NBSIR 73-318

MASS QUANTITY GAUGING BY RF MODE ANALYSIS

R.S. Collier

Doyle Ellerbruch

J.E. Cruz

Robert W. Stokes

Philip E. Luft

R. Gordon Peterson

A.E. Hiester

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1973

Interim Report

Prepared for

National Aeronautics and Space Administration

Johnson Space Center

Houston, Texas 77058

NBSIR 73-318

Mass Quantity Gauging by RF Mode Analysis

R.S. Collier

Doyle Ellerbruch

J.E. Cruz

Robert W. Stokes

Philip E. Luft

R. Gordon Peterson

A.E. Hiester

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

June 1973

Interim Report

Prepared for

National Aeronautics and Space Administration

Johnson Space Center

Houston, Texas 77058

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued February 1973

FOREWORD

Background - The RF technique is a gauging method which

samples all parts of the inside of a tank. The response is characteristic

of the total mass of fluid within a tank; this is one of the advantages of

the RF system over the standard capacitance system which is essentially

a local measurement of fluid density. Another advantage of the RF system is the simplicity of the hardware involved; a small grounded antenna

about the size of a paper clip is sufficient to communicate with the inside

of the tank.

Preliminary theoretical and experimental results on the RF gauging

idea were obtained by NBS in connection with a NASA sponsored contract

on slush hydrogen gauging. These results may be found in NBS Report

9793 dated June 1, 1971, on "Instrumentation for Hydrogen Slush Storage

Containers. "

Purpose - The purpose of this report is to summarize work done

under purchase order T-1738B from the NASA Johnson Space Center

Houston, Texas, to the National Bureau of Standards, Cryogenics Division,

Boulder, Colorado. Items covered include:

1) Phase I - Preliminary studies of the radio frequency (RF)

mass quantity gauging system for two phase and supercritical

fluids; and construction of experimental system for detailed

feasibility studies.

2) Phase II - Experimental evaluation of the system for supercritical nitrogen and hydrogen. (Oxygen is also included in

Phase II; the results will be reported separately upon the

completion of the oxygen testing. )

iii
Objective - The primary objective of this work is to design and

develop a breadboard system to verify that the radio frequency resonant

cavity mode analysis technique is conceptually sound for the fluid mass

quantity gauging of the Space Shuttle Orbiter PRSD (Power Reactant

Storage and Distribution) subsystem tankage, i. e. , supercritically stored

hydrogen and oxygen, in all gravity fields. The secondary program

objective is to analytically determine the applicability of the concept to

the quantity gauging of Shuttle Orbiter propulsion systems tankage,

(i. e. sub-critical fluids).

End Product - The end product of this contractual effort is to be

a breadboard RF mass quantity gauging system capable of gauging supercritically stored hydrogen and oxygen to an accuracy of one percent of

total tank quantity in any gravity environment. This should also include

the gauging of subcritical hydrogen and oxygen representative of fill and

drain operations on the PRSD tanks.

Acknowledgements - The authors wish to acknowledge D, B. Mann,

Chief of the Cryogenic Metrology Section, Peter V. Tryon and John F.

LaBecque of the NBS Statistical Engineering Laboratory, and Dan S. Trent,

the NASA, JSC technical monitor, for assistance and encouragement in

planning this program. The authors also appreciate the assistance of

Larry Lowe, Bill German, Bob Gray, Tim Hipsher, Ed Rogers, Brian

Nakagawa, and Jeannette Garing.

iv
CONTENTS

Page

INTRODUCTION 1 EXPERIMENTAL SYSTEM 3

THE RF ANTENNA 7

UNIFORM DENSITY FLUID 9

EFFECTS OF STRATIFICATION 17

NON-UNIFORM DENSITY FLUIDS - RF MODE ANALYSIS .... 19

CONCLUSIONS 31

REFERENCES 32

APPENDIX A - CRYOGENIC SYSTEMS (Robert W. Stokes). A-1

APPENDIX B - MASS TANK GAGING SIGNAL

CONDITIONER AND DATA ACQUISITION

SYSTEM (J. E. Cruz) B

APPENDIX C - Q MEASUREMENT SUMMARY

(Doyle EUerbruch) C-1

APPENDIX D - DATA REDUCTION FROM THE

MAGNETIC TAPE UNIT (A. E. Hiester). . D

APPENDIX E - UNIFORM DENSITY HYDROGEN DATA

ANALYSIS AND ACCURACY STATEMENTS

(R. S. Collier) E

APPENDIX F - TOTAL MASS GAUGING IN A SPHERICAL

RESONANT CAVITY (R. S. Collier) .... F

APPENDIX G - APPROXIMATE METHODS FOR AN INHOMOGENEOUS DIELECTRIC

(Philip E. Luft) G

APPENDIX H - NUMERICAL ANALYSIS OF THE SPHERICAL

CAVITY FOR THE LOWEST ORDER MODES

(R. Gordon Peterson) H

APPENDIX I - LIST OF FIGURES I -1

V
ABSTRACT

This is a summary report of work done to date on NASA (Johnson

Space Center) purchase order T- 1738B concerning Radio Frequency (RF)

Mass Quantity Gauging. Experimental apparatus has been designed and

tested which measures the resonant frequencies of a tank in the "time

domain. " These frequencies correspond to the total mass of fluid within the tank. Experimental results are discussed for nitrogen and hydrogen in normal gravity both in the supercritical state and also in the two

phase (liquid-gas) region. Theoretical discussions for more general

cases are given.

Key Words: Gauging; hydrogen; nitrogen; radio frequency; total mass.

vi
INTRODUCTION

When a small antenna is placed in a closed metal cavity, the

electromagnetic field pattern which the antenna generates inside the

cavity depends on the excitation frequency and the shape of the cavity.

At certain frequencies, f (called resonant frequencies), the field n patterns are standing waves (called resonant modes). These modes are

easily detected at the antenna since the impedance match of the antenna

to the cavity is more efficient at the resonant frequencies.

The presence of a dielectric fluid within the cavity will change

the resonant frequencies. The resonant frequencies will decrease with

an increasing amount of fluid because the presence of the fluid slows

down the propagation of the electromagnetic wave. This presents the

possibility of gauging the amount of fluid by measuring the resonant

frequencies

,

If the density of fluid is uniform throughout the cavity,

f =% (1)

where f^^ is the resonant frequency of the empty cavity for the nth

mode and e is the dielectric constant of the fluid. For many nonpolar

fluids of interest (including hydrogen and oxygen), the dielectric constant depends only on the density of the fluid. In this case there is a unique relationship between each resonant frequency and the total mass

within the cavity; and only one mode is necessary to determine the total

mas s.
If the density of the fluid is not uniform throughout the cavity

(either because of a two phase liquid-gas interface or a single phase

fluid with temperature gradients) the resonant frequencies depend on

the amount of fluid mass in the cavity and also somewhat on where the

dense portions of the fluid are located within the cavity (fluid geometry

effects). Since the geometry of each standing wave is different (the

modes are linearly independent functions of the space variables), it is

possible to partially compensate for the fluid geometry effects by comparing two or more modes. This process of comparison is called Mass

Quantity Gauging by RF Mode Analysis.

The purpose of this present work is to develop an experimental

system which will

(1) Provide a breadboard total mass gauge for measuring

uniform density fluids; developing the accuracy and data reduction of

the time domain technique for measuring a single resonant frequency,

and

(2) Determine the feasibility of RF Mode Analysis for nonuniform

fluids; to measure the total mass with sufficient accuracy for dynamic

Normal-g and Zero-g fluid geometries.

This report will emphasize item (1) above under the heading

Uniform Density Fluids and also give some preliminary discussion and

results of item (2) above under the heading Nonuniform Density Fluids.

2
EXPERIMENTAL SYSTEM

The experimental system consists of (1) an electronic signal

conditioner and data acquisition system which measures the resonant

frequencies in the time domain and (2) an 18 inch diameter spherical

cryogenic storage tank (experimental vessel) designed for testing nitrogen

and hydrogen, in both the two-phase and supercritical states; it is ex- pected that after further testing that the tank will also be suitable for

oxygen. The total fluid mass is determined by weighing with a calibrated

load cell; there are a number of thermocouples and resistance thermometers attached to the sphere to measure temperature gradients in the

fluid. There are several antenna locations for measuring the effect of

antenna orientation with respect to a non-homogeneous fluid. The details

of the cryogenic system are outlined in Appendix A.

The resonant frequencies may be detected by sweeping the antenna

with an RF sweep generator; where the generator frequency ranges between f and f and the resonant frequency (or frequencies) of interest A B

lies between f and f . When the generator frequency coincides with the A B

resonant frequency there is a decrease in the signal reflected from the

antenna which shows up as a spike in the detector output; this output may

be displayed on an oscilloscope (see figure 1).

If the sweep generator frequency output is linear (or at least

repeatable) in time then the resonant frequencies may be measured by

measuring the time interval between the output of a reference cavity

(tuned to a frequency f^) and the output of the experimental vessel (see

figure 1). For example, if the sweep rate r is linear, the resonant

frequency of the fundamental mode, f^, is given by

3
FREQUENCY

fB

U

TIME

RF SWEEP GENERATOR OUTPUT

AMPL TUDE

t t3 TIME

DETECTOR OUTPUT

Figure 1. Input and output of R F cavity.

4
Since and r are fixed, is determined by the time interval (t^ t^);

this then is a measurement of the resonant frequency in the "time domain".

The time intervals can be measured using a digital clock and a counter. The clock may be triggered in a start and stop process by

signals coming from the reference cavity and experimental vessel, respectively. This is shown schematically in Figure 2. The details of the

signal conditioner and data acquisition system are contained in Appendix B.

If the detectors are to start and stop the clock in a precise manner

then the pulse must be very sharp and narrow so that the signal in real

time comes precisely at the time the RF generator is at the resonant

frequency. The narrowness of the pulse is related to the Q of the cavity

which is defined by

f

Q . (3)

n of

n

where 6f is the width of the spike at the half power points. For

n example, if Q is 10, 000 then there will be about a 0. 01 percent un-

^ n

certainty in the measurement of f . n The Q for several of the resonant modes have been measured

n

in detail for a 19 inch diameter copper sphere, an 18 inch diameter

stainless steel sphere and a 5 foot diameter stainless steel sphere. The

measured Q values range between 6, 200 and 91, 000. The detailed

results are contained in Appendix C.

The conclusion of Appendix C is that the Q's are high enough to

accurately measure the resonant frequencies by the time domain technique

even for the large vessel; and it is reasonable to use small vessels for

scaling experiments on ultimate large tank configurations.

5
Swept Frequency

Signal Source

Directional

Coupler

Reference

Cavity

Detector

Annplif ier

Start

Directional

Coupler

Clock

Oscillator

Detector

^

Stop H. P. Filter

Amplifier

Counter

gure 2. Conversion of resonant frequencies to the "time domain.
THE RF ANTENNA

One of the attractive features of the RF technique is the

simplicity of the internal tank hardware, which is simply a small loop

of wire. Figure 3 shows two of the antenna configurations which have

been used in the experimental vessel; these are connected to high pressure coaxial feedthroughs.

The straight wire antenna is the "TM probe". It generates

only the TM modes. The straight wire is simply an extension of the

center conductor of the coaxial feedthrough.

In the loop antenna, the center conductor is bent into a U- shape

about 3/4 inch by 3/4 inch and the end is grounded to the outer conductor. This antenna will generate both TE and TM modes.

The coupling of the antenna to the cavity (and hence the amplitude

of the response) is changed only slightly by changes in the size and shape

of the antenna. There appears to be a wide variety of antenna size and

shapes which are acceptable for this gauging technique.

7
Figure 3. RF Antennas.

8
UNIFORM DENSITY FLUID

For a non-polar dielectric fluid of density, o. constant throughout the cavity), the resonant frequency of the nth mode f is given by^'''^

£ = (4)

n ye{p)

where f is the empty cavity frequency. Semi- emperically e(p) can be

.

, (2, 3, 4) given implicity by '

'

where A, B, and C are constants determined experimentally for each

fluid. The f are determined experimentally and serve to calibrate

on

the system, f depends on the size and shape of the cavity; if the cavity

changes size because of thermal contractions or pressure expansions f must be adjusted accordingly; f may also change if objects are placed

in the cavity. To a good approximation, for most situations of interest B

and C may be neglected in equation (5); if V is the total volume of the

cavity, the mass M in this case may be given by

Z _ ^

V / e(p)-l \ V / on" n \ (6) ^

A

/ e(p)-l \ V / ^on" ^n \

on n

corrections to this formula for non zero B and C may be applied if

necessary. It is seen that possible inaccuracies in the total mass come

from four sources:

1. The uncertainties in A, B, and C. This is not a serious

problem if the properties data taken by careful capacitance measurements

1, 2, 3, 4 See references on page 32.

9
are complete over the ranges of interest. This must be determined

for each application involving a specific fluid.

2. The volume of the container. This is usually inferred by

weighing a fluid with a known density.

3. The accuracy of f which can be measured accurately for an

on

empty cavity; however, if the cavity changes size or shape as a function

of fluid density the empty cavity value Is no longer valid in equation (6)

and corrections must be made to equation 6 to account for this fact.

4. The uncertainties in f^. For normal RF frequency ranges and

high Q cavities this measurement can be made very accurately. The inaccuracies in converting this frequency into a useful digital or analog signal

are presently about 0. 2 percent full scale; this can be improved if necessary.

It should be noted that the factors 1, 2, and 3 above may be bypassed by direct calibration of f vs M using a gravimetric weigh

n

system. However, this is not practical to do for every system. The

purpose of this work, then, is to examine the validity of equation (6) (and

possibly corrections to equation (6)) for general use in a system where

the four accuracy factors may be adequately evaluated. This will be

described for the cases of nitrogen and hydrogen. It is anticipated that

oxygen will also be evaluated in the near future. The general approach

will be to directly calibrate f vs M using a gravimetric weigh system

n

for measuring M and a calibrated reference cavity for measuring f^

and comparing these results with equation (6) using the "time domain"

method of measuring f . The amount of any possible density variations

n

will be inferred from an array of thermocouples attached to the cavity.

Data Reduction and Readout

Although equation (6) is fairly complicated in form it is surprisingly linear over the density range of interest. A least squares linear fit

10
to equation (6) for the density range from zero to the normal boiling

point shows a maximum deviation of 2. 45 percent of the total range for

oxygen, 2. 2 percent for nitrogen and 1. 3 percent for hydrogen. Thus

the frequency vs mass relation can be expressed to a good approximation by

f = f (a + Po) n on

(7)

where a and p are constants depending only on the fluid. This simple

linear readout may be sufficient for many applications. Quadratic fits

of the form

f - f (a + 3p + YD^) (8) n on

may be found which give less than 1 percent error for all fluids (see

Appendix E).

Experimental Data for Nitrogen and Hydrogen

Preliminary experiments were started with liquid nitrogen to

check out the experimental apparatus and the theory of uniform density

fluids. The pressure was raised to 500-700 psi and the experimental

vessel was agitated to achieve nearly uniform density. The frequencies

were measured directly using the reference cavity; the line of the reference

cavity was shifted to coincide with the resonant line on the face of the oscilloscope. Figures 4 and 5 show the data taken in this manner for the TMqh

and TMogi modes respectively. It is seen that the data are nearly linear

(slightly concave downward) and corresponds very well to the theoretical

form given by equation (6). We believe that the occasional point which

deviates from the theoretical curve is due to non-homogeneous density

"stratification. " We will give some preliminary results on stratification

for the case of hydrogen.

11
12
o

cvj

< s o Q

oro

or

LlI Q.

ID

CO

_ z CNJ LlI

H- <

O

o

o

00

o

o

o

lO

o

o

rO

O

CO

<

O

Io

H

0)

-(-> Ml

E!

• i-i CO

cv

>^

-(->

Xi

o

o

a

•i-i

CO

in

00 CO

in O

in

in o

O

in

1^

ZHIAI 'A0N3n03dd

o

in

ZS

•I-I

1

-3
The hydrogen data were taken with the data acquisition system

described in Appendix B. The resonant frequencies for the first five

fundamental modes were converted to milliseconds in the time domain

(as indicated in equation 2) to the nearest 0, 01 msec. These times

were digitized by an internal clock and stored on magnetic tape. The

tape was used to punch data cards which contain a run code, the time

conversion of the resonant frequencies, and the scatter in the time domain

data. The scatter in the time domain data (except for an occasional noise

spike) was ± 0. 01 msec corresponding to the first significant digit. The

data cards also contain the pressure, temperature and mass data corresponding to the run identification number. Computer plots of the data

for the TMq^li and TMq^i modes are shown in figures 6 and 7 respectively.

A least squares fit of these data to equation (6) gives a 3a (99. 9% confidence level) for the TMqii mode of 1. 2 percent, the maximum deviation

of the data points taken is 1. 10 percent. Further details of the data

analysis, and also the accuracy statements are contained in Appendices

D and E.

14
I

3D

At.' 2. "::e ; HlGri PPESS'-PE

0

0

0

0

:3.-

0

:2'

0

QE)

4- »•.•$ MASS. lbs

UfiirORM DEnSITY - HYDROGEN

Figure 6. Mass gauging of uniform density LHg using the TMqii mode

15
40

38--

3o"

£

Y

-

54..

32-'

1

' 1 1

AfiT 2, mc:e 2 h:&h press'.pe :a'*

%

If

s e

•

f

e

f

• •

c.

••

8

MASS, lbs

UfllFOPM DEriSITY - HYDROGErj

Figure 7. Mass gauging of uniform density LHg using the TMogj mode.

16
EFFECTS OF STRATIFICATION

In the data described above, every effort was applied to achieve

uniform density; however, a few preliminary data have been obtained

concerning the effects of stratification on a single resonant frequency.

The following cases (in which the top and bottom temperatures were not

equal) gives a rough idea of the effect of stratification on the TMq^x mode

for supercritical hydrogen at 350 psi:

Case II

J. \J LJ If^-L-LlL/v^X CL l< X ^ 47, 5 K

R(TflTiTTl f f^TTl rif^ ^1" n T — 2 7. 5 K

3

1 lb /ft

3

4 lb /ft

3

3 02 lb /ft Measured weight (load cell) - 5. 35 lbs

jjiierreQ weignt \xrom iig. — D. -) J IDS

% Full scale error 4%

Top temperature = 37. 5 K

Bottom temperature = 33 K

Top density ~

, 3

2. 25 lbs /ft

Bottom density ~ 3. 45 lbs /ft

Average density ~ 2. 00 lbs/ft^

Measured weight (load cell) = 3. 55 lbs

Inferred weight (from fig. 6) = 3. 45 lbs

% Full scale error a* 1. 5%

Top temperature = 31. 5 K

Bottom temperature = 53 K

Top density ~ 3. 8 lbs /ft"

Bottom density ~ 0. 8 lbs/ft^

Average density ~ 1. 07 lbs/ft^

17
Measured weight = 1. 9 lbs

Inferred weight (from fig. 6) = 1. 9 lbs

% Full scale error ~ 0, 0%

In Case I, the stratification is cold fluid on the bottom, warm on

top with a good share of the fluid in between at the colder temperature;

definitely not a linear thermal gradient.

In Case II, there is a cold fluid on top, a colder fluid on the bottom

and warrner fluid in the middle as inferred from the average density.

In Case III, there is a cold fluid on top, a warm fluid on the

bottom with most of the fluid in the cavity at the warmer temperature.

From these cases, it is seen that certain small amounts of stratification may be tolerable; this is because the antenna senses the entire

cavity and tends to have an integrative effect over all the mass within

the cavity. For larger amounts of stratification, one antenna and one

mode may not be sufficient to achieve the desired accuracy, and more

information from other antennas or modes may be necessary to properly

use the RF technique as a gauge. This situation is discussed in the next

section on non-uniform density fluids.

«

18
NON-UNIFORM DENSITY FLUIDS - RF MODE ANALYSIS

Non-uniform (inhomogeneous) density (or dielectric constant) may

occur either in a single phase fluid under temperature gradients or in a two phase fluid with a liquid-gas interface. The spacial gradients in the

dielectric constant have a diffractive effect on the propagation of the

electro-magnetic wave and this changes the shape of the standing wave

patterns of each resonant mode. These changes will be different for each

mode because of the dissimilar field patterns of the modes. The resonant

frequency of the nth mode will be given by

f = f + Af

n un n

Where f is the resonant frequency expected for the uniform density case

un

assuming that the total mass in the cavity is spread uniformly over the

cavity; Af is the change in resonant frequency due to the non-uniform

geometry of the fluid and will be different for each mode even to the extent

of being positive or negative depending on the mode.

It is natural to ask: what are the extreme limits of Af as the fluid

n

ranges over all possible configurations? This is a difficult and possibly

impractical question to answer. The reason is that for a given mass of

fluid, there are two fluid geometries which will give the maximum and

minimum values for Af ; but these geometries appear to be very complin

Gated and it is unlikely that they will occur in practice. It is more feasible to talk about the practical limits of Af which are determined by

n

experimentation and calculation of Af for likely fluid geometries.

n

Theoretical Approach

Theoretical work which was initiated in early stages of Phase I was directed along two lines. The first was to develop approximation

techniques for calculating how fluid geometry affects the resonant

19
frequencies; the second was to investigate the mode geometries to get a qualitative picture of how fluid location may affect the resonant frequencies.

Earlier work had calculated in closed form the resonant frequencies expected for a two phase fluid with a concentric spherical phase

boundary ("zero-g" geometry). This work is contained in Appendix F

for reference purposes. Further work on two phase geometry effects

must be handled by approximation or numerical techniques. Appendix G

is a survey of relevant approximation techniques and how they may be

applied to the cavity problem. Several of the examples are worked out

for the case of the normal two phase fill geometry. One of the conclusions

of this work is that the resonant frequencies are most affected when the

dense portion of the fluid moves in and out of the high field region. These

high field regions are different for different modes. The field profiles

for a few of the lowest order modes are plotted in figures 8 through 13.

It is seen that the modes partition the cavity into several distinct high

field regions; from this, it is expected qualitatively that an average of

the resonant frequencies of the lowest order modes may give a mass

value that is relatively independent of the location of the dense fluid.

The numerical derivation of these graphs as well as numerical solutions

of some of the approximation techniques are contained in Appendix H.

Experimental Approach

The experimental system described earlier is designed to record

on magnetic tape the resonant frequencies of the first five modes for

three different antenna locations; the data for each antenna is collected

every 0. 1 sec. thus making it possible to study dynamic effects where

the fluid is in motion. It is anticipated that this system will give useful

information in "zero-g" simulation experiments. Some preliminary data

20
R r MASS GAUGING-PLOT OF l£l , TM^n MOD€

Figure 8. Field magnitude contours for the TMon mode.

21
Figure 9. Field magnitude contours for the TMqsi mode.

22
R r MASS GAUGING-PLOT OF l£l , T£,„ M00€

Figure 10. Field magnitude contours for the TEon mode.

23
Figure 11. Field magnitude contours for the TMqsi mode.

24
R r MASS GAUGING'PLOT OF E , TE^s! MODE

Figure 12. Field magnitude contours for the TEqs: mode.

25
Figure 13, Field magnitude contours for the TMo4i mode.

26
has been taken for normal gravity LN^ two phase fill. Since the liquid

surface breaks the spherical symmetry, it is found that some of the

resonant lines split into two or more closely spaced lines; some of the

modes do not split. For example, the TMoii mode stays as a single line

and the frequency shifts as a function of total mass (see figure 14) giving

some idea of the magnitude of Af^. Figure 15 shows the response of the

TMosi r^ode which splits into three lines during normal fill. In this case,

a straight forward average of the three modes is very close to uniform

density curve for this mode as shown in figure 16,

These two cases indicate two alternate methods of gauging the

situation of normal fill together with uniform density:

(1) In the case of TMqii mode, the readout can be designed for

uniform density and then a correction factor can be applied for the normal fill condition,

(2) In the case of the TMqsi mode, the readout can be designed

for uniform density, with electronic averaging of the three split resonant

lines in the TMoj^i time frame.

It is reasonable to expect that both of these techniques could be developed to give readout accuracies on the order of 1 percent; however, the

averaging technique may be more useful If It can be generalized to tilt

geometries and "low gravity" geometries.

27
8-

8-

8-

« 8

O

CVJ

T—I—I—I—I—I—I—I—I—I

\—

-|—

o2

CVJo

E

<

o

Ji

C ^

I - £• - o) O

Si 4>

LlI CL

£ - E

o P o

c 5 c 3 2 3

I O <3 I

/

/

/

/

/

/

/

/

/

/

/

/

V

/

/

J L J I L

O

to CVJ O

CVJ

CVJ

8

00

>

E — _J

-J

UJ

o

to

CVJ If)

Figure 14. Comparison between supercritical and normal fill for

nitrogen, TMqu mode.

28
LJO

O_ Q

>

o

D

E ^ o

•••V

• • •

• • •

• • •

o

o

00

• • •

• • •

o

in

o

• • •

- • • •

o

ro

O

CJ

•• • •••

in

00

o

o

00

in o

in

1^

in

CVJ o

o

ZHlAi *A0N3n03dd lNVN0S3d

in

55

29
aOUO

<va

0)>o

1—1uou

to

c;a>

a>

a; o o ut ^ s :^ U H

30
CONCLUSIONS

We have demonstrated using hydrogen and nitrogen that the

RF Mode Analysis Technique is conceptually sound for uniform density

fluids and that there are encouraging results for non-uniform density

fluids. The results for uniform density fluids should apply to any size or

shape of tank as long as the factors listed on pages 8 and 9 can be adequately evaluated. Further work should be done on non-uniform density

fluids for the spherical tank and also on tank shapes which are not

spherical.

31
REFERENCES

1. See Appendix F.

2. Johnston, Andrews and Cole, J. Chem. Phys. 33 , 1310 (I960).

Orantt and Cole, J. Chem. Phys. 46, 697 (1967) (Nitrogen).

3. Younglove, B. A. , Journal of Research (NBS), Vol. 76A No. 1 Jan-Feb 1972. (Oxygen).

4. Stewart, John W. , J. Chem. Phys. 40_, 3297 (1964) (Hydrogen).

32
APPENDIX A

CRYOGENIC SYSTEMS

The basic system includes a pressure vessel, a heater, a piping

system and an insulation system. These components were common to

all the test configurations. The normal gravity calibrations utilized a load cell to measure the mass accurately. The design of the flight pallet

for the zero gravity tests included an additional storage dewar. The

major components of these systems are described.

The Pressure Vessel

The vessel is a sphere constructed of 304 stainless steel. It was formed by welding together two spun hemispherical heads. The inside

diameter is 17 - 1 3/ 1 6 inches. It was designed and constructed per

the A.S.M.E. Code for Unfired Pressure Vessels. The minimum wall

thickness is 0.21 inches and the maximum allowable working pressure

is 750 psig as defined in this code. The pressure vessel has been hydrostatically tested to l-l/Z times its working pressure at room temperature.

Access ports for instrumentation and piping are provided. All ports,

except the port on the bottom, are 3/4" female pipe threads. The bottom

port is l-l/Z" female pipe thread to accommodate the heater. All ports

are reinforced with welding spuds.

The Heaters

Two heater configurations have been used. The first was designed

for use with liquid nitrogen and the second for liquid hydrogen. Both were

designed to pressurize the fluids to their critical pressures in forty

minutes. Both were constructed with temperature independent resistance

wire. Pres surization lines as a function of heater power and current are

presented in figure Al.

A-1
M>1 'dBMOd d31V3H

(AQII = A®) ,.01 X V 'iNBydnO d31V3H

A-2
The nitrogen heater system consisted of two separate one kilowatt heaters that could be separately controlled. Each heater was

wound with ten feet of Z2 gage wire. Each had a room temperature

resistance of 1 Z ohms. The wire was wound on two separate coils.

The coils were then inserted in the bottom access port with the wire

in direct contact with the nitrogen.

The hydrogen heater was constructed differently because of

safety considerations with the flammable fluid. This heater consisted

of one 300 watt coil of wire. It was wound with four feet of 28 gage

wire with a room temperature resistance of 20 ohms. The wire was

covered with ceramic beads and inserted into a coil of l/4" O.D,

copper tubing. The tubing was then soldered on the outside of a brass

spindle which was screwed into the bottom port. The heater element

does not contact the hydrogen but heats the pressure vessel from the

outside. The copper tubing is purged with helium gas and is electrically grounded.

The Normal Gravity Calibration Tests

For these tests, the pressure vessel was hung in a cubical frame

constructed of aluminum angle. The piping system and the insulation

system was also supported from the frame. The frame, and the systems

it supported, was suspended from the load cell. All interface piping

and instrument lines were flexible to minimize their effect on the load

cell. Figure A-2 is a photograph of this system without the insulation.

The piping was designed for use with liquid hydrogen and was

used with liquid nitrogen as well. The system is equipped with a 750 psig

A-3
A-4
relief valve. The system has been pressure tested to 700 psig with

liquid nitrogen as the test fluid and it has been tested to 300 psig with

liquid hydrogen.

The insulation system was made up of six inches of fiberglass

batting. To prevent cryopumping the fiberglass was covered and

sealed with aluminized mylar. The insulation was then purged with

helium gas and a slight overpressure was maintained during the tests.

The load cell used was a strain gage type rated at 300 pounds

maximum load; it has a load sensitivity of 0, 1 mv/lb with a 10 volt

excitation voltage. Using a high precision digital voltmeter on the

output, the resolution is approximately 0. 01 lbs.

The pressure vessel was instrumented with several temperature

measuring transducers to determine the temperature and density gradients

during the supercritical tests. Nine copper- constantan thermocouples

were attached to the outside wall of the pressure vessel at six inch

vertical increments. Platinum resistance thermometers were inserted

into the fluid at the top and bottom to determine actual fluid temperature.

The Zero Gravity Tests

The design of this system utilizes the same pressure ve s s el, heater

and support frame as the normal gravity system. The piping and insulation

systems have been changed.

The piping system flow diagram is shown in figure A3. All the

valves are manually operated. The design utilizes a common overboard

vent and liquid dump, which will be compatible with the aircraft. The

system can be refilled and pressurized in flight if necessary.

A-5
0 - 1000 psig

0-10 psig

OVERBOARD

VENT a DUMP

PRESSURIZATION f GAS

MODEL

PRESSURIZATION

SUPPLY VESSEL

PRESSURIZATION

SUPPLY VESSEL

VENT

MODEL LOW

PRESS. RELIEF

MODEL VENT

MODEL FILL

AND DRAIN

SUPPLY VESSEL

a MODEL DUMP

SUPPLY VESSEL

FILL a DRAIN

ON GROUND

LN2 FILL CONTROL PANEL

Figure A3, Cryogenic flow system for zero-g simulation tests.

A-6
The complete system will be packaged in an aluminum paneled

container six feet long by three feet high and three feet wide. The

total weight will be approximately 700 pounds.

The system is designed to withstand an acceleration load of

16 g's in any direction. This will be accomplished by using a polyurethane foam insulation system. After the entire system has been assembled in the paneled container, the foam will be poured, filling all remaining space. The foam, then, will not only insulate but will support

the system. The outside paneling shall be 3/8 inch aluminum plate.

With this insulation system, the total heat leak will be approximately 15 BTU per hour. The rate of pressure rise, as a result of

this heat leak, will be 1 . 5 psi per hour.

A-7
APPENDIX B

MASS TANK GAGING SIGNAL CONDITIONER AND DATA

ACQUISITION SYSTEM

Purpose

To condition the resonant frequencies of the mass tank, to

measure these frequencies as a function of time and store the data in a magnetic tape unit.

Operation

A radio frequency generator is swept through a designated frequency spectrum of a starting frequency f , to an ending frequency f . A B

This generator is connected to a cryogenic mass tank. The mass

tank has resonant frequencies (modes) which are enhanced each time the

tank is energized with the generator sweep of selected frequencies.

The design objective of the data acquisition system and signal

conditioner is to measure these enhanced resonant modes with respect

to a known frequency and record them.

In the system this measurement is accomplished by using a time

measurement technique. The generator sweep is used as a time base for

all of these measurements.

A reference cavity tuned to a frequency, f , which is lower than

o the lowest cryogenic cavity mode, is the reference starting time for the

time measurement of the mass tank system.

When the generator energizes the reference cavity tuned to f , a pulse is generated by the cavity. This pulse is used to start a counting

sequence using a 100 kHz clock as the basic counting device.

The first resonant mode, fj^, from the mass tank strobes the

counter into a buffer register which loads the data to a shift register.

B-1
The shift register then sends the data to a magnetic tape unit. All these

data transfers occur before the next resonant mode of the mass tank is

generated.

The counter continues counting with its data being strobed into

the shift register every time a resonant mode is generated.

Upon completion of the generator sweep, the counter is reset,

another antenna in a different location on the mass tank is multiplexed,

and the generator initiates a new sweep. Beginning of the sweep initiates

a new set of data.

Using this technique, a measurement of mass in the cryogenic tank

is obtained as a function of time with respect to the reference cavity tuned

to f . o

B-2
Data Acquisition Schematic I

Functional Description

This schematic shows the timing sequences necessary to start

the counting sequence and data transfer functions. All timing events are

referred to the initiating pulse, t , generated by the reference cavity.

The pulse t , opens the gate to the counter and enables the real

time clock to send 100 KHz pulses to the six stage counter.

The next timing event occurs when the cryogenic mode pulse t

is generated. This pulse generates a strobe command to the storage

buffer, a parallel load command to the shift register, a serial-parallel

mode control pulse to the shift register, and a shift command gate enable

pulse to the serial shift clock.

The mode counter generates a clock disable signal at the end of

the last mode to be measured and resets itself.

B-3
n

to

fV\ODe

rw:)De

ounw

B-4

!• 106280"1
^\ooL- COUNTER

ClOCk:

(:(\20 "g

<:6IV\I\\«.N>C)

lO£MTvPiC(VTiOK

Di\nx

C40

S ? :^

a;^l^^;cu.tli

44

B-5

IB- 1062881
l

Data Acquisition Schematic II

Functional Description

This schematic shows the timing sequence necessary for control

of the mode counter, antenna multiplexer, sequence counter and magnetic

tape recording unit. All timing events are referred to the mass tank

pulse t generated by the modes f . n n

A recording sequence is initiated by depressing the manual reset

and manual start push buttons. The manual reset button resets all counters

and controls flip flops to initial status for recording and controlling. The

start push button starts the magnetic tape unit.

The t pulses are counted by the mode counter until a preselected

n

output is generated in the counter. This output multiplexes the antenna

or sends a counting pulse to the sequence counter if the antenna multiplexer is disabled.

The antenna multiplexer generates an output for every complete

cycle of antenna multiplexing. This output sends a counting pulse to the

sequence counter when multiplexing is enabled.

The sequence counter has a preselected number of counts it can

accept before it overflows and stops the magnetic tape recorder.

B-6
\MPUT

OUTPUT

CuTPUX

4^

B-7
•

^\ll>LrlPL£y

f

^—

PS /

AuTO _B£SEL

z

<C^ ^ 5 ^ e

SECUENCE

c t < VJ

M6Dt

tNO OFT^Pe

-Kg

3 -a

TO ^M^£w^JA

L» A2

-H§H§)

— seajeoEB)

^

ijtOiECieG "cm"

^ »j

!^

B-8 H- 1062B91
Counter -Smrr i?i£Si^ter C^^o

1

1 ,C i!/ '^J 'jy

20X22X24-X2

CuC^K _ I' \'i U

Q ft-9 (VCO

- 5^ S S X .,C .-^

Tfe Tip 12 15

-3 r r,

1 1 ,4

I

.An 1^.

rr

^

****

0,00 0000 .0000

l^r^ A7 Aio p-J All SrJ

g C 3J ^ I I A g C D J I ,4 g C D J I L A 3 C 0 J

^

I I Tit

I Ir-' li.I

1 I.? ( :

I TT' 1'=? I I I

r~ ^ * <<<<

^1 M f^io 4, 4, ;^ j^iV

All

L t < I <

A 14 1

'

AT Ik's A ,

I Ais'

Alt,

; AH

t

iff'

5 ? 4

2 ? ^ S-y

ilUv,!.

ooOO ? ? « s- —

A'' „Z „7- ATS 6000 ? 7 4 s-

; A2P

r

^1

AJ'i A21 ^JE 0000

. i ? " V All

L1

Alt AZI AT> >12g 0000

2 ? c V A12.

B-9

IB- 106290 I
COUNTER-SHIFT REGISTER CARD "A

From To Function Condition

A3 A56- C3 load shift register input

A5 Dl 5- C4 8|j.s serial shift register input

A2 B2 + 5 volt distribution

A4 Tens thumbwheel switch date

i nput

Ad Tens thumbwheel switch date

input

A8 Tens thumbwheel switc h date- 4 i nput

A I 0 Tens thumbwheel switch J i. o date- 8 input

AI 2 Ones thumbwheel switch date- 1 input

A14 Ones thumbwheel switch date-2 input

Al 6 Ones thumbwheel switch date

input

Al 8 Ones thumbwheel switch date- 8 i nput

A20 Tens thumbwheel switch run no. 1 input

A22 Tens thumbwheel switch run no. 2 input

A24 Tens thumbwheel switch run no. 4 input

AO/ A26 Tens thumbwheel switch run no. 8 input

A28 Ones thumbwheel switch run no. 1 input

A30 Ones thumbwheel switch run no . 2 input

Ai 2 Ones thumbwheel switch run no. 4 input

A34 Ones thumbwheel switch run no. 8 input

A36 C26 antenna identification 1 input

A38 C28 antenna identification 2 input

A40 A42- A60 antenna identification 4 ground

A42 A40-A60 antenna identification 8 ground

A44 D44 shift register output 1 A46 D46 shift register output 2 A48 D40 shift register output 4

A50 D42 shift register output 8 A52 C52 2(_is shift register mode control i nput

A54 D48 real time counter reset input

A56 A3-C3-G44 load buffer register input

A58 D58-G56 gated oscillator 100 K Hz clock input

A60 A40-A42-B60 common distribution

B- 10
Al 7490 decade counter

A2 7490 decade counter

A3 ' 7490 decade counter

A4 7490 decade counter

A5 7490 decade counter

A.6 7490 decade counter

A7^ 7495 4-bit right/left shift register

A8 7495 4-bit right/left shift register

A9 7495 4-bit right/left shift r e gi s te r AlO 7495 4-bit right/left shift r e gi s te r Al 1 7495 4-bit right/left shift register

A12 7495 4-bit right/left shift register

Al 3 7495 4-bit right/left shift r e gi s t e r A14 7495 4-bit right/left shift register

Al 5 7495 4-bit right/left shift register

A16 7495 4-bit right/left shift register

A17 7495 4-bit right/left shift register

A18 7495 4-bit right/left shift register

A19 7495 4-bit right/left shift register

A20 7495 4-bit right/left shift register

A21 7495 4- bit right/left shift register

A22 7495 4-bit right/left shift registe r A23 7495 4-bit right/left shift register

A24 7495 4- bit right/left shift register

A25 832 dual 4 input NAND g; A26 832 dual 4 input NAND g,ate

A27 832 dual 4 input NAND g,ate

A28 7404 HEX inverter

A29 7404 HEX inverter

A30 7404 HEX inverter

B-11
B-12
DATA CONVERTER-CONTROLLER CARD "B" AND CARD "D"

From To Function Conditi

B2 A.2.-<Z2 + 5 volt distribution

B30 F47 magnetic tape 1 output

B32 F45 " magnetic tape 2 output

B34 F43 magnetic tape 4 output

B36 F41 ' magnetic tape 8 output

B38 F39 magnetic tape A output

B40 F37 magnetic tape B output

B42 D56 100 K Hz clock free running output

B44 D32 shift register data 1 input

B46 D34 " shift register data 2 input

B48 D36 • shift register data 4 input

B50 D38 shift register data 8 input

B54 B56-B60 code conversion enable input

B60 A60-C60 common distribution

D28 C40 dummy data enable input

D30 C42 real data enable input

D32 B44 shift register data 1 output

D34 B46 shift register data 2 output

D3 6 B48 shift register data 4 output

D38 B50 shift register data 8 output

D40 A48 shift register data 1 input

D42 A50 shift register data 2 input

D44 A44 shift register data 4 input

D46 A46 shift register data 8 input

D52 G24 reference cavity input

D56 B42 100 K Hz clock free running input

D58 A58-G56 100 K Hz clock gated output

B-13
Bl 9311 one of sixteen decoder

B2 7430 8-input positive NAND gate

B3 7430 8-input positive NAND gate

B4 7430 8-input positive NAND gate

B5 7420 dual 4- input positive NAND gate

B6 7400 quad 2-input positive NAND gate

B7 2 capacitor cambion

B8 832 dual buffer

D17 2 K resistors

D21 7404 HEX inverter

D22 7403 quad 2-input positive NAND gate

D23 7403 quad 2-input positive NAND gate

D29 848 clocked flip-flop

D30 7400 quad 2-input positive NAND gate

B-14
B-15
CONTROL REGISTER-TIMING GENERATORS CARD "C"

From To Function Condition

C3 A3-G44 1 |js parallel load output

C5 D5 serial shift gated clock output

C49 D49-lO's tws pole sequence counter stop sweep input

C51 D54 15 ms T.D. start sweep input

C53 D53--N. O . SW manual start pushbutton input

C55 N. C . SW manual start pushbutton input

C57 G30 start sweep output

C4 D15--A5 8\As shift clock output

C6 G16 cryogenic cavity input

C36 D50 N. C. SW manual reset pushbutton input

C40 D28 dummy data enable output

C42 D30 real data enable output

C52 A52 2|j s shift register mode control output

C58 F55 recorder clock input

B-16
CI 7495 4-bit right/left shift register

C2 7495 4-bit right/left shift register

C3 7495 4-bit right/left shift register

C4 848 clocked flip-flop

C5 7404 HEX inverter

C6 848 clocked flip-flop

C7 RC network for one shot (C13)

C8 RC network for one shot (C14)

CI 1 7400 quad 2- input positive NAND gate

C12 848 clocked flip-flop

C13 9601 one shot multivibrator

C14 9601 one shot multivibrator

C16 7486 quad exclusive OR gate

C19 832 dual buffer

C26 848 clocked flip-flop

C29 7400 quad 2- input positive NAND gate

C30 848 clocked flip-flop

B-17
hV \)t -t S:ou£NC£ CouNireis r Antenna IWuLripuEKEt?

X

6 CP ^

7 8

lO

MOCfe COUNTER. DtCCDER

AAA%bBA« A? A C A C -€>^

-OB

-oc

q 10

I n I? CI& H<M

12."—-

jo- -

I?—

12- -3

13 tZ

r r-6-

C20

& 7 0 2 3 •7 10 2 3 C26

12 ^ 5

5" cn

3

1 1 '1 6 II

13

^

?ii is

0 0 0 0

A -g C P

sz- 5-

SetUENCe COUMTEa oeccc>t(i

—O CZ9-2

B-18 |» 106287 I
MODE COUNTER-SEQUENCE COUNTER-ANTENNA MULTIPLEXER

CARD "C"

From To Function Condition

C33 10 s tws pos 3 sequence counter 30 counts output

C37 10 s tws pos 5 sequence coixnter 50 counts output

C39 10 s tws / pos D sequence counter 60 counts output

C45 10 s tws pos 9 sequence counter 90 counts output

C6 D47 cryogenic cavity gated input

C8 N. O . sw manual reset pushbutton input

CIO D10-G44 reset and antenna multiplex input

CI 2 N. C . s w antenna multiplex enable input

C14 N. O . sw antenna multiplex enable input

CI 6 G46 mode 1 reset output

C18 G48 mode 2 reset output

C20 G50 mode 3 reset output

C22 G52 mode 4 reset output

C24 mode 5 reset output

C26 A36 antenna identification 1 output

C28 A38 antenna identification 2 output

C30 D20- E20- G40 antenna driver 1 output

C32 D16--E16- G36 antenna driver 2 output

C34 D12--E12- G3 2 antenna driver 4 output

C36 D50--N.C. sw manual reset pushbutton input

C44 D48 manual antenna advance input

B-19
C9 7490 decade counter

CIO 7400 quad 2- input NAND gate

cu 7400 quad 2- input NAND gate

C15 7400 quad 2- input NAND gate

C16 7486 quad exclusive OR gate

C17 848 clocked flip flop

C18 848 clocked flip flop

C20 cambion for wire connections

C22 7490 decade counter

C23 7400 quad 2- input positive NAND

C24 7400 quad 2- input positive NAND

C27 846 quad 2- input NAND gate

C28 7490 decade counter

B-20
D24

B-21
GENERAL PURPOSE TIMING EVENTS CARD "D"

From To Function Condition

D3

D5 C5 5|j,s serial shift gated clock input

Dl 5 A5-C4 8|js serial shift gated clock output

D19 F16 end of file 5|js output

D4I N.O. sw end of file pushbutton input

D43 N. C . sw end of file pushbutton output

D47 C6 cryogenic cavity gated 5ms output

D6 G16 cryogenic cavity gated input

DIO CIO reset, antenna multiplex and ISmsTD input

D48 A54-G54 real time counter reset output

D50 C36-N.C. sw manual reset pushbutton input

D54 C51 15ms time delay output

B-22
Dl RC network for one shot (10^f, 10 K)

D2 RC network for one shot (.001^f, 10 K)

D3 RC network for one shot (. 005^f, 10 K)

D5 RC network for one shot (.OOlMf, 10 K)

D6 RC network for one shot (10^f, 10 K)

D7 9601 one shot multivibrator

D8 9601 one shot multivibrator

D9 9601 one shot multivibrator

DIO 848 clocked flip-flop

Dll 9601 one shot multivibrator

D12 9601 one shot multivibrator

D18 7400 quad 2-input positive NAND gate

D24 7486 quad 2-input exclusive OR gate

B-23
(lihNtRKL FUR'PCJ^t hUNCTlOMS

B-24
GENERAL PURPOSE FUNCTIONS CARD "D"

From To Function Condition

Dl 1 Fl 1 tape running input

Did led 1 panel tape running indicator output

Dl 7 Fl 5 end of tape input

u6b XT' '2 C write command output

Di I Fd 5 - C 5o write shift clock input

D49 C49 lO's tws sequence counter switch pole i nput

D 5F53 file protect missing input

D 5 i C5i-N. O. sw manual start pushbutton input

D5 5 led 2 panel file protect missing indicator output

D57 N.O. sw recorder on clamp input

D8 C44 manual advance antenna output

D12 E12-G32-C34 antenna "0" input

D14 led ant "0" antenna "0" indicator output

D16 E16-G36-C32 antenna "1" input

D18 led ant "1" antenna "1" indicator output

D20 E20-G40-C30 antenna "2" input

D22 led ant "2" antenna "2" indicator output

D24 N.O. sw manual antenna advance input

D26 N. C. sw manual antenna advance Input

B-25
D18 7400 quad 2-input positive NAND gate

D19 7403 quad 2- input positive NAND gate

D20 848 clocked flip-flop

D24 7486 quad 2-input exclusive OR gate

D25 510n resistors

D26 7403 quad 2-input positive NAND gate

D27 7400 quad 2-input positive NAND gate

D28 848 clocked flip-flop

B-26
28 V o.c

TJX

28 V

^wwv2QK

2^

<5

?8 V DC

2<

AAAAA/-

20

2N?Q04

B-27
ANTENNA SWITCH DRIVERS CARD "E"

From To Function Conditi

E4 G28 + 28 volts input

ElO G34 antenna "0" output

E12 C34-D12- G32 antenna "0" input

E14 G38 antenna "1" output

El6 C32-D16- G36 antenna " 1

input

E18 G42 antenna "2" output

E20 C30-D20- G40 antenna "2" input

B-28
B-29
CARD "C

From To Function Condition

G6 connector 1 cryogenic cavity signal input

G8 amplifier "1" test point

GIO amplifier "2" test point

G14 amplifier "3" test point

G16 D6 cryogenic signal gated output

G18 cryogenic comparator signal test point

G20 connector 2 reference cavity signal input

G22 reference cavity amplifier "1" test point

G24 D52 reference cavity comparator output

G54 DlO-ClO mode counter reset signal input

G56 D58-A58 100 K Hz clock input

G58 reference cavity amplifier "2" test point

B-30
Gl LM301 operational amplifier

GZ IMD pot and IMn resistor

G4 LM301 operational amplifier

G5 IMQ pot and IMD resistor

G7 L.M301 operational amplifier

G8 capacitor cambion

G9 848 clocked flip-flop

GIO LM301 operational amplifier

Gil capacitor cambion

G13 L,M301 operational amplifier

G14 LM301 operational amplifier

G15 7400 quad 2- input NAND gate

G20 848 clocked flip-flop

B-31
B-32
From To Function Condition

F4 4-' ?8O vnH V (J ± L output

r;^ ^J ?c J_y DlX ?-t-< -1—* FlJ- ?-t-i C34 ^ni'^^T^'n^ ''0'' input

G34 ElO antenna ''0'' drive i1 Tl11Til 1 "f" L G36 D16-E16- C32 antenna "1

input

G38 E14 antenna "1" drive input

G40 D20-E20- C30 antenna "2" input

G42 E18 antenna "2" drive input

G44 C3-A3 parallel load input

G46 C16 mode 1 counter input

G48 CIS mode 2 counter input

G50 C20 mode 3 counter input

G52 C22 mode 4 counter input

G54 D48-A54 15 ms time delay reset input

B-33
G3 cambion for wire connections

G6 cable cambion

G12 7400 quad 2- input positive NAND gate

G17 cambion for wire connections

G18 7404 HEX inverter

B-34
B-35
6 STAGE DECADE COUNTER CARD

2/28/73 o n n p o T* "Pine:

1i + D V

2 + 5V

3 externa 1 reset

4 signal input

5 strobe input

6 segment test

56 push button reset (N. O

57

58 push button reset (N. C

59 common

60 common

B-36
INTERFACE CARD

2/28 /73 Connector Pins Function

1 + 5V

2 + 5V

3 100 KHz clock

4 common

5 : 6

7

8

9 strobe #1 ANT "0"

10 common

11 strobe #2 ANT "1"

12 common ,13 strobe #3 ANT "2"

14 common

15 external reset (all resets)

16 common

17

18

19 pa rallel load

20 common

59 common

60 common

B-37
Table B-I. Interface Connector Pin List

READ

ONLY w

F ACT.

PIN

1

SIG.

GRD.

SIGNAL (see notes 1 throu^ 3)

ACTIVE WIRE

COLOR

7-TRACK

r -J

1 05

BASIC O

w

H

2 READ

WRIT

RAW f

U

U

e.

TRANSPORT CONTROL SIGNALS

3 Run Normal Speed input level 80 X X X X X X X

5 Run Hiph Speed input level 30 X X X X X X X

7 Forward Select input level 70 X X X X X X X

9 Reverse Select input level 60 A A v AY AY AY

1

Tape Running output level 50 X X X X X X X

13 Load Point output level 8 X X X X X X X

15 End of Tape output level 6 X X X X X X X

17 Broken Tape output level 5 X X X X X X X

19 Unload Command input pulse 10 X X X X X X X

21 Rewind to Load Point input pulse 90 X X X X X X X

23 Off Line input pulse 9 X X X X X X X

25 Rewind in Process output level 20 X X X X X X X

27 On Line

Reser\'ed - Do Not Use

Reserved - Do Not Use

output level 40

0

X X X X X X X

X

X

29 Chassis Ground (see note 4) 1 X X X X X X X

31 Write Select input level 7 X X X X X

33 Wi-ite Parity Select (odd)

Reserved

input level 92

0 X

X X

X X X

X

35 Write Command input pulse 50 X X X X X

18 Data Channel 0 (Write) input level 97 X X X X

20 Data Channel 1 (Write) input level 90 X X X X

37 Data Channel 2-B (Write) input level 93 X X X X X

39 Data Channel 3-A (Write) input level 96 X X X X X

41 Data Channel 4-8 (Write) input level 95 X X X X X

43 Data Channel 5-4 (Write) input level 94 X X X X X

45 Data Channel 6-2 (Write) input level 9 X X X X X

47 Data Channel 7-1 (Write) input level 98 X X X X X

49 Data Channel P-C (Write) input level 30 X X ^ X

51 Write Status output level 90 X X X X X

53 File Protect Ring Missing output level 98 X X X X X

55 Write Clock output pulse 70 X X X X X

57 Write Clock Gate output level 60 X X X X X

22 Write Echo Error output pulse 10 X X X X

12 2X Write Clock input pulse 3 X X X

14 Write LHC input pulse 20 X X X X X

16 EOF Command input pulse 80 X X X X X

B-38
APPENDIX C

Q MEASUREMENT SUMMARY

The objective of the Q measurement task was to determine the

spherical vessel Q as a function of diameter and wall material. The

results of this measurement task are summarized in tabular form.

Spherical Vessels

Mode 19" Cu 18" SS 60" SS

TM^^ ^ ;

, 91, 000 1 1, 600 6, 200

TM . 85,200 10,900 13,600

J. TE^^ . 81,100 19,700 28,400

TM^^ 41,200, 10,900 17,500

Considering the lowest Q (TM^ ^ in the 60" SS), f^ is 172 MHz when

empty and would be 155 MHz when filled with boiling point LH2. The

total frequency shift from empty to full would be 1 7 MHz. The bandwidth at the 3dB points is 0.0 28 MHz, or approximately 0. 15% of the

empty-full bandwidth. For LOX the 3dB bandwidth is approximately

0. 10% of the empty-full bandwidth. Thus it appears that the Q will not

seriously degrade the RF resonance technique in stainless steel spherical vessels up to 5 feet in diameter.

Let us assume that the change in Q pattern measured with the 18"

and 60" stainless steel vessels continues to even larger diameters.

Then for a 15 foot diameter stainless steel vessel, the 3dB bandwidth

is 0. 3% of the empty full LH^ bandwidth. The LOX 3dB bandwidth is

approximately 0, 2% of the empty-full bandwidth.

C-1
gO.8

id

-J

UJ

0.6-

0.4-

UJ

S2

g 0.2

5

0

-30 -20

545,984 kHz

19" Dia. Copper Sphere

TM..

J L J L 10 0 10

FREQUENCY, kHz

20 30

Figure CI. Frequency response of the copper spherical vessel for

the TMq 1 1 mode.

C-2
1 r» I.U

Z

UJ 0.8

u.

0.6

z

o

hi

-J

U.

04

UJ

q:

UJ

0.2

s

0

1 1 1 1 1 1 1 1 1 1

• ••

•

•

•

• •

, 766,320 kHz

• •

• •

• • •

•

•

^* 19" COPPER

_ • •

• •

• •

1 1 1 1 1

>

1 1 1 1 1 -30 -20 10 0 10

FREQUENCY, kHz

20

Figure C2. Frequency response of the copper spherical vessel for the

TMo2 1 mode.

C-3
1.0

yo8

fc

UJ

8o.6h

Id

-I

UJ

0.4-

UJ

CD

^

0.2

1 1 1 1 1

1 1 1 1 1

••••

••

•• 892,230 kHz

• ••

•

•

• •

•

•

• •

•

• w

• • ••

l9"Dia. Copper Sphere

1 1 1 1 1 1 1 1 1 1 -30 -20 10 0 10

FREQUENCY, kHz

20 30

Figure C3. Frequency response of the copper spherical vessel for the

TE 1 1 mode.

C-4
W 0.8-

0.6-

jjJO.4

UJ

^ 02

"I r "I I I r

987,966 kHz

••^ • 1 9" Dia . Copptr Sphert

"^31

-30 -20 10 0 10

FREQUENCY, kHz

20 30

Figure C4. Frequency response of the copper spherical vessel for the

TMo3 1 mode.

C-5
99

6 kHz steps

172,454 kHz

-10 0 10 20

FREQUENCY, kHz

30 40

Figure C5. Frequency response of the 60 inch diameter stainless steel

sphere - TMqh mode.

C-6
4 kHz steps

284,276 kHz

-20 -10 0 10

FREQUENCY, kHz

20 30

Figure C6. Frequency response of the 60 inch diameter stainless steel

sphere - TEqh mode.

C-7
T

• 314,432 kHz

4 kHz steps

-10 0 10 20

FREQUENCY,kHz

30 40

Figure C7. Frequency response of the 60 inch diameter stainless steel

sphere - TMqsi mode.

C-8
APPENDIX D

DATA REDUCTION FROM THE MAGNETIC TAPE UNIT

The magnetic tape unit stores the information coming from the

data acquisition system described in Appendix B. Each data run lasts

about ten seconds; during this time the system records about 25 independent measurements for each mode or about 375 data points in all

(counting 3 antennas and 5 modes for each antenna). The data reduction

processes the raw tape, finds the average and standard deviation of the

2 5 data points for each mode and punches these numbers on to standard

data cards along with other identification codes. Data from other

sources are also punched on these cards i. e. , mass, pressure and

temperatures. The cards are then ready for plotting routines, and

other types of data analysis.

I. Program DMPMODES

Since the tape recorder in the laboratory is a continuous write

recorder and not a sequential recorder, the letter D is hard-wired to

the output whenever there is no data being recorded. Most of the tape

is, in fact, dummy data. DMPMODES accomplishes many functions.

It takes the data from the magnetic tape made in the laboratory, deletes

all D's and other non-essential characters, sorts the 12 character data

by antenna, lists this data and writes this data on a second magnetic tape.

Essentially, the program works as follows. A tape record is

read into array M. A pointer, KNT, moves down the array. If the

word in M(KNT) is all D's, KNT is incremented. When a word that is

not all D's comes along, the character: (or 15 octal) is searched for.

When a: is found, it and the characters surrounding it are placed into

the first two locations of the array IW. Thinking of these two 8 byte

words as one 16 byte word, it should look like d d r r a:t,t_t t t t.

D-1
DDDD, where all subscripted letters are integers, and D's are

dummy data. *^j^2 data code selected from the recorder front

panel. ^1^2 code, also selected from the front panel.

a is the antenna code, t -t, are the data. If this 16 byte "word" is

1 16

not correct, that is, if an alpha character appears where an integer is

supposed to be, the data point is rejected and the program starts searching for the next data point. If the data point appears to be all right, it is

decoded, and the data is printed under the appropriate antenna column.

Also, the 16 byte "word" is written on a second magnetic tape, thus

saving later programs from having to re-sort the mass of characters

on the input tape.

II. Program AVMODES

Basically, this program takes data from the second magnetic

tape, sorts it by antenna and mode, finds the average time and standard

deviation for each mode for each antenna, lists these averages and

standard deviations , and punches on standard data cards the averages

and standard deviations, along with the date, date code, run code,

antenna number, mode number, and the number of points used to calculate the average and standard deviation. The input magentic tape is

essentially a sequence of the magnetic tapes written by DMPMODES.

The tape has all the data from all the days of running the experiment.

Each day is separated from the others by an end of file marker, and

each run is separated from other runs by an inter-record gap. The

first record of each file contains the date of the run.

The program works as follows. The first record is read into

the first 10 words of array M. The date of the run is extracted from

these ten words and stored into the variable IDATE. Each of the other

records in the file are read into the array M, one at a time. Every

D-2
two words of the array M are the same as in the 16 byte "word" described

above in the discussion of DMPMODES. This 16 byte word is decoded,

or broken up, into the date code, IDC; the run code, IRC; the antenna

code, lA; the colon character, ICOL; and the time data, ITIME. The

time is checked to see which mode, if any, it belongs to. The point is

then used to calculate the average and standard deviation of that

particular mode of that particular antenna. Note that for each record

on the tape, or run, there is a possible total of 15 of these averages

and standard deviations to be printed and punched (corresponding to three

antennas with five modes each). The punch card output is used for

graph routines and other data analysis programs.

III. The following pages give a listing of these programs which are

being used on the CDC 3800 computer.

D-3
PROGRAM DMPMODES

D I MENS ION MSG(3)»IW(8)»IHZ(3)»IHZSO(3)»T(3)»TSQ(3)»K(3)»IOa(5 000)

COMMON M{32765)

1 FORMAT ( 15 ) 2 FORMAT ( IRECORD I5»* LONGER THAN 3275A* SOME LOST.*)

3 FORMAT (8R1)

4 FORMAT ( 2 12 » 1 1 »Rl 16 .4X

)

5 FORMAT (*1* . 57X TAPE RECORD NUMBER ,I3»3A8/*8*

151X»*DATE C0DE9 .I2»5X.*RUN C0DE9 *»I2/

255X .ANTENNA IDENTIFICATION CODE* / 3 3X » » 32 X * 1 » 32 X ^2* /

)

6 FORMAT ( 31X. 16

)

7 FORMAT ( 64X . 16

)

8 FORMAT

(

97X . 16

)

9 FORMAT (O AVG* »2 7X» F9.2 » 2 ( 23X F9.2 ) / S I GMA* 2 3X » E 1 2 . 6 » 2 ( 2 IX E12 .6

)

1/1X»I4,* SCANS DELETED IN THIS RECORD. '

NRUNO=909

CALL 09nVFP

CALL lOHCHECK

NREC=0

READ I.NSKIP

IF( E0F,60 ) 10.103

103 DO 105 I=1.NSKIP

BUFFERIN( 1.0) (M.M)

10^ IF(UNIT,1 ) 104,105

105 CONTINUE

10 BUFFERIN( 1.0) (M»M(32765 ) ) IF ( NREC.EO.O.OR.KCA.EO. 1 )G0 TO 115

BUFFEROUT ( 2 1 ) ( lOA , lOA ( KOA-l ) ) 115 K0A=1

MSG( 1 ) =MSG( 2 ) =MSG( 3 ) =8H

<NT=0

IHZ ( 1 )

IHZ( 2 ) =IHZ( 3 ) =0

ICCl )=K(2)=K(3)=0

IHZSQ( 1 )= IHZSQ( 2 )=IHZSQ( 3 ) =0

NREC=NREC+1

IDR0P=-1

LINE=75

KCODE=0

117 IF(UNrT, 2 ) 117.11

11 IFfUNlT.l )11. 15. 99.13

13 MSG ( 1 ) =8H CONTA IN MSG(2)=8HS PARITY

MSG(3)=8H errors

15 L=LENGTHF(1)

IF(L.GT.4)G0 TO 195

NREC=NREC-1

60 TO 10

195 IF(L. LT. 32764 )G0 TO 198

PRINT 2»NREC

198 IDROP= IDROP+1

205 KNT=KNT+1

D-4
IF(KC0DE.E0.1 )<NT=<mT-H

KCODE=0

20 IF(KNT.GT.L-2 )23»24

23 DO 242 I=l»3

FI=K( I ) T ( I )= IHZ( I

)

T ( I )=T { I ) /FI

IF (K ( I ) .GT. 1 )G0 TO 241

TSO(I)=0.

GO TO 24 2

241 TSQ( I )=IHZSQ( I

)

TSQ ( I) =SQRT( ABS( (FI*TSa(I)-T(I)**2)/FI/(FI-l.)))

242 CONTINUE

PRINT 9» ( T ( I ) 1 = 1 ^3 ) ( TSQ( I ) 1=1 O ) I DROP

GO TO 10

24 IF IM(KNT) .EQ.8HDDDDDDDD)G0 TO 27

DECODE(8»3»M(KNT) ) IW

25 DO 26 1=1*8

IF( IW( I ),NE.15B)G0 TO 26

NSH= 1-6

IF(NSH)29»37»30

26 CONTINUE

27 KNT=<NT+1

GO TO 20

29 KNT=KNT-1

NSH=NSH+8

KC0DE=1

30 MSK1=2*»( 6*NSH)

MMN1=MSK 1-1

MSK2=2»* { ( 8-NSH ) ^6

)

MMN2=MSK2-1

DO 35 I=l»2

IW(I)=M(KNT+I-1) .ANn.MMN2

IW( I ) = IW( I )*MSK1

IC=M(KNT+I ) /MSK2

3 5 IW( I ) = IW( I ) .OR. ( IC.AND.MMNl

)

GO TO 40

37 IW{1)=M(KNT)

I W { 2 ) =M

(

KNT + 1 ) 40 ITEST= IW( 2 ) •AND.7700000000B

IF( ITEST.EQ.8H000D0000 )G0 TO 198

ITEST= IW( 1 ) . AND.7700000000000000B

IF{ ITEST.EQ.SHDOOOOOOO ) IW(1 ) = IW(1 ). AND. 77 77777777777 7B

DECODE ( 16 »4 » I W ) ID»NRUN »NANT IC I DATA

IF( lOHERRCO^ > 198*43

D-5
43 IF{ IC.NE.15B)G0 To 198

IOA(KOA)= IW( 1

lOA ( KOA+1 ) = IW{ 2

K0A=K0A+2

LINE=L INE+1

IF ( NRUN.EO.NRUNO.AND.L INE.LE.67 )G0 TO

PRINT 5»NREC»(MSG(I)»I=I»3)» ID»NRUN

LINE=1

53 NANT=NANT+1

K(NANT)=K(NANT)+1

IHZ(NANT)=IHZ(NANT)+IDATA

IHZSQ(NANT) = IHZSQ(NANT ) + I DA T A * I 0 AT

NRUNO=NRUN

GO TO ( 55 » 56 57 ) NANT

PRINT 6»IDATA

GO TO 205

56 PRINT 7»IDATA

GO TO 205

57 PRINT 8,IDATA

GO TO 205

99 STOP

END

D-6
PROGRAM AVMODES

D I MENS ION V!(5000).TIME(5»3)»TIMESQ(5»3)»KNT(5»3) »MODE (5*2)

DATA( ( (MODE( I fJ) = ! »5 ) »J=1»2 ) = 1 1 00 2 900 420 0 . 5 1 00 » 6 5 00 » 165 5 , 3900 » 15100. 6060»7800)

CALL IOHCHECk

NF = 0

READ 1 »NFSK IP

1 FORMAT ( 15 ) IF( EOF, 60 ) 10,102

102 DO 105 I=1,NFSKIP

105 CALL SKIPFILE( 1

)

10 BUFFERINC 1»1) (M,M( 10)

NREC=0

NF=NF+1

11 IF(UNIT,1 ) 11,15,99,13

13 PRINT 2»NREC,NF

2 FORMAT(»OPARITY ERROR IN RECORD *,I3»* OF FILE *,I3)

15 IDATE = ^^( 1 ) .AND.7777777777B

IDATE= IDATE*1000000B

M(2 )=M(2) /lOOOOOOOOOOB

IDATE= IDATE.0R.(M(2).AND.777777B)

PRINT 6

6 FORMATdHl/)

16 DO 17 J=l,3

DO 17 1=1,5

KNT(I,J)=0

TIME(I»J)=0.

17 TIMESQ( I ,J)=0.

BUFFER IN ( l,l)(M,M(5n00))

NREC=NREC+1

21 IF(UNTT,1 )21.25,10,23

23 PRINT 2,NREC,NF

25 L=LENGTHF(1)

DO 40 J=1»L»2

3 F0RMAT(2I2,I1»R1»I6,4X)

DECODE ( 16 .3 'NK J )

) IDC , IRC » I A, I COL , I T IME

IF( I0HERR(0^ )40,26

26 IF( IC0L.NE.15B)G0 TO 40

IF( IA.LT.0.0R.IA.GT.2)G0 TO 40

IAC=IA+1

DO 30 1=1,5

IF( ITIME.GT.MODE( I ,2 ) )G0 TO 30

IF( ITIME.GE.M0DE( I ,1 ) )G0 TO 32

GO TO 40

30 CONTINUE

GO To 40

32 KNT( I,IAC)=KNT( I,IAC)+1

PN=ITIME

PN=PN/100.

TIME( I ,IAC)=TIME( I , lAO+PN

TIMESQ ( I , lAC )=T IMESQ (I lAC ) +PN**2

40 CONTINUE

50 DO 55 J=l,3

IA=J-1

D-7
DO 55 I=l»5

PN=KNT ( I , J

)

IF(KNT ( I ,J) .EQ.0)GO To 55

!F (<NT ( I , J) ,GT, 1 ) r-0 TO 52

TIMESO( I,J)=0.

GO To 53

52 T IMESQ (I»J)=SQRT(ABS(PN*TIMESQ(I.J)-TIME(I»J***2)/(PN*(PN-1.)))

53 TIME( I J)=TIME( I,J)/PN

PUNCH 4»IDATE*IDC»IRC»IA»I»TIME(I»J)»TIMESG(I»J)»KNT(I,J)

4 FORMAT (A8»2I3»2I2»F7,3,E12.4»I5)

PRINT 5»IDATE»IDC.IRC»IA»I»TIME(I»J)»TIMESQ{I»J1»KNT(I,J)

5 FORMAT (1X»A8»2I3»2I2»F7.3»E12.4»I5)

55 CONTINUE

GO TO 16

99 STOP

END

D-8
APPENDIX E

UNIFORM DENSITY HYDROGEN

DATA ANALYSIS AND ACCURACY STATEMENTS

The data analysis for uniform density hydrogen is based on 41

observations at tank pressures above 400 psi; the temperatures at the

top of the tank were slightly lower than the bottom temperatures (the

heater was at the bottom) indicating almost uniform temperature with a condition for convective mixing due to gravity; experimentally, these

conditions were necessary to achieve a near uniform density within the

experimental vessel.

The data for the TM^^^ mode, as plotted in Figure 6 of the main

text, is expected to follow equation (6), i. e. ,

(E-1)

Using equation (2), the expression for f^ in the time domain

=fo + r(tj - y (E-21

it follows after a little algebra

/i

1 0 r \/ , 2AM

or equivalently

E-1
V

where

f - f At^ = 01 0

(E-5)

(At^ is just the time interval at M = 0). The gauging function is then

between At and M; these quantities being related to three independent

fo 1 A

parameters At ,

, — through equation (E-4). These three param0 r V

eters may be determined by fitting the data to (E-4) or, alternately,

may be determined by separate physical measurement. For example,

the sweep rate, r, can be determined from ^t^, f^ and f^^ by

f - f

r=^V^ (E.6)

The measured values of these quantities were f^^ = 581. 9 MHz,

f^ =411 MHz and At^ = 16. 20 milliseconds giving r = 10. 54 MHz /msec.

V can be calculated from the inside radius of the tank R = 8. 906

4 3

inches (22. 62 cm) giving V = 4. 85 x 10 cm . A can be obtained

from other experimental data; ref . 4 of the main text gives A = 1 . 006

3

cm /gm. Table E-I gives the comparison between these values and the

values obtained by a nonlinear least squares fit of the data to equation

(E-4).*

C. Daniel, F. S. Wood. Fitting Equations to Data, Wiley Interscience,

N. Y. (1971) p. 320.

E-2
TABLE E-I

At,

01 A

V

-3 -

Measured value 16.20msec 55.15msec 9.42x10 lb

Fitted value 16. 19 52. 88 9. 44 X 10

-3

95% confidence

limit (lower) 16. 20 39. 20 6.81x10 -3

95% confidence

limit (upper) 16. 20 66. 50 12.21 X 10

-3

Although the confidence limits for the fitted values are relatively

large, the actual fitted values are quite consistent with the independently

determined values.

The semi-empirical gauge equation may then be given by

At^ =16.19 + 52.

1 - 9. 44 X 10 M

1 + 2 X 9. 44 X

10~^ M

1 (E-7)

where At^ is measured in milliseconds and M in pounds; it remains

to determine the accuracy of the data with respect to this equation.

Table E-II gives the observed value of At^ and the value of At^

calculated from (E-7) for each of the observed mass values. The

residuals are plotted in Figure E-1 as the Percent Full Scale Residual

E-3
TABLE E-II

Mass (lbs) At, (OBS) At, (CALC) Residual % Full Scale Residual

0 . 00 16. 200 16. 186 0. 014 0. 28%

0. 20 16. 018 16. 037 -0. 019 -0. 38

0. 44 15. 839 15. 859 -0. 020 -0. 41

0. 52 1 5

820 15. 799 0. 021 0. 43

0. 73 1 5

631 15. 644 -0. 013 -0. 26

0. 84 15. 550 1 5

564 -0. 014 -0. 29

0. 94 15. 460 15. 483 -0. 023 -0. 47

1 . 00 15. 500 15. 446 0. 054 1

10

1.15 15. 340 15. 337 0. 003 0. 06

1.15 15. 329 15. 337 -0. 008 -0. 16

1 . 16 15. 326 15. 330 -0. 004 -0. 08

1. 42 1 5

163 15. 141 0. 022 0. 45

1 . 42 15. 1 60 15. 141 0. 019 0. 38

1 . 54 15. 034 15. 054 -0. 020 -0. 41

1 . 54 15. 029 15. 054 -0. 025 -0. 51

1.56 15. 029 15. 039 -0. 010 -0. 20

1 . 57 1 5019 15. 032 -0. 013 -0. 26

1 . 58 1 5

030 15. 025 0. 005 0. 10

1.81 14. 842 14. 859 -0. 017 -0. 35

2. 06 14. 674 14. 687 -0. 013 -0. 26

1 85 14. 841 14. 83

0. 010 0. 20

2.11 14. 670 14. 645 0 025 0 5 1 2. 43 14. 415 14. 417 -0

002 -0. 04

2. 45 14. 413 14. 403 0. 010 0. 20

2. 82 14. 1 80 14. 142 0. 038 0. 78

3. 33 13. 820 13. 786 0. 034 0. 69

3. 70 13. 557 13. 530 0. 027 0. 55

4. 23 13. 141 13. 167 -0. 026 -0. 53

4. 25 13. 136 13. 154 -0. 018 -0. 38

4. 71 12. 815 12. 856 -0. 041 -0. 83

5.19 12. 501 12. 520 -0. 019 -0. 39

5. 20 12. 518 12. 514 0. 004 0. 08

5. 62 12. 221 12. 235 -0. 014 -0. 28

5 . 63 1 2 222 12. 228 -0. 006 -0. 12

5. 68 12 212 12. 195 0. 017 0. 35

6 43 L J. . 67Q 1 1 703 -0 024 -049

6. 44 1 1 686 11 697 -0. Oil -0. 22

6. 51 1 1 680 11 651 0. 029 0. 59

6. 77 11 480 11. 483 -0. 003 -0. 06

6. 79 1 1 480 11 470 0. 010 0. 20

7. 08 1 1 304 11 283 0. 021 0. 43

E-4
lO O ift nvnaisaa 3ivos nnnd iN30d3d
vs. At (calc); this graph shows a fairly random scatter of the data

between -0. 83% and 1.10% full scale.

The cumulative distribution of the residuals is plotted in Figure

E-2 vs. the expected intervals in a normal distribution. The S-shaped

trend of the data indicates that the frequency distribution is slightly

flatter than the normal bell-shaped error function. This is probably

due to the fact that the At^ are really only measured to four significant

figures, the most significant digit corresponding to 0. 2%; this would

cause a broadening of the frequency distribution on the order of ± 0. 2%.

The slope of the straight line through the tails of the curve gives an

estimate of the standard deviation o = 0.4%. This gives a 3a deviation

(the 99. 9% confidence interval) of 1. 20%.

Operational Readout

As indicated in the main text, a quadratic fit will be easier to

work with operationally and should give a sufficiently accurate gauging

function. The quadratic equation

2

Atj = 16.20 + AM + BM (E-8)

was fitted to the data in Table II by solving (E-8) simultaneously for

M = 6.77, At = 11.48 and M = 2.43, At = 14.415. The result is

- 3 A = -0.7554 and B = 8.61 x 10 . The residuals for this fit range

between -0.63 and 0.93 percent full scale, a slightly better fit than

the theoretical curve. This analysis is tabulated in Table III.

Chernov, H. and Lieberman, G. S. , J. Amer. Statist. Assn. 49 , 778-85 (1954).

E-6

TABLE E-III

Mass (lbs) A t

^ (OBS) At^ (CALC) % Full Scale Residual

0.00 16.20 16.2 0.00%

0.2 16.018 16.049 ' 0.63

0.44 15.839 15.869 0.62

0.52 15.82 15.809 0.21

0.73 15.631 15.653 -0.45

0.84 15.55 15.571 -0.44

0.94 15.46 15.497 -0.77

1.00 15.50 15.453 0.93

1.15 15.340 15.342 -0.05

1.15 15.329 15.342 -0.30

1.16 15.326 15.335 -0.19

1.42 15.163 15.144 0.37

1.42 15.160 15.144 0.31

1.54 15.034 15.057 -0.47

1.54 15.029 15.057 -0.57

1.56 15.029 15.042 -0.28

1.57 15.019 15.035 -0.33

1.58 15.03 15.027 0.04

1.82 14.842 14.853 -0.24

2.06 14.674 14.680 -0.13

2.11 14.67 14.644 0.52

2.43 14.415 14.415 0.01

2.45 14.413 14.400 0.25

2.82 14. 180 14. 138 0. 85

3.33 13.820 13.780 0.82

3.7 13.557 13.522 0.70

4.23 13.141 13.158 -0.36

4.25 13.136 13.145 -0.19

5.19 12.501 12.511 -0.21

4.71 12.815 12.833 -0.37

5.2 12.518 12.504 0.27

5.62 12.221 12.226 -0.12

5.63 12.222 12.220 0.04

5.68 12.212 12.187 0.51

6.43 11.679 11.698 -0.40

6.44 11.688 11.692 -0.09

6.51 11.68 11.647 0.67

6.77 11.480 11.480 , -0.01

6.79 11.480 11.467 0.25

7.08 11.304 11.283 0.42

E-8
APPENDIX F

TOTAL MASS GAUGING IN A SPHERICAL RESONANT CAVITY

Introduction

When a closed metal container is excited by an RF antenna probe

inserted through a hole in the container, theoretically there are an infinite

nuraber of excitation frequencies for which the container is strongly

coupled to the antenna; this means that energy can flow more freely

between the antenna and the container at these resonant frequencies.

The resonant frequencies correspond to standing wave patterns in the

cavity which are called resonant modes. The wave pattern of the mode

which occurs at the lowest possible resonant frequency is called the

fundamental mode. This mode and the modes of the next few higher frequencies are called lower order modes.

When the cavity is uniformly filled with a fluid, the resonant frequency changes because the velocity of propagation of the resonant

standing wave, c = 1 // p,e, depends on the dielectric constant, e, and the

magnetic permeability, [i, of the fluid. For example, in a spherical

resonant cavity uniformly filled, the resonant frequencies, f^ , are given

by

np Zn b/ |j,e

where b is the radius of the sphere, and n and p are subscripts which label

the different modes (these will be explained in detail). The u are

^ ' np

eigenvalues of the modes and are obtained in the process of finding solutions

to Maxwells equations. The resonant frequencies can then be related to

total mass by using the Clausius -Mos sotti relation

Pp = (F-2)

F-1
where P is the polarizability of the fluid which is a slowly varying function of the fluid density, P.

If the cavity is uniformly filled with a liquid the eigenvalues, u np

are just numbers independent of the fluid within the cavity and there is

therefore a simple relationship between resonant frequency and total mass.

However, if the container is only partially filled with liquid, the rest of

the cavity being a vacuum or a gas, then the values of u will depend on

np ^

\i and e of the liquid, the [i and e of the gas, and the geometry which

the liquid takes within the cavity; the resonant frequency, then, is no

longer an unambiguous function of mass but depends on the liquid geometry

as well. This is because the standing wave patterns are distorted because

of the boundary conditions at the liquid-gas interface. However, the

resonant frequency of each partially filled mode does lie between the completely empty and completely full values

u u

< f < , (F-3) Znb/TTe np 2nb/lL e o o

and varies continuously between these values as the cavity is filled. This

suggests that the resonant frequency at least approximately indicates total

mass independent of geometry.

The purpose of this note is to investigate the geometry effects for

a spherical cavity with spherical symmetry of the liquid gas interface.

This geometry is similar to a "zero-g" formation with the liquid clinging

to the walls and a gas bubble in the middle of the cavity. The reason for

choosing this geometry is that it is one of the few examples of a partially

filled cavity for which the Maxwell Equations can be solved in closed form.

Even though this gometry is particularly simple, it does give a reasonable

F-2
indication of the uncertainty which may be involved when the geometry is

not known. Numerical examples are calculated for cases in which the

liquid is hydrogen or nitrogen.

From a practical point of view, the spherical cavity is an ideal

container geometry for this method of mass gauging. The reason for this

is that the spherical symmetry of the cavity wall creates a degeneracy

in the modes. That is, there are a number of standing wave patterns

which have the same resonant frequency. This results in the fact that

the distinct resonant frequencies of the lower order modes are widely

separated and minimizes the effect of mode crossing in a partially filled

cavity. Mode crossing occurs when, for a particular liquid geometry,

the resonant frequency of a higher mode falls below that of a lower mode.

For example, if the liquid is nitrogen, mode crossing between the first

two modes is impossible and for the next few higher modes is quite

unlikely; this is established from the table of eigenvalues. Table 1 on

page F-12 and the inequalities expressed in (F-3).

The relative independence of the lower order modes suggests that

they can each be monitored independently. Since each mode has its own

geometry in the standing wave pattern, it seems reasonable that the

modes themselves may be used to at least partially determine the fluid

geometry. (Mathematically the problem reduces to this: Given some of

the eigenvalues of a boundary value problem, how closely can the eigenfunctions be approximated. ) In fact, it will be shown that for the spherical

symmetry considered in this analysis, that for a liquid of unknown density,

both the location of the liquid-gas interface and the density (hence the

total mass) can be determined uniquely if and only if five modes are

monitored simultaneously. The reason for this is that each mode determines exactly one independent relation between the resonant frequency

F-3
of that mode and the five unknown parameters e, a, e , \i , and a, o o where r = a is the liquid-gas interface. For most applications it is sufficient to assume that e [i fsa uss 1, leaving only two unknowns, namely

o O

o / 7

s and a. In this case, two modes will uniquely determine the total mass.

Solutions for Maxwells Equations in Spherical Coordinates

When the cavity is resonating at an angular frequency, , the

time phase of the electromagnetic field is the same at all points within

the cavity. Hence, for a loss free cavity the electric and magnetic fields

can be written as the real parts of Ee*^^^ and He'^'^^, respectively, where

E and H are vectors which depend only on the spacial coordinates. The

source free Maxwell Equations can then be written

curl E = - i(ju[i,H curl H = iuDeE

div eE = 0

div |iH = 0. (F-4)

It should be emphasized at this point that only two assumptions have

been made, the cavity is loss free and it is source free; in practice

these are usually very good assumptions for calculating resonant frequencies. A third assumption which we will now make, may have to be

justified more carefully in any given situation: we assume that there

are two regions within the cavity, each of which have uniform density.

The technical advantage of this assumption is that derivatives of [x and

e are not involved; the equation ( F-4) can be solved in each region where

[X and e are constant and the boundary conditions are then modified to

include the liquid-gas interface. The boundary conditions can be written

•|eE*n, [j,H-n, Exn and Hxn continuous at each boundary point j- (F-5)

where n is the unit normal vector to the surface at that point. Since

F-4
div E = 0 and div H = 0, both E and H can be expressed in terms of

vector potentials G and F,

E = curl F

H = curl G (F-6)

where the Maxwell Equations impose consistency conditions between

G and F. Two independent solutions may be obtained by choosing a coordinate direction, say f , the unit vector in the radial direction and

finding fields which are perpendicular to f. If E is perpendicular to

f we say we have a TE (transverse electric) mode. This situation may

be assured if F is chosen to be

F = fr (F-7)

where f is a scalar function of the spatial coordinates. In this case we

have from (F-4) and (F-7)

E = curl ff

- H = - curl curl fr . (F-8)

If H is perpendicular to r we say we have a TM (transverse magnetic)

mode. This situation may be assured if G is chosen to be

G = gf (F-9)

where g is a scalar function of the spacial coordinates. In this case

we have from (F-4) and (F-9)

\ E = T~— curl curl sf

I cue ^

H = curl gf . (F- 10)

The general solution for E and H may be obtained by a superposition

of (F-8) and (F-10)

F-5
E = curl ff + -—- curl curl gf

H = curl gf - T~— curl curl ff . (F -11)

10) [i

To find equations which f and g satisfy, we consider the TE and TM modes

separately. For the TM raode (F-10) and (F-4) imply that

curl E = - ioi |_L curl gf

or

curl (E + iua ugf ) = 0 . ( F- 12)

This last relation is satisfied only if

E + i(juM,gf = grad cp (F-13)

for some scalar function cp. Substituting ( F-13) into the second of equation (F-4) we have

curl curl gf = uo^|i,e gf + ioDs gradcp. (F-14)

Using the vector equation

curl curl gf = V^gf - grad (V-gf) (F-15)

and

we find that g satisfies the following equations

(V^ + k^) gf = 0

V gf = - iuo e cp . (F- 1 6)

A similar argument for the TE mode shows that f satisfies the following

equation

(V^ + k^) ff = 0

Vff = iuDM,Vili. (F-17)

F-6
Equations (A- 16) and (A- 17) are equivalent to the scalar Helraholtz equations for g/r and f/ r with standard solutions given by'''

B (kr) L (e, cp) , (F-18) n n

where the L ^(6, cp) are spherical harmonics and the B (kr) satisfy the n n

differential equation

_ dr r " J n B (kr) = 0 . ( F-19)

The general solution of equation ( F-19) can be given as a linear combination of 1 (kr) and y (kr) which are the Spherical Bessel Functions of

n n order n of the first and second kind respectively.

B (kr) = C kri (kr) + D kry (kr) (F-20)

n n n n n

where C and D are constants. The general solutions for f and g may

n n be written as an infinite series

f = S (C krj (kr) + D' kry (kr)^L ^(9, cp) V nm n nm n y n

n, m

CD g = y (c krj (kr) + D kry (krj) L "^(6, cp). (F-21)

_ Z_, V nm n nm n y n

n, m

The constants C , D , C' , and D' may be evaluated by substituting nm nm nm nm

(F-21) into ( F- 1 1 ) and applying the boundary conditions (F-5), Equation

(F-21) can be viewed as an infinite superposition of modes.

See R. F. Harrington, Time Harmonic Electromagnetic Fields, McGraw

Hill (1961).

See M. Abrahamowitz and I, A, Stegun,, NBS Handbook of Mathematical

Functions, p. 437.

F-7
TM Modes Under Spherical Symmetry

The TM modes are obtained by setting f = 0. If the liquid has

spherical symmetry the boundary conditions may be satisfied by using

only one value each for n and m in equation (F-21) and thus the series

for g contains at most two non-evanishing terms:

g = Tc krj (kr) + D kry (kr)"^ L "^(e,cp). (F-22) V nm n nm n y n

Using equations ( F- 1 1 ) and (F-16) along with ( F-22), the components of

the electric and magnetic fields may be written as follows:

r icjueVdr y louer

-1

g

luj er Br S6

-1

E

cp i 0) er sin 9 Br B cp H = 0 r H

r sin 9 Bcp

H = - If. (F-25)

cp r 69

The boundary conditions are applied by letting the container walls

exist at r = b and the liquid-gas interface at r = a <b (if a = 0 the

container is full and if a = b the container is empty. ). The conditions

liH-n and Hxn continuous at r = a and r = b imply continuity of and

H and hence that g is continuous at r = a and r = b. This is comcp patible with the continuity of eE^n. The condition Exn continuous implies

1 B that E and E is continuous and hence that — —— g is continuous at

9 -43 e Br ^

r = a and r = b. In summary the boundary conditions are completely

specified by

F-8
g continuous at r = a and r = b (p-26) —Id— g continuous at r = a and r = b. (F-27)

e or

Since the L "^(9, cp) are independent both of radial position and fluid properties, the condition (F-26) is equivalent to

k aj (k a) = C kaj (ka) + D kay (ka) (F-28)

o n o nm n nm n and

g(b) = C kbi (kb) + D kby (kb) (F-29) nm n nm n

where the coefficient of y (k a) in equation ( F_22) is zero because e must no

be finite at r = 0. (Here, k = cju/e ll applies to the region in the gas and

o o o k = '^/e\}. applies to the region in the liquid. ) Likewise condition (F-27)

is equivalent to

- ^[k aj (k a)] = -—Fd kaj (ka) + D kay (ka) edaLono_i eSaL nm n nm n

1 S

o and

(F-30)

1 B C kbj (kb) + D kby (kb)1. (F-31)

e 3b L nm n nm -^n

Equations (F-28), (F-30), and (F-31) are three independent relations in

the eight variables, C , D , a, uu , e, ll, e , and ix . The inhomonm nm o o geneous equations (F-28) and (F-30) can be solved uniquely for C and

nm

D and these values are substituted in equation (f-31) which then benm

comes a homogeneous relation in six variables a, uo , e, ll, e , and [o, . o o We will denote this relation by

F ((ju, a, e, |i, e , [i. ) = 0 (F-32)

n o o

or sometimes more simply by F (o), etc, ) = 0, For a given set of

values for a, e, \x, e^, and jj,^ (which is determined by conditions in the

container), it can be shown that F^ plotted as a function of cjd is oscillatory

F-9
and hence there are an infinite number of solutions to equation (F-32).

The solution for the pth zero of equation (F-32) is called uu and the ^ np

field pattern obtained by substituting ou and the values for C and ^ np nm

D into equations (F-22) and (F-25) is called the TM mode where

nm mnp

n = 1, 2, 3 , ^ p - 1, 2, 3 , , m = 0, ± 1, ± 2, . . . . ± n.

(The range on m comes from the properties of the spherical harmonics, ) Since uj is independent of m, we see that there are a number of modes

np

corresponding to the same ud . This number is called the degeneracy

np

of o) . For example, the fundamental frequency w-^^ corresponds to

np

three modes, TMq^^, TM_^^^, and TM^^j^;^ and hence has degeneracy 3. Sometimes the first subscript is dropped and the three modes are collectively referred to as the TM^;;^ mode (which is an abuse of the term "mode").

We now discuss the conditions under which the resonant frequencies

o) can determine the total mass. The total mass M is a function of

np

three of the above variables, a, e , and e. If the resonant frequencies,

o

UD , of the modes are known, then we have the following relations in the

np

five variables a, e , u , e, and a o o

0 = F^(ujii, ) = Fi((jui2. etc. ) =

= F2(oQ2^ , etc. ) = ^2(01 etc, ) =

= F (w , etc. ) (F-33)

n np

where each of the F (oj , etc. ) is a relation determined by measuring

n np

the resonant frequency of a TM mode. Since there are five variables

mnp

F-10
it is clear that at least five different modes are necessary to completely

determine the total mass. From the properties of the Spherical Bessel

Functions it can be shown that each of the above relations is also independent; therefore, five modes are also sufficient to determine the total

mass. If further assumptions are made, fewer modes may be sufficient.

For example, for most liquids \x ^ ^ reduces the number of necessary modes to 3; if it is further assumed that e 1, then the number of o necessary modes is two; finally if in addition e is knowA then only one

resonant frequency is necessary to determine the total mass.

Alternately, it may be that the interface, r = a, is known and e (hence the density) is unknown; if (O,^ p. j« 1, then the density and

hence the total mass may be determined by a single resonant frequency.

As a limiting case of this situation, the case a = 0 indicates a completely

full cavity and the resonant frequencies are given by

u

i (JU = ^ (F-34)

np b/eiJwhere u is the p^^ zero of equation (F-32) considered as a function

np

of the quantity kb, (The quantities u^^ are also known as eigenvalues

of the TM "mode". ) The measured frequency f is given by f = np np np

as The calculated values for u in the case a = 0 are listed in Table

2rr ' np

1 in increasing order for the lowest ten modes. (Table 1 also includes

results of a similar analysis for the TE modes. ) The resonant frequencies

f also plotted in Table 1 are for the specific case of a 48 -cm diameter

np

empty container.

We see from Table 1 that the resonant frequencies of the lower

order modes are widely spaced. This is primarily due to the degeneracy

and makes it feasible to simultaneously monitor several of the lower

order modes.

F-U
Table F-I.

r -r^- T Frequency Modes Eigenvalues Degeneracy

(48 cm dia. Sphere]

"ll

= 2. 744 3 = 0. 543 GHz

"21 = 3. 870 5 0. 766

"n = 4. 493 3 0. 889

"31 = 4. 973 7 0. 984

^^21 "21 = 5. 763 c ^21

- 1. 140

"41 = 6. 062 9 hi " 1. 200

"12 = 6. 117 3 1. 210

"31 = 6. 998 7 1. 384

TM51 "51 = 7. 140 11 1. 413

T^22 "22 = 7. 443 5 ^22

= 1. 472

Examples Using Hydrog en and Nitrogen

Equation (F--32) was solved for the four lowest order modes usin;

the FORTRAN prog ram listed in Table 2. For g;iven value s of a, e, \i,

e , and

o

[i , the program finds the zeros of F (uL), n

etc.plotted as a function of kb where

kb = (ju/lTe b.

The p^^ zero is

u =0) /[Te b. np np

The computer plots the quantity Ci'U^ vs. p which is essentially resonant

frequency, f , vs. total mass M. Here,

np

eo o

Written by A. E. Hiester.

F-12
Table F-II.

PROGRAM PL0T3

DIMENSION IFILM{13) ,ITITLE( 13) .X(IOO) »Y(1U0) »AL(3) .RH0(3)

DATA ( IFILM=24HART HIESTER, X347A ) A(N»U»F»AL) = (1./AL*PJ(N»AL*F*U) *YPP ( N »F*U ) -AL**2*PPJ ( N AL*F*U ) *YP

(

IN ,F*U ) ) / ( PJ ( N ,F*U ) *YPP ( N »F*U ) -PPJ (N,F*U)*YP(N,F*U))

B{N,U.F,AL) = ( AL**2*PJ(N,F*U) *PPJ (N,AL*F*U )-l ./AL*PPJ(N,F*U)*PJ(N,A

1L*F*U ) ) / ( PJ( N ,F*U ) ^<-YPP (N»F-U)-PPJ(N,F*U)*YP(N,F*U) ) FUN(N,U»F»AL)=A(N,U»F»AL)*PJ(N»U)+B(N»U,F»AL)*YP(N,U)

1 FORMAT ( 3F10.0

)

2 F0RMAT(»U*»I2»5H ) 3 FORMAT ( IHI » lOX .2A8//9X » IIHALPHA * UNP , 1 OX

t

*RH0BAR* / /

)

4 FORMAT(9X»F9.5»1UX.F10,7)

5 FORMAT(*0U NOT FOuND IN lOu I

ERAT I ONS* / / 1 X , 6 E2 2 . 8

)

6 F0RMAT(5HS1* U»I2,1H ) P=l.

ITITLE( 1

) =8H RESONAN

ITITLE(2)=8HT FREQUE

ITITLE(3)=8HNCY VS M

ITITLE(4) =8HASS - H2

ITITLE(7)=8HRS9H0BAR

ITITLEl 10)=8H SI

ITITLE( 11 )=8HA$9LPHA

ITITLE(5)=ITITLE(6)=ITITLE(8)=ITITLE(9)=ITITLE(13)=8H

READ 1»(AL( I ) ,1 = 1,3)

READ 1 , ( RHO( I ) , 1 = 1 ,3 ) CALL GRAPH! 1 ,1 ,3, IFILM,0,6)

DO 60 N=l,4

ID = P ID=ID+10»N

ENCODE ( 8,2, IFILM) ID

ENC0DE(8,6,ITITLE( 12) ) ID

DO 55 1=1,3

GO TO ( 7,8,9 ) , I

- 7 LTYP=8HTP SOLID

GO TO 95

8 LTYP=8HTP LIQ

GO TO 95

9 LTYP=8HNBP LIQ

95 PRINT 3,IFILM( 1 ) ,LTYP

LINE=0

DO 50 J=l,99

RHOBAR=J

RHOBAR=RHOBAR/100,*RHO( I

)

F=( l.-RHOBAR/RHO( I) )**( 1./3.

)

UB=7.5

US=2.5

FUS=FUN(N,Ub,F,AL( I )

)

FUB=FUN(N,UB,F,AL( I ) IT = 0

10 UM= (UB-US ) /2.+US

IT=IT+1

IF( IT.LE.lOO )G0 TO 15

PRINT 5,US,FUS»UM,FUM,UB,FUB

STOP

71 05 17 F-13
Table F-II. (Continued)

15 FUM = FUN(N»UM»F»AL ( I ) ) IF(ABS(FUM) .LT..O0OU1 )G0 TO 45

IF(FUM.GT.O..AND.FUS.GT.U..OR.FUM.LT.O..AND.FUS.LT.O. )G0 TO 2 0 UB = UM

FUB=FUM

GO TO 10

20 US=UM

FUS=FUM

GO TO 10

45 X(J)=RHOBAR

Y ( J

) =AL ( I )*UM

L INE=L lNE+1

IF ( LINE.NE.51 )G0 TO 50

L INE = 0

PRINT 3» IFILM( 1 ) ,LTYP

50 PRINT 4»Y( J) »X( J) IF ( I .NE. 1 ) GO TO 53

CALL LGRAPh(X,Y»99» ITITLE. IFILM)

CALL CPGRAPH(X(99) Y(99) ,l,»»I+4)

GO TO 55

53 CALL CLGRAPH(X»Y»99)

CALL CPGRAPH( X ( 99 ) » Y ( 99 ) »1 1+4

55 CONTINUE

IFILM(1)=8HS9, $1TP

IFILM(2)=8H SOLID"/

IF ILM ( 3 ) =8HS 1+ TP L IFILM(4)=8HIQUID» /i

IFILM(5)=8H1* NBP L IFILM(6) =8HIQUID

CALL C0MGRAPH( .75».75»6» IFILM)

CALL SKIPFRM

60 CONTINUE

STOP

END

FUNCTION SY(N.Z)

Y1(Z)=-C0S(Z)/Z

Y2 (Z) =-COS(Z ) /Z**2-SIN( Z )/Z

Y3(Z ) =

( -3./Z**3+l. /Z )»COS( Z )-3./Z»*2*SIN(Z)

GO TO ( 10.20,30,40t50)N+l

10 SY=Y1(Z)

RETURN

20 SY=Y2(Z)

RETURN

30 SY=Y3(Z)

RETURN

40 SY=5. /Z»Y3 ( Z

) -Y2 ( Z

RETURN

50 SY=7./Z*( 5./Z*Y3( Z)-Y2 (Z) )-Y3(Z)

RETURN

END

FUNCTION SJ(N»Z)

Jl ( Z ) =SIN (Z ) /Z

71 05 17 F-14
Table F-II. (Continued)

J2(Z)=SIN(Z)/Z**2-COS(Z)/Z

J3(Z) =( 3./Z**3-l./Z )*SIN(Z )-3./Z**2*COS(Z)

GO TO ( lO»2U,3U,40»5u)N+l

10 SJ=J1(Z)

RETURN : 20 SJ=J2(Z)

RETURN

30 SJ=J3(Z)

RETURN

40 SJ=5./Z*J3(Z)-J2(Z)

RETURN

50 SJ=7./Z*( 5./Z*J3(Z)-J2{Z) )-J3(Z)

RETURN

END

FUNCTION PJ(N,Z)

FN = N PJ=Z*SJ(N-1»Z)-FN*SJ (N,Z

)

RETURN

END

FUNCTION YP(N»Z)

FN = N YP=Z*SY(N-1»Z)-FN*SY (N,Z

)

RETURN

END

FUNCTION PPJ(N,Z)

FN = N PPJ=( FN*( FN+1. ) )/Z*SJ{N,Z)

RETURN

END

FUNCTION YPP(N,Z)

FN = N YPP=( FN*( FN+1. ) )/Z*SY(N»Z)

RETURN

END

71 05 17 F-15
f = ^

np Zirb/l uT ' o o and

M = pV

where V is the volume of the tank in cm^. The results may then be

applied to spheres of any size and to any dielectric fluid.

We have assumed that ll = u = e =1 and plotted the results

o o for three different densities corresponding to solid hydrogen, triple

point liquid, and normal boiling point liquid; this corresponds to about

22 percent range in density. The results for the first four modes are

shown in figures Al, A2, A3, and A4. It is seen that the uncertainty in

total mass is smaller for higher modes. Qualitatively this is because

the field patterns are spread more uniformly throughout the cavity for

the higher modes. For example, the uncertainty in mass vs. O'u^^ (or f^^)

is less than 5 percent over most of the range. This is to be compared

with a density change of 22 percent indicating that the resonant mode has

a tendency to integrate over the mass of the liquid rather than the volume.

F-16
CJ -- O —• 3 — 3 C5

_l O —

O —• _J

a.

a. o. (D CM

I

o ^

^ UJ

a:

o

r

- cv CM m

CD

F-17
F-18
F-19
F-20
APPENDIX G

APPROXIMATE METHODS FOR AN INHOMOGENEOUS DIELECTRIC

Introduction

The problem is to compute resonant frequencies of a microwave

cavity containing an inhomogeneous but isotropic dielectric. We suppose

the cavity wall is a perfect conductor, the dielectric dissipates no power,

and the dielectric has uniform magnetic permeability. Mathematically, we

are dealing with the boundary value problem posed by Maxwell's equations^

in the absence of sources and with the electric vector everywhere normal

to the wall of the cavity. For a Fourier component Ee of electric field

with (angular) frequency ou, the boundary value problem is

VXE = -jau^iH

VXH = jooeE (1)

E normal to boundary.

We will use the subscripts 0, 1 to denote quantities pertaining to the

corresponding mode for a cavity containing a uniform dielectric of permittivity Gq or , respectively. We assume that the permittivity e is piecewise continuous and satisfies eo ^ e ^ . Also we assume that sx - So is

small enough so that the set of modes for the permittivities eo, e, ei are

at most slightly different from each other in shape and can be put in one-to-one

correspondence. For convenience we will regard Gq as the permittivity

of free space.

^ We use the technique, notation, and units (MKSA) of Wolfgang K. H.

Panofsky and Melba Phillips, Classical Electricity and Magnetism , Addison- Wesley, 1955.

G-1
The resonant frequency cjuq of any mode in the empty cavity

may be computed from a standard solution of Maxwell's equations.

The resonant frequency uo of the corresponding mode when the cavity

is partially filled with dielectric is lower, and we wish to estimate its

value without further extensive calculations. We prefer estimating techniques which are insenstitive to the spatial distribution of dielectric

material in the cavity and which do not require further computation of

electromagnetic field strengths.

The first method considered is adapted from a technique of

approximation due to Rayleigh. Two approximations of this type are

formed, and it is shown that one is always at least as large as the other.

Then the larger is shown to be always at least as large as the true value

of ou. Finally, a refinement of the last method is described, known as

the Rayleigh-Ritz method. This first group of methods gives upper bounds

for the true resonance frequency. The lower Rayleigh estimate has not

been proved to be a lower bound. But the difference between the upper

and lower Rayleigh estimates is within 10 percent of the difference between

the empty and full cavity resonant frequencies, in the case of liquid nitrogen

in a sphere in a steady uniform gravitational field (See Figure Gl). One

should bear in mind that the upper Rayleigh estimate has been proved an

upper bound only for the fundamental (TMon) mode.

The moment methods improve on the Rayleigh methods in two ways.

First, they provide lower bounds as well as upper bounds; and second, they

are easier to apply to higher modes. The first order moment method yields

a lower bound for the fundamental mode which is comparable to the lower

Rayleigh estimate, and an upper bound which is identical to the upper Rayleigh

estimate. Higher moments have not yet been computed.

See Appendix F, Table I. G-2
The Green's function was investigated as a tool for obtaining

lower bounds. The resonance frequencies of the cavity are eigenvalues

of a certain linear operator L. The Green's function is used to compute

a norm for L~^. Preliminary results are inconclusive as to its value,

and careful estimates must be made with an automatic computer.

Rayleigh Methods

3 The Rayleigh method of equating "potential" and "kinetic" energies

in the perturbed field suggests the following heuristic procedure. We suppose

that E = a Eq for some number a independent of position and compute the

magnetic field from equations (1):

VXE aVXEo a(-juJo^xHo) o) p H = — = — = ; ^ a Ho.

Then we eliminate a by equating the time-average electric and magnetic

field energies'^.

^1 2

/" 1

a^ / eEo ^ a^ / u Hq''

But /eoEo^ = / M-Hq^ , so we conclude that o)^ is approximately equal to

Eo"

where K = e/so is the dielectric constant. Notice that the ratio b of H

to Hp is also independent of position, as a is.

3

G. Temple and W, G. Bickley, Rayleigh's Principle , Dover, 1956, pp. 1-24.

4

All integrals are volume integrals over the region of the cavity, unless

otherwise specified. G - 3
But if we suppose H = bHo foi" some number Independent of position,

equations (1) lead to

(JDo So

E = b Eo

and the ratio of E to Eq is now dependent on position because e is. A

computation like that of the preceding paragraph shows that o)^ is approximately equal to

uj = ujo • (3)

We show (Theorem 1) that aui < o) < cu ^ oUq for every mode and (Theorem Z)

that (juj < uu <uj for the fundamental mode. The remainder of this Appendix

describes ways to compute lower bounds for od in the fundamental mode.

The electric field E for any solution of ( 1

) for a given e and a> is

an eigenfunction of the differential operator

1 L = — V X V X

corresponding to the eigenvalue uu^ . Let S be the spherical cavity including

the inside region and also the boundary surface. Let A be the set of all

vector functions on S having continuous derivatives of all orders. Then A

is a pre-Hilbert space with respect to the inner product

(F, G) = (f - G , J

and we will denote by 0 the Hilbert space which is the (, ) - completion

of A. It is easy to show that L maps A into 0, and so L''"' is well-defined

on LA. But iS'^ is bounded (Theorem 3) and LA is dense so that L"''" has a

unique continuous extension to 0 which we also denote L

5 Now L is symmetric in the sense that

(LF, G) = (F, LG)
for every F, G in A and positive in the sense that (LF, F) > 0 for every

F in D. The same is true for Li, Lq, and the inverses of all three

operators. Moreover we have easily

(Eo, LEo) ^ (Eo, LqEo )

(El, L"' El) < (E,, L-^Ei) .

Now observe that

Lo - ouo = — ( L 1 - 0)^-^ — , eo ^

so that Eg, Ej belong to the same eigenspace of L^. But we have agreed^

that E Q, E, E^ shall correspond to exactly the same mode. This means that

Ej is a scalar multiple of E ^ and

(El, E^) (Eo, E^)

(Ej, L-i E^) (Eo, L'' Eo

Theorem 1: uui ^ ou ^ uu ^ uuo

^ f (E , E ) (E , E ) Proof ; 2 1 1 1 1

(JUl = = (El, (Jo{ El) (El, L 1 El)

(El, El) (Eo, Eo) (Eq, EqI

(Ei,L-'^ El) (Eq, L-^ Eq) (Eq,KL-^Eo)

(Eo- E^)

(Eo, Kujo'^Eo!

3

0)

5 D. A. Taggart and F. W. Schott, "Ferrite-Filled Cavity Resonator, " Applied Science Research 25 , November 1971, page 38,

^See the penultimate sentence of the first paragraph of the introduction.
(Eo.LoEp) (Eq.LEo) (Eq.^LoEo)

^ (Eo, Eo) ' (Eo, Eo) ^ (Eo, Eo

(Ep, — tuq"' E) a (Eq. Eq)

Finally, we prove ^ o) using the Schwartz inequality:

(Eo, Eo)^ = ("/k Eo, 7 Eo)"" > H /K Eo '

|| ^ Eo "

(Eo, Eo ) ("TV^

i Eo, -7^^ i Eo)

so that ^^^^^ <

^^^^

Theorem 2. If o) is the lowest resonant frequency, !X).^< uo < ao

Proof. Because oj*^ is minimal for L and Eg belongs to D, we have

(Eq, L Eq)

(Eq, (L-cju^) Eq) s 0 which implies cju^ ^ ~7^^ ^T-T" = o) ^

Similarly, we have udi'^ maximal for L^'Vnd

(El, (L r'- (joi^ El) < 0

(El, El) so (JUi ^ = a; (El , L'^ El)

Remark. The result uu ^ ou is often called Rayleigh's principle.

G-6
Figures Gl through G5 show how oj/ ouq and uo/ ooq vary as a spherical cavity is filled with liquid nitrogen in the presence of a steady

7

gravitational field. Preliminary data indicate that the true value lies

between these two estimates for the fundamental mode.

The Rayleigh- Ritz method is a refinement of the foregoing which

also produces an upper bound for uu. We will consider here only the

case of the fundamental mode. The force of Therorem 2 is that

2 . (F, LF)

(F, F)

when F is any continuous vector field on the cavity which is normal to

the walls and twice differentiable in the interior. Theorem 2 states this

for the case in which F is the electric field of the fundamental mode for

an empty cavity. We now consider the case in which F is a finite sum

N

F = ^ C. F.

n -1

Where is the electric field of the n th mode in an empty cavity,

II Fj, II = 1, Cn is a complex number to be determined later, and N is a positive integer. The preceding inequality then leads to

N N ^ ^'

1 A Z_ C„ C„a).| (F„,T7FJ

2 ^ n-1 n-i ^ - N

V" I ^ I 2

/ I '^n I

n = 1 where cUj, is the angular resonant frequency of the nth mode in the

empty cavity. Then

7 These results have been calculated numerically by techniques described

in Appendix H.

G-7
PERCENT FULL (NORMAL GRAVITY)

Figure Gl

Upper and lower Rayleigh approximations to the normalized frequency

cu/ouq as a function of fill fraction, for the TMqii mode using liquid

nitrogen.

G-8
PERCENT FULL (NORMAL GRAVITY)

Figure G2

Upper and lower Rayleigh approximations to the nornrialized frequency

cu/oDo as a function of fill fraction, for the TMqsi mode using liquid

nitrogen.

G-9
Figure G3

Upper and lower Rayleigh approximations to the normalized frequency

(jd/u)o as a function of fill fraction, for the TEqii mode using liquid

nitrogen.

G-10
Figure G4

Upper and lower Rayleigh approximations to the normalized frequency

(ju/cUq as a function of fill fraction, for the TM^a-^ mode using liquid

nitrogen.

G-11
Figure G5

Upper and lower Rayleigh approximations to the normalized frequency

m/cDo as a function of fill fraction, for the TM041 mode using liquid

nitrogen.

G-12
N N

^"^^^ 7. I C„C„ 03^^ (F„,

n - 1

where the minimum is taken over all choices of C, , C-, , . . , , C for which N

N

L I

n - 1

This is easily solved by Lagrange's method of multipliers , because the

coefficients (F^,— F^ ) can be computed from results already obtained for

the lower modes in the empty cavity. We propose not only to estimate

o) this way, but also to investigate the way this estimate depends on K.

The Moment Method

8 9 The moment method was pioneered by Temple , elucidated by Kato

and generalized to higher order by Stackgold^P It derives its name from

the set of numbers

_(L" Eq, Ep) n-1 2 3

called moments of the operator L with respect to the vector function Eq.

1

Stackgold showed that

G. Temple, "The Theory of Rayleigh's Principle as Applied to Continuous

Systems, " Proceedings of the Royal Society of London, vol 1 19 (1928),

pp. 276-293^

9

Tosio Kato, "On the Upper and Lower Bounds of Eigenvalues, " Journal

of the Physical Society of Japan, vol 4 (1949), pp. 334-339.

10

Ivar Stackgold, The Use of Moments in Estimating the Spectrum , Department of the Navy, O. N. R. London B ranch, August 1 968 , pp. 1 - 1 9, Accession

Number A0839570.

11

Lynn Taylor Wells, Extensions, Generalization, and Clarifications of Lower

Bound Methods Applied to Eigenvalue Problems of Continuous Elastic Systems , Dissertation for the PhD at Ohio State Univ. (1970), pp. 66ff.

G-13
^ < 03^" < m„ + ^ ^

n m„ -a'

where x is the angular resonant frequency of some empty cavity mode

whose electric field is Eq . (The numbers a, b will be defined shortly.

For our purposes it will be convenient to normalize the moments as

follows

M - - Eq, En

so that the inequality becomes

m:„ ^ °— ^1 — I < M„ + ^

The real numbers a, b are chosen as far apart as possible consistent

with the condition that ou^ be the only spectral point of L which lies strictly

between a, b. Then for the fundamental mode the best choice of a is - » and the best choice of b is the angular resonant frequency of the first

harmonic jT it were known. This leads to the simplification

(yJ-1)M. Uoj

for the fundamental mode, where

Thus for the simplest estimate (n= 1 ) the upper bound is the upper Rayleigh

e stimate

G-14
^ K

^° ^° ' ^° ^ _ / K

^ _ f^\^

(juf (Eo , Eo ) fpS V cjuo /

'

and

2 b

0)

For this simplest estimate we also need a value for

1 1 /* ^ 2 (LEq.LEq) {~ Lq Eq , ~ Lq Eq ) J

^2 "

UU^ (Eo,Eo) " uu^ (Eo, Eo) f

^

Consider the special case of a spherical cavity half-full of liquid

dielectric with the fundamental mode symmetrically oriented across the

interface. Then

LI(>UID GAS

so that for n = 1 , 2

T^n / E§ + / E§

2 ^ K"i

Then if the liquid is nitrogen (K^ = 1.44),

Ml = 0.8472 and Ms = 0.7411.

The data indicate the resonant frequencies of the lowest two modes

have the ratio

700 MHz ^

^ 510 MHz

so

G-15
-

uuo VM, Vo.8472 " ^'^^^ '

Hence our best possible estimate of this type is

= 0.824

tjj/ujo = 0. 920 > uu/uuo ^ 0. 908 > 0. 906 = o^/cuq .

If Yi = 1 . 3 we obtain tju/uuo ^ 0. 90 2 and if Yi = 1.2, uu/uUo > 0. 886 (see Fig. G6)

Oddly enough, the upper bound does not improve as n increases from 1 to 2:

= .920 and .928.

Calculating the lower bound corresponding to n = 2 is much more

difficult because

( Eq , Eq ) _ ( Eq , L Eg )

UUS (Eo , Eq ) UUo (Eq , Eq

involves calculating higher powers of L operating on Eq .

The principle disadvantage of this class of methods is that it requires approximate knowledge of the next eigenvalue higher than the

one being estimated. That is, we must estimate b or Yn • But the above

results for a spherical cavity suggest that the lower Rayleigh estimate

is indeed a lower bound for the fundamental mode. This hypothesis should

be investigated for a spherically syrametric distribution of dielectric

material

G-16
0 80^ ' 1 I I I I I I 1 0 10 20 50 40 50 60 70 80 90 lOJ PERCENT FULL (NORMAL GRAVIT) J

Figure G6 (Compare Figure Gl)

Lower bounds for uu/ob from the first order method of moments,

using three different values of Yi • The solid lines are the

Rayleigh upper and lower approximations.

G-17
Green's Function Methods

Lemma, For any cavity V of diameter D

/ (4n,-lR - R'l^ ^ — '«=^>'

V

and for a spherical cavity V of diameter D,

/ (Re V)

d-" R ^ _D_ / 1 +JZ

^

(4n)^

I R - R' 1^ ~ 4n V 4

Proof . Make a change of variable

S = R' - R.

Since Re V, the point S = 0 always belongs to the cavity.

Then (for any shape of cavity) as R""" varies over the cavity, S varies

over a region which is the cavity translated by the vector -R. So

regardless of R the region of integration is contained in a sphere of

radius D about -R. Hence

D 4TT f

d^ S ^ _J_ C d s f

_d^ _ D

V-R ° o

where U denotes solid angle.

In the spherical case, refer to Figure G7, which is a crosssectional view of the region of integration. If the R'-origin (marked by

the vector -R) is eccentric from the S-origin by a distance aD, then

the cavity will be wholly contained in a pair of hemispheres centered

at the S-origin and having radii ( + a ) D and bD, where a^ + b^ = (1/2)'

Then we have

G-18
Figure G7

A cross-sectional view of the region of S - integration for the Lemma.

The R'- origin is marked by the vector -R, and is eccentric from the

S - origin by a distance aD.

G-19
2TT bD 4TT /" S 1 f

ds r _d(^ /* ds f

dQ_

V-R ' ' o o o 2n

1

. ,l^a)D • i . bD • i

4TT ^ 2 ' 2 4n 2

where (2a)'^ + (2b)'^ = 1. Let 2a = cos 9, 2b = sin B and maximize

f(9) = 1 + sin 9 + cos 9. Since - cos 9 - sin 9 we have the maximum

of f (-^ ) = 1 + J2 and the lemma is proved.

Theorem 3 . L~ is bounded on LD and is extended by the operator M,

where

^ ' /' 4n R-R 1

Furthermore, M is bounded and

1/2

L-^ll ^ llMll ^ -V I /k^(R^) iI , , J II d^R'^ 1^

regardless of the distribution of £ or the shape of the cavity. Here c

is the speed of light and D is the greatest distance between any two points

1/2

in the cavity, K = e/so, and ||k|| [ f

K^d^R^] ^

.

Proof . First we compute LM using the identity

G-20
VxVx = VV- - and the fact that

..•MP,., . /"»;;"«- yr^)>-^'

f

^£(R')F(R')

4tt

- 4n 6(R-R')

J

R'

= Me(R)F(R),

where 6 is the Dirac distribution.

Represent the mth Cartesian component of F by Fj, , set g(R, R ') =

4tt [ R r

"^ ' show differentiation by subscripts, referring to primed

coordinates. Then we have the following identity in Cartesian tensors:

Now integrate both sides over the cavity volume in primed coordinates and

apply the divergence theorem to the left hand side to obtain

y^eF,g,nd^R' = y(^eF,),,g,,d^R' +y^eF,g,„,d^R' .

The left hand side vanishes if the surface of integration is described just

outside the (lossless) cavity. The result expressed in vector notation is

0 = J'iv' • ^eF) V'g d^R' +y^e F • v'v'g d^R'

where v' is the nabla operating on primed coordinates

G-21
Using this last result it is easy to show that

VV • MF(R) = / ^£ F- VVg

(Jc F- v' g

•u£ F)V' g ,

and it follows that

VxVxMF(R) = \ieY - j(v'.p,£F)v' g

or LMF(R) = F - ^ J

-sFjv' g .

We see that for every G in D,

v' -sLGCR') = v' • - v' xv' xG(R' ) = 0 ,

whence LMLG = LG. But the boundary value problem for L has a unique

solution, so L has a trivial kernel and L, exists. Thus

MLG = G,

where G is arbitrary in D. This shows that M extends L .

Since M extends LT , it will have at least as large a norm, and it

now suffices to compute the norm of M.

||MF(R),i. < J

ue(R')| F(R')| il r.r> T li, d= R'

implies the first inequality asserted for ||m1| . The weaker bound on || M|

comes from an estimate of

G-22
1 4n R-R (4-If J

d-'R ^ D

(R-R' f ~ 4tt

which leads easily to the second bound on |1 M|| stated in the theorem.

Corollary . Suppose £ = Gi > Sq a- fraction a of the volume of the cavity

and e - Cq for the remainder. Then the crude estimate of the theorem

takes the form

M <

1 /2

1 + a

(

Kf - 1 ) n

/ ^

J

Proof

,

K = / k! . / £ =£ £ =£,

so

Dl|K|

4n

[a Kf + (l-a)] ^

D^,

g Kf + (l-g)

24

and the conclusion follows.

Theorem 4. For the fundamental mode.

L-n||l/2n M

n||l/2n

for every positive integer n.

Proof.

2n _ (E, E) ^ ~ (L-"E,E) '

but (L"°E,E) ^ ||l""E|| • ||e|| < II 17" II

•

II eP < II M"

by the Schwartz inequality and Theorem 3 above.

G-23
Thus ^ M

- n \

\ ^ w 1 1 and the assertion follows

L M

-niil/n 1 li Remark . Since li^^rn || L i| is the spectral radius 'ju ^ H L

the reader may think the case n= 1 sufficient. But the more general

estimate is given in the hope that it may facilitate computation.

Corollary .

^ D L l+<x(Kf - 1) J

Proof . Immediate from the corroUary to Theorem 3.

Application . For an empty spherical cavity 48 cm in diameter (any shape)

a = 0 and D = 0.48 meter, so

X ^^24 X 300 X 10^ meter/sec

~:r~ ^ — — —tt, = 220 MHz.

2n 2n X 0 . 48 meter

But theoretical calculations give an exact theoretical value of 543 MHz.

and experimental verification using liquid nitrogen at atmospheric

pressure gives 546 MHz. So our crude bound is quite crude. But we

hope for a better result when

II , 1II " 4n I R-r'^ "

is calculated accurately as a function of r' .

1 2

See Appendix F, Table I.

G-24
A somewhat different group of techniques is associated with

higher powers of M.

4n I R- Ri I 4n I Ri - R2 I . . . 4tt

| R^^^ - R^

M"F(R)i|, < J- J

i n K(Rt ) . . . K(R, ) I F(R, ) I d^ Ri . . . d^ R.

4tt|R-Ri1 "r 4n| Ri -R2 I . . .4n(R„_ 1 -R„

^n"eS In^^K) [y...yF^(R„)d^R, ...d=R, ^^^^

. , r Cw 1 11^ K^(Ri)...K^(R„)d^R, ...d^R„ where I„ (K)

-J . . .j \\ \ (4n)^ | R, - R, | ^ .

| R„ _ , - R„

Continuing, we have for a sphere of diameter D

n-1

l|M"F(R)iU<^ In^^(K) [yd^Rij ^ [yF^(Rn)d^Rn]^/^ .

If we call the last integral |1fP, then for a sphere of diameter D

n-1

± J_ n

ii^^nii 2n ^ 4n f ^ '; 4n

uu > ||M"|| 2^ c I„ (K)

J

1 1

6 ^ 4n

To estimate 1^^ , observe that (Lemma to Theorem 3) for a spherical cavity

of diameter D,

G-25
f

K^(R. R, ^ K^D IV2

J

(R„- 1 )K^ (Rn Rn- 1 R,

(4-^)-^ (R,_ 1 -R, r (4n)- (R,_ , -R,

/

Kf_D lyz r d^ R„ _ 1 ^ r Kf D , 1V2 ^

4rr V 4 ^7 (4n )^ (R„_ 1 - R^ )^ ~

L 4n V 4 7 J

1/n, K?D ' \+JZ \ lim I.' (K).^

r 6 4tt

Uo > c c . 6 x'/^

nD^ KfD V IV2 J (D/2)./Ki V 1+^2

1/4 6 T~ ^ r^/T^ I TTTT ; = q\^i o x 1 . 254 - 208 MHz

2n ttDVKi \ 1+V2 y 3.14(.48)1.2

And replacing by 1 as an approximation for K in the estimate of I increases the answer by only a factor of 1 . 2 to the value 250 MHz. This

suggests the crudeness of the estimate is due primarily to the way I was

evaluated. Alternatively, set

d^R^

Jn(K) f J (4TT)^(R,^k' )^2 //i^\2 (4n)"(R„_i -R„ r

and observe that for a sphere of diameter D (see Lemma).

G-26
In(K)^-^ ) Kr Jn(K)

J_ _1_

r D / 1+J2 \ ^~4n „-l/2 Z 4n r 6 "i

Jl' > c D / 1V2 N -"4n -1/2 4n P 6 "i 4n

tu c / ^1/4 6 \ ^

- 44n

^ 7^ — lim J„ (K)

One final approach in this vein is only heuristic at this point. The

formula for M" F(R) is an integral whose integrand is overwhelminglyimportant when the variables

R, Ri , R2 , . . . , Rn

have all nearly the same value. This suggests the approximation

M" F(R) ^ ^" K° (R)F(R)W„ (R)

where W„ (R) ..f— • • • Rn ^ _

4tt R-Ri Un] R1-R2 1 . . .4nl R^. ^ -R^

This strictly positive function can be estimated by the technique of the

lemma:

G-27
J 4n| R-R^ I " J 4rTS

V R+V

(-+a)D 2tt bD 4rr / |s|d|s| /-^ . / |s|dls| / 4tt o o 2tt

1(1, .1 , 1,.^. . 1

4 DM T + a + + b^ ] 4 4

1 r>2 r 1 1 1 4

L 4 4J

2 y

because ^ ^ "2 D 2n Therefore W„ (R) ^ ( 2 ^

And since

2n (E, E) (E, E)

(M"E,E) ~ (c '''' K"W„E,E)

l/2n

c (E, E) ^ ^ (K° W„E, E)V^n

But for a two phase system

G-28
(K" W„ E, E) = r Kl f W,E^ + y W„ E^ ] - K^^

LIQUID CAS

2n 2n 1 / 2n

l\^U\D GAS

and this upper bound approaches kI^^ (-^ y as n increases without bound,

Hence

(D/2)yKi

So for an empty cavity

uu 300 X 10^

= 166 MHz 2n 2n (0.24) 1.2

Whether the validity of this approach can be established is unknown, and

even if it can be, there seems less likelihood of getting a good estimate

for Wn (R) than for J„ (K)

.

Recommendations for Further Study

1. Compute the Rayleigh estimates for several of the lower modes in

the case of a spherically symmetric distribution of liquid.

2. Compute the bounds

2

M„ - —^ ^ (-] — 1 ^=M.M„ + ^ ^

for n = 1, 2 in several of the lower modes for both normal gravity fill and

spherically symmetric distributions of liquid.

3, Investigate the Green's function method by computing || mI|, (K), Wn (K)

accurately.

G-29
APPENDIX H

NUMERICAL ANALYSIS OF THE SPHERICAL CAVITY FOR

THE LOWEST ORDER MODES

Electric Field Contours

Equation (F-22) was used to generate the graphs plotted by subroutine TMTEPLOT. These are plots of the electric field contours,

|E(r, e)|z: |e|, fortheTMoii, TMosi, TMqsi, TMo41, andTEon

modes. Choosing C^j as imaginary (n = 1, 2, 3, 4) in (F-22) the

equations used in TMORTE for the TM modes became

g = z j„(z) P„ (cos e) (H-1)

z = u., — for n = 1, 2, 3, 4

b

and Uni are the eigen values of the different modes. (The outside radius,

b, is normalized to 1. 0. P^ and are the Legendre and spherical

Bessel functions, respectively. ) With

I

' E I =VeF+E^T"?" ' e cp

being the magnitude of the electric field, and using (H-1), the components become

H-1
Also

r 3r ae

j^z j„ (z) p„ (cos

e)J

= 7

^ [Pn (cos

6)]

j

z (z) - n j„ (z)

dz

dr

and

E = 0.

cp

The analygous equations for the TE modes are

f = ^nl jn (Z) P„ (cos 0)

The magnitude of the electric field becomes |E| = ^ E^

cp The components

Ep = E^ = 0

and

Ejp= u., j, (z)

The plots of |eI for the first six modes are shown in figures 8

through 13 in the body of this report. Before plotting, |e| was scaled

by its largest value and normalized to 10. Hence, the contours are

numbered 1, 2, . . . , 10.

Upper and Lower Raleigh Approximations for the Normal Fill Geometry.

The Resonant Frequency vs. Percent Full (Normal Gravity)

graphs generated by subroutine LINPT were obtained numerically

integrating |e|^ over different portions of the resonating spherical

cavity for the various modes. The graphs display the upper and lower

H-2
Rayleigh approximations for the true solution. Working in spherical

coordinates, volume is a function of (r, 9, cp),

- ie, V = f (r, e, cp).

However, assuming symmetry in the cp direction, V = 2n g (r, 9).

Hence, the integrals for the different portions of the resonating cavity

become

R

[e (Vi)1 = Zrr ^ r^ sin 9 drd( ff |e|

TT -H/cOS 3

+ 2tt / / |e|® r^ sin 3 dr dp

where

J

<: 0 ^ 9 (see figure HI for 9 )

and

TT R

F jE {V^^)j = 2tt y y

|e|^ r^ sin 3 dr dp.

9, -H/cos 3 k

G defines an integral of the field magnitude in the empty part (Vi) of the

cavity and F defines an integral of the field magnitude in the part of the

cavity containing fluid (V ). Figure HI displays and V . The C F OF

upper and lower approximations were then computed by evaluating.

KF + G

H-3
I (R.O)

Figure HI, Coordinate system for the normal fill geometry.

H-4
at steps of AV^ = 1%, Here C = F + G and K is the dielectric constant CF

of the fluid in the cavity.

Figures G1-G5 show the upper and lower Raleigh approximations

for the first five spherical modes for the normal fill geometry.

Computer Routines

Ten computer routines were developed and coded in Fortran to

calculate the magnitude of the electric fields ( |E(R, 9) | ) and to integrate

over the volume of the spherical cavity. These routines are as follows:

TMORTE, main program, calculates |e| for the transverse

magnetic or transverse electric modes;

BJ, PN, and DPN are three function routines called by TMORTE

to evaluate spherical Bessel functions of the 1st kind and

Legendre polynomials and their derivatives, respectively;

TMTEPLOT, subroutine called by TMORTE, plots |e| for the

transverse magnetic and transverse electric modes;

RCTOUR, subroutine called by TMTEPLOT, searches for

specific contour values for TMTEPLOT to plot;

VOLUME, subroutine called by TMORTE, inegrates |E|^over

the sphere to evaluate functions of the resonating frequency

in the empty part and the full part of the spherical cavity;

RINTGL, subroutine called by VOLUME, performs that part of

the integration along the radial lines, R, of the cavity;

LINPT, subroutine called by VOLUME, plots the upper and

lower bounds of a Rayleigh approximation to the true solution

for a Frequency vs. Percent Full graph;

INTRPL, subroutine called by VOLUME, interpolates between

values of the percent full calculations to obtain required values.

A listing of the computer routines is given as follows:

H-5
PROGRAM TMORTF TMT 1 C TMT 2 DIMENSION UU). IFMT1(6)» IFMT2(6). V(4) TMT 3 C TMT 4 COMMON /DATA/ EMAGN ( 109 . 18 1 ) I RL AB ( 3 ) I CL AB ( 1 9 ) MODE • DEL T AR »DEL T AT TMT 5

1 .NRINC •NT INC .NRIPl .NT IPl .PI . JJ ( 181 ) .ESCALE TMT 6 COMMON /INDEX/ I NDE X . E F SW T . CAM A 1 , CAMA2 . GAMA3 TMT 7 C TmT 8 DATA (U = 2. 744. 3. 870.4,973.6. 062). (B=l. I. 'PI = 3. 14159265) .( I RLAB= 18HTMT 9 IRADIAL DIRECTION ) . ( I CLAB ( 1

) = IH ) . ( I C LAB ( 2 ) =

HT ) . ( I CL AB ( 3

) = 1 HH ) . I TMT 10

2 ICLAB (4 ) = IHE ) . ( ICLAB (5 ) = 1HT ) . ( I CLAB ( 6 ) = 1 HA ) .(ICLAB(7)=1H ).(ICLAB(TMT n

38 ) = 1H ) . ( ICLAB t9) = lHD).(ICLA3(10)=lHI).(ICLAB(ll) = lHR).(ICLAB(12) = TMT 12 41HE ) . ( ICLABl 13 ) =1HC ) . ( ICLAB( 14) = 1HT).(ICLAB(15) = 1HI).(ICLAB(16)=1HTMT 13

50; . (ICLABl 17)=1HN) . ( ICLAB( 18) = 1H ) . ( I CLAB 1 19 ) = IH ) . ( I NCR = 9 ) . ( J NC T = T M T 14 610) .( V = 4. 493 ,5.763 .6.998 .0.0 ). t rFMTl=47H( 8H1TM M0DE.2H ( . I 2 . 1 H ) / /5 T MT 15

7 7X.3A8/10X.13(I3.6X))).(IFMT2=25H(1X,A1.I3.1H0.2X.13E9.2/)) TMT 16

C TMT 17 INTEGER TMTE.EFSWT TMT 18

REAL KX TMT 19

C TMT 20

C TMT 21

C ROUTINE TO CALCULATE COMPONENTS OF THE ELECTRIC FIELD OF THE TMT 22

C TRANSVERSE MAGNETIC MODE UNDER SPHERICAL SYMMETRY, EQUATION TMT 23

C (A-22) OF NBS REPORT 9793 BECOMES TMT 24

C TMT 25

C G = P [COS(THETA)] Z * J (Z) TMT 26

C N N TMT 27

C TMT 28

C WHERE Z = U R/B, TMT 29

C Nl TMT 30

C TMT 31

C TMT 32

C ALSO CALCULATES COMPONENTS OF THE ELECTRIC FIELD FOR THE TMT 33

C TRANSVERSE ELECTRIC MODE UNDER SPHERICAL SYMMETRY, THE TmT 34

C ANALYGOUS EQUATION OF (A-22) THEN IS TMT 35

C TMT 36

C F = P [COS(THETA)] * K * J (Z) TMT 37

C • N N N TMT 38

C TMT 39

C WHERE Z = K R = V R/B AND B IS THE RADIUS OF THE SPHERE, TMT 40

C Nl Nl TMT 41

C TMT 42

C TMT 43

R=1,0 TmT 44

C ASSIGN THETA THE VALUE 180 DEGREES (IN RADIANS) TMT 45

THETA=PI TMT 46

C TMT 47

1 READ (60.19) TMTE.MODE.NRINC.NTINC.DFLDR.EFSWT.GAMA1.GAMA2.GAMA3 TMT 48

C TMT 49

C EFSWT = 1. CALCULATE E (AND NORMALIZE) EVERY TjME TMT 50

C 2. CALCULATE E (AND NORMALIZE) AND WRITE E ON MAG TAPE TMT 51

C ( LOG. UNIT NO. 1 ) TMT 52

C 3. READ (NORMALIZED) E FROM MAG TAPE (LOG, UNIT NO. 1) TMT 53

C TMT 54

H-6
IF (EOF. 60) 18.2 tmt 55

c tmt 56

? N=MODE/10 tmt 57

ROTATt=DFLDR»P 1/180.0 TMT 58

IF ( TMTE.EQ.2RTE ) GO TO 3 TMT 59

c TM MODE PARAMETERS TMT 60

INDEX=1 TMT 61

UX=U ( N

)

TMT 62

IFMTl ( 1 ) =8h( BHl TM M TMT 63

GO TO 4 TMT 64

c TE MODE PARAMETERS TMT 65

3 INDEX=2 TMT 66

UX=V( N

)

TMT 67

IFMTl ( 1 )=8H( 8H1TE M TMT 68

4 KX=UX/B TMT 69

INCR=9 TMT 70

INCT=10 TMT 71

IF (NRINC.NE.12 ) GO To 5 TMT 72 INCR=1 TMT 73

INCT=1 TMT 74

5 DELTAR=R/NRINC TMT 75

NRIP1=NRINC+1 TMT 76

DELTAT=THETA/NT INC TMT 77

NTIP1=NTINC+1 TMT 78

DO 6 I = 1 .NT I Pi TMT 79

JJ( I

) =1-1 TMT 80

5 CONTINUE TMT 8 1 IF (EFSWT.EQ.3) GO TO 15 TMT 82

C TMT 83

r 84

EMAX=0,0 TMT 85

DO 12 IR=1.NRIP1 TMT 86

R=( IR-1) DELTAR TMT 87

Z=KX»R TMT 88

DZDR=KX TMT 89

DO 12 IT=l.NTIPl TMT 90

THETA= ( IT-1 ) »DELTAT+ROTATE TMT 91

GO TO (7.10'. I NDEX TMT 92

C RADIAL (R) DIRECTION TMT 93

7 IF (R.NE.0.0) GO TO 9 TMT 94

IF (N.EO, 1 ) GO TO 8 TMT 95

EMAGN( IR. IT ) =0.0 TMT 96

GO TO 12 TMT 97

8 EMAGN( IR, IT' = (2.0»KX*UX/(3.0»B) ) TMT 98

GO TO 11 TMT 99

9 ER=-N*(N+l)*(KX«»2)»PN(rOS( THETA) .N ) »BJ ( Z .N ) /Z TMT 100

C TMT 101

C ANGULAR (THETA) DIRECTION TMT 102

ETHETA=DPN(COS( THE TA),-SIN( THETA) .N)*(DZDR/R)*(Z*BJ(Z.N-1) -N#BJ (Z.TMT 103

IN) ) TMT 104

C TMT 105

EMAGN( IR. IT)=SQRT(ER#*2 + ETheTA*»2 ) TMT 106

GO TO 11 TMT 107

C TE MODE TMT 108

H-7
10. EPHI=DPN(COS( THETA) .-SIN(THETA) .N)*<X»BJ(Z,N) TMT 109

C ER = ETHETA = 0. 0 TMT HO

EMAGNI IR.IT)=ABS(EPHI ) TMT 111

11 IF ( EMAGN ( I R I T ) .GT .EMAX ) EMAX = EMAGN ( I R I T ) TMT 112

12 CONTINUE TMT 113

C TMT llA

PRINT IFMTl. MODE* IRLAB , ( JJ( J ) »J= 1 ,NR IPI » INCR ) TMT 115

DO 13 IT=1.NTIP1.INCT ' TMT 116

I=IT/10+1 TMT 117

IF (INCT.EQ.l) I=IT TMT 118

'

, PRINT IFMT2» I C LAB ( I ) J J ( I) . ( EMAGN ( J » I T ) . J= 1 , NR I P 1 I NC R ) TMT 119

13 CONTINUE TMT 120

C TMT 121

C SCALE E ACCORDING TO LARGEST VALUE AND NORMALIZE TO 10. 0 TMT 122

ESCALF=10.0/EMAX TMT 123

DO 14 IR=1.NRIP1 TmT 124

DO 14 IT=lfNTIPl TMT 125

EMAGNI IR, IT ) =EMAGN ( IR» IT ) *ESCALE TMT 126

14 CONTINUE TmT 127

CALL TMTEPLOT TMT 128

IF (EFSWT.EU.l) GO TO 16 TMT 129

WRITE U) ( I EMAGN( IR I T ) . IK=1 .NRI Pl ) I T=l .NTlPl ) »ESCALE TMT 130

GO TO 16 TMT 131

C TMT 132

15 READ (1) ( (tMAGN( IR. IT ) .IR=1.NRIP1 ) .IT = 1.NTIP1) .ESCALE TMT 133

C TmT 134

16 PRINT IFMTl. MODE 1 1 RLAB » < JJ ( J ) , J= l ,NR I P 1 » I NCR ) TMT 135

DO 17 IT=1.NTIP1»INCT TMT 136

IxIT/10+1 TMT 137

IF (INCT.EQ.l) I=IT TMT 138

PRINT IFMT2* I C LAB ( I ) » J J < 1 ) » < EMAGN ( J . I T ) » J= 1 NR I P 1 . I NCR ) TMT 139

17 CONTINUE TMT 140

PRINT 20 TMT 141

C TMT lti2 CALL VOLUME TMT 143

GO TO 1 TMT 144

C TMT 145

18 PRINT 21 TMT 146

CALL EXIT TMT 147

C TMT 148

19 FORMAT ( R2. 13.2 I5.F5.0. 15 .3E10.3 ) TMT 149

20 FORMAT (IHD TMT 150

21 FORMAT (///IIH END OF JOB) TMT 151

END TMT 152

SUBROUTINE VOLUME VOL 1 C VOL 2 C SPECIAL ROUTINE TO EVALUATE (FUNCTIONS OF) VOLUMES CUT FROM VOL 3 C A SPHERE BY A PLANE PERPENDICULAR TO THE VERTICAL AXIS OF VOL 4

C SYMMETRY. HERE VOL 5 C VOL 6 C V = F(R»THETA.PHI ) = 2*PI *G ( R . THETA ) . VOL 7 C VOL 8 DIMENSION SVSSOEd). SVSSEdSO). SVICOEd). SVICEdSO). SVcFOd). VOL 9 1SVCF<180). XTRA(3). SVCF^Ed). SVCFEdSO). XTRAE(3)» XU d ) . X(30).VOL 10

H-8
2 Y0(1)» Y(3'-')» U(lOl), DE&(51)» FVCF0(1)» FVCF(3U), GVIO ( 1 ) . GVK 3V0L 1 1 30)» DFVCF(51). DGVKSl), YKIUI), Y2(1U1). YlLBUlUl ) . Y1LB2 ( 101

.VOL 12

4 Y1LB3 (101) VOL 13

c VOL 14

COMMON /DATA/ L MAGN 1 1 0 9 . 1 8 1 ) . I RL AB ( 3 ) » I CL AO ( 1 9 ) » MODE .DELTAR.DELTATVUL 1 5

1 .NR I NC »NT INC.NRIPl .NT I Pi ,P I , JJ ( 18 1 ) »E SCALE VOL 16

CUMMOW /INDEX/ INDEXfEFSWT tCAMAl .GAMA2 »CAMA3 VOL 1 7 c VOL 18

EQUIVALENCE (iNRlNC»NlR) , ( NR I P 1 . N I RP 1 ) » (NTINC»NIT). /mTtDi INI mttDi Ir^l.NIIrl 1 VOL 19

1 » VOL 20

FQUIVALEMCE (VOL»TVCLE) VOL 2 1 c VOL 22

DATA lF3a = 0.3Tj] VOL 2 3 c VOL 24

D2R=PI/180.0 VOL 25

TBX=PI VOL 26 RBX = 1 .0 VOL 27

TPI38=2.0*PI*F38 VOL 28

IPSWT=VOL 29

TVOL=4.0»PI/3.0 VOL 30

c RE-SCALE TO ORIGINAL EMAGN AND SQUARE VOL 3 1 RSCALE=1 .0/ESCALE VOL 32

DO 1 IR=1.NRIP1 VOL 33

00 1 IT=1.NTIP1 VOL 34

EMAGN ( I R . I T ) = ( EMAGN (IR.IT)*RSCALE)**2 VO L 3 5

1 CONT INUE VOL 36

c VOL 37

c SET THE INITIAL AND TERMINAL VALUES FOR R AND THETA VOL 38

RA=0.0 VOL 39

RB=1.0 VOL 40

KTHETA=90 VOL 41 TA=PI/2.0 VOL 42 TB = PI VOL 43

XYZ=0.0 VOL 44

c VOL 45

c CALCULATE NECESSARY INDICES FOR A SIMPSONS 3/dTHS INTEGRAT ION VOL 46

JB= (RA/RBX)»NIR+1.01 VOL 47

JMAX= ( RB/RBX ) *N IR+0.01 VOL 48

IB=(TA/TBX)»NIT+1,01 VOL 49

IE=(TB/TRX)*NIT+0.01 VOL 50

IEP1=IE+1 VOL 51

c VOL 52

c SOLVE FOR The VOLUME OF THE SPHERICAL SEGMENT. U, LE. THETA, LE. PI VOL 53

ISWT=1 VOL 54

DO 2 1=1,91 VOL 55

CALL RINTGL ( I SWT , I , JB . JMAX . X YZ » VSSE

VOL 56

ISWT=3 VOL 57

IM1=I-1 VOL 58

SVSSE( IMl )=F38*VSSE*SIN( IM1«DELTaT)*DELTAT VOL 59

2 CONT INUE VOL 60

VALU=0.0 VOL 61

DO 3 1=1.90.3 VOL 62

VALU=VALU+SVSSE( I-l ) +3 . 0* ( SVSSE ( I ) +SVS3E ( I +1 > )+SVSSE( 1+2

VOL 63

3 CONT INUE VOL 64

H-9
V0LSSE=TPI38*VALU VOL 65

TVOLE= VOLSSE+VOLSSE VOL 66

1K=91 VOL 67

C VOL 68

C SOLVE FOR THE REMAINING VOLUMES. P I / 2 . LE . THE T A . LE . P I VOL 69

N=-l VOL 70

DO 10 NTHETA=KTHETA, IE.3 VOL 71 N=N+1 VOL 72

Y ( 30-N

)

=NTHETA VOL 73

H=-RB»C0S(NTHETA*D2R ) VOL 74

KThP1=NTHETA+1 VOL 75

IK=KTHPl VOL 76

C VOL 77

C SOLVE FOR THE VOLUMES. N THE T A . L E . THE T A . LE . P I VOL 78

DO 7 I=KTHP1.IEP1 VOL 79

IM1=I-1 VOL 80

TIM1=SIN( IM1*DELTaT)*DELTAT VOL 81

IF (NTHETA-90) 5.4.5 VOL 82

4 R=0.0 VOL 83

GO TO 6 VOL 8 4

5 BETA=IM1»D2R vol 85

R=-H/C0S( BETA ) VOL 86

6 JE= ( R/RBX )#N I R+O.Ol +0.6 VOL 87

C VOLUME OF THE SPERICAL SEGMENT CONTAING FUEL VOL 88

CALL RINTGL ( 4 » I . JE + 1

.

JMAX . VC F . VC FE ) VOL 89

T38=F38*TIM1 VOL pO

SVCF( IMl ) =T38*VCF VOL 91

SVCFE ( IMI )=T38»VCFE VOL 92

7 CONTINUE VOL 93

C VOL 94

VALU2=0.0 VOL 95

VALU2E=0.0 VOL 96

DO 8 I=KTHP1»IE.3 VOL 97

VALU2 = VALU2 + SVCF( I-l )+3.0*(SVCF( I )+SVCF( I+l) )+SVCF( 1+2) VOL 98

VALU2E = VALU2E + SVCFE (I-1)+3.0*(SVCFE(I) +SVCFE ( I+l ) )+SVCFE ( 1+2 ) VOL 9 9

8 CONTINUE VOL 100

VOLCFrTPl 38*VALU2 VOL 101

V0LCFE=TPI38*VALU2E VOL 102

C VOL 103

PCTF=VOLCF/TVOL VOL 104

V0L1=TV0LE-V0LCFE VOL 105

C VOL 106

X ( 30-N) =PCTF»100.0 VOL 107

FVCF ( N ) =VOLCFE VOL 108

GV1(N)=V0L1 VOL 109

GO TO (9,10>» IPSWT VOL 110

9 WRITE (61»21) NTHETA»V0LSSE»V0L1.V0LCFE»PCTF.N VOL 111

10 CONTINUE VOL 112

C VOL 113

X ( 1

) =X ( 1 ) +1 .OE-8 VOL 114

X(2 )=X(2)+5.0E-8 VOL 115

X(3)=X(3)+10.0E-8 VOL 116

DO 11 1=1.101 VOL 117

(Ij^I-l VOL 118

f

H-10
1

1

12

13

14

15

16

CONTINUF VOL 119

CALL INTRPL ( 6 1 » 3 1 X 0 YO . 5 1 U » DEG

)

VOL 120

WRITE I61»22) ( U( I ) ,DEG( 1) » I = 1 .51 ) VOL 12 1 WRITE (61.23) MODE VOL 122

GV 1(29)= GV 1(29)+ i,0E-8 VOL 123

GVl ( 30 ) =GV1 ( 30 ) +5 . OE-8 VOL 124

DO 12 1=1.31 VOL 125

Y( I-l ) =270.0-Y( I-l 1 VOL 126

CONT I NUE VOL 127

RK=l.n/l,44 VOL 128

RKSQ=RK»*2 VOL 129

N1=N2=N3=0 VOL 130

DO 16 1=1.51 VOL 131

CALL INTRPL ( 6 1 . 3 1 . Y<^ . GV 1 0 . 1 . DE G ( I

I . DGV 1 ( I ) ) VOL 132

CALL INTRPL ( 6 1 . 3 1 . YO . FVCFO . 1 . DEG ( I ) . DFVC F ( I ) ) VOL 133

FI=I-1 VOL 134

FM1=(RK*DFVCF( I ) +DGV1 ( I ) ) / VOL VOL 135

Yl ( I ) =SK.FPGOC = SQRT ( FMl ) VOL 1 36

Y2 ( I ) =RlCFPGDC = SuRT ( VOL / (1.44*DFVCF(I) +DGV1 ( I ) ) ) VOL 137

FM1SQ = FM 1**2 VOL 138

FM2= ( RKSQ»DFVCF ( I )+DGVl ( I ) ) /VOL VOL 1 39

QLBl=FMl-(FM2-FMlSQ)/( ( GAMA 1**2-1 . 0 1 *FM1

)

VOL 140

FLBl=l.nF+50 VOL 141

IF (OLBl.LT.0.0 ) GO TO 13 VOL 142

N1=N1+1 VOL 143

YlLBl ( I )=FLB1=SQRT(QLB1

)

VOL 144

QLB2=FM1- ( FM2-FM1SQ ) / ( ( GAMA2*»2-1 .0 ' *FM1

)

VOL 145

FLB2=1 .OE+50 VOL 146

IF (OLB2.LT.0.0 ) GO To 14 VOL 147

N2=N2+1 VOL 148

Y1LB2 ( I ) =FLB2 = SQRT ( QLB2

)

VOL 149

QLB3=FM1- ( FM2-FM1SQ ) / ( ( GAMA 3**2-1 .0 ) *FM1

)

VOL 150

FLB3=1 .nF+50 VOL 151

IF ( QLB3. LT . 0 .0 ) GO TO 15 VOL 152

N3=N3+1 VOL 153

Y1LB3 ( I ) = FLB3=SQRT ( QLB3

)

VOL 154

WRITE (61.24) F I .DFVCF ( I

) .DGVl ( I ) .SKFPGDC .RKFPGDC FM2 .FLBl .FLB2 .FLVOL 15 5 1B3 VOL 156

CONT I NUE VOL 157

WRITE (61.25) VOL VOL 158

WRITE (61.23) MODE VOL 159

DO 20 1=52.101 VOL 160

FI = I-1 VOL 161

XDGV1=V0L-DGV1 ( 10 2- I

)

VOL 162

XDFVCF=VOL-DFVCF ( 102-1

)

VOL 163

FMl = ( RK*XDFVCF+XDGV1 )/VOL VOL 164

Yl ( I ) =SKFPGDC=SQRT ( FMl

)

VOL 165

Y2 ( I ' =R<FPGDC=SQRT ( VOL/ ( 1 .44* X DFVCF +X DGVl )

)

VOL 166

FM1SQ=FM1**2 VOL 167

FM2=(RKSQ*XDFVCF+XDGV1 ) /VOL VOL 168

QLB1=FM1- ( FM2-FM1SQ ) / ( ( GAMA 1**2-1 . 0 ) *FM1

)

VOL 169

FLB1=1 .OE+50 VOL 170

IF (OLBl.LT.0.0 ) GO To 17 VOL 171

N1=N1+1 VOL 172

H-U
V W l_ 1LID7

1 7 QL B2 = FMl-(FM2-FMlSCJ)/( (GAMA2**2-1«0J*FM1) VOL 174

VOL i ( -J VOL 1 76

N 2 = N2 + 1 VOL 177

YllRP(I)=FlB7 = S0RT(0LR?) VOL 178

1 8 QLB3=FM1-(FM2-FM130>/((GAMA3*»2-1.0)»FM1) VOL 1 79

Fl R3 = 1 .nF + SO VOL 180

IF (QLB3.LT»0.0) GO TO 19 VOL 18 1 N3=N3+1 VOL 18 2 Yll R'^( I) = FI B3 = SORT(OI R'^) VOL 18 3

1 9 WRITE (61»24) F I XDFVC F »XDGV 1 » SK FPGDC » RKFPGDC • FM2 FLB 1 t F LB2 FLB3 VOL ] 8 4

c.?nw rONT I NIJF VOL 18 5 VOL 18 6 Q VOL 18 7 XX= 0.

0

VOL 188

; CAUL LINPT ( XX » XX »XX » 1 U » U , 0 1 00 ,0 4 »0 . 8 » 1 .U » 1 » XX XX

)

VOL 189

CALL LINPT ( U » Y 1 » 1 U 1 »XX ,XX » XX » XX , XX »XX »2 1 » XX

)

VOL 19 0 CALL LINPT ( U Y2 10 1 »XX ,XX . XX . XX . XX .XX 1 2 1 tXX

)

VOL 19 1 N1X = 101-N 1 + 1 VOL 192

CALL LINPT ( U ( N 1 X ) » Yl LB 1 ( Nl X ) » N 1 XX . XX » XX » XX » X X t XX .2 » 2 » 1 W, ) VOL 193

N2X=101-N?+1 VOL 194

CALL LINPT ( U ( N2X ) » Y 1LB2 ' N2X ) »N2 » XX . XX tXX XX XX » XX » 2 » 2 1 R*

)

VOL 19 5 N 3X = 1 m -N 3+

VOL 196

CALL LINPT I U ( N3X ) Y 1LB3 ( N3X ) NS » XX . XX » XX . XX XX • X X » 2 2 1 R+

)

VOL 197

CALL LINPT ( XX » XX » XX f XX » XX XX » XX » X X » XX » 3 » X X » XX

)

VOL 198

rV. VOL 199

RETURN VOL 200

r^ VOL 201

2 1 FORMAT ( / I4.3F10.5 2PF8,2 . 15

)

VOL 202

2 2 FORMAT (F8.1»F8.2) VOL 203

2 3 FORMAT (lOhlTMMUDE ( » I 2 » 1 H ) / 3 X . 3HPC T . 4X . 6HF ( VCF ) 5X , 6HG ( V 1

)

.3X.17VOL 204

1HSURT((F/IC + G)/C) .3X»19HSURT (C/(K*F + G)) * » 1 2X . 2 HM2 . 9X , 3 HLB 1 9X . VOL 2 0 5 23HLB2»9X.3HLB3/

)

VOL 206

FORMAT (F6.1»F10.3»F10.3»2F16.6»8X»F17.6»3(4X»F8.6)) VOL 207

FORMAT (//9H * K=1.44/5H C=.F9.5) VOL 208

FND VOL ^ U 7 SUBROUTINE RINTGL ( I SWT , I , JB . JE . SUM , SUME

)

RIN I r RIN 2 c ROUTINE TO INTEGRATE ALONG THE RADIAL LINES» R» INVOLVED RIN 3 c IN A TWO DIMENSIONAL INTEGRAL TO SOLVE FOR (FUNCTIONS OF) R I N 4 c VOLUMES OF SPHERICAL SECTORS OR SEGMENTS RIN 5 c RIN 6 DIMENSION RO(l). RdOB). RE0(1)» RE(108) RIN 7

V- RIN 8 COMMON /DATA/ EMAGN (109 , 18 1) . I RLAB ( 3 > I CL AB ( 1 9 ) , MODE DEL TAR

.

DELTATR I

9 ltNRINC.NTINC.NRIPl.NTIPi.PI.JJ(181).ESCALE RIN 1 0

I. RIN 1 1 GO TO ( 1 .3»11 .3 ) . ISWT RIN 11 c.7 RIN 1

3

1 DO 2 J= 1.109 RIN 14

JM1=J-1 RIN 15

R ( JMl ) = (

(

JM1»DELTaR ) *»2

)

DELTaR RIN 16

2 CONTINUE RIN 17

H-12
GD TO 11 R I r^i 18

c R I N 1 9

.SUM = 0 .0 R I N 20

IF (JB-JF) 4,4*11 R I N 2 1 4 IJE=( JE/3)*3 R I N 22

IJB= ( ( JB+2 ) /3 ) *3-2 R I N 2 3 IJBX=IJB R I N 24

IF ( ( JB-IJB < -1 ) 7,5.6 R"I N 25

5 SUM = SUM+1 . 5* ( R ( IJB ) +R ( 1 jB+1 ) ) R I N 26

6 SUM = SUM+1 . 5*R ( I JB+1 ) +R ( lJB + 2 ) R I N 2 7 IJBX=IJB+3 R I N 2 8 IF (IJE-IJBX) 11.11,7 R I N 29

7 DO 8 J=IJBX.IJE,3 R I N 30

SUM=SUM+R (J-1)+3.0*(R(J)+R(J+1 ) )+R(J+2) R I N 3 1 8 CONT I NUE R I N 32

IF ((JE-IJE'-l) 11.10,9 R I N 33

9 SUM = SUM+1.5*(R( IJE + ?J+R( IJE + 1 )

) R I N 34

10 SUM = SUM+l .5*R ( I JE+1 )+R( UE) R I N 35

C R I N 36

C R I N 37

11 SUME=0.0 R I N 38

IF (JB-JE) 12,12.20 R I N 39

12 I JE= ( JE/3 )*3 R I N 40

I JB= ( ( JB+2 ) /3 ) *3-2 R I N 41

IJBX=I JB R I N 42

DO 13 J=IJB.109 R I N 43

JM1=J-1 R I N 44

RE( JMl )=R( JMl )*EMAr,N( J» I

) RI N 45

13 CONT I NUE R I N 46

IF ((JB-UB'-D 16.14,15 R I N 47

14 SUME = SUME + 1.5*(RE( IJB)+PE( I JB+1 ) ) R I N 48

15 SUMF SUME+1 . 5»RE ( IJB+1 ) +RE ( IJB+2

R IN 49

I JBX= I JB+3 R I N 50

IF (IJE-IJBX) 20.20.16 RI N 5 1 16 DO 17 J=IJBX,IJE,3 R I N 52 SUME = SUME+RE ( J-1 )+3.0*(RE(J)+RE(J+l) )+RE(J+2) R I N 53

17 CONT INUE R I N 54

IF {(J£-IJE)-1) 20,18,19 R I N 55

18 SUME = SUME + 1.5*(RE( IJE + 2)+RE( IJE + 1 )> RI N 56

19 SUME=SUME+1.5*RE( IJE+1 )+RE(IJE) RI N 57

C RIN 58

20 RETURN RI N 59

END R I N 60

SUBROUTINE LINPT(XI ,YI , NPT »iMXD , FMNX ,FMXX ,NYD,FMuf ,FMXY,N,LS , ISYCH) LI

1 C LIN 2 C LIN 3 C LINEAR PLOTTING ROUTINE LIN 4 C LIN 5 COMMON /DD/ IN, lOR , I T , I S . IC . ICC. I X

I Y LIN 6 COMMON /DDC/ LU.LUCIFL LIN 7 COMMON /LABEL/ LABELX ( 5 ) .LABEL Y ( 5 ) , LSWTX ,LSWTY LIN 8 COMMON /DATA/ EMAGN ( 109 , 1 8 1 ) , I RLAB ( 3 ) I CL AB ( 1 9 ) , MODE , DEL TAR ,DELTATL I

9 1.NRINC,NTINC,NRIP1,NT1P1,PI,JJ(181) ,ESCALE LIN 10

COMMON /INDEX/ INDEX ,EFSWT ,GAMA1 ,GAMA2 .GAMA3 LIN 1 1

H-13
c L I N 1 2 DIMLN5IUIN XillNrlJt YI(fNrl)» 1D(4) L I H 1 3 c L I N 1 A DATA 1 ID = 32n KG PETEkSON X318^ ) L I N 1 5 r\ATA /TvfcjiiM— ir\n^ /rw. tm— l/~\n>_/iC'.fTv — oI.mci./tn/— l_)AlA I IXMlnJ-iUU) »( IY"iliN-iC;U / (Lotwl X-^ ' tlLoWlY-^:' LIN 1 6 c

r\ATA UAIA /lAQCTlV (LAntLX-blon — CfOLJ 1^./tA14ClV— C/OU 1\ J^ttLAdriL. T=T5iorl J) L I N 1 7

1L I

1 MIN 1 o riATA (lAOPlY — /.OH PF'inMAMTFOf-^'lllPMfY 1 1 I M L 1 IN

1 Q

c L I IN "J n liN'COCK urOWi 1 I M c. i r L 1 IN !? c. c

1 T M L 1 IN 0 "a c i c

t T M L 1 IN 2 ^

1 1 UK - U

1 T M L ] IN i. 5

I N= 0 L I N 2 6

I C= 0 L I N 2 7 ic /PPcuT pn 'ai c h\ \ ^^^^TMTT //i.Tr\i L i IN 2 o

T T » r\

I T = 0 L I N 2 9 UKAW A DUa AKUUniL) KLO '

' lINo AKcA L 1 IN LALL UUdUa * U » iU<:3 U • X UZ ^ '

1 r M L 1 IN 3 1 C oKtLIAL LAbtL rLU 1 icD 1 T M L i IN 32

11*1 LIN

T C O 1 o * 3

1 T Kl L i IN 3 4

f V — "7 C L 1 N 3 5

T V — Q A 1 T - tJ (JO L 1 iN J D LAbtL-on 1 M MUUt LIN i 1 TPIr ^rwncv I pn iadpi— qljtp m/^hp 1 r>iUCA • cU • if I LADcL-oMit. MUUt 1LINT M 38

LALL UU 1 Ad 1LINT M "X Q

LALL UUIAdN"C3 ^l»LADtL»i' 1LINT M

1 T Kl LIN H i CiNLUUt I o » ^ 3 • LC3 MUUC. J MUDC-A 1LINT Ki

I Q » 1

1 o« i

1 T M L i N

A a

T V » "7 1 C

I A * ( \ 0

1 T KJ LIN 44

f V ^ "TO 1 T Kl LIN 4 5 L A L L \J\J \ At3 1LINT M HO

LALL UUIADNMo \ i»LDMULJL»i )

1LINT Kl A 7

1LINT M A Q N T * N T L)+ 1

1 T M L i NA47Q

L lYUl V=900/NYU 1 T Kl LIN 50

1 TPnAA - 1 TMIN + L 1 TUlv*nlTU LIN 1 T Kl R 1 0 L

11 A ri A i

1 T Kl L 1 Pi

1L Ty1 AL'IV r> I \/ =— onn 7vU/ilAL-' /MX n 1LINT M lAmMA— TXMAXsTyMTKi+i lAnini~i_ TxnTv/«Kixn 1 AL'*v'*n|AU-' -J H

V.

1LINT Kl c DRAW DIVlblONo OR TICKMARICb rOR X-AX15 LIN 56

I I YMAX L I N 57

I X*M I NX= I XM I

L I N 58

CALL DDBP L 1 N 59

TV— TVkiTkl 1 T Kl L 1 IN 6 U LALL UUVL LIN o 1 DO 4 1*2 N

LIN 62

IY*IYMAX LIN 63

IXS=IX= IXMIN+( I-l )*LTXDIV LIN 64

CALL DDBP LIN 65

H-14
IYS= IY=IYMAX-8 L I N 66

CALL DDVC L I N 67

I X= I XS L I N 68

I Y= I YS L I N 69

IF ( I .EO.NX ) GO TO ^ L I K 70

GO TO ( 3 ,2 ) LSWTX L I N 71

? I Y= I YM I N+8 L I N 72 CALL DDBP L I N 73

3 IY=IYMIN L I N 74

CALL DDVC L I N 75

4 CONT I NUE L I N 76

C L I N 77

C DRAW DIVISIONS OR TICI^MAKKS FOR Y-AXIS L I N 78

IX= IXMAX L I N 79

I Y = MI NY= I YM I iM L I N 80

CALL DDBD L I N 8 1 IX= IXMIN L I N H? CALL DDVC L I N 83

DO 7 I=2»NY L I N 84

IX= IXMAX LIN 85

IYS=IY=IYMIN+(I-1)*LTYDIV LIN 86

CALL DDRP LIN 87

1X5= IX= IXMAX-8 LIN 88

CALL DDVC LIN 89

IX= I XS L I N 90

I Y= I YS LIN 91

IF ( I .EQ.NY) GO TO 6 LIN 92

GO TO ( 6»5 ) LSWTY LIN 93

5 IX= IXMIN+8 LIN 94

CALL DDBP L I N 95

6 IX= IXMIN L I N 96

CALL DDVC LIN 97

7 CONTINUE LIN 98

C LIN 99

C NUMBER THE X-AXIS LIN 100

IS=1 LIN lUl

IT=0 LIN 102

IOR = n LIN 103

FINCX=(FMXX-FMNX) /NXD LIN 104

DO 8 I=1»NX LIN 105

IYS=IY=IYMIN-15 L I N 106

IM1=I-1 LIN 107

IXS= IX = IXMAX-IM1*LTXDI V-32 LIN 108

FNUMB=FMXX-IM1»FINCX LIN 109

ENCODE ( 8 ,24 tFNUMBX ) FNUMB LIN 110

CALL DDTAB LIN 111

CALL DDTABNA8 (1,FNUM8X,1) LIN 112

8 CONT INUE LIN 113

C NUMBER THE Y AXIS LIN 114

IS=1 LIN 115

IOR = 0 L I N 116

FINCY= (FMXY-FMNY) /NYD LIN 117

DO 9 1 = 1, NY LIN 118

IX = 60 LIN 119

H-15
IM1= [-1 " LIN 120

I Y= lYMAX- IM1»LTYDIV LIN 121

FNUMB = FMXY-IKl<fFlNCY LIN 122

ENCODE ( 8.25 FNUMBY ) FNUMB LIN 123

CALL DDTAB LIN 124

CALL DDTABNA8 (1. FNUMBY, 1) • LIN 125

Q CONTINUE LIN 126

C LIN 127

C LABEL X AND Y AXES LIN 128

I T = 1 L I N I 2 9 IS=3 LIN 130

IX=40 LIN 131

IY=30 LIN 132

CALL DDTAB LiN 133

CALL DDTABNA8 ( 5 . L ABEL X ( 1 ) » 1 ) LiN 13A

IX=30 LIN 135

IY=40 LIN 136

I0R=1 LIN 137

CALL DDTAB LIN 138

CALL DDTABNAB ( 5 . LABEL Y ( 1 ) , 1 ) LIN 139

C

' LIN 140

XMIN=FMNX LIN 141

XMAX=FMXX . , LIN 142

YMIN=FMNY LIN 143

YmAX=FMXY LIN 144

FMINX=IXMIN LIN 145

FmAXX=IXMAX LiN 146

FMINY=IYMIN LIN 147

FMAXY=IYMAX LIN 148

IS=0 LIN 149

IT=0 LIN 150

IOR=0 - LIN 151

GO TO 22 LIN 152

C LIN 153

10 IXL=IYL=0 LIN 154

JSWT=1 LIN 155

GO TO ( 11 .12 ) t LS LIN 156

C PLOT VECTORS LIN 157

11 CALL DDCONVEC LIN 158

GO TO 13 LIN 159

C PLOT SYMBOLS OR CHARACTERS LIN 160

12 ICC=ISYCH LIN 161

CALL DDSYMBOL LIN 162

13 DO 20 1=1. NPT LIN 163

X = X 1(1) LIN 164

Y = YI ( I ) LIN 165

IX= ( X-XMIN ) / ( XMAX-XM IN ) * ( FMAXX-FM INX ) +FMINX LIN 166

I Y= ( Y-YMIN ) /

(

YMAX-YMIN )

(

FMAXY-FMINY ) +FMINY LIN 167

SLOPE=( IY-IYL)/( IX-IXD LIN 168

IF ( IX.GE. IXMIN ) GO TO 14 LIN 169

IY = SLOPE*( IXMIN-IXD + IYL LIN 170

IX=IXMIN LIN 171

JSWT=2 LIN 172

GO TO 15 LIN 173

H-16
1 L I N if**

TY-*^i nPF*( TXMAX-Iyl ) + rYi L I N 175

I X = I X M A X L I N 1 76

JSW T = 2 L I N 17 7

1

IF (IY.GE»IYMIN) GO TO 16 L I N 178

r Y = T YM T N L I N 1 79

GO To 17 L I N 180

1 n IF (lY.IF.IYMAX) CiO TO 1 ft L I N 18 1

T

1 Y—T— T1 Y^^ I "^HA a Y L I N 18 2

1

1 7' IX=( lY-IYL'/SLOPE+IXL L I N 183

JSWT = 2 L I N 184

1

IF (I»EQ.l«AN0.jSwT.i_Q»2' GO TO 19 L I N 18 5 IF ( I X . EQ. I XL . AND. I Y . LQ.

Yl ) GO TO 20 L I N 186

CALL DDXY L I N 187

1 Q I XL= IX " L I N 188

I YL = I Y L I N 189

TON T I N 1 IF LIN 190

r LIN 191

CALL DDTAB L I N 192

GO TO 22 L I N 19 3 rv_ L I N 194

FRAME ADVANCE L I N 19 5

J 1 c. 1 TAI 1 DDFR L I N 1 ~ o

/-

v_ 1 T N 197

22 RETURN L I N 198

LIN 199

23 FORMAT (13) L I N 200

24 FORMAT ( F4.0 ) L I N 201

25 FORMAT ( F4 • 2 ) L I N 202

END LIN 203-

SUBROUTINE TMTEPLOT IMP 1 IMP 2 c SPECIAL PLOTTING ROUTINE TO PLOT MAGNITUDES OF ELECTRIC FIELDS TMP 3 c TMP

COMMON /DD/ I N » I OR » I T I S » I C I CC » I X I Y TMP 5 COMMON /DDC/ LU»LUC»IFL TMP 6 COMMON /DATA/ EMAGN ( 109 , 181 ) I RLAB ( 3 ) ICLABI 19 ) iMGDE »DELTAR DELTATTMP 7 l»NRINC»NTINC»NRIPl»NTIPl»Pl»JJ(181)»ESCALE TMP 8 COMMON /INDEX/ I NDE X E FSWT » GAM A 1 » GAMA2 GAMAS TMP 9 COMMON /SUB/ I RR ( 1 8 1 ) » I XS ( 1 8 1 ) » I YS ( 1 8 1 ) »P I D2 I DNT I »R fNP TMP 1

TMP 1 1 DIMENSION LABEL(5)» IXN(10)» RAD(36)» ID(4) TMP 12

c TMP 1 3 DATA (ID=32HRG PETERSON X3184 )»(LABEL=40H R F MASS TMP 14

IGUAG I NG-PLOT OF E • TM MODE ) » ( R AD= 20 ( 475 , 0 ) » 477 ,0 » 4 78 ,0

480 .0 t4TMP 1 b 281.0»483.0»485,0»488.0»490,0t488,0»485.0»483,0»481.0»480,0, 478 . 0 »4TMP 16

?77.0t47S.n) TMP 1 7 rV, T MD 1 fl IOR=0 TMP 1 9

I C = 0 TMP 20

IT = 0 TMP 21

CALL DDINIT (4. ID) TMP 22

C TMP 23

C DRAW A BOX AROUND PLOTTING AREA TMP 24

H-17
T M - I T MD 2 5

T MP o TMD ^1 ' PLOT A CONTINUOUS CiRCLt CtNTERFD AT RASTFR fOOR. (^17.Sf»Pl T MP ^? fio WITH RADIUS = hbO RASTER UNITS T MP P Q

I S = 0 TMP

rAl 1 DDCnfMVEC T MP J 1 P I DNT I =P

/nT I NC TMP J c TMP -J D DO 1 I = 1 . N T I P

TMP J *+ T = ( I - 1 ) «P I nN T

J MP TT=PID2-T TMP J D

I X = COS ( TT ) •450 • 0 + 5 1 2 •

0

TMP ?D 7f IY=SIN(TT)*450.0+532.n TMP 3 8 IXS(NT IPl-I + 1 ) = 1024-IX TMP 3 9 IYS(NTIPl-I+l)=IY TMP 40

CALL DDXY TMP 4 1 CONT I NUE TMP 42

DO 2 I=2»NTIP1 TMP 4 3

I X = I X S ( T ) TMP 4 4

I Y= lYS ( 1) TMP H IP CALL DDXY TMP H D CONTINUE TMP 7 NUMBER CIRCLE EVERY 10 DEGREES TMP Zifl DO 3 1=1.36 TMP H 7 IDEG= ( I-l )* 10 TMP

TT=PID2-IDEG*PIDNTI TMP COSTT = COS ( TT ) TMP 52

SINTT = SIN ( TT ) TMP 5 3 CALL DDCONVEC TMP =1J ^*+ IX=COSTT»450.0+512.0 TMP ^ 5

I Y=SINTT*450. 0+532.0 TMP IP u CALL DDXY TMP 7 IX=COSTT*458.0+512.0 TMP n J o IY=SINTT*458. 0+532.0 TMP

CALL DDXY TMP A 0 IX=COSTT»RAD( I )+512.0 TMP O 1

I Y=SINTT*RAD( I )+532.0 TMP O c IDEGX=l000+IDEG TMP O J ENCODE (8.13.NV) IDEGX TMP 64

CALL DDTAB TMP 65

CALL DDTABNA8 (1»NV,1) TMP 66

CONT INUE TMP 67

TMP 68

LABEL PLOTTED TMP A Q

IT=1 TMD 1 Mr 7U

IS=3 1 Mr 1 i IX=10 TMD 1 M r 1 z IY = 25 TMD 1 Mr ~T af i LABEL(5)=8hM MODE TMP 1 n r 7 A IF (INDEX. EQ. 2) LABEL(5)=8HE MODE TMP 7 c CALL DDTAB TMP 76

CALL DDTABNA8 (5»LABEL(1)» 1

) TMP 77

MODEX=1000+MODE TMP 78

H-i8
ENCODE ( 8 , 1 

»LBM0DE ) MODEX

T MP 79

IS=1 TMP 80 I X= 794 TMP 8 1 I Y= 1 5 1 K 8 2 CALL DDTAB 8 3 CALL UUIAdNAo li»LDM(JL)L»l' T VI D 8 A c T M D Q C o 5 I S= 0 T M D 8 6 N 1 1 Uz = N 1 1 / Z + i T M D c T M D c bhAKCH hUK LUiNIUUKb T M D Q Q T M D Q n D — M ^ T M D Q 1 fc_ pnP TU HP TP Q 9 T M D T "^1 P Q A TDDf TTU-il— O T M D TMP 7 D

*f TDDf iKKl TTMl-'7n J 1 n'— ZU TMP

Q 7 c

T)L UN 1 1 fN

U t T

MDQ P L T M P Q7 Q7 ITU-T TMP inn o TMP J 1 n D — I In TMP

r- TMP 1 1*1 

C KLU I 1 He (^UtNIUUKb TMP IDA TMP 1 n c; i IJ 5 CALL uUCONVtL TMP i U o DO ' I - 1 » NP T mi D 10 7 TV— fVC^Tl i A - 1 A b I I ) TMP i U o TMP 1 1" " 1 0 Q i U 7 TMP 1 1*1 r 1 1 fl f \ TMP 111 ill T YD— 1 rvO/i—T Y^f MD 1 TMP 119

Ir 11ao11)«LU»1aK) CiO IC) o TMP ( Mr 113

v_ALL UUdH TMP I Mr 11/. CALL UULONVLL T lUI D 1 Mr lit) 8 O T — 1 MO UU V I - i NP T dii n 1 Mr 116

INPrI IPl— NP~1 + 1 TMP 117

TY— lA-iUZH— lA'^A TVQfMDiulTDT 1 lAoifNr'rilr'i ) 1 Mr 1 1 o TV — TVC/kinkJtDT \ I T - 1 Y b I NPM I P i J Tmp 119 L A L L UU A Y 1 Mr ion 1 ^: U

Q /" r\M

TT III 1 1 p

L UN 1 1 NU 

TM P 1 Mr 1 O 1 Id I C_ALL DUdP 1 MP 122

C TMP 123 1 n nn 1 1 TTi_i— tTuid-mttDi T M D I Mr 19/. tcir /TDD/TTLj\ llKK(linJ»Lt»NKlPi) ICT KIDTDi\ (jU 6 T M D 1 Mr 12 5 1 

/-HMTT mi IC LUfN 1 1 rNUt

TMD 1 Mr 126

/^HMTTW 1 IC TMP 1 rlr 10 7 c TMP 128 c FRAME ADVANCE AND RETURN TMP 129

CALL DDFR TMP 130 c TMP 131

RETURN TMP 132

H-19
c TMP 133

13 FORMAT (13) TMP 134

END TMP 135-

SUBROUTINE RCTOUR RCT 1 C RCT 2

rv_- SPECIAL ROUTINE TO SEARCH FOR ALL VALUES BELONGING TO A CONTOURRCT 3 C RCT

COMMON /DATA/ t MAGN ( 1 u 9 , 1 8 1 ) » I RL AP ( 3 ) » I CL AB ( 1 9 ) . MODEDELTAK »DELTATRCT 5 1.NKINC»NTINC»NRIP1,NT1P1»PI»JJ(181) ESC ALE RCT 6 COMMON /SUB/ I RR ( 18 1 ) . I XS ( 18 1 ) » I YS ( 18 1 ) »P I D2 P I DNT I .R NP RCT 7 C RCT 8 NP = 0 RCT 9

' DO ^ ITH=I,NTIP1 RCT 10

IRB=IRR( ITh) RCT 1 1 IF ( IRB.GT.NRIPl ) GO TO A RCT 12 EMI =EMAGN ( I R6-1 , I TH

)

RCT 1 3 00 1 IR=IRBtNRIPl RCT 14 E=EMAGN ( I R I TH

)

RCT 15

IF (EMl.LE.R.ANO.R.LE.E) GO TO 2 RCT 16

IF (EMl.GE.R.AND.R.GL.E) GO TO 2 RCT 17

EM1=E RCT 18

1 CONT INUE RCT 19

RNUM=0.0 RCT 20

GO TO 3 RCT 21

2 RNUM=EM1-R RCT 22

RDEN=EM1-E RCT 23

RINTP=( IR-2.0+RNUM/RDEN)*DELTAR RCT 24

RADIUS=RINTP«A50.0 RCT 25

TT = PID2-( I TH-1 ) *PIDNT I RCT 26

NP=NP+1 RCT 27

IXS( NP ) =COS( TT ) *RADIUS+512 .0 RCT 28

IYS(NP)=SIN(TT) *RADIUS+S32.0 RCT 29

3 IRR( ITH)=IR+1 RCT 30

IF (RNUM.EQ.O.O.AND.NP.NE.O ) GO TO 5 RCT 31

4 CONTINUE RCT 32

C RCT 33

6 RETURN RCT 34

END RCT 39-

SUBROUTINE INTRPL( I U . L X . Y N » U . V

)

INTERPOLATION OF A SINGLE-VALUED FUNCTION

THIS SUBROUTINE I NT ERPOL AT E S » FROM VALUES OF THE FUNCTION

GIVEN AS ORDINATES OF INPUT DATA POINTS IN AN X-Y PLANE

AND FOR A GIVEN SET OF X VALUES (ABSCISSAS). THE VALUES OF

A SINGLE-VALUED FUNCTION Y = Y(X).

THE INPUT PARAMETERS ARE

lU = LOGICAL UNIT NUMBER OF STANDARD OUTPUT UNIT

L = NUMBER OF INPUT DATA POINTS

(MUST BE 2 OR GREATER)

X = ARRAY OF DIMENSION L STORING THE X VALUES

(ABSCISSAS) OF INPUT DATA POINTS

0001

0002

0003

0004

0005

0006

0007

0008

0U09

0010

0011

0012

0013

0014

0015

0016

H-20
C (IN ASCENDING ORDER) 0017

C Y = ARRAY UF DIMENSION L STORING THF! Y VALUES 0018

C (ORDINATES) OF INPUT DATA POINTS 0019

C N = NUMBER OF PUJNTS AT WHICH INTERPOLATION OF THE 0020

C Y VALUt (ORDINATE) IS DESIRED 0021

C (MUST BE 1 OR GREATER) 0022

C U = ARRAY OF DIMENSION N STORING THE X VALUES 0023

C (ABSCISSAS) OF DESIRED POINTS 0024

C 0025

r The OUTPUT PARAMETER IS 0026

C Ot-2 7 C V = ARRAY OF DIMENSION N WHERE THE INTERPOLATED Y 0028

C VALUES (ORDlNATcS) ARE TO dE DISPLAYED 0029

C _ 0030

C 0031

C DECLARATION STATEMENTS 0032

r 0 0 3 3 DIMENSION X ( 1 ) ,Y ( 1 ) tUt 1 ) .V( 1 ) 0034

EQU I VALENCE ( PO . X 3 ) . ( 00 . Y3 ) » ! 0 1 » T 3 ) ( UK » DX ) , ( I MN . X2 » A 1 FM 1 ) . 00 3 5

1 ( IMX »X5 • A5 'EMS ) » ( J .oW .SA ) . ( Y2 W2 »W4 .02 ) » ( Y5 . ^3 »Q3 ) 0036

C 0037

C PRELIMINARY PROCESSING 0038

C Uu39

10 LO=L Uu40

LM1=L0-1 0041

LM2=LM1-1 0042

LP1=L0+1 0043

NO=N 0044

IF(LM2)90.100.100 0045

ion IF ( NO ' 91 .91 • 105 0046

ins DO 11 1=2. LO 0047

I F( X ( I- 1

) -X ( I ) ) 11 .95.96 0048

11 CONTINUE 0049

IPV=0 0050

C 0051

C MAIN DO-LOOP 0052

C . 0053

DO 80 K=1.N0 0054

UK=U(K) 0055

C 0056

C ROUTINE TO LOCATE THE DESIRED POINT 0057

C 0058

20 IF(LM2)200.27,200 0059

200 IFtUK - X ( LO ) ) 205 .26 .26 0060

205 IF(UK - X (1 ) ) 25 .208 .208 0061

208 IMN=2 0062

IMX=LO 0063

21 1= ( IMN+IMX ) /2 0064

IF(UK - X( I ) )22 .23.23 0065

22 IMX=I 0066

GO TO 24 • 0067

23 IMN=I+1 0068

24 IF(IMX - IMN ) 240 .240 .21 0069

240 I=IMX 0070

H-21
r. n T n 3 0 0 0 7 1

I = 1 00 72 GO To "^0 00 7 3 26 I =LP 1 0 0 7 4 GO TO 30 00 7 5 27 I = 2 0 0 76

00 7 7 CHECK I r I = I pv 0078

0079

30 I F ( I - I PV ) 300 , 70 . 300 00 80

300 IPV= I 0081

0u82

ROUTINES TO PICiC UP NECESSARY X AND Y VALUES AND 0083

TO ESTIMATE THEM IF NECESSARY 0084

008 5 40 J = I 00 8 6 IF( J-1 )401 »400,401 0087

400 J = 2 0088

401 IF ( J-LPl ) 403 .402 403 0089

402 J = LO 0090

403 X3=X(J-1) 0091

Y3=Y(J-1) 0092

X4=X(J) 0093

Y4=Y(J) 0094

A3=X4-X3 0095

FM3= ( Y4-Y3 ) / A3 0096

IF(LM2)404»43»404 0097

404 2*^05»41»405 0098

405 X2=X ( J-2

)

0099

Y2=Y ( J-2 0100

A2 = X 3-X 2 0101

PM2= ( Y3-Y2 ) / A2 0102

IF( J - LO )41 ,42»4l 0 10 3 41 X5 = X ( J+1 ) 0 10 4 Y5=Y( J+1) 010 5 A4=X5-X4 0106

FM4= ( Y5-Y4 ) /A4 0107

IF( J - 2)45.4l0t45 0108

410 FM2=FM3+FM3-FM4 0109

GO TO 45 0110

42 FM4=FM3+FM3-FM2 0111

GO TO 45 0112

43 FM2=FM3 0113

FM4=FM3 0114

45 IF( J - 3)46. 46,450 0 1 1 450 A1=X2-X( J-3) nU 1i

1i AO FM1= ( Y2-Y( J-3 1 ) / Al n 1 1 7 GO TO 47 0118

46 FM1=FM2+FM2-FM3 0 119

47 IF( J - LMl )470.48 .48 0120

470 A5=X(J+2)-X5 0121

FM5= ( Y( J+2 ) -Y5 ) /A5 0122

GO TO 50 0123

48 FM5=FM4+FM4-FM3 0124

H-22
C U 125

C NUMERICAL DIFFEREiNT lATION 0126

C 012 7 sn IF ( I-LPl ) 500 .52 »500 0128

500 W2=ABS( FM4-FM3 ) 0129

W3=ABS ( FM2-FM1 ) 0130

bW=W2+W3 0131

I F ( SW ) 5 1 » 50 1 5 1

' 0 132

5nl W2=0.5 0133

W3=0.5 013A

SW=l.n 0135

51 T3= ( W2*FM2+W3*FM3 > /SW 0136

IF( I-l )52«54»52 0137

52 W3=ABS ( FM5-FM4 ) 0138

W4=ABS( FM3-FM2 ) " 0139

SW=W3+W4 0140

IF(SW)53»620.53 0141

520 W3=0.5 0142

W4=0.5 0143

SW=1.0 0144

53 T4= ( W3*FM3+W4»FM4 ) /SW 0145

IF( I-LPl ) 60»530.60 0146

«^^n T3 = T4 0147

SA=A2+A3 0148

T4=0. 5* ( FM4+FM5-A2* ( A2-A3 ) * ( FM2-FM3 ) / ( SA*SA )

) 0 149

X3=X4 0150

Y3=Y4 0151

A3=A2 0152

FM3=FM4 0153

GO TO 60 0154

54 T4=T3 0155

SA=A3+A4 0156

T3=0. 5*( FM1+FM2-A4* ( A3-A4 )

( FM3-FM4 ) / ( SA*SA ) ) 0157

X3=X3-A4 0158

Y3=Y3-FM2*A4 0159

A3=A4 0160

FM3=FM2 0161

C 0162

C DETERMINATION OF THE COEFFICIENTS 0163

C 0164

60 Q2=(2.0*(FM3-T3)+FM3-T4)/A3 0165

03= ( -FM3-FM3+T3+T4 ) / ( A3*A3 ) 0166

C 0167

C COMPUTATION OF THE POLYNOMIAL 0168

C 0169

70 DX=U<-PO 0170

80 V(K.)=Q0+DX«(Q1 + DX»(Q2+DX«Q3) ) 0171

RETURN 0172

C 0173

C ERROR EXIT 0174

C 0175

90 WRITE ( IU.2090) 0176

GO TO 99 0177

91 WRITE ( IU»2091 ) 0178

H-23
GO TO 0 179

OSWRITE(I1I»20^5) 0180

GO TO ^7 318 1 Oft WRITE ( I1J»2096 ) 0182

Q7 WRITF (IU.2097) I.X(I) 0183

P9 WRITE (IU.2U99) L'l.NO ' U184

KFTURN 0185

C 0 186

r FORMAT STATEMENTS 0187

C 0 18 8 2090 FORf-'AT ( 1X/22H L = 1 OR LESS./) 0189

2091 FORMAT ( 1X/22H N - 0 OR LESS./) 0190

2095 FORMAT ( 1X/27H »» IDENTICAL X VALUES./) OlQl

2096 FORMAT ( lX/3'iri X VALUES OUT OF SEQUENCE./! 0192

2097 F0RMAT(6H I = » I 7 . 1 OX , 6HX ( I ) =»E12.3/) 01Q3

2099 F0RMAT(6H L =.I7.10X.3HN ^.17/ 019A

1 36H ERROR DETLCTCD IN ROUTINE INTRPL/) 0195

END 0196

FUNCTION BJ(Z,N) OCl

C Ou2

C THIS ROUTINE COMPUTES THE SPHERICAL BESSEL FUNCTIONS 003

C OF The 1ST KIND JN(Z) FOR THE FIRST FIVE VALUES OF N 004

C 00 5 NP1=N+1 006

GO TO ( 10«20»30»19,19) .NPl 007

C 008

C JO(Z) 009

inlFIZ.NE.O.OlGOTOll - 010

BJ=1.0 Oil

GO TO 60 012

11 BJ=SIN(Z)/Z 013

GO TO 60 014

C , 015

19 ASSIGN 30 TO JNSWT 016

GO TO 21 017

C Jl (Z) 018

20 ASSIGN 60 TO JNSWT 019

21 IF(Z.NE.0.0*GO TO 22 020

BJ=0.0 021

BJ1=0.0 022

GO TO 23 023

22 BJ=(SIN(Z)-Z*COS(Z) )/Z*»2 024

BJ1=BJ 025

?3 GO TO JNSWT» (60.30) 026

C J2(Z) 027

30 BJ= ( ( 3 .0-Z*Z ) S IN ( Z

) -3.0*Z#COS ( Z )

) /Z**3 028

BJ2=BJ 029

IF(N-^ )60»40 .50 030

C

' 031

C J3(Z) 032

40 BJ=5.0/Z*BJ2-BJ1 033

GO TO 60 0 34

C J4(Z» 035

50 BJ=7.0/Z*( 5.0/Z*BJ2-BJl )-BJ2 036

H-24
APPENDIX I

LIST OF FIGURES

Figure 1. Input and output of RF cavity 4

Figure 2. Conversion of resonant frequencies to the "time . . domain . 6

Figure 3. RF Antenna 8

Figure 4. Mass gauging of uniform density LNg using the

TMoii mode 12

Figure 5. Mass gauging of uniform density LNg using the

TM021 mode 13

Figure 6. Mass gauging of uniform density LHg using the

TMoi 1 mode 15

Figure 7. Mass gauging of uniform density LHs using the

TM021 mode 16

Figure 8. Field magnitude contours for the TMqii mode ... 21

Figure 9. Field magnitude contours for the TM^^i mode ... 22

Figure 10. Field magnitude contours for the TEqii mode ... 23

Figure 11. Field magnitude contours for the TM^s^ mode ... 24

Figure 12. Field magnitude contours for the TEqsi niode ... 25

Figure 13. Field magnitude contours for the TMo4i mode ... 26

Figure 14. Comparison between supercritical and normal fill

for nitrogen, TMqii mode 28

Figure 15. Normal fill data for liquid nitrogen TMqs 1 mode . . 29

Figure 16. Comparison between supercritical and normal fill

(average) for nitrogen, TMog^ mode 30

Figure Al. Heater power and current vs. pressurization time

for nitrogen A-2

Figure A2. Experimental vessel A-4

Figure A3. Cryogenic flow system for zero-g simulation tests . A-6

I-l
C 037

60 RETURN 038

END 039

FUNCTION PN(Z,\) 001

C 002

C THIS ROUTINE COMPuTF.5 VALUtS OF THE LtGENDRE 003

C POLYNOMIAL PN(Z) FOK THE FIRST FIVE VALUES OF N OOA

C 0U5

GO TO ( 10»20.30»40.50) .N 006

C 0U7

C P 1 I Z ) 0 0 8 m PN=Z 009

GO TO 60 010

C P2(Z' Oil

20 PN=0.5*(3.0*Z»»2-1.0) 012

GO TO 60 013

C P 3 ( Z ) (J 1

30 PN=0.5»(5.0*Z**3-3.0»Z ) 015

GO TO 60 016

C P4(Z) 017

40 PN=0,125*(33,0*Z»»4-30.0*Z**2+3.0) 018

GO TO 60 019

C P5(Z) 020

50 PN=0.125*(63.0*Z»»5-70.0*Z**3+15.0*Z) 021

C 022

60 RETURN 023

END 024

FUNCTION DPN(Z»DZDTH.N) 0001

C 0002

C SPECIAL ROUTINE TO COi'^iPUTE THE VALUES OF THE DERIVATIVE WRT 0003

C THETA OF THE LEGENDRE POLYNOMIALS PN(Z) FOR THE FIRST FIVE 0004

C VALUES OF N 0005

C 0006

GO TO ( 10 ^20 t 30 »40 » 50 ) »N 0007

C 0008

C D'Pl (Z)Z/DTHETA 0009

10 DPN=DZDTH 0010

GO TO 60 0011

C D'P2(Z)Z/DTHETA 0012

20 DPN=3.0*Z*DZDTH 0013

GO TO 60 0014

C D'P3(Z)Z/DTHETA 0015

30 DPN=0.5»( 15.0»Z**2-3.0)*DZDTH 0016

GO TO 60 0017

C D' P4 ( Z )Z/DTHETA 0018

40 DPN=0.125*t 1^0.0*Z*#3-60.0*Z)»DZDTH 0019

GO TO 60 0020

'C D'P5(Z)Z/DTHETA 0021

50 DPN=0.125«(315.0*Z**^-210.0»Z*»2+15.0)*DZDTH 0022

C 0023

60 RETURN 0024

END 0025

H-25
Figure CI. Frequency response of the copper spherical vessel

for the TMoii mode C-2

Figure C2. Frequency response of the copper spherical vessel

for the TM021 mode C-3

Figure C3. Frequency response of the copper spherical vessel

for the TEii mode C-4

Figure C4. Frequency response of the copper spherical vessel

for the TMpai mode C-5

Figure C5. Frequency response of the 60 inch diameter stainless

steel sphere - TMoii mode C-6

Figure C6. Frequency response of the 60 inch diameter stainless

steel sphere - TEqii mode C-7

Figure C7. Frequency response of the 60 inch diameter stainless

steel sphere - TMogi mode C-8

Figure El. Residual vs At^ E-5

Figure E2. Cumulative distribution of residuals E-7

Figure Fl. Resonant frequency vs. mass, TM mode . . . F-17 mil

Figure F2. Resonant frequency vs. mass, TM ^, mode . . . F-18 m21

Figure F3. Resonant frequency vs. mass, TM ^, mode ... F-19 mSl

Figure F4. Resonant frequency vs. mass, TM mode . . . F-20

Figure 01. Upper and lower Rayleigh approximations to the

normalized frequency ou/udq as a function of fill

fraction, for the TMqh mode using liquid nitrogen . G-8

Figure G2. Upper and lower Rayleigh approximations to the

normalized frequency uj/cdq a-s a function of fill

fraction, for the TMqsi mode using liquid nitrogen , G-9

Figure G3. Upper and lower Rayleigh approximations to the

normalized frequency uu/cdq as a function of fill

fraction, for the TEqii mode using liquid nitrogen. G-10

1-2
Figure G4. Upper and lower Rayleigh approximations to the

normalized frequency cjd/cdo as a function of fill

fraction, for the TMqsi mode using liquid nitrogen . G-11

Figure G5. Upper and lower Rayleigh approximations to the

normalized frequency uu/udq as a function of fill

fraction, for the TMq^ ^ mode using liquid nitrogen . G-12

Figure G6. Lower bounds for cd/ouq from the first order method

of moments, using three different values of yi.

The solid lines are the Rayleigh upper and lower

approximations G-17

Figure G7. A cross-sectional view of the region of S- integration

for the Lemma. The R origin is marked by the

vector -R, and is eccentric from the S-origin by a distance aD G-19

Figure Hi. Coordinate system for the normal fill geometry . . . H-4

1-3
FORM HBS-IKA (1-71) U.S. DEPT. OF COMM. 1 . PUBLICATION OR REPORT NO. 2. Gov't Accesaion

BIBLIOGRAPHIC DATA ^^^^^^ ^3_3^g

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Mass Quantity Gauging by RF Mode Analysis

5. Publication Date

6. Performing Organization Code

7. AUTHOR(s) R. S. Collier, Doyle Ellerbruch, J. E. Cruz,

Robert W. Stokes, Philip K. T,nft, Cnvdnr, V>^ff^^^r^r, kr A tt

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

tS. Project/Task/Work Unit No. 2750563

11. Contract/Grant No. T-1738B

12. Sponsoring Organization Name and Address National Aeronautics and Space Administration

Johnson Space Center

Houston, TX

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

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.) This is a summary report of work done to date on NASA (Johnson

Space Center) purchase order T-1738B concerning Radio Frequency (RF)

Mass Quantity Gauging. Experimental apparatus has been designed and tested

which measures the resonant frequencies of a tank in the "time domain. " These

frequencies correspond to the total mass of fluid within the tank. Experimental

results are discussed for nitrogen and hydrogen in normal gravity both in the

supercritical state and also in the two phase (liquid-gas) region. Theoretical

discussions for more general cases are given.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Gauging; hydrogen; nitrogen; radio frequency; total mass

18. AVAILABILITY STATEMENT

[x] UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCL ASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

22. Price

USCOMM-DC 66244-P7I

ISCOMM • FRl