Design and Parametric Simulation of Radially
Oriented Electromagnetic Actuators
"^S
by
Will Bosworth
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
ARCHNES
Masters of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
@ Massachusetts Institute of Technology 2011. All rights reserved.
Author .....
...............
/I
...................
Department of Mechanical Engineering
May 20, 2011
/7
/
15>2
Certified by...................
Jeffrey H. Lang
Professor of Electrical Engineering and Computer Science
Thesis Supervisor
Certified by..........
.............
................
exander H. Slocum
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by ................
David E. Hardt
Professor of Mechanical Engineering
Graduate Officer
2
Design and Parametric Simulation of Radially Oriented
Electromagnetic Actuators
by
Will Bosworth
Submitted to the Department of Mechanical Engineering
on May 20, 2011, in partial fulfillment of the
requirements for the degree of
Masters of Science in Mechanical Engineering
Abstract
This thesis presents the design and simulation of an electromagnetic actuator system
capable of delivering large pulses of radial force onto a cylindrical surface. Due to its
robust design, simple control scheme, and large output force capability, the actuator
design is developed to be considered for wellbore manipulation and other downhole
oil exploration and production activities.
The complete simulation - including capacitor bank power supply, solid state switching circuit, transducer, and target formation - is a thirteen value lumped parameter
model. The simulation was used heavily in the design and refining of two experimental
prototype systems. These prototypes showed excellent model-experiment matching.
The experimental prototypes are 2.5" radius, 12" length cylindrical transducers that
exert nearly 10 psi onto a simulated rock formation with 2 MN/m radial stiffness,
increasing the formation radius 3.5 mm during 5 ms pulse events.
It is with this experimentally validated simulation that we project forward a manufacturable system capable of exerting pulses of hundreds of psi in magnitude over
durations of 1 - 10 ms onto wellbore sized cylindrical surfaces.
Thesis Supervisor: Jeffrey H. Lang
Title: Professor of Electrical Engineering and Computer Science
Thesis Supervisor: Alexander H. Slocum
Title: Professor of Mechanical Engineering
4
Acknowledgments
This work was funded by Schlumberger and performed at MIT, directly enabled
through the guidance of Professor Jeff Lang and Dr. Julio Guerrerro, and furthered
along with efforts from Professor Alex Slocum, Dr. Jahir Pabon, and Alec Resnick.
6
Contents
1 Introduction
2
3
1.1
The Pulsed Electromagnetic Radial Actuator System . . . . . . . . .
1.2
Motivation and Prior Art...
1.3
T hesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
Radially Oriented Electromagnetic Transducer
19
2.1
Lorentz Force Law on a Wire Review .......
. . . . . . .
20
2.2
Conductor Layout for Radial Actuation . . . . .
. . . . . . .
21
2.3
Magnetic Field of the Axial-Current Layout
. .
. . . . . . .
23
2.4
Self Induced Electromagnetic Force . . . . . . .
. . . . . . .
25
2.5
Inductance of the Axial Current Layout . . . . .
. . . . . . .
25
2.6
Transducer Resistance
. . . . . . .
27
. . . . . . . . . . . . . .
Power Supply, Electrical Energy Model, Mechanical Energy Model 29
3.1
Capacitor Bank and Power Circuit
3.2
Discharging Capacitor Circuit: Modeling Current Flow Through the
. . . . . . . . . . . . . . . . . . .
29
Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
3.3
Mechanical Force/Displacement State Equations . . . . . . . . . . . .
32
3.4
Estimating Formation Properties Downhole
33
. . . . . . . . . . . . . .
4 Assembling the Full Parametric Simulation
35
4.1
System Param eters . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.2
State Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
5
4.3
Energy Balance in the Simulation Output
. . . . . . . . . . . . . . .
37
4.4
Viewing Output Data. . . . . . . . . . . . . . . . . . . . . . . . . . .
38
The Radial Actuator Prototype and Characterization
5.1
. . . . . . . . . . . .
42
Parameter Search Using the Simulation . . . . . . . . . . . . .
44
Prototype Machine Design . . . . . . . . . . . . . . . . . . . . . . . .
48
5.2.1
Transducer and Formation Design . . . . . . . . . . . . . . . .
48
5.2.2
Capacitor Bank and Circuit Components . . . . . . . . . . . .
53
Alpha Prototype Characterization . . . . . . . . . . . . . . . . . . . .
54
5.3.1
Static M easurements . . . . . . . . . . . . . . . . . . . . . . .
55
5.3.2
Dynamic Measurements: Setup . . . . . . . . . . . . . . . . .
57
5.3.3
Dynamic Measurements: Alpha Prototype Experimental Results 59
Specifying Parameters for the Initial Prototype
5.1.1
5.2
5.3
6
The Second Radial Actuator Prototype
61
6.1
Litz Wire Transducer Design . . . . . . . . . . . . . . . . . . . . . . .
61
6.2
Characterizing the Beta Prototype
. . . . . . . . . . . . . . . . . . .
64
6.3
Comparing Performance of Alpha and Beta Prototypes . . . . . . . .
67
7 Simulation Results for Wellbore Applications
8
41
71
7.1
An Actuator Design for Increasing Wellbore Radius . . . . . . . . . .
72
7.2
An Actuator Design for Creating Impulse Waves for Seismic Analysis
75
7.3
Selecting the Simulation Parameters Used in This Chapter . . . . . .
75
Conclusion
A Matlab Simulation Implementation
81
85
List of Figures
1-1
PEM schematic......
2-1
Current carrying wire in a magnetic field . . . . . . . . . . . . . . . .
2-2
Radial electromagnetic force layouts. On the left, current moves axi-
. . . . . . . . . . . . . . .. . . .
. . . . .
ally and magnetic field circulates. On the right, magnetic field moves
axially and current circulates. . . . . . . . . . . . . . . ........
2-3
Axial and Azimuthal current layouts for radial force. The axial current
layout is the focus of this thesis . . . . . . . . . . . . . . . . . . . . .
23
2-4 Axial current layout for Ampere analysis . . . . . . . . . . . . . . . .
24
2-5
Faraday surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2-6
Cylinders with radius, thickness, and number of turns . . . . . . . . .
28
3-1
Energy flow overview . . . . . . . . . . . . . . . . . .
3-2
Power circuit
3-3
RLC circuit and KVL polarity definitions..
3-4
Mass spring damper model . . . . . . . . . . . . . . .
4-1
Simulated energy flow
4-2
Simulation output: (a) outer radius extension, (b) current
. . . . . . . . . . . . . . . . . . . . . .
. . ..
. . . . . . . . . . . . . . . . . . .
tance, and (d) voltage in simulation . . . . . . . . . . . .
induc.
.
.
.
.
.
. . . . . . . . . . . . . . . . . . .
5-2
Parameter Sweeping C and K...
5-3
Radial Displacement with Varying K . . . . . . . . . . . . . . . . . .
40
5-4 Cross section of transducer and radial stiffness . . . . . . . . . . . . .
5-5
Orhtographic view of experimental device........... . .
. ..
50
51
5-6 FEA result for 800N loads on each spring . . . . . . . . . . . . . . . .
52
5-7 Plastic connection detail feature view . . . . . . . . . . . . . . . . . .
52
5-8 Completed Alpha Prototype Transducer, Formation, and Capacitor Bank 53
5-9
Complete Alpha Prototype Experimental Setup
54
5-10 Measuring Capacitor R and C . . . . . . . . . . . . . .
56
5-11 Non-contact inductive sensor on prototype . . . . . . .
58
5-12 Schematic of measuring voltage across a real capacitor
59
5-13 Alpha Prototype Experimental and Simulated Actuator Event
60
6-1
Model of Beta Prototype.. . . . . .
6-2
Beta Transducer Assembly Line . . . . . . . .
6-3
Beta Transducer Assembly in Progress
6-4
Assembled Beta Transducer
. . ..
.
.
.
.
.
.
.
.
.
.
.
.
.
63
. . . . . . . . .
64
. . . . . . . . . . . . . . . . .
65
. . . . . . . . . . . . . . . . . . . . . . .
66
6-5
Complete Beta Prototype Actuator . . . . . . . . . . . . . . . . . . .
67
6-6
Experimental and Simulated Actuator Event - Beta-Prototype . . . .
6-7
Simulated Energy Transfer of Experimentally Validated Prototypes
7-1
Outer Radius Displacement
7-2
C urrent
7-3
Electromagnetic Force on Conductors . . . . . . . . . . . . . . . . . .
7-4
Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-6
Simulation of current using best parameters from sweeps of Figure 7-5
.
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
79 79 79
List of Tables
4.1
Simulation input parameters . . . . . . . . . . . . . . . . . . . . . . .
36
4.2
Simulation system states . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.3
System parameters used in Section 4.4 output viewgraphs . . . . . . .
38
5.1
Simulation input parameters . . . . . . . . . . . . . . . . . . . . . . .
42
5.2
Refining simulation input parameters... . . . . . . .
. . . . . . .
45
5.3
Simulation input parameters for parameter sweep . . . . . . . . . . .
45
5.4
Constrained input parameters for initial prototype design . . . . . . .
48
5.5
Statically measured values of alpha prototype.. . . .
. . . . . . .
57
5.6
Purchased parts..... . . . . . . . . . . . . . .
. . . . . . . . . .
57
6.1
Statically measured values of beta prototype . . . . . . . . . . . . . .
64
7.1
A Set of Simulation Parameters for Rock-Pushing Application . . . .
72
12
Chapter 1
Introduction
This thesis concerns the design and simulation of an electromagnetic actuator that
creates large magnitude short duration force pulses oriented radially outward. The
system was first proposed while brainstorming solutions to a wellbore manipulation
challenge. the actuator had inherent features that made it intriguing to apply to the
oil service industry: it could have a compact robust package suitable for harsh down
hole environments, a simple yet repeatable and precise control scheme, and could
achieve impressive power density. It seemed that this electromagnetic actuator could
compete with explosive charges and hydraulic actuators in some downhole applications, so we set out to provide tools and analysis to enable further development.
We created a nonlinear thirteen parameter electromechanical simulation of the system. The simulation was used to design prototypes which were used to experimentally
validate the simulation. The prototypes also enabled us to develop and refine techniques for designing and building such systems.
We used the experimentally validated simulation, along with estimates of downhole
formation properties and availiable electrical and mechanical components, to predict
the capability of such systems in the field. We show that manufacturable versions of
this system could apply hundreds of psi onto a cylindrical wellbore surface in 1 - 10
ms bursts, capable of recharging and refiring at an axial rate through the wellbore of
nearly 1 m/s.
The remainder of this chapter presents a brief technical overview of the actuator
system, describes prior art that motivated our design, and presents the layout of the
complete thesis document.
1.1
The Pulsed Electromagnetic Radial Actuator
System
The actuator systems described herein have a capacitor bank that delivers electrical
energy to an electromagnetic transducer where electrical energy is converted into
mechanical energy. The capacitor bank and switching circuit are called the power
supply, the device where electrical and mechanical energy conversion takes place is
called the transducer, and the mechanical system that is acted upon by the mechanical
force is called the formation. In this document, the formations of interest are the rock
formations in which a wellbore hole exists.
A lumped parameter schematic of the electrical and mechanical system is shown
in Figure 1-1.
A capacitor bank discharges current into the transducer, which is
represented electrically as an inductor and resistor, and mechanically as a force source.
Electrical energy is converted into mechanical energy inside the transducer and force is
delivered to the formation. In this thesis the formation is modeled as a linear massspring-damper system, though other physical models could be applied. From this
perspective, the system model is an LRC circuit connected to a mass-spring-damper
mechanical model through the electromagnetic transducer.
Electromagnetic actuators often use permanent magnets and/or soft magnetic materials in the generation of force which can lead to saturation issues at high current/force.
The actuator design presented herein does not use magnetic materials; instead, currents traveling through conductors induce magnetic fields that interact to generate
forces. The advantage of an all-copper actuator is that much greater force can be
K
MB
mechanical
system
charging
actuation
electromagnetic
circuit
circuit
force to/from transducer
Figure 1-1: PEM schematic
generated without saturation or damage to permanent magnets. Additionally, an allcopper actuator can handle more extreme temperature and chemical environments
than an actuator that includes permanent magnets. Though, over operating conditions in which permanent magnets can perform, the permanent magnet actuators
will typically require less current to achieve the same force output and thus generally
operate more efficiently.
1.2
Motivation and Prior Art
The actuator system developed in this thesis is a direct extension of a patented
work by Wilson and the Carrier Corporation [7] which describes an electromagnetic
actuator to plastically deform metal tubes and plates. This thesis builds on the radial
actuator concept, presenting a parameterized actuator system design and simulation
tool to enable design engineers to assess and optimize system performance for their
own applications.
In non-radial coordinates, the most common pulsed electromagnetic induction actuator is the rail gun. A rail gun uses induced magnetic fields to accelerate projectiles
in a straight line. The United States Navy has been developing rail guns for long
range missile launch and for ship based aircraft launch applications [8], [9], [1]. These
launchers are being considered because they can compete with explosive based, pneumatic, and hydraulic systems in pure kinetic-energy creation (exit velocity of projectile), but also because the electromagnetic system can be more robust, repeatable,
and offers the ability to modify/control launch parameters in real time (so, the same
launcher can be designed to optimally launch different types of projectiles).
Pulsed power electromagnetic actuators have also been used as the signal source for
non-contact vibration based sensors. A capacitor bank driven transducer system was
developed and used by Edgerton in the 1960's to map the bottom surfaces of lakes
and rivers that were too deep, too murky, and too varying to use vision or contact
based techniques [4], [5].
High speed/force electromagnetic actuators were presented as early as 1919 [2] but
generally have not made it into practical implementation until recently. A number
of technologies and realizations have come together recently that make electricity
based actuation a compelling option for high force applications. Solid state switches,
a technology that enabled the consumer electronics industry we know today, have
become capable of switching very large amounts of current very quickly, efficiently,
and predictably. Though, switches are still often the limiting factor in increasing
high power electromagnetic system performance [3]. Battery and capacitor performance continues to improve, increasing the achievable power and energy densities
that can be delivered electrically. Also, pervasive computation and numerical simulation techniques make system design and optimization a faster and intellectually more
manageable process.
1.3
Thesis Structure
This document takes the following structure:
* Chapter 1 presents the concept of using pulsed electromagnetic induction actuators to create high peak short duration mechanical force pulses and describes
the general features of such a system. Potential uses of a radially acting PEMI
system for oil exploration and completion activities is discussed and the need
for a parametric simulation to aid in assessing viability is presented.
" Chapter 2 describes the conductor layout of an induced magnetic field transducer that results in radial force. The relationship between electrical current,
induced magnetic field, and mechanical force is derived in terms of the relevant
parameters (conductor material properties and geometry) that will be used in
the simulation.
" Chapter 3 presents the design and modeling of the power supply/circuit and of
the mechanical formation. The relationship between electrical and mechanical
parameters to the time varying performance of the simulation are presented.
" Chapter 4 assembles the three modules (the transducer, power supply, and mechanical formation) into a single four-state time varying simulation and summarizes all of the relevant parameters that affect system performance.
" Chapter 5 describes the design and characterization of the first experimental
prototype. Design decisions and the use of the simulation to aid in parameter
selection are documented. The manufactured actuator system and the sensors
used to measure performance are presented. Finally, successful matching of
model and measured system performance is shown, and factors that limit system
performance are discussed.
" Chapter 6 describes an improved transducer design and additional experimental
validation. Here, a second method for conductor layout was developed; the
expected improvements of this modification were validated by the simulation
tool before manufacture, and confirmed experimentally. The transducer design
presented here is used as the basis for simulation results of the next chapter.
" Chapter 7 presents simulation results that demonstrate the viability of this
type of actuator to rapidly mechanically deform wellbores mechanically at a
high axial rate down the wellbore.
* Chapter 8 summarizes the development and performance of the parametric
simulation tool and describes some limitations and potential extensions of the
project.
Chapter 2
Radially Oriented Electromagnetic
Transducer
This chapter presents the design of a radially oriented electromagnetic transducer. In
overview, a transducer converts electrical and mechanical energies by the following
processes:
e When current flows through a wire, a magnetic field is created.
This phe-
nomenon is described by Ampere's Law and Gauss's Law.
e When the magnetic field from one wire interacts with current of another wire,
a force is created. The direction and magnitude of the force created by the
interaction of current in a magnetic field is described by the Lorentz Force Law.
* How a conductor or set of conductors creates a terminal voltage from its terminal voltage and motion affects how the device interacts with other electrical
components. This relationship is described by the device's inductance, defined
in Faraday's Law.
e The size and length of the wires in a device affect the resistance of that device,
which affects the electrical properties and performance of the system.
2.1
Lorentz Force Law on a Wire Review
The Lorentz force law relates how electric and magnetic fields interact with point
charges to create force. The force f on a single charge in free space is described by:
f = q[E + (u x po H)].
(2.1)
Here, f is the force, E the electric field, H the magnetic flux, yo the permitivity of
free space, q the electric charge of the particle, and u the velocity of the particle.
From this point forward, the following relationship will be used for magnetic flux
density B, magnetic field H, and pto:
B = po H.
(2.2)
A current carrying wire contains many individual moving point charges. When a
current carrying wire is in the presence of electric or magnetic fields, the total force
on the wire will be a summation of the individual forces on each point charge. This
total wire force is derived as follows, citing the geometry described in Figure 2-1.
Given a current flux J as
J = q Ndu,
(2.3)
and force distribution F defined as
F
f Nd,
where Nd is a volume distribution of charges
(2.4)
charge]) moving with velocity
u. Then, Equation 2.1 can be modified to become
f
Nd= qu Nd x B
F =JxB.
(2.5)
The electric force has been discarded under the assumption that the wire in Figure 2-1
is charge neutral.
Given the definition of F, the geometric relationship to the current carrying wire as
shown in Figure 2-1 is
f
=FAW.
(2.6)
Substituting Equation 2.5 into Equation 2.6 leads to
f
= (J x B)AW.
(2.7)
Since current i and current flux J are related as
i=JA.
(2.8)
Then, the macroscopic force on a wire within a magnetic field can be written as
fwire
2.2
= i W x B.
(2.9)
Conductor Layout for Radial Actuation
A uniform radially outward force can be achieved by either having a magnetic field
circulate rotationally while a current moves axially on the cylinder or having a magnetic field point axially while a current circulates around the cylinder. These two
orientations are shown in Figure 2-2.
Using copper wires, the axial current design could be achieved using a layout shown
on the left of Figure 2-3. In this figure current could travel 'up' the inner green wires
and 'down' the outer brown wires; the connections between the inner and outer are
not shown in the figure. Alternatively, the azimuthal-current layout would look like
magnetic field B
cross section A
lengthW
current flux J,
A
current i = J A
Figure 2-1: Current carrying wire in a magnetic field
Figure 2-2: Radial electromagnetic force layouts. On the left, current moves axially
and magnetic field circulates. On the right, magnetic field moves axially and current
circulates.
Figure 2-3: Axial and Azimuthal current layouts for radial force. The axial current
layout is the focus of this thesis
that shown on the right in Figure 2-3.
The axial current layout is more practical for creating a radial actuator that translates
radially; though one must design properly compliant connections between the inner
and outer sets of wires. The remainder of this thesis focuses solely on the axial current
layout.
2.3
Magnetic Field of the Axial-Current Layout
When current travels through the transducer a magnetic field is created that interacts
with the current to create force. The magnetic field is quantified by applying Ampere's
Law:
B -dl iA
enclosed.
(2.10)
A top view schematic of the axial current layout is shown in Figure 2-4. The inner and
outer wires are in series; if current is traveling up (out of the page) the inner conductor
loop, it is traveling down (into the page) the outer loop. The surface enclosing region
1 has zero current and thus no magnetic field. In region 2 the magnetic field includes
current of the inner conducting loop, and the magnetic field is evaluated by summing
outer conductor cylinder
(current traveling down)
inner conductor cylinder
(current traveling up)
region
2
1
3
Figure 2-4: Axial current layout for Ampere analysis
the enclosed current over the circular path:
2 7r
0
rO
enlsd
(2.11)
~
27 r
(2.12)
B = 901.
(2.13)
H=
27r r
Region 3 of Figure 2-4 encloses a total of zero current, since the axial currents of the
inner and outer wires cancel each other out. This is results in zero magnetic field
outside of the outer set of wires.
If the axial layout had many turns of wire in series, the resulting magnetic field would
include N turns (which is not the same as charge density Nd):
B B
0
2 1i
NI
rinner
-(2.14
(2.14)
2.4
Self Induced Electromagnetic Force
The Lorentz force law can now be applied to the axial-conductor layout. Neglecting
fringe effects, the magnetic field is constant along the length of the cylinder and
pointed radially outward at all points on the the outer set of wires (and radially
inward on the inner wires).
Fem,outer = -N iWB = poWN 2
2
2 7r router
(2.15)
where Wz is the axial length of the cylindrical actuator.
The
j term in above equation accounts for the tapering of the magnetic flux from the
inner-most point of the outer conductor to the outer most point - at the inner surface
of the outer conductor loop the enclosed current is
encloccd.
But, at the very outer
edge of the outer conductor, the enclosed current is zero, since the total current is
the current traveling up the inner conductor loop and back down the outer conductor
loop. Thus, over the thickness of the outer conductor loop, the average enclosed
current is one half the total enclosed current.
The force is evenly distributed radially outward on the surface of the cylinder, thus
the pressure exerted on the cylinder is
Pwellbore
2.5
Fem,outer
7 ro W
__
z
-2
oN
2
2
A ir2 r2.1'
Inductance of the Axial Current Layout
The inductance of the actuator is derived by applying Faraday's Law,
A = N<,
(2.17)
ri
ro
axis of rotation
Figure 2-5: Faraday surface
where
<D
J
B ds.
(2.18)
<D is the magnetic flux, and A is the flux linkage.
Figure 2-5 describes the surface used to analyze Faraday's law. Here, a cross section
view of the cylindrical actuator is shown lying with the z-axis horizontal. The Faraday
surface spans the length of the cylinder from the inner to outer surface (the red dashed
line).
A is further computed from Equations 2.17 and 2.18, substituting B from Equation 2.13, as
A=r
o
Lt.
27r r
(2.19)
Thus, inductance, as a time varying function of the outer radius r,(t) is
L(t) =
Ipo Wz N 2
2
2 7r
).
ln(
ri
(2
(2.20)
2.6
Transducer Resistance
The resistance of the transducer is determined by conductor geometry cross section -
length and
and conductor resistivity:
Rtransducer = Plc
(2.21)
where Rtransducer is the resistance, p the resistivity, lc the conductor length, and Ac
the conductor cross sectional area.
Total conductor length is a function of axial length Wz and turns N, as well as the
length of the end-turn wires that connect the inner and outer axial wires together.
One end turn connector is length t end. Therefore,
lc = 2 NWz + 2Nend.
(2.22)
The length of the end turn plays a limiting role in the maximum radial travel of the
transducer; increasing the maximum travel distance will increase resistance of the
transducer, which can limit current and maximum force.
The calculation of any real Ac depends on attainable conductor geometry but the
following example shows how Ac is a function of turns N, conductor thickness -y,and
transducer radius r. Figure 2-6 shows two cylinders with cross section, radius, and
number of turns. Each cylinder could represent either the inner or outer conductor
set of the transducer.
Ac = 7 ((r±+
N
r2)
(2.23)
The N = 4 case would have a quarter the cross sectional area and four times the
conductor length, so the ratio of the total resistance of the N = 4 cylindrical conductor
to the N
=
1 conductor is 16 -
or more importantly, N 2
N=1
N=4
(a)
(b)
Figure 2-6: Cylinders with radius, thickness, and number of turns
Chapter 3
Power Supply, Electrical Energy
Model, Mechanical Energy Model
This chapter specifies and simulates the electrical power system that delivers electrical
energy to the transducer, and the mechanical system (the formation) upon which the
transducer acts upon. A schematic of the energy domains modeled here are described
described in Figure 3-1.
3.1
Capacitor Bank and Power Circuit
As seen in Equation 2.15, force out of the transducer goes up with the square of
current. If the goal of the system is to create large forces, the goal of the electrical
system design is to create large currents. Discharging a capacitor into a low resistance
and low inductance circuit is a common method for creating a short burst of large
electrical
energy
mechanica
energy
Figure 3-1: Energy flow overview
Rtransducer
Cbank
Vsource
charging
circuit
r
Ltransducer
actuation
circuit
Figure 3-2: Power circuit
current.
The electrical circuit is shown in Figure 3-2. A charged capacitor bank is triggered by
a silicon controlled rectifier (SCR). An SCR is a solid state switch that can stand-off
large voltages until triggered by a low energy control signal. The SCR can carry
thousands of amps of current and typically has a fraction of a milliohm of resistance,
making it a nearly ideal, lossless switch. The capacitors are charged by closing the
switch to a DC voltage source; given proper duty cycle, the capacitors will reach the
voltage of the charging supply.
3.2
Discharging Capacitor Circuit: Modeling Current Flow Through the Transducer
The actuation circuit can be simplified to the RLC circuit shown in Figure 3-3. The
capacitor contributes the dominant capacitance term and the transducer the dominant
inductance term (every component has a little bit of inductance, capacitance, and
resistance). The transducer wires, capacitor bank, connecting wires, and SCR all
contribute to the circuit's parasitic resistance.
The discharge of the capacitor bank through the inductor and resistor is derived using
Vr
Rparasitic
i
-TCbank
+
Ltransducer o
actuation
circuit
Figure 3-3: RLC circuit and KVL polarity definitions
Kirchoff's voltage and current laws and introducing the flux linkage A, resulting in
Equations 3.1: through 3.6.
KVL:
Vc- V, - VL=0
KCL:
-c
=z=L
flux linkage: A- L i
dA
d=
dt
Vc-
VL
(3.1)
i
(3.2)
VL dt
(3-3)
AR
V
c-iR
=Vc
L
-A
= C
LC
dVe
- = -i-
dt
(3.4)
(3.5)
d
A
-
I
A
dt
v
-1
0
V
(3.6)
Flux linkage A is chosen as a state rather than current i since inductance L is not
constant throughout the time varying simulation. The state equations in Equation 3.6
are used in the time-varying simulation of voltage and current during the capacitor
discharge event. These state equations will be combined with the mechanical system
state equations to form the full electro-mechanical simulation in Chapter 4.
Two components in the actuation circuit schematic of Figure 3-2 are not included
in the simplified schematic (and related model) of Figure 3-3: the SCR and the
freewheeling diode. Once triggered, the SCR remains closed as long as current is
flowing from anode to cathode. The switch latches opens when current approaches
zero. Pure RLC circuits may exhibit large negative voltage and current swings; an
RLC circuit controlled by an SCR will not experience the same transient response
as a pure RLC circuit beyond the first zero crossing of current. Though, nearly all
useful energy transfer occurs before current swings to negative in the RLC circuit,
so the straightforward RLC model is valid for the important parts of the actuation
event.
A flyback diode is necasarry to protect the capacitor bank from large reverse voltages
which is necessary for the integrity of electrolytic capacitors used in this power supply.
A simulation using the state equations in Equation 3.6 will only resemble reality
through the initial voltage drop to zero.
3.3
Mechanical Force/Displacement State Equations
The actuator system is designed to do work on a formation by applying radial force.
Simulating the mechanical system is an important part of understanding system performance. Figure 3-4 shows the simplified formation model used in the simulation.
The spring constant and damping of the stiff outer formation are treated as linear,
therefore the state equations for formation displacement can be described through
simple application of Newton's first law, with the full state described in Equation 3.8.
Note that Equations 3.7 and 3.8 are written with variable r to denote radial displacement.
Fradial
- kr
- drm
= r
(3.7)
mechanical
system
force to/from transducer
Figure 3-4: Mass spring damper model
d
+
Fradial
(3.8)
A similar force will occur radially inwards on the inner wires of the transducer. The
transducer should be designed such that the inner wires are mounted in a very stiff
configuration and thus displace negligibly.
3.4
Estimating Formation Properties Downhole
The stiffness of wellbores downhole is estimated using Young's modulus values in
[10], combined with a thick walled cylinder approximation described in Equation 3.9
from [11]. In Equation 3.9, a is the wellbore diameter, and b is much greater than a.
p is pressure applied to the wellbore, E the formation Young's modulus, and v the
Poisson's ratio. The stiffness of the formation is given by Equation 3.16 and derived
as follows.
Aa
p) (a2
E(b2-
33
+
b2
2
V)
(3.9)
a2 + b2
b 2 -a 2
(3.10)
when b > a
define wellbore diameter: d_ = a
(3.11)
Adw = d, pv
E
(3.12)
p A = F = K Adw
(3.13)
A =7r dw Wz
(3.14)
K =
(3.15)
Adw
gW, EN
Kradial
-
V
=
(3.16)
m
The Young's modulus values of interest range from 0.01 to greater than 10 GPa,values
associated with soft/unconsolidated sandstone to highly consolidated sandstone. This
results in an estimated wellbore stiffness from about 107 to 1010
N/r.
Chapter 4
Assembling the Full Parametric
Simulation
This section documents the full parametric simulation of the radial actuator system.
This includes the system parameters (inputs), state equations, and useful simulation
outputs. A complete implementation of this simulation in MATLAB is provided in
Appendix A.
The conversion of electrical energy from the capacitor bank to mechanical energy in
the formation via the electromagnetic transducer is modeled as a four state set of
differential equations. All of the variables and equations in this section were derived
in the previous transducer, power supply, and formation modeling sections (Chapters
2 and 3).
4.1
System Parameters
Table 4.1 shows the input parameters that define the simulation. This list includes
relevant parameters of the transducer, the power supply, and the formation.
Note that -y is the conductor thickness. If conductor thickness is constant and the
Description
initial screen outer radius
conductor thickness
screen axial length
number of axial turns
conductor resistivity
Unit
m
m
m
V
F
Q
vol
initial capacitor voltage
capacitor bank capacitance
capacitor bank effective resistance
capacitor to actuator volume ratio
K
M
D
wellbore radial spring constant
system mass
wellbore damping constant
N/rn
Rscr
SCR on resistance
Q
Symbol
Wz
N
p
Resr
Qm
kg
k 9 /s
Table 4.1: Simulation input parameters
X1
X2
_
outer radius
outer radius velocity
X3
flux linkage
X4
capacitor voltage
ro
[ml
ro
[m/s]
A [V s]
V [V]
Table 4.2: Simulation system states
inner and outer radius are close to each other, then
Rrnne= Ro,j -T.
(4.1)
The system states that will be used to solve the ODE's are shown in Table 4.2. The
first two states are displacement and velocity of the mass spring damper system, and
the second two states are related to voltage and current of the electrical system.
4.2
State Equations
In numerical simulation, a state update is performed every time step dt. The state
update equations are shown in Equation 4.2, incorporating the global constants from
Table 4.1.
L
-
ptoW_ N 2 In
,i-x
3
.X3
L
i
B - poN
2 7r X1
FerM
-K
(X1 -
1
NiB
2
-
Ro,2 ) -
3 = X4-
DX
(4.2)
2 + Fern
Rtot i
X4 = -
Due to the electrical implementation (the SCR and the freewheeling diode), the set
of update equations in Equation 4.2 is valid until V = 0 or iL < 0.
4.3
Energy Balance in the Simulation Output
Energy must be conserved in each time step of a valid simulation result less rounding
errors, this relationship is given by Equation 4.3.
E[n] = Enitiai =
CV
(4.3)
(4
Param
Static Measured
Cbank
0.24
ESRcapBank
2.8
Ro'i
Ri
N
55
45
24
Rtransducer
1.8
Ltransducer,i
7
K
2.32
Unit
F
mQ
mm
mm
mQ
MN
Table 4.3: System parameters used in Section 4.4 output viewgraphs
The energy domains relevant to this system are described in Equation 4.4.
1 C V2
Ecapacitor =
1
-
12
Ar2
Espring =K
21
Eresistance
Edamper
(4.4)
M r
2
=
Ekinetic =
=
J
Ji
T
2
R dt
D dt
Accounting for energy flow across the various electrical and mechanical domains is
useful as a sanity check -
if Equation 4.3 is not satisfied something is very wrong.
And, since the point of this actuator system is to transfer electrical energy into mechanical energy, viewing the energy flows in a simulation result is useful for comparing
system designs across different size/energy scales.
4.4
Viewing Output Data
The graphs in Figure 4-2 show some of the key output data of the simulation. The
simulated flow of energy, as described in Equation 4.4, is shown in Figure 4-1. This
figure is from the same simulation result as Figure 4-2.
The system parameters used in this simulation are summarized in Table 4.3:
0.7
0.6
0.5
0.4
---
------
---- ---- ------ ---
0.3
-- - --
--
--- --
M ech Spring
---
- ------------ - -
Electromagnetic
Mechanical
- - ---
-*Resistive
-------- - - - ----±-
0.2
-
Total
ilip
0.1
U
1
2
3
4
5
6
7
8
Time [ms]
Figure 4- 1: Simulated energy flow
9
10
0
2 - -------- -- -.--- ---
..
- -- -
---
6
-- - -- -- --------------------- ------ - - - - - -
2
.
.. ..
-4
-2- -
2
3- -6
7
-----------------
|
1
2
3
4
5
6
7
Time[ims]
8
9
10
I
1
4
3
2
1
-
|
5
Time [ms]
7
6
10
8
9
--
-------
(b) System current
(a) Outer Radius Extension
e
07
0.07
-- --
7
I
.1
|
31
U0
.
8
--- -------
------
---------- ---........
I-- --- - ---
-- ------ -----
06---
0.
-- - --- - - - - - - - - ------ - --
-- - - - - -
--- -
--- - -- - - - ---- -- - - -----
0.
6 ------ ....
..---- - ---- +- - -
--
-
-...
--.
------.
-0 02 -
6-
0, 01
--
- ---- - - 4 - ---- ---
-.. . . . . . ..
---- - -
0 03
---------- ...- ...-- - -- -----5 - --...
-... . . . I.. . .I-
0.04 ------
-- -- - -- - - - - -
--
-- - +----+-------
----- -
---
- - - - -
- ---
- ------
--
-
- -----
---
---- - - -.-- --.- -- -- ------
0
-0 01
01
-+ -
--------
..
-.
-
-0
0
1
2
3
4
6
5
Time [ms]
7
8
9
2
10
m1
e 5[1
Time [ms]
(d) Capacitor voltage
(c) Transducer inductance
Figure 4-2: Simulation output: (a) outer radius extension, (b) current , (c) inductance, and (d) voltage in simulation
40
--
-
-
-- - - - - - - - -- - - -- - - - - - -
+---+-------------- ----
-
---
Chapter 5
The Radial Actuator Prototype
and Characterization
This chapter begins the second part of the thesis: creating and evaluating real actuation system designs using the design framework and parametric simulation tool
developed in Chapters 2 through 4. The necessity of building real systems is twofold. First, a real system must be made and characterized in order to validate the
simulation; that is, the simulation must be proven experimentally. Second, to move
beyond simulation and into functional device design, the simulation parameters must
be connected to a real system layout made of real materials and components.
Two prototype actuator systems were built and experimentally characterized. The
complete design of these actuators and the experimental setup and methods used
to characterize are described here. The story of this design, manufacture, and experimental characterization will be told chronologically. The author feels that many
insights of this work -
both as a designer of things and as a designer of pulsed
induction radial actuators -
will be gleaned most from the design story. See Sub-
sections 5.3 and 6.2 to jump straight to the prototype dimensions/parameters and
experimental results.
Symbol
RO'
-y
Wz
N
p
V
C
Resr
vol
K
M
D
Rsecr
Description
initial screen outer radius
conductor thickness
screen axial length
number of axial turns
conductor resistivity
initial capacitor voltage
capacitor bank capacitance
capacitor bank effective
resistance
capacitor to actuator volume ratio
wellbore radial spring constant
system mass
wellbore damping
stant
SCR on resistance
con-
Requirement
wellbore sized
small enough to allow thin conductor approximation
long enough to make edge effects negligible
small enough to be practical to assemble by hand
the smaller the better
high enough to be interesting, low enough to simplify
design and safety requirements
same as above
the smaller the better
1:1 desirable, do not want too large
stiff enough to be interesting, small enough that displacement can be measured without huge forces or
obscure sensors
something practical
for experiment: low
the smaller the better
Table 5.1: Simulation input parameters
5.1
Specifying Parameters for the Initial Prototype
As shown in Table 4.1 on page 36 the system model is defined by thirteen lumped
parameters related to the actuator. Five parameters completely define the transducer
design and performance, four define the capacitor bank performance, three define the
formation, and one the SCR electrical switch. Thirteen parameters is a lot of values
to choose at once; a starting point for guiding parameter specification is described in
Table 5.1 and discussed here.
A wellbore is commonly 5 to 36 inches in diameter, though the diameter of tools that
will be inserted into the well are typically about half the wellbore diameter, putting
a scaled prototype tool around 2.5 to 18 inches in diameter.
To avoid fringe effects in applying Ampere's and Faraday's Laws, the axial length
should be larger than the radius (W, > 3 R0,).
Avoiding fringe effects is useful for
model accuracy, but also because fringe magnetic fields do not contribute to useful
force generation at the transducer. Thus, if the axial length is set to 12 inches (large
enough to be interesting, but can still easily fit on a lab bench), then the radius may
be set to around 4 inches.
A rough sizing of the capacitor bank can be made by relating the expected volume
of the transducer and the energy density of the capacitors in the capacitor bank.
This tool prototype specifies that the volume of the capacitor bank be approximately
the same as the volume of the transducer itself. The energy density of capacitors is
estimated as 0.18 MJ/m
3
using a data sheet for high temperature capacitors from
Vishay [12]. Thus, the capacitance and maximum operating voltage of the capacitor
bank can be sized as a function of volume as in Equation 5.1. Specifying capacitor
performance with high temperature capacitors is important for this design because
the tool should be capable of operating in extreme environments downhole (even
though the actual experimental device does not use specialty capacitors).
V0ltransducer
volVo~cap
cap
2?r
I CV2 [j]
-~V
volcap
7
J
2Edensity
(5.1)
voltransducer
2
Edensity 7rr2 Wz
2
2(0.176e6)
C
~ r 0.055 (0.3)
752
0
=-0.18
Operating at high voltages increases system design complexity from a power system
and safety system perspective. So, the voltage of the capacitor bank was designed
to be 75 volts or less (there really are no safe voltages; 75 volts is still capable of
creating significant human harm). System design for optimized performance will be
discussed further in Chapter 7, where voltage will not be limited to 75 V.
A range of formation stiffness downhole is described in Section 3.4. The design goal
is to select an experimental formation stiffness that is stiff enough to be interesting,
but not so stiff that it is hard to measure or that requires extremely large forces.
Thus, one can use the parametric simulation, given the general dimensions known
and filling in the rest, to get a feel for a good target stiffness. For instance, a desired
range of motion might be half an inch, which would allow for the use of relatively
inexpensive contact sensors; a minimum range of motion of a few millimeters would
allow inductive or capacitive range sensors to be used. Further refining K will be
covered in Subsection 5.1.1.
Formation mass and damping must also be selected. The author decided to limit
damping in order to simplify the experimental system design, so for initial specifying,
D is set to zero. The mass may be no more than the outer loop of the actuator; in
a real formation the mass of the rock being pushed would have to be included. The
mass will be estimated at most as the mass of pure copper of radius r,, and thickness
-y, as in Equation 5.2
M
M = (8940
Pcopper r ((r o,i ± _)2 - r2,i) Wz
[kg/m
3 ])
r ((0.055 + 0.01)2 - 0.0552) 0.3
(5.2)
M ~ 10 [kg]
5.1.1
Parameter Search Using the Simulation
At this point, Table 5.2 provides an update on the range of values for various components. Though progress has been made, N, -y, K, and the relationship between C
and V of capacitor bank are yet to be determined. The parametric simulator will
help select parameters that result in a useful actuator system.
A parameter sweep that iterates the simulation across number of turns N and conductor thickness y is shown in Figure 5-1. To create this figure, 25 values of N and
25 values of 7 were selected within an interesting value range. All other input parameters of the simulation were held constant at the values in Table 4. A simulation
for each set of parameters was run and the maximum displacement of the formation
was noted.
Symbol
Roi
ly
Wz
N
p
V
C
Resr
vol
K
M
D
Rsecr
Description
initial screen outer radius
conductor thickness
screen axial length
number of axial turns
conductor resistivity
initial capacitor voltage
capacitor bank capacitance
capacitor bank effective
resistance
capacitor to actuator volume ratio
wellbore radial spring constant
system mass
wellbore damping
stant
SCR on resistance
con-
Requirement
~ 4 inches
less than 0.25 inches
12 inches
less than
30
copper: p
1.7e-8
n/m
75 V
- 0.2 F
the smaller the better
1:1 desirable, do not want too large
stiff enough to be interesting, small enough that displacement can be measured without huge forces or
obscure sensors
~10 kg
small
small
Table 5.2: Refining simulation input parameters
Symbol
Wz
N
p
V
C
Resr
vol
K
M
D
scr
Description
initial screen outer radius
conductor thickness
screen axial length
number of axial turns
conductor resistivity
initial capacitor voltage
capacitor bank capacitance
capacitor bank effective resistance
capacitor to actuator volume ratio
wellbore radial spring constant
system mass
wellbore damping constant
SCR on resistance
Requirement
0.055 m
sweep 0 - 12 mm
0.3 m
sweep 1 - 100 turns
copper: p = 1.7e-8 n/m
function of C and volume ratio
sweep 0.01 to 1 F
0001 /C
1:1
sweep 106
-
10 7 N/r
10 kg
Table 5.3: Simulation input parameters for parameter sweep
max deflection (mm) for K = 1000o0[N/m], C = 0.25[F]
max deflection (mm) for K = 200000[N/m], C = 0.25[F ]
6
90
16
E
BO
14
80
12
70
.70
4
E
10
0
8 02
50
ts600
6
2
10
0
0
15
20
10
thickness
[mm]
conductor
5
25
2
10
0
30
0
5
(a)
25
30
(b)
o[N/m], C = IF]
maxdeflection
(mm)forK = 2(D
10
15
20
conductor
thickness
[mm]
(mm)forK 1E00tXM[N/ml,
C= 1[F]
maxdelection
4.5
14
12
7
10
a
3.5
35
6702
40
E4
105
10
0
1.5
20
520
2
10
asM
0
0
5
10
15
20
[mm]
conductor
thickness
(c)
25
30
0
5
10
15
20
conductor
thickness
[mm]
25
30
(d)
Figure 5-1: Parameter Sweeping N and -y
Given the results of Figure 5-1 and the desired limits on N and Y-
that is, desiring
no more than about 25 turns of copper and no more than about 0.25 inches (6.4 mm)
of conductor thickness, N can be set to 24 turns and -yset to 6.4 mm. 24 turns could
easily be 25 or 26 turns; but 24 is divisible by a larger range of values, which may
come in handy when discretizing the radial stiffness field (see Section 5.2.1).
Having set N and 7, the result of sweeping K and C is shown in Figure 5-2. In this
figure, each vertical row (constant K, varying C) deflection result is normalized to
itself; for the same force capability a formation with stiffness K = 105
N/m
will have
orders of magnitude more deflection than a formation with stiffness K = 107 N/r.
The first subfigure shows normalized effectiveness of the actuator for a range of C
between 0.1 and 10 F, while the second subfigure refines this parameter range to 0.1
maxdeflection
(mm)forN= 24.th = 6.4mm
1
1.5
2
radial
stiffness
[N/m]
2.5
max deflection (mm) for N = 24. th = 6.4 mm
3
x 07
|
(a)
1.5
2
radial
stiffness[N/m]
2.5
3
x Id
(b)
Figure 5-2: Parameter Sweeping C and K
K parameter
sweep
10'
10
stiffness [N/mI
radial
Figure 5-3: Radial Displacement with Varying K
to 2 F because almost all of the interesting information is in this range. Though the
optimal capacitance varies with K, a general choice of 0.25 F gives good performance
through the K parameter sweep.
Now, a reasonable range of K can be expected given all other selected parameter
values. A desired K value will result in a change in radius of more than 6 mm but
less than 12 mm. Figure 5-3 shows the expected maximum displacement for the
actuator system with varying K and all other parameters as specified previous in this
section. A value in the range of 1 - 6 MN/m should be selected.
Finally, a rough system design for the alpha prototype is set and summarized in
Table 5.4.
Wz
N
Description
initial screen outer radius
conductor thickness
screen axial length
number of axial turns
Requirement
0.055 m
6.4 mm
0.3 m
24-26 turns
p
conductor resistivity
copper: p = 1.7e-8 0/m
Vo
C
initial capacitor voltage
capacitor bank capacitance
function of C and volume ratio
0.25 F
Resr
capacitor bank effective resistance
.0001/C
vol
K
M
D
capacitor to actuator volume ratio
wellbore radial spring constant
system mass
wellbore damping constant
1:1
1 - 6 MN/rn
10 kg
0
Rscr
SCR on resistance
0
Symbol
RO,
Table 5.4: Constrained input parameters for initial prototype design
5.2
5.2.1
Prototype Machine Design
Transducer and Formation Design
The first prototype actuator system was designed around the parameter values in
Table 5.4 from the previous section. This section describes the primary decisions
that led to a real experimental prototype design and manufacture.
The axial conductor layout in Figure 2-3 on Page 23 shows the general conductor
layout for the transducer. This layout must be mounted so that the inner coils do
not move and the outer coils are constrained to only move radially outward into an
experimental formation of specified stiffness. Furthermore, the experimental formation must be the dominant compliance of the system so that force and displacement
sensors can be fixed to a mechanical ground plane.
Simulations showed that system performance was optimized with conductor thicknesses of 0.25 to 0.5 inches. 0.5 in thick conductors seemed too costly and hard to
work with. It was decided to make the individual turns of conductor out of sheets of
0.25 in thick copper. Solid copper was chosen over stranded copper wire because copper bars could provide their own radial stiffness, whereas a radial actuator that used
compliant copper wire would require an additional mechanical shell around the outer
conductors for the conductors to maintain their alignment with respect to each other.
As will be shown later, the second prototype used stranded wire more effectively.
Due to the high radial stiffness of the experimental formation, the frame holding the
transducer, formation, and sensors should be metal (aluminum) rather than plastic.
This creates an additional challenge in making sure all of the coils are electrically
isolated from each other and the circuit ground plane. Thus, it is critical to mount
all copper conductors to a plastic (electrically insulated) interface before connecting
to the aluminum frame.
Figure 5-4 shows a cross section of the experimental frame, formation, and transducer
design. Moving from the outside in, thick aluminum bars (1" x 1.5" x 12") sandwiched
by a top and bottom plate (0.25" thick) create the stiff mechanical frame for the
transducer and formation. Stiff compression springs are held between the frame and
transducer by magnets. The outer loop of the transducer (the brown loop) connects to
the springs via a plastic (ABS) mechanical interface. Stranded copper wire electrically
connects the inner and outer conductor loops, while the inner conductor loops attach
to an aluminum cylinder through another set of plastic ABS spacers.
The inner
aluminum cylinder is attached to the bottom aluminum plate.
Figure 5-5 shows the same cross section view of the system from an orthographic
perspective.
Here one can see how sets of four outer conductors (one sixth of a
revolution) share an outer-frame bar and pair of springs. Additionally, at the top of
the screen there are load cells in series with one set of springs.
As seen in Figure 5-4, compression springs sandwiched between magnets provide the
radial stiffness required to operate the actuator system. Provided that compression
springs are evenly spaced around the cylindrical transducer, the total radial stiffness
is the sum of the individual spring elements:
Kradial
Nsprings Kspring
(5.3)
The magnet-spring interface and the body of the spring are in shear, holding up the
inner conductor loop
connectingwires between
inner and outer conductor
outer conductor loop
top plate
compression spring
magnet
outer loop
conductor-frame
interface
outer frame
innerframe
inner loop
conductor-frame
interface
bottom plate
Figure 5-4: Cross section of transducer and radial stiffness
mass of the outer loop of copper bars; the one inch long springs, which are 1000
lbs/inch, could handle a few kg of cantilevered mass on their ends without deflecting
in the transducer's axial direction.
#10 gauge stranded wire was used to connect inner and outer conductors together.
Three inch lengths of wire with spade terminals crimped to each end act as low
stiffness mechanical flexures between the thick copper bars. Early in the design process, high stiffness blade flexures that would provide radial stiffness for the actuator
and conduct current were considered; this plan was ultimately scrapped because the
dimensions of a suitable flexure element were too extreme. As will be discussed in
Chapter 6, the decision to use #10 gauge wire was not ideal because it contributed too
much parasitic resistance. This design decision was revised in the second prototype
actuator.
The aluminum bars that make up the outer frame are sized in order to withstand the
Figure 5-5: Orhtographic view of experimental device
loads expected by the actuator, up to 200 pounds through each spring, and also to hold
all components together. 1.5" x 1"cross section bars were selected to provide sufficient
surface area to create a two-bolt joint to the top and bottom plates. A justification
for having both a top and bottom plate is shown in Figure 5-6. Both FEA results
have 800 N loads along each contact area between the spring and bar; the cantilevered
beam deflects nearly 0.3 mm, while the top and bottom plate fixed bar deflects only
0.003 mm in this loading scenario. 0.3 mm could approach a percentage of the total
spring deflection, while 0.003 mm would be barely 0.01% of total deflection.
Specifying compression load cells was fairly easy at this load requirement and size
scale. The load cells are 250 pound max cells from Omega; one mounts in series with
a spring on one of the six bars. The plastic connectors that provide electrical isolation
for the copper bars are shown in Figure 5-7. The plastic pieces and copper bars are
water jetted and epoxied together on the joint.
LRES(mm)
URES(mm)
3.479e-003
3.189e-003
2899e-003
3.1778-001
2.912e-001
2.647e-001
2.383e-00
2.609e-003
2.118e-001
2.3198-0
2.029e4)03
1.739e-003
1.450e-003
1.180e-003
1.853e-001
1.588e-001
1.324e-001
1.059e-001
8.697e-004
7.942e-002
5.798e-004
5.295e-002
2.899e-004
2.647e-002
I 0008-030
1.000e-030
(a) Top and Bottom Fixed Bar
(b) Bottom Fixed Bar
Figure 5-6: FEA result for 800N loads on each spring
(a) outer conductor plastic feature (b) inner conductor plastic
feature
Figure 5-7: Plastic connection detail feature view
Figure 5-8: Completed Alpha Prototype Transducer, Formation, and Capacitor Bank
The completed transducer is shown in Figure 5-8. Similarly, the complete experimental system including charging circuit, power circuit, and data acquisition system is
shown in Figure 5-9.
5.2.2
Capacitor Bank and Circuit Components
Building a 0.25 F, 75 V capacitor bank by hand is most practically done using large
aluminum electrolytic capacitors. Other capacitor types are more practical for handling extreme environment operation and for decreasing ESR, but aluminum elec-
charger power supply
DMM displays
capacitor voltage
trigger power supply
transducer
DAQ breakout box
SCR
capacitor bank
data aquisition
experimental frame
Figure 5-9: Complete Alpha Prototype Experimental Setup
trolytics can be easily purchased with relatively high capacitance, making the part
count low.
As shown in Figure 5-8, the capacitor bank is made of eight 0.033 F, 75 V electrolytic
capacitors in parallel, leading to a total capacitance of about 0.25 F to be operated at
up to 75 V. The bus bar is made of 0.125" copper, the positive and negative terminals
are spaced about 0.0625" apart.
The voltage of the capacitor and the expected current from simulation will guide the
selection of the additional electrical hardware -
the freewheeling diode and the SCR
switch.
5.3
Alpha Prototype Characterization
This section describes the complete sensor system and experimental process used to
characterize the prototype actuator system. The characterization process consists of
two steps. First, static system parameters such as the system's total resistance, the
capacitor bank capacitance, the transducer inductance (which reflects geometry), the
formation stiffness, and the system mass must be measured statically in the initial
state to verify that actual parameters of the built system. Second, the system's electrical and mechanical performance must be measured dynamically in order to compare
the complete performance of the system to the expected simulation performance.
5.3.1
Static Measurements
Static electrical system properties were measured using an impedance analyzer, which
can alternatively provide R and C, or R and L values.
Figure 5-10 shows the
impedance analyzer measuring R and C of a single capacitor (the capacitor bank
is made of 8 capacitors in parallel); from this measured value, the total system capacitance and ESR are
CcapBank =
E0
NcapacitorsCsingiecap = (8) 30 [mF] = 0.24 [ F]
0.004 [Q]
DRsige
ESRcapBank
ingiecap
-
Ncapacitors
_
00 [8
8
-
=
(5.4)
Q
[Q]
(5.5)
The expected capacitance was 0.264 F and the expected ESR was 1.3 mQ max.
The resistance of the transducer measured 0.022 Q. This resistance value is much
larger than the 2.5 mQ expected resistance as calculated using copper resistivity and
geometry of the copper bars. Of course, until this point the author had not given
proper attention to the additional resistance of the 10 gauge connecting wires, nor
the relatively long 6 gauge wire connecting the capacitor bank to the power switch to
the actuator. Given that there were 24 turns of the transducer, and thus 48 count 75
mm long 10 gauge wire connectors, and an additional 750 mm of 6 gauge wire, the
additional resistance was explained.
Fortunately, simulations of the system with all resistance sources accounted for showed
that the system would still perform measurably well, that is, the expected forces and
capacitance [pLF]
frequency [kHz]
resistance [0]
measured capacitor
impedance
analyzer
Figure 5-10: Measuring Capacitor R and C
displacements would still be experimentally quantifiable using the existing experimental setup.
The actual geometry of the transducer resulted in a outer radius of 0.055 m and
an inner radius of 0.045 m, with a conductor thickness of 6.35 mm. The additional
spacing between the inner and outer conductors is due to the copper bars needing
free space between each other because the conductors were not electrically insulated.
This results in an estimated initial inductance of 6.9 pH and a measured inductance
of 7 pH.
The purchased springs were specified as 1080 lbs/in stiffness. The actual spring
stiffness was measured during load cell calibration using an force-displacement testing
machine. Measured stiffness was 1000 pounds per inch, resulting in a total radial
stiffness of 2.32 MN/m. The total mass of the copper bars, outer plastic connectors,
and half the total spring mass came out to 6 kg.
A summary of all the measured static input parameters is summarized in Table 5.5.
The following section provides the time-varying performance of the system.
~I
Param
Cbank
ESRcapBank
Ro'i
Ri
N
Rtransducer
Ltransducer,i
K
Static Measured
0.24
0.5
55
45
24
22
7
2.32
Initially Expected
0.264
max: 1.3
55
49
24
2,5
6.9
2.4
Unit
F
mQ
mm
mm
mQ
pH
MN/rn
Table 5.5: Statically measured values of alpha prototype
Component
Capacitor
Load cell
Load cell signal
conditioner
Load cell calibration machine
Displacement
sensor probe
Displacement
sensor
conditioner
DAQ Hardware
DAQ Software
SCR
Freewheel diode
Name
Vishay
LCROmega
250
Omega DMD465WB
ADMET 26xx
series
Lion Precision
U8C
Lion Precision
ECL133
Relevant Specs
0.033 F, 10 mQ max ESR
250 lb capacity load cell
strain gauge signal conditioner
-
2 mm range, 0.31% error band
-
16 inputs, 250 klz throughput, 12 bit A/D
NI 6036E
NI Labview 8.6
-
Powerex
9 kA surge, 1600 V stand off
T7H8650A
Vishay 45L(R)
mGis
voltage; 3 kA,
100 V reverse
surge
max
Table 5.6: Purchased parts
5.3.2
Dynamic Measurements: Setup
The dynamic experiments seek to measure mechanical force and displacement of the
transducer and voltage and current in the electric power system during an actuation
event. In simulation, a single data event -
from triggering the SCR to fully discharg-
ing the capacitor - will occur in a total of 10 ms, thus a data acquisition system
that samples at greater than 10 kHz is desired. The acquisition system is a National
Instruments pcmcia card and a PC running NI LabView 8.6, which can sample at a
total of 250kHz over many A/D converters. All sensors were acquired at 20 kHz.
All purchased parts
/
part numbers are summarized in Table 5.6.
Figure 5-11: Non-contact inductive sensor on prototype
Two load cells were placed in series with a pair of springs on one of the six compliant
regions of the radial stiffness frame. Initially, when the expected displacement of the
actuator was expected to be nearly 0.5 inches, inexpensive linear potentiometers were
specified for displacement sensing. These sensors were too jittery over a range of a
few millimeters, so non-contact inductive sensors with a 2mm range were used. This
non-contact sensor is shown in Figure 5-11.
The voltage across the capacitor bank was sent through a voltage divider into the NI
DAQ. This measurement is not the 'ideal' capacitor voltage, but is the 'capacitor +
ESR' voltage, as shown in Figure 5-12.
Cap bank
ESR
measured voltage
Figure 5-12: Schematic of measuring voltage across a real capacitor
5.3.3
Dynamic Measurements: Alpha Prototype Experimental Results
Figures 5-13(a) and 5-13(b) show the experimental and simulated mechanical and
electrical performance of an actuation event for the alpha prototype. The experimental performance reasonably matches that expected by the simulation.
time [s]
100
0
-100
|
0
1
2
i
I
I
3
4
5
time [s]
6
I
I
i
7
8
9
10
(a) Mechanical Measurements
iterating current once to account for ESR drop...
c 80
60
experimental
simulation
~
w
CO I:'
20 I
Ca
|
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
time [ms]
6
7
8
9
10
U
3000
;T 2000
1000
0
0
(b) Electrical Measurements
Figure 5-13: Alpha Prototype Experimental and Simulated Actuator Event
60
Chapter 6
The Second Radial Actuator
Prototype
While the first prototype was a success in terms of model and experimental agreement,
the excess parasitic resistance in the transducer compromised system performance.
The bulk of this excess resistance was contained in the 'end turns' - the flexible
10-gauge wires that connected the thick copper bars to each other. Thus, a lower
resistance transducer that had the same general design goal as described in Table 5.4
was built.
This section describes the Beta prototype design and characterization. Only a new
transducer was built; the power electronics, experimental frame, and data acquisition
system were reused with slight modification.
6.1
Litz Wire Transducer Design
In the initial transducer design, using copper bars as conductors seemed like a feature:
the bars could do double duty as conductors and as the rigid frame for the transducer.
But, while this choice made the rigid-frame design simpler, it was challenging to make
48 additional connections between the 48 copper bars. These connections could only
easily be achieved by using very small (thus higher resistance) wire. These connecting
wires contributed too much resistance, causing poor energy transfer to the formation.
Upon re-evaluating design trade-offs, it becomes obvious that a similar design as the
Alpha prototype can be achieved with a single length of flexible wire instead of rigid
copper bars and smaller connectors. In the experimental setup, a rigid shell could be
placed around the flexible wire to guide the cumulative electromagnetic force on each
wire into the radially-stiff springs.
A suitable wire type is rectangular cross-section Litz wire, which was sourced from
New England Wire Technologies [13]. The macro-wire is 0.25" by 0.5" in cross-section,
made up of many individually insulated 30 gauge wires. Though, a 0.25" x 0.5" cross
section Litz wire will not have the same resistance-per-unit-length that a solid copper
bar of the same cross section would have.
The resistance cost of the Litz wire was calculated by comparing the typical cross
section of a wire gauge [14] to the 'equivalent AWG' gauge and cross section from the
Litz wire catalog. The calculation,
TO hOresistance
=-
cross section of Litz wire
cross section of equivalent AWG gauge
was performed for wire gauges and geometries from 4/0 gauge to 8 gauge; the resulting
average ratio is 3.3; this factor is incorporated into the predicted resistance of a Litz
Wire transducer.
A transducer concept that uses Litz wire but fits into into the experimental frame
of the alpha prototype is shown in Figure 6-1.
The conductor layout has similar
geometry to the alpha prototype, but a shell sits between the magnet/spring assembly
and the compliant wires. The shell pieces are cut from schedule 80 PVC pipe, which is
available in discretized dimensions; the most suitable purchasable size is 6" diameter
pipe, which makes the radius of the second transducer is slightly larger than the first.
A few images of the second transducer manufacture are shown in Figures 6-2, 6-3,
and 6-4, progressing from the first set of turns to completed assembly. Laying out
compliant Litz wire
PVC sheell
spring
experimental frame
Figure 6-1: Model of Beta Prototype
the inner and outer wires was done by measuring off 13" of wire (the length of the
transducer plus one inch), bending the wire with a vice, zip-tying and gluing the
inner length down, measuring another 13 inches, bending the wire, and repeating (24
times!). Electrical tape was applied at the points of bending to keep the Litz wire
from splaying. After all of the wires were laid out, the outer PVC shell was zip-tied
and glued into place.
Removing insulation from the ends of the Litz wire was done by applying paint
remover to the ends, following the instructions of the paint remover (and using a fume
hood). The success of this process (removing insulation from seemingly hundreds of
30 gauge wires) was verified by using an audible continuity checker (as on a multimeter) and running the leads of the multimeter across all of the wires.
wire'bend'
spool of Litz wire
'tool'for measuring
transducer lengths
electrical tape to
prevent splaying
transducer
I
vice for bending wire
Figure 6-2: Beta Transducer Assembly Line
Param
Static Measured
Cbank
ESRcapBank
0.24
0.5
Initially Expected
0.24
1.3 max
Unit
F
Ro,i
Ri
N
mm
mm
Rtransducer
mQ
Ltransducer,i
K
yH
2.32
2.32
MN/m
Table 6.1: Statically measured values of beta prototype
The complete Beta prototype system is shown in Figure 6-5
6.2
Characterizing the Beta Prototype
The experimental characterization of the second prototype followed the same process
as described in Section 5.3. The static measurements of the Beta Prototype are shown
in Table 6.1.
The dynamic measurements of the Beta prototype are shown in Figure 6-6.
Ex-
Figure 6-3: Beta Transducer Assembly in Progress
periment and simulation agree to reasonable accuracy over the period of interesting
actuator performance. Beyond about 8 milliseconds the mechanical response of the
simulated and experimental system do not agree. This disagreement occurs because
the real actuator's outer radius cannot displace negatively, because the inner loops of
the transducer are rigid.
A displacement sensor was not used to measure motion because F = K Ar had been
verified in the first experiment. So, the 'experimental displacement' data in 6-6(a) is
inferred from the force sensor data.
PVC shell
Litz wire conductor
net
ma gnet
layout
Figure 6-4: Assembled Beta Transducer
66
charger power supply
/I
DMM displays
capacitor voltage
transducer
I
I
i
capacitor bank
/
SCR
i
experimental frame
Figure 6-5: Complete Beta Prototype Actuator
6.3
Comparing Performance of Alpha and Beta
Prototypes
The alpha and beta prototypes are similar in size and layout (the Beta prototype had
a slightly larger radius to enable the use of off-the-shelf PVC pipe for its outer shell).
The two actuators were deployed in the same experimental frame and powered by
the same capacitor bank power supply. The only/most significant design change was
replacing solid copper bars and 10-gauge end turns with a single length of .25" x .5"
cross section Litz wire, which reduced the total resistance.
The alpha prototype displaced the radially stiff formation about 1 mm at peak, while
the beta prototype displaced the frame about 3.25 mm at peak. The beta prototype transfered an order of magnitude more energy to the formation than the alpha
prototype:
Emechanicalpha
Emechanicalb, t
2
K
alphapeak
K br2etapeak
1
(.
The energy transfer/balance viewgraphs of both prototypes is not particularly impressive, as seen in Figure 6-7. Of course, the goal of these prototypes was to build a high
enough performance system to enable measurement/characterization of performance
in order to validate the model. In this sense, the performance of these prototypes is
very good.
0
1
2
3
4
5
time [ms]
6
7
8
9
10
600
400
time [ms]
(a) Mechanical Measurements
570060 -experimental
CD
C"
'0 40
10 20
0
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
time [ms]
6
7
8
9
10
6000.
4000
2000
(b) Electrical Measurements
Figure 6-6: Experimental and Simulated Actuator Event - Beta-Prototype
69
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Time [ms]
(a) Alpha Prototype
0.6
0.5
0.4
U~ 0.3
-----*-
0.2
................... . .... ..
0.1
*
Electromagnetic
Mechanical
Mech Spring
Resistive
+-- Total
1
2
3
4
5
6
Time [ms]
7
8
9
10
(b) Beta Prototype
Figure 6-7: Simulated Energy Transfer of Experimentally Validated Prototypes
Chapter 7
Simulation Results for Wellbore
Applications
Now that the simulation has been created and validated, the simulation can be used
to assess the actuator system in real applications. This section presents initial results
and thought processes for using the electromagnetic radial actuator system in the
following applications:
1. A rock-pushing radial expansion device to deploy sand screens or other structural elements radially into a wellbore.
2. A seismic source to create sequences of short duration, high magnitude mechanical force impulses.
This chapter does not present absolute performance metrics of the actuator system
for these applications, but makes some informed stabs on how an actuator system
made of attainable components and materials could perform downhole.
Param
Value
0.0024
1000
0.056
0.07
0.055
Unit
F
V
N
160
-
Rtransducer
0.135
1e7
Q
Cbank
V
ESRcapBank
ROi
Rj
K
Q
m
m
N/r
Table 7.1: A Set of Simulation Parameters for Rock-Pushing Application
7.1
An Actuator Design for Increasing Wellbore
Radius
The electromagnetic radial actuator system can be used to push radially against
wellbores in order to install screens or structural elements. The simulation below
describes a 12" long transducer that can increase the radius of a 6" diameter openhole wellbore by nearly 5 mm; the transducer can deploy and recharge better than
twice a second, enabling the drill string that the actuator would be installed on to
move at better than 2 feet per second through the wellbore.
The simulation result shown uses the parameters in the table. The radial stiffness
K is on the low end of unconsolidated wellbore stiffness, following the estimation
method shown in Section 3.4.
The value for ESR was derived from the ESR of the prototype capacitor bank described in Chapters 5 and 6. Similarly, the resistance of the transducer is calculated
assuming a Litz-wire fill ratio that is the same as in the beta prototype (resistance
is 3.3x the same length and cross section of copper). This could be an over-estimate
of resistance (and thus an underestimate of performance) if solid copper bars with
proper end turn design are used.
Figure 7-1 shows that the wellbore increases in radius by 4.5 mm at peak displacement.
Figure 7-4 shows that the initial 1200 J of energy in the capacitor (jCV 2 ) is converted
into peak mechanical energy (IKLr
2)
with an efficiency of nearly 10%.
-
3
E
E
r- 2
Wi
0o.1
0
2
1
3
4
5
6
Time [ms]
7
8
9
10
Figure 7-1: Outer Radius Displacement
1.4 1.2 -
-
-
--
1-6 ---
.
1t
0.2
6
.
-....
-...-..-...-
-
-...
-..
-.. ..-.
-...
-..
.. ..
- .... .
.-.--.--
-
--.-.----
-
--
-.-.
..
7
8
9
1
-..
.-..
Time [ms]
Figure 7-2: Current
It is commonly accepted that 3 kW is a reasonable amount of electrical energy available on a drill string for use by electrical components. An actuation event in this
simulation uses 1200 J, so the actuator could be recharged and actuated at better
than twice a second, enabling the drill string to move through the wellbore at an axial
rate of better than 2 feet per second.
As mentioned in Chapter 1, this actuator system is made of components that can
be suitably packaged for the extreme environments seen downhole. The copper (or
more exotic conductor material) will need to be coated properly for the chemical
2 C
1
0
0
1
2
3
4
7
6
5
time (ms]
8
10
9
Figure 7-3: Electromagnetic Force on Conductors
1.
0.8
-
Electromagnetic
---
-+-Mechanical
Mech Spring
..-..-..-.....
W0.6 --...
--+ Resistive
-+- Total
0.4- - -... .. - ... ..- ..........
00
1
2
3
4
6
5
Time (ms)
7
8
-
9
1
Figure 7-4: Energy Balance
environment downhole but should be able to perform suitably under the loads and
thermal effects. Similarly, high temperature capacitors exist and can be designed into
a package that shields the components from chemicals and/or excessive heat.
This example should not be taken as an upper limit of what can be achieved in
this application. Neither local nor global optimization is exhaustively proven in the
simulation, and selected components (the choice of copper conductor, capacitor bank
ESR, etc) may be improved.
7.2
An Actuator Design for Creating Impulse Waves
for Seismic Analysis
The same simulation result can be used to exemplify how this actuator system might
be used as a seismic source. In this use case, the goal is to create a sharp impulse
of mechanical force. The electromagnetic force profile of the simulation is shown in
Figure 7-3, this force is directly related to the current, shown in Figure 7-2. In this
simulation, the transducer achieves a peak force of 3.3e4 N (7500 lb) in a bit more
than a millisecond.
The current/force profile of the actuator can be shortened or lengthened by modifying
the number of turns of the actuator or the capacitance of the power supply. A single
transducer with a varying-capacitance power supply could be used to modify the
seismic source output parameters in real time. Additionally, multiple independent
capacitors could be used to actuate a single transducer in closely-spaced bursts.
Furthermore, the "tail" of the force profile could be shortened by using more precise
timing. Instead of depending on the SCR to open the circuit when voltage goes to
zero, a switching scheme that opened the circuit very soon after reaching peak current
could be created.
Electromagnetic force impulses of thousands of newtons peak magnitude and durations of sub-millisecond to a few milliseconds are possible using this actuator system
design.
7.3
Selecting the Simulation Parameters Used in
This Chapter
The parameters shown in Table 7.1 were selected with the following parameter input
expectations:
"
Transducer would be same volume as beta prototype (0.3 m length, 0.07 m
initial radius)
" Stiffness of formation would be unconsolidated sandstone, as estimated in Section 3.4.
" Capacitance and voltage would be selected to have reasonable volume ratio to
transducer volume (bigger than 1:1 ok, but 10:1 not ok).
" No limit on number of turns N (previously, turns were limited because the
author was assembling prototypes by hand).
" Limit capacitor voltage to 1000 V. This is a somewhat arbitrary limit to reduce the complexity/risk of designing the charging circuit, sourcing switching
components, and exceeding voltage limits of conductor insulation.
" Limit on conductor thickness is
"
radius
Capacitor ESR same as in experimental prototype - ESR scales linearly; more
capacitance means less resistance.
" Transducer resistance assumed to be copper with Litz fill ratio as in second
experimental prototype.
Lower resistance can probably be achieved using a
mix of solid copper and flexible stranded copper, but the Litz wire method is
very designable.
The parameter search process that ensued was:
" Set a capacitance-voltage combination.
" Run the simulation along a parameter sweep of number of turns N and conductor
thickness; graph the peak mechanical energy transfer for each simulation result
on a 2-d color map.
* iterate: continue to modify capacitance and voltage by hand, run turns-thickness
sweeps and learn trends in efficiency transfer.
Lowering capacitance and increasing voltage resulted in the highest performing simulation event. Figure 7-5 shows energy transfer efficiency maps for N--Y parameter
sweeps along three different values of capacitance and voltage. The capacitance and
voltage combinations all result in the same initial energy (1200 J).
Figure 7-6 also shows the simulated current signal of the best efficiency parameter
combination for each set of sweeps. Subfigure (a) of Figure 7-6 is the best simulation
from parameter sweep figure of Subfigure (a) of Figure 7-5. The highest energy transfer simulation from each voltage-capacitance combo results in very similar current
shapes. Clearly, the shape of the RLC discharge has a large effect on energy transfer
efficiency.
formation
eneravtransfer
%
thickness
[mm]
(a) C = 0.24 F, V
100 V
formation
energytransfer
%
thickness
[mm]
(b) C = 0.024F, V = 316 V
formation
enerav
transfer%
thickness[mm]
(c) C = 0.0024 F, V = 1000 V
Figure 7-5: Parameter sweeping transducer turns N and conductor thickness -y
6 -
-
0
1
2
3
4
5
Time[ms
100 V
C =0.24 F, V
= 20, -y = 14.3 mm
5
Time[ms]
(b) C = 0.024 F, V = 316 V
y = 14.3 mm
N = 53,
1,6
1,6
..
..
-.
-..-..
. ..
. ..
-..
-..
... .
..- . -
--
1.4
-
-..-.-.
-.
.-.-
.-
.--
-.
1.2
-
.. -
--
..-..
.
-
.-..-
. ..
0.8
-. - -
.-.
..
.. .
-.-.
.
-.
0.6
0.4
-.
-
-.
..
.
-..
-
- .
.
-.
--
-.-.
--
0.2
1
1
2
3
4
6
5
Time[Ims]
7
8
9
10
(c) C = 0.0024 F, V = 1000 V
N = 160, -y = 15 mm
Figure 7-6: Simulation of current using best parameters from sweeps of Figure 7-5
79
80
Chapter 8
Conclusion
An electromagnetic actuator design has been presented that can be made of physically
robust components and is well suited for providing short duration high force bursts
of radial mechanical force.
Historically, this style of radial actuator has been to
plastically deform metallic cylinders in industrial processes. Non-radial-coordinate
versions of the actuator have been used as the signal source for sonar-based mapping
applications.
Motivated by the robust physical design and previous successful application of similar
actuators, there was interest in exploring the actuator's capability to enable some oil
exploration and well completion applications. A numerical simulation of the electromechanical event was created to enable designers to evaluate the actuator system's
performance and make design decisions.
The simulation was created and experimentally validated, and the simulation played
a critical role in selecting parameters and components for the first experimental prototype. The simulation was again used to help explain and understand the initially
unexpected performance of the first prototype and to design an improved second
prototype.
The simulation, combined with estimates of wellbore formation properties and component parameters used in the experimental prototypes was used to describe an actuator
design that could be packaged onto a drill string and could significantly deform rock
formations while traveling at a reasonable axial rate through the wellbore. The system's ability to make tight impulse-like wave forms and the implications this has for
seismology purposes was also highlighted.
In designing prototypes and performing a coarse parameter search for high performing system parameters, a few themes that maximize electrical to mechanical energy
transfer were identified.
1. Reduce resistance. All else held equal, reducing resistance in any part of the
system will result in increasing peak current (and thus force) and decreasing
resistive losses. This comes into play when specifying capacitors (consider the
ESR), designing the transducer conductor layout, selecting switching components, and the wires connecting the power supply and transducer.
2. Reduce excess magnetic field. All else held equal, reducing the gap between the
inner and outer conductor turns means less magnetic fields must be generated
for the same force capability. Of course, this goal can clash with decreasing
total resistance (a smaller gap means a smaller conductor).
3. Tune the RLC circuit output to exploit the spring-mass-damper response. If
the goal of the actuator is to maximally displace a target formation, the timing
of the electric energy delivery to the transducer should be tuned to avoid being
'filtered' out by the spring-mass-damper system. The timing of the RLC circuit
is affected by resistance, inductance, and capacitance, so the designer does have
many options to tune the timing of the system by altering the layout of the
transducer or the layout of the capacitor bank.
The parameter search methods used by the author were too coarsely grained and too
small in scope to provide precise local and/or globally optimal solutions. Furthermore,
even the low order sweeping involved in the parameter search of Chapter 7 was time
consuming and prone to error. Although the creation of this simulation was a step
forward in enabling designers to find sets of parameters that perform well, applying
improved high-order optimization techniques would further enable the use of this
simulation tool to design high performing actuator systems.
84
Appendix A
Matlab Simulation Implementation
The matlab implementation includes three files: parameter.m, diffEqn.m, and simulation.m:
" parameters.m contains the system parameters that define the transducer, power
supply, and formation.
" diffEqn.m defines a single state update. It contains the equations summarized
in Equation 4.2.
* simulation.m contains the ode solver that calls diffEqn.m and generates a full
simulation output. Additional higher level scripts use the ouput of simulation.m
to evaluate system performance.
The code for these scripts is included in the immediately following pages.
parameter.m:
% COMMON PARAMETER SCRIPT
% % % TRANSDUCER
RO =
[];
outer
radius
[m]
RI =
[];
inner radius
[m]
% RI
=
WZ
[;
=
=
initial
conductor thickness
th =
N
...
RO
-
th;
RI may be defined in terms of RO and the
axial length [m]
[];
resO
number of conductor turns
conductor resistivity [ohms/m]
=
[;
Rtrans
=
%% %
POWER SUPPLY
transducer resistance
C
=
[;
capacitance
V
=
[;
voltage
[F]
[V]
capacitor bank resistance [ohm]
Resr =
Rscr =
[m]
[];
SCR resistance
[ohm]
% % % FORMATION
K
=
[];
formation radial stiffness
D
=
[;
formation damping
M= [];
muO
= 4*pi*le-7;
well bore mass
[kg/si
[kg]
% free space permeability
Rtot = Rtrans+Resr+Rscr;
[N/m]
diffEqn.m:
1 %
FUNCTION TO COMPUTE DERIVATIVES
2
3
%
STATE X
4
%
X(1)
= FILTER OUTER RADIUS
5
%
X(2)
= FILTER OUTER VELOCITY
6
%
X(3)
= FILTER FLUX LINKAGE
7
%
X(4)
=
IS AS FOLLOWS:
CAPACITOR VOLTAGE
8
9 function
[DX]
= diffEqn(T,X)
10
11
--
%
SIMULATION PARAMETERS
12
13
parameter;
15
%
% read global paramteters
COMPUTE INDUCTANCE, CURRENT, MAG FIELD, RADIAL FORCE
16
17
L = muO*
18
I = X(3) /
19
B = muO
20
Fem =
* N^2
WZ
* log(X(1)/RI)
/(
2
*pi);
L;
* WZ
0.5 *
N*
*
N
I/
I *
*
(2*pi*(X(l)));
B;
21
22
%
COMPUTE DERIVATIVES
----
23
24
DX =
zeros (4,1);
25
26
DX(1)
= X(2);
27
DX(2)
=
(-K*(X(l)-RO) + Fem)
28
29
DX(3)
= X(4)
3o
DX(4)
= -I/C;
31
32
% d/dt outer filter radius
end
/ M;
Rtot*I;
D*X(2)
...
% d/dt outer filter velocity
% d/dt filter flux linkage
% d/dt capacitor voltage
simulation.m:
1 %
SCRIPT TO SIMULATE SAND FILTER EXPANSION
2
3
%
STATE X IS AS FOLLOWS:
4
%
X(l) = FILTER OUTER RADIUS
5 %
X(2) = FILTER OUTER VELOCITY
6
%
X(3) = FILTER FLUX LINKAGE
7
%
X(4)
= CAPACITOR VOLTAGE
8
9
clear all
io close all
11
12
pause off
13
plotz
=
1;
14
15
SIMULATION PARAMETERS
%
16
parameter;
17
% read global paramteters
18
19
%
INITIAL CONDITIONS
20
XI =
21
zeros(4,1);
22
= RO;
% initial outer filter radius
23
XI(l)
24
XI(2) =
0;
% initial outer filter velocity
25
XI(3) =
0;
% initial filter flux linkage
26
XI(4) = V;
% initial capacitor voltage
27
28
SIMULATE
%
29
30
TF
=
0.01;
% simulation time length
31
dT
=
10e-7;
% simulation precision
32
TSPAN
=
0:dT:TF;
33
[T,X]
=
ode23s('diffEqn',TSPAN,XI);
35
%
DETERMINE CURRENT, INDUCTANCE, MAG FIELD, RADIAL FORCE,
36
* N^2
37
L
=
mu0* WZ
38
I
=
X(:,3)./L;
39
B
=
mu0
40
Fem
=
*
*
WZ
*
0.5
N
.*
.*
N
.*
log(X(:,l)/RI)
I/
I
.*
(2*pi
.*
/(2*pi)+Lparasite;
(X(l)));
B;
41
42
%
DETERMINE ENERGIES [J]
43
WI
=
0.5
*
(L. *I.*I) ;
WE
=
0.5
*
(C* X(:,4) .*X(:,4)
+ L.* I.*I)
WM =
0.5
*
(K* (X(:,1) -RO).^2
+ M*X~(:,2)
=
.5
Wmass
Wspring
=
inductor energy
*M* (X(:,2)).
.5
* K *
electrical energy
.^2)
mechanical energy
kinetic energy
-2;R
(X (: ,1)-RO).^2;
spring energy
NT = length(T);
% number of time steps
52
WR =
% resistor energy loss
53
for n =
50
51
2:NT
WR(n) = Rtot
54
55
zeros (NT,1);
*
(T(n)-T(n-1))
*
I(n)^2 + WR(n-1);
end
56
=
57
WD
58
for n = 2:NT
WD(n) = D*X(n, 2)
59
60
zeros(NT,1);
end
% damping energy loss
*
(X(n,l)-X(n-1,1))
+ WD(n-1);
90
Bibliography
[1] D. Patterson, A. Monti, C. Brice, R. Dougal, R. Pettus, Design and Simulation
of a Permanent-MagneticElectromagnetic Aircraft Launcher. IEEE Transaction
on Industry Applications, Vol 41, Issue 2, 2005, pg 566-575.
[2] A. Fauchon-Villeplee, Electric Apparatus for Propelling Projectiles. US Patanet
1,421,435. July 4, 1922.
[3] H. D. Fair, Advances in Electromagnetic Launch Science and Technolgy and Its
Applications. IEEE Transactions on MAgnetic, Vol 45, No 1, January 2009, pg
225 - 230.
[4] H. Payson, Investigation of Bottom Sediment Probing By 12 Kilocycle Sounce
Pulses Reflected From Shallow Water Bottom Sediment Layers. MS Thesis, MIT,
Jan 1963.
[5] H. E. Edgerton, H. Payson, J. Yules, W. Dillon, Sonar Probing in Narrafansett
Bay Science, New Series, Vol 146, No 3650, Dec 11 1964, pg 1459-1460.
[6] N. J. Hyne, Nontechnical Guide to Petroleum Geology, Exploration, Drilling, and
Production (2nd Ed) Pennwell Books, Dec 15 2001.
[7] G. Wilson, Method and Apparatus for Reforming a Tube. US Patent 5,457,977,
Oct 17, 1995.
[8] I McNab, PArameters for an Electromangetic Naval Railgun IEEE Transactions
on Magnetics, vol 37, no 1, january 2001, pg 223 - 228.
[9] J McFarland, I McNab, A Long-Range Naval Rail Gun IEEE Transactions on
Magnetics, vol 39, no 1, January 2003, pg 289 - 294.
[10] Fjar, Erling, R. M. Holt, P. Horsrud, A. M. Raaen, and R. Risnes. "Appendix
A - Rock Properties". Petroleum Related Rock Mechanics, 2nd Edition. Elsevier
Science and Technology Books, Inc.. 2008. Books24x7.
[11] A Boresi, R Schmidt. Advanced Mechanics of Materials. Wiley, Oct 2002.
[12] Vishay Sprague. We Tantalum HI TMP CapacitorsData Sheet. Document Number 40072, revision Jan 29 2009.
[13] New
England
Wire
Technologies.
Product
http://www.newenglandwire.com/catalog/pdfs/newtcatalog.pdf.
accessed Feb 2011.
[14] Wikipedia. American wire gauge
http: //en.wikipedia. org/wiki/American-wiregauge}
accessed Feb 2011.
Catalog.
pgs 129-137,