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MODELING AND PERFORMANCE EVALUATION OF A POWER
ELECTRONIC-BASED CONTROLLED FUEL CELL
SYSTEM FOR VEHICULAR APPLICATIONS
by
YASHAR EHSAN KENARANGUI
Presented to the Faculty of the Graduate School of
The University of Texas at Arlington in Partial Fulfillment
of the Requirements
for the Degree of
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
THE UNIVERSITY OF TEXAS AT ARLINGTON
AUGUST 2006
Copyright © by Yashar Ehsan Kenarangui 2006
All Rights Reserved
iii
ACKNOWLEDGEMENTS
I am extremely grateful to my advisor, Dr. Fahimi, for not only providing me
guidance in matters of engineering, but for being some one that I look up to. I am
thankful for his patience and encouragement at times when I was frustrated with my
work’s progress.
I would like to thank Dr. Lee for his continues support and encouragement
during the last six years. Since my undergraduate years, for me he has been a source of
great insights and information in all matters.
Furthermore, I would like to thank Dr. Dillon for accepting to be on my
committee despite of his busy schedule.
I would like to thank Mr. Shiju Wang for being a great partner and a friend
during the last twelve month. Working with him was always a lot of fun.
Finally, I thank my parents and my brother for their inexhaustible support.
July 22, 2006
iv
ABSTRACT
MODELING AND PERFORMANCE EVALUATION OF A POWER
ELECTRONIC-BASED CONTROLLED FUEL CELL
SYSTEM FOR VEHICULAR APPLICATIONS
Publication No. ______
Yashar Ehsan Kenarangui, M.S.
The University of Texas at Arlington, 2006
Supervising Professor: Dr. Babak Fahimi
In fuel cell applications, design of an appropriate controller and selection of an
Ultra-Capacitor require a suitable model of fuel-cell-systems that can adequately
represent the significant characteristics of the system. Such a model, using lumped
circuit elements, was developed in this thesis via a simple method without requiring fuel
cell design parameters. To validate the model, simulation results from the model were
compared against the results from experimental setups and satisfactory results were
obtained. Later, this model was used in order to design a controller that meets the
control objectives, namely, regulation of output voltage and attenuation of load
v
current variations at the output fuel cell system. Performance of the controller was
verified by simulation. High efficiency is an indispensable requirement for fuel cell
power conditioners. Finally, in view of this requirement, an analytical method was used
in order to calculate power losses in the boost converter and determine factors that
influence these losses. This method is helpful in understanding power loss mechanisms
in power electronic converters and thus designing high efficient power conditioners for
fuel cell systems.
vi
TABLE OF CONTENTS
ACKNOWLEDGEMENTS....................................................................................... iv ABSTRACT .............................................................................................................. v LIST OF ILLUSTRATIONS..................................................................................... ix LIST OF TABLES..................................................................................................... x Chapter 1. INTRODUCTION …………............................................................... 1 1.1 Need for alternate sources of energy ....................................................... 1 1.2 Potential of Fuel Cells as the Future’s Clean Energy Conversion Devices ............................................................................... 2 1.3 Components of Fuel Cell Applications.................................................... 4 1.4 Components of Fuel Cell Vehicles .......................................................... 5 1.5 Power Processing Systems in Fuel Cell Applications ............................. 6 1.6 Need for a Fuel Cell Model Suitable for Power Electronics Design....... 8 1.7 Outline of this Thesis…........................................................................... 9 2. LINEAR NETWORK APPROXIMATION OF FUEL CELL SYSTEMS.......... 10
2.1 Specifications for Ballard NexaTM Fuel Cell Power Module .................. 10 2.2 Relationship between Thermodynamic and Electrical Descriptions of Fuel Cells……………………………………………….………………. 13 2.3 Role of a Control Module in Fuel Cell Systems….................................. 16 2.4 Fuel Cell Current Ripple Considerations in Power Converter Design .... 18
vii
2.5 Modeling of Fuel Cell Systems .............................................................. 20 2.6 Steady State Characteristics of the Fuel Cell System ............................. 22 2.7 Linear Network Approximation of the Fuel Cell System……………… 27
2.8 Parameter Extraction .............................................................................. 29 2.9 Model Validation…………………………………..…………………… 30
3. CONTROL SYSTEM DESIGN FOR FUEL CELL BASED BOOST CONVERTER……………………………………………………..…… 33 3.1 Selection of a suitable converter for fuel cell application ....................... 33 3.2 Small-signal equivalent circuit model for the fuel cell based boost converter………………………………………………………….. 34 3.3 Control system design.............................................................................. 40 3.4 Performance verification of the controller .............................................. 44 4. AN INTUITIVE THERMAL-ANALYSIS TECHNIQUE FOR SWITCH MODE POWER CONVERTERS......................................................... 51
4.1 efficient converter design for fuel cell applications……………………. 51
4.2 Power electronics converter thermal performance – Simulation Tools…………………………........................................... 52
4.3 Power electronics converter thermal performance – Analytical Methods…………………………………………………… 54
4.4 Topology under Consideration…………………………………………. 55
4.5 Obtaining dynamic model of MOSFET from datasheet.......................... 55 4.6 MOSFET model via graphical transformation ....................................... 57 4.7 Equivalent circuit representation of hard switching boost converter ...... 60
viii
4.8 Deriving equations for power losses and power stresses......................... 63 4.9 Numerical method ............................................................................... 69 4.10 N Comparison of analytical result and hardware-test result ................. 70 CONCLUSION AND FUTURE WORK…………………………………………… 70 REFERENCES .......................................................................................................... 72 BIBLIOGRAPHICAL INFORMATION…………………………………………... 75
ix
LIST OF ILLUSTRATIONS
Figure Page 1.1 Components of a typical fuel cell application ................................................ 4 1.2 Schematic for a typical power system of a fuel cell based passenger car ...... 5 2.1 Interconnection of Fuel-Cell-System components (access points indicated) ..................... 10 2.2 NexaTM Power Module access points (corresponding to Figure 2.1).............. 12 2.3 FC polarization curve .................................................................................... 17 2.4 Control action in fuel cell system ................................................................... 17 2.5 Fuel Cell Current Ripple at 20 kHz ................................................................ 19 2.6 Fuel Cell Current Ripple at 200 kHz .............................................................. 19 2.7 Experimental setup-A to obtain steady state voltage versus stack current ... 22 2.8 Applying 43.38 Amps load ............................................................................ 23 2.9 Disconnecting 43.38 Amps load .................................................................... 23 2.10 V-I Characteristic of stack (polarization curve).............................................. 24 2.11 Current supplied to auxiliary devices versus load ......................................... 24 2.12 Experimental setup-B to study transient stack voltage .................................. 25 2.13 Ripple (25%) on top of 30A DC at 1 Hz ........................................................ 26 2.14 One cycle from the above waveform ............................................................. 26 2.15 Linear network approximation of the fuel-cell-system ................................. 28 2.16 Linear network approximation of the fuel-cell-system (simplified) ............... 29
x
2.17 Experimental and simulation results at 1 Hz load ripple ............................... 31 2.18 Experimental and simulation results at 10 Hz load ripple ............................. 31 2.19 Experimental and simulation results at 50 Hz load ripple ............................ 32 2.20 Experimental and simulation results at 100 Hz load ripple ........................... 32 3.1 Boost converter to step up the output voltage of the fuel cell system ............ 34 3.2 Linear network model of the fuel cell based boost converter ........................ 34 3.3 Analog implementation of the cascaded control system ............................... 40 3.4 Bode plot for )(~)(~ sdsiL with an integral compensator ................................ 42
3.5 Bode plot for )(~)(~ sisv Lo with a proportional integral compensator ...................................................................................... 42
3.6 Simulink block diagram of the fuel cell based boost converter under load disturbance ................................................................................... 43
3.7 Response of output voltage to a 5 Amps step disturbance in load ................ 45
3.8 Response of inductor current to load current variation of [ )..21000sin(0.1 tπ× at 5 < t < 15] ................................................................... 46
3.9 ZOOMED Fig.3.8 ........................................................................................... 46
3.10 Response of inductor current to load current variation of [ )..21000sin(0.1 tπ× at 1 < 1.1]…………………………………………………47
3.11 Response of inductor current to load current variation of [ )210sin(0.1 tπ× at 5 < t < 15] .......................................................................... 48
3.12 ZOOMED Figure 3.14 .................................................................................... 48
3.13 Response of inductor current to load current variation of [ )210sin(0.1 tπ× at 1 < t < 2]…………………………………………………...49
3.14 Response of inductor current to load current variation of [ )21sin(0.1 tπ× at 1 < t < 10]…………………………………………………...50
xi
3.15 Response of inductor current to load current variation of [ )21sin(0.1 tπ× at 1 < t < 10]…………………………………………………..50
4.1 Hard-switching boost converter ..................................................................... 55 4.2 Graphical transformation to obtain drain current vs. time ............................. 59 4.3 Equivalent circuit representation for the boost converter .............................. 61 4.4 Waveforms that determine the time intervals of equivalent circuits ............. 62 4.5 Numerical method flow chart ........................................................................ 69 4.6 Analytical method and hardware test results – hard switching topology ...... 70
xii
LIST OF TABLES
Table Page 1.1 Advantages of fuel cells ................................................................................. 2
1.2 Disadvantages of fuel cells ............................................................................. 3
1.3 Different Types of Fuel Cells ......................................................................... 3 2.1 Results from parameter extraction for NexaTM Power Module ...................... 30
4.1 Nomenclature for Fig.4.4 ............................................................................... 62
1
CHAPTER 1
INTRODUCTION
1.1 Need for alternate sources of energy
In today’s world, electricity is absolutely critical. This form of energy must be
converted from other sources of energy such as chemical energy of fossil fuels, nuclear
energy, or energy of moving water. Among different sources of energy, chemical energy
is viewed favorable in converting to electricity because of their abundance, and ease of
transportation and storage. There are several methods to convert chemical energy into
electricity. Thermal generating stations that use fossil fuels, account for almost 80% of
the electric energy generated in the United States [1]. In addition to this, almost all cars
today still use the old fashion combustion engine that also utilizes fossil fuels to converter
chemical energy to mechanical energy. In both cases (thermal generating stations and
combustion engines), burning of fossil fuels generates harmful by products such as
carbon monoxide CO, carbon dioxide CO2, and nitrous oxide NOx. The well known
green house effect is the result of a canopy that is formed by these gases near the ozone
layer and has lead to an overall increase in the temperature of earth. According to U.S.
Environmental Protection Agency (EPA), vehicles account for about 75% of CO
emissions, about 45% of NOx emissions, and about 40% of other organic compound
emissions. In view of the ever increasing evidence regarding the dangers of such
emissions, the automotive industry has invested much effort in developing new
2
technologies, namely, the Hybrid Electric Vehicle (HEV) and Fuel Cell Vehicle (FCV)
[2].
1.2 Potential of Fuel Cells as the Future’s Clean Energy Conversion Device
Fuel cells have numerous outstanding characteristics that make them attractive for
several crucial applications such as transportation, power generation, and portable
devices. Some of the prominent features of these energy conversion devices include non-
toxic emissions, application versatility, and relative high efficiency. [3]-[4] Ironically fuel
cells are not newly invented devices; however recent innovations in material science
have made these devices viable for commercialized. The fuel cell was first demonstrated
by Sir William Grove in 1839. Later in 1950s it was further developed and was
successfully used in the American Manned Space Program. During the last ten years, due
to the environmental concerts, an effort to advance fuel cell technology to a next level has
begun. [5] Table 1.1 and Table 1.2 have listed some of the advantages and disadvantages
of fuel cells in general. Table 1.3 has listed different kinds of fuel cells that are
commercially available today.
Table 1.1 Advantages of fuel cells
ADVANTAGES of FUEL CELLS Unparalleled environmental performance Operates on hydrogen, thus, water is the only by-product.
Also these systems have quiet operation which makes its overall impact on the environment minimal. [6]
High efficiency These systems are nearly double the simple-cycle efficiency of conventional gas turbine and reciprocating engine power generation technologies. Similarity between the efficiencies of small systems and large ones is another advantage point. [7]
Continuous output Their ability to continuously produce electrical output through replenishing their reactants (hydrogen and oxygen). [5]
3
Table 1.1 – Continued
Fuel diversification Hydrogen can be produced not only from fossil fuel sources but also from biomass and other sources. [3]-[4]
Reliability and flexibility Composed of very few moving parts; therefore these systems have much higher reliability than combustion engines, turbines or combined-cycle systems. [4]
High power density New technologies in material science and novel fuel delivery mechanisms have allowed power density of fuel cells to exceed that of lithium ion (Li-ion) batteries.
Wide ranges of applications Stationary power generation, mobile applications, and automotive applications such as Fuel Cell Vehicles
Table 1.2 Disadvantages of fuel cells
Disadvantages of Fuel Cells High cost Catalysts (such as platinum) are relatively expensive. The
cost of auxiliary devices (e.g. compressor) and power conditioners (e.g. converters and inverters) are also high.
Short lifetime Experimental fuel cell systems such as Nexa™ Power Module have a life span of only 1500-hours. [8]
Wide fluctuating low dc-output-voltage The fuel cell output is characterized by a low voltage which is current and temperature dependant (full-load to no-load ratio of around ½). [3]-[5]
Slow dynamic responses under sudden load changes Due to fuel transport delay to the site of reaction [5] and also mechanical components that are involved in fuel cell operation, fuel cell systems are slow to respond to faster load changes.
Relatively long startup process Generally today’s commercially available systems take 2 min. to achieve rated power from a cold start, therefore need a back up energy source such as batteries. [4]
Table 1.3 Different Types of Fuel Cells [5]
Different Types of Fuel Cells Fuel Cell Type Mobile ion Operating
Temp Application
Alkaline (AFC)
OH−
50–200C
Used in space vehicles, e.g. Apollo, Shuttle.
Proton exchange membrane (PEMFC)
H+
30–100C
Vehicles and mobile applications, and for lower power CHP systems
Direct methanol (DMFC)
H+
20–90C
Suitable for portable electronic systems of low power, running for long times
Phosphoric acid (PAFC)
H+
~220C
Large numbers of 200-kW CHP systems in use
Molten carbonate (MCFC)
CO32-
~650C
Suitable for medium- to large-scale CHP systems, up to MW capacity
4
1.3 Components of Fuel Cell Applications
The design of a complete fuel-cell (FC) application, represented in Figure 1,
encompasses ideas from diverse disciplines such as chemistry, material science,
mechanical engineering, and electrical engineering. The FC system, shown in Figure 1
(block-C), consists of an FC stack (block-B), a control module (block-C), and other
auxiliary devices (e.g., compressors, valves).
Figure 1.1 components of a typical fuel cell application
FC stack is the core of this system where the energy conversion takes place.
Inside each fuel cell, chemical energy is produced from the reaction and is directly
converted into electrical energy. In order for any chemical reaction to result in the desired
outcome, specific pressure, and temperatures, along with right amounts of reactants need
to be provided. Therefore, additional equipment is necessary to support Nexa™ system’s
operation; that is to provide pressurized air (compressor/air pump) and regulate reaction
temperature (cooling fan). Also a control module is utilized to optimize the operation of
the FC system (sensors, actuators and controllers). Auxiliary equipments consume power
for their operation and as a result, introduced losses reduce the total efficiency of the
system. [8]
5
1.4 Components of Fuel Cell Vehicles
There are two different loads in an FCV, traction load and hotel loads. Traction
load consisting of an electric motor consumes about 100kW of power. On the other hand
various hotel loads (e.g., air conditioning unit, radio) consume 10KW of power. As
mentioned before, the output voltage of a fuel cell system is low (~50 V) and varies
widely based on the load current and temperature. Therefore, power conditioning
systems are a major component of any fuel cell application, especially fuel cell vehicles.
[9]
Different arrangements exist to connect the main fuel cell output to various loads,
namely, traction loads, hotel loads, and fuel cell support systems. For a fuel cell based
passenger care, typically an arrangement such as Figure 3 is utilized.
Figure 1.2 Schematic for a typical power system of a fuel cell based passenger car [10]
6
The battery is also an important part of this system that addresses some of the
short coming of fuel cells as the main power source. For example, the battery provides
the power during warm up of the fuel cell. After the process of warm up, the battery is
cut out and the fuel cell provides the power for traction motor. Also notably, the same
battery provides power during fast load changes. Acceleration, change in road conditions
and stop at red lights are some of the examples of fast load changes in vehicles. Ultra
capacitor can also be utilized to provide power during these transient power demands.
[9]-[10]
Vehicle traction controller is the brain for the entire system. It receives feedback
signals from the traction motor and command signals from the deriver. Then it sends
control signals to the converters, inverters, and the fuel cell system. Thus the operation
of fuel cell system is optimized according to the desired speed and torque of the traction
motor.
1.5 Power Processing Systems in Fuel Cell Applications
The electrochemical reaction in fuel cells directly generates electric power. In
order to capture the usefulness of this energy, output voltage and current of fuel cell
systems must match the input requirements of our load. This presents several challenges
in the application of fuel cell systems. The fuel cell output is characterized by a low
voltage which is current and temperature dependant (full-load to no-load ratio of around
½). This requires power processing modules to convert the output of fuel cells into a
specific voltage and current that meets the load requirements; at the same time regulation
of the output is necessary to provide a stable power source to the load. Furthermore, fuel
7
cell systems are unable to deliver the required power in the face of fast load dynamics
and continuous high current ripple will deteriorate their life span; this imposes additional
requirements on fuel cell power converters. [11]-[12]
To design a cost-effective and highly efficient dc-dc converter, a proper topology
must be selected that is a good match with fuel cell characteristics. Compromises should
be made in considering size, efficiency, input voltage range, amount of input current
ripple, and other parameters when selecting a converter topology. Below is a list of some
general requirements for fuel cell based power processing systems and possible methods
of meets some of these requirements.
• Step-up voltage – 4 to 6 times the input voltage, use of step of converters
such as boost and inverter with a step up transformer
• High efficiency – low switching losses that can be achieved by zero
voltage switching (ZVS) and zero current switching (ZCS) schemes
• High Power Density – by operating at high switching frequency requires
smaller energy storage components
• Minimize current ripple of fuel cell – by controlling the input current of a
converter and/or operation at high frequencies
• Improved system lifetime – by minimizing fuel cell current ripple and
power stress in semiconductor devices
• Low cost – smaller energy storage components, simple and effective
controller
• Low EMI – decreased dtdi / and dtdv / in semiconductor devices during
commutation
8
Boost topology will be used in this thesis as demonstrate some of the interactions
between fuel cell systems and their associated power conditioners. This choice was
based on several reasons such as boost converter’s structural simplicity and its input
inductor. When the boost converter is connected to a fuel cell system, it draws
continuous current owing to the input inductor. As it will be explained in more details,
this is significant because drawing continuous current suits the dynamic characteristic of
fuel cells. In conclusion, boost converter is a simple power circuits in which low cost,
high efficiency and high reliability can be achieved. Therefore, this appears to be the
best choice for fuel cell applications amongst many topologies.
1.6 Need for a Fuel Cell Model Suitable for Power Electronics Design
Fuel cell systems are complex devices that involve technologies from various
disciplines such as chemistry, mechanical engineering, material engineering, and
electrical engineering. In order to design a power electronics converter that meets the
requirements for optimal operation of fuel cell systems, an understanding of the system is
crucial. However, learning all the electrochemistry and the science behind PEM is a
daunting task and also might be a waste of effort. Therefore a different approach is used
in order to understand the chemistry aspect and thermodynamics of fuel cell systems in
terms of familiar concepts in electrical engineering such as voltage, current and
impedance [13]-[16]. Active and passive components can be utilized to construct a fuel
cell model to describe the essential structure of the system so that the significant
characteristics of its performance can be adequately represented. In addition, these
circuit based models can be used in designing control systems for power electronics
9
converters and simulations of the performance of the entire system (fuel cell system
connected to power electronics converters).
1.7 Outline of this Thesis
In Chapter 2, first a description of fuel cell systems is presented in terms of
thermodynamic terms such as concentrations, gas flow rates, and temperature. Then its
electrical counterpart will be described in terms of voltages, currents, and impedances.
Here the main objective is to translate the thermodynamics description of the fuel cell
system to an electrical description. The result of this will be a lumped element circuit
model of the system. This linear network model will then be used in Chapter 3 to design
an appropriate control system for the fuel cell based boost converter. This is
accomplished by deriving a transfer function of the entire system (fuel cell system
connected to the boost converter). Small signal modeling method (Taylor series
approximation) is used to linearize the system. To avoid crossing the current ripple
limitations of the fuel cell system, an appropriate control system is design for the boost
converter. A cascaded controller is used so as to reduce the converter’s speed in
responding to faster load changes by controlling the input inductor current. As
mentioned, high efficiency of the fuel cell power conditioner is a crucial design goal.
Finally in Chapter 4 an analytical method is used to examine the power losses in the
boost converter. Use of analytically derived equations allows designers to determine
parameters that contribute to losses in a power electronics circuit. Based on this one can
take measures to improve the efficiency of the power converter.
10
CHAPTER 2
LINEAR NETWORK APPROXIMATION OF FUEL CELL SYSTEMS
2.1 Specifications for Ballard NexaTM Fuel Cell Power Module
NexaTM Components:
Figure 2.1 Interconnection of Fuel-Cell-System components (access points indicated)
Fuel Cell Stack:
• 47 Cells
• Power Capacity = 1.45 kW
• Operating Voltage = 22 V – 48 V
• Max Current Ripple = 24.7 RMS = 35% P-P @120 Hz
Air Compressor:
• Power = 100 W @ 100% duty cycle
• Draws pulsed current (pulse frequency increases with load)
11
Cooling Fan:
• Max Power = 100 W
• Draws steady current (current level increases with load)
Control PCB:
• Max Power = 5 W
• Close to constant power consumption
• Responds to load variations by adjusting the flow of hydrogen and air
Switch:
Input power supply for auxiliary devices (Compressor and cooling fan) is
automatically switched to the fuel cell stack from the auxiliary power
supply at the end of start up process
Relay:
Protective relay, triggered by excessive current (e.g., short circuit)
Access points to the system:
Several access points facilitate voltage and current measurements without the
need to open up the system. These are indicated in Figure 2.1 by A0, A1, A2, A3, V1,
V3, ∆Vcell.
A0: Current supplied by the stack
A1: Load current
V1: Stack Voltage
A2: Sum of currents drawn by compressor, cooling fan, and control PCB
(supplied by stack after startup)
12
A3: Sum of currents drawn by compressor, cooling fan, and control PCB
(supplied by auxiliary power sup. )
V3: Input voltage from auxiliary power sup.
∆Vcell: Potential different across cells
Figure 2.2 NexaTM Power Module access points (corresponding to Figure 2.1)
Current supplied by the stack can be measured at point A0. Current supplied to
the Compressor, Fan, and Control-board can be measured together at point A2. To
measure these current separately the system needs to be taken apart. Therefore, further
considerations are necessary to determine which component of the current corresponds to
each device (Compressor, Fan, and Board). This can be accomplished by separately
exciting the fan and the compressor to observe their individual current profile.
13
2.2 Relationship between Thermodynamic and Electrical Descriptions of Fuel Cells
Recent years has witnessed significant research and development on fuel cells.
Due to its interdisciplinary nature, various arts of engineering are involved in optimal
design and operation of fuel cell. In electrical engineering, the focus is placed on
determination, and control of voltages, currents, and impedance that characterizes their
relationship. Therefore there has been an effort to describe fuel cells in terms of the
mentioned concepts.
Internal Voltage: A central part of this description is the Nernst equation. This
equation expresses electric potential induced across anodes and cathodes (internal
voltage) in terms of reactant pressures, temperature, and a few constants as indicated in
Equation-2.1, internal-voltage.
Fuel Transfer Delay: Concentrations of reaction species (reactants and products)
along with the equilibrium constant determine the rate of chemical reactions. In the case
of reactions in solution the available reactants are present in the site of reaction. In the
case of fuel cells, the reactants are flowing into the site of reaction and therefore are not
immediately available. Only part of the reactants is available on the catalysts (site of
reactions). This implies that if the load suddenly increases there will be a shortage of
hydrogen due to increase in the rate of reaction. This shift in the equilibrium can be
explained based on Le Chatelier’s principle which states that if a stress occurs in a
reaction, then the reaction will respond towards relieving that stress. As apparent from
the below equations, when load increases, the electrons are removed from the right hand
side of the equation. Le Chatelier suggests that more hydrogen has to react in order to
replace the removed electrons. An increase in the load means a decrease in products in
14
the first reaction and an increase in reactants for the second equation. Therefore in the
second equation, according to Le Chatelier’s principle, the reaction will speed up to
decrease the amount of reactants and thereby relief the stress. The net result of this will
be an increase in the reaction rate or speed.
2H2 4H+ + 4e- (anode)
O2 + 4H+ + 4e- 2H2O (cathode)
2H2 + O2 2H2O (net reaction)
Now there are two effects that are competing against each other. One is the rate
of reaction and the other is the flow rate of reactants coming into the site of reaction. As
discussed before, the rate of reaction increases in response to the increase in load. It is
important to consider what happens if reactant flow rate can not keep up with the reaction
rate. When the flow rate is unable to keep up with the increased reaction rate the partial
pressure of the reactant (H2) will drop because flow rate is what sustained its partial
pressure. According to the internal-voltage term in Equation 2.1, the internal voltage of
fuel cell will drop due to a drop in the partial pressure of hydrogen. As indicated from
the fuel-transfer-delay term in Equation 2.1, the voltage drop will settle in a new
equilibrium and it will reach a steady state value equal to i⋅λ (indicated in Equation 2.1).
One of the main functions of the fuel cell control module is to increase the flow rate such
that it matches the reaction rate. Therefore, the control action will decrease the steady
state value of this voltage drop.
Ohmic Voltage Drop: This voltage drop is simply understood as the resistance of
the electrodes and resistance to the flow of electrons in PEM (or electrolyte). In this case
the voltage drop is proportional to the fuel cell current. [16]-[18]
15
Activation Voltage Drop: Activation losses are common for chemical reactions
and they become more severe in slow reactions. Tafel’s equation is an experimentally
determined expression that is used to calculate the value of this voltage drop. [16]-[18]
Concentration Voltage Drop: The area on the catalyst (the site of reaction) is
limited. This puts a limitation on the maximum amount of current that can be drawn
from fuel cells. At high load current this area gets saturated and an increase in current
output results in an increased voltage drop. Current utilization will also suffer because of
concentration effects since more reactants will not be able to react. The equation for this
term is given in Equation 2.1, [16], [19].
For ease of observing important characteristics of the Equation 2.1 several
constants were redefined, such as zFRTK = .
Next page will provide the nomenclature for Equation-2.1.
( )
⎪⎪⎪⎪⎪
⎭
⎪⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−
Τ⋅Κ−Τ−⋅−⋅+−
⋅−−⋅⋅Τ⋅Κ+
=Ω
−
44 344 2143421444 3444 21
44344214444 34444 21
DROPVOLTAGE
IONCONCENTRAT
l
DROPVOLTAGE
ACTIVATION
DROPVOLTAGEOHIMIC
DELAYTRANSFER
FUEL
t
VOLTAGEINTERNAL
OH
stack
II
IIIbITkIkR
eiippE
nV1ln)ln()(
)(ln
210
/220
τλ
(2.1)
16
Nomenclature
0E Open Circuit Potential or Reference Potential n Number of Cells in FC Stack T Temperature in Kelvin of FC Channel [K]
2Hp Effective Partial Pressure of Hydrogen
2Op Effective Partial Pressures of Oxygen iI , I: FC Stack Current, i : current increase
lI Limiting Current (A) R Ideal Gas Constant [ ( )KmolJ ⋅/3143.8 ]
τ Fellow Delay
2,1 kk Empirical Constant to Calculate ROhmic
z Number of Electrons Participating F Faraday’s Constant λ Constant Factor in Calculating Transfer Delay b Constant Terms in Tafel Equation
0ΩR Constant component of resistance 2.3 Role of a Control Module in Fuel Cell Systems
This topic is not clearly and thoroughly discussed in the literature. Most of the
ideas in this section are from the experience of the authors working with NexaTM system.
It is important to distinguish the output characteristics of a fuel cell stack from that of a
fuel cell system. Fuel-Cell-Systems incorporated a control module in which the input
fuel and oxygen is adjusted continuously according to the output load. This process
involves varying the oxygen and hydrogen flow rates that will result in an alteration of
the polarization curve (steady-state V-I characteristic) of fuel cells. This is because the
polarization curves, shown in Figure 2.3, are usually described under constant flow rates
of reactant gases. Without a control module, the output of fuel cell stacks drops more
drastically than it does in an open loop system when the load increases [22]. Also, if fuel
flows are controlled according to the current then fuel utilization (percent of reactants
17
that participate in the reaction) will remain constant with the current. As shown in Figure
2.4, the control action results in shifting of polarization curves. The resulting curve,
indicated by a bold line, represents the output voltage versus the fuel cell system load.
Figure 2.3 and Figure 2.4 are not to scale and are only used as a tool to demonstrate the
functionality of the control module.
Figure 2.3 FC polarization curve [16]-[18]
Figure 2.4 Control action in fuel cell system [22]
18
The control module responds inadequately and slowly to fast transient loads.
This is because the actuation of control signals is carried out by compressors and
valves. Also, reactant gasses need to travel through channels to reach fuel cells in which
further delay is imposed. The consequence of this is the appearance of voltage dips at
any time sudden load increases occur.
2.4 Fuel Cell Current Ripple Considerations in Power Converter Design
Fuel cell current ripples consist of both high and low frequency components.
High frequency ripples could be a result of a converter switching action or fast load
variations. On the other hand, low frequency ripples could be a result of slower load
variations or inverter switching action [23]. Electrochemical reaction responds to slower
variations in the load (about dcfkHz >>10 ) [25], even if the fuel cell voltage never
reaches steady state for frequencies higher that 1Hz. An important point here is that, in
fuel cells continuous variations in reaction conditions result in mechanical stresses that
will decrease the life span of the system [23], [25]. Therefore, the fuel cell system should
respond only to load shifts (about fHz >5.0 ) and not to load variations
(about HzfkHz 5.010 >> ). Current ripples with frequencies higher than about 10 kHz are
filtered out because of the presence of double layer capacitors. This implies that current
ripples at these higher frequencies have no significant effect on the chemical reaction and
therefore do not have the adverse effects of the lower frequency ripples. This is verified
also for high peak-to-peak values of ripple (up to 50%).
In can be concluded that a range of frequencies roughly between Hz5.0 and
kHz10 should be avoided by utilizing a proper ultra-capacitor and implementation of a
19
suitable controller. The control system should restrict the transfer of load variations in
this range (at the output of a converter) to its input inductor (connected to the output of a
fuel cell system). Current ripples of the fuel cell system with a boost converter was
examined under various loads and switching frequencies. As shown in Figure 2.5 and
Figure 2.6, the output voltage of the fuel cell system is not impacted by the high
frequency current ripple.
Figure 2.5 Fuel Cell Current Ripple at 20 kHz
Figure 2.6 Fuel Cell Current Ripple at 200 kHz
20
2.5 Modeling of Fuel Cell Systems
In recent years there has been an effort by many people to develop a model for
fuel cells in terms of electrical engineering concepts (e.g., resistance, capacitance,
voltage, current), with an addition of some nonlinear elements [19], [23]-[24]. This type
of model benefits an electrical engineer at least in two ways. First, with a reasonable
effort, the performance of the electrochemical device can be evaluated in terms of
electrical characteristics. Second, it will endow a power converter designer with a fuel
cell model that can be utilized in controller design.
In references [19], [24], a lumped electrical model of fuel cell stack is developed
via thermodynamic and mechanical concepts (e.g., concentration, flow rate, temperature,
and pressure). The central part of this modeling is Nernst’s equation (Equation 2.2) in
which reversible potentials of fuel cells are expressed in terms of reactant effective partial
pressures and operating temperatures. E0 is the constant value of this voltage and it is
called to as reference potential.
( )220 ln OHcell ppEE ⋅⋅Τ⋅Κ+= (2.2)
Next, voltage drops due to activation losses, ohmic losses, and concentration
losses are expressed in terms of current and operating temperatures. The most important
part of the model in determining dynamic characteristics of fuel cells is the double-layer
charging effect. As the name suggests, two charged layers (cathode and anode) of
opposite polarity are formed that allow charges to accumulate.
21
Even though the end result of this type of modeling is an electrical circuit, there
are a few significant practical issues with this procedure. The first issue is the
requirement of design parameters which are unavailable due to proprietary nature of this
information. This leaves no choice for the power converter designers but to
experimentally extract these parameters. For this purpose, frequency response analyses
technique can be utilized to develop a fuel cell equivalent circuit [23], [25]. This is
reported to be an effective modeling method; however it requires a Frequency-Response-
Analyzer and a programmable-electronic-load. The second issue is that the effect of fuel
cell control module is not taken into consideration.
Fuel-Cell-Systems incorporate a control module in which the input fuel and
oxygen are adjusted continuously according to the output load. This process (varying the
oxygen and hydrogen flow rates) will modify the polarization curve (steady-state V-I
characteristic of fuel cell) which introduces additional nonlinearities between the output
voltage and current. The third issue is the omission of auxiliary loads (e.g., compressor,
fan, control board) from the model. The current drawn by the compressor depends on the
value of output load current and its responds to the load variation with some delay. On
the other hand, the current drawn by the fan is temperature dependent and its delay is
relatively long. Next, a much simpler method will be presented that looks at the step
response of the system rather than its frequency response. The satisfactory results
indicate that fuel cell may be considered or modeled as a linear system at an operating
point of interest. The operation of the compressor is also taken into consideration in this
model.
22
2.6 Steady State Characteristics of the Fuel Cell System
The experimental setup shown in Figure 2.7 was used to obtain the steady state
characteristics of the fuel cell system.
Experimental Setup-A: Steady-State Characterization
Figure 2.7 Experimental setup-A to obtain steady state voltage versus stack current
• Loads at the range of 1.6 – 43.4 Amps were applied to terminals of the FC system • Measurements: (A0, A1, A2, V1) • Steady-state V-I characteristic • Falling-time of the output voltage (with rising of load current) • Rising-time of the output voltage (with falling of the load current) • Fuel consumption at each load read from the fuel cell system interface exaMon
OEM 2.0)
Examples of captured waveforms from Experiment-A with 43.38 Amps load is
shown in Figure 2.8 and Figure 2.9.
23
Figure 2.8 Applying 43.38 Amps load
Figure 2.9 Disconnecting 43.38 Amps load
As shown in Figure 2.10 the steady-state output voltage versus the output load
current of the NexaTM power module was experimentally determined with load
resistances 0.7 to 66.5 ohms. The slope of this curve represents the value of the series
resistor in steady-state conditions. Figure 2.11 shows the dependency of current drawn
by auxiliary devices on the output load which is close to linear.
24
0 5 10 15 20 25 30 35 40 4526
28
30
32
34
36
38
40
42
Fuel-Cell-Stack Current (A)
Fuel
-Cel
l-Sta
ck V
olta
ge (V
)
Figure 2.10 V-I Characteristic of stack (polarization curve)
0 5 10 15 20 25 30 35 40 450.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
Fuel-Cell-Stack Current (A)
Mea
n C
urre
nt S
uppl
ied
to A
uxili
ary
Dev
ices
(A
)
Figure 2.11 Current supplied to auxiliary devices versus load
25
The experimental setup shown in Figure 2.12 was used to obtain the dynamic
characteristics of the fuel cell system.
Experimental Setup-B: Transient Characterization
Figure 2.12 Experimental setup-B to study transient stack voltage
• Load ripple at different frequencies was added on top of a DC current • Simultaneous voltage and current profiles of stack (V1) and auxiliary
devices was captured for analysis • Fuel consumption at each load was recorded from the fuel cell system interface
NexaMon OEM 2.0)
Examples of captured waveforms from Experiment-B with a 25% ripple on top of
a 30 Amps DC current at 1 Hz is shown in Figure 2.13 and Figure 2.14.
26
Figure 2.13 Ripple (25%) on top of 30A DC at 1 Hz
Figure 2.14 One cycle from the above waveform
27
An experimental setup in Figure 2.12 consisting of a chopper load and a resistive
load was used in order to obtain the transient response of the fuel-cell-system. In this
experiment, the fuel-cell-system response is examined under various load static and
dynamics (dc component, ripple frequency, peak-to-peak value).
2.7 Linear Network Approximation of the Fuel Cell System
A fuel-cell-system model was developed shown in Figure 2.14. Active and
passive components were integrated into the model to describe the essential structure of
the system so that the significant characteristics of its performance can be adequately
represented. The stack model reported in [19] and [24] was employed and parameters of
the model were extracted from experimentally obtained waveforms (Experiment–A and –
B). This procedure is explained in Section 2.8. The compressor is modeled as a current
controlled current source in parallel with a constant current source, I0 (no-load
compressor current). As the load increases the current drawn by the compressor will also
increase. It is important to note that the compressor draws a pulsed current that adds a
parasitic load on the overall system. Only the average value of this current is considered
in the model. All the delays and the current sensor filter are approximated as first order
delays (R1C1, R2C2, and R3C3). K2 and K3 are the controlled source gains and K1 is
the sensor gain.
28
R3
C3 C1
is
+-
Compressor
K2+-
Current SensorK1
E
R-act
ci
R-ohmic
R1
C
0
ti
tvR2
C2
Control Loop
+-
K3
Reaction Regulation
0Io
+
-
R-conc
Figure 2.15 Linear network approximation of the fuel-cell-system
Ract, Rconc, Rohmic, C network is the stack model reported in [19], [24]. E, K3, R3,
C3 represnet the internal voltage in Equation 2.1. Part of this voltage depends on the
concentrations of reactions. Current drawn by the compressor is a good measure of how
reactant concentrations changes. K1, R1, C1 network represents the sensor filter. K2, R2,
C3 network models the operation of the compressor. Compressor can be modeled as
current controlled current source based on Figure 2.11. K2 denotes that linear
relationship between the current draw by the compressor and the stack current. R1, C1
represent the delay associated with the compressor in responding to changes in stack
current.
For control system design purposes some assumptions can be made to reduce the
complexity of the model in Figure 2.15. It is assumed that the internal voltage is constant
and does not change with reactant concentrations. Also the sensor filter is neglected.
With these assumptions the linear network in Figure 2.15 is reduced to the one in Figure
2.16.
29
Figure 2.16 Linear network approximation of the fuel-cell-system (simplified)
2.8 Parameter Extraction
First Rohmic is determined based on the step responses shown in Figures 17-19.
The voltage jump in the waveforms is related to Rohmic. Rohmic is the ratio between the
voltage jump and the value of step current. This is because at the moment that the load is
applied voltage across C can not change instantaneously. Therefore any voltage drop will
occur across Rohmic. To find Ract + Rconc we let the system go to steady state so that the
capacitor can be considered as open. We also know the no-load voltage of the stack,
output voltage of the system (with load), stack current, and Rohmic that was found in the
pervious step. Hence using KVL we obtain the value of Ract + Rconc. Double layer
charging effect (C) can be obtained from the time constant associated with the voltage
rise in the step response (Figures 2.17 – 2.20). K1.K2 is simply the ratio between Ic and
Is. R2, C2 network represents the delay between a change in stack load and compressor
response. This can be obtained by additional experiments (by examining the response of
the compressor to a step change in stack current) or by trial and error to match the
is
+-
Compressor
K2+-
Current SensorK1
E1
R-act
ci
R-ohmic
0
C
t i
t v R2
C2
0
Io
-
+R-conc
v
Rb Ra
- +
30
experimental waveforms with the simulated ones. Results from parameter extraction for
NexaTM Power Module are shown in Table-2.1 (at the operating point of 30A stack
current). It is important to note that the accuracy of the parameter extraction depends on
the amount of noise present in the measurement.
Table 2.1 Results from parameter extraction for NexaTM Power Module
FC 04.0= Ω=+ 55.0ba RR 1.021 =kk FCR Ω= 1.022 VE 5.40=
2.9 Model Validation
Simulation results from this model were compared with waveforms obtained
from the experimental setup B. The linear network representation captures the essential
features of the fuel-cell-system, as shown in Figure 2.17 to Figure 2.20. Red lines show
the SIMULINK results of the model. Blue lines show the experimental results that were
obtained from the NexaTM system. Simulation and experimental results match well. It is
important to note that Ract, Rconc, and Rohmic are current and temperature dependant,
therefore the fuel cell system model is most accurate at the operating point of interest (dc
component of stack current at the operation temperature).
31
Figure 2.17 Experimental and simulation results at 1 Hz load ripple
Figure 2.18 Experimental and simulation results at 10 Hz load ripple
32
Figure 2.19 Experimental and simulation results at 50 Hz load ripple
Figure 2.20 Experimental and simulation results at 100 Hz load ripple
33
CHAPTER 3
CONTROL SYSTEM DESIGN FOR FUEL CELL BASED BOOST CONVERTER
3.1 Selection of a suitable converter for fuel cell applications
Output voltage of fuel cell systems are low however most adjustable speed motor
drives and appliances require 200-500V dc or ac voltage to operate. A power electronics
converter is required in order to transform the low dc output voltage of fuel cells to a
desirable high dc voltage. A typical fuel cell based power converter has two parts: First
part is a dc/dc converter, which converts the variable low dc output voltage of the fuel
cells to a regulated high dc voltage. The second part consists of either a battery and/or an
ultra-capacitor to improve the output dynamic response and also serve as an energy
storage backup. In selection of a DC-DC converter cost, efficiency, output characteristics
of fuel cells are of main considerations. Boost converters possess several important
features that make them attractive in fuel cell applications. A boost converter draws
continuous input current due to its input inductor. As stated in Chapter 1 this feature of
boost converter is significant because high current ripple leads to mechanical stresses,
poor fuel utilization, and reduced life span in fuel cell systems. In this chapter a boost
converter, shown in Figure 3.1, will be used to demonstrate a control design for fuel
based power converters.
34
Figure 3.1 Boost converter to step up the output voltage of the fuel cell system
3.2 Small-signal equivalent circuit model for the fuel cell based boost converter
An equivalent circuit model for the fuel cell system was derived in Chapter 3 that
consisted of linear elements. Next an averaged small signal model will be derived for the
fuel cell system – converter arrangement shown in Figure 3.2. The ultimate analysis
objective is to obtain a transfer function for the control-to-output, line-to-output, and load-to-
output (output impedance). This will allow for frequency domain control design methods.
Figure 3.2 Circuit based model of the fuel cell based boost converter
35
State-space averaging method:
The averaging method approximates the time-variant non-linear boost converter
as a time-invariant linear system [26]. The boost converter shown is a second order
system; however fuel cell system is also represented with a second order system. This
will make the order of the entire system as forth order. State equations of the system are
given below.
⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
−⋅
−=
⋅+−⋅−−=
++⋅−
−+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+−=
+⋅
−=
oL
o
o
L
o
Lo
oos
bL
oL
sbs
s
a
CRv
CDi
Ci
dtdv
LDv
Lvi
LR
Lv
LE
dtdi
LE
CRIv
LD
CRii
LR
CRkk
Lv
dtdi
Ci
RCv
dtdv
22
0
2222
21 )1(1
(3.1)
Small-signal linearization by Taylor series approximation:
The average model obtained in the pervious step is a non-linear representation of
the system. This is because the duty ratio, D, is multiplied by state variables and
therefore results in a multiplicative non-linearity [27]. For this reason a linearization step
is required which will unfortunately restrict the validity of the resulting model to a single
operating point. This local model can be developed via a common method in which
Taylor series approximation is utilized [28].
36
These definitions apply: x is the state vector, u is the input vector, y is the output
vector, and (X, U) is the modeling point of interest. Terms with tilde represents perturbed
variables about the operating point.
),(~~~~~~
),(),(
UXatuDxCyuBxAx
onAproximatiseriesTaylorbyionlinearizat
uxhyuxFx
+=+=
== &&
(3.2)
The above approximation results in a model that is linear in terms of x~ andu~ .
Then based on this a number of important transfer functions can be derived to examine
the effects of variation of variables of interest on the output of the system. Any variable
can be considered as an input or an output depending on our choice of u and y. Here, it is
desired to obtain transfer functions for )(~)(~ sdsvo and )(~)(~ sisv oo at the operating point of
interest (Vo = converter output voltage, D = duty cycle, IL = inductor current, Is = stack
current, V=voltage across double layer capacitive effect) which are obtained from steady-
state model of the system and design specifications. Below is a list of equations which
was used to obtain the operating point of interest, where Vo and RL (load resistance) are
from design specifications and fuel cell parameters were obtained in Chapter 2.
L
oL R
VD
I ⋅−
=1
1 (3.3)
( ) L
ooLos RD
kVIkIII−
+=+=1
(3.4)
( )( )( ) RkRD
RDRIEVL
Loo +−
−−= 21
1 (3.5)
37
L
oL R
VD
I ⋅−
=1
1 (3.6)
)( bas RRIV += (3.7)
For cascaded control implementation, instead of )(~)(~ sdsvo we are interested
in )(~)(~ sdsiL and )(~)(~ sisv Lo . This is because it is desired to control the inductor current
in order to prevent fast current variations at the fuel cell output. These definitions apply:
x1=v, x2=is, x3=iL, x4=vo. For )(~)(~ sdsiL the state space matrices are as follows, where
u=D, and y=x3:
( )
( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−−−−
−−−
−−
⋅−
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
Loo
b
b
a
RCCu
Lu
LR
L
Lu
CRLR
CRkk
L
CRC
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
A
1100
101
1111
0011
2222
21
4
4
3
4
2
4
1
4
4
3
3
3
2
3
1
3
4
2
3
2
2
2
1
2
4
1
3
1
2
1
1
1
(3.8)
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⋅−
⋅
⋅
=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂∂∂∂∂∂∂
=
o
L
o
o
o CIL
VL
V
xC
xL
xL
ufufufuf
B
0
1
1
10
3
4
4
4
3
2
1
(3.9)
[ ]01004321
=⎥⎦
⎤⎢⎣
⎡∂∂
∂∂
∂∂
∂∂
=xy
xy
xy
xyCT (3.10)
38
To obtain state space matrices for line-to-output transfer function, )(~)(~ sisv oo , an
extra step is required to decouple vo and io. In other words the linear relationship between
vo and io that is imposed by RL must be disregarded to be able to examine the independent
variation of load and its effect on the output of the system. This is accomplished by
replacing oLo CRv in last statement of Equation 3.1 by oo Ci . For )(~)(~ sisv oo the state
space matrices are the following, where u=io, and y=x4:
( )
( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−−−
−−−
−−
⋅−
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
0100
101
1111
0011
2222
21
4
4
3
4
2
4
1
4
4
3
3
3
2
3
1
3
4
2
3
2
2
2
1
2
4
1
3
1
2
1
1
1
o
b
b
a
CD
LD
LR
L
LD
CRLR
CRkk
L
CRC
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
xf
A (3.11)
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂∂∂∂∂∂∂∂
=
oC
ufufufuf
B1
000
4
3
2
1
(3.12)
[ ]10004321
=⎥⎦
⎤⎢⎣
⎡∂∂
∂∂
∂∂
∂∂
=xy
xy
xy
xyCT (3.13)
39
Derivation of transfer functions from linearized state space representation:
Transfer functions of )(~)(~ sdsiL and )(~)(~ sisv oo are obtained by substituting sate
space matrices into: ( ) ( ) BAsICsH T 1−−= (3.14)
The transfer function of )(~)(~ sisv Lo is obtained directly from the state equations
in (3.1) since it is already linear. These transfer functions are presented here and will be
used in the next section for control system design [29]. The parameter values and
operating points in Table-3.1 were used. The other the operating points such as D, IL, Is,
and Io are given by Equations 3.3 – 3.7.
Table 3.1 Parameters and operation points used in deriving transfer functions
FC 04.0= Ω= 45.0aR Ω= 1.0bR FCR Ω= 1.022 HL 610140 −×= FCo
610470 −×= Ω= 40LR 1.021 =kk VE 5.40= VVo 200=
87234
10102836
109914.31007429.5777.83341803276.8321059780.71005239.11044178.21042857.1)(~)(~
×+×+++×+×+×+×
=ssss
ssssdsiL
19.53638.410)(~)(~
+=
ssisv Lo
10927344
11102735
103303.1108202.1107224.3106605.347101825.2102590.2107884.7101)(~)(~
×+×+×+×+×−×−×−×−
=ssss
ssssisv oo
10927344
10928
103303.1108202.1107224.3106605.347108929.6108994.8103786.1)(~)(~
×+×+×+×+×+×+×
=ssss
sssisi oL
40
3.3 Control system design
The control system for the boost converter of the fuel cell is designed to meet two
main control objectives. One objective is to regulate the converter output voltage.
Simultaneously, it is necessary to control the input inductor current to avoid crossing the
current ripple limitations of the fuel cell system. As mentioned before crossing the
limitation will results in mechanical stresses and inefficient fuel utilization. To
accomplish these objectives a cascaded controller is used with an outer voltage control
loop (PI compensator) and an inner current control loop (integral compensator). This
configuration is shown in Figure 3.3 in analog implementation.
R-i
+3
-2
OUT1
R-v 1
1 2L1
C-v 1
R1
R-v 2
D1
+5
-6
OUT
C-i
M1
v-ref
0
Boost Converter
Fuel
Cel
l Sy
stem
Hv Hi
PWM Generator & Gate Drivers
i-ref
Load
v-sense
i-sense
Current CompensatorVoltage Compensator
Figure 3.3 Analog implementation of the cascaded control system
41
Bode plots of gain loops ( diL~~ & Lo iv ~~ ) were used to analyze stability of the
system. Nyquist’s criterion of sufficient phase margin is applied to establish stability.
Another important criterion in frequency domain design is to provide sufficient
bandwidth in order to speed up the response of the system to disturbances. But,
bandwidth can not be increased indefinitely because it will lead to noise amplification
and thereby system will become unstable. Here, limiting the bandwidth will serve
another key objective which is to reduce the current ripple of the fuel cell system. A
detailed description of the current ripple frequencies that need to be attenuated is given in
Chapter 2, Section 2.4. In summery, a range of frequencies approximately between
Hz5.0 and kHz10 should be avoided since the fuel cell system is negatively affected by
load variation in this range. Here the term avoiding refers to the requirement to attenuate
current ripple at a particular frequency.
Bode plots of )(~)(~ sdsiL and )(~)(~ sisv Lo satisfy the Nyquist’s stability
criterion and also the bandwidth requirements of the design, shown in Figure 3.4 and
Figure 3.5 respectively. As it is evident from Figure 3.4, the response of the inductor
current is restricted to control signals of no more than 1Hz. This will ensure that any
current ripple or disturbance at the load side is filtered out from the fuel cell output.
42
-120
-100
-80
-60
-40
-20
0
20
Mag
nitu
de (d
B)
100 101 102 103 104 105-180
-135
-90
-45
0
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
Figure 3.4 Bode plot for )(~)(~ sdsiL with an integral compensator
10
20
30
40
50
60
70
80
Mag
nitu
de (d
B)
10-1 100 101 102 103-90
-60
-30
Phas
e (d
eg)
Bode Diagram
Frequency (rad/sec)
Figure 3.5 Bode plot for )(~)(~ sisv Lo with a proportional integral compensator
43
The Simulink block diagram of the fuel cell based boost converter is shown
below. It is necessary to simulate the effects of load variation on the output voltage and
ultimately on the inductor current. The output impedance, )(~)(~ sisv oo , gives the
relationship between the output current and the output voltage. Therefore, any variation
in the output voltage caused by the load current can be obtained via the transfer
function, )(~)(~ sisv oo . The effect of the load disturbance on the inductor current (fuel cell
output current) is of main interest. One can also directly use the transfer function
between the load current and the inductor current, )(~)(~ sisi oL , to analyze the effect of an
output load disturbance on the inductor current in an the open loop system. This can
become very useful in ultra-capacitor design.
Figure 3.6 Simulink diagram of the fuel cell based boost converter under load disturbance
44
3.4 Performance verification of the controller
Next it is verified that the controller meets both of the control objectives, namely,
output voltage regulation and current disturbance rejection at the fuel cell. To establish
this, sinusoidal load current disturbances of magnitude 1 Amp with different frequencies
were applied to the system. It is shown that only attenuated versions of high frequency
load current ripples appear in the inductor current. In other words, control action serves
as a low pass filter and attenuates load disturbances of higher frequencies. Also
regulation is achieved at the same time but with a long delay and huge overshoot. The
long delay and the overshoot is a consequence of the compromise that was made to
control the speed at which the inductor current changes in response to load variations.
The overshoot can be reduced to an acceptable amount with utilization an ultra-capacitor
that keeps the voltage steady while supplies the transient loads. The results for sinusoidal
load current disturbances of magnitude 1 Amp with frequencies 1000, 10, and 1 Hz is
shown in figures below. All disturbances start from t=5sec and last till t=10.
Comparison of the open loop and closed loop responses to disturbances clarifies the
performance of the controller. From Figure 3.7 it is seen that even though the regulation
takes place but the overshoot is enormous. This highlights the necessity for an ultra-
capacitor that can supply the transient currents and keep the voltage steady.
45
0 5 10 15 20-80
-60
-40
-20
0
20
40
time (sec)
Out
put V
olta
ge (v
olt)
Figure 3.7 Response of output voltage to a 5 Amps step disturbance in load
46
0 5 10 15 20 25 30-8
-6
-4
-2
0
2
4
6
8x 10-4
time (sec)
Figure 3.8 Response of inductor current to load current variation of [ )..21000sin(0.1 tπ× at 5 < t < 15]
CLOSED LOOP RESPONSE
14 14.005 14.01 14.015 14.02 14.025 14.03 14.035 14.04 14.045 14.05-1.5
-1
-0.5
0
0.5
1
1.5x 10-5
time (sec)
Figure 3.9 ZOOMED Fig.3.8
47
1 1.02 1.04 1.06 1.08 1.1 1.12-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
time (sec)
Figure 3.10 Response of inductor current to load current variation of [ )..21000sin(0.1 tπ× at 1 < t < 1.1]
OPEN LOOP RESPONSE
0 5 10 15 20 25 30-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
time (sec)
Figure 3.11 Response of inductor current to load current variation of [ )210sin(0.1 tπ× at 5 < t < 15]
CLOSED LOOP RESPONSE
48
11 11.5 12 12.5 13 13.5 14-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
time (sec)
Figure 3.12 ZOOMED Figure 3.14
0.8 1 1.2 1.4 1.6 1.8 2 2.2-5
-4
-3
-2
-1
0
1
2
3
4
5
time (sec)
Figure 3.13 Response of inductor current to load current variation of [ )210sin(0.1 tπ× at 1
< t < 2]
OPEN LOOP RESPONSE
49
0 2 4 6 8 10 12 14 16 18-1
-0.5
0
0.5
1
1.5
time (s)
Figure 3.14 Response of inductor current to load current variation of [ )21sin(0.1 tπ× at 1 < t < 10]
CLOSED LOOP RESPONSE
0 2 4 6 8 10-6
-4
-2
0
2
4
6
time (sec)
Figure 3.15 Response of inductor current to load current variation of [ )21sin(0.1 tπ× at 1 < t < 10]
OPEN LOOP RESPONSE
50
CHAPTER 4
AN INTUITIVE THERMAL-ANALYSIS TECHNIQUE FOR SWITCH MODE POWER CONVERTERS
4.1 Efficient converter design for fuel cell applications
Efficiency is one the hallmarks of switch-mode power supply design. With the
immergence of alternative energy sources, high efficiency, high power density, and
reliability have become indispensable design goals for high power converters. To
achieve these objectives, an assessment of the thermal performance of converters is
necessary. Losses in semiconductor devices constitutes the most important and complex
part of losses in power electronics circuits. These losses have two aspects, one is
determined by power semiconductor device characteristics, and the other is determined
by the converter topologies. Power semiconductor manufacturers continue to improve
on-resistance and dynamic performance of their products which helps to reduce losses.
On the other hand, power electronics engineers utilize techniques such as ZVS and ZCS
to make the best use of available devices in terms of reducing power losses and device
power stresses, and adding reliability.
In this chapter, a brief overview will be given on some of the tools and methods
for thermal analysis in power devices. Then, a straight forward analytical method will be
discussed, that will achieving two objectives. First, it provides an estimation of power
losses and power stresses in power devices. Second, it points out to factors that influence
51
power losses and power stresses in a converter circuit. One of the strong points of this
method is gaining an intuition on how device characteristics and circuit topology each
contributes to power losses in power electronics circuits. With this knowledge one can
determine the available degrees of freedom to reduce these losses. Also a straightforward
graphical technique is introduced in order to obtain dynamic models for MOSFETs
without a need to use MOSFET equations which makes deriving loss equations even
more complicated. The main shortcoming of this method is the accuracy that is limited
by the use of graphical technique and uncertainties regarding the information in
datasheets. Here, hard switching boost converter with a MOSFET device is used as an
example and a similar approach can be extended to other power electronics circuit
topologies.
4.2 Power electronics converter thermal performance – Simulation Tools
Designers of power semiconductor devices use process and device simulators,
such as TCAD synopsys®, to perform FEA on the basis of equations from semiconductor
device theory. The simulation inputs are physical device parameters such as doping
profiles, impurity concentrations, and layer thicknesses. Various electrical and thermal
analyses can be performed under different conditions. These types of tools are utilized at
first development time before device fabrication. [32] The simulation results generally
have errors of several orders of magnitude. Therefore in developing device datasheets,
parameter extraction is accomplished by experimental setups rather than simulation tools.
Parameter extraction can also be performed by end users, however, due to proprietary
52
nature of some structural parameters (e.g., channel length and width), some rules of
thumb are necessary. [34]
Next SPICE models are developed based on electrical parameters, extracted from
pervious step. The problem in developing SPICE models for Power MOSFET is the
limitations of the standard low-voltage level-1, 2, and 3 models. These models were
originally intended for small signal lateral MOSFETs. Due to structural differences
between small signal FETs and large geometry vertical FETs, the model is unable to
accurately simulate the power behavior of power MOSFETs, especially nonlinear device
capacitances [31], [33]. To overcome this problem, over the years, many power MOSFET
models have been developed by manufacturers, as well as end users. In recent years,
thermal models are also provided by some manufacturers that can be coupled with the
electrical model in order to take into account the effects of temperature variation on some
parameters such as carrier mobility.
SPICE tools allow for estimation of power losses and junction temperatures of
power devices in circuits and yield acceptable results if provided with a sufficiently
accurate model. However they have several disadvantages. First, model validation is
necessary to be certain that the model is acceptable [31], [34]. This requires some
experience and is time consuming. Second, after power losses are obtained, it provides
the circuit designer with a little idea on how the power losses are related to voltages,
currents, and circuit parameters in the converter circuit. This is because no analytical
expression is obtained from simulations.
53
4.3 Power electronics converter thermal performance – Analytical Methods
In analytical method loss equations are derived based on circuit analysis, and
MOSFET models (e.g., quadratic, linear). Drain current, Id, and drain-to-source voltage,
Vds, are expressed in terms of circuit and MOSFET currents, voltages, and parameters. In
this method either extensive MOSFET models or simpler ones can be used based on how
much accuracy is required. Reference [34] is an example of an analytical method where
important parasitic effects have been taken into account. For a typical application, this
method yields an acceptable estimation of power losses and also enables a designer to
determine ways to reduced losses in a converter circuit. This is possible because
analytical equations are available with all the variables and parameters that influence
power losses. However this approach requires a MOSFET equation to relate Vgs, Vds, and
Id. This makes the process of deriving power loss equations an enormous task. In this
chapter a straightforward method will be demonstrated in which power loss equations
will be derived with the help of datasheet curves rather than MOSFET equations. This
will simplify the task of deriving equations for Id and Vds and provides insight into how
device characteristics, deriver circuit, and topology influence losses and power stresses.
54
4.4 Topology under consideration
Figure 4.1 Hard-switching boost converter
A boost converter is shown in Figure 4.1 with a fluctuating input voltage of 22-
60V and a regulated output voltage of 200V (1±1 %). This topology has two
semiconductor devices. To analyze this circuit, understanding physics of these devices is
not necessary. A circuit based models are sufficient in order to conduct a reasonably
accurate analysis of power electronics circuits. Models are necessary in order to perform
a variety of analysis such as the calculation and estimation of losses, efficiency, junction
temperature, and device power stresses, thereby enabling the designer to optimize their
designs.
4.5 Obtaining dynamic model of MOSFET from datasheet
Every semiconductor manufacturer provides datasheets with detailed
explanations of their device characteristics. It is necessary to determine which
parameters influence the thermal performance and power losses in a device. In addition,
55
power losses are not limited to device characteristics, and are also greatly influenced by
the choice of topology. Power losses in semiconductor devices can be classified as two
types: conduction losses which occur in the saturation and cutoff regions, and switching
losses which occur in the commutation regions. The following discussion will consider
deciding factors that influence power losses in each of the operating regions.
Power loss in the cutoff region: In this region the power loss is caused by the
leakage current and voltage across a MOSFET device (drain-to-source). In this region
both the MOSFET and diode can be modeled as resistors. Values of these resistances can
be obtained by considering the worst case scenario when the junction temperature is at
125°C. In the MOSFET, when VVGS 0= , we have VVDS 400= and uAIDSS 1000= [36],
therefore the equivalent resistance is Ω= kRDSS 400 . In the case of diode, for reverse voltage
we have VVRM 400= and for reverse leakage current we have uAIR 1000= [37], therefore the
reverse equivalent resistance is Ω= kRR 400 . Due to the high values of these resistances,
these devices are considered open circuits when operated in these regions.
Power loss in the saturation region: In this region the power loss is caused by the
current through a MOSFET device and forward voltage across it (drain-to-source).
Device characteristics and the current will determine the magnitude of this loss. In this
region power MOSFETs can be modeled as a small resistor and diodes can be models as
a small dc-voltage source. The value for this resistor is obtained by considering the worst
case scenario when the junction temperature is at 125°C. When the gate voltage is
VVV GS 1020 ≥≥ , it can be obtained from the datasheet that the equivalent resistance is
56
Ω=− 076.0onDSR [36]. In the case of the diode, it is obtained from the datasheet that the
forward voltage is VVF 7.1= [37].
Power losses in the commutation regions: A MOSFET device has two
commutation regions, one is when the device goes from cutoff to saturation region and
the other is when the device goes from saturation to cutoff region. In commutation
regions (turning-on and off processes), voltages and currents exist simultaneously which
leads to major power losses. It is important to note that turning-on process is different for
different topologies. There are usually two cases when considering this transition. In the
first case, the current starts rising as the voltage simultaneously starts to drop. In the
second case the current starts rising even though the voltage is still fixed at the cutoff
region. Then the current must rise to the peripheral current before the voltage starts to
drop. In the turning-off process, the current starts to drop as the voltage starts to rise, and
then they reach their final values simultaneously. Turning-on and turning-off times
become particularly important when employing hard switching. This sets an inevitable
restriction in increasing the frequency of operation in hard switching applications.
4.6 MOSFET model via graphical transformation
A graphical method will be introduced to obtain an approximation for the drain
current verses time, Ids vs. t. Datasheets only provide graphs for the drain current verses
gate voltage, Ids vs. Vgs, which the manufacturers obtain from experiment. Therefore it is
clear that we do not have an exact mathematical equation for Ids vs. Vgs. It is also
important to note that one of the major tasks of a power electronics engineer is designing
a proper gate deriver circuit which determines Vgs vs. t. Therefore, it is essential to obtain
57
Vgs vs. t either analytically or experimentally. After obtaining Ids vs. Vgs and Vgs vs. t, Ids
vs. t is obtained graphically, shown in Figure 4.1. As it will soon be clear, even though
this method is not exact however it provides a reasonable view of the relationship
between the gate signal and drain current. The graph in Figure 4.1 consists of four
regions which are as follows:
Quadrant-I: Drain current versus gate voltage.
This is obtained from the datasheet [36].
Quadrant-II: Gate voltage versus time
Gate deriver signal, Vdr, can be considered as a pulse. When Vdr is applied, Vgs
does not reflect this voltage simultaneously. This is because of an RC circuit formed by
the deriver loop. Input capacitor, Ciss, must be charged with time constant issg CR ⋅ before
Vgs reaches Vdr. Accordingly, the Graph in region-II can be obtained by linearizing:
)1( / issg CRtdrgs eVV ⋅−−= .
Gate to source voltage can also be captured on an oscilloscope if linearization
step needs to be verified or avoided all together.
Quadrant-III: Effect of common source inductance
This inductance is the parasitic inductance mainly due the bonding wire. Its value
is related to the type of package and is usually in the range of 4 to 10 nH [33]-[34]. This
inductance increases the duration of commutation and its effect becomes more evident at
high frequencies [34]. This effect influences the waveforms of Id and Vds, but has little
effect on Vgs. When neglecting this effect, the slope should be one. As the slope is
58
decreased (counter-clockwise direction), larger commutation times are realized. Wire
inductances can also be represented by this curve.
Quadrant-IV: MOSFET drain current versus time
The outcome of this graphical transformation is the drain current as a function of
time.
Figure 4.2 Graphical transformation to obtain drain current vs. time [30]
Figure 4.2, demonstrates the turning-on case only. Drain current can be obtained
similarly for device turning-off region. Next, the waveforms of Ids vs. t are linearized to
obtain the constants, kdtdids ≈ and 'kdtdids −≈ , as the rising and falling slopes
respectively. Since the falling and rising rates of the drain current are comparable, we
can assume that their magnitudes are equal.
59
4.7 Equivalent circuit representation of hard switching boost converter
The analysis of hard-switching boost converter will be established. The
following equivalent circuits represent the behavior of the topology in Figure 4.1, at
different time intervals. In majority of the literatures, only cutoff and saturation intervals
are considered in analyzing the time varying circuits. Here a slightly different approach
is taken and commutation intervals are also incorporated in representing the time varying
circuits. The equivalent circuits are established by the following considerations:
Drain current as a function of time was obtained and then linearized in section IV.
This current is represented as a voltage controlled current source during the commutation
regions. Constant, k, was also obtained in section IV.
Drain-to-source voltage is determined by the topology since it is the voltage
across a current source.
Diode is modeled as a voltage source during conducting region. During turning-
off and turning-on regions diode is represented as a ramp voltage source while reaching
its final value.
Time intervals, ta, tb, tr, tf is shown in Figure 4.4 and will be obtained in sections
VIII & X.
60
Interval 1: turning on MOSFET 0<t<tr Interval 2: turning off diode tr<t<tr+ta
Interval 3: recovering diode tr+ta<t<tr+ta+tb; Interval 4: charging inductor tr+ta+tb<t<DT
Interval 5: turning off MOSFET DT<t<DT+tf Interval 6: Charging Cap DT<t<DT+tf
Figure 4.3 Equivalent circuit representation for the boost converter [30]
In Figure 4.4, the waveforms for MOSFET, diode, gate-to-source voltage, and
inductor current are shown. These waveforms are developed according to MOSFET and
diode datasheets and results from section IV. These waveforms indicate the time
intervals for equivalent circuits in Figure 4.3.
61
Figure 4.4 Waveforms that determine the time intervals of equivalent circuits [30]
Table 4.1 Nomenclature for Fig.4.4
IL1: Minimum inductor current IL2: Maximum inductor current D: Duty cycle T: Period Vin: Input Vo: Output voltage RL: Resistance load vds: Drain voltage of MOSFET
ids: Drain current of MOSFET tr: Raise time of MOSFET tf: Fall time of MOSFET RON: Turn-on VF: Forward voltage drop of diode Irr: Reverse peak current of diode trr: Reverse recovery time of diode Qrr: Reverse recovery charge
Diode waveform in Figure 4.4 is drawn according to [37]. Several characteristics
of diodes are important for the power loss and stress analysis that will follow. In most
power electronics applications, a diode acts as a capacitor during the turning-on and
turning-off intervals, except in tb interval. It is also important to note that the switching
losses only occur in tb interval of the reverse recovery [38]-[39]. Therefore, reducing the
switching time is essential for minimizing switching losses. However, this will have a
62
downside of creating huge voltage overshoots if the rate of reverse recovery, dIR/dt, is too
high. This voltage overshoot can cause failures in converter circuits and may also
destroy the diode.
4.8 Deriving equations for power losses and power stresses
For this analysis, the following assumptions are made:
• Only operation of the circuit in steady state is considered
• The output capacitor is large enough to hold the output voltage as constant
• The inductor and capacitor are considered to be ideal
• The inductor current is continuous
Switching and conduction losses of the MOSFET
a. Turn-on loss: This loss is represented in two parts. In the first
part )0( rtt ≤< the drain voltage remains constant (at the output voltage) while the drain
current rises (to reach the inductor current). In the second part )( arr tttt +≤< , the drain
current keeps rising (torrL II +1
) while the drain voltage drops to the saturation voltage.
Constant, k, is obtained from section-IV.
Drain current in the first part: ktids =1 )0( rtt ≤<
Drain voltage in the first part: ods Vv =1 )0( rtt ≤<
Drain current in the second part: ktids =2 )( arr tttt +≤<
Drain voltage in the second part: ( )ra
oods tt
tVVv −−=2
)( arr tttt +≤<
Average MOSFET turn-on power loss:
63
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −−+= ∫ ∫
+r ar
r
t tt
ta
rooQturnon ktdt
tttVktdtV
TP
011
633 22
aarro ttttT
kV ++×= (4.1)
at and rt are obtained as follows:
ktids = for )0( ar ttt +≤< , when rtt = , we have rLds ktIi == 1 ; therefore kIt Lr /1= .
Also, when ar ttt += , we have ( )arrrLds ttkIIi +=+= 1 , therefore kIt rra /= .
Substituting tr and ta in equation-1 yields:
( )kT
IIIIVP rrrrLLoQturnon 6
33 21
21 ++
= (4.2)
Approximation of turn-on power stress:
( )( )
( )6
36
33)/( 1
1
21
21 rrLo
rrL
rrrrLLoarQturnonstressQon
IIVII
IIIIVttPTP +≈
+++
=+=−
Turn-off loss: During the interval tf, the drain current drops to the leakage current
while the drain voltage increases to the output voltage. Moreover, we can assume that the
slope of the drain current is (–k).
Drain current: ktIi Lds −= 2 )0( ftt << ; Drain voltage: ttV
vf
ods = )0( ftt <<
Average MOSFET turn-off power loss:
( ) dtkttItV
TP ft
Lf
oQturnoff ∫
⎥⎥⎦
⎤
⎢⎢⎣
⎡−=
0
22
1⎟⎠⎞
⎜⎝⎛ −= 2
2 31
21
ffLo kttI
TV (4.3)
When ftt = , drain current 02 =−= fLds ktIi ; therefore kIt Lf /2= .
Substituting ft in equation-3 yields: kTIVP Lo
Qturnoff 6
22=
Approximate turn-off power stress: 6
/ 2LofQturnoffstressQoff
IVtPTP ==− (4.4)
64
Conduction loss: Conduction loss depends on the drain current and MOSFET
structure. This loss has to be calculated in order to determine the efficiency of the
system. Moreover, knowledge of this loss will enable designer to estimate the junction
temperature and design appropriate heat sink.
Drain current: tttDT
IIIiar
LLLds −−
−+= 12
1 )0( ar ttDTt −−<<
Average turning-off power loss of MOSFET:
dttttDT
IIIRT
P ar ttDT
ar
LLLonQon
2
012
11∫
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−+= ( ) onLLLL
ar RIIIIT
ttDT 2112
223
++−−
= (4.5)
Substituting kIt Lr /1= and kIt rra /= into the above equation:
( ) onLLLLrrL
Qon RIIIITk
IIDTkP 2112
22
1
3++
−−= (4.6)
Switching and conduction losses of the diode
a. Switching loss: As mentioned before, this loss only occurs during the tb interval.
Reverse recovery current: ttIi
b
rrrr −= )0( btt <<
Reverse voltage: oreverseD Vv −=− )0( btt <<
Average switching power loss of the diode: dtttIV
TP
b
rrt
oDswitchingb
⎟⎟⎠
⎞⎜⎜⎝
⎛−−= ∫0
1T
tIV brro
2= (4.7)
Reverse recovery charge: ( )2
barrrr
ttIQ += ,
arr
rrb t
IQt −=
2 , kIt rr
a = , Therefore: k
IIQ
t rr
rr
rrb −=
2
Substituting tb into Equation-7: ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
kIQ
TVP rr
rro
Dswitching 2
2 (4.8)
b. Conduction loss: This loss is due to the forward current and constant forward voltage.
65
Forward current: ttDTT
IIIif
LLLD −−
−+= 21
2; Forward voltage: DD Vv =
Average conduction power loss of diode:
dtttDTT
IIIVT
P ftDTT
f
LLLDDon ∫
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−−
+=0
212
1 ( )( )T
tDTTIIV fLLD
221 −−+
= (4.9)
Substituting kIt Lf /2= into Equation 4.9: ( )( )T
kIDTTIIVP LLLDDon 2
/221 −−+= (4.10)
Average inductor current
Now the average inductor current is calculating by taking the time average of
inductor current in each of the intervals indicated below.
Turning-on process of the MOSFET: this process occurs in three intervals( tr, ta,
tb ), but interval tr can be added to the interval that the diode is on ( T(1-D)-tf ), and
interval tb can be added to the interval that the MOSFET is on ( DT-tr-ta-tb ) in order to
simplify the calculation. Therefore turning-on process of the MOSFET only has the
interval ta and all the expressions need to be shifted by time, tr.
Refer to the Figure 4.3 Interval-2 topology ( )att <<0 .
⇒=−+dtdiLV
ttVV L
oa
oin( ) ( )⇒+
−+= 0
2)(
2
Loin
a
oL i
LtVV
LttVti ( ) ( ) ( )0
22
Laoin
aL iL
tVVti +−
=
Contribution of the average inductor current due to the interval ( )att <<0 :
dttiT
i at
LL ∫=01 )(1 ( ) ( )⎥
⎦
⎤⎢⎣
⎡+
−= 0
6231 2
Laaoin it
LtVV
T (4.11)
On-state of the MOSFET (including tb interval): in the interval tb, the drain
current of the MOSFET is the sum of reverse recovery current of the diode and the
66
inductor current. The voltage drop which is caused by the reverse recovery current also
influences the inductor current, but it is disregarded here because of the small values of tb
and Ron.
Refer to Figure 4.3 Interval-4 topology ( )ra tDTtt −<< :
⇒+= onLL
in RidtdiLV ( )⎥
⎦
⎤⎢⎣
⎡−
⎭⎬⎫
⎩⎨⎧ −−−= aL
on
in
on
a
on
inL ti
RV
RLtt
RVti
/exp)( ( ) ( ) ( )
⎥⎦⎤
⎢⎣⎡ −−+−≈
LttRtitt
LV aon
aLain 1
⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ −−≈
⎭⎬⎫
⎩⎨⎧ −−
on
a
on
a
RLtt
RLtt
/1
/exp ( ) ( ) ( ) ( )
⎥⎦⎤
⎢⎣⎡ −−−+−−=−
LttDTRtittDT
LVtDTi aron
aLarin
rL 1
Contribution of the average inductor current due to the interval ( )ra tDTtt −<< :
dttiT
i r
a
tDT
t LL ∫−
= )(12
( ) ( )[ ] ( ) ( )TL
tittDTLtiRVttDT aLaraLoninar
222 −−+−−−
= (4.12)
Turning-off process of MOSFET:
Refer to Fig4.3 Interval-5 topology ( )frr ttDTttDT +−<<− :
( )⇒=
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+⎥⎥⎦
⎤
⎢⎢⎣
⎡ −−−−−
dtdiLV
ttDTtVV L
of
roin 1 ( )
dtdiL
ttDTtVV L
f
roin =
−−−
( )[ ] ( )[ ]⇒−+
−−+
−−−= )(
2)(
2
rLinr
f
roL tDTi
LVtDTt
LttDTtVti )(
2)2(
)( rLoinf
frL tDTiL
VVtttDTi −+
−=+−
Contribution of the average inductor current due to the
interval ( )frr ttDTttDT +−<<− :
( )dttiT
i fr
r
ttDT
tDT LL ∫+−
−=
13
( ) ( )⎥⎥⎦
⎤
⎢⎢⎣
⎡−+
−= frL
foin ttDTiL
tVVT 6
31 2 (4.13)
Off-state of the MOSFET (including the ta interval):
Refer to Fig4.3 Interval-6 topology ( )TtttDT fr <<+− :
( ) ( ) ( ) ⇒−−=
+−−
+−−oDin
fr
frLL VVVttDTt
ttDTitiL ( ) ( ) ( )[ ] ( )frLfroDin
L ttDTiL
ttDTtVVVti +−+
+−−−−=
67
( ) ( ) ( )[ ] ( )frLfroDin
L ttDTiL
ttDTTVVVTi +−+
+−−−−=∴
Contribution of the average inductor current due to the interval ( )TtttDT fr <<+− :
( )dttiT
iT
ttDT LLfr
∫ +−=
14
( ) ( )[ ] ( )( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+−+−++−−−−
= frfrLfroDin ttDTTttDTi
LttDTTVVV
T 21 2
(4.14)
According to the above expressions for the inductor current, two approximations
are made:
the minimum inductor current, ( )01 LL iI ≈
the maximum inductor current, ( )rLL tDTiI −≈2
Summery of equations to be solved by a numerical method:
Average inductor current: 4321 LLLLL iiiii +++= (4.15)
Inductor continuous current condition: ( ) ( ) 00 ≠= Tii LL (4.16)
Power Balance equation: DonDswitchingQonQturnoffQturnon
L
oLin PPPPP
RViV +++++=
2 . (4.17)
68
4.9. Numerical method
Start
δ = iL step sizeε = D step size
For f=25k to f=100k
iL(0)=0
D=0
D=0+δ
Solve for: iL(T)
iL(0)=iL(T)
iL(0)=iL(0)+ε
Solve for: avg. iL, iLmax, iLmin, Pin, Ploss for all devices
D<1 N
Y
Pin>Po+PlossN
Y
∆P=Pin -(Po+Ploss)
∆P < β
END
N
Y
N
reduce step-size
Figure 4.5 Numerical method flow chart
Numerical method is implemented to obtain the power losses in the
semiconductor devices (MOSFET and diode). All the necessary equations were derived
in section-VIII. Power loss equations are in terms of the duty cycle, and minimum and
maximum inductor current. In order to solve for the unknowns, power balance equation
is used that requires the average inductor current. In section-VIII the average inductor
current was derived. In addition to these equations, the continuous condition for inductor
current is also needed. Figure 4.5 depicts a numerical method that was implemented to
69
solve all these equations simultaneously. First guess values are set for IL(0) and D.
Other guess values can be set based these two guess values according to the
corresponding equations. Power loss values are continuously tested in the power balance
equation and if this equation is satisfied then the solution is reached. This can be
repeated for different operating frequency by utilizing a loop.
4.10. Comparison of analytical result and hardware-test result
For calculations, the following parameters were used:
L=140μ Vin=60V Vo=200V
RL=66.5Ω K=200A/μ.sec
Irr=2x4.6 A
Qrr=100nC Ron=76mΩ Vd=1.7V
20 40 60 80 100 120 1400.955
0.96
0.965
0.97
0.975
0.98
0.985
0.99
0.995
Frequency (kHz)
Efic
ienc
y (%
)
Hardware TestAnalytical Method
Figure 4.6 Analytical method and hardware test results – hard switching topology [30]
Figure 4.6 shows the efficiency of the hard switching converter, which is
obtained from the analytic method and hardware tests. The graph for the calculated
efficiency is a linear function of the frequency. This is because nonlinear effects of
frequency on the power loss are not considered in our analytic method. These effects
show up on the graph obtained from the hardware tests. Based on the information
70
represented on the above graph, a designer is able to estimate losses for each of the
semiconductor devices for the purpose of determining the efficiency of power electronics
converters. Also this information can be used in selecting a heat sink with an appropriate
thermal resistance so as to avoid crossing the maximum allowed junction temperature.
CONCLUSION AND FUTURE WORK
In fuel cell applications, to design an optimized power electronics converter, it is crucial
to consider the output electrical characteristics of the fuel cell system. A methodical way
to accomplish this is to develop a suitable model for the fuel cell system. Such a model
was developed and validated in this thesis. It is shown that this model accurately
simulates the electrical output characteristics of fuel cell systems. Using this model, a
controller design for fuel cell power conditioner is successfully demonstrated. Future
work in required to more accurately extract the fuel cell system parameters. One problem
is the measurement noise that for the most part is due to pulsed currents drawn by the
compressor.
71
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75
BIBLIOGRAPHICAL INFORMATION
Yashar Kenarangui was born in Des Moines, Iowa. He lives in Arlington, Texas
since 1995 where he went to high school. He received his Bachelors degree in Electrical
Engineering in 2004 from the University of Texas at Arlington. He is currently pursing a
Masters degree in Electrical Engineering also at UTA. His areas of interests include
circuit design, renewable energies, and semiconductor devices such as photovoltaic cells
and detectors.
