HARMONIC ANALYSIS OF A PARALLEL-LOADED RESONANT CONVERTER
KHAIRUL BARIYAH BINTI ISA
This thesis is submitted in part fulfillment of the requirements for the awarding of Degree in Bachelor of Electrical Engineering
Faculty of Electrical Engineering Universiti Teknologi Malaysia
MAY 2008
ii
“I declare that the content of this thesis is my original with the exclusion of references that have been appropriately acknowledged.”
Signature
:
Name
:
KHAIRUL BARIYAH BINTI ISA
Date
:
7 MAY 2008
iii
Specially dedicated,
To my lovely family especially my beloved mother and father Thanks for your morale support and understanding
To my lovely friends Thank you for all your help and kindness
iv
ACKNOWLEDGEMENT
In the name of Allah The Most Gracious and Most Merciful. Within the Salawat and Salam to Prophet Muhammad S.A.W.
I would like to express my thanks to my supervisor, Dr. Naziha Ahmad Azli because of his supervision, advice and instructions in the completing this project and report. I also want to thanks to Miss Norjulia Mohamad Nordin my ex-supervisor because of her helpfulness and kindness.
To my family, I wish millions of thanks with loves. Thanks for their moral support all the time. My sincere appreciation also extends to my best friends, who have been there to help me whenever I needed them.
Last but not least, I would like to express millions of thanks again for all of them who are supporting and helping me directly or indirectly in the completion of this project and report.
v
ABSTRACT
A method is presented for calculating the harmonic components of the current and voltage in a parallel-loaded resonant converter using frequency domain techniques. Imperfect switching is a major contributor to power loss in converter because switching devices absorb power when they turn on or off if they go through a transition when both voltage current and voltage are nonzero. High switching frequencies are otherwise desirable because of the reduced size of filter components and transformers, which reduces the size and weight of the converter. In resonant switching circuits, switching takes place when voltage and/or current is zero, thus avoiding simultaneous transitions of voltage and current and thereby eliminating switching losses. This type of switching is called “soft” switching. The average, rms and peak values of the voltages and currents in the resonant converter may be determined from steady-state solution for the converter using both time and frequency domain techniques. Through this project, the steady-state response of the converter may be determined using frequency domain techniques by solving for the unknown coefficients in the Fourier series instead of solving the differential equations (time domain techniques). The unknown coefficients in all Fourier series are calculated using the harmonic model and Kirchhoff’s laws. Analysis of the harmonics inside the circuit will be done by adjusting the values of load and frequency. In this project a circuit of a parallel-loaded resonant converter will be designed and simulated using MATLAB/SIMULINK software and compared favorably with those obtained using frequency domain techniques. However, these two results obtained are not the same because of inaccurate setting while simulating using MATLAB/SIMULINK.
vi
ABSTRAK
Satu kaedah di persembahkan untuk mengira komponen harmonik bagi arus dan voltan dalam penukar salun beban selari menggunakan teknik domain frekuensi. Pensuisan tidak sempurna adalah penyumbang terbesar kepada kehilangan kuasa di dalam penukar kerana alat pensuisan menyerap kuasa apabila operasi buka atau tutup jika melalui peralihan ketika kedua-dua voltan dan arus tidak sifar. Sebaliknya frekuensi pensuisan tinggi adalah diperlukan kerana mengurangkan saiz komponen penapis dan pengubah, sekaligus mengurangkan saiz dan berat penukar. Di dalam litar pensuisan salun, pensuisan berlaku ketika voltan dan/atau arus adalah sifar, lalu mengelakkan peralihan serentak pada voltan dan arus, sekaligus menghapuskan kehilangan pensuisan. Jenis pensuisan ini dipanggil pensuisan “lembut”. Nilai purata, rms dan puncak bagi voltan dan arus di dalam penukar salun boleh ditentukan dari penyelesaian keadaan tetap untuk penukar menggunakan kedua-dua kaedah domain masa dan frekuensi. Melalui projek ini, tindak balas keadaan tetap bagi penukar akan ditentukan menggunakan kaedah domain frekuensi dengan menyelesaikan pekalipekali tidak diketahui berbanding dengan persamaan pembezaan (kaedah domain masa). Pekali-pekali tidak diketahui di dalam semua siri Fourier dikira menggunakan model harmonik dan hukum Kirchhoff. Analisis harmonik di dalam litar akan dilakukan dengan menukar nilai beban dan frekuensi. Di dalam projek ini, litar penukar salun beban selari direkabentuk dan disimulasi menggunakan perisian MATLAB/SIMULINK dan dibandingkan dengan kaedah domain frekuensi. Bagaimanapun, kedua-dua keputusan ini adalah tidak sama kerana penetapan simulasi pada MATLAB/SIMULINK adalah kurang tepat.
vii
TABLE OF CONTENTS
CHAPTER
CONTENTS
PAGE
APPROVAL OF SUPERVISOR TITLE
CHAPTER 1
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOL
xiii
INTRODUCTION
1
1.1 Objectives of Project
2
1.2 Scopes of Project
3
1.3 Outline of Thesis
3
viii CHAPTER 2
RESONANT CONVERTER
5
2.1 Introduction
5
2.2 Components of Resonant Converter
5
2.3 Soft-switching DC-DC Converters
7
2.4 Full-Bridge Parallel Resonant Inverter
8
2.4.1
Analysis of PRI
9
2.4.2
Voltage Transfer Function
12
2.5 Full-Wave Bridge Rectifier
CHAPTER 3
2.5.1 Voltage Transfer Function
14
2.5.2 Fourier Series Analysis
16
2.6 The Parallel Resonant Converter
21
2.6.1 Voltage Transfer Function
22
2.6.2 Resonant Converter Analysis
23
HARMONIC ANALYSIS
25
3.1 Introduction
25
3.2 Frequency Domain Technique
26
3.3
CHAPTER 4
13
3.2.1 Harmonic Analysis of PRC
27
Simulation using MATLAB/SIMULINK program
30
RESULTS AND DISCUSSIONS
32
4.1 Introduction
32
4.2 MATLAB/SIMULINK Simulation
33
4.2.1 Switching Frequency, fS Variation
33
4.2.2 Load, RL Variation
39
4.2.3 Discussion
43
4.3 Harmonic Analysis
44
ix CHAPTER 5
REFERENCE
CONCLUSION AND RECOMMENDATION
47
5.1 Conclusion
47
5.2 Recommendation for Future Work
48
49
x
LIST OF TABLES
TABLE
TITLE
PAGE
3.1
Magnitude and phase of Vi
29
4.1
Comparison of output voltage between QL = 4.36
39
and QL = 6.55 for switching frequency above resonant. 4.2
Comparison of output voltage between QL = 4.36
39
and QL = 6.55 for switching frequency below resonant. 4.3
Output voltage for variation of loads
43
4.4
Rectifier Input Voltage Coefficients
44
4.5
Rectifier Input Current Coefficients
45
4.6
Rectifier Output Voltage Coefficients
45
4.7
Rectifier Output Current Coefficients
45
xi
LIST OF FIGURES
FIGURE
TITLE
PAGE
1.1
Current and voltage waveforms of hard and
2
resonant switching systems 2.1
General block diagram of a DC-DC resonant
6
converter 2.2
Types of Resonant Inverter
6
2.3
Types of High-frequency Rectifier
7
2.4
Circuit of a Full Bridge Parallel Resonant Inverter
9
2.5
VT versus time
9
2.6
Circuit of a Full Wave Bridge Rectifier
13
2.7
Current and voltage waveform for Bridge Rectifier
15
2.8
Resonant converter model
16
2.9
Rectifier input voltage
17
2.10
Rectifier output voltage
17
2.11
Frequency domain model for rectifier output
19
2.12
Rectifier output current
20
2.13
Rectifier input current
20
2.14
(a) Circuit of a Parallel-Loaded Resonant
22
Converter, (b) Equivalent ac circuit for parallel resonant converter. 3.1
Final year project outline
25
xii 3.2
Frequency domain harmonic model for resonant
26
converter 3.3
Circuit of a Parallel-loaded resonant converter
31
4.1
Output waveform for QL = 4.36 and fS above
34
resonance. (a) fS = 35 kHz (b) fS = 40 kHz (c) fS = 45 kHz 4.2
Output waveform for QL = 4.36 and fS below
35
resonance. (a) fS = 15 kHz (b) fS = 25 kHz 4.3
Output waveform for QL = 6.55 and fS above
36
resonance. (a) fS = 35 kHz (b) fS = 40 kHz (c) fS = 45 kHz 4.4
Output waveform for QL = 6.55 and fS below
38
resonance. (a) fS = 15 kHz (b) fS = 25 kHz 4.5
Output waveform for QL > 2.5 and fS = 35 kHz (a) RL = 2.5 Ω (b) RL = 5 Ω (c) RL = 7.5 Ω (d) RL = 10 Ω (e) RL = 12.5 Ω (f) RL = 15
40
xiii
LIST OF SYMBOLS
SYMBOL
DESCRIPTION
FDT
-
Frequency domain technique
SMPS
-
Switching modes power supply
ZCS
-
Zero Current Switching
ZVS
-
Zero Voltage Switching
SCR
-
Series Resonant Converter
PRC
-
Parallel Resonant Converter
SPRC
-
Series-Parallel Resonant Converter
PRI
-
Parallel Resonant Inverter
SPRI
-
Series-Parallel Resonant Inverter
Vin
-
Input voltage
Zin
-
Input impedance
RL
-
Load resistance
V0
-
Output voltage
I0
-
Output current
DC
-
Direct current
AC
-
Acceleration current
Qr
-
Resonant quality factor
QL
-
Loaded quality factor
fo
-
Corner frequency
fS
-
Switching frequency
fr
-
Resonant frequency
xiv Cr
-
Resonant capacitance
Lr
-
Resonant inductance
Cf
-
Output filter capacitance
Lf
-
Output filter inductance
L
-
Electrical inductance in Henries
C
-
Electrical capacitance in Farads
Z0
-
Characteristic impedance
Ri
-
Equivalent rectifier resistance
Vrms
-
Rms value of input voltage
V1
-
Input voltage
MVs
-
Voltage transfer function of inverter input
MVr
-
Voltage transfer function of resonant circuit
MVI
-
Voltage transfer function of the inverter
-
Voltage transfer function of the rectifier
Vfilter
-
Voltage across filter component
VS
-
Input voltage of rectifier
VSm
-
Peak value of input voltage of rectifier
IS
-
Source current
α
-
Phase shifted
VRI (t)
-
Fourier series for input voltage of rectifier
VR0 (t)
-
Fourier series for output voltage of rectifier
-
Voltage across resonant capacitance
-
Input voltage of resonant circuit
-
Complex Fourier coefficient of input voltage
-
Complex Fourier coefficient of output voltage
IR
-
Rectifier input current
I0
-
Rectifier output current
DFT
-
Discrete Fourier transform
-
Convergence tolerance
k
-
Accelerating factor
n
-
Odd harmonic order
m
-
Even harmonic order
F
-
Farad—unit of electrical capacitance
xv H
-
Henry—unit of electrical inductance
Ω
-
Ohm—unit of electrical resistance
A
-
Ampere—unit of electrical current
V
-
Voltage—unit of electrical voltage
CHAPTER 1
INTRODUCTION
Increasing the frequency of operation of power converters is desirable, as it allows the size of circuit magnetic and capacitors to be reduced, leading to cheaper and more compact circuits. However, increasing the frequency of operation also increases switching losses and hence reduces system efficiency. One solution to this problem is to replace the chopper switch of a standard switching modes power supply (SMPS) topology such as Buck, Boost and etc. with a resonant switch, which uses the resonances of circuit capacitances and inductances to shape the waveform of either the current or the voltage across the switching element, such that when switching takes place, there is no current through or voltage across it, and hence no power dissipation as shown in Figure 1.1. A circuit employing this technique is known as a Resonant Converter (or, more accurately, a quasi-resonant converter, as only part of the resonant sinusoid is utilized). A Zero Current Switching (ZCS) circuit shapes the current waveform, while a Zero Voltage Switching (ZVS) circuit shapes the voltage waveform.
2
Figure 1.1 Current and voltage waveforms of hard and resonant switching systems
Resonant converter topologies can be categorized into three groups that are Series Resonant Converter (SRC), Parallel Resonant Converter (PRC) and SeriesParallel Resonant Converter (SPRC). In an SRC, power from source to the load is done by rectification of the resonant current. On the other hand, the capacitor’s voltage transfers power from source to load in a PRC. Meanwhile, the SPRC combines both the behavior of SRC and PRC [6].
1.1
Objectives of Project
The objectives of this project are as follows:¾ To calculate the steady-state response of the parallel-loaded resonant converter using frequency domain techniques and MATLAB/SIMULINK software ¾ To obtain the steady-state solution by solving for the unknown coefficients in the Fourier series ¾ To compare the solution obtained using frequency domain techniques with those obtained using MATLAB/SIMULINK.
3 1.2
Scopes of Project
This project is based on work presented in [3]. However, simulations in this project are made using MATLAB/SIMULINK rather than PSPICE that has been employed in [3]. The steady-state solution for a parallel-loaded resonant converter is calculated using frequency domain techniques instead of using time domain techniques that are based on the differential equations which describe the converter. These techniques utilize Fourier series descriptions for all voltages and currents in the converter. The steady state solution is obtained by solving for the unknown coefficients in the Fourier series instead of solving the differential equations [3]. The resonant converter is divided into an inverter section and a rectifier section. For this project, the parallel-loaded resonant converter is obtained by cascading a full-bridge parallel resonant inverter to an ideal full-wave bridge rectifier. The rectifier section includes an output filter and a load resistance. The solutions obtained from frequency domain
techniques
are
compared
favorably
with
that
determined
using
MATLAB/SIMULINK.
1.3
Outline of Thesis
This thesis consists of five chapters. The first chapter discusses the objective and scope of this project. Chapter 2 will discuss more on the theory and literature review that have been done. This includes the types of DC-AC resonant inverters, high frequency rectifiers, and DC-DC resonant converters that are the basic building blocks of various high-frequency, high efficiency energy processors.
4 Chapter 3 explains the methods or approaches to implement the proposed objectives. The results and discussions of the project will be presented in Chapter 4. Last but not least, Chapter 5 discusses the conclusion of this project and future work that can be done.
CHAPTER 2
RESONANT CONVERTER
2.1
Introduction
In this chapter, it will discuss on components of resonant converter which are resonant inverter and high frequency rectifier, soft switching DC-DC converters, Full-Bridge Parallel Resonant Inverter (PRI) and also High Frequency Rectifier analysis. Last but not least, this chapter will discuss on Fourier series analysis of PRC.
2.2
Components of Resonant Converter
Cascading a PRI with a High Frequency Rectifier can produce a PRC as shown in the Figure 2.1. The resonant inverter can be classified into three groups which are Series Resonant Inverter (SRI), Parallel Resonant Inverter (PRI) and Series-Parallel Resonant Inverter (SPRI) [2]. This can be shown in Figure 2.2. In this
6
project, harmonic analysis will focus on the PRI which consists of either a halfbridge inverter or a full-bridge inverter. Whereas, the high-frequency rectifier can be half-wave rectifier, full-wave bridge rectifier or a transformer center-tapped rectifier as shown in Figure 2.3.
Figure 2.1 General block diagram of a DC-DC resonant converter
Figure 2.2 Types of Resonant Inverter
7
Figure 2.3 Types of High-frequency Rectifier
An input DC voltage will be inverted to produce a sinusoidal voltage and subsequently be rectified to produce an output voltage of DC level. The PRC can be used as both a step-down and step-up converter.
2.3
Soft-switching DC-DC Converters
New classes of DC-DC converters known in the literature as soft-switching resonant converters have been thoroughly investigated in recent years. Generally speaking, soft switching means the one or more power switches in a DC-DC converter have either turn-ON or turn-OFF switching losses eliminated. This is in contrast to hard-switching when both turn-ON and turn-OFF of the power switches are employed at high current and high voltage levels. One approach is to create a full resonance phenomenon within the converter through series or parallel combinations
8
of resonant components. Resultant converters are known generally as resonant converters. Two major techniques are generally employed to achieve soft switching [1]: •
Zero-Current-Switching (ZCS)
•
Zero-Voltage-Switching (ZVS)
Unlike linear and switched-mode converters, the potential advantage of softswitching resonant converter are reduced power losses, therefore, achieving high switching frequency and high power density, while maintaining high efficiency. Moreover, due to the higher switching frequency, such converters exhibit faster transient responses. Today’s soft switching techniques are used in the design of both high frequency DC-DC conversion and high frequency DC-AC conversion.
2.4
Full-Bridge Parallel Resonant Inverter (PRI)
The inverter section consists of the dc source, the four transistor switches S1S4, the four antiparallel diodes D1-D4, and the resonant inductor and capacitor. The circuit of a Class D Full-Bridge Parallel Resonant Inverter is shown in Figure 2.4. There are total 4 switches; S1, S2, S3 and S4. S1 and S4 will be turned on simultaneously with the duty cycle of about 50%. After turning off S1 and S4, S2 and S3 will be turned on. This process will produce a rectangular wave with high level of V1 and low level of –V1 across the resonant circuit as shown in Figure 2.5 [3].
9
S1
S3
V1
Ri
S2
S4
Figure 2.4 Circuit of a Full-Bridge Parallel Resonant Inverter
VT V1
T/2
T
t
‐V1
Figure 2.5 VT versus time
2.4.1 Analysis of PRI
The following assumptions are made for the analysis of the PRI:
All switches are ideal. Short circuit when ON and open circuit when OFF
The loaded quality factor at resonant frequency, Qr of the resonant circuit is high enough (Qr > 2.5) so that the currents through the resonant circuit are sinusoidal.
10
It is important to mention that the loaded quality factor at resonant frequency, Qr should be large enough (Qr > 2.5) in order to produce a nearly sinusoidal current in the resonant circuit. If Qr < 2.5, the current waveform is different from a sine wave hence analysis will become inaccurate [2].
The resonant circuit in the inverter of Figure 2.4 can be described by the following normalized parameters:
The corner frequency
The characteristic impedance
The loaded quality factor at the corner frequency fo
The input impedance of the resonant circuit shown in Figure 2.4 is
Where
11
At f = fo,
The resonant frequency fr is defined as a frequency at which the phase shift
is zero.
Hence, the ratio of the resonant frequency fr to the corner frequency fo is
To achieve high efficiency and reliability, f should be higher than fr under all operating conditions [2]. The following conclusions can be drawn from (2.10): 1) For QL ≤ 1, the resonant frequency fr does not exist and the resonant circuit represents an inductive load at any operating frequency. 2) For QL > 1, fr / f0 increase with QL. For f > fr, , the transistors are loaded by an inductive load and current i lags the voltage across diode switch 2 vDS2, resulting in desirable operation. For f < fr,, the transistors are loaded by a capacitive load and the current i leads the voltage vDS2, causing spikes in the switch currents due to the reverse recovery of the antiparallel diodes at turn-off.
12
2.4.2
Voltage Transfer Function
All equations and derivations below are based on [2]. The input voltage of the resonant circuit v is a square wave of magnitude V1, given by
The fundamental component of this voltage is
where the amplitude of vi1 can be found from Fourier analysis as
Hence, one obtains the rms value of vi1
which leads to the voltage transfer function from V1 to the fundamental component at the input of the resonant circuit
The magnitude of the voltage transfer function of the resonant circuit is
Finally, the magnitude of the DC-AC voltage transfer function of the Class D inverter is
13
The range of MVI is from zero to ∞. The magnitude of the voltage transfer function, MVI is related to 3 most important parameters, that are switching frequency, f, loaded quality factor, QL and corner frequency, fo. In other words, varying these parameters can control the output voltage of a PRI. The corner frequency is always fixed as it is determined by the value of L and C [2].
2.5
Full-Wave Bridge Rectifier
A high-frequency rectifier is an AC-DC converter that is driven by a highfrequency AC energy source. The input source may be either a high-frequency current source or a high-frequency voltage source [2]. Rectifier that are driven by a current source are called current-driven rectifier. Specifically, the diode current waveform is a half sine wave and the diode voltage waveform is a square wave. The on-duty cycle of each diode is 50% [3]. Therefore, these rectifiers are referred to as Class D rectifier. The circuit of a bridge rectifier is shown in Figure 2.6.
Vs
D1
D3
D2
D4
‐
+ ‐
‐
Figure 2.6 Circuit of a Full-Wave Bridge Rectifier
14
2.5.1
Voltage Transfer Function
The current and voltage waveforms of a bridge rectifier are shown in Figure 2.7. The bridge rectifier exhibits the same behavior as the half-wave rectifier except that the bridge rectifier performs full-wave rectification rather than half-wave rectification. All equations and derivations below are based on [2]. During positive half cycle of VS (VS > 0), D1 and D4 are turned on while during negative half cycle (VS < 0), D2 and D3 are turned on. This will produce the voltage at the input of the output filter as
The output voltage is
The source current is a square wave and is determined as
The fundamental component of IS is obtained by performing the Fourier series analysis
This will produce the fundamental component of the input current IS as
Finally, the input resistance and voltage transfer function of the rectifier can be determined
15
VS VSm 0
IS, ISI I0
ωt
π
2π
ISI IS ωt
‐I0 VD2, VD4
ωt
‐VSm ID2, ID4 I0 ωt
Vfilter V0 ωt
Figure 2.7 Current and voltage waveforms for Bridge Rectifier
16
The AC-DC voltage transfer function of the rectifier,
2.5.2
Fourier Series Analysis
This Fourier series analysis is based on the work presented in [3] which include all equations and derivations as elaborated in this section. The rectifier section is represented by a voltage-dependent current sink in Figure 2.8. This rectifier representation is developed beginning with an examination of Figures 2.9 and 2.10. Figure 2.9 shows arbitrary rectifier input voltage which is an odd function and may be expressed by a Fourier series containing only odd harmonics. Note that this is phase shifted from origin by α radians. The Fourier series for the input voltage may be expressed as
RL
Lr A
+ VT
Cr
VRI ‐ B
Figure 2.8 Resonant converter model
iR
17
In (2.25)
is the nth complex coefficient in the Fourier series of the input
voltage. Assuming ideal rectifier operation, Figure 2.10 shows the output voltage of the rectifier. The Fourier series for this output voltage contains a DC component and even harmonics because it is an even function. The Fourier series for the output voltage is
is the mth complex coefficient in the Fourier series of the output
In this equation voltage.
vRI
π‐α
2π‐α
ωt
‐α
Figure 2.9 Rectifier input voltage
vR0
‐α
π‐α
ωt 2π‐α
Figure 2.10 Rectifier output voltage
18
The Fourier coefficients of the output voltage can be related to the coefficients of the input voltage. The
may be calculated using the following
equation:
Examination of Figures 2.9 and 2.10 shows that over the interval of interest, Equation (2.25) is substituted into equation (2.27) yielding
The summation and integration operations may be interchanged because the Fourier series is uniformly convergent on
. Equation (2.28) now becomes
Evaluating the integral and noting that m is even and n is odd yields the following expression relating the output voltage coefficients to the input voltage coefficients
Given an input voltage which can be expressed as a Fourier series with odd harmonics only, the Fourier series for the output voltage which contains even harmonics can be calculated using (2.30). In addition to (2.26), the output voltage may be expressed in the following form
19
The terms in (2.31) are defined as
The Fourier series for the output current of the rectifier may now be calculated using the circuit in Figure 2.11. In this circuit, the output filter components and the load resistance have been combined into single impedance and the voltage source
represents the output voltage of the
rectifier. Each term in the series for the current is calculated using the following expression
Figure 2.11 Frequency domain model for rectifier output
The expression for the output current is determined by summing all current components calculated using (2.32) and is
where (2.32).
is the DC component of the output current and is also calculated using
20
The last step in the development of the model for rectifier section is the calculation of the rectifier input current. Figure 2.12 and 2.13 are plots of the rectifier output current and input current, respectively. Because of ideal rectifier operation the input current is equal to the output current for one-half of the period and is equal to the negative of the output current for the other half of the period. Using this property of ideal rectifiers and (2.33), a time series is constructed for the input current, and a discrete Fourier transform (DFT) algorithm is then utilized to calculate the Fourier series for the input current. This current is then used in the circuit of Figure 2.8 for solution of the resonant converter.
I0
‐α
π‐α
ωt 2π‐α
Figure 2.12 Rectifier output current
IR
π‐α
2π‐α
ωt
‐α
Figure 2.13 Rectifier input current
All of the analysis in this section requires determination of the angle α. This angle corresponds to the zero crossing of the input voltage given in (2.25), and it is
21
solved for using the Newton-Raphson numerical technique. The iterative equation utilized to determine α is
is the new estimate of the angle α calculated from the jth
In this equation, estimate
. The quantities
manner that until
2.6
and
and
are determined from the
are calculated from
in (2.25) in the same
. Equation (2.34) is solved repeatedly
is less than a predefined convergence tolerance.
The Parallel Resonant Converter
Both the inverter and rectifier sections of the parallel resonant converter have been discussed at this point. The solution of the harmonic model for the parallel resonant converter in the frequency domain is presented in Figure 2.14 (b). Since the voltages and currents in the parallel resonant converter are represented by Fourier series, the circuit in Figure 2.14 (b) is utilized to perform the calculations at each frequency. The parallel resonant converter is shown in Figure 2.14 (a) with the capacitor Cr placed in parallel with the rectifier bridge rather than in series. An output filter inductor Lf produces essentially a constant current from the bridge output to the load. The switching action causes the voltage across the capacitor and bridge input to oscillate [2].
22
+ V b ‐
+
+
vx
Vo
‐
‐
(a)
+
+
Va ‐
Re
Vb ‐
(b)
Figure 2.14 (a) Circuit of a Parallel-Loaded Resonant Converter, (b) Equivalent AC circuit for PRC.
2.6.1
Voltage Transfer Function
The PRC can be analyzed by assuming that the voltage across the capacitor Cr is sinusoidal, taking only the fundamental frequencies of the square-wave voltage input and square-wave current into the bridge. The equivalent resistance for this circuit is the ratio of capacitor voltage to the fundamental frequency of the squarewave current. Assuming that the capacitor voltage is sinusoidal, the average of the
23
rectified sine wave at the bridge output (vx) is the same as Vo. All equations and derivations below based on [1].
The equivalent resistance is then
Solving for output voltage in the phasor circuit of Figure 2.14 (b),
Since Vo is the average of full-wave rectified value of Vb,
Va is the amplitude of the fundamental frequency of the input square wave
Combining (2.38) and (2.39) with (2.37), the relationship between output and input of converter is
The DC-DC voltage transfer function is
24
2.6.2
Resonant Converter Analysis
The voltage across the capacitor, which is also the input voltage to the rectifier, is designed as
The resistance RL in Figure 2.14 (b) is the series
resistance of the inductor. Because of the nonlinear nature of the rectifier, an iterative solution procedure is utilized to solve for the unknown Fourier coefficients. The iterative procedure begins with an initial guess at the input voltage to the rectifier [3]. An initial guess may be calculated using the following expression
which assumes that the current drawn by the rectifier is zero. Since the transformation between the input voltage coefficients to the output voltage coefficients was derived assuming that the input voltage was phase shifted α radians from the time origin, the zero crossing of the input voltage is next solved for using (2.34). Equation (2.30) is then utilized to calculate the output voltage Fourier coefficients. The output current is calculated next using (2.32) and (2.33). The Fourier series for the input current is determined by constructing its time series from the output current and by using a DFT algorithm.
CHAPTER 3
HARMONIC ANALYSIS
3.1
Introduction
There are two methods used to accomplish this final year project as shown in Figure 3.1. It is divided into two major parts, which includes frequency domain technique and simulation using computer program which is MATLAB/SIMULINK.
Frequency Domain Technique Harmonic analysis of a parallel‐loaded resonant converter Simulation using MATLAB/SIMULINK computer software
Figure 3.1 Final year project outline
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3.2
Frequency Domain Technique
The average, rms, and peak values of the voltages and currents in the resonant converter may be determined from steady-state solution for the converter. The steady-state solution can be obtained using both time and frequency domain technique [3]. For this final year project, a frequency domain technique is selected since time domain techniques are based on the differential equations and may be solved to obtain closed-form steady-state equations for various voltages and currents in the converter. A method is presented for calculating the harmonic components of the currents and voltages in a parallel-loaded resonant converter using frequency domain techniques. These techniques utilize Fourier series descriptions for all voltages and currents in the converter. The steady-state solution is obtained by solving for the unknown coefficient in the Fourier series instead of solving the differential equations. The unknown coefficients in all Fourier series are calculated using the harmonic model and Kirchhoff’s laws [4]. Because of the nonlinear nature of the rectifier section, an iterative technique must be utilized to find the unknown Fourier coefficients.
For the frequency domain solution presented here, the resonant converter is divided into two parts which are an inverter section and a rectifier section. A harmonic model is developed for the resonant converter in which the rectifier section is treated as a voltage-dependent current sink [3]. In both sections, all transistors and diodes are assumed to be ideal. Harmonic models are developed for both sections and are combined to form the harmonic model for the resonant converter in Figure 3.2.
+
‐
Figure 3.2 Frequency domain harmonic model for resonant converter
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3.2.1
Harmonic Analysis of PRC
The frequency domain technique for the harmonic analysis of parallel-loaded resonant converter is utilized to solve an example converter based on [3]. The circuit parameters for this converter are Lr = 5μH, Cr = 5.63μF, RL = 0.1Ω, Lf = 50μH, Cf = 5μF, and R = 5Ω. The input voltage for this converter is 10V and the switching frequency is 20 kHz. Using a convergence tolerance of
= 0.01 and an accelerating
factor of k = 0.1, the frequency domain technique reached a solution in 5 iterations. The Fourier series for the converter voltages and currents is also calculated using MATLAB/SIMULINK rather than using PSPICE that has already done in [3] and used as reference for this project. The coefficients of the Fourier series are calculated from a transient analysis of the converter [3].
The iterative procedure is now summarized:
1) Calculate initial guess for capacitor voltage using (2.42). 2) Solve for α using (2.34). 3) Calculate Fourier coefficients for output voltage using (2.30). 4) Calculate the output current using (2.32) and (2.33). 5) Calculate input current coefficient using DFT.
VT is the amplitude of the fundamental frequency of the input square wave using (2.39)
where
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The iterative procedure begins with an initial guess at the input voltage to the rectifier. An initial guess may be calculated using (2.42) When n = 1,
= 22.56 – j 2.87 V =
When n = 3,
= -1.406 – j 0.099 =
When n = 5,
= -0.25 – j 0.0088 =
The second step is to calculate α using equation (2.34). Table 3.1 shows the magnitude and phase Vi for odd order that has been calculated from the rectifier input voltage coefficient (2.25) in the same manner that from
.. This value is used for calculating α angle.
and
are calculated
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Table 3.1 Magnitude and phase of Vi
n
Vi
1
45.48
7.25
3
2.82
4.03
5
0.5
2.02
For making the calculation of α more simple and easy, Microsoft Excel has been used for the calculation and had determined that α is equal to -5.4208.
Given an input voltage which can be expressed as a Fourier series with odd harmonics only, the Fourier series for the output voltage which contains even harmonics can be calculated using (2.30).
The Fourier series for the output current of the rectifier may now be calculated using the circuit in Figure 2.11. In this circuit, the output filter components and the load resistance have been combined into single impedance and the voltage source rectifier.
For m = 2,
represents the output voltage of the
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For m = 4,
Each term in the series for the current is calculated using the following expression. For m = 0,
For m = 2,
For m = 4,
3.3 Simulation using MATLAB/SIMULINK Software
MATLAB/SIMULINK has been chosen as the simulation tool for this final year project. Simulink, developed by The MathWorks, is a tool for modeling, simulating and analyzing multidomain dynamic systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries. It offers tight integration with the rest of the MATLAB environment and both drive MATLAB or are scripted from it. Simulink is widely used in control theory and digital signal processing for multidomain simulation and design. Figure 3.3 show the
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circuit of the parallel-loaded resonant converter that combines the inverter section and rectifier section, with the resonant circuit in between that will be designed and simulated using MATLAB/SIMULINK program. Finally, the frequency domain technique for the harmonic analysis of the parallel-loaded resonant converter is utilized to solve an example converter.
Figure 3.3 Circuit of a Parallel-loaded resonant converter
The solutions obtained from the frequency domain techniques are then compared with that determined using MATLAB/SIMULINK.
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1
Introduction
The topology of a PRC described in Chapter 2 has been simulated using MATLAB/SIMULINK. Analysis has been carried out as well for the PRC to investigate the output of the PRC under load variation and switching frequency variation. In the following sections, the simulation results will be shown. The simulated output waveform of a PRC will be illustrated. Harmonic analysis of the PRC using FDT and MATLAB/SIMULINK also showed and compared with results from PSPICE software [3].
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4.2
MATLAB/SIMULINK Simulation
The analysis of the PRC will be carried out using MATLAB/SIMULINK simulation results to investigate the performance of the PRC under several circumstances.
4.2.1
Switching Frequency, fS Variation
Figure 4.1 shows the simulated output waveforms when QL = 4.36 for switching frequency variation analysis in which the switching frequency is higher than the resonance frequency, fr = 30kHz (f > fr).
Meanwhile, Figure 4.2 illustrates the simulation output waveforms for the operation below resonance frequency (f < fr). Figure 4.3 and Figure 4.4 show the output waveforms of the PRC when QL = 6.55 for operation above resonance and operation below resonance.
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Current, I0 = 1.1 A
(a)
Current, I0 = 1.06 A
(b)
35
(c)
Figure 4.1 Output waveform for QL = 4.36 and fS above resonance. (a) fS = 35 kHz (b) fS = 40 kHz (c) fS = 45 kHz.
(a)
36
(b)
Figure 4.2 Output waveform for QL = 4.36 and fS below resonance. (a) fS = 15 kHz (b) fS = 25 kHz.
(a)
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(b)
Current, I0 = 4 A
(c)
Figure 4.3 Output waveform for QL = 6.55 and fS above resonance. (a) fS = 35 kHz (b) fS = 40 kHz (c) fS = 45 kHz.
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Voltage, V0 = 26 V
Current, I0 = 5.2 A
(a)
Voltage, V0 = 20.5 V Voltage, V0 = 19 V
Current, I0 = 4.1 A Current, I0 = 3.8 A
(b)
Figure 4.4 Output waveform for QL = 6.55 and fS below resonance. (a) fS = 15 kHz (b) fS = 25 kHz.
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All the output voltage values from the figures above have been compressed in Table 4.1 and Table 4.2. It is noticed here that when QL is increased, the output voltage will increase as well. When the switching frequency is above resonant, it shows that the output voltage is almost the same but it is not so for switching frequency below resonant.
Table 4.1 Comparison of output voltage between QL = 4.36 and QL = 6.55 for switching frequency above resonant.
fS (Hz)
Output Voltage, Vo (Volt) QL = 4.36
QL = 6.55
35
5.3
21
40
5.5
20.2
45
5.7
20
Table 4.2 Comparison of output voltage between QL = 4.36 and QL = 6.55 for switching frequency below resonant.
fS (Hz)
Output Voltage, Vo (Volt) QL = 4.36
QL = 6.55
15
6
26
25
15.5
19
4.2.2 Load, RL Variation
Loaded quality factor will be initially set to QL > 2.5 and switching frequency is fixed at 35 kHz (above resonance) with corner frequency, fo = 30 kHz. Simulation
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will be run for various value of load in which RL is varied from 2.5 Ω to 15 Ω with the increment of 2.5 Ω. Figure 4.5 shows the simulated output waveform.
(a)
(b)
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(c)
(d)
42
(e)
(f) Figure 4.5 Output waveform for QL > 2.5 and fS = 35 kHz. (a) RL = 2.5 Ω (b) RL = 5 Ω (c) RL = 7.5 Ω (d) RL = 10 Ω (e) RL = 12.5 Ω (f) RL = 15
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From Table 4.3, it is noticed here that when the load is increased, the output voltage will increase as well. If the output voltage of the PRC is to be regulated, the controller should be able to shift the load resistance to appropriate value when switching frequency interruption occurs.
Table 4.3 Output voltage for variation of loads
RL (Ω)
2.5
5
7.5
10
12.5
15
Output Voltage (Volt)
10.5
20.5
30.5
40
50
60
4.2.3
Discussion
From the simulation results, the following summary can be drawn: •
The parallel-loaded resonant converter can be used as both a step up and step down converter.
•
Varying the switching frequency, fS and load resistance, RL can control the output voltage of a PRC.
•
When RL increase, QL will increase as well. If the output voltage is to be regulated, the controller should adjust the switching frequency to the desired point.
•
The PRC contains an inductive output filter and, thereby, the current through the filter capacitor is low, reducing conduction loss in the ESR and ripple voltage.
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4.3
Harmonic Analysis
The frequency domain technique for the harmonic analysis of parallel-loaded resonant converter is now utilized to solve an example converter. The coefficients of the Fourier series are calculated from a transient analysis of the resonant converter. Table 4.4 to Table 4.7 give the Fourier coefficients obtained by solving the example resonant converter using both the frequency domain technique (FDT) and MATLAB/SIMULINK, and also results from PSPICE software [3]. These tables contain the number of the harmonic and the magnitude and phase for the corresponding coefficient. Since the rectifier input voltage and current are odd function, they are represented by Fourier series which contain only sine terms.
Table 4.4 Rectifier Input Voltage Coefficients
FDT
MATLAB/SIMULINK
PSPICE
n
Mag
Phase(°)
Mag
Phase(°)
Mag
Phase(°)
1
22.738
-7.258
32.22
95.32
20.62
-15.75
3
1.411
-175.954
0.5714
82.36
1.187
154.2
5
0.252
-177.997
0.498
179.5
0.079
136.3
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Table 4.5 Rectifier Input Current Coefficients
FDT
MATLAB
PSPICE
n
Mag
Phase(°)
Mag
Phase(°)
Mag
Phase(°)
1
3.385
-28.94
5.087
89.75
3.375
-28.6
3
1.094
-30.8
1.743
-59.8
1.116
-29.8
5
0.644
-74.4
1.063
131.1
0.656
-72.3
Table 4.6 Rectifier Output Voltage Coefficients
FDT
MATLAB
PSPICE
n
Mag
0
12.496
2
9.58
149.9
13.95
99.97
9.6
149.8
4
1.22
103
2.239
-70.4
1.21
102.6
Phase(°)
Mag
Phase(°)
20.45
Mag
Phase(°)
12.895
Table 4.7 Rectifier Output Current Coefficients
FDT
MATLAB
PSPICE
n
Mag
0
2.575
2
0.8127
60.5
0.7106
10.6
0.8138
60.4
4
0.0494
13.1
0.0965
-159.2
0.0489
12.3
Phase(°)
Mag
Phase(°)
3.976
Mag
Phase(°)
2.577
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Table 4.4 to Table 4.7 compares the results obtained based on FDT and MATLAB/SIMULINK. The results obtained using MATLAB/SIMULINK are not in good agreement to that obtained based on FDT. This may be due to inaccurate settings that have been used while simulating using the MATLAB/SIMULINK software. PSPICE results however, are satisfactory as presented in [3].
CHAPTER 5
CONCLUSION AND RECOMMENDATION
5.1
Conclusion
The objectives of this project as outlined in Chapter 1 have not been achieved. The reason for this has been highlighted in the previous chapter.
However, while simulating the PRC using MATLAB/SIMULINK, analysis has been carried out to investigate the performance of output voltage of the PRC under two circumstances:
(a) Switching frequency, fS variation (b) Load resistance, RL variation
It is found that when QL is increased, the output voltage will increase as well. When the switching frequency is above resonant, it shows that the output voltage is almost the same but it is not so for switching frequency below resonant. When the load is increased, the output voltage will increase as well.
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5.2
Recommendation for Future Work
This project shows that the results obtained are not satisfactory, there are a number of area in which the overall performance can be further improved. These are the suggestions:
(a) The processes while simulating the PRC using MATLAB/SIMULINK should be improved, in terms of more proper work and design. (b) The power stage of the PRC was implemented only in simulation. Therefore the results of the harmonic analysis are just estimation. Extended research should include hardware implementation of the PRC circuit for better harmonic analysis.
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REFERENCES
[1]
Daniel W. Hart (1997). “Introduction to Power Electronics”, Indiana: Prentice Hall International Inc.
[2]
Marian K. Kazimierczuk & Dariusz Czarkowski (1995). “Resonant Power Converter”, New York: John Wiley & Sons Inc.
[3]
R. M Nelms (1991). “Harmonic Analysis of a Parallel-Loaded Resonant Converter”, Auburn University; IEEE Transactions On Aerospace & Electronic Systems, Vol. 27, No. 4 July 1991
[4]
Mohd. Nor Mohamad (1990). “Siri Fourier”, Malaysia: Dewan Bahasa dan Pustaka.
[5]
Simon Haykin & Barry Van Veen (2003). “Signals and Systems”, United States of America: John Wiley & Sons Inc.
[6]
Norjulia Mohamad Nordin (2006). “Analysis of a Series Parallel Resonant Converter”. Universiti Teknologi Malaysia: Thesis Degree.
