1
U
NIVERSITÀ DI
P
ISA
S
CUOLA DI
I
NGEGNERIA
C
ORSO DI
L
AUREA IN
I
NGEGNERIA
E
LETTRONICA
Laurea Magistrale
Design and Optimization of Planar
Magnetics Integrated on a
Multilayer PCB
Candidato: Vittorio Pascucci
Relatore:
Prof. Luca Fanucci
……….
3
Abstract:
Planar magnetics represented an interesting alternative to wire-wound components for an increasingly wide range of applications over the last two decades, especially for power electronics. The reason of this success are the remarkable advantages in terms of cost, low profile, efficiency and repeatability.
This technology is becoming more and more interesting for space applications too, where its peculiar aspects make it an ideal choice, in a field where reliability, efficiency and size are the key aspects for any design.
Planar technology however does not have advantages only over the classic approach. The flat tracks forming the windings are naturally prone to capacitive coupling. Parasitic capacitance can be a critical aspect in some cases, especially for complex components like multi-coil transformers.
The technological challenge of the times to come is making planar magnetics usable in as many contexts as possible, solving their criticalities while maintaining their positive aspects.
4 Table of Contents: ABSTRACT 3 TABLE OF ABBREVIATIONS 8 TABLE OF KEY-WORDS 8 TABLE OF CONSTANTS 9 TABLE OF FIGURES 9 LIST OF TABLES 12 INTRODUCTION 14
1) THE SINGLE ENDED FORWARD CONVERTER 15
2) REAL COMPONENTS 17 2.1 The transformer 17 2.1.1 Magnetization inductance 17 2.1.2 Parasitic Capacitance 17 2.1.3 Stray inductance 18 2.1.4 Series Resistance 18 2.1.5 Core Losses 18
2.2 The Output Inductor 18
2.3 The Rectifiers 19
2.4 The Output Capacitor 19
3) THE REAL CONVERTER CIRCUIT 21
3.1 Demagnetization Coil 22
3.2 Resonant Demagnetization 23
3.3 Synchronous Rectifiers 24
4) THE CONTROL LOOP 25
4.1 Voltage-mode Control 25
4.2 Current-mode Control 26
4.3 Averaged Value Transfer Functions 27
5) THE COUPLED OUTPUT CHOKE 28
5.1 Main Disadvantages of Independent Inductors 28
5.2 Main Advantages of a Coupled Choke 29
5.3 Main Disadvantages of a Coupled Choke 31
6) THE GYRATOR-CAPACITOR APPROACH 34
6.1 Reluctance-Resistance and Reluctance-Inductance Models 34
6.2 Reluctance-Capacitance Model 35
6.3 The Gyrator 36
7) PARASITIC MODELING WITH GYRATOR-CAPACITOR APPROACH 39
7.1 Turn and Coil Modeling 39
7.2 Series Resistance Modeling 41
7.3 Capacitance Between Turns of the Same Winding 42
7.4 Capacitance Between Turns of Different Windings 42
7.5 Stray Inductance 43
7.5.1 Stray Inductance in Classic Two-Coil Transformers 44
7.5.2 Stray Inductance in Multi-Coil Transformers 45
8) PARASITIC EFFECTS 50
8.1 Transformer’s Series and “Hysteretic” Resistance 50
8.2 Transformer’s Inter-Winding Capacitance 51
5
8.4 Transformer’s Stray Inductance 52
8.4.1 Single-Output Transformers 52
8.4.2 Multiple-Output Transformers 54
8.5 Transformer’s Series Resistance (2) 54
8.6 Output Choke’s Stray Inductance 56
8.7 Output Choke’s Series Resistance 57
8.8 Output Choke’s Parasitic Capacitance 57
9) SELECTION AND DESIGN OF MAGNETIC COMPONENTS 60
9.1 Transformer Core 60
9.2 Inductor Core 61
10) PLANAR MAGNETICS 63
10.1 Types of Planar Magnetics 63
10.2 Peculiar Aspects of Planar Technology 64
11) PARASITIC COMPONENTS ESTIMATION 66
11.1 DC Series Resistance 66
11.2 Self Capacitance 66
11.3 Inter-Winding Capacitance 68
11.4 Stray Inductance 71
11.4.1 Approximated Estimation 71
11.4.2 Exact Estimation and Modeling 73
12) MINIMIZATION OF PARASITIC COMPONENTS IN PLANAR MAGNETICS 77
12.1 Dc Series Resistance 77
12.2 Winding Self Capacitance 77
12.2.1 Reduction of the “Turn-to-Turn” component of Self Capacitance 78 12.2.2 Reduction of the “Core-Coupling” component of Self Capacitance 79
12.2.3 Interleaved Scenario 80
12.3 Inter-Winding Capacitance 80
12.3.1 Reduction of the “Direct” Capacitance 81
12.3.2 Reducing the Effective Coupling by Voltage-Swing Matching 81 12.3.3 Managing the “Core-Coupling” component Inter-Winding Capacitance 82
12.3.4 Managing Divergent Voltage-Swing Interfaces 82
12.3.5 Using Shields 84
12.3.6 Turn Shifting 85
12.4 Stray Inductance 85
12.5 Reciprocal Relations Between Parasitic Components 86
12.5.1 Self Capacitance 86
12.5.2 Inter-Winding Capacitance 86
12.5.3 Stray Inductance 87
13) THE CSO CONVERTER 88
13.1 Circuit Description 88
13.2 Circuit Operation 90
13.3 Circuit Analysis 91
13.4 Magnetics’ Operation 92
14) CSO CONVERTER PROBLEMS AND IMPROVEMENT FRONTS 93
15) ORIGINAL COUPLED CHOKE ANALYSIS 95
15.1 Stackup and Winding Arrangement 95
6
15.2.1 Resonance Frequency (Capacitance) 97
15.2.2 Stray Inductance 97
15.2.3 DC Series Resistance 97
15.3 Capacitive Model and Estimation Validation 98
15.4 Inductive Model and Estimation Validation 99
15.5 DC Resistance Estimation Validation 101
15.6 Comments on the Original Choke 102
15.7 Defects of the Original Choke 102
15.8 The Substitutive “Wire-Wound” Choke 103
16) COUPLED CHOKE OPTIMIZATION 104
16.1 Qualitative Winding Stack 104
16.2 PCB Selection 105
16.3 Coil DC Resistance Determination 106
16.4 The “Ideal” Capacitive Equivalent Structure 107
16.5 Optimization Process 110
16.6 Final Stackup 111
16.7 Stray Inductance Control 113
17) THE LAYOUT 116
17.1 Inter-Board Connections and Clearances 116
17.2 The Bottom PCB 118
17.3 The Motherboard 120
17.4 The Top PCB 123
18) THE FINISHED PROTOTYPE 126
18.1 The Prototype 126
18.2 Parameter Measurement 127
18.2.1 Resonance Frequency (Capacitance) 127
18.2.2 Stray Inductance 129 18.2.3 DC Series Resistance 130 18.3 Final Considerations 130 19) FUTURE CHALLENGES 131 19.1 Capacitive Compensation 131 CONCLUSIONS 134 APPENDICES: A: RELATIONSHIP BETWEEN PARASITIC CAPACITANCE AND RESONANCE FREQUENCY 135
A.1 Self Capacitance 135
A.2 Inter-Winding Capacitance 136
A.2.1 Transformer Example 136
A.2.2 Coupled Choke Example 138
A.3 Inter-Winding Capacitance and Miller Effect 141
A.4 Core-Coupling 141
B: EFFECT OF OUTPUT CHOKE’S PARASITIC CAPACITANCE ON TIME DOMAIN BEHAVIOR 143
C: EMI FILTERS 146
C.1 Common-mode Filtering 146
7
D: WINDING ARRANGEMENT EXAMPLES 147
D.1 Divergent Voltage Swing Interfaces 147
D.1.1 V-Shaped Divergent Voltage Swings 147
D.1.2 Parallel Divergent Voltage Swings 147
D.2 Core-Coupling Effects Depending on the Facing Coil 148
D.3 In-Line vs. Stacked Arrangement 148
D.3.1 In-Line Solution 149
D.3.2 Stacked Solution 149
D.4 180° Coil Rotation Effects on the Resonance Frequency 151
D.4.1 Convergent Voltage Swings 151
D.4.2 Parallel Divergent Voltage Swings 152
D.4.3 V-Shaped Divergent Voltage Swings 153
E: DIFFERENCES IN THE MODEL WITH FLOATING OR GROUNDED CORE 154
AKNOLEDGEMENTS 156
8
Table of Abbreviations
Finite Elements Modeling
Motherboard
Printed Circuit Board
Magneto-Motive Force Voltage/Current-Mode Control Thickness Width Height Distance Average Length Table of Key-Words:
Planar Magnetic A magnetic component (transformer or inductor)
made with planar technology. The component’s coils are made of flat printed copper tracks in a PCB, routed around a ferrite core
Wire-Wound Magnetic A classic magnetic component where the coils are made of round copper wire, mechanically wound around a ferrite core
Self-Capacitance That amount of parasitic capacitance seen by a coil
that is exclusively due to itself. Its origin is the capacitive coupling between different parts of the same coil
Inter-Winding Capacitance That amount of parasitic capacitance seen by a certain coil which is exclusively due to the capacitive coupling with other coils
In-Line Arrangement Also called “Spiral” arrangement. A planar turn arrangement where all the turns of a coil lay on the same layer, forming a spiral around the magnetic core
Stacked Arrangement Also called “Helical” arrangement. A planar turn arrangement where each turn of a coil lays on a different layer
Stackup The disposal of copper and isolation layers in a
circuit board
Overlapping Ratio When a track on a certain layer faces two or more tracks on an adjacent layer, the overlapping ratio approximately defines the percentage of area of the largest turn overlapping each other
Power/Output Line In a single or multiple output switching converter, it is the circuit that goes from a transformer’s secondary coil to rhe trlative output port
9
Table of Constants:
Name Symbol Value
Vacuum Dielectric Constant
FR4 Relative Dielectric Constant
Vacuum Magnetic Permeability
Copper-Air-FR4 Relative Magnetic permeability
Table of Figures:
Figure 1 - The classic two-coil transformer (image taken from travel.wikinut.com) ... 15
Figure 2 - The ideal single-ended forward converter ... 15
Figure 3 - Forward converter typical waveforms ... 16
Figure 4 - DCM output voltage behavior ... 16
Figure 5 - Magnetization inductance ... 17
Figure 6 - Two-Coil transformer equivalent circuit ... 18
Figure 7 - Real Inductor ... 18
Figure 8 - Synchronous and normal rectifiers ... 19
Figure 9 - Real Output Capacitor ... 20
Figure 10 - Output Voltage Ripple ... 20
Figure 11 - The real forward converter ... 21
Figure 12 – Transformer’s Magnetization Current ... 21
Figure 13 - Demagnetization Coil ... 22
Figure 14 - Demagnetization Coil Waveforms ... 22
Figure 15 - Resonant Reset Waveforms ( normalized to primary) ... 24
Figure 16 – Voltage-Mode control ... 25
Figure 17 - Output LC Filter ... 25
Figure 18 - Peak Current-Mode Control ... 26
Figure 19 - Average Current-Mode Control ... 27
Figure 20 - Multiple Output Converter ... 28
Figure 21 - Non-Sensed Output Load Step Reaction ... 28
Figure 22 - Three Single Chokes vs. Coupled choke Resonance ... 29
Figure 23 – Coupled Output Choke ... 29
Figure 24 - Stray Inductance Effects on a Coupled Choke ... 30
Figure 25 - Double Primary Transformer and Waveforms ... 31
Figure 26 - Double Primary Transformer Equivalent Circuit ... 32
Figure 27 - Series Parasitics’ Filtering Action in a Double Primary Transformer ... 32
Figure 28 - Inductance-Reluctance Equivalent Model ... 35
Figure 29 - Capacitance-Reluctance Equivalent Model ... 36
Figure 30 - The Gyrator ... 36
Figure 31 - Coil-Gyrator Equivalence ... 37
Figure 32 - Gyrator Series Connection ... 37
Figure 33 - Gyrator-Capacitor Modeling Example ... 38
Figure 34 - Gyrator Electrical Equivalent ... 38
Figure 35 – Voltage Swing Shape Along a Turn ... 39
10
Figure 37 - Turn Modeling with Gyrators ... 40
Figure 38 - Coil Modeling with Gyrators ... 41
Figure 39 - Series Resistance Modeling ... 42
Figure 40 - Turn-to-turn Capacitance ... 42
Figure 41 - Inter-Winding Capacitance ... 43
Figure 42 - Turn-to-Turn Overlapping Ratios ... 43
Figure 43 - Stray Magnetic Flux ... 44
Figure 44 - Stray Magnetic Flux Confined ... 45
Figure 45 - Stray Inductance Equivalent Model ... 45
Figure 46 – One-Inductance-per-Coil Model ... 46
Figure 47 - Reluctance-Resistance Model for Stray Inductance ... 46
Figure 48 - Reluctance-Inductance Model for Stray Inductance ... 47
Figure 49 - Reluctance-Capacitance Model for Stray Inductance ... 47
Figure 50 - Stray Inductance Example 1 ... 48
Figure 51 - Stray Inductance Example 2 ... 49
Figure 52 - Simplified Reluctance-Inductance Model for Stray Inductance ... 49
Figure 53 - One-Inductance-per-Coil Model for Stray Inductance ... 49
Figure 54 - Core's Hysteresis Cycle ... 50
Figure 55 - Transformer Inter-Winding Capacitance Effects ... 51
Figure 56 - Transformer Inter-Winding Capacitance Effects 2 ... 52
Figure 57 - Transformer Stray Inductance Effects 1 ... 53
Figure 58 - Transformer Stray Inductance Effects 2 ... 53
Figure 59 - Transformer Stray Inductance Effects 3 ... 54
Figure 60 - Transformer Series Impedance Mismatch ... 55
Figure 61 - Mismatch Effects With no Choke's Stray Inductance ... 55
Figure 62 – Transformer Mismatch Effects Filtered by Choke's Stray Inductance ... 56
Figure 63 - Choke's Stray Inductance Affecting Cross Regulation ... 57
Figure 64 - Output Filter Frequency Response ... 57
Figure 65 - Choke's Parasitic Capacitance Effects... 58
Figure 66 - Output Capacitive Spikes Qualitative Example ... 58
Figure 67 - Core B-H Curve ... 60
Figure 68 - Gapped Core B-H Curve ... 62
Figure 69 - Core Air Gap... 62
Figure 70 - Different Types of Planar Components ... 64
Figure 71 - DC Series Resistance Estimation ... 66
Figure 72 - Self Capacitance Estimation ... 66
Figure 73 - Self Capacitance Estimation With FEM ... 67
Figure 74 - Parallel Layers Multiply the Self Capacitance ... 67
Figure 75 - Folded Coil Capacitive Coupling ... 67
Figure 76 - Core Capacitive Coupling ... 68
Figure 77 - Final Equivalent Capacitance ... 68
Figure 78 - Inter-Winding Capacitance Estimation ... 69
Figure 79 - Inter-Winding Capacitance Distribution ... 69
Figure 80 - Different Relative Coil Orientations ... 70
Figure 81 - Overall Inter-Winding Coupling ... 70
Figure 82 - Core Window Magnetic Flux Path ... 71
11
Figure 84 - Simplified Magnetic Field Plot ... 72
Figure 85 - Final Stray Inductance Modeling ... 73
Figure 86 - Stray Flux Outside the Core Window ... 74
Figure 87 - Inter-Winding Flux Path Estimation... 74
Figure 88 - External Flux Path Estimation ... 75
Figure 89 - Positioning the External Path Equivalent Capacitors in the Gyrator Model ... 75
Figure 90 – “Stacked” and Parallel "In-Line" Arrangements (turns of the same coil) ... 77
Figure 91 - "In-Line" or “Spiral” Self Capacitance ... 78
Figure 92 - Folded Spiral Winding Self-Capacitance ... 78
Figure 93 - Full “Stacked” or “Helical” Arrangement... 78
Figure 94 - Folded vs. In-Line Core Coupling ... 79
Figure 95 – Optimised Winding Stack Example ... 80
Figure 96 - Interleaving Example ... 80
Figure 97 – Inter-Winding Coupling ... 81
Figure 98 - Voltage Swing Matching ... 82
Figure 99 - Divergent Voltage Swing Scenario ... 83
Figure 100 - Divergent Voltage Swing Possibilities ... 83
Figure 101 - Shielding ... 84
Figure 102 - Coil Shifting ... 85
Figure 103 - Core Window MMF Plot ... 85
Figure 104 - Interleaved Arrangement Advantages ... 86
Figure 105 - Basic Converter Circuit Diagram ... 89
Figure 106 - Reducing Output Capacitor Series Impedance ... 94
Figure 107 - Reducing Output Capacitor Series Impedance 2 ... 94
Figure 108 – Original CSO Converter Winding Arrangement ... 96
Figure 109 - CSO Converter Coupled Choke’s Resonance Frequency ... 97
Figure 110 - Stray Inductance Measurement ... 98
Figure 111 – Original CSO Choke Estimated Resonance Frequency ... 99
Figure 112 – Original CSO Choke Stray Inductance Estimation Circuit ... 100
Figure 113 - Original Planar vs. Wire-Wound Choke ... 103
Figure 114 - Optimized Qualitative Winding Arrangement ... 105
Figure 115 - New Core ... 105
Figure 116 - PCB Stack Example... 106
Figure 117 - Equivalent Structure for Capacitive Optimization ... 107
Figure 118 - Final Equivalent Structure For Optimization ... 108
Figure 119 – Optimized Choke Capacitive Coupling Map ... 110
Figure 120 - Estimated "Ideal" Optimised Choke Resonance Frequency ... 111
Figure 121 - Air Gap Placement ... 111
Figure 122 - Ideal Output Filter First-Order Transfer Function ... 114
Figure 123 - Stray Inductance Effects Compensation for Regulation Loop ... 114
Figure 124 - Normal vs. Compensated Output Filter Transfer Function ... 115
Figure 125 - Board Interconnection Foil ... 116
Figure 126 - Motherboard Slot Hole/Mounting Hole ... 116
Figure 127 - Via-Pad Interconnection for Top & Bottom PCB ... 117
Figure 128 - Adjacent PCB Vias Shifting ... 117
Figure 129 - Bottom PCB Plan ... 118
12
Figure 131 - MotherBoard Plan ... 120
Figure 132 - Motherboard Exploded. Different shades of the same colors represent parallel layers ... 121
Figure 133 - Depth Milling Plan ... 122
Figure 134 - Depth Milling ... 122
Figure 135 - Top PCB Plan ... 123
Figure 136 - Top PCB Exploded. Different shades of the same colors represent parallel layers ... 124
Figure 137 - Complete Device's Top View ... 125
Figure 138 – Optimized Choke Pin Connection Diagram ... 126
Figure 139 - The Optimized Coupled Choke Prototype ... 126
Figure 140 - Measured Optimized Choke's Resonance frequency ... 127
Figure 141 - Separate Boards for Measurement ... 128
Figure 142 - Terminal capacitance ... 128
Figure 143 - Corrected Resonance Frequency Estimation ... 129
Figure 144 - Internal Capacitive Compensation ... 131
Figure 145 - Non Compensated Output Current ... 131
Figure 146 - Partially Compensated Output current ... 132
Figure 147 – Maximum Output Current Compensation ... 132
Figure 148 - Compensation Capacitor Connection ... 133
Figure 149 - Magnetics' resonance frequency ... 135
Figure 150 - Self Capacitance Examples ... 135
Figure 151 - Transformer Inter-Winding Capacitance ... 136
Figure 152 - Transformer Inter-Winding Coupling Currents ... 137
Figure 153 - Transformer Inter-Winding Parallel Equivalent Capacitance ... 137
Figure 154 - Divergent Voltage Swing Inter-Winding Coupling ... 138
Figure 155 - Equivalent Capacitance Transformation ... 138
Figure 156 - Coupled Choke Inter-Winding Coupling ... 138
Figure 157 - Coupled Choke Input Impedance on Different Driven Ports ... 139
Figure 158 - Overall Coupled Choke Input Impedance Calculation ... 140
Figure 159 - Miller-Transformed Inter-Winding Capacitance ... 141
Figure 160 - Core coupling Example ... 142
Figure 161 - Coupled choke Circuit Transformation 1 ... 143
Figure 162 - Choke Impedance Simplification ... 143
Figure 163 - Completely Unified Coupled Choke ... 144
Figure 164 - General Output Filter Behavior ... 144
Figure 165 - Stray Inductance Effects on Circuit Transformation ... 145
Figure 166 - Typical Filtering Stages' Position ... 146
Figure 167 - Common Mode Filtering ... 146
Figure 168 - "V-Shaped" Divergent Voltage Swings ... 147
Figure 169 - Parallel Divergent Voltage Swings ... 147
Figure 170 - Topology Convenience Diagram ... 148
Figure 171 - Influence of the Turn Number on Core Coupling Effects ... 148
Figure 172 - "In-Line" Core Coupling Example ... 149
Figure 173 - Stacked Coupling Example ... 149
Figure 174 - 180° Coil rotation... 151
Figure 175 - 180° Rotation for Parallel Divergent Swings ... 152
Figure 176 - 180° Rotation for V-Shaped Divergent Swings ... 153
13
List of Tables:
Table 1 - Classic Approach Dimension Equivalence Table ... 34
Table 2 - Gyrator Approach Dimension Equivalence Table ... 36
Table 3 - Planar Magnetics Design Advantages ... 64
Table 4 - Planar Magnetics Manufacturing Advantages ... 65
Table 5 - Planar Magnetics Limitations... 65
Table 6 - CSO Converter General Specs 1 ... 88
Table 7 - CSO Converter General Specs 2 ... 88
Table 8 - Output Choke Data ... 90
Table 9 - Choke's Core Data ... 90
Table 10 - Transformer Data ... 90
Table 11 - Output Capacitor Series Impedances ... 94
Table 12 - CSO Converter Qualified Board Stackup ... 95
Table 13 - Choke's Measured Stray Inductance ... 97
Table 14 – Choke’s Measured DC Series Resistance ... 98
Table 15 – Original CSO Choke Estimated Stray Inductance ... 101
Table 16 – Original CSO Choke Estimated DC Resistance ... 101
Table 17 - Number Of Parallel Layers Estimation ... 107
Table 18 - Equivalent Coil Thickness ... 108
Table 19 - Total Estimated Copper Thickness ... 108
Table 20 - FEM Capacitance Estimation ... 109
Table 21 - Coil Vertical Position Ranges ... 110
Table 22 - Final Optimized Coil Positions ... 110
Table 23 - Final Complete Optimized Stackup ... 112
Table 24 - Ideal vs. Post-Fitting Coil Vertical Positions ... 113
Table 25 - Estimated Choke Stray Inductance After Optimization and Fitting ... 113
Table 26 - Compensation Network Values ... 114
Table 27 - Layout Rules ... 117
Table 29 - Measured Stray Inductance ... 129
14
INTRODUCTION:
The following study starts from the necessity to optimize the “coupled” output inductor of a 30W multiple-output forward-converter for space application, based on planar technology.
The high complexity of the component, along with planar technology natural tendency to high capacitive coupling, result in a considerable parasitic capacitance affecting the output inductor. Such a high parasitic capacitance causes, for several reasons that will be discussed in detail in this Thesis work, high output voltage
spikes occurring at switching instants. Such a highly disturbed output voltage does not match the customer’s
specifications, hence the need of optimizing the inductor’s parasitic capacitance to reduce the output voltage spikes.
Main goal:
The main goal of this Thesis is reducing the parasitic capacitance of the given component as much as possible, while keeping the other non-idealities below a certain critical value. The previous component’s planar size shall be approximately maintained too, in order to substitute the component in the original converter’s layout without any sensible modification.
To achieve this goal, the following steps have been followed:
Detailed analysis of magnetics’ non idealities and their effects on the device’s performance.
Improvement of the existing “Gyrator-Capacitor” electrical-equivalent model (as an alternative to classic “reluctance-resistance” and “reluctance-inductance” models), in order to integrate the modeling of all the non-idealities in a single “unified” equivalent circuit with no mathematical approximation. The structure of such an equivalent circuit is also tightly bound to the physical geometry of the modeled component, with great benefit for understanding and optimization purposes.
Development and improvement of simple ways to accurately estimate the parasitic components of a planar magnetic component from its geometry, without the use of heavy FEM simulations.
Development of a set of qualitative design rules for capacitive coupling reduction in planar magnetics. Design, production and measurement of an optimized prototype of the inductor, in order to validate the
15
1) THE SINGLE-ENDED FORWARD CONVERTER:
The forward converter is a switched mode DC-DC converter that features a transformer to provide galvanic insulation between primary and secondary side, which is an important requirement for power applications.
Figure 1 - The classic two-coil transformer (image taken from travel.wikinut.com)
The working principle:
For the moment only components’ ideal behavior will be considered for better understanding.
N1
N2
D1
D2
L
C
Load
Vin
Figure 2 - The ideal single-ended forward converter
On the primary side a voltage square wave is generated with a certain duty cycle δ. This voltage is transformed to the secondary side, multiplied with the turn ratio. Assuming the circuit reached the steady state condition, and that load voltage and current are constant (reasonable assumption if the output capacitor is big enough and, as in most cases, the load has an inductive behavior) there will be two working phases.
On-Phase: During the first phase ( ) the voltage across the output inductor is constant and
positive. The current in the inductor is increasing linearly, flowing through the diode .
Off-Phase: During the second phase, the voltage on the secondary winding is negative, so is in blocking mode. The “inertial” current of the output inductor starts flowing through , also called “freewheeling diode”. This imposes a negative voltage across the output inductor, equal in value to the output voltage ( ), so the current is linearly decreasing until it reaches the initial value at , assuming the typical triangular shape. The cycle then starts again.
𝑉
𝑠𝑉
𝑝∗
𝑁
𝑠𝑁
𝑝𝐼
𝑠𝐼
𝑝∗
𝑁
𝑝𝑁
𝑠𝛷
𝑉
𝑝𝑑𝑡
𝑁
𝑝16 Vin ID1 ILoad ILoad IL IL t t t ID2 ton T toff
Figure 3 - Forward converter typical waveforms
When the current in the inductor is above or below the load current, the difference flows through the output capacitor, generating a small ripple voltage in the order of 1-3% of the dc output voltage. The frequency of the ripple is equal to the frequency of the input square wave. The average current in the output inductor is obviously equal to the load current, and the triangular current ripple is added to this value. The previous calculations and formulas are only correct if the device is working in “continuous current mode”, this means that the output inductor current shall never reach zero. For this reason the load current value must be at least half of the peak-to-peak amplitude of the ripple. This is the preferred working mode for most forward converters. The output choke has on one end a pulse modulated voltage, while the other end (output voltage) is nearly a DC-voltage. The output DC-voltage of the converter is the average of the PWM signal only in case of continuous conduction mode. Discontinuous current mode operation (DCM) occurs when the output inductor current reaches zero before the end of the cycle. The ripple current in this case is higher than two times the load current. In this operation mode the output voltage is not the average of the PWM signal on the left end of the choke. This mode occurs at light load only in case rectification is done with normal diodes. Synchronous rectifiers do not allow discontinuous conduction mode, as will be explained in chapter 2.
When DCM occurs the output voltage tends to raise over the PWM average (fig 4), potentially leading to destruction of the load or other components if no regulation loop overcomes this effect. This working mode is usually undesired even if the output is sensed. Fig. 4 below shows the non-sensed output voltage raise when the load decreases instantaneously and the circuit starts working in discontinuous mode.
Figure 4 - DCM output voltage behavior
∆𝐼𝑙(𝑡𝑜𝑛) ∆𝐼𝑙(𝑡𝑜𝑓𝑓) (𝑉𝑖𝑛 𝑉𝑜𝑢𝑡)𝑡𝑜𝑛 𝐿 𝑉𝑜𝑢𝑡𝑡𝑜𝑓𝑓 𝐿 (𝑉𝑖𝑛 𝑉𝑜𝑢𝑡)𝛿 𝑉𝑜𝑢𝑡( 𝛿) 𝑉𝑜𝑢𝑡 𝑉𝑖𝑛𝛿
𝐼
𝐿𝑉
𝑜𝑢𝑡17
2) REAL COMPONENTS: 2.1 The transformer:
The non-ideal parasitic elements of a transformer are: Magnetization inductance Parasitic capacitance Stray inductance Series resistance Core losses 2.1.1 Magnetization Inductance:
An ideal transformer is supposed to have an infinite winding inductance, so only the transformed secondary current is flowing through the primary winding. A real transformer’s coils have finite inductance instead. When voltage is supplied on the primary coil, current starts flowing through it, besides the secondary transformed one. This effect is represented by a so called “magnetization inductance” connected in parallel with the ideal transformer. It’s value is equal to the inductance of the winding it is connected in parallel to. The value of the magnetizing inductance ( ) depends from the “inductance factor” of the core ( ), provided by the ferrite manufacturer.
Lm
Iin Iout ImagN1
N2
IpVin
Figure 5 - Magnetization inductance
The current flowing in the magnetization inductance is called “magnetizing current” because it forces the orientation of magnetic dipoles in the core. The magnetization of the core has a limit above which it “saturates” losing its magnetic properties. To avoid this limit to be reached the core shall be carefully dimensioned so that during the entire working cycle the magnetic field doesn’t raise over the critical value (see chapter 9) . The magnetization must be taken to its initial value at the end of each cycle, during the so called “demagnetization phase” for the circuit to reach a true steady state condition.
2.1.2 Parasitic Capacitance:
Another important fact is that the distance between different turns and different windings is finite, thus parasitic capacitive coupling will occur between turns of the same winding, and this is represented connecting a parasitic capacitor in parallel to each ideal winding ( fig. 6). Similarly, capacitive coupling between different windings will occur, represented by a capacitor across the windings ( fig. 6). The so called “inter-winding
capacitance” between windings is distributed and the exact positioning of the corresponding capacitor depends on the real winding arrangement.
𝐿𝑀 𝐴𝐿∗ 𝑁12
𝐼𝑀𝐴𝐺
𝑉𝑖𝑛𝑑𝑡
18
2.1.3 Stray Inductance:
The inescapable distance between turns and windings causes the magnetic flux generated by each winding not to be completely coupled with all the others. This phenomena can often be represented with a series “uncoupled” inductor on one or more ports ( fig. 6).
2.1.4 Series Resistance:
In addition to these parasitic components it’s important to consider the series resistance of the winding, due to the finite resistivity of copper and increased by skin effect and proximity effect at high frequency operation. This is a dissipative parasitic component and it’s represented by connecting a resistor in series with the winding (
fig. 6).
2.1.5 Core Losses:
Another source of dissipation are the losses in the core which are due to hysteresis losses and eddy currents induced in the core. These losses, at a given input voltage and duty cycle, have a fixed value each cycle and, for power and efficiency consideration, can be taken into account by connecting a resistor in parallel with the ideal winding ( fig. 6).
The parasitic components on the secondary side can then be transformed and added to the primary to achieve the following circuit (The equivalent circuit is often simplified and can be changed to highlight different phenomena in different situations).
Lm
Cp
Rs1
Ls1
Cx1
Cx2
Rp
Rs2'
Ls2'
Np
Ns
Figure 6 - Two-Coil transformer equivalent circuit
Each of these parasitic components will be discussed in detail for planar magnetics in the relative chapter.
2.2 The output inductor:
Real inductors consist of a coil wound on a magnetic core. Similarly to transformer coils, these components have parasitic series resistance and parallel capacitance.
Figure 7 - Real Inductor
𝑍𝑝 𝑍𝑠∗ 𝑁𝑝 𝑁𝑠 2 L Rs Cp
19
In multiple output converters several different output inductors may be wound on the same core, to produce a so called “coupled choke”. Using a coupled choke considerably improves output cross regulation and simplifies the regulation loop (see Chapter 5). Coupled chokes behave, form the coupling point of view, like transformers and present the same parasitic components.
2.3 The Rectifiers:
In the previous discussion of the converter’s working principle we assumed to have ideal rectifiers, with no series
resistance, no parallel capacitance and no voltage drop, but this is not the real case, and real rectifiers (diodes)
have all these three parasitic elements. The voltage drop in particular causes considerable power dissipation and reduces the output voltage.
The problem can be partially avoided using synchronous rectifiers (MOS), that are turned on and off at the right time by a control network to emulate a diode behavior. Synchronous rectifiers have no voltage drop across the terminals except for the one due to its series resistance, improving efficiency and regulation on low voltage outputs. The resistive behavior is highly repeatable on each output and will improve cross regulation performance.
Another important advantage of synchronous rectifiers is that they allow the current flow in both direction as long as they are “ON”. This prevents discontinuous conduction mode operation because the inductor current can
reach negative values. Normal diodes would block the inductor current as it tends to get below zero.
The fact that synchronous rectifiers need a gate-drive circuit makes it desirable to connect their sources to a common ground with the driving circuit. This introduces some topology limitations that will be discussed in detail later on, referring to the particular circuit analyzed. The drain-bulk parasitic diode ensures the rectifier works properly even if for some reason the gate is not driven.
Figure 8 - Synchronous and normal rectifiers
2.4 The output capacitor:
As any real capacitor, the output one will unescapably have a series resistance ( ) and a series inductance ( ) that can only be decreased by connecting many smaller capacitors in parallel (see chapter 14). These components are responsible for the output voltage ripple even if the capacitance is ideally infinite, in fact, still
assuming that the load only draws its average DC current, the inductor current ripple must flow through the
capacitor, causing a voltage drop on the series resistor and inductor.
20 Cout ESR ESL L Load
Figure 9 - Real Output Capacitor
The effect of series components is also mainly related to the interaction with the output choke’s parasitic capacitance and will be explained in detail (see Chapter 5 and 8.7).
t t t IL ICout Vesl Vesr ΔVout Iload
Figure 10 - Output Voltage Ripple
The fact that the output capacitance is not infinite determines an additional component of the ripple consisting in the capacitor voltage variations due to the periodic charge drain and injection [15].
∆ ∫ ∆
This is one of the main parameters used to determine the size of the output capacitor. 𝑉𝐶𝑜𝑢𝑡 𝐶𝑜𝑛𝑠𝑡 𝐼𝐶𝑜𝑢𝑡 𝐼𝐿 𝐼𝑙𝑜𝑎𝑑 𝑉𝐸𝑆𝑅 𝐸𝑆𝑅 ∗ 𝐼𝐶𝑜𝑢𝑡 𝑉𝐸𝑆𝐿 𝐸𝑆𝐿 𝑑𝐼𝐶𝑜𝑢𝑡 𝑑𝑡
21
3) THE REAL CONVERTER CIRCUIT:
Vin
Lm
Cp
Cd
M1
L
Cout
Load
D1
D2
Figure 11 - The real forward converter
As represented in fig. 11, in the real implementation of the device, the DC input voltage is transformed in a square wave for the primary winding by using a power MOSFET in series with it. Since the average value of such a square wave is ( ) instead of zero, the magnetization of the core ( ) tends to increase. A demagnetization
technique is required to take the magnetic field back to the initial value at the end of each cycle. The fig. 12 shows that, when a transformer is driven by a non-zero-average signal, it can’t reach the steady state condition because the magnetization current is not periodic. The current, instead of returning to zero within the duration of the single cycle, raises, inevitably leading to the saturation of the core (see chapter 9) and possibly to unsafe operation, because high current may be destructive if not limited.
VLm VLm Imag Imag t t
22
3.1 Demagnetization Coil:
A simple and straight forward way of demagnetizing the core is using an auxiliary winding [1]. Reset Vin L Cout Load D1 D2 D3 M1
Figure 13 - Demagnetization Coil
Referring to the waveforms in fig. 14, during the “active phase” ( ) the demagnetization winding is acting
as a secondary, transforming the primary voltage but without any current flowing through it because the diode is blocking. Except for parasitic components, it doesn’t influence the behavior of the transformer. The primary magnetization current increases linearly, summing to the secondary current which is linearly increasing too, transformed to the primary with the turn ratio.
The demagnetization phase starts with the OFF-Phase ( ), when opens, interdicting current flow. This
causes an “instantaneous” increase of its drain voltage because the inertial primary demagnetization current charges the small drain capacitance. As the drain voltage reaches the value , so there is no voltage drop on the primary, the secondary voltage drops to zero too, interdicting diode . The output inductor current starts to flow in the freewheeling diode, and primary (secondary transformed component) and secondary currents are forced to zero. The demagnetization current keeps flowing in the primary coil charging the drain capacitance of
. This imposes increasing negative voltage on all the coils, definitively interdicting .
Vgs Vprim Iprim Isec Iaux t t ΔIMag-p ΔIMag-aux
Figure 14 - Demagnetization Coil Waveforms
The demagnetization current keeps flowing in the primary winding until it charges the drain capacitance to . At this time the voltage across the windings is – (normalized to the primary turn ratio) and the diode
23
opens, letting the magnetization current “jump” to the demagnetization coil ( , fig. 14). The auxiliary
winding has voltage across its terminals as long as some current is flowing in , so demagnetization current decreases linearly until it gets to zero again. For the rest of the cycle current and voltage across each winding are zero, and the core magnetization is the same as at the beginning of the cycle. It is clear from the fig. 14 that the average value of the primary voltage is now Zero, and the circuit can reach periodic operation.
This is an interesting solution but it has got two main disadvantages: first, adding an additional winding to the transformer core adds its parasitic elements to it, and it cumbers useful space on the PCB. The second disadvantage is the fact that the duty cycle is limited to 50% in order to have a complete demagnetization of the core, because demagnetization always takes the same time as magnetization if the drop on is neglected.
∆ ∗ ∗ 2∗ ∆ ∗ ∗ ∗ 2 ∗
To provide total reset: ∆ ∆
3.2 Resonant Demagnetization:
The second demagnetization technique, the one chosen in the present project, is called “resonant
demagnetization” [12]. Instead of an additional winding, a resonating voltage peak on the drain of MOS is used to invert the primary voltage and get the current back to its initial value. The magnetization phase is exactly the same as in the previous case. When opens the primary circuit, the magnetization current, flowing through the primary coil, charges the total capacitance seen from the drain of to “variation-ground” ( ) (fig. 11), until that node reaches voltage. represents the sum of the primary and transformed secondary coils’ equivalent parasitic parallel capacitance. From now on has a negative voltage across its terminals and interdicts secondary current flow. Demagnetization current can then only keep on flowing in the primary winding, still charging and generating a resonance peak at the frequency given by the magnetization inductance and
itself (fig. 15).
Since the magnetization current is the integral of the voltage across the leakage inductance, during the resonant peak it decreases to the value with sinusoidal shape. Considering that during the resonant peak the entire energy stored in the inductor is passed to the equivalent capacitor and back, it is possible to deduce the following formulas:
√ ∗
1√
∗
24 t t Vcc Vds-max t Vprim Vgs Vds Imag I1 -I1 Tpeak
Figure 15 - Resonant Reset Waveforms ( normalized to primary)
When the primary coil’s voltage gets to zero again (and magnetization current is back to the initial value) it gets clamped to zero by the simultaneous conduction of diodes and , which respectively are conducting ( ) and ( ), in fact, since the primary current is now forced to zero, can only flow in
the secondary winding (transformed with the turn ratio). For a proper behavior of the circuit it’s important that ( ) is positive, so the freewheeling diode is still conducting.
With this technique some space on the PCB can be saved and the duty cycle is not necessarily limited to 50% but to the value ( ⁄ ). To get a higher duty cycle span a smaller drain capacitance can be used, even no
additional capacitor on drain may be used, only relying on parasitic capacitance. This causes an increased amplitude of the resonant peak with consequent stress of the switching MOS . The best tradeoff shall always be found.
Both techniques have their positive and negative sides; the first one cumbers more layers and decreases the general performance, but allows to use cheaper components since involved voltages are never higher than the DC supply voltage. The resonant technique however is more suitable in performance critical applications and when the transformer already has several windings, as in the present case. Having more freedom on duty cycle regulation helps a lot when dealing with transformer’s stray inductance effects, as will be clarified later on.
3.3 Synchronous Rectifiers:
A couple of mosfets will also be substituted to the rectifier diodes to avoid their power loss and voltage drop, also preventing DCM operation. These mosfets will need their source terminal to be connected to ground to work properly, and this means that the output inductors must be placed in the positive output line. The final look of a real circuit is represented in fig. 11.
25
4) THE CONTROL LOOP:
[3], [4], [5] The reaction loop will be now explained shortly for better comprehension of some problems and choices in the actual project.
A forward converter can be controlled in “voltage” or “current” mode.
4.1 Voltage-Mode Control:
The voltage control is implemented by measuring the output voltage and subtracting it to the expected voltage. The error signal is then amplified with a specifically designed PI function, and then compared with a fixed frequency saw-tooth signal. The output of the comparator delivers the PWM signal for the primary switch with the right duty cycle.
Figure 16 – Voltage-Mode control
This technique is rather simple but slow, in fact any change in input voltage must first propagate to the output to be sensed and corrected. Plus, control changes must propagate through the two poles output filter to reach the output, resulting in poor dynamic response and compensation difficulties.
L
Cout
Vout
Vin
Figure 17 - Output LC Filter
Basically, the output inductor and capacitor should be big enough to properly filter the input PWM signal, and properly dimensioned to achieve the desired loop corner frequency (bandwidth) and output impedance.
4.2 Current-Mode Control: 𝑉𝑖𝑛 𝑎𝑣𝑔 𝛿𝑉𝑖𝑛 𝑚𝑎𝑥 𝑉𝑜𝑢𝑡 𝑉𝑖𝑛 𝑎𝑣𝑔 𝐿𝐶𝑜𝑢𝑡𝑠2 𝑉𝑜𝑢𝑡 𝛿 𝑉𝑖𝑛 𝑚𝑎𝑥 𝐶𝑜𝑢𝑡𝑠2
26
In (Peak) current mode control, the output voltage is still sensed, subtracted to the expected voltage and compared with a fixed frequency ramp signal, but in this case the ramp is mainly the inductor ripple current, translated into a voltage by a current sensing resistor. When the current raises to the control voltage value the switch is turned off, ending the active phase and defining then the “peak” current in the inductor.
Figure 18 - Peak Current-Mode Control
The ramp representing the inductor current is fed back to the comparator forming an inner current control loop, the outer voltage control loop programs the inductor current via the inner loop while the current loop directly controls the duty cycle.
Unlike VMC, CMC has inherent voltage feedforward because directly influences the inductor current. The device responds instantaneously to input voltage changes due to a change in inductor’s current raise time. A behavior similar to true-average current mode control can be achieved for applications with a maximum duty cycle of 50% if a suited value of slope compensation [5] is added. The inductor pole is now defined by the current loop. Instead of the two pole second order filter of the VMC loop the outer voltage loop has now a single pole (the filter capacitor) because the Choke’s output behaves now as a current source, so the inductor’s impedance
is blanked, greatly simplifying the loop compensation.
The peak-CMC can be upgraded to average-CMC by introducing an additional “ ” error amplifier in the current loop that increases the open loop gain (ideally to infinite) for the sensed signal DC component. This, via the
27
negative feedback, sets the average DC value of the current in the inductor. The zero of the error amplifier shall be lower than the switching frequency in order to provide a flat gain for all the harmonics.
Figure 19 - Average Current-Mode Control
4.3 Averaged-Value Transfer Functions:
Switching devices have sample & hold characteristics due to a digital modulation, so they cannot be directly described by classic transfer functions. However the behavior of different stages of the control loop can be described by transfer functions between the “averaged” input and output signals. For example, the averaged transfer function of the switch-transformer stage would be:
̅̅̅̅̅̅̅̅̅̅̅̅̅
Such an approach is valid for frequencies that are much lower (10 times) than the switching frequency. In that frequency range the system can be considered linear or be mathematically averaged.
Normal transfer functions can be considered for those stages that have no switching components inside, like for example the output filter. These functions are valid for all frequencies and can be integrated with the averaged ones at low frequencies.
28
5 THE COUPLED OUTPUT CHOKE:
[6] Output voltage regulation of a single output transformer is rather simple to design and really high precision
can be achieved. In multiple output converters regulation is not as simple because of “cross regulation” errors. A multiple output converter can feature independent output inductors or a “coupled output choke”, which behaves as a multiple winding transformer since the coils are wound on the same core.
Figure 20 - Multiple Output Converter
5.1 Main Disadvantages of Independent Inductors:
In most multiple output converters only one output is sensed to “close” the regulation loop. The remaining non-sensed outputs are not directly regulated and so they are more prone to output voltage variations due to load-steps. While the sensed output reacts as fast as possible to “little” and “big-signal” variations, thanks to the intervention of the control loop, the non-sensed output variations hardly cause any control reaction, meaning low voltage stability and long recovery times, depending of course from the control method.
Fig. 21 shows the load-step response of a power stage driven with a fixed duty cycle. The instantaneous load current variation is held by the output capacitor because the inductor current is inertial. This current causes a high voltage peak (negative) on the capacitor’s ESL because of the high current derivate, and a proportional voltage drop on the ESR. The current step also starts a damped resonant oscillation between the output capacitor and the inductor.
Figure 21 - Non-Sensed Output Load Step Reaction
Independent output chokes of different power lines generate multiple resonance frequencies with the corresponding output capacitors which, causing significant loop gain irregularities and phase shifts, are difficult to control without too much loss in performance (regulation bandwidth). *6+”With the transformer secondaries normally closely coupled, all outputs in the small signal loop gain model are driven in parallel at the input of their respective inductors”. Multiple resonances appear in every transfer
Np N1 N2 L1 L2 Vin Cout 1 Cout 2 Load 1 Load 2
𝑡
𝑉
𝑜𝑢𝑡29
function that involves the transformer because it connects the secondary power lines in parallel via cross-coupling, making them “feel” the impedance of each other. For example, fig. 22-a shows the multiple resonances of the (averaged) conductance seen from the transformer’s primary of a three-output converter with three independent output chokes.
( ) ( ) ( )
a) b)
Figure 22 - Three Single Chokes vs. Coupled choke Resonance
In particular, *6+ “the sensed output is shunted by all the other outputs at the common driving point. The LC filters of these shunt outputs soak up much of the source current at their respective series resonance frequencies, causing reduced gain and significant phase shifts at these frequencies. This effect is
especially severe with current mode control because of its characteristic high impedance at the driving point”.
5.2 Main Advantages of a Coupled Choke:
Vin Lm Cp M1 L2 Cout2 Load2 D3 D4 L1 Cout1 Load1 D1 D2 Np N1 N2
Figure 23 – Coupled Output Choke
A first important advantage of this technique is that the coupled choke will only have a main resonance
frequency with the output capacitors (the same on every output, see appendices A, B) which makes
closed loop compensation much easier, as it behaves like a single inductor connected to a single output capacitor (fig. 22-b). The output inductor is in fact dynamically in common with all outputs combining them into a single circuit with a single resonance frequency. Some spurious high frequency resonances
30
will be present (fig. 24), due to the windings’ stray inductances that resonate with the output capacitors, but if the stray inductance stays below a certain percentage of the corresponding winding inductance, it’s resonance frequency will be higher enough than the main resonance, not affecting the control loop stability. Stray inductance effects can usually be compensated up to 10%. A further explanation of this phenomena will be given in Chapter 16.7.
Figure 24 - Stray Inductance Effects on a Coupled Choke
The main advantage though is in dynamic cross regulation. It’s important that the turn ratio of the choke windings is exactly the same as the transformer secondary ratios. Assuming for the moment that the coupling is ideal, an instantaneous voltage drop on a non-sensed output will cause the voltage across the corresponding winding of the coupled choke to increase in the ON phase, or decrease in the OFF phase. Since the coupled choke behaves like a transformer, the superimposed voltage increase (or decrease) on a winding will be transformed to the other windings with the turn ratio, including the sensed output one. Now all the outputs have the same percentage error on their output voltage. So the control loop senses the error on the sensed output and adjusts the duty cycle by the same percentage of the output error, taking all the outputs simultaneously again to the right voltage. The circuit ideally behaves like it only had one output. The AC cross-regulation is now excellent because all outputs are dynamically coupled. A clearer view of this behavior is given in Appendix B. The cross regulation is only limited by the choke’s stray components, as explained in chapter 8.5.
Since the choke’s coils are wound on the same core, they will always have the same magnetic flux-per-turn coupled: 1 2 1 2
A coupled choke only works properly if its turn ratio is exactly the same as the transformer’s one (fig. 23): 1 1 , 2 2 On phase: 1 1 1 1 1 1 ( ) 2 2 2 2 2 2 ( ) 1 2 1 1 2 2 ( )
Simulation of a three output converter with 10% series stray inductance on each winding.
31 Off phase: 1 1 1 2 2 1 1 2 1 1 2 2
This is the only situation that allows the single windings to work as separate chokes in normal operation because in every moment they induce exactly the same field in the core so, ideally, they don’t “feel” the presence of the other coils. The presence of diodes, instead of synchronous rectifiers, does not affect the operation as long as both components on the same line have the same .
The single filter inductor is lower in cost and has smaller volume and mounting area compared to independent inductors.
5.3 Main Disadvantages of a Coupled Choke:
The superimposed voltage on the winding terminals is not in every moment matched with the turn ratio because of transformer’s stray inductance (see chapter 8), series resistance or a difference in rectifiers’ voltage drop. The voltage on the windings of an ideal transformer is always imposed by the transformation rules. When the voltage on different coils is not matched, the actual windings get to an intermediate matched voltage, while the rest drops on the parasitic series components, which include the coil series resistance as well as the rectifiers’ one and the output capacitor’s ESL and ESR. The smaller these components are, the higher will be the “compensation” current flowing through them and through the windings. The following example (fig. 25) shows the behavior of a “double-primary” transformer, where missing voltage matching causes high compensation currents.
a)
V1
V2
R1
R2
I1
I2
N1 N2 V1* V2* b) t t t V2 V1 V1-V1' V1'Figure 25 - Double Primary Transformer and Waveforms
{ 1 1 ∗ 1 1 2 2∗ 2 2 { 1 1∗ 1 1 2 1∗ 1 2 { 1∗ 1 1 1 2 1 1 1 1 2 1. 2 1/ 2 1 , 2 1
32
The last equation reveals that as long as no current flows in the windings except for the magnetization current which in the case of a coupled choke is the load current flowing through the output inductor with its triangular ripple. If the input voltages are not matched so the second term of the equation is not zero, a huge current spike will flow through both windings, higher the lower the series parasitic components are. The phenomena is easier to understand referring to the equivalent circuit below (fig. 26), where power line 2 from fig. 23 is entirely transformed to power line 1 by the coupled choke. As long as the voltages are matched, ( ⁄ ) , no disturbance current flows in the circuit, but if these voltages mismatch for a certain time, the circuit behaves like it was driven by the voltage with consequent disturbance current flow, and voltage noise on the output. The
magnetization inductance can be ignored because its value is much higher than the rest of parasitic components. V1 V2' ESL2' ESR2' ESL1 ESR1 Vout1 Vout2'
Figure 26 - Double Primary Transformer Equivalent Circuit
In this case, since nothing is perfect in reality, a certain amount of series stray inductance on the coupled choke windings would help slowing down the current raise and bucking the output voltage spikes as it behaves like a voltage divider with the output capacitor’s ESL and ESR. In the following equivalent circuit (fig. 27) the voltage mismatch is represented by a square voltage step, stray components from line 2 are transformed to line 1 by the coupled choke [6]. The dual circuit would be considered for line 2.
Ls2' ESL2' ESR2' Ls1 ESL1 ESR1 Vstep Vout Vout_2'
Figure 27 - Series Parasitics’ Filtering Action in a Double Primary Transformer
For control loop simplicity and best cross regulation it’s better not to exceed 10% stray inductance on the output choke otherwise, the spurious resonances of these parasitic inductors with the output capacitor may come too near to the control loop crossover frequency, making compensation difficult.
Another important disadvantage of this technique is that the resonance frequency of the coupled choke
with its parasitic capacitance is lower than any of the single output ones. All the windings’ parasitic
capacitances contribute together to form an equivalent parallel capacitance, bigger than the single ones. 𝑉𝑠𝑡𝑒𝑝 𝑉1 𝑉2
33
This is due to the fact that the coupled choke behaves like a transformer, so each coil sees its parasitic capacitance plus all the other coils’ parasitic capacitances transformed in parallel (see Appendices A, B). This is particularly undesired because output choke’s parallel capacitance lets driving signal’s high frequency harmonics pass through instead of being filtered by the inductor, causing output spikes as will be explained in detail in Chapter 8.
34
6) THE GYRATOR-CAPACITOR APPROACH:
For optimization purpose, instead of a finite element simulator, p-Spice simulator will be intensively used. Using a circuit simulator is much lighter and, once the electric equivalent model of the magnetics has been chosen, it’s very easy to make chances to the topology. For this reason, choosing the best possible electric model is fundamental. The chosen model is the so called “Gyrator-Capacitor” model, a rarely used approach, which on the other hand is extremely realistic and accurate [7][13].
6.1 Reluctance-Resistance and Reluctance-Inductance Models:
The traditional “Reluctance-Resistance” approach is based on the analogy between magneto-motive force and
voltage, as well as magnetic flux and current, to generate an electric equivalent circuit. The produced by a
winding carrying a current * + is {Ampere-turns} and is represented by a voltage source. Magnetic flux and are proportional according to Ohm’s law: where the reluctance is the counterpart of the resistance ( )⁄ . If the currents in the windings are known the fluxes in the core can be easily found. However, since in most cases magnetic and electric circuits interact, this kind of equivalent circuit can be transformed in a corresponding inductive circuit, which can be connected directly to the rest of the surrounding circuit.
Table 1 - Classic Approach Dimension Equivalence Table
The conversion procedure comprises two steps (fig. 28):
1) *7+“The dual of the resistance model is formed by applying the laws of electrical voltage-current duality (b, c). Network loops become nodes and vice versa; each voltage source * + becomes a current source of * +, and each resistance of * + is replaced by a conductance of * +”.
2) *7+”A particular winding is chosen as the reference winding, having turns, say. The associated current
source is replaced by a couple of terminals. The current source associated with any other winding of turns is replaced by an ideal transformer of ratio feeding a pair of terminals. Each conductance of
35
Figure 28 - Inductance-Reluctance Equivalent Model
It can be demonstrated however, that not every network has a dual, in particular the so called “non-planar” networks cannot be converted in such a way. A network is “non-planar” if it can’t be drawn without crossovers. Here is where the correction resistance to inductance fails so the designer would be forced to stick to the resistive model. Although this model is very simple and straight forward, it can’t be directly connected to the surrounding circuit, plus it doesn’t take into account that magnetic reluctances “store” energy instead of dissipating it as a resistor does. This asymmetry is particularly critical in power electronics where energy relations are of prime importance.
In the electric equivalent in fact, the chosen fundamental variables are voltage (effort variable) and current (flux variable), the product of which gives power * +. In the real device instead the product of the corresponding fundamental variables (effort) and magnetic flux (flux) gives an energy * +, highlighting some asymmetry between reality and equivalent model.
6.2 Reluctance-Capacitance Model:
Among the different possible ways to correct this asymmetry the so called “Gyrator-Capacitor” approach has been chosen for its greater capability of being integrated with parasitic elements calculation. This approach is based on considering the derivate of the magnetic flux as the “flux” fundamental variable instead of the magnetic flux itself, so that the product of the two fundamental variables gives a Power as in the electric equivalent; so in the end:
The magnetic flux then, as the “integral of its derivate” is represented by a charge in the electric equivalent. The permeance of the magnetic core , which is calculated as will be represented by a capacitor in the electric equivalent instead of a resistor as in the traditional approach. The charge stored in the capacitor represents the flux stored in the core and the corresponding energies are exactly the same. From these considerations a complete table of magnetic-electric equivalences is derived:
a)
b)
c)
36
Table 2 - Gyrator Approach Dimension Equivalence Table
1
2 3
5
4
1/
1
1
/
2 4 / 1 3/
1
5/
1
1 1I
N
N
2I
2N
1I
1N
2I
2Figure 29 - Capacitance-Reluctance Equivalent Model
In a normal core, where the coils are wound on the same leg, the core can be simply represented with one equivalent capacitor (permeance), and the generators representing the coils are connected in series (fig. 33).
6.3 The Gyrator:
Not only the gyrator model provides a more realistic way of modeling the core and in general the permeance paths of the device, but it also allows an extremely useful representation of the windings which is extremely advantageous when dealing with parasitic modeling. A winding with a certain voltage imposed will force a certain magnetic flux variation rate in the core, as well as the generated by this flux rate needs to be “supported” by a certain current in the winding:
{ { 1 2 1 2 * +
From this simple system an equivalent for an N-Turn winding can be found as the so called “Gyrator”
37 Where the parameter is called “module”
of the gyrator. A gyrator is a more fundamental component than the ideal transformer used in the traditional approach, in fact a transformer can be produced by cascading two gyrators but a gyrator cannot be produced from transformers.
When an impedance is connected to one of its ports an impedance of ⁄ is seen on the other port. The inductance of a winding may then be determined by finding the effective permeance loading the corresponding gyrator; the inductance is then calculated as .
It’s important to note that a winding can be represented by an arbitrary number of gyrators connected in series, even dividing a single turn in several segments, which is the real strength of this model. It is possible in fact to divide a winding until the desired granularity is reached to represent and properly connect the parasitic elements to intermediate nodes among the winding, simulating their distributed nature.
N
N/2
N/2
Figure 32 - Gyrator Series Connection
Different coils that are wound on the same leg are similarly connected in series on the magnetic side, while they have separated input voltages on the electric side.
Figure 31 - Coil-Gyrator Equivalence
a)
38 V1 V2 Req N1 N2 N1 N2 1/Req V1 V2 Es Ms
Figure 33 - Gyrator-Capacitor Modeling Example
The circuital equivalent of the gyrator con be deduced from the characteristic equations outlined at the beginning of Chapter 6.3, and can be “built” combining two current controlled voltage sources or two voltage controlled current sources:
N
rI2 rI1 V2/r V1/r
V1 V2
I1 I2
