Master’s Thesis
January 2014
Department of Applied Signal processing
Blekinge Institute of Technology
SE – 371 79 Karlskrona
Sweden

This thesis is submitted to the School of Engineering at Blekinge Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science in Electrical
Engineering. The thesis is equivalent to 20 weeks of full time studies.
Contact Information:
Author(s):
Srinivasa Kranthi Kiran Kolachina
E-mail: kolachina.kranthi@gmail.com
Nishu Reddy
E-mail: nishureddy87@gmail.com
University advisor(s):
Anders Hultgren
University Examiner
Sven Johansson
Department of Applied Signal processing
Blekinge Institute of Technology
Blekinge Institute of Technology
SE – 371 79 Karlskrona
Sweden www.bth.se/com
Phone : +46 455 38 50 00
Fax : +46 455 38 50 57 ii

Context : This thesis is a part of collaborated project between Alstom and Blekinge
Institute of Technology. In this thesis a fifth order non- linear Hamilton observer is applied on a series resonant converter. Two models for individual modes are given for a resonant power converter, one is suitable for simulation and other is suitable for simulation and analysis. The circuit is run in eight modes. A switched model of a fifth order DC/DC converter consisting of eight different switching modes has been derived and the performance of the circuit is studied under several conditions by simulation.
Tools: MATLAB, SIMULINK
Keywords: Modeling, Hamiltonian Observer, Switching Modes.
1

We sincerely express our immense gratitude to our supervisor, Anders Hultgren of
Blekinge Institute of Technology for his constant support and encouragement from the initial level to the final level that enabled us to develop a good understanding of the subject. Without him this thesis work would not have been completed or written.
One just simply could not wish for such a friendlier supervisor. We have learnt a lot from Anders Hultgren . It is a pleasure to thank many people who made this thesis possible.
We would like to express our deepest sense of gratitude to our GOD, Almighty who gave us strength and courage to complete this gigantic task.
We warmly thank Pradeep Kumar Kambhampati , for his valuable advice and friendly help.
I Srinivasa Kranthi Kiran Kolachina , very grateful to my parents Mrs. Revathi
Kolachina & Mr. SanyasiRao Kolachina . This thesis is only possible because of their consistent sacrifices and sustained devotion to provide me a brighter future.
They encouraged me a lot and made me the person I am today.
I would also like to thank my brother Mr. Ramakrishna Kolachina , who is my role model and my constant encouragement in the course of my Masters.
I wish to express gratitude to my sister Mrs. Lakshmi Sri Sudha for helping me get through the difficult times and for all the emotional support.
2

ABSTRACT .................................................................................................................................... 1
ACKNOWLEDGEMENTS ............................................................................................................ 2
1 INTRODUCTION .................................................................................................................. 5
1.1
Problem Statement ........................................................................................................... 5
1.2
Thesis Outline .................................................................................................................. 5
2 BACKGROUND AND RELTAED WORK ........................................................................... 6
2.1 Function of a DC/DC switching power converter …………………………………………...6
2.2
Hamiltonian Modeling ………………………………………………………………… .........8
2.3
Related Work ................................................................................................................... 8
3 MODELLLING OF A FIFTH ORDER OBSERVER ……………………………………….16
3.1
Modelling of a fifth order observer circuit ………………………………………………...1
6
3.1.1
Modelling for Mode 1 switching state ………………………………………………………..1
7
3.1.2
Mode 1 model for simulation …………………………………………………………………..
19
3.1.3
Mode 1 skew symmetric model for simulation and analysis ……………………………….
19
3.2.1
Modelling for Mode 2 switching state ………………………………………………………..2
0
3.2.2
Mode 2 model for simulation …………………………………………………………………..
.22
3.2.3
Mode 2 skew symmetric model for simulation and analysis ……………………………….2
2
3.3.1
Modelling for Mode 3 switching state ………………………………………………………..2
3
3.3.2
Mode 3 model for simulation …………………………………………………………………..2
4
3.3.3
Mode 3 skew symmetric model for simulation and analysis ……………………………….2
4
3.4.1
Modelling for Mode 4 switching state ………………………………………………………..2
4
3.5.1
Modelling for Mode 5 switching state ………………………………………………………..2
6
3.5.2
Mode 5 model for simulation …………………………………………………………………..
29
3.5.3
Mode 5 skew symmetric model for simulation and analysis ……………………………….
29
3.6.1
Modelling for Mode 6 switching state ………………………………………………………..3
0
3.6.2
Mode 6 model for simulation …………………………………………………………………..3
1
3.6.3
Mode 6 skew symmetric model for simulation and analysis ……………………………….3
1
3.7.1
Modelling for Mode 7 switching state ………………………………………………………...3
2
3.7.2
Mode 7 model for simulation …………………………………………………………………..3
2
3.7.3
Mode 7 skew symmetric model for simulation and analysis ……………………………….3
3
3.8.1
Modelling for Mode 8 switching state ………………………………………………………..3
3
4 DESIGN AND IMPLEMENTATION OF SIMULINK ……………………………………...
36
5 PROBLEMS IN THE THESIS WORK ……………………………………………………… 48
6 SIMULATION RESULTS …………………………………………………………………….
49
7 CONCLUSION AND FUTURE WORK …………………… ..
……………………………...5
4
8 REFERENCES …………………………………………………………………………………5 5
9 APPENDIXES ………………………………………………………………………… 64
3

Figure 2.1 The Switched resonant converter circuit ............................................................ ..6
Figure 2.2 Two-element RNs …………………………………………………………………9
Figure 2.3 Threeelement VNV RNs Topologies…………………………………………...11
Figure 2.4 Threeelement VNI RNs Topologies………………………………………… .
…12
Figure 2.5 HigherOrder RNs………………………………………………………………..14
Figure 3.1 Spanning graph for mode 1 …….
..........................................................................17
Figure 3.2 Spanning graph for mode 2...................................................................................20
Figure 3.3 Spanning graph for mode 3...................................................................................23
Figure 3.4 Spanning graph for mode 4...................................................................................25
Figure 3.5 Spanning graph for mode 5...................................................................................27
Figure 3.6 Spanning graph for mode 6...................................................................................30
Figure 3.7 Spanning graph for mode 7....................................................................................32
Figure 3.8 Spanning graph for mode 8....................................................................................33
Figure 4.1 The overall Simulink system design .....................................................................36
Figure 4.2 Significant system blocks .....................................................................................37
Figure 4.3 System Block 1...
………………….
.....................................................................37
Figure 4.4 Signal Generator for Block 1.................................................................................38
Figure 4.5 Signal Gain for Block 1..
………… .......................................................................39
Figure 4.6 System Block 2……………..
................................................................................40
Figure 4.7 S-function settings for Block 2...
…………..
.........................................................40
Figure 4.8 System Block 3 .....................................................................................................41
Figure 4.9 System Gain for Block 3.
………….
.....................................................................42
Figure 4.10 System Block 4.
……….
.....................................................................................43
Figure 4.11 Code Inside the Matlab Embedded Function in Block 4……………………….
44
Figure 4.12 System Block 5………………………………………………………………… 44
Figure 6.1 System Voltage Input at 540Volts…..
..................................................................49
Figure 6.2 System Responses……………………… ..............................................................50
Figure 6.3 Process values for fifth order model, K=0.............................................................51
Figure 6.4 Process values for fifth order model, K=0.............................................................51
Figure 6.5 Modes of operation, Switch State (qd)..................................................................52
Figure 6.6 Modes of operation, Switch State (wd)..................................................................53
4

1.1
Problem statement
The thesis work makes an attempt to answer the below research questions: x The converter is switching due to transistors and diodes and that the resulting model will be a piecewise model. x How to analyze different switching modes or operations on a fifth order
Hamiltonian observer?
1.2 Thesis outline
We have tried efficiently to illustrate the background and design and implementation discussed and the conclusion regarding the thesis work.
Chapter 1 makes an attempt to illustrate the problem statement.
Chapter 2 gives a brief idea about the background and introduction to a switched
Hamiltonian observer.
Chapter 3 explains about the modeling of a fifth order Hamiltonian observer.
Chapter 4 design and implementation of Simulink.
Chapter 5 Problems in the Thesis work.
Chapter 6 Simulation Results.
Chapter 7 Conclusion and Future Work.
Chapter 8 References.
5

2.1 Functioning of a DC/DC switching power converter in an Industrial application
A switched resonant converter circuit is shown in Figure 2.1
Figure 2.1 The switched resonant converter circuit
In the above Figure 2.1, ܸ
௖௖
has a magnitude of 540 volts as input voltage signal to our DC-DC converter. The input voltage is a square wave input based from the magnitude defined by ܸ
௖௖
. With this input, when the voltage magnitude is -540 voltage to diode which is reverse-biased will conduct to let the current pass while the other two diodes will conduct only when the input voltage is 540V. The combination of those diodes is called a full-wave rectifier to have a constant DC output with a magnitude based from the designed DC voltage output. The inductor Lr and the capacitor Cr are energy storing components. The Cs1 and Cs2 are snubber capacitors. They are there to decrease the switching losses in the transistors.
6

During switching the transient system response will be formed hence the combination of capacitors and inductor in the circuit will try to compensate those values for our system design operates at resonant.
A design method is applied to a switched system which is based on Hamiltonian modeling from the viewpoint of Passive systems [1]. In order to determine the states, the observer is analyzed which a non-linear observer is called as a switched
Hamiltonian observer. The observer is applied to a series loaded resonant converter.
Due to the switching nature of the converter, the regions x is divided into three subspaces {-1, 0, 1}[2].
The four switches or transistors ܼ
ଵ
to ܼ
ସ
help to generate the voltage at the resonant circuit between the node A and node B [3]. The full wave bridge DC/DC converter is controlled by the four switches. It can have two switching states, one is open state and the other is the closed state. The basic analysis of the resonant power converters is described in [4].
The purpose of the anti-parallel diodes is that the switch conducts in the forward biased or in the reverse biased state. Because of the anti-parallel diodes the current can pass in the opposite direction.
The Snubber Capacitor ܥ
௦
is used for switching losses during transient response. A small voltage fluctuation may affect the system devices which needs protection especially at the time of switching. At the time of switching, there is always a transient response in the system that may cause expected increase of the system voltage or current which is very critical to the system components. Therefore the snubber capacitor is to minimize or filter the unwanted signal during switching.
The scope of this thesis work is to derive and model the fifth order Hamiltonian observer. The results achieved through this study can be used further in the industrial applications and determine whether modifications and replacements are beneficial or necessary.
7

Hamiltonian modeling is based on the energy properties of the system. It preserves the internal structure of the system and this can be utilized in the analysis of switched systems. Hamiltonian modeling described in local coordinates, x, is used.
This kind of Hamiltonian modeling allows the number of states to be odd. The
Hamiltonian model can be given by [2]
ݔሺݐሻ ൌ ሾܬሺݔǡ ݐሻ െ ܴሺݔሻሿ
߲ܪ
߲ݔ
ሺݔሻ ൅ ܩሺݔǡ ݐሻݑሺݐሻሺͳሻ
ݕሺݐሻ ൌ ܩ ் ሺݔǡ ݐሻ
߲ܪ
߲ݔ
ሺݔሻሺʹሻ
Where ܪ ( ݔ ) is the energy storage function, ܬሺݔǡ ݐሻ is a skew-symmetric interconnection matrix, ܴሺݔሻ is a positive semi-definite symmetric dissipation matrix and ܩሺݔǡ ݐሻ is a skewsymmetric control matrix. The power balance of the system is given by [2]
݀ܪ
݀ݐ
൫ݔሺݐሻ൯ ൌ ݕሺݐሻ ் ݑሺݐሻ െ
߲ܪ
߲ݔ
்
൫ݔሺݐሻ൯ܴ൫ݔሺݐሻ൯
߲ܪ
߲ݔ
൫ݔሺݐሻ൯ሺ͵ሻ
The model structure given by equations (1) and (2) is used for modeling the converter as well as for defining a Hamiltonian observer.
A systematic investigation of RC topologies with two, three and four energy storage elements is given in [6], [7]. Various types of Resonant Converters are formed by using different Resonant Networks, suitable sources and loads. In further discussion, only Resonant Networks are shown instead of complete circuit diagram for each converter. [5]
Two-element Resonant Networks, shown in Figure 2.2, are the simplest of various possible configurations. Figure 2.2(a) shows the Resonant Network of
Series Resonant Converter (SRC). It is one of the oldest Resonant Converter topology [8] – [12], primarily used for commutation of thyristors in bridge inverters.
Modeling and analysis of SRC is described in [13] – [19]. A complete steady state analysis of SRC for all operating modes has been given by A. Witulski, [20] which is then applied for its design with minimum component stress [21]. The advantages of
SRC are: simple topology, inherent dc blocking of isolation transformer, high part-
8

load efficiency, and possibility to integrate leakage inductance of transformer as the resonant inductor. The disadvantages of SRC are: lack of no-load regulation, large frequency variation for output control and high output ripple current. It is not inherently short-circuit proof. [5]
Figure 2.2 [5] Two-element RNs. (a) SRC, (b) PRC, (c) CS-PRC, (d) CS-SRC
However, overload protection can be achieved by clamping the resonant capacitor voltage to the supply voltage [22], [23] using diodes. It is primarily suitable for high-output-voltage and low-output-current applications.
The Resonant Network of a Parallel Resonant Converter (PRC) is shown in Figure. 2.2(b) [24] – [27]. A complete steady state analysis of PRC for all operating modes has been given by S. Johnson, et. al. [28]. Analysis of PRC for operation above the resonant frequency, which is only possible with forced commutated switches, has been given by A. Bhat, et. al. [29]. An effect of the leakage and magnetizing inductance of transformer on the operation of PRC is analyzed [30] and it is suggested that by placing the resonant capacitor on the secondary side of the transformer [31], or on the tertiary winding [32], the transformer leakage inductance can be used advantageously as the part of Resonant
Network. A comparison among conventional PRC, the PRC with secondary side resonance and the PRC with tertiary side resonance are reported by M. Swamy, et. al.
[33]. PRC offers load independent output current, which makes it a suitable topology for application as a constant-current power supply [34]. The advantages of PRC are: simple topology, inherently short-circuit proof, capability of no-load regulation, low output ripple current and gainful utilization of transformer parasitic components as a part of RN. The disadvantages of PRC are: lack of inherent dc blocking of isolation transformer and poor part-load efficiency. It is primarily suitable for low-outputvoltage and high-output-current applications. [5]
9

The two input-current-fed two-element Resonant Converter topologies which are Current-Source Parallel Resonant Converter (CS-PRC) [35] – [37], and
Current-Source Series Resonant Converter (CS-SRC) [38], [39], who’s Resonant
Networks are shown in Figure 2.2(c) and Figure 2.2(d), respectively. Since a large inductance is needed at the input to realize input current source the current-fed topologies are less popular. However, a current-fed converter has the following advantages: non-pulsating input current, less input filter requirement, simplified driving requirement for low-side switches in pushpull [40] configuration and inherent immunity to shoot-through failures. Due to this, they have been used for applications such as induction heating [41], ozone generation [42] and lamp ballasts
[43]. Detailed analysis using fundamental frequency approximation is given by M.
Kazimierczuk, et. al. [44]. It is not guaranteed during start-up conditions while ZVS operation is achieved in the steady-state A. P. Hu, et. al. [45], recently proposed a solution to this problem using forced dc current. [5]
Figure 2.3 shows three-element Voltage Source-resonant networkvoltage sink (VNV) which are Resonant Networks. Out of nine VNV networks only four topologies have been further analyzed in the past. The Resonant Network of
Figure 2.3(b) is termed as the LLC-type SRC [46], [47]. Inorder to overcome the inability in SRC to achieve no-load regulation by placing an inductor in parallel with the resonating capacitor, this network was developed. Control characteristics are modified in such a way that under any loading condition with moderate sweep of switching frequency, the output voltage can be varied from zero to maximum. [5]
LCL-type SRC or modified SRC [48] – [50] is shown in Figure 2.3(e), which is one of the most popular and useful topology that has SRC modified with an additional inductor in parallel with the output. With the above modifications, control characteristics are altered when it offers load independent output voltage when operated at resonant frequency and when no-load regulation is possible. [5]
10

Figure 2.3 [5] Three-element VNV RNs. Topologies shown in (a), (b), (c), (d), (e),
(f), (g), (h), and (i)
The analysis and operation of the topology with fixed frequency pulse width modulation (PWM) control is derived by A. K. S. Bhat [51] and it is shown that without losing ZVS of the switches the converter can operate over the entire range. Recently, there has been enhanced interest in this topology (which is also termed as the LLC RC) for application as voltage regulator modules, front-end converter [52], [53] and power factor correction [54]. Series and shunt inductor along with the transformer can be integrated in a single magnetic component [55], [56].
The converter has various over-current protection schemes which are discussed in
[57], and is inherently short-circuit proof. [5]
A T-type Resonant Network of Figure 2.3(h) has been extensively studied in the past as its exhibits interesting properties. It displays load independent output voltage suitable for voltage regulator modules [58]. When operated at resonant frequency the high ratio of load current to source current makes it attractive for induction heating [59] and inductive power transfer [60]. When operated at resonant frequency the topology offers load independent output current which renders it useful as a current source [61], [62] with power factor correction [63].
11

Inherent constant-voltage property to the converter [64] is added by the addition of clamp diodes, which is useful for capacitor charging applications [65]. [5]
The other modified form of SRC is CLL RC, which is shown in Figure
2.3(i). It is a potential candidate for voltage regulator type of applications [66], [67] as this topology offers load independent output voltage and no-load regulation. The topology is also agreeable for full integration of magnetic components. The application of topology for single phase power factor correction has been derived by
C. Chakraborty, et. al. [68]. [5]
Figure 2.4 shows topologies of three element VN1 RNs, out of which three topologies have been analyzed in the past. Topology of Figure 2.4(b) has been termed as the LLC-type PRC [69], [70]. It is deemed to be advantageous in highvoltage power supplies since, the leakage and magnetizing inductance as well as the winding capacitance of transformer is absorbed in the RN.
Figure 2.4 [5] Three-element VNI RNs (a) PRC with input inductor, (b) LLC-type
PRC, (c) LCC or series-parallel RC, and, (d) hybrid RC.
The topic of extensive investigation [71] – [78] is Series-parallel resonant converter (SPRC) or LCC RC, shown in Figure 2.4(c). The LCC RC, can be reported with capacitive output filter which is also normally a VNI topology [79],
[80]. It is mostly advantageous and eliminates drawbacks if it is a combination of
SRC and PRC. It has been profitably applied also to lighting ballast applications
[81]; power factor correction [82]; arc welding [83]; capacitor charging [84]; high frequency ac power distribution [85] etc. SPRC offers both load independent output voltage and load independent output current [86], executing it suitable for both types of applications. Three-phase SPRC [87], [88] is suitable for high power applications
12

because of increased output ripple frequency and easier filtering. The clamped-mode
[89] and phase controlled SPRC [90] is useful for fixed-frequency operation. [5]
Desirable properties of SRC and PRC are combined in the Resonant
Network of Figure 2.4(d) which is termed as the hybrid Resonant Converter [91].
Other than providing commutating current under all conditions it also operates into short-circuit and offers load independent output voltage and current - the latter property has made this topology useful for ballast applications [92] and corona discharge process [93]. Hybrid RC can also be useful in singlephase ac-dc converter with power factor correction [94], [95]. [5]
The number of discrete components, size, cost and analytical complexity of an RC increases with increasing order of RN. In the past, higher-order
RC topologies have been explored and studied mainly because of their ability to absorb circuit parasitic components and peculiar characteristics. RNs of these circuits are summarized in Figure 2.5. The 4th order LCLC RC of Figure 2.5(a) can be viewed as a combination of LCL RC and LCC RC. It can also be viewed as a combination of LLC-type PRC and LCC RC. Therefore, it is claimed to have the merits of all these topologies. Winding capacitance is absorbed in the RN Leakage and magnetizing inductance of transformer. Therefore, it is considered suitable for high voltage application by P. Jain et. al. [96]. Analysis and design methodology of this topology is studied in [97] – [101]. Issues involved in its application as lamp ballast is described by R. Liu, et. al. [102]. It is also found appropriate for compensation of leakage and magnetizing inductance of loosely coupled transformer
[103]. The topology of Figure 2.5(b) is a double tuned network with two series LC arms [104]. Full-range control of output voltage under any loading condition can be achieved with moderate sweep of resonant frequency by properly selecting the resonant frequencies of two series arms. The topologies of Figure 2.5(c) and Figure
2.5(d) were developed to compensate a loosely coupled transformer which is characterized by large leakage inductance and small magnetizing inductance used in transcutaneous energy transmission system for medical application using a series capacitor [105]. All magnetic components can be integrated when used in conventional dc-dc converter [106]. [5]
13

Figure 2.5 [5] Higher-order RNs
The RN of Figure 2.5(e) is LCC RC with transformer leakage inductance [107]. Since transformer leakage inductance can be used as the part of resonant inductor in basic LCC RC by placing the parallel capacitor on the secondary side, this topology does not hold any advantage over LCC RC.
Figure 2.5(f) is Series-loaded series-parallel 4th order RN, offers voltage boosting property for which, it is used for high power factor single-phase acdc interface with high-frequency isolation [108]. M. A. Choudhary, et. al. described a three-phase version of this higher order RN [109] for high-power application.[5]
The five element RN of Figure 2.5(g) was suggested, as a modification over the topology of Figure 2.5(b) and Figure 2.5(c), for medical application to compensate large-air-gap transformer in transcutaneous energy transmission system
[110] by placing the compensating capacitors in series with both the windings. This topology results in more complete compensation with less reactive power loading on the source, less circulating current and better efficiency. [5]
Figure 2.5(h) is a Six-element T-type RN which exhibits an interesting property: the output power can be continuously varied, while the circuit remains in
14

resonance (in-phase input voltage and current) with minimum reactive current loading and ZCS operation over a wide frequency range [111]. This property is exploited by H. Pollock, et. al. for developing a power supply for electric arc welding
[112]. [5]
A Simulink model has been designed and implemented for a third order and a fourth order converter in [3]. The performance of the two converters under different conditions has been investigated. The simulations are based on modeling of a DC/DC power converter and simulation of the whole system by using Simulink and
Matlab. The performance of the circuit is estimated under several conditions.
In [2], a fifth order Resonant Converter is realized and studied. Five states used are resonant current, the current through the magnetizing inductance, the voltage across the series capacitor, the voltage across leakage capacitance, and the voltage across the capacitive load. In this thesis, the five states are used, the only difference here is instead of magnetizing inductance the snubber capacitance is used.
15

The models for individual modes are given for a resonant power converter, one is suitable for simulation and the other is suitable for simulation and analysis. The modeling for eight switching modes haven been given below.
3.1 Modeling of a fifth order observer circuit
In Figure 2.1 an electrical circuit is given in parallel with the switches ܼ
ଵ
and ܼ
ଶ there are modeled capacitances, ܥ
௦
.
The process values : ܷ
஼
ೝ
is the resonance capacitor voltage, ܷ
஼
ೢ
is the voltage of winding capacitor across the transformer, ܷ
஼
ಽ
is the output load voltage , ܷ
஼
ೄ
is the snubber capacitor voltage, ݅
௅
ೝ
is the resonance inductive current.
The circuit is run above the resonance frequency, and will then have eight switch modes. The switch modes are characterized in Table I.
Switch mode Control signal, s
1
2
3
4
5
6
7
8
1
-1
-1
-1
-1
1
1
1
Normalized potential at leg
A
1, ܵ
஼
= 1
Cs recharging,
ܵ
஼
= 0
0, ܵ
஼
0, ܵ
஼
0, ܵ
஼
= -1
=-1
= -1
Cs recharging
ܵ
஼
1,
= 0
ܵ
1, ܵ
஼
஼
= 1
= 1 resonant circuit conduction
+
+
+
-
-
-
-
+ rectifier conduction
+
+
+
0
-
-
-
0
16

3.1.1 Modeling for Mode 1 switching state
In switch mode 1 the transistor switches are in s = 1, the snubber capacitors charged, ܵ
஼
= 1, ܷ
஼
ೞభ
= 0 and ܷ
஼
ೞమ
= E. This implies that the supply voltage is applied across node A and B, ܷ
஺஻
= E. The resonant current is positive, ݅
௥
൐ Ͳ and the winding capacitance is fully charged, ܷ
஼
ಽ
ൌ ܷ
஼
ೈ
, and the rectifier is conducting in positive direction.
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 3.1
Figure 3.1
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ =
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
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Ͳ
Ͳ
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Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
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Ͳ
Ͳ
Ͳ ͳ ͳ
െͳ െͳ ͳ
Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ۑ
െͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ ے
ۑ
ۑ
ۑ
ۑ
ې
ۑ
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
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ܷ
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ې
ۑ
ۑ
By simplifying the matrix in the equation (3.1), the matrix can be split as
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
݅
݅
݅
݅
ܷ
ܷ
஼
ೝ
஼
ೢ
஼
ೞ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
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Ͳ
Ͳ Ͳ ͳ
െͳ െͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ۑ
െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ ے
ې
ۑ
ۑ
ۑ
ۑ
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
ܷ
ோ
ೝ
݅
݅
ோ
ಽ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
൅
ێ
ێ
ۍ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
ۑ
ۑ
Ͳ
Ͳے
ۑ
ۑ
Ͳ
Ͳ
Ͳ
Ͳ
ܷ
஺஻
By further splitting the fourth and fifth rows, we get
(3.1)
(3.2)
17

ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ െͳ ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
Ͳ Ͳ
െͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.3)
൤
݅
ܷ
ோ
ೝ
ோ
ಽ
Ͳ ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
Ͳ Ͳ
ቃ ൤
ܷ
ோ
ೝ
݅
ோ
ಽ
Ͳ
ቃ ܷ
஺஻
And the constitutive equation for the resistors is given by
(3.4)
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
቉ ൤
݅
ܷ
ோ
ೝ
ோ
ಽ
൨
Performing the matrix manipulation, we get
(3.5)
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ಽ
௅
ೃ
݅
݅
஼
ೝ
݅
஼
ೢ
ܷ
஼
ೞ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ ے
ۑ
ۑ
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅ ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
By further manipulation of R matrix i.e., writing the R matrix in the form of 5 X 5 matrix in the equation (3.6), we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ې
ۑ
ۑ
ۑ
ێ
ێ
ۍ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ܷ
஺஻
By simplifying the equation (3.7), we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
ܴ
௅
Ͳ Ͳ
Ͳ െͳ ͳ ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ
Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ۑ
ۑ
ې
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ۏ
ܷ
ܥ
ݎ
ܷ
ܷ
ܥ
ݓ
݅
ܥ
ܥ
ݏ
ܮ
݅
ܮ
ݎ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ێ
ێ
ۏ
ۍ
Ͳ
Ͳ െͳ ܴ
Ͳ
ൗ
Ͳ
ܮ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ െܴ
ݎ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
ۑ
ۑ
ې
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ۏ
ܷ
ܷ
ܥ
ܥ
ݎ
ݓ
ܷ
ܥ
ݏ
݅
ܥ
ܮ
݅
ܮ
ݎ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
(3.8)
(3.6)
(3.7)
18

Since the voltages in parallel circuits are equal for all the elements, from Figure 3.1, we get
ܷ
ௗ
஼
ಽ
ௗ௧
௜
ܷ
಴ಽ
஼
ಽ
ൌ
஼
ಽ
ൌ
ܷ
஼
ೢ
ൌ
஼
ೢ
ௗ
ௗ௧
௜
಴ೢ
ܷ
஼
ೢ
Re-arranging the above equation, we get
݅
஼
ಽ
ൌ
ܥ
௅
ܥ
௪
݅
஼
ೢ
(3.9)
Substituting the equation (3.9) in the equation (3.8), we get
ێ
ێ
ێ
ۏ
ۍ
ێ ݅
஼
ೢ
݅
൅
஼
ೝ
஼
ಽ
஼
ೢ
݅
ܷ
஼
ೞ
ܷ
஼
ಽ
௅
ೃ
݅
஼
ೢ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
Ͳ
ܴ
௅
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.10)
3.1.2
Mode 1 model for simulation
Rearranging the second and the fourth row in equation (3.10), a suitable model for simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
ሻ
ோ
ಽ
ሺ஼
ಽ
െͳ
ା஼
ೢ
ሻ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ Ͳ Ͳ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
െͳے
Ͳ
Ͳ
Ͳ
Ͳ
ې
ۑ
ۑ
ۑ
3.1.3
Mode 1 skew symmetric model for simulation and analysis
ܷ
஺஻ (3.11)
Using the freedom based on ܷ
஼
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ಽ
௅
ೃ
݅
݅
஼
ೝ
݅
஼
ೢ
ܷ
஼
ೞ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
ൌ ܷ
஼
ೢ
஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
Ͳ
ሻ
Ͳ
ሺͳ െ ߙሻ Ͳ
Ͳ
ோ
ಽ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
ሺͳ െ ߚሻ Ͳ െ
െͳሺͳ െ ߛሻ Ͳ
, the system can be described as
ି஼
ೢ
Ͳ
ሺ஼
ಽ
ା஼
ೢ
ሻ
஼
ಽ
Ͳ
ሺ஼
ಽ
ା஼
ೢ
െߛ
ሻ
ሺߙሻ
ሺߚሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
Choosing ߙ ൌ
஼
ಽ
஼
ೢ ǡ ߚ ൌ Ͳǡ ߛ ൌ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
ሻ
19

The system for the mode 1 is given by the skew symmetric system
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
െ
Ͳ
ሺ஼
ೢ
ି஼
ಽ
ሻ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
Ͳ െ
െͳ െ
ோ
ಽ
ሺ஼
ಽ
஼
ೢ
ା஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
ሻ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
ି஼
ಽ
ோ
ಽ
ሺ஼
ಽ
Ͳ
ା஼
ೢ
ሻ
Ͳ െ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
3.2.
1 Modeling for Mode 2 switching state:
ܷ
஺஻
(3.12)
Figure 3.2
In switch mode 2 the transistor switches are switched and are in s = -1, the snubber capacitors is recharging, ܵ
஼
= 0, 0 < ܷ
஼
ೞభ
< E and 0< ܷ
஼
ೞమ
< E. This implies that the voltage applied across node A and B, -E < ܷ
஺஻
E. The resonant current is positive,
݅
௥
൐ Ͳ and the winding capacitance fully charged, ܷ
஼
ಽ
ൌ ܷ
஼
ೈ
, and the rectifier is conducting in positive direction.
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.2
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ۍ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
ێ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ െͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ െͳ െͳ െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
(3.13)
20

By simplifying the matrix in the equation (3.13), system can be split as
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ۍ
ێ
ێ
ۏ
݅
݅
݅
݅
ܷ
ܷ
஼
ೝ
஼
ೢ
஼
ೞ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ െͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
Ͳ െͳ െͳ െͳ െͳ Ͳ Ͳ Ͳ ے
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
ܷ
ோ
ೝ
݅
݅
ோ
ಽ
݅
஼
ಽ
௅
ೝ
൅
ێ
ێ
ۍ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
ۑ
ۑ
Ͳ
Ͳے
ۑ
ۑ
Ͳ
Ͳ
Ͳ
Ͳ
ܷ
஺஻
By further splitting the fourth and fifth rows, we get
ۍ
ێ
ێ
ێ
ێ
ۏ
݅
݅
݅
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
஼
ಽ
௅
ೃ
൤
݅
ܷ
ோ
ೝ
ோ
ಽ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ ͳ
Ͳ ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ െͳ Ͳ Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
Ͳ Ͳ
െͳ Ͳ
Ͳ Ͳ
ቃ ൤
ܷ
ோ
ೝ
݅
ோ
ಽ
ې
ۑ
ۑ
ۑ
ے
൤
ܷ
ோ
ೝ
݅
ோ
ಽ
൨ ൅
Ͳ
ቃ ܷ
஺஻
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
Ͳ
Ͳے
ۑ
ۑ
ܷ
஺஻
And the constitutive equation for the resistors is given by
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
൨ ൤
ܷ
݅
ோ
ೝ
ோ
ಽ
቉ ൤
݅
ܷ
ோ
ೝ
ோ
ಽ
Performing the matrix multiplication, in the similar manner as in mode 1, we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
ܷ
஼
ೞ
஼
ಽ
௅
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
ې
ۑ
ۑ
ۑ
Ͳے
ۍ
ێ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
െͳ ͳ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ െͳ Ͳ Ͳ
ܷ
஺஻
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
Further simplifying the above equation, we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
Ͳ ͳ
ܴ
௅
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
Ͳ ͳ ͳ ͳ
Ͳ
െͳ െͳ െͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ۍ
ێ
Ͳ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
൅
Ͳ
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
ې
ۑ
ۑ
Ͳ
Ͳ ۑ
Ͳے
Ͳ
Ͳ
ൗ
௅
Ͳ
ܷ
஺஻
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
(3.14)
(3.15)
(3.16)
(3.17)
(3.19)
(3.18) .
21

Since the voltages are equal in parallel circuits for all the elements, therefore from
Figure 3.1, we get
ܷ
஼
ಽ
ௗ
ൌ
ܷ
஼
ಽ
ܷ
஼
ೢ
ൌ
௜
ௗ௧
಴ಽ ൌ
௜
಴ೢ
ௗ
ௗ௧
஼
ಽ
஼
ೢ
ܷ
஼
ೢ
Re-arranging the above equation, we get
݅
஼
ಽ
ൌ
ܥ
௅
ܥ
௪
݅
஼
ೢ
(3.20) .
By substituting the equation (3.20) in the equation (3.19), we get
ێ
ێ
ێ
ۏ
ۍ
ێ ݅
஼
ೢ
൅
݅
஼
ೝ
஼
ಽ
஼
ೢ
݅
ܷ
஼
ೞ
ܷ
஼
ಽ
௅
ೃ
݅
஼
ೢ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
Ͳ ͳ
ܴ
௅
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ ͳ ͳ
Ͳ
െͳ െͳ െͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
ې
ۑ
ۑ
ۑ
Ͳے
ܷ
஺஻ (3.21)
3.2.2 Mode 2 model for simulation
Rearranging the second row and the fourth row in the equation (3.21), a suitable model for simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
ሻ
ோ
ಽ
ሺ஼
ಽ
െͳ
ା஼
ೢ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ ͳ
஼
ಽ Ͳ Ͳ
ሺ஼
ಽ
ା஼
ೢ
െͳ Ͳ െܴ
௥
ሻ
ሻ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
ې
ۑ
ۑ
Ͳ
Ͳ ۑ
Ͳے
ܷ
஺஻
3.2.3 Mode 2 skew symmetric model for simulation and analysis
Using the freedom based on ܷ
஼
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
ൌ ܷ
஼
ೢ
, the system can be described as
஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
ሻ
Ͳ
ሺͳ െ ߙሻ
Ͳ
Ͳ
Ͳ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
ሺͳ െ ߚሻ
െͳሺͳ െ ߛሻ
Ͳ
െͳ
െ
ି஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
െߛ
ሻ
ሺߙሻ
ሺߚሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
ې
ۑ
ۑ
Ͳ
Ͳ ۑ
Ͳے
ܷ
஺஻
(3.22)
22

Choosing ߙ ൌ
஼
ಽ
஼
ೢ ǡ ߚ ൌ Ͳǡ ߛ ൌ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
,
The system is given by the skew symmetric system
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ோ
ಽ
Ͳ
ሺ஼
ೢ
ି஼
ಽ
ሻ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
ሻ
Ͳ െ
ێ
ۏ
െͳ െ
ோ
ಽ
ሺ஼
ಽ
ା஼
஼
ೢ
ೢ
ሻ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
ோ
ಽ
ሺ஼
ಽ
Ͳ
ା஼
ೢ
ሻ
െͳ െ
Ͳ
ି஼
ಽ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
ې
ۑ
ۑ
ۑ
Ͳے
ܷ
஺஻
(3.23)
3.3.1 Modeling for Mode 3 switching state
In switch mode 3 the transistor switches are in s = -1, the snubber capacitors is charged, ܵ
஼
= -1, ܷ
஼
ೞభ
= E and ܷ
஼
ೞమ
= 0. This implies that the voltage applied across node A and B, ܷ
஺஻
= -E. The resonant current is positive, ݅
௥
൐ Ͳ and the winding capacitance fully charged, ܷ
஼
ಽ
ൌ ܷ
஼
ೈ
, and the rectifier is conducting in positive direction.
Figure 3.3
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.3.
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ െͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ ے
ۑ
ۑ
ۑ
ۑ
ې
ۑ
ۑ
Which is similar to Mode 1 hence it has a similar solution.
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
(3.24)
23

3.3.2
Mode 3 model for simulation
Rearranging the second row and the fourth row in equation (3.24), a suitable model for simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
ሻ
ோ
ಽ
ሺ஼
ಽ
െͳ
ା஼
ೢ
ሻ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ Ͳ Ͳ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
3.3.3
Mode 3 skew symmetric model for simulation and analysis
ܷ
஺஻
(3.25)
Using the freedom based on ܷ
஼
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ಽ
௅
ೃ
݅
݅
஼
ೝ
݅
஼
ೢ
ܷ
஼
ೞ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Choosing ߙ ൌ
ൌ ܷ
஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
Ͳ
ሻ ǡ ߚ ൌ Ͳǡ ߛ ൌ
Ͳ
ሺͳ െ ߙሻ Ͳ
Ͳ
ோ
ಽ
ோ
ಽ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
ሻ
ሺͳ െ ߚሻ Ͳ െ
െͳሺͳ െ ߛሻ
஼
ಽ
Ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
,
஼
ೢ
, the system can be described as
ି஼
ೢ
Ͳ
ሺ஼
ಽ
ା஼
ೢ
ሻ
஼
ಽ
Ͳ
ሺ஼
ಽ
ା஼
ೢ
െߛ
ሻ
ሺߙሻ
ሺߚሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
The system is given by the skew symmetric system,
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
െ
Ͳ
ሺ஼
ೢ
ି஼
ಽ
ሻ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
Ͳ െ
െͳ െ
ோ
ಽ
ሺ஼
ಽ
஼
ೢ
ା஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
ሻ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
ோ
ಽ
ሺ஼
ಽ
Ͳ
ା஼
ೢ
ሻ
Ͳ െ
Ͳ
ି஼
ಽ
Ͳ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.26)
3.4.1 Modeling for Mode 4 switching state
In switch mode 4 the transistor switches are in s = -1, the snubber capacitors is charged, ܵ
஼
= -1, ܷ
஼
ೞభ
= E and ܷ
஼
ೞమ
= 0. This implies that the voltage applied across node A and B, ܷ
஺஻
= -E. The resonant current is negative, ݅
௥
൏ Ͳ and the winding capacitance voltage is െܷ
஼
ಽ
൏ ܷ
஼
ೈ
൏ ܷ
஼
ಽ
, and the rectifier is not conducting.
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.4
24

ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ێ
ێ
ێ
ێ
ۍ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
െͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
ܷ
݅
݅
݅
ோ
ೝ
ோ
ಽ
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
(3.27)
Figure
By simplifying the matrix in the equation (3.27), the matrix can be split as
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
݅
݅
݅
݅
ܷ
ܷ
஼
ೝ
஼
ೢ
஼
ೞ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
=
ۍ
ێ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
݅
ோ
ಽ
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
൅
ۍ
ێ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
ێ
ۏ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ۑ
ۑ
ې
ۑ
ܷ
஺஻
By further splitting the fourth and fifth rows, we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ ے
ۑ
ۑ
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ
െͳ Ͳ ے
ۑ
ۑ
ې
ۑ
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൅
൤
݅
ܷ
ோ
ೝ
ோ
ಽ
Ͳ Ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
Ͳ Ͳ
ቃ ൤
ܷ
ோ
ೝ
݅
ோ
ಽ
The constitutive equation for the resistors is given by
Ͳ
ቃ ܷ
஺஻
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.28)
(3.29)
(3.30)
25

൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
቉ ൤
݅
ܷ
ோ
ೝ
ோ
ಽ
൨
By substituting the equation (3.30) in equation (3.31), we get
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
Ͳ Ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
By matrix multiplication of equation (3.32)
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ۑ
ۑ
ۑ
ے
ې
ۑ
By substituting equation (3.32) in the equation (3.29), we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ
Ͳ
Ͳ
Ͳ Ͳ Ͳ Ͳ െܴ
௥
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ܷ
஺஻
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ې
ۑ
ۑ
ۑ
Performing the matrix manipulation, we get the below which is the model for
ܷ
஺஻
(3.31)
(3.32)
(3.33) simulation of Mode 4 switching state
ۍ
ێ
ێ
ێ
ێ
ۏ
݅
݅
݅
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
஼
ಽ
௅
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻ (3.34)
3.5.1 Modeling for Mode 5 switching state
In switch mode 5 the transistor switches are in s = -1, the snubber capacitors is charged, ܵ
஼
= -1, ܷ
஼
ೞభ
= E and ܷ
஼
ೞమ
= 0. This implies that the voltage applied across node A and B, ܷ
஺஻
= -E. The resonant current is negative, ݅
௥
൏ Ͳ and the winding
26

capacitance voltage is fully negatively charged ܷ
஼
ಽ
ൌ െܷ
஼
ೈ
, and the rectifier is conducting in negative direction.
Figure 3.5
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3. 5
݅
݅
஼
ೢ
஼
ೞ
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
ێ
ێ
ێ
ێ
ێ
ۏ
ێ
ێ
ۍ
݅
݅
௎
ಲಳ
஼
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ െͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ
Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ െͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
Ͳ
Ͳ
ۑ
ۑ
ۑ
ۑ
ې
ۑ
ۑ ͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ ے
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
By simplifying the matrix in the equation (3.35), the matrix can be split as
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ۍ
ێ
ێ
ۏ
݅
݅
݅
݅
ܷ
ܷ
஼
ೝ
஼
ೢ
஼
ೞ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
െͳ െͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ۑ
െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ ے
ې
ۑ
ۑ
ۑ
ۑ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ۍ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
ܷ
ோ
ೝ
݅
݅
ோ
ಽ
݅
஼
ಽ
௅
ೝ
൅
ێ
ێ
ۍ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
ۑ
ۑ
Ͳ ͳے
ۑ
ۑ
Ͳ
Ͳ
Ͳ
Ͳ
ܷ
஺஻
By further splitting the fourth and fifth rows, we get
ۍ
ێ
ێ
ێ
ێ
ۏ
݅
݅
݅
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
஼
ಽ
௅
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ െͳ ͳ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
Ͳ
Ͳ
Ͳ
െͳ
ێ
ۏ
Ͳ Ͳ
Ͳ Ͳ
െͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൅
ێ
ۏ
ێ
ێ
ۍ
Ͳ
ې
Ͳ
Ͳ
Ͳ ۑ
ۑ
ۑ ͳے
ܷ
஺஻
(3.35)
(3.36)
(3.37)
27

൤
݅
ܷ
ோ
ೝ
ோ
ಽ
Ͳ ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
Ͳ Ͳ
ቃ ൤
ܷ
ோ
ೝ
݅
ோ
ಽ
Ͳ
ቃ ܷ
஺஻ and the constitutive equation for the resistors is given by
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
቉ ൤
݅
ܷ
ோ
ೝ
ோ
ಽ
൨
Performing the matrix manipulation, we get
(3.38)
(3.39)
൤
ܷ
ோ
ೝ
݅
ோ
ಽ
൨ ൌ ቈ
ܴ
௥
Ͳ
൤
ܷ
ோ
ೝ
݅
ோ
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
Ͳ ͳ
ൗ
ܴ
௅
Ͳ ͳ Ͳ Ͳ Ͳ
ቃ
൨ ൌ ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
Ͳ ͳے
ۑ
ۑ
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ െͳ ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ܷ
஺஻
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
Ͳ Ͳ
െͳ Ͳ
ۑ
ۑ
ۑ
ے
ې
ۑ
ې
ۑ
ۑ
ۑ
ے
ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
௥
ൗ
௅
Ͳ Ͳ Ͳ ቉
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
By further manipulation of the R matrix i.e., writing R matrix in the form of 5 X 5
(3.40)
൅ matrix in the equation (3.41), we get
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ െͳ ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
Ͳ ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅ ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
ۍ
ێ
Ͳ
By simplifying the above equation, we get
Ͳ
Ͳ
Ͳ
ൗ
௅
Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
Ͳ ͳے
ۑ
ۑ
ܷ
஺஻
(3.41)
28

ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
Ͳ ͳ
ܴ
௅
Ͳ Ͳ
Ͳ െͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ ͳ
Ͳ
Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ێ
ۍ
Ͳ
Ͳ
Ͳ
ێ
ۏ
Ͳ ͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
Since the voltages in parallel circuits are equal for all the elements, from Figure 3.1 we get,
ܷ
஼
ಽ
ௗ
ൌ െܷ
஼
ೢ
ܷ
஼
ಽ
ൌ െ
௜
ௗ௧
಴ಽ
஼
ಽ
ൌ
ି௜
಴ೢ
஼
ೢ
ௗ
ௗ௧
ܷ
஼
ೢ
Re-arranging the above equation we get
݅
஼
ಽ
ൌ െ
ܥ
௅
ܥ
௪
݅
஼
ೢ
Substituting the equation (3.43) in the equation (3.42), we get
ۍ
ێ
ێ
ێ
ێ
ۏ
݅
஼
ೢ
െ
݅
஼
ೝ
஼
ಽ
஼
ೢ
݅
ܷ
஼
ೞ
ܷ
஼
ಽ
௅
ೃ
݅
஼
ೢ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
െͳ
Ͳ
ൗ
Ͳ ͳ
ܴ
௅
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ ͳ
Ͳ
Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
Ͳ ͳے
ۑ
ۑ
ܷ
஺஻
3.5.2
Mode 5 model for simulation
Rearranging the second and the fourth row in equation (3.44), a suitable model for
(3.42)
(3.43)
(3.44) simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ೢ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ
஼
ೢ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
஼
ಽ Ͳ Ͳ
ሺ஼
ೈ
ି஼
ಽ
Ͳ Ͳ െܴ
௥
ሻ
ሻ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
Ͳ ͳے
ۑ
ۑ
ې
ۑ
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ܷ
஺஻
3.5.3 Mode 5 skew symmetric model for simulation and analysis
Using the freedom based on that ܷ
஼
ಽ
ൌ െܷ
஼
ೢ
, the system can be described as
(3.45)
29

ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
= ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
െ
஼
ೈ
Ͳ
ሺଵାఈሻ
ோ
ಽ
ሺ஼
ೈ
஼
ಽ
ି஼
ಽ
Ͳ
ሺଵାఉሻ
ሻ
Ͳ
Ͳ െ
Ͳ
Ͳ െ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ െ
െͳ െͳሺͳ െ ߛሻ Ͳ
Choosing α =
஼
ಽ
஼
ೢ
, β =0, γ =
ሺ஼
ಽ
஼
ಽ
ି஼
ೢ
ሻ
ோ
ಽ
ோ
ಽ
,
Ͳ
஼
ೈ
ሺିఈሻ
ሺ஼
ೈ
஼
ಽ
ି஼
ಽ
Ͳ
ሺିఉሻ
ሺ஼
ೈ
െߛ
ି஼
ಽ
ሻ
ሻ ͳ
஼
ೢ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െܴ
௥
The system is given by the skew symmetric system
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
െ
Ͳ
ሺ஼
ೈ
ା஼
ಽ
ሻ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ Ͳ െ
െͳ െ
ோ
ಽ
ሺ஼
ೈ
஼
ೢ
ି஼
ಽ
ሻ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
ି஼
ಽ
ோ
ಽ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ ͳ
஼
ೢ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
ሻ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
൅
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ۍ
Ͳ
Ͳ
Ͳ
ێ
ۏ
Ͳ ͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
൅
ێ
ۏ
ێ
ێ
ۍ
Ͳ
ې
Ͳ
Ͳ
Ͳ ۑ
ۑ
ۑ ͳے
ܷ
஺஻ (3.46)
3.6 .1 Modeling for Mode 6 switching state
In switch mode 6 the transistor switches are switched so s = 1, the snubber capacitors is recharging, ݏ
௖
ൌ Ͳ , 0 < ܷ
஼
ೞభ
< E and 0 < ܷ
஼
ೞమ
< E. This implies that the voltage applied across node A and B, -E < ܷ
஺஻
< E. The resonant current is negative,
݅
௥
൏ Ͳ and the winding capacitance voltage is fully negatively charged ܷ
஼
ಽ
ൌ െܷ
஼
ೈ
, and the rectifier is conducting in negative direction.
Figure 3.6
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.6.
30

ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ۍ
ێ
ێ
ێ
ێ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ െͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
Ͳ
ێ
ۏ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ െͳ െͳ െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
(3.47)
Which is similar to Mode 2, therefore we get the same skew symmetric system for mode 6.
3.6.2 Mode 6 model for simulation
The suitable model for simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ಽ
Ͳ
஼
ಽ
ା஼
ೢ
ோ
ಽ
ሺ஼
ಽ
െͳ
ା஼
ೢ
ሻ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ ͳ
஼
ಽ Ͳ Ͳ
ሺ஼
ಽ
ା஼
ೢ
െͳ Ͳ െܴ
௥
ሻ
ሻ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ۏ
ێ
ێ
ۍ
Ͳ
ې
Ͳ
Ͳ
Ͳ ۑ
ۑ
ۑ
Ͳے
ܷ
஺஻
3.6.3 Mode 6 skew symmetric model for simulation and analysis
Using the freedom based on ܷ
஼
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Choosing ߙ ൌ
ൌ ܷ
஼
ೢ
஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
஼
ಽ
ା஼
ೢ
ሻ
Ͳ
ሺͳ െ ߙሻ
Ͳ
Ͳ
Ͳ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
ሺͳ െ ߚሻ
െͳሺͳ െ ߛሻ
Ͳ
െͳ
െ
ି஼
ೢ
Ͳ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
െߛ
ሻ
஼
ಽ
஼
ೢ ǡ ߚ ൌ Ͳǡ ߛ ൌ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
,
, the system can be described as
ሺߙሻ
ሺߚሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
The system is given by the skew symmetric system
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ோ
ಽ
Ͳ
ሺ஼
ೢ
ି஼
ಽ
ሻ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
஼
ಽ Ͳ െ
ێ
ۏ
െͳ െ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
ோ
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
Ͳ
െͳ െ
Ͳ
ି஼
ಽ
Ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ೢ
ሺ஼
ಽ
ା஼
ೢ
ሻ ͳ
஼
ಽ
ሺ஼
ಽ
ା஼
ೢ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
൅
ێ
ۏ
ێ
ێ
ۍ
Ͳ
ې
Ͳ
Ͳ
Ͳ ۑ
ۑ
ۑ
Ͳے
ܷ
஺஻
ۍ
Ͳ
Ͳ
Ͳ
ێ
ێ
ێ
ۏ
ې
ۑ
Ͳ
Ͳے
ۑ
ۑ
ܷ
஺஻
(3.48)
(3.49)
31

3.7.1 Modeling for Mode 7 switching state
In switch mode 7 the transistor switches are in s = 1, the snubber capacitors charged, ݏ
௖
ൌ ͳ , ܷ
஼
ೞభ
= 0 and ܷ
஼
ೞమ
= E. This implies that the voltage applied across node A and B, ܷ
஺஻
ൌ ܧ . The resonant current is negative, ݅
௥
൏ Ͳ and the winding capacitance voltage is fully negatively charged ܷ
஼
ಽ
ൌ െܷ
஼
ೈ
, and the rectifier is conducting in negative direction.
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.7.
Figure 3.7
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ =
ۍ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ െͳ
ێ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ െͳ െͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ Ͳ Ͳ Ͳ Ͳ Ͳ ͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
ܷ
஼
ೞ
ோ
ೝ
݅
݅
ோ
ಽ
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
݅
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
Which is similar to Mode 5, therefore we get the same skew symmetric matrix for
Mode 7 as well.
3.7.2
Mode 7 model for simulation
The suitable model for simulation is given by
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
Ͳ
Ͳ
Ͳ
െ
ێ
ۏ
Ͳ
െͳ
െ
Ͳ
஼
ೢ
ோ
ಽ
ሺ஼
ೢ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ
஼
ೢ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
஼
ಽ Ͳ Ͳ
ሺ஼
ೈ
ି஼
ಽ
Ͳ Ͳ െܴ
௥
ሻ
ሻ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ێ
ۍ
Ͳ
Ͳ
Ͳ
ێ
ۏ
Ͳ ͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
(3.50)
(3.51)
32

3.7.3 Mode 7 skew symmetric model for simulation and analysis
Using the freedom based on ܷ
஼
ಽ
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ൌ
ێ
ێ
ێ
ۏ
ۍ
ێ
Ͳ
Ͳ
Ͳ
െ
஼
ೈ
Ͳ
ሺଵାఈሻ
ோ
ಽ
ሺ஼
ೈ
஼
ಽ
ି஼
ಽ
Ͳ
ሺଵାఉሻ
ሻ
Ͳ
Ͳ െ
Ͳ
Ͳ െ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ െ
െͳ െͳሺͳ െ ߛሻ Ͳ
െܷ
ோ
ಽ
ோ
ಽ
஼
ೢ
, the system can be described as
Ͳ
஼
ೈ
ሺିఈሻ
ሺ஼
ೈ
஼
ಽ
ି஼
ಽ
Ͳ
ሺିఉሻ
ሺ஼
ೈ
െߛ
ି஼
ಽ
ሻ
ሻ ͳ
஼
ೢ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െܴ
௥
ێ
ێ
ێ
ۏ
ۍ
ێ
ۑ
ۑ
ۑ
ے
ې
ۑ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ێ
ۍ
Ͳ
Ͳ
Ͳ
ێ
ۏ
Ͳ ͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
Choosing α =
஼
ಽ
஼
ೢ
, β =0, γ =
஼
ಽ
ሺ஼
ಽ
ି஼
ೢ
ሻ
,
The system is given by the skew symmetric system
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
െ
Ͳ
ሺ஼
ೈ
ା஼
ಽ
ሻ
ோ
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ Ͳ െ
െͳ െ
ோ
ಽ
ሺ஼
ೈ
஼
ೢ
ି஼
ಽ
ሻ
ሺ஼
ೈ
ି஼
ಽ
ሻ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
ି஼
ಽ
ோ
ಽ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
ሻ
Ͳ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ ͳ
஼
ೢ
ሺ஼
ೈ
Ͳ
ି஼
ಽ
ሻ
஼
ಽ
ሺ஼
ೈ
ି஼
ಽ
ሻ
െܴ
௥
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ێ
ۏ
ێ
ێ
ۍ
Ͳ
ې
Ͳ
Ͳ
Ͳ ۑ
ۑ
ۑ ͳے
ܷ
஺஻
3.8.1 Modeling for Mode 8 switching state
In switch mode 8 the transistor switches are in ݏ ൌ ͳ , the snubber capacitors charged, ݏ
௖
ൌ ͳ , ܷ
஼
ೞభ
= 0 and ܷ
஼
ೞమ
= E. This implies that the voltage applied across node A and B, ܷ
஺஻
= E. The resonant current is positive, ݅
௥
൐ Ͳ and the winding capacitance winding capacitance voltage is – ܷ
஼
ಽ
൏ ܷ
஼
ೈ
< ܷ
஼
ಽ and the rectifier is not conducting.
The circuit diagram and a possible optimal spanning graph according to suggested modeling method is given in Figure 2.1 and Figure 3.8.
Figure 3.8
33

ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
݅
݅
ܷ
ܷ
ோ
ೝ
ோ
ಽ
஼
ಽ
ܷ
௅
ೝ
݅
௎
ಲಳ
஼
ೝ
݅
݅
஼
ೢ
஼
ೞ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
=
ێ
ێ
ێ
ێ
ۍ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ
െͳ െͳ െͳ Ͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ێ
ێ
ێ
ێ
ۏ
ۍ
ܷ
௎
ಲಳ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
ܷ
݅
݅
݅
ோ
ೝ
ோ
ಽ
஼
ಽ
௅
ೝ
ې
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ۑ
ے
By further splitting the fourth and fifth rows, we get
൤
݅
ܷ
ோ
ೝ
ோ
ಽ
Ͳ Ͳ Ͳ Ͳ Ͳ
ቃ ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
Ͳ Ͳ
ቃ ൤
ܷ
ோ
ೝ
݅
ோ
ಽ
The constitutive equation for the resistors is given by
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
൨ ቉ ൤
݅
ܷ
ோ
ೝ
ோ
ಽ
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
ܴ
Ͳ
௥
Ͳ ͳ
ൗ
ܴ
௅
Ͳ Ͳ Ͳ Ͳ Ͳ
ቃ
൤
ܷ
݅
ோ
ೝ
ோ
ಽ
൨ ൌ ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
஼
ಽ
݅
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
Performing the matrix manipulation
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ ے
ۑ
ۑ
ې
ۑ
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ
Ͳ Ͳ
െͳ Ͳ ے
ۑ
ۑ
ې
ۑ
Ͳ
ቃ ܷ
஺஻
ቈ
Ͳ Ͳ Ͳ Ͳ ܴ
Ͳ ͳ ܴ
௅
Ͳ Ͳ
௥
Ͳ ቉
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.52)
(3.53)
(3.54)
By further manipulation of R matrix i.e., writing the R matrix in the form of 5 X 5 matrix in the above equation, we get
34

ێ
ێ
ێ
ۏ
ۍ
ێ
݅
݅
݅
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
஼
ಽ
௅
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ ͳ
Ͳ Ͳ ͳ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ Ͳ
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
ܷ
஼
ೝ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
Ͳ Ͳ Ͳ Ͳ Ͳ
ێ
ۏ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ െܴ
௥
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ې
ۑ
ۑ
ۑ
ܷ
஺஻
By simplifying the equation (3.55), we get the below equation which is the skew symmetric system of mode 8
ێ
ێ
ێ
ۏ
ۍ
ێ
݅
஼
ೝ
݅
݅
஼
ೢ
ܷ
஼
ೞ
ܷ
௅
஼
ಽ
ೃ
ۑ
ۑ
ۑ
ے
ې
ۑ
=
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
Ͳ Ͳ
Ͳ Ͳ ͳ ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
Ͳ Ͳ Ͳ Ͳ Ͳ
െͳ െͳ Ͳ Ͳ െܴ
௥
ې
ۑ
ۑ
ۑ
ے
ێ
ێ
ێ
ۏ
ۍ
ێ
ܷ
஼
ೝ
ܷ
ܷ
஼
ೢ
஼
ೞ
݅
݅
஼
ಽ
௅
ೝ
ۑ
ۑ
ۑ
ے
ې
ۑ
൅
ۍ
ێ
ێ
ێ
ۏ
Ͳ
Ͳ
Ͳ
Ͳ
െͳے
ۑ
ۑ
ې
ۑ
ܷ
஺஻
(3.55)
35

The DC/DC switched power converter is simulated based on a fifth order model. The only measurement signal that the observer receives is the resonant current and the other states will be estimated in the observer [3].
The fifth order Hamiltonian observer is designed and implemented by using Matlab software and Simulink toolbox shown in Figure 4.1. Simulink® is a block diagram environment for multi-domain simulation and Model-Based Design. It supports system-level design, simulation, automatic code generation, and continuous test and verification of embedded systems. It also provides a graphical editor, customizable block libraries, and solvers for modeling and simulating dynamic systems. Furthermore, it is integrated with MATLAB® that incorporates MATLAB algorithms into models and export simulation results to MATLAB for further analysis (www.mathworks.com/products/simulink).
Figure 4.1 The simulation of the whole system
36

Figure 4.2 shows the specific Simulink block(s) used in the design for the 5 th order
Hamiltonian filter.
Figure 4.2 Significant System Blocks
For Block 1:
Figure 4.3 System Block 1
The Signal Generator block shown in Figure 4.3 can produce one of four different waveforms: sine wave, square wave, sawtooth wave, and random wave. The signal parameters can be expressed in Hertz as default or radians per second. In default parameter values, any one of the following waveforms such as sine wave, square wave, sawtooth wave, and a random wave can be obtained or otherwise waveforms can be specified by the user.
37

A negative amplitude parameter value causes a 180-degree phase shift. A phase shifted wave other than 180 degrees can also be generated in many ways. Connecting a clock block signal to a Matlab function block and by specifying or writing the equation for a specific wave desired is one way. Another way is through connecting another continuous source blocks depending on the users design.
The output settings of the Signal Generator block can be varied while a simulation is in progress. This is useful to determine quickly the response of a system to different types of inputs.
The Amplitude and Frequency parameters determine its respective output signal. The parameters must be of the same dimensions after scalar expansion. If the
Interpret vector parameters as 1-D check box are cleared, the block outputs a signal with the same dimension as the Amplitude and Frequency parameters (after scalar expansion).
However if the check box is selected or checked, the block outputs a vector (1-D) signal and if the Amplitude and Frequency parameters are row or column vectors, that is, single row or column 2-D arrays will result. Otherwise, the block outputs a signal with the same dimension as the parameters.
In this study, Signal Generator with the rating below Figure 4.4 and a gain block was utilized . A signal generator generates a square wave with the amplitude ‘E’ and a frequency of ‘fcontrol’.
From fifthOrderAH.m;
Figure 4.4 Signal Generator for Block 1
Line 9: E = 540;
Line 10: fcontrol = 30e6; % for 30MHz Sampling time
38

On the other hand, the gain block properties are shown in Figure 4.5. It uses a variable ‘B’ based from line 20 of fifthOrderAH.m
Line 20: B=[0;0;0;0;1];
The Gain block receives decision coverage. Through a gain block, the
1 ൈ ͳ matrix signal produced by the signal generator will now become a 5 1 matrix data. If the Saturate on integer overflow parameter is selected, the decision coverage measures the number of times the block saturates on integer overflow. It indicates a true decision and the number of times the block does not saturate on integer overflow, indicating a false decision. If at least one time step is true and at least one time step is false, decision coverage for the block is 100%. If no time steps are true, or if no time steps are false, decision coverage is 50%. The software treats each element of a vector or matrix as a separate coverage measurement
(www.mathworks.com).
Figure 4.5 Signal Gain for Block 1
For Block 2:
Figure 4.6 shows the s-function block in Matlab Simulink for 5 th order hamiltonian filter implementation. S-function helps to solve various kinds of problems when working with the Simulink and Simulink Coder products. These problems includes extending the set of algorithms (blocks) provided by the Simulink and Simulink Coder
39

products, interfacing legacy (hand-written) code with the Simulink and Simulink Coder products, interfacing to hardware through device driver S-functions, generating highly optimized code for embedded systems, and verifying code generated for a subsystem as part of a Simulink simulation. An application program interface (API) was used to write Sfunctions that allows the user to implement generic algorithms in the Simulink environment with flexibility.
Figure 4.6 System Block 2
Although S-functions provide a generic and flexible solution for implementing complex algorithms in a Simulink model, the underlying API incurs overhead in terms of memory and computation resources. Most often the additional resources are acceptable for real-time rapid prototyping systems.
Figure 4.7 shows the s-function block settings for parameter initialization. S-function name is the fields were the user can specify the corresponding code for its overall functionalities..
Figure 4.7 S-function Settings for Block 2
40

For Block 3:
Figure 4.8 System Block 3
As illustrated in Figure 4.8, A gain and demux block evaluate every element in the array for switching conditions. The first element output of the demux is terminated by the terminator block since it is not needed for switching. This signal terminator is used to cap blocks whose output ports do not connect to other blocks which prevents Simulink to issue warning messages.
The Demux block extracts the components of an input signal and outputs these components as separate signals. The output signals are ordered from top to bottom output port. The number of output parameters allows the user to specify the number and occasionally, the dimensionality of each output port. If the dimensionalities of the outputs are not specified, the block will determine the dimensionality of the outputs instead. Also the Demux block operates in either vector mode or bus selection mode. The two modes differ in the types of signals they accept. Vector mode accepts only a vector-like signal, that is, either a scalar (oneelement array), vector (1-D array), or a column or row vector (one row or one column 2-D array). Bus selection mode accepts only a bus signal. In this study, the researcher used the vector mode operations.
41

Figure 4.9 System Gain for Block 3
From fifthOrderAH.m
;
At line 35 to line 40, the system matrix parameters were defined. Variable ‘D’ parameter in Figure 4.9 was initialized using the code that follows. At line 35 to line
40, the system matrix parameters were defined.
D1=[Cr 0 0 0 0;
0 Cw 0 0 0;
0 0 Cs 0 0;
0 0 0 CL 0;
0 0 0 0 Lr];
D = inv (D1);
Where,
CL=1e-3;
Cw=210e-9;
Cr=1.75e-6;
Lr=32e-6;
Cs = 40e-9;
42

For Block 4:
Figure 4.10 System Block 4
Figure 4.10 is a matlab embedded function to implement the
Hamiltonian filter switching conditions and in figure 4.11 shows the implementations for the eight (8) modes of operation for the 5 th order Hamiltonian filter. The functions in figure 4.10 can be altered or customized depending on the users preferences.
Matlab embedded function has the advantage to simplify complicated system especially for switching logic conditions. In addition, Figure 4.11 shows the Matlab environment for the embedded system. In here, the used can initialize or customize the switching functions.
43

For Block 5:
Figure 4.11 Code Inside the Matlab Embedded Function in Block 4
Figure 4.12 System Block 5
Figure 4.12, uses two (2) memory toolbox to get the previous switching condition and switching state. This is useful in Matlab simulation since the simulation is a straightforward process and without the memory toolbox, the chronology of the
44

process might not be followed and will not result to the desire output. Below is the code for the S-function of Figure 4.6.
For the s-functions:
The fifthhamiltonAH1.m
the continuous Hamilton filters.
It receives 6 vector input and gives and output of 5 vectors. Again, the 6 th input vector is a field for modes of operation detections coming from the designed controller.
@sys=mdlDerivatives in line 76 of the fifthhamiltonAH1.m
Below is the code for this function from line 105 to 125 modes = u(2); xc=x(1:5); v = u(1); if modes == 1 %mode 1
sys = A01*xc + B01*v; elseif modes == 2 %mode 2
sys = A02*xc + B02*v; elseif modes == 3 %mode 3
sys = A03*xc + B03*v; elseif modes == 4 %mode 4
sys = A04*xc + B04*v; elseif modes == 5 %mode 5
sys = A05*xc + B05*v; elseif modes == 6 %mode 6
sys = A06*xc + B06*v; elseif modes == 7 %mode 7
sys = A07*xc + B07*v; else %mode 8
sys = A08*xc + B08*v; end
where A01-A08 and B01-B08 is defined by fifthOrder.m
at line 41-108.
@sys=mdlOutputs in line 107 of a fifthhamilton.m
Below is the code for this function from line 109-126 modes = u(6);
%mode 1 if modes == 1
sys = A01'*x; elseif modes == 2 %mode 2
sys = A02'*x;
45

elseif modes == 3 %mode 3
sys = A03'*x; elseif modes == 4 %mode4
sys = A04'*x; elseif modes == 5 %mode 5
sys = A05'*x; elseif modes == 6 %mode 6
sys = A06'*x; elseif modes == 7 %mode 7
sys = A07'*x; else %mode 8
sys = A08'*x; end
Where A01-A08 is defined at fifthOrderAH.m at line 44-115.
%8 MODES of Operation
%MODE 1
A01 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 0;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) 0 -CL/(Cw+CL) -Rr]/D1;
B01 = [0;0;0;0;1];
%MODE 2
A02 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 1;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) -1 -CL/(Cw+CL) -Rr]*D;
B02 = [0;0;0;0;0];
%MODE 3
A03 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 0;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) 0 -CL/(CL+Cw) -Rr]*D;
B03 = [0;0;0;0;1];
%MODE 4
A04 = [0 0 0 0 1;
0 0 0 0 1;
0 0 0 0 0;
46

0 0 0 -1/RL 0;
-1 -1 0 0 -Rr]*D;
B04 = [0;0;0;0;1];
%MODE 5
A05 = [0 0 0 0 1;
0 -(Cw+CL)/(RL*(Cw-CL)) 0 -CL/(RL*(Cw-CL)) Cw/(Cw-CL);
0 0 0 0 0;
0 -CL/(RL*(Cw-CL)) 0 0 CL/(Cw-CL);
-1 -Cw/(Cw-CL) 0 -CL/(Cw-CL) -Rr]*D;
B05 = [0;0;0;0;1];
%MODE 6
A06 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 1;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(CL+Cw);
-1 -Cw/(CL+Cw) -1 -CL/(CL+Cw) -Rr]*D;
B06 = [0;0;0;0;0];
%MODE 7
A07 = [0 0 0 0 1;
0 -(Cw+CL)/(RL*(Cw-CL)) 0 -CL/(RL*(Cw-CL)) Cw/(Cw-CL);
0 0 0 0 0;
0 -CL/(RL*(Cw-CL)) 0 0 CL/(Cw-CL);
-1 -Cw/(Cw-CL) 0 -CL/(Cw-CL) -Rr]*D;
B07 = [0;0;0;0;1];
%MODE 8
A08 = [0 0 0 0 1;
0 0 0 0 1;
0 0 0 0 0;
0 0 0 -1/RL 0;
-1 -1 0 0 -Rr]*D;
B08 = [0;0;0;0;1];
47

1.
Below are the list of problems which we faced while doing simulation:
¾ Expected output in the s-function in the system is not yet met.
¾ Since the expected output from the s-function is not yet met, the system cannot trigger the switching actions due to unmet switching conditions.
¾ Modes of Operations can be determined if the switching conditions are met.
¾ Since timed modes of operations is not a good solution, we need to have a working equations before our controller can react and trigger its switches (S & Sc) to turn on/off.
¾ Unstable system response due to unhandled system conditions for switching.
48

Results are plotted for the fifth order Hamiltonian Observer using Matlab and
Simulink shown in Figure 4.1
Fifth order model, uAB
600 u
400
-200
-400
200
0
-600
0 0.5
1 1.5
2 2.5
3 3.5
x 10
-5
Figure 6.1
System Voltage Input at 540 Volts
Figure 6.1 show the system input signal for the fifth order Hamiltonian. The voltage magnitude is 540V at a system frequency of 30MHz.
49

. ro e lue for Fifth rder model
-200
-400
200
0
600
400 u r u i r
-600
0 0.5
1 1.5
2 ime
2.5
3 3.5
x 10
-5
Figure 6.2 shows the system response when all the modes of operations were completed.
These values show that the system responses correctly as the eight modes of operations were completed in one complete cycle
50

ro e lue for Fifth rder model, 0
3000
2500
2000
1500
1000
500
0
-500
-1000
0 u
0.5
1 1.5
2 ime
2.5
3
Figure 6.3 Behavior of the Snubber Capacitor
3.5
x 10
-5 ro e lue for Fifth rder model, 0 u
51.5
51
50.5
50
4 .5
0
53.5
53
52.5
52
0.5
1 1.5
2 ime
2.5
Figure 6.4 Load Voltage
3 3.5
x 10
-5
51

Fifth order model, it h t te d
6
5
4
3
2
1
0 0.5
1 1.5
2 2.5
3 3.5
x 10
-5
Figure 6.5 Modes of Operations
Figure 6.5 show the modes of operations for Z1 and Z2 Switches. The controllers response based on the specific values of , , , , and .
52

Fifth order model, it h t te d
6
5
4
3
2
1
0
0 0.5
1 1.5
2 2.5
3 3.5
x 10
-5
Figure Operations
Figure 6.6 show the modes of operations for Z1 and Z2 Switches. The controllers response based on the specific values of f f f f f f f f f f f f f f f f f f f f f , , , , and . In one complete system cycle all modes of operations were activated to meet the succeeding system conditions.
53

In chapter 2, a Hamiltonian model was proposed based on fifth order model of the resonant series loaded power converter. The load in the resonant power converter was modeled by a load capacitance, C
L
, see Figure. 2.1. The capacitor in the converter was modeled by the Snubber capacitor, C s
.
Two types of models for each of the eight switch modes are given for the resonant power converter, one is suitable for simulation and other is suitable for simulation and analysis. The circuit was run in eight modes. For this new configuration of the system, we proposed the Hamiltonian model based on fifth order model of the system.
The Hamiltonian model is shown to work in a proper way in simulations.
The state trajectory follows the expected trajectory.
The proposed model for the power converter will be used for design of an
Hamiltonian Observer. The simulation shows that the behavior of the model is in accordance with the physical system. It is judged that the model can be used for design of a Hamiltonian observer.
The model can be used for designing a switched Hamiltonian filter, see [2].
54

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63

factor operation," IEEE Trans. Power Electron., vol. 12, no. 1, pp. 103-115,
January 1997.
%close all; clear all ;
%%%%%%%%%%%%%Fifth Order Model%%%%%%%%%%%%% clc; s = 1; sc = 1;
E=540; fcontrol=40e3;
CL=1e-3;
Cw=210e-9;
Cr=1.75e-6;
Lr=32e-6; %11.5e-6;
%Cs=396e-9;
Cs = 40e-9;
RL=1000;
Rr = 1;
B=[0;0;0;0;1];
C = 2e-6;
L = 32e-6;
TStop = 5*0.5e-4;
Rs = 0.01; flag1 = 0;
TStep = 3.33e-08; %30MHz Sampling time
%TStop = .8e-4; %end of simulation
%TStop = 6.76e-005;
%initialization
D1=[Cr 0 0 0 0;
0 Cw 0 0 0;
0 0 Cs 0 0;
0 0 0 CL 0;
0 0 0 0 Lr];
D = inv(D1);
64

%8 MODES of Operation
%MODE 1
A01 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 0;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) 0 -CL/(Cw+CL) -Rr]/D1;
B01 = [0;0;0;0;1];
%MODE 2
A02 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 1;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) -1 -CL/(Cw+CL) -Rr]*D;
B02 = [0;0;0;0;0];
%MODE 3
A03 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 0;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(Cw+CL);
-1 -Cw/(Cw+CL) 0 -CL/(CL+Cw) -Rr]*D;
B03 = [0;0;0;0;1];
%MODE 4
A04 = [0 0 0 0 1;
0 0 0 0 1;
0 0 0 0 0;
0 0 0 -1/RL 0;
-1 -1 0 0 -Rr]*D;
B04 = [0;0;0;0;1];
%MODE 5
A05 = [0 0 0 0 1;
0 -(Cw+CL)/(RL*(Cw-CL)) 0 -CL/(RL*(Cw-CL)) Cw/(Cw-CL);
0 0 0 0 0;
0 -CL/(RL*(Cw-CL)) 0 0 CL/(Cw-CL);
-1 -Cw/(Cw-CL) 0 -CL/(Cw-CL) -Rr]*D;
B05 = [0;0;0;0;1];
%MODE 6
A06 = [0 0 0 0 1;
0 -(Cw-CL)/(RL*(Cw+CL)) 0 -CL/(RL*(Cw+CL)) Cw/(Cw+CL);
0 0 0 0 1;
0 -CL/(RL*(Cw+CL)) 0 0 CL/(CL+Cw);
-1 -Cw/(CL+Cw) -1 -CL/(CL+Cw) -Rr]*D;
B06 = [0;0;0;0;0];
%MODE 7
A07 = [0 0 0 0 1;
0 -(Cw+CL)/(RL*(Cw-CL)) 0 -CL/(RL*(Cw-CL)) Cw/(Cw-CL);
65

0 0 0 0 0;
0 -CL/(RL*(Cw-CL)) 0 0 CL/(Cw-CL);
-1 -Cw/(Cw-CL) 0 -CL/(Cw-CL) -Rr]*D;
B07 = [0;0;0;0;1];
%MODE 8
A08 = [0 0 0 0 1;
0 0 0 0 1;
0 0 0 0 0;
0 0 0 -1/RL 0;
-1 -1 0 0 -Rr]*D;
B08 = [0;0;0;0;1];
%END of 8 modes of operation
D = inv(D1); h=3.33e-08; %30MHz
[F0,dG0]=c2d(A01*D,B01,h);
[F1,dG1]=c2d(A02*D,B02,h);
[F2,dG2]=c2d(A03*D,B03,h);
[F3,dG3]=c2d(A04*D,B04,h);
[F4,dG4]=c2d(A05*D,B05,h);
[F5,dG5]=c2d(A06*D,B06,h);
[F6,dG6]=c2d(A07*D,B07,h);
[F7,dG7]=c2d(A08*D,B08,h); maxiter = 3.8e-5;
%---------------------------% opt=simset( 'MaxStep' ,1e-9); tic; sim( 'fifthmdl' ,maxiter,opt); %runs the Simulink file elapsetime=toc; ys=yd/D1;
%---------------------------------Graphs----------------------------
-----% figure(1),plot(t,ud); title( 'Fifth order model, uAB' ); legend( 'u' ); xlim([0 maxiter]); grid on figure(2),plot(t,[ys(:,1:2) ys(:,5)] , 'LineWidth' ,1.5); title( 'Fifth order model, states' ) legend( 'uCr' , 'uCw' , 'iLr' ); title( 'Process values for Fifth Order model' ); xlabel( 'Time s' ); xlim([0 maxiter]); grid on figure(3),plot(t,[ys(:,3)] , 'LineWidth' ,1.5); title( 'Fifth order model, states' ) legend( 'uCs' );
66

title( 'Process values for Fifth Order model, K=0' ); xlabel( 'Time s' ); xlim([0 maxiter]); grid on figure(4),plot(t,[ys(:,4)] , 'LineWidth' ,1.5); title( 'Fifth order model, states' ) legend( 'uCL' ); title( 'Process values for Fifth Order model, K=0' ); xlabel( 'Time s' ); xlim([0 maxiter]); grid on figure(5),plot(t,qd, 'LineWidth' ,1.5); title( 'Fifth order model, switch state' ) xlim([0 maxiter]); legend( 'qd' ) grid on
%
% figure(6),plot(t,wd,'LineWidth',1.5);
% title('Fifth order model, switch state')
% xlim([0 maxiter]);
% legend('wd')
% grid on
function [sys,x0,str,ts] = fifthhamiltonAH1(t,x,u,flag,A01,A02,A03,A04,A05,A06,A07,A08,B01,B02,
B03,B04,B05,B06,B07,B08)
%CSFUNC An example M-file S-function for defining a continuous system.
% Example M-file S-function implementing continuous equations:
% x' = Ax + Bu
% y = Cx + Du
% Copyright 1990-2009 The MathWorks, Inc.
% $Revision: 1.1.6.1 $ switch flag,
%%%%%%%%%%%%%%%%%%
% Initialization %
%%%%%%%%%%%%%%%%%% case 0,
[sys,x0,str,ts]=mdlInitializeSizes();
%%%%%%%%%%%%%%%
% Derivatives %
%%%%%%%%%%%%%%% case 1, sys=mdlDerivatives(t,x,u,A01,A02,A03,A04,A05,A06,A07,A08,B01,B02,B03
,B04,B05,B06,B07,B08);
%%%%%%%%%%%
67

% Outputs %
%%%%%%%%%%% case 2,
sys = mdlUpdate(t,x,u); case 3,
sys=mdlOutputs(t,x,u); case 4,
sys=mdlGetTimeOfNextVarHit(t,x,u);
%%%%%%%%%%%%%%%%%%%
% Unhandled flags %
%%%%%%%%%%%%%%%%%%% case 9,
sys = [];
%%%%%%%%%%%%%%%%%%%%
% Unexpected flags %
%%%%%%%%%%%%%%%%%%%% otherwise
DAStudio.error( 'Simulink:blocks:unhandledFlag' , num2str(flag)); end
% end csfunc
%
%===================================================================
==========
% mdlInitializeSizesss
% Return the sizes, initial conditions, and sample times for the Sfunction.
%===================================================================
==========
% function [sys,x0,str,ts]=mdlInitializeSizes(t,x,u) sizes = simsizes; sizes.NumContStates = 5; sizes.NumDiscStates = 0; sizes.NumOutputs = 5; sizes.NumInputs = 2; sizes.DirFeedthrough = 1; sizes.NumSampleTimes = 2; sys = simsizes(sizes); uCL=50; uCw=uCL; uCs=-540; uCr=100; iLr=000;
%x0 = [uCr*2e-6;uCw*0.2e-6;uCs*0.4e-6;uCL*1e-3;iLr*32e-6;1];
Cr=1.75e-6;
Lr=32e-6;
Cw=0.2e-6;
68

CL=1e-3;
Cs=40e-9; x0 = [uCr*Cr;uCw*Cw;uCs*Cs;uCL*CL;iLr*Lr]; str = []; ts = [0 0
1e-7 0];
% [Cr 0 0 0 0;
% 0 Cw 0 0 0;
% 0 0 Cs 0 0;
% 0 0 0 CL 0;
% 0 0 0 0 Lr];
% The state vectors, X and X0 consists of continuous states followed
% by discrete states.
% end mdlInitializeSizes
%
%===================================================================
==========
% mdlDerivatives
% Return the derivatives for the continuous states.
%===================================================================
==========
% function sys=mdlDerivatives(t,x,u,A01,A02,A03,A04,A05,A06,A07,A08,B01,B02,B03
,B04,B05,B06,B07,B08) modes = u(2); xc=x(1:5); v = u(1); if modes == 1 %mode 1
sys = A01*xc + B01*v; elseif modes == 2 %mode 2
sys = A02*xc + B02*v; elseif modes == 3 %mode 3
sys = A03*xc + B03*v; elseif modes == 4 %mode 4
sys = A04*xc + B04*v; elseif modes == 5 %mode 5
sys = A05*xc + B05*v; elseif modes == 6 %mode 6
sys = A06*xc + B06*v; elseif modes == 7 %mode 7
sys = A07*xc + B07*v; else %mode 8
sys = A08*xc + B08*v; end
%disp('Simulating..........');
% end mdlDerivatives
%
%===================================================================
====
% mdlUpdate
69

% Handle discrete state updates, sample time hits, and major time step
% requirements.
%===================================================================
====
% function sys = mdlUpdate(t,x,u); sys = [];
%end mdlUpdate
%===================================================================
==========
% mdlOutputs
% Return the block outputs.
%===================================================================
==========
% function sys=mdlOutputs(t,x,u) sys=x;
% end mdlOutputs
%
%===================================================================
==========
% mdlGetTimeOfNextVarHit
% Return the time of the next hit for this block. Note that the result is
% absolute time.
%===================================================================
==========
% function sys=mdlGetTimeOfNextVarHit(t,x,u) sys = []; %t + 3.33e-08;% 30Mhz Sampling time
% end mdlGetTimeOfNextVarHit
70

Signal
Generator
1
Clock fifthhamiltonAH1 continuous myfunc1 t ud yd qd u *D
Gain1 u uCw uCs uCL fcn q iLr z w
Embedded
MATLAB Function
Memory1
Memory wd
function [q,z] = fcn(u,uCw,uCs,uCL,iLr,w) if w ==1 && u >= 0 %switch state 1
q = 1; elseif w==1 && u <= 0 %condition for entering switch state 2
q = 2; elseif w==2 && u <= 0 && iLr > uCw %switch state 2
q = 2; elseif w==2 && u <= 0 && iLr <= uCw %condition for entering switch state 3
q = 3; elseif w==3 && u <= 0 && iLr >= 0 %switch state 3
q = 3;
71

elseif w==3 && u <= 0 && iLr <= 0 %condition for entering switch state 4
q = 4; elseif w==4 && u <= 0 && uCw >= -uCL %switch state 4
q = 4; elseif w==4 && u <= 0 && uCw <= -uCL %condition for entering switch state 5
q = 5; elseif w==5 && u <= 0 && iLr <= 0 %switch state 5
q = 5; elseif w==5 && u >= 0 %condition for entering switch state 6
q = 6; elseif w==6 && u >= 0 && uCs >= 540 %switch state 6
q = 6; elseif w==6 && u >= 0 && uCs <= 540 && iLr <= 0 %condition for entering switch state 7
q = 7; elseif w==7 && u >= 0 && iLr <= 0 && uCw >= -uCL %switch state 7
q = 7; elseif w==7 && u >= 0 && iLr >= 0 %condition for entering switch state 8
q = 8; elseif w==8 && u >= 0 && uCw <= iLr %switch state
8
q = 8; elseif w==8 && u >= 0 && uCw > iLr %condition for entering switch state 1
q = 1; else
q = 1; %system forced to switch state 1, u=540, ... end z = q;
72