Linearly-Acting Variable-Reluctance Generator for
Thermoacoustic Applications
MASS,
by
CHUSETTS INSTJTULf
OF T c'
AUG 15 2014
Philip Clinton Knodel
B.S., United States Air Force Academy (2012)
LIBRARIES
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
......................
Department of Mechanical Engineering
May 9, 2014
A uthor ...
Signature redacted
Certified by....
Signature redacted
John G. Brisson
Professor
Thesis Supervisor
....................
David E. Hardt
Chairman, Department Committee on Graduate Theses
Accepted by ....
This work is sponsored by the Department of the Air Force under Air Force Contract
#FA8721-05-C-0002. Opinions, interpretations, conclusions and recommendations are those of the
author and are not necessarily endorsed by the United States Government.
2
Linearly-Acting Variable-Reluctance Generator for
Thermoacoustic Applications
by
Philip Clinton Knodel
Submitted to the Department of Mechanical Engineering
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
Advances in battlefield equipment have created a demand for portable power systems
with greater power density and more flexibility than current battery sources alone
can provide. One potential solution lies in portable, high-frequency thermoacoustic
engines, which can provide a battery recharge station or direct power supply for
soldiers in-field, with the flexibility of operating on any quality heat source such as a
butane heater. In this work, a linearly-acting variable-reluctance generator (VRG) is
developed to act as a high-frequency (250 Hz) electroacoustic transducer and extract
electric power from a proposed thermoacoustic engine design.
A computational model of the thermoacoustic engine was developed using DeltaEC
to determine the feasibility of the concept and the necessary characteristics of the
transducer element. The unique requirements of the high-frequency thermoacoustic
engine led to the design, optimization, fabrication, and testing of the VRG, which
is designed to operate resonantly in the thermoacoustic system and supported by a
self-pumping gas bearing system. The VRG is uniquely suited to efficiently convert
small amplitude mechanical oscillations (5 mm in this work) into electric power.
Linear and nonlinear saturation models of the transducer were developed to optimize the VRG design and predict transducer performance in the thermoacoustic
engine. The accuracy of these models was established by comparing simulation results to static experimental data and dynamic experimental data taken at 50-60 Hz
oscillation frequency, testing one half of the full transducer. Experimental testing
resulted in a maximum power output of 5.74 Watts with an efficiency of 55.8%. The
results led to the conclusion that the transducer would function as designed in the
thermoacoustic engine. Recommendations for future work and guidelines for future
development of the engine, transducer, and bearing system are provided based on the
design and results presented in this work.
Thesis Supervisor: John G. Brisson
Title: Professor
3
"Man's flight through life is sustained by the power of his knowledge."
-Austin 'Dusty' Miller
4
Acknowledgments
I would like to extend a special thank you to Sumanth Kaushik for giving me this
opportunity to work and study at MIT the past two years. I would also like to thank
Lincoln Laboratory and the Advanced Concepts Committee for funding my studies
and research.
I have an amazing family and I have to thank them and especially my parents
Bryan and Lael for their continued encouragement, especially through the toughest
times. I would not be where I am today without their love and support even from
afar.
There are a number of other people who have been particularly important in my
academic, personal and professional development. Professor John Brisson has been
my research adviser the past two years and his mentorship and direction in my thesis
work has been greatly appreciated. Professor Jeffrey Lang was also instrumental in
my education and helping me come up with a research path while at MIT. I cannot
thank them enough for their patience and instruction.
The team that formed around the thermoacoustic project became invaluable to
me. To my fellow graduate student, Claudio Hail, I thank you for your dedication to
the project and everything you brought to the team. I wish you all the best in your
academic studies and beyond. I also thank the UROP, Niharika Bhargava, for her
assistance on the project.
The Cryolab has been a great place to conduct research for the past two years
and I would like to thank Doris Elsemiller, Michael Demaree, Paul Flinn, and Don
Strahan for their insights and ability to make the lab run so smoothly. Also a thank
you to my labmates Nick Roche, Martin Segado, Victoria Lee, and Melissa Ireland
for making the lab an enjoyable place.
Finally, I would like to thank my friends at MIT for making MIT a great place
to work and Cambridge a fun place to live, and most importantly, I thank God who
gives all things to all people including the strength to persevere and the will to look
ever skyward.
5
DISCLAIMER: The views expressed in this article are those of the author and
do not reflect the official policy or position of the United States Air Force, Department
of Defense, or the U.S. Government.
6
Contents
Abstract
3
Acknowledgements
5
M otivation . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
13
1.2
Background . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
14
1.3
Standing and Traveling Wave Engines.
. . . . . . . . . . . . . . . .
17
1.4
Literature Review . . . . . . . . . . .
. . . . . . . . . . . . . . . .
25
. . . . . . .
.
. . . . . . . . . . . . . . . .
25
.
. . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . .
33
.
.
1.4.1
Thermoacoustics
1.4.2
Electroacoustic Transduction
.
Thesis Overview . . . . . . . . . . . .
Thermoacoustic Generator Description
34
2.1.1
DeltaEC Model . . . . . . . .
. . . . . . . . . . . . . . . .
39
2.2
Variable Reluctance Generator Design
. . . . . . . . . . . . . . . .
48
2.3
Gas Bearing Design . . . . . . . . . .
. . . . . . . . . . . . . . . .
52
2.3.1
Design Iterations . . . . . . .
. . . . . . . . . . . . . . . .
54
2.3.2
Final Gas Bearing Concept
.
. . . . . . . . . . . . . . . .
57
. . . . . . . . . .
. . . . . . . . . . . . . . . .
63
2.4
Chapter Summary
.
.
.
.
. . . . . . . . . . . . . . . .
.
34
Engine Design . . . . . . . . . . . . .
2.1
Design of Variable Reluctance Generator
3.1
D esign . . . . . . . . . . . . . . . . . . .
.
3
.
1.1
1.5
2
13
Introduction
.
1
7
64
64
. . . . . . . . . . . .
69
3.2.1
Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.2.2
Nonlinear Saturation Model . . . . . . . . . . . . . . . . . . .
75
3.2.3
Loss Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.4
Flux Tube Analysis and Fringing
. . . . . . . . . . . . . . . .
82
3.3
Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.4
Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.5
VRG Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . . . . . . .
92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
3.2
3.6
VRG Model ..
..
. . ..
..
......
3.5.1
Laminated Components
3.5.2
C oil
Chapter Summary
....
99
4 Experimental Design Verification
Saturation Characterization
4.2
Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
4.3
Dynamic Experimental Results
. . . . . . . . . . . . . . . . . . . . .
112
Frequency Scaling and Losses . . . . . . . . . . . . . . . . . .
125
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
4.3.1
4.4
5
. . . . . . . . . . . . ........... 99
4.1
Chapter Summary
130
Conclusions and Future Work
5.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
5.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
. .... . . . . . . . . .
133
5.2.1
M odeling
5.2.2
System and Fabrication Improvements
A Flux Tube Analysis
134
B Model Code and Experimental Data
137
C Matlab Code for VRG
141
D VRG Component Drawings
146
8
List of Figures
1-1
Sondhauss-Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1-2
R ijke-Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1-3
Standing Wave Engine Description
. . . . . . . . . . . . . . . . . . .
18
1-4
Brayton Cycle T-s Diagram and Modified T-s Diagram . . . . . . . .
20
1-5
Effect of Compresion Ratio and TH on the Brayton Cycle . . . . . . .
22
1-6
Traveling Wave Engine . . . . . . . . . . . . . . . . . . . . . . . . . ..
24
1-7
Swift/Backhaus Traveling Wave Engine . . . . . . . . . . . . . . . . .
26
1-8
NASA HEPS Engine . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1-9
Yazaki Looped Tube Thermoacoustic Engine . . . . . . . . . . . . . .
28
1-10 Score Looped Tube Engine . . . . . . . . . . . . . . . . . . . . . . . .
29
. . . . . . . . . . . . . . .
30
2-1
Proposed Thermoacoustic Engine Schematic . . . . . . . . . . . . . .
35
2-2
Double Helmholtz Resonantor . . . . . . . . . . . . . . . . . . . . . .
36
2-3
Resonant Engine with Nonlinear Transduction Proposed by Swift
. .
36
2-4
DeltaEC Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
2-5
Pressure, Volume Flow Rate, Acoustic Power and Total Power Profiles
1-11 Aster Thermoacoustic Multi-stage Engine
in the Thermoacoustic Engine . . . . . . . . . . . . . . . . . . . . . .
42
2-6
Stack Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2-7
Effect of Stack Location
. . . . . . . . . . . . . . . . . . . . . . . . .
45
2-8
Geometry Optimization
. . . . . . . . . . . . . . . . . . . . . . . . .
46
2-9
Effect of Stack Gap Size . . . . . . . . . . . . . . . . . . . . . . . . .
47
2-10 Radial vs Axial Air Gap Linear Alternator . . . . . . . . . . . . . . .
51
9
52
............................
2-11 VRG Concept ......
53
2-13 Gas Diode Based Bearing Design . . . . . . . . . . . . . . . .
55
2-14 Piezoelectric Gas Diode Pressures . . . . . . . . . . . . . . . .
56
2-15 Unsuccessful Sliding Piston Check Valve Gas Bearing . . . . .
58
2-16 Schematic and Pressure Profiles of Gas Bearing Breakdown . .
59
. . . . . . . . . .
60
2-18 Relative Piston Position for Charging and Discharging Plenums
60
2-19 Gas Bearing Restoring Force . . . . . . . . . . . .
62
. . . . . . . . . . .
65
3-2
VRG Design Diagram
. . . . . . . . . . . . . .
. . . . . . . . . . .
65
3-3
VRG Geometry . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
66
3-4
VRG Assembly Including Piston and Piston Cap
. . . . . . . . . . .
68
3-5
Magnetic Circuit Diagram ...............
. . . . . . . . . . .
69
3-6
Inductance versus Air Gap Length
. . . . . . .
. . . . . . . . . . .
72
3-7
Cyclic Inductance Motoring Versus Generating .
. . . . . . . . . . .
73
3-8
Ideal Flux Linkage-Current Profiles . . . . . . .
. . . . . . . . . . .
74
3-9
Flux Linkage-Current Plots with Saturation
. .
. . . . . . . . . . .
76
. . . . . . . . . . . . . . .
. . . . . . . . . . .
77
3-11 Potential Current Waveforms and Cyclic Flux Linkage Current Loops
78
3-12 Minor Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . .
81
3-13 Magnetic Circuit Diagram Including Fringing and Leakage Fields
.
83
3-14 Flux Tube Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
. . . . . . . . . . . . . . . . . . . . . . . .
86
3-17 Drive Circuitry PCB Schematic . . . . . . . . . . . . . . . . . . . .
87
3-18 Potential Excitation Stages . . . . . . . . . . . . . . . . . . . . . . .
88
3-19 Winding Height Dimension Optimization . . . . . . . . . . . . . . .
90
3-20 Optimimum Air Gap Geometry . . . . . . . . . . . . . . . . . . . .
91
3-21 Laminated Stator and Piston Components . . . . . . . . . . . . . .
92
.
.
.
.
.
.
.
.
.
.
.
3-16 Drive Circuit Schematic
.
.
.
3-10 Magnetic Force Plot
.
Stator Design . . . . . . . . . . . . . . . . . . .
.
3-1
.
.
2-17 Schematic of Proposed Gas Bearing System
.
.
.
.
.
.
2-12 Gas Bearing Voltage Divider Analogy . . . . . . . . . . . . . .
10
3-22 Materials and Jig Used for Stator Assembly
. . . . . . . . . . . . . .
93
3-23 Epoxy Cure Bake Assembly . . . . . . . . . . . . . . . . . . . . . . .
94
3-24 Stator Assembly with Brackets
95
. . . . . . . . . . . . . . . . . . . . .
3-25 Assembly of Piston Laminated Components
. . . . . . . . . . . . . .
95
3-26 Coil Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
3-27 Stator Assembly with One Phase Coil . . . . . . . . . . . . . . . . . .
97
4-1
Static Experiment Air Gap Shims . . . . . . . . . . . . . . . . . . . .
100
4-2
Circuit Diagram for Generating Flux Linkage-Current Plots
101
4-3
Current and Voltage Profiles for Generating Flux Linkage-Current Plots102
4-4
Flux Linkage-Current Plots from Experiment Data
. . . . . . . . . .
103
4-5
Correction to Fringing Fluxes Based on Experimental Data . . . . . .
104
4-6
Comparison of Approximate B-H Curves to Published Data
. . . . .
106
4-7
Experimental Setup Schematic . . . . . . . . . . . . . . . . . . . . . .
110
4-8
Dynamic Test of VRG . . . . . . . . . . . . . . . . . . . . . . . . . .
110
4-9
Experiment Drive Circuitry and Power Dissipation
. . . . . . . . . .
112
4-10 Predicted Efficiency and Power Curves . . . . . . . . . . . . . . . . .
113
4-11 Predicted versus Measured Flux Linkage-Current Loops . . . . . . . .
116
4-12 50 Hertz Data Point for Power Output . . . . . . . . . . . . . . . . .
117
4-13 50 Hertz Data Point for Efficiency . . . . . . . . . . . . . . . . . . . .
118
4-14 60 Hertz Data Point for Power Output . . . . . . . . . . . . . . . . .
120
4-15 60 Hertz Data Point for Efficiency . . . . . . . . . . . . . . . . . . . .
121
4-16 60 Hertz Comparison of Model to Data Power Output . . . . . . . . .
122
4-17 60 Hertz Comparison of Model to Data Efficiency . . . . . . . . . . .
123
4-18 Effect of Minimum Air Gap on Efficiency and Power
4-19 Operating Frequency Effect on VRG Losses
. . . . .
. . . . . . . . . 124
. . . . . . . . . . . . . . 126
4-20 Frequency Scaled True Power Output and Efficiency . . . . . . . . . . 128
A-1 Flux Tube Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
11
2.1
.
List of Tables
Thermoacousic engine parameters . . . . . . . . . . . . . . . . . . .
40
2.2
Gas Bearing Operating Parameters . . . . . . . . . . . . . . . . . . .
61
3.1
VRG final design dimensions . . . . . . . . . . . . . . . . . . . . . . .
67
3.2
Drive Circuitry PCB Components . . . . . . . . . . . . . . . . . . . . .
86
4.1
Estimated VRG Core Loss . . . . . . . . . . . . . . . . . . . . . . . .
127
12
Chapter 1
Introduction
1.1
Motivation
Soldier pack weight has become an increasingly important issue as technological advances in battlefield equipment have broadened the capabilities of the war-fighter, but
also increased the electrical power demands in remote locations. With these advances,
soldiers have been asked to carry more and more weight in the form of batteries, in
addition to all of their other gear. Currently, soldiers in the field carry between 10
and 16 pounds of batteries, which is roughly 20 percent of their pack weight. The
rate at which batteries are used is also an alarming figure. An infantry company,
roughly 150 soldiers, will use 6,600 batteries amounting to 1,400 pounds in 72 hours
of operation. The army reports that yearly spending on batteries for a battalion is
roughly $150,000 and is second in magnitude only to munitions.
There are a number of ongoing research efforts aimed at solving the portability of
electrical power for soldiers. A few of these attempts include portable photovoltaics
known as the Rucksack Enhanced Portable Power System (REPPS) and fuel cell
research being conducted by CERDEC and DARPA. However, both of these systems
have serious limitations. The weight of the REPPS system is in the range of 4-7
pounds and produces 55 Watts in optimal conditions.
The power density of this
system is not unreasonable, but the photovoltaics systems themselves are limited
to day use, in an open/exposed area, and are climate dependent.
13
These systems
should not be discounted, but must be augmented if the soldier is expected to rely
on rechargeable batteries.
Fuel cell research is a promising technology for replacing conventional batteries.
The energy density of hydrogen used in fuel cells is nearly four orders of magnitude greater than the best batteries to date. However, current fuel cells require a
highly purified hydrogen source to operate. This is the current technological shortcoming, particularly for operation as a deployed power source. Further research will
be required before fuel cells become a viable alternative to conventional batteries for
portable soldier power.
An alternative to these technologies is thermoacoustic engine based technology.
Thermoacoustic engines have many desirable attributes because the engine operates
with the application of any quality heat source and requires few to no moving components. These attributes make the engine both flexible and robust as a portable
power source. The challenges associated with applying thermoacoustic technology to
meet portable power demands are scaling down existing technology, increasing the
power density in terms of watts per pound, and efficiently coupling acoustic to electric
power. This collaborative program between Lincoln Laboratory and Massachusetts
Institute of Technology (MIT) aims to make advances toward solving each of these
challenges. This work details the design of the Portable ThermoAcoustic Generator
(PTAG), and to greater detail, the acoustic-to-electric transducer component within
the thermoacoustic engine.
1.2
Background
Thermoacoustic technology has been applied to a number of different applications
including: refrigeration, cryogenics, natural gas liquefaction, space power, remote
power generation, waste heat recovery, cooling of microelectronics, and solar cooling
of homes. Some of the most significant and foundational research in thermoacoustics
was led by Greg Swift at Los Alamos National Laboratory.
In the scope of thermoacoustics, there are two forms of engines: standing wave
14
Figure 1-1: The Sondhauss-tube is the earliest investigation of a thermoacoustic type device.
Figure originally from Rott [19].
and traveling wave. At the most basic level, both standing and traveling wave engines
operate by correctly phasing heat transfer to/from a fluid that is also undergoing the
compression and rarefaction of acoustic type waves. A resonant system and a temperature gradient are used to correctly phase the heat transfer in all thermoacoustic
devices, where the temperature gradient is maintained along the same direction as
bulk fluid motion at a specific location(s) in the resonant system.
Different methods of properly phasing the heat transfer and setting the resonance
of the system are what separates standing wave engines and traveling wave engines,
and has led to the design of numerous forms of thermoacoustics devices, several of
which are discussed in Section 1.4. The details of the difference between the standing
wave and traveling wave engines is discussed in more detail is Section 1.3.
The thermoacoustic phenomenon was first observed by glass blowers when a hot
bulb was placed against a cold tube as shown in Figure 1-1. In certain dimensions and
temperature differences between the hot bulb and cold tube, this interaction would
cause the spontaneous generation of pressure oscillations (sound) from the end of the
cold tube, later deemed a Sondhauss-tube after the first experimenter [19]. In this
configuration, the temperature gradient was established by conduction from the hot
bulb to the cold tube, and the specific geometries of the bulb and tube are what set
the acoustic resonance of the system. The sound would quickly fade from these tubes
as the temperature gradient diminished and equilibrium between the bulb and tube
15
Sound
v~m ,pip(geeraon
Gauze heated
Jkjnveoflow
Figure 1-2: The Rijke-tube, which was the first thermoacoustic engine to use a stack (the
steel gauze) to maintain the temperature gradient. Figure originally from Matveev [12].
was established.
The Sondhauss-tube was followed by the Rijke tube in which steel gauze was used
to maintain the temperature gradient as shown in Figure 1-2. The use of a porous
matrix to establish the temperature gradient became known as a stack or regenerator
in thermoacoustic devices. "Stack" is typically reserved for standing wave engines,
while "regenerator" is used to describe the porous medium of traveling wave engines,
which is discussed further in the following section. The acoustic resonance of the
Rijke tube, like that of the Sondhauss-tube, was maintained by the geometry of
the acoustic wave enclosure, labeled "vertical pipe" in the Rijke tube figure. Lord
Rayleigh provided the qualitative description of the engine in his 1877 text, The
Theory of Sound. Lord Rayleigh writes for the description of this engine [16]
For the sake of simplicity, a simple tube, hot at the closed end and getting
gradually cooler towards the open end, may be considered... If heat be
given to the air at the moment of greatest condensation, or be taken from
it at the moment of greatest rarefaction, the vibration is encouraged.
Following Lord Rayleigh's qualitative description, Nikolaus Rott developed the
foundations of thermoacoustics, which are used for the analysis of thermoacoustic
16
engines today. Rott's work is based on linear acoustic theory, which Swift describes
in great detail in his Thermoacoustics text [24].
1.3
Standing and Traveling Wave Engines
Standing Wave Engine
An example of a simple standing wave engine is shown in Figure 1-3 part (a) with a
resonant tube, closed at both ends, and a stack centered at a distance L, from the left
tube wall. Acoustically, this configuration is known as a half-wavelength resonator.
The acoustic wavelength being given by the equation
a
A= -
(1.1)
f
where A is the wavelength, a is the speed of sound in the working fluid, and
f
is the
frequency of oscillation. The resonant frequency of the half wavelength resonantor is
set by the speed of sound in the fluid and the length of the resonant chamber, L,.
The half wavelength is formed by a rightward traveling wave and a leftward traveling
wave, which upon reflection off of the ends of the tube forms a half of a single standing
wave with a velocity node and a pressure antinode at each tube end, as shown in part
(b) of Figure 1-3. Inherent in a standing wave is a 90 degree phase shift between the
pressure and velocity oscillations, which is particularly important when considering
the thermodynamics of the gas parcels within the stack.
With the pressure and velocity oscillations 90 degrees out of phase, the bulk gas
motion is in phase with the pressure oscillations such that gas parcels within the stack
are at their leftmost extreme (hot side) when the pressure is high in the stack and at
their rightmost extreme (cold side) when the pressure is low in the stack. To meet
Rayleigh's criterion for the standing wave engine, heat transfer to the gas must occur
at high pressure and heat transfer from the gas must occur at low pressure as shown
in part (c) of Figure 1-3. To achieve this phasing, poor thermal coupling between the
stack and gas is required, such that the gas experiences adiabatic compression and
17
Lc LA/2
=
A/20
=L
0Resonant
Tube
Gas Particle
Motion
....
Figure Part (c)
Stack Cold Heat
Exchanger
Hot Heat
Exchanger
(a)
PMean
+ i
~~
Volume Flow
---
- - --~- --~
Rate
+lull
-
7e'ur7e
I
-'phl
-lU1l
Volume Flow Rat--'--..
- -'
!ressure_- ---
(b)
O, =o*, Oe =
O,= 90*, eu= 90*
*
Brayton Cycle P-V Diagram
C
B
0
-
I
-- -- -
AS=O
Expansion
-Ipl
-
- - --
C
AS=O
compression
-- ~~~~~~~~~ --- ~~ A
D
Qut
<
_
---------
_
__
A
I
.
---
cr
Volume (V)
(C)
(d)
Figure 1-3: Shown in (a) is a simple example of a standing wave engine in a half wavelength
resonant tube with the thermoacoustic stack located a length L, from the left wall. (b) The
phasing of a half wavelength standing wave is shown. The phasing of the wave is necessary
for understanding (c) the relative phasing of heat transfer, pressure and particle motion in
the stack. (d) The relative phasing of the thermodynamic interactions of the gas parcels
and porous stack produces a thermodynamic cycle within each gas parcel nearly identical
to the Brayton cycle shown.
18
expansion and heat transfer is limited to isobaric conditions at the extremes of the
gas parcel motion. Imperfect thermal contact between the gas and the solid walls of
the stack is accomplished by making the stack gap spacing or pore hydraulic radius,
rh,
perpendicular to the bulk fluid motion approximately the size of the thermal
penetration depth [2]. The thermal penetration depth is given by
6k
2k
WPCp
(1.2)
where k is the thermal conductivity of the gas, p is the gas density, c, is the specific
heat of the gas, and w is the angular frequency of the bulk gas oscillation. With the
stack plates, honeycomb or other porous material spaced in this fashion, the majority
of heat transfer occurs at the extremes of fluid motion and Raleigh's criterion is met.
The adiabatic compression, isobaric heat transfer to the fluid, adiabatic expansion, and isobaric heat heat transfer from the fluid is characteristic of the Brayton
cycle, which is shown on a pressure-volume diagram in part (d) of Figure 1-3. The
actual motion of the gas parcels is sinusoidal, and thus the true P-V diagram is not
discrete thermodynamic steps, but the idealized form is useful for both conceptual
understanding, and examining the limits of the standing wave engine.
The standing wave engine cannot exceed the Carnot efficiency,
77c
= 1 - --
TH
(1.3)
In addition to this limit, standing wave engines are inherently inefficient due to the
imperfect thermal contact between the stack and gas. As the contact between the
gas and stack -increases, such that the lateral spacing in the stack decreases below
the thermal penetration depth of the gas, the adiabatic compression and expansion
assumptions are no longer valid and Rayleigh's criterion is no longer met. The effect
of increasing the thermal contact becomes particularly clear when the cycle is plotted
on a T-s diagram as in Figure 1-4, where the increase in entropy during compression
and decrease in entropy during expansion is caused by heat transfer to and from the
19
Ideal Brayton Cycle
T-s Diagram
Modified Brayton Cycle
T-s Diagram C
C
AS=0
4-
Expansion
0.
E
a)
40.
E
D
B
D
B
a)
AS=O
Compression
A
A
Entropy (s)
Entropy (s)
(b)
(a)
Figure 1-4: (a) The ideal Brayton T-s digram is shown. (b) In contrast to the ideal Brayton
cycle, increasing the thermal contact between the gas and the stack causes non-adiabatic
compression and expansion. In the extreme of perfect thermal contact, no asymmetry exists
and no net work is produced by the cycle.
gas respectively. As the contact between the gas and stack becomes perfect, there no
longer exists thermodynamic asymmetry, and the temperature and entropy of the gas
follow the dotted black line in the figure with no net work transferred to the gas over
the course of an oscillation. Therefore, the imperfect thermal contact is fundamental
to the standing wave engine.
Although imperfect thermal contact is required for a standing wave engine to
operate, the efficiency can still theoretically approach the Carnot efficiency. In the
limit that the temperature gradient of the stack exactly matches the temperature
increase of the gas due to adiabatic compression, the heat transfer to the gas parcel
occurs over an infinitesimally small temperature difference, which indicates no entropy
generation. However, with an infinitesimally small temperature difference, the heat
transfer is also infinitesimally small. This limit is known as the critical temperature
gradient and is expressed as
VTcrt
Pm
-
I
P.cA|1|
20
(1.4)
where 1p, I and IjI are the magnitude of the pressure and displacement oscillations
respectively, and Pm is the mean density of the gas. Therefore, the standing wave
engine can only approach Carnot efficiency in the limit that no acoustic power is
generated. This is of little practical use, so the heat transfer must occur over a large
temperature difference to produce meaningful acoustic power. Entropy generation
increases with the square of the temperature difference, which indicates that the
standing wave engine must operate with an efficiency significantly below the Carnot
efficiency. Due to this inherent entropy generation, the actual efficiency of standing
wave engines to date is approximately 10-25% of Carnot's efficiency [24].
Other prime movers operating from the Brayton cycle largely overcome the poor
efficiency characteristics of the cycle by increasing the compression ratio
P2
rP
PL
(1.5)
where PL is the minimum pressure before compression and P2 is the pressure of
the working fluid immediately following compression. By increasing the compression
ratio, the temperature difference, AT, during heat transfer to the fluid is minimized
as shown in Figure 1-5. In part (a) of the figure, three Brayton cycles are shown
with increasing compression ratios. The heat transfer to the fluid, which occurs from
points B to C in the figure, occurs with a smaller temperature difference between the
gas and the the hot side temperature at larger compression ratios.
For thermoacoustic engines, the compression ratio is given in terms of a drive ratio
defined as
Dr =
PM
(1.6)
where Pm is the average pressure of the working fluid. The standing wave engine
drive ratio is limited to approximately 0.1 before the acoustics become significantly
nonlinear and the efficiency drops. The drop in efficiency has been suggested to occur
due to increased power dissipation in acoustic harmonics formed within the resonant
system[23]. Therefore, the compression ratio for the standing wave engine is limited
to approximately 1.3, which is significantly smaller than typical Brayton cycle engines
21
Brayton Cycle T-s Diagram
Brayton Cycle T-s Diagram
(Varying Compression Ratio)
(Varying TH)
4
TH
rp =8
C3
C
TH
-
r,=15
.-------- ---- ----- ---- ----- ------
3
AT
D3
C2
TH
CL
M
2
B
E
E
a)
9
C1
TH1
T-
C1T
D2
D1
A
Entropy (s)
Entropy (s)
(b)
(a)
Figure 1-5: (a)The three Brayton cycles shown are for increasing compression ratios, rp.
The larger compression ratios experience heat transfer (points B-C for the middle cycle
shown) across a smaller temperature difference, AT, thus limiting entropy generation. (b)
The only way to decrease entropy generation in the standing wave engine is to decrease the
temperature at which heat transfer occurs, which comes at the expense of power output
(the area enclosed in the cycle loop).
operating with compression ratios of up to 30-40. The smaller compression ratio of
the standing wave engine necessarily leads to heat transfer across a larger temperature
difference for all meaningful acoustic power outputs. This is the primary limitation
of the standing wave engine and has been the impetus for research in the area of
traveling wave thermoacoustic engines.
Traveling Wave Engine
Traveling wave thermoacoustic engines rely on a feedback path, such as a looped tube,
to provide acoustic power to the cold side of the regenerator, which is subsequently
amplified while traveling through the regenerator toward the hot side.
A simple
example of this type of engine is shown in Figure 1-6, where the standing wave stack
has been replaced by the traveling wave regenerator.
In contrast to a stack, the
regenerator is made such that the gas remains in intimate thermal contact with the
22
surrounding porous solid
(rh <<
6
k).
While this configuration produces no power
output in a standing wave engine, the excellent thermal contact is necessary for
the operation of the traveling wave engine because of the difference in pressure and
velocity phasing between a standing wave and a traveling wave.
Unlike the standing wave engine, a traveling wave engine has pressure and velocity
oscillations that are nearly in phase in the regenerator. The pressure is therefore high
while the gas moves toward the hot heat exchanger and low while it moves toward
the cold heat exchanger. Examining the behavior of a single gas parcel within the
regenerator, as shown in part (b) of Figure 1-6, the gas is:
1. Compressed nearly isothermally while at its leftmost extreme position.
2. Moves laterally toward the hot side while accepting heat transfer from the
porous regenerator, which was transferred there a half cycle earlier.
3. Expands isothermally given the excellent thermal contact at the parcels rightmost extreme.
4. Moves laterally toward the cold end while conducting excess thermal energy to
the porous regenerator material.
First noted by Ceperly, these cyclic conditions for the individual gas parcels is nearly
identical to the Stirling cycle, which is shown as a p-V diagram is part (c) of Figure 16. The actual motion of the particles as in the standing wave engine is sinusoidal and
thus the discrete cycle shown is an approximation of the true cycle of the individual
gas parcels, but is useful for both developing understanding and analyzing the engine
thermodynamically.
The limit of the traveling wave engine is also set by Carnot's efficiency, but the
excellent thermal contact leads to (ideally) reversible heat transfer. These conditions
are not limited to a certain amplitude of pressure oscillation, and therefore meaningful
acoustic power can be generated more efficiently than in a standing wave engine, with
the highest efficiency to date reaching 41% of Carnot from thermal to acoustic power
[3].
23
Figure Part (b)
Acoustic
Power In
Acoustic
Power Out
__
Cold Heat Regenerator Hot Heat
Exchanger
Exchanger
Resonant Looped Tube
(a)
Isothermal Compression
Regen. Heat In (+ P , +| UI)
'B
II
K
'loo*p,*
rh«&5k
OUt
A
6k
C
Regen. Heat Out
(-IPi, -IU, 1
Isothermal Expansion
)
(b)
Stirling Cycle T-s Diagram
Stirling Cycle p-V Diagram
T
P
D
T+ .C--------C
0.
B
_0
12i
0 ---..
-1pil ------------- CI
--A
A
c B C-----
~~~ ~--
S
V
(d)
(c)
Figure 1-6: (a) An example of a traveling wave engine in a looped tube is shown.
(b)
The gas parcels within the regenerator undergo the cycle shown, which effectively amplifies
acoustic power as the wave travels from the cold side to the hot side. (c) The phasing shown
in the second part of this figure is essentially a Stirling cycle shown on a p-V diagram and
(d) on a T-s diagram. The heat transfer occurs across (ideally) no temperature gradient.
24
Although the efficiency of the traveling wave engine can be greater than the standing wave engine, the traveling wave engine is far more complex from a design standpoint. Additional limitations include viscous losses in the small channels of the regenerator, time averaged DC streaming around the loop (Gideon streaming), and
extracting power from a traveling acoustic wave. The regenerator viscous losses and
Gideon streaming are mentioned here only for completeness, but extracting power
from the acoustic waves is discussed further in Section 1.4.2 and Section 2.2.
1.4
1.4.1
Literature Review
Thermoacoustics
The Condensed Matter and Thermal Physics Group at Los Alamos National Laboratory conducted research in a number of areas for thermoacoustic engines, including
both standing and traveling wave type engines as well as combinations of the two [11]
[23]. The most well known of Swift's engines is his "traveling wave engine," which is
actually a standing wave engine with a looped gas RLC circuit in order to shift the
pressure and velocity nodes to achieve traveling wave phasing inside the regenerator
[3]. A schematic diagram of the engine is shown in Figure 1-7.
One of the characteristics of engines of this type is that the acoustic and thermal
components are significantly shorter than an acoustic wavelength. This is what allows
the components to be treated as lumped circuit elements and analyzed using the
acoustically analogous electric circuit parameters. Examining Swift's traveling wave
engine, the resonator establishes a half wavelength resonance in the engine, but the
loop at the left end, referred to as a torus, shifts the pressure-velocity phasing such
that they are in phase within the regenerator. The traveling wave condition is met
only at one specific location in the acoustic circuit, while the rest of the engine is,
acoustically, a standing wave.
This engine was ultimately capable of converting heat into acoustic power with
an efficiency of 41% of Carnot and no moving components [3]. However, the engine
25
Mlain ambient
heat exchanger
Regenerator
[lot heat
exchanger
Secondary ambient
heat
exchanger
and flow
Feedback inertancenr
20cm
ie.
naor
junm
To rsonAtor
o
M
%1
P
Variable acoustic load
P We,,
Resonator
P
Figure 1-7: This figure shows the Swift/Backhaus traveling wave engine, which reached a
thermal efficiency of 30%. The engine operated at 80 Hz and produced 71OW of acoustic
power. Figure adapted from [3].
did not produce any electrical power. Without an electroacoustic conversion mechanism this engine configuration is only useful for directly driving other systems that
require acoustic power, such as a thermoacoustic refrigeration cycle. Combinations
of thermoacoustic engines and refrigerators have been used in a number of application areas, including natural gas liquefaction [26], pulse-tube refrigeration [24], and
microelectronics cooling [27].
Building off of Swift's traveling wave design, a similar engine configuration was
investigated by NASA as a spacecraft radioisotope power source [15].
This engine,
known as the High Efficiency Power Source (HEPS), was scaled down significantly in
size and used a mechanical, instead of an acoustic, resonator for both compactness
and to provide an acoustic-to-electric transducer. As shown in Figure 1-8, the same
RLC phasing was used around the torus, but instead of the acoustic resonator the loop
was attached to twin opposed linear alternators. These alternators were connected at
the point labeled "Alternator Interface" on the diagram, and oscillated in a direction
26
Flat Plate Hot Heat Exchanger
Thermal
Buffer
Tube
egenerator
R ject HX
it Pump
00
omplinCe
o
Altrnator
Inteface
0
0
0
0
0
0
0
0
0
II nertance -ine
Figure 1-8: This figure shows the High Efficiency Power Source (HEPS) system, which was
designed for operation on a spacecraft with a radioisotope heat source. The engine reached
a thermal efficiency of 18%, operated at 120 Hz, and produced 58 Watts of electric power.
The power density for the engine was 8.3 W/kg. Figure adapted from [15].
perpendicular to the loop as it is depicted in the figure.
However, reports on the
HEPS engine detail significant issues acoustically matching these linear alternators.
With the torus and mechanical resonance, the HEPS engine was capable of producing 58 Watts of electric power with a thermal efficiency of 18%. The research
undertaken in the current project is largely an extension of the HEPS engine, with
more emphasis on robustness, higher power density, and alternative geometries, and
less emphasis on high efficiency.
More recently, a large collaborative effort in Europe has been undertaken by
SCORE (Stove for Cooking, Refrigeration and Electricity) in the UK, the FACT
Foundation, and Aster Thermoacoustics in the Netherlands to develop an electricity
producing thermoacoustic generator (TAG) for use with cooking stoves in develop-
27
Stack
LDV
Glass tube
e
e aa by t
P ressu re tra nsd
T
TC
PhotomultiplierK
Heat exchangers
,=0 or 1
uce rs
A
.......-
Looped tube
Figure 1-9: This figure shows te experimental setup of Yazaki et al to produce traveling
waves in a looped tube. Figure taken from [32].
ing countries. Their initial efforts were to develop a standing wave engine, but were
turned away by the pressure amplitudes and displacements necessary for the linear
alternator [10].
This led to the development of what can be deemed a pure trav-
eling wave engine. The pure traveling wave engine was first investigated by Yazaki
et al, building off the work done by Ceperley
[32], where the regenerator is placed
in a looped tube as shown in Figure 1-9, and similar to the traveling wave example
discussed in Section 1.3.
While Yazaki's looped tube engine was able to spontaneously produce acoustic oscillations, the viscous losses in the regenerator were significant, due to large acoustic
velocities, owing to very low acoustic impedance in the regenerator. There is a bal-
ance in terms of the regenerator impedance for traveling wave engines, because lower
impedance means higher acoustic velocities. Larger acoustic velocities can increase
the amplification of acoustic power in the regenerator but this subsequently increases
viscous losses as well [33].
The SCORE engines are an advance over that of Yazaki's looped tube engine,
where the impedance in the regenerator is carefully controlled. Both the placement
of the regenerator in the loop and an increase in the cross-sectional area of the loop
are used to increase the pressure oscillations within the regenerator while decreasing
28
P2
it
COk He
T4
T3
Ragenwratr
T2
T
lEXcr (CyX)
Hot Hoe Exco
Aftmtor
w
ih
c(HH)o
~ftFeedback p*e
Figure 1-10: This figure shows the design and experimental setup of a looped traveling wave
thermoacoustic engine with a loudspeaker acoustic-to-electric transducer developed by the
SCORE initiative. The engine produced 10.5 watts of acoustic power with an efficiency of
1.93%. The acoustic-to-electric conversion efficiency was 52.5%. Figure taken from [1].
the velocity.
The reduction in velocity helps to mitigate the viscous losses which
dominated Yazaki's engine.
The thermoacoustic traveling wave in the SCORE engine is coupled with a loudspeaker operating in reverse to convert acoustic to electric power as shown in Figure 1-10. The low-impedance loudspeakers are used to match the traveling acoustic
wave
[1]. However, to date, these eni
a
very limited in their efficiency.
This low efficiency is due to both the near atmospheric mean operating pressures, and
the low transduction efficiency of loudspeakers, converting at roughly 50% acoustic
to electric compared to typical 80-90% of higher mass, higher cost linear alternators.
Other methods for power conversion are under development, which will be discussed
in greater detail in section 1.4.2. Both the mean pressure limitation and the use of
loudspeakers for transduction are limits associated with the low-cost objectives of the
SCORE project and are not inherent limitations in the thermoacoustic engine.
For the application of a portable power source, these looped tube engines are still
not feasible due to the large acoustic wavelengths and subsequent lengths of tubing
required for the feedback path. Reduction in the size of the system is possible at higher
29
load
#1load
#4
24
load
#
#
SHeat
in at Thigh
C]Heat out at T,,
load
7
Acoustic
loop power
Figure 1-11: This figure shows the design of the waste heat recovery system developed by
de Blok for a paper manufacturing plant. The engine produced 1.64 kW of acoustic power
with an efficiency of 38% of Carnot. The engine was not used with the linear alternators
because they were not correctly designed with the integral system. Figure taken from [1].
frequencies, but at higher frequencies the mass of the electroacoustic transducer acts
to lower the frequency. This paradox is discussed further in Section 2.1.
Kees de Blok, the founder of Aster Thermoacoustics, has generated a number of
unique system geometries and has developed the use of multiple stages to lower the
oscillation onset temperature [4]. The primary effort of de Blok has been to develop
alternate feedback geometries that do not require resonators because of the large
acoustic losses they generate. This has led to a similar looped tube concept as what
is currently being investigated by SCORE. One of the largest projects completed by
Aster Thermoacoustics was for waste heat recovery using a multistage pure traveling
wave engine shown in Figure 1-11. The engine achieved an onset temperature differ-
ence of only 45K, but the engine also experienced significant issues with the 8 twin
opposed 1.25 kW linear alternators. These 4 sets of linear alternators are shown as
the acoustic load elements in the figure. Due to the failure of the linear alternators,
the engine was tested only for acoustic power production and not electrical power
production.
The works described heretofore do not constitute the whole body of literature, es-
30
pecially when considering the large number of simulation based papers and proposals,
as well as literature that this researcher may be unaware of in coming to understand
thermoacoustics and the state of the art (SOA). Design choices for the current project
were based on optimizing the system for power density, robustness, and compactness,
while attempting to incorporate the lessons learned and design guidelines of previous
researchers.
1.4.2
Electroacoustic Transduction
A thermoacoustic engine that produces acoustic power does not solve the objective
of creating a portable power source for soldiers.
A common failure/limitation of
the thermoacoustic engines described in the previous section is the electroacoustic
transducer. The primary issue found in the literature is the poor coupling between the
acoustic waves and the transducer in terms of both the local pressures/displacements
at the interface and in coupling of resonance between the acoustics and transducer
suspension. For this reason, this project focused on building a thermoacoustic engine
concept around the transduction mechanism.
The transducer is the focus of this
thesis, but the full engine concept is described in Chapter 2.
The primary incentive for research into thermoacoutic engines is their potential
for long lifetime and low maintenance due to removal of moving components. Swift
made an attempt to completely remove oscillating components through magnetohydrodynamic [14] transduction.
Unfortunately, the magnitude of the temperatures
required to make this feasible, as well as the requirement to use an electrically conducting fluid, ruled this out for further investigation in this research. Additionally,
the efficiency for the engine described by Swift and Migliori was less than 2%.
Another method for transduction that has been investigated is direct coupling
to piezoelectric material [13] [20] [27].
However, there is an impedance mismatch
between the relatively compliant acoustic waves and the stiff piezoelectric material.
Placing this material on a diaphragm can increase the compliance marginally, but
typical transduction efficiencies to date have been less than 10% [13] [20]. This is not
a realistic option when combined with a thermoacoustic engine because the thermal31
to-electric efficiency would be in the low percents. This efficiency was considered
too low to be a practical portable power source and was not pursued further in this
research.
One of the other significant developments by de Blok is the concept of using a
bidirectional turbine for converting traveling wave acoustic power to electrical power
[5].
This concept appears to hold a significant amount of merit, and may, in the
case of this research, be applied to scaling down the size of traveling wave technology.
However, the width of the blades must be smaller than the displacement amplitude of
the gas parcels, and therefore, there is a limit to the frequency at which these turbines
could be applied.
Ultimately, the engine described herein relies on the resonant
mass of the piston and gas springs; and therefore, a bidirectional turbine, while an
interesting idea, was not applied to this research.
The SCORE project places a primary emphasis on the cost of the thermoacoustic
waste heat engine. Therefore, their investigation has been limited to loudspeaker
type transducers for converting acoustic to electric power [1] [5] [33]. Unfortunately,
these transducers are 30-40% less efficient than the more costly linear alternators,
can withstand only small pressure differentials across the thin diaphragm, have a low
power density, and have reliability problems associated with the flexible connectors
[17]. For example, in one of the SCORE looped tube engines, the conversion efficiency
from acoustic to electric was 52.5% to produce 10.5 watts of electricity using a 0.0132
mr2 (20.5 in2 ) diaphragm [1]. The proposed cross sectional area for the engine in this
research is 0.00456 m 2 (7 in2 ). These design issues are exacerbated in standing wave
type engines as discussed in [10], and therefore, the loudspeaker type transducer was
not used in this research.
When searching for alternative transduction methods, the concept of oscillating
capacitor plates was also investigated. However, an analysis showed that thousands
of capacitive plates would be required to generate meaningful forces with an assumed
breakdown voltage of 106 V/m. This was considered an unrealistic option for the
purposes of power transduction.
Other linear alternator designs are moving magnet and moving iron type linear
32
alternators.
Typically, these type of transducers are more efficient, higher mass,
and have a radial iron-comb structure such as described in [17].
However, these
generators do not handle small oscillations well; because for complete flux reversal,
the iron or magnets must move past several coil windings. This becomes very difficult
when oscillation amplitudes are only a few millimeters because the size of the wires
must be extremely small. The suspension and mass requirements to operate at higher
frequencies in order to increase the power density are also not feasible [17]. Therefore,
for this research, a moving iron type application is introduced with axial air gaps
instead of radial air gaps; and the gas springs of the engine and air bearing provide
all suspension requirements, thus reducing both losses and structural limitations. A
more detailed description of the transducer designed for this research is provided in
the following chapter.
1.5
Thesis Overview
The basic operating fundamentals, history of thermoacoustic engines, and SOA have
been provided in Chapter 1, as well as typical electroacoustic transduction methods in
thermoacoustic applications. Chapter 2 provides a description of the thermoacoustic
engine which is proposed for this research. Also in Chapter 2 is a description of the
electroacoustic transducer in terms of the overall engine and how it is coupled to the
proposed air bearing design. A detailed description of the modeling and design of
the variable-reluctance generator, as well as the fabrication of the generator system
is provided in Chapter 3. Experimental design and results for the static and dynamic
tests of the variable-reluctance generator and subsequent analysis of the collected data
is detailed in Chapter 4. Conclusions and suggestions for future work are provided in
Chapter 5.
33
Chapter 2
Thermoacoustic Generator
Description
The project has been divided into three sub-components: the thermoacoustic engine,
acoustic-to-electric transducer, and the bearing system for the transducer. A large
portion of this thesis was dedicated to generating a viable concept for conversion
from heat to electricity.
Numerous design iterations took place in developing an
understanding of these research areas. Ultimately, a standing wave engine with a
linearly-acting, variable-reluctance generator is proposed while using self-pumping gas
bearings for the linear alternator for both lifetime and improved suspension design.
2.1
Engine Design
A schematic of the proposed thermoacoustic engine and electroacoustic transducer
design is shown in Figure 2-1. The resonator uses a single piston to establish a
mechanical resonance instead of an acoustic resonance. The single piston is used as
the electroacoustic transducer converting the mechanical oscillation to electric power.
In this configuration, the engine is simple and robust. The design is compact since
it uses a mechanical resonance instead of the typical acoustic resonance. The rest of
this section is dedicated to describing the design choices made for the thermoacoustic
engine.
34
Heat In
Heat In
Heat Out
Heat Out
&I
Bounce
Volume
Bounce
Volume
Pressure Vessel Wall Z
Stack
Stack
I
-
I
-
Piston
Stator
W - -
Flux Loop
Windings
- -
- -
-
- -
- -
-
Piston Rightmost Position
Piston Leftmost Position
Gas Bearing Surfaces
It
I
Piston Steel
Stator Fixed Inside Piston
Minimum Air Gap
Figure 2-1: This figure provides a diagram of the proposed thermoacoustic engine and
acoustic-to-electric transducer. The acoustic wavelengths are significantly longer than any
of the components with the piston acting as a center mass between two gas springs. The
magnified views of the generator system shows the piston at the leftmost and rightmost
extreme positions.
35
IC
L
I
I
C
Figure 2-2: This figure gives the approximate representation of the thermoacoustic engine
acting as a double Helmholtz resonator with a center mass and two bounce volumes. The
engine components are significantly shorter than an acoustic wavelength, and therefore the
lumped impedance model is applicable. This simplification does not model the impedance
to flow from the heat exchangers and stack.
Figure 2-3: This figure is taken from [22] where Swift describes a mechanically resonant
standing wave engine using an axial air gap based electroacoustic transduction system.
The engine concept presented for this research is essentially a double Helmholtz
resonator where the pressure amplitudes across the piston are 180 degrees out of
phase. Figure 2-2 gives the electrical analog for the resonator, where the capacitors
are the gas springs, labeled "Bounce Volumes" in Figure 2-1, and the piston is the
inductance.
This mechanical resonance and transducer concept is similar to one proposed by
Swift where the mass of the transducer is, as he says, "resonated away," and the
highly efficient transduction from acoustic to electric power can occur through the
nonlinear, inductance based transducer [22]. Figure 2-3 depicts Swift's concept of
the mechanical resonant system with nonlinear transduction. Thus, the mass of the
piston is necessary to the operation of the engine, and sets the resonant frequency
with the gas springs.
36
Traveling Wave Design Constraint
Contrary to what has just been described as this works ultimate engine design, the
original engine concept was to use a traveling wave scheme such as those being designed for the SCORE project ([1] [5]) because of their potentially more efficient
operation than standing wave engines as discussed in Section 1.3. These engines consist of a looped tube, traveling wave, regenerator and inline electroacoustic converter
as previously shown in Figure 1-10. To decrease the size of the looped tube thermoacoustic engine and make it a portable system, either the tubing diameter or length
must be decreased. Decreasing the diameter of the feedback tubing decreases the
acoustic power for a given drive ratio and also increases the relative viscous losses
due to the increased surface area to acoustic volume ratio. In short, both the power
density and efficiency are significantly reduced by decreasing the size of the tubing to
make a portable engine. Alternatively, decreasing the length of the tubing increases
the frequency given the integral number of wavelengths required for a traveling wave
around the engine loop. At higher operating frequencies, the mass of the electroacoustic transducer becomes an increasingly significant impedance, because the pressure
difference required to move the piston with the same amplitude increases with the
square of the frequency. Assuming a sinusoidal motion, the acoustic displacement
(i), velocity
(u), and acceleration (a) can be defined as
(t) = Re[1ei" t ]
(2.1)
u(t) =- = Re[iw6iewt ]
dt
(2.2)
(t(t)
=
2
=
e6(-21ei
(2.3)
where Re[] denotes the real portion of the complex number and 61 is the complex
amplitude and phase of the displacement such that
1
= 6eeo
37
(2.4)
where &, is the amplitude and
#
is the phase. This is the same notation as used
in [24], and the "1" subscript will continue to represent the complex amplitude and
phase of acoustic variables. In steady state, the pressure amplitude across the piston,
Ap, is related to the displacement amplitude as
2
Ap A= w- mp~
AA
(25
(2.5)
where mp is the mass of the piston and A is the cross-sectional area of the piston.
Typically, pressure and velocity variations across a lumped acoustic element are given
in terms of an impedance defined as
A(2.6)
Z =zU
1
or
Z=
AU1
(2.7)
depending on whether the component causes a change in pressure or volume flow rate.
The mass of the electroacoustic transducer in a looped tube cannot be resonated
away as in the standing wave type engine. The transducer is an acoustic impedance,
which increases with the square of the frequency. This impedance causes a shift in
phase between pressure and volume flow rate which must be corrected in order to
maintain traveling wave phasing in the regenerator. This correction in the SCORE
engine is done using a "stub" element following the transducer as shown in Figure
1-10, which is meant to tune the volume flow rate and pressure back into the appropriate phase [1]. This acoustic element creates a change in volume flow rate at
constant pressure, where as the transducer changed the pressure at constant volume
flow rate. However, the stub represents an additional loss mechanism proportional
to the pressure times the change in volume flow rate. As the impedance of the electroacoustic transducer increases at higher frequency, the change in volume flow rates
across the stub must also increase, with an additional increase in the losses of the system. Continuing to increase the frequency or mass of the piston causes the piston to
38
become the equivalent of a wall to the high frequency oscillations, and the resonance
pattern in the tube becomes that of a standing wave.
2.1.1
DeltaEC Model
To get a more accurate basis for the design of the engine and requirements for the
power transducer, the Design Environment for Low-amplitude ThermoAcoustic Energy Conversion (DeltaEC) was used [311. This software, developed by Los Alamos
National Laboratory, is specifically designed for numerical integration of the momentum and continuity equations for acoustics using predefined or user-defined acoustic
segments and ensuring continuity of the segment boundaries. DeltaEC is useful for
making design decisions, but provides only moderately accurate predictions of engine performance (usually within 10-20%). This is especially true for large amplitude
oscillations, where non-linearities become more significant [23].
A DeltaEC schematic of the engine is provided in Figure 2-4. The engine has
two sides that are symmetric about the piston power transducer. The components
Hot".
?tsKHX
WmdHX
-
"-m
COM Dud
Sectrscousic w
T#rsdwv
Figure 2-4: A schematic of the engine as designed in the Design for Low-amplitude ThermoAcoustic Energy Conversion (DeltaEC) program. Additional labels are included to identify components. The engine is symmetric around the center piston transducer.
39
Table 2.1: Thermoacousic engine parameters
Component
Description
Variabl e
Value
Global Parameter
Mean Pressure
Frequency
Gas
Drive Ratio
Engine Length
Hot Duct Length
Hot Duct Area
HHX Temperature
HHX Gas Area/Total Area
Heat Transfer per Side
HHX Length
HHX Area
Stack Gap Thickness
Stack Length
Stack Area
-HX
Temperature
CHX Gas Area/Total Area
CHX Length
CHX Area
Cold Duct Length
Cold Duct Area
Piston Mass
PM
30 bar
250 Hz
Helium
0.05
.26 m(10 in)
25 mm
2
4.56. 10-3 m
Hot Heat Exchanger (HHX)
Stack (Parallel Plates)
Cold Heat Exchanger (CHX)
Cold Duct
Transducer
Dr
LHD
AHD_
LHHX
668 K
0.67
194.12 W
5 mm
AHHX
3.8. 10-3
TH
#
Hot Duct
f
Qrn
y/o
Latack
AStack
4
LCHX
ACHX
LCD
ACD_
mp
m
2
0.14 mm
60 mm
2
3.8 -10-3 m
340 K
0.253
6 mm
2
3.8- 10-3 m
8 mm
2
4.56. 10-3 m
0.2 kg
on each side of the transducer include a hot duct, which constitutes a large portion
of the bounce volume, hot and cold heat exchangers (HHX and CHX respectively),
and a stack. In the schematic, two stacks are shown on each side, which was done
for modeling purposes so that the properties at the center of the stack could be readily identified and used for engine characterization and optimization.
The "Begin"
segment in the DeltaEC program initializes certain variables such as the mean pressure, and the "Hardend" segment allows the user to dictate that no acoustic power
is flowing past that point (i.e. the acoustic impedance for both real and imaginary
components are set to zero). The DeltaEC program is based on guesses, targets, and
fixed engine parameters. For more information on the DeltaEC program see references [24] [31]. The DeltaEC code used for the design of this engine is provided in
Appendix B. Design decisions were then made based on the DeltaEC simulations.
The operating parameters of the model are provided in Table 2.1. These operating parameters are for a thermoacoustic engine that is 0.26 m (10 in) long and
0.0762 m (3 in) in diameter. This was determined to be a reasonably portable size.
40
These operating conditions were chosen after a number of optimizations were done using the DeltaEC program. The details of the engine design optimization are provided
later in this section.
The system, according to DeltaEC, behaves very similarly to the double-Helmholtz
resonator depicted in Figure 2-2. This is clearly seen in Figure 2-5, where the signed
amplitude of the pressure is shown and the magnitude of the volume flow rate. The
pressure amplitudes are nearly 180 degrees out of phase on either side of the transducer, while the volume flow rate is a maximum at the transducer and nearly linearly
decreases to zero at the ends of the gas springs. This is conceptually the same as the
phasing described in the example standing wave engine shown in Figure 1-3, except
that the piston mass has discretized the otherwise sinusoidal appearance of acoustic
resonance in a tube closed at both ends.
Also shown in Figure 2-5 are the acoustic and total power flows within the engine.
Acoustic power is important because it gives the power available in the standing
wave that can be extracted by the transducer element, and gives a reference for what
dissipation elements are of greatest importance in the engine. The acoustic power is
the time averaged power over an integral number of cycles given by the equation [24]
2
(x)
Re[p1(x)ew t ]Re[Ui(x)ewt dt
=
1 p1 ||U1 I coskPU
2
where
4pu
(2.8)
(2.9)
is the phase angle between the complex pressure and volume flow rate.
Therefore, examining the figure moving left to right, acoustic power is dissipated in
the hot duct and HHX as the power drops below zero and then increases traveling
through the stack portion of the engine. Acoustic power is then dissipated in the
CHX and cold duct and then drops significantly across the VRG transducer element.
The same acoustic power trend is mirrored on the other side of the thermoacoustic
engine, but is negative given the 180 degree phase shift in pressure while the volume
flow phase angle remains the same.
Referencing Equation (2.9), the acoustic power is clearly dependent on the phase
41
Cold i
I
i Cold
d I VRG| Duct |CHX|Stack|HHX
Stack CHXXDuct-
HHX
Hot Duct
-- - -
--- --
I
I
S
200
Hot Duct
--
Vessel Wall:l
LPressure
Pressure and Volume Flow Rate in Thermoacoustic Engine
150
100
50
0
'-
-50
-100
-150
Pressure
-200
Acoustic Power and Total Power
C.)
200
U
0
4-J
0
Total Power
100
0
-100
-200
-300
-400
05.
15
1
2)5)r
2
.
.4-
400
300
DistanceAlong Thermoacoustic Engine (m)
.
Figure 2-5: This figure provides the predicted pressure and volume flow rate profiles for the
thermoacoustic engine, which indicate that the system behaves very similarly to a double
Helmholtz resonantor given the 180 degree phase shift in pressure across the VRG. Pressure
is given as the real component, Re[pieiwt}. Volume flow rate is given as the magnitude, I Ui
Also shown is the acoustic and total power in the engine. The acoustic power only increases
in the regenerator and decreases due to losses elsewhere. The total power is only changed
by power entering or leaving the control volume of the thermoacoustic engine such as in the
HHX, CHX, and VRG.
42
angle between the pressure and volume flow rate,
#pu.
For standing wave phasing, the
phase angle between pressure and volume flow rate is nearly 90 degrees. The phasing
cannot be exactly 90 degrees, because then no acoustic power would be generated in
the stack. However, with a phase angle of nearly 90 degrees, the acoustic pressures or
volume flow rates must be very large for meaningful acoustic power to be generated.
This is another method of understanding the limitations of the standing wave engine
compared to the traveling wave engine where acoustic pressure and volume flow rates
are in phase and thus meaningful acoustic power can be generated at lower acoustic
amplitudes.
The total power flow in a thermoacoustic engine is also a time averaged value over
an integral number of cycles [24], and is significant in thermoacoustic engines since,
treating the engine as a control volume, it represents the net flow of power in and
out of the engine. In more precise terms, the total power flow is the time averaged
enthalpy flux and conduction through the solid/gas in acoustic segments where a
temperature gradient is relevant. The total power term is then given by the equation
ft 2 (x)
=
-pmRe[h101] - (Ak + Asolidksoid) dT
2
dx
(2.10)
where h1 is the enthalpy, C1 is the complex conjugate of the volume flow rate, A
and Aolid are the cross-sectional areas in the stack of the gas and solid respectively,
and k and k,olid are the thermal conductivity of the gas and solid respectively. For
the model of the standing wave engine in this research, it was specified that energy
could only be provided to or extracted from the engine through the heat exchangers
or the electroacoustic transducer. For this reason, the total power must be constant
everywhere else. In Figure 2-5, the total power is seen to be constant in the hot duct,
increases in the HHX's, and decreases in both the CHX's and the electroacoustic
transducer element where power was extracted from the engine. The efficiency of the
engine, ratio of power extracted to the thermal power provided to the engine, is also
apparent from this plot comparing the increase in total power across the HHX segment
(approx. 200 Watts per side) to the decrease in total power (power generated) across
43
Figure 2-6: This figure shows the important geometric variables of the stack gas flow,
where y, is the half width of the gas passage dimension and h" is the half width of the plate
material.
the VRG segement (approx. 50 Watts).
The stack is the core of the thermoacoustic engine, where a portion of the total
power is converted from thermal to acoustic power. Therefore, the parameters of the
stack, including the stack placement and design/fabrication, significantly effect the
power output and efficiency of the standing wave engine. For this engine, the stack
was modeled as a series of parallel plates as shown in Figure 2-6, where h" represents
the half thickness of the solid plates and yo is the half thickness of the gas passage
gap.
Typically, standing wave engines have stacks located at approximately A/20 from
the pressure node located at either end of the half-wavelength resonator. This location
has been determined to be the approximate required location to maximize the ratio
of power output to viscous losses [24]. Shifting the stack closer to the velocity node
Ij, and increases the pressure amplitudes,
1pi
,
decreases the acoustic displacements, I
which effectively increases the critical temperature gradient as defined in Equation
(1.4). Increasing the temperature gradient can result in increased conduction and
diffusion losses and less power output.
Conversely, shifting the stack closer to the
velocity antinode results in larger volume flow rates and subsequently larger viscous
losses.
For this engine, which is designed to operate at 250Hz, the acoustic wavelength
is between 4-6 meters using Helium as the working fluid. The range is given because
the speed of sound, "a", has a significant dependence on the gas temperature, and
44
Effect of Stack Location on Efficiency
TH
210
-
Effect of Stack Location on
900
Max Efficiency
Point7
S205
HD
z
200
X 700
600
X195
5
10
15
20
190
5
10
15
20
Cold Duct Length (mm)
Cold Duct Length (mm)
(b)
(a)
Figure 2-7: The stack location is varied by changing the hot and cold duct lengths, but
maintaining a constant total length. (a) The temperature significantly increases as the cold
duct length increases. (b) Engine efficiency is also coupled to the placement of the stack to
balance viscous losses with temperature gradient losses.
the engine itself contains a large range of temperatures between 300-700 Kelvin. The
acoustic wavelength would place the stack at approximately 0.25 meters from the end.
However, because this system is being designed for portability, a 0.5 meter system
would be ungainly. Therefore, the heat exchangers and stack were placed as close to
the piston power transducer as possible. To verify that the stack should be placed
near the piston and close to the A/20 value, a number of DeltaEC simulations were
conducted to vary the length of the hot and cold ducts, while maintaining the overall
length of the system. The piston displacement and transducer power output were
also fixed for this optimization. Figure 2-7 shows the effect of moving the stack in the
bounce volume while keeping the overall length constant, where the x-axis represents
the length of the cold side duct. Thus, decreasing the length of the cold side duct is
the same as shifting the relative position of the stack away from the velocity node.
As expected, there is a local minimum in heat transferred to the engine (maximum in
efficiency), and the hot side temperature continues to drop as the critical temperature
decreases. The chosen design point was to place the stack at the location for maximum
efficiency.
Another way to ensure the proper acoustic impedance in the stack is to neck down
the diameter of the engine at the location of the stack and heat exchangers. Figure
45
Geometry Optimization
222
Hot Duct Length
0.043m
221
0.04m
220
S219~
Z
0.036m
219
218
217
I Smallest
Geometry
Most Efficient Geometry
216
L9
15.2
11.4
7.6
3.8
0
Precent Reduction in HHX, Stack CHX Cross Sectional Area
Figure 2-8: The diameter of the cross-section is necked down at the stack and heat exchangers to maintain the proper gas displacement amplitudes while minimizing engine size. The
dimensions for maximum efficiency and minimum overall size are labeled for reference.
2-8 provides a summary of the geometry optimization, where the cross-sectional area
for the stack and heat exchangers was varied, while also varying the length of the hot
duct in order to determine an optimum geometry. In this simulation, the smaller the
hot duct, the smaller the overall engine was, which is ideal for portability. The design
points for minimum size and minimum HHX heat transfer (maximum efficiency) are
labeled in the figure.
Therefore, there is a balance that must be struck between
minimizing the size of the engine and the engine efficiency.
In addition to the placement and size of the stack, another factor in the efficiency
and operation of the standing wave engine is the gap spacing in the stack, labeled
"yo" in Figure 2-6. To maximize the engine efficiency, this stack spacing should be
matched with the thermal penetration depth, which is inversely proportional to the
square root of the frequency.
Therefore, an optimal spacing was searched for in
DeltaEC as shown in Figure 2-9, where the two factors considered were again the size
of the engine and the hot side temperature in relation to the gap spacing.
There is a clear efficiency optimum for the stack gap spacing at the point in
part (a) of the figure where the HHX heat transfer to the engine is a minimum.
This is a maximum efficiency because the power output from the transducer was
46
Stack and Geometry Optimization
Stack and Geometry Optimization
800
240
Smallest Geometry/
235
Hot Duct Length
0.05m
Highest Frequency
r230
22S
0.02m
750
Hot Duct Length
O.03m
O.04m
O.05m
0.04m
2
2250.02m
.550
220
521500
210
550
S205
-00
Most Efficient
0.1
0.12
0.14
0.16
Geometry
.18
q
500
.2
0.1
Half Width of Stack Spacing (mm)
.18
0.14
0.16
Half Width of Stack Spacing (mm)
0.12
.2
(b)
(a)
Figure 2-9: The efficiency of the standing wave engine is particularly sensitive to gap spacing
in the stack. (a) The efficiency have local maximums where the heat transfer to the HHX
is lowest. (b) As the hot duct length decreases and the frequency increases the hot side
temperature increases significantly to compensate.
held constant as the other variables were changed.
The half gap stack dimension,
y0 is strongly coupled to the efficiency, and therefore should be strictly controlled
in the manufacturing process. However, manufacturability of the stack to exacting
tolerances becomes increasingly challenging as the gap spacing decreases [24]. Further
research is necessary for precisely controlling the plate spacing during fabrication to
increase the actual efficiency of standing wave engines, especially for higher frequency
applications.
In part (b) of Figure 2-9, the hot side temperature is shown for different hot
duct lengths. This again shows the effect of the critical temperature gradient as the
acoustic displacements decrease in the stack and the pressure amplitudes increase as
the stack is moved closer to the end of the hot duct.
The length of the stack can also be used to optimize the engine. If the stack is
lengthened, the critical temperature gradient remains approximately the same, but
the actual overall temperature difference across the stack, set by TH and TC, must
increase. If T0 is fixed, this indicates that the hot side temperature must increase as
it did when moving the stack toward the the pressure antinode. However, with the
increased length, more surface area is presented to the gas for heat transfer, and the
47
same acoustic power can be produced across a smaller temperature difference. This
can increase the efficiency as discussed in Section 1.3, but also leads to additional
viscous losses because the restriction to flow increases. This is another optimum that
was performed in DeltaEC, but is not shown in a figure here.
A number of iterations for each of these optimizations were performed in order to
develop a system that would be compact, have a high power density with a reasonable
drive ratio, and have a reasonable efficiency as predicted by DeltaEC. The final design
is summarized in Table 2.1, but the full design (including dimensions, power, phase
angles, etc.) is included in Appendix B.
The power transducer used in the DeltaEC program was an "IESPEAKER" element placed in series between the two cold duct segments. This element is ideally
suited for incorporating the moving coil type linear alternator. It became apparent
from the DeltaEC model that the piston/gas displacement amplitudes would be quite
small for a higher frequency/high mean pressure thermoacoustic engine. For this reason and others that will be outlined in Section 2.2, an alternative form of power
transducer is investigated and constitutes the majority of the work for this thesis.
A direct coupling of the transducer designed for this application and the DeltaEC
software was not performed. The linear alternator segment of the model was therefore used to model as accurately as possible the mass of the piston and power output
characteristics, without trying to incorporate the highly non-linear transducer that
was actually developed during this project. Now that the transducer is built, a more
accurate transducer DeltaEC segment could be defined using a user defined "RPN"
segment, which is suggested for future work on this project, to more accurately predict
the operating characteristics of the PTAG system.
2.2
Variable Reluctance Generator Design
The DeltaEC models coupled with the project objectives led to very clear design
requirements for the electroacoustic transducer. These requirements were low oscillating mass, small displacement amplitude, high power density, and efficient conver48
sion from mechanical oscillations to electric power. The system becomes necessarily
higher frequency with small displacement amplitudes for three reasons: the internal
gas volumes must be made as small as possible to make the system portable, the
drive ratio (pi Il/Pm) must be less than 0.1 as discussed in Section 1.3, and the power
output of the transducer is proportional to the operating frequency.
To make the system portable, the "bounce volumes," as labeled in Figure 2-1, need
to be as small as possible. Considering the thermoacoustics as merely gas springs for
the moment, the effective spring constant for an isentropic compression is
pm7IA
,
K=
(2.11)
where Ap is the cross-sectional area of the piston interfaced with the gas spring and
V is the volume of the gas spring.
Thus, the effective spring constant increases
proportionally with the mean pressure and inversely proportional to the volume. The
resonant frequency of the double Helmholtz gas spring-mass system is then simply
. (2.12)
F
=
Therefore, decreasing the gas spring bounce volume, as well as increasing the mean
gas pressure, increases the resonant frequency of the system.
Combining Equations (1.6), (2.5), (2.11), and (2.12), the drive ratio is
Dr = 2
V
1j
.Ap
(2.13)
In developing Equation (2.13), AP across the piston is assumed to be 2Ip1I, which
is indicative of standing wave phasing, and
I1i
is taken as the gas displacement
at the piston interface. Examining Equation (2.13) it becomes apparent that the
drive ratio is not affected by the mean pressure, but only by the piston area, piston
displacement, and bounce volume. Therefore, as the bounce volume decreases, either
the piston area and/or piston displacement must also decrease. The force exerted by
the gas on the piston decreases in proportion to the piston area, so it was determined
49
that the transducer should be designed to handle small oscillations to keep the drive
ratio within the typically linear region.
The piston mass does not affect the drive ratio, but it does lower the resonant
frequency of the engine. Given the small oscillations of the transducer in order to
couple with the thermoacoustics, higher frequency operation is necessary to extract
meaningful electric power from the oscillations. This is because transducer power
scales proportionally with frequency and approximately the square of the piston displacement amplitude. From this analysis, the conclusion was drawn that a portable
thermoacoustic engine would require a small displacement, high frequency electroacoustic transducer.
These requirements led to the design of a linearly-acting variable-reluctance generator (VRG) with an axial air gap. The axial air gap concept is different than typical
linear alternator systems which rely on flux reversal and a constant radial air gap,
"G" as shown in Figure 2-10 part (a). Shown in the figure, typical linear alternators
use permanent magnets which travel past a full winding to completely reverse the
flux passing through the winding. This provides efficient transduction using magnetic shear forces and a constant radial gap if the displacement amplitudes are large
enough to completely reverse the flux. For smaller amplitudes, it is difficult to fabricate a a system where complete flux reversal is possible. This led to the alternative
VRG geometry shown in Figure 2-10 part (b) where the flux does not reverse, but
instead the axial air gap is changed, which causes a change in inductance.
The VRG concept, at the most basic level, is to apply a current to the winding
at large inductance and extract a larger current when the inductance is small. The
increase in current is driven through the winding by the mechanical forces acting on
the piston. This geometry is beneficial for small displacements because the inductance
changes rapidly with an increase in air gap, "G". However, for the same reason, the
VRG concept is not very good for large displacements because the magnetic forces
drop with the square of the air gap length. Because there are two air gaps in this
design for a given flux loop, the drop in force occurs even more rapidly. This leads
to relatively poor coupling with the mechanical motion because the greatest velocity
50
Piston Leftmost Extreme
4
uo
Piston Leftmost Extreme
G
winding
Flux Loop
IIJLH
Piston Rightmost Extreme
I
G
-
sttow
winding
Pisston Rightmost Extreme
G
C
n
windi ing
Flux Loop
06
G
(b)
(a)
Figure 2-10: (a) Typical flux-reversing radial air gap linear alternator. (b) Small displacement axial air gap variable reluctance generator.
occurs at a 90 degree phase shift from the piston extreme locations and minimum
air gaps where the largest magnetic forces are generated.
However, the magnetic
normal forces can be an order of magnitude larger than the shear forces as described
in Section 3.2. Therefore, for small oscillations, the large magnetic forces can be
maintained efficiently through the length of the oscillation.
The diagram shown in Figure 2-10 is one half of the VRG concept, which actually
has two connected sides so that the transducer extracts power during both halves of
the piston sinusoidal motion. A more detailed drawing of the VRG system shown in
Figure 2-1 is provided in Figure 2-11.
It is relevant to note that the generator can only experience attractive forces
between the stator and piston components. Therefore, the mechanical action is always
to push from one side of the piston to increase the air gap on the opposing side. The
51
Piston
Piston Steel
Stator
Windings
Stator Fixed Inside Piston
Figure 2-11: The VRG component from Figure 2-1 is shown again here for reference. The
sinusoidal pressure acting on the piston steel surfaces are 180 degrees out of phase as the
piston oscillates around the fixed internal stator changing the air gap, "G", on both sides.
force is transmitted through the piston support structure, labeled "Piston" in Figure
2-11, which holds the two sides of the piston together. Additionally, the stator is fixed
internally to the piston so that the piston oscillates around the stator. The modeling,
construction, and experimentation of the VRG are described in much further detail
in Chapter 3. This description is provided to give an understanding of the proposed
engine design and to give a basis of understanding of what is needed for the gas
bearing, which is the topic of the following section.
2.3
Gas Bearing Design
The easiest way to describe the operation of a gas bearing is with a voltage divider
analogy as shown in Figure 2-12.
In the figure, a pressure reservoir at pressure
PHigh supplies gas through an orifice to the gap between the piston and the cylinder.
The gas flows through this gap to the edge of the piston to a low pressure reservoir
)
at PLow. The fluid flow resistances provided by the orifice (R1 ) and the gap (R 2
constitute a "voltage" divider. The pressure at the orifice-gap interface, Pmeaing, is
52
Piston
wall
Orifice
0
R1
PHigh
R2
RR
Figure 2-12: This figure depicts the voltage divider analogy for gas bearing design. This
was the basis of analysis for evaluating self-pumping gas bearing concepts.
easily calculated using the analogy of the voltage divider,
PBearin
ge
2
R2a+nR1
(PHigh - PLow)
+ PLow
(2.14)
The value of PBearing is dependent on the gap dimension "g". Should the piston move
towards the cylinder wall, the resistance to flow in the gap, R 2 , increases. Since the
orifice flow resistance, R 1 , and reservoir pressures, PHigh and PLO, remain unchanged,
Equation (2.14) requires that PBearing increases.
The reverse is also true. Should the piston move away from the cylinder wall,
the gap resistance decreases. This drop in resistance leads to a subsequent drop in
bearing pressure in accordance with the voltage divider. In a piston configuration,
the reduction in gap on one surface of the cylinder wall is accompanied by an increase
in gap on the opposing surface. Since the overall pressure in the gap has increased
on the side with a smaller gap and decreased on the side with a larger gap, the net
pressure force on the piston tends to restore the piston back to the original centered
position. This restoring effect is most responsive when the orifice resistance, R 1 , and
gap resistance, R 2 , are approximately equal in the centered piston position.
To approximate the flow resistances for the purpose of modeling specific bearing
53
geometries, circular Poiseuille and plane Poiseuille flow were used for the orifice and
gap respectively. The plane Poiseuille resistance is approximated by the equation
(2.15)
R = 12 1L3
wg
where p, is the viscosity of the gas and w and L are the width and length of the
channel respectively. The circular Poiseuille flow resistance was approximated by the
equation
R =
_128iLc
rD4
(2.16)
7rD4
where D and L, are the diameter and length of the orifice respectively.
Typical gas bearing systems are fed directly by an external pressure supply or
compressor to operate continuously.
For a portable power system, the aerostatic
bearing cannot be fed by an auxiliary system.
Therefore, a number of methods
were considered to produce self-pumping gas bearings, which relied on the oscillating
pressures to feed high and low pressure reservoirs or plenums. The design concepts
and design chosen for further development are described in the following sections.
2.3.1
Design Iterations
The idea of self-pumping gas bearings was initially considered after reading through
the Swift and Backhaus paper on gas diodes [25]. The gas diodes have been used both
in jet pumps for preventing Gideon streaming and for self-circulating heat exchangers.
The concept is to use non-symmetric gas passages to change the resistance to flow
between the forward and reverse flow directions. This is done by effectively changing
the minor loss coefficients, K+ and K_, in the forward and reverse directions respectively. The minor loss coefficients express the irreversible turbulent pressure drop
across an impedance element as defined by
12
AP = K-pu2
2
54
(2.17)
*Diodes treated as nonlinear resistors
Po~sc
WW
ating
Piston
Motion
PISTI
Piston
Rs*
PLow
PHigh
Bearing Surface
Stator
Bearing Surfaces
Figure 2-13: This figure depicts the gas diode based bearing design for both the nonlinear
end effects and piezoelectric check valve concepts. The resistance R 2 constitutes the orifice
resistance, and the resistances R 3 and R 4 in parallel constitute the second voltage divider
resistance depending on whether POcillating is high or low.
where K is equal to K+ when the flow velocity, u, is positive, and K is equal to K_
when the velocity is negative. In oscillatory flow, this leads to a net DC flow in one
direction if K+ is not equal to K_ [25]. This DC flow component is exactly what is
necessary to produce a gas bearing system.
Common to all the designs presented here are internal pressure reservoirs to act
as accumulators to rectify the AC pressure oscillations to DC flow. In the electric
circuit analogy, these reservoirs act as "capacitors".
One inherent reservoir in the
design presented for the VRG is the volume inside the piston, which also contains
the stator. The second reservoir presented in these designs is a secondary chamber
fabricated into the cylinder wall.
A schematic of a bearing concept using the gas diodes similar to the Swift gas
diodes is shown in Figure 2-13. This figure shows a section of the piston, where the
piston has a bearing surface on which the pressure, PBearing acts, and changes with
respect to piston motion off its center-line axis. The bearing surface is a ring around
the circumference of the piston, and although only one is shown in the inset of the
figure, a second bearing surface is present on the lower half of the piston as well. In
55
Pressure Regions as a Function of Time
Pressure Regions as a Function of Time
3.15
3.1v
75%I61mm
-
2.__5
- -
M
High
PinmCtkn"
P_ [
1
26
2.9
2.86
-
s nn
"""
Hih
t
P,41h,
9
2.9
0
D002
004
0.006
0.008
0.01
0.02
299
0
0
0 6
Time (s)
Time (s)
(a)
(b)
0
0.01
1
0012
Figure 2-14: (a)This figure shows the oscillating pressures achieved with a piezoelectric gas
diode and a passive end effect diode on the piston. Pressures are the same as indicated in
Figure 2-13. Shown in (b) is the variation in bearing pressure as the piston moves off its
center axis to increase or decrease the radial gap between the piston and the chamber wall.
the figure, the resistances R1 and R5 use asymmetric channels to produce a net DC
flow, which tends to "charge" or "discharge" the high and low pressure plenums. With
the high and low pressure plenums established, the orifice resistance, R 2 , and bearing
gap resistance, R 4 , constitute the voltage divider necessary for the operation of a gas
bearing.
Thus, a decrease in the bearing gap causes the resistance R 4 to increase
resulting in an increase in bearing pressure, PBearing. The reverse is also true, and
the piston tends to remain centered due to these changes in bearing pressure. The
connection of the resistance, R 4 , to the reservoir inside the piston labeled PL,, is
accomplished by the same ports in the piston through which the stator is connected
externally to the pressure vessel wall.
Unfortunately, for the passive DC flow design based on asymmetric end effects,
the reverse bias on the diode is quite poor. For example, the minor loss coefficient
in the forward and reverse directions, K+ and K_, can be changed by a factor of
approximately 2-3 depending on the chosen geometry. For large pressure oscillations,
this leads to a significant breathing effect for gas flowing in and out of the plenums.
This was calculated to dissipate several watts of power based on the pressure drops
56
and volume flow rates. For this reason, it was proposed to introduce a piezoelectric
element to further increase the difference in minor loss factors between forward and
reverse flow. This concept holds merit, but the strain values for piezoelectric material
are on the order of 500pm/V. This indicates that extremely small holes would be
necessary for the piezoelectric strain to cause a change in the gas flow resistance.
These small holes would significantly decrease the maximum pressures obtainable in
the plenums. Additionally, the gas diode located on the piston as shown in Figure
2-13 could not easily be of the piezoelectric type, because flexible wires would need
to be connected to the piston and represent a failure mechanism.
The piezoelectric concept still appears to be a valid solution if the piezoelectric
material can change the minor loss coefficient by a factor of 10 and a passive end effect
diode is used on the piston. Figure 2-14 shows what plenum pressures and bearing
pressures that could be obtained with 30 bar mean pressures and 1.5 bar amplitude
oscillations if the piezoelectric material changes the minor loss coefficient by a factor
of 10, and the engineer is willing to accept a few watts of dissipation to make the
bearing operate. However, these limitations and losses coupled with the additional
complexity, control and necessity for a high voltage across the piezoelectric material
led this research toward other passive gas diode concepts.
2.3.2
Final Gas Bearing Concept
The gas bearing design chosen, which has become the topic of another master's thesis
in the Cryogenics Laboratory at MIT, uses the oscillating motion of the piston to act
as the check valve for pressurizing the gas bearing system. A literature review of this
concept yielded little in the form of papers, but did reveal a few patents related to
this concept [7]. Typically these bearing systems have been suggested for free piston
Stirling engines such as those developed by NASA for solar electric power [8]. For
this research, two concepts for the sliding piston check valve were introduced. The
first was ultimately a poor design, but led to the conclusion that both a high pressure
and low pressure plenum were necessary for continuous bearing operation. Figure
2-15 depicts the original sliding piston check valve bearing system.
57
Bounce Volume 1
Piston
ROscillatingl
Motion
R
R5
Piston Port
R4
High Pressu re
PHigh
Plenum
*Stator Not S iown
POscilIating2 (1810
Shift from Poscating)
Bounce Volume 2
Figure 2-15: This figure shows the piston and cylinder wall where the internal portion of the
piston is used as a high pressure plenum. The resistance R 2 provides the constant resistance
pressure drop in the voltage divider analogy. Resistances R 3 and R5 increases or decreases
as the piston moves closer or farther away. The resistance R 4 decreases to a near zero value
as the piston moves up. At this point, the value of POscillating is at a maximum and gas
flows from bounce volume 1 to the high pressure plenum, charging it to high pressure.
This bearing system tends to charge the high pressure plenum, represented as
a capacitor in Figure 2-15, because the resistance labeled R 4 decreases to a near
zero value as the piston travels upward.
In the piston's extreme upward position,
the pressure, Posciiating1, is at a maximum and charges the plenum through the low
resistance path labeled R 1 . The "voltage divider" for the journal bearing operation is
provided by the constant resistance R 2 in the piston wall and the two resistances R 3
and R5 , which vary with radial piston displacement. This bearing system works for
nearly the full piston cycle, but breaks down when the oscillating pressure, Posciiiatingi,
is greater than the high pressure plenum, which occurs at the extreme positions of
the piston. A schematic of the flow which causes bearing failure is shown in Figure
2-16 and plots the bearing pressures at various radial piston displacements. When the
58
Bearing Pressure
6
POscillatingl > PBearing1>PHigh
x10
.
When
3.14
Oscillatingi
PISTON
/
'7
B
3.13-
/
R3
3.12
Fhat
causes bearing to fail
CL3.11
/Bearing
Failure
3.1
Baing1
3.09F
33% Clearance Decrease
Aligned
n
Clearance Increase
-33%
3.08
. 0
~2
3
4
5
6
7
8
9
10
Time (ms)
(b)
(a)
Figure 2-16: (a)This figure shows a schematic of the flow which causes the bearing to
breakdown. A flow reversal in R 2 reverses the voltage divider and makes the bearing
unstable. (b) Shows the variation in bearing pressure as the piston moves off its center axis.
The bearing becomes unstable as shown in the figure when an increase in the clearance gap
increases the bearing pressure. This effectively forces the piston to touch down.
oscillating pressure is larger than the bearing pressure, the flow is able to reverse and
flow into the high pressure plenum through resistance R 2 . This effectively reverses
the voltage divider and causes the bearing pressure to increase as the clearance gap
increases causing unstable bearing operation.
From this initial sliding bearing design, it was determined that a second low
pressure plenum was necessary to ensure continuous bearing operation. A diagram
of this bearing scheme is illustrated in Figure 2-17, where the high pressure plenum
is still internal to the piston, but a second low pressure plenum is introduced into
the wall surrounding the piston. The high pressure and low pressure plenums are
both controlled by selectively opening and closing channels when the bounce volume
pressure is at a maximum or minimum pressure. The relative phasing of the charging
and discharging of the plenums is shown in Figure 2-18.
In the figure, when the
piston is in its lowest position as shown on the page, the pressure, POscillting,is at a
minimum and PLO, drops to match this pressure. As the piston moves back up past
59
POscillatingl
POscillatingl
PISTON
Rc
R3
-
PLow
In
Bearing3
. R6
aril10
bR2
R
(
2High
PHigh
(a)
(b)
Figure 2-17: This figure shows: (a) the proposed gas bearing system with both high and
low pressure plenums (b) the variable references for the electric circuit analogy used in the
analysis of this bearing system.
"Capacitors" Charging
"Capacitors" Discharging
POscOlatngl
POscillatingl
..*Resistance goes to zero
Ptow
R3
PBearing
PISTON
PBearing1
Beancg2
=
goth
*Resistance goes to zero
--
PHigh
T
(a)
High
(b)
Figure 2-18: This figure provides the relative piston position for charging and discharging
of the plenums. Both plenums charge and discharge at the same time. Charging of the
high pressure plenum occurs when Poscillatingl is high and the resistance R4 goes to zero.
Discharging of the low pressure plenum occurs when Poscillatingl is low and the resistance
R3 goes to zero.
60
Table 2.2: Gas Bearing Operating Parameters
Component
Description
Variable
Value
Global Parameter
Mean Pressure
Frequency
Gas
Density
Viscosity
Temperature
Radial Clearance
Piston Displacement_
Volume
Pm
30 bar
250 Hz
Helium
4.6 kg/M 3
2. 10-5 Pas
300 K
12.7 /im
5 mm
3
10-5 M
Plenum (PHigh)
Plenum (PL,,,)
Resistance (R4)
)
Resistance (R 2
)
Resistance (Re)
Re-si stanJe _(R 3
)
)
)
Resistance (R 4
Resistance (R 5
Resistance (R 6
Volume
Length
Diameter
Length
Diameter
f
p
AI
g
Li
D1 _
Length
Effective Width
(Same for R 4 -R6)
Length
Length
Length
_ _
10_6 m
4 mm
.8 mm
1mm
80 pm
3
L3
w
0 - 4.9 mm
L4
0
L_6 _ _ _
1 - 5.5 mm
1_- 5.5 mm
3 mm
-
4.9 mm
the low pressure port, the resistance R 3 increases rapidly and the flow rate into the
port is less than in the discharge configuration.
The flow path through the RI? resistance from the high pressure plenum to the
low pressure plenum provides a constant bearing surface and ensures no flow reversal
through the R 2 resistor, which would cause the bearing to fail. Therefore, a continuous
gas bearing is formed on both the upper and lower bearing surfaces of the piston.
A number of assumptions were made for the analysis of this bearing system. The
plane Couette flow was not included even though the oscillating piston velocity would
cause some flow due to the no slip boundary conditions. Additionally, the gas compressibility was excluded from the analysis because the relative changes in pressure
were small compared to the mean system pressure.
Table 2.2 lists the pertinent
dimensions for the final bearing system shown in Figure 2-17.
Gas bearings typically require tight tolerances and honing of surfaces. The radial
gap dimension given in Table 2.2 was based on discussions with Professional Instruments. Typical radial bearing clearances were said to be within 12.7 pm and 25.4 pm
(0.5-1 thousands of an inch) on the diameter for the 0.0762 m (3 in) diameter piston.
61
Restoring Force
Gas Bearing Spring Constant
45
40
30 -
35
frs= 900-1100 Hz
30
020-~
25
U020
10-
-
~30
ift
- 50% Shift
70%SN
-4-Low Force
-0
Hg
oc
-High Force
Linear (Low Force)
10
Shi
-90%
-
5
5-
1
0
0
0
0.5
1
1.5
2
Time (s)
2.5
3
3.5
X1-
2
4
3
Linear (High Force)
5
6
7
Piston Displacement From Center (pm)
4
(b)
(a)
Figure 2-19: (a) The restoring force acting on the piston for different radial displacements
is shown over one cycle. The restoring force changes over the cycle due to the fluctuating
bounce volume pressures, resistance lengths and plenum pressures. (b) The restoring force
effective spring constant is calculated for both a best and worst case scenario based on the
restoring forces shown in part (a). The spring constant leads to resonant frequencies significantly higher than the proposed 250 Hz piston operating frequency so lateral resonances
should not be an issue.
The pressure was assumed to vary linearly along the piston face. The pressure
profile is dependent on the bounce volume pressure and the pressure PL"". Thus,
at each point in time, the three point pressures PBearingl, PBearing2, PBearing3 were
determined and integrated along the length between the points to determine an approximate restoring force per unit width for the piston. A 3-dimensional model was
not completed as part of this analysis, and therefore, the force per unit width was
multiplied by the radius to get an estimate of the piston restoring force. This restoring force could be determined as a function of piston radial displacement from its
centered position. The restoring force as a function of time at specific radial displacements and the effective spring constants obtained from these radial displacements is
presented Figure 2-19. As shown in part (a) of the figure, the restoring force from
the bearing can reach 20 - 30 N, which is more than enough for the 0.2 kg piston.
Additionally, the resonant frequency of the bearing system must necessarily be much
larger than the oscillation frequency of the piston. Shown in part (b) of the figure,
62
the resonant frequency of the bearing spring constant is approximately 5 times the
oscillating frequency of the piston. Therefore, harmonics in the lateral motion of the
piston should not be an issue with the design.
It is proposed, but not sufficiently analyzed, that this same bearing system would
act as a thrust bearing. This effect arises from the shared high pressure capacitor
between the two gas springs. Should the piston drift off axial center, the high pressure
plenum will be open to the gas spring longer, and the mean resistance to the low
pressure plenum will me less than on the other side. This trust bearing application in
addition to the analysis of bearing start-up and lift-off should be considered in future
work.
2.4
Chapter Summary
A full thermoacoustic engine and electroacoustic transduction method has been proposed in this chapter. The thermoacoustic engine has been modeled and to an extent
optimized for the PTAG system. The DeltaEC program predicts an efficiency of approximately 12.5% from thermal to electric power. The electric power is generated
by a variable-reluctance generator, which will be discussed in length in the following
chapters. Finally, a gas bearing system has been proposed that can be coupled with
the oscillatory motion of the power transducer piston in order to create a self-pumping
gas bearing system.
63
Chapter 3
Design of Variable Reluctance
Generator
The Variable Reluctance Generator (VRG) design as briefly introduced in Section 2.2
was chosen for its potential to efficiently convert small-amplitude mechanical oscillations to electric power. This chapter presents the design, model, and fabrication of
the electroacoustic transducer necessary for converting the thermoacoustic oscillations
into electric power.
3.1
Design
The stator for the electric transducer is shown in Figure 3-1 part (a). The stator
consists of four laminated steel wedges as shown in part (b) of the figure.
Four
laminated piston wedges align with each of the stator wedges on each side of the
stator as shown in Figure 3-2, giving a total of eight piston wedges and four stator
wedges per VRG system. The two sets of piston laminations (one set on each side of
the stator) complete the magnetic flux loops established by the two coils set into the
stator. One set of piston laminations and the corresponding coil constitute one phase
of the two phase generator system.
During the operation of the VRG system, the two sets of piston laminations are
driven sinusoidal closer and then farther away from the stator. This motion changes
64
(a)
(b)
Figure 3-1: (a) The stator design is shown, which consists of (b) the four laminated steel
wedges.
Stator Lamination
-
Phase 1
Piston Laminations
Center Air Gap
Surface
Phase 1
Flux Loop
Outer Air Gap
Surface
Phase 2
Piston Laminations
-
Phase 1 Coil
-
Phase 2 Coil
Figure 3-2: The VRG magnetic core is shown consisting of 4 laminated steel stator wedges
and 8 laminated steel piston wedges. The two sets of piston wedges align on either side of
the stator to complete the magnetic flux loops. The VRG consists of two phases, which
generate power by changing the inductance of the flux loop by moving the piston closer and
farther away from the stator in a sinusoidal motion.
65
Ip
Piston
Laminations
1W
hwPhase ICON
_A,
Winding
Phase 2 CoHl
Section AA'
(a)
(b)
Figure 3-3: This figure depicts (a) the top down view of the stator with the associated
geometry variables used for modeling and optimization. The piston is not shown because
it exactly overlays the stator. (b) The side view of one of the four wedge stator pieces is
shown and the corresponding piston wedge piece.
the distance between the piston lamination face and the air gap surfaces labeled
"Center Air Gap Surface" and "Outer Air Gap Surface" in Figure 3-2. The change
in the length of this air gap changes the inductance of the VRG in a predictable way
such that electric power can be extracted from the mechanical motion. The model of
this change in inductance and the subsequent electric power generation are presented
in Section 3.2.
The dimensions of the system effect the inductance power output capability, and
efficiency of the VRG system. The important dimensions of the VRG system used for
modeling the transducer are shown in Figure 3-3. These dimensions were also used
for optimization of the VRG system, which is presented in Section 3.4. For reference,
the final dimension values selected after optimization are presented in Table 3.1.
To interface with the thermoacoustic system, a continuous surface must be presented to the bounce volume as shown previously in Figure 2-1. Additionally, the two
sets of piston laminations must be connected so that the gas pressures in one bounce
66
Table 3.1: VRG final design dimensions
Component
Variable
Value
Stator
l4
w
z
hw
7.62 mm
22.22 mm
7.62 mm
11.81 mm
2.18 mm
15.88 mm
-h
1P
hp
w
6.35 mm
35.56 mm
3.81 mm
11.81 mm
1W
10
Piston
volume force the piston laminations on the opposite side away from the stator as
discussed in Section 2.2. Therefore, the piston laminated steel components are inset
into a piston cap, and the piston cap is fixed to a honed aluminum piston as shown in
Figure 3-4 (a). The piston cap provides the pressure bearing surface interfacing with
the bounce volumes, and the piston provides the connection between the laminated
piston sets, as well as the required gas bearing surfaces as described in Section 2.3.
When the full system is put together, as shown in Figure 3-4 parts (b) and (c),
the stator sits internally to the piston and is held fixed by a screw or pin connected to
the four "Stator Brackets" as labeled in the figure. These brackets protrude through
the piston through the four ports in the piston. The piston mid-section between the
gas bearing surfaces is relieved to allow machining of the ports without concern for
the micron type tolerances required on the gas bearing surface.
The VRG system may then be fixed between the two bounce volumes such that
the piston oscillates with the stator enclosed within the piston. The following section
presents the model used to predict the performance of the VRG when the piston is
forced to mechanically oscillate in resonance with the thermoacoustic system.
67
(a)
Piston Cap
Phase 1 Piston
Laminations
Stator
Laminations
Stator Bracket
Coil
Phase 2 Piston
Laminations
(b)
(c)
Figure 3-4: (a) An exploded view of the piston, piston cap, and laminated piston steel
components is shown. The piston assembly is designed in three parts so the stator can be
fixed internally to the piston. (b) A radial cross-section of the full VRG system is shown
where the stator is fixed externally inside the piston by the stator brackets. This figure
includes all of the structural non-magnetic components that were not included in Figure
3-2, which leads to (c) the final VRG design. Not shown is the cylinder in which the VRG
is fixed.
68
3.2
VRG Model
3.2.1
Linear Model
Figure 3-5 is a radial cross-section of one phase of the proposed electroacoustic converter. It consists of laminated steel stator and piston elements and a copper coil as
described in Section 3.1.
Although saturation effects in a high power density device will be important, great
insight and guidance towards a final design can be obtained by a simple linear model
of the magnetic circuit used in this device.
A magnetomotive force, F, equivalent to voltage in a circuit analogy, is generated
by the net current flowing inside the magnetic circuit loop or
F = NI
(3.1)
where N is the number of turns penetrating the loop and I is the current in each of
those turns. The magnetic reluctance of a material in the loop is
Rm =
(3.2)
[LA
where 1 is the length of the magnetic flux path, A is the cross-sectional area of the
material through which the flux passes, and p is the magnetic permeability. Often,
15
vVAv
__
Piston Steel
=Air Gap
Air Gap
Winding
^w~w^4stator
Figure 3-5: This figure shows the magnetic circuit analogy used for the initial analysis of
the VRG. Only half of the VRG is shown and is a 90 degree rotation of Figure 2-11.
69
the magnetic permeability of a material is given as a relative permeability such that
A = pro where bo is the permeability of free space.
The magnetic flux 0 is the
equivalent of current in a circuit analogy and is given as
.F
m=
Z Rm
(3.3)
where E R, is the sum of magnetic reluctances in a series circuit of reluctances.
For linear magnetic materials, the flux linkage is linearly dependent on the current
in the windings or
A = LI
(3.4)
where L, the proportionality constant, is the inductance of the device. The voltage
that appears across the terminals, V, of a device with no internal resistance depends
only on the flux linkage as
(3.5)
V = -dt
For the linear model of the electroacoustic transducer, the inductance can be
described with a fixed steel reluctance and a variable air gap reluctance such that
L=
N
Rsteei + Rgap
(3.6)
where Rgteel is the reluctance of the steel and Rgap is the sum of both the inner and
outer air gap reluctances. Therefore, the inductance is only a function of the air gap
length, G, which is given by the equation for sinusoidal piston motion as
G=
2
cos(0) +
2
1+ Gmin
(3.7)
where 111 is the amplitude of the piston displacement,9 is the cyclic angular displacement, and Gmin is the air gap length when the piston laminations are at their closest
position to the stator air gap surface, also referred to as the minimum air gap.
To calculate the constant reluctance of the steel and the variable reluctance of
the air gaps, the geometry of the generator must be known. Applying the notation
70
defined in Figure 3-3 and making the simplifying assumption that both the inner and
outer air gaps have the same area, the following geometric relations can be made:
A = 4(l - z 2 )
(3.8)
w = 2(l4 - z)
(3.9)
10
Ag
= 4w
(3.10)
The reluctance of the steel and air gaps are then approximated by
Rsteei = 2h' 1p
1p
pAg phew phpw
Rgap
2G
-
poAg
(3.11)
(3.12)
where Rgap is the sum of both the inner and outer air gap reluctances.
With the reluctances set by the physical dimensions of the generator, the inductance can be plotted as either a function of air gap length (piston displacement), G,
or (using Equation (3.7)) the angular displacement, 0, as shown in Figure 3-6 and
Figure 3-7 respectively.
Since the energy stored in the magnetic circuit is
A =
1 L12
2
(3.13)
the force at constant current can be determined
F
dA =I1 2dL
dG
2 dG
(-4
Ultimately, the most important factor for the engine design is the power output
capability. This is determined by the area of the flux linkage-current profile. The
idealized current profile for the linear model can be determine by applying equations
71
Inductance vs Air Gap Length
5-
4-
3 --
2
1
2
Air Gap Length (m)
6
x
10-
Figure 3-6: This figure depicts the inductance for one phase of the generator as a function
of the air gap length. This is based on the final design with 148 turns.
(3.4), (3.5), and (3.7) to get
1(6) =
VdO
j""L(9)
(3.15)
where Oon is the phase angle at which the voltage is first applied across the winding.
This leads to current profiles that qualitatively look like Figure 3-7 part (b), and fluxlinkage-current profiles that look like Figure 3-8, where the shaded area represents
the work output per cycle converted from mechanical to electrical energy.
Saturation and eddy current losses shift the flux-linkage-current profiles away from
the ideal ones shown in Figure 3-8. These effects, discussed in Section 3.2.2, reduce
the power output from the power outputs predicted by the simple linear model.
Rotational Stability
An additional benefit of the wedge design, as presented in this thesis, is the rotational
stability provided by the cross shaped laminated steel components. Because the stator
is held internally to the piston, if the piston rotates about its center axis, rubbing
between the piston and stator brackets could occur. However, in this design, should
72
Cyclic Inductance
Generating
Motoring
5
4
C
3
2
0
1
5
4
3
2
6
Theta (rad)
(a)
1 (0) Current: Motoring
Oof
1(0) Current: Regeneration
Oaff
eon
0
0
(b)
Figure 3-7: This figure depicts (a) the cyclic inductance with the assumed sinusoidal motion.
Also shown are the regions for motoring versus generation. This is based on the final design
with 148 turns. Provided in (b) are qualitative waveforms for the system operating as a
motor and a generator. Adapted from [30].
73
A
L~e)I
I
Figure 3-8: This figure depicts ideal flux linkage-current profiles for operating the VRG.
Current is provided at the maximum inductance and extracted at the minimum. Figure
adapted from [30].
the steel components become unaligned, a net restoring torque will be generated to
minimize the reluctance of the flux path given by
1 2 dL
2 dOr
T-
(3.16)
where 0, describes the rotation of the piston about its center-line axis. Neglecting
the reluctance of the steel and any fringing or leakage, the inductance is simply
L =
N 2 poA
2G
(3.17)
and the area as a function of 0 , as approximately
A = 4(w - Orlp)lo.
(3.18)
Therefore, the restoring torque is approximately
T-
__I
2
N 21 ull
*
G
*
N
74
(3.19)
The restoring torque is highly dependent on the gap displacement and current, but
reasonable torque values were calculated to be between 0.1 - 0.9 N -m. Given that
there should be no external forces causing rotation, this was determined to be more
than enough to keep the piston aligned for the rotational degree of freedom. Therefore, the stator support structure could be passed through the ports in the piston
assembly without concern of wear due to rubbing between a misaligned piston and
stator bracket.
3.2.2
Nonlinear Saturation Model
Saturation effects can be accounted for by using a piecewise linear model, an approach
similar to that taken in [30]. In this piecewise linear model, the same linear inductance
methods are assumed until the steel becomes saturated. Saturation occurs-for fields
of 1.5 to 2.4 Tesla, depending on the material. In this study, a value of 1.7 Tesla
was used for the initial generator design. After saturation, the saturated incremental
inductance, L., is assumed to be constant and a factor of 1000 greater than the
permeability of free space. This was to approximate the knee seen in the B-H curves
for the saturation of steels. The steel saturates at the same flux linkage given by
A = NBmin[Astee]
(3.20)
where min[Asteel] corresponds to the smallest cross-sectional area of steel along the
flux path. This occurs in the piston steel because of the emphasis to minimize the
overall piston mass. With this model, the flux linkage is described by the piecewise
equation:
A
0 < I < i(3.21)
L(0)I(0)
L(0)Is + Ls(I() - Is) Is < I.
This leads to the characteristic flux linkage-current plots shown in Figure 3-9. These
plots are for various piston displacements and are significantly different than the
plots obtained by Vallese for his rotary variable reluctance motor (VRM) because the
75
Flux Linkage- Current Plot
X
0.5unsaturated
region
Saturated Region
.045-
M 035
Unsaturated
0.5mm
Region
0)"
-.
saturnted
region
-
.04
I
M
03-
025
--
X
4mm
's
La
M
Lmax
u'.02-
5.5mm
.015.01-L
.005L
095
10
15
Current (A)
(b)
(a)
Figure 3-9: This figure depicts (a) the modeled flux-linkage-current plots at various piston
displacements. This is in contrast to (b) which shows the model done by Vallese for a rotary
motor, where the steel saturates at constant current [30].
steel becomes saturated at a constant current, rather than at a constant flux linkage
as is the case for the linearly-acting VRG [30]. This is because the air gap area is
fixed for the linearly-acting VRG machine where it is varying for the rotary type
motor/generator.
The force can then be calculated from the coenergy expression
d
F = -O
i
A(I, O)dI.
(3.22)
Having determined the flux-linkage-current characteristics of the generator, the force
takes the same piecewise form dependent on whether the generator is saturated or
not. The force is given by
F= f
I tI- I)Lcf
0
1
I
1I.
(3.23)
From these equations, the magnetic force on the piston can be calculated as a function
76
Force vs Displacement at Constant Currents
-50--
0
-100.
-l=1A
-1=3A
-150
-1=5A
--
-1=7A
1=9A
-1=11A
-=13A
-200
-1=1
5A
-250-
-300
Air Gap Length (m)
X 10-
Figure 3-10: This figure depicts the the magnetic attractive force between the piston and
stator at the various piston displacements with lines of constant current.
of the position and current. Figure 3-10 shows what the static forces on the piston
should look like along its range of motion at constant current profiles. These forces
were used in subsequent design of components and experimental apparatuses to ensure
proper mechanical strengths, deflections, and deformations. As seen in Figure 3-10,
the force at saturation is constant (approximately 275 Newtons). This is true when
neglecting fringing. A flux tube analysis was performed to determine the relevance
of fringing for this generator, which will be discussed further in Section 3.2.4.
While the linear model also assumed negligible wire resistance as in Equation (3.5),
the actual voltage applied across the winding at a specific phase angle contributes to
both changing the flux linkage and driving current through the winding resistance.
For sinusoidal motion, this is given as
dA
Vp = I(6)R + WO
(3.24)
where the time derivative of flux linkage has been converted to
d\ = dA dO
d d
dt
77
(3.25)
Flux Linkage-Current Cycle Profiles
Saturated and Unsaturated Current Waveforms
zt.
U.UD)
Saturation Current Peak
18-
0.045-
16-
0.04
14
Saturation Current Peak
0.035
Min Air Gap
.
120
10
.0
0.025-
0
cm
x
8-
6
0.02
0.02-
0.015-
4-
0.01 -
2-
2
-Saturated
Unsaturated
.-
0.005
--
0
0
1
2
3
4
Phase Angle (rad)
5
6
S
7
2
(a)
4
Current (A)
6
8
10
(b)
Figure 3-11: Shown in the figure is (a) potential current waveforms for one phase of the
generator. Examples of current waveforms for both saturated and unsaturated operation are
shown, and (b) the flux-linkage-current cycle profiles for two potential excitation schemes
(the same as in part (a)). Shown in the background are the plots of flux linkage and current
at fixed axial piston displacements.
and the time derivative of 6 is simply the angular frequency, w. Solving Equation
(3.24) for flux linkage gives
A(0)
1 rO
[V - I(9)R]d6.
(3.26)
To determine the current profile, an initial current was assumed and the flux
linkage was calculated. The current was then calculated using the piecewise equation
-M
()
0< I< i
(3.27)
L()
I(M)-L()1 8 + is
is <I.
This process was iterated to determine the current profile accounting for saturation
and the effect of the winding resistance.
Figure 3-11 part (a) gives examples of
two current waveforms, one in the linear region and one in the saturated region.
The current spike occurs because the flux linkage continues to increase; but in the
78
saturated region, the inductance is very low, necessitating a rapid increase in current
to match the volt-seconds applied to the winding. This current increases until the IR
drop matches the applied voltage. The actual current waveform would be smoother
than depicted in the model, owing to the model's piecewise linear assumption.
If the applied voltage across the winding minus the voltage drop due to the winding resistatnce is integrated over time, the flux linkage is obtained. Plotting the flux
linkage and current on separate axes, the cyclic flux linkage-current profile is obtained
as shown in Figure 3-11 part (b), where the saturated and unsaturated profiles correspond to the current waveforms plotted in part (a) of the figure. The area within
these loops is the net work per cycle.
The flux linkage-current profile (net cycle work) is determined by the turn on, 0"",
and turn off, 9Of, angles. Determining the optimum excitation is dependent upon
the balance between generating larger forces with larger current and dealing with the
PR losses associated with those higher currents.
3.2.3
Loss Mechanisms
The efficiency of the transducer can be estimated by modeling the various loss mechanisms.
The three primary loss mechanisms are the winding, eddy current, and
hysteresis losses. The inverter losses are neglected for this analysis. Typically, the
eddy current and hysteresis losses are significantly smaller than the winding loss and
are lumped together as core losses. Early in the design of this generator, neglecting
the effects of eddy currents resulted in poor design decisions. The original design
used a ferritic stainless steel stator and piston in an axisymmetric geometry, which
would be simple and relatively inexpensive to fabricate. Although this design appeared attractive because of its simplicity, it was determined impractical due to the
high eddy current losses, even though the electrical resistivity of ferritic stainless steel
is approximately 50% higher than typical silicon iron or iron cobalt laminations. The
skin depth for the steel is given by
6=
2psleel
W7
79
(3.28)
where Pateel is the resistivity of the steel. This led to a calculated skin depth of
approximately 0.5 mm, which indicated that the thickness dimensions of the steel
should be smaller than 1 mm to avoid eddy current losses. This was not feasible for
the axisymmetric design, and therefore, the generator was redesigned using laminated
electric steel components. The generator built used 29 Gauge M19 electric steel to
minimize the eddy current losses. Higher flux carrying materials such as iron cobalt
(Hyperco) laminations could be used to increase power density but at a significantly
higher price. The fabrication of the generator is discussed further in Section 3.5.
The core loss is difficult to determine analytically. Typically, the core loss can be
estimated by the power loss per kilogram of laminated steel based on the frequency
of operation and maximum flux density. However, applying the core loss from the
published B-H loop data to the generator in this research overestimates these losses
because published core loss data assumes that the magnetic flux in the core switches
direction. In this VRG design, the flux does not change direction since the magnetic
force direction between the stator and piston components does not depend on the flux
direction. The entire hysteresis loop experienced by the VRG core extends only in the
positive magnetic field intensity (H) range. Consequently, the total core loss (proportional to the area of the hysteresis loop) is smaller than what would be exhibited
by a machine that drives the core over the entire range of magnetic field intensities.
The relative sizes of the hysteresis loops for complete magnetic flux reversal and the
minor hysteresis loop associated with the VRG in this research are shown in Figure
3-12, where the shaded region represents the area within the minor hysteresis loop.
An estimate of the core loss can be obtained by applying
Pe =]
W2
t2
fvoi p eel
dB
I - | dV
d
(3.29)
to the model of the VRG [30], where tiam is the thickness of the laminations. The
approximate flux density is calculated at each time step of the VRG simulation with
the equation
B -
NA
80
(3.30)
aI
Figure 3-12: This figure depicts qualitatively the minor hysteresis loop traveled for the
unipolar excitation of the core material. This leads to less hysteresis losses than in flux
reversing cores. Figure taken from [30].
where A is the steel's cross-sectional area. Since the cross-sectional area of the piston
steel is half that of the stator, the flux density was calculated separately for the
stator and piston and then summed to determine the total core loss. These equations
underestimate the core loss for saturated steel but was included in the model to get
an estimate for the core loss and more accurately optimize the generator system.
The gauge of the motor wire and number of turns for determining the wire resistance is directly tied to the geometry of the generator shown in Figure 3-3. These
variables are only relevant when determining what voltages and currents should be
used for driving the VRG. The choice of wire gauge and turns were determined by
selecting drive circuitry components that are within the range used in the automotive
industry to keep the electrical component costs low. This will be discussed further in
Section 3.3. The resistance of the wire is
R = pwLw
Aw
(3.31)
where pw is the wire resistivity, Lw is the length of the wire, and Aw is the area of a
81
single wire. The approximate length of the wire was calculated by
LW =
8lwhw(li + 1w)(
A
2
(3.32)
This gives the length of wire, based on the volume available, for the coil compared
to the volume of the coil corrected by a packing factor, (. The ideal packing factor
for windings stacked directly on top of each other is pi/4. This is the value that was
used for the analysis. The number of turns is then
N = (Wh
A
(3.33)
W
Solving these equations together gives an approximate resistance for the coil;
R
8(l + E')N 2
(3.34)
2
The winding losses are then calculated for the two generator phases such that
2
Pw = E RI2
(3.35)
n=1
where In refers to the current of phase n of the two phase generator. The winding
losses are the most significant loss contribution particularly in high power applications.
3.2.4
Flux Tube Analysis and Fringing
For most generators, particularly the rotary type, the air gaps are very small throughout the duration of the cycle. This leads to nearly negligible fringing effects. Due to
the large air gaps of this engine, it was determined that fringing would start to have
a significant effect particularly at larger air gap lengths.
The addition of fringing
into the model effectively raises the minimum inductance. This increase in minimum
inductance can decrease the work output per cycle depending on the excitation. For
this reason, fringing was included in the model so that the power output capability
82
Piston Steel
Outer Air Gap
RFringe
Inner Air Gap
Reap
RFringe
RGap
NI
coil
Stator
Figure 3-13: This figure depicts the magnetic circuit used for the more detailed analysis
including fringing and leakage flux paths. The parallel reluctances RFringe and RLeakage
have a significant effect on the magnetic circuit particularly at larger air gaps.
would not be overestimated. The addition of fringing and leakage fields create par-
allel reluctances in the magnetic circuit analogy. The magnetic circuit used for the
flux tube analysis is shown in Figure 3-13.
The flux tube analysis done for this generator follows the work of Herbert Roters
[18].
The method for calculating the fringing and leakage fields is based on what
Roters calls estimating the"permeances of probable flux paths." The permeance is
simply the inverse of reluctance. The solutions are geometric in nature, and having
established the generator geometry, the probable flux paths could be calculated using
the same geometric variables as given in Figure 3-3. The air gaps were separated into
an inner air gap and an outer air gap, and each air gap was broken into simple-shaped
volumes that contain all of the potential flux paths. These simple-shaped volumes
have permeances that are easily approximated. The volumes used for the flux tube
analysis include partial cylinders, partial annuli, spherical segments, and quadrants of
spherical shells. An example of two of these flux path volumes is provided in Figure
3-14, where the half cylinder represents the flux from edge AB to edge GF, and the
half annulus represents the flux from surface B to surface F. In total, 15 different
83
H
E
Figure 3-14: This figure shows two of the simple-shaped volumes used for the flux tube
analysis. The half cylinder represents the flux from edge AB to edge GF, and the half
annulus represents the flux from surface B to surface F.
permeances were calculated for the possible flux paths between the air gaps, and the
permeance of the leakage path was also determined. The leakage flux is the flux path
from surface "H" to surface "C" in the figure which completely bypasses the piston.
Appendix A includes the equations used for determining the different permeances,
which change as the air gap length changes.
The permeances for the outer air gap and inner air gap were summed in parallel
to ultimately obtain values for the inner and outer air gap fringe reluctances labeled
RFringe
in Figure 3-13.
The reluctance, RGap, is the reluctance of the flux path
perpendicular to the stator and piston air gap surfaces labeled "A" and "G" in Figure
3-14 for the outer air gap and "I" and "G" for the inner air gap.
The addition of fringing and leakage fields tends to decrease the current at which
saturation occurs in the laminated steel. This can be seen clearly when looking at a
plot of the magnetic forces as a function of displacement at constant current. This is
shown in Figure 3-15 part (a). The effect of the fringing and leakage fluxes can be
seen by comparing Figure 3-10 part (a) with Figure 3-15.
84
0
Force vs Displacement at Constant Currents
v-s0.045
Fringing Effects
on
Flux Linkage-Current Plots
0.04-500.
0.035
0.03
-100
0.025
S=34
1=5A
-l=7A
-=9A
3
0.02
-1=11A
LL
0.015-
--
S-150
0
-200
I=1 3A
-
1=15A
-250
0.0050
1
w/o Fringe Effects
W/ Fringe Effects
5
Air Gap Length (m)
x 10-
10
15
Current (A)
(a)
(b)
Figure 3-15: This figure shows (a) the force at constant current for the range of air gap
lengths. This figure should be compared with Figure 3-10 to see the effects of fringing
and leakage fluxes. (b) Shows the effect of fringing and leakage fluxes on the flux linkagecurrent plots. The inductance is increased as the total reluctance of the flux linkage path
has decreased. This effect is particularly pronounced at larger air gaps and limits the work
output per cycle.
Excluding the fringing and leakage would overestimate the power output capability of the generator because the magnetic forces are significantly smaller at larger
air gaps than would be predicted otherwise. Figure 3-15 part (b) shows the effect of
fringing and leakage fields on raising the minimum inductance. Therefore, the flux
linkage-current profile will enclose a smaller area and thus have a lower work output
per cycle. The results of this analysis led to the conclusion that to produce an accurate model of the VRG, the effects of fringing and leakage fluxes should be included
due to the potentially large air gap lengths inherent to the design.
3.3
Power Electronics
There are multiple configurations for excitation of the winding.
There are trade-
offs for each design as described in references [6] [28] and [30]. For the design of this
VRG, only two phases exist, and thus, the number of MOSFETs was not an important
85
Gate Driver,
Gate Driver,J
Wnding 2
Winding 1
JLGate Drlverj
A
Gate Driej
Figure 3-16: This figure shows a schematic of the drive circuitry used for the VRG. The
two windings are driven 180 degrees apart during the piston cycle.
consideration. Additionally, the simplification of having only one power supply and
only one winding led to the implementation of the drive circuitry as shown in Figure
3-16, where the gate driver for the MOSFETs provides the on/off signal and ensures
fast switching to reduce circuit losses.
Only one phase of the VRG was used for experimentation as will be described in
Section 4.2. The necessary power electronics were placed on a printed circuit board
(PCB). The PCB was specifically designed to limit the physical loops between the
gate driver and high-side/low-side MOSFET transistors. This is to prevent any issues
with ringing in the circuitry while switching the power MOSFETs on and off. The
schematic used for the PCB is shown in Figure 3-17.
A number of considerations went into the design of the drive circuitry. A bootstrap
circuit was used to maintain the charge on the high-side MOSFET. Additionally, a
fuse, labeled F1 in the figure, was included to ensure if the VRG were driven too
Table 3.2: Drive Circuitry PCB Components
Circuit Components
Gate Driver (MDr)
Resistors (RI, R2, R3)
MOSFETs (MOS1, MOS2)
Diode (D1)
Diodes (D2, D3)
Capacitors (C1, C2)
86
FAN7360
1OQ
NDB5060L
SK31OA-LTP
SBR15U50SP5-13
3pF, 1pF
Fl30-
~
1 41t
.......
L
s..
RE
Figure 3-17: This figure shows a schematic of the drive circuitry used for testing the VRG
system. The winding was soldered across connections labeled W2 and W3. A LabVIEW
pulse signal was applied to the connector labeled Coni. More detail on the experimental
setup is described in Section 4.2.
heavily into saturation and large currents were experienced, the circuit components
and measurement equipment would not be damaged.
The coil ends of the VRG
were soldered to points W2 and W3 as labeled on the schematic, and traces that
-
experienced the same currents as the winding were made between 5 - 6.5 mm (0.2
0.25 in) wide. Table 3.2 lists the components used for the drive circuitry.
These components were specified for the original operation of 250 Hertz with a 50
volt driver and expected peak currents up to 15 amps. The number of turns in the
coil and the size of the wire were targeted to keep the voltage and amperage ratings
of the MOSFET components within the automotive industry's standard. Without
considering the limitations of electrical components, the design optimization may
provide unrealistic current profiles or applied voltages.
This is especially true for
higher frequency operation because of the large voltages necessary to quickly drive
current into the windings.
There are potentially three stages to motor excitation: charging, freewheeling, and
87
Charging
Freewheeling
Generating
Figure 3-18: The three excitation stages are the charging of the winding, freewheeling,
and generating. The freewheeling stage was not included in the analysis of this VRG. The
charging stage drives a small current into the winding and the generating stage drives a
larger current back into the power supply delivering a net electrical power output.
generating. These three stages are depicted in Figure 3-18, where the charging stage
occurs when both MOSFETs are operated as closed switches, the freewheeling stage
occurs when the high side MOSFET is an open switch but the low side MOSFET
is a closed switch, and the generating stage occurs when both MOSFETs are open
switches.
The charging stage is used to drive current into the winding and provides input
to what is essentially a parametric amplifier, driving more current out of the winding
during the generating phase. The use of the freewheeling stage was not considered for
this work. Other potential excitation schemes and methods for determining optimum
excitation are described in [6] and [28].
3.4
Design Optimization
Geometric optimization was completed for the VRG by calculating the power output
and efficiency of the generator and performing a grid search method for the geometric
variables provided in Figure 3-3. A number of variables were fixed, including the
outside diameter of the generator (.0762 m) and the inner and outer air gap areas
were kept equal.
The dimension "z," as labeled in Figure 3-3, was held constant
at 0.25 mm to keep individual laminations from having thin long sections, which
would greatly increase fabrication difficulty and cost.
The h, dimension was also
left fixed for structural purposes and was significantly thicker than the piston steel
88
so that saturation limitations would not be an issue in this segment of the stator
core material. This left only 3 geometric optimization variables: h", 1j, and hp. The
design choices were based on the transduction efficiency, normalized power output,
and inverter rating. The inverter rating being important to preclude unrealistic or
costly drive circuit components as discussed in the previous section. The efficiency of
transduction is defined as
IPO
1P01+ jPWI + IN
(3.36)
where PO is the power output of the generator, Pw is the power loss due to winding
resistance, and P. is the core loss.
For each set of VRG power performance calculations as a function of coil height,
piston lamination height and air gap dimensions, there exists a peak power point.
Since the power and efficiency trends are of the most interest here, the results are
graphically presented using normalized power output, PO* defined as
P* =
where
IPol
I(3.37)
max[|Po|]
is the power output and max[Pol] is the power corresponding to the
maximum power point of the calculated curve.
The variable h, represents the maximum height of the coil winding. As the height
of the coil increases, the efficiency increases because the same number of turns can be
made with larger and larger diameter wires, thus reducing the PR losses. However,
because the leakage flux was included in the model, there exists a maximum power
output because as h, increases the reluctance of the leakage path (described in Section
3.2.4) decreases. This reduces the useful flux traveling through the air gap to the
piston, which provides the magnetic attraction forces between the piston and stator.
The normalized power and efficiency curves as a function of the winding height are
shown in Figure 3-19, which were produced by fixing the excitation angles and all
other variables except the winding height. The chosen design dimension lies between
the maximum efficiency point and the maximum power point. The winding height
becomes more relevant considering the power density of the device because of the
89
Design Optimization Curves for Coil Height Dimension
0.95
0.9
0.9
0.85
0--
-M 0.8
Efficiency
-Normalized
0
Z
0.7
0.75
:Chosen
Desin
-
8.
Power
10
16
14
12
18
20
22
24
Length Dimension for Coil Height, hw (mm)
Figure 3-19: This figure shows the chosen design point for the coil height based on the
efficiency and normalized power output of the generator. The power output is normalized
such that the maximum is set to one.
significant weight of copper windings.
The variable 1i is a more complicated variable in terms of optimization. All of the
air gap dimensions are driven by this length dimension. To get an accurate prediction
of the optimum air gap design, the volume of magnetic material in the piston was
fixed and the length dimension, 1j, was varied. The volume of magnetic material in
the piston had to be fixed in order to decouple the piston saturation effects with the
air gap geometry.
A value for this dimension was again chosen between the power
and efficiency maximums. The results of this design study and the selected design
dimension are shown in Figure 3-20.
The larger the mass/volume of the piston steel, the larger the cross-sectional
area available for the magnetic flux, thus increasing the current at which the steel
saturates. Increasing the piston dimensions increases the overall piston mass, but can
also increase the power output capability of the system. The same design study for the
dimension 1i was repeated, but the piston mass was allowed to increase. The results
are exactly as expected and shown in Figure 3-21 part (a). The normalized power
output of the generator increases until the point again where leakage flux begins to
dominate. However, shown in part
(b) of the figure, the necessary voltage to drive
90
Design Optimization for Stator Air Gap Dimensions
0.9-
-
0.8.-
-
Efficiency
Normalized Power
0.70
(-
0.60
Z
:Chosen
Desin
0.5-
0.4
1'0
11
12
Driving Length Dimension for Stator Air Gap Geometry, Ii (mm)
Figure 3-20: This figure shows the chosen design point for 1i length dimension, which was
set as the driving dimension for the air gap geometry. The piston volume was held constant,
and the current waveform was kept constant by driving at the same turn on and turn off
angles and peaking at the saturation current.
the VRG also increases because the overall inductance of the generator increases
with larger piston steel volume, which complicates the drive circuitry and increases
the required inverter rating. The maximum drive voltage selected was 50 volts and
peak currents of 15 amps. The importance of the inverter rating was overestimated in
this research, and the drive voltages should be reconsidered for the power electronics
developed for the complete portable power system. A value of 3.8 mm for the piston
height, hp, was chosen to reach the desired 50 watt output with the stated voltage and
current limits, but kept as small as possible to keep the mass of the piston low, which
is desirable for higher frequency operation as discussed in Section 2.2. The power
output of the generator could be increased by simply increasing the dimensions of the
piston steel.
Apart from the geometric optimization, the turn on and turn off angles significantly affect the efficiency and power output of the generator. The geometric optimization was done around a single turn on and turn off angle based on an approximate
peak in power and efficiency regardless of the geometric dimensions. For experimental
validation of the model, the VRG was tested at different turn on and turn off angles
91
Excitation Voltage for Stator Air Gap
Design Optimization for Stator Air Gap Dimensions
0.95-
Optimization
90 --
-
4) 0.9
80
0
4.8>
0.8-
C
_
oi
Dimension
0
-
.Efficiency
ccNormalized Power
Z
70-
Chosen
Design
60
0.7 -0.65-
'Chosen
Design
50
Dimension
6
8
10
12
14
16
Driving Length Dimension for Stator Air Gap Geometry, li (mm)
(a)
6
8
10
12
14
16
Driving Length Dimension for Stator Air Gap Geometry, li (mm)
(b)
Figure 3-21: Shown in (a) is the effect of increasing the air gap dimensions and subsequently
decreasing the available volume for the coil. The optimum is shifted when the volume of the
laminated steel piston components is allowed to increase. (b) Shows the increase in applied
voltage necessary to drive the higher inductance generator to the same steel saturation
point.
to determine optimum power and efficiency points. This data as well as the model
determination of the optimum turn on and turn off angles is presented in Section 4.3.
3.5
3.5.1
VRG Fabrication
Laminated Components
The mechanical design incorporated the model optimizations and was done with an
emphasis on the final product, which will ultimately be the portable thermoacoustic
engine. As previously discussed in Section 3.1 and Section 3.2.3, the core material had
to be laminated steel to suppress significant eddy currents, which would otherwise
dominate the system.
This led to the design of the laminated steel wedges.
All
laminated components were manufactured by Polaris Laser Laminations. The number
of wedges was chosen so that the 90 degree cuts made by the laser cutter would align
during assembly of the wedges to make a single component. This worked to an extent,
92
(a)
(b)
Figure 3-22: Shown in (a) are the materials used for assembling the stator, and in (b) is a
closer view of the stator pieces as they were set in the delrin jig.
but the tolerance on the 45 degree wedge angle left a small gap where the wedge faces
met in the center of the inner air gap.
Engineering drawings for all components used in the VRG are provided in Appendix D. The dimensional accuracy held to the drawings for the laminated steel
components was
0.025 mm for critical dimensions. The thickness, as defined by the
dimension "w" in Figure 3-3, was within
0.15 mm". The poor dimensional toler-
ancing of the thickness dimension made fabrication of the VRG more difficult and
should be more precisely accounted for in future designs.
The four stator pieces were then assembled using Stycast 2850 epoxy and a right
angle delrin jig fabricated for this purpose as shown in Figure 3-22. The epoxy was
placed on the wedge faces and then pressed together.
A urethane release agent,
Camie 980, was used on the jig to ensure any epoxy that left the wedge faces would
not adhere to the jig. Delrin was used for the jig material because it has a natural
resistance to the epoxy adhesion. To ensure parallelism and flatness of the air gap
surfaces, the surfaces were held against a smooth glass surface. The epoxy was cured
in an oven at 200'F for 2 hours. A flat plate of aluminum was used as a weight to
ensure the air gap surfaces were held against the glass. The rubber gasket material
was placed between the aluminum and the stator pieces to ensure even distribution
93
Aluminum
Plate
Gasket
Material
Stator
PiecesC-lm
-lm
D elr in
Glass
r ac
Surface
Epoxy
Sample
Figure 3-23: This figure shows the stator assembly fixture system as it was baked to cure
the epoxy. The C-clamp was necessary for the fabrication of this stator to hold an accurate
stator width dimension.
of force as shown in Figure 3-23.
Type 316 stainless steel brackets were made to hold the stator as shown in Figure
3-24. These brackets, in the complete system, protrude through the piston ports,
allowing the stator to be held internally to the oscillating piston as discussed in Section
3.1. The stainless steel brackets were annealed to remove residual magnetism left from
the machining process. In the pre-annealed state, a small magnetic attraction could be
observed when a magnet was held against the brackets. The brackets were brought
above 10000C using a propane torch.
This method caused slight oxidation of the
stainless steel, but was a very quick method for getting rid of residual properties when
it was unimportant whether it oxidized or not. The post-annealed brackets revealed
no detectable magnetic attraction. This was done to prevent any magnetic flux from
permeating the stainless steel bracket, which would result in eddy current dissipation
due to its relatively large thickness dimensions compared to the laminated steel sheets.
Unfortunately, because of the poorly toleranced overall thickness dimension of the
laminated stator components, brass shims were necessary to fit between the set screw
and outermost steel laminations.
This had the added benefit, however, of allowing
the set screws to be set with a large amount of force without concern of damaging
94
Outer Air Gap
Surface
Stator Bracket
V
Shims
Figure 3-24: This figure shows the stator assembly with all components except the windings.
Glass
Surface
Gasket
Material
Stator
Assembly
Piston
Laminations
Piston Laminate
Fixture -"Piston Cap"
(a)
(b)
Figure 3-25: Shown in (a) is the method for ensuring parallelism between the stator and
piston laminations. Not shown in this figure is the gasket material which was placed between
the piston laminations and the piston cap. (b) Shows the piston laminations after assembly.
the laminations.
Thread-locking set screws were used to prevent vibrations from
loosening the brackets over time. One design change that should be implemented is
the use of one additional set screw on each side of the bracket in order to prevent
any rotational motion of the brackets. For this design, the steel laminations overhung
the bracket to prevent the bracket from sliding. This significantly strengthened the
brackets from sliding axially, but shims were again necessary above and below the
bracket because of the
0.025 mm tolerance. It was decided to not attempt a press-
fit for the brackets, which may have damaged the laminations or affected the more
important stator length dimension.
95
The piston laminations were set into an aluminum fixture. In the full design, this
aluminum, referred to as the piston cap, would provide the continuous surface necessary for the thermoacoustic gas spring pressures to act on as discussed in Section 3.1.
The aluminum fixture shown in Figure 3-25 part (a) is over-sized relative to the full
piston design and was used for experimental testing of one half of the VRG system.
To ensure optimum parallelism with respect to matching stator and piston components, strips of 1.5 mm rubber gasket material were placed between the laminated
piston components and the aluminum holding piece. The stator laminations were
then aligned to the piston laminations and weighted so that the compliant elastomer
allowed the piston to exactly conform to the stator air gap faces as shown in part (b)
of Figure 3-25 while the epoxy set and fixed the laminations in place. Rubber gasket
material was again used for pressure application above the stator assembly to ensure
even distribution of force on the piston laminations. Post fabrication analysis using
feeler gauges revealed that the minimum effective air gap length was 0.1 mm" when
the stator sat directly on the piston laminations.
3.5.2
Coil
The coil was wound using a mandrel fabricated out of nylon. Nylon was used to
prevent the adhesion of the epoxy, and was used in combination with a urethane
mold release. It was estimated that 160 turns could be made out of 16 gauge motor
wire, but ultimately 148 turns were all that could be fit to preserve a modest clearance
between the coil and laminated steel components. With more experience and precision
winding methods, the 160 turns is definitely attainable considering the gap that was
left between the coil and laminations. The coil was turned slowly on a lathe as shown
in part (a) of Figure 3-26, and a foot operated switch was used to manually turn
the lathe chuck on/off. A cycle counter was fixed to the end of the lathe to keep
track of the number of revolutions. The coil was wound while Stycast 2850 epoxy was
simultaneously applied to each layer of the coil. This particular epoxy was chosen
because of its excellent thermal properties.
After completing the winding process, a nylon sleeve was placed over the coil to
96
(b)
(a)
Figure 3-26: Shown in (a) is the lathe and a completed coil with the nylon cover over the
winding. (b) Shows the winding after the epoxy has cured and the cover has been removed.
The coil shown was a first attempt. In subsequent attempts, additional epoxy was used
leading to a significantly more continuitous epoxy coating.
Figure 3-27: This figure shows the stator assembly with one of the two phase coils. This
assembly was used for experimental validation of the VRG.
ensure that the coil was within the allowable dimensions set by the laminated stator
assembly. The lathe was then run continuously for several hours to ensure the epoxy
was uniformly distributed throughout the coil and did not shift due to gravity. After
97
16 hours of curing, the nylon cover was removed and the mandrel was separated into
its two pieces. The coil and mandrel are also shown in Figure 3-26 part (b).
The coil was then placed on the stator and epoxied in place with 2850 epoxy.
Only one side of the VRG was necessary for the experimental validation. In the full
generator, a second coil would be placed on the opposing side of the stator. The
second coil would be wound in a manner such that magnetic flux is driven in the
same direction as the first coil. Otherwise, significantly larger core losses will be
experienced as the full hysteresis loop will be traced in sections of the stator. The
full assembly of the stator used for experimental validation is shown in Figure 3-27.
The resistance of the coil was measured using a Hewlett Packard 34401A multimeter and a resistance of R = 0.253 Q was found. This value was subsequently
verified in future experimental tests by measuring the applied voltage and current in
steady state.
3.6
Chapter Summary
A design model of the VRG that accounts for saturation effects of the laminated
steel, the leakage and fringing fluxes, and the losses due to both resistance in the
coil and the core losses was generated. This model allowed the prediction of current
waveform and drive voltages that could be used to specify and fabricate the required
power electronics.
The losses associated with the electronic components including
the diodes and MOSFET transistors was not included in the model.
The model
was then used to optimize the specific geometry of the VRG system using laminated
steel components. Based on the optimization results, structural, and power output
requirements, the stator and piston were designed and fabricated. Chapter 4 provides
the experimental validation of the model and the overall VRG system.
98
Chapter 4
Experimental Design Verification
The models developed in the previous chapter were experimentally validated in two
ways: a static analysis for generation of flux linkage-current plots, and a dynamic
test to verify the power output projections of the model. Section 4.1 describes the
experiment setup and data collected for the generation of the flux linkage-current
plots. Section 4.2 describes the experiment design used for the model verification,
and Section 4.3 presents the results of the dynamic experiment.
4.1
Saturation Characterization
The static experiment was conducted to produce the flux linkage-current characteristics of the generator at fixed air gap lengths. The air gap is the free-space between
the piston face and stator pole as shown in Figure 3-5 labeled as the inner air gap
and outer air gap. Displacement of the piston causes an an equivalent increase or decrease in the air gap. The flux linkage-current plots are necessary for understanding
the magnetic forces and work output capability of the generator as discussed in Section 3.2. To generate these plots, the generator stator and piston were aligned, and
plastic shims were used to accurately gauge the air gap thickness as shown in Figure
4-1. A voltage was then applied across the windings and the winding current and
voltage were simultaneously measured and used to determine the flux linkage-current
characteristics at each air gap length.
99
Stator
Winding
Plastic Shims
Piston
F _____________________
Laminations
Figure 4-1: This figure shows the setup used for generating the flux linkage-current plots
for different air gap lengths set by the plastic shims.
To ensure accuracy of the measurements sources of potential error were identified
and mitigated if possible. Plastic shims (orange/yellow in Figure 4-1) were necessary
to prevent the generation of eddy currents in the shim material, which would cause
the maximum inductance to appear larger than the true maximum inductance. Another source of error for the experiment may have come from elastic deformation of
the plastic shims due to the magnetic forces The elastic deformation would tend to
decrease the thickness of shims to less than their stated thickness.
The error due
to elastic deformation was calculated to be less than 1% by assuming a 275 Newton compression distributed across the air gap surfaces. This error was considered
negligible and was subsequently neglected. If the shim materials were not initially
flat, the air gap would be larger in some sections than the stated shim thickness until
the magnetic attraction forces between the piston and stator poles flattened each gap
to exactly the shim thickness. To minimize the effect of "bowed" shims, approximately 10 pounds of pre-load was applied to compress the gap uniformly to the shim
thickness.
With no shim in place, the stator poles rest directly on the face of the piston
laminated pieces. However, due to small deviations in the laminations themselves
or limitations in the manufacturing process, the plane formed by the stator poles is
not directly parallel to the plane formed by the faces of the piston laminations. For
100
Gate Driver
2-1
Winding
Resistance
Supply___V
V.
Phane
Winding
Figure 4-2: This figure depicts the circuit used for driving the currents through the winding
to measure the flux linkage-current characteristics at various air gap lengths. The diode
was necessary to allow the current to freewheel after the MOSFET "switch" was opened to
prevent charge buildup and extreme voltages across the MOSFET component.
this reason, the piston and stator were placed in direct contact, and the resulting
gap between each stator pole and piston face was measured using feeler gauges. The
measured gap was found to be up to 0.1 mm. The tolerance of the laminations themselves were +0.025 mm, which indicates the fabrication process described in Section
3.5 potentially added an additional 0.075 mm to the gap between the stator and
piston. This gap dimension also represents the smallest possible air gap used in the
calculation of the air gap reluctance. This corresponds to the maximum inductance
and is referred to as the zero air gap in this document. This is different than the
term minimum air gap, Gmin, which is used in Sections 4.2 and 4.3 to refer to the
minimum gap achieved during the cyclic motion of the piston as it moves closer and
further from the stator.
The circuit shown in Figure 4-2 was used to drive the current through the winding
at fixed gap displacements. A LabVIEW program was used to send a single pulse
to the gate driver, which subsequently drove the MOSFET to act as a closed switch.
Current was then driven through the winding by the power supply until the LabVIEW
pulse ended and the MOSFET switch was "opened". Due to the inductance of the
winding, the current in the coil then freewheeled around the path constituting the
diode and winding until the current was dissipated in the winding.
The same gate driver, MOSFET, and diode were used for both the static exper101
Current Profile for Flux Linkage-Current Plots
Voltage Profile for Flux Linkage-Current Plots
6.5
12-
6-
10
5.5
Interal Impedance
Voltage Drop
8
5--
4.5.
4
4-
2
3.5
0
0.002
0.004
0.006
0.008
Time (s)
0.01
0.012
0.014
0
(a)
0.002
0.004
0.006
0.008
Time (s)
0.01
0.012
0.014
(b)
Figure 4-3: This figure depicts (a) the applied voltage to the phase winding as a function
of time. The integral of this voltage minus the IR component provided the flux linkage.
(b) The current profile for the static test at zero shim gap is shown. The current changes
slowly until the steel begins to saturate at about 6 ms.
iment and the dynamic drive circuitry presented in Section 3.3. An HP 6428B DC
power supply was used to drive up to 14 amps of current through the winding at 6.38
Volts. The voltage was chosen so that the maximum current measured would be approximately 20% greater than the expected peak current for the dynamic operation of
the generator. Additionally, the voltage drop due to the resistance of the winding was
less than the inductor emf driving voltage for all currents. Therefore, the experiment
was terminated when the voltage drop across the winding was approximately equal
to the IR drop, where I is the current and R (0.25 Q) is the winding resistance.
A Tektronix MSO2014B oscilloscope with a passive voltage probe and a TCP0020
current probe were used to simultaneously measure and record the voltage and current. The current data was put through a low-pass (fcutoff = 1 kHz on 300 kHz
sampled data), zero-phase shift filter to smooth out the high frequency components
due to the ADC resolution of the oscilloscope.
The voltage applied minus the IR
voltage component was integrated to determine the flux linkage. Figure 4-3 shows
the voltage and current profiles typical for the experiment. It can be seen that the
102
Flux Linkage-Current Plots
0.07
0.06r
-
0"
0.001"
-
0.002"
-
0.004"
0.0075"
0.01"
-
0.05
-
0.015"
0)0.04
0.02"
0.025"
0.03"
-
0.03
-
0.04"
-
0.05"
-
0.06"
--
0.02
-
-
0.08"
0.1"
0.12"
0.15"
0.18"
0.22"
0.01
U O 0 2
4
6
8
en
2p4 6
Current (A
Figure 4-4: This figure shows the flux linkage-curr ent plots at a range of air gap displacements measured experimentally.
voltage measurement is significantly more noisy than the current measurement, but
this does not significantly alter the flux linkage measurement due to the smoothing
effect of the integration. Also shown in part (a) of the figure is a slight voltage drop
(6.4 -> 4.8 V), which can be attributed to the internal impedance of the power supply.
Plotting the integrated voltage data with the filtered current data at every point
led to the flux linkage-current profile for a single air gap length. Data was collected
for 37 air gaps lengths, 19 of which are shown plotted in Figure 4-4. As predicted,
the steel saturates at approximately a constant flux linkage instead of a constant
current, which can be seen in the figure noticing that each line becomes nonlinear at
approximately the same flux linkage value (0.038 Wb).
A comparison of the actual flux linkage-current plots to that of the model revealed
that at larger air gaps the model deviated substantially from the actual data. The
model predicted a minimum inductance (LMin
=
1.75 mH with air gap G = 5.6 mm)
that is significantly less than the experimentally measured minimum inductance
103
Flux Linkage-Current Plots Predicted vs Measured
F
0.08-
0.08---
---Predicted
rd.ed
0.07-
---Measured
-- -Measured
0.04
0.07-
0.07 -**
0.07
-d
-
0.05
0.05
c-0.02
0.01
Flux Linkage-Current Plots Predicted vs Measured
0.0
*
0.0
0.02
--
J.0.01
0
10
15
Current (A)
(a)
0
5
10
15
Current (A)
(b)
Figure 4-5: This figure depicts (a) the original model compared to the experimentally determined flux linkage-current plots. The model significantly underestimates the magnitude
of the fringing fluxes. The flux linkage plots are for gaps ranging between 0 - 3.81 mm. (b)
This plot shows the model with the permeances of the flux path increased by 40%, which
far more accurately models the true flux linkage characteristics.
(LMi, = 2.6 mH with air gap G = 5.6 mm). This suggests that the effect of the
fringing fluxes is larger than what was predicted by the flux tube analysis. Increasing
the effective permeance of each flux path by 40% allows the model better match the
measured flux linkage characteristics of the generator. These results are shown in
Figure 4-5, where part (a) of the figure shows the experimental data and the model
predictions without the fringing correction factor and part (b) shows the experimental
data and the model predictions with the fringing correction factor. The accuracy of
the model is significantly improved in the linear region as evidenced by the closeness
of the predicted and measured lines in the figure, but there are clear limitations of
the piecewise-linear approximation in the saturated region. The rounded saturation
region is poorly matched by a single saturated incremental inductance, which leads to
inaccuracies when predicting power output, efficiency, magnetic forces, and current
waveforms using the piecewise-linear model.
It is now possible to create an improved model for predicting the generator charac-
104
teristics using the measured flux linkage-current data. The data can be fit and a new
model generated based on the actual flux linkage-current characteristics of the generator. This approach is described in detail in [29]. From this model, the power output,
forces, and efficiency can be predicted with greater accuracy demonstrated in [29].
The piecewise linear model is useful for making the correct design decisions before
the flux linkage-current data is available, but now that the data is available, it is left
as future work to generate a new model based on the measured flux linkage-current
data.
To verify the accuracy of the data collected, the approximate B-H curves for the
steel used in the magnetic core were generated based on the flux linkage-current measurements. This was done by using the zero air gap static test, where the piston steel
was placed in direct contact with the stator poles. In this case, the flux loop is nearly
continuous but with a small additional air gap reluctance due to the manufacturing
limitations of the magnetic core. The B-H curve produced by this method is only an
approximation because the cross-sectional area of the flux loop is not constant in this
design of the generator core and the small unavoidable air gap still has a significant
effect on the flux loop compared to that of a continuous toroid. The flux density, B,
and magnetic-field intensity, H, are given by the equations
A
B
(4.1)
NA
H =
(4.2)
H1
where A is the cross-sectional area of the steel, lc is the mean core length, N is the
number of turns in the coil, A is the flux density, and I is the current in each wire of the
coil. The B-H curves calculated by this method and published data for M19 29-gage
steel are provided in Figure 4-6. The green plot represents the calculated values for the
B-H curve assuming the magnetic loop has everywhere the cross-sectional area of the
stator (A = 2.71
i 2 ).
The red plot represents the calculated values for the B-H curve
assuming the loop cross-sectional area is that of the piston (A = 1.63
2
), and the
blue plot is the published B-H curve data. As expected, the actual B-H saturation for
105
= -R)
Comparison of 0 Air Gap B-H Data to Published Data
3
2.5-
2
1.5-
0
-
0.5-
0
4000
8000
12000
16000
H (A/m)
Figure 4-6: A plot of the calculated B-H curves based on the zero air gap test of the
flux linkage-current plot for both the maximum and minimum cross-sectional areas of the
laminated generator components. The data cannot be fit directly to the published data
because the cross-sectional area is not constant, and therefore, the published data lies at
some mean cross-sectional area. Additionally, the approximate minimum air gap can also be
calculated based on the difference in inductance between the published and acquired data.
This minimum air gap was calculated to be 0.066 mm, which is reasonable considering the
physical measurements done for this zero air gap discussed earlier in this section. Published
curve from [9].
published data lies somewhere between the maximum and minimum cross-sectional
areas of the generator data because the saturation is ultimately affected by both steel
sections.
It is also possible to calculate an approximate minimum effective air gap when the
stator and piston are pressed directly together. The inductance in the linear region
of the B-H curve is relatively unaffected by the cross-sectional areas. Therefore, the
inductance of the published data and the experimental data were calculated, and
using Equation 3.6, the air gap reluctance was also calculated. The approximate air
gap length is then given by
(4.3)
32
N
L
where Ag is the area of the inner and outer air gaps and R is the reluctance of the
106
steel as calculated from the inductance of the published data. The approximate minimum air gap calculated from this analysis is 0.066 mm. This value fits well with
the measured 0.1 mm air gap using the feeler gauges, and is reasonable considering the laminations themselves were manufactured with dimensional tolerances of
+0.025 mm. From this analysis, it was concluded that the static experiments were
accurate and the data could ideally be used for more accurately modeling the dynamic
performance predictions for the generator.
4.2
Experiment Design
To experimentally validate the electroacoustic transducer, the added complexity and
control of the thermoacoustic engine was removed. The initial proposed design for
testing the VRG at the 250 Hertz operating point was to set up a resonant system
using two gas springs and the center piston, driven to resonance by a shaker table.
Mechanical springs are not an acceptable substitute for gas springs because the spring
constants required for 250 Hertz vibrations and 5 mm peak-to-peak oscillation amplitudes of a 0.2 kg piston are on the order of 250, 000 N/m. The size of the spring
necessary for this spring constant creates a situation where the spring mass is as
significant as the piston mass for the oscillation. The resonant frequencies are very
difficult to obtain in this situation. For this reason, it was determined that the gas
spring system was necessary for testing the electroacoustic transducer.
The resonant system was necessary to test the VRG system at the proposed 250
Hertz operating frequency, because the shaker table cannot directly drive the generator at the required amplitudes at that frequency. However, without a gas bearing
system, another type of bearing and clearance sealing method would be required.
This added complexity was not feasible for the time-scale of this project and was not
necessary for initial validation of the VRG system and VRG model. For this reason, a simpler and less costly experiment was developed to test the VRG by directly
driving the VRG system with the shaker table. With this method only one phase of
the two-phase generator was tested and at a significantly lower frequency than for
107
the original design operating point. The lower frequency experiments were ultimately
used to test the model, the power output of the generator and the efficiency. The
final power output and efficiency could be reasonably determined by scaling the experimental data to higher frequency using the known loss mechanisms as discussed
in Section 4.3.1.
The dynamic tests were done such that the stator was fixed above the shaker
table, and the piston laminations were driven with a sinusoidal motion by the shaker
to change the inductance of the magnetic flux path. A detailed schematic of the VRG
dynamic test setup is shown in Figure 4-7. The shaker used for the test was a Ling
Dynamic Systems (LDS) model V722. The shaker runs off of a closed-loop control
cycle with the accelerometer readout fed back to the digital sine controller.
Figure 4-8 shows the physical setup of the dynamic tests. The Z-axis control of
the shaker, which is the axis running along the center-line axis of the shaker shown
in the figure, was used to manually set the minimum air gap by changing the gas
pressure below the moving shaker mass. Increasing the pressure below the shaker
mass raises the mean position of the shaker element and attached generator piston.
This decreased the gap between the piston and stator and subsequently decreased the
minimum air gap. The term "minimum air gap" is meant to indicate the distance
between the piston and stator air gap surfaces when the piston is at its extreme
position closest to the stator.
This minimum air gap strongly affects the power
output and efficiency of the generator because the magnetic forces are much greater
with the same winding current for smaller air gaps compared to larger air gaps.
More precise control of the Z-axis control of the shaker and subsequent minimum
air gap parameter would have been beneficial but was not possible using the shaker
system. To compensate for the poor precision of this variable, data was taken over
a range of minimum air gap lengths and data was then compared with other data at
similar minimum air gap lengths.
The displacements and corresponding air gap lengths were measured using a
Philtec D170 fiber-optic displacement sensor. The voltage readout of the displacement sensor was input to a PCIe-6361 National Instruments data-acquisition (DAQ)
108
board. The PCI board was necessary to achieve low latency measurement and control
of the generator. The pulse for the gate driver discussed in Section 3.3 was triggered
by a rising analog level trigger on the DAQ board. The trigger occurred because the
voltage of the fiber-optic sensor increased toward its optical peak as the piston moved
closer to the stator. Therefore, the voltage was switched on across the winding when
the piston reached a certain displacement on its path moving toward the stator. This
trigger point effectively set the turn-on angle, 0n, of the generator
The pulse duration was set in the LabVIEW program and precisely controlled by
the on-board 100 MHz time-base counter on the DAQ board. The turn-off angle,
Gff, is set by the pulse duration. The pulse is sent to the MOSFET gate driver so
that the MOSFET is conducting when the pulse is high and an open circuit when
the pulse drops to low. Both high and low sides of the circuit where driven with the
same pulse signal.
Figure 4-8 also shows the method by which the plane formed by the stator poles
were made parallel to the plane of the piston laminations. An XY stage and shims
placed between two flat aluminum plates above the stator were used to fix four degrees
of freedom including the X and Y translation and rotation about the X and Y axes.
The rotational degree of freedom about the Z-axis was adjustable by screws on the
aluminum component which held the piston laminations. The Z-axis translation was
the degree of freedom along which the shaker motion occurred, with the mean position
of the plate set by adjusting the mean gas pressure under the shaker plate.
Four analog measurements and one digital signal were measured simultaneously
using the mixed signal oscilloscope. The digital signal measured was the LabVIEW
pulse signal sent to the gate driver on the PCB drive circuit. This signal was used as
the trigger for the oscilloscope measurements. The first channel of the oscilloscope was
used to measure the voltage readout of the displacement sensor directly so that the
displacement amplitude and minimum air gap could be calculated for data analysis.
To determine the displacement from the voltage readout, the exact characteristics of
the displacement sensor had to be measured. A stainless steel mirror disk was placed
on the shaker element as the target for the fiber-optic sensor. The voltage readout of
109
r
mr -
Physical Connection
- - - - - Measurement
Set Test Parameter
S-r.
m Magnetic
Unity Gain
Differential
Force
A
>
Mixed Signal
Oscilloscope
(Tektronix MS2014B)
-
---
- -- -- -- --
lIf;m per (1INA105)
Ca pacitor Bank
(90000 VF)
Current Probe
(TCP0020)
Set Frequency
& Amplitude
V
Wi
A
W2
AArWwA'
Vl
LDS Digital Sine
Controller (DSC4)
Drive Circury
Conl (1,2)
Cn
,2
W3
Set Displacement
Trigger and
Pulse Duration
i
LDS Power
Charge
Amplifier (PA1000)
Accelerometer
(PCB J357B01)
Gener ator
Pisto.n I
Fiber-optic
Displacement
Sensor (D170)
LDS Vibrat )r (V722)
LDS Field Power
Supply (FPS1)
Connector Block
(NI SCC-68)
LabVl EW
Software
DAQ Board
(NI PCIe-6361)
Desktop Computer
(T5600 Dell)
Vibrator Manual
Z-axis Control
Figure 4-7: This figure details the experimental setup for testing the VRG dynamically.
The data from the oscilloscope was saved and exported directly from the oscilloscope to
a computer for analysis. The orange lettering refers to the connection points on the PCB
schematic shown in Figure 3-17.
-
XY Stage
-
Current Probe
Shims
Z-axis Stage
M PCB
Fiber-Optic
Sensor
-
Mirror Target
Drive Circuitry
Stator/Winding
Piston
M Accelerometer
Shaker Plate
Figure 4-8: This figure shows the physical setup of the dynamic test used for validation of
the VRG. The XY stage and shims were used for aligning the piston and stator. The Z-axis
stage was used to calibrate the displacement sensor.
110
the optical peak of the fiber-optic sensor was set to 9 volts. The zero air gap between
the piston and stator was set to 8.75 V, which is at the upper end of the linear backslope region of the fiber-optic sensor. The slope of the linear region was measured
to be 481.68 mV/mm. The sensor experienced some drift and it was necessary to
allow approximately 30 minutes for the sensor to warm-up , and periodic checks were
necessary to make sure the optical peak and zero air gap voltages had not shifted so
that data was comparable from test to test. This displacement measurement and the
corresponding true minimum gap dimension were the greatest source of uncertainty
because the drift of the displacement sensor could be up to
50 mV, which is ap-
proximately +0.1 mm. This is particularly relevant when comparing experimental
data to the model, where the minimum air gap has a significant affect on the power
output and efficiency of the generator.
The second channel of the oscilloscope was used to measure the voltage across
the generator winding terminals. This measurement was a floating voltage with no
reference to ground, considering that the applied voltage would switch from positive to
negative each cycle. The voltages were applied across a unity gain differential amplifier
(INA105), which allowed the voltage difference between the winding terminals to
be measured with respect to ground.
This was necessary because the 4 channels
of the oscilloscope must share the same ground reference. The third channel of the
oscilloscope was used to measure the current through the winding using the TCP0020
current probe, and the fourth channel was used to measure the voltage on the bank of
capacitors, which were used to store the energy necessary to drive the initial current
in the winding.
To dissipate the power converted from the mechanical motion to electric power,
5 watt zener diodes were reverse biased from the capacitor bank to ground as shown
in Figure 4-9. In this configuration, the zener diodes acted as voltage limiters and
dissipated any excess charge driven into the capacitors. A heat sink was manufactured
to prevent the diodes from overheating.
From this experimental setup, the work output per cycle was determined by calculating the area of the flux linkage-current loop based on the winding applied voltage
111
Gate Driver
Wnding
Zener
5W
a citor Bank
Zener
5W
Gate Drivere
Figure 4-9: The zener diodes were used in this configuration to dissipate the electric power,
while maintaining a constant drive voltage.
and the current passing through the winding as measured with the current probe.
The true power output was calculated based on the winding current and the voltage
of the capacitor bank. These measurements were taken for various turn-on and turnoff angles, and frequencies to characterize the power output potential and conversion
efficiency of the VRG. These results are presented in the following section.
4.3
Dynamic Experimental Results
For the dynamic tests of the VRG, the piston was oscillated at 50 and 60 Hertz with
a peak-to-peak amplitude of 5 mm. The power output is significantly lower for these
frequencies than the proposed 250 Hertz thermoacoustic engine operating frequency,
but the model and VRG performance can still be characterized. Using the trigger
voltage and pulse duration as the control variables, the model predicts power and
efficiency curves as shown in Figure 4-10. Although the clockwise integration of the
flux linkage-current loop produces a negative work output per cycle, the total power
output referred to in this thesis is the absolute value of the work output for one cycle
multiplied by the operating frequency. This was done to eliminate ambiguity with
negative powers. Thus, power output is the generated power that is converted from
mechanical oscillations to electric power.
The turn-on angle is set by the trigger voltage and maximum/minimum voltages
112
Predicted Efficiency Curves for Fixed Turn-On Angles
.I
3.
Predicted Power Curves for Fixed Turn-On Angles
5
.
S7.
Turn-On Angle=2.71
Turn-On Angle=2.54
Turn-On Angle=2.41
Turn-On Angte=2.29
- Turn-On Angle=2.18
Turn-On Angle=2.09
-
0.6
0.5k
rad
rad
rad
rad
rad
rad
-
3 _
Turn-On Angle=2.71
Turn-On Angle=2.54
Turn-On Angle=2.41
Turn-On Angle=2.29
Turn-On Angle=2.18
Turn-On Angle=2.09
rad
rad
rad
rad
rad
rad
2.5
0.4
2
0
w 0.3
0
0.2-
01
0.1
0
1.5
0.5
10
20
30
Pulse Duration (% of Cycle)
40
0
50
10
(a)
30
20
Pulse Duration (% of Cycle)
40
50
(b)
Figure 4-10: This figure depicts (a) the predicted efficiency of the VRG operating at 50Hz
with a minimum air gap of 0.5 mm and a 10 volt driving voltage. The curves are shown
for fixed turn-on angles while varying the pulse duration. (b) Shows the predicted power
curve for the same operating conditions. There are clear peaks in both of these plots with
the maximum efficiency occurring at shorter pulse durations than the maximum power.
for one cycle as determined by the displacement sensor. The turn-on angle (in radians)
is then given by
GOn = 7r - COS 1 ((VTrigger
VMax + VMin
2
2
VMax - VMin
(4.4)
The pi term comes from the 180 degree shift between the displacement sensor and the
minimum gap, where the maximum displacement voltage corresponds to the minimum
air gap.
As seen in Figure 4-10, there is a local optimum for power and efficiency for each
turn-on angle. The shorter the pulse duration and the closer the turn-on occurs to
the minimum air gap (Gmin occurs at 0 = r) the less power is lost due to the winding
resistance, but this also means the magnetic attractive forces are lower and power
output is diminished because there is less time for current to build up in the coil.
When the pulse duration is long and the turn-on angle is significantly before the
113
minimum air gap occurs, the winding losses dominate additional power output gains
and an overall optimum in power output can be found. This can be seen in part
(b) of the figure moving from the peak output power of the purple line (Turn-On
Angle = 2.18 radians) to the peak output power of the yellow line (Turn-On Angle =
2.09 radians), where the additional winding losses have caused the maximum output
power to decrease even though larger currents (larger magnetic forces) are present.
The model does not include the power loss due to the drive circuit components,
such as the MOSFETs and diodes, and also neglects the resistance of the wires and
PCB traces, with the exception of the copper wire used in the winding itself. The
core losses were neglected for the data analysis in this section, but are considered in
greater detail in Section 4.3.1. For this reason, to get an accurate comparison of the
model to the experiment data, the power output is considered the total power output
minus the winding losses, where total power is the area of the flux linkage-current
loop (cycle work output) multiplied by the frequency. The predicted power output
is the model predicted total power output minus the model predicted winding losses.
This method provides a direct comparison of experimental data to model predictions
by essentially neglecting losses in the experimental data not accounted for in the
model. The efficiency is then defined as the power output divided by the total power
output, and the predicted efficiency is the model predicted power output divided by
the predicted total power output. When necessary, the actual power dissipated by
the zener diodes is specified as true power output, and the true efficiency is the true
power output divided by total power output.
Shown in Figure 4-11 are a few of the characteristic flux linkage-current loops for
the collected data as well as the model predicted loops. These plots were generated
by integrating the winding differential voltage to get flux linkage values and the flux
linkage and current data were then plotted at each incremental measurement, which
traces a clockwise path around the loop for each cycle. Comparison of the model and
data was done by specifying identical parameters in the model as those measured in
the data. These parameters included the turn-on angle, pulse duration, frequency,
applied voltage, and minimum air gap.
114
There are a few possibilities for the deviation between the predicted and actual
flux likage-current loops. For the 50 Hertz tests, the model tended to overestimate
the maximum flux linkage. This was caused by the capacitor bank shown in Figure
4-9 having a limited capacitance. The voltage drop during the winding charging stage
of the cycle shown in Figure 3-18 caused by the finite capacitance led to a smaller
maximum flux linkage than predicted by the model. To try and reduce this error, the
capacitance of the capacitor bank was increased from 50, 000 to 90, 000 pF between
the 50 and 60 Hertz tests.
Shown in part (b) of Figure 4-11 is the flux link-current loop for a 60 Hertz test
which was driven into saturation. The divergence between the model and the data in
this instance is clear in the saturation region, where the inductance of the model is
less than the actual inductance. This error is caused by the piecewise linear approach
as shown in Figure 4-5, and is an inherent limitation of the model approach used in
this thesis.
The minimum air gap was the third major contributor to differences between
the predicted and actual flux linkage-current paths. This was particularly true for
measurements with a short pulse duration because small deviations in the maximum
inductance significantly changed the initial trajectory and subsequent path around
the flux linkage-current loop. These instances are relatively unimportant because
little power can be generated with short pulse durations.
The total power, power output, and efficiency were then calculated for each measurement. The results of the 50 Hertz tests for power and efficiency are shown in
Figures 4-12 and 4-13 respectively. The data is categorized by the minimum air gap,
where the red circles are data points for when the piston at its closest point to the
stator during the cycle was less than 0.4 mm from the stator air gap surfaces. The
green X's are for minimum air gaps between 0.4 mm and 0.45 mm, the black crosses
for minimum air gaps between 0.45 mm and 0.5 mm, the blue stars for minimum air
gaps between 0.5 mm and 0.55 mm, the blue squares for minimum air gaps between
0.55 mm and 0.6 mm, and the magenta diamonds for minimum air gaps greater than
0.6 mm. This notation is constant in all of the plots of experimental data on power
115
Predicted vs. Measured Flux Linkage-Current Loop
Predicted vs. Measured Flux Linkage-Current Loop
0.045
0.035-
0.04
0.03
0.035
0.025
0.03
---
Predictd
0.025
-
-
0.02
X
X
x
x 0.02-
0.015
-
0.01
0.01
0.005
0.005
0
4
00
2
6
4
Current (A)
Current (A)
(a)
(b)
8
10
12
Figure 4-11: This figure shows (a) the measured versus predicted flux linkage-current loops
for a 50 Hz test at a turn-on angle of 2.65 radians and a 22.5% duty cycle, and (b) a 60 Hz
test driven to saturation with a turn-on angle of 2.1 radians and a 27% duty cycle.
and efficiency.
Looking at the first plots in Figures 4-12 and 4-13, there are clear bands of power
and efficiency with only a few outliers. These bands show the trend that both the
power and efficiency increase as the minimum air gap is decreased. This suggests that
the minimum air gap has a strong effect on the power output and efficiency of the
engine, which makes sense considering the magnetic forces drop with approximately
the square of the air gap length. Therefore, if the gap is on average smaller, the
generator will convert more of the mechanical oscillation to acoustic power and will
do so at smaller winding currents. The effect of the minimum air gap is considered
in more detail later in this section.
Also shown in these plots are the local maximums in power output and efficiency
as predicted and shown in Figure 4-10. The efficiency peak occurs at a pulse duration (duty cycle) shorter than that of the peak power point, which is also expected
considering the increase in winding losses when larger magnetic forces are generated.
For the first plots in Figures 4-12 and 4-13, the peak power (2.1 W) occurs at a
duty cycle of 22.5%, whereas the maximum efficiency (0.677) occurs at a duty cycle
116
50 Hertz Power Output Data
Turn On Angle (Approx. 2.8 rad)
Local Power Maximum
2. 5
x *-Small
2
0.
Gmin
x
2
+
2. 5
Turn On Angle (Approx. 2.65 rad)
3
3.
1. 5
1. 5.
0
1
1X
0. 5
0. 5
Large Gmin
K,,
+
01
-0.
10
15
20
-0. 5
30
25
0
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.55 rad)
3
2. 5
2 .5
+
3
1. 5
1 .5*
.
2
2.
+
0
15
20
30
25
20
15
10
Turn On Angle (Approx. 2.2 rad)
+
*
1.5
1
0. 5
0
0
20
25
+
2
0.5
15
-0.5
30
Mn Ar Gap <0.4mm
0.4mm <Min Air Gap <0.45mm
0.45mm <Min Air Gap <0.5mm
0.5mm -Min Air Gap <0.55mm
0.55mm <Min Air Gap <0.6mm
0.6mm <Min Air Gap
.
oX
2.5
1
10
0
Turn On Angle (Approx. 2.3 rad)
1. 5
-0. R
0
X
Pulse Duration (% of Cycle)
2
0
0
Pulse Duration (% of Cycle)
Power Maximum---*-
2. 5,
-0 .51
30
25
+
10
3.
0
A-
0 .5
0 EE
CL
Turn On Angle (Approx. 2.41 rad)
1
-
1
0. 5.
-V.
30
25
20
15
10
Pulse Duration (% of Cycle)
*
L
0
0
I
8
10
15
0
10
0
20
25
30
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
Figure 4-12: The power output for a range of turn-on angles and pulse durations are shown
for the dynamic testing of the VRG at 50 Hertz. The power output is the absolute value of
the work per cycle minus the winding losses multiplied by the operating frequency. Data
is grouped by minimum air gap to show the correlation between the minimum air gap and
the power output. The legend in the bottom right figure applies to all plots.
117
50 Hertz Efficiency Data
Turn On Angle (Approx. 2.8 rad)
Turn On Angle (Approx. 2.65 rad)
Local Efficiency Maximum
0.8
Efficiency Maximum
0.8
0.6
0.6
0.4
E
Minimum Air
Gap Bands
0.4
0.2
0.2
0
0
Outlier
-0.2
-0.2
10
15
20
25
30
10
15
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.55 rad)
1
20
25
30
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.41 rad)
1
0.8
0.6
0.6
0.4
0.4
+
0.8
0.2
0.2
0
0
-0.2
10
15
20
41--
25
0
-0.2
30
15
10
Pulse Duration (%of Cycle)
20
25
30
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.3 rad)
Turn On Angle (Approx. 2.2 rad)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
0
O
+
Gap <0.4mm
0.4m <Min Air Gap '045mm
0.45mm <Min Air Gap '0.5mm
0.5mm -Min Air Gap <055m
0.55mm 'Min Air Gap '06mm
0.6nmm -Min Air Gap
Min Air
-0.2
10
15
20
25
30
10
Pulse Duration (% of Cycle)
15
20
25
30
Pulse Duration (% of Cycle)
Figure 4-13: The efficiency for a range of turn-on angles and pulse durations are shown
for the dynamic testing of the VRG at 50 Hertz. The efficiency is the useful work output
per cycle over the total cycle work. Data is grouped by minimum air gap to show the
correlation between the minimum air gap and the efficiency. The legend in the bottom
right figure applies to all plots.
118
of 15%. These local maximums are shown for specific turn on angles in each plot,
but overall maximums can be seen as predicted by the model and shown in Figure
4-10. Therefore, the maximum power for the 50 Hertz tests (2.52 W) occurred with
a turn-on angle of 2.3 radians and a pulse duration of 27.5% of the cycle. The maximum efficiency point (0.69) occurred with a turn-on angle of 2.65 radians and a pulse
duration of 17.5% of the cycle. These points are labeled in the Figures 4-12 and 4-13.
The 60 Hertz data shown in Figures 4-14 and 4-15 shows the same trends as the
50 Hertz data but with an increase in both power and efficiency. For an increase in
frequency from 50 to 60 Hertz, an increase of greater than 20% in the power output of
the engine is expected. The increase is expected to be greater than 20% because the
generator losses such as the winding loss and circuit board component losses remain
constant while the net power output gain scales linearly with the operating frequency.
This is discussed further in Section 4.3.1. Comparing the data, there is approximately
a 50-60% increase between the 50 and 60 Hertz operation. For example, comparing
point "A" labeled in Figure 4-12 and point "B" labeled in Figure 4-14, the points are
at the same turn on angle and have approximately the same duty cycle and minimum
air gap. For these two points, the 60 Hertz power output (3.3 W) increased by 51%
compared to the 50 Hertz power output (2.18 W).
Also of interest is the comparison of the experimental results to the model predictions. To compare the results to the model, the results of data points that fell within
0.075 mm of the 0.5 mm minimum air gap were averaged and individual power and
efficiency curves were created for each turn-on angle. Figure 4-16 shows the difference
for the 60 Hertz tests between the predicted power output, the power output, and the
true power output. These results show qualitative agreement for the peaks in power
output. The quantitative agreement is decent between the predicted power output
and the power output and suggests that including the drive component loss and a
more accurate core loss prediction would predict the true power output with much
greater accuracy. The average error between the model predicted and actual power
output is 29.7%. This percent error for the generator output power is exaggerated
because at the 50 and 60 Hertz frequencies where tests were conducted, the winding
119
60 Hertz Power Output Data
Turn On Angle (Approx. 2.75 rad)
Turn On Angle (Approx. 2.52 rad)
.
6
5.
5
4-
4.
3
0
2
0
3.
0
-
0.
0
2
1
0
0
10
20
15
30
25
6.
6
5.
5.
4.
25
30
35
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.41 rad)
CL
20
15
10
35
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.3 rad)
.5
+
B
0
0.
3
[7
0
0
6)
,
2.
3
2
0
0~
,
1
0
0
10
15
20
25
30
15
10
35
Turn On Angle (Approx. 2.19 rad)
6.
o
5
X
+
*
4.
30
35
0.
3.
0
2
(D
3t
0.4mm
x
Mi Air Gap
0.4mm <Min Air Gap <0.45mm
0.45mm <Min Air Gap <0.5mm
0.5mm 'Min Air Gap <0.55mm
0.55mm -Min Air Gap <0.6mm
0.6mm <Min Air Gap
2
-
4)
25
Turn On Angle (Approx. 2.1 rad)
6.
5.
0
20
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
0
0
1
0
0
10
15
20
25
30
10
35
15
20
25
30
35
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
Figure 4-14: The power output for a range of turn-on angle and pulse durations are shown
for the dynamic testing of the VRG at 60 Hertz. A comparison of this figure to Figure 4-12
reveals that the power output is greater for the 60 Hertz tests as expected. The data is
again separated by the minimum air gap length.
120
60 Hertz Efficiency Data
Turn-On Angle (Approx. 2.52 rad)
Turn-On Angle (Approx. 2.75 rad)
0.8
0.8
0.6
r-
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
10
15
20
25
-0.2
30
S+
15
10
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
10
10
15
15
20
20
25
25
30
Turn-On Angle (Approx. 2.3 rad)
Turn-On Angle (Approx. 2.41 rad)
1
25
20
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
-0.2
30
-~
15
10
25
20
30
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
Turn-On Angle (Approx. 2.1 rad)
Turn-On Angle (Approx. 2.19 rad)
1
0.8
0.8
0.6
0.6
-
1
O
0.2
0.2
X
+
*
E
0
0
-0.2
$1
0.4
10
15
20
25
-0.2
30
10
+
I
0.4
Min Air Gap <0.4mm
0.4mm <Min Air Gap <0.45mm
0.45mm <PMn Air Gap <0.5mm
0.5mm <Mn Air Gap <0.55mm
0.55mm <Min Air Gap <0.6mm
0.6mm <Min Air Gap
15
20
25
30
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
Figure 4-15: The efficiency for a range of turn-on angle and pulse durations are shown
for the dynamic testing of the VRG at 60 Hertz. Comparing the data in this figure to
Figure 4-13 reveals that for a given turn-on angle, the efficiency is slightly greater for the 60
Hertz operation than the 50 Hertz operation. This is also expected and discussed further
in Section 4.3.1
121
60 Hertz Power Output Comparison
Turn On Angle (Approx. 2.75 rad)
6
6
5.
5
4.
4
0.
3
3
0
2.
0
0-
Turn On Angle (Approx. 2.52 rad)
2
a. 1
0
0
5
10
20
15
25
10
15
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.41 rad)
6
6
5.
Turn On Angle (Approx. 2.3 rad)
-
4
3.
0
2
a0 2
0
16
1
3
0
18
20
22
24
26
28
15
20
Pulse Duration (% of Cycle)
6
5
30
25
5
4
0
20
Pulse Duration (% of Cycle)
30
25
Pulse Duration (% of Cycle)
Turn On Angle (Approx. 2.19 rad)
Turn On Angle (Approx. 2.1 rad)
6
Power Output
5
Predicted Power Output
True Power Output
4.
3
0.
2
0 2
3
n
1
1
10
0
-
0
15
20
25
30
20
Pulse Duration (% of Cycle)
22
24
26
28
30
32
Pulse Duration (% of Cycle)
Figure 4-16: In this figure, the data for each pulse duration is averaged to produce experimental results plots for power output. The same is done for the model and true power
output, which includes the core and inverter losses not included in the model. The model
shows good qualitative accuracy to the experimental data and reasonable quantitative accuracy for the power output.
122
60 Hertz Efficiency Comparison
Turn-On Angle (Approx. 2.75 rad)
0 .0
0.8
0 .6.
0.6
0 .4
W.
0 .2
Turn-On Angle (Approx. 2.52 rad)
0.4
0.2
01
5
20
15
10
1 0)
Turn-On Angle (Approx. 2.41 rad)
0 .0
.4-
C.)
II
0 .40..2
0..20
16
Turn-On Angle (Approx. 2.3 rad)
0 .6-
0..60
20
18
22
26
24
0
15
23
30
25
20
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
0 .8
30
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
0..8
25
25
20
20
15
15
0L
25
Turn-On Angle (Approx. 2.1 rad)
Turn-On Angle (Approx. 2.19 rad)
A
Efficiency
Predicted Efficiency
True Efficiency
0.6
.
0 .6
C,
0 .4
0.4
0 .2
0.2
'0
n
10
15
20
25
220
30
22
24
26
28
30
32
Pulse Duration (% of Cycle)
Pulse Duration (% of Cycle)
Figure 4-17: In this figure, the data for each pulse duration is averaged to produce experimental results plot for the efficiency. The same is done for the model and true efficiency
of the VRG, which includes the core and inverter losses not included in the model. The
model shows excellent qualitative and quantitative accuracy for efficiency, suggesting that
the model predicts the current waveforms quite accurately for a given applied voltage.
123
True Power Output vs. Minimum Air Gap
True Efficiency vs. Minimum Air Gap
0
0
x(
5-
+
*
0
0
0
Min Air Gap <0.4mm
04mm <Mln AirCGa 6m
0.45mm <MinAirGap<0.5mm
0.5mm <Min Air Gap <0.55mm
0.55mm <M nArGap <0.6mm
0.6mmn<MinAir Gap
.
.
'
x
0.5
0
*
0
>
0
MmA i AGap <0.4mm
04mm <Mi Air Gap <0.45mm
.
m n r
ap . mm
0.5mm <MinAirGap<0.55mm
0.55mm <Mn Air Gap <0.6mm
0.6mm <Min Air Gap
0.4*
4-
X
x
-
*+
w
0.3
+
++
ft+
2
0.2-
2
1
0
E
*
3
0.1
0.2
0
0.4
0.6
Minimum Air Gap (mm)
0.8
1
01
0
0.2
0.4
0.6
Minimum Air Gap (mm)
(a)
0.8
1
(b)
Figure 4-18: (a) The true power output of the engine is shown as a function of the minimum
air gap for a turn-on angle of approximately 2.3 radians and a 27% duty cycle. The colors
are shown so that the data is comparable to Figures 4-14 and 4-15. The maximum true
power output (5.74 Watts) occurs at the smallest minimum air gap (Gmin = 0.325 mm).
(b) Shows the true efficiency of the generator as a function of minimum air gap spacing for
the same tests as in (a). There is a very strong correlation between the minimum air gap
and both power and efficiency. The maximum true efficiency (55.8%) also occurs at the
smallest Gmin.
losses are nearly as large or larger than the actual power output. This means the
power output measurement is very sensitive to winding losses and the accuracy of
data measurement.
The same comparison of the efficiency between the model and experimental data
was done and shown in Figure 4-17.
The efficiency predicted by the model and
the experimental efficiency are qualitatively very similar.
The average difference
between the measured and predicted efficiency is 4.6%. This quantitative accuracy
suggests that the model predicts the current waveform quite accurately. The efficiency
including the inverter losses and core losses is qualitatively similar, but again the
difference is exaggerated because of the lower operating frequency at which the testing
was done.
To examine more closely the effect of the minimum air gap, a set of data points
124
was collected at a range of minimum gap lengths while maintaining the same pulse
duration and trigger voltage. The trigger voltage is referenced to the displacement
sensor, and therefore, the change in minimum gap spacing results in a slight shift of
the turn-on angle. This is seen by Equation 4.4, where VMa, and VMin shift up or
down slightly when the piston on average is closer or further away from the stator
resulting in smaller or larger minimum air gaps respectively. However, for a shift
of 0.2 mm the change in turn-on angle is about 0.1 radians for the 5 mm piston
displacement. Therefore, the data is treated as essentially a constant turn-on angle.
The data for the various minimum air gap spacings was taken at 60 Hertz, with a
trigger voltage of 8.1 volts, and a pulse duration of 4.5 milliseconds. This corresponds
to a turn-on angle of approximately 2.3 radians with a 27% duty cycle. This data
is presented in Figure 4-18. This operating point was chosen because it was a clear
local maximum in power as seen in Figure 4-14.
Therefore, for this axial gap design to be effective, the minimum air gap, which
results in the maximum inductance, should be as small as possible. For this generator, the maximum true power output generated by the 5 mm piston displacement was
5.74 Watts with a true efficiency of 55.8%. The model underestimates this operating
point by 1.5 watts, which suggests that the maximum inductance and saturation characteristics of the piecewise linear model do not exactly align with the true generator
characteristics. This would lead to an underestimate in the total work per cycle and
overestimate the currents driven through the winding.
4.3.1
Frequency Scaling and Losses
The proposed operating frequency of the generator is approximately 4-5 times the
frequencies used for model validation and testing of the VRG system. Given the
operating frequency of the engine, w, the power dissipated by eddy current losses is
proportional to w 2 , the power output is proportional to w, and the winding losses are
not affected by frequency [30]. Qualitatively this can be described as in Figure 4-19,
where the plotted variables are the fraction of power dissipated in the winding, P.,
125
k
x
Figure 4-19: This figure depicts qualitatively the fractional core losses and winding losses
to the power output as a function of frequency. This is particularly important considering
the scaling of the engine to higher frequencies where eddy current losses become more
significant. Figure adapted from [30].
to power output, P, or
Xw
(4.5)
-
PO
and the fraction of power dissipated by core losses, P, to power output, P., or
Pe
Xe = Pe .(4.6)
PO'
Therefore, the minimum fraction of power dissipated to power output occurs when
the power dissipated through winding and core loss are equal [30], given by
X = Pe+Pw
PO
(4.7)
The core losses for the M19, 29-gage steel used in the generator are currently being
characterized, but the testing is not complete at the time of the writing of this thesis.
However, a good approximation can be made based on published data of M19 electric
steel. Considering the VRG system designed, the core loss as a function of frequency
is shown in Table 4.1. The core loss data comes from data supplied by Polaris Laser
Laminations in reference [21]. The core loss was estimated by assuming a saturation
of 1.5 Tesla in the piston laminations and 1 Tesla in the stator laminations, given that
126
Table 4.1: Estimated VRG Core Loss
Core Loss (per phase)
Frequency (Hz)
50
60
100
150
200
300
Piston (W)
0.02
0.07
0.13
0.22
0.34
0.61
Stator (W)
0.05
0.06
0.12
0.20
0.31
0.55
Total (W)
0.07
0.13
0.25
0.42
0.65
1.17
the cross-sectional area of the stator is approximately 50% greater than the piston.
The core loss is given in terms of watts per kilogram, so that core loss could be
approximated based on the known mass of the steel laminations (181.6 gram stator,
42.8 gram piston). The core loss of the published data was subsequently multiplied by
one half because the minor hysteresis loop traveled is roughly half the full hysteresis
loop as discussed in Section 3.2.3.
Given that the winding loss for the 50 and 60 Hertz tests was approximately
3-4 watts when driven near the the maximums of power output, the eddy current
loss does not become as significant as the winding loss until frequencies significantly
greater than the proposed 250 Hertz operating point even when the steel is driven
into saturation. For this analysis, it was determined that the model underestimates
the magnitude of the eddy current loss by up to a factor of 2 using Equation (3.29).
Nonetheless, by scaling the results of the dynamic testing and increasing the effect of
eddy current by a factor of two, the model predicts the VRG is easily capable of the
desired 25 watts per phase power output with a conversion efficiency in the range of
80-85%, with potential for greater efficiency if less than 0.5 mm can be held for the
minimum air gap.
The experimental data can also be scaled to higher frequency with the known
scaling proportion of losses and power output.
Figure 4-20 shows the true power
output and true efficiency by scaling the 60 Hertz data to 250 Hertz. The data shown
is for a turn-on angle of 2.3 radians. The data was scaled by increasing the total
power in proportion to the frequency, increasing the core losses in accordance with
127
Scaled 250 Hz True Power Output
.
.
.
.
.
Scaled 250 Hz True Efficiency
0.02
.
n.
70
0.8F
60
-
0.78P-
Tum-On Angle = 2.3 radl
50
0
0
C
40
0.76
0
4E
---- Tum-On Angle = 2.3 rad
0
2
0.7220
0.7-
10
0.18
0.2
0.28
0.26
0.24
0.22
Pulse Duration (% of Cycle)
0.18
0.3
0.2
0.22
0.24
0.26
0.28
0.3
Pulse Duration (% of Cycle)
(b)
(a)
Figure 4-20: (a) The 60 Hertz data (0o0 = 2.3 rad) for true power output is shown scaled
to the 250 Hertz operating point by appropriately scaling the power output and generator
losses. (b) The true efficiency is shown for the same operating points as in (a). These figures
show that the generator is capable of producing the desired 50 Watt output at the higher
operating frequency. Additional power output may be possible by increasing the duty cycle
(pulse duration), but the generator efficiency begins to drop off quickly.
Table 4.1, and assuming all other losses are constant. The data was also multiplied
by two to give true power output when operating with both phases of the generator.
The scaled data shows that the VRG will be able to generate greater than 50 Watts
of electric power with an efficiency of 80-85%.
If the peak operating point for the generator shown in Figure 4-18 (Gmrn =
3.2 mm) is scaled to 250 Hertz, the true power output of the generator is 74.8 Watts
with a true conversion efficiency of 87.3%. This again shows the importance of the
minimum air gap and predicts that the generator can be highly efficient for convert-
ing high frequency, small amplitude oscillations if the minimum air gap is precisely
controlled.
128
4.4
Chapter Summary
The VRG system was tested both statically and dynamically to characterize the power
output of the system and to determine the accuracy of the model by which the VRG
was designed. The results of the static testing may be used to create a more accurate
model using similar methods as [29]. The qualitative predictions of the model matched
very well with the collected data. The model's quantitative predictions were less
accurate, which is attributed to the inaccuracy of the measurement of the minimum
air gap, the drop in applied voltage across the capacitor during the charging of the
winding, and to the model's piecewise-linear approximation, which is particularly
relevant when the steel saturates. The maximum output of the VRG at 60 Hz was
measured at 5.74 watts with an efficiency of 55.8%. From the experimental data,
it is concluded that the VRG system will be able to produce the desired 50 watt
output from the 250 Hertz thermoacoustic engine oscillations with an efficiency of
approximately 80-85%.
129
Chapter 5
Conclusions and Future Work
5.1
Conclusions
This work has covered a full thermal-to-electric power system concept based on thermoacoustic technology. The three separate parts of the design are the thermoacoustic
engine in which thermal energy is converted into mechanical pressure oscillations, an
electroacoustic transducer, and a self-pumping gas bearing system. The engine was
designed to produce 50-100 Watts of electric power from a robust, portable system
to serve as a remote power source for soldiers.
The thermoacoustic system was designed to operate as a double Helmholtz-like
resonator with a mechanically resonant piston as the inductance element of the resonator. This allows for a compact design and the piston to act as an electroacoustic
transducer that is strongly coupled to both the electromagnetic system as well as
the acoustic system. The thermoacoustic engine was designed and optimized using
DeltaEC software.
The electroacoustic transducer based on variable reluctance principles was designed, fabricated, and tested. The cross-shaped design of the transducer allowed the
suppression of eddy currents and allowed for rotational stability of the piston assembly. The impetus for the new linear alternator design was the necessity of coupling the
transducer with higher-frequency and smaller displacement amplitudes than typical
linear alternators are optimized for. The design was evaluated and optimized for the
130
thermoacoustic system based on a piecewise linear saturation computational model
of the VRG system.
The model and VRG system were then validated based on static and dynamic
tests. The static tests led to a reevaluation of fringing effects in the generator. The
dynamic tests revealed excellent qualitative agreement between the model predictions and the experimental data. The quantitative predictions were accurate within
approximately 15% for the high power tests of the VRG system. The minimum air
gap length was shown to have a significant effect on both the power output and efficiency of the generator. This led to the conclusion that for the VRG system with
an axial air gap to be effective, the minimum air gap should be kept as small as
possible and must be less than approximately 0.6 mm to be a realistic option for
power transduction. At its peak operating point, the VRG system generated 5.74
Watts of electric power with an efficiency of 55.8% at a frequency of 60 Hertz. Given
the scaling of power output to losses of the VRG with increasing operating frequency
to 250 Hertz, the VRG will be capable of producing at least 50 Watts of electric
power with an efficiency of 80-85%. Scaling the peak operating point to 250 Hertz
suggested a true power of 74.8 Watts is possible with a conversion efficiency of 87.3%
if the minimum air gap can be held to 0.3 mm, and potentially better operation if
the minimum gap can be kept smaller.
The gas bearing system designed for the VRG is beneficial for both engine operating lifetimes and removing suspension resonance and displacement limitations. The
bearing system proposed relies on the motion of the piston to selectively open and
close ports with a specific timing to produce high and low pressure reservoirs. In this
configuration, a passive bearing system is produced that keeps the piston centered.
The primary limitation of the bearing system is the necessity for extremely tight tolerances of machined surfaces to minimize the gap between the piston and cylinder
wall. This bearing system as well as other components of the engine have been left
as future work for the project, which is discussed in the following section.
131
5.2
Future Work
The compilation of the thermoacoustic engine, linearly-acting variable-reluctance generator, and the gas bearing system presented in this thesis is left as future work for
the project, as well as the other recommendations described in this section.
5.2.1
Modeling
There are a few improvements to the models that should be completed to better
predict the performance of the full thermoacoustic generator system. For the thermoacoustic portion of the engine, incorporating the nonlinear transducer with the
DeltaEC code or other thermoacoustic coupling analysis should be completed. This
would present more accurately the magnetic forces and there effect on the motion of
the piston given the relative phasing of the pressure oscillations. The DeltaEC model
should also be updated with specific component parameters as a more complete design
of the stack, cold heat exchanger, and hot heat exchanger are completed.
Additionally, the piecewise-linear approach for modeling the VRG can be improved by using the flux linkage-current data produced for the generator. This would
eliminate errors in the projected performance particularly when operating the generator in high power applications where the steel becomes saturated. This would also
present a more effective method for determining the optimum turn-on and turn-off
angles for the VRG system.
For the bearing system, additional modeling should be done including better approximations of the flow resistances, the inclusion of gas compressibility, and potentially turbulence in orifice type restrictions. Additional consideration should be given
to the thrust bearing application of the bearing system so that the piston operates on
average in exactly the center position (axially) of the generator. Finally, an analysis of
bearing start-up and liftoff are important modeling considerations for the application
of the gas bearing system.
132
5.2.2
System and Fabrication Improvements
The thermoacoustic system is capable of operating from any quality heat source.
How the thermoacoustic hot heat exchanger interfaces with the heat source should be
considered, whether that heat source is a butane heater or a solar type application.
The vision of this researcher is to produce a thermoacoustic engine with an interface
similar to the commercial Jetboil system but with the aim of producing electric power
instead of cooking food. Additionally, the cold heat exchanger was modeled as operating at 340 Kelvin to account for an air cooled cold heat exchange system because
a more effective water cooled system is not a practical option for a portable system.
Further design of the cold heat exchanger is required.
The control of the VRG system is an important consideration of the final system.
Sensorless control of the VRG (using the coils to determine the piston of the piston)
should be considered to remove the required displacement sensor that was used in the
experimental testing of the VRG system. Additionally, using the timing of the VRG
turn-on and turn-off angles to precisely control the location of the piston should be
considered. This piston control is a secondary option to using the bearing system in
a thrust bearing type application.
This engine should be built and tested, but as a long term improvement on the
thermoacoustic system, the standing wave bounce volumes could be operated as torus
type traveling wave bounce volumes as in the Swift traveling wave engine. This would
be an important improvement if the efficiency of the system becomes a more important
consideration.
133
Appendix A
Flux Tube Analysis
The estimation of fringing fluxes was necessary in the design of the VRG system because of the large air gaps where fringing effects significantly increase the minimum
inductance. Figure 3-14 is repeated here in Figure A-i for clarity when describing in
more detailed the flux tube modeling done for the generator. The fringing types are
divided into surface-to-surface simple shaped volumes, edge-to-edge simple shaped
volumes, and corner-to-corner simple shaped volumes. Surface-to-surface are referenced by the two letters defining the two surfaces, edges are defined by the surfaces
which meet to form the edge, and corners are defined by the three surfaces which
meet to form the corner. For derivations of these equations and for more description
on the specific simple shaped volumes see [18].
The following equations were used to calculate the flux path permeances. For
clarity, there is a "G" that corresponds to a surface in the figure, but in the equations
"G" refers to the air gap length.
Outer Air Gap
PA=>G =
PAB=>GC =
G
2(0.26Iu(lo + z))
(A.1)
(A.2)
where the multiplication by 2 corresponds to the 2 sides of the stator. Numbers in
front of the equation will continue to reference the number of times that geometry
134
F
0
A
H
I
0
D
Figure A-1: This figure shows two of the simple-shaped volumes used for the flux tube
analysis. The half cylinder represents the flux from edge AB to edge GF, and the half
annulus represents the flux from surface B to surface F.
occurs for one stator piece-piston piece pair.
26 w
PAD=>EG =0.
PB=>F =2
PD=E =
+ G ))
(A.4)
G1))
(A.5)
(L/r n (I +
PABD=>EFG =
PBD=>EF =
0
PAC=G
PC=>G
2hp
z) 1
(l+
w
(A-3)
4(.077pG)
(A.6)
2
(A.7)
(4)
.
1
135
5 2 tpw
1P
(A.8)
(A.9)
Inner Air Gap
(A.10)
PI=>- =
PH=G = 0.52pw
+L
w1
PH=>-G
PBHI=>FG=
(A.11)
-A12)
2(0.077 1 tG)
(A.13)
(Lv)
-2
(A.14)
PBH=>F
Leakage Permeance
PL = p(A.15)
1w
Each of these permeances were present on each of the 4 stators, so the permeances
here were multiplied by 4 to calculate the fringing and leakage fluxes for one phase
of the full generator.
136
Appendix B
Model Code and Experimental
Data
The DeltaEC model used for analysis and optimization of the thermoacoustic engine
is provided here.
Three-Inch Thermoacoustic Engine
TITLE
21-Feb-2014 with DeltaEC version 6.3b11.12!under win32,
!CreatedQ03:47:33
using Win 6.1.7601
(Service Pack 1) under Python DeltaEC.
!--------------------------------- 0 --------------------------------Initial
BEGIN
3.0000E+06 a Mean P Pa
253.51
b Freq
Hz
659.54
c TBeg
K
G
Ip1
Pa
G
1.5727E+05 d
0.0000 a Ph(p)
0.0000
:
IUI
dog
m^3/s
0.0000 g Ph(U)
helium
G
dog
Gas type
!--------------------------------- 1 --------------------------------SURFACE
Hot End
4.5600E-03 a Area
1.5727E+05 A IpI
m-2
0.0000 B Ph(p)
1.3099E-05 C
180.00
IUI
D Ph(U)
0.0000 E Htot
ideal
Solid type
-1.0301
?---------------------------------
2
deg
m-3/s
deg
W
W
---------------------------------
Hot Duct
DUCT
sameas
la a Area
0.23938 b Perim
2.5000E-02 c Length
m-2
Mstr
1.5722E+06
m
a
2a
2.2548E-04 B Ph(p)
deg
5.7284E-03 C 1UI
m^3/s
-90.303
ideal
3
A IpI
D Ph(U)
Pa
deg
0.0000 E Htot
W
F Edot
W
-2.382
Solid type
!--------------------------------HX
F Edot
Pa
---------------------------------
Hot HX
3.80OOE-03 a Area
m-2
1.5717E+05
A IpI
Pa
137
0.6700
b GasA/A
5.0000E-03 c Length
m
5.2540E-04 d
194.12
e
m
580.00
yO
HeatIn W
G
Solid type
!---------------------------- -----
STKSLAB
Stack
sameas
4a a Area
0.8200
4
m-2
D Ph(U)
194.12
E
W
F Edot
W
659.54
G GasT
K
668.11
H SolidT K
---------------------------------
1.1175E-02 C IUI
5.0000E-04 e Lplate mn
Solid type
Stack
sameas
4a a Area
sameas
6b b GasA/A
sameas
6c c Length mn
sameas
6d d yO
m-2
m^3/s
D Ph(U)
dog
194.12
E Ntot
W
F Edot
W
659.54
G
TBeg
K
446.22
H TEnd
K
1.5578E+05 A Ipi
0.31691 B Ph(p)
1.6047E-02 C IUI
m
5.0000E-04 e Lplate m
Solid type
*--------------------------------- 6
Pa
deg
m-3/s
-88.179
D Ph(U)
dog
194.12
E Htot
W
F Edot
W
446.22
G TBeg
K
346.98
H TEnd
K
32.812
stainless
deg
---------------------------------
!---------------------------------5
STKSLAB
Pa
-88.824
15.785
stainless
dog
Htot
0.14281 B Ph(p)
m
m
1.4000E-04 d yO
m3/s
-90.887
1.5672E+05 A Ip1
b GasA/A
3.0000E-02 c Length
deg
6.4358E-03 C IUI
-7.852
f SolidT K
ideal
3.0308E-03 B Ph(p)
---------------------------------
Ambient HX
HX
sameas
4a a Area
0.2530
m-2
1.5504E+05 A Ip1
b GasA/A
0.34036 B Ph(p)
6.0000E-03 c Length mn
4.0600E-04 d yO
e
-169.12
340.00
-88.288
Is
HeatIn
W
f SolidT K
ideal
1.6355E-02 C IUI
m^3/s
dog
25.000
E
-13H
30.341
F Edot
W
346.98
G GasT
K
340.00
H SolidT K
!---------------------------------
7
Htot
deg
G
Solid type
DUCT
D Ph(U)
Pa
W
---------------------------------
Ambient Duct
samea.
la a Area
0.23938 b Perim
m-2
m
1.5484E+05 A Ip1
0.33884 B Ph(p)
8.0000E-03 c Length m
1.8159E-02 C IUI
Solid type
ideal
!--------------------------------RPN
Pa
dog
m'3/s
-88.435
D Ph(U)
deg
25.000
E Htot
w
30.096
F Edot
w
8
Piston Disp
2.5000E-03 a G or T
-16A
A Piston
2.5000E-03
/
Ul mag la / v
I--------------------------------IESPEAKER
sameas
la a Area
0.2500
b R
0.0000 c L
18.000
0.2000
0.0000
m-2
1.5484E+05 A Ipi
ohms
-179.66
H
1.8159E-02 C IUI
B Ph(p)
Pa
deg
m'3/s
d BLProd T-m
-88.435
D Ph(U)
e M
kg
-25.00
E
Htot
w
f
N/m
-30.096
F Edot
-50.00
G WorkIn
w
W
K
0.1000
g Rm
N-s/m
8.1227
h III
A
i
deg
-10.00
9
Change Me
Ph(I)
G
71.300
8.1227
99.943
H
Volts
I Amps
dog
V
A
3 Ph(V/I) deg
138
3.0968E+05 K IPxI
-179.66
Solid type
ideal
10
RPN
Pa
L Ph(Px) deg
---------------------------------
ChangeMe
-50.00
A ChngeMe
-50.00
-18A
a G or T
17G
!---------------------------------
11
---------------------------------
Change Me
DUCT
sameas
la a Area
0.23938 b Perim
sameas
m-2
Mstr
1.5504E+05 A
a
19a
-179.66
14c c Length a
1.6355E-02 C lUl
Solid type
ideal
IpI
B Ph(p)
i---------------------------------
Pa
deg
m^3/s
-88.288
D Ph(U)
deg
-25.00
E Htot
w
-30.341
F Edot
w
12
Ambient2
HX
sameas
1.5578E+05 A IpI
Pa
sameas
13b b GasA/A
-179.68
deg
sameas
sameas
13c c Length a
1.6047E-02 C IUI
m-2
4a a Area
m
13d d yO
-88.179
-169.12
e HeatIn W
G
340.00
f SolidT K
-20H
-194.12
-32.812
Solid type
ideal
!---------------------------------
B Ph(p)
D Ph(U)
F Edot
346.98
G GasT
340.00
H SolidT K
K
13 ---------------------------------
Me
Change
sameas
4a a Area
sameas
sameas
6b b GazA/A
-179.86
6c c Length a
1.1175E-02 C 1Ut
m-2
1.5672E+05 A Ip1
M
5.0000E-04 e Lplate m
-88.824
6d d yO
!---------------------------------
B Ph(p)
Pa
deg
m^3/s
D Ph(U)
deg
E Htot
v
-15.785
F Edot
w
346.98
G TBeg
K
446.22
H TEnd
K
-194.12
Solid type
stainless
14 ---------------------------------
Change He
STKSLAB
m-2
1.5717E+05 A IpI
sameas
4a a Area
sameas
sneas
sameas
6b b GasA/A
-180.0
6c c Length a
6.4358E-03 C 1UI
m
21d d yO
-90.887
6.0000E-04 e Lplate m
-194.12
Solid type
stainless
I---------------------------------
15
B Ph(p)
Pa
deg
m^3/s
D Ph(U)
deg
E Htot
w
F Edot
w
446.22
G TBeg
K
659.54
H TEnd
K
7.8520
HX
deg
E Htot
STKSLAB
sareas
m-3/s
---------------------------------
Hot2
m-2
Ip1
sameas
4a a Area
sameas
4b b GasA/A
-180.0
sameas
4c c Length a
5.7284E-03 C 1U1
samoas
4d d yO
sameas
sameas
4e e HeatIn W
ideal
1.5722E+05 A
M
-90.303
4f f SolidT K
Solid type
Change
sameas
Pa
deg
M^3/s
deg
2.3820 F Edot
w
w
659.54
G GasT
K
668.11
H SolidT K
16
Me
Ia a Area
0.23938 b Perim
sameas
D Ph(U)
9.1518E-12 E Htot
!--------------------------------DUCT
B Ph(p)
m^2
m
2c c Length a
Mstr
25a
1.5727E+05 A 1p1
180.00
B Ph(p)
1.3099E-05 C IUI
-180.0
D Ph(U)
9.1518E-12 E
Htot
Pa
deg
mr3/s
deg
w
139
ideal
Solid type
1.0301
F Edot
SURFACE
Change
sameas
la a Area
Me
m-2
1.5727E+05 A Ip1
180.00
B Ph(p)
9.3406E-15 C
-81.797
ideal
Solid type
!--------------------------------HARDEND
W
17 ---------------------------------
I---------------------------------
IUI
D Ph(U)
Pa
deg
mi3/s
deg
9.1518E-12 E Htot
W
-1.0480E-10 F Edot
W
18 ---------------------------------
target this to seal the end
0.0000 a R(1/z)
-27G
0.0000 b I(1/z)
-27H
0.0000 c Htot
W
1.5727E+05 A Ip1
180.00
-27E
B
9.3406E-15 C
-81.797
Pa
Ph(p)
deg
IUI
m^3/s
D Ph(U)
deg
9.1518E-12 E Htot
W
-1.0480E-10 F Edot
W
-6.1486E-15 G R(1/z)
4.2653E-i4 H I(1/z)
I The restart information below was generated by a previous run
I and will be used by DeltaEC the next time it opens this file.
guesez
Ob
Oc
Od
4e
13e
xprecn
2.0190E-03 -5.8246E-04
targe
13f
mstr-slave
16a
18a
20f
27a
17h
20e
3.9218
27b
9.2722E-04 -9.2722E-04 -2.8523E-06 -9.2722E-04
27c
3 2 -2 19 -2 25 -2
I Plot start, end, and step values.
Outer Loop:
May
be edited if you wish.
I Inner Loop
140
Appendix C
Matlab Code for VRG
The Matlab model of the VRG transducer is provided here.
Variable Initialization
clear all
%Circuit Time
.Number of divisions per cycle
n-10000;
Angle
theta-linspace(0,2*pi,n)';
XPhase
Freq-250;
%Operating frequency
omega-Freq*2*pi;
XDisplacement,
Angular
frequency
%Capacitor bank capacitance
C-50000;
Clearance, Position, Velocity
Disp-0.0060;
Peak-to-peak displacement amplitude
Clear-0.0006;
%Minimum Air Gap
x-Disp/2*cos(theta)+Disp/2+Clear;
Y.Air gap length (all lengths in meters), "G" used in thesis
%VRG Design Constants
ItM-.0254;
%Inch to Meter Conversion
muo-4*pi*10^-7;
XPermeability
mur-3000;
of free
space
Approx. steel relative permeability
mu-muo*mur;
%Eddy Current Parameters
rhos-5*10-7;
tlam-3.46*10^-4;
%Steel resistivity (Ohm m)
Laination Thicknes
%Component Dimensions (Same notation as in thesis)
li-.31*ItM;
z-. 1*ItM;
w-2*(li-z);
Ag-4*li2-4*z-2;
lo-Ag/(4*v);
lp-sqrt(.0367^2-v^2);
lv-lp-(li+.021*ItM)-lo-.0018923;
Y.Constant used for
winding
clearance
hc-.25*ItM;
hv-.626*ItM;
hp-.1 *ItM;
As-4*hc*w;
Ap-4*hp*w;
A-[Ap, As, Ag];
Amin-min(A);
141
Leakage and Fringing Permeances
% Leakage and Fringing Permeances
CF-1.4;
ZFringing correction Factor
Pi-4muo*lo*w./x*CF;
P2-2*4*.26*muo*lo*CF;
P3-4*.26*muo*w*CF;
P4-2*4*muo*lo/pi*log(1+2*hp. /x)*CF;
P5-4uo*w/pi*log(1+2*hp. /x)*CF;
P6-4*4*.077*no.*x*CF;
P7-4*muo*hp/4*CF;
P8-4*.52*muo*w*CF;
P9-4*2*muo*w/pi*log(i+hp./x)*CF;
PL-4*muo*hw*w/lw;
P11-muo*Ag./x*CF;
P12-4*.52*muo*w*CF;
P13-4*2*muo*w/pi*log(i+hp./x)*CF;
P14-2*4*.077*muo.*x*CF;
P15-2*4*muo*hp/4*CF;
%Reluctances
Ri-2*hw/(mu*Ag);
R2-lp/(mu*hp*w);
R3-R1;
R4-lp/(mu*hc*w);
RL-2/PL;
Rf-1i./(Pi2+Pi3+P14+Pi5);
Rg1-1./P11;
Rf2-i. /(P2+P3+P4+P5+P6+P7+P8+P9);
Rg2-1./P1;
Rp-./(/RL+i./(R2+1./(i./Rfi+i./Rgi)+i./(i./Rf2+1./Rg2)));
Rtot-R1+R3+R4+Rp;
Wire Resistance
XWire
Resistance/Number of turns based on wire gauge
N-148;
Number
rhow-16.8*10^-9;
Winding resistivity
of turns
Ff-.pi/4;
%Wire packing factor
Aw-(lv-.032*ItM)*(hw-.032*ItM);
XConstant
Lw-8*(li+lw/2);
XWire
AWGA-Aw*Ff/N;
%Wire area
AWG-sqrt(4/pi*AWGA);
Y.American Wire Gauge
R-rhow*Lw*N.^2/(Ff*Aw);
%Wire resistance
used for winding clearance
length
Material Saturation Characteristics
UInductance
L-N^2. /(Rtot) ;
UInductance (H)
Ls-10*pi*10^-4;
XConstant
saturated inductance
%Gap Saturation Current
Bs-1.6;
%Assumed saturation flux density (T)
Is-Bs*Amin*Rtot/N;
%Saturation Current (A)
Drive Circuitry Variables
Batt-50;
ThetaOn-2.6;
%Capacitor Bank Voltage
Turn on angle
LVTimeOn-.0009;
%Pulse duration
ThetaOff-LVTimeOn*omega+ThetaOn;
Y.Turn off angle
142
Initializing Cyclic Current and Flux Linkage Profiles
%Calculating current vaveform
Isat-ones(n, 1);
I-ones(n,i);
Lambda-ones(n,i);
Vb-ones(n,
1);
V-ones(n,i);
Bp-ones(n,
1);
Bc-ones(n,1);
Theta-ones(n,i);
dBpdth-ones(n, 1);
dBcdth-ones(n,1);
dLdxcur-ones(n,1);
L-ones(n,i);
vel-ones(n,i);
F-ones(n,i);
Test-ones (n,1);
%Loop to determine initial current waveform/ flux linkage
for m-i:n
angl 0
Theta(m)-(m-1)/n*2*pi;
XPhase
xp-x(m);
Y.Air gap/pi ston displacement
vel(m)--Disp/2*omega*sin(Theta(m));
%Piston vel oicty
% Leakage and Fringing Permeances
Pi-4*muo*lo*w/xp*CF;
P4-2*4*muo*lo/pi*log(1+2*hp/xp)*CF;
P5-4*muo*w/pi*log(1+2*hp/xp)*CF;
P6-44*.077*muo*xp*CF;
P9-4*2*muo*w/pi*log(i+hp/xp)*CF;
P1i-muo*Ag/xp*CF;
P13-4*2*muo*w/pi*log(1+hp/xp)*CF;
P14-2*4*.077*muo*xp*CF;
%Reluctances
Rf 1-1/(P12+P13+P14+Pi6);
Rg1-1/P11;
Rf2-1/(P2+P3+P4+P5+P6+P7+P8+P9);
Rg2-1/Pi;
Rp-1/(i/RL+1/(R2+1/(i/Rfi+/Rg)+1/(1/Rf2+1/Rg2)));
Rtot-R+R3+R4+Rp;
L(m)-N-2/Rtot;
Isat (m)-Bs*Ap*Rtot/N;
if Theta(m)<-ThetaOn
Lambda(m)-0;
I(m)-0;
Vb(m)-0;
V(m)-0;
else if (Theta(m)>ThetaOn U Theta(m)<-Thetaff)
Vb(m)-Batt;
Lambda(m) -Vb (i) /omega* (Theta(m) -ThetaOn);
I(m)-Lambda(m)/L(m);
V(m)-Vb(m)-I(m)*Ract;
if V(m)<O
Lambda(m)-Lambda(m-1);
I(m)-Vb(m)/Ract;
end
else if (Theta(m)>ThetaOff U Lambda(m-1)>O)
Vb(m)--Batt;
Lambda(m) -Batt/omega* (ThetaOff-ThetaOn) +Vb(m) /omega* (Theta(m) -ThetaOff);
I(m)-Lambda(m)/L(m);
V(m)-Vb(m)-I(m)*Ract;
if V(m)>O
Lambda(m)-Lambda(m-1);
143
I(m)--V(bm)/Ract;
end
else
Lambda(m)-Lambda(m-1);
I(m)-I(M-1);
Vb(m)-0;
V(m)-O;
end
end
end
if (I(m)<-Isat(m))
I(m)-Lambda(m)/L(m);
else
I(m)-(Lambda(m)-L(m)*Isat(m))/Ls+Isat(m);
end
end
Iterative Calculation of Current and Flux Linkage Profiles
% Loop to find corrected current/flux linkage with winding resistance
for o-1:5
for m-i:n
if Theta(m)<-Thetan
Lambda(m)-0;
I(m)-0;
Vb(m)-0;
V(m)-0;
Bp(m)=O;
Bc(m)-O;
dBpdth(m)-O;
dBcdth(m)-O;
else if (Theta(m)>ThetaOn U Theta(m)<-Thetaaff)
Vb(m)-Batt;
Lambda(m)-Lambda(m-)+V(m)*(Theta(m)-Theta(m-1))/omega;
V(m)-Vb(m) -I (m)*Ract;
Bp(m)-Lambda(m)/(N*Ap);
Bc (m)-Lambda(m)/(N*Ag);
dBpdth(m)-abs((Bp(m)-Bp(m-1))/(Theta(m)-Theta(m-1)));
dBcdth(m)-abs((Bc(m)-Bc(m-1))/(Theta(m)-Theta(m-1)));
else if (Theta(m)>ThetaOff &k Lambda(m-1)>O)
Vb(m)--Batt;
Lambda(m)-Lambda(m-i)+V(m)*(Theta(m)-Theta(m-1))/omega;
V(m)-Vb(m) -I (m)*Ract;
Bp(m) -Lambda(m)/(N*Ap);
Bc(m)-Lambda(m)/(N*Ag);
dBpdth(m)-abs((Bp(m)-Bp(m-1))/(Theta(m)-Theta(m-1)));
dBcdth(m)-abs((Bc(m)-Bc(m-1))/(Theta(m)-Theta(m-1)));
else
Lambda (m)-Lambda(m-1);
I(m)-I (m-1);
Vb(m)-0;
V(m)-O;
Bp(m)-O;
Bc(m)-O;
dBpdth(m)-O;
dBcdth(m)-O;
end
end
end
if (I(m)<-Isat(m))
144
I(m)-Lambda(m)/L(m);
else
I(m)-(Lambda(m)-L(m)*Isat(m))/Ls+Isat(m);
end
end
end
Calculation of Cyclic Work, Power Output, and Losses
XEddy
Current Power Loss
%Eddy Current Power Loss Piston
Pep-tlam^2*omega'2*dBpdth. ^2/(12*rhos) *lp*v*hp*4;
%Eddy Current Power Loss Stator
Pec-tlam^2*omega^2*dBcdth. -2/(12*rhos)*(4*(li+lo)*w*(hc+hv)+4*lw*hw*w);
PeCyc-Pep+Pec;
Pe-trapz(Theta/omega,PeCyc)*Freq;
V.Eddy current power loss total
Y.Winding Power Loss
PwCyc-I.^2*R;
Pv-trapz(Theta/omega,PwCyc)*Freq;
XPower
Generated
net-vork - 1/2*sum(I.*Lambda([2:end,1])-Lambda.*I ([2:end,1D));
Power-abs (net-work*Freq);
PowOut-abs (Power) -abs (Pw) -abs (Pe);
%Efficiency
Eff-abs(PowOut)/abs(Power);
145
Appendix D
VRG Component Drawings
The two engineering drawings on the following two pages are for the VRG stator and
piston laminations. These are the magnetic core components used for testing the
VRG system. The components were fabricated by Polaris Laser Laminations. The
stator and piston laminations are the essential components to the operation of the
VRG system, while the other components fabricate (with the exception of the coil)
are support structure components. Dimensions are in inches.
146
6
.
IN
PR"$HIBR
Inslitule
5
IN
OR
OF
THE
CONTAINED THIS
DRAWING THE
PROPERTY OF
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of Technology. ANY
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REPRODUCTION PART
WITHOUT THE WRITTEN PERMISSION
INFORMATION
IS SOLE
PROPMETARY AND CONFIDENTIAL
LO'
0.205
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APPLICATION
1.170
1.380
0.875
10
USED ON
A nnealedAI
DO NOT SCALE DRAWING
FINISH
M1 9 29 GaugeSIED
MATERIAL
TOLERANCING PER:
Q.A.
COMMENTS:
EN A PPR.
N
INTERPRET GEOMETRIC
CHECKED
MFG APPR.
BEND*2
FRACTONAL0.001
ANGULAR: MACH
TWO PLACE DECIMAL 10.01
THREE PLACE DECIMAL 20.001
0.090
2
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Stator Assembly1
SCALE: 2:1 WEIGHT:
SZ
SHEET 1OF
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Sta tor A sse m b ly
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IS
5
PROPRIETARY AND CONHDENTIAL
THE INFORMATION CONTAINED IN THIS
DRAWING THE SOLE PROPERTYO2
Mass. Institute of Technology. ANY
REPRODUCTION IN PART OR AS A WHOLE
WITHOUT THE WRITTEN PERMISSION OF
Mass. Institute of Technology 15
PROHIBITED.
1.189
4
APPLICATION
1.400
USED ON
-----------
UNLESS OTHERWISE SPECIFIED:
31
DRAWN
MFG APPR.
ENG APPR.
CHECKED
DIMENSIONS ARE IN INCHES
TOLERANCES:
FRACTIONAL
ANGULAR: MACHt: BEND
TWO PLACE DECIMAL t0.01
THREE PLACE DECIMAL 0.001
Q.A.
COMMENTS:
COEMG.NSE
0.00I
INTERPRET GEOMETRIC
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MATERIAL
FINISH
Mn
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NAME
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