits. Most of the conventional transformer models fail when applied to the multi-winding case due to the effects of cross-coupling between auxiliary windings. The multi-winding models that have been proposed earlier either fail to consider high-frequency parasitic effects or involve complex mathematical formulation in their parameter extraction procedure and analysis. With these factors taken into consideration the Mesh model was developed. The Mesh model has the advantage of having easily extractable parameters and can also accurately model high frequency operation of multi-winding transformers.

20 CHAPTER 2 MULTI-WINDING TRANSFORMER MODELING METHODS 2.1 Introduction Various multi-winding transformer modeling approaches have been proposed over recent years to characterize the transformer windings behavior and its dependence on frequency and geometry effects. The models are utilized for designing the transformers to be built as well as for analyzing the transformers already constructed. An accurate model of the transformer reduces the number of prototypes built during the design phase. This chapter provides a brief introduction to the recent developments in multi-winding magnetic models. Though a wide variety of models are in existence the models most relevant to the one proposed in this study are only presented in the following sections in the chronological order in which they were developed Model Description 2.2 General Multi-Winding Transformer Equivalent Circuit One of the earliest models for multi-winding transformers was proposed by MIT EE staff [MIT43]. An n-winding transformer can be analyzed either as an n-loop linear element circuit or as a circuit with n nodes if the winding capacitances are neglected. The performance of the transformer as an n-loop circuit with each loop electromagnetically coupled to all other loops can be represented by a set of n equations given by Equation 2-1. V = Z I + Z I + L+ Z I + Z I + L+ Z j j1 1 j2 2 jj j jk k jn n For 1 j n (2-1) I 8

21 9 Where the impedances Z jj represent the self impedance in each winding and the impedances Z jk represent the mutual impedance between windings j and k. All the impedances in the equations have an open circuit character and hence can be determined experimentally by appropriate open circuit tests. But they need to be calculated with great accuracy since the voltage drops in the transformer windings depend on usually small differences among self and mutual impedances. Solving the voltage equations for the currents we obtain the equations to analyze the multi-winding transformer as an n-node circuit given by Equation 2-2. I = Y L L+ j j1 V1 + Y j2v2 + + Y jjv j + Y jkvk + Y jnvn For 1 j n (2-2) Where Yjk is called the short-circuit transfer admittance and Yjj is called the short-circuit driving-point admittance. These admittance parameters can be extracted by performing short-circuit tests on the transformer Equivalent Circuit Though these equations can be used to analyze the performance of the transformer, it is often more convenient to represent it by an equivalent circuit which allows for its simulation on a network analyzer. The equivalent circuit of a 4-winding transformer is shown in Figure 2-1. The equivalent circuit contains the same number of independent branches as there are independent coefficients in Equation 2-1 and Equation 2-2. The equations contain a total number of n(n+1)/2 admittances with n self admittances and n(n-1)/2 independent mutual admittances. The admittances Y 10, Y 20, Y 30 shown in Figure 2-1 represent the excitation characteristics of the transformer while the admittances Y 12, Y 23, Y 34, Y 13, Y 24, Y 14 represent the effects of winding resistance and magnetic leakage. An approximate

22 10 equivalent circuit (Figure 2-2) can be obtained if the effects of the excitation currents, which are usually small, are neglected. Figure 2-1. Equivalent circuit of a four winding transformer Figure 2-2. Equivalent circuit of a four winding transformer neglecting the excitation currents

23 11 The relations between the coefficients of the Equation 2-2 and the equivalent circuit parameters as shown in Figure 2-2 are defined by Equation 2-3 and Equation 2-4. Y = (2-3) jk Y jk n Y jj = Y ji i= 1; i j This circuit is one of the simplest equivalent circuits to represent a multiwinding transformer with negligible excitation current due to its optimum number of terminals and branches. 2.3 Cross-Coupled-Secondaries Model Model Description The cross-coupled secondaries model [Nie90] is a multi-winding magnetic model capable of predicting the effects of cross-coupling among the windings, frequency dependent leakage inductance and winding resistance on transformer operation. The values of the leakage inductances and winding resistances are dependent on frequency and can be extracted either from a set of short-circuit impedance measurements or from calculations based on winding geometry. The frequency dependence of the model parameters can be attributed to skin and proximity effects. Cross-coupling among the windings causes the terminal voltage of a secondary winding to change because of the loading on another secondary winding. The model assumes that the transformer is linear, an infinite magnetizing inductance and zero core losses. Distributed capacitive effects are not included in the model. (2-4)

24 12 Figure 2-3. Cross coupled secondaries winding model of 4-winding transformer Parameter Extraction A CCS model for a four winding transformer comprising of an ideal transformer with each winding containing self impedances Z Cjj and cross coupling impedance voltage drops represented by controlled voltage sources is presented as shown in Figure 2-3. The stray winding resistance and inductance parameters of the K th winding are included implicitly in the other K-1 windings. The cross coupled secondaries impedances of the model can be calculated from short circuit impedance measurements using Equation 2-5 and Equation 2-6. The self Impedances Z Cjj can be obtained by a series of tests in which the impedance across an excited winding j is measured with the winding K short circuited and all other windings carrying zero current for j = 1to K-1.

25 13 Z Cjj = Z jk (2-5) The mutual cross-coupling impedances are obtained from the short circuit tests by measuring the impedance Z jk seen across short circuited winding k by exciting the winding j and the cross-coupling impedance is calculated as, 1 N N k j Z C, jk = Z C, kj = ( Z jk Z jk ) + Z kk (2-6) 2 N j N k At high frequencies the numerical values of the parameters are valid for only the frequency at which they are extracted. The model is further developed to take into account the magnetic energy stored, power losses in core, electric energy stored in winding space [Owe92] and to make the model frequency independent [Nie92] Model Description 2.4 Frequency Dependent Fully Distributed Model A Multi-winding magnetic model characterizing the frequency behavior of the windings taking into account geometry and frequency effects was proposed by Asensi et al [Ase94, Ase99]. The model is calculated from its geometrical description using Finite element analysis (FEA). Figure 2-4. Flow Chart for generating the fully distributed model

26 14 An FEA tool is used to analyze the geometric model and to simulate a set of open circuit tests, electrostatic analysis and obtain a model by a systematic approach represented by the flowchart as shown in Figure 2-4. Figure 2-5. Frequency dependent fully distributed model of a 3-winding transformer The 3-winding model developed using the above approach is shown in Figure 2-5. In this model ZCORE represents the non linear core behavior. In order to get the proper magnetizing inductance the core model represented by ZCORE is connected to the circuit using a reluctance adaptor Parameter Extraction An FEA tool is used to calculate the losses and energy storage in the windings. An AC analysis is performed on the model to calculate the magnetic fields and a DC analysis is performed to calculate the electric fields. From these analyses the model parameters are calculated using Equation 2-7, Equation 2-8, Equation 2-9, and Equation R i 1 = 2 I N r Re( J ki σ k = 1 Dk k r J * ki ) dv (2-7)

27 15 H r i 1 r r * L i = µ 0 Re( H i H i ) dv (2-8) 2 I V r r N * 1 Re( J ki J kj ) Rij = dv (2-9) 2 I k = 1 σ Dk k 1 r r * M ij = µ 0 Re( H i H j ) dv (2-10) 2 I V Where, I is the rms current injected in each winding during the open-circuit test, is the magnetic field in the component, and J r ki is the current density in winding k with a non-zero current through winding i. The magnetizing inductance is calculated from the magnetic fields in the core using the Equation r r * L m = µ Re( H1 H1 ) dv (2-11) 2 I C The electric field calculations obtained from the DC analysis of the geometric model using FEA tools are used to calculate the parasitic capacitances in the model using Equation 2-12 and Equation r r * C i = ε ( Ei Ei ) dv 2 U V (2-12) 2 r r * C ij = ε ( Ei E j ) dv 2 U (2-13) V The frequency dependent model characterizes the mutual flux between the secondary windings in addition to the magnetizing and leakage flux. This facilitates using the model for simulating cross-regulation effects in the transformer windings. 2.5 Extended Cantilever Model Model Description The extended cantilever model [Eri98] is a popular model due to its simplicity and ease of parameter extraction. It has directly measurable parameters, which can be extracted by performing open and short circuit measurements and solving a set of

28 16 non-linear equations. Figure 2-6 shows the extended cantilever model (ECM) for an N-winding transformer with N(N+1)/2 independent parameters. Figure 2-6. Extended cantilever model of an N-winding transformer The inductance L 11 models the self-inductance of the primary winding W 1. The ratios of winding voltages under open circuit conditions are given as effective turns ratios n 1, n 2, n 3 n N. The model also has an effective leakage inductance l jk between each pair of windings Parameter Extraction The self inductance is measured with all the windings open circuited and the inductance across winding W 1 measured. Similarly, the turns ratios are measured by applying a voltage across the primary winding W 1 and measuring the voltages induced across the open circuited secondary windings. The effective turns ratio of a winding k is given by Equation vk n = (2-14) k The effective leakage inductance between windings is calculated from a series of v 1 short-circuit measurements which are made by applying a voltage v j across winding W j

29 17 with all other windings short-circuited and measuring the short-circuit current i k in winding W k. The expression for the leakage inductance is given by Equation l jk = v j ( s) s n n i ( s) (2-15) j Where s is the Laplace transforms operator. The disadvantage with the ECM model arises due to the error introduced due to the short circuit current measurement which cannot be done accurately with out an ideal short circuit. This problem is discussed and a method for calculating the model parameters without having to take short-circuit current measurements is presented by Ngo and Gangupomu [Ngo03]. k k Figure 2-7. Broadband Extended Cantilever Model A broadband representation of the ECM model shown in Figure 2-7 is discussed by Ngo et al [Ngo01]. The extended cantilever model provides a simple circuit with easily extractable parameters which is suitable for modeling close-coupled transformers in multiple output converters and coupled inductors. The model is shown to predict the converter waveforms in various regions of operation (such as continuous and discontinuous modes), cross regulation of output voltages, the winding current ripples, and converter small-signal dynamics with reasonable accuracy.

30 Ladder Model Model Description An electric transformer model known as the Ladder model which is derived from the physical structure of the transformer is proposed by Wang et al [Wan99]. In this model the leakage inductance is modeled as an incremental leakage which increases with every secondary winding wound concentrically away from the primary, thus accurately accounting for the geometry of the windings. A magnetic circuit is first derived from the physical structure of the transformer. Using the electric-magnetic duality theorem an equivalent electric model (Figure 2-8) is derived with inductances replacing the reluctances of the magnetic model. Figure 2-8. Ladder Model of a multiwinding transformer Parameter Extraction The leakage inductance values are obtained by evaluating the magnetic energy storage in the windings of the transformer. The model parameters can also be experimentally extracted using a set of open-circuit and short-circuit tests. The transformer turns ratios can be obtained by applying a voltage across the primary winding and measuring the induced voltage across the open circuited secondaries. The Leakage inductances can be measured using a short-circuit test setup (Figure 2-9).

31 19 Figure 2-9. Leakage inductance measurement setup The expression for the leakage inductance is given as in Equation L N V 2 P in ij = (2-16) N Si N Sjiin Due to its derivation from the transformer structure, the Ladder model has a direct correlation between the model parameters and the physical attributes of the transformer. The model can be used both as a design tool and for analysis of a transformer after its construction. 2.7 Leakage Impedance Model Model Description The Leakage Impedance (LI) model [Nie00] is a frequency dependent model that accounts for ac winding and core effects in multi-winding transformers. It has a relatively low number of elements that can be calculated from short circuit and open circuit impedance measurements. Only three extra elements are required to be included to account for magnetizing inductance, core loss, and inter-winding capacitance. Most of the

32 20 elements of the LI model cannot be directly associated with specific physical attributes. A multi-winding transformer with N windings has a model with N(N-1)/2 impedances calculated from short circuit measurements. The model is valid only for sinusoidal waveforms of the frequency at which the model parameters are extracted and is not suitable for analyzing transformer behavior in switched mode power supply circuits. Figure Leakage Impedance Model for a three winding Transformer Parameter Extraction The Figure 2-10 shows the LI model for a three winding transformer with lumped elements for core loss, magnetizing inductance and intra-winding capacitance drawn with dotted lines. Cross coupling between secondary windings is not considered in the three winding model. The impedances of this model can be calculated using the Equation 2-17, Equation 2-18, and Equation N 0 Z L 00 = Z Z 0 2 Z1 2 (2-17) 2 N1 2 1 N 1 Z L 11 = ( Z 0 1 Z 0 2 ) + Z1 2 (2-18) 2 N 0

33 21 ( ) + = Z N N Z Z N N Z L (2-19) The Figure 2-11 shows a four winding transformer leakage impedance model with impedance blocks and auxiliary loops. Figure Leakage Impedance model of a four winding transformer. The Model parameters are calculated from short circuit impedance measurements using the Equation 2-20, Equation2-21, and Equation 2-22.

34 22 2 N 3 L33 ( Z 0 3 Z L00 N 0 Z = ) (2-16) N 3 N 1 Z L31 = ( Z L11 Z 1 3 ) Z L33 N + 1 N 3 (2-21) N 3 N 2 Z L32 = ( Z L22 Z 2 3 ) Z L33 N + 2 N 3 (2-22) The losses in the transformer are estimated by summing the average power dissipation in all resistive elements except the auxiliary loop elements. The LI model has a small number of elements to model as winding resistances and leakage inductances in multi-winding transformers. The model can also be used to characterize the transformer performance in PWM power converters if the switching frequency is the same as the frequency at which the model is extracted. 2.8 Synopsis All the models described in this chapter have certain disadvantages. While some models are suitable only for analyzing sinusoidal excitation at a particular frequency due to the frequency dependence of their parameters, others do not include the high frequency effects and most of the models need detailed geometrical information, complex mathematical calculations and finite element analysis tools to extract the model parameters. Some of the models need short circuit current measurements which introduce a small error in the model while some have parameters which do not have a direct correlation with the transformer properties. These limitations in existing models highlight the need for a broadband, multi-winding model with a relatively small number of easily extractable parameters that can be directly correlated with the transformer magnetic properties and can be easily extended to transformers with a higher number of windings.

35 CHAPTER 3 MESH MODEL 3.1 Introduction This work deals with the development of a new winding model based on modeling the mutual coupling among the windings as opposed to the leakage inductances in ECM with parameters that can be extracted without short circuit current sensing. The model is named mesh model due to its property of representing an n windings of a transformer or inductor with an electric network containing n loops. The derived mesh model has the same number of n(n+1)/2 parameters as the ECM for an n-winding transformer. The Mesh model parameters can also be obtained from the ECM model and vice-versa by using the parameter extraction procedure described in the present chapter. 3.2 Model Description The Mesh model block (Figure 3-1) contains n(n+1)/2 parameters for the n windings of a transformer or inductor. Figure 3-3 shows the equivalent electric circuit Mesh model for the n windings of a transformer or inductor. The effective turns ratios, N 2, N 3...N n are equal to the ratios of the winding currents under short circuit conditions. In general, the effective turns ratios are not equal to the physical turns ratios. The model also has impedance Z jk shared between every two windings, which models the mutual coupling between the windings j and k, j 1, k 1, k j. The impedances Z 1j, 2 j n model the magnetizing inductance in the transformer. The model is symmetric, i.e. Z jk is equal to Z kj. A mesh model circuit in a three-winding case is shown in Figure

36 24 Figure 3-1. Block Diagram of a Mesh Model for the n windings of a transformer or inductor. Figure 3-2. Mutual Coupling Based Mesh Model for a three-winding transformer. Under ideal conditions all the parameters can be extracted independently. All the turns ratios N k, 2 k n, can be extracted by short circuiting windings k, 2 k n, and applying a current i 1 across winding 1 and measuring the short circuit currents flowing in the windings k, 2 k n. Under these conditions, all the impedances except Z 11 carry no current. Thus the turns ratios can be found by,

37 25 N k i1 = For v k =0, 2 k n (3-1) i k N 1 =1 (3-2) Figure 3-3. Mesh Model for the N windings of a transformer or inductor. Also impedance Z 11 can be obtained under the same test conditions as above by measuring the voltage v 1 across the winding 1. v 1 Z 11 = For v k =0, 2 k n (3-3) i1

38 26 The impedances Z 1k can be obtained by open circuiting all windings k, 2 k n; applying a current i 1 to the winding 1 and measuring the voltages v k across the windings k, 2 k n. Hence, Z 1k vk = For i k =0, 2 k n (3-4) N i k 1 Similarly the impedances Z jk, 2 k n, 2 j n, j k, can be found by open circuiting all windings k, 2 k n; applying a current i j to the winding j and measuring the voltages v k across the windings k, 2 k n, k j. Thus, Z jk k = For i k =0, 2 k n, 2 j n, k j. (3-5) N j v N k i j The extraction of the turns ratios by the procedure discussed above requires short circuit current sensing. The difficulty in doing this is discussed by Shah et al [Sha00] and Ngo et al [Ngo01]. The current sensing impedance introduces an error in the measurement. In the next section a parameter extraction procedure which does not need short circuit current sensing is discussed. 3.3 Parameter Extraction Procedure Extraction Of Open-Circuit Parameters Z jk, J K. The extraction of all the model impedances can be done by open circuit measurements. The impedances Z 1k, 1 k n can be obtained by open circuit measurements as discussed in the previous section with the transformer circuit as shown in Figure 3-4. Z 1k vk = For i k =0, 2 k n (3-6) N i k 1

39 27 Figure 3-4. Extraction of Z 1k for the n winding mesh model of a transformer or inductor Figure 3-5. Extraction of Z 12 and Z 13 for a three-winding transformer Similarly the impedances Z jk, 2 k n, 2 j n, j k, can be found by open circuiting all windings k, 2 k n; applying a current i j to the winding j, and measuring the voltages v k across the windings (Figure 3-6). Z jk k = For i k =0, 2 k n, 2 j n, k j. (3-7) N j v N k i j

40 28 Figure 3-6. Extraction of Z jk for an n-winding mesh model Figure 3-7. Extraction of Z 23 for a three-winding mesh model Extraction Of Short-Circuit Parameters N k, Z 11. The extraction of Z 11 does not need short-circuit current sensing and hence it can be extracted by short circuiting windings k, 2 k n, and applying a current i 1 across winding 1 and measuring the voltage v 1 across the winding 1 (Figure3-8).

41 29 v 1 Z 11 = For v k =0, 2 k n (3-8) i1 Figure 3-8. Extraction of Z 11 for the n winding mesh model of a transformer or inductor Figure 3-9. Extraction of Z 11 for a three-winding mesh model The extraction procedure for the turns ratios discussed in the previous section needs short circuit current sensing. The turns ratios can be extracted by an alternate procedure which does not need current sensing. In this procedure extraction of each turns ratio requires two voltage measurements.

42 30 Figure Extraction of Z kks for the n winding mesh model Figure Extraction of Z 22s for a three-winding mesh model Firstly the impedance Z kks looking into the winding k, 2 k n, is measured as shown in Figure 3-10 by applying a current i k to winding k with winding 1 open and all other windings j, 2 j n, j k, short circuited and measuring the voltage v k across the winding k. Then the impedance Z k1s is measured by applying a current i 1 to winding 1 with all other windings j, 2 j n, j k, short circuited and measuring the voltage v k

43 31 across the winding k (Figure 3-12). The turns ratios N k, 2 k n, can be obtained from Z kks and Z k1s as given by Equation 3-9, Equation3-10, and Equation3-11. vk Z kks = For v j =0, i 1 =0, 2 j n, j k (3-9) i k vk Z k 1s = For v j =0, i k =0, 2 j n, j k (3-10) i 1 N k Z kks = For 2 k n (3-11) Z k1s N 1 =1 (3-12) Figure Extraction of Z 1ks for the n winding mesh model of a transformer or inductor The expression for N k is obtained from the circuits as shown in Figure 3-10 and Figure 3-12 from which Equation 3-13 and Equation 3-14 are derived. Z kks = N 2 k Z THk (3-13) Z k 1 s = N 1 N k ZTHk = N k ZTHk (3-14)

44 32 Figure Extraction of Z 12s for a three-winding mesh model Where Z THk is a given combination of the impedances Z jk, j k, for each turns ratio N k. For the circuits shown in Figure 3-11 and Figure 3-13, the relation given in Equation 3-15 gives the value of Z TH2. Z TH = Z + Z ) (3-15) 2 12 ( 13 Z Comparison Of Mesh Model With ECM Parameter Extraction The main advantage of the Mesh model over other models is that its parameters can be extracted without the need for taking any short-circuit current measurements. The parameter extraction procedure for the ECM given by Erickson et al [Eri98] requires short circuit current measurement. An improved procedure for extracting ECM parameters without taking any short-circuit measurements is proposed by Ngo and Gangupomu [Ngo03]. Both the ECM and Mesh models for the n windings of a transformer or inductor have (n-1) turns ratios, and (n 2 -n+2)/2 impedance parameters. Using the procedure given by Ngo and Gangupomu [Ngo03] to extract the ECM parameters, the (n-1) turns ratios

45 33 the admittance Y11 are obtained using n open circuit measurements. The remaining (n 2 -n)/2 impedance parameters are obtained using two measurements for each parameter. Hence the total number of measurements needed for the ECM extraction will be given by Equation ( n n) m ECM = = n + 2 n (3-16) Using the parameter extraction procedure described in section 3.3 the Mesh model impedances are extracted using (n 2 -n+2)/2 open circuit measurements. The (n-1) turns ratios are extracted using two measurements for each turns ratio. Hence the total number of measurements needed for the complete parameter extraction of a Mesh model is given by Equation ( n n + 2) ( n + 3n 2) m Mesh = + 2 ( n 1) = (3-17) 2 2 Table 3-1. Comparison of number of measurements for Mesh model and ECM Number of Windings Mesh Model Measurements ECM Measurements From Equation 3-16 and Equation 3-17 it can be observed that for a transformer with greater than two windings the number of measurements needed to extract its complete Mesh model is less than the number of measurements needed to extract the corresponding ECM. Table 3-1 gives a comparison between the number of measurements required for the Mesh model extraction and ECM extraction.

46 Extraction Of ECM Parameters From Mesh Model The ECM parameters of a transformer can be derived from its Mesh model. Figure 3-14 shows the Saber schematic of a Mesh model for a three-winding pot-core transformer built in the lab. The mesh model parameters were extracted using the procedure given above. The ECM parameters of the transformer can be extracted from the mesh model by simulating the ECM parameter extraction procedure given by Erickson and Maksimovic [Eri98] on the Mesh model. Figure Saber Schematic of Mesh Model derived for the test specimen It can be observed from the schematic that the model parameter Z 23 has a negative reactance. This signifies that the flux linkage due to the mutual coupling between windings W 2 and W 3 opposes the leakage flux in the windings. Thus the polarity of the mutual coupling parameters depends on the nature of the coupling among windings.

47 Synopsis A multi-winding transformer model based on Mesh topology was proposed in this chapter. A simple procedure to extract the complete set of model parameters without having to take any short-circuit current measurements was outlined. The model is observed to have the same number of parameters as the extended cantilever model but needs a comparatively lesser number of measurements and calculations.

48 CHAPTER 4 EXPERIMENTAL VERIFICATION OF THE MESH MODEL 4.1 Introduction In order to verify the Mesh model theory proposed in the previous chapter a simple three-winding pot core transformer was constructed in the laboratory. A ferrite 0P42616 core was used with no air gap. The three windings were wound according to the following specifications: Primary winding W1: 12 turns, #24 AWG Secondary winding W2: 6 turns, #20 AWG Secondary winding W3: 6 turns, #20 AWG The primary winding was wound in the center section of a three-section bobbin with the secondary windings wound on either side of the primary winding. 4.2 Model And Measurements The mesh model parameters of the transformer were extracted using the procedure described in the previous section. The impedance measurements required for the extraction were done with a HP 4194A impedance/gain-phase analyzer [Hew00]. These parameters shown were extracted in the frequency range of 1 KHz to 100 KHz with the upper limit set by the useful bandwidth of the P-type ferrite core. The waveforms as shown in Figure 4-1 to Figure 4-16 show the measurements taken with the HP4194A impedance/ gain-phase analyzer. 36

49 37 Figure 4-1. Magnitude variation of Z 11 over frequency Figure 4-2. Phase variation of Z 11 over frequency

50 38 Figure 4-3. Magnitude variation of Z' 12 over frequency Figure 4-4. Phase variation of Z' 12 over frequency

51 39 Figure 4-5. Magnitude variation of Z' 13 over frequency Figure 4-6. Phase variation of Z' 13 over frequency

52 40 Figure 4-7. Magnitude variation of Z' 23 over frequency Figure 4-8. Phase variation of Z' 23 over frequency

53 41 Figure 4-9. Magnitude variation of Z 22S over frequency Figure Phase variation of Z 22S over frequency

54 42 Figure Magnitude variation of Z 12S over frequency Figure Phase variation of Z 12S over frequency

55 43 Figure Magnitude variation of Z 33S over frequency Figure Phase variation of Z 33S over frequency

56 44 Figure Magnitude variation of Z 13S over frequency Figure Phase variation of Z 13S over frequency

57 45 The schematic of the mesh model for the test transformer is shown with the extracted parameters as shown in Figure The circuit shown in Figure 4-17 was simulated in SABER circuit simulator [Sab97]. Waveforms as shown in Figure 4-18 to Figure 4-25 show the measurements extracted from the SABER simulation which are comparable to the actual measurements shown in Figure 4-1 to Figure Figure Saber Schematic of Mesh Model derived for the test specimen

58 46 Figure Simulation of Z 11 variation over frequency

59 47 Figure Simulation of Z' 12 variation over frequency

60 48 Figure Simulation of Z' 13 variation over frequency

61 49 Figure Simulation of Z' 23 variation over frequency

62 50 Figure Simulation of Z 22s variation over frequency

63 51 Figure Simulation of Z 12S variation over frequency

64 52 Figure Simulation of Z 33S variation over frequency

65 53 Figure Simulation of Z 13S variation over frequency The transformer was tested under different conditions with loading. The results from measurements with the secondary windings subjected to different loading conditions at two different frequencies of 50 KHz and 25 KHz are compared with the model predictions in Table 4-1 and Table 4-2 and the results of load test under similar conditions on the ECM derived from the Mesh model at operating frequencies of 50 KHz and 25KHz are shown in Table 4-3 and Table 4-4. It can be seen that the agreement is good with the transformer under load.

66 54 Table 4-1. Results of Load test on Mesh model at 50 KHz Load Voltage From Theory From Measurement Condition Measured Magnitude (V) Phase ( ) Magnitude (V) Phase( ) R2=open R3=30Ω V R2=open R3=30Ω V R2=30Ω R3=open V R2=30Ω R3=open V R2=30Ω R3=30Ω V R2=30Ω R3=30Ω V R2=30Ω R3=75Ω V R2=30Ω R3=75Ω V R2=75Ω R3=30Ω V R2=75Ω R3=30Ω V R2=75Ω R3=75Ω V R2=75Ω R3=75Ω V Table 4-2. Results of Load test on Mesh model at 25 KHz Load Voltage From Theory From Measurement Condition Measured Magnitude (V) Phase ( ) Magnitude (V) Phase( ) R2=open R3=30Ω V R2=open R3=30Ω V R2=30Ω R3=open V R2=30Ω R3=open V R2=30Ω R3=30Ω V R2=30Ω R3=30Ω V R2=30Ω R3=75Ω V R2=30Ω R3=75Ω V R2=75Ω R3=30Ω V R2=75Ω R3=30Ω V R2=75Ω R3=75Ω V R2=75Ω R3=75Ω V

67 55 Table 4-3. Results of Load test on ECM at 50 KHz Load Voltage From Theory From Measurement Condition Measured Magnitude (V) Phase ( ) Magnitude (V) Phase( ) R2=open R3=30Ω V R2=open R3=30Ω V R2=30Ω R3=open V R2=30Ω R3=open V R2=30Ω R3=30Ω V R2=30Ω R3=30Ω V R2=30Ω R3=75Ω V R2=30Ω R3=75Ω V R2=75Ω R3=30Ω V R2=75Ω R3=30Ω V R2=75Ω R3=75Ω V R2=75Ω R3=75Ω V Table 4-4. Results of Load test on ECM at 25 KHz Load Voltage From Theory From Measurement Condition Measured Magnitude (V) Phase ( ) Magnitude (V) Phase( ) R2=open R3=30Ω V R2=open R3=30Ω V R2=30Ω R3=open V R2=30Ω R3=open V R2=30Ω R3=30Ω V R2=30Ω R3=30Ω V R2=30Ω R3=75Ω V R2=30Ω R3=75Ω V R2=75Ω R3=30Ω V R2=75Ω R3=30Ω V R2=75Ω R3=75Ω V R2=75Ω R3=75Ω V

68 Flyback Transformer Example Figure Mesh Model schematic of 4-winding flyback transformer In order to test the usefulness of the Mesh model in predicting the behavior of practical circuits employing multi-winding transformers, the flyback converter circuit described in section 4 of [Eri98] was simulated using the equivalent mesh model for the flyback transformer. Figure Mesh Model schematic of 4-winding flyback transformer

69 57 Figure Mesh Model schematic of 4-winding flyback transformer Figure Flyback converter example circuit

70 58 First, the equivalent mesh model of the flyback transformer shown in Figure 4-26 was obtained by simulating in Saber the extraction procedures described in the previous section on the ECM of the transformer (Figure 4-27). The equivalent mesh model was obtained (Figure 4-28). This mesh model was implemented in a simulation of the flyback converter circuit (Figure 4-29) using Saber. Figure Waveforms from Flyback converter circuit simulation Simulated secondary winding current waveforms are given in Figure The simulated waveforms correctly predict the following behaviors of the secondary winding currents: continuous conduction mode with regular (negative) slope in winding W2, discontinuous conduction mode in winding W3 and continuous conduction mode with inverted (positive) slope in winding W4.It was observed that the simulated waveforms (Figure 4-30) have good agreement with the measured waveforms (Figure 4-31).