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.

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...................................................................................................... 4Kinetic Energy 4Potential Energy 5Lagrangian Equations 7

Discretized Equations ....................................................................................................7Kinetic Energy 8Potential Energy 8Lagrangian Equations 12

Linearized Equations ................................................................................................... 13

Magnetostatic Levitator ..................................................................................17

Apparatus ......................................................................................................................... 17

Equations of Motion ..................................................................................................... 18Nonlinear Equations 18Discretized Equations 21Numerical Comparison 21

Static Displacement ..................................................................................................... 22

Magnetostatic Pendulum ...............................................................................24

Apparatus ........................................................................................................................ 24Pendulum 25Electromagnets 25Power Supplies 26Sensors & Computer 26Calibration 27

Equations of Motion .................................................................................................... 29Nonlinear Equations 29Discretized Equations 34

Linearized Equations 34

Natural Frequencies ......................................................................................................35Apparatus 35Equations of Motion 36Experiment 36Results 38

Tracking ............................................................................................................................41Apparatus 41Experiment 41Results 44

Regulation ....................................................................................................................... 45Apparatus 45Equations of Motion 45Experiment 49Results 49

Conclusions ...................................................................................................... 53

Works Consulted.............................................................................................. 55

Appendix: A Physics Primer for Magnetostatic Energy............................. 58

Classical Physics............................................................................................................ 58

Electromagnetism ........................................................................................................ 59Conservation of Charge 59Maxwell’s Equations 60

Induction ..........................................................................................................................61Flux 62Electromotive Force 63Faraday’s Law 66

Filamentary Approximation ..................................................................................... 67

Magnetostatics .............................................................................................................. 68Magnetic Vector Potential 69Biot-Savart Law 70

Energy of Circuits ..........................................................................................................72Power in Circuits 72Work Done by Circuits 76Energy and Virtual Work 79Energy of a Magnetic Field 80Field 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 4–5. 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 (a, b) ............................................................ 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 a angle versus the two base electromagnets’

currents....................................................................................................... 42

viii

Figure 4–14. Pendulum’s b 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)

(2–1)

where is the mass density at position and is the velocity of the posi-

tion vector

T1

2-- m r( )r r? v,d

V

e5

m r( ) r, r

r.

4

Nonlinear Equations

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

(2–2)

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). is the self-adjoint, positive, semi-definite stiffness operator,

is the strain component of the position vector and is the (constant)

gravity vector.

The magnetostatic potential energy is composed of the integral of the dot prod-

uct of the current density and the magnetic vector potential (Jackson 216;

see also Eq. (A–25) on page 81†):

(2–3)

This integral is over the space 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 from a current density at a point can be

defined as (Jackson 176; see also Eq. (A–15) on page 70)

(2–4)

again where the integration need only be over the region where the current

source exists. The constant is the magnetic permeability of free space and

has the exact value of

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.

U1

2-- rs Krs? vd

V

e m r( )r g? vdV

e21

2-- J r( ) A r( )? v,d

V

e15

K

rs r, g

J A

Umag

1

2-- J r( ) A r( )? v.d

V

e5

V

r J r9

A r( )m0

4p------- J r9( )

r r92------------------ v9d

V9

e5

V9

m0

4p72×10 H/m.

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

It is important to remember that the current densities are defined as the density

of current at a particular position and the vector potential is defined as the

potential measured at a particular position 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

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:

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)

the potential energy becomes

J r( ) J1 r( ) J2 r( )1 J1 J2,15 5

A r( ) A1 r( ) A2 r( )1 A1r A2r.15 5

J r,

A r( )

Umag

1

2-- J1 J21( ) A1r A2r1( )? vd

V

e5

1

2-- J1 A11? J1 A21? J2 A12? J2 A22?1 1 1( ) v,d

V

e5

A12 A r2( )m0

4p-------

J r1( )

r2 r12-------------------- v,d

V1

e5 5

}12

1

I1I2

-------- A12 J2? v,dV2

e5 +11

1

I12

---- A11 J1? v,dV1

e5


Discretized Equations

where and are the currents flowing through circuits 1 and 2 respectively.

(Note that the two mutual inductances are equal:

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

(2–5)

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

Lagrangian Equations With the Lagrangian defined as

(2–6)

Lagrange’s equations of motion are simply

(2–7)

where the position vectors are now functions of the generalized coordinates

(The vectors are discussed in more

detail in the following section.)


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

density is assumed constant. The center positions of these subvolumes are

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, which also serve as the

generalized coordinates described above. Expressing the position vectors and

their time derivatives in terms of the generalized coordinates gives

(2–8)

Umag

1

2--+11I1

2}12I1I2

1

2--+22I2

2,1 15

I1 I2

}12 }21.)5

1

Umag

1

2-- };'I;I',

' 15

1

^; 15

1

^5

};; +;;5

L T U,25

tdd

ul

L

ulL

2 0,5 l 1 … M,, ,5

r

ul: ri r u1 u2 … uM, , ,( ),5 i 1 … N., ,5 ri

N

r1,

r2 … rN., ,

u1 u2 … uM,, , ,

ri ri u1 u2 … uM, , ,( ),5 ruj

ri uj,

i 15

M

^5 i 1 … N., ,5



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

(2–9)

where is the mass of the 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 is divided into subintervals of length and

the strain potential energy is

(2–10a)

(2–10b)

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

The matrix 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

(2–11)

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.

T1

2-- miri ri?

i 15

N

^ 1

2-- mi uj

ri

uk

ri? ujuk,

k 15

N

^j 15

N

^i 15

N

^5 5

mi i th

L h,

Ustrain

1

2-- KSijuiuj,

j 15

M

^i 15

M

^5

Ks

EI

h-----

1 12 0 ª 0

12 2 12 0 ª 0

0 12 2 12 0 0

Á ¢ ¢ ¢ ¢ Á

Á ¢ 12 2 12

0 ª ª 0 12 1

,5

EI

Ks

Ugrav mirig? .

i 15

N

^25



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),

(2–12)

with a summation

(2–13)

where and are the paths around the two circuits and respectively,

and are the differential length elements along the paths, and are

the position vectors for the differential elements, is the finite length

element (a vector) of circuit is the finite length element (also a vec-

tor) of circuit and and are the positions of the two finite elements.

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

ments and each of the elements, must be written in terms of the general-

ized coordinates 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, and from Eq. (2–13) becomes

(2–14)

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

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

the loops are approximated by and 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-

};'

m0

4p-------

<;d <'d?

r' r;2---------------------- ,

C'

rC;

r5

};'

m0

4p-------

<;zD <'jD?

r'j r;z2---------------------------- ,

jz

5

C; C' ; '

<;d <'d r; r'

<;zD z th

;, <'jD j th

', r;z r'j

<;zD

<'jD

ul.

; ',

};'

m0

4p-------1;1'

<;zD <'jD?

r'j r;z2---------------------------- ,

j 15

n'

^z 15

n;

^5

1; 1'

n; n'

n;- n'-



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 in cylin-

drical coordinates is (Nayfeh and Brussel 253)

where the current density is considered to be uniform across the cross-section

of the wire, i.e., The self-inductance is defined as (Smythe 313; see

also Eq. (A–31) on page 84)

(2–15)

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

(2–16)

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

(2–17)

R

Bm0Jr

2-----------f,5

J I pR2./5

+1

1

I12m0

---------- B1 B1? vd .V

e5

L;

+;;int

1

I2m0

---------- B2

vdV

e m0

8p-------L;.5 5

+;;ext

m0

4p-------1;

2<;za

D <;zsD?

r;zsr;za

2------------------------------- ,

zs 15

n;

^za 15

n;

^5



where is the number of loops for the circuit is the number of discret-

ization elements for the loop, and refer to the finite length elements

along the axis and outer surface of the wire respectively, and and are

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, apart from each other.

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

of loops of wire is approximately

(2–18)

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

(2–19a)

where, using Eqs. (2–14) and (2–18),

(2–19b)

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

nates if needed: e.g., Note that except for the inter-

nal self-inductance term, 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):

1; ;, n;

<;zaD <;zs

D

r;zar;zs

R,

L;

1;

+;;

m0

8p-------L;

m0

4p-------1;

2<;za

D <;zsD?

r;zsr;za

2------------------------------- .

zs 15

n;

^za 15

n;

^15

Umag

1

2-- };'I;I',

' 15

1

^; 15

1

^5

};'

m0

4p-------1;1'

<;zD <'jD?

r'j r;z2---------------------------- ,

j 15

n'

^z 15

n;

^ ; 'Þ

m0

8p-------L;

m0

4p-------1;

2<;za

D <;zsD?

r;zsr;za

2------------------------------- .

zs 15

n;

^za 15

n;

^1 ; '5

5

<D

<;zaD <;za

D u1 … uM, ,( ).5

m0 8p/( )L;,



Note also their equation (5a),

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

(2–20)

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),

(2–21)

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

(2–22a)

Uelec

1

2-- PijQiQj.

j 15

N

î 15

N

^5

Vi PijQi,

j 15

N

^5 i 1 … N,, ,5

':

^' };'I;,

; 15

1

^5 ' 1 … 1., ,5

L T U25

1

2-- mi uj

ri

uk

ri? ujuk

k 15

M

^j 15

M

î 15

N

^ 1

2-- KSijuiuj

j 15

M

î 15

M

^25

1 miri g?

i 15

N

^ 1

2-- };'I;I'.

' 15

1

^; 15

1

^2

ulL

mi uj

ri r2

i

uk ul--------------- ujuk?

k 15

M

^j 15

M

î 15

N

^ KSljuj

j 15

M

^25

1 mi ul

rig?

i 15

N

^ 1

2--

};'

ul-------------I;I',

' 15

1

^; 15

1

^2


Linearized Equations

(2–22b)

(2–22c)

Substituting into Eq. (2–7) gives the discretized, nonlinear system of ordi-

nary differential equations describing the motion of the structure:

(2–23)


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

ting and where Following Silverberg and

Weaver (385), a system of coupled, nonlinear, algebraic equations that govern

the equilibrium positions are obtained:

(2–24)

where the 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

(2–25)

ul

Lmi uj

ri

ul

ri uj,?

j 15

M

î 15

N

^5

tdd

ul

L mi uj

ri

ul

ri uj˙?

j 15

M

î 15

N

^5

1 uj

ri r2

i

uk ul---------------?

r2

i

uj uk---------------

ul

ri?1

ujuk

k 15

M

^

M

mi uj

ri

ul

ri uj˙?

r2

i

uj uk---------------

ul

ri? ujuk

k 15

M

^1

j 15

M

î 15

N

^ KSljuj

j 15

M

^1

2 mi ul

rig?

i 15

N

^ 1

2--

};'

ul-------------I;I'

' 15

1

^; 15

1

^1 0,5 l 1 … M., ,5

uk 05 uk 0,5 k 1 … M., ,5

M

KSliui0

i 15

M

^ mi ul

ri

0

g?

i 15

M

^21

2--

};'

ul-------------

0

I;0I'0

' 15

1

^; 15

1

^1 0,5

l 1 … M,, ,5

0

hl t( ) ul t( ) ul02 ,5 l 1 … M, , .5



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

(2–26)

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

(2–27a)

(2–27b)

(2–27c)

By changing the currents and 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

(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):

mljhj aljhj bljhj1 1[ ]j 15

M

^ 05 , l 1 … M, ,5 ,

mjl mlj hj

fl

0

mi ul

ri

0uj

ri

0

?

k 15

N

^ ,5 5 5

alj hj

fl

0

};'

ul-------------

0

I;

uj-------

0

I'0,

' 15

1

^; 15

1

^5 5

blj hj

fl

0

KSlj mk ul uj

2

rk

0

g?

k 15

N

^25 5

1 };'

ul-------------

I;

uj-------I'

1

2--

ul uj

2

};'I;I'1

0

.' 15

1

^; 15

1

^

I; I'

Ir Ir u1 u2 … uM u1 u2 … uM, , , , , , ,( ),5 r 1 … 1., ,5



(2–29)

Because of the similarity of the inductance coefficients of Eq. (2–19b), the equa-

tion for the partial of is found by substituting

and of course in Eq. (2–29). This is assuming that the

lengths of the circuits, 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):

(2–30)

};'

ul-------------

m0

4p-------1;1'

<;zD <'jD?

r'j r;z23

---------------------------- r'j r;z2( )ul

r'j r;z2( )?2j 15

n'

^z 15

n;

^5

1

<;zD

ul-------------- <'jD? <;zD

<'jD

ul---------------?1

r'j r;z2------------------------------------------------------------------------- , ; '.Þ

};; 'j ;zs,5 ;z ;za,5

z za,5 j zs,5 ' ;5

L;,

ul uj

2

};' m0

4p-------1;1' 3

<;zD <'jD?

r'j r;z25

---------------------------- 3j 15

n'

^z 15

n;

^5

r'j r;z2( )ul

r'j r;z2( )? r'j r;z2( )

uj

r'j r;z2( )?

2

<;zD

uj-------------- <'jD? <;zD

<'jD

uj---------------?1

r'j r;z23

------------------------------------------------------------------------- r'j r;z2( )ul

r'j r;z2( )?

2 <;zD <'jD?

r'j r;z23

----------------------------uj

r'j r;z2( )ul

r'j r;z2( )?

1 r'j r;z2( )ul uj

2

r'j r;z2( )?

2

<;zD

ul-------------- <'jD? <;zD

<'jD

ul---------------?1

r'j r;z23

------------------------------------------------------------------------- r'j r;z2( )uj

r'j r;z2( )?

1 1

r'j r;z2-------------------------

ul uj

2

<;zD <'jD?<;zD

ul--------------

<'jD

uj---------------?1

1 <;zD

uj--------------

<'jD

ul---------------? <;zD

ul uj

2

<'jD?1 , ; 'Þ



where again, assuming that the circuit lengths, are fixed, the double partial

derivative of is found by substituting

and of course in Eq. (2–30).

In Eq. (2–26) the coefficients constitute a positive definite, symmetric mass

matrix. The coefficients can be split into a symmetric matrix called the

damping matrix, and a skew-symmetric matrix called

the gyroscopic matrix, Similarly, the coefficients can

be separated into a symmetric matrix, called the stiff-

ness matrix and a skew-symmetric matrix, 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).

L;,

};; 'j ;zs,5 ;z ;za,5 z za,5

j zs,5 ' ;5

mlj

alj

clj 1 2/( ) alj ajl1( ),5

glj 1 2/( ) alj ajl2( ).5 bij

klj 1 2/( ) blj bjl1( ),5

hlj 1 2/( ) blj bjl2( ),5


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.

Figure 3–1. Magneto-static levitator

-0.05

0

0.05x @m D

-0.05

0

0.05

y @m D

0

0.1

z @m D

-0.0

0x @m D

upper coil

lower coil

plastic cylinder

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

(3–1)

where is the mass of the upper coil (coil #2), and 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

(3–2)

where 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:

T1

2--mz

2,5

m z

Ugrav mgz,25

g z

Magnetostatic Levitator 18

Equations of Motion

(3–3)

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

(3–4)

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

(3–5)

where and are the bounding contours for the circuits that make up the

lower (#1) and upper (#2) coils, and are the differential length elements

along the contours, and and 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

and 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

Umag

1

2-- };'I;I'.

' 15

1

^; 15

1

^5

Umag }12I1I2,5

1 25 }12 }21.5

Umag I1I2

m0

4p-------

<1d <2d?

r2 r12--------------------- ,

C2

rC1

r5

C1 C2

<1d <2d

r1 r2

C1

C2


Equations of Motion

(3–6)

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

The Lagrangian for the system is

(3–7)

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

(3–8)

where 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

(3–9)

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

(3–10a)

Umag 11I112I2

m0

4p-------

<1d <2d?

r2 r12--------------------- ,

C2

rC1

r5

11 12

L T U25

1

2--mz

2mgz I1I2

m0

4p-------

<1d <2d?

r2 r12--------------------- .

C2

rC1

r215

Q cz,25

c

tdd

z

L

zL

2 Q.5

mz1

R1 R21( )2z

21

---------------------------------------11I112I2m0

R1 R22( )2z

21

-------------------------------------------

1

3 R12

R22

1( )E m( ) R1 R21( )2K m( )2 z E m( ) K m( )2 z

31

1 R1 R21( )2mg cz1( ) mg cz1( )z

21

0,5


Equations of Motion

where and are the radii of the two coils, is the complete elliptic

integral of the second kind, is the complete elliptic integral of the first

kind, and the elliptic integral parameter is

(3–10b)

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

(3–11)

As in Eq. (2–14) on page 9, and are the numbers of finite length elements

by which each circuit is approximated, is the finite length element (a

vector) of circuit 1, is the finite length element (also a vector) of circuit

2, and and are the positions of the two finite elements.

Substituting Eq. (3–11) into 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 The currents were set to

and 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

R1 R2 E m( )

K m( )

m4R1R22

R1 R22( )2z

21

--------------------------------------- .5

Umag 11I112I2

m0

4p-------

<1zD <2jD?

r2j r1z2--------------------------- .

j 15

n2

^z 15

n1

^5

n1 n2

<1zD z th

<2jD j th

r1z r2j

L T U,25

c 0.005 kg/s.5

I1 0.6 A5 I2 0.6 A25


Static Displacement

(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-

ments and 15 segments 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

signs, 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.

Figure 3–2. Comparison of nonlinear and dis-cretized simulated dynamic solutions

0.2 0.4 0.6 0.8 1 1.2 1.4t @sD

0

0.01

0.02

0.03

0.04

0.05

0.06z @m D Upper Coil Position

Nonlinear 30 Segments 15 Segments

(n1 n2 30)5 5 (n1 n2 15).5 5

I1 I2).25


Static Displacement

The numerical solutions come close to the measured value. Please note, how-

ever, that the uncertainty of the measurement was approximately

so even the 15-segment discretization 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%.

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

Numerical Solutions

Experimental

Measurement

Nonlinear

Solution

30-Segment

Discretization

15-Segment

Discretization

Equilibrium

Position0.027m 0.02621m 0.02613m 0.02597m

% Error of

Solutiona

a. The error of each numerical solution relative to the experimental measurement.

— –2.90% –3.17% –3.77%

% Error of

Discretizationb

b. The error of each discretized solution relative to the nonlinear solution.

— — –0.28% –0.90%

0.0016 m,6

(n1 n2 15)5 5


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.

Figure 4–1. Magneto-static pendulum

base

sensors

pendulum

sensor reflectors

U-joint support

electromagnets

24

Apparatus

Pendulum A photograph of the pendulum can be seen in Fig. 4–1. The base of the pendu-

lum apparatus is Centered on the base is an grid of tapped

holes, each row and column spaced apart. The sides are 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

The pendulum’s mass is 0.142kg without an electromagnet attached.

The maximum angular displacement of the pendulum without touching the

side of the apparatus is 0.165rad, which is approximately π⁄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 an inner diam-

eter of and a thickness of

180 180.3 11 113

10 33⅜0

29½0.

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

4½0,

3½0, ¹¹⁄₁₆0.

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 and the pendulum coil has an outer

diameter of approximately 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

4⅜0,

4½0.

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

U-joint support

wires to pendulum’s electromagnet

sensor reflector “vane”

infrared sensor

top of pendulum tube


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 array of tapped holes with each row

and column set apart. A particular position on the grid, 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, 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.

Figure 4–4. Pendulum support reference frame (X, Y, Z) and its relation to generalized coordi-nates (a, b)

ab

ab

atanY

Z2-------- ,5

btanX2

Y2

Z2

1

------------------------5

Y2

Z2

1

Opendulum pivot

Y

X

Z

pendulum tip (–X, Y, –Z)

11 113

10 X Y,( ),

0 0,( ),


Apparatus

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

Figure 4–5. Pendulum’s voltage calibration sur-face for first sensor

-5

0

5

X @inD

-5

0

5

Y @inD

4.2

5.2

V0 @VD

-

0X @inD

Figure 4–6. Pendulum’s voltage calibration sur-face for second sensor

-5

0

5

X @inD

-5

0

5

Y @inD

4.2

5.2

V1 @VD

-

0X @inD


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

With the position of the pendulum known, the pendulum angles are

found geometrically:

where 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 in the inertial

system is related to a position in the body’s coordinate system by

X Y,( ).

X Y,( )

atanY

Z2-------- ,5 btan

X2

Y2

Z2

1

----------------------- ,5

Z

a

b

r9

r

r9 RT

r,5


Equations of Motion

where 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

where is the volume element at position is the mass

density, and is the volume of the part. The final tensor for the part is

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

(4–1)

where is the angular velocity and the angular momentum, is defined as

Figure 4–7. Pendulum’s coordinate frames

a

b

Generalized

Coordinates:

Inertial

Reference Frame:

Body-Fixed

Reference Frame:

a b,( )

x9 y9 z9, ,( )

x y z, ,( )

a

a

b

b

x

y

z

x9

y9

z9

R

Iij m r( )dij xk2

xixj2

k 15

3

^ vdV

e5 ,

vd x1d x2d x3d5 r, m r( )

V

I

I11 I12 I13

I21 I22 I23

I31 I32 I33

.5

T1

2--v L? ,5

v L,


Equations of Motion

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

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 where 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

Using either method and setting (since the pendulum cannot rotate

about that axis), the angular velocity is just

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

where is the mass of the object, and is the acceleration due to gravity. For

an idealized spherical pendulum the gravitational potential energy is just

L I v.?5

vbcos gcos gsin 0

bcos gsin2 gcos 0

bsin 0 1

a

b

g

,?5

g

V Rd td/( ) RT

,?25 R

v V32 V13 V21

T.5

g 05

v a bcos b a bsinT

.5

mg rd?1

2e U1 U22 U,D25 5

m g

Ugrav mgr u,cos25


Equations of Motion

where is the mass of the pendulum, is the magnitude of the acceleration

due to gravity, 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 so that the

gravitational potential energy of the pendulum is

(4–2)

where 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-

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

-p��2

0

p��2

a @radD

-p��2

0

p��2

b @radD

-2

0

Ugrav @ JD

0

p

adD

-

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

-p

��20

0

p��20

a @radD

-p

��20

0

p��20

b @radD

-2

0

Ugrav @ JD

0

p

dD

-

m g

r u

a b.

ucos acos b,cos5

Ugrav mgr acos b,cos25

m r


Equations of Motion

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

the possible range of motion, ±π⁄₂₀rad, in Fig. 4–9.

Magnetostatic Potential. The potential energy of a magnetostatic field is

(Eq. (2–5) on page 7)

(4–3)

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

mutual inductance between electromagnets and and and are the

currents flowing through electromagnets and respectively. When

is the self-inductance of the 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 the magne-

tostatic potential energy of the pendulum, Eq. (4–3), reduces to the form

(4–4)

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

(4–5)

a b

Umag

1

2-- };'I;I',

' 15

1

^; 15

1

^5

1 };'

; ', I; I'

; ' ; ',5

};; ; th

};' }';,5

Umag };'I;I',

' 15

; 12

^; 15

1

^5

L T U,25


Equations of Motion

where and are the kinetic and potential energies. The potential energy, as

stated above, has a gravitational component and a magnetostatic component:

(4–6)

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

equations of motion are just

(4–7)

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:

(4–8)

where and are the number of loops for each of the coils, (a vector)

is the finite length element of circuit (also a vector) is the finite

length element of circuit and and are the positions of the two finite

elements. Each coil has only one loop each, which is discretized into and

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:

(4–9)

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.

T U

U Ugrav Umag.15

aL

tdd

aL

2 0,5b

L

tdd

b

L2 0.5

};'

m0

4p-------1;1'

<;zD <'jD?

r'j r;z2---------------------------- ,

j 15

n'

^z 15

n;

^5

1; 1' <;zD

z th ;, <'jD j th

', r;z r'j

n; n'

Umag

m0

4p------- 1;I;1'I'

<;zD <'jD?

r'j r;z2----------------------------

j 15

n'

^z 15

n;

^

.' 15

; 12

^; 15

1

^5


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 the pendulum

coil has an outer diameter of and both have a thickness of 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.

The coils have resistances and where and

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)

4½0,

3⅞0, ¹¹⁄₁₆0.

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

IR1

I1

L1

coil 1

M12 L2

R2

I2

coil 2

R1 20.375 V5 R2 40.5 V,5 R1 R2


Natural Frequencies

where is the total current delivered by the power supply, and and 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 (two coils):

(4–10)

where and

Eq. (4–10) is plotted over the range of possible pendulum motion and

varying over ±π⁄₂₀rad) in Fig. 4–11 for a total current of 0.6A and

and in Fig. 4–12 for a total current of 1.0A and

In both plots and for the experiment the base coil is in the

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,

I1

R2

R1 R21-------------------I,5 I2

R1

R1 R21-------------------I,5

I I1 I2

1 25

Umag

m0

4p-------11I112I2

<1zD <2jD?

r2j r1z2--------------------------- ,

j 15

n2

^z 15

n1

^5

11 600,5 12 300,5 n1 n2 10.5 5

(a b

(I1 ⁴⁄₁₀ A,5

I2 ²⁄₁₀ A)5 (I1 ⅔ A5

I2 ⅓ A).5 0 0,( )


Natural Frequencies

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

Figure 4–11. Pendulum’s magnetostatic potential energy for 0.6A total cur-rent to two electromag-nets (one base and one pendulum coil)

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

-p

��20

0

p��20

a @radD

-p

��20

0

p��20

b @radD

-0.002

0

Umag @ JD

0

p

dD

Figure 4–12. Pendulum’s magnetostatic potential energy for 1.0A total cur-rent to two electromag-nets (one base and one pendulum coil)

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

-p

��20

0

p��20

a @radD

-p

��20

0

p��20

b @radD

-0.002

0Umag @ JD

0

p

dD

-

a b

a b

a b


Natural Frequencies

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, 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

a b

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

simulated measured % error simulated measured % error

sma

ll in

itia

ld

ispl

ace

men

t

[rad⁄s] 3.78533 3.77811 +0.19% 3.78395 3.774 +0.26%

T [s] 1.65988 1.66305 –0.19% 1.66049 1.66486 –0.26%

larg

e in

itia

ld

ispl

ace

men

t

[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%

asim aexp

asim

aexp---------- % bsim bexp

bsim

bexp---------- %

v

v



sma

ll in

itia

ld

ispl

ace

men

t

[rad⁄s] 4.15099 4.21782 –1.58% 4.14506 4.21405 –1.64%

T [s] 1.51366 1.48968 +1.61% 1.51582 1.49101 +1.66%

larg

e in

itia

ld

ispl

ace

men

t

[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%

asim aexp

asim


bsim

bexp---------- %

v

v

a b

v,

T,


Natural Frequencies

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



sma

ll in

itia

ld

ispl

ace

men

t

[rad⁄s] 4.44644 4.529 –1.82% 4.43302 4.5415 –2.39%

T [s] 1.41311 1.38734 +1.86% 1.41737 1.38351 +2.45%

larg

e in

itia

ld

ispl

ace

men

t

[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%

asim aexp

asim


bsim

bexp---------- %

v

v



sma

ll in

itia

ld

ispl

ace

men

t

[rad⁄s] 4.77958 4.89641 –2.39% 4.80069 4.9164 –2.35%

T [s] 1.31465 1.28324 +2.45% 1.30886 1.27801 +2.41%

larg

e in

itia

ld

ispl

ace

men

t

[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%

asim aexp

asim


bsim

bexp---------- %

v

v


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

each circular loop was approximated by decagons Although

not shown, several simulations were run comparing the results given by setting

and setting 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.

(n1 n2 10).5 5

n1 n2 105 5 n1 n2 20.5 5


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

the pendulum coil has an outer diameter of and both have a thick-

ness of 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

resistances and where

and 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.

a b,

4⅜0, 4½0,

¹¹⁄₁₆0.

R1 19.6 V,5 R2 20.2 V,5 R3 24.5 V,5 R1, R2,

R3


Tracking

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.

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

00.2

0.40.6

0.8

I1 @AD0

0.2

0.4

0.6

0.8

I2 @AD

0

0.01

0.02

0.03

a @radD

00.2

0.40.6I1 @AD

0

Figure 4–14. Pendulum’s b angle versus the two base electromagnets’ currents

00.2

0.40.6

0.8

I1 @AD0

0.2

0.4

0.6

0.8

I2 @AD

-0.03

-0.02

-0.01

0

b @radD

00.2

0.40.6I1 @AD

-

a b,


Tracking

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, 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 b,( ),

a a b

b

I1 I2,( ), a b,( ).

Figure 4–15. Example comparison of actual versus desired path for the first tracking experi-ment

-0.015

-0.0125

-0.01

-0.0075

-0.005

-0.0025

0

b@r

adD

0.01 0.015 0.02

a @radD

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

-0.015

-0.0125

-0.01

-0.0075

-0.005

-0.0025

0

b@r

adD

0.01 0.015 0.02

a @radD


Tracking

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 for and

approximately for These rotations amount to approximately

along the +X-axis and approximately 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:

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)

Mean % Error 3.07% 3.81% 3.38% 4.27%

aerror berror aerror berror

normalized rms% error measured desired2( ) range of values./5

a b

a b

0.025 rad a

0.015 rad2 b. ½0

⁷⁄₁₀0


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

(4–11)

Umag

m0

4p------- 11I112I2

<1zD <2jD?

r2j r1z2---------------------------

j 15

n2

^z 15

n1

^

5

+ 11I113I3

<1zD <3jD?

r3j r1z2---------------------------

j 15

n3

^z 15

n1

^

+ 12I213I3

<2zD <3jD?

r3j r2z2---------------------------

j 15

n3

^z 15

n2

^

,


Regulation

where is the number of loops, is the current, is the number of discreti-

zation segments, is the finite length element, and is the finite length

element position vector for the coil respectively (where For

this experiment and, as stated under “Apparatus” on

page 41, loops and loops.

Eq. (4–11) is plotted over the range of possible pendulum motion and over

±π⁄₂₀rad) in Fig. 4–17 for a current of 0.6A and in

Fig. 4–18 for a current of 1.0A Again, in both plots

and for the experiment the base coils are in the and 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

1i Ii ni

<izD rij

i th i 1 2 3)., ,5

n1 n2 n3 10,5 5 5

11 12 4505 5 13 5505

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

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

-p

��20

0

p��20

a @radD

-p

��20

0

p��20

b @radD

-0.02

0Umag @ JD

0

p

dD

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

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

-p

��20

0

p��20

a @radD

-p

��20

0

p��20

b @radD

-0.02

0Umag @ JD

0

p

dD

-

(a b

(I1 I2 I3 0.6 A),5 5 5

(I1 I2 I3 1.0 A).5 5 5

3 0,( ) 0 3,( )


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 and 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, and yields the equations

governing the equilibrium positions for the system (equivalent to Eq. (2–24) on

page 13). New generalized coordinates are introduced,

and the equations are expanded in a Taylor’s series about the new equilibrium

position Retaining only the linear terms gives the linearized equations

of perturbed motion for the system (equivalent to Eq. (2–26) on page 14).

As an example, setting the system currents to and

the pendulum’s equilibrium position changes from

to the new equilibrium

The linearized equations of motion reduce to

(4–12)

where the currents have been written as

I1 I2,, I3

a a 05 5 b b 0,5 5

h1 t( ) a t( ) a0,25 h2 t( ) b t( ) b0,25

a0 b0,( ).

I1 0.85 A,5 I2 0 A,5

I3 0.75 A,5

agrav bgrav,( ) 0.002262 0.00226,( ) rad,5

a0 b0,( ) 0.001842 0.02222,( ) rad.5

0.2205 0

0 0.2206

h1 t( )

h2 t( )

3.777 0.04402

0.04402 2.589

h1 t( )

h2 t( )1

1 I1 t( ) 0.0015292

0.08817

0.8341 0.05175

0.05175 0.56322

h1 t( )

h2 t( )1

1 I2 t( ) 0.060952

0.015162

1.442 0.54382

0.54382 0.4805

h1 t( )

h2 t( )1

0

0,5

I1 t( ) I1 t( ) I1 0,2 ,5 I2 t( ) I2 t( ) I2 0,2 ,5


Regulation

(remember that and are the equilibrium currents

above). Note that since the current 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,

(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

(4–14)

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

(4–15)

I1 0, 0.85 A5 I2 0, 0 A5

I3 0.75 A5

I1 t( ) h2,5 I2 t( ) h2 1,5

b

a

0.2205 0

0 0.2206

h1 t( )

h2 t( )

0.06095 0.0015292

0.01516 0.08817

h1 t( )

h2 t( )1

2 1.442 0.54382

0.54382 0.4805

h1 t( )h1 t( )

h2 t( )h1 t( )

0.8341 0.05175

0.05175 0.56322

h1 t( )h2 t( )

h2 t( )h2 t( )1

1 3.777 0.04402

0.04402 2.589

h1 t( )

h2 t( )

0

0.5

0.2205 0

0 0.2206

h1 t( )

h2 t( )

0.06095 0.0015292

0.01516 0.08817

h1 t( )

h2 t( )1

1 3.777 0.04402

0.04402 2.589

h1 t( )

h2 t( )

0

0.5


Regulation

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

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 and

to adjust the currents to the base coils, and

where and 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

By setting and new equilibrium positions are created for the pendu-

lum. Regulation was performed about two equilibrium positions with varying

gains: and

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

det s2 0.2205 0

0 0.2206s 0.06095 0.0015292

0.01516 0.08817

3.777 0.04402

0.04402 2.5891 1

05

s 0.19862 3.419i6 0.13952 4.137i,6,5

I1 t( ) I2 t( ),

I1 t( ) g1 V0D ,5 I2 t( ) g2 V1D ,5

I1 t( ) I2 t( ),

I1 t( ) I1 0, I1 t( ),15 I2 t( ) I2 0, I2 t( ),15

g1 g2

V0D h2,~ V1D h12 .~

I1 0, I2 0, ,

I1 0, I2 0,,( ) 0.3 0.7,( ) A5 I1 0, I2 0,,( ) 0.6 0.6,( ) A.5


Regulation

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

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.

Tab e 4–6. Pendulum’s approximate settling times for various gains

Equilibrium for (0.3, 0.7)A

(see Fig. 4–19 on page 51)

Equilibrium for (0.6, 0.6)A

(see Fig. 4–20 on page 52)

Gai

ns

16¼ s 13½ s 18½ s 20 s

12½ s 12½ s 13 s 14½ s

10 s 8¼ s 6½ s 9 s

4¼ s 4¼ s 4 s 4¼ s

a b a b

g1 g2 05 5

g1 g2 225 5

g1 g2 825 5

g1 g2 1525 5

g1 g2 15).25 5

a


Regulation

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

g1 = g2 = -15

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = -8

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = -2

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = 0

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD


Regulation

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

g1 = g2 = -15

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = -8

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = -2

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD

g1 = g2 = 0

5 10 15 20t @sD

0.005

0.015

0.02

0.025a @radD

5 10 15 20t @sD

-0.02

-0.015

-0.01

-0.005

b @radD


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

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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.

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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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puter Control of a Two-Dimensional Hyperbolic System.” IEEE Transactions

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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

Law of Motion

Law of Gravitational Force

Law of Electromagnetic Force

Maxwell’s Equations

Ftd

dp5 p

mv

1 v2

c2

/2

-----------------------------5

F Gm1m2

r2

--------------er25 er r r/5( )

F q E v B31( )5

D=? r5 B=? 05

E=3t

B 25 H=3 J

tD

15

58

Electromagnetism

Electromagnetism

Electromagnetic phenomena are governed by the law of electromagnetic force,

(A–1)

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

Conservation of Charge

Implicit in Maxwell’s equations is another fundamental tenant: the conservation

of electric charge. Taking the partial time derivative of yields

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

Quantity Symbola Unitsb

Force

Momentum

Velocity

Position

Unit Position Vector —

Time

Mass

Electric Charge

Electric Field Intensity

Electric Flux Density

Volume Charge Density (free charges)

Magnetic Flux Density

Magnetic Field Intensity

Volume Current Density (free currents)

Gravitational Constant

Velocity of Light

a. Bold letters represent vectors and italic letters represent scalars.

b. All units are Système International (SI)—also known as MKSA.

F N

p m s2

?

v v, m s2

/

r r, m

er

t s

m kg

q C

E V m/

D C m2

/

r C m3

/

B T

H A m/

J A m2

/

G 6.6726 310112> N m

2? kg

2/

c 3 3108> m s/

F q E v B31( ),5

D=? r5

t

D=? ( )t

D=?

tr

5 5

Appendix: A Physics Primer for Magnetostatic Energy 59

Electromagnetism

because the divergence operates only on spatial coordinates. Substituting this

into the divergence of gives

Since the divergence of the curl of a vector is always zero, the result is the con-

servation of electric charge:

(A–2)

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

or more simply

This shows that the current leaving the volume through the surface is equal

to the negative rate of change of the total charge in the volume 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 to and to For homogeneous, iso-

tropic, linear materials (i.e., not ferroelectric or ferromagnetic materials) at low

electric fields,

where is the electric permittivity of free space 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-

cern is air, which has a dielectric constant of approximately 1.00059, so .

H=3 J D t/15

H=3( )=? J=? t

D=? 1 J=?

tr

.15 5

J=? t

r.25

J=? ( ) vdV

e J ad?S

rt

rvd

V

e2td

dr vd

V

e2td

dQ,25 5 5 5

J ad?S

rtd

dQ.25

V S

V.

D E H B.

D e0KeE eE,5 5

e0 (e0 8.85122×10 F/m),> Ke

e

e e0>


Induction

The magnetic field intensity becomes

where is the magnetic permeability of free space

is the relative permeability, and is the permeability of the material. The per-

meability of air is approximately the same as that of free space, so

Using the constitutive relations, Maxwell’s equations simplify to

(A–3)

(A–4)

(A–5)

(A–6)

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).

H1

m0Km

-------------B1

m----B,5 5

m0 (m0 4p72×10 H/m),; Km

m

m m0.>

E=? re0

---- ,5

E=3t

B,25

B=? 0,5

B=3 m0J m0e0 tE

,15


Induction

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 represents the velocity of

a fluid, then the flux of 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

(A–7)

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

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

circuit 1, and is the differential unit normal to surface The sign for the

flux is chosen so it is positive when in the direction of the normal of the surface.

v

v

^12 B12 a2,d?S2

e5

^12 S2

C2, B12

ad 2 S2.

Figure A–1. Magnetically coupled circuits

I1C1

S1

C2

B12

S2


Induction

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, that is stationary in the reference

frame of the observer. Now consider another circuit defined by the closed curve

moving through the static field The curve has the shape at time

and changes to the shape at time as shown in Fig. A–2.

B,

C B. C t( ) t,

C t td1( ) t td1

Figure A–2. Conducting filament in motion

C t( )

C t td1( )

S1 S t( )5

S2 S t td1( )5

<drd

d2a


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 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

since 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:

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, known as the Lorentz force, which pushes them

around the circuit. (Note that it is assumed that This generates the emf

where is the velocity of the element The scalar triple product

so

The displacement of the filament in time is so This gives

r

E F,=25 F

% E <,d?C t( )r5

E F q/5

% E <d?C t( )r F= <d?

C t( )r2 Fd

C t( )r2 0.5 5 5 5

F q v B3( ),5

v c.)!

% t( ) v B3( ) <,d?C t( )r5

v <.d v B3 <d?

<d v? B35 <d v3( ) B,?5

% t( ) <d v3( ) B.?C t( )r5

td r,d v rd t.d/5

% t( ) <drd

td-----3

B.?C t( )r5


Induction

The area swept out in time by the element is and

is just the flux through this area (see Fig. A–2 on page 63). Therefore

where is the flux of through the total area swept out by in

time Note that the normal of the surface has been chosen as so

that the flux will be positive when it is in the direction of

With this choice of sign for the flux, the induced electromotive force becomes

As a check it will be shown that really is the change in flux through the loop

between times and The flux passing through the surface composed

of and the “sides” is zero at all times because

† But this integral can also be written as

The sign for the second term is required because of the sense chosen for

The third term, is just the flux through the “sides” as shown above. Thus

or

It is clear that really is the change in flux through the circuit between

times and

Induced EMF. Again consider a circuit producing a magnetic field 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:

td <d d2a <d r,d35

B d2a?

% t( )1

td----- d

2a( ) B?

C t( )r 1

td----- ^d2( ),5 5

^d2 B rd2a, C

dt. r <,d3d

d2a2 r <.d3d5

%td

d^.25

^d

C t t t.d1

S t( ), S t td1( ), rd2a r

SB ad?

0.5

rS

B ad? eV

B=? ( ) vd eV

0( ) vd 0.5 5 5

0 B ad?S

r B ad?S t td1( )e B ad?

S t( )e2 ^.d25 5

C t( ).

^,d2

0 ^ t td1( ) ^ t( )2 ^,d25

^d ^ t td1( ) ^ t( ).25

^d C

t t t.d1

B,


Induction

Eq. (A–4). Integrating Eq. (A–4) over the surface bounded by another nearby,

fixed circuit gives

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

Since the curve and the surface are fixed in space, neither of them depend

upon so this becomes

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 So the above

equation reduces to

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:

(A–8)

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

S

C

E=3 ad?S

et

Ba.d?

S

e25

E <d?C

rt

Ba.d?

S

e25

C S

t

E <d?C

rt

B a.d?

S

e25

%.

%td

d^.25

%td

d^.25


Filamentary Approximation

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 where is the cross-sectional area of the wire and

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,

Finally, because is the amount of charge flowing through a unit sur-

face area per unit time, is just the total charge flowing through the surface

area of the wire per unit time, i.e., So, for filamentary currents

(A–9)

vd A ,,d5 A

,d

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

J

<d

I

A

J vd

JA <.d5 J

JA

JA I.5

J vd I <.d5


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

and 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, 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),

(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:

(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:

E t/

B t/

B,

B=? 0.5

B=3 m0J.5


Magnetostatics

(A–12)

The second integral equation is obtained from Eq. (A–11) by taking the scalar

surface integral and using Stoke’s theorem to arrive at

But the last term above is just the flux of current through the surface. Thus the

integral from of Ampere’s circuital law is

(A–13)

Magnetic Vector Potential

The divergence free postulate of Eq. (A–10) means that can be expressed as

the curl of another vector field. This new field is the magnetic vector potential:

(A–14)

The definition of a vector field requires specifying not only its curl but also its

divergence. The choice of divergence for 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

Recalling the vector identity gives

The divergence of is now chosen to simplify this further. The most obvious

choice is the Coulomb gauge:

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

B ad?S

r 0.5

B=3( ) ad?S

e B <d?C

r m0 J a.d?S

e5 5

B <d?C

r m0I.5

B

B A.=35

A

A=3( )=3 m0J.5

A=3( )=3 A=? ( )= A=225

A=? ( )= A=22 m0J.5

A

A=? 0.5

A,


Magnetostatics

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

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

which has a particular solution of

where the voltage is measured at point and the integration is over the vol-

ume of the source charge distribution,

Thus, the solution for the magnetic vector potential is

(A–15)

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

(A–16)

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:

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

=2A m02 J.5

=2Ax m0Jx25 , Ay=2 m0Jy25 , Az=2 m0Jz25 .

F r( )=2 r r( )e0

--------- ,25

F r( )1

4pe0

------------ r r9( )r r92------------------ v9,d

V9

e5

F r,

V9.

A r( )m0

4p------- J r9( )

r r92------------------ v9.d

V9

e5

A r( )m0

4p------- I r9( )

r r92------------------ <9.d

C9

e5

E r( )1

4pe0

------------ r r9( )r r92( )r r92

-------------------- v9.dV9

e5


Magnetostatics

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 at the point The

integral operates on the coordinates of the source points, i.e., the points The

curl, then, operates on the integrand Using the vector identity

the curl operation can be written as

The primed and unprimed coordinates are independent, however, so the second

term is zero. The divergence of the first term gives

The integrand of the magnetic flux density given above is then

Figure A–4. Magnetic field from a wire

I

<d

r9

r

r r92( )

O

field point

source point

B r( ) A r( )=3m0

4p------- J r9( )

r r92------------------ v9d

V9

e .=35 5

A r.

r9.

J r9( ) r r92 ./

fA( )=3 f= A3 f A=3 ,15

J r9( )r r92------------------=3

1

r r92------------------= J r9( )3

1

r r92------------------ J r9( ).=315

1

r r92------------------= ex x

1

r r92------------------

ey y 1

r r92------------------

ez z 1

r r92------------------

1 15

ex x x92( ) ey y y92( ) ez z z92( )1 1

x x92( )2y y92( )2

z z92( )21 1[ ]

3 2/-------------------------------------------------------------------------------------------------25

r r92( )r r92

3-------------------- .25

J r9( )r r92------------------=3

r r92( )r r92

3--------------------2 J r9( )3 J r9( ) r r92( )

r r923

-------------------- ,35 5


Energy of Circuits

and the magnetic flux density itself is just

(A–17)

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

(A–18)

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

B r( )m0

4p------- J r9( ) r r92( )3

r r923

--------------------------------------- v9.dV9

e5

B r( )m0

4p------- I r9( )

<9d r r92( )3

r r923

------------------------------------C9

e .5

E =F,25


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

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

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,

inside the source which generates an electromotive force (emf) defined as

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:

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.

F

W F <d?1

2

e q E <d?1

2

e q =F <d?1

2

e2 q Fd1

2

e2 qF.25 5 5 5 5

E d<?C

r 0.5

E,

% E <.d?–

+

e2

inside the source

5

E <d?C

r E <d?+

–

eoutside the source

E <d?–

+

einside the source

1 0.5 5


Energy of Circuits

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 where is the force on each charge and is the

charge’s velocity through the wire. If there are free charges per unit length

moving through the wire, then the power delivered to an element of length is

since for a thin wire is in the same direction as (i.e., The

total power delivered to the complete circuit is then

Remembering that and this is just

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 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

2) by the loop (circuit 1), and using Faraday’s law, Eq. (A–8), the power

equation becomes

% E <d?+

–

eoutside the source

F.5 5

F v,? F v

n

,d

F vn ,d? nvF <d? ,5

v <d v ,d v <).d5

power delivered to charges nvF <d?C

r .5

qnv I5 rC F q/( ) <d? rC E <d? %,5 5

power delivered to chargesWd

td-------- %I.5 5

`2

%12,


Energy of Circuits

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

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

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

Figure A–5. Coil moving toward a loop

I1

I2

%2

%1B1

B2

v

power delivered to charges in coilW2d

td---------- %2I2 %12I215 5

%2I2 I2

^12d

td------------ .25

power delivered to charges in loopW1d

td---------- %1I1 %21I115 5

%1I1 I1

^21d

td------------ .25

power delivered to all chargesW1d

td----------

W2d

td----------15

%1I1 %2I2 I1

^21d

td------------2 I2

^12d

td------------ .215

^21 B21 a1d?S1

e5

A21=3 a1d?S1

e5


Energy of Circuits

where . (It is important to realize that the notation used here,

and should be read as “the effect caused by #2 at position #1.” For

example, 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 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

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

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

sources is:

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

to be held constant, the source emfs, and must increase or decrease to

A21 <1d?C1

r5

m0

4p-------I2

<2d

R21

--------C2

r

<1d?C1

r5

m0

4p-------I2

<2d <1d?

R21

--------------------- ,C2

rC1

r5

R21 r1 r225

^21, B21, A21,

B21,

R21,

^12

m0

4p-------I1

<1d <2d?

R12

--------------------- ,C1

rC2

r5

I1

^21d

td------------ I2

^12d

td------------ .5

power delivered to all chargesWd

td-------- %1I1 %2I2 2I1

^21d

td------------ .215 5

%1 %2,


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:

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

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

(A–19)

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 free charges

per unit volume of the wire, then the force on a small volume of the wire is

But recall that is just the current volume density, so

The force per unit volume is then just Recall that for a thin wire with a

current that is uniform over its cross-sectional area so

where is the length of the volume element. The force on the element is then

But for a very small length of wire (and if the wire is thin), So for a

differential length of wire, the force from the magnetic field on the current is

power needed to maintain constant currentsWconst. currentsd

td--------------------------------- 2I1

^21d

td------------ .5 5

Wconst. currents 2I1^21.5

^21

Wconst. currents 2m0

4p-------I1I2

<1d <2d?

R12

--------------------- .C2

rC1

r5

N

FD Nqv B VD .35

Nqv J,

FD J B3 V.D5

J B.3

I JA,5

FDI

A--- B3 A L,D5

LD

FD I LD B3 .5

I LD I L.D5

Fd I <d B.35


Energy of Circuits

Now, if circuit 1, is in the magnetic field created by circuit 2, then the

complete force on circuit 1 is

(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:

(A–20)

where is the vector from to The force between the

circuits can be simplified by expanding the triple vector product

so that the double closed line integral can be broken into two parts. The first is

where the relations and have been

used. Since the closed-line integral (with identical upper and lower bounds) of

vanishes:

The negative sign indicates a force of attraction.

C1, B21,

F21 I1 <1d B21.3C1

r5

F21

m0

4p-------I1I2

<1d <2d R213( )3

R213

---------------------------------------------- ,C2

rC1

r5

R21 r1 r22( )5 <2d <1.d

<1d <2d R213( )3 <2d <1d R21?( ) R21 <1d <2d?( ),25

<2d <1d R21?( )

R213

-----------------------------------C2

rC1

r <2dC1

r <1d R21?

R213

--------------------- C2

r?5

<2dC1

r <1d1

R21

-------=2 ?

C2

r5

<2dC1

r 1

R21

------- d

C2

r5

0,5

Vd <d V=?5 1 R21/( )= R21 R213

/25

1 R21/( )d

F21

m0

4p-------I1I2

R21

R213

-------- <1d <2d?( ).C2

rC1

r25


Energy of Circuits

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

to the position near circuit 2 during the time t is

(A–21)

Energy and Virtual Work

As seen in the previous section, circuits carrying currents produce forces on

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,

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:

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:

(A–22)

`2

R21

Wmech F21 R21d?`2

R21

e m0

4p-------I1I2

<1d <2d?

R21

--------------------- .C2

rC1

r5 5

Wconst. currentsd 2 Wmechd( ).5

Wconst. currentsd Wmechd Ufield.d15

Ud field Wd mech.5


Energy of Circuits

The mechanical work can be written as a magnetic force acting on the circuit

in question as Therefore

and

(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:

Energy of a Magnetic Field

The previous section showed that the energy of the magnetic field for a system

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 Eq. (A–21) becomes

(A–24)

F

Wmechd F r.d?5

Ufieldd Wmechd F rd? Ufield=( ) r,d?5 5 5

F Ufield= .5

Fu

Ufield

u-------------- .5

I <d JA <d5 JA ,d5 J v,d5

Ufield

1

2--

m0

4p------- J vd J9 v9d?

r r92-------------------------- ,

V9

eV

e?5


Energy of Circuits

where is the distance between the volume elements and and

and 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

(A–25)

The volume may be extended to include all of space, since including regions

where will not change the value of the integral. Substituting Eq. (A–11),

into this equation gives

Using the identity this simplifies to

By the divergence theorem, this can be written as

where the surface encloses the volume If the currents are finite in extent

then the volume and hence the surface can be taken to be very large so that

all points on are great distances from the currents. At the surface of the con-

tribution from the surface integral will tend toward zero because falls off as

and falls off as as seen in Eqs. (A–15) and (A–17). Therefore, the

magnitude of falls off at the rate of whereas the surface area of

is increasing at the same time at the rate of So as approaches infinity, the

surface integral vanishes. This leaves

(A–26)

r r92 vd v9,d J

J9

Ufield

1

2-- J A? v.d

V

e5

V

J 05

B=3 m0J,5

Ufield

1

2m0

--------- B=3 A? v.dV

e5

A B=3? B A=3? A B3( )=? 25

Ufield

1

2m0

--------- B B? vdV

e 1

2m0

--------- A B3( )=? v.dV

e25

Ufield

1

2m0

--------- B B? vdV

e 1

2m0

--------- A B3( ) a,d?S

r25

S V.

V S

S S

A

1 R/ B 1 R2

/

A B3 1 R3,/ S

R2. R

Ufield

1

2m0

--------- B B? v.dV

e5


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):

Field Energy and Induction Coefficients

Mutual Induction. Consider again the two circuits of Fig. A–1 on page 62. A

current flowing around the circumference creates a magnetic field. Some

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

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

in the circuit 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

(A–27)

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

Substituting the filamentary solution of the vector potential, Eq. (A–16), yields

the Neumann formula for mutual inductance (Fig. A–6 shows the geometry):

(A–28)

where 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

mutual inductance between circuits 1 and 2, is the same as that between

circuits 2 and 1, i.e.,

Ufield

e0

2---- E E? vd .

V

e5

I1 C1

S2

C2. C2

C1

}12

1

I1

----^12

1

I1

---- B12 a2d .?S2

e5 5

}12

1

I1

---- A12=3( ) a2d?S2

e 1

I1

---- A12 <2d? .C2

r5 5

}12

m0

4p-------

<1d <2d?

R12

--------------------- ,C2

rC1

r5

R12 r2 r12 ,5

}12,

}21, }12 }21.5


Energy of Circuits

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

vector potential, multiply by and then use the filamentary approximation

of Eq. (A–9), to get

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

(A–29)

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

(described below): 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

Figure A–6. Mutual inductance geometry

R12

<1d

<2d

C1

C2

I1

I2

I2 I2/ ,

I <d J v,d5

}12

1

I1I2

-------- A12 I2 <2d?C2

r 1

I1I2

-------- A12 J2? v2.dV2

e5 5

}12

1

I1I2m0

--------------- B1 B2? vd ,V

e5

}12 +12.5


Energy of Circuits

(A–30)

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:

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

by gives

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

(A–31)

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

for the mutual inductance: 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

+11

1

I1

----^11

1

I1

---- B11 s1.d?S1

e5 5

+11

1

I1

---- A11=3( ) a1d?S1

e 1

I1

---- A11 <1d? .C1

r5 5

I1 I1/

+11

1

I1I1

-------- A11 I1 <1d?C1

r 1

I12

---- A11 J1? v1.dV1

e5 5

+1

1

I12m0

---------- B1 B1? vd ,V

e5

+1 }11.5


Summary

two circuits, the magnetic flux density of the entire system is

Substituting into Eq. (A–26) gives

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

(A–32)

For circuits this can be generalized to

(A–33)

where the notation 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:

Finding the mechanical forces between the circuits, then, is equivalent to taking

the positive gradient of the field energy:

B B1 B2.15

Ufield

1

2m0

--------- B1 B21( ) B1 B21( )? vdV

e5

1

2m0

--------- B12

vdV

e 2 B1 B2? vdV

e B22

vdV

e1 1 .5

Ufield

1

2--+1I1

2}12I1I2

1

2--+2I2

2.1 15

1

Ufield

1

2-- };'I;I',

' 15

1

^; 15

1

^5

};;

Ud field Wd mech.5


Works Consulted for Appendix

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

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.

F Ufield= .5

Ufield

1

2-- };'I;I'.

' 15

1

^; 15

1

^5


dynamics and control of magnetostatic structures

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