Report 2030
ELECTROMAGNETIC SOIL PROPERTIES
IN THE VHF/UHF RANGE (PHASE 1)
by
Robert A. FallsAndrew Cuneo,jr. Jr•
Henry Knauf "
May 1972
Approved for puhi releae; distribution ..nmited.
S. S. ARMY MOBILITY ESUIPIENT RESARC! HO DEVELOPMENT CENTERFORT RELVOIR, VIRGINIA
NATIONAL TECHNICALINFORMATION SERVICE
V
SIC) IN ICATT I.......... ..........
list. WAIL x L
Destroy this r•.lort wihen no longer needed.SDo not n:turn it to thel originator.
----------- ~ ~ -- -
L I UNCI.AMSIFIKI)Securfty Classification
B DOCUMENT CONTROL DATA.- R 1. D(.SecwitY classifitcation of title. body of abs tract on.j indexing annotation must be enteted when the overall report to clasal flod)
I ORIGINA TING AC TIVI TY (Cotpordt author) Iza.0REPORT SECURITY CLASSIFICATION
U:. S. Arniq Mob~ility Lqiiipinvnt Research.nid Deiielopment C:enter I tJeelassifiedFort lI.jVirgriiia 22060 jib. GROUPs
3 REPORT TITLE
EIA::i'RMAGNTICSOIL PROPERTIES IN THlE VIIFI/UIIF RIAN(;E (PhIASE 1)
4 oESCRIPTIVE NOTES (7y.- olosieptaodnglncuetv dittos)
Fincal
Robet A Fals.Andrew Ctuneo, Jr., and hliettr F. Knatif
* REPORT DATE 78. TOTAL NO. OF PAGES Tb. NO. OF REo's
MNay 1972) 64 11e.CONTRACT OR GRANT NO. e.ORIOINATORWS REPORT HU&OUERIS)
&. PROJECTPNO. IJ062712AJ22 20:10
c. Sb. OmECN REPORT NoMS (Any Wata numberS M~et ae" be d.atelesioShia report)
10. OISTRIU11UTION STATEMENT
Approved for pubhie rele-as: d~ktriblittion unlimeited.
It- SUPPLEUMETARY NOTES 12. SPONSORING MILITARY ACTIVITY
USANMERIXCFort ltelIir. Virginiii 22060
IS. ASISTAIACT
This report describies measureients of the eleetromnagnetir paramleters of -;oil ýanipelvs in the VIIF/UIlI Frange the purpose- (f A-icd ~h tas oarieeitiiilatt' basir duta (pin thv IroiIN'rties that limit substirfare target
detection byv KNS wethods. A sectionl of coax~ial % :awegiide filled with .oil is treated a.-a lengthl of letytraii.-
mirbinin linet. Cl.assical trawsmnission-litie tlleorv 6~ iiseti in ihet-erni iiing the eo pl Jr(ijagato oeletistatit ofthe xilisamiple fromt meastireenit-ison ft(- electtrival jphlIIamlfile- lii' M I.Tesdtar f'hlioelseLqeeations which were proggrallineed for copiflhleer t41hil~ioiI.
Also included in the report for comphleeness; aned eonpari~soii are thee, re-stlts or twii other teeheeciqecet. formcastiring the real part oif lthe complino propagationi viiestanst. Thei first (of thiese Ite'teiiiii(It. rertiiire~. elveerieal
pleasve and VSWR 1 inewatar('men ts anld the 'ilitltionl is ohetailmd graphically. Thi.evsenud Itech ini(tiv re-quires theemjeasuiremen-tt of a pio~er ratio ietiter tnatceld rominlhionib of the linle cindfer test.
ACESSO 0KM 4,5 I AN 06. W*4ICN ISDD 'F..,1473 room P0 ARM*Y (15 t;N V..;S M h
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14.yCasiiain e OD LINK A LINK * LINK C
POLU WT mOLa WT R041t WT
Soil E.Itwiromiagmicet Pro jwrtievs
V'SR Mea-mirement.,
UNCLASSIFIED
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U. S. ARMY MOBILITY EQUIPMENTRESEARCH AND DEVELOPMENT CENTER
FORT BELVOIR, VIRGNIA
Report 2030
ELECTROMAGNETIC SOIL PROPERTIES
IN THE VHF/UIIF RANGE (PHASE i)
Project IJ662712AJ22
May 1972
Distributed by
The Commanding OfficerU. S. Army Mobility Equipment Research and Development Center
Prepared by
Robert A. Falls, Andrew Cmico, Jr., and llenry KnaufMine Delection D)ivision
" Countermine/Counter Intrusion Department
Appme.d for public nhl.ase: distribution unlimited.
SM
SUMMARY
This report describes measurements of the elcetromagicetic parameters of soilsamples in the VI1 l"/Ull F range made to accumulate basic data on the properties thatlimit subsurface target detection by electromagnetic (EM) methods. A section of co-axial waveguide filled with soil is treated as a length of lossy transmission line. Classi-cal transmiission-line theory is used in determining the complex propagation constantof the soil sample from measurements on the electrical phase antd the volLage standing-wave ratio (VSWIR). This data is fed into classical equations which were programmedfor compuder solution.
Also included in the report, for completeness and comparison. are the results oftwo other techniques for measuring the real part of the complex propagation constant.The first technique requires electrical phase and VSWR measurements, and the solu-tion is obtained graphically. The second technique requires tihe measurement of apower ratio uinder matched conditions of the line under test.
FOREWORD
This work was done in support of Project IJ662712AJ22, Barrier DetectionResearch.
SThe following people have contributed to this report: Robert !L. Brooke, whoencouraged and suggested the "power ratio under matched condition technique,"furnished the critical equipment, and served as a consultant; Benjamin Fletcher, whoperformed most of the early measurements using the "Ginzton Technique"; Walter.. Scott, who performed the "power ratio under matched condition technique" mca-surements: Ingrid Scharn, who programmed the equations for the "inp)ut impedancetechnique"; and Charles N. Johnson, Jr., for his ieview.
Nil
iz
Il :
ii
CONTENTS
Section Title Page
SUMMARY ii
FOR EWOR 1)iii
ILLUSTRATIONS v
TABLES vi
INTRODUCTION
1. Purpose of Program 12. Known Measurement Problem 1
II INVESTIGATION
3. Review of Known Measurement Techniques 14. Methods Utilized in This Investigation 45. Sources of Error (Kirkscether's Technique) 86. Discussion of Techniques and Results 16
III CONCLUSIONS
7. Conclusions 26
APPENDICES
A. Computer Program and Printout 27
B. Laboratory Procedure for K irkseether's Traismission.Line Technique 34
C.. Modification of Transmission-Line Equation forInput Impedance 41
D. Solht ion of Loss Tangent Equation
iv
._-;....
ILLUSTRtATIONS
v9e Title Page
1 Laboratory Setup for Measuring F.Ii Parameters of Soil 6
2 Laboratory Equipment 9
3 Coaxial Line Soil-Insertion Device 10
4 Setting Up Insertion Device: Tamping Soil Into Air Line 11
5 Tamping Soil Into Air Line 12
Removing Filled Coaxial Line 13
7 Inherent System Mismatches 17
8 Attenuation vs Frequency (Ginzton Technique) 18
9 Power Ratio Measurement U nder Matched ConditionsTechnique 20
10 Attenuation vs Frequency (Power Ratio Technique) 22
I Attenuation vs Frequency (Input Impedance Technique) 23
12 Attenuation vs Frequency-Vietnam Soil 24
13 "Window" Effect of Vietnam Soil 25
14 Location of Voltage Null 35
15 System Calibration 36
16 Accurate Adjustrment Procedure for Short or Open ModeBehind Soi! Sample 37
17 Location of Vollage Minimum 38
18 Ten-Times Mipinum Method for Measuring lfigh VSWR 40
7• ...
TABLES
Table Title Page
I Summary of Measurement Techniques 3
II Errors i-sue to Equipment 16
III Effort vs Re'sults 19
IV Co)-.-,ison of Power Ratio/Input Impedance Techniques 21
V Data as Lisk.d in Notebook 40
-m4
.4.
ELECTROMAGNETIC SOIL PROPERTIES
IN THE VHF/UHF RANGE (PHASE I)
I. INTRODUCTION
1. Purpose of Program. The primary purpose of the initial phase of tile pro-gram is to select measurement techmiques and to measure and record the values of theelectromagnetic properties of soils with a description of the methods used. The long-range objective of these measurements is to gain a clearer understanding of what rolethle soil environment plays in subsurface target detection.
2. Known Measurement Problem. i'be electromagnetic properties of a soilplay an important role in determining whether subsurfice targets can be detected.The preferred way for determining these properties would be to bring the laboratoryto the sites and measure the soil in situ. This method is not the normal and practicalway to measure soil properties, at least, not in the early stages of an investigative pro-gram. This method would limit the types of soils to the inimediate laboratory location.
The second method. which is the most practical for the initial stages, is toobtain small samples of soil from many parts of the world and to bring them to thelaboratory. It has the disadvantage that the soils are disturbed and properties therebyare possibly altered. However, this method offers the advantage of testing a large num-ber of soil samples rather easily and the flexibility of a fully equipped laboratory inselecting the measuring techniques. The ultimate technique would incorporate thebest laboratory method into a mobile, rapid procedure to test soil in situ.
II. INVESTIGATION
3. Review of Known Measurement Techniques. Various methods by which theelectrical properties of soil can be measured canl be grouped under two major hIeadings:those that employ a radio ground wave and those that do not. The two groups are asfollows:
SA. Methods using radio ground waves (in situ measurements):
"* Attenuation of (;round Wave"* Wave Tilt"* N Magneto-Telluric"" Reflection Coefficient
B. Methods using signal generators, etc.:
"* Electrode Array (in situ)"• Bridge Substitution"* Intrinsic (one-way) Loss of 4-Terminal Network:
0 Ginzton Technique• Power Ratio Technique0 Kirthscether's Transmission-Line Method
The methods which use a radio ground wave for the measurement of theelectrical properties of soil are all large scale, field in situ methods, whereas the secondgroup contairns methods which can be carried out in a much more limited area and,with the exception of the electrode array, can be carried out in the laboratory.
A summary of measurement techniques is given in Table 1. A brief reviewof the known methods follows:
a. Attenuation of Ground Wave. The attenuation vs distance techniquerequires a high-power transmitter and extensive field strength measurements over alarge area. Conductivity measurements of a specified small area are not possible withthis method.
b. Wave lilt. The wave tilt method is, peaiaps, the most used method ofdetermining the effective soil-conductive and dielectric-constant values. A transmitter,transmitting antenna, receiver, and two receiving antennas are included in the system.An electromagnetic field is produced by the transmitter. The amount of tilt of theelectromagnetic wave across the surface is a function of the effective ground constantsand frequency used.
C. Magneto-Telluric. This method measures the naturally occurring E/Mfield at the earth's surface to determine the surface impedance and subsurface strata.The subsurface has to be effectively homogeneous in a horizontal direction for dis-tances much greater than the wavelength employed.
d. Determination of Reflection Coefficient. This method computes theconductivity and dielectric constant of a soil by propagating an electromagnetic wavebetween two nonconducting towers and by comparing the phase change between thedirect and ground-reflected waves. Tower heights and separation have to be of theorder of a wavelength or greater to avoid near-field phenomena.
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e. Electrode Array. Wenner describes a four-in-linc electrode method ofmeasuring earth resistivity.' The measuring technique is simple. The current source isapplied to two electrodes, and a voltage is read on an~other pair of electrodes. The soilconductivity call be computed from a formula.
f. Bridge Substitution-Resistivity. Thec resistivity (low-audio.freqlitency)bridge method is generally employed in the laboratory to obtain the electrical proper.
ties of soil samples. It can, however, be used in the field to obtain in situ measure-ments. The relative dielectric constant and coonductivity can be obtained with thismethod. Some disadvantages are: the surface reactance between the electrodes andsoil; and the breaks, cracks, or discontinuities produced in the soil by the insertion ofthe electrodes. These can lead to errors in the value of the properties.
g. Intrinsic (One-way) Loss of a 4-Terminal Network.
(1) Ginzton Technique, The intrinsiec loss of a soil packed in a co-axial line can be found by measuring the input impedance of the network for aseries of positions of a movable short circuit at the output terminals. This is alaboratory method and reveals only the attenuation at frequencies at and above100 mliz. Details are given in paragraph 4a.
(2) Power Ratio Technique. This technique is based on the fact thatthe one-way loss through a network containing a soil sample matched both at theinput and output is given by a simple equation. This method is faster and lessprone to human mistakes than the Ginzton Technique. It is a laboratory methodonly. Details are given in paragraph 4b.
(3) Transmission-Line Method (Kirkscether). The soil sample ispacked in a suitable transmission line (coaxial line). At, adequate length of thesoil is selected (15 to 30 centimeters). The input impedance of the lint, " th thesoil is measured with the far end of the sample line short circuited, thlen opencircuited. The method can be used to determine the conductivity, dielectric Con-stant, attenuation, and velocity of propagation. This method has very good possi-bilities for field in situ measurements. Details are given in paragraph 4c.
4. Methods Utilized in This Investigation.
a. Ginzton Technique. The first in-house attempt to measure attenuation(of soil) employed a technique described by Ginzton.2 According to Ginzton, "The
~.2IF. Wenner, "A Method of Measuring Earth Rcsistivity'," Bulletin of the Bureau of Standards, Vol. 12 (1915).2E. L,. (;inzton, ltkirowave Aeawurements, pp. 465,473,474, McGraw Hiill Book Co., Inc., New York, 1957.
4
intrinsic (one.way) loss, L, of an arbitrary four-terminal network (coaxial line) can befound b) measuring the input impedance of thl network for a series of positions of amovable short circuit at the output terminals..." It should be emphasized that the de-termination of the intrinsic loss, Ili, by this method does not require the network to bematched at either end. The necessary data can be taken for a series of positions of themovable short circuit. The position of the short circuit need not be measured. If inputimpedance locus is plotted on a Smith Chart, the intrinsic loss of the network, L,, is
given by:
li= 10 Log J(1+R)2 ..p2 +/(1_R)2 p2 (db)
(I+R)2 -P - ,/(I-R)' -p2
where R is the radius of the impedance circle and p is tie distance from the center ofthe impedance circle to the center of the Smith Chart. R and P are normalized to unitywith respect to the center of the Smith Chart. In our case, the 4-terminal network isthe soil.filled section of coaxial line, and (X = L/sample length). The actual coaxial re-ceptacle for the solid was fabricated from 3/4-inch I) by 1/16-inch wall thickness brasspipe with a 0.322-inch-diameter brass center conductor. Alford #11890 reducers wereused on both ends of the pipe. The rest of the system was made up of a signal genera-tor, a low-pass filter, a pad, a slotted line, a VSWR meter, an adjustable line, and a shortcircuit.
b. Power Ratio Under Matched Condition Technique. Another techniquefor measuring soil ,attenuatioa was the power ratio under matched condition technique.A block diagram of the setup is shown in Fig. I. The theory behind the technique isbased on the fact that the one-way loss through a network which is matched both onits input and output ports is given by the simple equation,
Li = 10 log 0(db),Pin
where Pin is the input power to the network and Put is the power delivered to amatched load, and again li = L/samplc length. These two power levels are measuredby using directional couplers on the input and output of the network (coaxial linefilled with soil). The network is matched on its input and output by using stub tunrsand adjusting them for maximum power transfer. The detectors are matched over afrequency range which includes the range of interest. An Alford Network AnalyzerModel 7051 is used to take the ratio of the detected levels.
The Power Ratio Technique using the Vector Voltmeter lip 8405A is avariation of the previously mentioned method. Essentially, the Alford Analyzer and
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detectors are replaced by the iHewlett Packard Vector Voltmeter. Tae advantage ofusing the V :ctor Voltmeter is that it provides not only the direct readout of attenua-tion on a meter but also yields the phase angle of the sample. This mcthod for soilsmeasuremen's was originally conceived in the latter part of 1970.
The attenuation values obtained from this method correlate closelywith those values obtained by an independent contractor laboratory and by the Inputimpedance Technique.
C. Kirkscether's Transmission-Line Technique. The first attempt to mea-sure the electromagnetic properties of dielectrics by inserting the material into a wave-guide structure was reported by S. Roberts and A. von Hippie in 1946.3 The measure-ments were made at the centimeter wavelengths in a rectangular waveguide. It waspointed out in the paper by Roberts and von Hippie that "by limiting the clectromiag-netic field to the closure of a hollow pipe or coaxial line, all boundary and stray effect'disappear automatically and small amounts of any dielectric can be measured with pre-vision." The values of the properties obtained using such a technique must be theactual free-space values in order for the technique to be useful. In a paper by T. W.Dakin and C. N. Works, use is made of the standing-wave measurement technique de-veloped by Roberts and von Hippie and it is stated by Oakin and Works that, "In actualpractice, the wave and the dielectric sample are restricted to an enclosed hollow or co-axial waveguide, although this is not in principle a necessary restriction. The sameequations are valid in principle for a measurement using lecher wires or free space witha parallel beam of radiation, although it is more difficult in practice to do measure-ments under those conditions." 4 In the current report, classical theory, as stated in apaper by E. J. Kirkscether,5 is employed.
According to theory, the propagation constant of a length of transmis-sion line can be determined by a knowledge of the input impedance of the line with theoutput open circuited (Zoc), then short circuited (Z.). The input impedance of thetransmission line is related to the VSWR and the position of the voltage minimum ofthe standing-wave pattern setup on the input side of the transmission-line section underconsideration. The attenuation (a) of a line is solved using the equation:
ar= I/In [W.s, Zo,) f]42nIZ, 0 )j
S Roberts, A von llipple. "A New Method for Measuring Dielectric Constant and Lom in the Range of Centimeter
Wiaves,"J. App. Phys., 17, 610(1946).
4T. W. Dakin, C. N. Works, "Microwave Dielectric Nlcasurements,' J. App, Phys.. 18, 789 (1947).5 E. J. Kirkscether, "Ground Constant Measurenents Using a Section of Balanced Tio.Wirc Tranrmnission Line,"IRE Trans on Ant. and Prop.. AP.8, 307 (1960).
• _i - = • •= . .•.•• •;• • • ••• •- o;. :- - •, ..- •. •• L•• = • -J •;•• - - • "•" ,_
"TThe phase constant (•) is solved by using the equation
"2 tal' [g(ZSC, Zoe)I +li.
By use of the instrumentation shown in Figs. I and 2, measurementscan be made on a coaxial transmission line which is partially filled with a sample ofsoil whose EM properties are to be found.
The soil is prepared by packing it in a coaxial line (20 or 30 cm long)with the device shown in Fig. 31 4, 5, and 6. The connectors are replaced, and theline containing the soil is connected to the slotted line as sltown ill Fig. 1. All adjust-able, coaxial air line of exactly one-half wavelength is placed behind the sample. Acalibrated short is placed on the end of the one-half-wavelength line and the VSWRreading, and the distance from the first null to the sample input is measured and re.corded. The calibrated short is replaced with a calibrated open, and the procedure isrepeated.
The four measurements and the frequency at which they were takenare programmed into a computer to obtain the following properties: attenuation,phase constant, velocity of propagation, dielectric constant, and conductivity (seeAppendix A).
A detailed procedure for this technique can be found in Appendix B.The mathematical solutions to the transmission-line equations can be found it'Appendices C and 1).
5. Sources of Error (Kirkucether's Technique).
a. Presence of TE, TM Modes. The presence of higher modes (TE, TM)will invalidate all equations used, since they are derived from Maxwell's equations onthe assumption of TENt. It will be shown L )w that, for the frequency range used.TE and TM modes cannot exist either in the air-filled or soil-filled coaxial line.
First, %(c shall show that even for the case of very lossy soil the wave-length is given very accurately by the simple relationship (for good dielectrics).
X, = X0
where W. is the wavelength in the soil, X. is the wavelength in air. and e, is the relative
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dielectric constant of the soil, Von Hipple, in 1954, reported for a clay soil (20%k ~~moi'sture r'ontent) at 300 ONl a value of loss tangent, 6, equal to 0.5 and dielectric•I constant, 6r0 equal to 20.0. Using these values, we can compute beta (and, therefore,
the wavelength) in the soil since this case represents an extremely lossy soil. Startingwith the basic relationship for beta,6
( I + tan2 8 + 1
and inserting the value for the loss tangent we obtain,
-=6=
(A; ( 11.251 +Y
1. 030=~~ 1.03
c
which can be written approximately as,
2v
2ir
6Simon Rarno. John Wliinnery, F ieds and Waves in Mlodern Radio, p. 306. John Wiley and Sons. Inc.. New York, 1960.
14
From the above expression, we see that the wavelength in the soil is given by Xo/Ve-since beta is by definition equal to 21r/wavelength.
In order to insure that no higher mode will be propagated, the wave,length in the medium in question must obey the inequality.'
,> r (b + a)
t where b is the radius of the outer conductor and a is the radius of the inner conductor.For the coaxial line being used, b = .007 meter and a = .003 meter, Substituting tiesevalues in the above equation,
X > 7r (0.007 + 0.003)
> 3.14 (0.01)
> 0.03 meter
- 0: ~X= X,
1/r
X0o 0.03 V-er "meter
and setting Er 20 (probable uppter limit for typical soils),
X= 0.03 1/20 = 0.03 (4.47)
= 13.4 x 10.2
S= 0.134 meter.
This corresponds to a frequency of 2.24 gllz. U;nder the assumption of Er 20. themeasurements would be valid up to a frequency of 2.2 gIz. Our measurements wentno higher than 1.0 gllz.
h.odore Miori'no. /huurne. Transmrfision Desiks Dat/•. i)9.69, )ver lPuhijlicoti• I ih-.. NenA YorI.. 1958.
5
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b. Inherent Errors Due to Equipment. When calibrated terminations con.neeted to the output end of the Slotted Line were used, the accuracy of the instrumentwas checked over a VSWR ranging from 1.00:1 to 6.00:1. The mnaximuni error foundwas 5%. The actual components (Fig. 7) connected to the end of the Slotted Line have•i maximum eo!l!ctive VSWIl of (1.31) (1.01) (1.03) = 1.05.8 The estimated effect thatthis VSWR has on ineasuremients is given in Table H1.
Table 11. Errors lDue to E,(uipnilent
VS\VR of Sample i•i Ideal Line VSW\R Range of Sample Error( Inheient VS\VR) (l)ue Wo Inherent V'SWR of Line)
1.5 1.44-1.36 ± 4%2.0 1.92 - 2.08 ± 4%4.0 3.84 -4.16 ± 46.0 5.76 -6.25 ± 4%
When the 5W error of the slotted line itself is considered, the total esti-mated( maximrerm error turns out to be ± 9%. Since calibrated mismatches were availableonly up to 6.00: 1, no calibration of the slotted line was inade above this value. llow-ever, standard. accepted procedures for measuring high V'SWVICs (greater than 10.00:1)wvr" followed. The above technique (Kirkscether s) yields the value of a number ofelectromagnetic parameters, i.e.. diehectric. propagaltion and phase constants as well asattenuation and permeIability. The techniques dhescribed in paragraph 6 yield only thevalue of attenuation.
6. Discussion of Techniques and Results. The advantages an(l disa(lantag(Cs ofthe thre(:e techniques and the results of Ast,, described in this report ar(e gih en ini the fol.lowing paragraphs.
a. The Ginzton Technique. The (;inzton Terhni(lue was the first attemptat IlERI)C to measure the one-was attenuation through soil. This technique required
approxiniately nine V'SW\I readings on one sample at one frequenncy. The readingsw,,re plotted on a Smith Chart. and impldance values from the chart were tediouslyhaind calculated to arrive at the attenualiomi.
The G;inzton Technique i6- a laborious mnehod to obtain the attenuationvalue of a sample. ThI, hand 'alcuhlations can easily lead to niistakcs in Ihlie finial aisler.
Soil att('uaiii(io %s l're4( 1ui% for this inethod appears in Fig. 8.
licrnrwave Engineer's Tefchnicna andi uyens GUide, 1). 11. I loriton Ilh iM' a,.4., 1967.
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tb. The Power Ratio Technique. The Power Ratio Technique uising theAlford Network Analyzer was the second mntlhod employed in the quest for attenua-tion data from soils. This technique requires that the soil sample input and output"ports he tuned for maximum power transfer as indicated on a network analyzer oscillo-scope (Fig. 9). Once this is done, the measurement of the attenuation is obtained by
-ciretly reading, in decibels, the control knob on the analyzer.
This method is the least laborious and has the minimum calculationsnecessarv to arrive at an attenuation value. It has the disadvantage of not being able togo below approximately 190 mllz because of the short length of the tuners which areused to match the impedance of the line to the soil sample.
c. The Input Impedance Technique. The Input Impedance Technique wasemployed as a method to obtain not only attenuation but also velocity of propagation,conductivity, and dielectric constant. This technique, therefore, had greater potentialin vielling more properties and their values from one initial set of readings. Ilowever,except for the attenuation, the other properties proved to be more clusive than at firAtthought. The equation finally used to obtain these properties proved to b e multi-valued:and, unaless thte experimenters had previous knowledge of the approximate dielectricvalue of the soil under test, there was no way to identify the properties except by usingtwo different lengths of the same soil.
d. Results. In weighing the results and polential (yielding other propertiesbesides attenuation) of each technique against the effort, a conclusion ca.nit bv reachedas to which is the most effectivc" technique and which is the least effective.
Assuming that the shortcomings of tithe Input Impedance Technique aresolhed (Phase (Constant. p) 'abhle Ill ranks the techniques in effort vs resulls.
'T dbl" Ill. Eifforl vs I estills
14111111114 HaL l1l441 li'..ilt.Use for Other"• 'i','T('hniqfu,' Rank IEflfort lt,'..lt.- I', fr()h r ,mar,,-£:: ~Soil .Me.miroemnt.s
C(;into, ia.ir 'T'4 Eliots Atlimat ion No , --
l'owr Ratio (Cod I.m \-t1t,'at ion None -
Poiir Ratio V\rb (Good A\ ,rag.- Allen eLation I )ielcri,• ith \',,eelur '•/l'hia-, , ( :,..,t~g t\ nithVru% oli
I11n11 \er1 (;, Good 'r',di in- . A '-u ati je e ha-.e. \. ,'I. t-1 oft gN nool.hl ,.daech to I ,'-I Projp.. I)idelcrie for high
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There is good correlation bctwecn tite PIower Ralio and tithe Input Ian-pe(lance Techniques as indicated by Figs. 10 and II Table IV shows the results of a"recent comparison between the Power Ratio (Vector Voltmeter) Method verslts theInput Impedance Technlque on the same sa-nple of soil at two differetit frequencies(i.e., density and moisture content wert identical ini the samples for both techniques).
Table IV. Comparison of Power Ratio/Input Impcdn-c,. Techniqties
Frtequency Moisture % Wet Density Input Pnipt-dalle'- Ttuhiqu' Power Ratio (Vector(11llz) of D)ry Wt (gm/cc) (db/mcter) Voltmneter) (db/mcter)
600 12 1.33 19.1 17.31000 12.8 1.38 2:1.7 24.9
Note: Laguana joyuda. P.R. (Tunnel site) sample from 56" h.•tI.
Mefore turning to the conclusions. the reader is directed to Figs. 12 and13 which show the attenuation versuls freqtutncy of a Vietnam soil (silt clay). Note adlecrease in attenuation across several hundred megahertz. The rate of decrease in atten-ttation appears to be a function of moisture. This soil is the onh) one of several dozensoils observed that behaves in this fashion. No clear explanation can be givett at thistime for this odd behavior.
21
.. .... H it ji;l IW !!H :11... ... . ... .... lit
. .... ... ..W., Nit - ý.Yef
T7 7-M ..... ..... .... .... ....
.... .... .... ..90 .. ..... .... ..... ... .... ... ... ..... ... .... ....
.... .... ... .... ......... ...... .... I:j
.... .... ....... .. ..... .... .
... .... ... ... ...... ... ..... .... .... .. .... .... ..... .... ....
I E -, ii --X . ... .... ... ... .... ... .... ............. .... .. .... .... ....... ... .... .... .. ..... ... .... ...
.... ... . ... .... .. ...... .... .... ..... .... .... .... ...... .. .1:.*, ,:::. ..... .... .... .. ...... ..... .... ....
so ..... ..... ......... .. ..... .. ...... .....
V,
40.... ...... ......
30....... .... ............ .... .........
.... ......... . .... ... .... ...
... .... .... ......... ......... ........ ......... ......... .......
10........ ....... . .
.... ........ .. !Am
S"
Fig. 10. Attenuation vs fre(pictiv% (Power Ratio Techniquv).
I01 itI....... .... .... ...... .... .... ... ... ... .... .... .... .... .... ....
.... .. ........ .... .... .... .... ... ....... .... .... .... .... .... .... .... .... .... .....
.. .... ...... .... ...... .... .... ............ ....... ... .. ... ... .... ... ... .
..... ... .... ... ... ..... .... ... ......... ........................... ... ...... .... .... ..... ...... .... ..... ......... ........ .... .... .... .. ....
... .... ..... .... .... ............. ... .... .... ..... ... ... .... ..... .... .... ..... .... .... ..... .........:::: ::::i .... .... ...
.... .... .... .... ........ ......... .... ... ........ .........k 4 .: ... .... .... .... .... .... ... .. .... ..... .. ........
... .... .... ..... ........20 7 ... ... ... ..I !R . ... ... ... .... .... ....
.. ........ .
... .......... .... ... .......... rF.'-.... ..... .... ..... .... . . .... ..... ... ........ .. .... .... ... .... ...V
...... .......... ......
. .... .. ...
30 464
Ig -nuation vs frequenry (11111111 11111m.41.111ev Technifiliv).
23
)o0 .... .... .. G
i ~MONO
~I .. ...... .. ..... ......... .. .... ... ..
r7o .. ....
.. ... .... ... .... ... ...too Ii q. .........
A - ... .. ... .... ....
'.....4....t..t. i........
424
...... ..... ~4 1 ..... .... ........ . .
it .. .... ...
: .O
21 .. . . . . . . . : I :
.. .... .. .. .. .. ....
10~
25)
1II. CONCLUSIONS
7. Conclusions. It is concluded that:
a. An increase in moisture content of soils will generally increase attenla-tion. (However, there are exceptions as noted in Figs. 12 and 13 in that windows occurin the attenuation curve above 500 mtlz. The transmission windows are accentuated bythe percentage of moisture occurring in the soil.)
b. Correlation is obtained between the Input Impedance Technique andthe Power Ratio M'thod (+ 9%) if the density and moisture content are the same inboth methods.
26
APPENDIX A
COMPUTER PROGRAM AND PRINTOUT
(FORTRAN IV)
I
L
•" 27
CASE LAGUNA JCYUOA
VSWR(OC)= 6°50
VSWR(SC)= 2.80
L(OC)= e1600
L(SC)= .1850
LAMBDA= .1667
LENGTH= oC730
SF= 1800 MCS
SZOC= 8.18664005 t1251443551
ZSC= 27.66337154 -33049033998
ZCH= 25.37679253 1.46736654
"ALPHA= 3.94698271 ALPHA P= 34,28349180
"0tOCGA= 1.lioq7337E+1O
Z 3,734&7L47 -. 41#596331 ZO 50.0
F- E'ETA LC.'STA14 SIGHAtI0 CPSILGN
0 6.6157 1*8526 3.e7K6E-04 .0198
1 28.1335 .2S6? 1.5626E-03 °-j1g
2 4396513 .1600 2#7573E-03 1.7236
3 71.t699 .1113 3oS530E-03 ?.5529
4 •92.685 .0853 5,1481E-03 6.0337
5 114.2045 .0e692 6.3433E-33 9.1E61
6 135e7223 .0582 7o5385E-03 12°9501
7 157.240t .050? 8.7337E-03 17.3856
8 178e7578 .0442 9,9288E-03 22.4727
9 200.2756 .0394 1.1124E-02 23.2114
t10 2?17933 .0356 1°231IE-02 34.6017
it 2'o3,3111 -0325 loZ514E-02 4t.6435
12 Z6,48?08 .0233 1*4710E-02 49.3369
13 28663166 .0276 1°5905E-02 57.6819
t4. 3G7.864,4 .-9256 1.71 W.l 66.6784
218
CASE LAGUNA JCYUDA
VSWR(OC)= 1.70
VSWR(SC)= 3.20
L(OC)= .2330
L(SC)= .1690
LAMEDA= .1667
LENGTH= .1730
SF= 1800 MCS
ZOC= 38.38609755 20.45308475
ZSC= 15.73457789 -3.97596541
* ZCH= 26.37420836 3.20765759
ALP14A= 3,43363526 ALPHA P= 29.52455586
OMEGA= 1i.13097337E+10
Z = 3.43839237 -. 84891835 Zo 50.0
SN l1'TA LCSSTAN SIGMA EPSILO4
0 -1.7691 1.4029 -8.548:SE-05 -. 0061
1 7.3106 1.2052 3.5325E-04 o0293
2 16o3904 .4382 7.9197E-04 .1807
3 25.4701 .2746 I.P30?E-03 .4482
"4 34o5499 .2007 1,6694E-03 .8316
S5 436296 .1584 2°1082E-03 1.3311
6 5?.7094 .1308 2,5469E-03 i.94b6
7 F1o7191 .1115 2.9856E-03 2.6780
8 71.8•.9 .0971 3.4243E-03 3.5256
9 79o9486 .0861 3.8631E-03 4•.891
£0 89o0284 .0773 4.3013E-03 595686
11 98.1081 .07*.1 4.7405E-03 6.7642
12 107,1179 .0641 5.1793E-03 8.0758
13 115.2676 .0591 5,6180E-03 9.5033
14 125.347'. .0548 6.0567E-03 11.0469
29
CASE LAGUNA JOYUOA
VSWR(OC)= 1*44
VSWRCSC)= 2*30
L(OC)= .1700
L(SC)= .1610
LAMPOA= .1667
LENGTH= .2730
F= 1800 tCS
ZOC= 35,aOA93333 -3o24535620
ZSC= 22.56150637 8.71889857
SZCH= 28.88312784 4o01621628
ALPIIA= 3,49092984 ALPHA P= 30*32221663
OMEGA= 1.13097337EO10
Z = 2.d28•318 -. 80208519 70 sp.°
N P'ETA LOSSTAN SIGfIAEp
0 1.6159 -1.1732 7o I3s 6- 7
1 7,3697 1,2214 3.U,?9'0-04
2 13.1235 .5725 6. Is4f'.':-C 4
3 1iq8774 3 A",Y,073E2E-04 2
4 24.6312 e2893 1.2100•-
5 33.3850 .2329 1.4c927E-G3
6 36,1389 61'50 1.7753E-93 5 4
7 41s8927 .1678 ?.0585F-03
8 47.6465 .1473 2.,I'C7'E-03 1.5188
9 53.4004 .1313 2.6233E-C3 1.9919
10 59.i542 .1184 2,90601-03 ?.Y535
11 64.9080 .1079 3.117.-031 i3a
12 70.6619 .0090 Z*4?13E-C3 3.5241
13 75,4157 .0916 3,7540E-33 4.tu1
14 82.1695 00851 4oli3hGE-03 4.7421
v 30
)I
CASE TEFLON
VSWR(O(;.b. 43,UbO ZoC 1.19714719 -103.8518543
L(OC) 9 ,2450 ZCH 34*68348262 ,81443966
VSWR(SC) = 39.6000 7SC 8,21415972 11/°07662124
L(SC) = .1360 ALPHA °3Lhd06703 ALPHA F 3oC23 1(:Il
DELTA L = .2710 OHFGA 3o76991124E+09
LAMODA = o5000 Z 2.0748008 -90974S492
LENGTH = .0730 ZO = 50.0
FREGUENCY 600 MCS
N BETA LOSSTAN SIGMA EFSILON
0 -3.9235 -.*18d -5.7654E-u4 .0967
1 17.5942 .635E6 ?.d54E-uZ 1.9595
2 39.1120 .0178 5.7473E-03 9.68b5
3 60.6297 .5115 6o9092E-b3 23.2776
5 103.6b53 .0067 1.5233E-02 61.0521
6 125.1830 .O65E 1.8395E-02 99.2357
7 146.7008 .6647 2.1557L-G2 136*2133
8 J.68o2185 .0041 2.4719E-02 179.1951
9 189.7363 .0037 2.788!E-02 227.9710
10 211.2541 ..J33 3.1343L-02 62.62111
11 232.7718 o0030 3.4204E-U2 343.1151
12 254.2b9G ,5u27 3.?366E--2 409*4136
13 275.8073 .0025 ,.052AL-J2 431.7161
14 297.3251 .J023 4.°590E-0O 553,8127
5i 31dJ8428 93U22 4.6852E-02 E43°7735
16 340o3606 U21'3 5.0014E-62 7J3.5984
18 38 3 *3zlbl 18 D, s 3 - . d z, 3u
19 40L,.S139 #"J17 115 2 -. 31251
31
- ,-?--
LM ~ I ItrLUg'NVSWR(OC) 43.5b00 ZOC 1.l97Ia~yb -10.18518301L(OC) .0 9 5d ZCH 35oS.340633 °77646024VSWR(SC) 39.6000 ZS = 9o279'44658 125.69372525L(SC) .2340 ALPHA = .3443U545 ALPHA F 2F9 =63719
DELTA L = .1710 CEF'GA = 3-?E991124L+O9
LAtEDA = .5000 Z = .937074 --01374163
LENGTH = .0730 ZO 50.0
FRECUENCY = 600 MCS es, a ed,
N BETA LOSSTAN SIGMA EPSILON0 -3.7925 -. 1831 -5.5126-EJ4 .09031 I7.7753 .038S 2.576:;E-d3 1,98
39°2430 o91 7 5 5°704ZC-0. 9.7515
3 6097608 °C113 8°8319E-53 23.37d3
4 82.?785 08 119J-j426?5 103.7963 ,006E 1.5078?-02 68*224.6 125.3141 09055 1,8215U-02 99.4435
7 146.8318 .0047 2°1343E-02 135e52698 168o31496 ,0001 2 .4471-02 17994744
9 189,8673 .aoc . 8- 221.236010 211.3851 .Cd33 3°072LE-02 282.9618
ii 232.9028 c03J 363854C-02 343.541712 254.4206 .ý27 1506962E-02 409.93o
13 275.9354 °JJ25 400109E-42 48e.174ý
14 297.4561 .,)2.s 4.32-7c-02 5a,30E315 318.9739 ..322 1,063UýE-V2 E44.3023
16 340.4916 .0U20 4o'19?c-d2 134.163417 362.0b94 ,001S •°262.3..02 123.8851
18 333.5Z72 ,ruilp 5.5741L-62 S31.4770
19 405.0449 -3;,17 1
32
CASE TEFLONVSWR(OC) 40.060G IOC 116.Si3t6835 *.6L.7U513901
L(aC) .2120 ZCH 320224'.q69± 11-15q1~c0 4
rVSWR(SC) =43.5000 ZSC 1,1515,5956 -1.8b485340L(SC) = 3320 ALPHA = .15081760 A~LPHA P 13OC6
DELTA L .1710 OMEGA = 3.76991124E~+09
LAMBODA = .5000 z 2oZ4573265 -*701519'.8
LENGTH .73 zaS0* 1 7 3 z 05 0 .0 ý k ep ý'lc e dFREQUENCY 600 mcs 'to
N BETA LOSSTAN SIGMlA EPSILGN
0 -.3610 -14iJ123 -2o2983F.-L5 000
1 8.7188 *0346 5*51.sE-i4 .148122 17,7985 *0±bS lo.1S3?E03 2.000!.)~
.3 26.8783 .31 d14 34. 5748S
5 45,.0378 .6,06? ?8676bL-03 12,844i9
6 54,1175 06056 .3.44'.7.?-u3 18,5461
7 63.1973 .~.349u2ý,8E-03 2i.2915
a 72.2770 .0042 4*6019L-C,3 33,0F3109 81s3568 .0037 5.1801L-,s 4.1a9147?
10 90.4365 .J033 5.7562L-03 51.792:)
11 99.5163 lu0ia a.33P63E-~jj b2.71'.4
12 1055960 -ý2 b.9144;--u3 .35
13 11746758 *002E 7.4.925L-03 b 7 ,U~g07
14 1?6.7555 *L024 ~117.~15 135.8352 ..JJ22 'J.C488EL-03 11*5
16 144.9150 oC021 ý.22(69---u3 132098r63
17 153.99'4? jJf2L- 9.'61FCE-u3 5.7918 163,074.5 .11,18 l...&3o-
33
APPENDIX B
LABORATORY PROCEDURE FOR KIRKSCETHER'S
TRANSMISSION-LINE TECHNIQUE
Procedure
The sample of soil is packed into a (;eneral Radio ((;R) Type 87 Air Line with theapparatus shown in Figs. 3 and 4. The center conductor exiension of the apparatus isscrewed to the center conductor of [lie air line. The line is now ready to receive the soilsam ple.
The soil sample is funneled into the air line a little at a time. The soil is tainpedbetween levels of soil until the entire air line is filled (Fig. 5). The center conductor ex-tension of the apparatus is unscrewed from the (enter conductor of the GR Line. Thiscompletes the filling (Fig. 6).
The air line containing the soil sample is conneeted to the end of the Alford slottedline with a suitable connector. The tunable probe should be inserted in the line and ad-justed for maximum signal out.
The frequency to be used in the attenuation measurements is selected on the Hew-
left Packard 5105A. Frequenc" SynthesLer.
The Alford slotted line utilized for these experiments was 5 feel long, thus limitingthe lowest frequency to 100 nillz without the use of extensions. A (R Constant-Impedance Adjustable Line (874-I.K) may be used to obtain an accurate one-half.wavelength line (Fig. 1) for each frequency to be used. The one-half-waveiength linesmay be obtained in the following manner:
a. The wavelength in meters can be determined by using the following formula:
X.= i- 0 avelength in mecters• • f mllzf m~l (assuming C. = Co)
b. Before the one-half-wavelength line is attached to the slotted line, a calibratedshort circuit is connected and the first null on the slotted line is located as shown inFig. 14.
~~Ar'4 I ~
I q MOO. 21-81-6
GENERAL RADIO SHORT(874-WN3)
S~HP 41SESWR METER
Fig. 14. Location of voltage null.
C. Once the null point has been located, the detector probe is then at one-halfwavelength from the short affixed to the end of the slotted line. The short is removed,and a Telonic calibrated VSWR standard is placed on the end of the slotted line (Fig. 15).The probe is moved along the line to seek the maximum voltage; and, when this is ob-tained, the gain of the VSWR meter is adjusted for 0 dl. The probe is then movedalong to find the voltage minimum. The VSWR is indicated by the meter. This readingcompared to the standard on the end of the line is the error. The error (foes not exceed
e 5% on a consistent basis.
d. The standard is removed and the Gil short is replaced on the end of the slotted
line. The probe is again moved along the line away front the short and is stopped at thefirst voltage minimum as shown in Fig. 16A.
e. The short is removed from the slotted line and )placed on the end of the one-half-wavelength, adjustable line. The line is adjusted so that a minimum occurs at thesame place as it did with the short connected directly to the slotted line as shown in"Fig. 16B.
35
SAVANTEK AMP=FE ADp.o
ALFORD TUNED PROBE
10 db PAD
TELONIC CALIBRATED ALFORD SLOTTED LINE,VSWR STANDARD MODEL 21.81-6
UHP 415E VSWRI METER
Fig. 15. System calibration.
f. The adjustable line must now be shortened by 4.6 centimeters to move theplane of the short 4.6 centimeters toward the genrator as shown in Fig. 16C. Thereason for this is to account for the spacing inherent in the General Radio connectors.It ensures that when the adjustable line is connected to the output end of a section ofGR line filled with soil, the open or short circuit will be traisformed exactly to air-soilinterface (Fig. 16D).
g. It may be well at this point to recheck along the slotted line the distance be-tween the nulls to make sure it corresponds to the right frequency. It can happen that,if the tuning knob on the probe is unintentionally turned, the distance between nullswill not be related to the generator frequency.
h. The carriage containing the probe/detector is moved along the slotted line tofind the voltage maximum.
i. Once this voltage maximum is found on the VSWR meter, the gain of the in-strument is increased so that the meter indicator reads a VSWR of 1 .00:1 (0 db).
36
71Wý
rI
I-%
x 0
C4 w
-,00
z wU
w ww
ca X0 0 W
us I-
z z)0LU) IU L..).
CA W
0CIS
00.. 0..
us 0z LU
z
37
.4 - -. L .p 4u
Im.:LU
.-i
I- zz
00
Lui
0 - i
a.a
00
C--
38D
j. The operator should now record the following:
(1) The VSWR with the soil-fillcd line electrically short circuited at itsoutput.
(2) The VSWR with the soil-filled line electrically open circuited at itsoutput.
(3) The distance expressed in meters from a voltage minimum to the inputair-soil int'erface with the short circuit in place (Fig. 17).
(4) The distance expressed in meters from a voltage minimum to the inputair-soil interface with the open circuit in place (Fig. 17).
k. When data is obtained from a slotted-line system, one of the best aids for de-termining the normalized input impedance of the open- and short-circuited, soil-filled,coaxial line is the Smith Chart, and one proceeds to calculate the complex propagationconstant by hand. However, this is tedious and not recommended. It is advisable touse the computer program.
I. The Smith Chart approach works well for soils which have average to highattenuation. Normally, this produces VSWR's that are less than 10.0:1.0; however.low-loss materials such :as sands produce VSWR's which are quite high.
m. Accurate, high VSWR readings for low-loss soils are best obtained by usingthe "Ten-Times-Minimum Method." 9 Measure the distance (d) between positions onthe standing wave pattern where the voltage is 10 db above the voltage at the minimumor null point (Fig. 18). Substituting the value obtained from the slotted line at the10-db points with the wavelength used in the following formula results in an accurateVSWR reading:
VSWR -A x
Recording the position of the 10-db points and the null point (Alford Slotted Line hasa centimeter scale) in this method is important because it enables the operator to recheckhis work if the need arises. This method is the same for both the open and shortedmodes. The distance from the null point to the front face of the soil sample must berecorded. An example of a record is given in Table V.
9Hewlett Packard Operating and Service Manual l6, SWR meter 415E, Section 3, par. 3.29.
39
1."
VIVWi
Fig. 18. Ten-times minimum method for measuring high VSWR.
Table V. Data as Listed in Notebook
FREQ. SHORT XI X2 X3 Ax d(mHz) (cm) or OPEN (cm) (cm) (cm) (cm) VSWR (i)
100 300 SHORT 61.05 65.35 69.00 7.95 35.8:1.0 1.116100 300 OPEN 23.95 25.30 26.65 2.70 105.0:1.0 1.515
where:
S= 10-db point
X2 = null pointX3 = 10-db pointd = distance from the probe to nearest soil face
The above values of VSWR (short and open) and their respective "d" values arc enteredinto the computer.
40
g=•
[. APPENDIX C
r . MODIFICATION OF TRANSMISSION-LINE EQUATION FOR INPUT IMPEDANCE
The Smith Chart greatly simplifies calculations of impedance from the measure-ment of VSWR and electrical length. This length can be either toward or away fromthe generator. Calculation 4f the impedance from the transmission-line equations de-mands that this distance from the voltage minimum be measured in wavelengths towardthe generator, since this is the premise on which the equations were developed.
When a measurement of voltage minimum is made for a sample, it is ordinarilytaken in distance from the air-soil interface-the distance away from the generator,that is, toward the load. The following manipulation serves to put the transmission-!ineequation in the proper form for transformation of impedance away fron) ti, gm-41ator.
From the literature:
Zc Z. [ Zd cosPd+jZosinPd d
ZSC Zo cos P d +j Zd sin3P d
at a voltage minimum
ZoZmn- VS--WR (2)
Zd = Zmin (3)
Scosjdd+jZosin dZ[_ Z VSWR (4)
8c Zo cosIpd+j VSW sing3 d
__ - [vs~ cospd+jsinpd 1Zoe_ Z VSWR(5ZSC cosp3 d +j I sin5)Z 1V sinj~d I
V SWRJ
41
where, now, Zoc or Zsc is calculated from a measurement of the VSWR (under open orshort circuit condition) and the corresponding electrical length from the voltage mini-mum to thc input air-soil interface
where:
Zo =50 ohms characteristic line impedance (6)
and
=--; X measured in meters. (7)
Using a diagram to illustrate:
*Psell
The calibrated open circuit is placed at the end of the line and the respective opencircuit parameters are measured. Next, the calibrated short circuit replaces the opencircuit ind the short circuit parameters are measured. The electrical length in both casesis measurt wi vw from the generator (toward the load).
Consider the sine wave below which would be a plot of the variation of voltage am-plitude vs distance toward the load:
42
S -
A voltage minimum is found to exist at point x2 . If the load exists at point x3 , then acorresponding point is found at x, ; and the distance from x, to x3 is one wavelength.The distance measured toward the load is d; thus, the corresponding distance toward
the generator, L, is
L = A/2 - d in meters. (8)
Now, d in equation (5) must be replaced by L from equation (8). Thus, equation (5)becomes, with this substitution,
• ~[1VSW-R1 cos 6 (X/2-d) + j sin ( (X/2-d)
•:~~ = or Z P c (X/2-d) +j .. l--.sin P (A2-d)
where d phase constant of the soil-filled line (radians/meter), and substituting forfrom equation (7):
Z d 1W- cos 2.! (V~2-d) + j sin (X/2-d)0CVW 2 25 cosr (V~2-d) + j ~W sin L~ (X/2-d) (0
where:
SZ• = input impedance of a soil-loaded line terminated in a short circuit(ohms)
Zo --input impedance of a soil-loaded line terminated in an open circuit(ohms)
d = distance from a voltage minimum to the input terminals of the soil-loaded line (meters)
V SWR = voltage standing-wave ratio = Vmax /Vmin
X = free space wavelength (meters)
Z = characteristic impedance of the air-filled line = 50 ohms.
43
APPENDIX D
SOLUTION OF LOSS TANGENT EQUATION:
"~o 0_
zotanhZyR (1)
where ,y = propagation constant of the soil-filled line (complex quantity).
LetZ = c+jd (2)
and Zoc = g +jh (3)
th c+jdten _- +jh(4)zoc g~j
R ationalizing yields
c+jd g-jh cg+hd (dg-ch)g-+jh "-jh g2 + h2 + g2 + h2
cg+ hdlet a = (6)
andb dg-chad gb +h 2 (7)
then ' = a+jb (8)z-c
From Fig. D-1, it can be seen that a vector,'v, in the complex plane can be represented
by the notation 'a+ jl•
44
•"I
*'•• "* ' ;° "-*<P *. "',•4A>'i' *,...... .. .. .... . . . .. -,.,',! " '. .. i"• .'••... . .. . . . .. .. ---... .. .. ... I'•
V- - -
Fig. D-1
"Thus a + jb can be equated to a product of an amplitude times the phase angle. If welet this phase angle be represented by ei'P, then:
v = a +-ju- la+jb leiO (9)
From Fig. D-l, we have
I a,+ j-ri= Na2 + b2 (10)
tan ip = b/a (11)
arid cos V a (12)S•/ a2 + b2
or '+j=Va2 +b2 ei' (13)
-a7+ -b= (a2 +b2)Y4 ei0/2 (14)
From Eulcr's theorem:
eiP/2 = cos 0p/2 + j sin 0/ 2 (15)
so that V"ja =(a 2 + b2 )¼ [cos ýp/2 + j sin ip/2 1 (16)
45
- &-
cos 0/2 = 1/2 (I + os ip) (17)and sia 0/2 /1/ 2 (1 =cos ýp) (18)
thus: V"a+ = (a, +b2 )'•4 [\i/2(l +cos•) + j ýV/2(l -cosp) 1 (19)
From equation (12):
Via ( a 2 + b 2 ) V 1[ l / 2 ( 1+ a / + [ 1 /2 ( 1 a Y .)1/2 (20)
\fa + f b 2 +a + j[N/ a ýJV (21)2 2 -2 2 (1
S~1Va + a 22
and B = aL2 2 (23)
"a-+"= A + jB (24)
From equations (1) and (8):
tanh- tqo = A+ :B (25)
YR tanh'! (A +jB) (26)
1/2 "1-- (A + jB) (27)
Rationalizing yields
46
,y (l - A)2 + B2 (I - A)2 +
•,2,yR0 R n I - (A2 +B)+J 2 (29)( (- A? +B2 (I - A) 2 +B 2
.e2'R, 1 - (A2 + B2 ) + J 2B (30)e2'V•( -(1 -A)+2 B -(l-A) +B2
By definition, •y = c + jB (31)
-e2l = e2(a+iB)Ro =ea% eji2P°O (32)
Using Euler's relation again yields:
e•�'yRo = e2RO (cos 2P3 +J sin 213 2,) (33)
Equating real and imaginary components of equations (30) and (33) yields:
e2Qo° cos 2 O= 1 - (A2 + B2 ) (34)(1-A) 2 +2
e2Qoa sin 2 R0 3 2B (35)(I - A) 2 + B2
Squaring equations (34) and (35) and adding the resultant yields:
Se cos 22£3+e£oO~sin22£ = 1-(A2 +B2) 2 4B2-42C 224a2 + J- + (36)( -A) 2 +B2 1(1-A) 2 + B212
47?.
Since sin2 0 + cos2 0=1
4a= I-(A 2 +B 2 )1 2 +4B2 (37)
[(1 - A)2 + B212
Finally, has the solution,
n [I-(A2 +B 2 )12 +4B 2i• a =42o [(1 - A)2+ B21]2 (
where a = attenuation constant of soil-filled line (dib/meter) and
A and B are functions of a and b which, in turn, are functions of Zc and Zo.
a = -- 2n [ f (Zsc, Zoc) I which is the result shown. (39)
In order to solve for P, we again make use of equations (34) and (35).
If we divide equation (34) into (35). we have,
e2Qoa sin 2 o A 2 B(I_ 0 -A)'+ B2 (40)
C2 2i cos 2 go I -(A 2 + B2 )
(1- A) 2 +B 2
!i 2Btan 2 2o -- (41)
1 - (A2 + 12)
=- tan'l (42)221- -(A 2 + B2 )
.4,•
48
S. . .•. •. • .... .. .. . .... .. .. .. • .... •. .... -•. i • •,Ol i-.• . . . .. .i • ,,.. .• .•,,. .,, _ . ... .. . ...•.• . .. .A
However, since the tangent is a multival'ied function, equation (42) presents onlythe first-order solution and not necessarily the correct solution. Thus, the solution for(is not unique, and the correct value must be determined by other measurements.Valups obtained from the equation,
______1_[_2 1 (43)I •n tan- 2B +n
(3-+-(A2 +B 2 )
Or - tan [g (ZS, Z,,) + +n r (44)
are tabulated for several values of n as an output of the computer program. Whereg (Z8C, Z..) are known functions, P. is the nth solution of a multivalucd function andRO is the physical length of the soil-filled section of coaxial line.
I
49
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