Sensor and Simulation Notes

Note 257

October 1976 .

A Study of Charge Probe Response ona Junctionof Unequal Radii Thin Wires

Santiago SingarayarB. M. Duff

University of Mississippi CLEAREDFORPil&LICRELEME

p@39 5%+97

Abstract

The experimental measurement of charge density induced on conducting

surfaces often makes use of a short monopole as the charge probe. Measure-

ments on structures containing junctions of unequal radii cylinders have

been hampered by incomplete knowledge of the probe characteristics. An,

investigation of charge probe response as related to the radius of the

cylinder on which it is mounted is reported here. An experimental determin-

ation of the ratio of probe response when mounted on two different radii

cylinders has been conducted using a coaxial line geometry with a step

change in radius of the inner conductor. In addition, a theoretical solution

in numerical computations of the probe response in relation to the radius of

the cylinder upon which it is mounted has been conducted. The results of

these studies are presented in the following report.

o

TABLE OF CONTENTSN

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . .“.3~

CHAPTERw

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . ...5

II. MEASUREMENT TECHNIQUE . . . . . . . . . . . . . . . . . . . @*

lntroduc.tLon . . . . . . . . . . . . . . . . . . . . . . 8

Analysis of Coaxial Transmission Line . . . . . . . . . . 12

Description of the Measurement System . . . . . . . . . . 1.6

Determination of Probe Calibration Factor . . . . . . . . 21

Curve Fitting: Modified Prony’s Method . . . . . . . . . 2~”

111. NUMERICAL ANALYSIS. . . . . . . . . . . . . . . . . . . . .28

Charge Probe as a Scatterer: Theory and Solution . . . . 28 m

Formulation of Straight Wire Integral Equation . . , . . 29

Charge Probewith50flLoad . . . . . . . . . . . . . ..32

Numerical Solukion Technique . . . . . . . . . . . . . . 33

Expansion of Current and Formulation of Matrix . . . . . 35

Determination of Probe Calibration Factor , . . . . . . . 39

IV. PRESENTATION OF DATA AND CONCLUSION . . . . . . . . . . . .42

Introduction . . . . . . . . . . . . . . . . .’. . . . .4z

Measured Data for the Charge Distribution . . . . . . . .42

Probe Calibration Factor . . . . . . . . . . . . . . . .~2

Conclusions. . . . . ... . . . . . . . . . . . . . . . . .64

“

2

1

oLIST OF FIGURES

FIGURE*

2.1

8 2,2

2.3

2.4

2..5

2.6

3.1

3.2

3.3

3.4

3.5

3.6

4.1

4.2

4.3

4.4

4*5

4.6

Charge Probe . . . . . . . . . . . . . . . . . . . . . . . . .

I/p Variation as a function of (p-a) . . . . . . . . . . . . .

Coaxial System with Step Radius System . . . . . . . . . . . .

(a) Zero Order . . . . . . . . . . . . . . . . . . . . . . .

(b) Approximate actual distributions of Charge per unit

length and Radial E field in Coaxial Line . . . . . . . .

Coaxial Transmission Line . . . . . . . . . . . . . . . .

Block Diagram of Measurement System . . . . . . . . . . .

Charge Probe on a Charged Conductor . . . . * . . . . . .

Charge ProbeasaScatterer . . . . . . . . . . . . . . .

Lumped LoadZ1oadati= 0 . . . . . . . .. . . . . . . . .

Triangles in Piecewise Linear Testing Scheme . . . . . . .

Pulse Arrangement for Approximating current on Straight

Wire. . . . . . . . . . . . (l. . . . . . . . . . . . . .

Relative locations of-Pulses (Bases Set) and

Triangles (Testing Set) on Straight Wire . . . . . . . . .

Maximum Charge Condition - Probe Height 0.10”, al = 5/16”

Medium Charge Condition - Probe Height 0.10”, al = 5/16” .

Minimum Charge Condition - Probe Height 0.10”, al = 5/16”

Max5.mumCharge Condition - Probe Height 0.08”, al = 5/16”

Medium Charge Condition Probe Height 0.08”, al = 5/16” . .

Minimum Ch&rge Condition - Probe Height 0.08”, al = 5/16”

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

● ☛

✎ ✎

✎ ✎

✎✌

PAGE

9

11

13

14

14

15

27

30

31

31

34

38

40

43

44

45

46

47

48

3

-!- =

.

FIGURE

4.7

4,8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

Maximum Charge Condition - Probe Height 0.06”, al = 5/16” .

Medium Charge Condition - Probe Height 0.06”, al = 5/16?’. .

Minimum Charge Condition - Probe Height 0.06”, al = 5/16” .

Maximum Charge Condition - Probe Height 0.10”, al = 3/161’ .

Medium Charge Condition - Probe Height 0.10”, al = 3/16”. .

Minimum Charge Condition - Probe Height O.10~’,al = 3/16” .

Maximum Charge Condition - Probe Height 0.08”, al = 3/16” .

Medium Charge Conditicm - Probe Height 0.08”, al = 3/16”. .

Minimum Charge Condition - Probe Height 0.08”, al = 3/16” .

Maximum Charge Condition - Probe Height 0.06”, al = 3/16” .

Medium Charge Condition - Probe Height 0.06”, al = 3/16!’. ,

Minimum Charge Condition - Probe Height 0,06”, al = 3/16” .

Probe Calibration Factor for 5/16’~to 1/8” ratio of radii .

Probe Calibration Factor for 3/16” to 1/8” ratio of radii .

PAGE

49

50

51

52

53

54

55

56

57

58

59

60

61

63

.

o

●

●

0

The

antennas

standing

measurement of

and scatterers

Chaptez I

INTRODUCTION

the current and

is important in

charge distributions on

order to gain an under-

of their electromagnetic behavior and to aid in developing

and checking of theoretical solutions, The basic techniques for\

making these measurements have been established for some time and

used by many investigators. However, during a recent experimental

project, to be described below it became obvious that more accurate

knowledge of the response characteristics of charge probes was

necessary. The resulting investigat-ionof charge probe response

forms the basis of this thesis,

The experimental project which leadsto the investigat~on re-

ported here, concerned the measurement of

induced on structures containing junction

radii. The purpose of that investigation

charge distribution

of thin wires of unequal

was to experimentally

determine the appropriate condition on charge which should

applied in theoretical solution of this type of structure.

A theoretical analysis and discussion of the junction

be

has been

presented by T.T. Wu and R.W.P. King [3]. The probe used for the

measurement of charge distribution consisted of.a very short mono-.

pole mounted on the surface of.the cylindrical structure. For

the experimental determinati.onof the charge-condition at the

5

-junction,an accurate knowledge of the probe response as a function

of the radius of the cylinder upon which it is mounted is necessary.

A charge probe mounted on a cylindrical structure has been analysed

theoretically by R.W.P. King [1]

However, these investigations do

solutions for the probe response

and experimentally by Whiteside [2].

not provide sufficiently accurate

as a function of the radius of the

cylindrical stru~tures. An independent experimental and theo~’etical

investigation of the charge probe response has been conducted, the

results of which are presented in this thesis.

Chapter II deals with the measurement technique by which the

probe calibration factor can be determined. This probe calibration

factor has been determined for 3 different probe heights for 2

junctions of thin wires of 5/16” to 1/8” radii and 3/16” to 1/8”

radii. To make sure that the probe calibration factor is the same

for all charge conditions at the junction, the probe calibration

factor has been determined for 3 different charge conditions namely

,

8

maximum, minimum, and a condition in between maximum

The experimental data was analysed by using modified.

A numerical solution for the current induced in

and minimum.

Prony’s method.

the probe

has been carried out using moment method techniques [4,5]. The

analysis models the probe as a monopole receiving antenna perpen-

dicular to an infinite plane condcctor, but includes the incident

~ field variation which is to be expected when the probe is mounted

on a cylinder. Chapter 111 deals with the methods used for this

numerical analysis.

6

Finally, in Chapter IV, the experimental data, the results and

the conclusion are presented.

w

.

●

CHAPTER 11

MEASUREMENT TECHNIQUE

(i) Introduction

The measurement of the surface charge density on a conductor may

be related to ameasurementof ~ field by using the boundary condition,

fi.cE = rjo

(2.1)

where &o

is the relative permittivity, 6 is the unit normal vector to.

tfiesurface of

surface charge

charge density

the test structure under consideration and q is the

density. It is therefore possible to obtain the surface

by using a sensor whose output is proportional to the

normal component of the E field at the surface. The most common method

uses a short monopole receiving antenna, as shown in Fig. (2.l),which

is perpendicular to the surface of the test structure. The signal

induced in the load is however related to the ~ field along the entire

length of this monopole rather than simply the field at the surface.

If the normal component of the ~ field is constant over the length of

the probe there is no problem. But generally this is not the case.

In order to relate the monopole probe response to the charge density

on the test structure it is therefore necessary to have some knowledge

of the manner in which ~ varies over a short distance away from the

surface.

.

8

w

‘8 f

;.

Coaxial cable

-.

f 1

--L_+--—.—. .—.! .—. —. —.—. -(!9* .—.—I*(a) Probe carriage for smaller radius tube.

a *

.—— —7T’—

. — . —. —.— . —.t“ ‘—1t1

.

(b) Probe carriage for

—.—. e ,’.—— .1 .—1!Ilarger radius tube.

Fig. (2.1) CHARGE PROBE.

I

The present work considers only test structures composed of

electrically thin conducting cylinders. The field produced by a

1uniformly charged infinite cylinder is known to vary as —. If the

P

test structure (a cylindrical antenna) is of electrically thin radius

and the probe is electrically short, then ~ over the length of the

probe is controlled primarily by the local charge distribution. The

field then becomes.essentially the same as that of an infinite cylinder

and ~ is very nearly

near discontinuicies

of this variation on

1proportional to –. This assumption is not valid

P

of the conducting surface, however. The effect

the response of the probe depends on the radius

of the test structure. To illustrate this effect, Fig. (2.2) presents

~plotted as a function of (p-a) for three different cylinder radii,

For a given probe height, the variation of EP

over the length of the

probe is less when the probe is mounted on a large radius cylinder.

The effect of the: variation ntaybe reduced by reducing the

probe height. However, the signal induced across the load is also

reduced and so the signal to noise ratio. A second effect of changing

the radius of the test structure is that the image plane for the

monopole probe is of different radius of curvature. The fact that the

charge probe is of finite size and it responds to fields away from the

suiface of the test structure makes it necessary to determine a probe

calibration factor as a function of cylinder radius in an exact manner.

To determine the probe calibration factor experimentally, a system

containing cylinders of different radii and for which the variation of

the charge density as a function of cylinder radius can be predicted

o

t . *

e

,. wP (b

●

0.04 0.06 0.08 0.10 0.120.0 0.02

Fig. (2.2) ; VARIATION AS A

(p-a) in inches.

FUNCTION OF (p-a).

1’

I

is needed. The system chosen was a coaxial line with step change in

the radius of the inner conductor. This system is shown in Fig. (2.3)

and has been analysed by Marcuvitz[6]. The TEM mode ~ field dis-

tribution of the above system is shown in Fig. (2.4). The ~ field is*

uniform and proportional to ~ inside the coaxial line except near theP

junction. The local effect near the junction can be replaced by a

shunt capacitance.so that the Kirchoff~s current law is satisf~ed.

At the junctton the voltage is continuous making it easy to analyse

the charge condition at the step in the following way.

(ii) Analysis of Coaxial Transmission Line

Fig. (2.5) illustrates a coaxial transmission line with a TEM

mode propagation. In cylindrical coordinates the two dimensional

Laplace equation is,

1 32@~*(p?&=-= oP a~2

For a potential function independent of angular coordinate+,

eq. (2.2) reduces to

The solution of equation (2.3) may be expressed as,

(2.2)

(2.3)

-(2.4)

●

Imposing the bo!.mdaryconditions @ = VO at p = a and @ = O at p = b,

eq. (2.3) yields the final solution for the potential as,*

12

● 1

e

1,,

I

PLJ

.

4I P’

Charge Probe I $dvable Short “1a ( .--, --- ---I

II

a

1---- ---- ---

I 4/

iJ _ .If.—.—.—.—-—.—“z

I----- ---- -- ---,I ------ ---

/ 4

Step radius system/ I Coaxial system ~

Ito#

t1

I,,

;1 ,1

1’

Fig. (2.3) COAXIAL SYSTEM WITH STEP RADIUS SYSTEM.

F.b

●

m“

------- .

---- ---

.— --- -

-------- --

------ --

-“

------ -

.z----.----

.

r

.

---’--------—

-----------

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

----- ----”

---- ---- -

+n>

4.

t

+

+

+4.

+

+

+

+

+—

,------ ---

----- --

-.-...+_ --

----% ---

-------

..-.-------m-------

-. . .------ -

--‘. ------ -

-- --~------

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

----- ----

[

r

------ ---

----- --- -.----- -------- ___

---—- ------- ---------- -_-,----- -------- -------- --—_ .

1

Iv!-cr~”

I

_____ ____

—--- --- ---

------- ----

--- ------ _.

---- - ----

1I

i1

1IIIII1

.a>.!

I

I1III

III

;T

*NN

0w ;

P.-—-- ---

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

----- --~

--- -— --”

---- —-

---- ----

-- -----

----- ---

---- --- -.- —- --- ----

----- ----

---- -----

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

---- ------ -

N&nN g

ri

-,,, ,

i

●

●

----

---

,)-

k field

~ field_—

Fig. (2.5) Coaxial transmission line.

an associated charge density which differs from that of a TEM mode as

indicated in Fig. (2.4)(b).

The voltage

different radii,

at the junction.

the junction are

is continuous at the junction but because of the

a difference in the magnitude of charge density exists

The ratios of linear and surface charge densities at

given by L

L .LL1 L,-=— .

(iii) Description of the Measurement System

A block diagram of the measurement system

(2.9)(a)

(2.9)(b)

.

.

is given in Fig. (2.6).”

The commercial instruments used in the system are given in the block

diagram itself.

The power at 300 MHz is fed to the amplifier. The amplifier sig-

nal is fed to the coaxial system through a balanced feed to avoid

higher order mode propagation. Matching of the amplifier output to

the coaxial system is done through a single stub tuner. The reference

signal to the vector voltmeter reference channel is taken from the

amplifier outpu~ with proper attenuation. The signal to the frequency

meter is also is taken from the amplifier output with proper attenua-

tion. The signal from the charge probe is fed to the test channel

16

o

~ field of TEM mode in +Z direction,

where k is the propagation constant.

Using eq. (2.5) ~ may be expressed as,

The surface density of

at z=O may be obtained

eq. (2.1) as,

‘O 2Pe-jkz——In(&) p

al

chafge ?IIon an inner conductor

from eq. (2.7) and the boundary

(2.5)

(2.6)

(2.7)

of radius a,,1.

condition

(2.8)

-.Equation (2.8) can be written as,

q~ = c1‘o

where Cl is the capacitance per unit length and ql = 2na1Tllis the

linear charge density. The same.relationship applies to the inner

conductor of radius a2“

It must be remembered that the above analysis assumes that only

TEM mode fields exi’stand that the junction is represented by a lumped

shunt.capacitance. The effect of the step change of radius is actually

distributed.over some short distance on either side of–the junction with.

#

17

of the vector voltmeter. The power level is monitored by the broad-

band sampling meter to compensate for the drift in power level.

The magnitude or the phase of the signal is switched to the.

digital voltmeter by means of the relay actuator. The calculator

samples this output 5 to 10 times and takes the average to minimize .

the effect of the minute-fluctuations in the probe signal. The magni-

tude and phase quantities are stored in the calculator for lat~r

plotting and for further use.

The stepped radius coaxial line was

brass tubing. The three different inner

constructed from cylindrical

conductor radii 0.3125’!

(0.79375 cm), 0.1875’’(0.47625cm), and 0.125’’(0.3175cm) were selected

as being available readily. Two inner conductor systems were con-

structed one composed of 0.3125T’and 0.1251’radius tubing, the other

of 0.1875” and 0.125’!radius tubing.

was used as the outer conductor. The

conductor was filled with styrofoam.

A tube of 1.0” inside diameter

space between inner and outer

A slot of 1/19’’(0.13368cm) was

cut axially in the inner conductor tubing to allow movement of the

charge probe along the structure. The probe carriage was designed

to slide inside of the inner conductor tube and extend through the

slot to the surface of the tube so that the structure is electrically

continuous. The charge probe was constructed on a probe carriage

using a semi rigid coaxial cable. The charge measurement was con-

ducted on two different radii tubes. If two different probes had been

used it would have been necessary that they be constructed identically.

.

0

.-

@Othe~ise, it would have become necessary to calibrate them indepen-

dently. To avoid this problem, a single charge probe with two different

probe carriages was used. The wall thickness of the tubes had to be

taken into consideration while making the probe carriage. For example.

for the 0.3125” radius tube wall thickness was 0.06’’(0.1524cm) and the

0.125” radius tube was 0.04”. The probe carriage fit exactly inside.

of the tube and through the slot to the surface so that the charge

probe was effectively mounted on the surface of the inner conductor

tube. The cable used for making .the probe was UT 20 manufactured by

Uniform Tubes, Inc., with the following specifications.

Impedance: 50 Ohms

o

Outside diameter: 0.023’’(0.05842 cm)

Center conductor diameter: 0.0045’’(0.011.43cm)

The outer–conductor of–the cable was soldered to

cut at the surface. The dielectric insulator on

was removed so that the inner conductor extended

the probe carriage and

the inner conductor

beyond the surface to

form the monopole antenna. It was necessary that this monopole be

straight and perpendicular to the test structure. The signal from the

probe was carried by a coaxial cable contained inside of the inner

conductor tube. Care was taken to see that stray pickup did not affect

the signal by shielding the signal cables. A 1/8” diameter tube

attached to the probe carriage was used both to protect the signal

cable and to provide a means of positioning the probe.

aThe

actual E

measured

charge density measurement was made at intervals of 1 cm. The

field distribution near the step due to local effect made the

result unreliable in this region. Moreover the charge probe

was mounted at the center of the probe carriage which made it impos-

sible to reach the step in the large radius tube. For analysis of the

data, afew data points were eliminated near the junction to eliminate

the local effect and the data was analysed by using a modofied Pronyfs

method to fit curves to the TEM mode distribution. This procedure

will be discussed separately.

Measurements were made for probe heights of 0.10’’(0.2540cm) to

0.06’’(0.1524cm) at an interval of 0.02’’(0.0508cm). The charge

condition at the junction was varied by using the movable short and

the experiment was carried out for three charge conditions at the

junction, namely charge maximum, charge minimum and a case in between

maximum and minimum. The experiment was repeated for consistency.

Even though frequency drift was observed, the change in electric length

of the test structure was of only of the order of ~ 0.00IA which was

considered negligible. Any change in the power output

later was

broadband

Care

compensated for

sampling meter.

w’astaken while

probe carriage fits flush

by monitoring the oscillator

making the probe to see that

with the test structure. To

of the oscil-

output using a

the top of the

achieve good

elect~ical contact between the probe carriage and the test structure,

.

0

silver conducting grease (Eccoshield, manufactured by Emerson & Cuming)

20

o was applied along the sides of the probe carriage and the sidewallsof the probe. The signal carrying probe tube was coated with an

absorber (Eccosorb by Emerson & Cuming) to avoid the possibility that

. the probe tube would become the center conductor of a TEM structure

inside of the inner conductor of the test structure and give rise to

.propagation and unwanted resonances.

After obtaining the extrapolated value of the probe reading at the

junction the probe calibration factor was obtained as described in the

following section.

(iv) Determination of Probe Calibration Factor

Let PI and P2 be the probe reading obtained by extrapolating the

measured data to the junction of–the large and small tubes of radii a1

0

-.

*

and a .2

Let N1 and N2 be the normalizing factors.

‘1 = Nlrj

‘2= N2?12

therefore

~1 = ‘1 ‘2.—~ ‘2 ‘1

nlfrom eq. (2,9)(b) the ratio of— is given by,

~2

a2—

al

(2.10)

(2.11)

21

N,‘HIe

(v)

probe calibration factor, $ is then given by,QL

Curve Fitting: Modified Prony~s Method

,

(2.12)

The experimental data obtained has magnitude and phase so that it

is a complex quantity. Curve fitting to the experimental data is done

by using a modified Prony’s method. Prony’s method [7] fits a set of

data to a series of exponential. By assuming a function of the form

f(z) = Aleyz + A2e-yz

because of the TEM mode transmission,

the exponential series of the general

Y= (a +

only two terms

Prony’s method

j $)

(2.13)

are selected from

with

(2.14)

.

.

.

where a is the attenuation constant, B is the propagation constant.

= A e(a+j6)z+A2e-(a+j6)’f(z) ~ (2.15)

where Al and A are complex constants, A2

~, A2, a, 6 are to be determined.

Let Az be the spacing between the data points and

~yAz‘1 =

(~.16)

-yA!z .

‘2=e (2.17)

220

then

Assume U1 and U2 are

f ~+l(!dz) = A1u1k+A2u2k

two roots of

U2+(%U+UO=01

where aoand a

1are unknown.

Equation (2.18) can be expanded to

‘1=A1+A2

‘2 = ‘cl 4“A2U2

2 2

‘3 = ‘I”l ‘-‘2U2

‘k= A1ulk--l+A2U;-1

(2.18)

(2.19)

(2.18)(a)

(2.18)(b)

(2.18)(c)

(2.18)(d)

Since the above set of equations satisfy equation (2.19)

‘3 + ‘2al ‘E‘Iao =o

‘4 + ‘3al ‘p‘2ao = 0

The unknowns an, a, can be determined from the above equations. Onceu L

a.‘ al are‘nom’the roots Ul, U2 can be determined from equations(2.19). Once U1 and U2 are known, y may be obtained from equations

(2.16) or (2.17) or,

Y = ln(ul)/Az

-y = kn(u2)/Az

23

(2.20)

(2.21)

●

With y known Al and A2

Since the roots U1 and

stituting in (2.19) we

are then determined from (2.18}(a) and (2.18)(b).

1

%!are reciprocal by definition, U2 = ~ . Sub-1get

2‘1 ‘W+ao= 0

~2+ct ~+ao=o1U2

‘2

J.

(2.22)

(2.23)21 + alu2 ‘%U2 = 0

Comparing (2.19j and (2.23) we obtain,

ao=l

U*+alu+l= o (2.24)

If we determine al, then the roots can be found easily in the following

way,

fl+f2~1+f3=o ; f2a1 = -(fl + f3)

f2+f3c11+f4=o ; ‘3% =-(f2 + f4j

‘3+f4%+f5=0; f4cx1= -(f3 + f5)

With three datapoints one can easily determine al from which U1 and U2

can be determined.

The method descrj>ed so far is not satisfactory for the set of

data’where experimental noise is present. A larger number of datapoints

24 ●

used and a least square curve fitting technique isis employed.

3M=N, where NLet MAz be the spacing between datapoints. When

is —,.th_etotal_nurnber_ofdata_po_ints,

6f ti+f = om+l 2m+l

fl +

‘2 +

f3 -1-

fm +

; fm+la=-(fl+ f2m+1)

.

; f~+2~ =-(f2 + f )2m+2f~,a + f

2m+2 =o

fm+3‘+f2m+3=0 ; ‘m+3

~ =-(f3 + f )2m+3

f2ma -1-f3m = o .$ f2mcl=-(fm + f3m).,

In general

[F]cx= -(G)

.[FTF]cx= -[FT] [G] whereo

[F’] is the transpose matrix of-[F]

Therefore u = - [FTF]-l [FT] [G] (2.25)

The residues Al and A2 can be determined in the following way

‘1 = Al -1-A2

A ;yAz ●1

A e-yAz2‘2 =

i

A1(eyAz)N-l +A2(e-yAz)N-lfN =

[F] = [G] [A]

[A] = [GTG]-l [F][GT] (2.26)

25

0

.-

with Al, A2, Y1 and Y2 thus determined, a continuous curve may then

be computed from equation (2.15) and extrapolated to the junction. The

value of P and P in equation (2.12) were obtained by this procedure.1 2 .

a

.

.

26

:

. ‘1

‘1

I‘1,!II,:

1’,,

II Nd

,’,,

II ,/

nFREQUENCY r SINGLEMETER STUBHP 5345A TUNERBROADBANDSAMPLINGVOLTMETER ~HP 3406A1

50L?TEEf

1 5ofJ

r--

HP 778A DUALDIRECTIONAL COUPLE

OSCILLATOR AMPLIFIER

GR 1362 BOONTON 230A ---- -. ---- -.

i 4 50QTEE

=4MAG.

VECTORVOLTMETERHP 8405A

H&E

I I & 50Q I4 1

11% I FEEDS CHARGE PROBE

1AII f

REI?.

PEST.1

poWER

DIVIDERa- ! COAXIAL MOVABLE SHORr7

~1’STVM

‘3A -E!ZI”FFig. (2.6) BLOCK DIAGW OF MEASUREMENT SYSTEM.

“uPLOTTERHP 9862A

#

CHAPTER 111

NUMERICAL ANALYSIS

.

(i) Charge Probe as a Scatterer: Theory and Solution

The preceding chapter discussed the experimental determination of

the response of a charge probe. A theoretical analysis and numerical

solution for the probe response and calibration factor are presented

in the present chapter. For the purpose of this analysis, the probe

is modeled as a monopole scatterer perpendicular to an infinite plane

conductor and planar

Chapter II the major

the probe is mounted

image theory is employed.

effect of the cylindrical

1is the ~ variation of the

As discussed in

conductor upon which

~ field produced byIJ

the charge on the test structure. In the treatment of the probe as a

scatterer the correct ~ dependence of the incident field is imposed.

It is recognized, however, that some error is introduced by using

planar image theory rather than the Green’s function for a source ex-

terior to a cylindrical conductor.

From the specification given in Chapter 11, for the cable from

which the probe is constructed, the electrical radius, ka, is found to

be .00036 for an operating frequency of 300 MHZ. It is therefore

appropriate to make the usual thin wire approximations.

.

a

.,

28

(ii) Formulation of Straight Wire Integral Equation

The charge probe of length ~ slides in a charged conductor of

radius a~ in a homogeneous medium characterised by (u, e, u = O) is-.

shown in Fig. (3.1). The z axis of a cylindrical coordinate system

lies at the center of the charged conductor as shown in Fig. (3.1).

The probe is perpendicular to the charged conductor and extends frcm

PH=

al ‘0 p.= al+z”For convenience a coordinate

used with a new variable < = (p-al) measured along

shown in

The

ES(C)on

where Ac

—J-

Fig. (3.2).

current and charge on the probe produce an:,

the probe surface which has a L component,

— ~;(~) ..5U!! (t

k2 3

9

0

4PI,

slot.- . ..- ------ ---

Coaxial Line 1

1-. —.- ----- . . . —--+

Charged Conductorz

Fig. (3.1) CHARGE PROBE ON A CHARGED CONDUCTOR.

*.I t

.

e

,’

I

a

L.-lF-J

+H/2

o

III

-H/2 i

II‘1

‘:

1III

I

Fig. (3.2) CHARGE PROBE AS

+H/2

0

-H/2

I

.

50S250Q

:

.> A

7

+H/2 1

Ak

-H/2

Fig. (3.,3) LUMPED LOAD ‘LOAD AT C=O.

*

‘,

‘,

t “1’

‘x”

I

where a = radius of probe, and @ = angle variable’of a cylindrical

coordinate system with C as the axis.

Applying the boundary condition that the tangential electrical

field on the probe surface be zero, we obtain,

+0 +E;(G) = Oand therefore

@ = -E;(C)

Thus the integro-differential equation becomes,

(,iii) Charge Probe with 50Q Load

(3.4)

The charge probe sees the 50 0 input impedance of the coaxial

cable, so that loading on the probe should be considered. For a

lumped load of impedance Zload inserted at L = O as illustrated in

Fig. (2.3), the current IC in ZIoad causes a scalar potential drop

A@g across the load of

The approximate electric field on the surface of

which must be added to the Es is,

the load element ,

To achieve more useful representation for the electric field due

to the IC in Zload

, we look upon the effect of the load at K = O as a

32

—

local behavior at C = O and write,

.

In the case of a.scatt$rerz

As explained in the

tional to ~ where p

(3.5)

beginning E;(3) has been selected to be propor-

= (~+al). Equation (3.5) with (3.2) is the final

form of the integral equation for the probe current. Equation (3.5)

has been solved numerically by the method described in the next section.

(iv) Numerical Solution Technique

In this section a technique for solving the integro-differential

equation is presented. This technique uses piecewise linear testing

and pulse expansion. One performs the testing of the equation to be

solved with elements t

,

ww-

C-m c-m+l c-m-l-2

-H/2

Fig. (3.4) TRIANGLES IN PIECEWISE LINEAR TESTING SCHEME.

e,..,,....._,-----------.....-.———...II II

o L—___ . . . . . . -t .,f

0. . . ... ,— I

@

(B

‘(3.6)(a,))AC+Z ~oadIc(o)8(c),t><

E; ,m tm >

.where the scalar product is

-- ..:-+, $=J+H:2,*(3.6)(b)

type considered here where

.

-{Jreal testing set t ofmFor atin(c)

the subdomain

is zero outside

. .Clz

Equation (3.6) becomes

“(cm-l,Cm+l)

!-

<

Jm+l

.=

E;(c) tin(c)dc

c =Cm-l

A A-1 C- gm ISetting tm =Am =A

in the above, the equation becomes

(3.7)

(3.8)

(Lm)

‘r

+

%1

+-“JIG(0)

35

with xn=O~l,~2,~3,~ 4,.., ~(M-l) - (3.10)(a}

and

For a function f sufficiently

one employs the approximation

%-f-l

(3.10)(b).

smooth over the interval (Gm-l,~til)

The final equation becomes

+-AZ I (0)60m= AE; (Qm)load G

where 6 is the kronecker delta

&=~{

1, m=nO, m#n

m= 0, ~1, -& 2,~ 3, +4,...,& (M-1)—

(3.11)

(3.12) ● -

(v) Expansion of Current and Formulation of Matrix

The current Icis represented by expanding in known expansion

functions each weighted with an unknown coefficient.

coefficients ultimately determine I . In terms of aL

uPn , as the known functions and the complex unknown

These unknown

pulse basis set,

coefficients In?

the current becomes

.

36

J

,

0= —

(0

“

IC(i) = ~ InPn(G)n=-N

(3.13)

The pulse arrangement for approximating the current on straight wire

is shown in Fig. (3.5)..

The vector potential can be written as

o

where

‘c!:)= f InALn(L)n=-N.

J

.H2

‘ALn(c) =% pn(C’)K(C-C’)dC’

G’=-H/2

(3.1.4)

(3.15)

@ a partial vector potential contributed by the pulse basis element

Pn(c)* In convenience, the following quantities are defined:

z =mn

+

Vm =

In terms of.the above definition equation

N~ ZmnIn = V

mn=-N m=O, +1, +2,. ... ~(1)l)——

37

(3.16)(a)

(3.16)(b)

(3.17)

LJm

I I 1,1,

IL(O

‘“w

G-N C-N+l ‘-N+2

-H/2

Lb

-.””.-

A

‘\ $.J+\

‘N-2 ‘N-1 ‘N

Fig. (3.5) PULSE ARRAIJGEKENTFOR APPROXIMATING CURRENT ON STRAIGHT WIRE.

,, ,, 1,,

+H/2

*

@

ends

The boundary condition that the current goes to zero at the

is incorporated by directly setting p_N = pN = O. Equation

(3.17) can

Therefore

be conveniently written in matrix form as,

‘“[Z][1] = [V] (3.18)

[1] = [Z]-l[V]

The total scheme of testing antiexpansion is shown in Fig. (3.6).

(vi) Deterrni.nationof Probe Calibration Factor

Once the current induced in the load can be found as a function

of a the probe calibration factor can be determined as follows.1’

Let 11 be the current induced in the load when the probe is mounted

on the larger radius tube and I~ be the current induced in the load

when the probe is mounted on the smaller radius tube, then relating

these to surface charge density

11 = ‘1’h

12‘:‘2%

If the linear density of charge is the same for both tubes then

q~ =q2

.

nl a2-=’ (—)‘2 al

39 L

(3.19)

(3.20)

.

4(9TestingTriangl I_A

C-5 C-4

-H/2

‘Opo

*-2

&‘-3

‘-3 c-z

k!!L

Note: m=~=o, ~l,~z,...

L

L-J A

13 “’

a“‘4 15=~ (Ha~f_pulse)‘3 ‘4

@

From eq. (3.19) and (3.20~, ‘1the probe calibration factor ~ isN 2

obtained as,

‘1 11 al—=— ._ (3.22)‘2 12 a2

The computed value of this ratio for different probe heights for two

different radii of cylinders are shown in Fig. (4.19) and (4.20).

41

CHAPTER IV

PRESENTATION OF DATA AND CONCLUSION

(i) Introduction

Chapter 11 and 111 have discussed the measurement technique and

the numerical analysis respectively. It is the purpose of the

present chapter to present the data resulting from both approaches

and discuss these results.

(ii) Measured Data for the Charge Distribution

As explained in Chapter 11 on measurement the charge density

measurement was made for 3 charge conditions at the junction near

maximum near minimum and a condition in between minimum and maximum.

The figures (4.1) to (4.9) present the charge distribution of

the step radius system with al= 5/16” and a~= 1/8” for 3 different. .

probe heights 0.10”, 0.08”, and 0.06”. For each probe height,

measured data for 3 different charge conditions at the -junctionare

presented along with the curve determined from the modified Prony’s

method for the magnitude of the measured data. Similarly the

Figures (4.10) to (4.18) present the data for the step radius system

with a~= 3/16” and a2= 1/8”.

(iii) Probe Calibration Factor

The probe calibration factor (N1/N2) as a function of probe

height as computed from the measured data is shown in Fig. (4.19) for

al= 5/16” and a2= 1/8”. The theoretical probe calibration factor

42

:

4-u

,,

CHFIRGE PROBE RESPONSE: PROBE HEIGHT 0.10”c1 PHFtSE 91 = 5/16” 92 = 1/8”x MRGNITUDE

o ,—.+ CURVE FIT

0.00 -50.00 -io.oo .10.00 30.00 :X10”

Z(MMI

Fig. (4.1) MAXIMUM CHARGE CONDITION - PROBE HEIGHT = O.1O”, al= 5/16”

o(3

G’N,

o0

G-+Hod

x’

fg-(3.

gel .LLJC3

o-‘?LIJo~~a

sL’

o0

&N

!00

. ..- ,-.

11,-

1

— —

tIII

Fig. (4.2)

CHRRGE PROBE RESPONSE:cl PHRSEx MFIGNITUIIE

o CURVE FIT

(9

PROBE HEIGHT 0.10”91 = 5/16n 82 s 1/8”

0.00

MEDIUM CHARGE

-30”00 -’20.00 . lb.oo 3i?.oo)kl~i

Z[MMjCONDITION - PROBE HEIGHT = 0.10”, al= 5/16”

5

.—..._ %. . ~

I

,.) ,

:. ,1

I

,1’

,’

1,

CHflRGE PROBE RESPONSE:

% MRGNITUDE

‘z CURVE FIT

o0

G’

Q

PROBE HEIGHT 0.10”fll = 5/16” W = 1/8”

.,

0.00 -30.00 -io.oo - lb.oo 3tl.oo 5

o’0 1,

>4 -iN

,1

00.

.s:00

)KltllZ(rfrl)

Fig. (4.3) MINIMUM CHARGE CONDITION - PROBE HEIGHT = 0.10”, a,= 5/16”.L .—.

1,

‘k-0

1’

0

CHftRGEax

9 .—.*

RESPONSES PROBE HEIGHT 0.08”PHflSE RI = 5/16” f%? s 1/8”MRGNITUIIECURVE FIT

1

0=00 -30 ●OO -io.oo lt).oo 30.00 5M@

Z(Mtl)

Fig. (4.4) ?4AXl141JlfCHARGE CONDITION - PROBE HEIGHT = 0.08”, al==5/16”

0

c)SJ

;0LLJo

:

CHFIRGE PROBE RESPONSE: PROBE HEIGHT 0.08” I

c) PHFISE ,Rl = 5/16” f12 = 1/8”x tlRGNITUllE I

o* CURVE FIT ~

I Io’0

.

I

m.l--m -d)

,!

-1-*-~w

,,I!’

>’w

-1

-1’0-INx “.-f=

0

I

t

(-J-

0.GoLIJ0

,,

(3

-c---l h.1

Io-:WIom ,i I

-—,

oc)

t4- 1.

1I

1 1 I

0.00 -30.00 -Lo’-ou-- .10.00 30.00 50’!003KmL

i!trml

Fig. (4.5) MEDIUM CHARGE CONDITION - PROBE HEIGHT = 0.08”, al= 5/16”

I

(4

mal-(-o

LLJ

00

2’.

PROBE RESPONSE: PROBE HEIGHT 0.08”PHI?SE 91 = 5/16” 82 = 1/$”MflGNITUOECURVE FIT

Fig. (4.6) MT.NIMUM

a

0.00 -30.00 -10.00 10.00 30.00

1 1,

IEIQ1Z(MMI

CONDITION - PROBE HEIGHT = 0.08”, al= 5/16”

m a

q ‘,!1

:

.c-W

b .

:

.

,1

CHflRGE PROBE RESPONSE: PROBE HEIGHT 0.06”a PtifW3E 91 = 5/16” fl12s 1/$”% I’IRGNITUDE

~,~

-1(!)

I I

m

.,

,,

In

on

09

~c5I.LJo

0-

31fll)1Z?(MM]

,,

I

Fig. (4.7) MAXIMUM CHARGE CONDITION - PROBE HEIGHT = 0.06”, al= 5/16”

I

u

CHRRGE PROBE RESPONSE: PROBE HEIGHT 0.06”Q PHRSE 81 = 5/16” W? = 1/8”x MRGNITUDE

cl_v

CURVE FIT

c?)

0.00 -!30 .00 -ioaoo . lb.oo 30.00 53E101

Z(Mtll

J?ig. (4,8) MEDIUM CHARGE CONDITION - PROBE HEIGHT = 0.06”, al” 5/16”

O . ,

ao

&N

o0

●

r-l

x

Q

o-%W

1!00

*

..

i .:

CHRRGE PROBE RESPONSE: PROBE HEIGHT 0.06”a PHfiSE fll = 5/16” ff2 s 1/8-# MFIGNITUDiE

o* CURVE FI’T 1:

‘o

-OJE“-0

IJJo

~o

(2)

o

0

IS”

(3

(3

c1

I

v K0.00 -30.00 -10.00 10*OO 30.00 &c

XIO1

Z[MM)

Fig. (4.9) MINIMUM CHARGE CONDITION - PROBE HEIGHT = 0.06”, al= 5/16”

I

o

CH9RGE PROBE RESPONSE:(9 PHRSEM tlflGNITUOE

o

N— CURVE FIT

Fig. (4.10)

,

V@

PROBE HEIGHT 0.10”fal = 3/16” R2 = 1/8”

Q

II 1

0.00 -50 ●OO -io.oo .10.00 30=00 53E101

Z(MMI

MAXIMUM CXMRGE CONDITION - PROBE HEIGHT = 0.10”, al= 3/16”

@!

II II . 1,

ml .. : *

.,

CHRRGE PROBE RESPONSE: PROBE HEIGHT

c1 PHflSE “ ,91 = 3/16”

% Mf!GNITUOE

o*lo- 1

92”= 1/8”

E CURVE FIT I

0.00 -30.00 -io=ooI I

10*OO 30.00 5IE1(I1

Z(MMI

I?ig, (4.11) MEDIUM C~GE CONDITION - PROBE HEIGHT = 0.10”, a~= 3/16”

,’!

0’<0:

.&do4

N

0

O(J-J

Q

o

0.

,R1!00

I

—-- n..

WI.P-

Ct!FIRGE PROBE RESPONSE: PROBE HEIGHT 0.10”

a PHRSE 91 = 3/16” F12 = 1/8”

x MRGNITUDEo@J — CURVE FIT

t?)

@)

;0.00 -30.00 -io.oo lb=oo 30.00 Ixmr

Z[MM)

Fig. (4.12) MINIMUM CHARGE CONDITION - PROBE HEIGHT = 0.10”, al= 3/16”

,=*

, ,

I ,,I

m’.+ ●

:

1!

CHflRGE PROBE RESPONSE: PROBE HEIGHT 0.08”c) PHflSE .Ifll = 3/16n 92 = Ii/8”% MRGNITUDE

oN CURVE FIT

-1

wo

o

0.

0

c)

0.00 -30.00 -io.oo lU.00 30.00 5MIO1

ZIMMI

1’,

‘1

o~iIOi!

0-

Dc)

.

DN

!00

Fig. (4.13) MAXIMUM CHARGE CONDITION - PROBE HEIGHT = 0.08”, al= 3/16”

,

●

CH13RGE PROBE RESPONSESc1 PHWSEx flRGNITUilE

aml CURVE FIT

PROBE HEIGHT 0.08”RI = 3/16” R2 = lf8”

c)

c1

Q

C9

(3

0.00 -30.00 -iOaOO .10.00 30.00*IQ’

2(flM)

Fig. (4.14) MEDIUM CHARGE CONDITION - PROBE HEIGHT = 0.08”, al= 3/16”

* , 1.

. w

m-J

:

CHf3RGE PROBE RESPC?NSEX PROBE HEIGHT 0.08”c) PHf7SE 91 = 3/16” 92 = 1/8”x tlfiGNITUDE

~R— CURVE FIT .

.CJ

-

Coml-.-l”C!3°

‘>n-1

wo3

I~l.g

b-i.

~ 0“(3

o0

&’

‘o

I1’

(

)c

I I I I

0.00 -30.00 -10.00 10.00 30.00 5*101

Z(MMJ

Fig. (4.15) MINIMUM CHARGE CONDITION - PROBE HEIGHT = 0.08”, al= 3/16”

~

,,‘1

uw

aa

.

*

-.

I

,1.,

CHRRGE PRi3BE RESPONSE: PROBE HEIGHT 0.06’a PHFISE 91 = 3/16” f12 = 1/8”x MRGNITUDE

ww CURVE FIT

4

m

. . I

?%lO1Z(MM1

- PROBE HEIGHT = 0,06”, al= 3/16”Fig. (4.16) MAXIMUM CHARGE CONDITION I I

●b +

.—,“

,

* ..

I

II I

1,,, I

,!z

:

CHflRGE PROBE RESPONSE: PROBE HEIGHTa PHfWEx MRGNITUDE

oN CURVE FIT

III

I

,,

.0

n

mml--d

.go

>-

-1.-l1--1:z“-0

I.lJn

~o

C5

o0

.0

91 = ,3/16=0.06”

R2 = 1/0”

c1

I v1 I 1 I

0.00 -30.00 -10.00 .10.00 30900 53K10X

Z(MMJ

Fig. (4.17) MEDIUM CHARGE CONDITION - PROBE HEIGHT = 0.06”, al= 3/16”

.0N

o0

.

,=

1!00

,,

‘1,

I ,)

Fig.

.

1.’1’. .1, .

(4.18)

<

PROBE RESPONSE: PROBE HEIGHT 0.06”PHRSE R1 = 3/16” F12 = 1/8”

tli3GNITUOEC3N CURVE FIT

Q

(!)

MH!

i I I i0.00 -30.00 -10”00 .10.00 30.00

MINIMUM CHARGE

MI(Y2(MMJ

CONDITION - PROBE HEIGHT = 0.06”, al= 3/16”

o“

C20

●

CJ(N

.

c)-?Ll

r2-

08

:

1.4

1.32

1.24

N1/N2

1.08

1.0

I

,! I, I Charge Condition at the’1 :,

I Junction ~1,

1,

4~I I I

JI l,’

PROBE HEIGHT IN INCHES

MinimumMedium

Maximum

Theoretical

,,,,II

,,

I

.. ,

0.04 0.06 0.08 0.10 0.12

Fig (4.19) PROBE CALIBRATION FACTOR FOR 5/16” to 1/8” RATIO OF RADII

obtained by the technique explained in Chapter III is also shown in

Fig. (4.19). Similarly Fig. (4.20) presents the probe calibration

factor for al = 3/16” and a2 = 1/8”.

Examination of Figs. (4.19) and (4.20) reveals differences of about

8% in the experimental values of N1/N2 for different charge conditions

are the junction. With the assumption that the probe responds to the

local charge density at its location on the cylinder N1/N2 should be a

function of probe height and cylinder radii only. The observed effect

of the position of the junction with respect to the standing wave pattern

is believed to result from inaccuracies in the extrapolation of the data

to the,junction. The extrapolation,

is accomplished by fitting the curve

data 5s available over a distance of

the accuracy in determination of the

limited. For these reasons the data

and hence determination of N /N12

to the measured data. Since the

only A/2 each side of the junction

parameters for this curve fs

obtained for the case of maximum

charge at the junction is the most reliable since the slope of the curve

in the extrapolation region is near zero. In future use of these experi-

mentally determined probe calibration factors greatest weight will, there-

fore, be given to the values determined for the maximum charge condition.

Examination of Figs. (4.19) and (4.20) also reveals a consistent

difference between the measured and theoretical values of N1/N2. The

reasons for this difference are not fully understood; however, it must

be remembered that the theoretical solution is based on a planar image

theory. The effects of the cylindrical geometry are the most likely

62

..= .-

m w : ●✌‘e

1.25

I

I 1.20

1.15

N1/N2

1.10

1.05

I

),,

!

Charge Condition at;theJunction

I

I

PROBE HEIGHT IN INCHES

1 * #

Medium

Minimum

Maximum

Theoretical

0.04 0.06 0.08 0.10 0.12

Fig. (4.20) PROBE CALIBRATION FACTOR FOR 3/16” tO 1/8” RATIO OF RADII

.

●source of the observed differences between experimental and theoretical

values. Until more accurate data and theoretical solutions are available,

the experimental value of N1/N2 will be used as the

(iv) Conclusions

The results of this investigation have clearly

effect of cylinder radius on the response of charge

most reliable.

demonstrated the

probe and have taken

a considerable step toward better understanding of charge measurements.

The probe calibration factors obtained can now be used to correct charge—

. distribution measurements on structures containing junctions of unequal

radii cylinders.

Further work, however, is needed for complete understanding of the

response of monopole probes on nonplanar surfaces. In”particular, an.—

effort should be made to include the cylindrical image plane geometry o

into the theoretical treatment presented here. In addition, other probe

configurations should be examined and evaluated with respect to their

response on nonplanar surfaces.

64

—

[El

[2]

[3]

[4]

[5]

[6]

[7]

REFERENCES

R.W.P. King, “Quasi-Stationary and Nonstationary Currents inElectric Circuits,” pp. 267-270, Encyclop-ediaof Physics,Edited by S. Flfigge,Vol. XVI: Electric Fields and Waves,Springer-Verlag, 1958.

H. Whiteside, “Electromagnetic Field Probes,”Technical Report SO. 377, Harvard University,setts, 1962.

T.T. Wu and R.W.P. King, “The Tapered Antenna

Cruft LaboratoryCambridge, Massachu-

and its Applicationto the Junction Problem for Thin Wires,” IEEE Transactions onAntenna and Propagation, Vol. AT-24, pp. 42-45, January 1976, andalso Interact~on Note 69, January 1976.

C.M. Butler, K.R. Umashankar and C.E. Smith, “Theoretical Analysisof Biconical Antenna,” report submitted to the U.S. Army,STFL4TCOM,1975.

R.F. Barrington, Field Computation by Moment Methods,Company, New York, 1968,

Macmillan

N, Marcuvitz, Waveguide Handbook, McGraw-Hill Book Company,New York, 1957.

Hildebrand. Introduction to Numerical Analvsis. McGraw-HillBook Company, New York, 1956.

65

!-