I mil J.W R-I.C. PUBLICATIONS

NBSIR 80-2143

A11 1'05 3533flfl I

An Analysis of the Effects of

Dynamic and Static Forces

Present in the NBS SI Volt

Experiment

r;-

M. E. Cage

Electrical Measurements and Standards Division

Center for Absolute Physical Quantities

National Bureau of Standards

U.S. Department of Commerce

Washington, DC 20234

Issued September 1 980

Final Report

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AN ANALYSIS OF THE EFFECTS

DYNAMIC AND STATIC FORCES

PRESENT IN THE NBS SI VOLT

EXPERIMENT

CiCIOD

. U S~6

86 -9/lJ2

M. E. Cage

Electrical Measurements and Standards Division

Center for Absolute Physical Quantities

National Bureau of Standards

U S. Department of Commerce

Washington, DC 20234

Issued September 1980

Final Report

U S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity. Technology, and Innovation

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

i
, -
TABLE OF CONTENTS

Page

SUMMARY i

1. INTRODUCTION 1

1.1 Objectives and order of the analysis 1

1.2 Description of the SI volt experiment 3

2. CAPACITANCE AND ELECTRICAL FORCE EQUATIONS 7

2.1 Coordinate system for the balance 7

2.2 Capacitance equations for individual electrode

displacements or tilts 9

2.3 Coordinate system for the relative electrode positions ... 10

2.4 The capacitance equation 12

2.5 The force equations 13

3. THE EQUATIONS OF MOTION AND THE 111 SOLUTIONS 16

3.1 Generalized torques 16

3.2 The system in static equilibrium 18

3.3 Equations of motion and their solutions 19

3.4 Verification of the equations of motion 23

3.5 Dynamical capacitance measurements ... 24

4. THE SLOPE AND PATH INTEGRAL METHODS WITH POSSIBLE ERRORS .... 28

4.1 The slope method 28

4.2 The path integral method . 33

5. RESULTS AND CONCLUSIONS 38

A

5.1 Problems in measuring F

z 38

A

5.2 Problems in measuring C 45

5.3 Problems in measuring z

L 50

5.4 Problems in the slope and path integral methods 52
Page

ACKNOWLEDGEMENTS 54

REFERENCES 55

APPENDIX A. CAPACITANCE OF RADIALLY DISPLACED CYLINDRICAL

ELECTRODES 57

APPENDIX B. CAPACITANCE OF TILTED CYLINDRICAL ELECTRODES 61

APPENDIX C. VARIATIONS AND UNCERTAINTIES OF z,_ AND C 66

a. The laser interferometer alignment 66

b. The uncertainty in measuring Az^ 66

c. The variation of z^ for a freely-swinging balance beam .... 67

d. The variation of C for a freely-swinging balance beam ..... 67

e. The variation of z^ and C for a servoed balance beam 67

f. The electrode shape-dependence of C 68

APPENDIX D. VARIATIONS OF C WITH TEMPERATURE 71

a. Variation of C with uniform temperature changes 71

b. Variation of C as the carriage moves through a vertical

temperature gradient * 72

c. Variation of C for a radial temperature gradient across

the electrodes

APPENDIX E. DERIVATION OF THE FORCE EQUATIONS

APPENDIX F. COORDINATES OF POINTS UNDERGOING ONLY y and 3 MOTIONS

motions . . 0 . 0 y, . . y . . . . APPENDIX G. COORDINATES OF POINTS UNDERGOING ONLY y and 3

motions . . . x

. . . y ..

APPENDIX H. PROOF THAT h

1 = h° and h

1 = h°

x x y y

APPENDIX I. GENERALIZED TORQUES FOR y

y

AND 3

y

MOTIONS ONLY . . . .

a. Derivation of Q

Y,

b. Derivation of Q^

APPENDIX J. GENERALIZED TORQUES FOR y AND 3 MOTIONS ONLY . . . . x y

a. Derivation of Q

b. Derivation of Q

Yv

3 .

73

74

76

77

79

81

81

82

84

84

85
Page

APPENDIX K. BALANCE BEAM STIFFNESS AND SENSITIVITY 86

APPENDIX L. KINETIC ENERGY FOR y

y

AND 3

y

MOTIONS ONLY 89

APPENDIX M. KINETIC ENERGY FOR yx AND 3

y

MOTIONS ONLY 90

APPENDIX N. D'ALEMBERT'S PRINCIPLE FOR y AND 3 MOTIONS ONLY ... 91

-J J a. Equation of motion for the suspended electrode system 91

b. Equation of motion for the balance beam 92

APPENDIX 0. D'ALEMBERT'S PRINCIPLE FOR yv AND R MOTIONS ONLY ... 95

x y

a. Equation of motion for the suspended electrode system 95

b. Equation of motion for the balance beam 96

APPENDIX P. SOLUTIONS OF THE EQUATIONS OF MOTION FOR yw AND 3

OSCILLATIONS ONLY y

. . . y. . . . 93

a. The suspended electrode motion 98

b. The equilibrium angle y

y

100

c. The period P 101

Y

y

d. The amplitude modulations of y

y

102

e. The balance beam motion 103

A

f. The equilibrium angle 3

y

104

g. The force determination 106

h. The period P 0 109

ey

i. The amplitude modulations of 3

y

110

APPENDIX Q. SOLUTIONS OF THE EQUATIONS OF MOTION FOR y AND 3

OSCILLATIONS ONLY X

. . . y

. . . . 112

a. The suspended electrode motion 112

A

b. The equilibrium angle y

c. The period PYx

d. The balance beam motion . 114

A

e. The equilibrium angle 3

y

116
Page

f. The force determination H?

g. The period P' I 18

y

h. The amplitude modulations of By

.

.. 18

APPENDIX R. DYNAMICAL FORCES ON THE SUSPENDED ELECTRODE AS DETECTED

BY THE BALANCE 121

APPENDIX S. DYAMICAL CAPACITANCE VARIATIONS 128

APPENDIX T. ERROR CALCULATIONS FOR THE SLOPE METHOD 131

a. Method used to estimate the voltage ratio uncertainty 131

b. Calculations for Type I scans with a brass electrode at V=9 kV . 132

c. Calculations for Type I scans with an aluminum electrode at

V=4. 02 kV 138

d. Calculations for Type II scans with a brass electrode at V= 9 kV

and 6

y

=5 n 141

e. Calculations for Type II scans with an aluminum electrode at

V=4.02 kV and 0 =5" 144

f. Calculations for Type III scans with a brass electrode at

V=9 kV 146

g.

Calculations of Type III scans with an aluminum electrode at

V=4.02 kV 149

APPENDIX U. DERIVATION OF THE PATH INTEGRAL EQUATION 152

APPENDIX V. THE ADJUSTMENT OF F TO F' IN THE PATH INTEGRAL 155

APPENDIX W. A CHECK THAT THE PATH INTEGRAL IS CONSERVATIVE 156

APPENDIX X. THE UNKNOWN PART OF THE PATH INTEGRAL 159

APPENDIX Y. ERROR CALCULATIONS FOR THE PATH INTEGRAL METHOD 161

a. Method used to estimate the voltage ratio uncertainty 161

b. Calculations for Type I scans with a brass electrode at

V n =9 kV 162

c. Calculation for a Type I scan with an aluminum electrode at

V n

= 4.02 kV 167

d. Calculations for Type II scans with a brass electrode at V PI =9 kV

and e =5" 168
Page

e. Calculations for Type II scans with an aluminum electrode at

V

n

= 4.02 k V and ey

=5" 170

f. Calculations for Type III ‘scans with a brass electrode at

V

n =9 kV 171

g. Calculations for Type III scans with an aluminum electrode at

V

n

= 4.02 kV 172

FIGURES . . 174

TABLES 196

SUMMARY

An analysis of the NBS SI volt experiment has been made first to

obtain algebraic expressions for the electrical forces present on the

suspended electrode of the electrometer. These forces, and the

gravitational forces, are used in the Principle of Virtual Work and

D'Alembert's Principle to obtain the second order, non-linear,

inhomogenous , coupled differential equations of motion for the balance.

Exact, analytical solutions of these equations of motion are obtained

using small angle approximations and perturbation methods. Estimates

are then made of the uncertainties that might result in both the slope

and path integral methods in order to determine what requirements must

be satisfied to reduce the systematic and random errors of the force

determination and the capacitance measurements to acceptable levels so

that the SI volt can be determined to within a few ppm.

The mutual alignment of the four electrode surfaces is critical in

reducing the size of the horizontal forces between the electrodes. They

must be oriented at some position and then maintained over at least a

2.5 cm carriage scan such that the relative electrode tilts and

displacements vary by no more than a few seconds of arc (20 yrad) and a

few micrometers from perfect alignment in each of two orthogonal

directions.

A brass suspended electrode remains in stable equilibrium at V = 9 kV,

with a resulting vertical electrical force that is equivalent to a

gravitational force due to a 5 gram mass (m). The present aluminum

electrode could only be used at V = 4.02 kV (m = 1 gram).

i
The suspended electrode will execute simple harmonic motion about

an equilibrium angle, and the amplitude of this motion needs to be less

than 6" (30 yrad) during both the force and the capacitance

measurements. The balance beam will also execute simple harmonic motion

about its equilibrium angle. This equilibrium angle should be such that

the plane defined by the balance beam knife edges is determined to be

within 16" (80 yrad) of gravitational horizon when the masses of both

sides of the balance are matched to within 150 yg. These conditions are

for the aluminum suspended electrode at V = 4.02 kV. They are 7" (35

yrad) and 65 yg, respectively, for a brass electrode at V = 9 kV.

The capacitance changes by 1 ppm if the electrode overlap changes

by 0.04 ym. Therefore, the balance beam cannot be allowed to swing

during the capacitance measurements; it must either be mechanically

braked or electronically servoed. The errors involved in measuring the

electrode overlap are much smaller if the upper part of the laser

interferometer is attached to the suspended electrode system. The

average temperature, and the radial and vertical temperature gradients

need to be maintained to within a few millidegrees Celsius during the

measurements.

The capacitance values are very sensitive to the electrode shapes;

so there may be large curvatures and changes in shape of the capacitance

versus electrode overl ap pi ots. Thus, the slope might not be determined

with sufficient accuracy to use the slope method.

The forces are measured at high voltages; so, in the presence of

horizontal forces, the suspended electrode equilibrium angle becomes

greater - resulting in capacitance values that are actually larger than
those measured at the low voltages of the capacitance bridge. Also, the

contribution of the horizontal components of the path integral are not

measured in the present experiment, and thus they contribute

significantly to the uncertainty. It is estimated that the uncertainty

in comparing the SI volt value with the NBS volt value might be ±4 ppm.

Experience gained with actual data could increase or decrease this

uncertainty estimate by several ppm.

iii
'
1. INTRODUCTION

1.1 Objectives and order of the analysis

The motivation for this work is first to determine what static and

dynamic external forces could be present on the suspended electrode of

the cylindrical capacitor used in the NBS SI volt experiment of Dr.

F. K. Harris. (This experiment is briefly explained in Section

1.2.) Then, to examine how the presence of these forces will affect

the force determination measurements of the balance, the capacitance

measurements, and the path integrals, and to determine what

requirements must be satisfied in order to reduce the resulting

systematic errors in these measurements to acceptable levels.

The first step towards achieving these goals is presented in

Section 2, in which an algebraic equation is developed that adequately

predicts the capacitance values as a function of the electrode

overlap, the electrode tilts, and the relative electrode radial

displacement. The electrical forces between the electrodes are then

obtained from partial derivatives of this capacitance equation. These

forces are used in the Principle of Virtual Work and D'Alembert's

Principle [1,2] in Section 3 to obtain the equations of motion for the

balance. Analytical solutions of these second order, non-linear,

inhomogeneous, coupled differential equations are obtained using small

angle approximations and perturbation methods. Systematic errors in

the balance measurements due to unwanted electrode forces immediately

follow from the solutions to these equations of motion. The

1
electrical forces are integrated over possible paths of the moveable

electrodes in Section 4 to obtain an algebraic expression for the

electrical work done by the system. This path integral equation is

equal to the change in energy stored in the electric field between the

capacitors, which is a function of the voltage applied to the

electrodes. Systematic errors in the force determination measurements

and in the capacitance measurements are then incorporated in path

integral calculations to estimate the effects of these errors in the

voltage determination. Section 5 then summarizes the results of this

work.

This paper pertains to a particular experiment. However, the

algebra in Section 3 can be applied to any balance measurement in

which horizontal forces are present on the object to be weighed; the

object may be at rest or swinging.

Liberal use is made of appendices so that the sections will be

more straightforward and less cluttered with derivations and

equations. The figures and tables are grouped together because many

of them are continually referred to in the text.

SI units are used throughout the paper; but, to avoid

entering powers of ten into numerical calculations, and to readily

comprehend the magnitudes of physical quantities present in the volt

experiment, multiples and submultiples of SI units are employed. As a

result, an equation may contain lengths expressed in cm and relative

displacements in ym. The multiples are clearly stated in the

equations, so they can be readily transformed to whatever multiples

the reader finds appropriate.

2
1.2 Description of the SI volt experiment

The objective of this research is to realize the SI volt in terms

of the units of mass, length, time (kg, m, s) and the assigned

magnetic constant (4tt x 10~ 7 H/m) to an uncertainty of 1 to 3 parts in

10^ - or equivalently, to determine the NBS volt in terms of the SI

volt to this accuracy. This experiment requires a considerable effort

in accurate and precise electrical and dimensional metrology.

The present assignment of the volt in terms of the SI ampere and

ohm determinations is uncertain by more than 5 ppm [3]. A direct

determination of the volt should not have the same systematic errors

as those present in the SI ampere experiments - and, since no

electrical energy is converted to heat as in the latter experiments,

it may be possible to achieve smaller random uncertainties. An

improvement in the assignment of the SI volt could provide an absolute

value of 2e/h, which would be useful for Josephson effect

volt-maintenance applications throughout the world. It also might

resolve the inconsistencies present among the measured values of a

number of fundamental constants: for example, the Avogadro and

Faraday constants [3].

The method uses a cylindrical electrode suspended from a balance

beam as shown in Figure 1. The suspended electrode occupies the

annular space between two coaxial cylinders that form the high voltage

electrodes. When voltage is applied between the concentric electrodes

and the grounded suspended electrode there is an electrical force on

the suspended electrode that acts vertically downward in the z

3
direction. This force, F

z

, results from fringing of the radial

electric field lines at the lower rim of the suspended electrode, and

_2

for the NBS apparatus has a value of 4.91 x 10 N at 9 kV. (There i

also a net force -F acting upward at the upper rims of the high

voltage electrodes which need not be considered.) This electrical

force has the same magnitude as that of a gravitational force due to

increasing the weight of the suspension system by the addition of a

mass m of 5 grams to a weight holder with voltage-off. The electrical

force, F , can therefore be compared with a known weight, mg, by a

substitution procedure where g is the local value of the acceleration

of gravity at the weight holder.

The 2-knife balance beam generally follows the design of the NBS

No. 2 kilogram comparator [4] - except that the inter-knife separation

has been increased to 7.5 cm. The suspension knife shown in Figure 1

actually consists of a load knife and an intermediate knife crossed ir.

the same horizontal plane; so the suspension system is free to pivot

with the same lever arm about any direction. A resolution and

stability of 1 yg has been achieved by Dr. Harris with this balance

for voltage-off when using an aluminum electrode with a resulting

suspension system mass of 2.35 kg.

The two concentric cylinders (henceforth referred to as the carriagelectrodes) rest upon an electrically insulated stand on a moveable

carriage that can be raised or lowered by a decoupled elevating shaft.

Vertical alignment of the carriage is maintained by means of teflon

bearings riding on three stainless steel guide rods positioned in a

ring spaced 120° apart. The carriage displacement, az, is monitored

4
by a laser interferometer system having a reproducibility of

±0.025 ym.

The entire apparatus is enclosed in an 0-ring sealed container and

will be maintained with an overpressure in an environment of either

dry air or sulfur hexafluoride. The container is surrounded by

thermal insulating material to minimize temperature fluctuation

effects.

There are two ways of obtaining the SI volt - the slope method and

the path integral method. In the slope method the electrical force at

any point z is

F z = mg = -(1/2) V

2 (9C/9z),

where C is the capacitance, 9C/9z = 12.12 x 10

10 F/m (12.12 pF/cm)

,

and at V = 9,000 V, F

z = 4.91 x 10" 2

N and F

z /g = m = 0.005 kg.

Therefore, V can be obtained by measuring m, g, F

z and 8C/9z at point

z. In the path integral method the electrical work required to move

the electrodes equals the change of electrical potential energy stored

in the field between the electrodes; in an ideal situation

mgdz = (1/2) V

2 (C

f

and V is obtained from the path integral and C. and C^. The three

terminal capacitance value, C, is measured at 100 volts a.c. in both

methods with a transformer ratio-arm bridge [5] having a resolution of

5
o one part in 10 . The capacitance has been previously related to the

SI unit of length by using a computable cross-capacitor [6].

The SI value of the applied voltage can thus be determined in both

methods from the measured quantities. (The same quantities are used

in both methods, but in different ways.) The comparison between the SI

value of the volt and the NBS legal volt, maintained at the one-volt

level via the Josephson effect, will be made using a resistance

voltage divider whose ratio uncertainty is 0.2 ppm [7] and a group of

calibrated standard cells.

The above equations are for perfect right circular cylinders -

exactly centered, and aligned parallel to the gravitational field. I

these conditions are not met then F

z is not a constant at constant

voltage and 3C/3Z also is not a constant. The analysis will usually

assume that the electrodes are right cylinders, but not necessarily

perfect, and will study the effects of electrode misalignments.

Arguments will be given that the path integral method may be the more

promising one.

The electrodes are gold plated to provide an inert, oxide-free

surface. There may be migrating dielectric films on the electrode

surfaces that could affect the 100 volt a.c. capacitance

measurements, which in turn might be different from the assumed same

capacitance values for the kilovolt d.c. force measurements. There

could also be corona discharge from dust on the electrode surfaces,

and there may be resonances in the capacitance values for capacitance

measurements using frequencies near a mechanical resonance of either

the electrodes or their support system [8]. The possible problems of

this paragraph will not be addressed in this paper.

6
2. CAPACITANCE AND ELECTRICAL FORCE EQUATIONS

2.1 Coordinate system for the balance

There are two coordinate systems used in the calculations. The

one for the balance is presented in this section, while the one used

to completely describe the combined relative electrode orientations is

given in Section 2.3. Figure 2 shows the balance coordinate system,

with the origin located at the center knife position (C). The z axis,

which is parallel to the gravitational field, points downward so that

increasing vertical electrode separations correspond to increasing

laser interferometer readings. The y axis is parallel to the

direction that the balance beam would have if the plane defined by the

two knife edges at contact points (P) and (C) were at gravitational

horizon. The x axis is directed into the figure so that the

coordinate system will be right-handed. Cartesian coordinates are

used because the cylindrical symmetries are broken by the balance

beam.

Point masses are shown for the suspension system, the balance beam

and the counterweight; they will be given distributed masses when

required. Equations for the electrical forces F

z

, F^ and F^ will be

derived in Section 2.4. The superscri pts , I and 0, refer to the inner

and outer electrode surfaces respectively.

Normally a gimbal would replace the connecting ring shown in

Figure 1: but, the equations of motion would be unneccessari ly complicated for that case. Therefore the equations of motion in Section 3

involve a single, eliptical pendulum rather than a double pendulum.

This simplification will not affect the final conclusions.

7
Each electrode surface has an electrical symmetry axis, which is a

straight line for a right cylinder or a conical section. Figure 2

shows the combined symmetry axis of the four electrode surfaces when

they are all aligned parallel to the z axis and when 3 y

equals zero.

If any electrode surface significantly deviates from radial symmetry

then the combined symmetry axis is a curved, three-dimensional line.

The carriage electrodes can be displaced from the symmetry axis b>

the distances d and d and tilted about the symmetry axis by e and

x y x

0y. The suspended electrode can be displaced from the symmetry axis

by Dy as a result of the balance beam being at an angle 3 yfrom

gravitational horizon. The suspended electrode can also tilt by the

angles a y> and y . Note that the quantities 3 wJ y , y , D w , 0,6, x 'y ' y

* 1 x ’ y

* y x * y

d

x

, dy, and all have signs associated with them, and of those

shown in Figure 2, all are given positive signs except for d

y

- which

is negative there - because carriage electrode displacements have

the opposite effect of suspended electrode displacements. The

displacements and tilts are greatly exaggerated in this figure, which

is also not drawn to scale.

Table 1 lists the length and mass values of quantities shown in

Figure 2, assuming either an aluminum or a brass suspended electrode.

The value of the electrode overlap for 0 , 0 , y and y equal to zero

x y x y

(L

q

=L^=L^) is assumed to be L Q

= 10 cm at C = 100 pF. Measurements

have not been made of this value, but it is actually quite close to 10

cm - and the calculations are insensitive to changes in L Q

. Note that

the carriage position is indirectly given in this coordinate system

via L

1 and L*^. o o

8
The effects that the knives at points (C) and (P) of Figure 2 have

on the balance arm lengths and Lg are shown in Figures 3 and 4;

they are assumed to have radii of curvature p

c and pp

, respectively.

The best knife edges have radii of 0.25 ym, but they can be larger [9].

-5

The value 5 x 10 cm (0.5 pm) will be used for both knives; also,

they are assumed to be symmetrically honed. Note in Figure 3 that the

origin of the balance coordinate system moves in the ±y direction with

the center knife contact point. This choice simplifies the algebra,

but the coordinate system is not an inertial frame of reference;

however, since the center knife horizontal accelerations are so small,

it is essentially an inertial frame.

2.2 Capacitance equations for individual electrode displacements or

tilts

Equations are derived in Appendices A and B for the capacitance of

the electrodes used in the SI volt experiment when either the carriage

electrodes or the suspended electrode are displaced from, or tilted

about, the combined electrode symmetry axis. The results for

individual electrode displacements or tilts are

C(d,z

L

) • C(z

L )[l + (6.5 x 10'9

)d 2

] (2-1)

C(D

y

,z

L

) = C(z

L )[l + (6.5 x 10

-9

)D

y

] (2-2)

C(6, z L

) = C(z

L )[l + (6.5 x 10‘ 9

)d

2

] (2-3)

C(y,z

L

) = C(z

L )[l + (6.5 x 10" 9

)d

9

] (2-4)

9
where

i

(2-5)

(2-6)

(2-7)

(2-8)

[C in pF; d, D , d

t>

d

T

in um; z L

,

l_ A

, t, L, L Q

in cm; B y

, 6, T in

rad]

with d

2 » d

2 + d2

e 2 = e 2 + e

2

, and y

2

= y

2 + y2

. The quantities ar

a y a y x y

defined in Appendices A and B and in Figures 2 and 5.

2.3 Coordinate system for the relative electrode positions

Section 2.2 gave the capacitance equations when either the

suspended electrode is displaced or tilted, or the carriage electrodes

are displaced or tilted. The problem is how to combine these effects.

If, for example, d^ = -D (as defined in Figure 2) then Equations 2-1

and 2-2 would predict that C * C(z

L )[l + 2(6.5 x 10

-9

)dy], when

actually C = C(z

L

) because all the electrodes have been translated in

the positive y direction by the same amount D . If, on the other

hand, d^ = then Equations 2-1 and 2-2 would again predict that C ~

C(z

L )[l + 2(6.5 x 10

-5

)d^], when actually C = C(z

L )[l + 4(6.5 x _9 2 -. 10 )dy] because the electrodes have been radially separated by the

C(z

L

) = [100 pF - (12.12 pF/cm)z

L

]

Dy = 1°

4

(LA/2)B

2

d

t = 10

4

[* - L 0 (z

l )/2]

0

d

T

= 10

4

[(L - L 0 (z

L )/2]y

10
amount 2d^. Clearly, the electrode displacements and tilts must be

expressed in relative coordinates.

Figure 6 shows the suspended electrode and the outer carriage

electrode each displaced from, and tilted about, the combined

electrode symmetry axis. It is shown in Appendix B that the

effective displacements due to electrode tilts, d

t

and dy, can be

vertically located at half the electrode overlap - the same as for d^

and Dy. We can therefore add these quantities to obtain the total

relative horizontal displacements, h^ and h[J, of the electrodes from

the combined symmetry axis:

h° = 10

4

[L - L

q

(z

l )/2]Yx + 10

4

C*. - L

0

(z

l )/2]9x + d

x

h° = 10

4

(LA/2)g2 + 10

4

[l - L

o

(z

L )/2]Ty +

10

4

ft - L 0

(z

L )/2]e

y

+ d

y

[h, d in ym; L, £, LQ

, L^, in cm; 3, Y, 6 in rad].

Note that the angles 3, Y and 6 all have signs associated with them as

defined in Figures 2 and 6; so the signs of individual displacements

are automatically accounted for.

If the inner and outer surfaces of the suspended electrode and the

carriage electrodes are parallel, then it is clear from the form of

Equations 2-9 and 2-10 that

h x

= h x

= h

x and h

y

= h

y

= h

y

. (2-11)

(2-9)

( 2 - 10 )

11
These equalities will be more rigorously derived in Appendix H.

We now have a coordinate system, shown in Figure 7, that

completely describes the relative orientations of the electrodes in

terms of the relative coordinates h

x

, h^ and z L

, where z L is the lase

interferometer reading, z^, relative to the initial value z^ at C

=

100 pF and h

x = hy = 0. This coordinate system should not be thought

of as being attached to any object in Figure 1; all the coordinates

in Figure 7 are relative quantities derived from Equations 2-9 and

2-10, and from the relationship z L

= z

£

- z^. The forces shown in

Figure 7 will be derived in Section 2.5.

2.4 The capacitance equation

The capacitance, as a function of h x

, h^ and z^, immediately

follows from Equations 2-1, 2-9, and 2-10:

C(h,z

L

) = C(z

L )[l + (6.5 x 10" 9

)h

2

] (2-12)

C(z

L

) = [100 pF - (12.12 pF/cm)z

L

] (2-13)

[C in pF; h in pm; z

L in cm]

? 2 2 where tr = hj + hj.

This equation ignores any effects due to the rounded electrode

rims that might not be included in the 100 pF calibration point at L

Q

=

10 cm. It also assumes that the suspended electrode does not

12
significantly interact with the horizontal surface located at the

bottom of the carriage electrodes. All that will be required for this

paper is to predict the magnitude of the capacitance to within

10 - 20% of the actual value, so that the effects of dynamical forces

can be studied. It is the capacitance variations that will be of

concern - not the absolute values.

Appendix C demonstrates the sensitivity of the C(h,z^) values to

changes in z^. If the electrode overlap changes by ±0.041 ym when

C = 50 pF, then the value of C(h,z

L

) changes by ±1 ppm. So the

balance beam cannot be allowed to swing during capacitance

measurements. This appendix also shows that the capacitance values

are very sensitive to the shapes of the electrode surfaces. The

calculations of Appendix D further suggest that the temperature

distribution inside the 0-ring sealed container must be maintained to

within a few millidegrees Celsius during all of the data collection

periods in order to keep the resulting capacitance variations below

the 1 ppm level.

2.5 The force equations

The equations for the h , h and z, components of the electrical

x y l

force are derived in Appendix E. Using Equations E-3 to E-5 and

Equations 2-12 and 2-13, and converting the quantities to similar

dimensions yields the results

F

h (h,z

L

) = (3.25 x 10'9

)V

2

(h,z

L )C(z

L )h

x (2-14)

X

13
L C(z

L

) = [100 pF - (12.12 pF/cm)z^]

[F in N; V in kV; C in pF; h in pm; z

L in cm]

in the relative coordinate system of Figure 7; they will be used in

evaluating the path integrals in Section 4.

In order to solve the equations of motion in Section 3 we will

need expressions for the electrical forces which act on the suspended

electrode. Referring to Figure 2, they are:

these are the net forces which remain after summing the incremental

forces over the surfaces of the suspended electrode.

The vertical force F

z is placed at point 4 in Figure 2 at the

center axis of the suspended electrode rim since the vertical forces act

respectively, at one half the overlap distance of each electrode

because this was found in Appendix B to be the centroid locations of

(2-17)

(2-18)

(2-19)

where F° = F

1 because h® = hj and C°(z. ) C

1 (z. ). Note that

*>y x *y x jy l l

primarily at the rim. F^ and F^ are placed at points 5 and 6,

14
the inner and outer capacitance contributions. F and F are placed

at analogous points in the x-z plane.

We can now express the vertical electrical force F

z in terms of

the independent variables y , y and 3 of Figure 2 by using Equations

x y y

2-16, 2-19, 2-9, 2-10 and 2-11. We will next solve the equations of

motion in Section 3 to see if the dynamical balance measurements of

F

z are affected by the presence of the horizontal forces.

15
3. THE EQUATIONS OF MOTION AND THEIR SOLUTIONS

3.1 Generalized torques

The equations of motion for the balance and suspended electrode

system will be obtained using the Principle of Virtual Work and

D'Alembert's Principle [1,2]. The first step is to determine the

(x • , y.j , z.j) coordinates of the points in Figure 2 which locate the

positions of the gravitational and electrical forces M^g, M^g, M^g,

F z j F^, F*, fJ and F y

. These coordinates are listed (using Figures 2,

3, 4, 8 and 9) in Appendix F for and 3

y

motions only, and in

Appendix G for y and 3 motions only.

* y

In general there would be simultaneous y

x

, y

y

and 3 ymotions for

particular K values of d v , d w , 6 and 0 - and the variable y would be

x’ y* x y

more appropriate than y and y . However, an expression such as y,- in

x y s Appendix F would be much more complicated, since one would have to

trace from point 5 around the outer suspended electrode surface until

it intersected the y-z plane at x = 0. To avoid this solid geometry

problem, we will assume special cases in which only F* and F^ or F^

and Fy are present, and will ignore the D displacements when

considering the y and 3 V motions. These special cases are similar to

x y

the normal mode motions, and should adequately cover the physics of

the real system.

Appendix H uses Figure 2 and coordinates in Appendices F and G to

more riguorously prove that h^ = hj = h and that h^ = h.

1 axa . = h . It

y y y

then follows from Equations 2-14, 2-15, 2-17 and 2-18 that F^ = fL A A and that F y

= F y

. The forces, and their locations, are thus known in

16
terms of either the independent variables Y and 3 or the variables

y y

Y and 3 . We will next use this information in the Principle of

x y

Virtual Work to obtain the generalized torques.

The Principle of Virtual Work states that:

6

6W = £ C ( F-j

)

x 6x

i + ( F i) y

+ ( p i) z 6z

i^

( 3-1 ) i=l

where for Y

y

and 3

y

motions x

i = x

i (Y

y

,B

y

), y^ = y.j(Y

y

, 3

y

), z. =

z^ (

Y

y

» 3

y

) , and the index i represents any of the six points shown in

Figure 2; so

6x

i = (8xi /9y

y

)6Y

y

+ ( 3x

i /93

y

) 63

y

, etc. (3-2)

Substituting Equation 3-2 into Equation 3-1 yields

6

6W C(F

i ) x

(6 x

1-/<SY

y

) + (F

1

) y

(3y

1 /aYy ) + (F

1

) z (3z

i /3Yy )]6Yy + £ C(F

i

) x (ax

i /93

y

) + (

F

n

- ) y

( ay-j /93

y

) + ( F

i

) z

( 3Z

i / 83

y

) H 6B

y

(3-3)

i=l

or

6W = + qey

6e

y

(3-4)

where Q and Q. are the generalized torques. Equations 3-3 and 3-4

Y

y

6 y

can also be used for y^ and 3

y

motions by replacing y

y

with The

generalized torques for y

and 3 y

, and y

x and 3 motions are given ir y y y

Appendices I and J respectively.

17
3.2 The system in static equilibrium

We can use the generalized torque equations derived in Appendices

I and J to study several properties of the system when both the

balance beam and the suspended electrode are in static equilibrium,

A A A with equilibrium angles 3 , Y and y respectively. For this

y X y

situation Q = Q = Q

= 0.

Y x

Yy y - Equations 1-3 and J-3 can be used to solve for y and y

- X y

A /\ /\ which have identical forms if 3 =0. Values of y and y are listed

y x 'y

for various combinations of d and 6 or d w and 0 in Tables 3-7,

x x y y

using either an aluminum cylinder in the suspended electrode system

(Ma

= 2.35 kg), or a brass cylinder (MA= 5.47 kg). The voltage

values for these tables are such that the electrical forces, F

z

, are

equal to the gravitational forces, mg, for integer masses between 1 and

5 grams. The suspended aluminum electrode is in unstable equilibrium

for voltages larger than 6.3 kV; j_.e. it would swing outward until it

collided with the carriage electrodes. The aluminum electrode is

stable at V = 4.02 kV, but the 1 gram mass could only be determined to

within 1 ppm. It is therefore preferable to use the brass electrode,

which is stable at V = 9 kV. L in this case is 2 cm below the top

of the suspended electrode rather than 6 cm above the top.

A A /\ The h and h (j_.e. h ) values listed in these tables are

A y

x ,y obtained from Equations 2-9, 2-10 and 2-11; the 2F^ values from

X 5j

Equations 2-14, 2-15, 2-17 and 2-18; the A

F

z /Fz values from Equations

A 2-16 and 2-19; and the aC/C(z

l

) values from Equations 2-12 and 2-13.

The 6(C/C(z^)) numbers are the differences in the AC/C(z^) values

between the appropriate kilovolt values used in the force

18
determinations and the V pms

= 100 volt value used in the

A A a.c. capacitance measurements (in which y and y are essentially

, x y

zero, so that the 10 (L - L Q /2)yx term does not contribute to h

x or

* ^0

h w . F and F are large fractions of F_ because most of the

y x y

z electric field lines are radial; thus, small horizontal displacements

produce large horizontal forces.

Autocollimator measurements of e and 6 each yield smooth

x y

variations of several seconds of arc over a carriage travel span of a

few centimeters, with a 1" repeatability. The variations of d and

dy are not presently known. It will be assumed in many of the sample

calculations which follow that a brass suspended electrode is used,

with V = 9 kV, d

x

= d = 0 and e

x or 0 = 5". This may yield values

of h or h that are better or worse than the actual relative

x y

displacements, depending upon the sizes of d and d , but it should

x y

provide reasonable information about any potential problems.

Appendix K derives the balance beam sensitivity equations, and

shows that there should be no difficulty in obtaining a 1 yg

resolution when using the brass suspended electrode at V = 9 kV.

There would, however, be more stresses induced on the balance when

converting between the voltage-on and the voltage-off configurations

with these larger masses for and M

g

. The F^ dependency of the

balance sensitivity can be avoided if the balance is used as a null

device by adjusting the voltage so that 0

3.3 Equations of motion and their solutions

D'Alembert's Principle states that

19
(d/dt ) ( 3T/3x) - (8T/3x) = Q

X

(3-5)

where x represents the generalized coordinates y , y or $ . The

x y y

generalized torques, Q , were found in Appendices I and J. The

X

kinetic energy, T, is also required - and is evaluated in Appendices L

and M for y and r or yx and R motions, respectively. Equation 3-5

y <y y

is used in Appendices N and 0 to obtain the equations of motion for

p small angles; the factor 10 appears on the l.h.s. of Equations N-l,

N-4, 0-1 and 0-4 in order that both sides of the equations will have

the dimensions N-cm. The equations of motion (Equations N-2, N-5, 0-2

and 0-5) are second order, non-linear, inhomogenous , coupled

differential equations.

Appendices P and Q give the solutions of these equations by use of

a perturbation method. The suspended electrode executes simple

/\ A harmonic motion about the equilibrium angles y and y , with very

x y

small cos2oj t and cosoo t amplitude modulations. The period, Py , of

%

y

this pendulum-like motion is voltage-dependent, and as is shown in

Figure 10 for a brass suspended electrode, increases from a value of

1.48 s at V = 0 kV to a value of 2.82 s at V = 9 kV.

The balance beam also executes simple harmonic motion about the

A

equilibrium angle 8 > with very small cos2ooy t and cosw t amplitude

y

y y

modulations. The period, P. , of this motion is also

ey

voltage-dependent, and for a balance sensitivity of O.ll'Vug increases

from a value of 44.9 s at V = 0 kV to 46.8 s at V = 9 kV for a brass

electrode. Since the pendulum and balance beam periods are so

different, only very small energy transfers occur between these

20
motions. Hence, the y and 3

y

amplitudes do not alternately, and

periodically, increase and decrease with time.

A A

The equilibrium angle 3 depends upon F , and therefore upon

s ^ y z Yx and y for voltage-on measurements. There is a dynamical

/N /V systematic error, 63y, in 3^ which depends upon the voltage, the

~2 ~2 ~2

electrode overlap, and upon y

= Yx + Yy. Figure 11 shows a plot of

(5 ^ (ON) versus y for a b rass suspended electrode at V = 9 kV and Y

x =

j y

0; y must be less than 2.5" in order to measure m 5 the mass

/A equivalent of F , to within 1 yg at V = 9 kV. This systematic error is

zero for voltage-off measurements.

A

F

z is measured by first adding mass m to the weight holder and

determining 3

(OFF) for voltage-off. Mass m is then removed and the

voltage adjusted until 3 differs from mg by two

systematic error terms. The dynamical error term involves the

~2

quantity y

- and, as was discussed above, y should be less than 2.5"

for the brass electrode at V = 9 kV if m is to be determined to within

1 yg.

/v /\ ( OFF

)

The relative static error for F7 is (2F„/mg)3 v ' rO . F

y

15

z

-- , -y-.'-y

predetermined by the relative electrode alignments and displacements

* 2

( i . e

by h ) and by V . Therefore, this error term must be minimized

by adjusting 3, (OFF) to be near zero. If the brass electrode is used

at V = 9 kV, and if d = d = 6 =0 and 9 = 5", then the plane

x y x y

defined by the balance beam knife edges must be determined to be

parallel to within ±7" of gravitational horizon in order for the

/N static error term of F

z to be less than 1 ppm - and, if the

counterweight adjustment for obtaining this 3

(OFF) reading is incorrect

by 64 yg, then an additional maximum 1 ppm error could occur. In order to

21
achieve a 1 yg in 5 gram precision for this example, the knife edges

plane must be determined to within ±1", and Mg (or M^) must be

adjusted to within 10 yg to obtain the balance beam readout angle

value that corresponds to gravitational horizon. This adjustment

could be made electronically if the balance beam was servoed.

This ±1 yg in 5 gram precision would be very difficult to achieve

with the brass suspended electrode at V = 9 kV. If an aluminum

electrode was used at V = 4 kV and 6y

= 5", then Table 3 provides the

information that the knife edges plane must be determined to within

16" of gravitational horizon in order to determine F

z to within 1 ppm.

If Mg was out of adjustment by 150 yg then another 1 ppm error could

occur. An additional 1 ppm dynamic systematic error arises if y

= 6"

for this case. Results for a brass electrode at V = 4 kV and

6^

= 5"

are only marginally better than these aluminum electrode requirements

.

Because of these systematic error terms, these sample calculation

suggest that little, or nothing, may be gained in the force determinati

measurements by increasing the masses and Mg. The brass

electrode would, however, provide a better safety margin against

unstable equilibrium, and it also would enable data to be obtained at

voltages larger than 4 kV.

The exact requirements for the readout angle determination and the

Mg or M^ adjustment depend upon the largest possible combination of

d

x

, dy, @ x and

6^ that occurs during a az^ scan. These = 5"

examples may, or may not, give an upper limit for the 3

(OFF)

requirements: 6 and 6 may each be about 3", and d and d w are

x y

J x y

unknown. Also, all the electrodes are assumed to have radial

symmetry, and to have been perfectly aligned at some point z^; i .e.

22
the capacitance value at that point was at the absolute minimum and

not at a local minimum. The horizontal forces would thus be zero at

that point. If this were not the case, then a A0

y

= +3" variation, for

example, would produce 0^ values larger or smaller than 3", depending

upon the initial value of 0 .

The systematic error terms of F

z = mg do not occur for normal

mass intercomparisons because mg is constant for a swinging balance

pan and there are no horizontal forces. In this experiment, however,

s

F

z is not constant and horizontal forces do occur in the voltage-on

measurements.

3.4 Verification of the equations of motion

The solutions to the equations of motion are verified using

Newton's second law of motion in Appendix R by investigating the

time-dependence of the vertical components of the external forces

induced on the moving suspended electrode as shown in Figure 12. The

shapes and frequency dependence of these forces (plotted in Figures

13-17 versus time for a brass electrode at V = 0 kV and at V = 9 kV)

agree with the predictions of Appendix P, but the predictions from

Equations P-21 and P-22 for the magnitudes of the Am mass variations,

as measured by the balance, do not agree with those of Figure 18

because the balance acts like a low-pass frequency filter since

to »oo

B

. This frequency filtering explains why the amplitude modulations

7 y

of the simple harmonic 8

y

waveform are so small. Therefore, the only

significant problem with the pendulum being in motion during the force

23
measurements is with the dynamical error term of F^ due to the Y

quantity affecting the value of 3^^.

3.5 Dynamical capacitance measurements

The suspended electrode executes simple harmonic motion about the

A

equilibrium angle Y. At the V rms

= 100 V voltage value for the

A A capacitance measurements, Y

x and Y are essentially zero.

The time variation of capacitance is given by Equation S-2 of

Appendix S, and from that expression the quantity 6C(t)/C(z^) = [C( t

)

- C(z^)]/C(z^) is obtained in Equation S-3. This quantity is plotted

versus time in Figure 19 for a brass suspended electrode at C(z^) =

100 pF and d = d w = 0 =0, assuming various values of Y and 0 . K x y x

’ 3 y y

The only difference for' an aluminum electrode would be that the

pendulum period decreases from 1.5 s to 1.3 s.

Perfect electrode alignment produces a pure cos2ul, t signal.

y

The cosooy t component becomes more enhanced with increasing electrode

y

misalignments and relative displacements. Therefore, the capacitance

A will increase symmetrically about Y = 0 only for perfect electrode

alignments. So, dynamical capacitance measurements could be used for

a deliberately swinging suspended electrode to find the absolute

minimum in capacitance at some point z

L by maximizing the cos2w t y

signal via adjustments of d x

, d

y

, 0

x and 0 .

The pendulum amplitude must be quite small when making actual

capacitance measurements - and, the larger the electrode

misalignments, the smaller this amplitude must be. In general, y

probably needs to be less than 5" in order to measure the static

24
capacitance value to within 1 ppm if the bridge detector averages

C, or less than 10" if the minimum value of C is used.

The dynamic capacitance values could only be safely used if the

pendulum amplitudes were similar to those used in the force

determination measurements; e.g. a 2.6 ppm error would occur if

By = 5", and if y

= 10" in one type of measurement and

y

= 0 in the

other type.

We found in Appendix C that the balance beam must be either

mechanically braked or electronically servoed so that the electrode

overlap does not change by more than +0.041 pm in order not to change the

capacitance to within ±1 ppm at C = 50 pF. If a mechanical brake is

used with its axis parallel to the center knife axis, and if it should

rotate the balance beam about the z axis, then this rotation will

produce a suspended electrode displacement, D , of ±0.36 pm for each

A arcsecond of rotation, with the sign depending upon the direction of

rotation.

This effect, if present, could be monitored by mounting the

balance readout angle mirror in the x-z plane rather than the x-y

plane; any balance beam rotation would thus cause a parallel shift in

the light beam at the readout detector position. This mirror

arrangement could also detect any balance beam rotation that might

result from the balance being placed in arrestment [4] and then

released.

If the upper part of the laser interferometer is not connected to

the suspended electrode system, and if the brake stopped the balance

beam to within ±0 . 11 " of the 3

'

' readout angle for the capacitance

measurements, then there would be a ±0.041 pm uncertainty in the z L

25
position of this capacitance reading, in addition to the ±0.025 ym

laser interferometer reproducibility. In a Az^ = 2.5 cm scan between

capacitance measurement endpoints, this could give a maximum error of

±5 ppm in az^. The error could be as large as ±3 ppm for az

l

= 4 cm.

The most probable errors for these two examples would be ±3 ppm and ±2

ppm respectively.

The same errors would also occur for a balance beam servoed to

within ±0 . 11 " of the 3^^) readout angle - providing the zero-point

level of the servo was known be stable enough to not cause additional

shifts on the order of 0 . 11 " during the course of the measurements.

Therefore, it would be preferable to mount the upper part of the laser

interferometer on the suspended electrode system.

It does not matter if the capacitance values are measured at

exactly the same z

L points as those used in the force determination

measurements - as long as the force is nearly constant, or is slowly

varying over reasonable changes in z L

. Therefore, a capacitance

measurement could be made at any braked position, and the z^ value

correctly measured if the laser interferometer has the latter

configuration. The capacitance could also be measured while using a

servoed balance and this interferometer configuration, with the

possible ±0.041 ym variations in the z^ readings being monitored by

the interferometer; z

L could thus be determined to within ±0.025ym.

Any zero-point drifts of the servo would automatically be accounted

for.

Similar z^ variations could occur during the force determination

measurements when using a servoed balance beam. If the balance beam

A was freely-swinging with a

3^ = 4" amplitude, then z^ would vary by

26
±1.5 ym (±150 times the 0.01 ym laser interferometer resolution)

during the 45 s period. If the force varies slowly with z

L

, then the

assigned value of z

L for each measurement is not very critical.

Therefore, the balance might be allowed to swing freely during the force

A

determination measurements provided that is small enough, and

provided that the electrodes are nearly perfect right circular

cylinders (so that the capacitances and the forces are not too

dependent upon the electrode shapes).
4. THE SLOPE AND PATH INTEGRAL METHODS WITH POSSIBLE ERRORS

4.1 The slope method

Using Equations 2-19 and E-5 - and converting to appropriate

multiples and submultiples for V, C and z L

- yields the equation

F 7 (h,z.) = -(1/2)(1 x 10"4

)V

2[3C(h,z. )/3z. ]r

j; (4-1)

n x

,ny

[F in N; V in kV; C in pF; z L in cm; h in urn].

F

z is the equilibrium value for the vertical component of the electrical

A

force induced on the suspended electrode at the equilibrium point (h,z^;

of the relative coordinate system. It is assumed to be equal to mg in

the force determination measurement when the measured voltage, V is

adjusted until g^N) = g^FF) resu iting (by using Equations 2-16 and

y y

2-19) in the equation

F ( h,z. ) = (6.06 x 10' 4

)V 2 (h,z, )[1 + (6.5 x 10

_9

)h

2

] = mg (4-2)

z L meas *-

[F

z

, mg in N; V in kV; z

L in cm; h in ym].

V mG as

1S obtained via a resistance voltage divider and a group of

standard cells which have been calibrated in terms of the NBS

as-maintained volt. The mass m can be determined from mass

28
intercomparisons, and the gravitational acceleration, g, has been

accurately measured at the suspended electrode position.

A A

The quantity 3C/8z

L in Equation 4-1 is the slope of C versus

A

Z|_ evaluated at point (h, z^) of the relative coordinate system. The

capacitance can be measured as a function of z

L by translating the

carriage to different positions and monitoring z L via the laser

A interferometer system. The slope at point (h, z L

) can then be

determined from the derivative of an nth-order polynomial which provides

A

the best least-squares fit to the C versus z L data. Ideally, this slope

A A should be obtained from carriage scans that keep h and h constant

x y

while changing z^, but experimentally this is probably not possible.

All the quantities in Equation 4-1, except V, can therefore be

measured in SI units. So the SI voltage, V -j , can be calculated from

this equation and then compared with the measured voltage, Vmeas , which

is expressed in terms of the NBS as-maintained value.

There are, however, errors associated with the measurements of

s /s F,, C and z, - and these errors affect the voltage ratio V_ al _/Vm ___.

Sample calculations are presented in Appendix T to gain an impression of

how much this ratio could be changed by those errors. These

calculations are not intended to be an error analysis, but rather to

indicate how large the deviations in the voltage ratio might be. Part a

of Appendix T explains why this approach is used.

The individual errors used in these calculations are: ±2 ppm for

the combined static and dynamic systematic error of F in the force

determination measurement (as discussed in Section 3.3 and part g of

Appendix P); a ±0.025 ym uncertainty in the measurement of z

L (as

29
discussed in Section 3.5); a +1 ppm dynamic systematic error in the

A

measurement of C (as given in Section 3.5 and Appendix S); and

A

sometimes a ±1 ppm error in C due to the electrode overlap not being

correctly monitored by because of temperature fluctuations (as

discussed in Appendix D).

Various combinations of these errors are used in the examples of

Appendix T, but one combination is, rather arbitrarily, chosen to obtain

a possible spread in the voltage ratio values. This combination is: a A ±2 ppm systematic error for F , a ±2 ppm random error for Az^ , and a +1

A

ppm dynamic systematic error for C. No errors due to temperature

fluctuations are assumed. (Hence, the slope error is determined from

capacitance endpoint measurements in the examples rather than from a

/\ /v. linear fit to a plot of C versus zL in which the values of C deviate

from the straight line because of temperature fluctuations and dynamical

measurement errors. The values of z^ are all uncertain by ±0.025ym, so

the slope of the linear fit might thus have an uncertainty of ±2 or 3

ppm, but one would require real data before actually assigning an

uncertainty.

)

These errors are combined in such a way as to give the largest

effects. This combination is perhaps pessimistic. However, only the

three errors given in the above paragraph are involved in the

combination and some of them are random, some are systematic and some

are correlated; so the errors are just as likely to add as to cancel.

Also, an ideal measurement situation is assumed with perfect right

A cylinders, no balance zero-point drifts, and no curvature in the C

versus z L function. In addition, no experimental errors are included

30
for due to the voltage divider step-down or to the comparisons

me a s with the NBS volt via the group of standard cells, or to the ±1 ppm

random uncertainty in m at V = 4.02 kV.

Three different carriage paths are assumed for the calculations of

Appendix T. They are shown in Figure 20. The carriage moves along the

symmetry axis from initial values of = 0 cm and C(z

L

) = 100 pF to

final values of z

L = 2.5 cm and C (

L

) = 69.7 pF in the Type I scan. In

/\ the Type II scan the carriage moves such that h^ is constant at low

voltages ( i.e. for the capacitance measurements). (Note from Tables 9

and 10 that h

y

is not constant at high voltages because the values of

A

Yy decrease with increasing z^ values. Thus, the horizontal forces, and

A A

F

z also decrease.) Finally, h^ increases or decreases linearly with

increasing z L at low voltages in the Type III scans. There are

/\ additional errors introduced in the Type II and III scans because F is

A

measured at high voltages, while C is measured at low voltages where the

suspended electrode remains nearly vertical - and also the slope is not

determined at constant h.

Examples 4, 8, 17-20, and 25-28 of Appendix T imply that the

measured ratio of the SI volt and the NBS volt would be uncertain by

+3.5 ppm and -2.5 ppm for a brass suspended electrode at V = 9 kV,

resulting in a 3 ppm measurement. Additional data would not reduce that

A part of the uncertainty due to systematic errors - such as the F

z measurements; also, the nearly repeatable carriage movements would

always apply the same bias to the results, but this bias cannot be

corrected unless y , y , 0 , 6 and d and d are known at each z, x y x y x y l point, so that h could therefore be determined.

31
Examples 10-11, 21-24, and 29-32 of Appendix T imply that the

measured ratio of the SI volt and the NBS volt would be uncertain by +2

ppm and -2.5 ppm for an aluminum suspended electrode at V = 4.02 kV,

resulting in a 2 or 3 ppm measurement. (Note that is assumed to be

equal to V SI

in the examples. It does not matter if the experimental

voltage ratio is not actually equal to one - the above uncertainty still

appl ies.

)

Another approach to estimating the uncertainty in the voltage

/s ratio is to assume a ±2 ppm error in F

z and a ±2 or 3 ppm error in the

A

slope 2C/9 z l

, and to combine the errors by the square root of the sum

of their squares. This yields an uncertainty of ±1.5 or 2 ppm for both

the brass and aluminum electrodes. Or, if the two errors are combined

by addition, the uncertainty in the voltage ratio is ±2 or 3 ppm. This

uncertainty does not include errors due to F

z being measured at high

A

voltages while C is measured at low voltage, or the uncertainty in m.

It would appear to be difficult to obtain a value of the voltage

ratio with an uncertainty less than ±2 or 3 ppm by the slope method

under ideal conditions. The uncertainty could be much larger in reality

A because of the very strong electrode shape-dependence of C (as discussed

A

in part f of Appendix C). The plot of C versus z

L could have a large

curvature - with a shape that might change along the carriage path - and

would thus greatly increase the uncertainty in the derivative of the

best lease-squares fit of the nth order polynomial to the data. Also,

A the laser interferometer uncertainty in the position of at which F

z A is measured becomes more important if the slope of C versus z^ is

rapidly changing.

32
4.2 The path integral method

Appendix U derives the path integral for the mechanical work

A A

required to move the electrodes from an initial point (h , h z, ) in

^

x

i yi Li the relative coordinate system to a point (h , h , z, ) . The

j j

expression for the path integral is greatly simplified if the measured

A A A A

force F(h, z^) at each point is adjusted to a new value F'(h,z

L ), where

/\ A A

F' is the value that F would have had if V„__(h, z. ) were constant at

. mcaS L A every point of the carriage scan. The adjustment factor, €(h,z^) =

P 2 * * *

v /Vmeas » z l) » for z

i_) is obtained in Appendix V; V is either

A the average of the Vmeas

(h,z^) values, or any other arbitrary normalization

value - such as the nominal voltage V^.

The combination of Equations U-5, V-4, 2-14, 2-15 and 2-16 gives

W e

= (6.5 x 10" 13 y h xi ah,z

L

)v‘ eas

(h,z

L )C(z

L )h x dh

x + (6.5 x 10- 13 )/h

yjS(h.z

L

)V

3eas

(i;.z

L )C(zL)hydh

y

-(6.06 x 10-4 )/^LjUhTz

L

)V

3eas

(K. z

L

).

(1 + (6.5 x l(f

9

)(hj; + h^)}dz,

*(1 X 10

- ^) (1/2) ¥

2CCj(h ,Z L

) - C.(h,z

L

)] (4-3)

[We in N* cm; V in kV; C in pF; z

L in cm; h in ym].

Note that since h x

, h^, F^ and F^ contain information about e^, 6 , x y

y and y , that the mechanical work integral W

e includes not only the

x y ^ a

translational work -/F - d£ due. to linear motion of the carriage

electrodes, but also the rotational work integrals - /r ‘dy and -/r

0

*d6

33
due to changes in tilts of the suspended and carriage electrodes,

respectively.

The path integral represented by Equation 4-3 must be

conservative. Appendix W verifies that this is indeed the case. Any

path may be used - with any normalization voltage V - but, in order to

compare several paths, the value of V must be the same for each path.

So, the normalization voltage might just as well be the nominal value, V

f of the voltage applied to the electrodes, i .e. = 9 kV for m = 5

grams and V

N = 4.02 kV for m = 1 gram.

Equation 4-3 was mathematically useful to show in Appendix W that

the path integral is conservative, but, experimentally, the more

convenient combination of Equations U-5 and V-4 is

w

e

w i+f

(2xl0-4 )/>f C<vg)/(£

as (h.zL

))]F

h

(h.z

L

)dh,

/s /s

-/

Lf

[<V|j)/(V^(h.z

L ))]|F (h,zL )|dz

L

L

i

(ixio4)(i/2) v|alcCCf

(hf

,z Lf

) - qfh-.z^)] (4-4)

[W

e in N-cm; V

N

, Vmeas , V ca]

- in kV; F

h , F7 in N; C in pF; h in urn; h’ z,

Z|_ in cm]

Some problems in evaluating this equation are immediately apparent:

A /s. /\ and F„ cannot be measured with the balance, and h

\

and h are also

y

34
unknown at each point (h , h , z. ) of the relative coordinate system

x y l

because only 9 , 6 , y and y could be conveniently monitored via

x y x y

autocol 1 imators. The measurements of d and d are difficult and

x «y involve expensive laser interferometry techniques. Therefore, the

horizontal parts of the path integrals are unknown. Calculations of

Appendix X show that the missing horizontal integral contributions to

the path integral can be quite large, with the amounts depending upon

the particular path taken. The horizontal integral contributions are 11

ppm and 15 ppm for the two paths used in that appendix.

A

Another problem arises from F

z being measured at high voltages

where the suspended electrode has, in general, swung out to some

/\ /v equilibrium angle y - whereas, C is measured at low voltages where the

suspended electrode is essentially vertical. So, the capacitance

measurement is not made with the same electrode configuration as that

for the force measurement.

Appendix Y provides sample calculations of Types I, II and III

scans for a brass suspended electrode at = 9 kV and for an aluminum

electrode at = 4.02 kV. The errors chosen to obtain a possible

spread in the voltage ratio values Vca -| c

/ V

N are: a ±2 ppm systematic

error for F

z and none for m, a ±2 ppm random error for Az^, and a +1 ppm

dynamic systematic error or temperature fluctuation effect error for one

of the capacitance endpoint measurements. (More emphasis must be given

to the possible capacitance measurement errors in the path integral

method because this method uses the capacitance endpoint values, and

therefore one cannot average the capacitance deviations over the Az^

scan as in the slope method.) (The largest canacitance errors might be

35
+2 ppm at one endpoint and -1 ppm at the other, but this combination is

very unlikely.)

The errors are combined additively in the examples of Appendix Y

because some of them are random, some are systematic and some are

correlated - and because perfect right cylindrical electrodes are

assumed, along with no balance zero-point drifts. The ratio v ca ^ c

/ v

n evaluated with the experimentally unknown horizontal parts of the path

integral not included, and with the capacitance values not being

corrected to the values that they would have had at high voltages; i .e.

only measured quantities, and their assumed experimental uncertainties,

are used in the voltage ratio calculations. Part a of Appendix Y

explains the reasons for using this approach.

Examples 2-3, 8-9, and 13-14 of Appendix Y imply that the measured

ratio of the SI volt and the NBS volt would be uncertain by +4.5 ppm and

-3.5 ppm for a brass suspended electrode at V

N = 9 kV, resulting in a 4

ppm measurement. Additional data would not reduce that part of the

uncertainty due to systematic errors. Also, the nearly repeatable

carriage movements would always apply the same bias to the results, but

this bias cannot be corrected unless y , v , 6 , 6 and d and d w are

'x* T y* x’ y x y

known at each point z^.

Examples 3, 6, 11-12, and 15-16 of Appendix Y imply that the

measured ratio of the SI volt and the NBS volt would be uncertain by

+3.5 ppm and -4 ppm for an aluminum suspended electrode at = 4.02 kV,

resulting in a 3 or 4 ppm measurement.

36
Another approach for estimating the uncertainty in the voltage

A

ratio is to assume ±1 ppm systematic static and dynamic errors in F , a

±1 ppm random error in m, a ±1.4 ppm random error in az

l

, and a ±1.4 ppm

A A

error in (Cf-C.j) - and to then combine these errors by taking the square

root of the sum of the squares of the individual errors for each type of

scan given in Appendix Y. The ratio would be uncertain in this approach

by +2.5 ppm and -1.5 ppm for both a brass electrode at V

N= 9 kV and for

an aluminum electrode at Vjy|=4.02 kV - resulting in a 2 ppm measurement.

This uncertainty is, however, probably optimistic. A more

conservative estimate might be that 3 or 4 ppm measurements could be

obtained with either a brass electrode at = 9 kV or an aluminum

electrode at V

N = 4.02 kV.

Unlike the slope method, it is not essential that C versus z

L be nearly a straight line with no changes in its shape, (or therefore

/V /\ that versus z^ be nearly constant). If F^ changes dramatical ly , then

A

F

z must be measured at more points so that the area under the curve of

A

F

z versus z L (analogous to that of Figure 22 for perfect right

cylinders) can be obtained with sufficient accuracy.

37
5. RESULTS AND CONCLUSIONS

5.1 Problems in measuring F^

This section summarizes the results of the calculations presented

in Sections 2 through 4 and Appendices A to Y, and also states the

requirements or conditions that probably must be satisfied in order

to reduce the systematic and random errors of the measurements to

acceptable levels so that the SI volt can be determined to within a

few ppm.

If the four electrode surfaces are not perfectly aligned then

large horizontal forces occur on the suspended electrode at high

voltages because most of the electric field lines are radially

directed. These horizontal forces may typically be on the order of a A

few percent of the vertical electrical force value f .

The relative alignments of the three cylinders are critical in

reducing the size of the horizontal forces. They must be properly

oriented at some point z

L and then maintained over a azl span of

about 2.5 cm such that the relative electrode tilts and displacements

deviate by no more than a few seconds of arc and a few micrometers

from perfect alignment at each z

L value for both the x and y

components of d, 0 and y.

Note that the calculations assumed that the carriage electrodes

were perfectly aligned with respect to each other so that h

1 = h^,

C

1 = and F* = F®. The same alignment requirements of a few

38
seconds of arc and a few micrometers also occur if h

1 f h®. The

deviation of any of the four electrode surfaces from the combined

symmetry axis results in horizontal forces.

The carriage platform presently changes its tilt by several

seconds of arc in both the 0 v and 0 w directions during az. scans.

x y L The d and d changes are unknown. It is also presently unknown

x y

just how well the four electrode surfaces can be mutually positioned

and oriented with respect to the combined electrodes electrical

symmetry axis. Many of the sample calculations have used d = d

=

x y

0, yx = y =0 for voltage-off, and 0 x

= 5" or 0 = 5" to provide h x y y or hy values that might experimentally occur at some points of the

carriage scans.

A brass suspended electrode remains in stable equilibrium at

V = 9 kV (m = 5 grams); whereas an aluminum suspended electrode is

sufficiently stable at V = 4.02 kV (m = 1 gram), but its equilibrium

angle becomes too large at V = 5.68 kV (m = 2 grams).

In general, the suspended electrode executes simple harmonic

motion about an equilibrium angle y. This motion can be that of an

elliptical pendulum, a conical pendulum, a simple pendulum whose

swept-out plane can be perpendicular to any direction in the x-y

plane, or it can be at rest at the angle y.

The pendulum period is voltage-dependent, and for a brass

suspended electrode increases from a val ue of 1.5 s at V = 0 kV to a

value of 2.8 s at V = 9 kV. The aluminum electrode period increases

from a value of 1.3 s at V = 0 kV to 2.0 s at 4.02 kV.

39
The balance also executes simple harmonic motion about an

A

equilibrium angle 3 . The period of this motion is

voltage-dependent, and for a balance sensitivity of 0.11"/ug

increases from a value of 44.9 s at V = 0 kV to 46.8 s at V = 9 kV

with a brass suspended electrode. Since the pendulum and the balance

beam periods are so different, only very small energy transfers occur

between the y and 3 motions. Therefore, the pendulum and balance

beam amplitudes do not alternately, and periodically, increase and

decrease with time.

The balance acts as a low-pass frequency filter. Thus, the

amplitude modulations (due to the suspended electrode motion) are

undetectable on the simple harmonic

3y waveform for small pendulum

amplitudes since V

> “b-

(Note that if the pendulum amplitude y is

greater than 7" then the 3

y

waveform becomes more like a Mathieu

function rather than a simple sine or cosine function.)

A systematic error does occur in the force determination

measurements, however, because the balance beam equilibrium angle

~2

depends upon y at high voltages. A 1 ppm dynamical error in the

value occurs in both a brass suspended electrode at V = 9 kV and an

aluminum electrode at V = 4.02 kV if y

= 6". If this dynamical error

A

in F z were to be less than 0.2 ppm (so that advantage were taken of

the 1 yg resolution in 5 grams at V = 9 kV) then y must be less than

2.5".

40
The sign of the error is such that F

z always appears larger than

mg for a swinging suspension system. This geometry-dependent

systematic error would still occur at high voltages for perfect

electrode alignment, but it does not occur for normal mass

intercompari sons.

No damping terms were included in the calculations, but the

pendulum amplitude would decay exponentially if there are no air

currents. It is not presently known what the time constant would be.

It could be quite long because of the small friction of the load

knives and the small velocities for the viscous damping forces, but

only experience will tell.

Another systematic error can also occur in the force determination

measurements if the y component of the horizontal electrical force on

^1

the suspended electrode is nonzero (F + Fy

f 0) and if the

equilibrium angle of the plane defined by the contact points of the

balance beam center knife and the load knife is not parallel with

^ t n ff l ^ gravitational horizon (3y t 0). A torque, (F + Fy

)l_ Asin 3

y

,

occurs about the center knife in this case, and - since this torque

A A A A is linear in $

y

for small 3

y

- there is no minimum for F

z versus 3 .

This problem does not occur in normal mass intercomparisons because

there are no applied horizontal forces.

If the carriage platform is displaced or changes its tilt during

carriage scans then horizontal forces are present on the suspended

electrode. The only way to reduce this static error term (without

adjusting d , d , 6 and e in order to minimize the horizontal force

x y x y

41
"(OFF)

at each position of the carriage scan) is to assure that 3

y

is

nearly zero.

If a brass suspended electrode is used at V = 9 kV, and if

d =d v = e

Y = 0 and 0 = 5", then a systematic ±1 ppm error would

x y x y A A occur in the F measurement at C = 100 pF if the knife edges plane

"(OFF)

( i . e

.

3' ) was ±7" from gravitational horizon. The same error

"(OFF)

would occur for an aluminum electrode at V = 4.02 kV if 3

y

= ±16".

The plane of the knife edges would have to be determined to within +1"

/A gravitational horizon in order to obtain a ±0.2 ppm error in F^ for

the brass electrode at V = 9 kV. This does not seem possible.

"(OFF)

A 3

y

'= ±16" requirement corresponds to determining that the

vertical height of the contact point (P) of the load knife in Figure

2 is within ±6 ym of the height of the contact point (C) of the

center knife. (Six micrometers is about ten wavelengths of red

"(OFF) light.) The 3

y

*0 determination would surely need to be done with

the balance under load in order to account for knife edge and balance

beam deformations. This determination would provide a balance beam

"(OFF) readout angle that corresponds to 3

y

*0. The mirrors for the

readout optics should not tilt after this readout angle

"(OFF) determination, or a systematic error in 3

y

would result.

The suspension system mass + m and the counterweight mass

Mg must be adjusted until the net torque about the center knife is

"(OFF) close enough to zero, with the voltage turned off, that 3

y

~0;

i .e. the desired readout angle value is obtained. This corresponds

/s to another ±1 ppm systematic error in F^ if M^ + m differs from by

42
about ±65 u9 for a brass suspended electrode at V = 9 kV, or by about

±150 wg for an aluminum electrode at V = 4.02 kV if L A

= Lg and the

mass distribution of the balance beam is symmetrical about the center

knife.

If the balance beam was servoed then the balance could be

trivially adjusted to the desired readout angle if the zero-point

level of the feedback circuit was sufficiently stable. If the

balance was not servoed, then either + m or Mg would have to be

adjusted to obtain the desired readout angle for a freely-swinging

A

balance. The systematic mass adjustment error in F

z could

either add to or subtract from the error associated with the knife

edges plane determination.

The above discussion of possible static and dynamic systematic

A A errors in suggests that the uncertainties in the F

z determination

would not be reduced by using higher voltages because the y and 3

(OFF)

conditions become too stringent. However, it is important to

compare the SI volt with the NBS volt at different electrode

voltages. Therefore, a brass suspended electrode, or an aluminum

electrode with additional counterpoise mass placed along its symmetry

axis near the lower rim is ultimately required because an aluminum

electrode is not stable enough at voltages larger than V = 4.02 kV.

The equilibrium angle y would also be smaller with more mass for a A

given horizontal force, thereby reducing the static error term of F .

43
Another possible systematic error in F , which has not been

mentioned, arises if net torques about the x axis are induced on the

suspended electrode. This could occur if the carriage electrodes

pivoted about a point in the overlap region of the electrodes, so

that the y components of the inner and outer horizontal forces

distributed along the suspended electrode formed a couple, and

therefore a torque about the x axis.

/s, This r torque on the suspended electrode would appear as a

force on the load knife at point (P), and it would essentially be in

the y direction. Thus, an additional torque would be induced about

the balance beam center knife if the knife edges plane was again not

at gravitational horizon. (Torques about the y axis of the suspended

electrode would try to twist the load knife and might thus change the

length l_ A

.)

These torques on the suspended electrode could be detected if the

connecting ring of Figure 1 was replaced with a gimbal device, and if

the angles of both parts of the suspension system were monitored by

autocollimators. This suspension system would act as a double

pendulum, however, so the motion would be more complicated.

A The T torque could be eliminated at only one point of the

A A carriage scan. Therefore, this systematic error in must be

^fOFFl minimized by again assuring that 3^

~0 as was the condition for

A

reducing the static error term of F . z

44
F

z is assumed to be equal to mg when = 3 ^^. However,

y y

the balance resolution and stability is 1 yg. Therefore, there is a A

±1 ppm random uncertainty in the value of F

z at V = 4.02 kV

(m = 1 gram). This error decreases at higher voltages.

5.2 Problems in measuring C

/v /v The equilibrium value of the capacitance C(h,z

L

) at any point

/v (h h , z

L

) is very sensitive to the value of z

L

, where z

L is a A measure of the electrode overlap. The value of C changes by ±1 ppm

at C ~ 50 pF if z

L changes by ±0.041 ym. (Note that 0.041 ym is

about one tenth a wavelength of blue light.)

The electrode overlap changes by ±0.041 ym if the balance beam

angle changes by ±0.11". So, the balance beam cannot be allowed to

swing during the capacitance measurements. On the other hand, the

/\ suspended electrode system must remain free to swing; otherwise C

would be measured for an electrode configuration that was different

A

from that for F .

There are two ways of not allowing the balance to swing:

electronically servoing the balance beam to within a random stability

of ±0 . 11 ", or mechanically braking the balance beam along an axis

parallel to, and above, the center knife axis. A balance beam servo

could not be driven via the voltage applied to the electrodes because

the beam must also be servoed for voltage-off measurements. A

mechanical brake would have to virtually stop the balance beam from

45
moving. If, in addition, it rotated the beam about the x axis, the

balance readout optics could be used to detect it.

If the mechanical brake rotates the balance beam about the z axis

by ±1" then the suspended electrode would be displaced in the x

direction by ±0.36 ym, thus changing the electrode configurations

during the capacitance measurements. This effect could be checked by

mounting the balance beam readout mirror in the x-z plane rather than

in the x-y plane. This mirror arrangement would also detect any

balance beam rotations that might occur whenever the balance is

placed in arrestment and then released.

When the voltage is off, the center of mass of the suspended

electrode system (point 1 in Figure 2) will, by definition be

directly below the pivot point (P). The suspended electrode

surfaces, however, will not necessarily be parallel with g if there

are any radial mass asymmetries. Therefore, there is a spider

arrangement located at the top of the suspended electrode that allows

the electrode surfaces to be oriented so they are nearly parallel to

A

The suspended electrode equilibrium angle y will then be zero with

/s the voltage turned off - and y will remain essentially at zero at the

100 V a.c. used in the capacitance measurements because the

horizontal forces are small. The capacitance measurements are thus

only affected by changes in the displacements d and tilts 0 of the

/s /\ carriage electrodes if y = 0. At the higher voltages used in the F

z

/v measurements, however, y f 0 if there are horizontal forces on the

46
suspended electrode. So, the actual capacitances are larger at high

voltages than the measured capacitance. For example, if

d = d = 0 =0 and e = 5", then the measured capacitance

x y x y

equilibrium values are systematically too small by 4.4 ppm for a

brass suspended electrode at V = 9 kV, and by 0.6 ppm for an aluminum

electrode at V = 4.02 kV.

The balance beam must not be allowed to swing during the

capacitance measurements, but the suspended electrode system should

be free to swing so that it is not artifically displaced or tilted.

If the suspended electrode is swinging during the capacitance

measurements then C will oscillate with time during the 1.5 s period

of a brass electrode or the 1.3 s period of an aluminum electrode.

(Some examples of these oscillations are shown in Figure 19.)

The capacitance bridge detector needs to have a short enough time

constant so that it can correctly respond to these oscillations. If

it does respond quickly enough, then the minimum reading, C

.j , is

/\ probably the best one to use for the static capacitance value C.

There might be a ±1/2 ppm random error in determining Cm ^ p if the

•N/ ^ pendulum amplitude y is 10", and a -1/2 ppm systematic error due to C

actually being larger than Cmi

- n if 0 ~ 5" and d ~ 0.

If the bridge detector averages the signal, then y probably should

/s be less than 5" in order to measure C to within ±1 ppm. The average

value of the signal might be larger or smaller than the actual value

of C.

47
The capacitance oscillation shapes could be utilised for aligning

the electrodes. Perfect electrode alignment produces a pure cos2w t

signal. There is also a cosoyt component at imperfect alignment -

and this component becomes more enhanced with increasing relative

electrode misalignments and displacements. This effect could

therefore be used to align the electrodes by delibrately swinging the

pendulum and adjusting the displacements and tilts of the four

electrode surfaces such that the cos2oyt component was maximized.

Friction in a gimbal might dampen the pendulum amplitude too

quickly, and thus prevent this alignment technique from being used

for a double pendulum configuration. Also, there would be two

fundamental pendulum frequencies w and oj and alternately

' 1 y

2 , increasing and decreasing pendulum amplitudes y

and y^, so it might

be too difficult to decipher the proper capacitance signal waveform

for perfect electrode alignment when using a double pendulum.

Another test of electrode alignment would be to observe the

pendulum equilibrium angle y at high voltages. If the electrodes are

A

misaligned, then the horizontal forces would cause y to be nonzero.

This might not be as sensitive a check as the capacitance signal

shape measurement because the carriage electrodes could be tilted in

some direction and then displaced in the opposite direction such that

the resulting horizontal forces (or torques about point (P)) were

equal and opposite, such that y = 0. The value of C would not be at

its absolute minimum, however; so a cosoyt component would be

present for a high enough capacitance bridge detector sensitivity.

48
Part f of Appendix C demonstrates that the value of C, at any

/s point (h, z

L

), is very sensitive to the overall shapes and the

macroscopic bumps and valleys of the four electrode surfaces. If the

average radius of the overlap region of any electrode surface changes

by ±1 ym from the radius of a perfect right circular cylinder then

the capacitance value changes by ±54 ppm from that of a right

cyl inder

.

This is a real effect, not an error. However, it could introduce

large curvatures and changes of shape in plots of C versus z

L

,

thereby producing greater uncertainties in the slope determination,

A

8C/^z

l

, for the slope method and causing fluctuations in the values

A

of F

z versus z^.

If the temperature distribution within the 0-ring sealed vessel

changes by a few millidegrees Celsius while data are being obtained

/A then the capacitance values assigned to each point (h, z L

) might vary

by ±1 ppm. These variations could be due to changes in the overall

temperature or to changes in the vertical or the radial temperature

gradients.

A

Fluctuations in the values of C versus z

^ would increase the

uncertainty of the slope assignment, but they might be partially

compensated with enough data. It would be more critical in the path

integral method, however, because it is the capacitance values at the

endpoints of the carriage scans that count.

49
5.3 Problems in measurina z,

The corner cube mirror of the laser interferometer is attached to

the moveable carriage in Figure 1, while the upper part of the

interferometer might, or might not, be attached to the suspended

electrode system. The interferometer random uncertainty is ±0.025

urn, and its resolution is ±0.01 ym.

If the laser interferometer direction makes an angle of 90" with

respect to the combined electrodes symmetry axis then the resulting

error in measuring changes of electrode overlap is only 0.1 ppm. It

should not be difficult to attain this alignment condition.

If the balance beam is at rest and if the upper part of the

interferometer is attached to the suspended electrode system, then

the random uncertainty in z

L is ±0.025 ym. If the upper

interferometer part is not attached, then an interferometer

calibration of the change in electrode overlap for a change in

balance beam readout angle must be made. This would allow a

correction (with any resulting systematic errors in the calibration)

to be made to z^ if the balance beam was not mechanically braked at

''(Off) the readout angle corresponding to

3^

'= 0. (No correction need be

applied to z L if the balance beam is at the desired readout angle.)

In either event, another ±0.041ym random uncertainty is combined with

the ±0.025 ym interferometer uncertainty of z^ if the readout optics

uncertainty, or the balance servo uncertainty,^ ±0.11" and if the

upper interferometer part is unattached. This would result for

capacitance measurements in a maximum random error in az^ of ±5 ppm,

50
or a probable error (using /T times the square root of the sum

of the squared uncertainties) of ±3 ppm for a az

l = 2.5 cm scan. If

the upper interferometer part is attached then the maximum error in

Az

l

is ±2 ppm and the probable error is ±1.4 ppm for a az

l = 2.5 cm

scan.

It is therefore preferable to attach the upper part of the

interferometer to the suspended electrode system. However, its

present mass is about 830 grams. So, it would be best for mechanical

stability reasons to increase Lcmby using this interferometer component

counterpoise mass located on the suspended electrode symmetry axis

near the lower rim, and to move the corner cube mirror to the bottom

of the carriage platform. (A swinging electrode would cause parallel

shifts in the return laser beam, but this intensity decrease is

negligible for the small y amplitude restrictions in the capacitance

and force measurements.)

If the upper part of the interferometer is not attached to the

a suspended electrode during the F

z measurements and if the voltage is

~(0N) ''(OFF) adjusted until 8y

- 8^ , then the maximum uncertainty in is

±0.066 ym and the more probable uncertainty is ±0.048 ym, whether the

balance beam is servoed or is freely-swinging. If the upper

interferometer part is attached and the balance beam is servoed then

the uncertainty in z

L is ±0.025 cm. If it is attached, but the

balance beam is freely-swinging with a 8^

= 4" amplitude, then z^

would vary by ±1.5 ym during the 45 s balance beam period. This

might limit the accuracy in determining the value of z^ corresponding

A A to F

z

, but it is not a problem if F

z is constant or only a slowly

51
varying function of z L

. If F varies rapidly with z L

( i .e. 8C/8z

L versus z

L is very nonlinear at this point) and if the upper part of

the i nterferometer is attached to the suspended electrode, then

either the balance beam amplitude must be extremely small, or the

balance must be servoed.

5.4 Problems in the slope and path integral methods

The slope at point (h, z

L

) would be determined in the slope

method from the derivative of the nth-order polynomial which provides

A A the best least-squares fit to the C versus z

L data. If C varied

/\ linearly with z^ and the values of C fluctuated by ±1 ppm due to

temperature variations and dynamical measurement errors, then the

uncertainty in the slope for a az

l = 2.5 cm scan might be ±1 or 2 ppm

if the upper part of the interferometer was attached to the suspended

electrode and ±3 to 5 ppm if it was not.

A If the curvature of C versus z L is large then, presumably, the

uncertainty in the slope of the nth-order polyminal increases.

/\ Changes in shape of the C versus z

L function during a scan would

require different polynomial fits over each region - further

increasing the slope uncertainties because az

l

would be smaller for

A

each region. The great sensitivity of C to the shapes of the four

electrode surfaces may preclude the use of the slope method.

52
In the path integral method it is the difference in capacitance at

the two endpoints of the scan that is important. There might be a

±1 ppm uncertainty at each endpoint due to dynamical measurement

errors and temperature fluctuations. The uncertainty in az

l

between

the endpoints of the path integral for 2.5 cm scans would be ±1.5 to

2 ppm if the upper part of the interferometer was attached to the

suspended electrode and ±3 to 5 ppm if it was not.

A

F

z does not have to be measured at the endpoints of the z

L A scan if F

z varies slowly enough with z^ so that it can be

extrapolated, or interpolated, to the endpoint values. The path

A

integral method will still work if C versus z

L has large curvatures

A and shape changes as long as enough F

z measurements are taken. The

/\ number of F

z measurements needed for the path integral depends upon

A

the extent of the F

z variations.

A ±1 ppm static systematic error and a +1 ppm dynamic systematic

A

error could occur in the F

z determinations - in addition to a ±1 ppm

random error in mg at V = 4.02 kV.

Only the vertical component of the path integral is measured. The

horizontal components due to relative electrode tilts and

displacements are unknown. Also, the capacitance is measured at low

voltages, and is thus smaller than what it would have been at high

voltages if horizontal forces are present.

The errors mentioned in this section might lead to a comparison of

the SI volt with the NBS volt to within an uncertainty of ±3 or 4

ppm. This is only an estimate. Experience gained with actual data

might increase or decrease the uncertainty.

53
ACKNOWLEDGEMENTS

The author gratefully acknowledges the invaluable assistance of

R. D. Cutkosky who contributed many hours of discussion about the

physical aspects of the problem. He was the vital link with reality.

R. S. Davis devoted long hours of discussion on how balances work,

critically read the manuscript, and provided the yeoman task of verifying

the equations. E. R. Williams also critically read the manuscript, and

along with W. D. Phillips, provided many key suggestions - particularly

about the path integrals. The author also wishes to thank Mrs. B. L. Main,

who has so carefully and cheerfully prepared the manuscript for

publication.

54
References

[1] Goldstein, H., Classical Mechanics, pp. 14-18 ( Addi son-Wesley

Publishing Co., Inc., Reading, MA., 1950).

[2] Wells, D.A., Theory and Problems of Lagrangian Dynamics, pp. 58-62

(Schaum Publishing Co., New York, N.Y., 1967).

[3] Taylor, B. N., Is the Present Realization of the Absolute Ampere in

Error?, Metrologia 12, 81-83 (1976).

[4] Aimer, H. E., National Bureau of Standards, One Kilogram Balance,

NBS No. 2, J. Res. Nat. Bur. Stand. (U.S.), 76C (Phys. and

Chem.), No. 1, 1-10 (Jan. 1972).

[5] Cutkosky, R. D., Capacitance Bridge -- NBS Type 2, Nat. Bur.

Stand. (U.S.), Report 7103, 10 pages (March 1961).

[6] Cutkosky, R. D., New NBS Measurements of the Absolute Farad and

Ohm, IEEE Trans. JM-23, No. 4, 305-309 (Dec. 1974).

[7] Childers, C. B., Dziuba, R. F., and Lee, L. H., A Resistive Ratio

Standard for Measuring Direct Voltages to 10 kV, IEEE Trans. IM-25, No.

4, 505-508 (Dec. 1976).

[8] Kusters, N. L. and Petersons, 0., The Voltage Coefficients of

Precision Capacitors, IEEE Trans. IM-11 , No. 4, 300-318 (Dec. 1962).

55
[9] Hodsman, G. F. and Chappell, F. A., The Effective Radius of

Curvature of Knife Edges, J. Sci . Instr. 29, 330-332 (Oct. 1952).

[10] Smythe, W. R., Static and Dynamic Electricity, pp. 74-76

(McGraw-Hill Publishing Co., Inc., New York, N.Y., 1939).

[11] McLachlan, N.W. , Ordinary Non-Linear Differential Equations in

Engineering and Physical Sciences, pp. 113-144 (Clarendon Press,

Oxford, England, 1956).

[12] Abramowitz, M. and Stegun, I. A., Handbook of Mathematical

Functions, pp. 68, 87, 722-743 (Dover Publications, Inc., New York,

N.Y., 1968).

[13] Murphy, G. M., Ordinary Differential Equations and Their

Solutions, pp. 82-85, 311-316 (D. Van Nostrand Co., Inc., Princeton, New

Jersey, 1960).

56
APPENDIX A

CAPACITANCE OF RADIALLY DISPLACED CYLINDRICAL ELECTRODES

The capacitance per unit length, T, of three concentric,

infinitely long, right circular cylinders is

T = 2u€

q

/1 n(R^/r£ ) + 2*e

0/ln(r°/Rs) (A-l)

where R^ , R^, rj and r^ are shown in Figure 2 - with their values

2 -7

listed in Table 1, and e Q

= 1/ (uQ c ), with y

0 = 4tt x 10 H/m and c

=

8

2.99792458 x 10 m/s. Using these values in Equation A-l one obtains

T = T

1 + T° = 6.0686879 x 10" 10 F/m + 6.0511887 x 10' 10

F/m.

Smythe [10] has derived an expression for infinitely long right

circular cylinders that are radially displaced a distance d from

their symmetry axes:

T=2we

0 /cosh

-1

{[ ( r^)

2

+(Ft| ) 2-d

2 D/(2r^R| )>

+2re 0/cosh'

1 {[(R°) 2+(r°) 2-d2 ]/(2r°R°)} (A-2)

2 2 2

where d = d^ + d . Using the approximations [12]

cosh

-1

(u) = 1 n ( u + /u^-l) and

1 n(u) - 2{ (u-l)/(u+l ) + (1/3 )[(u-1)/(u+1)]

3

+ (1/5)[(u-1)/(u+1)]

5 +

57
Dr. F. K. Harris has numerically computed T for small values of d,

obtaining the result

T (d) = [T

1 (d=0) + T°(d=0)][l + (6.5 x 10" 9

)d

2

] (A-3)

where d is now in ym rather than cm. At d=0, Equation A-3 yields

T

1

( d=0 ) + T°(d=0) = 6.06868392 x 10" 10 F/m + 6.05119039 x 10" 10 F/m.

These values differ slightly from those obtained from Equation A-l,

but it is the change of T with d that concerns us, not the magnitude

of T.

The radii R*, R^, rjand r^were chosen so that T

I (d=0) ~

T^(d=0) ~ 6.06 x 10~ 10

F/m (6.06 pF/cm) ; therefore,

T (d) = (12.12 pF/cm)[l+(6.5 x 10" 9)d

2

]. (A-4)

This equation is for infinitely long cylinders with radially directed

electric field lines. This is not the case for Figure 1, where

fringe fields occur. However, given sufficient electrode overlap

L q

, the fringe fields of the suspended and carriage electrodes do

not interact, so that T = 12.12 pF/cm remains valid - and has been

experimentally verified.

The capacitance, however, is not equal to TL

Q

because of the

fringe fields. A magnitude calibration point (chosen to be 100 pF at

L

q = 10 cm and d = 0) is needed to account for fringing, yielding the

equation

58
C(d,z

4

) = CC(z

1

> - (12.12 pF/cm)(zr

z

1 )][l + (6.5xlO' 9

)d

2

]

where and are the laser interferometer readings at an initial

point i and at a point a respectively. Subtracting z^ from each

point z^ (thus yielding a relative vertical displacement value z

^ =

z%

- z^), and choosing the capacitance magnitude calibration point as

the value for z^ (where z^ = 0) gives

C(d,z

L

) = C(z

l

)[ 1+(6.5 x 10"9

)d

2

] (A—5)

[C in pF; z^ in cm; d in ym]

where

C(z

L

) = [100 pF - (12.12 pF/cm)z ]. (A-6)

The displacement, D , for the suspended electrode, using Figures 2-4,

is

[Dy in ym; L A

, Pp , p

c in cm; ey9 Y

y

in rad].

We will only need D to hundredths of micrometers, so D ~ lo\*(l-cosB ),

J J i* J and for small gy

59
Dy = 10 (L A/2)0y (A-7)

and

C(D

y

, z

L

) = C(z

L

)[l + (6.5 x 10'9 )C^] (A-8)

[C in pF; z L

, in cm; D in ym; 8 in rad]

60
APPENDIX B

CAPACITANCE OF TILTED CYLINDRICAL ELECTRODES

There are at least seven possible locations in Figure 1 about which the

carriage electrodes could pivot - any of the six teflon

bearings if they tended to bind on the stainless steel guide rods,

and the ball bearing located in crossed v-grooves at the top of the

elevating shaft. The ball bearing location will be used for the

calculations since it is a likely pivot point, and is also

geometrically representative of the other pivot points.

Figure 5 shows the outer carriage electrode tilted about the ball

bearing (point 3) by the angle 0 . The figure is drawn in the y-z

plane (defined in Figure 2) at x = 0, where the overlap area A (swept

out by the carriage electrode as its rim goes from point 1 to point

2) is at a maximum. This area monotonical ly decreases to zero as one

moves around the electrode surface to the x-z plane at y

= 0, and

then increases again to another maximum at y

= -r£. The resulting

volume is similar to that swept out by the outer carriage electrode

when it is radially displaced, except in that case the cross section

is rectangular.

The overlap area B is what would have been swept out if the

electrode were tilted about point 4 rather than point 3. This area

could be used to calculate the magnitude of the capacitance if B = A,

61
which is equivalent to assuming that B is much greater than the upper

part of A for small 0^:

[L

0

*e

y

- (1/2) L^0

y

]»[(l/2)tr°e^] or L 0 [*-L

0

/2]»[(l/2)Hr°6

y

].

Letting L Q

= 10 cm and 0^= 4.85 x 10 ^ rad (10") yields 300 » 0.009 (cm^

Therefore the pivot will be transposed to point 4.

We know from Equations A-5 and A-6 of Appendix A how to calculate

the capacitance for radial displacements, and could use those results

if electrode tilts were expressed as equivalent displacements.

Therefore, divide Cq(<1 , z

l)

into infinitesimal sections

C°(dy,z

L )/L

0 (zL

) and sum them to get C^(0

y

,z

L

):

4

where d

y

= 10 A^.0 and A^ is in cm. Therefore,

C°(e

y

,z

L

) = C°(z

L ){l+(0.217)[d 3

-Jt ,3 (z

L

)e

y

/ l

0

(z

l )]} ( B— 1

)

[C in pF; i , i' , Lq

, z

L in cm; 0

y

in rad].

62
This equation could be used to compute the capacitance due to a

tilted outer carriage electrode, but it will instead be reduced to a

much simpler form. The centroid of area B, if measured from point 4,

is

a £

Mzi ) = f MA/ f dA =

1 L A'(z

l

) 1 J £'0'(,(z \ L) J

l 1

x tey(dx

t

) = £'(z

l

)

(2/3){[*3 -£' 3 (z

L )]/[t^t ^(zL

)]}

a' (z

L

)

2 „

1 2 ,

Ut

, I , £' in cm]

(B-2)

or, if measured from the gravitational horizon of the balance beam,

is

Lt(z

L

) = L + Jt'(z

L

) - *t

(z

L

) = L + l

- L

q

(z

l

) - *

t (z

L

). (B-3)

Table 2 lists l_ t

versus z

L for an initial value of C = 100 pF and L Q

= 10 cm. Note that at the most Lt

varies from I_ d by 0.5%, where l_ d =

Lq = L - L q

/ 2. Therefore, let l_ t = L

d

, so L + i - L q

- n

t

= L -

L 0/2, or

*t( z

L>

= * - L

o< zL )/2 - (B-4)

63
0 -5

(This equation assumes that L o>>r

c 0^, which for 0^

= 4.85 x 10 rad

(10") is 10 cm»(10.74 cm) (4.85xl0

-5

rad) = 5.21 x 10"4 cm.)

If Equation B-2 is substituted into Equation B-l, then

C°(0

y

, z 0

= C°(z

L ){l + (0.217)[(3/2)(«-t')(m , )/Lo]«

t

ey}.

Noting that L = £ - £', C^(z^) = C

1

( )

,

and using Equation B-4:

C

t (0,z

L

) = C(z

L ){l + (O.65 )U-Lo (z

l )/2]

2

0

2

}. (B-5)

Comparing Equations B-5 and A-5 we see that the capacitance due to

tilting both carriage electrodes can be written as an equivalent

radial displacement, d^, of the centroid £^ of the area B, where

d

t

= 10

4

[t - L 0(z

L )/2]6 (8-6)

[d

t in ym; £, L , z

L in cm; 0 in rad],

2 2 2 0 = 0 x +6y’ and C ( Z |J 1S de fi ned by Equation A-6.

An exactly analogous derivation for the capacitance due to the

suspended electrode pivoting about point P of Figure 2 yields the

resul ts

C

t (y,zl

) = C(z

l )(1+(0.65)[L-L

o

(z

l )/2]

2

Y

2

) (B-7)

64
and d (B-8)

T

= 10

4

[L - L

0 (zl )/2] y

[dy in ym; L, L Q

, z

L in cm; C in pF; yin rad],

2 2 2

where y = Yy + and C(z.

)

is again defined by Equation A-6.

x y l

65
APPENDIX C

VARIATIONS AND UNCERTAINTIES OF z

L AND C

a) The laser interferometer alignment.

The axis is defined by the direction of the laser

interferometer, whereas the z axis (which is also the direction of the

combined electrodes symmetry axis) is parallel to the gravitational

field. How well must the z^ axis be aligned with the z axis in order

for az

l

to be within 1 part in 10

7 of az?

2 Az = UzL

) z = az

l cos <J> l = AZ|_( 1-4>l/2 + ....)

.\<J)jy2 = 1 x 10~ 7

, so

4

>l = 4.47 x 10‘ 4

rad (92").

This should not be difficult to attain.

b) The uncertainty in measuring Az^.

The laser interferometer system can measure distances with a

reproducibility of 6^ = ±0.025 urn. How large must Az^ be in order for

it to be known to within 1 ppm?

^L 7 AZ|_ = ^ ( ±0.025um)/Az^ = ±1 x 10"^, so Az^ = 3.5 cm.

66
c) The variation of for a freely-swinging balance beam.

_3

It is estimated that the values of were ±2.4 x 10 rad for

the turning points of the balance beam when the 1 yg reproducibility

was achieved with the voltage off. Therefore

Az

L

~-LAsin3y*-L

A3y=-(7.5 cm) (±2.4x10 rad)

Az

l

=±0.018 cm (±180 ym)

.

d) The variation of C for a freely- swinging balance beam.

Using Equation 2-12 and the result of part c:

(9C/9z

l

) = -12.12 pF/cm [1 + (6.5 x 10“9

)h

2

] » -12.12 pF/cm

SC/C = (9C/9 z

l

)az

l

/C = (-12.12 pF/cm) (±1.8xl0

_2

cm)/50pF

SC/C = ±4.36 x 10“ 3 = ±0.44% at C = 50 pF.

The value of C = 50 pF was chosen because it, along with C = 100 pF,

will be the values compared with reference capacitors. Clearly the

balance beam cannot swing during the capacitance measurements. It

will either have to be mechanically braked or electrically servoed to

gravitational horizontal.

e) The variation of z

L and C for a servoed balance beam.

How well must 3

y

be servoed for SC/C = ± 1 x 10

Az

l = (±1 x 10" 6 )C/(3C/9 z

l

) = (±1 x 10"6

) (50 pF)/(-12. 12 pF/cm)

Az^ = ±4.12 x 10" 6 cm (±0.041 ym).

67
The capacitance will change by 1 ppm if the electrode overlap L q

changes by a tenth of a wavelength of blue light. The laser

interferometer could monitor this via z

L however (if the upper part

were attached to the suspended electrode system) because 6^

=

±0.025 ym.

= az

l

/I_ a =(±4.12 x 10" 6 cm)/(7.5 cm) = ±5.50 x 10~ 7 rad (±0.11").

Therefore would have to be servoed to ±0.11" in order to measure

the capacitance at 50 pF to within ±1 ppm.

f) The electrode shape-dependence of C.

We can use Equation A-l to find the capacitance per unit length,

T' , if the average radius of any electrode surface varies

from the value for a right circular cylinder:

T' = 2*e

0/{ln[Rl(l + 6^) /r^(l + 6jE)]} +

2ire

0 /{ln[r°(l + «°)/R°(l + 6 °]}

where tsj = = &Rj/rJ, etc. Using the expansion

ln(l + u) = u - u

2

/2 + u

3

/3 u,

we obtain the result

68
AT = T' - T = 2ne

0

(6^ - 6 j)/(ln ^/rj ) 2 +

2ire

0 («s

- 6

C )/ 0 n r^R®) 2

,

where T= 12.12 pF/cm. Using the values from Table 1:

6 C/C = AT/T= (4.59 x 10" 2

)[(119.0)(6j - 6*) +

(118.3 ) (63 - 6 °)]. (C-l)

If the outer surface of the suspended electrode was lapped more at

the rim than in the middle, then it would deviate from a right

circular cylinder. Assume this surface has a slope of 1 ym/cm over

the last 10 cm, so that the radius at the rim is 10 ym smaller than R^.

The average value of <5^ depends upon the electrode overlap, and has

the value -5.10 x 10~^ at C = 100 pF. Let the other three electrodes

be right cylinders, so <$J= <5$ = =0. Then 6 c/C = -277 ppm at C =

100 pF and 6C/C = -388 ppm at C = 51.54 pF. These variations from the

values that the capacitance would have for right cylinders are linear

with Z|_ over the az

l = 4 cm span. If the slope was not a constant 1

ym/cm, then a plot of C versus z^ would have curvature. These

variations of C are real effects, not errors.

Let all the electrode surfaces be lapped more near their rims,

with constant slopes of 1 ym/cm over the last 10 cm. Then 6C/C =

-1139 ppm at C = 100 pF and 6C/C = -1595 ppm at C = 51.54 pF. A plot

of C versus z^ would be a straight line only if all four electrode

surfaces had constant slopes.

69
Let all four electrode surfaces be right circular cylinders, but

assume there is a bulge extending around the outer surface of the

suspended electrode. Let the bulge be symmetrically shaped with a 5 ym

height and a 4 cm base, and place the peak 8 cm from the electrode

rim. Then 6C/C varies from about 37 ppm to about 35 ppm in the az

l =

2 cm distance between C = 100 pF and C = 75.76 pF, and then decreases

to about 0 ppm in the az

l = 2 cm span between C = 75.76 pF and C

=

51.54 pF.

These examples demonstrate that the value of C, at any particular

value of z L

, is very sensitive to the shapes of the electrode

surfaces. If the average radius of any surface, for a given electrode

overlap, changes by ±1 ym from the value for a right circular

cylinder, then 6C/C ~ ±54 ppm.

70
APPENDIX D

VARIATIONS OF C WITH TEMPERATURE

a) Variation of C with uniform temperature changes.

The corner cube mirror in Figure 1 is mounted near the carriage

electrode rims, so it can monitor the carriage electrodes thermal

expansions. The upper part of the laser interferometer, however, is

about 40 cm from the rim of the suspended electrode, so the electrode

overlap LQ

could change due to expansion of the suspended electrode

system below the upper part of the interferometer and not be monitored

via the z

L measurement. (Only the overlap change due to the carriage

electrodes expansions would be monitored.) The upper part of the

suspension system is essentially temperature compensated, whether or

not the upper part of the interferometer is attached. to it, because

the coefficients of thermal expansion of the aluminum, brass and

-5

stainless steel guide rod components are all about « = 2 x 10 /C°.

Thus

6z

l

/aT = L<* = (40 cm) (2 x 10" 5 /C°) (°C/10

3

m°C) = 8 x 10" 7 cm/m°C.

Using a result from part e of Appendix C:

<5 C/C = (8 x 10~ 7 cm/m°C)(l m°C) (ppm/4. 12 x 10~6cm) = 0.19 ppm

71
for a 1 m°C temperature increase at C = 50 pF.

This result is for a uniform temperature increase everywhere in

the 0-ring sealed container. If the temperature varies during data

collection, then the capacitance measurements will account for the

6C/C changes, but the z L measurements will be in error.

b) Variation of C as the carriage moves through a vertical temperature

gradient.

A higher temperature is maintained at the top of the container to

minimize convection currents, which would affect the balance

measurements. Assuming a constant temperature gradient of 1 m°C/cm,

the average temperature change of the carriage electrodes is

AT = -4 m°C during the AZ|_ ~ 4 cm carriage translation from a

capacitance value of C = 100 pF to C = 50 pF.

The = 25 cm heights of the carriage electrodes have thus

changed by

A£

c

= * c

ccaT = (25 cm) (2 x 10

_8/m°C)(-4 m°C) = -2 x 10'6

cm,

giving the result

SC/C = (-2 x 10~ 6cm) (ppm/4.12 x 10" 6 cm) = -0.49 ppm

for a constant temperature gradient of 1 m°C/cm at C = 50 pF.

72
This is a real effect, not an error, because the change is

correctly monitored via the z

L measurement. There would be an error,

however, if the temperature gradient changed during data collection

because of the effect presented in part a.

c) Variation of C for a radial temperature gradient across the

electrodes.

Let the inner carriage electrode be 1 m°C warmer than the

suspended electrode, and the outer carriage electrode be 1 m°C cooler

than the suspended electrode. Then <$£ = ArJ/rJ = aAT = 2 x 10"^, 6* =

6^ = 0, and 6^

= -6^. Using these values in Equation C-l yields

6 C/C = 0.22 ppm

for a -2 m°C radial temperature gradient across the electrodes.

There would be errors if this temperature gradient changed during

data collection because the values of C would change for a given z L value. Also, the lengths and ^

would change by different amounts,

so the corner cube mirror would not correctly monitor individual

variations of the inner and outer electrode overlaps, resulting in

errors of the type presented in part a.

73
APPENDIX E

DERIVATION OF THE FORCE EQUATIONS

The electrical potential energy, U e

, stored in the electric field

of a capacitor is U

e

= (1/2)CV , and the change in U e from an initial

state i to a state j, using conservation of energy, is

AU

e

= (1/2)C.V^ - ( 1/2 )

J J ‘i- *

/•" J F ' d£ + / Vdq (E-l)

q

i

[U in N*m; C in F; V in V; F in N; S, in m; q in C]

where F is the electrical force and q is the charge on the electrodes.

Using the coordinate system of Figure 7, and anticipating that

F

h = F

h and F

h = F

h

( because h

v = h

x and hv

= h

v)’ the

x x y y

y y

differential of Equation E-l becomes

dU

e

= -2F. dh - 2F. dh - F 7 dz, + Vdq (E-2)

n xx y

y

Z

L

L

dU e = OUe /8h

x )dh

x + (3U

e /8h

y

)dh

y

+

(9U

e /9z

L )dz

L

dq = Oq/3h

x )dh

x

+ ( 9 q/9h

y

)dh

y

+ (9q/9z

L )dz

L

.

74
Substituting these equations into Equation E-2, and realizing that h ,

h

y

and z

^ are independent variables - so that the equation must be

satisfied for each variable - we obtain:

(3U

e /9h

x

) = -2F

h + V(3q/9h

x

)

(9U

e /9h

y

) = -2F

h + V(9q/9h

y

)

(9U

e/9z

L

) = -F + V(9q/9z

L

).

Using U e

= ( 1/2 ) CV^ and q = CV in these three equations thus yields

= (l/4)V

2

[3C(h,z

L )/3h

x

] h jZ ^ (E-3)

= (l/4)V

2

[3C(h,z. )/3hl . y VZ

L (E-4)

» (l/2)V

2 [3C(h,z )/3z ] L L h

x

,h

y

(E-5)

V in V; C in F; h, z, in m].

75
APPENDIX F

COORDINATES OF POINTS UNDERGOING ONLY Y

y

and 3 yMOTIONS

This appendix lists the y and z coordinates of the six points,

shown in Figure 2, which locate the positions of the gravitational and

electrical forces when only y and 3 motions occur. Figures 3, 4, 8

y y

and 9 are also needed to determine these coordinates. The units are

given at the end of the list.

y1 =-L Acos3yp

p sin$y-ppsin y

y

+Lcm

sin Y

y

z

1 = “L

Asin3y-PpCOSYy+p

p

cos3y+L cmcosYy

y2

=-A

b

sin3

y

z

2= *bC0Sf5

y

y3

=L

B cosfy*

B sin3y

z 3

=L

B sin3

y

+Jl Bcos3y

•Y4 = - L Acose y“ P

p si n(3 y- p

p si nY

y

+Lsi n y

y

z 4

= ‘ L A

sin3y“ p

P cos Vp

Pcose

y

+LcosY

y

y5 = ‘ L AC0S VP

P

S1 ney" P

P

S1 n Y

y

+Lsi n Y

y

+R

S

C0S Y

y

-(l/2)[L

o -R

3

sinY

y

-L(l-cosYy)+r^siney-ji(l-cos0y)]tanYy

z 5 =-L

A

si n e y

-PpCOSY

y

+Ppc os3y+LcosYy-RsSi ny

y

~ ( 1 /2 ) [ L Q

-R ^s i nYy-L(l-cosYy) + rQsin0y-ji( l-cos0y)]

y6 =_L

AC0sey" P

P Sin3y' P

P SinY

y

+LsinTy" R

S

C0SY

y

-( 1/2 ) [L

0

+R 3si nYy-L ( 1-cos Yy

) -r Jsin 0y-A( 1-cos 0y)] tan

Yy

Z 6 =_L

A sin6 y“ P

p cosY

v+P

P COS6

v+Lc0sYv+R SSlnYv

I 1 Y

-(l/2)[L

o+R 3sinY -L( 1-cos Y )-r

csin 0-t(l -cos 0 )]

j y y y

[y, z, La>

p

p

, L cm, £

b

, £g, L B

, L, Lq

, £, R

s

, R

s

, r

c

, r

c in cm; y

y

> 3

y

,

0y in rad].

76
APPENDIX G

COORDINATES OF POINTS UNDERGOING ONLY Y

x and 3

y

MOTIONS

This appendix lists the x and z coordinates of the six points,

shown in Figure 2, which locate the positions of the gravitational

and electrical forces when only y and 3 motions occur. The points

x y

5 and 6 are, however, in the x-z plane, rather than the y-z plane.

Figures 3, 4, 8 and 9 are also needed to determine these coordinates.

The Dy

displacements, due to the 3

y

motion, is ignored; so there are

no F yand F y

forces, and the y coordinates are therefore not

required. The units are given at the end of the list.

x l“L cm sinYx

zr-LA

sin Vp

P

C0SVLcmC0SY

x X2=0

z2

=£

bcos 3y

x

3

=0

z3=L

B

sinVJl B

cose

y

x4=Lsiny

x

z 4

=-LA sin3y+P

p cos3y+LcosY

x

x

5

=Lsi ny

x+RsCOSYx -( 1/2) [L

0

-R

5Sinyx-L( 1-cos Yx

)

+r

Csin 6

x"

Jl ^ 1 ‘ cosex^tanY

x

z

5 __L

A

s

'i np

y

+ RpCOS3y+ Lcos Yx-Rg si n yx -(l/2)[L

o-R^sinY

x -L(l-cosY

x )+r^sin0y-Jl(l-cose

x

)]

77
Xg=LsinY

x -R^cosYx -(l/2)[L

0+R^siny

x -L(l-cosy

x

)

-r Jsin6

x -£(l-cos0

x )]tany

x

z 6=-L

A

si ney

+ppCOse

y

+Lcosy

x+R^si nyx -(l/2)[L

o +R^siny

x -L(l-cosy

x )-rJsin0x -Jl(l-cos0

x )]

Cx, z, l~ A

, Pp > Lqjji

, £g > Lg , L , Lq

Jl

, R^» Rg>

^c* ^ cm, Y^»

ByS ©x in rad].

78
APPENDIX H

PROOF THAT h

1 = h° AND h

1 xx = h°

y y

Using Appendix F and Figure 2, and recalling that hj and d^ are

in pm, then the relative displacement, hj, of the suspended electrode

and the outer carriage electrode is:

10"4h°={y

5 -(Rj-LA )}+{(r°-LA )-[-LA

-10"4

d

y

+r°cos e -*sin8

y

+(L°/2)tan9

y

]} (H-l)

where

L o=L

o"

R

S

si nYy-L( 1 -cosY

y

)+rgsin0

y

-t( 1 - c ° sey)

.

Using the small angle approximations sin'f ~ tan'F 155 y and cos'F * 1 -

^/2, and keeping only the predominant terms, yields the result

h°=10

4

(L A/2)6

y

+10

4

(L-L

o

/2) Yy+10

4

(«-L

o/2)8

y

+d

y

(H-2)

[h, d in pm; L, Si , L Q

,

l_ Ain cm; e, y, 6 in rad].

This is the same expression as that derived in Section 2.3.

If the expression analogous to Equation H-l is written for the

relative displacement, h^, of the suspended electrode and the

inner carriage electrode - and similar approximations are made to this

expression - then one finds that h

1 = h°.

79
The expression for is:

lO’

4 h^={x

5 -R^}+{r^-[-lO

_4

d x+r^cos6

x -Jlsin0

x +

(L^/2)tan6

x ]} (H-3)

where

Lo

= L o"

R

s

sirTY

x"

L ^ 1 “ c ° SY x^ +r

C sin0

x _£ ^ 1 " cos6

x^ *

Making the same approximations as above, we obtain the equation

h“=10

4

(L-L

o/2)Y x+10V-Lo/2)e

x+d

x (H-4)

[h,d in ym; L, 1, L in cm; y, 9 in rad],

which is the same expression as Equation 2-9. A similar procedure

for h l gives the result h

1 = h

0

. x

3 xx

80
APPENDIX I

GENERALIZED TORQUES FOR y AND e MOTIONS ONLY

J J

a) Derivation of 0 . T

y

Using Equations 3-3 and 3-4:

Qy

=M

A

g(9z

1 /8Y^)+Mb g(8z

2 /8Y

y

)+M

B g(9z

3 /9Y

y

)

+F

z Oz4/3y

y

)+F°Oy5 /3Y

y

)+F^(3y

6 /3Y

y

). (I-D

Substituting the coordinates given in Appendix F into Equation I - 1

,

using the small angle approximations tany

y

~ sinYy~Yy and cosY

y

» 1,

and recalling that Fy

= Fy

yields:

\

= " M

A

9LcmVF

z

LV2F

y

(

L

‘ L o

/2 >• (I " 2)

Using Equations 2-10, 2-15, 2-16,2-18 and 2-19 in Equation 1-2, and

2

ignoring the h

y

term in F z gives:

qY

= -[M

A

gL cm+(6.06xl0'4

)V

2 L-(6.5xl0' 5

)V

2C(z

L )(L-Lo/2)

2

]Y

y

+[(6.5xl0‘ 9

)V

2 C(z

L )(L-Lo/2)][10

4

(L

A/2)6

y

+104

(Z-L

o/2)e

y

+d

y

] (1-3)

[Q in N*cm; M

Ag in N; V in kV; C in pF; d in urn; L cm , L, L Q

, L A>

l, z^ in cm; y» 3» 0 in rad].

81
b) Derivation of Q

Using Equations 3-3 and 3-4:

Q

6 =MAg(3z

1 /3B

y

)+Mbg(3z

2 /3e

y

)+M

BgOz3/3B

y

)

+F

z (3z

4/33

y

)+F

y

(3y5/33

y

)+F

y

(3y6/33

y

). (1-4)

Substituting the coordinates given in Appendix F into Equation 1-4,

realizing that small angle approximations cannot be used, again

recalling that Fy

=F

y

, and keeping only the significant terms yields:

V

C2F

°

L

A'

M

A

9P P"

M

b

9VVVF

z‘>P ]sin ey

+[M

B gL

B -M

A

gL A-F

z

L A]cos3

y

. (1-5)

Using Equations 2-10, 2-15, 2-16, 2-18, and 2-19 in Equation 1-5, and

2 keeping the h

y

part of F

z in the F

z

L Acos3

y

term of Equation 1-5

gives:

Q

0

=[(3.25xl0' 5

)V

2

C(z

L )L^6y+(6.5xl0'

5

)V

2

C(z

L )L A

(L-L

o/2)Yy

+(6.5xlO' 5

)V 2 C(z

L )L A

(z-L

o/2)0

y

+(6.5xlO" 9

)V

2

C(z

L )L A

d

y

- M

A9Pp-Mbgzb-M

B gz

B-(6.06xl0"4

)V

2

p

p ]sins

+CM

B

9L

B'

M

A

9L

A' (6 - 06xl0

’4)v2l

-A‘

(3 - 94xl0

" 12

)',2L

A

d

y

-(9.85xl0' 5)V

2

L

3

By

-(3.94xl0'4)V

2

L

A

(L-L

o/2)

2

y

2

-(3.94xl0'4

)V

2

L A

(z-L

o/2)

2

e

2

-(7.88xl0'8)V

2

L A (z-L

o/2)6

y

d

y

-(7.88xl0'8

)V

2

LA

(L-L

o/2)dyYy-(3.94xl0'8

)V

2

L2dy6

2

82
-(3.94xl0'4

)V

2

L 2

(*-L

o/2)e

y

fi 2

-(7.88xlO‘

4

)V

2

L

A (L-Lo/2)(*-L

o/2)0

y

Y

y

-(3.94xl0'4

)V

2

L

2

(L-L

o /2) Yy$

y

]cos6

y

[Q in N*cm; M^g, M^g, M^g in N; V in kV; C in pF; L

Lg, A, A

b

, Jig, p pj z L in cm; d in ym; y, 3 , 0 in rad]

(1-6)

cm’

L ’ L o’L A’

83
APPENDIX J

GENERALIZED TORQUES FOR Yy AND e MOTIONS ONLY

X Jr

a) Derivation of Q . Y x

Using Equations 3-3 and 3-4:

Qy

= Ma9(9z

1 /8y

x

) + M

b g(3z

2 /3Y

x

) + M

B g(9z

3/3Y

x

)

+F

2

<3z4/3Y

x

) + F°(3 x 5/3t

x

) + fJ(3x6/3t

x

). (J-l)

Substituting the coordinates given in Appendix G into Equation J-l,

using the small angle approximations tanYv * sinY Y ~ Y v and cosy y ~

1, and recalling that F* = Fx yields:

V-M

A9

L cm

Y

xF

z LV2Fx(L-L0/2). (^)

X

Using Equations 2-9, 2-14, 2-16, 2-17, and 2-19 in Equation J-2, and

2 ignoring the hv term in F, gives:

x Z

Q

y

=-[M

A

gLcm+(6.06xl0'4)V

2L-(6.5xl0'5

)V

2C(z

L )(L-L

o/2)

2

]Y

x +[(6.5xlO' 9

)V

2

C(z

L )(L-L

o /2)][lO

4

(z-L

o/2)0

x+d

x

] (J-3)

[Q in N-cm; H^g in N; V in kV; C in pF; d in ym; L cm , L, L Q

, L A

,

i, z

L in cm; y» e in rad].

84
b) Derivation of Q

Using Equations 3-3 and 3-4:

Q

3

=M

Ag(3z

1 /3e

y

)+M

b g(3z

2 /93

y

)+M

B g(9z

3 /33

y

)

+F

z (3z

4/3e

y

)+F°(3 x5/3$

y

)+F'(3 x6/3B

y

). (J-4)

Substituting the coordinates given in Appendix G into Equation J-4,

realizing that small angle approximations cannot be used, and again

recalling that fJ=F^, yields:

Q

e =[-M

A9Pp-Mb

9i! b-M

B

gJl

B

-F

z pp]si n(Sy

+[M

B gL

B

-M

A

gLA

-F

z LA]cos$

y

. (0-5)

Using Equations 2-9, 2-16 and 2-19 in Equation J-5, and keeping the

2

hx part of F

z in the F

z L Acos3

y

term of Equation J-5 gives:

Q

6 = [-MAgpp-M

bg*b-M

Bg*B-(6.06 x10'4

)V

2

pp]sin$

y

+CM

BgLB-MA

gLA

-

(

6 .6xl°"

4

)

V

2

L A

-(3.94x10' 4

)V

2

(L-L

o/2)

2

L

AY

2 -(3.94xl0' 12

)V

2

L A

d

2

-(3.94)(10'4

)V

2

(«-L

o

/2)

2

L

A

6

2

- ( 7 . 88xl0‘4

) V

2

( L-L

q

/2 )

( Jl-L

0/2 ) L

A

0 xyx -(7.88x10'8

)V

2

(l-L

o /2)L

A

d

xYx -(7.88xl0

_ 8

)V

2

(A-L

o/2)LA

e x

d x ]cose

y

( 0

- 6 )

[Q iN N* cm; M

A

g , M

b g, M

g

g in N; V in kV; L cm , L, L Q,

l_ A

, L

B

, l ,

A

b

> £

B

5 p

P in cm; d in Y > 6 in rad l*

85
APPENDIX K

BALANCE BEAM STIFFNESS AND SENSITIVITY

Let the balance beam and the suspended electrode system be in

static equilibrium. Therefore Q„ =0, and for very small angles

ey

Equation 1-5 reduces to

[M

B

gL

B

-(M

A

g+ Fz

)LA>[-2F0L

A+

(MA

g+ P

z )Pp+Mbg*b-M

Bg*

B]V

A A A This equation has the form r =XS , where r is the torque in N-cm,

^y y ey

and V is the beam stiffness in N-cm/rad;

^

= [-2^L

A+(M

Ag+Fz

) P p+M

b g£ b

+M

B gAg] (K-l)

[V in N-cm/rad; F , F

z

, M

A

g, M

b g, M

g

g in N; L A

, p p

, * b, £

B in cm].

The balance beam sensitivity, S, is

/s /s S=AB /ATg =1/K

y

y

(K-2)

[S in rad/N«cm;

3^

in rad; in N; )i in N-cm/rad]

We found in part e of Appendix C that the balance beam angle has

to be held to within ±5.50x10 ^ rad (±0.11") in order to measure the

86
capacitance at 50 pF to within ±1 ppm. Therefore, in order to

monitor the minimum sensitivity for the addition of a 1 pg mass

to must be

S=A0/Ar =-(5.50xl0" 7 rad)/-[(lxl0’ 6

g) (kg/10

3

g) (9.8m/s 2

) (7 .5 cm)]=

y ey

7.48 rad/N-cm,

~ ~0

orR=0.134 N-cm/rad. With voltage off, is replaced by mg, and = 0,

so Equation K-l becomes

X

(0FF) =[(M

A+m)gPp+M

bg*b

+M

B g«

B

].

Using the brass electrode, and the values K^FF ^=0. 134 N-cm/rad,

_3 -5

Ma

= 5.47 kg, m=5xl0 kg, and p

p =5xl0 cm, this equation reduces to

18.6 Jt b

+53.6 i

B=0.131, where Jl b

and are in cm, and can be positive

or negative. (They are both drawn with positive signs in Figure 2.)

One has no idea experimentally what the combination Ji b

and £g

_3

actually is, but if Jl b

=0 then a

b=2 .44x10 cm (24.4 pm), or if £g=0

_3

then Ji b

=7. 04x10 cm (70.4 ym). Clearly, with a reasonable

combination of fc b and Jig, the sensitivity screw on the balance can be

adjusted to obtain a value of S=7.48 rad/N-cm for the brass electrode

with voltage off.

If V=9 kV and ey

=5" then F

z=4.91xl0

-2

N and 2F°=140.7xl0~ 5 N from

Table 7, and Equations K-l and K-2 yield the values

X( 0N )=0.123 js|. Cm/ rad and S^

0N ^=8.13 rad/N-cm for 18.6 *

b+53.6£g=0. 131

87
N •cm/rad. So depends upon F

1^ (but not upon F^) - and is 9%

y * greater than for this example. This sensitivity variation

need not affect us, however, because the voltage can be adjusted so ^(OFFl that /=

8y , i .e. the balance can be used as a null device.

(Although we will see in part g of Appendix P that an error results

if S<

0N)

- B< 0FF)

, 0 and fJ t 0.)

88
APPENDIX L

KINETIC ENERGY FOR Y AND a MOTIONS ONLY

y y

Using Figure 2:

T= T 1+T

2+T3

T=(l/2)M

A (y|+z|)+{l/2)Mb (y!+z| ) +( 1/2)M

B (y|+i|>.

Substituting the coordinates listed in Appendix F into the above

equation yields:

>(l/2)M

A[L2(6y)

2+P^ (;

y

) 2^(iy) 2

+P ^

y

) 2

- 2LAPpSin6

y

COS Yy ByYy

+2L

A

L

cm

s1nV0SYyVy

+2PpCOSByCOSYy6yTy

- 2LcmPpCOSB

y

COSY

yVy-2LcmPp ( fy*

-2L

APpC0S$ySinYyBy-Vy

+2LA

L

cm

COSB

y

sinYyVy

-2P pSi n$

y

si nifyByiy

+2L

cmp

P si n6

y

s 1 nY

yVy]+ ( 1 /2 > Vb< B

y

>

'

2

+(l/2)M

B

[L

2

(g

y

) 2 n

2

(By)

2

] (L-l)

2 2

[T in kg^cm /s ;

By, y in rad/s;

M

A’

M

b> M

B in k 9i L A> Pp> L cm , z b

, Lg, £

b

in cm;

By, Y

y

in rad].

I

I

I

89
APPENDIX M

KINETIC ENERGY FOR y AND b MOTIONS ONLY

x y

Using Figure 2:

T=Wt

3

T=(l/2)M

A {x^+2^)+(l/2)M

b z|+(l/2)M

B

i|.

Substituting the coordinates listed in Appendix G into the above

equation yields:

T-=(l/2)MA

CL^

m

{Yx) 2

+L

Acos

2

By

(6

y

) 2+p2s in

2

6

y

(6

y

) 2

+2L

A

p

p sin3

y

cosey

(6

y

)

+2L

cm

L

A

sinVos6yVx

+2L

cmp

P

sin6

y

sinYxVx]

+(l/2)M

b

t 2

sin

2

sy

(8

y

) 2

+(l/2)M

B

[LgCOS2

8

y

(6

y

) 2

-2L

B

«

B sins

y

cos$ (8

y

) 2

+Jt|s-in

2

By( §

y

) 2

] (M-l)

[T in kg.cm2

/s

2

; M

ft

, M

fc> Mg in kg; Lfl

, L cm , p p> t b

, Lg, « B in cm

By, Y

y

in rad/s; e y> y in rad].

90
APPENDIX N

D'ALEMBERT'S PRINCIPLE FOR y AND a MOTIONS ONLY

J J

a) Equation of motion for the suspended electrode system.

Writing Equation 3-5 so that it has the dimensions N*cm, yields

10

-2 {d/dt(3T/3Y

y

)-(3T/3Yy ) }=Q (N-l)

Substituting the expressions for Q and T given by Equations 1-3 and

Y

y

L-l into Equation N-l, using the small angle approximations siny «y ,

cosy wl, sin3y«3y and cos3

y

»l, keeping only the significant terms,

and replacing the point mass M^ with a distributed mass, yields the

result

10' 2l pY/CMA

gL

crn+(6.06xl0'4

)V

2

L-(6.5xl0' 5

)V

2C(L-L

o

/2)

2

] Y:y -[(6.5xl0'9

)V

2C(L-L

o/2)][10

4

(*-L

o/2)e

y

+d

y

]=

+[(6.5x10' 9

)V

2

C(L-L

o/2)10

4

(L a/2)]6 2

(N-2)

2 2

[I

p in kg. cm ; M^ in kg; g in m/s ; L cm , L, L Q

, i , L A, p

p in cm;

V in kV; C in pF; d

y

in ym; Yy , 6

y

in rad].

91
where I p is the moment of inertia of the suspended electrode system

about the pivot point P of Figure 2, which from the parallel axis

theorem is equal to the moment of inertia about the center of mass of

the suspension system plus the mass times distance squared from the

2

center of mass to the pivot point: I p =( I s)cm+M

A

L cm*Usin 9 the brass

4 2 cylinder in the suspension system gives I *1. 39x10 kg*cm , which is

2

15% larger than the value of The off-diagonal, product of

inertia terms are represented by the point mass expression M

A

L cm

L A

in

Equation N-2; this is a very good approximation. The

10

" 2

M

A

L cm

L A^y) 2 and 10

" 2M

A

L cmL A^ ey

+Yy^y terms are the torques

induced about point P due to the respective centripetal and

tangential accelerations of the suspended electrode system resulting

from the motion of the balance beam about point C of Figure 2.

Equation N-2 has the form:

aVbVc “' e< ®y

)2' e(6

y

+Y

y

)s

y

+fVke

y

(N' 3)

where the coefficients are defined in Equation N-2.

b) Equation of motion for the balance beam.

Writing Equation 3-5 so that it again has the dimensions N.cm,

yields

10

_2 {d/dt(3T/3e

y

)-OT/38

y

)}=Q

6

. (N-4)

92
It is clear from inspection of Equation 1-6 that it would not be

possible to find an analytical solution of Equation 1-4 without

simplifying the terms involving By. We want to detect 1 ug in 5.47

kg; therefore the balance must have a resolution of at least 1.8

10 2

parts in 10 . If we approximate cosp =1 by ignoring the -0 /2 term,

y y

then 0^ must be less than or equal to 1.94x10

5 rad (4") in order to

keep the errors at the 1.8 parts in 10*° level. So, let the balance

swing freely with an amplitude and equilibrium angle such that By £

4".

We can then obtain an analytical solution; this solution should also

be valid for larger values of By, except, possibly, for

amplitude-dependent errors, which could be corrected by extrapolating

the balance measurement values to a zero-amplitude result.

Substituting the expressions for Q and T given by Equations 1-6

8

y

and L-l into Equation N-4, using the small angle approximations

sin0y~By, cos0y~l, sinYy^Yy and cosYy*'l, keeping only the significant

terms, and replacing the point masses with distributed masses, yields

the result

10' 2

I C

By+{M

A

gp

p+M

b

9Jl b

+M

B

gt

B

+(6.06xl0‘

4

)V

2

p

p -(6.5xl0' 5

)V

2

CL

A

[(n-L

o/2)e

y

+10‘4

d

y

]}6y

-[(6.5xl0' 5

)V

2

CLA

(L-L

o/2)]Y

y

6

y

- {M

BgLB

-M

A

gLA

- { 6 . 06xl0"4

)

V

2

La -(3.94xl0"8

)V

2

L A

tl0'

4

d

2

+10

4

(j-L

o/2)

2

6

2

+2(*-L

o/2)e

y

d

y

]}

=- 10

' 2MALA

Lcm

(V

y

) 2_ 1 o2

MAU ALcm ( ey+T)T

y r y

+10

' 2

M

A

L cmPpVC ( 3-94x 10

_4

)V

2

L a

( L-L

0/2)

2

]y

y

93
-{(7.88xl0 (N-5)

_4

)V

2

L

A (L-L

o /2)C(i-Lo/2)e

y

+10

_4

d

y

]}Yy

2 2 [I

c in kg*cm ; M

b

, in kg; g in m/s ; L A

, *

b

, * B

, Pp , i, L,

L o

s L cm 5 L

B in cm; V in kV; C in pF; d

y

in V B y’ 6

y

in

rad]

where the moment of inertia, 1^, about point C of Figure 2 is

2 2

I C

=M

A

LA

+ ( I cw)B+M

B

L B + ( I b)c’

and ^cw^B and 'Vc are the moments of

inertia of the counterweight about point B and of the balance beam

about point C, respectively. Using the brass cylinder ( M^=Mg=5 . 47

kg) gives I c«683 kg •cm

2

. The 10~ 2

M

A

L A

l_ cm (Yy) 2 and

_2

10 M/\L^L cm

(3

y

+Y^)Yy terms are torques induced about point C due to

the respective centripetal and tangential accelerations of the

suspended electrode system about point P of Figure 2.

Equation N-5 has the form:

ney

+ (o-PY

y

)6

y

-q=-e(;

y

) 2 -e(6

y

+Y

y

)Yy

+fY

y

-rYy-sYy (N-6)

where the coefficients are defined in Equations N-2, N-3 and N-5.

94
Appendix 0

D'ALEMBERT'S PRINCIPLE FOR Y AND fi MOTIONS ONLY

x y

a) Equation of motion for the suspended electrode system.

Writing Equation 3-5 so that it has the dimensions N*cm, yields

10" 2 {d/dt(3T/9Yx )-(3T/3Y„)}=<) . (0-1)

x Yx

Substituting the expressions for Q and T given by Equations J-3 and

Yx M-l into Equation 0-1, using the small angle approximations sinyY ~yv ,

cosy

x ~

1, sinewy and cosey

~l, keeping only the significant terms,

and replacing the point mass M^ with a distributed mass, yields the

result

10‘ 2

I pv[MA

9Lcm+(6 - 06xl0

’ 4)v2l-' (6 - 5xl0

’ 5

) v2c(L ' L o/2)2:lY

x -[(6.5xl0‘9

)V

2C(L-L

o/2)][10

4

(t-L

o/2)6x+d

x

]

“- 10

' 2¥»laV, <°- 2 >

2 2

[I

p in kg-cm ; M^ in kg; g in m/s ; V in kV; L cm

, L, L

Q

, £, L A

in

cm; C in pF; d

x in ym; yx

, 6 x

, 3

y

in rad]

4 2 where I p is defined in Appendix N, and is 1.39x10 kg «cm for the

-2

brass cylinder. The 10 M/\*- cm

LAYx

3

y

term is the torque induced about

95
point P of Figure 2 due to the tangential acceleration of the

suspended electrode system resulting from the balance beam motion

about point C. The centripetal acceleration term is negligible.

Equation 0-2 has the form:

a Yx+bY

x'

c

' =-eYx3y

(°~ 3 )

where the coefficients are defined in Equations 0-2, N-2, and N-3.

b) Equation of motion for the balance beam.

Writing Equation 3-5 so that it again has the dimensions N*cm,

yiel ds

10" 2 {d/dt(9T/96 )-(9T/9e ) }=Q . (0-4)

y y Py

-5

We assume, for the reasons given in Appendix N, that B <1.94x10

rad (4")

.

Substituting the expressions for Q and T given by Equations J-6

and M-l into Equation 0-4, using the small angle approximations

sinBy~By, cosb *1, sinY

x ~Y

x and cosy

x ~1, keeping only the significant

terms, and replacing the point masses with distributed masses, yields

the result

10' 2

I c By+[M

A

g Pp+H

b

g)t b+M

BgJ

B + (6.06xl0‘

4

)V

2

p

p 3B

y

96
' {M

B

9L

B'

M

A

gL

A'

(6 ' 06x10

" 4)v2l

A

-(3.94 X 10‘8

)V

2

L

A

[10'4

d

2

+lO

4

U-Lo/2)

2

e 2

+2U-L

o/2)0

x

d x]}

“- 10

" 2M

A

Lcnl LA(Yx

) 2 -10' 2

M

A

LcmL AYx Yx -(3.94x10'4

)V

2

(L-L

o/2)

2

L ay

2

-{(7.88xl0'4

)V

2

(L-L

o/2)LA[(4-L

o/2)e

x+10'4

d

x ]}y

x (0-5)

[I

c=kg-cm2

; M

fc

, M

g

in kg; gin m/s 2

; LA

, « b

, « B> p

p

, i, L, L Q

,

L cm , L B in cm; V in kV; C in pF; d

x in pm; yx

, 0 , 6x in rad]

2 where 1^ is defined in Appendix N, and is 683 kg*cm for the brass

cylinder. The 10 M

A

l- cm

L A (Yx

) and 10

“ M

A

Lcm

LAYxYx terms are torques

induced about point C due to the respective centripetal and

tangential accelerations of the suspended electrode system about

point P of Figure 2.

Equation 0-5 has the form:

•* # n p n8y+vey-q'=-e(Y

x

) - eYxYx- rYx

- s, Yx (0-6)

where the coefficients are defined in Equations 0-2, 0-3, 0-5, N-2,

N-3 and N-5.

97
APPENDIX P

SOLUTIONS OF THE EQUATIONS OF MOTION FOR y AND 6 OSCILLATIONS ONLY

y y

a) The suspended electrode motion.

We found in Equation N-3 of Appendix N that

Y

y

+(b/a)y

y

-(c/a)=-(e/a)(B

y

r

- ( e/ a ) ( B y

+

Y

y

) 8

y

+ ( f

/

a )

8

y

+ ( k/ a )

(

By

2

) (P-1)

where the coefficients are defined in Equation N-2. The homogenous

equation y +(b/a)y -(c/a)=0 has a periodic solution, which depends

y y

upon the starting conditions. If y

y

happens to be at its miminum

value at time t=0, then

y =-y coso) t + y

y y y

y

y

(P-2)

where y

y

is the amplitude of the oscillation and y

y

is the

equi 1 ibri urn angl e.

We will find in part e that the approximate solution of the

equation is also periodic - and, if 3 y

is at its mimimum value at

t=0, then

Wos\t + v

( P-3)

98
Equations P-2 and P-3 are approximate solutions for the y and r y y

motions. We next treat the r.h.s. of Equation P-1 as a perturbation

of the l.h.s. of the equation [11], with 3

y

having the form of

Equation P-3. Taking the appropriate time derivatives of 3

y

, and

noting that cos a-sin a=cos2a and that cos a-(l/2) ( l+cos2a) yields

the result

Y

y

+[(b/a) + (e/a)3

y

a)g cosoo

3

t]y

y

-[(c/a)+(k/a) (6

y

+6

y

/2)]

«y y

=[(e/a)3

y

Wg +(k/2a)3

y

]cos2o)

e t

y y

+[(f/a)3

yw

3 -(e/a^y^ -(2k/a)3

y

3

y

]cosoo

g t. (P-4)

The homogenous equation for the l.h.s. of Equation P-4 has the

2

form of Mathieu's equation [12,13] unless b»e3yog . Using values

given in Table 8 for V=9 kV and 3

y

=2. 40x10

3 rad (rather than

1.94xl0~ 3 rad), yields 689 N *cm » 0.001 N.cm; so, the

~ 2 (e/a)3 oog cosoo t term can be ignored in Equation P-4, and Y is

y y 7

y

primarily a cosine function rather than a Mathieu function.

The perturbation solution of Equation P-4 is then exact and, if

w fuQ or w t 2to0 , the solution is [13]: Y

y

ByY

y

6

y

V¥0SY

+ Y

y

+[(e/a)SyU? +(k/2a)?]cos2co t/(¥ -4oj? ) y

y

y

Yy y

99
+[(f/a) ^oig -(e/a) -(2k/a)B

y

B

y

]cosw^t/(w2 -w

2

). (P-5)

Therefore, the suspended electrode system executes simple harmonic

A

motion about the equilibrium angle y with cos2o)

6 t and cosw t y

y y

amplitude modulations.

b)

/v The equilibrium angle yy*

It follows from Equations P-4 and P-5 that

Y

y

=[c+k(f^+(l/2)B

y

)]/b

;? y

=(6.5xl0'

9

)V

2 C(L-L

o/2)[10

4

(j-L

o/2)e

y

+d

y

-H0

4

(L A /2)(B^B

y

/2)]/

[ M

A

g L cm

+( 6 . 06xl0

' 4

)V 2L-( 6 . 5xl 0

' 5

)V

2

C( L -L

o

/ 2)

2

] (P-6)

[V in kV; C in pF; M„g in N; L, L„, i, L a , L cm

in cm; d„ in pm;

V V V ey

in rad] ‘

So, y depends upon d , 0 , 3 and 6 for a given V, C and L . Using

y y y y y s values from Table 8 in Equation P-6 results in a change in y of 0.1"

-v « A A y

when 8 =2.40x10 rad and 8 ~0. Therefore, y is not affected very

y y y

much by the balance beam motion.

100
c) The period P . y

y

It follows from Equations P-4 and P-5 that:

w

2 =b/a=[M

A

gLcm+(6.06xl0'4

)V

2L-(6.5xl0' 5

)V

2

C(L-L

o/2)

2 ]/[10- 2

I p

] (P-7)

O

y

in rad/s; M^g in N; L cm , L, L 0

in cm; V in kV; C in pF; I p y 2

in kg-cm ],

and

P =2tt/oj

y

r y

( P-8)

[P in s; to in rad/s].

y y

P and a) are plotted versus V in Figure 10 using values from

y

y

y

y

Table 8 in Equation P-7. The points shown in this figure are at the

values of V which provide electrical forces that are equivalent to

gravitational forces due to masses of 0, 1, 2, 3, 4 and 5 grams.

P increases from a value of 1.48 s at V=0 kV to a value of 2.82 s y

at V=9 kV. If the voltage were increased to the point of unstable

equilibrium (10.56 kV for this example) then P -*», and the suspended

y

y

electrode would collide with the carriage electrode in a time

interval which depends upon the initial position of the suspended

electrode and its horizontal velocity component.

101
d) The amplitude modulations of y .

The amplitude of the cos2o)0 t modulation term in Equation P-5 is

[ 10

' 2

M

A

Lcm

LA"p

+(3.25x10' 9)V

2 C(L-L

o/2)10

4

(La/2)]B^/

[10-2

I p

( U

2

-4co

2

)] (P-9)

Y y y

[M

a

in kg; I p in kg -cm

2

; L cm

, L A

, L, L Q

in cm; V in kV; C in pF;

a) a) in rad/s; 3 in rad],

Yy y

y

-3

and using values from Table 8 and 3y=2. 40x10 rad, has the value 7.7

x 10” 7 rad (0.16"). So, the y

y

cosine waveform of 2.82 s period

would have a ±0.16" modulation imposed upon it, with a period

P =46.8 s for this example. A sufficiently sensitive autocollimator

py

that monitored the angle of a mirror mounted on top of the suspended

electrode could detect this amplitude modulation of y .

The amplitude (for sufficiently small

modulation term in Equation P-5 is

3y) of the cos

“is.

[M

A

L cmPp“g V/[I

P

( V“B )3

y

Yy

py

(P-10)

[M

a

in kg; I p in kg -cm

2

; L cm , Pp in cm;

rad]

,

a).. ,o^ in rad/s; 3 W in

Y

y y

and has a value 4.1x10 for the values from Table 8. So, the

cos 03 t modulation would not be detectable.

p»/

102
e) The balance beam motion.

We found in Equation N-6 of Appendix N that

3

y

+[(o/n)-(p/n) Y

y

]$

y

-(q/n)=-(e/n)(Y

y

) 2 -(e/n)(e

y

+y

y

)Y

y

+(f/n)Y

y

-(r/n)Y

y

-(s/n)Y

y

(P-11)

where the coefficients are defined in Equations N-2, N-3 and N-5.

The r.h.s. of Equation P-11 can be treated as a perturbation of the

l.h.s. of the equation. Using the approximate solution for y

y

given

by Equation P-2, taking the appropriate time derivatives of y , and

2 2 2 ^ noting that cos ot-sin cf=cos2a and that cos a=(l/2) (l+cos2a) yields

the result

By

+[(l/n) (o-pYy) + (y

y

/n) (p+ea)

2

Jcosa^ t]e

y

y y

-[ (l/n) (q-ry

y

-sy

y

-(r/2)y2

)]

=[(Yy/n)(ea) 2 -r/2)]cos2aj t

y Yy

y y

+C (

Y

v/n) ( fa)J +2ry +s-ey a) 2 )]cosw t. (P-12)

y 'y y y 'y Yy

The homogenous equation for the l.h.s. of Equation P-4 has the form

of Mathieu's equation [12,13] unless (o-py )=)^

0N ) »y ( p+e(Z ).

y ^ y Y

y

Using values given in Table 8, and letting Y =Y =6.8", results in the

y y

103
values 0.123 N •cm/ rad >0.011 N-cm/rad. So, if Y

y

<6.8", the

p

(l/n)(p+eco )y coso3 t term can be ignored in Equation P-12, and

Y

y

y Y

y

3^

is primarily a cosine function rather than a Mathieu function.

The perturbation solution of Equation P-12 is then exact - and

if 3y happens to be at its minimum value at time t = 0, and if

03 7*03 or 03 f 2oj , the solution is [13]:

y

Y

y ^y Y

y

V-Vos \

t+ ^y

+[(Yv /n)

2

(eo3^ -r/2)]cos2o3 t/(o3

2 -4o3

2

) y

y

Yy y

Yy

+[(

Y

v/n) (fo)

2

+2ry +s-eY„w

2

)]coso3 t/

(

oj? -oj2

). (P-13)

Y y

y Y y

Yy y

Y y

Therefore, the balance beam executes simple harmonic motion about the

A

equilibrium angle 3 with cos2o3 t and cosoj t amplitude modulations.

y Y

y

Y

y

/s f) The equilibrium angle 3 .

It follows from Equations P-12 and P-13 that

6 y

=[q-rTy-SY

y

-(r/2)Y^]/(o-pY

y

) =

[q-rY

y

-SY

y

-(r/2)Y^]/K

6

y

= {MBgL

B -M

A

gL A

- (6 . 06xl(T4

) V

2

L A

-(3.94xl0'8

)V

2

L

A

[10'4

d

y

+104

(«-L

o/2)

2 ey

+2(«-L

o /2)6

y

d

y

]

-(3.94xlO- 4

)V

2

L

fl (L-L0/2)

2

[^

2 + (1/2)y

2

]

104
-(7.88xlO"4

)V

2

L A (L-L

o/2)[(*-Lo/2)0

y

+lO"

4

d

y]y/

{MA

9P P +M

b

9VM

B

9 V( 6 • 06x10

’ 4

) v%

-(6.5xl0' 5

)V

2

CL

A[(n-L

o/2)e

y

+10’ 4

d

y

]

-(6.5xlO“ 5 )V

2

CL

A (L-L0/2)^

y

} (P-14)

CMA

g , M

b g, M

B

g in N; V in kV ; C in pF; l_ A

,

l_ B

, L, Z, L Q, p p

, *

b

, A A ^

£

b

in cm; d

y

in ym; 8

y

, 6 , y , Y

y

in rad],

/\ A ^ So, 8

y

depends upon 0 , d yJ y

y

and y

y

for a given V, C and L Q

.

The variation in 3 due to y is

J J

dB

y

=-(1.97xl0"

4

)V

2

L

A

(L-L

o/2)

2

Y^/K (P-15)

A ^ [V in kV; L fl , L, in cm; Kin N*cm/rad; 8 , y in rad].

m u y y

where X is defined by Equations K-l, 2-10, 2-15, 2-16, 2-18 and 2-19,

Using values from Table 8 in Equation P-15 gives, for V=9 kV:

«B

y

°N)=-(l-64xlO" 2

)Y

y

(P-16)

C fy.

Y in "]

where 8

(ON) and y are now expressed in arcseconds in order to get a j y . ""(ON) feeling for the magnitude of this effect. S8

y

is plotted versus

y

y

in Figure 11 for V = 9 kV. If y=9", then 3

y

^

is 1.33'' less

105
than its value for Y =0; so, F

z would appear to be larger by an

amount equivalent to a 12 ug mass increase because the balance would

have been calibrated using sensitivity weights with the voltage off -

''(OFF)

yielding a A

8y /Am calibration value of 0.11"/yg.

The value of Yy depends upon the starting condition. If, in this

example, the suspended electrode was released, at rest, from

gravitational vertical (Yy=0), then Y^=Yy=6.8" , making F

z appear to

be larger by an amount equivalent to a 7 ug mass increase. Figure 11

shows that Yy has to be less than 2.5" in order to measure the mass

equivalent of F

z to within 1 ug at V = 9 kV. Note, from Equation

A P-15, that the systematic error, 68y, is zero for the voltage-off

measurements.

The amplitude, y , of the suspended electrode will decrease with

time; if only velocity-dependent damping occurs, then the Yy

waveform given by Equation P-5 would have an exponential decay

envelope superimposed upon it - and, since the velocities are small,

the decay time-constant would be long. The actual rate at which Yy

decreases is not yet known, but it could be monitored by

autocollimator measurements via a mirror mounted on top of the

suspended electrode.

g) The force determination.

The voltage will be adjusted in the force measurements until

106
(P-17)

'(ON) _ '(OFF)

ey

‘ 6

y

Using Equations P-14, 2-10, 2-15, 2-16, 2-18, 2-19 and K-l, and

a ^0

realizing that F

z=F =0 and that for V=0, Equation P-17

-4 2

becomes, when neglecting the (6.06x10 )V p

p term:

[M

B

gL

B

-M

A

gLA

-F

z

L A-(1.97xl0'

4

)V

2

L A (L-L

o

/2)

2

Y^)/

[^

(0FF) -2^L

A

]

=[M

B

9LB-(MA

+"')gL A]/K^

0FF

*

,

which can be rewritten in the form:

F

z=mg[l+(2F°/mg)g(°FF) -(l/mg)(1.97xlO'4

)V

2

(L-L

0/2)

2^2

] (P-18)

A AA -A. ^ [F , F^, mg in N; V in kV; L, L in cm; $ , y

in rad].

y j y

Therefore, F

z=mg only if F^=o or By^^=0, and if yy=0.

The dynamical error term -(1/mg) (1.97x10 ^)V^( L-L

q /2)^y^ leads to

A

a systematic error in F

z of -1.2 ppm at V = 9 kV and C = 100 pF for

A/ fV Yy=6.8", and a systematic error of -0.2 ppm for Yy=2.5"; i .e. the

A voltages were lower than they should have been if F^ were to be equal

to mg. This error term can be experimentally monitored by measuring

Yy with the suspended electrode autocollimator. It always has a

negative sign.

107
OFF) Values for the static error term (1/mg) 2F^6y are listed in

Tables 3-7 in ppm/", where the " symbol refers to the 3

(OFF) angle in

arcseconds. If $^^=-10" with the brass suspended electrode at

V= 9 kV, d^=0 and 6^=+5", then Table 7 provides the resulting

A systematic error in F

z of -1.4 ppm.

The angle is measured from gravitational horizon. So, a

method will have to be devised to determine, with sufficient

accuracy, the balance beam readout angle that corresponds to the

plane defined by the contact points (P) and (C) of the pivot and

center knives (see Figures 2-4) being parallel with gravitational

horizon. In the above example the horizontal plane determination

would have to be better than 7" in order for the static error term to be

less than 1 ppm.

Either the counterweight mass, Mg, or M

A

will then have to be

adjusted until the correct readout angle is obtained with

voltage-off. For a balance mass sensitivity of 0.11"/yg, this means

that the Mg or M

A adjustment must be to within 7" (or 64 yg) for a

combined possible error in F

z of 2 ppm. More than the usual care must then

be taken after adjustment, and the readout angle mirror mounts must

be mechanically stable.

The sign of F can be determined by monitoring y with the

^ “Toff) suspended electrode autocollimator, but the sign of $ could only

be determined if gravitational horizon were accurately known.

Therefore, the sign of the systematic static error term is unknown.

108
Neither error term occurs for normal mass intercomparisons. The

dynamic term is zero because V=0, and the static term is zero because

F^=0, independent of the value of 3^*^.

y y

h) The period P Q . 6

y

It follows from Equations P-12 and P-13 that

cog =(l/n) (o-py )=X/n.

«y * =[l/(10' 2

I c )][MA

gp p+M

b g£b

+M

B g^B +(6.06xl0' 4

)V

2

p

p -(6.5x10' 5

)V

2

CLa

[(£-L

o

/ 2) 0

y

+lO‘4

d

y

+(L-L

o/2h

y

] (P-19)

[co 3

in ra d/s; M

A

g, M

b g, M

g

g in N; V in kV; C in PF; p , * B

,

Ln,

y

l, L in cm; d in ym; e , y

/so in rad; I r in kg *cm ] ; h u y y y

c

and

P

Q

=2-tt/ to (P-20)

[P

e in s; o)fi in rad/s].

y y

A

P depends upon 6 , d and y for a given V, C and L . Using

Py y y y

values from Table 8 in Equations P-19 and P-20, P Q increases from a ey

value of 44.9 s at V=0 kV to a value of 46.8 s at V=9 kV.

109
i) The amplitude modulations of 3 .

The amplitude of the cos2w t modulation term in Equation P-13 is

Y

y

^[10-2

M

ALALcm“Y'( 1 - 97xl0

‘ 4

) v2L

A(

L -Lo/ 2 ) 2] /

y

[10' 2

I r (4 -4 m

2

)] (P-21)

By

Y

y

2

[M

a

in kg; I c in kg*cm ; l_ cm , L A

, L, L Q

in cm; V in kV; o> B

,

in rad/s; y

y

in rad].

Using values from Table 8 in Equation P-21, and assuming that

Y

y

=Y

y

=3.30xl0~^rad (6.8") at V=9 kV; then the amplitude is

2.6x10 ^ rad (0.00054"). So, the 3

y

cosine waveform of 46.8 s period

would have a ±0.00054" modulation imposed upon it, with a period

P =2.82 s for this example. The balance would not be sensitive

T

y

enough to detect this modulation.

The amplitude of the cosw t term in Equation P-5 is

V

V[10

' 2i

C

( “

6 -“y

y

)] fl0 "Vcm

Pp Y

10

' 2 WcmyY

y

+(7.88x10 )V

2

L A (L-L

o /2)[(t-L

o/2)e

y

+(L-L

o/2h

y

+10-4

d

y

]> ( p

- 22 )

[M

a

in kg; I c in kg*cm2

; l_ cm , Pp , L A

, L, L Q

, i in cm; V in kV;

^ A

» W

Y

in rad/ s ; d

v in Y v

s Y u > 6

v, in rad ^- y ' y

” y y y

110
-8

Using the above example, the amplitude is -7.2x10 rad (-0.015"),

which is a factor of 7 smaller than the O.H"/yg mass sensitivity of

the balance, and is therefore undetectable.

Ill
(

APPENDIX Q

SOLUTIONS OF THE EQUATIONS OF MOTION FOR y and fi OSCILLATIONS ONLY.

x y

a) The suspended electrode motion.

We found in Equation 0-3 of Appendix 0 that

yx+(b/a)y

x -(c7a)=-(e/a)y

x

3

y

(Q-l)

where the coefficients are defined in Equations N-2, N-3 and 0-2.

The homogenous equation y +(b/a)y -(c'/a)=0 has a periodic solution,

which depends upon the starting conditions. If y is at its minimum

X value at time t=0, then

A Y x

= -Y x C0Sw

y

t + Y x (Q-2)

X

A where y is the amplitude of the oscillation and y is the equilibrium

X X angl e.

We found in part e of Appendix P that the approximate solution of

the 3 y

equation is also periodic - and, if is at its minimum value

at t=0, then

By-Byeoa^t * B

y

. (Q-3)

112
Equations Q-2 and Q-3 are approximate solutions for the Y and B x y

motions. We next treat the r.h.s. of Equation Q-l as a perturbation

of the l.h.s. of the equation [11], with By having the form of

Equation P-3. The result is

2 Yx +[(b/a)+(e/a)a)

e coso)

g t]y

x -(c'/a)=0. (q-4)

This has the form of Mathieu's equation [12,13], but we found in part

~ 2

a of Appendix P that b>>e3yOJ^ , so

Yx+(b/a) Y

x -(c'/a)=0 (q_ 5 )

The solution of Equation Q-5 is

V-V0S V + v (Q- 6 ) X

Therefore the suspended electrode system executes simple harmonic

/\ motion about the equilibrium angle y • X

/s b) The equilibrium angle y . A

It follows from Equations Q-5 and Q-6 that

Yx=cVb=(6.5xlO" 9

)V

2

C(L-L

o/2)[lO4

(t-L

o/2)0x +d

x ]/

[M

A

9Lcm

+( 6 - 06xl 0

' 4

)v

2

L'

( 6 - 5 xl 0

"

5 )v

2c(L ‘ l-o /2)2] (Q~ 7)

113
[V in kV; C in pF ; M^g in N; L, L Q

, l, L cm

in cm; d

x in pm, y

x

,

6x in rad].

A

So, y

x depends upon 6x and d

x for a given V, C and L Q

.

c) The period P . Y

x

It follows from Equations Q-5 and Q-6 that

J =b/a=aj^ =[M.gL +(6.06xl0'

4

)V

2 L-(6.5xl0" 5

)V

2

C(L-L /2)

2 ]/[10‘ 2

I p

]

Yx

Yy

[ oo in rad/s; M„g in N; L , L, L in cm; V in k V; C in pF; I p x

2

in kg*cm ] , and

P =P =2tt/ to (Q-9)

'x 'y y x

[P , P ins; oj in rad/s].

rxYy

Yx

So, P^ =P , and for the case given by Table 8, P Y

x y

y

Yx value of 1.48 s at V=0 kV to a value of 2.82 s at V=

increases from a

9 kV.

d) The balance beam motion.

We found in Equation 0-6 of Appendix 0 that

114
3y+(v/n)g (Q-10)

y

-(q7n)=-(e/n)( Yx

) 2

-(e/n)Y

xY

x -(r/n)y^-(s7n)Y

x

where the coefficients are defined in Equations N-2, N-3, N-5, 0-2,

0-3 and 0-5. The r.h.s. of Equation Q-10 can be treated as a

perturbation of the l.h.s. of the equation. Using the solution for

Yx given by Equation Q-6, taking the appropriate time derivatives of

Yy » and noting that cos a-sin a=cos2a and that cos a = ( 1/2) ( l+cos2a)

A

yields the result

3

y

+(v/n)e

y

-[(l/n)(q'-rY

x-s'Yx -(r/2)Yx

]

=[ (

x /n) (ew

2 -r/2)]cos2aj t A 'X +[(Yx/n)(2ry -ey

x u +s')]cosoo t. (Q-ll)

x Y x

Y x

The perturbation solution of Equation Q-ll is then exact - and if

is at its minimum value at time t=0, and if to' fa or to' /2to , y By

Yx ®y V

the solution is [13]:

By

=-ByC°!M^t + e;

+[(

x /n)(e w

2

-r/2)]cos2to t/(to'

2

-4o

2

) Y x

Y x

p y

Y x

+[(

x /n)(2ry -eYx w

2

+s')]costo t/(to'

2

-to 2

). (Q-12)

Y x

Yx

py

Yx

115
Therefore, the balance beam executes simple harmonic motion about the

A equilibrium angle $' with cos2w t and cosw t amplitude modulations

Y Yx Yx where u> = oo . Yx Y

y

e) The equilibrium angle 3y.

It follows from Equations Q-ll and Q- 12 that

3y=[q' -ry^-s'

x -(r/2)Y^]/v=[q' -ry^-s' yx

~( r/2)y^]/

[K* 0FF) +(6.06xlO'4

)V

2

p

p

]

By- {M

B

gL

B

-M

A

gL

A

- ( 6 . 06xl0'

4

)

V

2

L

ft -(3.94xlO"8

)V

2

L A[lO'

4

d

2

+lO

4

(«-L

o/2)

2

6

2

+2(Jl-L

o/2)0

x

d

x

]

-(3.94x10'

4

)V

2

(L-L

o/2)

2

L a

[^

2

+y

2

/2]

-(7.88xl0

_4

)V

2

(L-L

o /2)L

A

C(<l-Lo /2)e

x +10

_4

d

x ]Yx }/

{MA

9p

P+M

b 9VM

B

9 V(6 - 06xl0

~ 4)v2p

P

} (Q ' 13)

[M

a

9> M

b g, M

g

g in N; V in kV; C in pF; L A

, Lg, L, )l, L Q

, p

p

, * 5,

/\ *

B in cm; d

x in ym; By , 0 X

, r x

, Y

x in rad].

A /V So, 3' depends upon 0 , d , y and y for a given V, C and L . y x x x x ^ Equations P-14 and Q-13 imply that when y , y

and 6 motions all x y y

A occur that there would be a single value of By which depended upon

A /\ 9

x’ V d

x’

dy’ Y

x> Y y’ Y x and Y

y

for a 9iven V ’ C and L o‘

116
The variation in 3' due to y is

y x

«6y =-[(1.97xlO'

4

)V

2

L

A

(L-L

0 /2)

2

y^]/

[K (0FF *+(6.06xl0"4

)V

2

p

p

] ( Q- 14)

/s [V in kV; l_ A

, L, L Q, p

p in cm; K in N*cm/rad; 8y, , Y

x in rad]

where X^

0FF ) is defined by Equations K-l, 2-9, 2-14, 2-16, 2-17 and

2-19. Using values from Table 8 in Equation Q- 14 gives, for V=9 kV:

66;

(ON) =-(1.51xlO' 2

)y

2

(Q-15)

y *

[b;.yx in "].

This is slightly different from Equation P-16 for y , motion

because the balance beam stiffness is different in the two cases.

f) The force determination.

The voltage is adjusted in the force measurements until

;,(0NLy y

(OFF)

(Q— 16)

117
Using Equations Q-13, 2-9, 2-14, 2-16, 2-17, 2-19 and K-l, and

A

realizing that F

z =0 and that for V=0, Equation Q- 16 becomes,

_4 9

when neglecting the (6.06x10 )V p

p term:

[M

B 9L

B

-M

A

gL A

-F

z

L A-(1.97xl0'

4

)V

2

L A (L-L

o/2)

2Y^/K(0FF)

I [M

B

g L B

-(M

A

+m )9 L A]/^

0FF ^

F

z =mg[l-(l/mg)(1.97xl0'4

)V

2

(L-L

o/2)

2

Y

2

] (Q-17)

A

[F, mg in N; V in kV; L, L in cm; y in rad].

U A

Equations P-18 and Q-17 imply that when y , y

and ft motions all

- ~0 occur that the static error term for F depends only upon F and not

"0 -2

upon F x

, but that the dynamical error term depends upon both Yx and

~2

v

g) The period Po . y

It follows from Equations Q-ll and Q- 12 that

“'e =v/n=[K(0FF) +(6.06xl0‘4

)V

2

Pp ]/n

y

K

<*'1 =[l/(10“ 2I c )]{M

A

g Pp+M

b

gii b +M

B

gil B+(6.06xl0'4

)V

2

p p

} (Q-18)

118
[cog in rad/s; M^g , M

R

g , M

b

g in N ; V in k V; £

b

, £

R

, p

p in cm,

y o Iq in kg*cm ],

and

P

I (Q-19)

[Pg in s oo

;

in rad/s].

For the case given in Table 8, p^^)=p^*

r

*r )=p^

fr ^)= 44 .g s , and

^

0N W0FF > =oo0

^ OFF ^ =0.140 rad/s. A comparison of Equations P-19

By

By

By

and Q- 18 implies that when dnu tSy mu l i uns y^

Equation P-19 would represent the correct expression for co0 because

3 v Equation Q- 18 is independent of e , d and XX y . X

Note that the solutions to the equations of motion do not predict

large energy transfers between the e motion and the y and y

y x y

motions for the sample calculations. Therefore, yx

, y and 3^ are

not time dependent (except for damping effects). This is because

oo and oo are very much greater than oo . If the angular

Yx Y

y

3

y

frequencies were nearly equal, then there would be large amplitude

modulations on the y^ and waveforms as predicted by Equations P-5,

P-13 and Q-12.

119
h) The amplitude modulations of 3 .

The amplitude of the cos2u t modulation term in Equation Q-12,

Y

remembering that co =oo , is

Yx y

y

^[ 10‘ 2

M

A

LcmL A ^-(1.97xl0"4

)V

2

L A

( L -L

o/2)

2

]/

[10“ 2

I C

( u> 2

-4 oi 2

)]

7 7

(Q-20)

[Ma

in kg; I c in kg-cm ; Lcm ,

l_ A

, L, L Q

in cm; V in kV; oj , w

y y

in rad/s; y in rad].

A

The example given in part i of Appendix P shows that this amplitude

modulation of the 3^ cosine waveform would be undetectable.

The amplitude of the cosw t modulation term in Equation Q-12,

Y

x again remembering that oo =co , is

Y

x Y

y

\/C 10

' 2l c(%-“y

y

)] { -10' 2MALA

L cmVT

y

+ (7.88xl0- 4

)V

2

L A (L-L

o/2)[(4-L

o /2)6x+(L-Lo/2)Y

x +10‘ 4

d

x ]} (Q-21)

[M

a

in kg; I c in kg.cm ; L cm , L A

, L, L Q, i in cm; V in kV; ^ , ~ y w in rad/s; d in urn; y , y , 0 in rad].

Y x x’ x

The example given in part i of Appendix P shows that this amplitude

modulation of the $ cosine waveform would also be undetectable for

A reasonable values of 6 x> y

x and d and for a smal1 enough v

120
APPENDIX R

DYNAMICAL FORCES ON THE SUSPENDED ELECTRODE AS DETECTED BY THE

BALANCE

The algebra involved in deriving the equation of motion for the

balance beam, using the Principle of Virtual Work and D'Alembert's

Principle, was straightforward, but somewhat lengthy. It would be

useful to check parts of the result by using Newton's second law of

motion.

/\ The equilibrium angle g , given by Equation P-14, agrees with the

generalized torque Equation 1-6 if the balance beam is in static

equilibrium (Q0 =0), if 3 is a very small angle, and if the

sy

y

suspended electrode is in static equilibrium (Yy=0). Equation P-14

also agrees with the Newtonian prediction of the torque induced about

point (C) of Figures 2-4 when Yy= 0*

The Newtonian prediction for the period of the balance beam is

P =2tt

/

Ir/xi*. Using Equations K-l and P-19, this equation becomes

P =2tt/o) , which agrees with Equation P-20.

^y e y

The amplitude modulations of

3^ can be checked by investigating

the time variations of the external forces induced on the suspended

/\ electrode. Assume that 3^*0, and that either the plane defined by

the balance beam knife edges is servoed to gravitational horizon, or

that the balance beam freely swings about 3 =o with a very small

amplitude (3^-0) when Yy=0. The suspended electrode is + herefore not

121
significantly accelerated by the balance beam. Figure 12 shows the

remaining external forces that are induced on the moving suspended

electrode.

It would be difficult to predict how the tension in the electrode

support system gets translated to the knife edge point (P) via the

stresses induced in the support rods. So, we will only look at the

vertical components of the external forces shown in Figure 12, and

will then assume that the balance experiences these vertical forces

at the knife edge.

The vertical force due to the radial acceleration is

( R- 1

)

Remembering that

(R-2)

and using the small angle approximation cosy^l, Equation R-l becomes

(R-3)

[f

R in N; in kg; l_ cm

in cm; y in rad; a) in rad/s; t in s].

Y

y

The vertical force due to the tangential acceleration is

122
- 2. f

T

( I ) =M

A

a Tsi n Yy=10 M

A

Lcm

Y

y

si n Y

y

(R-4)

Using Equation R-2 and the small angle approximation siny^^,

Equation R-4 becomes

fT(t) = 10

2

T

M.L

A uP Y.COS00 t(-y cosu) t + y) cm Yy y Yy y Y

y y

(R-5)

[fy in N; M. in kg; Lcmin cm; co in rad/s; y , y in rad; t in

Yy y y

si:

p Adding Equations R-3 and R-5, and using the identity 2cos cp(l+cos2a)

yields

f

R (t)+f

T

(t) = l(f

2

M

a

L oo 2 [-y^cos2w t + y y cosoo t] (R-6)

Yy Y Yy J J Yy

where the units are defined in Equations R-3 and R-5.

The quantities f

R

, fy and f

R+fy are plotted versus time in Figures

13 and 14 for V=0 kV, (in which case y =0). The time span covers one

pendulum period - which is P =1.5 s for a brass suspended electrode

Yy

with voltage-off.

The torque about the balance center knife is then

- L A(fR+f

T

)= +1 0' 2M

A

L cmL

A“Y

y

Y

COs2

“Y

y

t '

123
This torque should be - and is - proportional to e , which is defined

by Equations P-13, P-21 and P-22. Only the cos2u) t term is present;

T

y

this is also always the case for a normal balance pan swinging in the

absence of horizontal forces. If 7^=3.30x10 8 rad (6.8") and if the

mass sensitivity is 0.11"/yg, then would appear to vary by

+5.3 yg.

The quantities f

R

, fy and f

R+fy are then plotted versus time, for

V=9 kV, in Figures 15-16. P has increased to 2.8 s for the brass

Y

y

suspended electrode. The shape of f

R+fy depends upon the relative

A

magnitude of y and y , and is described by both the cos2oo t and

y y Yy

cosoo t terms of Equation R-6. If d =0 and 0 =5" , then

^ Yy y y

y =3.30xl0~ b rad (6.8"), and if y =y then f

p +fy=-2.76xlO'8

N at

v-7, t= ( 1/2 ) P , giving a torque of -2.07x10 N*cm. If the mass

T

y

sensitivity is O.ll'Vyg, then would appear to be 3.2 yg lighter at

this instant.

The electrical force, F z

, is defined by Equations 2-9, 2-10, 2-16

and 2-19:

F

z (t)=(6.06xl0

_4

)V

2 [l+(6.5xl0' 9

)h

2

].

If h =0 and & is very small, this equation becomes

* y

F

z (t)=(6.06xl0'4

)V

2 +(3.94xl0" 12

)V

2 [10

8

(L-L

o /2)

2 y2

+108

(<t-L o/2)

2e^dY(2xl08

)(L-L

o/2)( )1 -L

o/2)e t

J J +(2xl0

4

)(L-l

o /2)dyYy+(2xl0

4

)(>-L

o/2)d

y

ey

].

124
Substituting Equation R-2 for y t using the identity

o cos of(1/2) (l+cos2a) , and gathering the constant and time-dependent

terms into groups gives

F

2 (t) = {(6.06xl0~4

)V

2+(3.94xl(f 12

)V

2

[10

8 U-Lo /2)2 6y+dy

+ (2xl0

4

)U-Lo/2)d

y

8

y

+10

8

(L-L

o/2)

2

(^

2

+^2)

+(2xl0

8

) (L-L

Q/2) («-L0

/2)

6

y

y

y

+(2xl0

4

) (L-L

Q/2)d

y

Y

y

]

}

+{(3.94x10' 12)V

2[10

8

(L-L

o/2)

2 (y^/2)cos2w

y

t

-(2xl0

8

)(L-L

o/2)

2

Y

y

yosu>

Y

t *

-(2xl0

4

)(L-L

0/2)d

y

Y

y

cosio

y

t

-(2xlO

8

)(L-L

o/2)(4-Lo/2)0

y

Y

y

coS(l)y

t] } (R-7)

[F in N; L, i, L in cm; d in ym; V in kV; w in rad/s; 6 ^ y Y

y

y

Yy» Yy in rad; t in s]

or

F

z (t)=f

c +fz (t) (R ' 8)

where f is the constant term in Equation R-7 and f is the

time-dependent term.

A The equilibrium angle, $ , for the balance beam should depend upon

A the net, time-independent, torque, P , about the center knife, and

6

y

upon the balance beam stiffness, X:

125
%

=r

g/

K

' CM

B

9L

BMA3LA

-f

c

L A

]/k.

This equation agrees exactly with Equation P-14, and thus verifies

~2

the presence of the (l/2)y

y

quantity in that equation.

The quantity f is plotted versus time in Figure 17 using

Equations R-7 and R-8 for the brass suspended electrode at V=9 kV

with d =d = 0 =0 and 6 =5". The amplitudes of oscillation would be

x y x y zero if either V=0 or if y =0.

The total time-dependent torque about the balance center knife,

-L

A

(f

R +fT+fz

) , is proportional to Substitution of Equations R-6 -

R-8 into this equation gives agreement with the numerators of the

modulation terms of Equations P-5, which are defined by Equations P-21

and P-22. So, the amplitude modulations of arise from the

time-dependent vertical forces f

R

, fy and f .

It is useful to express the time-dependent torque in terms of the

mass variation of the mass M^. Assuming that the mass sensitivity is

calibrated using sensitivity weights with voltage-off, then the mass

variation, Am, is

Am=-L

A

(f

R+f

T

+f

z )(l/K

(0N) )("/4.85xl0“6 rad) (yg/0.11") (R-9)

[Am in yg, l_ A

in cm; f

R

, fy, f in N; X in N* cm/ rad],

which for the brass electrode at V=9 kV is

126
Am=(l. 14x10 )(f ( R-10)

R +f

T

+f

z

)

[Am in yg; f

R

, fy, f

z in N]

Figure 18 shows Am versus time for a brass electrode at V=9 kV

and Qy

= 5". This figure was obtained by adding the curves in Figures

16 and 17 and then using Equation R-10. It predicts that, for this

example. Am varies from -30.7 yg to +41.3 yg every 2.8 s; but the

prediction for this same example in part i of Appendix P is of a Am

variation between -0.13 yg and +0.14 yg. However, if, somehow, the

period of the pendulum, P , could be made very long compared with

(ON) the P' =46.8 s balance beam period (so that o)g >>go ), then

y

Y

y

Equations P-21 and P-22 also predict the same Am variation as that of

Figure 18 because U^N ^ = 10 .

The balance acts as a low-pass frequency filter; the 2.8 s period

of the pendulum vertical force variations is too short for the 46.8 s

balance beam period to respond to, and thus these variations

are attenuated. So, the amplitude modulations of the

3y cosine waveform

/N are very small. The equilibrium angle 8 , however, is directly

2

affected by the amplitude squared (y ) of the pendulum motion.

127
APPENDIX S

DYNAMICAL CAPACITANCE VARIATIONS

The capacitance is defined by Equations 2-9, 2-10, 2-12 and 2-13

as

C(t)=C(z

L )[l+(6.5xlO"

9

)h

2

].

If h =0 and 3 is very small, this equation becomes

x y

C(t)=C(z

L )+C(2L )(6.5xl0'

9

)[10

8

(L-L

o /2)

2

Y^+10

8

(£-L

o/2)

2

e^

+d

2

+(2xl0

8

)(L-L

o/2)U-Lo/2)eyYy

+ (2xlO

4

)(L-L

o/2)d

y

y

y

+(2xlO

4

)U-L

o/2)0

y

d

y

]. (S-l)

^ /N But Y =-Y wcosw t + y , and y*0 for the = 100 V value at which

y y Y

y

y

'y rms the capacitance measurements are made. Therefore, using the identity

2 cos or(l/2)(l+cos2a) , and gathering the constant and time-dependent

terms into groups gives

C(tHC(z

L

) + (0.325)C(z

L )(L-L

o /2)

2

Y

2

+(0.65)C(z

L )U-Lo/2)%

2 + (6.5xl0~9 )C(z

L )d

2

+(1.3xlO' 4

)C(z

L )(z-L

0/2)6

y

d

y

}

+{(0.325)C(z )(L-L

0/2)

2

y

2

cos2u t

-(1.3)C(z

L )(L-L

o/2)(z-Lo/2)0

y

Y

y

cosai

T

t

128
,-4.

(1.3x10 )C(z

L )(L-L0/2)d

y

Y

y

cosa)^ t> (S-2)

[C in pF; z

L

, L, i, L Q

in cm; d

y

in urn;

rad; t in s].

a) in rad/s;

Y

y

in

So, the fractional change in C(t) compared with C(z

L

) is

{C(t)/C(z

L )-l}={[(0.65)U-L

o/2)

2 ey+(6.5xl0‘ 9 )dy

+(1 .3xl0'

4

) (Z-L

q

/ 2)

6

y

d

y

]+(0.325) (L-L

0

/2)

2

y

2

}

+ {(0.325) (L-L

0

/2)

2

Y

2

cos2 Wy t-( 1 . 3) ( L-L

Q/2) ( *-LQ/2 ) 6

y

Y

y

cos Wy t

-(1.3x 10' 4

)(L-L /2)d y cosu t) (S-3)

y y Y

y

or

6C(t)/C(z

L )-e

c + et

(t) (S-4)

where is the constant term in Equation S-3 and is the

time-dependent term. It is assumed that the amplitude of the Yy

motion decreases due to damping at a rate which is long compared with

the pendulum period. Therefore, the (0.325) (L-L

Q /2) y term is

included in e . c 6C(t)/C(z^) is plotted versus time in Figure 19 for C(z

L )=100 pF

and d =d =e =0. An analogous plot occurs for pendulum motion in the

x y x x-z plane. A small time constant and a fast enough recording device

must be used for the capacitance bridge detector in order to

129
respond correctly to the P =1.5 s period of a brass electrode. If

Y y

the time constant is too long, then the detector will begin averaging

the signal - thus resulting in large systematic errors.

The signal shape depends upon the relative electrode equilibrium

alignments and upon the pendulum amplitude. Perfect alignment yields

a pure cos2co t signal. The cosoo component becomes more enhanced

Y

y

y

y

with increasing electrode misalignments and relative displacements.

The static equilibrium values of 6C/C are +0.34 ppm and 0 ppm for

0^=5" and 0y=O, respectively. Therefore, the minimum reading of 6C/C

-A. /\ is the correct one only when h =h =0, but it provides the best

x y

estimate of any of the simple options. The pendulum amplitude must

be quite small when making the capacitance measurements - and, the

larger the electrode misalignments and relative displacements, the

smaller this amplitude must be.
APPENDIX T

ERROR CALCULATIONS FOR THE SLOPE METHOD

a) Method used to estimate the voltage ratio uncertainty.

The voltage ratio, V /VmQ3p , which can be obtained from

Cd I C iMca 5 Equation 4-4, is

Vcalc/Vas

=

( 1 / V

meas)C-( 2xl°

4

)V(^/^L)h

'x> ^

1/2

1n N; Vcalc* Vas in kV; C in Pp; h in

^

2

L

(T— 1

)

in cm].

The quantities in Equation T-l that have errors associated with

/\ /\ them are Vmoao , F , C and z. . There is an error involved in

meas z l

measuring the capacitance at low voltage rather than at the high

A

voltages, which must be included in C (along with the dynamical

measurement errors and any errors resulting from temperature

fluctuations)

There is a random error in determining m, which (according to

A Equation P-18) should be included in the uncertainty of Fz

- along

with the systematic static and dynamic errors. If there are large

A

curvatures and changes in the functional form of C versus z^ then the

uncertainty in the slope will increase - in addition to the error

involved in not determining the slope at constant h.

131
The static systematic error in F

z is correlated with the low

A voltage error in measuring C, and also with the error involved in not

determining the slope at constant h. The uncertainties of Equation

T-l are therefore correlated mixtures of random and systematic

errors.

Rather than attempting to assign correlation coefficients and

deciding how to combine random and systematic errors for nonexistent

data, only measured quantities and their assumed measurement

uncertainties will be used. The voltage ratio will be calculated for

a variety of carriage scans, using the low-voltage capacitance values

and neglecting the fact that the slope should be determined at

constant h. The correlation effects and the effects due to the

unmeasurable quantities should thereby be reflected in this

uncertainty estimate.

b) Calculations for Type I scans with a brass electrode at V= 9 kV.

Example 1. Perfectly aligned right circular cylinders with no measurement

errors.

A A A

Let mass m be such that F

z =mg at V= 9 kV when h=0. Since h=0,

Equation 4-2 becomes

F

z =mg=(6.06xl0'

4

)(9 kV)

2 =4.9086xl(f

2 N.

132
Since the path is linear, and F^ is assumed to be equal to mg when

"'(OFFl

y y

in t *16 ^orce determination measurement. Equation 4-1

becomes

V

2

, =-(2xl0

4

)mg(Az. /AC)=-(2xl0

4

) (4.9086xl0' 2 N) (2.5 cm)/

Ca I C L (69.7 pF-100 pF)

or V calc 9 k V= V,meas* Therefore, there is no error in this example.

Example 2. A -2 ppm error in F^ and no other errors.

The static and dynamic errors are such that F appears to be 2

ppm smaller than mg when F^ actually equals mg. F

z would thus be

increased by 2 ppm until , so that

V

2

a =(4.9086xl0' 2 N)[l+(2xl0" 6 )]/(6.06xl0"4

)

or

Vmoa =9.000009 kV. meas

V ca ic

is the same as that for Example 1; therefore, V ca

-| c is 1 ppm

smaller than V . meas

133
Example 3. A -2 ppm error in F' , the i nterferometer errors, but no

other errors.

The smallest value of az

l

that might be measured by the laser

inteferometer system is [2.5 cm -(5xl0~ 8 )cm] or (2.5 cm) [1 -(2xlCf 8

)],

so

V calc

= " (2xl °

4 ^ 4,9086x10

"

2 N ) (2 ' 5 cm)[l-(2xl0"6

)]/

[69.7 pF-100 pF]

or

vcalc

=8 ' 999991 kv ‘

V meas

=9 * 800889 kV; therefore V ca -j c is 2 ppm smaller than Vmeas

.

A Example 4. A -2 ppm error in F

z

; all other errors except the

temperature effects.

V

2

. =-(2x10

4

)(4.9086x10"2

N) (2.5 cm)[l-(2xl0"6

) ]/ ca i c{[69.7 pF-100 pF] ( 1+10

-6

)}

or

V calc

=8. 9999862 kV.

134
Vmoaf.=9. 000009 kV; therefore V„ al „ is 2.6 ppm smaller than

iTIcdS Ca I C lilcab

A

Example 5. A -2 ppm error in F

z and all errors included.

V

2

. =-(2x10

4

)(4.9086x10' 2

N) (2.5 cm)[l-(2xl0" 6

)]/ ca i c {[(69.7 pF) ( l-10" 6

)-( 100 pF)(l+10

_6

)](l+10”

6

)

}

or

V calc=8. 9999609 kV.

Vmea s = 9«000009 kV; therefore Vca

-| c is 5.4 ppm smaller than Vmeas w^ en ^calc actually equals Vm _._. It is not likely, however, that

the errors would all combine in this manner. The combination given

in Example 4 is a more likely choice for the maximum error in

determining V calc .

A

Example 6. A +2 ppm error in F

z and no other errors.

A The static and dynamic errors are such that F

z appears to be 2 ppm

larger than mg when F

z actually equals mg. F

z would thus be

decreased by 2 ppm until 3y^s 3y^^» so that

V

2 =(4.9086xl0” 2 N)[1-(2x10'6 )]/(6.06x10"4

) me as

or

135
W8 - 999991 kv -

Vcal

c

=9 ' 0000000 kV; therefore V calc is 1 ppm larger than Vmeas

.

A Example 7. A +2 ppm error in F ; the i nterferometer errors, but no

other errors.

The largest value of az

l

that might be measured by the laser

inteferometer system is [2.5 cm+(5xl0

_6 )cm] or (2.5 cm) [l+(2xl

0

-6

)cm]

,

so

V^ 1 =-(2xl0

4

)(4. 9086X10- 2

N) (2.5 cm)[l+(2xl0"6

) ]/ Ca I C

[69.7 pF-100 pF]

or

V calc

=9 ‘ 000009 kV *

V meas

= ^* 99 "91 kV; therefore V ca

-| c is 2 ppm larger than Vmeas

Example 8. A +2 ppm error in F ^

all other errors except the

temperature effects.

Vcalc

= - (2xl0

" )(4 - 9086x10

" 2 N)(2 ‘ 5 cm)[l+(2xlO" D )]/

{[69.7 pF-100 pF] ( l+lO

-6

)

}

»- 6 ,
or

Vcalc

=9 * 0000043 kV *

Vm =8.999991 kV; therefore is 1.5 ppm larger than VmQac . mGaS Ca I C INGaS

Example 9.

A

A +2 ppm error in and all errors included.

Vcalc

= “( 2xl °

4

) (4. 9086x1 °“ 2 N ) (2.5 cm)[l+(2xl0" 6)]/

{[(69.7 pF) ( 1+10~ 6

)-(100 pF) ( l-10

-6

)]( 1+10" 6)}

or

Vcalc

=9 * 0000339 kV *

Vmna =8. 9999908 kV; therefore V is 4.7 ppm larger than VmQ a{.. InGaS Ca I C iTIGaS

However, it is again unlikely that the errors would all combine

in this manner.

A reasonable choice of the maximum uncertainty in Vca -| c for a Type I

scan might be best given by Examples 4 and 8: V ca

-| c can be

determined to within +1.5 ppm and -2.5 ppm for a scan known to be

along the combined electrodes symmetry axis.

137
c) Calculations for Type I scans with an aluminum electrode at

V=4.02 kV.

A

Example 10 . A -2 ppm error in and all errors except the

temperature effects.

Let mass m be such that F z =mg at V=4.02 kV when h=0. Since h=0.

Equation 4-2 becomes

F

z =mg=(6.06xl0'4

)(4.02 kV)

2 =9.7932024xl0' 3

N.

/\F

z appears to be 2 ppm smaller than this value for mg; therefore,

the voltage is adjusted until

V

2meas

= (9 .7932024x10 N)[1+(2x10

_6

)]/(6.06x10” 4

)

or

Vmo;, =4.020004 kV.

Ultra b

V calc

= "^ 2xl °4 ^ 9 * 7932024x10

" 3 N ^ 2 - 5 cm)[l-(2xl0"6

)]/

{[69.7 pF-100 pF](l+10" 6

)}

or

138
V calc=4. 0199937 kV.

Therefore, V ca -j c is 2.6 ppm smaller than Vmeaswhen V ca -j c actually

equals Vmooc. when there are no errors.

Mica 5

A

Example 11 . A +2 ppm error in and all errors except the

temperature effects.

A

F

z appears to be 2 ppm larger than mg; therefore, the voltage is

adjusted until

V meas

=(9 * 7932024x10

"

3 N)[l-(2xl0' 6 )]/(6.06xl0~4

)

or

Vmeas=4. 019996 kV.

V^aic= -( 2xl ° 4

)( 9 * 7932024xl0

" 3N )( 2 - 5 cm)[l+(2xl0"6

)]/

{[69.7 pF-100 pF]( 1+10

-6

)

}

or

V calc =4. 020002 kV.

Therefore, V is 1.5 ppm larger than . Ca I C Mica

S

139
Examples 10 and 11 agree with Examples 4 and 8, as expected, because

it should not matter which electrode is used, at what voltage, as long

as the carriage is translated along the combined electrode symmetry

axis. The voltage, of course, must be such that the suspended

electrode system is in stable mechanical equilibrium.

A

Example 12 . A +2 ppm error in F

z

; the interferometer errors, and a

dynamical capacitance error at C=69.7 pF.

V calc

= " (2xlo4)(9 * 7932024x10

" 3 N H 2 * 5 cm)[l+(2xl0"6

)]/

[(69.7 pF) ( 1+10

-6

) - 100 pF]

or

V calc

=4. 0200078 kV.

Vmeas-4. 0199958 kV; therefore Vca -| c is 2.9 ppm larger than Vmeas -

as compared with 1.5 ppm larger in Example 11. However, the

capacitance would be measured at many points in order to find the

best least-squares fit of the straight line in this example.

Therefore, it is more appropriate to include the dynamical

capacitance error in all the measurements for the slope method.

140
d) Calculations for Type II scans with a brass electrode at V= 9 kV and

V5

''-

Example 13 . F

z determined at C(z^)=l00 pF with no measurement errors.

Table 7 lists the actual value of F^ as being 4.7 ppm larger than

the value for mg. Thus the voltage must be adjusted until

V

2meas=(4.9086x10

-2

N)[1-(4.7x10"6 )]/(6.06x10‘ 4

)

or

Vmo =8. 9999787 kV. meas

The capacitance, from Equation 2-12, is

C(h,z

L )=C(z

L )[l+(6.5xlO" 9

)h

2

]

/s where, from Equation 2-10, h is

yio4u-L

0

(2

L )/2]Sy

because y

~0 at V =100V. L (z, )=10 cm at C(z, )=100 pF, so h=7.275 pm

y rms o L L and AC/C=+0. 34 ppm at z

L =0 cm, whereas C(

z

L )=69 . 7 pF, L Q

(z

L )=7.5 cm

h=7.58 ym and AC/C=+0.37 ppm at z

L =2.5 cm. Therefore the

capacitance will be approximately +0.3 ppm larger than the C(z

L

)

value at every z

L point.

141
v calc

= " (2xl °4)(4 - 9086x10

" 2 N H 2 * 5 cm )/

{[69.7 pF-100 pF][l+(0.3xl0'6

)]}

or V_i =8.9999985 kV. calc

Vmoaf.=8. 9999787 kV ; therefore V_ al „ is 2.2 ppm larger than VmQ3c

with no experimental errors present. This discrepancy arises because

the force is measured at 9 kV (so that the suspended electrode has

swung out to an equilibrium angle of Yy=6.8", causing a 4.7 ppm

increase in the actual capacitance), but the capacitance is measured

at V rms =100 V (resulting in only a 0.3 ppm capacitance increase). No

error would occur in this example if the capacitance could be

measured at V= 9 kV.

A

Example 14 . F

z determined at C(z

L )=69.7 pF with no measurement

errors.

The value of L 0 (z^) is smaller at z^=2.5 cm than at z^=0 cm.

A Therefore, the value of h^, as calculated from Equation 2-10, is also

/v smaller - which then results in a smaller value for F . Table 9

lists the relevant quantities for this Type II scan with a brass

el ectrode.

142
V

2 =-(2x10

4

)(4.9086x 10‘ 2 N)(2.5 cm)/{[69.7 pF-100 pF] X

Cct I C

[l+(0.3xl0" 6

)]}

or

V calc =8. 9999985 kV.

Vmoac =8. 9999923 kV; therefore V_ al _ is 0.6 ppm larger than Vmoac at

IMG do Cd I C IMG do

A C=69.7 pF. The discrepancy is less than at C=100 pF, but F^ should

be determined at many points along the scan.

A

Example 17 . A -2 ppm error in F

z at C(z

L )=100 pF and all

except the temperature effects.

V

2

. =-(2x10

4

)(4.9086x10“ 2N) (2.5 cm)[l-(2xl0‘6

)]/ ca l c {[69.7 pF-100 pF][l+(1.3x!0" 6

)]}

or

V ,=8.9999848 kV ca I c

VmQ2 =8. 9999878 kV; therefore V_ al „ is 0.4 ppm smaller than VmQao . iMGdS Cd I C iMGdS

A

Example 18 . A +2 ppm error in F

z at C(z^)=100 pF and all errors

except the temperature effects.

143
V cal c= "( 2xl ° 4

)

(

4 * 9086xl0

~ 2

N) (2.5 cm)[l+(2xl0" 6

)]/

{[69.7 pF-100 pF][l+(1.3x!0'6

)]}

or

V calc =9,0000029 kV *

Vmeas =8,9999698 kV; therefore v

C alc is 3,6 ppm lar9 er than vmeas*

A

Example 19 . A -2 ppm error in F^ at C(z

L )=69.7 pF and all errors

except the temperature effects.

V calc

=8 * 9999848 kV and Vmeas=9 ‘ 00000 13 kV; therefore v ca lc

is 1,9

ppm Kr smaller than \T . meas

A Example 20 . A +2 ppm error in F at C(z

L )=69.7 pF and all errors

except the temperature effects.

V calc

=9. 0000029 kV and Vas"8 * 9999833 kV; thereto re V cak is 2.1

ppm larger than Vmeas

.

e) Calculations for Type II scans with an aluminum electrode at

V=4.02 kV and 0

y

=5".

144
A Example 21 . A -2 ppm error in F^ at C(z

L )=100 pF and all errors

except the temperature effects.

V calc

= " (2xl °4 ^ 9 * 7932024x10

" 3 N H 2 * 5 cm)[l-(2xl0"6

)]/

{[69.7 pF-100 pF][l+(1.3xlO" 6

)])

or

Vcalc

=4 - 0199931 kV '

Vmea s =^*{)2 0001 9 kV (see Table 10); therefore V ca

-| c is 2.2 ppm

smaller than Vmea$ .

Example 22 . A +2 ppm error in F^ at C(z

L )=100 pF and all errors

except the temperature effects.

V calc

= " (2xl °4)(9 * 7932024x10

” 3 N H 2 - 5 cm)[l+(2xl0" 6

)]/

{[69.7 pF-100 pF][l+1.3xlO" 6 ]}

or

vcaic

=4 - 0200013 kv *

V meas

=4 * 0199939 kV; therefore vca i c is 1 - 8 PPm !arger than Vmeas

.

145
Example 23 . A -2 ppm errors in F

z at C(z^)=69.7 pF and all errors

except the temperature effects.

Vcalc

=4 ' 0199931 kV and

e=4.0200025 kV; meas

therefore V calc is 2.4 smaller than Vmeas.

Example 24 . A +2 ppm error in F

z at C (

z

L

) =69 . 7 pF and all errors

except the temperature effects.

Vcalc =4,0200013 kV and

vmQ ,c=4. 0199944 kV; meas *

therefore V_ al _ is 1.7 ppm larger than VmQ3C . Ca I C MlGa S

f) Calculations for Type III scans with a brass electrode at V=9 kV.

A Example 25 . 6y varies linearly from 0" to 5", and F

z is determined

with a -2 ppm error at C(z^)=69.7 pF with all errors except the

temperature effects.

A C versus z

L plot would have a parabolic shape, but since the

maximum change in AC/C is only +0.3 ppm, it can safely be fitted by a

straight line for perfect right cylindrical electrodes.

146
V

cal

c

=_

( 2xl °

4

)

(

4 * 9086xl0_2 N) (2.5 cm)[l-(2xl0”6

)]/

{(69.7 pF)[l+(0.3xl0"6 )]-100 pF}[l+(lxlO" 6

)]

or

V calc

=8 ‘ 9999892 kV *

Vmeas

=9 *0000013 kV (see Table 11); therefore Vca

-| c is 1.4 ppm

smaller than V

meas

.

Example 26 . ey

varies linearly from 0" to 5", and F z is determined

with a +2 ppm error at C(z^)=69.7 pF with all errors except the

temperature effects.

V

2

. =-(2xl0

4

) (4.9086xl0~2 N)(2.5 cm)[l+(2xl0" 6

)]/ ca I c {(69.7 pF)[l+(0.3xl0"6 )]-100 pF}[l+( lxlO" 6

)

]

or

Vcalc

=9 * 0000073 kV

Vmeas=8,9999833 kV; therefore v

C alc is 2,6 ppm lar9 er than vmeas*

147
Example 27 . 6^ varies linearly from 5" to 0" between z L =0 cm and

/\ z

l

=2.5 cm, and F

z is determined with a -2 ppm error at C(z

L )=100 pF

with all errors except the temperature effects.

V

2

. =-(2x10

4

)(4.9086x10" 2

N) (2.5 cm)[l-(2xl0

-6

) ]/ ca i c {69.7 pF- ( 100 pF)[l+(0.3x!0” 6

) ] }[l+(lxl0" 6

)]

or

Vcalc

=8 * 9999817 kV *

Vmoac =8. 9999878 kV; therefore V„ al „ is 0.7 ppm smaller thaan VmQ3C . Illcdo Ca I C MlcaS

/N Example 28 . 0^ varies linearly from 5" to 0", and F

z is determined

with a +2 ppm error at C(z

L )=100 pF with all errors except the

temperature effects.

V calc

= “( 2xl °

4

)

(

4 - 9086x10

" 2

N) (2.5 cm)[l+(2xl0"6

) ]/

{69.7 pF- ( 100 pF)[l+(0.3xl0"*

6

)] }[l+(lxl0

-6

)

]

or

V calc =8,9999998 kV ‘

Vmea s =8 * 9999698 kV; therefore Vca -j c is 3.3 ppm larger than Vmeas

.

148
g) Calculations of Type III scans with an aluminum electrode at

V=4.02 kV.

Example 29 . 0^ varies linearly from 0" to 5", and F

z is determined

with a -2 ppm error at C(z

L )=69.7 pF with all errors except the

temperature effects.

V calc=_(2xl ° 4)(9 * 7932024x10

” 3 N M 2 - 5 cm)[l-(2xl0" 6

)]/

{(69.7 pF)[l+(0.3xl0~6 )]-100 pF}[l+(lxlO

-6

)]

or

vcalc

=4 - 0199951 kv '

Vmoa ,.=4. 0200039 kV (see Table 12); therefore V_ al „ is 2.2 ppm

iM6aS Ca I C

smaller than Vmmeas

/s Example 30 . 9^ varies linearly from 0" to 5" and F

z is determined

with a +2 ppm error at C(z^)=69.7 pF with all errors except the

temperature effects.

V calc

= " (2xl °

4 ^ 9 * 7932024x10

" 3 N ^ 2 * 5 cm)[l+(2xl0" 6

)]/

{69.7 pF)[l+(0.3xl0'6 )]-100 pF}[l+(lxlO" 6

)]

or

149
=4.0200032 kV.

V =4. 0199958 kV; therefore V_ al _ is 1.8 ppm larger than Vmoao . meas caic mcas

A Example 31 . 0^ varies linearly from 5" to 0", and F

z is determined

with a -2 ppm error at C(z

L )=100 pF with all errors except the

temperature effects.

V calc

= " (2xl ° 4)(9 * 7932024x10

" 3 N)(2 * 5 cm)[l-(2xl0"6

)]/

(69.7 pF-(100 pF)[l+(0.3xl0”6 )]}[l+( lxlO

-6

)]

or

Vca ,c=4.° 199917 k».

VmQa _=4. 0200019 kV; therefore V_ al „ is 2.6 ppm smaller than Vm . a _.

iilcab Ca I C m6d

S

A Example 32 . varies linearly from 5" to 0", and F

z is determined

with a +2 ppm error at C(z^)=100 pF with all errors except the

temperature effects.

V calc

= ' (2xl °4 ^ 9 * 7932024x10

" 3 N H 2 - 5 cm)[l+(2xl0" 6

)]/

(69.7 pF-( 100 pF )[ 1+ ( 0.3xl0

-6

)]}[!+( lx 10~ 6

)]

or

150
=4.0199998 kV calc

Vmeas

=4 * 0199939 kV; therefore Vcalc is 1.4 ppm larger than Vmeas

151
APPENDIX U

DERIVATION OF THE PATH INTEGRAL EQUATION

The change in electrical potential energy, AU

e

, from an initial

state i to a state j is

AU

e=(l/2)CjVj - ( 1/2)C

1 V? = -J

j F-d«

£

i

[U in N'cm; C in F; V in V; F in N; l in m; q in C]

Vdq (U-l)

where F is the electrical force between the electrodes and q is the

charge on the electrodes: q

= C V.

Substituting the derivative of the equation q=CV (dq=CdV+VdC) into

Equation U-l gives

(1/2)C.V? - (1/2)C- V? = -f F-d£ + f

J CVdV + f

j V

2 dC (U-2)

J J 11 */ £

i J\I. */ C

i

The last two integrals on the r.h.s. of this equation would be

difficult to evaluate unless V were constant. Therefore, assume that

Visa constant value 7, and arbitrarily choose this value V to be

either the average value of V if V varies between states i and j or

the nominal voltage. Also, adjust each value of F to a new value F'

,

where F' is the value that F would have been if V were equal to V.

152
(This adjusted value for F will be defined in Appendix V.) Thus,

Equation E-2 reduces to

. 1 . (1/2) V

2

( Cj

- C

i

) = -J

3 F'-dt + V

2

(C

j

-C

i

) Z •l

or

f

J F'-d£ = ( 1/2 ) V

2

( C - -C - )

JZ,

3 (U-3)

Using the relative coordinate system of Figure 7, Equation E-3

becomes

h x h^ z L I

j2VV/ j *Wf ^ F z

L

dz

L

=(1/2 ^

2

( Cj-C

i) ' U

- 4 > x

i L

i

[F‘ in N; h, z

L in m; V in V; C in F].

So, the mechanical work, W

e=y^F' • dts to move the system from

point (h

x

, h , z. ) to point (h x , h , z, ) is equal to the change

i y i L

i j y

3 j

in electrical energy stored in the field. This work and energy is

expressed in N*m in Equation U-4. It is more convenient to express

these terms in N*cm, and to use multiples and submultiples of h, V

and C. In equilibrium, Equation U-4 then becomes

W

e

= (2xl0‘4 )/"

Xj

F' dh

x

+ (2xl0

-4

) r y

j F' X

dh

153
dz

L = (l/2)(lxlO'

4

)V

2

(C

j

-C

i

) (U-5)

[W

e in N*cm; F' in N; h in ym; z

L in cm; V in kV; C in pF]

.

A

Note in Figure 7 that F^ is actually negative, i .e. its direction is

opposite to that for z^.

154
APPENDIX V

THE ADJUSTMENT OF F TO F' IN THE PATH INTEGRAL

The partial derivatives of Equation U-4 are

= (l/4)V2 [3C(h,z

L )/3h

x

] h >z y l

(V-l)

=(l/4)V

2[3C(h,z

L )/3h

y

] h >z

^

(V-2)

= (l/2)V

2

[3C(h,z )/3z,]

L h h x

,ny

( V— 3)

[F in N, V in V; C in pF; h, in m],

It is clear when comparing Equations V-l to V- 3 with Equations

E-3 to E-5 that

F'(h,z

L

) = [V2 /V2

(h,z

L )]F(h,zL

) = 5(h,z

L )F(h,z

L

) (V-4)

[ F

'

,

F in N; V, V in kV; h in ym; z

L in cm].

where £

= V^/V^ is the factor by which F is adjusted to a new value

F' , such that F' is the value that F would have been if V v.ere equal

___ to V. Equation V-4 holds for each component of F, i .e . for F

h ,

F. and F . h

y

z

l

155
APPENDIX W

A CHECK THAT THE PATH INTEGRAL IS CONSERVATIVE

The path integral must be conservative. If the integral is

conservative then it should not matter which path is taken between the

initial state i and the final state j shown in Figure 21. To

simplify the calculations, the two paths indicated in the figure are

sel ected.

Note that the values of h^ for these two paths are for those at

high voltages not for those at low voltages. Therefore, in that part

e of the path used to calculate W

4

, d^ and/or 6^

are not constant, but

A instead vary in such a manner that h^ is constant. Table 9 shows

that this would not be the case for a Type II scan with d^=0 and

A

0 =constant because for that case, h

v varies at high voltages.

y y

If Equation 4-3 is conservative then

W

1 + W

2 = W

3 + W4

or

-( 6 . 06x 10

" 4

)V^ +2

z

l ^+ (6.5xlO‘

13 )V^

+2 C(z

L ^)(hy/2) =

(6.5xl0" 13 )V

3

2+4C(0)(h

3

/2) - (6.06xl0’4

)V

2 +4 [l+(6.5xl0" 9 )h

2 ]z

L ^.

156
If V

^ + 2 = V

3 +4

> then this equation becomes

C ( z,

) = C ( 0) - ( 12 . 12 ) z. f

L f

or

C(z, ) = [ lOOpF- (12.12 pF/cm)z, ].

L f

L f

This agrees with Equation 2-13, and therefore the path integral is

conservative - but only if the normalization values and V^+4 are

chosen to be the same for each path. This normalization value is

completely arbitrary (as shown in Example 2 of Appendix Y), so it

might just as well be the nominal value of the voltage applied to the

electrodes, i ,e. V=V

N

, and V^=9 kV for m=5 grams, or V

N =4.02 kV for

m=l gram.

The other necessary condition for the path integral is that the

mechanical work required to translate and/or rotate the electrodes

must equal the change in electrical energy stored in the field

between the electrodes, i .e.

W® + W® = {l/2)(lxlO

_4

)V^|ICf-C

1.]

or

-(6.06xl0'4 )V^z

L ^+ (6.5xl0' 13

)vjjjc(z )(hy/2) =

157
(1/2)(1x10' 4 )v2{C(z, )[l+(6.5xlO

-9

)h^] - C(0)

}

y

or

C(z, ) = C(0)-( 12. 12)z.

L f

L f

C(z, ) = [100 pF-( 12- 12 pF/cm)z ].

L f

L f

This again agrees with Equation 2-13.

\

158
APPENDIX X

THE UNKNOWN PART OF THE PATH INTEGRAL

The horizontal integrals /2 F

' dh and /2F' dh are unknown in „ ' hx

x J h

y

y

Equation 4-4 because F^ and F^ cannot be measured with the balance,

and measurements of the d^ and d w contributions to h and h would

x y x y

require expensive laser inteferometry techniques.

This appendix will determine the size of these missing integrals

for the two paths shown in Figure 21 with a brass suspended electrode

at V

n=9 kV. Let C(0)=100 pF and C(z

L )=C(2.5 cm)=69.7 pF. Also, let

/\ f /\ /V dy=0 pm and 0^=5" at point (0,h^,0); so Yy=6.80" and h^=26.8 pm at

^ G that point. Remember that h is constant for the integral in this

/s /\ figure; therefore, 0 =8.35", y =4.84" and h =26.8ym at point (0,

y y y

A

hy» Z

L

)•

Equations 4-3 and V-4 yield the results

W®/W *

= (6.5x10~

13 )V^(69.7 pF) (1/2) (26.8 ym)

2

/

[-(6.06x 10" 4

)V

2 (2.5 cm)]

W|/W® = -1.07xl0"

5

or the path integral should be 10.7 ppm less negative than the W®

integral for the first path. For the second path

159
W

e

3

/W

e4

= (6.5xl0~ 13

) V^( 100 pF) ( 1/2) (26.8 ym)

2

/

{-(6.06xl0~4

)V

2 [l+(6.5xl0"9

)(26.8 ym)

2 ](2.5 cm)}

W

3

/W

4

=-1.54xl0‘ 5

0 or the path integral is actually 15.4 ppm less negative than the W

4

i ntegral

.

This error can therefore be quite large. Its magnitude depends

upon the particular path taken during a scan, and it will have the

same sign, independent of the carriage translation direction (i .e.

the absolute value of the path integral is actually less than

160
APPENDIX Y

ERROR CALCULATIONS FOR THE PATH INTEGRAL METHOD

a) Method used to estimate the voltage ratio uncertainty.

The voltage ratio, V ca i c /V^, which can be obtained from Equation

4-4, is

V ral /VmQa = ([-(2xl04

) /i/vLJF, l d + 4 A/vL.cFudhJ/CC.-C,)}^2 (Vcalc meas ' J meas 1 z

j_ J meas h

J v f

[F

z l

’ F

h

in N; V

calc> V

N>

Vmeas1n kV; C in pF; h in um; Z

L in

cm].

The quantities in Equation Y-l that have errors associated with

A A A /\ them are V

meas , F

z

, z

L

, F

h

, C. and C^. The horizontal component of

the path integral is experimentally unknown - so it can be treated as

A an uncertainty, rather than F^ and h individually. Also, the errors

involved in measuring the capacitance at low voltage rather than at

A A

the high voltages must be included in C. and C^. (along with the

dynamical measurement errors and any errors resulting from

temperature fluctuations.) For example, if the carriage tilts

linearly from 0=0" to 5" between C=100 pF and 69.7 pF for a brass

suspended electrode at V

N= 9 kV, then the path integral is too large

A by 4 ppm when the horizontal component is not included, and C^ is 1.4

ppm too small because of the low voltage measurement problem. If 0

161
varies from 5" to 0" between C=100 pF and 69.7 pF then the path

integral is 16 ppm too small, and is 4.4 ppm too small. (Notice

that the horizontal component of the path integral and the low

voltage capacitance errors are correlated.)

There is a random error in determining m, which (according to

Equation P-18) should be included in the uncertainty of F - along

Z

L with the systematic static and dynamic errors. The static systematic

A

error is also correlated with the low -voltage error in measuring C

and with the horizontal component of the path integral. Therefore,

the uncertainties of Equation Y-l are correlated mixtures of random

and systematic errors. Rather than attempting to assign correlation

coefficients and deciding how to combine random and systematic errors

for nonexistent data, only measured quantities and their assumed

measurement uncertainties will be used. The voltage ratio will be

calculated for a variety of carriage scans, using the low voltage

capacitance values and neglecting the horizontal path integral

components. The correlation effects and the effects due to the

unmeasurable quantities should thereby be reflected in this

uncertainty estimate.

b) Calculations for Type I scans with a brass electrode at V^= 9 kV.

Example 1. Perfectly aligned right circular cylinders with no

errors.

162
Let mass m be such that F,=mg at V^=9 kV when h=0. Since h=0,

Equation 4-2 becomes

F7 =mg=(6.06xl0'

4

)(9 kV)

2=4.9086x!0" 2

N

and V.

A /\ meas

=Vj

V|=9 kV. Also, I

F

z | = | F | and |F'Z | = |F because £(h,z

L )=l,

Therefore, Equation 4-4 becomes

v ^aic

= -( 2xl0

't

) m9( AZ

L/AC ) =_

( 2xl °

lt )( 4 * 9086x10

" 2

N) (2.5 cm)/

(69.7 pF-100 pF)

or V ca i c =9 kV=V

meas

. Therefore, there is no error in this example.

A

Example 2. A -2 ppm error in F z and a possible combination of other

errors.

The static and dynamic errors are such that F

z appears to be 2 A /S ppm smaller than mg when F^ actually equals mg. F

z would thus be

increased by 2 ppm until 3^^ =3^^ , so that

V* =(4.9086xl0" 2 N)[1+(2x10" 6 )]/(6.06x10"4

)

or

Vm_ =9.000009 kV.

meas

163
', calc

= ' (2xlo4)(ms)V

N/Vmeas

(iz

L /AC)= ' (2xl °4)(m9)[1 ' (2xl0

" 6)](AZ

L /4C)

V

2

, =-(2x10

4

)(4.9086x10‘ 2 N)[1-(2x10" 6 )](2.5 cm)[l-(2xlO' 6

)]/ ca I c {69.7 pF-( 100 pF)[l+(lxlO" 6

)]}

Vcalc

=8 * 9999668 kv *

Therefore, V ca -| c is 3.7 ppm smaller than V^.

Rather than using as the normalization value, let us use

V=Vmeas

=9 - 000009 kV -

V

calc

= - (2xl°4) Cmg)

V

2/ v2eas

(Az

L

/ ac )=-(2xl °4

) ( mg)( 4z

L

/ Ac )

V 2, =-(2xl0

4

) (4.9086xl0

-2

N) (2.5 cm)[l-(2xl0' 6

)]/ ca i c {69.7 pF - ( 100 pF)[l+(lxlO" 6

)]}

or

Vcalc

=8 * 9999758 kV *

Therefore V ca -| c is 3.7 ppm smaller than V, which is the same result

as that for a nominal voltage (V^=9 kV) normalization factor. The

normalization factor is completely arbitrary as long as it is used

164
consistently and correctly. The nominal voltage will be used as the

normalization value in the examples.

Example 3. A +2 ppm error in F and a possible combination of other

errors.

V calc

=9 * 0000267 kV *

Therefore, V ca -| c is 2.9 ppm larger than V^. Note that if a -1 ppm

error is assumed for 100 pF rather than the +1 ppm error for 69.7 pF,

then V ca

-| c would be 3.7 ppm larger than V^. The +1 ppm error is

chosen because this is the sign for a dynamical capacitance

systematic measurement error.

Example 4. A -2 ppm error in F z and a maximum combination of other

errors.

V

2

. =-(2x10

4

)(4.9086x10" 2 N)[1-(2x 10' 6 )](2.5 cm)[l-(2xl0"6

)]/ calc

=l+(2xl0” 6

)

V

calc

= ' (2xl ° 4)(4 - 9086x10

" 2 N)[1+(2x10" 6

)](2.5 cm)[l+(2xl0' 6

) ]/

{(69.7 pF)[l+( lxlO

-6

) ]- 100 pF}

or

165
{(69.7 pF)[l-( lxlO

-6

) ]-( 100 pF)[l+(2xlO

-6

)]}

or

V calc

=8 ’ 9999507 kV *

Therefore, V ca -| c is 5.5 ppm smaller than V^.

A

Example 5. A +2 ppm error in F z and a maximum combination of other errors.

V

2

, =-(2x10

4

)(4.9086x10"2 N)[1+(2x 10"6 )](2.5 cm)[l+(2xl0

-6

)]/ Ca I C

{(69.7 pF)[l+(2xlO~6

) ]-(100 pF)[l-( lxlO

-6

) ] }

or

V calc

==9 - 0000443 kV *

Therefore, V ca -| c is 4.9 ppm larger than V^.

Examples 4 and 5 are very pessimistic. Examples 2 and 3 provide a

better estimate of the uncertainty involved in using the path integral

method for a Type I scan - which is +3 ppm and -3.5 ppm, or a ±3 ppm

measurement.

166
c) Calculation for a Type I scan with an aluminum electrode at V^= 4.02

kV.

Example 6. A -2 ppm error in and a possible combination of other

errors.

Let mass m be such that F

z =mg at V N=4.02 kV when h=0. Since h=0,

Equation 4-2 becomes

F

z =mg=(6.06xl0'4

)(4.02 kV)

2 =9.7932024xl0

3

N.

A

F

z appears to be 2 ppm smaller than this value for mg; therefore, the

voltage is adjusted until

V

2meas =(9.7932024x10 N)[1+(2x10"6 )]/(6.06x10" 4

)

Vm. =4.020004 kV. meas

V^ic= -

( 2 xlo

4

)(m9 )(V

N/vLas )(AZL /AC)=-

( 2 xl°

4

)(m9 )[ 1 '

( 2xl 0

" 6 )](AZ

L /AC)

V calc

= “( 2xl °4

)

(

9 ’ 7932024x10

” 3 N)[M2x 1 °- 6

)](2.5 cm)[l-(2xl0' 6

)]/

{69.7 pF-( 100 pF)[l+( lxlO

-6

)]

>

167
=4.0199851 kV.

Therefore, Vca ^ c is 3.7 ppm smaller than V^.

This example agrees with Example 2 for the brass suspended electrode,

at V

n=9 kV, as expected, because it should not matter what electrode is

used, at what voltage, as long as the carriage is translated along the

combined electrode symmetry axis.

d) Calculations for Type II scans with a brass electrode at V

N=9 kV and

0 =5"

.

y

Example 7. No measurement errors involved in the experiment.

A /\ Figure 22 shows a plot of 6F'/(mg) versus z^, where 6F ' / (mg ) is

derived from the VmQao values listed in column 7 of Table 9. The

illca o A straight line at 6F

1 / (mg ) =+2.8 ppm yields the same change in path

integral as the curved line. Therefore ,^F 'dz

L=(mg)[l+(2.8xlO" 6 )]Az

L

.

Example 13 of Appendix T shows that the capacitance is +0.3 ppm larger

than the C(z^) value at every point z^.

V calc

= “( 2xl ° 4

)

(

4 - 9086xl0

~ 2 N)[1+(2.8x 10‘ 6 )]{2.5 cm)/

{(69.7 pF)[l+(0.3xl0~6)]-( 100 pF)[l+(0.3xl0

-6

) ] }

or

V calc =9. 0000109 kV.

168
Therefore, V calc is 1.2 ppm larger than V N with no experimental errors

present. This discrepancy arises because the horizontal parts of the

path integral are unknown, and because the force is measured at 9 kV

where the suspended electrode has swung out to some equilibrium angle

(causing a 1.7 ppm to 4.7 ppm increase in the actual capacitance,

depending upon the value of z^) , but the capacitance is measured at

V rms =108 V ( resu l tin9 i n on ^ a 0*3 ppm capacitance increase).

A Example 8. A -2 ppm error in F

z and a possible combination of other

errors.

V

2

. =-(2x10

4

)(4.9086x10"2 N)[1+(0.8x 10" 6 )](2.5 cm)[l-(2xl0" 6

)]/ ca I c {(69.7 pF)[l+(0.3xl0~6

) ]-( 100 pF)[l+(1.3xlO

-6

)]}

or

V calc

=8 * 9999811 kV *

Therefore V calc is 2.1 ppm smaller than V

N

.

A

Example 9. A +2 ppm error in F

z and a possible combination of other

errors.

V

2

-| =-(2x10

4

)(4.9086x10"

2 N)[1+(4.8x 10' 6

)](2.5 cm)[l+(2xl0" 6

) ]/ ca i c {(69.7 pF)[l+(1.3xlO" 6

)]-(100 pF)[l+(0.3x!0"6

)]}

or

169
=9.0000393 kV.

Therefore, Vca -j c is 4.3 ppm larger than V^.

e) Calculations for Type II scans with an aluminum electrode at

V

n= 4.02 kV and 0=5".

Example 10 . No measurement errors involved in the experiment.

A

A plot of the (SF'/mg values obtained using column 7 of Table 10

/N versus z

L yields a straight line at <5F' /mg=+0.8 ppm.

V(L 1c=-(2x10

4

)(9.7932024x10’3 N)[1+(0.8x 10' 6

)](2.5 cm)/

{(69.7 pF)[l+(0.3xl0"6)]-(100 pF)[l+(0.3xl0

-6

)]

}

or

V calc

=4 * 0200008 kV *

Therefore, V ca -| c is 0.1 ppm larger than with no experimental errors

present.

/\ Example 11 . A -2 ppm error in F

z and a possible combination of other

errors.

V cal

c

= ~( 2xl °

4

)

(

9 * 793 2°24x1 °“ 3 N)[1-(1.2x 10~ 6

)](2.5 cm)[l-(2xl0“ 6

)]/

{(69.7 pF)[l+(0.3xl0

_6

)]-( 100 pF)[l+(1.3xlO'6

)]}

170
or

Vcalc =4. 0199861 kV.

Therefore, Vcalc is 3.5 ppm smaller than V

N

.

A Example 12 . A +2 ppm error in F^ and a possible combination of other

errors.

V calc

= " (2xl°

4

) (9 - 7932024x10

" 3 N)[1+(2-8x10'6

)](2.5 cm)[l+(2xl0'6

)]/

{(69.7 pF)[l+(1.3xl0' 6

)]-(100 pF)[l+0.3x!0' 6

)]}

or

Vcalc

=4 - 0200135 kV *

r Therefore, V ca -| c is 3.3 ppm larger than V^.

f) Calculations for Type III scans with a brass electrode at V

N =9 kV.

/\ Example 13 . e varies linearly from 5" to 0"; F

z is determined with a

-2 ppm error, and a possible combination of other errors.

A

A plot of the 6F'/mg values obtained using column 7 of Table 11

A versus z L yields a straight line at 6F'/(mg)=+1.4 ppm.

V

2

, =-(2xl0

4 )(4.9086xl(f 2

N ) [

1 - ( 0 . 6x 10“ 6

) ] ( 2 . 5 cm)[l-(2xl0" 6

) ]/ ca i c {69.7 pF- ( 100 pF)[l+(1.3xlO~ 6

)]}

171
or

Vcalc

=8 * 9999687 kV *

Therefore, V ca -| c is 3.5 ppm smaller than V^.

A Example 14 . 6y

varies linearly from 5" to 0" ; F

z is determined with a

+2 ppm error, and a possible combination of other errors.

V

2

. =-(2x104 )(4.9086x10"2 N)[1+(3.4x10"6

)](2.5 cm)[l+(2xl0

-6

) ]/ ca I c {(69.7 pF)[l+(lxl0'6

)]-(100 pF)[l+(0.3xl0"6

)]}

or

V calc

=9. 0000286 kV.

Therefore, V ca

-| c is 3.1 ppm larger than V^.

g) Calculations for Type III scans with an aluminum electrode at

V n= 4.02 kV.

A

Example 15 . varies linearly from 5" to 0"; F

z is determined with a

-2 ppm error, and a possible combination of other errors. A plot of the

A <5F'/(mg) values obtained using column 7 of Table 12 versus z

L yields a

/\ straight line at 6F'/(mg)=+0.3 ppm.

V

2

. =-(2x10

4

)(9.7932024x10' 3 N)[1-(1.7x10

_6

)](2.5 cm)[l-(2xl0“ 6

) ]/ ca I c

172
or

{69.7 pF-( 100 pF)[l+(1.3xlO" 6

]}

Vcalc

=4 * 0199838 kV *

Therefore, V ca -| c is 4.1 ppm smaller than V^.

Example 16 . ey

varies linearly from 5" to 0"; F

z is determined with a

+2 ppm error, and a possible combination of other errors.

V

2

, =-(2xl0

4

) (9.7932024xl0~ 3 N)[l+(2.3xl0'6

)](2.5 cm)[l+(2xl0" 6

) ]/ ca i c {(69.7 pF)[l+(lxl0‘6 )]-(100 pF)[l+(0.3xl0" 6

)]}

or

Vcalc =4. 0200105 kV.

Therefore, V ca -j c is 2.6 ppm larger than V

N

.

173
SUSPENSION YOKE

BALANCE BEAM

SUSPENSION

WEIGHT HOLDER

CONNECTING RING

LASER INTERFEROMETER

SUSPENDED ELECTRODE

CARRIAGE ELECTRODES

GLASS INSULATOR

CARRIAGE

ELEVATING SHAFT

Fig, 1 Relevant features of the arrangement for the SI volt experiment.

174
(symmetry

axis)

x

Fig. 2 Coordinate system and quantities required for the forces, the

capacitances, and the equations of motion.

175
Fig. 3

z

Effect on balance beam of center knife rolling on its flat.

176
£>P

COSYy

-Q

pCOS#

epsin/3y

>

Fig. 4 Effect on balance beam of pendulum flat rolling on the load knife

edge.

177
(symmetry

axis)

Fig. 5 Effect on capacitance of outer carriage electrode pivoting about

point 3. *

178

.j2

t

(zi-)

--

L

t

(zi_)
CM

CM

Fig. 6 Relative displacements of the suspended electrode and the outer

carriage electrode.

179
Fig. 7 Coordinate system for the relative electrode positions and the

resulting electrical forces.

180
(symmetry

axis)

x

Fig. 8 Location of a point on the outer rin of the suspended electrode.

181
- L

q

(z

l)

(P) (gravitational horizon)

x

Fig. 9 Location of a point on the rim of the outer carriage electrode.

182

i(1

-COS(9y)
and

co-v.

vs V

(s/pej)

AV> (s)

A/L

d

183

and

io

versus

V

for

a

brass

suspended

electrode.
I

(„) INO>%

oo

in

=3toS-a

>

^3

< ca

to4-o

+-> o

D_

CD

184

for

values

from

Table
MAg

Fig. 12 External forces induced on the suspended electrode when the balance

beam is at rest.

185
10

2

{Ma

+

m)

L

CM

co 2

y

7

2y

10

2

(Ma

+

rn)

l-CM^YyTy

2

1.5 s

V = 0 kV

frit)

t(s)

f-r(t)

Fig. 13 Plots of f

R and versus tine, with voltage off, for one pendulun

period, which is 1.5 s for the brass electrode. f

R = +5.08 x 10~ 8

N at t = ( 1 / 4 ) P if y

= 3.30 x 10' 5 rad (6.8").

Yy

y

186
f

R<t)

-h

fj(t)

</> ID

rII

O

II>

A^mlAlOTiui + V|Aj)z _oi.

187
V = 9 kV

P

7y — S

Fig. 15 Plots of fD and fT versus time at V = 9 kV for one pendulum period,

K I _o

which is 2.8 s for the brass electrode, fp = +1.38 x 10 N at t =

( 1/4 ) P if y

y

= y

y

= 3.30 x

10“ 5 rad (6.8").

188
V = 9 kV

Fig. 16 Plot of f

R + f

T versus time at V = 9 kV for one pendulum period,

which is 2.8 s for the brass electrode.

189
V = 9kV

P-yy — 2.8 S

fz(t)

Fig. 17 Plot of f versus tine at V = 9 kV for one pendulum period,

which is 2.8 s for the brass electrode.

190
V = 9kV

2.8 s

Fig. 18 Plot of Am versus time at V = 9 k V for one pendulum period, which

is 2.8 s for a brass electrode. Am is defined by Equation R-10.

191
Vrms = 100 V

P7y = 1.5 s C(zL) = 10" pF

Fig. 19 Plot of 6C(t)/C(z

L

) versus time at V rms

= 100 V and C(z

L

) -100 pF

for one pendulum period, which is 1.5 s for a brass electrode.

6C( t) /C(

)

is defined by Equations S-3 and S-4.

192
TYPE I

TYPE

li

TYPE

III

Scan

Scan

Scan

193

Throe

different

paths

assumed

for

the

calculations

of

Appendices T

and

Y.

The

paths

have

these

shapes

for

low

voltages,

but

not

necessarily

for

high

voltages.
( 0, 0, 0)

ZL

W-, e

(0, 0, z

Lf

)

Fig. 21

(0, hy, 0)

w4

e

(0, hy, Zl^

)

i

I

I

Two different paths used in Appendix W to

verify that the path integral is conservative.

These are the values of fi at high voltages,

so d

y

and/or 0 are not constant for the path

integral W^.

194
5

E

a

a

ra

E

V N

<LL

OO

4

3

2

+ 2.8 ppm

i 2.0

Z|_(cm)

Fig. 22 Determination of the straight line which yields the same change in path integral as the curved line used in Example 7 of Appendix Y.

195
Table 1. Length and mass values of quantities shown in Figures 2-4.

L b

= 7.5 cm L=64 cm £=35 cm

L<~=19 cm £^=25 cm L-L^=45 cm

rj=8.5725 cm R*=9.3955 cm R°=9.7980 cm r°=10.7415 cm

M

A

~M

B =

a 2.35 kg orb

5.47 kg M

b

=1.90 kg

L =

a 39 cm or

b

47 cm cm

L =10 cm @ C=100 pF

o

r _5

pp~P

c=5xl0 cm

a Value obtained for an aluminum suspended electrode.

b

Value obtained for a brass suspended electrode.

196
Table

2.

Variations

with

z.

of

quantities

defined

in

Appendix

_l r—i cn oo ld —

N ' r—1 CO O CO ID O

E

I— a cn cti o o o i—

i

_i — tn in ^ u)

_i O O O LD O o N ' O LO O CM LD O

Q

EEO cri

• cti•••••

o o o i—

i

1 LD LO VO CO IX) CO

o o o un o o un co un o ^

E

-p u O') CT> O O O '—

—i un un co co co co

_J co oo co o r-» t-H N -—

co co !—i co <Ti E

4-> O CO CTi CTi O O O 1 LD lo lo vo <o <o

1 oo co o co cri N CO C"- rH VO O — E

+-> u (*~D CD rH t-H f—H C\j CO CO CO CO CO CO

__i N o o o un o o

E

O un VO UD f"» CO

1 un un un un un un

_l un N • E in ID N N CO c

a CO CO CO CO CO CO

o?

_l N ' o o o un o o

E

o a o cti co r^. uo _i '

rH

o ^

O 00 VO O ^ vT _i • co r-. c- vd un N U_ O

Q. o r-" un <n co t-H O v— r—i co vd vo un

un

• -i E O I—1 CO CO CO *3- N O

197
Table

3.

Static

equilibrium

values

when

V=4.02

kV, Fz

=9.

79x10

N,

C(z^)-100

pF, LQ10

cm

and 3

The F

value

is

equivalent

to

the

gravitational

force

due

toa1

gram

mass. o

ii

si

<U_

C\J

fO

<c_> <

<o<

<Li— <

<u_

CM

CM

CD

-o

cn

— VO CO r-H cn CO rx E E o r-H o r-H o o o o >1 CLCL o o o o o o o o >> . X >. —i E N CL vo VO CO 00 r-H LO r—H

'Xt" CL

O o CM o r-H o o o o D^—i E N CL o cn LD 00 LO cn CM —' CL O - r-H CO o CM O r-H o r-H ^ 

E

si Ixl CL o cn LO 00 LO cn CM

Ll_ CL • • • • • • • • — t“H CO o CM o r-H o t-H

>)

c\ X”

—X

X

N

Li- x— CO VO cn CM cn co VO LO

r-H CM o CM o r-H o t—H • > >sLf) -VC

#» 1 cn 00 oo O- o cn r-H i—1 X o • • • • • • • • 1 1 r-H CM LO 00 r-H cn rx vo LO •

V. r-H CM CM i—

H

r-H o

II CO >> X X CO VO rx LO f—H

00 E r\ E s-

>< CM co o co IX LO > v r-H CM CM 1 1 r-H -oc

>> ^ LO CTi o CO r-x o CO CO

r\ IX CM cn oo CO IX X z > *

r-H CO r-H CM o o o o -VC CM ^ -V O

XJ • ra >> S- cn oo co r-H CM <vi- LO II #» VO > X 1 co vo LO *Xt“ CM t—H

CO o r-H r-H c

i—H CD X , Cl) s+-> >) ^—.

CL) LO o O LO LO o o LO -Q

X z 1—1

r-H > CO

CD

13 s r—

-O CO

1X3 >

S- LO o LO LO o LO

#» VO CM LO CM CM LO CM ^—

X

X 1 • • • • • • o 00 O 00 o ISI 1—1

CM *=3- CM CM CM

XX

—

c_> <o

>>

#s

X < E X CS- O o LO LO O o LO LO c —

•r—

CD ^—X

C_> = E on cn cn cn rx rx rx IX c u u CO CO co ro •xf «vl- CD

X. ^ SCD 4- „ lo LO LO LO IX IX. [X. 4-

X. cn CO CO co CO •r— o —

CM CM CM CM LO LO LO LO 03

198
oII < OCL

>>

aE

fTD Eaor-H

II oo

fsl CD

CMI oX

CD

CD

II fsl

00

CO

LO

II E

CL) -Eco

a)

fO>E3

•r—E

-Q3

CT

aio

*r—

+->

<T3 +-> O0

PC

cu -Q

fC

CO

co

fO

fOE

CD CM

fOO

Q)3

-o

a>oEO

4-

»aEo

-M

fa

4->

•i

>

faE

CD

a> E+-> o

+->

+-> E

CL)

fa>

•r-3

cr

<D

co

CL) 3

r—

fa>

ISI Ll_

<D

Ll_

Ll_ *—.. O v z CD \ o 1—H

pC CM LD r-H PC CD <ca E E CM PC r-H CO O r-H o O o :>i CL • • • • • • <u_ CL o o o o o o o O CM

ta^ »—v

_l ^ fsl E

CCD

— Q. CD CM LD o CO CM

O

<\ o a. • m • • • • ' CD oo pc CO o r-H o i—1

°o CO CM

-l E N CL CD CD r-s

O rs* co CO o CO -— CL • m • • • • • • < O CD CD pc pC o CM o CM

CO CM ^ -s fsl ISI E

CLl_ Ll. CL CD CO

O c-. 00 CO

O < CL • • • • • • • • CD CD pc PC o CM o CM

CO CM

>1

fsl

X LJ- ^ CM PC CD LD 1—1

CM CO CD

C Ll.

v—

r • • • • • 9 • CM CO CM CD rH CM o rH „ „ • >

>>lo

r» 1 o o CM r- CD CM 00 PC r-H X O • • • • • • • • cu_ i—1

CM pc CD i"- r-H CO CD o CM -—r

CO CD LD CM CM PC 1—1CO II r-H i—1

(/> <

>> r-H r-H CO CO CO CD o CO

c=E 0% E • • • • • • • • > X P- CD 00 CO o o o 1"- C^ CD — CO CM co r—l CM r-H COE

ra -

O o r-H o PC 00 rH LD

r» r-H CM CD C's

O o t"- C^ > X — • • • • • • • • -X < >- v * rH CM r-s CO r—1

C\J o r-H

r-H CM 1—1 00

CD ^ N »- -a LD

ta II >> E > CD CD 00 CD 00 o i-H LD LD X 1 • • • • • • • • E < >- O CO c- CD o LD o CO CO CL) i—i

CD o CO CD r-H a)

—* rH s4-> ai

>» jp X = CD o O LO LO o o ID L0 CD v ^ I—t

r-H CD3

r—

r—.

ta -O >

fa

>> E LD o LD LO o LD

rv

•N CO CM LO CM CM ID CM —I X 1 • • • • • • N

CD o PC CO pC PC 00 o PC •—- I—1 — CM 'O’ O CM CM PC CM C_>

<C_) <1

>1 ^ .

#» E E 5< p. O o LD LO o o LD LD * f

-a -

CDO . E £ E CD CD CD CD rs- C'- 1"- r-s CD 3 u CO CO CO CO pc pC PC PC E

_i CD 4- .—,

LO LD LD LD I"* C-s fs. 4- C CD CO CO CO CO pC PC PC pC •r— s: DC • • • • • • • • Q — CM CM CM CM LD LD LD LD ra

199
Table

5.

Static

equilibrium

values

when

V=6.96

kV, F

=2.94x10

N,

C(z. )

=100

pF, L10

cm

and

3-0.

The j

to

to

rerO

S-CD co

reo

-l->

CDD

•U

CD

c_> S-o

4—

raco

+-> ra>

ra

SC7>

cd JC

+-> o

+->

+-> c

a»

ra>

cr

cd

cu

cj

ra> < ca r"~ CO LO c~H E o ! 1 o r—1 o ;>J Q- • • • <u_ Q- o o o o CM IE

ra

x-

—s^ 

_l , s <c_> N E

<1 CL r^. o tH O Q. • • • • o CO o <XJ N

O

X

<u_cm

>,Ln

•> 0X0i

<Lt_ i—l

CM

N

>. =

<o

<1

IM

C_>

Q_

Cl

CL

Nl Q. N ^ E X CL

<-£C

x

<>-

~a

ra

s_ to

X

<?-

>>

r* x

CD

-o

ra

»> to X I CD O

>>

—" E X CL O O

<=C CD E -X

'O’ LO

'0-* o CO

rH <d- LO rH • • • • i-H •d* o CO

00 o CO • • • r-H CNJ 1—1 CXI

00 LO 03 C". > • • • .X o r-H r^. co

^d" 00 CXI to I—1 o

II CT> CT> cr> CO to • • • E

OsJ LO 00 1—1

ir-H C\J CM >

•a

'd" LO CXI sc

CT> CO CO ra • • • • rH CO rH CO >

to i—1to r-H • • • • c

03 CTi rH

to LO

r-H cd

CD2+-> CD JO

LO orH o LO

to

CD

=3

ra LO

CM oLO

•d"

Oxl > • • • X -N •a- 00 o *d“ CM «d" OsJ N o o LO LO c

•r—

CDa

r"> r^s c: *0- ‘d- «a- d- CD

SCD 4-

I'v. r^. 4—

'O' *d" • i— • • • • Q

LO LO LO LO ra

to

03toM

c_> o

«]

200
Table

6.

Static

equilibrium

values

when

V=8

kV,

F

z

=3.

88x10

N,

C(z^)=100

pF, L

Q

=

10

cm

and

$y=0.

The

F

value

is

equivalent

to

the

gravitational

force

due

to

a

4

gram

mass.

vj

Nl I

O X

“ll.

CXJ

< ca

>> = O >J CL

<Ll_ | CD CL

C\J I !

03

<t_> <3 N E kO CXJ —

CL • • • C_> CL r—

H

to o

<o

<o

<

N CL (T> to ax CL • . • • O rH r'- o LO

e

CL N CL

N S-S

>>LO CXI to

ft i • • • O X o r-H CXI CO

<Ll_ 1—1 =ct •3-

CXJ "

r-H

>) , ^ H CXJ

ft E • • • X CL r^ «3- r-H <-C —

rH CO i—

1

>> „ CO to LO

ft _ 00 CO X • • • <>- — CO to CXI

-u

03

S- >>LO

•> I X o <>-

CD

T3

03 >> S-

•* CO X I CD O

^ 'e

X 3L

E E

a o

C CD E ^

cn 00 to LO o rH o r—

1

• • • • o o o o

^3"

CTi co cr> ^ • • • • «—I O LO

co ro t—i • • • • r-l CO r—I CO

OCVI

CO

CO

CXI

00

I"-*

• LO

to CO o • • • • to CO 1—

1

CO

T—H CO r-H CXJ

LO o o LO

I—

1

to o LO

CXJ LO CXI

. • 00 o CXJ <3- CXJ

o o LO LO

r-. r^.

'd- =d‘

r^

. • . • LO LO LO LO

oII toE

sT3C

ra

CO

II

CD

<u2+J

CL) JO

(Oa>

03

OJ>

N

<_>

< O<3

0)o

c

d)

Sa> O

03

201
Table

7.

Static

equilibrium

values

when

V=9

kV, Fz

=4.

91x10

N,

C(z^)=100

pF, LQ10

cm

and

3^-0.

The

F

value

is

equivalent

to

the

electrical

force

due

toa5

gram

mass. < ca

>> = O >J CL

<u_ | cr> Cl CM I !

ro < o<3

Nl CJ

<o

Q.

Q. < C_) <3 E

CL

CL

CL

N CL o X

<Li_ CM

>,Ln

€\ Oxo| < L_L_ i—I

CM

<JC

X P-

>.

—

X < >-

-O

fO >> S-

•> CD X I < >- o

CD

-a

(O

S- >> CD * I X o

>>

—" E X p. Eo E

(_>

=C cn

e: -x

CO CD CO t-H 00

O CO o O o i—H

i“

H

CM r-H CM o O o o o O o O

CM I-'- CD co *=d- CM o CM O O I 1 CM CM CM

r-H r-H

CM I''.

O i^. CD CM CM

O o T—H

CO *=d- 00 CM CO

r-H i-H

CM o CO CM CM O o l—1 CO co CM CO

r-H r—H

CD CM CO cn I-'-

O co

O i—

H 1—1 CM CM LD CM

1—1

CO LD r-'- LD r-H CO CD 'vf CM o r-H CD r-'. > CM LD co r-H 00 O') co

r-H r-H CM CM

r-H o I''. o co LD CO r-H ii </) LD O CD r-H CD CO co LD E

r-H r-H CM CM LD r-H s- > CD CM rH LD O r-H CO CD

CO r^. o 00 CD CD "Oc

T—H

CM *=3" LD CD CO r-H ro

r-H r-H >.x

cn

II CD CM 00 O o CD CD >

CO CO cn CD CO CD CM LD £=

r-H r-H CM CO CD CM LD CL) <us+->

CL) -Q 1 1 CM CO LD O O LD

r-H CO<D

=3 1 ro>

LD

O o O LD o LD

co I''. CO CM ID CM ^ 

• _j cn cn co O N

r-H i—1

CM CM *» " CD <o

<3 o o o o O o LD LD c

CL) O

n- c.CD

£_<U 4—

i". 4-

*1Q

CD ID LD LD LD LD LO LD fd

202
Table

8.

Values

of

quantities

that

are

derived

from,

or

used

in

the

sample

calculations

for

a

brass

suspended

electrode.

203

=0.134

N-cm/rad

K

vun

'=0.123

N-cm/rad

S

(0FF)

=7.48

rad/N-cm

(0.11"

/yg)
Table

9.

Static

equilibrium

values

fora

Type

II

scan

with a

brass

suspended

electrode

when

V=9

kV,

1X3 E

to>3

scu>do

4-> o

1X3

CL) .a

CU

(Jd

03

1X3 jQ

CL)

-od

CO

CMIor-4X

tocoo03

II

N

CO r-^ 03 i-H o oo

r**. CM un 00 o r-H 00 03 03 03 o o 03 03 03 03 o o co 03 03 03 03 o o CO ^ 03 03 03 03 o o 03 > 03 03 03 03 o o o E > v— CO CO co 00 03 03

00 r-^ 03 1 1 o oo

03 r-. o CM oo

to co co co 03 03 03 03 03 03

to 03 03 03 03 03 03

1X3 ^—-N 03 03 03 03 03 03

03 > 03 03 03 03 03 03 -a E • > — 00 co co 00 oo CO

CO r^s 03 1 1 o oo

co CO to 03 r-H CM

co co co 03 03

03 03 03 03 03 03

CO 03 03 03 03 03 03

03 03 03 03 03 03 03

O) > 03 03 03 03 03 03

1X3 E > — co co CO 00 oo oo

d sc O o

•r— *r—

—-N 4-3 4-)

N

N E • 03 CO

<Ll_ Ll_ CL to 03 o r-.

d

dd

<1 Cl o *P“ *r— oo CM CM CM f—1

•1— E E

4-3

dd

1X3 cu cu d 4-3 4-3 •r— CU CU E “O O S- ^ •> CO to 1 1 r-H LO X—1

CU cu cu >3 E • • • • • • 4-> cj a

Cd CL to oo

H 03 r"- to CU

dd * CM CM CM r-H f“H rH -a o o

4- 4-

cu

CJ CU CU d -d _d o 4-3 4-> 4- d d

O LO CM CM t-H CU •r— • 1

>3 co to r-^

O 03 -d < >- “ • 4-3 s_

d ' to LO OO CM o o d d d

*r—

d

d

cu cu CO

i~ CJ

a o *r— •r— d 4-3 4-3 s i~ 1X3 03 >3 cu E E

CD

_ LO LO LO LO LO LO cu cu V u 44 4-> •1“ CO CO

4-> >3 >3

1X3 CO CO E

CU E E

oo CM toO 4-> CL CL *—

o 03 00 co r^. CO Q- CL ^ -N >3 N Ll. o oo r^. i—1

LO 03 CO CM CM - CL o 03 co 00 to + | t_> ' «—1 od CO 03 03 03 03 d SC d

•r— •r— •P- E E E

=3 =3 Z3 ^ -X CO CO co _l E o LO o LO o LO CO CO CO N a • <c < c o o i—H

r-H CM CM 1X3 JO

a

204
Table

10.

Static

equilibrium

values

for a

Type

II

scan

with

an

aluminum

suspended

electrode

when

V=4.

02

kV, F

=9.7932024 x

10

-3N,

and

the

balance

beam

motion

is

very

small. o f—

H

CM CO *3" LD

C\J CM CM CM CM CM o o O O O O o o O O O o CO o o o O o o ro ^ >x C\J CM CM CM CM CM

CD > o o O

OO

O o E > o o r-H CM CO

'vt- on cr> 03 03 O'! cn

CT 03 03 03 cn 03

CO cr CTi 03 O'! O'! 03

ro t—

H

i—H f—

H

rH rH r-H CD > o o o o o o

-Q E > v—

«vt* •vt* o o t-H CM CO

co oo CO 00 co oo cr cn 03 CT) 03 CT

O'! cr cn O'! 03 <T

CO cr> CT 03 O'! CT CT

H3 rH rH rH t-H

H rH

CD > o o o o o o <T3 E • • > V—<x e E o o

•(

•r—

4-> +-> • fO n3 E e E o •l

•r*

x^ -N •r— E E N M E 4-> E E

<U- U- Q_ o cn on CO 00 fC CD CD <1 CL E 4-> +J " r-H O o o o o •r— CD CD E -o -o E

CD CD CD

+-> CJ

U

CD e E

•o o O . >. CO 03 LO CO o co 4- 4-

Do E CD <-E CL C\J r-H rH I—1

r-H o C_> CD (D

*> -» r-H r-H t-H rH rH rH E .e -E o +-> 4-> 4- E E

CD •r— •1“ -E

+-> e E

X-

—-X LO o LO CO rH CT o O

>) r-x CD CO CM

O e E E < >- •r- E E '—

rH rH rH tH rH rH

CO

CD CD E O a o •1— •1— e +J +-> e ro (T3 CD E E

>)

_ CD O) CD LO LO uo LD LD LO O +-> 4-> V <x •r- CO CO >> DO

(T3 CO to E

CD E E

+-> CL CL CO CM LO O CO CL CL

X

—V

O 03 00 co Cx Cx DO _l x- x. CO CM CM

Nl U_

O CO rx rH LO cn + | V ^ CL O 03 oo 00 rx LO o O '

rH e H3 fO cn cn cn e e E

•r— •r— •I— E E E

=3 13 C3

N. CO CO CO E o LO o LO o LD CO CO CO

N

u <c c =E o o rH rH CM CM 03 _Q o

205
Table

11.

Static

equilibrium

values

for

Type

III

scans

with a

brass

suspended

electrode

when

_ 9

V=9

kV, F

=4.9086x10

N,

and

the

balance

beam

motion

is

very

small. o 1—t

03 i—1

ro 00 CO 03 CO

O 03 00 i-^ ro pH I1". co ro r-^ CO 03 o o o o o o co 03 o o o O o o o o o o 03 03 o o o O CO o o o o o o 03 03 o o o o ^ % o o o o o o 03 03 o o o o CD > o o o o o o 03 03 o o o o o E > V " 03 03 03 03 03 03 00 CO 03 03 03 03 o <vj- i—1

03 pH ro CO CO 03 CO o

pH o 03 CO LO ro 03 o LO 03 o pH

03 03 co 00 CO co CO CO co co 03 03 03 03 03 03 03 03 03 03 03 03 03 03

CO 03 03 03 03 03 03 03 03 03 03 03 03

cO 03 03 03 03 03 03 03 03 03 03 03 03

CD > 03 03 03 03 03 03 03 03 03 03 03 03 -o E > V ' 00 CO CO 00 CO oo* CO co CO co CO CO o 03 pH 03 pH ro co CO 03 CO 03 o o 03 CO cn =a- CM co 03 co 03 o o 03 03 03 03 03 oo 03 03 03 o o 03 03 03 03 03 03 03 03 03 03 o CO o 03 03 03 03 03 03 03 03 03 03 o 03 ^ >N o 03 03 03 03 03 03 03 03 03 03 o CD > o 03 03 03 03 03 03 03 03 03 03 o 03 E > 03 CO CO 00 CO CO* CO* 00 CO co CO 03 M N E

<Li_ Ll_ Q. o r—I

03 ro I-"- ro pH ro « 1 o < CL o o o O pH 1—1

CM pH o o o

LO 03 o >> E o o LO o pH co co r"- 03 o <-C v—-» o 1 1 *=3" CO CO co CM co ro o r-H *—1 1—1

CM t—1

pH o CO o r-H ro pH O r-H LO r-'. 00 o

>> o pH 03 r-~ 03 co LO OO ro CO o <>- “ o t—H pH CM CM CM CO CM pH o o

>3

CD - o r-H CM ro LO LO ro CM pH o o *sf CO CM CO O o co CM CO o X ^ o 03 co oo r-'~ o 03 00 00 1"- ^ N u_ o CO r-. r-H LO 03 o ro r-^ r-H LO 03 V Q. o 03 oo 00 r-^ CO o 03 co CO CO

<_> 1—1

pH E o cn o LO o LO o LO o LO o LO

N ov—

o o t-H pH CM CM o’ o 1—1

pH CM CM

4-> roE

SCL) +>

CD "D

<Uu

s-o

CD -CC

t/3 S-o

i-Scuo

raE

qj

+-> CO>1

(/> oc

CD

ZJCOCO

+->

S-cu

4-> O) O

a>a

s-o

at -C

+->

s-o

s-Scdu

+-> raE

cu

+-> to>5

to

a.

Q. CM+

raCO

ZJ

to

to cJO CJ I 206

Assuming a

-2

ppm

systematic

error

in

the

force

determination.
Table

12.

Static

equilibrium

values

for

Type

III

scans

with

an

aluminum

suspended

electrode

when

V=

4.02

kV, F

=9.7932024 x

10” 3

N,

and

the

balance

beam

motion

is

very

small. 1 o 00 CO cn LO o r^. co CO

O 'O- PO co CO CM CM CM CM CO CO CO o o o o o o o o o o o o o o o o o o o o o o o o IS) o o o o o o o o o o o o CO X -s C\J CM CM CM CM CM CM CM CM CM CM CM

CD > o o o O

O o o o o O o O CJ E > o CO LO CM cn LO o LO CM LO CO o lc LO LO LO 'tf- •3- LO LO LO LO

O'! CT) cn <n cn cn cn <n cn cn cn cn

CTl cn cn cn cn cn cn cn cn cn cn cn

CO CTi cn cn cn cn cn <n cn cn cn cn cn

CO \—1

r-H r—H r-H t—H t-H r-H i—

H

rHr-H »—H r-H CD > o o o o o o o o o o o o

-Q E JXL > v o CO r'- CO CO LO o r^. CO CO o o cn cn cn CO CO co 00 cn cn cn o o cn cn cn cn cn cn cn cn cn cn o o cn cn cn cn cn cn cn cn cn cn o CO o cn <n m cn cn cn cn cn cn cn o co ^—N o r—1

r-H r-H r-H r-H r-H i—1

i-H r-H r-H o CD > CM o o o o o o o o o o CM

CO E > *0-

N1

N E

<u_ Ll. Q_ o o l“H CO LO o LO CO r-H o o < CL o o o o o o r-H o o o o o

>> E o LO CO CO CO CO LO o LO CM o < -C d- v—-• o CM LO CO o CM cn CM o r-H r-H o CM CO o cn LO 00 co CO o

>> o CO LO CO cn o r^. CM CO LO CM o < >- - o O o o o r-H r-H r-H o o O o ^

CD = o CM CO d- LO LO CO CM r-H o

00 CM LOO CO CM LO o ^—. o cn CO 00 r-- O <n CO CO r-- _j —v N Ll_ o co I-'- i—H

LO cn

O CO I-'N r-H LO cn

CL o cn 00 00 r^. LO o cn co co LO CJ r-H r-H

_l E o LO o LO o LO o LO o LO o LO N CJ o o r—H r-H CM CM o o rH «—H CM CM co

fOc

*E

s-CD

+-> O) •o

CDCJ

S-o4—

CD -c

-t-> c

C/> S-o

s_s_

CD

CJ

CO

CD 4->

(/) >-> LOoc

cncE3

c/> co

CO co

f—

+-> COc* 1—E

S-CD 4-> CD ~a

CDo

5-o4-

CD .c

+->c

s-o

ssCD

CJ

•r* 4-> COE

CD

+-> co>>

CO

QQCO

CTC

CO

CO C CJ Il

207

Assuming a

-2

ppm

systematic

error

in

the

force

determination.
H oS-1 1 4A . V. 9-76)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-2143

sion No. 3. Recipient's Access* if?

4. TITLE AND SUBTITLE

An Analysis of the Effects of Dynamic and Static Forces Present

in the NBS SI Volt Experiment

5. Publication Date September 1980

7. AUTHOR(S)

M. E. Cage

V mm*

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (street. City, state, zip) 13. Type of Report & Period Covered

Fi nal

( 1 1 ip

: 15. SUPPLEMENTARY NOTES

1 | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of most significant information . If document includes a significant bibliography or

literature survey, mention it here.) An analysis of the NBS SI volt experiment has been made first to obtain algebraic

expressions for the electrical forces present on the suspended electrode of the electrometer. These forces, and the gravitational forces, are used in the Principle of

Virtual Work and D'Alembert's Principle to obtain the second order, non-linear,

inhomogeneous, coupled differential equations of motion for the balance. Exact,

analytical solutions of these equations of motion are obtained using small angle

approximations and perturbation methods. Estimates are then made of the uncertainties

that might result in both the slope and path integral methods in order to determine

what requirements must be satisfied to reduce the systematic and random errors of

the force determination and the capacitance measurements to acceptable levels so that the SI volt can be determined to within a few ppm.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless q proper name;

separated by semicolons)

Absolute volt; balance equations of motion; D'Alembert's principle; metrology;

precision balance; precision electrical measurement

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(THIS REPORT) PRINTED PAGES

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