Dynamics and Control of Magnetostatic Structures

Abstract

Stock-Windsor, Jeffry Clifton

Dynamics and Control of Magnetostatic Structures

Under the direction of Larry Silverberg.

The equations governing the dynamics of magnetostatic structures are formu- lated using Lagrangian mechanics. A potential energy function of gravitational, strain, and magnetostatic components is defined. The Lagrangian equations of motion are discretized and then linearized about equilibrium points created by the additional magnetostatic energy, leading to a linear system of ordinary differential equations. These equations are characterized by mass, stiffness, damping, gyroscopic, and circulatory effects.

Four experiments are conducted. Using the one-degree-of-freedom magneto- static levitator, the measured static displacement is compared to those pre- dicted by the exact nonlinear solution and the discretized approximate solution.

Three experiments are performed with the two-degree-of-freedom, spherical, magnetostatic pendulum: The natural frequencies of the pendulum are pre- dicted and compared with measurements; the pendulum is made to track a desired path using electromagnets to control the motion; and the pendulum’s oscillations about new equilibrium points are regulated using electromagnets and velocity feedback to control settling time. In the last experiment, the stabil- ity of the controlled system is proven by examining the eigenvalues about the new equilibrium position.

©1999 Jeffry Clifton Stock-Windsor

Dynamics and Control of Magnetostatic Structures

Jeffry Clifton Stock-Windsor

A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy.

Mechanical Engineering

Raleigh 1999

Approved by:

Dr. Ethelbert Chukwu

Dr. Eric Klang

Dr. Larry Royster

Dr. Larry Silverberg, Chair of Advisory Committee

Dedication

To my wife, Christina Stock-Windsor.

You are both my advocate and my inspiration.

Vita

I was born to John and Gilda Windsor on March 15, 1968 and grew up in the vil- lage of Oak Ridge, a cotton farming community in northeast Louisiana. After attending Riverfield Academy in Rayville through the tenth grade I was accepted at the Louisiana School for Math, Science, and the Arts in Nat- chitoches where I graduated from high school in 1986. With an academic schol- arship, I then attended Louisiana State University in Baton Rouge and in 1990 earned a Bachelor of Science degree in physics with a minor in mathemat- ics. In August 1991 I accepted a teaching assistantship in the Department of Mechanical and Aerospace Engineering at North Carolina State University where I concentrated in controls and minored in mathematics. During my second year of graduate work I received a research assistantship at the Mars Mission Research Center. In August 1993 I earned a Master of Science degree in mechanical engineering. That Fall I began work on my doctoral degree with Dr. Larry Silverberg. I examined the dynamics and control of magnetostatic structures with funding from the Mars Mission Research Center. The night of Thursday, September 26, 1996, under a total lunar eclipse I married Christina Sophia Stock, and we currently live in Raleigh, North Carolina.

Acknowledgements

I wish to thank my advisor Dr. Larry Silverberg for all of his guidance, support, and humor during the years finishing my degree. I also want to thank my gradu- ate committee, Dr. Ethelbert Chukwu from the Mathematics department and Drs. Eric Klang and Larry Royster from the Mechanical and Aerospace Engi- neering department, for their insightful questions.

Thanks are also in order for the staff of the Mechanical and Aerospace Engineer- ing machine shop, Mike Breedlove and Skip Richardson. Their expert crafts- manship in constructing the magnetostatic spherical pendulum made my life much simpler.

Finally, I would like to thank my family. My parents John R. Windsor, Sr. and Gilda Tyson Massingill have always supported and encouraged me in my efforts. My wife, Christina Stock-Windsor, in particular has been a constant source of reassurance and assistance while finishing this dissertation. Thank you.

Table of Contents

List of Tables...... vii

List of Figures...... viii

Introduction...... 1

Governing Equations...... 4 Nonlinear Equations...... 4 Kinetic Energy 4 Potential Energy 5 Lagrangian Equations 7 Discretized Equations...... 7 Kinetic Energy 8 Potential Energy 8 Lagrangian Equations 12 Linearized Equations ...... 13

Magnetostatic Levitator ...... 17 Apparatus...... 17 Equations of Motion...... 18 Nonlinear Equations 18 Discretized Equations 21 Numerical Comparison 21 Static Displacement...... 22

Magnetostatic Pendulum ...... 24 Apparatus...... 24 Pendulum 25 Electromagnets 25 Power Supplies 26 Sensors & Computer 26 Calibration 27 Equations of Motion...... 29 Nonlinear Equations 29 Discretized Equations 34

Linearized Equations 34 Natural Frequencies...... 35 Apparatus 35 Equations of Motion 36 Experiment 36 Results 38 Tracking ...... 41 Apparatus 41 Experiment 41 Results 44 Regulation...... 45 Apparatus 45 Equations of Motion 45 Experiment 49 Results 49

Conclusions ...... 53

Works Consulted...... 55

Appendix: A Physics Primer for Magnetostatic Energy...... 58 Classical Physics...... 58 Electromagnetism...... 59 Conservation of Charge 59 Maxwell’s Equations 60 Induction...... 61 Flux 62 Electromotive Force 63 Faraday’s Law 66 Filamentary Approximation ...... 67 Magnetostatics...... 68 Magnetic Vector Potential 69 Biot-Savart Law 70 Energy of Circuits ...... 72 Power in Circuits 72 Work Done by Circuits 76 Energy and Virtual Work 79 Energy of a Magnetic Field 80 Field Energy and Induction Coefficients 82 Summary ...... 85 Works Consulted for Appendix...... 86

List of Tables

Table 3–1. Levitator’s experimental and predicted displacements...... 23

Table 4–1. Pendulum’s mean natural frequencies for 0.0A total current..... 38

Table 4–2. Pendulum’s mean natural frequencies for 0.6A total current..... 38

Table 4–3. Pendulum’s mean natural frequencies for 0.8A total current..... 39

Table 4–4. Pendulum’s mean natural frequencies for 1.0A total current..... 39

Table 4–5. Pendulum’s mean normalized tracking error for two paths...... 44

Table 4–6. Pendulum’s approximate settling times for various gains...... 50

Table A–1. Fundamental Equations of Classical Physics ...... 58

Table A–2. Quantities, Symbols, and Units used in Table A–1 ...... 59

vii

List of Figures

Figure 3–1. Magnetostatic levitator ...... 17

Figure 3–2. Comparison of nonlinear and discretized simulated dynamic solutions...... 22

Figure 4–1. Magnetostatic pendulum...... 24

Figure 4–2. Close-up of magnetostatic pendulum’s electromagnetic coils.25

Figure 4–3. Close-up of magnetostatic pendulum’s sensor reflectors...... 26

Figure 4–4. Pendulum support reference frame (X, Y, Z) and its relation to generalized coordinates (␣, ␤) ...... 27

Figure 4–5. Pendulum’s voltage calibration surface for first sensor...... 28

Figure 4–6. Pendulum’s voltage calibration surface for second sensor...... 28

Figure 4–7. Pendulum’s coordinate frames ...... 30

Figure 4–8. Pendulum’s gravitational potential energy over large range of motion...... 32

Figure 4–9. Pendulum’s gravitational potential energy over possible range of motion...... 32

Figure 4–10. Electromagnet circuit diagram for natural frequencies experiment ...... 35

Figure 4–11. Pendulum’s magnetostatic potential energy for 0.6A total current to two electromagnets (one base and one pendulum coil) ...... 37

Figure 4–12. Pendulum’s magnetostatic potential energy for 1.0A total current to two electromagnets (one base and one pendulum coil) ...... 37

Figure 4–13. Pendulum’s ␣ angle versus the two base electromagnets’ currents...... 42

viii Figure 4–14. Pendulum’s ␤ angle versus the two base electromagnets’ currents...... 42

Figure 4–15. Example comparison of actual versus desired path for the first tracking experiment...... 43

Figure 4–16. Example comparison of actual versus desired path for the second tracking experiment...... 43

Figure 4–17. Pendulum’s magnetostatic potential energy for 0.6A current to three electromagnets (two base and one pendulum coil)...... 46

Figure 4–18. Pendulum’s magnetostatic potential energy for 1.0A current to three electromagnets (two base and one pendulum coil)...... 46

Figure 4–19. Regulation of pendulum motion with various gains at the equilibrium created by setting (I1, I2) = (0.3, 0.7)A ...... 51 Figure 4–20. Regulation of pendulum motion with various gains at the equilibrium created by setting (I1, I2) = (0.6, 0.6)A ...... 52 Figure A–1. Magnetically coupled circuits ...... 62

Figure A–2. Conducting filament in motion ...... 63

Figure A–3. Wire approximated as a filamentary current...... 67

Figure A–4. Magnetic field from a wire...... 71

Figure A–5. Coil moving toward a loop...... 75

Figure A–6. Mutual inductance geometry...... 83

ix Introduction

The systematic study and modification of structures began with Archimedes’ first formulations of the mechanical workings of pulleys, levers, and screws dur- ing the third century bc. In the mid-nineteenth century Maxwell’s equations summarized more than a hundred years of theoretical and experimental work in electricity and magnetism. Aside from simple examples such as electric motors and generators, and electromagnetic relay circuits, the fields of electro- magnetism and structures have grown almost exclusively of one another.

In this century, structures and electromagnetism started to converge. This includes the coupling of electrostatics and structures, such as electrostatic audio speakers (Streng “Charge” and Streng “Sound”), some scientific instru- ments (Rhim et al. “Positioner” and Rhim et al. “Levitation”), and electrostati- cally controlled space-based reflector antennas (Yam et al.; Lang and Staelin; Mihora and Redmond). But these examples lacked a strong unifying mathemat- ical framework until Silverberg and Weaver showed how to treat general struc- tures with attached or embedded electrostatic charges.

The coupling of magnetism and structures, likewise, can benefit from a general formulation. Several new areas of research rely on this convergence, such as magnetic resonance imaging, magnetically assisted medical devices (Kovacs), vibration control (Kojima et al.), magnetic bearings (Di Gerlando), and magnet- ically supported robot hands (Higuchi et al.). But the research tends to treat spe- cific examples of structures and not structures in a general manner.

1 Typically, the research takes a force equation derived from a specific electro- magnetic actuator and then applies this to a particular device and/or control algorithm. Kojima et al., for example, look only at vibrations in a beam and use a magnetic actuator designed to produce a linear force response. Similarly, Bagryantsev and Tyurin look at the dynamic stability of a vehicle with an elec- tromagnetic suspension that produces a specific force response. In a more gen- eral approach, Brauer et al. compute the dynamic stresses in a magnetic actuator. Their analysis computes the coupled structural and electromagnetic time-varying stresses in a two-dimensional plane of the actuator, but dynamic motion is not studied.

The research presented here develops the dynamic equations of motion for mag- netostatic structures in a general formulation and examines controlling the motion of a specific magnetostatic structure. By keeping the formulation gen- eral, the system designer can see the qualitative effect of the electric currents on the structure’s dynamic response. This in turn enables the designer to develop tracking and regulation problems for magnetostatic structures.

The following chapter, “Governing Equations,” describes the process of deriving a structure’s equations of motion. Starting with expressions for the kinetic and potential energies of the structure the nonlinear Lagrangian equations of motion for the system are written. The structure is then discretized, which sim- plifies the equations considerably. To aid the application and design of control algorithms, the discretized equations of motion are then linearized about the system’s equilibrium points.

This method is applied to two specific structures in the chapters that follow. “Magnetostatic Levitator” describes a one-degree-of-freedom device composed of two electromagnetic coils, one that levitates above the other. The accuracy of the discretized solution is compared with the nonlinear solution, and both are compared with the experimental measurement. “Magnetostatic Pendulum” describes three experiments with the two-degree-of-freedom, spherical, magne- tostatic pendulum. The natural frequencies of the system were measured and

Introduction 2 compared with theory when the system is both under and not under the influ- ence of magnetic forces. A tracking experiment is performed using open-loop controls. And finally, a regulation experiment is conducted with closed-loop controls to damp out vibrations.

The last chapter, “Conclusions,” provides closing remarks. The results of the pre- sented research are summarized, suggestions for further research are given, and potential applications are briefly discussed.

Introduction 3 Governing Equations

The assumption that a magnetic field is static, i.e., it is a magnetostatic field, implies a number of assumptions: that the volume of space under consideration contains homogeneous, isotropic, linear materials; that the magnetic fields are relatively low in strength; and that the current densities are constant over time. The last assumption can actually be relaxed to include quasi-static current den- sities. If the current and charge densities vary slowly over time and the dimen- sions of the structure are small when compared to the wavelength of the electromagnetic radiation produced from the changing densities, then the equations derived for static conditions are still valid (Cheng 277; Feynman et al. 2: 17–9).

This chapter derives the equations of motion for a quasi-magnetostatic struc- ture using Lagrangian dynamics. First the nonlinear equations for a general sys- tem are examined. These equations are then discretized and then linearized about equilibrium positions.

Nonlinear Equations

Kinetic Energy The structure’s kinetic energy is (Silverberg and Weaver 383)

1 T ϭ -- ͐m()r r˙ и r˙dv, (2–1) 2 V

where m()r is the mass density at position r, and r˙ is the velocity of the posi- tion vector r.

4 Nonlinear Equations

Potential Energy The potential energy for a magnetostatic structure is given by

ϭ 1͐ и Ϫ ͐ и ϩ 1͐ ()и () U -- rs Krsdv m()r rgdv -- Jr Ardv, (2–2) 2 V V 2 V

where the first term is the strain potential, the second term is the gravitational potential, and the last term is the magnetostatic energy (Silverberg and Weaver 383; Jackson 216). K is the self-adjoint, positive, semi-definite stiffness operator,

rs is the strain component of the position vector r, and g is the (constant) gravity vector.

The magnetostatic potential energy is composed of the integral of the dot prod- uct of the current density JA and the magnetic vector potential (Jackson 216; see also Eq. (A–25) on page 81†):

ϭ 1͐ ()и () Umag -- Jr Ardv. (2–3) 2 V

This integral is over the space V in which the current exists, but may be expanded to include all space since including regions where there is no current will not affect the value of the integral.

The vector potential at a point rJr from a current density at a point Ј can be defined as (Jackson 176; see also Eq. (A–15) on page 70)

␮ Ј ϭ 0 ͐ Jr() Ј Ar() ------␲------Ϫ Ј-dv (2–4) 4 VЈ rr

again where the integration need only be over the region VЈ where the current ␮ source exists. The constant 0 is the magnetic permeability of free space and Ϫ has the exact value of 4␲×10 7H/m.

Typically, the magnetostatic field is produced by several current carrying regions or circuits. In this case the field energy can be written as the sum of the

† Please see “A Physics Primer for Magnetostatic Energy” on page 58 for a detailed explanation of the derivation of the magnetostatic potential energy.

Governing Equations 5 Nonlinear Equations contributions from the various regions. Using two regions as an example, the current density and vector potential of Eq. (2–3) can be expanded as

ϭϭϩ ϩ Jr() J1()r J2()r J1 J2, ϭϭϩ ϩ Ar() A1()r A2()r A1r A2r.

It is important to remember that the current densities are defined as the density of current Jr at a particular position , and the vector potential is defined as the potential measured at a particular position Ar() that is created by a current density distribution over a region of space away from the position of measure- ment (as shown in Eq. (2–4) above). Thus, when using subscript notation it is useful to write two subscripts for the vector potential, the first being the posi- tion of the source and the second being the position of measurement.

Eq. (2–3), then, expands to

ϭ 1͐()ϩ и ()ϩ Umag -- J1 J2 A1r A2r dv 2 V

ϭ 1͐()и ϩϩϩи и и -- J1 A11 J1 A21 J2 A12 J2 A22 dv, 2 V where the second line now includes the positions of measurement in the sub- scripts for the vector potentials. One of these is written out for clarity:

␮ Jr() A ϭϭAr() ------0-͐------1 dv, 12 2 4␲ r Ϫ r V1 2 1 where this can be read as “the vector potential measured at position #1 as cre- ated by source #2.”

This energy can also be written in terms of inductance coefficients. Since the mutual and self-inductances for circuits are defined as (see “Field Energy and Induction Coefficients” on page 82)

1 1 ᏹ ϭ ------͐A и J dv, ᏸ ϭ ----͐A и J dv, 12 I I 12 2 11 2 11 1 1 2 V2 I1 V1 the potential energy becomes

Governing Equations 6 Discretized Equations

1 2 1 2 U ϭ -- ᏸ I ϩϩᏹ I I -- ᏸ I , mag 2 11 1 12 1 2 2 22 2

where I1 and I2 are the currents flowing through circuits 1 and 2 respectively. ᏹ ϭ ᏹ (Note that the two mutual inductances are equal: 12 21.)

For ᏺ circuits, then, the potential energy of the magnetostatic field is

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ, (2–5) mag 2 ᐇ ϭ 1 ᐈ ϭ 1

where ᏹᐇᐇ ϭ ᏸᐇᐇ (Nayfeh and Brussel 375; see also Eq. (A–33) on page 85).

Lagrangian Equations With the Lagrangian defined as

LTUϭ Ϫ , (2–6)

Lagrange’s equations of motion are simply

Ѩ Ѩ d L Ϫ L ϭ ϭ ,,… Ѩ␪ 0, l 1 M, (2–7) dt Ѩ␪˙ l l

where the position vectors r are now functions of the generalized coordinates ␪ ϭ ␪ ,,,␪ … ␪ ϭ ,,… l: ri r()1 2 M , i 1 N. (The vectors ri are discussed in more detail in the following section.)

Discretized Equations

The volume of the structure is discretized into N regions over which the mass

density is assumed constant. The center positions of these subvolumes are r1, ,,… r2 rN. This set of vectors may or may not form a basis (i.e., they are linearly independent) for the space that they span. In either case the set can always be ␪ ,,,␪ … ␪ written in terms of the space’s basis set, 1 2 M, which also serve as the generalized coordinates described above. Expressing the position vectors and their time derivatives in terms of the generalized coordinates gives

M Ѩr ϭ ␪ ,,,␪ … ␪ ˙ ϭ i ␪˙ ϭ ,,… ri ri()1 2 M , r Α Ѩ␪ j, i 1 N. (2–8) j i ϭ 1

Governing Equations 7 Discretized Equations

Kinetic Energy Using the discretization of Eq. (2–8), the kinetic energy of Eq. (2–1) becomes

N N N N Ѩr Ѩr ϭϭ1 ˙ и ˙ 1 i i ␪˙ ␪˙ T -- Α miri ri -- Α Α Α miѨ␪ и Ѩ␪ j k, (2–9) 2 2 j k i ϭ 1 i ϭ 1 j ϭ 1 k ϭ 1

th where mi is the mass of the i subvolume.

Potential Energy Strain Potential. As an example of how to discretize the strain potential energy, consider the bending one-dimensional, hinged-free beam of Silverberg and Weaver (384). The beam of length Lh is divided into subintervals of length , and the strain potential energy is

M M 1 U ϭ -- Α Α K ␪ ␪ , (2–10a) strain 2 Sij i j i ϭ 1 j ϭ 1

11Ϫ 0 ۙ 0 Ϫ1 21Ϫ 0 ۙ 0 EI 01Ϫ 21Ϫ 00 K ϭ ----- , (2–10b) ۘۛۛۛۛۘ s h Ϫ1 21Ϫ ۛ ۘ 01Ϫ 1 ۙۙ 0

where here EI represents the flexural rigidity (Meirovitch Computational 235).

The matrix Ks represents a positive, semi-definite, symmetric stiffness matrix.

Gravitational Potential. Substituting Eq. (2–8) into Eq. (2–2) yields the dis- cretized gravitational potential energy

N ϭ Ϫ и Ugrav Α miri g. (2–11) i ϭ 1

Magnetostatic Potential. The magnetostatic potential energy as given by Eq. (2–5) can also be approximated by discretization. Here the circuits over which currents flow are discretized along their paths. After the magnetostatic potential has been discretized, the coordinates used can be written in terms of the generalized coordinates.

Governing Equations 8 Discretized Equations

Looking at Eq. (2–5) it is clear that both the mutual and self-inductance coeffi- cients must be discretized. The mutual inductance is discretized by approximat- ing Neumann’s formula (from Eq. (A–28) on page 82),

␮ dᐍ и dᐍ ᏹ ϭ 0 ͛͛ ᐇ ᐈ ᐇᐈ ------␲------Ϫ , (2–12) 4 Cᐇ Cᐈ rᐈ rᐇ

with a summation

␮ ⌬ᐍ и ⌬ᐍ ᏹ ϭ 0 ᐇ␨ ᐈ␰ ᐇᐈ ------␲-ΑΑ------Ϫ , (2–13) 4 ␨ ␰ rᐈ␰ rᐇ␨

where Cᐇ and Cᐈ are the paths around the two circuits ᐇᐈ and respectively,

dᐍᐇ and dᐍᐈ are the differential length elements along the paths, rᐇ and rᐈ are th the position vectors for the differential elements, ⌬ᐍᐇ␨ is the ␨ finite length th element (a vector) of circuit ᐇ, ⌬ᐍᐈ␰ is the ␰ finite length element (also a vec-

tor) of circuit ᐈ, and rᐇ␨ and rᐈ␰ are the positions of the two finite elements.

In general each finite length element of each circuit, i.e., each of the ⌬ᐍᐇ␨ ele-

ments and each of the ⌬ᐍᐈ␰ elements, must be written in terms of the general- ␪ ized coordinates l. Most electromagnetic actuators consist of coils of wire, i.e., each circuit is numerous loops of wire lying more or less along the same path. Modeling each loop of each coil results in Lagrangian equations of motion with so many terms that it becomes computationally prohibitive. To simplify matters each coil is modeled as numerous loops lying along the same path. The mutual inductance of two coils, ᐇᐈ and , from Eq. (2–13) becomes

n n ␮ ᐇ ᐈ ⌬ᐍ и ⌬ᐍ 0 ᐇ␨ ᐈ␰ ᏹᐇᐈ ϭ ------ᏺᐇᏺᐈ Α Α ------, (2–14) 4␲ rᐈ␰ Ϫ rᐇ␨ ␨ ϭ 1 ␰ ϭ 1

where ᏺᐇ and ᏺᐈ are the number of loops for each coil. Thus each coil has only

one unique loop, which is discretized into nᐇ and nᐈ elements respectively (i.e.,

the loops are approximated by nᐇ- and nᐈ- sided polygons).

The self-inductance can also be approximated by discretization, but the process is slightly more involved. Smythe (314) and Grover (7) show that the self-induc-

Governing Equations 9 Discretized Equations

tance of a circuit may be calculated by treating separately the inside and outside regions of the wire. If the radius of the wire is small compared to the dimensions of the circuit, then it may be assumed that the magnetic field inside the wire is the same as it would be inside a long straight wire of the same size and same cur- rent. Also, the field outside the wire may be assumed the same as if all the cur- rent were concentrated along the axis of the wire.

Since the magnetic flux density inside a long straight wire of radius R in cylin- drical coordinates is (Nayfeh and Brussel 253)

␮ Jr B ϭ ------0 -␾ˆ , 2

where the current density is considered to be uniform across the cross-section of the wire, i.e., JIϭ ր ␲R2. The self-inductance is defined as (Smythe 313; see also Eq. (A–31) on page 84)

1 ᏸ ϭ ------͐B и B dv. (2–15) 1 2␮ 1 1 I1 0 V

The internal self-inductance of a wire of length Lᐇ is then

␮ 1 2 0 ᏸᐇᐇ ϭϭ------͐B dv ------Lᐇ. (2–16) int 2␮ 8␲ I 0 V

For the external self-inductance, note that the magnetic flux density near the surface of the wire very nearly forms circles perpendicular to the axis of the wire. (See Eq. (A–7) on page 62 for the definition of flux.) Therefore, any line drawn parallel to the axis of the wire and along its surface will link all of the external flux produced by the axial filament of current. Finding the self-inductance out- side the wire, then, is the same as finding the mutual inductance between two parallel curvilinear circuits, one along the axis of the wire and one along the outer surface of the wire. Using Eq. (2–14), the external self-inductance is just

nᐇ nᐇ ␮ ⌬ᐍᐇ␨ и ⌬ᐍᐇ␨ 0 2 a s ᏸᐇᐇ ϭ ------ᏺᐇ Α Α ------, (2–17) ext 4␲ rᐇ␨ Ϫ rᐇ␨ ␨ ϭ ␨ ϭ s a a 1 s 1

Governing Equations 10 Discretized Equations

where ᏺᐇ is the number of loops for the circuit ᐇ, nᐇ is the number of discret-

ization elements for the loop, ⌬ᐍᐇ␨ and ⌬ᐍᐇ␨ refer to the finite length elements a s

along the axis and outer surface of the wire respectively, and rᐇ␨ and rᐇ␨ are a s the distance between the two elements. It is important to note that the denom- inator of Eq. (2–17) causes no problems since the two finite length elements always remain a finite distance apart—specifically, they remain at least the radius of the wire, R, apart from each other.

Thus, the total self-inductance of a circuit of total length Lᐇ that is composed

of ᏺᐇ loops of wire is approximately

nᐇ nᐇ ␮ ␮ ⌬ᐍᐇ␨ и ⌬ᐍᐇ␨ 0 0 2 a s ᏸᐇᐇ ϭ ------Lᐇ ϩ ------ᏺᐇ Α Α ------. (2–18) 8␲ 4␲ rᐇ␨ Ϫ rᐇ␨ ␨ ϭ ␨ ϭ s a a 1 s 1

Using Eq. (2–5), the discretized magnetostatic potential energy can be written

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ, (2–19a) mag 2 ᐇ ϭ 1 ᐈ ϭ 1

where, using Eqs. (2–14) and (2–18),

 n n  ␮ ᐇ ᐈ ⌬ᐍ и ⌬ᐍ  0 ᐇ␨ ᐈ␰ ------ᏺᐇᏺᐈ Α Α ------, ᐇᐈÞ  4␲ rᐈ␰ Ϫ rᐇ␨ ␨ ϭ ␰ ϭ ᏹᐇᐈ ϭ  1 1 (2–19b)  nᐇ nᐇ  ␮ ␮ ⌬ᐍᐇ␨ и ⌬ᐍᐇ␨ 0 0 2 a s  ------Lᐇ ϩ ------ᏺᐇ Α Α ------. ᐇᐈϭ 8␲ 4␲ rᐇ␨ Ϫ rᐇ␨  ␨ ϭ ␨ ϭ s a a 1 s 1

where all four of the ⌬ᐍ vectors can be written in terms of generalized coordi-

nates if needed: e.g., ⌬ᐍᐇ␨ ϭ ⌬ᐍᐇ␨ ()␪ ,,… ␪ . Note that except for the inter- a a 1 M ()␮ ր ␲ nal self-inductance term, 0 8 Lᐇ, the inductance coefficients of Eq. (2– 19b) have exactly the same form.

It is informative to compare these equations with Silverberg and Weaver’s equa- tions (4)–(6) for electrostatic structures (384). Eq. (2–5) above has the same form as their equation (6):

Governing Equations 11 Discretized Equations

N N 1 U ϭ -- Α Α P Q Q . elec 2 ij i j i ϭ 1 j ϭ 1

Note also their equation (5a),

N ϭ ϭ ,,… Vi Α PijQi, i 1 N, j ϭ 1

which corresponds to Eq. (A–27) on page 82 and Eq. (A–30) on page 84 written in a more general form for the total flux through each circuit ᐈ:

ᏺ

Ᏺᐈ ϭ Α ᏹᐇᐈIᐇ, ᐈ ϭ 1,,… ᏺ. (2–20) ᐇ ϭ 1

It is clear then that keeping voltages constant for an electrostatic structure is analogous to keeping the fluxes constant for a magnetostatic structure. And, keeping the charge density constant for an electrostatic structure is analogous to keeping the current density constant for a magnetostatic structure.

Lagrangian Equations Following from Silverberg and Weaver (384), the discretized equations of motion are now written. The Lagrangian of Eq. (2–6) in discretized form is, by using Eqs. (2–9), (2–10a), (2–11), and (2–19a),

LTUϭ Ϫ

N M M M M Ѩr Ѩr ϭ 1 i i ␪˙ ␪˙ Ϫ 1 ␪ ␪ -- Α Α Α miѨ␪ и Ѩ␪ j k -- Α Α KSij i j 2 j k 2 (2–21) i ϭ 1 j ϭ 1 k ϭ 1 i ϭ 1 j ϭ 1 N ᏺ ᏺ 1 ϩ Α m r и g Ϫ -- Α Α ᏹᐇᐈIᐇIᐈ. i i 2 i ϭ 1 ᐇ ϭ 1 ᐈ ϭ 1

The derivatives needed for Lagrange’s equations, Eq. (2–7), are

N M M M Ѩ Ѩr Ѩ2r L ϭ i и i ␪˙ ␪˙ Ϫ ␪ Ѩ␪ Α Α Α miѨ␪ Ѩ------␪ Ѩ␪ j k Α KSlj j l j k l i ϭ 1 j ϭ 1 k ϭ 1 j ϭ 1 (2–22a) N ᏺ ᏺ Ѩr Ѩᏹ ϩ i и Ϫ 1 ᐇᐈ Α miѨ␪ g Α Α ------Ѩ␪ IᐇIᐈ, l 2 l i ϭ 1 ᐇ ϭ 1 ᐈ ϭ 1

Governing Equations 12 Linearized Equations

N M Ѩ Ѩr Ѩr L ϭ i и i ␪˙ Α Α miѨ␪ Ѩ␪ j, (2–22b) Ѩ␪˙ j l l i ϭ 1 j ϭ 1

N M Ѩ Ѩr Ѩr d L ϭ i и i ␪˙˙  Α Α mi Ѩ␪ Ѩ␪ j dt Ѩ␪˙ j l l i ϭ 1 j ϭ 1 (2–22c) M Ѩr Ѩ2r Ѩ2r Ѩr ϩ i и i ϩ i и i ␪˙ ␪˙ Α Ѩ␪ Ѩ------␪ Ѩ␪ ------Ѩ␪ Ѩ␪ Ѩ␪ j k j k l j k l k ϭ 1

Substituting into Eq. (2–7) gives the discretized, nonlinear system of M ordi- nary differential equations describing the motion of the structure:

N M M M Ѩr Ѩr Ѩ2r Ѩr i и i ␪˙˙ ϩ i и i ␪˙ ␪˙ ϩ ␪ Α Α mi Ѩ␪ Ѩ␪ j Α Ѩ------␪ Ѩ␪ Ѩ␪ j k Α KSlj j ϭ ϭ j l ϭ j k l ϭ i 1 j 1 k 1 j 1 (2–23) N ᏺ ᏺ Ѩr Ѩᏹ Ϫ i и ϩ 1 ᐇᐈ ϭ ϭ ,,… Α miѨ␪ g Α Α ------Ѩ␪ IᐇIᐈ 0, l 1 M. l 2 l i ϭ 1 ᐇ ϭ 1 ᐈ ϭ 1

Linearized Equations

The equilibrium positions of the structure are found by using Eq. (2–23) and set-

ting ␪˙ k ϭ 0 and ˙˙␪k ϭ 0, where k ϭ 1,,… M. Following Silverberg and Weaver (385), a system of M coupled, nonlinear, algebraic equations that govern the equilibrium positions are obtained:

ᏺ ᏺ M M Ѩ Ѩᏹ ri 1ᐇᐈ Α K ␪ Ϫ Α m и g ϩ Α Α ------Iᐇ Iᐈ ϭ 0, Sli i0 i Ѩ␪ Ѩ␪ 0 0 (2–24) l 2 l 0 i ϭ 1 i ϭ 1 0 ᐇ ϭ 1 ᐈ ϭ 1 l ϭ 1,,… M,

where the 0 subscript denotes evaluation of a quantity at the equilibrium point.

After solving Eq. (2–24) for the equilibrium position a new set of generalized coordinates relative to this position can be defined as

␩ ϭ ␪ Ϫ ␪ ϭ ,,… l()t l()t l0, l 1 M. (2–25)

Governing Equations 13 Linearized Equations

Expanding Eq. (2–24) in a Taylor’s series about the equilibrium position and dis- carding nonlinear terms yields the linearized equations of perturbed motion

M []␩ ϩϩ␩ ␩ ϭ ϭ ,,… Α mlj˙˙ j alj ˙ j blj j 0, l 1 M, (2–26) j ϭ 1

where each coefficient is a different partial derivative of Eq. (2–23):

N Ѩf Ѩr Ѩr ϭϭl ϭ i и i mjl mlj Ѩ␩ Α mi Ѩ␪ Ѩ␪ , (2–27a) ˙˙ j l j 0 k ϭ 1 0 0

ᏺ ᏺ Ѩf Ѩᏹ ѨI ϭϭl ᐇᐈ ᐇ alj Ѩ␩ Α Α ------Ѩ␪ ------Iᐈ0, (2–27b) ˙ j l 0 Ѩ␪˙ 0 ᐇ ϭ 1 ᐈ ϭ 1 j 0

N 2 Ѩf Ѩ r ϭϭl Ϫ k и blj Ѩ␩ KSlj Α mk Ѩ␪ Ѩ␪ g j l j 0 k ϭ 1 0 (2–27c) ᏺ ᏺ 2 Ѩᏹ ѨI Ѩ ᏹ ϩ ᐇᐈ ᐇ ϩ 1 ᐇᐈ Α Α ------Ѩ␪ Ѩ------␪-Iᐈ -- Ѩ␪ Ѩ␪ IᐇIᐈ . l j 2 l j ᐇ ϭ 1 ᐈ ϭ 1 0

By changing the currents Iᐇ and Iᐈ the behavior of the magnetostatic structure can be controlled. Typically the currents will be prescribed by some control algo- rithm in terms of the generalized coordinates and their time derivatives, so that

ϭ ␪ ,,,␪ … ␪ ,,,,␪˙ ␪˙ … ␪˙ ϭ ,,… ᏺ Ir Ir()1 2 M 1 2 M , r 1 . (2–28)

The control algorithm, then, will determine what form the partial derivatives of the currents take in Eqs. (2–27b) and (2–27c).

The partial derivatives of the inductance coefficients in Eqs. (2–27b) and (2–27c) are found from Eq. (2–19b):

Governing Equations 14 Linearized Equations

nᐇ nᐈ Ѩᏹᐇᐈ ␮ ⌬ᐍ и ⌬ᐍ ------ϭ ------0-ᏺ ᏺ ᐇ␨ ᐈ␰ Ѩ Ѩ␪ ␲ ᐇ ᐈ Α Α Ϫ------()rᐈ␰ Ϫ rᐇ␨ и ()rᐈ␰ Ϫ rᐇ␨ l 4 3 Ѩ␪ ␨ ϭ 1 ␰ ϭ 1 rᐈ␰ Ϫ rᐇ␨ l Ѩ⌬ᐍ Ѩ⌬ᐍ (2–29) ᐇ␨ и ⌬ᐍ ϩ ⌬ᐍ и ᐈ␰ ------Ѩ␪ ᐈ␰ ᐇ␨ ------Ѩ␪ - ϩ ------l l - , ᐇᐈÞ . rᐈ␰ Ϫ rᐇ␨

Because of the similarity of the inductance coefficients of Eq. (2–19b), the equa- ᏹ ᐈ␰ ϭ ᐇ␨ ᐇ␨ ϭ ᐇ␨ tion for the partial of ᐇᐇ is found by substituting s, a, ␨␨ϭ ␰␨ϭ ᐈᐇϭ a, s, and of course in Eq. (2–29). This is assuming that the lengths of the circuits, Lᐇ, are not functions of the generalized coordinates or their time derivatives. In that case their partial derivatives with respect to the generalized coordinates would also need to be calculated.

The second partial derivative of the inductance coefficients in Eq. (2–27c) is found from Eq. (2–29):

2 nᐇ nᐈ Ѩ ᏹᐇᐈ ␮ ⌬ᐍ и ⌬ᐍ ϭ 0 ᏺ ᏺ ᐇ␨ ᐈ␰ Ѩ␪ Ѩ␪ ------␲- ᐇ ᐈ Α Α 3------ϫ l j 4 5 ␨ ϭ 1 ␰ ϭ 1 rᐈ␰ Ϫ rᐇ␨ Ѩ Ѩ ()Ϫ и ()Ϫ ()Ϫ и ()Ϫ rᐈ␰ rᐇ␨ Ѩ␪ rᐈ␰ rᐇ␨ rᐈ␰ rᐇ␨ Ѩ␪ rᐈ␰ rᐇ␨ l j

Ѩ⌬ᐍᐇ␨ Ѩ⌬ᐍᐈ␰ ------и ⌬ᐍ ϩ ⌬ᐍ и ------Ѩ␪ ᐈ␰ ᐇ␨ Ѩ␪ Ѩ Ϫ ------j j -()r Ϫ r и ()r Ϫ r 3 ᐈ␰ ᐇ␨ Ѩ␪ ᐈ␰ ᐇ␨ rᐈ␰ Ϫ rᐇ␨ l

⌬ᐍᐇ␨ и ⌬ᐍᐈ␰ Ѩ Ѩ Ϫ ------()rᐈ␰ Ϫ rᐇ␨ и ()rᐈ␰ Ϫ rᐇ␨ Ϫ 3 Ѩ␪ Ѩ␪ rᐈ␰ rᐇ␨ j l (2–30) Ѩ 2 ϩ ()Ϫ и ()Ϫ rᐈ␰ rᐇ␨ Ѩ␪ Ѩ␪ rᐈ␰ rᐇ␨ l j

Ѩ⌬ᐍᐇ␨ Ѩ⌬ᐍᐈ␰ ------и ⌬ᐍ ϩ ⌬ᐍ и ------Ѩ␪ ᐈ␰ ᐇ␨ Ѩ␪ Ѩ Ϫ ------l l -()r Ϫ r и ()r Ϫ r 3 ᐈ␰ ᐇ␨ Ѩ␪ ᐈ␰ ᐇ␨ rᐈ␰ Ϫ rᐇ␨ j Ѩ 2 Ѩ⌬ᐍ Ѩ⌬ᐍ ϩ 1 ⌬ᐍ и ⌬ᐍ ϩ ᐇ␨ и ᐈ␰ ------Ϫ Ѩ␪ Ѩ␪ ᐇ␨ ᐈ␰ ------Ѩ␪ ------Ѩ␪ - rᐈ␰ rᐇ␨ l j l j Ѩ⌬ᐍ Ѩ⌬ᐍ Ѩ 2 ϩ ᐇ␨ и ᐈ␰ ϩ ⌬ᐍ и ⌬ᐍ ᐇᐈÞ ------Ѩ␪ ------Ѩ␪ - ᐇ␨ Ѩ␪ Ѩ␪ ᐈ␰ , j l l j

Governing Equations 15 Linearized Equations

where again, assuming that the circuit lengths, Lᐇ, are fixed, the double partial ᏹ ᐈ␰ ϭ ᐇ␨ ᐇ␨ ϭ ᐇ␨ ␨␨ϭ derivative of ᐇᐇ is found by substituting s, a, a, ␰␨ϭ ᐈᐇϭ s, and of course in Eq. (2–30).

In Eq. (2–26) the coefficients mlj constitute a positive definite, symmetric mass matrix. The coefficients alj can be split into a symmetric matrix called the ϭ ()ր ()ϩ damping matrix, clj 12alj ajl , and a skew-symmetric matrix called ϭ ()ր ()Ϫ the gyroscopic matrix, glj 12alj ajl . Similarly, the coefficients bij can ϭ ()ր ()ϩ ff be separated into a symmetric matrix, klj 12blj bjl , called the sti - ϭ ()ր ()Ϫ ness matrix and a skew-symmetric matrix, hlj 12blj bjl , called the circulatory matrix.

These equations have the same form as those found for electrostatic structures (Silverberg and Weaver 385). Likewise, their stability characteristics are also the same. Namely, the definiteness properties of the stiffness and damping matrices and the presence of the circulatory matrix determine whether the system is sta- ble. If it is possible to construct a control algorithm that removes the circulatory matrix then the structure will be stable as long as both the mass matrix and damping matrices are positive, semi-definite (Meirovitch Computational 289).

Governing Equations 16 Magnetostatic Levitator

The magnetostatic levitator is a one-degree-of-freedom device consisting of two electromagnetic coils. When current flows through the two coils they generate magnetic fields that repel one another causing the upper coil to levitate.

Both the nonlinear equations of motion and the discretized equations of motion were solved for the magnetostatic displacement of the upper coil and compared with the experimental measurement. The error was approximately 4%.

Apparatus

The apparatus consists of a plastic cylinder of radius 0.04396m, about which the two electromagnets are placed. A diagram is shown in Fig. 3–1 with the cylinder shown in outline. The cylinder and the lower coil rest on a table-top with the small upper coil initially resting upon the lower coil.

plastic cylinder

0.1 upper coil

z @m D -0.05 0 0 y @m D -0.050.0 0 0.05 x m Figure 3–1. Magneto- 0.05 @ D lower coil static levitator

17 Equations of Motion

The lower coil (coil #1) is made of 500 loops of 28 gauge (AWG) wire and has a radius of approximately 0.04422m.The center of the coil lies approximately 0.009525m above the table surface. The small upper coil (coil #2) is 15 loops of 36 gauge (AWG) wire with a radius of 0.04503m.

The two coils are powered by a BK Precision Triple Output DC Power Supply 1660 which can deliver a constant current of 2A at 60V by manual control. The coils were attached in series to the power supply, with the upper coil “flipped” electrically so that the generated magnetic fields oppose one another (in the equations of motion this means that the currents were opposite in sign).

Equations of Motion

Using the formulation of “Governing Equations” on page 4, the nonlinear model for the levitator is defined and then discretized. Unlike most magnetostatic structures, the nonlinear, analytical solution to the system’s equilibrium posi- tion is possible. Using the nonlinear model, a discretized model is then devel- oped for comparison.

Nonlinear Equations The kinetic energy of the system is simply

1 2 T ϭ -- mz˙ , (3–1) 2

where m is the mass of the upper coil (coil #2), and z˙ is the time rate of change of the upper coil’s position above the center of the lower coil (the origin of the coordinate system).

The potential energy consists of gravitational and magnetostatic energies. The gravitational potential is

ϭ Ϫ Ugrav mgz, (3–2)

where gz is the (constant) acceleration due to gravity, and is the distance between the upper and the lower coils’ centers.

The magnetostatic potential is given by Eq. (2–5) on page 7:

Magnetostatic Levitator 18 Equations of Motion

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ. (3–3) mag 2 ᐇ ϭ 1 ᐈ ϭ 1

The self-inductances must be known to calculate the total magnetostatic poten- tial energy for a general structure. Like the mutual inductances, the self-induc- tances are purely geometrical relationships. If the self-inductances do not change, however, they need not be calculated to determine the equations of motion. This is because the potential energy is only used in the Lagrangian, and since the Lagrangian is differentiated with respect to time, constant self-induc- tances have no effect on the dynamics of the system. If the geometries of the coils themselves do not change, then, the self-inductances will remain constant and only the mutual inductance between the coils is needed in the magnetostatic potential energy expression. In such cases, the self-inductances can be skipped when calculating the magnetostatic potential energy.

Therefore, Eq. (3–3) can be written as

ϭ ᏹ Umag 12I1I2, (3–4)

ᏺ ϭ ᏹ ϭ ᏹ since 2 and 12 21. Using Eq. (2–12) on page 9, this can be written

␮ ᐍ и ᐍ 0 d 1 d 2 U ϭ I I ------͛͛------, (3–5) mag 1 24␲ r Ϫ r C1 C2 2 1

where C1 and C2 are the bounding contours for the circuits that make up the ᐍ ᐍ lower (#1) and upper (#2) coils, d 1 and d 2 are the differential length elements

along the contours, and r1 and r2 are the positions for the differential elements.

Eq. (3–5) is “lumped” in the nonlinear solution. Specifically, each coil (both lower and upper) is approximated as having all of its loops lying along the same

path. In this way only one loop is modelled for each coil, and the contours C1

and C2 each make only one complete “trip” around their respective circuits. The resulting integral is then multiplied by the number of loops for each coil. Eq. (3– 5) becomes

Magnetostatic Levitator 19 Equations of Motion

␮ ᐍ и ᐍ 0 d 1 d 2 U ϭ ᏺ I ᏺ I ------͛͛------, (3–6) mag 1 1 2 24␲ r Ϫ r C1 C2 2 1

ᏺ ᏺ where 1 and 2 are the numbers of loops for coils 1 and 2 respectively.

The Lagrangian for the system is

LTUϭ Ϫ

1 2 ␮ dᐍ и dᐍ (3–7) ϭ -- mz˙ ϩ mgzϪ I I ------0-͛͛------1 2. 2 1 24␲ r Ϫ r C1 C2 2 1

In a slight departure from the equations of motion given by Eq. (2–7) on page 7, damping is introduced by the non-conservative function

Qczϭ Ϫ ˙, (3–8)

where c is the damping coefficient. This allows comparison of the dynamic solu- tions for both the nonlinear and the discretized formulations. Please note, how- ever, that only the final resting positions given by the solutions of the equations of motion are compared with the experimental measurement of the levitator’s static displacement. No comparison is made between the experimental dynamic motion and the dynamic solutions obtained from the equations of motion.

The Lagrangian equation of motion is then

Ѩ Ѩ d L Ϫ L ϭ  Q. (3–9) dt Ѩz˙ Ѩz

Substituting Eqs. (3–7) and (3–8) gives

1  ᏺ I ᏺ I ␮ mz˙˙ ϩ ------------1 1 2 2 0 ()R ϩ R 2 ϩ z2 ()Ϫ 2 ϩ 2 1 2 R1 R2 z

ϫ()2 ϩ 2 ᒊ Ϫ ()ϩ 2 ᒊ ϩ ᒊ Ϫ ᒊ 3 (3–10a) R1 R2 E() R1 R2 K()zE() K()z

 ϩ ()R ϩ R 2()mgϩ cz ϩ ()mgϩ cz˙ z2  ϭ 0, 1 2 

Magnetostatic Levitator 20 Equations of Motion

ᒊ where R1 and R2 are the radii of the two coils, E() is the complete elliptic integral of the second kind, K()ᒊ is the complete elliptic integral of the first kind, and the elliptic integral parameter is

Ϫ 4R1R2 ᒊ ϭ ------. (3–10b) ()Ϫ 2 ϩ 2 R1 R2 z

Discretized Equations Using Eq. (2–14) on page 9 the mutual inductance is discretized to give the mag- netostatic potential energy of Eq. (3–6) as

n n ␮ 1 2 ⌬ᐍ и ⌬ᐍ ϭ ᏺ ᏺ 0 1␨ 2␰ Umag 1I1 2I2------␲- Α Α ------Ϫ . (3–11) 4 r2␰ r1␨ ␨ ϭ 1 ␰ ϭ 1

As in Eq. (2–14) on page 9, n1 and n2 are the numbers of finite length elements ⌬ᐍ ␨ th by which each circuit is approximated, 1␨ is the finite length element (a ⌬ᐍ ␰ th vector) of circuit 1, 2␰ is the finite length element (also a vector) of circuit

2, and r1␨ and r2␰ are the positions of the two finite elements.

Substituting Eq. (3–11) into LTUϭ Ϫ , and then this Lagrangian into Eq. (3– 9) gives the discretized equation of motion for the levitator.

Numerical Comparison It is now possible to compare the dynamic responses of the nonlinear and the discretized equations of motion. Both the nonlinear equation of motion, Eq. (3– 9), and its corresponding discretized version discussed in “Discretized Equa- tions” above can be solved numerically. The upper coil’s initial position was set to the cross-sectional radius of the lower coil (in effect letting it rest upon the lower coil), the initial velocity was set to zero, and the damping coefficient was set (somewhat arbitrarily) to c ϭ 0.005 kg/s. The currents were set to ϭ ϭ Ϫ I1 0.6 A and I2 0.6 A just as in the actual experiment.

With the model defined, the solutions to Lagrange’s equations were found using a numerical partial differential equations solution routine in Mathematica® 3.0. As explained in Wolfram (1143) the function NDSolve uses an adaptive step- size and automatically switches between a non-stiff implicit Adams method

Magnetostatic Levitator 21 Static Displacement

z @m D Upper Coil Position 0.06 0.05 0.04 0.03 0.02 0.01 t @sD Figure 3–2. Comparison 0 0.2 0.4 0.6 0.8 1 1.2 1.4 of nonlinear and dis- Nonlinear 30 Segments 15 Segments cretized simulated dynamic solutions

(with order between 1 and 12) and a stiff Gear method (backward difference for- mula method with order between 1 and 5) to find a numerical solution.

Fig. 3–2 shows the simulated motion given by solving the nonlinear equation of motion Eq. (3–9) and the corresponding discretized versions for both 30 seg- ϭϭ ϭϭ ments (n1 n2 30) and 15 segments (n1 n2 15). All three solu- tions give approximately the same static displacement, and for the most part both discretized solutions give similar dynamic responses. Note, however, that the 15-segment example appears to lack enough discretization to closely repro- duce the nonlinear solution early in the simulation. Despite this, the more coarsely discretized solution’s frequency of oscillation matches well with the other solutions.

Static Displacement

As stated in “Apparatus” above, the coils were attached in series to the power supply. Electrically, the upper coil was connected so that the magnetic fields of the two coils opposed each other (this amounts to the currents having opposite ϭ Ϫ signs, I1 I2). With the upper coil resting upon the lower coil, the power supply was set to 0.6A and the upper coil was allowed to come to rest at its new equilibrium position. The static displacement of the upper coil was approxi- mately 0.02699m. Table 3–1 compares this result with both the nonlinear solu- tion and two discretized solutions.

Magnetostatic Levitator 22 Static Displacement

Table 3–1. Levitator’s experimental and predicted displacements

Numerical Solutions Experimental Nonlinear 30-Segment 15-Segment Measurement Solution Discretization Discretization Equilibrium 0.027m 0.02621m 0.02613m 0.02597m Position % Error of — –2.90% –3.17% –3.77% Solutiona % Error of — — –0.28% –0.90% Discretizationb a. The error of each numerical solution relative to the experimental measurement. b. The error of each discretized solution relative to the nonlinear solution.

The numerical solutions come close to the measured value. Please note, how- ever, that the uncertainty of the measurement was approximately Ϯ0.0016 m, ϭϭ so even the 15-segment discretization (n1 n2 15) falls within the range of uncertainty. What is remarkable, however, is that the numerical solutions are close at all, considering the approximations that were made. Even the nonlinear solution has some degree of discretization as described in “Nonlinear Equa- tions” on page 18: all the loops for each coil were “lumped” together at the cross- sectional center of each coil. Then the discretized version breaks this perfectly circular loop into straight line segments. Despite these changes the resulting displacements are still in the right range of answers, and even the 15-segment discretization differs from the nonlinear solution by only 1%.

Magnetostatic Levitator 23 Magnetostatic Pendulum

The magnetostatic pendulum is a two-degree-of-freedom spherical pendulum with an electromagnet attached to its tip and other electromagnets located below the pendulum on the base of the apparatus. Fig. 4–1 shows the apparatus with two base electromagnets.

Three experiments were conducted with the magnetostatic pendulum: compar- ison of the pendulum’s predicted and measured natural frequencies (“Natural Frequencies” on page 35); prescribing the pendulum to track a desired path (“Tracking” on page 41); and regulating the pendulum’s motion to dampen oscillations (“Regulation” on page 45).

Apparatus

Some of the measurements that follow are approximate; precisely measured quantities were used in the mathematical model.

U-joint support sensors sensor reflectors pendulum electromagnets base

Figure 4–1. Magneto- static pendulum

24 Apparatus

Pendulum A photograph of the pendulum can be seen in Fig. 4–1. The base of the pendu- lum apparatus is 18Љ ϫ 18Љ. Centered on the base is an 11ϫ 11 grid of tapped holes, each row and column spaced 1Љ apart. The sides are 33⅜Љ tall. The U- joint is located over the center of the base and its grid of holes. The pendulum itself is two parts: an upper aluminum rod that attaches to the U-joint and a lower acrylic tube that attaches to the rod by an aluminum collar and set screw, allowing the length of the pendulum to be adjusted (see Fig. 4–3 on page 26). The length of the pendulum (from the U-joint pivot to the center of the pendu- lum’s electromagnetic coil) for all of the experiments was approximately 29½Љ. The pendulum’s mass is 0.142kg without an electromagnet attached.

The maximum angular displacement of the pendulum without touching the π side of the apparatus is 0.165rad, which is approximately ⁄₂₀rad.

Electromagnets Two sets of electromagnets were used. The first set consists of two electromag- nets: the base coil and the pendulum coil, used in the natural frequencies exper- iment (see page 35). The second set consists of three electromagnets: two base coils and the pendulum coil, used in the tracking (page 41) and regulation (page 45) experiments. A photograph of the coils used for the last two experi- ments can be seen in Fig. 4–2. All coils are constructed of flat acrylic spools upon which wire is wound. The spools have an outer diameter of 4½Љ, an inner diam- eter of 3½Љ, and a thickness of ¹¹⁄₁₆Љ.

Figure 4–2. Close-up of magnetostatic pendu- lum’s electromagnetic coils

Magnetostatic Pendulum 25 Apparatus

The two coils used for the natural frequencies experiment are almost identical except for the number of loops: the base coil consists of 600 loops, and the pen- dulum coil consists of 300 loops. Likewise, the coils for the tracking and regula- tion experiments are almost identical except that the base coils consist of 450 loops each and the pendulum coil consists of 550 loops. The base coils have outer diameters of approximately 4⅜Љ, and the pendulum coil has an outer diameter of approximately 4½Љ. All coils use 26 gauge (AWG) wire. The mass of the pendulum coil is 0.361kg.

Power Supplies Two power supplies were used for the experiments: a BK Precision Triple Out- put DC Power Supply 1660, and a Hewlett-Packard e3631a Triple Output DC Power Supply. The BK Precision power supply can deliver a constant cur- rent of 2A at 60V by manual control. The Hewlett-Packard can supply 1A at 25V, 1A at –25V, and 5A at 6V, via manual or computer control.

Sensors & Computer Two Idec Analog Distance sa1d–lk4 infrared sensors measure the distances to the two aluminum “vanes” (sensor reflectors) attached to the pendulum (see Fig. 4–3). The sensors produce continuous analog voltages that are updated every 50ms. These voltages are measured and recorded by the computer and correlate to the pendulum angles as explained in “Calibration” below.

An Apple Macintosh IIci computer running National Instruments Lab- VIEW performed data acquisition and control functions. The IIci has two

U-joint support

wires to pendulum’s electromagnet

sensor reflector “vane”

infrared sensor

Figure 4–3. Close-up of top of pendulum tube magnetostatic pendu- lum’s sensor reflectors

Magnetostatic Pendulum 26 Apparatus

expansion cards installed: a National Instruments nb-mio-16l-9 Multi- Function i/o data acquisition card which reads the sensor voltages and a National Instruments nb-gpib/tnt card which communicates with the pro- grammable Hewlett-Packard power supply via its gpib interface. With this configuration, the IIci can acquire data from the sensors, compute a response and, if desired, send control commands to the power supply to adjust the base electromagnets’ currents.

Calibration The pendulum angles of rotation are calculated from the voltages recorded. Unfortunately, this is not a straightforward process since many factors can affect the calculation. The sensor reflectors are rigidly attached to the pendulum, but the reflectors themselves are not precisely perpendicular to the sensors, nor are they precisely flat. In addition, the pendulum itself is not attached to the U-joint such that the reflectors would be precisely perpendicular to the sensors.

The pendulum and sensors are calibrated to accommodate for these errors. The base of the pendulum contains an 11ϫ 11 array of tapped holes with each row and column set 1Љ apart. A particular position on the grid, ()XY, , corresponds to the hole X inches along the X-axis, and Y inches along the Y-axis of the base (see Fig. 4–4). The origin of the grid, ()00, , is directly underneath the U-joint (to within a small distance that is measured and accounted for). Using this grid, the pendulum is placed into 121 unique, rigid positions, and distance measure- ments are made over the entire possible range of pendulum motion.

Y tan␣ ϭ ------, ϪZ Z 2 ϩ 2 Y Z ϪX ␤ ϭ ------tan 2 2 Y ϩ Z O pendulum pivot ␤ ␣ ␤ Y ␣ pendulum tip (–X, Y, –Z) X Figure 4–4. Pendulum support reference frame (X, Y, Z) and its relation to generalized coordi- nates (␣, ␤)

Magnetostatic Pendulum 27 Apparatus

5.2

V @VD -5 0

4.2 Y @inD 0 -5 0 Figure 4–5. Pendulum’s X @inD voltage calibration sur- 5 5 face for first sensor

The calibration technique is as follows: A socket-head machine screw, the “base screw,” is placed through a stand-off and screwed into one of the grid holes (the stand-off determines that the socket-head is always at the same height above the base). A screw with a ball bearing affixed to its head, the “pendulum screw,” is screwed into the bottom of the pendulum (instead of an electromagnet). Align- ing the length of the pendulum, the pendulum screw is screwed in or out so that the ball bearing fits snugly into the base screw’s socket. With the pendulum rig- idly fixed, distance measurements are recorded. This procedure is performed for all 121 grid positions.

The calibration measurements produce distance versus voltage surfaces for each sensor as shown in Figs. 4–5 and 4–6. Interpolating between the voltage

5.2

V @VD -5 1

4.2 Y @inD 0 -5 0 Figure 4–6. Pendulum’s X @inD voltage calibration sur- 5 5 face for second sensor

Magnetostatic Pendulum 28 Equations of Motion

values produces continuous functions relating grid position and sensor voltage. For a specific set of voltage pairs (one voltage from each of the two sensors) the interpolation functions are used to find the corresponding grid position ()XY, . With the ()XY, position of the pendulum known, the pendulum angles are found geometrically:

Y ϪX tan␣ ϭ ------, tan␤ ϭ ------, ϪZ 2 2 Y ϩ Z

where Z is the distance in inches from the U-joint to the plane formed by the pendulum screw’s ball bearing as it is moved around the grid. See Fig. 4–4.

Equations of Motion

A nonlinear model is developed based on Lagrangian dynamics, the magneto- static potential is then discretized, and these discretized equations are then lin- earized. The discretized model was used to predict the natural frequencies of the pendulum (in “Natural Frequencies” on page 35). The linearized model was used to pick an appropriate control method and to assure the controlled system’s sta- bility when regulating the pendulum’s motion (in “Regulation” on page 45).

Nonlinear Equations When no current is flowing through the electromagnets the derivation is straightforward: an inertia tensor is determined for the pendulum and is then used in expressions for the kinetic and potential energies. When there is a cur- rent through the electromagnets, however, then the magnetostatic potential energy created by the magnetic fields of the coils must be included as well.

The pendulum has two angles of rotation, the rotation ␣ about the x-axis and the rotation ␤ about the carried y-axis, which serve as the generalized coordi- nates for the system (see Fig. 4–7). These rotations also correspond to the first two angles of rotation in a standard Euler 1–2–3 rotation, so transforming from body coordinates to inertial coordinates is simple. A position rЈ in the inertial system is related to a position r in the body’s coordinate system by

rЈ ϭ RTr,

Magnetostatic Pendulum 29 Equations of Motion

Generalized Coordinates: zЈ ()␣␤,

z ␣ ␤ Inertial y Reference Frame: ␣ ␤ ()xЈ,,yЈ zЈ ␣ ␤ yЈ Body-Fixed xЈ Reference Frame: x ()xyz,, Figure 4–7. Pendulum’s coordinate frames

where R is the Euler 1–2 rotation matrix that carries the inertial system into the rotated body system.

Inertia Tensor. The inertia tensor for the pendulum is calculated as the sum of the inertia tensors for each constituent part. Using detailed measurements of each part, the entries for its inertia tensor are found from

3 ϭ ͐ ␦ 2 Ϫ Iij m()r ij Α xk xixjdv, V k ϭ 1

ϭ where dxv dx1dx2d 3 is the volume element at position r, m()r is the mass density, and V is the volume of the part. The final tensor for the part is

I11 I12 I13 މ ϭ I21 I22 I23 .

I31 I32 I33

Summing all the tensors yields the inertia tensor for the entire pendulum.

Kinetic Energy. Once the inertia tensor is known, the kinetic energy is calcu- lated using the formula

1 T ϭ -- ␻ и L, (4–1) 2 where ␻ is the angular velocity and L, the angular momentum, is defined as

Magnetostatic Pendulum 30 Equations of Motion

L ϭ މ и ␻.

The angular velocity for a system undergoing an Euler 1–2–3 rotation is given by

cos␤cos␥ sin␥ 0 ␣˙ ␻ ϭ Ϫcos␤sin␥ cos␥ 0 и ␤˙ , sin␤ 0 1 ␥˙ where ␥ is the rotation about the carried z-axis resulting from the first two rota- tions (Meirovitch Analytical 143).

The angular velocity can also be derived from constructing the skew symmetric matrix ⍀ ϭ Ϫ()dtR ր d и RT, where R is the Euler 1–2–3 rotation matrix that carries the inertial frame into the rotated body frame (Meirovitch Analytical 108); the angular velocity is then

T ␻ ϭ ⍀ ⍀ ⍀ 32 13 21 .

Using either method and setting ␥ ϭ 0 (since the pendulum cannot rotate about that axis), the angular velocity is just

T ␻ ϭ ␣˙ cos␤ ␤˙ ␣˙ sin␤ .

Gravitational Potential Energy. The gravitational potential energy of an object is defined as the work done in moving the object from one point to another (with no change in the kinetic energy) in a gravitational field. From Eq. (2–2) on page 5 this is just

2 ͐ и ϭϭϪ Ϫ⌬ mgrd U1 U2 U, 1 where m is the mass of the object, and g is the acceleration due to gravity. For an idealized spherical pendulum the gravitational potential energy is just

ϭ Ϫ ␪ Ugrav mgrcos ,

Magnetostatic Pendulum 31 Equations of Motion

0 p - €€€€€€€ Ugrav @JD 2 -2 p a @radadD 0 €€€€€€€ 2 0 Figure 4–8. Pendulum’s p b @radD gravitational potential €€€€€€€ p 2 - €€€€€€€ energy over large range 2 of motion

where mg is the mass of the pendulum, is the magnitude of the acceleration due to gravity, r is the magnitude of the center of mass vector, and ␪ is the angle of the pendulum from the z-axis.

For the magnetostatic pendulum this expression for the gravitational potential energy must be rewritten in terms of the generalized coordinates ␣␤ and . Without much difficulty it can be shown that cos␪ ϭ cos␣cos␤, so that the gravitational potential energy of the pendulum is

ϭ Ϫ ␣ ␤ Ugrav mgrcos cos , (4–2) where mr is the mass of the pendulum and is again the magnitude of the cen- ter of mass vector. The gravitational potential energy for the pendulum is plot-

0 p - €€€€€€€€€€€ Ugrav @JD 20 -2 p a @raddD 0 €€€€€€€€€€€ 20 0 Figure 4–9. Pendulum’s p b @radD €€€€€€€€€€€ p gravitational potential 20 - €€€€€€€€€€€ energy over possible 20 range of motion

Magnetostatic Pendulum 32 Equations of Motion

π ted over the large range of motion ± ⁄₂rad for both ␣␤ and in Fig. 4–8, and over π the possible range of motion, ± ⁄₂₀rad, in Fig. 4–9.

Magnetostatic Potential. The potential energy of a magnetostatic field is (Eq. (2–5) on page 7)

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ, (4–3) mag 2 ᐇ ϭ 1 ᐈ ϭ 1

where here ᏺᏹ is the number of electromagnets for the experiment, ᐇᐈ is the

mutual inductance between electromagnets ᐇ and ᐈ, and Iᐇ and Iᐈ are the currents flowing through electromagnets ᐇᐈ and respectively. When ᐇᐈϭ , th ᏹᐇᐇ is the self-inductance of the ᐇ circuit.

Almost all parts of the pendulum are constructed from acrylic Plexiglas to pre- vent mechanical damping from eddy currents (see Feynman et al. 2: 16–6). Only the aluminum rod attached to the U-joint, the attachment of the pendulum tube to the aluminum rod, the sensor reflector vanes, the electromagnetic coils, and a few screws are conductive (the screws holding the coils in place are nylon). Since the only conductive materials near the electromagnetic coils are the coils themselves, eddy currents can be ignored in the model.

Since the self-inductance of each coil is constant and ᏹᐇᐈ ϭ ᏹᐈᐇ, the magne- tostatic potential energy of the pendulum, Eq. (4–3), reduces to the form

ᏺ ᐇ Ϫ 1 ϭ ᏹ Umag Α Α ᐇᐈIᐇIᐈ, (4–4) ᐇ ϭ 1 ᐈ ϭ 1

as explained in “Nonlinear Equations” on page 18. Eq. (4–4) assures that each unique mutual inductance is only calculated once and that self-inductances are never calculated for the pendulum.

Lagrangian Equations. The Lagrangian is

LTUϭ Ϫ , (4–5)

Magnetostatic Pendulum 33 Equations of Motion

where TU and are the kinetic and potential energies. The potential energy, as stated above, has a gravitational component and a magnetostatic component:

ϭ ϩ UUgrav Umag. (4–6)

Dissipative forces are ignored, so from Eq. (2–7) on page 7 the Euler-Lagrange equations of motion are just

ѨL d ѨL ѨL d ѨL Ϫ ϭ 0, Ϫ ϭ 0. (4–7) Ѩ␣ dt Ѩ␣˙ Ѩ␤ dt Ѩ␤˙

Discretized Equations While it is not productive to discretize the physical structure in the case of the pendulum (as outlined in “Gravitational Potential” on page 8), the mutual inductance is discretized according to Eq. (2–14) on page 9:

n n ␮ ᐇ ᐈ ⌬ᐍ и ⌬ᐍ 0 ᐇ␨ ᐈ␰ ᏹᐇᐈ ϭ ------ᏺᐇᏺᐈ Α Α ------, (4–8) 4␲ rᐈ␰ Ϫ rᐇ␨ ␨ ϭ 1 ␰ ϭ 1

where ᏺᐇ and ᏺᐈ are the number of loops for each of the coils, ⌬ᐍᐇ␨ (a vector) th th is the ␨ finite length element of circuit ᐇ, ⌬ᐍᐈ␰ (also a vector) is the ␰ finite

length element of circuit ᐈ, and rᐇ␨ and rᐈ␰ are the positions of the two finite

elements. Each coil has only one loop each, which is discretized into nᐇ and nᐈ elements respectively. While this discretization does reduce the accuracy of the model, it still yields predictions that closely match experiment, as shown in “Natural Frequencies” on page 35.

Substituting Eq. (4–8) into the potential energy, Eq. (4–4), gives the discretized magnetostatic potential energy of the pendulum:

ᏺ ᐇ Ϫ n n ␮ 1 ᐇ ᐈ 0 ⌬ᐍ и ⌬ᐍ U ϭ ------Α Α ᏺ I ᏺ I Α Α ᐇ␨ ᐈ␰ (4–9) mag 4␲ ᐇ ᐇ ᐈ ᐈ ------Ϫ . ᐇ ϭ 1 ᐈ ϭ 1 ␨ ϭ 1 ␰ ϭ 1 rᐈ␰ rᐇ␨

Linearized Equations Following the same procedure outlined in “Linearized Equations” on page 13, the linearized Lagrangian equations for the pendulum are formed. The equa- tions of motion were only linearized for the last experiment, “Regulation” on page 45, so details are given in that section.

Magnetostatic Pendulum 34 Natural Frequencies

Natural Frequencies

Natural frequencies of oscillation about both angles of rotation for the pendu- lum were measured. Large and small initial displacements were used, and four different values of electric current were used for the system’s electromagnets.

Computer simulations of the pendulum’s response yielded predictions of the natural frequencies within 4% of those measured.

Apparatus Electromagnets. Two electromagnets were used, one attached to the center of the base (the base coil or coil #1), and one attached to the pendulum’s tip (the pendulum coil or coil #2). The base coil has 600 loops of wire and the pendulum coil has 300 loops. The constructed base coil has an outer diameter (measured from the spool center to the outer edge of the wire coil) of 4½Љ, the pendulum coil has an outer diameter of 3⅞Љ, and both have a thickness of ¹¹⁄₁₆Љ. Both coils use 28 gauge (AWG) wire. The mass of the pendulum coil is 0.225kg.

Power Supply. Both electromagnets were powered with a constant DC current by the BK Precision power supply. Fig. 4–10 shows the circuit diagram for the electromagnets and power supply.

ϭ ⍀ ϭ ⍀ The coils have resistances R1 20.375 and R2 40.5 , where R1 and R2 are the resistances of coils 1 and 2 respectively. Because the coils are in parallel, current is split according to the standard formulas (O’Malley 34)

I1 I2

I R1 R2

coil 1 coil 2

L1 M12 L2 Figure 4–10. Electro- magnet circuit diagram for natural frequencies experiment

Magnetostatic Pendulum 35 Natural Frequencies

R R ϭ 2 ϭ 1 I1 ------ϩ I, I2 ------ϩ I, R1 R2 R1 R2

where II is the total current delivered by the power supply, and 1 and I2 are the currents through the two coils.

Equations of Motion Discretized Equations. The nonlinear, discretized equations of motion were used to predict the natural frequencies of the pendulum for various electromag- net currents. While the rest of the equations of motion are detailed in “Equa- tions of Motion” on page 29, the magnetostatic potential energy is given by Eq. (4–9) with ᏺ ϭ 2 (two coils):

n n ␮ 1 2 ⌬ᐍ и ⌬ᐍ ϭ 0 ᏺ ᏺ 1␨ 2␰ Umag ------␲- 1I1 2I2 Α Α ------Ϫ , (4–10) 4 r2␰ r1␨ ␨ ϭ 1 ␰ ϭ 1

ᏺ ϭ ᏺ ϭ ϭϭ where 1 600, 2 300, and n1 n2 10.

Eq. (4–10) is plotted over the range of possible pendulum motion (␣␤ and π ϭ varying over ± ⁄₂₀rad) in Fig. 4–11 for a total current of 0.6A (I1 ⁴⁄₁₀ A, and ϭ ϭ I2 ²⁄₁₀ A) and in Fig. 4–12 for a total current of 1.0A (I1 ⅔ A and ϭ (), I2 ⅓ A). In both plots and for the experiment the base coil is in the 00 grid position directly under the pendulum’s gravitational equilibrium position.

Numerical Solution. Just as in “Numerical Comparison” on page 21, the solu- tions to Lagrange’s equations were found using NDSolve in Mathematica®.

Initial conditions for the numerical solution were found from the experimental sensor data. Due to the limited sensor resolution, the first six measured posi- tions and their differences from each experimental run were curve fit, and these values were used for the starting position and velocity.

Experiment The experiments were designed to measure the natural frequencies about both of the pendulum’s angles of rotation under varying circumstances. Both large and small initial displacements were used about both angles, and for each of these, four different values of current were used for the electromagnets (0.0A,

Magnetostatic Pendulum 36 Natural Frequencies

Total Current =1.0 A, Base Coil at H0, 0L

p 0 - €€€€€€€€€€€ Umag @JD 20 -0.002 p Figure 4–11. Pendulum’s a @raddD 0 €€€€€€€€€€€ magnetostatic potential 20 0 energy for 0.6A total cur- p b @radD rent to two electromag- €€€€€€€€€€€ p nets (one base and one 20 - €€€€€€€€€€€ 20 pendulum coil)

0.6A, 0.8A, and 1.0A). Each of these experiments was performed four times. In total sixteen experiments were run, and sixty-four data sets were taken.

Before any experiments were run, the pendulum was calibrated as detailed in “Calibration” on page 27. Then for each experimental run the current was set with the power supply, the pendulum was displaced then released, and the sen- sor voltages were recorded as the pendulum oscillated. By using the calibration data the ␣␤ and angles corresponding to the recorded voltages were calcu- lated for each data set. From these values the experimental rotational periods of ␣␤ and for the first 10s of recorded motion were measured.

To simulate the observed motion of the pendulum, the ␣␤ and angles for each data set were used to arrive at initial conditions for the numerical solution to

Total Current =1.0 A, Base Coil at H0, 0L

p 0 - €€€€€€€€€€€ Umag @JD 20 -0.002 p Figure 4–12. Pendulum’s a @raddD 0 €€€€€€€€€€€ magnetostatic potential 20 0 energy for 1.0A total cur- p b @radD rent to two electromag- €€€€€€€€€€€ p nets (one base and one 20 - €€€€€€€€€€€ 20 pendulum coil)

Magnetostatic Pendulum 37 Natural Frequencies

Table 4–1. Pendulum’s mean natural frequencies for 0.0A total current

simulated measured % error simulated measured % error ␣ ␤ ␣ ␣ sim ␤ ␤ sim sim exp ------␣ % sim exp ------␤ % exp exp l ␻ [rad⁄s] 3.78533 3.77811 +0.19% 3.78395 3.774 +0.26% initia ll T [s] 1.65988 1.66305 –0.19% 1.66049 1.66486 –0.26% sma displacement

l ␻ [rad⁄s] 3.78402 3.7747 +0.25% 3.78375 3.77025 +0.36%

T [s] 1.66045 1.66455 –0.25% 1.66057 1.66652 –0.36% arge initiaarge l displacement

Lagrange’s equations of motion. The nonlinear equations of motion were then solved, and the simulated rotational periods of ␣␤ and for the first 10 s of sim- ulated motion were determined.

Results The simulated and experimental periods for rotation about ␣␤ and are com- pared for each experiment performed. The individual runs of each experiment have been averaged together. The results for 0.0A, 0.6A, 0.8A, and 1.0A total cur- rent are listed in Tables 4–1 to 4–4 respectively. The mean frequency, ␻, in radi- ans per second, the mean period, T, in seconds, and the percentage error between experiment and simulation are listed.

As expected, the most accurate simulations are those without magnetic effects (total current is 0.0A) as seen in Table 4–1. Note that the frequency is virtually

Table 4–2. Pendulum’s mean natural frequencies for 0.6A total current

simulated measured % error simulated measured % error ␣ ␤ ␣ ␣ sim ␤ ␤ sim sim exp ------␣ % sim exp ------␤ % exp exp l ␻ [rad⁄s] 4.15099 4.21782 –1.58% 4.14506 4.21405 –1.64% initia ll T [s] 1.51366 1.48968 +1.61% 1.51582 1.49101 +1.66% sma displacement

l ␻ [rad⁄s] 4.04692 4.09579 –1.19% 4.0594 4.10878 –1.20%

T [s] 1.55259 1.53406 +1.21% 1.54782 1.52921 +1.22% arge initiaarge l displacement

Magnetostatic Pendulum 38 Natural Frequencies

Table 4–3. Pendulum’s mean natural frequencies for 0.8A total current

simulated measured % error simulated measured % error ␣ ␤ ␣ ␣ sim ␤ ␤ sim sim exp ------␣ % sim exp ------␤ % exp exp l ␻ [rad⁄s] 4.44644 4.529 –1.82% 4.43302 4.5415 –2.39% initia ll T [s] 1.41311 1.38734 +1.86% 1.41737 1.38351 +2.45% sma displacement

l ␻ [rad⁄s] 4.2334 4.32039 –2.01% 4.25795 4.33868 –1.86%

T [s] 1.48421 1.45431 +2.06% 1.47564 1.44819 +1.90% arge initiaarge l displacement

identical for both large and small initial displacements. This matches the well- known result that the period of a simple pendulum does change with amplitude of oscillation when the small angle approximation is made.

In contrast, note that as the total current through the electromagnets is increased, the difference between the frequencies of the large and small initial displacements also increases. From Tables 4–2 to 4–4 it is seen that on average the natural frequency of the large displacement experiments increases over that of the small displacement experiments by 0.113rad/s for 0.6A, 0.207rad/s for 0.8A, and 0.302rad/s for 1.0A total current.

This effect seems to result from the nonlinear nature of the magnetostatic potential as the current increases. Figs. 4–11 and 4–12 on page 37 show this non- linearity: while the edge of the potential well does not appreciably move, the

Table 4–4. Pendulum’s mean natural frequencies for 1.0A total current

simulated measured % error simulated measured % error ␣ ␤ ␣ ␣ sim ␤ ␤ sim sim exp ------␣ % sim exp ------␤ % exp exp l ␻ [rad⁄s] 4.77958 4.89641 –2.39% 4.80069 4.9164 –2.35% initia ll T [s] 1.31465 1.28324 +2.45% 1.30886 1.27801 +2.41% sma displacement

l ␻ [rad⁄s] 4.46316 4.58751 –2.71% 4.4836 4.62723 –3.10%

T [s] 1.4078 1.36964 +2.79% 1.40137 1.35788 +3.20% arge initiaarge l displacement

Magnetostatic Pendulum 39 Natural Frequencies depth of the well increases by almost an order of magnitude as the current is increased from 0.6A to 1.0A.

Also apparent is that as the current increases the simulation becomes less accu- rate. This is probably a result of the simplifications made to the magnetostatic potential function to aid computation. By modeling more than one loop of the coil (see Eq. (2–14) on page 9), it is likely that the accuracy will increase.

Another simplification made in the magnetostatic potential function is that ϭϭ each circular loop was approximated by decagons (n1 n2 10). Although not shown, several simulations were run comparing the results given by setting ϭϭ ϭϭ n1 n2 10 and setting n1 n2 20. The error between the two differed by an average of 6%. The larger number of segments took more than 500% as long to compute, and consumed more than an order of magnitude more computer memory. Clearly, the small increase in accuracy does not justify the extra effort.

The method of modeling magnetostatic effects outlined in the chapter “Govern- ing Equations” on page 4 and detailed in “Equations of Motion” on page 29 pre- dicts the natural frequencies of the magnetostatic pendulum accurately. It clearly reproduces the nonlinear nature of the magnetostatic potential, as shown by the small prediction errors for both the large and small initial dis- placements and the increase of total current through the electromagnets.

Magnetostatic Pendulum 40 Tracking

Tracking

Maps of the values of the pendulum’s angles of rotation, ␣␤ and , versus the currents in two base electromagnets were made. Using these data, current val- ues were calculated to make the pendulum track a desired path.

For two different paths, the pendulum’s motion stayed within approximately 4% of that desired.

Apparatus Electromagnets. Three electromagnets were used, two attached to the base (the base coils or coil #1 and coil #2) at grid positions (3, 0) and (0, 3) respectively, and one attached to the pendulum’s tip (the pendulum coil or coil #3). (See Fig. 4–2 on page 25 for a photograph of the setup.) The base coils each have 450 loops of wire and the pendulum coil has 550 loops. The base coils have an outer diameter (measured from the spool center to the outer edge of the wire coil) of 4⅜Љ, the pendulum coil has an outer diameter of 4½Љ, and both have a thick- ness of ¹¹⁄₁₆Љ. All coils use 26 gauge (AWG) wire. The mass of the pendulum coil is 0.361kg.

Power Supply. The pendulum coil was powered with a constant DC current by the BK Precision power supply. The two base coils were powered by separate constant current outputs of the programmable Hewlett-Packard power sup- ply. The Macintosh computer was used to control the Hewlett-Packard power supply and thus the currents to the base coils.

All three coils were attached to separate power supply sources. The coils have ϭ ⍀ ϭ ⍀ ϭ ⍀ resistances R1 19.6 , R2 20.2 , and R3 24.5 , where R1, R2,

and R3 were the resistances of coils 1, 2, and 3 respectively.

Experiment This experiment consists of two steps: first the current to the base coils was stepped through a series of values and the position of the pendulum was mea- sured. With this “map” the required currents to make the pendulum track a desired path were calculated. Second, the programmable power supply was used to deliver these currents to the base coils in sequence and the pendulum’s actual positions were compared with those desired.

Magnetostatic Pendulum 41 Tracking

0.03 0.02 a @radD 0.01 0.80 0.6 0 0.4 @ D I2 A Figure 4–13. Pendulum’s 0.2 ␣ 0.4 0.2 angle versus the two @ D I1 A 0.6 base electromagnets’ 0 0.8 currents

Current vs. Position Maps. The computer was used to control the program- mable power supply to step the currents supplied to the two base electromag- nets from 0.0A to 0.8A in 0.1A increments. The current to the pendulum’s coil is set to a constant 0.8A with the other power supply. The base coils were con- nected so that any current would produce an attractive force between them and the pendulum coil. At each of the two current values (eight values for each base coil equals 64 individual current settings) the position of the pendulum is mea- sured. This produced two maps of the pendulum’s angles of rotation, ␣␤ and , versus the two base coils’ currents as shown in Figs. 4–13 and 4–14. As can be seen from the figures, the dependence of the angles of rotation on the two cur- rents is slightly nonlinear.

0 -0.01 b @radD -0.02 - 0.80.03 0.6 0 0.4 @ D I2 A Figure 4–14. Pendulum’s 0.2 ␤ 0.4 0.2 angle versus the two @ D I1 A 0.6 base electromagnets’ 0 0.8 currents

Magnetostatic Pendulum 42 Tracking

a @radD 0.01 0.015 0.02 0 -0.0025 -0.005 D -0.0075 rad @ b -0.01 Figure 4–15. Example - comparison of actual 0.0125 versus desired path for -0.015 the first tracking experi- ment

Interpolation functions were then found that “surface-fit” the 64 points of each map. A path, i.e., a series of sequential values for each angle of rotation ()␣␤, , was then chosen. Two simultaneous equations were then solved for each posi- tion on the path—␣␣␤ equal to the interpolation function from the map and equal to the interpolation function from the ␤ map to arrive at the unique set (), ()␣␤, of two base coil currents, I1 I2 , that would give the desired position . Thus, a series of two current values was found for each pendulum position of the path. In this manner, two paths were constructed, one “square” and one “dia- mond” shown as a series of gray points in Figs. 4–15 and 4–16.

Tracking Experiment. Once the list of current values was found for each path, the currents to the base coils were set by the computer controlling the program-

a @radD 0.01 0.015 0.02 0 -0.0025 -0.005 D -0.0075 rad @ b -0.01 Figure 4–16. Example - comparison of actual 0.0125 versus desired path for -0.015 the second tracking experiment

Magnetostatic Pendulum 43 Tracking

Table 4–5. Pendulum’s mean normalized tracking error for two paths

First Path (shown in Fig. 4–15) Second Path (shown in Fig. 4–16) ␣ ␤ ␣ ␤ error error error error Mean % Error 3.07% 3.81% 3.38% 4.27%

mable power supply. At each current value pair, the pendulum was allowed to come to rest at the new equilibrium position and the computer recorded the position via the sensors. For each path the experiment was run several times.

Results The list of measured angles was then compared with those desired. Fig. 4–15 shows the actual positions (in black) of one example experimental run com- pared with the desired positions (in gray) for the first path. Likewise, Fig. 4–16 shows the actual positions of an example experimental run versus desired posi- tions for the second path. At each point of the path the normalized rms error was calculated for each measured angle with each desired angle.† And the nor- malized rms errors for all points of the path were then averaged. The mean errors for both paths are approximately 3% for ␣␤ and approximately 4% for as shown in Table 4–5.

Note in Figs. 4–15 and 4–16 that the sensor values are clearly “quantized,” i.e., they form rows and columns of discreetly spaced values with no intermediate values. This arises from the finite resolution of the sensor measurements and clearly affects the tracking errors listed in Table 4–5. When averaged for all experiments, the sensor resolution errors for ␣␤ and is approximately 1.49% and 2.68% respectively. This makes sense because the range of motion of the pendulum for either path is quite small: approximately 0.025 rad for ␣ and approximately Ϫ0.015 rad for ␤. These rotations amount to approximately ½Љ along the +X-axis and approximately ⁷⁄₁₀Љ along the +Y-axis of the base.

† This is the simply the rms error “normalized” to the range of possible motion shown in Figs. 4–15 and 4–16. Instead of dividing the difference between the measured and the desired values by the desired value, the normalized error divides by the possible range of values: normalized rms% errorϭ () measuredϪ desired ր range of values.

Magnetostatic Pendulum 44 Regulation

Regulation

The discretized equations of motion for the pendulum are linearized about new equilibrium positions created by the system’s three electromagnets. Although the pendulum’s electromagnet was maintained at a constant current, the effect of the base electromagnets’ currents on the linearized equations of motion was examined. It was determined that when using velocity feedback to control the base electromagnet’s currents, the real part of the eigenvalues for the linearized equations was negative, ensuring a stable system.

Using this control law for the currents, the pendulum’s oscillations about two newly created equilibrium positions were regulated to control settling time.

Apparatus This experiment used the exact same apparatus as that used in the tracking experiment described under “Apparatus” on page 41. In particular note that the same three electromagnets were used and that the base electromagnets were in the same positions as before: one coil at grid position (3, 0) and one coil at grid position (0, 3) as shown in Fig. 4–2 on page 25.

The computer was used to take sensor readings, compute angular velocities, cal- culate the appropriate currents for the base coils, and then send the necessary commands to the programmable power supply.

Equations of Motion Discretized Equations. The magnetostatic potential energy of Eq. (4–9) on page 34 for three coils becomes

n n ␮  1 2 ⌬ᐍ и ⌬ᐍ ϭ 0 ᏺ ᏺ 1␨ 2␰ Umag ------ 1I1 2I2 Α Α ------4␲ r ␰ Ϫ r ␨  ␨ ϭ 1 ␰ ϭ 1 2 1

n n 1 3 ⌬ᐍ и ⌬ᐍ ᏺ ᏺ 1␨ 3␰ (4–11) + 1I1 3I3 Α Α ------r ␰ Ϫ r ␨ ␨ ϭ 1 ␰ ϭ 1 3 1 n n 2 3 ⌬ᐍ и ⌬ᐍ  ᏺ ᏺ 2␨ 3␰ + 2I2 3I3 Α Α ------ , r ␰ Ϫ r ␨ ␨ ϭ 1 ␰ ϭ 1 3 2 

Magnetostatic Pendulum 45 Regulation

All Currents =0.6 A, Base Coils at H3, 0L & H0, 3L

p 0 - €€€€€€€€€€€ Umag @JD 20 -0.02 p Figure 4–17. Pendulum’s a @raddD 0 €€€€€€€€€€€ 20 magnetostatic potential 0 energy for 0.6A current p b @radD €€€€€€€€€€€ p to three electromagnets 20 - €€€€€€€€€€€ (two base and one pen- 20 dulum coil)

ᏺ where i is the number of loops, Ii is the current, ni is the number of discreti- ⌬ᐍ zation segments, i␨ is the finite length element, and ri␰ is the finite length element position vector for the i th coil respectively (where i ϭ 123).,, For ϭϭϭ this experiment n1 n2 n3 10, and, as stated under “Apparatus” on ᏺ ϭϭᏺ ᏺ ϭ page 41, 1 2 450 loops and 3 550 loops.

Eq. (4–11) is plotted over the range of possible pendulum motion (␣␤ and over π ϭϭϭ ± ⁄₂₀rad) in Fig. 4–17 for a current of 0.6A (I1 I2 I3 0.6 A), and in ϭϭϭ Fig. 4–18 for a current of 1.0A (I1 I2 I3 1.0 A). Again, in both plots and for the experiment the base coils are in the ()30, and ()03, grid positions.

Linearized Equations. Substituting the magnetostatic potential given by Eq. (4–11) into Eq. (4–6) on page 34, and then Eq. (4–6) into the Lagrangian

All Currents =1.0 A, Base Coils at H3, 0L & H0, 3L

p 0 - €€€€€€€€€€€ Umag @JD 20 -0.02 p Figure 4–18. Pendulum’s a @raddD 0 €€€€€€€€€€€ 20 magnetostatic potential 0 energy for 1.0A current p b @radD €€€€€€€€€€€ p to three electromagnets 20 - €€€€€€€€€€€ (two base and one pen- 20 dulum coil)

Magnetostatic Pendulum 46 Regulation

equations of motion, Eq. (4–7) on page 34, yields the discretized, nonlinear equations of motion for the pendulum. (These equations have the same general , form as Eq. (2–23) on page 13.) When the currents I1 I2, and I3 are applied to the three electromagnetic coils (the two base coils, coil #1 and coil #2, and the pendulum coil, coil #3), the system’s equilibrium position is altered because of the magnetostatic potential energy of Eq. (4–11). Setting the velocities and accelerations to zero, ␣˙ ϭϭ␣˙˙ 0 and ␤˙ ϭϭ␤˙˙ 0, yields the equations governing the equilibrium positions for the system (equivalent to Eq. (2–24) on page 13). New generalized coordinates are introduced,

␩ ϭ ␣ Ϫ ␣ ␩ ϭ ␤ Ϫ ␤ 1()t ()t 0, 2()t ()t 0,

and the equations are expanded in a Taylor’s series about the new equilibrium ()␣ , ␤ position 0 0 . Retaining only the linear terms gives the linearized equations of perturbed motion for the system (equivalent to Eq. (2–26) on page 14).

ϭ ϭ As an example, setting the system currents to I1 0.85 A, I2 0A, and ϭ I3 0.75 A, the pendulum’s equilibrium position changes from

()␣ , ␤ ϭ ()Ϫ , grav grav 0.00226 0.00226 rad,

to the new equilibrium

()␣ , ␤ ϭ ()Ϫ , Ϫ 0 0 0.00184 0.0222 rad.

The linearized equations of motion reduce to

␩ ␩ 0.2205 0 ˙˙ 1()t ϩ 3.777 0.04402 1()t ␩ ␩ 0 0.2206 ˙˙ 2()t 0.04402 2.589 2()t  Ϫ0.001529 0.8341 0.05175 ␩ ()t ϩ ˜I ()t ϩ 1 (4–12) 1 Ϫ ␩ 0.08817 0.05175 0.5632 2()t

Ϫ Ϫ Ϫ ␩ ()t ϩ ˜I ()t 0.06095 ϩ 1.44 0.5438 1 ϭ 0 , 2 Ϫ Ϫ ␩ 0.01516 0.5438 0.4805 2()t 0

where the currents have been written as

˜ ϭ Ϫ ˜ ϭ Ϫ I1()t I1()t I10, , I2()t I2()t I20, ,

Magnetostatic Pendulum 47 Regulation

ϭ ϭ (remember that I10, 0.85 A and I20, 0A are the equilibrium currents ϭ above). Note that since the current I3 0.75 A is held constant it does not appear symbolically in Eq. (4–12).

Stability and Control. From Eq. (4–12) it is seen that using velocity feedback,

˜ ϭ ␩ ˜ ϭ Ϫ␩ I1()t ˙ 2, I2()t ˙ 1, (4–13)

will add a linear and a nonlinear term. Do not be confused that the subscripts of Eq. (4–13) seem to be switched; this is because base coil #1 creates an attractive force that decreases the pendulum’s ␤ coordinate (coil 1 is along the base’s X- axis) and base coil #2 creates an attractive force that increases the pendulum’s ␣ coordinate (coil 2 is along the base’s Y-axis). See Fig. 4–4 on page 27.

Substituting Eq. (4–13) into Eq. (4–12) gives

␩ Ϫ ␩ 0.2205 0 ˙˙ 1()t ϩ 0.06095 0.001529 ˙ 1()t ␩ ␩ 0 0.2206 ˙˙ 2()t 0.01516 0.08817 ˙ 2()t

Ϫ1.44 Ϫ0.5438 ␩ ()t ␩˙ ()t 0.8341 0.05175 ␩ ()t ␩˙ ()t Ϫ 1 1 ϩ 1 2 (4–14) Ϫ ␩ ␩ Ϫ ␩ ␩ 0.5438 0.4805 2()t ˙ 1()t 0.05175 0.5632 2()t ˙ 2()t

␩ ()t ϩ 3.777 0.04402 1 ϭ 0 . ␩ 0.04402 2.589 2()t 0

The new linear term has the effect of adding artificial damping to the system. The nonlinear terms complicate matters, but recalling Liapunov’s theorem on the stability in the first approximation (Meirovitch Analytical 227), the stability characteristics of the linearized system are the same as for the complete system.

Writing Eq. (4–14) and ignoring the nonlinear terms gives

␩ Ϫ ␩ 0.2205 0 ˙˙ 1()t ϩ 0.06095 0.001529 ˙ 1()t ␩ ␩ 0 0.2206 ˙˙ 2()t 0.01516 0.08817 ˙ 2()t (4–15) ␩ ()t ϩ 3.777 0.04402 1 ϭ 0 . ␩ 0.04402 2.589 2()t 0

Magnetostatic Pendulum 48 Regulation

Solving the eigenvalue problem for Eq. (4–15) gives

Ϫ dets2 0.2205 0 ϩϩs 0.06095 0.001529 3.777 0.04402 ϭ 0 0 0.2206 0.01516 0.08817 0.04402 2.589

s ϭ Ϫ 0.1986 Ϯ 3.419i, Ϫ 0.1395 Ϯ 4.137i,

which is stable since the real part of the eigenvalues is negative.

Experiment The effect of using the control law of Eq. (4–13) on the motion of the pendulum about two equilibrium points was examined. Implementing the control law of Eq. (4–13) is straightforward since the difference between two consecutive sen- sor readings is directly proportional to the time rate of change (i.e., the velocity) of the coordinate that the sensor measures. Thus, the computer was pro- grammed to take the difference between consecutive readings for each sensor, multiply these differences by gain values to arrive at the control currents, and then instruct the programmable power supply to change the base coils’ current.

˜ ˜ More explicitly, the computer calculates I1()t and I2()t ,

˜ ϭ ␥ ⌬ ˜ ϭ ␥ ⌬ I1()t 1 V0, I2()t 2 V1,

to adjust the currents to the base coils, I1()t and I2()t ,

ϭ ϩ ˜ ϭ ϩ ˜ I1()t I10, I1()t , I2()t I20, I2()t ,

␥ ␥ where 1 and 2 are the gains for the feedback. Note that because of the partic- ulars of the sensors, the gains must both be negative to create damping since

⌬ ϰ ␩ ⌬ ϰ Ϫ␩ V0 2, V1 1.

By setting I10, and I20, , new equilibrium positions are created for the pendu- lum. Regulation was performed about two equilibrium positions with varying (), ϭ (), (), ϭ (), gains: I10, I20, 0.3 0.7 A and I10, I20, 0.6 0.6 A.

Results As can be seen from Figs. 4–19 and 4–20 (on pages 51 and 52 respectively) the settling time of the system was controllable with appropriate gains. Settling time was defined as the time for the pendulum’s motion to decrease to 10% of its

Magnetostatic Pendulum 49 Regulation

Tab e 4–6. Pendulum’s approximate settling times for various gains Equilibrium for (0.3, 0.7)A Equilibrium for (0.6, 0.6)A (see Fig. 4–19 on page 51) (see Fig. 4–20 on page 52) ␣␤␣␤ ␥ ϭϭ␥ 1 2 0 16¼ s13½ s18½ s 20 s ␥ ϭϭ␥ Ϫ 1 2 2 12½ s 12½ s 13 s 14½ s

Gains ␥ ϭϭ␥ Ϫ 1 2 8 10 s 8¼ s6½ s 9 s ␥ ϭϭ␥ Ϫ 1 2 15 4¼ s 4¼ s 4 s 4¼ s

original amplitude. Table 4–6 lists approximate settling times about both equi- librium positions for various gains. Even though velocity feedback introduces more nonlinearities into an already nonlinear system, the linear terms dominate the motion and stability of the structure, as expected.

Note that in some plots there is “chatter,” especially the last plot of Fig. 4–20 (for ␥ ϭϭ␥ Ϫ 1 2 15). This is due to sensor fluctuations. There is inherent uncer- tainty in reading the position of the pendulum, and this error, if large enough, can cause the computer to “over control” the system—adjusting its position because of sensor errors, not because the pendulum is moving. Note that the equilibrium position of the system also drifts because of sensor instabilities. This is most apparent in the ␣ plots of Fig. 4–19 on page 51 where the final posi- tion of the pendulum changes from plot to plot.

Magnetostatic Pendulum 50 Regulation

g1 =g2 = 0

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-2

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-8

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-15

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

Figure 4–19. Regulation of pendulum motion with various gains at the equilib- rium created by setting (I1, I2) = (0.3, 0.7)A

Magnetostatic Pendulum 51 Regulation

g1 =g2 = 0

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-2

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-8

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

g1 =g2 =-15

a @radD b @radD 0.025 t @sD 5 10 15 20 0.02 -0.005 0.015 -0.01 t @sD -0.015 5 10 15 20 0.005 -0.02

Figure 4–20. Regulation of pendulum motion with various gains at the equilib- rium created by setting (I1, I2) = (0.6, 0.6)A

Magnetostatic Pendulum 52 Conclusions

This research developed the governing equations for magnetostatic structures. Using a Lagrangian mechanics approach the magnetostatic potential energy was derived, and then nonlinear, discretized, and linearized equations of motion were formulated. With these equations of motion four experiments were per- formed: one with a one-degree-of-freedom magnetostatic levitator, and three with a two-degree-of-freedom, spherical, magnetostatic pendulum.

The first experiment compared the measured static displacement of the magne- tostatic levitator with that predicted by both the exact nonlinear and also the approximate discretized equations of motion. The predicted displacement was within 4% of the experimental measurement.

The second experiment compared the natural frequencies of the magnetostatic pendulum with the predicted values for both angles of rotation. The nonlinear solutions gave predicted frequencies within 4% of those measured for cases of varying currents and initial displacements.

By controlling the currents in the electromagnets the pendulum was made to track a desired, arbitrary path for the third experiment. Despite limited sensor resolution, the pendulum’s motion was repeatable with approximately 4% error.

Finally, the linearized equations of motion were used to show that velocity feed- back can create a stable, damped system about the magnetostatic equilibrium points of the pendulum. Using this control law, the settling time of the pendu- lum’s motion was regulated when moved to each of two equilibrium positions.

53 This research described general methods to understand and control the behav- ior of magnetostatic structures. The experiments performed showed that the developed governing equations can be used to predict dynamic properties of, and formulate control laws for, quasi-magnetostatic structures.

Using these methods, future research could study the dynamics and control of more general structures. For example, the shape of even highly precise surfaces such as mirrors might be controllable. It is possible that the effects of gravity could be overcome, or that adaptive optics could be used for very large optical surfaces. Light and flexible surfaces for use in space as telecommunications antennas or for power beaming applications might be feasible.

The use of embedded electromagnets might provide a means of creating an entirely new class of customizable smart materials. No longer constrained to use materials that have inherent shape-changing properties, structural members with required physical characteristics could be embedded with the needed elec- tromagnetic circuits to provide the desired “smart” properties. Clearly there are many applications to the methods described herein.

Conclusions 54 Works Consulted

Bagryantsev, V. I. and Yu. V. Tyurin. “Dynamic Instability of a Track Structure.” Power Engineering—Journal of the Academy of Sciences of the USSR 22 (1984): 16–21.

Boas, Mary. Mathematical Methods in the Physical Sciences. 2nd ed. New York: Wiley, 1983.

Brauer, J. R., J. J. Ruehl, M. A. Juds, M. J. Vander-Heiden, and A. A. Arkadan. “Dynamic Stress in Magnetic Actuator Computed by Coupled Structural and Electromagnetic Finite Elements.” IEEE Transactions on Magnetics 32 (1996): 1046–9.

Cheng, David. Field and Wave Electromagnetics. 2nd ed. New York: Addison, 1989.

Di Gerlando, A. “Design Characterization of Active Magnetic Bearings for Rotating Machines.” International Conference on Electrical Machines in Aus- tralia Proceedings. Vol. 3. Adelaide: U of South Australia, 1993. 606–11.

Feynman, Richard, Robert Leighton, and Matthew Sands. The Feynman Lec- tures on Physics. 3 vols. New York: Addison, 1964.

Ginsberg, Jerry. Advanced Engineering Dynamics. 2nd ed. New York: Cambridge UP, 1995.

Grover, Frederick W. Inductance Calculations: Working Formulas and Tables. [New York]: Nostrand, 1946. New York: Dover, n.d. Research Triangle Park: Instrument Society of America, 1973.

Higuchi, Toshiro, Masahiro Tsuda, and Shigeki Fujiwara. “Magnetic Supported Intelligent Hand for Automated Precise Assembly.” IECON ’87: 1987 Interna- tional Conference on Industrial Electronics, Control, and Instrumentation. New York: IEEE, 1987. 926–33.

Jackson, John David. Classical Electrodynamics. 2nd ed. New York: Wiley, 1975.

55 Kojima, Hiroyuki, Kosuke Nagaya, Humihiko Niiyama, and Katsumi Nagai. “Vibration Control for a Beam Structure Using an Electromagnetic Damper with Velocity Feedback.” Bulletin of the Japan Society of Mechanical Engineers 29 (1986): 2653–9.

Kovacs, S. G. “A Magnetically Actuated Left Ventricular Assist Device.” Fron- tiers of Engineering and Computing in Health Care—Proceedings of the Fifth Annual Conference. New York: IEEE, 1983. 442–6.

Lang, J. H., and D. H. Staelin. “Electrostatically Figured Reflecting Membrane Antennas for Satellites.” IEEE Transactions on Automatic Control AC-27 (1982): 666–70.

Marion, Jerry, and Stephen Thornton. Classical Dynamics of Particles and Sys- tems. 4th ed. New York: Harcourt Brace, 1995.

Meirovitch, Leonard. Computational Methods in Structural Dynamics. Alphen aan den Rijn, Neth.: SijthoV, 1980.

– – –. Methods of Analytical Dynamics. New York: McGraw, 1970.

Mihora, D. H., and P. J. Redmond. “Electrostatically Formed Antennas.” General Research Corporation, Internal Memo 2222, Santa Barbara, CA. March 1979.

Nayfeh, Munir, and Morton Brussel. Electricity and Magnetism. New York: Wiley, 1985.

O’Malley, John. Schaum’s Outline: Basic Circuit Analysis. 2nd ed. New York: McGraw, 1992.

Rhim, W. K., M. Collender, M. T. Hyson, W. T. Simms, and D. D. Elleman. “Devel- opment of an Electrostatic Positioner for Space Material Processing.” Review of Scientific Instruments 56 (1985): 307–17.

Rhim, W. K., S. K. Chung, M. T. Hyson, E. H. Trinh, and D. D. Elleman. “Large Charged Drop Levitation Against Gravity.” IEEE Transactions on Inducstry Applications IA-23 (1987): 975–9.

Silverberg, Larry, and Leslie Weaver, Jr. “Dynamics and Control of Electrostatic Structures.” Transaction of the ASME—Journal of Applied Mechanics. 63 (1996): 383–91.

Smythe, William. Static and Dynamic Electricity. New York: McGraw, 1939.

Works Consulted 56 Streng, J. H. “Charge Movements on the Stretched Membrane in a Circular Elec- trostatic Push-Pull Loudspeaker.” Journal of the Audio Engineering Society 38 (1990): 331–8.

– – –. “Sound Radiation from Circular Stretched Membranes in Free Space.” Journal of the Audio Engineering Society 37 (1989): 107–18.

Udwadia, Firdaus, and Robert Kalaba. Analytical Dynamics: A New Approach. New York: Cambridge UP, 1996.

Wolfram, Stephen. The Mathematica Book. 3rd ed. New York: Cambridge UP, 1996.

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Works Consulted 57 Appendix: A Physics Primer for Magnetostatic Energy

This primer derives the energy of a magnetic field due to a system of steady elec- tric currents (a magnetostatic field) using Maxwell’s equations as the starting point. It is not an exhaustive derivation but is intended as a quick introduction to the equations and assumptions necessary to calculate the energy of a magne- tostatic field. Please note that parts of this treatment follow similar derivations as can be found in “Works Consulted for Appendix” on page 86.

Classical Physics

All classical physical phenomena can be described with the use of the equations listed in Table A–1. Table A–2 lists the symbols and units used in Table A–1.

Table A–1. Fundamental Equations of Classical Physics mv dp ϭ ------Law of Motion F ϭ p dt 1 Ϫ v2 ր c2 m m ϭ Ϫ 1 2 ()ϭ ր Law of Gravitational Force F G------er er r r r2

Law of Electromagnetic Force F ϭ q()EvBϩ ϫ

D ϭ ␳ ٌؒ B ϭ 0 ٌؒ Maxwell’s Equations ѨB ѨD ϫE ϭ Ϫ ٌϫH ϭ J ϩٌ Ѩt Ѩt

58 Electromagnetism

Table A–2. Quantities, Symbols, and Units used in Table A–1

Quantity Symbola Unitsb Force F N

2 Momentum p msи 2 Velocity v, v msր Position r, r m

Unit Position Vectorer — Time t s Mass m kg Electric Charge q C Electric Field Intensity E Vmր

2 Electric Flux Density D Cmր 3 Volume Charge Density (free charges) ␳ Cmր Magnetic Flux Density B T

Magnetic Field Intensity H Amր 2 Volume Current Density (free currents) J Amր Ϫ11 2 2 Gravitational Constant G Х 6.6726 ϫ10 Nmи ր kg 8 Velocity of Light c Х 3 ϫ10 msր a. Bold letters represent vectors and italic letters represent scalars. b. All units are Système International (SI)—also known as MKSA.

Electromagnetism

Electromagnetic phenomena are governed by the law of electromagnetic force,

F ϭ q()EvBϩ ϫ , (A–1)

and Maxwell’s equations given in Table A–1.

Conservation of Implicit in Maxwell’s equations is another fundamental tenant: the conservation Charge of electric charge. Taking the partial time derivative of ٌؒ D ϭ ␳ yields

Ѩ ѨD Ѩ␳ D ϭϭٌؒ ٌؒ() Ѩt Ѩt Ѩt

Appendix: A Physics Primer for Magnetostatic Energy 59 Electromagnetism

because the divergence operates only on spatial coordinates. Substituting this into the divergence of ٌϫH ϭ JDϩ Ѩ ր Ѩt gives

ѨD Ѩ␳ . ϫH ϭϭٌؒ J ϩ ٌؒ ٌؒ J ϩٌ() ٌؒ Ѩt Ѩt

Since the divergence of the curl of a vector is always zero, the result is the con- servation of electric charge:

Ѩ␳ (J ϭ Ϫ . (A–2 ٌؒ Ѩt

Eq. (A–2) relates the charge density at a point to the current density at that point. Applying the divergence theorem to the volume integral of Eq. (A–2) gives

Ѩ␳ ϭϭ͛ и Ϫ͐ ϭϪ d ͐␳ ϭϪ dQ ٌؒ()͐ J dv Jad Ѩ dv dv , V S V t dt V dt

or more simply

dQ ͛Jaи d ϭ Ϫ . S dt

This shows that the current leaving the volume VS through the surface is equal to the negative rate of change of the total charge in the volume V. Thus electric charge is conserved.

Maxwell’s Equations Maxwell’s equations can be simplified with the use of the electric and magnetic constitutive relations, which relate DE to and HB to . For homogeneous, iso- tropic, linear materials (i.e., not ferroelectric or ferromagnetic materials) at low electric fields,

ϭϭ⑀ ⑀ D 0KeE E,

⑀ ⑀ Х × Ϫ12 where 0 is the electric permittivity of free space ( 0 8.85 10 F/m), Ke is the relative permittivity or dielectric constant, and ⑀ is the permittivity of the material in which the electric field exists. For this derivation the material of con- ⑀⑀Х cern is air, which has a dielectric constant of approximately 1.00059, so 0 .

Appendix: A Physics Primer for Magnetostatic Energy 60 Induction

The magnetic field intensity becomes

ϭϭ1 1 H ␮------B ␮----B, 0Km

␮ ␮ ϵ ␲× Ϫ7 where 0 is the magnetic permeability of free space ( 0 4 10 H/m), Km is the relative permeability, and ␮ is the permeability of the material. The per- ␮␮Х meability of air is approximately the same as that of free space, so 0.

Using the constitutive relations, Maxwell’s equations simplify to

␳ ϭ ٌؒ E ----⑀ , (A–3) 0

ѨB (ϫE ϭ Ϫ , (A–4ٌ Ѩt

(B ϭ 0, (A–5 ٌؒ

ѨE (ϫB ϭ ␮ J ϩ ␮ ⑀ , (A–6ٌ 0 0 0Ѩt

Eq. (A–3) is the differential form of Gauss’ law; Eq. (A–4) is the differential form of Faraday’s law; Eq. (A–5) states there are no magnetic monopoles†; Eq. (A–6) is the differential form of Ampere’s circuital law as modified by Maxwell.

Induction

Faraday and Henry discovered independently that when the magnetic flux through a closed conducting circuit changes, a current is generated in the cir- cuit. This phenomenon is called induction and is the principle behind a wide array of technologies such as electric motors, electric generators, transformers,

† A magnetic monopole would be the magnetic equivalent of the electric charge, i.e., magnetic north poles (sources of magnetic field lines) would be free and separate from magnetic south poles (sinks of magnetic field lines).

Appendix: A Physics Primer for Magnetostatic Energy 61 Induction

B12

S2

C2

S1

I C1 1 Figure A–1. Magnetically coupled circuits

and radio. The magnetic flux through a circuit can change due to a magnetic field surrounding a fixed circuit changing its strength, the circuit moving or changing its shape in a region of space where there is a steady magnetic field, or a combination of these two.

First flux will be defined, then the induction forces that cause the electrons in the circuit to move (producing a current) will be examined, from which Fara- day’s law of induction will emerge.

Flux Flux is a simple mathematical idea: if a vector field v represents the velocity of a fluid, then the flux of v through a surface is the volume of fluid that passes through that surface per unit time. Flux is a more general concept than this example, however, and applies to any vector field, not just velocities. The flux of a magnetic field through a circuit, as depicted in Fig. A–1, is written as

Ᏺ ϭ ͐ и 12 B12 da2, (A–7) S2

Ᏺ where 12 is the flux from circuit 1 through circuit 2, S2 is the surface bounded

by circuit C2, B12 is the magnetic flux density at circuit 2 due to the current in

circuit 1, and da2 is the differential unit normal to surface S2. The sign for the flux is chosen so it is positive when in the direction of the normal of the surface.

Appendix: A Physics Primer for Magnetostatic Energy 62 Induction

ϩ Ct()d t ϭ ϩ S2 St()d t

ϭ S1 St()

2 d a dr Ct() dᐍ Figure A–2. Conducting filament in motion

Electromotive Force A current in a circuit results from the motion of the free charges through the wire of the circuit. There must be some push on the charges to start and keep them in motion along the wire. As the charges move they will also be pushed by atoms in the wire and by one another. What is important, however, is the net push the charges receive around the entire circuit—i.e., the tangential force along the wire per unit charge integrated around the entire length of the circuit. This net push around the circuit is called the electromotive force or emf.

What Faraday and Henry discovered was that an electromotive force can be gen- erated, or induced, in a circuit by changing the magnetic flux through the cir- cuit. The flux through a circuit can change as a result of two separate phenomena as stated above: the circuit moving through or changing its shape in a region of space where there exists a magnetic field, or the magnetic field that surrounds the circuit changing its strength.

Motional EMF. First the case of a circuit that changes its position, shape, size, or orientation in a static field will be considered. Suppose there is a circuit pro- ducing a time-constant magnetic field, B, that is stationary in the reference frame of the observer. Now consider another circuit defined by the closed curve C moving through the static field B. The curve has the shape Ct() at time t, and changes to the shape Ct()ϩ d t at time ttϩ d as shown in Fig. A–2.

Appendix: A Physics Primer for Magnetostatic Energy 63 Induction

In general, the force on a particular free charge in the circuit is given by Eq. (A– 1). Any electric field present must result from a charge distribution ␳ as shown in Eq. (A–3), and not from Eq. (A–4) since only a static magnetic field is being considered (time-varying fields are considered in the next section). It is well- known from the study of electrostatics that such a distribution will produce an electric field such that E ϭ⌽Ϫٌ⌽, where is the scalar potential field for the distribution. By definition, then, this electric field is conservative. The emf pro- duced by this field is just

Ᏹ ϭ ͛ E и dᐍ, Ct() since EFϭ ր q is the force per unit charge on each free charge in the circuit. But because this field is conservative, it must be that the emf produced is zero since the the circuit starts and ends at the same point:

.Ᏹ ϭϭ͛ E и dᐍ Ϫ⌽͛ ٌ⌽ и dᐍ ϭϭϪ ͛ d 0 Ct() Ct() Ct()

The electromotive force that the free charges feel, therefore, must come from their motion through the magnetic field. This motion produces a force on the charges, F ϭ q()vBϫ , known as the Lorentz force, which pushes them around the circuit. (Note that it is assumed that vcӶ .) This generates the emf

Ᏹ()t ϭ ͛ ()vBϫ и dᐍ, Ct() where v is the velocity of the element dᐍ. The scalar triple product vBϫ и dᐍ ϭ dᐍ и v ϫ B ϭ ()dᐍ ϫ v и B, so

Ᏹ()t ϭ ͛ ()dᐍ ϫ v и B. Ct()

The displacement of the filament in time dt is dr, so vrϭ dtր d . This gives

Ᏹ ϭ ᐍ ϫ dr и ()t ͛ d ----- B. Ct() dt

Appendix: A Physics Primer for Magnetostatic Energy 64 Induction

The area swept out in time dt by the element ddᐍ is 2a ϭ dᐍ ϫ dr, and B и d2a is just the flux through this area (see Fig. A–2 on page 63). Therefore

1 2 1 Ᏹ()t ϭϭ----- ͛ ()d a и B -----()ϪdᏲ , dt Ct() dt where ϪdᏲ is the flux of B through ͛d2a, the total area swept out by C in time dt. Note that the normal of the surface has been chosen as dr ϫ dᐍ, so that the flux will be positive when it is in the direction of Ϫd2a ϭ dr ϫ dᐍ. With this choice of sign for the flux, the induced electromotive force becomes

dᏲ Ᏹ ϭ Ϫ . dt

As a check it will be shown that dᏲ really is the change in flux through the loop Ct between times and ttϩ d . The flux passing through the surface composed ϩ ͛ 2 ͛ и of St(), St()d t, and the “sides” d a is zero at all times because S Bad ϭ 0. † But this integral can also be written as

0 ϭϭ͛Baи d ͐ Baи d Ϫ ͐ Baи d Ϫ dᏲ. S St()ϩ d t St() The sign for the second term is required because of the sense chosen for Ct(). The third term, ϪdᏲ, is just the flux through the “sides” as shown above. Thus

0 ϭ Ᏺ()ttϩ d Ϫ Ᏺ()t Ϫ dᏲ, or

dᏲ ϭ Ᏺ()ttϩ d Ϫ Ᏺ()t .

It is clear that dCᏲ really is the change in flux through the circuit between times ttt and ϩ d .

Induced EMF. Again consider a circuit producing a magnetic field B, but one that is now fixed in the reference frame of the observer. If the magnetic field changes with time, a changing electric field will be produced according to

† This is seen by taking the volume integral of Eq. (A–5) and using the divergence theo- .rem: ͛ Baи d ϭϭϭ͐ ()ٌؒ B dv ͐ ()0 dv 0 S V V

Appendix: A Physics Primer for Magnetostatic Energy 65 Induction

Eq. (A–4). Integrating Eq. (A–4) over the surface S bounded by another nearby, fixed circuit C gives

Ѩ ϫ и ϭ Ϫ͐ B иٌ͐ E da Ѩ da. S S t

Using Stoke’s theorem, the left hand side of this equation can be rewritten:

Ѩ ͛ и ᐍ ϭ Ϫ͐ B и E d Ѩ da. C S t

Since the curve CS and the surface are fixed in space, neither of them depend upon t so this becomes

Ѩ ͛ и ᐍ ϭ Ϫ ͐ и E d Ѩ Bad . C t S

The right hand side is, by Eq. (A–7), the negative of the flux through the circuit. The left hand side is just the tangential push per unit charge integrated around the entire circuit—in other words, the electromotive force Ᏹ. So the above equation reduces to

dᏲ Ᏹ ϭ Ϫ . dt

Faraday’s Law As seen above, two separate phenomena produce the same results: When a cir- cuit moves through a magnetic field or a stationary circuit is subjected to a time- varying magnetic field, an electromotive force is produced. These two separate phenomena are contained in what is known as Faraday’s law of induction:

dᏲ Ᏹ ϭ Ϫ . (A–8) dt

In words, the induced electromotive force around a circuit is equal to the nega- tive time rate of change of the flux through the surface bounded by the circuit. Note that the induced emf is such that it sets up flux that opposes the change: if the flux through a circuit in a certain direction is increasing, the induced current

Appendix: A Physics Primer for Magnetostatic Energy 66 Filamentary Approximation

I A

J

dᐍ Figure A–3. Wire approx- imated as a filamentary current

sets up flux in the opposite direction, and vice versa. It can be shown that Eq. (A–8) is equivalent to the differential form given in Eq. (A–4).

This result is quite amazing. There is no other single principle in physics that requires the understanding of two different phenomena to interpret its results. In all other cases such a general result stems from a single unifying principle.

Filamentary Approximation

In many cases the magnetic field of interest is produced from circuits of wire in which the diameter of the wire is very small compared to the dimensions of the circuit as a whole. For such systems the filamentary approximation can be used to simplify the equations for the magnetic flux density.

For thin wire dAv ϭ dᐉ, where A is the cross-sectional area of the wire and dᐉ is a differential length of the wire (see Fig. A–3). Since for very thin wire the current flowing would also be in the same direction as a differential length, Jdv ϭ JAdᐍ. Finally, because J is the amount of charge flowing through a unit sur- face area per unit time, JA is just the total charge flowing through the surface area of the wire per unit time, i.e., JAϭ I. So, for filamentary currents

JdIv ϭ dᐍ. (A–9)

Appendix: A Physics Primer for Magnetostatic Energy 67 Magnetostatics

Magnetostatics

As can be seen from Maxwell’s equations, Eqs. (A–3)–(A–6), the electric and magnetic fields are generally coupled. When all electric charge densities and all current densities are constant the electric and magnetic fields created are also constant. Because the fields do not change with time, all terms involving ѨEրѨt and ѨBրѨt are zero and disappear from Maxwell’s equations. The fields are then said to be static; a static electric field is called an electrostatic field, and a static magnetic field is called a magnetostatic field. In such cases the electric and mag- netic fields are decoupled, greatly simplifying analysis. Studying the magneto- static field requires considering only the magnetic flux density, B, which is completely defined by its curl and divergence.

The two fundamental postulates of magnetostatics specify the divergence and curl of the magnetostatic field. The first postulate is just Eq. (A–5),

(B ϭ 0. (A–10 ٌؒ

The second postulate comes from Eq. (A–6). For constant currents this equa- tion reduces to the differential form of Ampere’s law:

ϫϭ ␮ٌ B 0J. (A–11)

In fact, Eq. (A–11) is valid not only for static magnetic fields but also for quasi- static fields. It is sufficient for currents to vary slowly with time and that the dimensions of the circuits carrying the currents be very small in comparison to the wavelength of the electromagnetic radiation produced. For example, an alternating current with a frequency of 60Hz produces a 5,000km wavelength electromagnetic wave; a 1MHz current produces a 300m wavelength wave. These assumptions in effect ignore the finite speed (the speed of light) of the propagation of the changes in the electric and magnetic fields.

These postulates can also be written in integral form. Integrating Eq. (A–10) over a volume and using the divergence theorem gives Gauss’ law of magnetism:

Appendix: A Physics Primer for Magnetostatic Energy 68 Magnetostatics

͛Baи d ϭ 0. (A–12) S

The second integral equation is obtained from Eq. (A–11) by taking the scalar surface integral and using Stoke’s theorem to arrive at

ϫ и ϭϭ͛ и ᐍ ␮ ͐ иٌ()͐ B da B d 0 Jad . S C S

But the last term above is just the flux of current through the surface. Thus the integral from of Ampere’s circuital law is

͛ и ᐍ ϭ ␮ B d 0I. (A–13) C

Magnetic Vector The divergence free postulate of Eq. (A–10) means that B can be expressed as Potential the curl of another vector field. This new field is the magnetic vector potential:

(BAϭ ٌϫ . (A–14

The definition of a vector field requires specifying not only its curl but also its divergence. The choice of divergence for A is called a gauge and is usually cho- sen to simplify the mathematics of whatever is being considered.

Substituting Eq. (A–14) into Eq. (A–11) yields

ϫ()ٌϫ ϭ ␮ٌ A 0J.

Recalling the vector identity ٌϫ()ٌϫA ϭ ٌ()ٌؒ A Ϫ ٌ2A gives

Ϫ␮ٌ2 ϭ ٌؒ()ٌ A A 0J.

The divergence of A is now chosen to simplify this further. The most obvious choice is the Coulomb gauge:

.A ϭ 0 ٌؒ

Applying Coulomb’s gauge results in a vector Poisson’s equation for A,

Appendix: A Physics Primer for Magnetostatic Energy 69 Magnetostatics

ϭ Ϫ␮ 2ٌ A 0J.

In Cartesian coordinates it is equivalent to the three scalar Poisson’s equations

ϭ Ϫ␮ ٌ2 ϭ Ϫ␮ ٌ2 ϭ Ϫ␮ 2ٌ Ax 0Jx, Ay 0Jy, Az 0Jz.

These are each the same as Poisson’s equation in electrostatics for free space:

␳ 2⌽ ϭ Ϫ ()rٌ ()r ------⑀ , 0

which has a particular solution of

␳ Ј ⌽ ϭ 1 ͐ ()r Ј ()r ------␲⑀ ------Ϫ Ј-dv , 4 0 VЈ rr

where the voltage ⌽ is measured at point r, and the integration is over the vol- ume of the source charge distribution, V.Ј

Thus, the solution for the magnetic vector potential is

␮ Ј ϭ 0 ͐ Jr() Ј Ar() ------␲------Ϫ Ј-dv . (A–15) 4 VЈ rr

For filamentary currents, Eq. (A–15) reduces to

␮ Ј ϭ 0 ͐ I()r ᐍЈ Ar() ------␲------Ϫ Ј-d . (A–16) 4 CЈ rr

The geometry for Eqs. (A–15) and (A–16) can be seen in Fig. A–4.

Biot-Savart Law In the study of electrostatics there exists an equation that gives the electric field intensity based upon a given charge distribution:

()Ϫ Ј ϭ 1 ͐␳ Ј rr Ј Er() ------␲⑀ ()r ------Ϫ Ј-dv . 4 0 VЈ rr

There is a similar expression for the magnetic flux density written in terms of current distributions. Substituting Eq. (A–15) into Eq. (A–14) gives

Appendix: A Physics Primer for Magnetostatic Energy 70 Magnetostatics

I field point

()rrϪ Ј

r dᐍ source point rЈ O Figure A–4. Magnetic field from a wire

␮ Ј ϭϭٌϫ ٌϫ 0 ͐ Jr() Ј Br() Ar() ------␲------Ϫ Ј-dv . 4 VЈ rr

It is important to note that the curl operation means differentiation with respect to the space coordinates of the field point, i.e., the value of Ar at the point . The integral operates on the coordinates of the source points, i.e., the points rЈ. The curl, then, operates on the integrand Jr()Ј ր rrϪ Ј . Using the vector identity ϫ()fA ϭ ٌf ϫ A ϩ fٌϫA, the curl operation can be written asٌ

Jr()Ј 1 1 . ϫ------ϭ ٌ------ϫ Jr()Ј ϩ ------ٌϫJr()Јٌ rrϪ Ј rrϪ Ј rrϪ Ј

The primed and unprimed coordinates are independent, however, so the second term is zero. The divergence of the first term gives

1 Ѩ 1 Ѩ 1 Ѩ 1 ------ϭ e ------ϩϩe ------e ------ٌ rrϪ Ј xѨxrrϪ Ј yѨyrrϪ Ј zѨzrrϪ Ј

e ()xxϪ Ј ϩϩe ()yyϪ Ј e ()zzϪ Ј ϭ Ϫ ------x y z - 32ր []()xxϪ Ј 2 ϩϩ()yyϪ Ј 2 ()zzϪ Ј 2 ()rrϪ Ј ϭ Ϫ ------. rrϪ Ј 3

The integrand of the magnetic flux density given above is then

Jr()Ј ()rrϪ Ј ()rrϪ Ј , ------ϫ------ϭϭϪ ------ϫ Jr()Ј Jr()Ј ϫٌ Ϫ Ј 3 3 rr rrϪ Ј rrϪ Ј

Appendix: A Physics Primer for Magnetostatic Energy 71 Energy of Circuits

and the magnetic flux density itself is just

␮ 0 Jr()Ј ϫ ()rrϪ Ј Br()ϭ ------͐------dvЈ. (A–17) ␲ 3 4 VЈ rrϪ Ј

In the case of filamentary currents, Eq. (A–17) can be written as

␮ 0 dᐍЈ ϫ ()rrϪ Ј Br()ϭ ------͐I()rЈ ------. (A–18) ␲ 3 4 CЈ rrϪ Ј

Eqs. (A–17) and (A–18) are know as Biot-Savart laws. The geometry is shown in Fig. A–4.

Energy of Circuits

This rather long section is the objective of the primer. First the power required to produce and maintain currents is examined. Then the work required to main- tain constant currents and the work required to move circuits relative to one another are derived. With this knowledge it is possible to derive the energy con- tained in a magnetic field and thus compute the forces between circuits. Finally, a method of writing the field energy using induction coefficients is given.

Power in Circuits Source of Voltage. Current flowing through a circuit requires a source of energy to put the free charges of the wire into motion. The energy source accomplishes this by creating a voltage difference across its terminals. When a circuit is attached to these terminals, the voltage difference that the free charges of the wire experience impels them to move around the circuit. The energy source acts in many ways as a pump to move the charges.

But how does this potential difference move the charges? From electrostatics it is known that the electric field can be defined as the negative gradient of a potential

,E ϭ Ϫٌ⌽

Appendix: A Physics Primer for Magnetostatic Energy 72 Energy of Circuits where ⌽ is the electric potential and is measured in volts. The electric potential is related to the work required to move a charge from one point to another since

2 2 2 2 .W ϭϭ͐F и dᐍ q͐E и dᐍ ϭϪqٌ͐⌽ и dᐍ ϭϪq͐d⌽ ϭϪq⌽ 1 1 1 1

A static electric field, however, is conservative, so the following must be true:

͛E и dᐍ ϭ 0. C

This would seem to be a contradiction since there needs to be a potential differ- ence around the loop for a current to exist. The answer lies in the source men- tioned above.

The source of potential is typically a battery or generator, inside of which chem- ical or mechanical forces create a separation of positive and negative charges. This creates a non-conservative† electric field inside the source supplying the potential needed to move the free charges. Inside the source positive charge accumulates at the positive (+) terminal and negative charge accumulates at the negative (–) terminal. This potential difference creates an electric field, E, inside the source which generates an electromotive force (emf) defined as

+ Ᏹ ϭ Ϫ͐E и dᐍ. – inside the source

The electric field exists both inside and outside the source, however, so this emf will also create a current in an attached circuit, much the same as an emf induced by changing magnetic flux through the circuit creates a current. Inte- grating around the complete circuit, the emf is just zero:

– + ͛E и dᐍ ϭϭ͐E и dᐍ ϩ ͐E и dᐍ 0. C + – outside the source inside the source

The emf outside the source, then, is just the potential between the terminals

† At least in the sense that here the energy used to separate the charges is not included in the calculation of the energy of the system.

Appendix: A Physics Primer for Magnetostatic Energy 73 Energy of Circuits

– Ᏹ ϭϭ͐E и dᐍ ⌽. + outside the source

Thus, the source introduces an emf into the connected circuit despite that the source is non-conservative. It is the voltage difference of the source’s terminals that pushes the free charges around the circuit. (Obviously, there may also be a current due to induction effects.)

Power in One Circuit. The rate at which the source provides energy to the free charges in the wire is Fvи , where Fv is the force on each charge and is the charge’s velocity through the wire. If there are n free charges per unit length moving through the wire, then the power delivered to an element of length dᐉ is

Fvи ndᐉ ϭ nvF и dᐍ, since for a thin wire v is in the same direction as dᐍ (i.e., vdvᐉ ϭ dᐍ). The total power delivered to the complete circuit is then

power delivered to chargesϭ ͛ nvF и dᐍ. C

ϭ ͛ ()ր и ᐍ ϭϭ͛ и ᐍ Ᏹ Remembering that qnv I and C F q d C E d , this is just

dW power delivered to charges ϭϭ------ᏱI. dt

Power in Two Circuits. Now consider two circuits, each with its own current. Circuit 1, the “loop,” is stationary and circuit 2, the “coil,” moves from very far away at Ϫϱ to nearby the loop (as shown in Fig. A–5). As the coil moves into the region of stronger magnetic flux density, the changing flux will induce an emf around the coil. This induced emf will either add to or take away from the power delivered by the coil’s source to the free charges in the coil—changing their velocities and hence the current. Including the emf induced in the coil (circuit Ᏹ 2) by the loop (circuit 1), 12, and using Faraday’s law, Eq. (A–8), the power equation becomes

Appendix: A Physics Primer for Magnetostatic Energy 74 Energy of Circuits

I1

Ᏹ B1 1 v

Ᏹ 2 B2

I2 Figure A–5. Coil moving toward a loop

dW power delivered to charges in coil ϭϭ------2 Ᏹ I ϩ Ᏹ I dt 2 2 12 2 dᏲ ϭ Ᏹ I Ϫ I ------12-. 2 2 2 dt

Likewise, the power equation for the loop’s source is

dW power delivered to charges in loop ϭϭ------1 Ᏹ I ϩ Ᏹ I dt 1 1 21 1 dᏲ ϭ Ᏹ I Ϫ I ------21-. 1 1 1 dt

For a system of just two circuits, the total power to all the free charges is

dW dW power delivered to all charges ϭ ------1 ϩ ------2 dt dt dᏲ dᏲ ϭ Ᏹ I ϩ Ᏹ I Ϫ I ------21- Ϫ I ------12-. 1 1 2 2 1 dt 2 dt

Substituting the magnetic vector potential solution for filamentary currents, Eq. (A–16), into the definition of flux, Eq. (A–7), and using Stoke’s theorem gives the flux through circuit 1 caused by the magnetic field from circuit 2

Ᏺ ϭ ͐ и 21 B21 da1 S1

ϭ ٌ͐ϫ и A21 da1 S1

Appendix: A Physics Primer for Magnetostatic Energy 75 Energy of Circuits

ϭ ͛ и ᐍ A21 d 1 C1 ␮ dᐍ ϭ ͛------0-I ͛------2- и dᐍ 4␲ 2 R 1 C1 C2 21 ␮ dᐍ и dᐍ ϭ ------0-I ͛͛------2 1, 4␲ 2 R C1 C2 21

ϭ Ϫ where R21 r1 r2 . (It is important to realize that the notation used here, Ᏺ 21, B21, and A21, should be read as “the effect caused by #2 at position #1.” For

example, B21, means the magnetic field created by a current in circuit 2 as mea- sured at the position of circuit 1, or “the field created by 2 at position 1.” For the

distance R21, it is best to read it as “the magnitude of the vector from position 2 to position 1.”)

Likewise, the flux created by circuit 1 through circuit 2 is just

␮ dᐍ и dᐍ Ᏺ ϭ ------0-I ͛͛------1 2, 12 4␲ 1 R C2 C1 12

so the change in power from each source due to the changing fluxes is equal:

dᏲ dᏲ I ------21- ϭ I ------12-. 1 dt 2 dt

Therefore the total power delivered to all charges in the system by the emf sources is:

dW dᏲ power delivered to all charges ϭϭ------Ᏹ I ϩ Ᏹ I Ϫ 2I ------21-. dt 1 1 2 2 1 dt

Work Done by Circuits Work to Maintain Constant Currents. Continuing with the circuits of the previous section: for the currents in the system as a whole to remain constant, the power delivered to all charges must also remain constant. Otherwise, the charges would gain or lose speed from their change in energy, and thus the cur- rents would change. Studying the equation above it is clear that for this power Ᏹ Ᏹ to be held constant, the source emfs, 1 and 2, must increase or decrease to

Appendix: A Physics Primer for Magnetostatic Energy 76 Energy of Circuits balance the induced emfs. The last term of the equation represents the excess power needed by the combined sources to keep the currents constant:

dW dᏲ power needed to maintain constant currents ϭϭ------const. currents 2I ------21-. dt 1 dt

If the motion of the coil near the loop takes place over a period of time t, then the work required of the sources to hold the currents steady during that period is

ϭ Ᏺ Wconst. currents 2I1 21.

Ᏺ Substituting the double-line integral expression above for the flux 21 yields

␮ ᐍ и ᐍ 0 d 1 d 2 W ϭ 2------I I ͛͛------. (A–19) const. currents 4␲ 1 2 R C1 C2 12

Work to Move a Circuit. When a wire carrying a current is in a magnetic field, a force is exerted on the wire according to Eq. (A–1). If there are N free charges per unit volume of the wire, then the force on a small volume of the wire is

⌬F ϭ NqvBϫ ⌬V.

But recall that NqvJ is just the current volume density, , so

⌬F ϭ JBϫ ⌬V.

The force per unit volume is then just JBϫ . Recall that for a thin wire with a current that is uniform over its cross-sectional area IJϭ A, so

I ⌬F ϭ --- ϫ BAL⌬ , A where ⌬L is the length of the volume element. The force on the element is then

⌬F ϭ I⌬L ϫ B.

But for a very small length of wire (and if the wire is thin), I⌬L ϭ I⌬L. So for a differential length of wire, the force from the magnetic field on the current is

dIF ϭ dᐍ ϫ B.

Appendix: A Physics Primer for Magnetostatic Energy 77 Energy of Circuits

Now, if circuit 1, C1, is in the magnetic field created by circuit 2, B21, then the complete force on circuit 1 is

ϭ ͛ ᐍ ϫ F21 I1 d 1 B21. C1

(Again, this is read as the force on circuit 1 by the field from circuit 2.) Substitut- ing the Biot-Savart law, Eq. (A–18), yields Ampere’s law of force between two cur- rent-carrying circuits:

␮ ᐍ ϫ ()ᐍ ϫ 0 d 1 d 2 R21 F ϭ ------I I ͛͛------, (A–20) 21 4␲ 1 2 3 C1 C2 R21

ϭ ()Ϫ ᐍ ᐍ where R21 r1 r2 is the vector from d 2 to d 1. The force between the circuits can be simplified by expanding the triple vector product

ᐍ ϫ ()ᐍ ϫ ϭ ᐍ ()ᐍ и Ϫ ()ᐍ и ᐍ d 1 d 2 R21 d 2 d 1 R21 R21 d 1 d 2 , so that the double closed line integral can be broken into two parts. The first is

dᐍ ()dᐍ и R dᐍ и R ͛͛------2 1 21- ϭ ͛dᐍ и ͛------1 21 3 2 3 C1 C2 R21 C1 C2 R21

1 ------ϭ ͛dᐍ ͛dᐍ и Ϫٌ 2 1 R C1 C2 21

1 ϭ ͛dᐍ ͛d ------2 R C1 C2 21 ϭ 0,

ϭ ᐍ и ٌ ٌ()ր ϭ Ϫ ր 3 where the relations dV dV and 1 R21 R21 R21 have been used. Since the closed-line integral (with identical upper and lower bounds) of ()ր d 1 R21 vanishes:

␮ R F ϭ Ϫ------0-I I ͛͛ ------21-()dᐍ и dᐍ . 21 4␲ 1 2 3 1 2 C1 C2 R21

The negative sign indicates a force of attraction.

Appendix: A Physics Primer for Magnetostatic Energy 78 Energy of Circuits

So, finally, the work done by the magnetic forces in moving circuit 1 from Ϫϱ

to the position R21 near circuit 2 during the time t is

R21 ␮ ᐍ и ᐍ 0 d 1 d 2 W ϭϭ͐ F и dR ------I I ͛͛------. (A–21) mech 21 21 4␲ 1 2 R Ϫϱ C1 C2 21

Energy and Virtual As seen in the previous section, circuits carrying currents produce forces on Work each other, and these forces do work in moving the circuits. It was also shown that work must be done against the induced electromotive forces in order to maintain steady currents in the circuits as they move toward one another. Con- sider now that both circuits are held rigidly in place. Let one of the circuits make a rigid virtual displacement while the currents in both circuits are held constant by its source. Because the displacement is virtual there is no loss of energy due to heating or cooling of the circuits. As a result of the virtual displacement, mechanical work is done by the magnetic forces, and electrical work is done by the sources against the induced emf to maintain the constant currents. Com- paring the mechanical and electrical work, Eqs. (A–19) and (A–21), from the previous section shows that only half of the work performed by the sources is used by the system to perform mechanical work. That is,

ϭ () dWconst. currents 2 dWmech .

Because the only difference between the initial and final states of the two circuits is the magnetic field surrounding them, the remainder of the energy must be in the magnetic field:

ϭ ϩ dWWconst. currents dUmech d field.

The change in field energy, then, must equal the amount of mechanical work done. Thus, when two constant-current circuits are moved relative to each other, the mechanical work done and the energy of the magnetic field increase or decrease together and at the same rate:

ϭ dUfield dWmech. (A–22)

Appendix: A Physics Primer for Magnetostatic Energy 79 Energy of Circuits

The mechanical work can be written as a magnetic force F acting on the circuit ϭ и in question as dWmech Frd . Therefore

ϭϭϭи ()ٌ и dWUfield d mech Frd Ufield dr,

and

ٌ ϭ F Ufield. (A–23)

So the mechanical force (or torque) trying to increase any coordinate of a partic- ular circuit can be found by taking the positive partial derivative of the field energy with respect to that coordinate:

ѨU ϭ field F␪ ------Ѩ␪ .

Energy of a Magnetic The previous section showed that the energy of the magnetic field for a system Field of circuits with steady currents is equal to the mechanical work done to bring the circuits into proximity. But what about the energy of a single circuit?

Consider building up a single circuit carrying a steady current by bringing together many infinitesimal current filaments. In this way Eq. (A–21) can be used to find the energy of a circuit. If the final current density is nowhere infinite, the denominator of this equation causes no problems because all the filaments carry a finite amount of current and the filaments themselves remain a finite distance apart. Realize, however, that when using Eq. (A–21) for a single circuit a factor of ½ must be included. This is because Eq. (A–21) gives the energy for a pair of circuits. The total energy of one circuit requires the sum of all such pairs. Instead of keeping track of all the pairs, the integral is the complete sum over all the filaments—counting the energy for each pair twice.

By writing IdJAᐍ ϭ dᐍ ϭ JAdᐉ ϭ Jdv, Eq. (A–21) becomes

1 ␮ и Ј Ј ϭ -- и 0 ͐͐Jdv J dv Ufield ------␲------Ϫ Ј , (A–24) 2 4 V VЈ rr

Appendix: A Physics Primer for Magnetostatic Energy 80 Energy of Circuits where rrϪ Ј is the distance between the volume elements dvv and d Ј, and J and JЈ are the current densities in these elements. The integration is performed twice throughout the space where the currents exist.

Substituting Eq. (A–15) into Eq. (A–24) gives

ϭ 1͐ и Ufield -- JAdv. (A–25) 2 V

The volume V may be extended to include all of space, since including regions where J ϭ 0 will not change the value of the integral. Substituting Eq. (A–11), ϫϭ ␮ٌ B 0J, into this equation gives

ϭ 1 ٌ͐ϫ и Ufield ------␮ B Adv. 2 0 V

Using the identity ABи ٌϫ ϭ BAи ٌϫ Ϫ ٌؒ ()ABϫ this simplifies to

ϭ 1 ͐ и Ϫ 1 ٌ͐ؒ ()ϫ Ufield ------␮ BBdv ------␮ ABdv. 2 0 V 2 0 V

By the divergence theorem, this can be written as

ϭ 1 ͐ и Ϫ 1 ͛()ϫ и Ufield ------␮ BBdv ------␮ ABda, 2 0 V 2 0 S where the surface SV encloses the volume . If the currents are finite in extent then the volume VS and hence the surface can be taken to be very large so that all points on SS are great distances from the currents. At the surface of the con- tribution from the surface integral will tend toward zero because A falls off as 1 ր R and B falls off as 1 ր R2 as seen in Eqs. (A–15) and (A–17). Therefore, the magnitude of ABϫ falls off at the rate of 1 ր R3, whereas the surface area of S is increasing at the same time at the rate of R2. So as R approaches infinity, the surface integral vanishes. This leaves

ϭ 1 ͐ и Ufield ------␮ BBdv. (A–26) 2 0 V

Appendix: A Physics Primer for Magnetostatic Energy 81 Energy of Circuits

It is interesting to note that while this equation was derived under the assump- tion of steady electric currents, it can in fact be shown to be true for dynamic fields as well (but this is left as an exercise for the reader).

Eq. (A–26) for the energy of a magnetic field is analogous to that for the energy of an electric field (which is also valid for time-varying fields):

⑀ ϭ 0͐ и Ufield ---- EEdv. 2 V

Field Energy and Mutual Induction. Consider again the two circuits of Fig. A–1 on page 62. A Induction Coefficients current I1 flowing around the circumference C1 creates a magnetic field. Some

of the flux due to this magnetic field will pass through the surface S2 that is

bounded by C2. The amount of flux linkage with circuit C2 due to a unit current

in the circuit C1 is called the mutual inductance. Using the definition of mag- netic flux, Eq. (A–7), the mutual inductance between circuit 1 and 2 is written as

1 1 ᏹ ϭϭ----Ᏺ ----͐B и da . (A–27) 12 I 12 I 12 2 1 1 S2

Using the vector potential definition, Eq. (A–14), and Stoke’s theorem, yields

1 1 . ᏹ ϭϭ----͐()ٌϫA и da ----͛A и dᐍ 12 I 12 2 I 12 2 1 S2 1 C2

Substituting the filamentary solution of the vector potential, Eq. (A–16), yields the Neumann formula for mutual inductance (Fig. A–6 shows the geometry):

␮ ᐍ и ᐍ 0 d 1 d 2 ᏹ ϭ ------͛͛------, (A–28) 12 4␲ R C1 C2 12

ϭ Ϫ where R12 r2 r1 , the distance between the differential length elements of the two circuits. Eq. (A–28) shows that the mutual inductance between circuits is a purely geometrical relationship. Also, it is clear from this equation that the ᏹ mutual inductance between circuits 1 and 2, 12, is the same as that between ᏹ ᏹ ϭ ᏹ circuits 2 and 1, 21, i.e., 12 21.

Appendix: A Physics Primer for Magnetostatic Energy 82 Energy of Circuits

I1 C1 C2

I2 ᐍ d 1 ᐍ d 2 R12 Figure A–6. Mutual inductance geometry

The mutual inductance can also be written in terms of the magnetic fields cre- ated by the two circuits. Take the mutual inductance given above in terms of the ր vector potential, multiply by I2 I2, and then use the filamentary approximation of Eq. (A–9), Idᐍ ϭ Jdv, to get

1 1 ᏹ ϭϭ------͛A и I dᐍ ------͐A и J dv . 12 I I 12 2 2 I I 12 2 2 1 2 C2 1 2 V2

By the same technique used to transform Eq. (A–25) into Eq. (A–26) this is

ᏹ ϭ 1 ͐ и 12 ------␮- B1 B2dv, (A–29) I1I2 0 V where the notation for the fields has been simplified. Take particular note that the volume integral of Eq. (A–29) is over all space as required by the technique used before. Again, it is clear that the two mutual inductances are equivalent.

Occasionally the same notation will be used for mutual and self-inductance ᏹ ϭ ᏸ (described below): 12 12. The subscripts make it clear what is meant.

Self-Inductance. Just as the current of one circuit creates magnetic flux that links with others, the circuit’s flux links with the circuit itself. This is called the circuit’s self-inductance and is defined as the magnetic flux linkage per unit cur- rent in the circuit itself. For circuit 1, the self-inductance is written as

Appendix: A Physics Primer for Magnetostatic Energy 83 Energy of Circuits

1 1 ᏸ ϭϭ----Ᏺ ----͐B и ds . (A–30) 11 I 11 I 11 1 1 1 S1

Like the mutual inductance this relationship can be rewritten using the defini- tion of the vector potential, Eq. (A–14), and then Stoke’s theorem:

1 1 . ᏸ ϭϭ----͐()ٌϫA и da ----͛A и dᐍ 11 I 11 1 I 11 1 1 S1 1 C1

Unlike the mutual inductance, the solution to the vector potential cannot now be substituted. The solutions to the magnetic vector potential, Eqs. (A–15) and (A–16), assume that the vector potential is measured away from the source point. Clearly the denominator of the integrand tends toward infinity as the field point and source point become closer. Thus, the mutual inductance solution of Eq. (A–28) is only an approximation that is valid when the cross-sectional areas of the wires are small when compared to the distance between the circuits.

Using the filamentary approximation, Eq. (A–9), and multiplying the integrand ր by I1 I1 gives

1 1 ᏸ ϭϭ------͛A и I dᐍ ----͐A и J dv . 11 I I 11 1 1 2 11 1 1 1 1 C1 I1 V1

Again, using the same technique used to transform Eq. (A–25) into Eq. (A–26):

1 ᏸ ϭ ------͐B и B dv, (A–31) 1 2␮ 1 1 I1 0 V where the notation has been simplified. Again note that the volume integral of Eq. (A–31) is over all space.

The self-inductance is sometimes written with the same notation as that used ᏸ ϭ ᏹ for the mutual inductance: 1 11. The subscripts make the meaning clear.

Field Energy in Terms of Inductances. The field energy of a system of circuits can now be expressed in terms of the inductance coefficients. For a system of

Appendix: A Physics Primer for Magnetostatic Energy 84 Summary

ϭ ϩ two circuits, the magnetic flux density of the entire system is BB1 B2. Substituting into Eq. (A–26) gives

ϭ 1 ͐()ϩ и ()ϩ Ufield ------␮ B1 B2 B1 B2 dv 2 0 V

ϭ 1 ͐ 2 ϩϩ͐ и ͐ 2 ------␮ B1dv 2 B1 B2dv B2dv . 2 0 V V V

Using Eqs. (A–29) and (A–31) this becomes

1 2 1 2 U ϭ -- ᏸ I ϩϩᏹ I I -- ᏸ I . (A–32) field 2 1 1 12 1 2 2 2 2

For ᏺ circuits this can be generalized to

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ, (A–33) field 2 ᐇ ϭ 1 ᐈ ϭ 1

where the notation ᏹᐇᐇ is used for the self-inductances.

It is important to realize that the field energy for a system of circuits with con- stant currents, as seen in Eq. (A–33), will only change if the inductance coeffi- cients change. If the circuits themselves are rigid, then only the relative motion of the circuits, and hence the changing mutual inductances, will contribute to a change of field energy.

Summary

For homogeneous, isotropic, linear materials in low electric and magnetic fields that form a system of circuits with constant electric currents, the work done by the magnetic forces is equal to the change in the magnetic field energy:

ϭ dUfield dWmech.

Finding the mechanical forces between the circuits, then, is equivalent to taking the positive gradient of the field energy:

Appendix: A Physics Primer for Magnetostatic Energy 85 Works Consulted for Appendix

ٌ ϭ F Ufield.

And the field energy is ½ the double sum of the inductances times the currents:

ᏺ ᏺ 1 U ϭ -- Α Α ᏹᐇᐈIᐇIᐈ. field 2 ᐇ ϭ 1 ᐈ ϭ 1

Works Consulted for Appendix Cheng, David. Field and Wave Electromagnetics. 2nd ed. New York: Addison, 1989.

Feynman, Richard, Robert Leighton, and Matthew Sands. The Feynman Lec- tures on Physics. Vol. 2. New York: Addison, 1964.

Nayfeh, Munir, and Morton Brussel. Electricity and Magnetism. New York: Wiley, 1985.

Smythe, William. Static and Dynamic Electricity. New York: McGraw, 1939.

Appendix: A Physics Primer for Magnetostatic Energy 86