NAT'L INST. OF STAND & TECH

A111DL 3311MM

7 V~\

\

*<"*AU O* * NBS TECHNICAL NOTE 1054

REFERENCE

NBS

PUBLICATIONS

U.S. DEPARTMENT OF COMMERCE / National Bureau of Standards

Input Impedance of a Probe

Antenna Exciting aTEM Cell
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and

the Institute for Computer Sciences and Technology.

THE NATIONAL MEASUREMENT LABORATORY provides the national system of

physical and chemical and materials measurement; coordinates the system with measurement

systems of other nations and furnishes essential services leading to accurate and uniform

physical and chemical measurement throughout the Nation's scientific community, industry,

and commerce; conducts materials research leading to improved methods of measurement,

standards, and data on the properties of materials needed by industry, commerce, educational

institutions, and Government; provides advisory and research services to other Government

agencies; develops, produces, and distributes Standard Reference Materials; and provides

calibration services. The Laboratory consists of the following centers:

Absolute Physical Quantities 2 — Radiation Research — Chemical Physics —

Analytical Chemistry — Materials Science

THE NATIONAL ENGINEERING LABORATORY provides technology and technical ser- vices to the public and private sectors to address national needs and to solve national problems; conducts research in engineering and applied science in support of these efforts; builds and maintains competence in the necessary disciplines required to carry out this research and technical service; develops engineering data and measurement capabilities;

provides engineering measurement traceability services; develops test methods and proposes

engineering standards and code changes; develops and proposes new engineering practices;

and develops and improves mechanisms to transfer results of its research to the ultimate user. The Laboratory consists of the following centers:

Applied Mathematics — Electronics and Electrical Engineering2 — Manufacturing

Engineering — Building Technology — Fire Research — Chemical Engineering2

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts

research and provides scientific and technical services to aid Federal agencies in the selection, acquisition, application, and use of computer technology to improve effectiveness and

economy in Government operations in accordance with Public Law 89-306 (40 U.S.C. 759), relevant Executive Orders, and other directives; carries out this mission by managing the Federal Information Processing Standards Program, developing Federal ADP standards

guidelines, and managing Federal participation in ADP voluntary standardization activities; provides scientific and technological advisory services and assistance to Federal agencies; and

provides the technical foundation for computer-related policies of the Federal Government.

The Institute consists of the following centers:

Programming Science and Technology — Computer Systems Engineering.

'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. 'Some divisions within the center are located at Boulder, CO 80303.
Input Impedance of a Probe Antenna

Exciting a TEM Cell

P. F. Wilson |

D. C. Chang t

M. T. Ma

Electromagnetic Fields Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

fDepartment of Electrical Engineering

University of Colorado

Boulder, Colorado 80309

J*""**

r*TCi

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued April 1982
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1054

Nat. Bur. Stand. (U.S.), Tech. Note 1054, 52 pages (April 1982)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982
CONTENTS

Page

FOREWORD i v

1

INTRODUCTION 1

2. VARIATIONAL FORMULA FOR THE PROBE INPUT IMPEDANCE 3

3. THE GREEN'S FUNCTION K(x,x') 4

4. THE INPUT IMPEDANCE Z 6

5. NUMERICAL RESULTS AND DISCUSSIONS 9

6. CONCLUDING REMARKS 12

7

ACKNOWLEDGMENTS 12

8REFERENCES 13

APPENDIX A. Derivation of k| j) and K^ 14

APPENDIX B. Evaluation of Z

1 .17

APPENDIX C. Evaluation of Z

2

24

APPENDIX D. Checks of R and Zj 27

APPENDIX E. Subroutine ZIN 31

FIGURES 41

iii
FOREWORD

This report describes theoretical and experimental analyses developed by

staff of the University of Colorado at Boulder in collaboration with the

Electromagnetic Fields Division of the National Bureau of Standards (NBS),

under a contract sponsored by NBS. Professor David C. Chang heads the

University team. Dr. Mark T. Ma of NBS serves as the technical contract

monitor. The period covered by this report extends from July 1980 to July

1981.

The work presented in this report represents a further aspect of

establishing a theoretical basis for the technical analyses of transverse

electromagnetic (TEM) transmission line cells. The general purpose of

pursuing theoretical studies is to evaluate the use of TEM cells for (1)

measuring the total rf radiated power by a device inserted into the cell for

test, or (2) performing necessary susceptibility tests on a small electronic

device.

The particular topic addressed herein is to determine the input impedance

of a probe antenna which may be inserted into a TEM cell to measure or excite

fields. The formulation of the problem is based on a variational principle,

which insures that the input impedance so obtained is stationary for small

arbitrary variations in the probe current distribution about its true value.

The evaluation of the variational integral requires a knowledge of the field

distribution inside the cell excited by a vertical short electric dipole.

This latter problem was the subject of a previous report by the same authors

[4], the results of which are readily adapted to the present problem. Thus,

the input impedance treated in this report represents an extension to the

previous studies.

The analysis of the present problem is significantly simplified by

assuming that the gap between the inner septum and the outer walls of the cell

is electrically small, and that the operating frequency is such that no higher

order modes are excited inside the cell. These assumptions are consistent

with the practical cell utilization and design principle.

The final impedance expression is shown to consist of two distinct terms,

which can be clearly identified as the respective contributions by the

ordinary rectangular waveguide and the gap perturbation.

IV
Previous publications under the same effort include:

Tippet, J. C. and Chang, D. C, Radiation characteristics of

dipole sources located inside a rectangular coaxial transmission

line, NBSIR 75-829 (Jan. 1976).

Tippet, J. C, Chang, D. C, and Crawford, M. L., An analytical

and experimental determination of the cut-off frequencies of

higher-order TE modes in a TEM cell, NBSIR 76-841 (June 1976).

Tippet, J. C. and Chang, D. C, Higher-order modes in rectangular

coaxial line with infinitely thin inner conductor, NBSIR 78-873

(March 1978).

Sreenivasiah, I. and Chang, D. C, A variational expression for

the scattering matrix of a coaxial line step discontinuity and its

application to an over moded coaxial TEM cell, NBSIR 79-1606 (May

1979).

Tippet, J. C. and Chang, D. C, Dispersion and attenuation

characteristics of modes in a TEM cell with a lossy dielectric

slab, NBSIR 79-1615 (Aug. 1979).

Sreenivasiah, I., Chang, D. C, and Ma, M. T., Characterization of

electrically small radiating sources by tests inside a transmission line cell, NBS Tech Note 1017 (Feb. 1980).

Wilson, P. F., Chang, D. C, and Ma, M. T., Excitation of a TEM

cell by a vertical electric Hertzian dipole, NBS Tech Note 1037

(March 1981).

Sreenivasiah, I., Chang, D. C, and Ma, M. T., A method of

determining the emission and susceptibility levels of electrically

small objects using a TEM cell, NBS Tech Note 1040 (April 1981).

INPUT IMPEDANCE OF A PROBE ANTENNA EXCITING A TEM CELL

P. F. Wilson, D. C. Chang and M. T. Ma

The input impedance of a probe antenna exciting a transverse

electromagnetic (TEM) transmission line cell is formulated by a variational approach. The formulation also utilizes the results

from a previous work on the field distribution inside a TEM cell

excited by a vertical electrical Hertzian dipole. The final

result of imoedance is shown to consist of two distinct terms,

which are respectively contributed by the ordinary rectangular

waveguide and the gap perturbation. Numerical results for both

the real and imaginary parts of the impedance are given. The

resistive part is found to be proportional to the square of the

probe length, and the reactive part largely capacitive.

Key words: Green's function; input impedance; probe antenna;

radiation resistance; rectangular coaxial transmission line; TEM

cell; variational method.

1. INTRODUCTION

The National Bureau of Standards (NBS) has been interested in developing a method for testing the emission and susceptibility properties of electronic

equipment. One approach to this problem is the transverse electromagnetic

(TEM) transmission line cell. The TEM cell, shown in figure 1, consists of a section of rectangular coaxial tramission line (RCTL) coupled at each end to

standard 50ft cylindrical coaxial line via a tapered section. The RCTL section

supports a dominant TEM mode and provides an isolated, standard test field

environment. The TEM cell results may be related to the free space

environment via results given by Tippet [1] and Sreenivasiah [2],

This report concerns the manner in which an idealized coaxial probe

excites an infinite section of RCTL. The results may be used to study the

effect of coaxial probes inserted into the TEM cell either to excite, or to measure fields. In addition, the configuration studied may be useful in

modeling feed lines to electronic equipment. The method of solution is to

formulate a variational expression for the input impedance of the probe at the

aperture plane, as was done by Collin [3] in analyzing the coaxial probe

excitation of a rectangular waveguide. The evaluation of the variational

integral requires that the Green's function corresponding to the fields

excited in an RCTL by a vertical electric dipole (VED) be known. The VED

excitation of an RCTL was the subject of a previous report by the authors [4],

»
the results of which will be adapted to the present problem. The analysis of

the Green's function is significantly simplified by assuming that the gap

between the inner septum and the outer walls, is electrically small, and that

the operating frequency is not too large. Both these premises are consistent

with the TEM cell usage.

Evaluating the variational impedance integral results in two physically

distinct terms, which may be referred to as the ordinary rectangular waveguide

contribution, and the gap perturbation contribution. The ordinary term gives

the input reactance due to a coaxial probe in a rectangular waveguide with the same dimensions as that of the RCTL's upper chamber. Since the dimensions of

the cell are such that the rectangular waveguide modes in the upper chamber

would be cutoff, the result is a highly capacitive term for which the analysis

yields an approximate analytic expression. The gap perturbation term

represents the change from the ordinary result due to the presence of the gap

fields, and contains the radiation resistance resulting from the dominant TEM

mode. The perturbation term, as derived, remains in an integral form which

must be treated numerically. However, if the TEM mode radiation resistance is

extracted, the remaining reactive integral is negligible when compared to the

highly capacitive ordinary contribution. Thus neglecting the perturbation

reactance results in a computationally simple algebraic form for the probe

input impedance.

Section 2 of this report will define the variational impedance integral to

be evaluated and discuss the necessary probe idealizations. The Green's

function associated with the VED excitation of an RCTL is examined in Section

3. Section 4 gives the result of the evaluation of the impedance integral and

examines the distinct physical nature of the resulting terms. Certain special cases are used to verify the solution. These checks are examined in Section 5

along with numerical results. Many details of the analysis and a listing of

the FORTRAN program written to evaluate the impedance terms are left to the

Appendices.
2. VARIATIONAL FORMULA FOR THE PROBE INPUT IMPEDANCE

The RCTL cross section and the coaxial-line probe antenna are depicted in

figure 2. The septum, of width 2w, is allowed to be vertically offset with

the heights of the upper and lower chambers given by b-., and bp,

respectively. The width of the upper chamber is 2a-i, and the width of the

lower chamber is 2ap. The gap between the septum and the outer wall is g.

The probe, of length d and radius t, is located in the upper chamber. A TEM

cell would have an RCTL cross section with equal chamber widths and previous

discussions have assumed this geometry. However, in order to avoid any

degeneracies caused by letting a-i = ao = a, the analysis will initially assume

general chamber widths. The time convention will be exp(-icot).

If the coaxial -line aperture is small, it is reasonable to neglect the

higher order modes excited at the aperture discontinuity and assume only

incident and reflected TEM modes exist in the coaxial-line. As shown by

Collin [3] in his analysis of the coaxial-line probe antenna excitation of an ordinary rectangular waveguide, the above approximation allows the antenna

input impedance Z to be expressed as follows

z = 4" / / JfxJ-KCx.x'-O'JCx'Jds'ds (1)

I S S'

where x and x' denote source and observation points, J(x) is the unknown

current density on the probe, K(x,x') is a dyadic Green's function

corresponding to the electric field excited by a unit strength dipole source,

and I t is the total current at the aperture plane. S denotes the probe

surface plus the aperture plane which carries an equivalent electric current

J(x) = nxH(x) where n is a unit vector in the outward normal direction.

Experimental studies have shown that Z is effectively independent of the

aperture size for electrically small apertures [3]. Therefore it is reasonable

to idealize the probe as driven by a y-directed E-field concentrated at the

base. Thus we need consider a y-directed current only, which reduces

J(x) to a J(x), and K(x,x') to a y K(x,x')a

y with a y being the y-directed unit

vector.
The above impedance expression may be shown to be variational in J(x).

That is, a first order approximation for J(x) leads to a second order

approximation for Z. At normal TEM cell operating frequencies, the length of

the probe will be small compared to a wavelength. Therefore, it is reasonable

to assume that the current distribution is approximately sinusoidal,

I J(x) = -^t sin k(b r

d-y)6( P -t) (2)

where the delta function is interpreted in the sense of a generalized

function. The form of the Green's function K(x,x') will be considered next.

3. THE GREEN'S FUNCTION K(x,x')

The appropriate Green's function for the special case of equal chamber

widths has been derived previously [4]. However, the modifications due to

generalizing the chamber widths are not difficult. Since a VED in an RCTL

will excite both transverse electric (TE or h-type), and transverse magnetic

(TM or e-type) modes, both cases need to be treated. For each modal type we begin by formulating an integral equation for the unknown field in the gap.

These integral equations could be solved by Chebyshev polynomial expansion

methods [5], However, a less cumbersome solution may be achieved by making

the following pair of assumptions. First, the gap is assumed to be small, and

second, the frequency is not allowed to be too high. These assumptions lead

to an approximate integral equation which may be solved directly. The small

gap assumption amounts to discarding terms which are of the order G JinG, where

G is a normalized gap parameter defined by G=7Tg/2a, a=min(a-^,a2) . Up to the

order of this approximation only the TE type modes contribute significantly to

the gap field.

The derivation of the Green's function begins with the removal of the z- dependence via a Fourier-transform defined by

K (j)(x,x') = — / K

(j)

(x" ,x')e

laZ

da (3)

2tt t t
where j=l ,2 refer to the upper, and lower chambers respectively, the subscript t

denotes the transverse plane, a is the transform variable representing a continuum of propagation factors, and the superscript tilde signifies a function

in the transform domain. The analysis outlined above yields the following

result

where K represents the Green's function expected if there were no gap, i.e.,

the ordinary rectangular waveguide result, and i< 2 gives the gap field

perturbation contribution. K and K

2 derived in Appendix A are found to be

2 2

2 A [M +a ] ~(j) - - ia»M a - - n j (j) - - K x ,x' = -6 — 6 x -x' + I g x ,x')

1 t t lj 2

l 2 t t

L 2 2 2 mn t t

J k c m,n=0 M + N -? (5)

j j

^»(x x^-J^U.Jf (x)f (x;> (6)

2aiY

where u> = 2irf is the angular frequency, p is the permeability, k = ui(\ie)'2 is

the TEM mode propagation constant, 6^ is the Kronecker delta function, rj- -

k

2 -a , M- = nrn/2aj, and Nj = nir/bj. The remaining functions are defined by

(i) - - 2 9 (x tS x!) = (——)sinM.(x+a.)sinM.(x'+a.)cosN.ycosN.y' (7) mntt a.b. jj J J J J

( 8/a, a, 2 , cotK b. , ) L(o , .L—Ll t I I 1 I %+ + +[ (8)

o m
M.cosK

J (b.-y)

f.(xj = I

J ,., m

, ."? sin^-cosM.xJ (M.g) (9) m K sinK b. on m j

Ki

j) = ?2

" M

j

(10)

with m denoting the summation over odd m only (m=l ,3,5. . .) , and J is the

zeroth order Bessel function. Again the delta function should be understood

in the sense of a generalized function.

4. THE INPUT IMPEDANCE Z

4.1. The Green's function terms K,

J and k! j will lead to two distinct

impedance contributions Zj and Z2 which represent the ordinary and

perturbation contributions to the probe input impedance. The total impedance

is the sum of these two terms. Combining the form of the Green's function

given by (4), with the impedance integral (1), yields Z = Z^ + 1^, where

00 Z = —^

/ / J ^) / k!

1 ^

,x')e

loZ

J(x') da ds' ds (12)

2ttI. S S'

Z

2

=-^- / / J(x) / K^^x^x^e^vKx') da ds' ds (13)

2TrI t S S'

The integrals defined for Z^ and Z^ are evaluated in Appendices B and C for a centrally located probe. The approximate form found for Z^ depends upon two

assumptions: (1) the probe is taken to be thin, i.e., irt/2a.<<l, and (2) the

frequency is restricted so that (ka^) 2 is small. More precisely, terms in t

of the order (irt/2a

1

) 2 are discarded and terms in ka^ of the order (k/2M

1

) 4
are discarded. The evaluation of Z2 involves no approximations. However, Z

7 remains in an integral form. The results of Appendices B and C are as follows

2 2

Z „ J3L_ tan

2J^£n (-4-1 +^V (4.207175)

1 ^TTb^2 2

! rt /

9 » , sin

2 N.d/2 \ 2 K [t(N

2 -K2 ^ ) -2k

2

I (l- ^ °

2

1 2 (14)

n=l \ sin kd/2 / fq - r J

2 »

1 Z = ^^-

CSC

2 kd / -^^h ,(o,d) I h (a,d)J [tfa^V2 2,2 ] da (15)

2 m mo 1 4a, -« c m m

1 00

(16

M. sin^f-J (M.g) (cos kd - cos K (1) i/i\i ^01 d) m Va ' d) " ,(1) . „(1) . 7T—-2

Km

S1 n Km

b

l M

l + a

where K

Q (x) is a modified Bessel function.

The perturbation term 1^ contains the radiation resistance due to the

propagating TEN! mode, which may be extracted by evaluating the half residues

at a = ±k due to the forward and backward propagating TEM modes. The TEM

mode radiation resistance is found to be:

,2

771-1 k

7 ? ? 1/ R = -—— esc kd L(k) I h ,(k,d) I

h (k,d)J [t(k +M V2

] (17)

2 m

' ' L m ' 4a, m m

1 00

where ti = 12(k.

If we now let a^^^ s0 that we are considering a normal TEM cell

r by [4]geometry, and recall that the characteristic impedance 2

Q

of an RCTL is given

n it Z

c =-g-L(k) (18)
The radiation resistance may be written

?k 2 2 2 l/o R=^-Z esc kd ^ h ,(k,d) \ h (k,d)J [t(k +M )'2

] (19)

a 2 c mmm mo o o

Because only the TEM mode is allowed to propagate, R represents the total

resistance. We may also express Z as the sum of the radiation resistance R and a reactance i(X+6X), with X and 6X defined by

n ~ . i e << 2,2

o 2 kd ) ,4a. a k , . „„_,,_,,_. X = ^-r-j- tan — Un(—

+ ——- (4.207175)

2ttd. k 2 ( ut 2

1 TT

9 - / sinVd/2 > 2 K [t(N

2

-k

2

) /2

] j -* * t --

1-) -w-

}

(2o)

n=l v sin kd/2 J N - k

J

2 «> V2

6X = ^_ csc

2

kd P / t^- J. h ,(o,d) Ih (a,d)J [t(a

2

+M

2

) ] da .2 J 2 m m

4a -°° c m 00m

(21)

where the integral is now understood in the principal value sense. The

dominant reactive contribution comes from the sum over n and therefore the

ractance is highly capacitive. The capacitive behavior is expected from

comparisons to either a monopole in free space, or a similar probe in a rectangular guide. For typical TEM cell geometries, the principal value

integral does not significantly add to the reactance. Therefore, a yery good

algebraic approximation for the reactance is simply X which allows us to write

Z as follows:

Z * R + iX, (22)

where R and X are given respectively in (19) and (20).
5. NUMERICAL RESULTS AND DISCUSSIONS

In order to check our result we would like to compare our expression to

certain special cases for which solutions are known. The impedance Z consists

of three terms: (i) the radiation resistance R due to the propagating TEM

mode, (ii) the reactive ordinary rectangular waveguide contribution Z-^, and

(iii) the gap field perturbation reactance principal value integral. Ideally,

since each of these terms has a distinct physical interpretation, each term

should be checked individually. Because the probe is electrically short, the

probe radiation resistance may be compared to that of a Hertzian dipole via a result given earlier by Tippet [1]. His expression for the radiation

resistance of a Hertzian dipole involves the square of IE

Q

dJ£ where I is the

current magnitude, E Q

is the TEM mode electric field distribution, and di is

the effective dipole length. In the monopole case, the dipole quantity

(IE

Q

dJi), may be replaced by an effective value which results from integrating

the product I(y) E Q (y) over the monople length. The two results agree up to

the order of our approximations. Next, by letting the gap width go to zero

(g+0) thereby reducing the upper chamber to an ordinary rectangular waveguide,

Z-l may be compared to a result given by Collin [3]. If we allow the frequency

to be large enough that the ordinary rectangular waveguide TE^q mode will

propagate, then our result reduces exactly to that found by Collin. Both

these checks are detailed in Appendix D for the special case when a^=a

2 =a. The perturbation reactance term is more difficult to verify and is not

comparable to any simple special case result. However, a good check of the

self consistency of our expression exists. If we had allowed k to be

sufficiently large, such that k 2 >M^, the expression (12) for Z^ would contain

poles at cx=+bIq , where &! Q

= (k -M^r 2 . This would imply an apparent mode

propagating the same as an upper chamber TE^q mode. Equation (15) for Z

2 would also imply the existence of this mode. However, the guide as a whole

supports no such mode, therefore, we expect that the contributions from Z^ and

Z

2

, to this evident Te[q mode, should cancel out to the order of our previous
approximations. It is found in Appendix D that the radiation resistance due

to a propagating TEJq' mode in a rectangular waveguide of the dimensions of

the upper chamber, is given by

n n o . 2 kd ,„,

R = pry tan —7y (23)

2

»i

b

i kBiS

Now consider the perturbation contribution given by Z

2 in (15). In order to

avoid the degeneracy due to an additional TEJq' mode also propagating in the

lower chamber, we refrain from letting the chamber widths be equal until after

we have evaluated the pole contribution at a = +$! n. As a approaches

l\\ (\\ (i \o o 1 /2 iU

tPiJ , we see that V. '= (b{q - a ) ' +0. Evaluating the half residues

at a = ±$ 10 , we find

2a 1 b

1 (3 1

( n

)2

" <?)

L( ) +

l l 1U

(24)

£ + (7i/2a

1

) 2

(25)

2a

i bi k (3lo - a) l

JoEtfM^+a

2

)

1^ + J Q

(kt) (27)

Because both irg/2a

1 and kt are small, the Bessel functions may be replaced by

unity. Collecting results gives

h

-lwyk csc kd -2th) in 1) 7—zr—ry h , (e._ ,d)

(1) --—2 «*±3

1Q

U72i^ 1 10

a=±3

1Q

4a

x

(28)

" n

o .2 kd

2a

l bl k 3 10

or tan ~2

10
which indeed will cancel the unperturbed result given by expression (23).

Thus the poles at a = ±3J , that apparently imply a physically meaningless

mode, do not contribute up to the order of our approximations. The impedance

expressions will next be used to generate numerical data for typical TEM cell

geometries.

The radiation resistance R given by (19) is plotted in figure 3 for

g/a=0.1 and 0.2. The dimensions of the RCTL cross section are a^=a2=1.0 m,

and b-j_=b2=1.0 m. The probe varies in length from d/a=0.0 to d/a=0.95, and is

of radius t=l mm. The frequency is set at 1 MHz. The curves show a basically

parabolic dependence on d with the main difference between the two curves resulting from the logarithmic dependence of Z

c on the gap size. This d^

variation is significantly larger than the free-space probe radiation

resistance which is proportional to (kd) , a frequency dependent quantity.

This difference between the radiated power by a short monopole in free space

and that delivered to a TEM mode has also been encountered in other types of

problems such as the coupling between a dipole antenna and an infinite cable

[8]. Figure 4 gives the dependence on d of the ordinary rectangular waveguide

reactance X. The same set of parameters is used to plot X, given by (20), as that used to plot R. The curve shows a highly capacitive contribution which

is independent of the gap size. Again, this result may be compared to the

free-space result for an electrically short monopole which is known as proportional to -i(kd)" 1 [9]. From (20), the dominant reactance behavior for

small kd is more like -i(kd) . Thus, we find that the reactance of a monopole in a TEM cell is more highly capacitive than that in free space.

Finally, the perturbation reactance 6X is plotted in figure 5 using (21),

again for the same parameter set. We find that |6X|<lfi for the entire range

of probe lengths. Thus 6X is at least three orders of magnitude less than

X. Therefore, neglecting 6X will yield excellent results. The FORTRAN

program used to compute Z is listed in Appendix E.

11
6. CONCLUDING REMARKS

This report has detailed the manner in which a coaxial probe excites a TEM

cell. The analysis utilizes three simplifying assumptions, which are (i) the

gap is small, that is G £nG<<l where G=irg/2a, (ii) the probe is thin,

or irt/2a<<l where t is the probe radius, and (iii) the frequency is not too

high so that only the TEM mode propagates and (ka)^ is small when compared to

unity. None of the requirements is restrictive in terms of common TEM cell

usage. The resulting expression for the probe input impedance consists of

three distinct terms R, X, and 6X, these being the radiation resistance due to

the propagating TEM mode, the ordinary rectangular waveguide reactance, and

the reactance perturbation due to the gap fields. The first two terms, R and

X, are computationally simple, with R showing a basically <r dependence on probe length and X giving a highly capacitive reactance. Although 5X consists

of a nontrivial principal value integral, numerical results show

that |6X|<<X, and therefore 6X may be neglected. Thus a reasonably

straightforward form for Z is achieved which may be used to analyze probes

inserted into a TEM cell either to measure, or excite fields as well as other

types of feed lines.

7. ACKNOWLEDGMENTS

The authors are grateful to Dr. M. Kanda for his technical discussions

during the course of this work. They also wish to thank C. K. S. Miller and

F. X. Ries for their continuous interest and support.

12
8. REFERENCES

[1] Tippet, J. C. Modal characteristics of rectangular coaxial strip line,

Ph.D. Thesis, University of Colorado, Boulder, Colorado, 1978.

[2] Sreenivasiah, I.; Chang, D. C; Ma, M. T. Characterization of electrically

small radiating sources by tests inside a transmission line cell. Nat.

Bur. Stand. (U.S.) Tech. Note 1017; 1980.

[3] Collin, R. E. Field theory of guided waves. New York: McGraw-Hill; 1960.

[4] Wilson, P. F.; Chang, D. C; Ma, M. T. Excitation of a TEM cell by a vertical electric hertzian dipole. Nat. Bur. Stand. (U.S.) Tech. Note

1037; 1981.

[5] Tricomi, F. G. Integral equations, chapter IV. New York: Interscience

Publishers, Inc.; 1957.

[6] Abramowitz, M.; Stegun, I. A., ed. Handbook of mathematical functions with

formulas, graphs, and mathematical tables. Nat. Bur. Stand. (U.S.) Appl . Math. Ser. 55; 1964.

[7] Gradshteyn, I. S.; Ryzhik, I. M. Tables of integrals, series, and

products. New York: Academic Press; 1965.

[8] Hill, D.A.; Wait, J.R. Coupling Between a dipole antenna and an infinite

cable over an ideal ground plane. Radio Science, 12, No. 2: 231-238; 1977

March-April

[9] Tai , C.T. Characteristics of linear antenna elements, Chapter 3 of Antenna

Engineering Handbook, H. Jasik, Ed. New York, NY: McGraw-Hill; 1961.

13
APPENDIX A

DERIVATION OF k{^ and K^

Generalizing the chamber widths involves some modifications in the

expressions derived in NBS Technical Note 103 [4]. For the most part, this

involves nothing more than replacing a by a.. The details will be briefly

outlined beginning with the TE type mode kernel which becomes

Q

(h) (t,f ) = Q^h) (t,f) - Q|

h) (t,f) (a-1)

Q^

h) (t,t') s -I*n [(^(4^) It

2

- t'H] (a-2)

Q^

h) (t,t') , f

1 I j=l J mo

cot rJ'V -, JO J_ + J_

K

(j) Mj m

J

= Q

f

(a-3)

where t = a-x, t' = a-x', a = min(a l5 ao), V^ ' = c 2

- M.» M. = mTr/2a.,

and mo refers to the summation over odd m. This kernel will result in the

following solution for the unknown gap field

f (h)(f) =

- Id * L(a)f!(x') (a-4)

2a 1 /g 2 -t' 2 l

^- M^J + h]"

1 (a - 5)

M. cos K^(b. -y)

f.(xj =

I

-1 m

m fa sin ^ cos Mx J (Mg) (a-6)

J z mo K u; sin K U V ^

J ° J m m j

where we recall that x! is the dipole location. The above solution assumes

that the dipole is centrally located.

The y-component of the electric field in the upper, and lower chambers

may now be found. Both TE and TM type unperturbed (ordinary) modes, as well as TE type perturbed modes will contribute significantly to E

J '

. The

unperturbed modes exist only in the upper chamber. Denoting the unperturbed

14
contribution by Ri J '

, and the gap perturbation contribution by Ko , and

referring to Appendix C of NBS Technical Note 1037 [4], we find

K

l Ut'V 6 ij ^

3 x z U

t' V c

2

3 yE z Ut'V

Suitably modifying equation (49) of NBS Technical Note 1037 [4] yields

(a-7)

K^^, X

t

)

^ ^

2a iaC

2~ Idfi, L(a) f (x

t

) f

1 (x

t

) U-8)

Taking the proper derivatives required in (a-7) and substituting the

proper forms for E^ , and H^ ' allows K,^' to be written as follows.

A

n M

j + 4 N

j

K| j) (x

t

,x;) = 6 M Id£ J \

k

3 g(j)(xtJ x;) (a-9)

1 ' r U C m,n=0 M. + N^ -£ mn r

J U

g (x.,x') = ( —r—) sin M.(x + a,-) sin(x'+a.) cos N .y cos N.y' (a-10) mn t V v a_.b_. v j' v 3 J

J J where

A

=

n

\ n =

1 n > 1 (a-11)

and N. = mr/b.. It will prove convenient to extract the delta function

J ~(i) J - -• nature in IOJ/ which occurs when x. -> x . lotice that

A m

2 + -2L N2 = A \ [M

2 + N2

- ?

2 + % M2 + C 2

] n J 7 J n

k 2 j j a

2 j

(a-12)

Therefore K; becomes

Ki

j) (x + ,x:)

1 -t^-lj^^^^/n^^)

I 9

J > ? g

(j) (x.,x:)

m.n=0 M

2 ^ - s

2 """> l l

(a-13)

If we expand a delta function in a double Fourier series over a chamber cross

section we will find

m,n=0

(a-14)

15
Therefore K

(J)

1

(j)

becomes

icuy

K

l Ut' V "

6 lj ,2

Id£

a

C

, «(x

t *;>

°° A [M

2

+a

2

]

/ -v X

" Jo o o 9

(j) (x

+

, x ) m,n=0 M

2 +N2 -c2 mn l l

J J

(a-15)

R{J> and K (j) represent the y-component of the electric field excited

by a symmetrically located VED at x, . There remain only two small changes

to adapt this result to the probe excitation problem. First, the probe

carries a negative y-directed current. Thus Id£ -* - idl . Second, the

Green's function represents results from a unit strength dipole. Therefore

-Id£ -> -1 Therefore K, and Ko finally become

i 'vV "

lj

k

2

A [M

2 + a

2

] n

L j

J

m,n=0 M

2 + N 2

C

V(*t- x P

g

(j)

3 (x., x!) mn t t

K

2 (x

t'

X

t

} lojyiT

2a

l 3

j C

2 L(a)f

J.(x

t

)f

1 (xt

)

(a-16)

(a-17)

16
APPENDIX B

EVALUATION OF Z

The integral defined for Z, is to be evaluated.

z

l =772

2ttI

'V\ i uO)iv v<\J°<z

t s s

J(x) K) u (x

t ,x^)e

iaz

J(x

, )dads' ds (b-1)

(1) Kr ; (x,, xj = - J wy

,2 ? Ut X

t

j

2 2 M

i +a rn - -i

m ;n=0 n M

2+N

2 -?2 mn l l

The a-integration may be performed first, which requires evaluating the

following pair of integrals.

(b-2)

I

I,

2 a .,— —. , iaz,

-~2 o(x. - x )e da

2 2

I A 1 . . g (1) (x

t ,x:)e

laz

da

m,n=0 n M

2+N

2 -C2 mn t l

(b-3)

(b-4)

In each integral we will deform into the upper a half plane for z >

(forward waves), and into the lower a half plane for z < (backward waves).

iklzl

For I, , we find

I, = -iirk6(x.

f

- x. )e

The denominator of I~ may be factored as follows

m2 m 2 r 2 2 r (l) 2 11 mn

T (1)

= [M

2 + N? - k 2

]

2 mn

L

1 J

(b-5)

(b-6)

(b-7)

17
Thus we see that a simple pole arises in the I_ integrand when ct - +ir

2

3 mn Again deforming as for I, gives the following result.

k

2 -N? m -r

(1) |z|

l 2

= * I ^-TTT^Vxpe

^

m,n=0 n r u; mn L z mn

We next proceed to the integration over the current distributions

J(x) and J(x'). The source current J(x') will be concentrated on an idealized linear antenna located at the origin, i.e. x' = (0,y ' ) . Thereto

(b-8)

re x. f

x, i on S and the delta function term will not contribute to

Z-, • Thus Z-, reduces to

I

= ML

o,272 _

L_

n "

n JJT

1 2 M

2

k - N-,

2k I" m,n=0 r J(x)g (1)

v /3 (x,,x:)j(x')e

mn t V v ' mn '

mn S S

ds'ds

(b-9)

The simpler integration over the source current will be considered first,

We recall that the assumed current density on the probe will be

I J(x) = ^t sink ( b

! -d-y)6(p-t)

For the 6* integration we let t' + 0, and therefore

(b-10)

-r

(1) |z

I 3

(^-)I sin M^x + a^ sin f- cos N n y e

'V (b-11)

where the integral I~ is given by

Io

f b

l

b rd

, , , , x .. ,, . , k cos n-n-(cos kd - cos f^d ) ,, 19 \ sin k (b.-d-y'jcos N, d' dy' = ^ ~ 1—*- (b-12)

1 ' k -N^

Because of the factor sin rmr/2, only the summation over odd m will contribute.

•, ta

loop I

Therefore Z-, takes the form

A„ cos niT Z, = —= %-

I —

pry- (cos kd - cos N,d) J(x)cosM,xcosN,y

a

l b

l kI

t m0 ' n=0 rin M ;

JS -r

(1) |z|

xe mn

ds (b-13)

18
The integration over S may be split into separate integrations over y

and . The y-integration has already been encountered and we find

-r^iz I Io P -riPtlsine

J(x)cos M,x cos N-,y e mn ' ds = ~-%-4- I cos(M,t cose)e ~ mn

t de

2fTt

J

v 1 ' o (b-14)

If we now interchange the remaining theta integration with the double

summation, Z, may be written

z

i =

ioou I'

Tra

1 b1 I t

(cos kd - cos N-j d)

n=0 k 2 -N^

x I

,, x cos(M-.t cos 6 )e

r^t sin mn mo r mn

de (b-15)

where we have noted that the theta integration is symmetric about tt . Let

us now consider the sum over m

-r^ 'tsin 6 S (9) = I pjy-COs(M,tCOSe)e

mo r mn

mn (b-16)

The dominant series will be for n = 0. S (e) may be expressed as follows.

S (0) = Re Imo

For large m we see tha

iM,t cose - rijt sini

e e

r nr

mn

(b-17)

n=0

t r'

1 '-'^ M, . Adding and subtracting the asymptotic

mo 1 3 J form of the summand will thus give

S (0) = Re Imo

+ Re I e mo

M,t(i cos0 - sin 0)

iM,t cos -r^t sin0

mn M,t sin

mn

'1 Jn=0

(b-18)

where the first series represents the major contribution. The first series

may be summed

19
Imo

iM^e 19

) •a, , 10 ia,

-Un [tan(^f-)] + ^i

4a (b-19)

We now assume that the probe is thin, irt/4a, << 1, which allows us to

replace the tangent by its argument

i ft an[tan(^|f-)]z ^ (^ e

ie

) = in (jjj-) + ie

If we now take the real part of the first sum in (b-18) we see that

(b-20)

a, 4a, iM,tcos0

S n (6) ~ ln{-±) + Re I e mo

TTt

(1). -T v 't sin -M, t sin

« mn ~ 1 e e

r nr

mn

The correction series will be considered next. Again consider F

"1

(1) mo

J n=0

(b-21)

r

tio )= Ml (1 " k2/ M

l)

2 * V1 -k2

/ 2 ^) = V1 - £ ) (b-22)

2 2 where e = k /2M, is assumed to be small. The correction series may be

written

imo

M, t(i cos0 - sin )

1

M, t esin

ej - 1

1 -e

(b-23)

1

For e small, we have (1 - e) « 1+e. This series converges very rapidly,

thus only a few significant terms may be considered. Because Trt/2a, «1

for these few terms, the exponents may be replaced by unity with little error giving

, 2

y — -

K

y ^ M, ~ T L M

3 mo 1 c mo M,

a 3,2

4a, k I

} (b-24) mo m

The sum of the inverse cube over odd m may be done in terms of the Riemann

Zeta function. Recalling that

C(s) =

I k

k=l

-s (b-25)

For Re(s) > 1 , we see that our expression may be written

l 1 I

1 mo m

- 8 c(3) (b-26)

m=l m m=l (2m)

20
The Riemann-Zeta function is well tabulated, and from [6] we find

c(3) z 1.20205 (b-27)

Collecting these results we have the following for the dominant series

s (e)

3 2 a, 4a a,k

S (6) = -L In (-T1

) + -^5- (4.207175)

O TT TTt 3

V ' (b-28)

The remaining series S (0) for n > 1 will be summed in a manner 3 n —

similar to that used by Collin [3] in treating the probe excitation of a rectangular waveguide. We will make use of the following Poisson summation

formula: if f(x) is a square integrable function, then

I f(na) = - I F(^)

n= -o° n= -°° (b-29)

where F(co) is the Fourier-transform of f(x) defined by

IcoX F(co) = f(x)e dx (b-30)

Our particular problem involves the function f(x) defined by

ixt cos 9 -1^ (x)t sin

f(x) = e mn

/r (1) (x) mn

v '

T (1)

(x) = [x

2 + N2

- k

2

]

2 mn

V L

(b-31)

(b-32)

This function is clearly Lp so Poisson's summation formula applies. The

Fourier transform is given by

F(o>)

0° -r^'(x)t sin e

' e mn ix(tcose+o))

» mn

(b-33)

,(D

If we now rewrite the complex exponential, and notice that r^ (x) is even in x, then F(w) takes the form

°o -r^ (x)t sine

F(uO = 2

f e mn

mn

cos((t cos + co)x)dx (b-34)

21
This integral may be evaluated in terms of the modified Bessel function

K [7] with the result

o

7 V

F(co) = 2K

Q

[(N

2 -kV (t

2

sin

2

9 + (tcose + w)V] (b-35)

If we now return to the definition of S (0) we see that

n

s(e) = 1

+

l f(^) = 1 I If(^L) - f(f)} (b-36)

Thus applying Poisson's summation formula we have

S(e)=| I {-J-F(4a

1 m)--J-F(2a m)} (b-37) m= -oo

which in terms of modified Bessel functions K becomes

o k o ok

S (0) = — J 2K [(N? - K

2

)^((4a 1 m)

2 + 8a n mtcos e + t

2

)

2 ] n ir ^ j ol 1 1 m= -°°

- K

q

[(N

2 -k2 )** ((2a

1 m)

2 + 4a

1 mtcos e + t

2 )^]

j (b-38)

The modified Bessel function decays very rapidly so the only significant

contribution comes from m=0. Thus

S

n (9) ,^Ko [(N

2 -k2 )\] (n>l)

^

(b-39)

Returning to the integral for Z, in (b-15) we have the following

2 TT 2

icou I f (/, . ,n2 °° (cos kd - cos N n d) \ V—H ( 2 Skd)

S o (6)+ I ' 2 r

1 Ve > de

1 ] t ol 1

J (b _ 4Q)

Note that the approximate form for S (0) in (b-39) is now independent of

0. Thus combining all these results, we have

2 / 2 2

icoy r 4a, a^k"

Z, = ^V (1 - cos kdT In {—± ) + -~- (4.207175)

' Zvb^l[ |

TTt ^2

- - / sin

2N

1 d/2 x 2 K [t(N

2 -k2

) %

] -2k2

I 1 ^l— ) — i

1-^— (b " 41)

n=l V sin kd/2 / N^ - \C

22
Note that Z, given in (b-41) is approximate, based on two assumptions

that the probe is thin (Trt/2a-, « 1) and that the frequency is not too high

(kaj « 1).

Finally, we note that the total current at the aperture (y = b, ) is

given by I.to= - I sinkd. Thus,

2 2

1 z ^M tan 2 W in (i)+ i> (4.207175)

1 2TTb,r l irt / (b _ 42)

1

o / sin

2 N, d/2 \ K [t(N2

- k 2 ft

2k I V-—"H

^ 1 2 2-

n=l ^ sin kd/2 ' N^ - VT

23
APPENDIX C

EVALUATION OF Z,

The integral defined for 1

? in (13) is to be evaluated

Z

2 " „_ T 2 2ttI

0) ^ IT' \o iaz

J(x) lO, u (x

t

, x

t )e' uzJ(x )dads'ds (c-1)

t S S

where

and

4

1)( \' \] "TTZ ^^V^V't1 2a 1?

'8/a,a \ 2

frg a . j = l j mo L K

cot K^b. ,

(j) M,

W= I (1) "

. sln^costyj^g)

mo K

v ; sin V 'b, m m 1

(c-2)

(c-3)

(c-4)

If we define two integrals I. and l^ via

—i \ i

i J(x ) f

1 (xt )ds (c-5)

J(x) f

1 (xt )e

iotZ ds (c-6)

and interchange the a- and surface integrations, we may write Z~ in the

following form

7 -lojy

? 2 2 4a

l ! t

' Uo)

2

! 4 hda (c-7)

The integrals I. and I,- now need to be evaluated,

24
The assumed current distribution is given by (2). Therefore I- becomes

V

1 " T J o

(M

l 9) ^

h = I I 'o ± M) ein „(U mo 10 y sin IO y b, m m 1 b, -d

sink(b

1 -d-y') cos K^1) (b

1 -y')dy

(c-8)

Performing the y-integration gives

rmr

4 o

M

1 sin-^- J (M-,g)

K^ sin K^ b, m m

(cos kd - cosK^d)

m

M

2 X2 M, + a (c-9)

The integral Ir is similar to I, except that an additional azimuthal

integration around a probe must be performed. The y-integration is the

same as in I, so Ir becomes

kl

I - y rt L

M.sin

^ J (M,g) (cos kd - cos K^d)

1 2 oM 3/ v m

'

mo K

v 'sin 10 'b, M, + a

m m 1

x cos(M

1 tcose)e 1atsin9 tde (c-10)

o If we designate the theta integration by H (t), then letting -»- 2tt - 9

in the interval [tt ,2tt] allows H (t) to be written

H (t) = 2t cos(M,t cos 9 )cos(at sin 6 )d9 (c-11)

o The second cosine term may be expanded in terms of Bessel functions [7]

oo cos(at sin e) = J.(at) + 2 \ J

9

. (at)cos 2ke (c-12)

k=l

^

Interchanging the sum and integral gives

H (t) = 2t jj (at) cos(M

] t cos0)de

TT f ) + 2 I

J

9 ,(at) cos 2k6cos(M,t cos e)dej (c-13)

k=l J

Again consulting [7] allows us to evaluate these integrals with the result

that

Hm

(t) = 2Trt{ J o (at)J

Q

(M

1 t) + 2 I (-l)

k

J 2k (at)J

2k

(M

1 t)J (c-14)

25
This is of the form described by Gegenbauer's Addition Theorem [6], which

for the special case that the function being expanded is a zero th order

Bessel function, becomes

J (pR) = J (pq)J (pr) + I cos k<j> J. (pq)J. (pr)

o o (c-15)

k=l

7 7 7 where R = [q + r - 2qr cos $ ]

2

. If we let p = t, q

= a, r = M, , and

T7

<p = j , we f ind

R = [a

2 + M

2 ]^ (c-16)

Therefore, H (t) in (c-14) becomes

m

v

'

Hm

(t) = 2^tJ

Q

[t(a

2 + M2

) %] (c-17)

If for simplicity, we define the following quantity

h (a,d)

M

1 sin^ J ( M

1 9) (coskd " cos K

md

} )

rnir

V** sin K^ b, m m

2 2

1^ + a

(c-18)

the above results for I. and I c combine to give Z as

4 5

3 2

h

=

2 2 -iwyk I

a 2 T 2 4a

l J t ^ T h ,(a,d) I

h (a,d)J [t(a2 +M

2

) %

](d

2 _ • m „., m o 1 1 mo mo

a

(6-19)

If we now recall that I , the total current at the aperture, is given by

-I sin kd, Z becomes

o 2

7 -iojyk „„Jl. [ \ L(ot)

Z 9 = £— esc kd

4a

i

2.„2^- f I

hm

,( a ,d) £ h (a,d)J [t(aSM^) ]

l, mo mo

da

'(c-20)

Unfortunately, the a-integration is sufficiently complex that some numerical

approach has to be used in evaluating this integral. Notice that for large

m, since d < b-, , h (a,d) will contain an exponential decay according to

exp(-a(b-,-d)). Therefore, the integrand will decay rapidly with a and a numerical routine should not prove time consuming.

26
APPENDIX D

CHECKS OF R AND Z

This appendix will detail checks of two of the impedance contributions.

First, the radiation resistance will be checked against a result given by

Tippet [1], and second, Z, will be compared to the rectangular waveguide

result given by Collin [3]. The radiation resistance r' of an electric

dipole inside a TEM cell due to the TEM mode, as given by Tippet [1], is

2Z IE cose 2 Re

= —\ (d£ °2V } (d_1)

where Z is the characteristic impedance of an RCTL, I is the magnitude

of the current, d£ is the effective length of the dipole, E is the TEM

mode electric field distribution, 9 is the angle between the dipole direction

and the y axis, (for a vertical dipole 9= 0), and V is the voltage asso- ciated with the TEM mode. Because the probe is an electrically short monopole (kd«l) the radiation resistance should be nearly that of a dipole

with the same current distribution. Expression (d-1 ) assumes an elementary

dipole, however. We may replace IE^d£ with an effective value defined by

(IE v d£) ,,

o eff

o h

I(y)E(y)dy (d-2)

-d

where I(y) = sin k (b -d-y), and E(y), for a centrally located dipole, is

found in [4] to be

?u cosh M(b,-y)

E(y) = — I -tJ sin

^ J (Mg) (d-3) VJ;

a

L sin h Mb, 2 o

v a/ x ' mo 1 Integrating yields the following

(IE d£) « = 2B.

i

(cos kd- cosh Md)

sin

rmr j ( } (d _

v 4)

o

y eff a

L sin h Mb, 2 o

v 3/

which, if we recall the definition of hm (a,d), may be written

mo

(IE da) .-

= -^2Vk.

y h ((< d) (d _

v 5)

o

7 eff a Jtl mx '

27
If we now normalize by the magnitude of the current, |l| = sin kd, we find

R becomes

2Z k' K = —>r csc^kd (I h_(k,d)f (d-6)

a mo m

97 k2 9 2 c 2, - ^

This may be compared to the radiation resistance R due to the TEM mode

given by

2Z k 2

R = -f-csc2 kd £ h , (k,d) I h(k,d)J [t(k

2+M

2 )^] (d-7)

a mo

m

mo m °

The term h (k,d) has a strong decay in M according to

hm (k,d)^e ' (d-8)

M

OnlyJ a few terms will be significant. 3 For these few terms J will remain near unity since we have assumed that 7Tt/2a « 1. Thus, the radiation

resistance may be approximated by

U/ ? I \

2 R = —V- csc^kd I

h (k,d) = R' (d-9)

a \mo /

and the results indeed do check.

Next the case of no gap will be considered. As g -t 0, L(a) ->

so only Z-, contributes. Collin discusses a rectangular waveguide with

a short circuit at a distance i from the probe [3]. Thus we need to

modify his result by letting £ -> °°. Making the appropriate changes, the

radiation resistance for a probe in a rectangular waveguide becomes

R 2ib\k ta "

2 f < d - 10 >

The equivalent reactance is given by

X = 2^k ta"

2 ¥ U(£) + ^(.2072)-2(l-i)

' ^ k

7

I sin

2N

1 d/2? Kn[t(N

2 -k2

)

2 ] ) -2k2 I fi-_l )-5 1 (d-11)

n=l \ sin kd/2/ N^ - r )

In order to compare our expression to that of Collin, we must return to

Appendix B (b-4) and let k be larger than M (a-. =a

2 =a). This will create

a pole on the real axis when a = 3-in- Thus I ? becomes

28
2 2B 1Q

9 10 l V t'

e

^

2 -n? -r<

1 >Ui

m,n=0 " T KU "in t x mn (d-12)

where the primed summation indicates that the term m = "I , n = is missing,

Performing the current integrations yields

.2

Z

l = o nab I 2t o

(1 - c os kd) „„ /Tit o A

v —^ —L- cos(~— cos 9)e

i 3

Q

t: sin

2ik

2 e 10

°° (cos kd - cos N n d)'

t^ An -

mo,n=0

2 .,2 k - N -7-yyC0s(Mt cos 0)e

-ril't sine

mn

de mn (d-13)

The sum over m and n is precisely that encountered by Collin [3] with

the result that

, -r^'tsine

I A L. _ r mo,n=0

(cos kd - cos N,d) ~~

2 2~ yy cos(Mt cos e)e

mn

mn

2 _ (1 - cos kd) a_ 2k

2

2 2 in (TM) + i<_A_ (.2072) -2(1 - -)

Tit 2 a

v (cos kd - cos N-,d)' + i. L n=l k

2 -N

2 K

Q [t(Nf - k Z

) ]

The significant contribution for Trt/2a « 1 may be approximated via

1

, i3 int sin

cos(^ cose )e

IU

(d-14)

(d-15)

Combining these results and integrating over , which simply introduces a factor of tt, yields

.2

Z

l - iwy I

(

-nab I!

(1-coskd) 2 ^ a(l-coskd) 2

' r

n( 4i

} + kY (t2072)

2ik 3,~ 2k^ /

2(1 --)

10

2 P

<*> (cos kd - cos N,d) ., r ., 2 ,2 N 2

-, + a I , ?

] K

o [t(N

1 " k } ] n=l k -N^

(d-16)

29
Finally, letting I = -I sin kd, and separating Z, into real and imaginary parts R and X respectively, we obtain

n, R

=

X

=

o 2abk3

n,

10

. 2 kd

tan -=-

'o . 2 kd

^Tbk

tan ~1

2 2 £n(% v + M- (.2072) - 2(1 - ^)

7Tt 2 a

7T

(d-17)

_ 2k

2 ?L sm^d/z x

2 K

Q

[t(N

2 -k2

)

2

] j

n=l\ sin

2

kd/2 / N

2

- k

2

J

which are the precise results found by Collin in (d-10) and (d-11)

(d-18)

30
APPENDIX E

SUBROUTINE ZIN

SUBROUTINE ZIN is used to compute the input impedance Z of a probe

inserted into a TEM cell. ZIN calls upon four major subroutines. These

are (1) SUBROUTINE ZC which computes the characteristic impedance of a section of RCTL according to (18); (2) SUBROUTINE HMSUM which sums

hm (a,d), and hm (a,d)J

o [t(a

2 + M^] ; (3) SUBROUTINE XSUM which finds the

reactance X given by (20), and (4) SUBROUTINE INTEGR which integrates

a complex function over the real axis. INTEGR is called twice

in order to evaluate the principal value integral defined by (21). The

input to ZIN is mostly via a common block PARAM which contains the cell

dimensions a-,, a

?

, b., , b

?

, and g, the probe dimensions d and t, the

constant it , and the TEM mode propagation constant k. Figure 6 gives a flow chart of SUBROUTINE ZIN. The only direct input to ZIN is e which

sets the desired accuracy. ZIN outputs Z , R, X, and 6X as well as the

total impedance Z.

The constant e is used to set the upper summation limit in the

various sums evaluated by the subroutines. The infinite integral in 6X

is also truncated according to e. There are three basic sums; the sums inherent in L(a) and X, and the summation of the h^c^d). We will

2 ? 2 begin with the last. For large m, that is M n » k

c - a , we have

(1)

K U MM, , and

sin °^ cosh NLd sin °^ -Mjb.-d)

h(a,d)^—5

^

l— * ^-e ' ' (e-1) m

M^ sin h M

] b

] M^

We wish to truncate the summation so that the remainder is less than e . Thus we need to look at the remainder sum

- 4a?(-l)

m

-(2m-l)Tr(b 1 -d)/2a 1 I —! o-2 e <e (e " 2)

m=K (2m-irTT

Clearly the sum is dominated by the following much simpler sum.

31
n / \ v r t\M -(2m-l)x Mx ) =

L H) e K m=K

so we will set M by requiring that

4a? Tr(b,-d)

K ( o!— ) < e 2 K

v 2a,

IT I

(e-3)

(e-4)

Summing R,.(x) is straightforward and yields

R K (x)

2e -2(K-l)x

cosh x

Thus inserting this result into (e-4) and solving for K we find

2

K >

1 rr(b r

d)

In

8a

ett cosh

(b~-d)

~2lu

+ 1

(e-5)

(e-6)

This condition will be used to truncate the h (a,d) summation.

m Next consider the summation in L(a) given by

C( - I

J mo

:ot K^ -, 1 m + -L f Km

J .

(e-7)

For large m the remainder will behave according to

2 I

* m=K.

(2nl " 1)

1 - coth

(2m-l) rrb \

2a

(e-8)

Again we choose a simpler dominating sum. For large m, such that

(2m-l) rrb./2aj >> 1, the coth term may be replaced by

coth

(2m-lH b

J 2a.

= 1 + 2e

-(2m-l)7rb./a.

J J (e-9)

Therefore we need to consider

4 \ _i_ -(Zm-lhbj/aj

tt m=K. 2m-l

(e-10)

32
Clearly we have

4

°°

-, -(2m-l)TTb./a.

4 « -(2m-l)^b /a - I ~—-y e JJ <-£e JJ

m=K. m=K.

(e-11)

Summing the R.H.S. and requiring that the result is less than e/2 yields

a . K. >

j 27Tb.

ix\

16

- £tt sin h

rb.

J

+ 1 (e-12)

This condition will be checked for both j =1 and 2, and the maximum value

K. chosen.

J We next consider the series contained in X. Because both kd and kt

2 are small, the decay of the series will be primarily due to the N-, in the

denominator. Thus, we will consider the remainder series dominated by

I

1 n=N N

(N

1 = niT/b.,) (e-13)

This series in turn may be bounded by a simple integral

n=N N J n im i^ x N

_-, x

So requiring that the R.H.S. be bounded by e implies that

L 2

1 £TT + 1

(e-14)

(e-15)

What remains is to bound the truncation of the principal value integral

-2

In terms of a we see h (a,d) behaving roughtly as a . Therefore, the

111 /- total integrand behaves approximately as a~ . Thus, the truncation gives a remainder integral similar to

-6 , C -5 a da = -p x

o

2 2 2 where C = -couk esc kd/4a-, . If we bound this result by e/2, x must

satisfy

(e-16)

x > (2C/5e)

1/5

(e-17)

33
which is the criterion used to set the upper limit of integration.

A listing of the subroutines follows. They were run at the University

of Colorado using the MNF compiler, Version 5.4. It should be stressed that

e sets a percentage error rather than an absolute error. That is £ = 0.01

implies 1% error rather than accuracy to two digits beyond the decimal point,

34
COz

o

COoz

o

UiD

Z

3

O

o

10

UIX

o

COo\

01N

CD

- o

10 q<

z a: o o - _i

co o

a: o

LU> U. - O

ce >

ui i-

_i —

- co

a. cc z ui o >

o -

z

Z 3

<

CK i K C£ O LU

U. h-Z

< Id

I- O

D

CO CD

LU Z

z -

Z h- - 3

Z Q-Z

U O

o o

> >

CM

CD

<

\ ^ COO z ~ o \

< Z + Li. a: o < uj * < ^ n h to co —

a. - < o o o_ \ — X .- * + N * ZQ-WOOOO OOlllfflO o _i • • - cm o • o <n z _i o_ - - - - . .^r .

. Z<E 11 11 'I II O O 11 CO CM D W O «- CM — CM

N

ocN

o o"

- N

CD O - CO TT CO -\ 0-

vr — LU

CO Z ~

.- II w z LU on— 3

O - < N Z -OOO —

CM II <— I II

n CO U. _J Z Q

O 0- O 11 < o z

OCEO<<C0ajCDI-CLU.O:^UIO.OOOUJ

CQC0c0cDC0flj(Oc0fljC0cQfljfljflj(QQjfljfljflj 0<- — .-CO^TLOLOCDOCOIOCD — CMLOCDCMT

ococococococococo'sr'q-rr'q-ifimioiotQiD 0__,-.-__,____,-__,-.-.-.-,-

OCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM OOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOO

N

oN

CO

a.uizN

LUZ

3

O

tren3

CO

LU Z

CO <

O X O

a: h LU Q_ CO CO CO < CO <

lli <l

Li. _l D w LU

a: N CO <

a. LU LU CO

ft Z CO LU < - H

a lu LUZZ

0- - <

z n ct: -> < o a. 1- H

3 LU

a. \~ a Z 3 - — 0. 3

Z CD UJ — 1 LU

I- I- X

O ICO LU LU m - 3 Q O

a. <

z > cl O _J 3

o z o o o z <

— LU N I O

I- UI

ui en

CO CO Z I

— I- < Q

I CO Z 3 H

D 1 I

O U CD

_l —

X < UI O X X x <r oc 2 ui ui

CO CO z z z < < <

< o o

ce ce ^ ui ui 032 000 _i _i _i

CO O Q

Z Z Z X

O < < H z o z cl a: z

O UI UI UI O 0. Q I

UI 3 3 UI X CO I- UI UI O IIK <hhQ.

LU

CD<I

Q_

CO3

X Q

I- <

a ct

ui ui x x

- _J CO h _i ui 3 ui o OO > CM CM UI ce ui < to x

co z x z - — 1— —

XX> Lm_I I 1- t- COI- N<OQ(Dh

< ~ D

UI H- z z z

- o -

^ ILU Z

LU <

CC h-

<D COz

LU O

X o

H

Z

u. o

o -

I- cc <

UI o

I- <

I- 0.

UI o

_i a:

a. X

I- UI Z O

UI o

UI Z

H

x z ui — ui ct: CO 1- <

UI UI CO X X h- 1-1-3

CL — Ol0- * 3

O

UIX

z

oH

o

LJ

CO

o

cc

UIX ~

H 01

U, ~

O

UI

UI O

o z

z <

< H

a co UJ _

CL CO z ui - re o z

- o

ICO H

- <

oc —

ui a

H <

a: <

at _i < <

1 LU

QC

UI UI 1 X

I- hO CM ~

CM w —

— N

UI _l

UI CJ UI O Z Q

Z < + < I HON

O < - < UI *

LU CC -)

a: + z at UI O M O — N

— h- 3 < - CD CO UI

UI QC O > 3 Z < I- <

3 CC Q

UI UI > 0. CL

0t Z

< Q_ - z <

— (3 -I O <

CC UI I- OXOI- H

UIX - UJ

I- N X

O CK — UI N N N O N

CL

I-

(DO ~> cmV

CO h- ^ Z

CO 3

C\To" < ce -ui ^ N

< s

\ z

z <

< a:

cl < o

< CL *

CL O - N N. •— Z Z CL O D

Z Z _l z z <

O O UI o o oc

<

>

z

X

o

UI CD N CD X _l

UI <

_l o

CL —

z o

o o

O _J

CM

o

CO<o 00 * <— • o

u o o ^

o - - ^ * X O O O T —

• - - CM CL CO CO _JO>->-0* £LQ_ <ww ,^0 UIUI Z 11 11 O 11 - O 1 +

re > • a. o • UIOHOUICOO^^ 1— a: — w 1— 11 11 H 11 XLUZ II COO — CMCO UJN3nZXXXX

00000000000000000

co co o o

o o

o o

o o

o o o o

ajfljQjfljfljfljCOCOcQQjflJCOfljCOCO OOOOOOOO — ^-COOO — CM — — — CMC0C0O OOOOOOOOOOOOOOO

OOOOOOOOOOOOOOO OOOOOOOOOOOOOOO

CO CO

cc at

UI UI > >

Z z

3 3

— CMCOTIOCONCOOTO — CMCO^TIOCOr^COCn — CM COTlOCOI^COCnO'- CMCOTlOCDN

35
II II .- OJ < ca

M n o ~

N N <- _J X

LJ CO Q - CO

CM

CD CD

CD CD

> >

_l _l

UJ Ul

on on X X

^ c\T _l _1 < <

> >

< <

U. U. O D

X X

Q. D.

UJ u

H H

CO CO 2 2

CO CO

Q. Q.

UJ UJ

CO 0_

a. uj

UJ w

\ z

03Z

X

XzX ~ OJ ^T -XX

N -

^ <

X o

CO —

x x 2

on on co

cd cd x

UJ uj <- H i- <

2 2 X — — O

I i

—

J

<

< >

UJ +

on -- * _i o < N >

x w

<- _1 H <

CO UJ x CC

>— II h- —

CO N

- CD N - _l CM

UJ £0 Q - + r- - CO N - CM<

X X xV

X X

x CD CO - 3 OJ O -

_l u. _l x o

U. II

CO £0 <

UJ

on<

X

CO - 2 X o <-

CO x x 2 CD X

UJ • CD Z OJ - — — OJ Q U. -

X U_

UJ x a n — OJ II 3 < CD

CD

UJI

H

xxx

o

_i

_Jo

Ll

CO<

UJ

on<

COz

D

CO - Z X

UJ X

Z CD N

a oj

UJ u_

co * o

CC ll 0. H

x OJ X IZ

i- X X x X -> X «- +

to - 2 X x D CD - _l - CD

_1 OJ O «- OJ U. U. <- X U. CO x < M— ii UJ N N

on< UJo

to z

UJ <

O X Q

2 -UJ < X Q. Q - Z

UJ - —

0_ X

Z CD _J - • <

OJ H

CD <- O

2 L. hLUX

H

crN

x O

XXX -XX

QDCCh -WN

— UJ N CC

CE UJ NX x

-I- x - Ox - X

N -X «- -X - - OJ rI I -— II I < < H

O O to

CL UJ N Q

2 2

r> ~

ii on N Q_

(- X X

< CD CD Z • OC OJ OJ o «- «-

u. u. u.

«- OJ

H X

\- < CO z z • — OL C\J a: o —

a. u. u.

H

I- < z z

— a:

a: o

a. u.

- \ X X

CD CD

OJ OJ

on

r>

h- Q

UJ 2

or uj

o

- _j

CD - -OJ a _i — OJ ^

X CD _l

X

w

CLLUZ

a:o

z

oN

<

ui on 2 < — Q_ I- X

3 2

© D

ct z

CO

— o

CD O

OJ OJ < z

-y

X

UJ

I «- OJ -> < < — XXX «- OJ ~

co cd a.

o o

OJ OJ X X

CO T

X X

x x

co \r o o x x

X. X

OJ CM x \ a. q. - to co OJ 0. Q_ X UJ UJ

I X X w o o -»

Q_ - - CM <- X CD CD _l +

UJ- CD CD _J wooX _J _l —

w < <

- OJ < <

x \ o o

CM OJ

O z -

- OJ

\- \~ to to X X

2 2 -- < < '

inP=CO <

COX

z -

x cQ O x

X X

o o

CM

CO •-

Q_ O x x X h- < 2

ll OJ z «- o o

OJ

O h- <- O CO < + X

CO UJ - =5

CD Q_ W

x

ii ii ii ii ii ii < ii DOUO co o on a

< ii ii ii ii ii ii ii 21 —

«- OJ CO XXX _l _l _1 z

o —

<- z

oQ

cm co \r \n ii hHHh CO CO 00 CO

o —

a h

O D

G O

0. o * o o

_l • 2

< o on

II ll CM D

CD |x — I- Q

I- h ll Ul z

co to x a: uj

CO

CD

CDCO

COCO

CO

I COo

CM

ID

IT) CD

CM

O x — >- > ~-> I (Q €> ix

\- 0) X 01 — O CD O CD CO

-> T

CQ — OJ

UJ CO OJ z • ~

— UJ x H CD >

D - X - i O CO X I- X

q; x - x - ^_

CO X CO H ii 3 ii n ii U. ~-> tO I- N >- — £D

CDCDCDcDcDcDCDCDCDcDCDCD

co — mo^rNCMNr^ojcoLO

tf T to to lO CD CD o o o o o o o

o o o o o o o

o o o o o o o

o o o o o o o

N O - CM o — — .- o o o o

o o o o

o o o o

CD CO

lO CM

OJ CO

CD CO

CM CM CO -sT o o o o

o o o o

o o o o

cd en

cm \r o o

o o

o o

cDaOcDcDcOcDcOcDcDcDcDcDcDcDcOcDcQcOcQcOcDcOcDcDcDcDcD OOOOCD — COOJCMCOCOlfilx^lOOCMCOO^-CDCM^rCMCO — CO CO CO CD CO CD CD

CM CO o o o o

o o o o

•q- Ifi IO lO CD CD o o o o o o

o o o o o o o o o o o o

o o o o o o

OOO<--~CMCMC0C0

ooooooooo

ooooooooo

o o

o o o

o o o

o o o

o o o o o o

OJ CO -7 o o o

o o o

o o o

o o o

o o o

cocno^ojcotTiocDr^cocj)

'- — OJOJCMOJOJOJOJCJOJOJ o *-

CO CO CM CO

CO CO "7 IDCO CO — CMCOTIOCDIXCOOO CMCO'^IOCDIXCOOIO — CMCO^LOCDN

^^^^--.-^.-CMCMOJCMOJOJOJOJ

<- 0J CO ^ \D CD

36
co

C.J

co

+

(J) to

CO

CO o + co o

CO

CM

CO o N w r^ N o i x- ID CJ1 ^, CO N

i CM

CD

ID

CD *> CM CJ CM + CM *— o CO CD

LO o " O ID O CO -—

w N w— O w D * CJ. O o * * 3 CD o N *— + o o o i ~

Z CM

(J) COCM ^^ * w to *~* v^ n ^-s N —

* 0) ^ 00 N o ^. i *-. N CO -— CO D CO ^> CO ~

+ o -—

H o o * ^-^ CO —

o o CO CO o CM > ^r ID ~

ST o to -» 1 + N 00 — N o ID O CM ^ r- CM O CO CO CM CO " ^ h- Q w ^T O — CM

to w «— CO - 00 x * CO O CM CO IO K o Q_ - -I o 3 N o O ID o N o LU 0_ <^ ^ + CM o • O o

—. +|x o

-> h- s - CO CM CM o

O O w O *-^ *"-* 0) X m - •* - o \r ^-v * * * ^ CO N Z CD 1 <- o 01 ^-v O > N O ^ N •-> CO 1 X - CO H * CO C5) ^^ + ^ + ^ o CO -Q "0 z «— 00 > t— H CO ^^ X ID 00 X

*-* N CO *— CO z - -. X o < i H h- 00 o ^-* > N ID N *-* N N \r co X CM > i N. O — oo \ CO + o CO i ^ O ^r o en -CO h- z CM o o *— ^ Z * K CO CD CM -3- o o ^ CM < - ~- I "X CM z * D *-v -o o *^ CO CD o CO ID o o r- <^ O X <- z - *-> X CM CM * O ~ «— CO *— * LO LO o CO — *— o o » o 0. CO 3 -0 *-^ \ N. X. <— ^ K QJ * .-s 2 ^r c— o h- ID

CM o o N o _l - -. n *— «— —

t— — z ^- *-. K ~ *-^ CO « >- o + o CJ O CM O o o i 2 < CM D > < X 0- X LL + * >— -— O H n 1- <J) o r^ o • O ^^ o CO ^ \ w < or z h \ 1 \ \ * < > < z 1- -o "0 00 CQ X CJ N w O O X

o ^ * •— * -^ Z LU — — o w 00 ^^ ^^ X H * ^ K CO CO \ a ^^ Q CM o LO * w

* N CO N < 3 — N _l z 0_ a. CO «— O 0. ~ ^~ ^^ *-* -. -. CO CJ O CD CO * > * o 0. * N i © + H 00 < \ 3 CM X X + z _l z z z T— ':T LO -o 1- ^ to 3 o + N ~ X

> I IO CO LO 111 Z \ Z «— -. X LU s. w <-^ — «— < D — — h- H H CO 00 -> S-. «— + 1 00 i * LU

1 CO o in CM I x z < z o ^ + "- O Z _! i * - Z 00 C0 10 CO X + CO O LO CO D en ^ N \

1 id DO CO — 1- < 0T or Q ^ < o " z z < 1 o — ^^ — CM "> CD * CO ,— i ^

sr T CM |x ^ w LU QT < o Ld i r— X _1 _J -, a x or to * o h- o o H CO LU LU CM CO N .^ CM T CM X

\T ^ CO f- to CO 2 < Cl *: N o i— X «— < w <— o Q_ O x *— * or ~) -i oo z => z s^ o N CO >- N

0) X 00 O i o o — 0. o or m w < + H II _l 00 - z «— o OJ QJ + X Z — LU LO * in *^y CM O CO h- n ^ r^ o X o o h- \ \ — X D Ld — Q_ * o Z Z CM < o z i<r z 00 z II •—

H CD O N 3 CO CJO O X CO N or o z h- o> o II o X z 3 Z z LL LU or N x X o ^^ w ii n ii II II ii _i J I ~) H Z _0 + o o X ID CO o z

o Qi or N < Ll or O O D _l Ld ll — LU e— II o II *— - CM CO ^ LO _i _J II co z or O X CM N CO H O _J • or o CM ID oo or D o ^ h- w < 3 or s: Z _l a. r j n n_ w CO II z r— <— 1- Z H 1- h- H < < z z O 3 or w \ * v. X o < O 3 - O X 3 ^ 1- to ii X LU * ll 1- Q co z

^- < z ii cCi ii ll II z — Z 00 ^ 00 CO to co CJ o I X O H Q CO X N X H f— ^> i ~ 1- CO r- O 3 H Q

X Ld ii b_ N I N -i LU Z DOOliJOE ^r «— CM CO •— _l o LU Z 3 Ll ll II ii ii II * II x LU II II ll LU Z

> CC 2 < 1 H I QJ Or LU 00 o O or CJ XXXXX-IZQ or LU 00 — N>H233:>>0rN3O>QrLU

CO CD CD CO CQ CO CO

ID O CO ^r N CO CD — CM CM CO ^T ID ID o o o o O O O

o o o o O O O

o o o o o o o

o o o o o o o

n co a

cocOQjfflooaoicQajajcQcoiiicOcQcQcQcococoajcoajcOcQcOcOcO OOOOOO — C0r-tDCMN<-C0|x<-^N^-C0CD--C0ONCM ,T — — — CMCOCO^r^r^-lDCDOCMCMCMCOCOIDCDNN OOOOOOOOOOOOOOOOO^ — ^- — ,-,-^ ^ ^^ ooooooooooooooooooooooooooo ooooooooooooooooooooooooooo ooooooooooooooooooooooooooo

^CMOTIDCD|xc000O--CMC0^rL0CDNC000O'-CMC0TlOCDr^ — — <- — r-^-.-^^-^CMCMCMCMCMCMCMCM

ojcococoajcoco QJ CQQJCO qjojcq O O CM O T ID CD N CD O — CO CM ID O O O O O O O <— ni7 ID CD CD O O O O O O O o o o o o o o

o o o o o o o o o o o o o o

o o o o o o o o o o o o o o

o o o o o o o o o o o o o o

— CM CO ^ ID CD N CO OOO— CMCOT

37
z

3

<- CM z e:

CO X X X

* M > >

o CM H H ^ l- •_< MM * co 2 Z o M D D o ^ O <- + + ^~. O

MM X -- H — CM o Mt o CO o o z: z MM ^"N — - ^ ^ CM MM

0_ o CM 0_ Z CM CM — ^ X X X 0.

-» i D X X ^ ^ ^. ^. z M K CM io - h- CO CO T -- CM «- CM D h-

* X N CO - X X Z Z CO CO CO M cd Q r- 0- CD CM X X X X X X X - CD

1 * N Ul - z ,—s ^-. — CM — CM MM CO - Q O O -Q -> Z ^ CO T z z z z a. a. a -» _m CM X - -~ - O O X X 1 1 ^ ^ + UJ - ^-v CJ ~ O o -CM > _J .- • -XX CM CM w w CD ^- -CM CO CO z - <7 -_ < CO 1- -Z < CM CM 1— cm < : CM X CO > CO < 0-

^-* M — CM X X I -— CM ^ X X X Q_ 0_ < < H MM (- M M X UJ X *- CO * CO CD f— 0. <- 2 _l - Cl ~ —• X X X X UJ UJ (0 CO X a. CO < «- c_ - -CO ~ X X «-» h h- CO _l CO 3 --> -J - - CO CO o o X s^ X DO >- _J <

w - Q. ,— ^- .— H CO co x <,,,-., < ^- CM Q_ 0_ — — X X H ^^ o Q. - -< _J

0- CM X (Q O <- -_ x X r^ - WO J> X X X UJ UJ CM CM x x -* — CO CM O -J CM n x - UJ < Z CO \ - h- O 1- CD H CD — h- < < c_ -k < 1 1 X X X X >- > - CM + < -J < < CO < < w ^ — 0- — CM CO ^ * ._ H O CO X -UJ CM - X w _ o o •— MM —

1

1- 1- CO CO — X < V z x x Z -_l LU I— 0- X X X ~ \T CO H CO - X 0. -- N Z 2 CL o_ o_ - - CM — 0- Cl — — X x < r— X < -- X 0. D_ 3 < - X + E * Q — o CO H X X + CM CO _J < X -D _J X X CD CD _l + £. X X 2 Z .- CM X < > u. < -_J _J

CO x .- «— *% — — * 1- ^ 1- CO CM mm -> X 1- < \ z ^- - < UJ UJ - *- -. /-v ^- .— *-* D D Z Z - v-> 1- O x z < <

X Z 2 CD Z -I 2 ^- CO 1 CO ~ H Cl — ID Q CO JE<ED i 1 1 w -^ r-. E JZE " — ^ -_ H H MM X z X i ~

< - * m- z w z i M- w o CO X a. H X X < c_ -q. O — CM — - CD O _J ^^ s: ^- <~^ 1- h- w w co a: z < -o <

UJ ct o T— _l 1 1- ~ O 2 1- _^ + UJ <— x CO O CO UJ C_ < O UJ -_ < < •- CM O O <~> _l - HI-KKWIO + UJOD UJ C_ O < __ X

z < __ CO w I— < 2 - X o_ -> z :_> < o w y |- 2 < 0_ ^ N X X X X X _J _l •— *-> «— < <OOOOZD« ^^ Z<mJ*0. — Q. - + 1- II q m. _- <— a CO 3 2 x ^ CO w CO — 0. o - o O — CM V_ w < < X h- II OOC0C0OO3Z X X MM Q_ O _J h\-OOZZJ(0- 2 CO CO - O X D 2 X i-\v — xcc:^cocoolQ.+ + <zzjjoooo«- o > 1- X — X -_ < 3ZQ. — Ll ii II ll ii _i ll h- «— _l < O 2 3 2 2 0. UJ UJ II X X X X O O £ 1

u. u. ii ll ii M ll 1— H Z d z a. UJ ii _J o o o •— II o ii *-cMC0'q-_jz2T<<h- • c_ ODD -J N CM — MM UJ UJ - - < II o II II -- CM -- CM Z Z CO M. Cm o o _J CM

0_ Z -I II II Z w- -l-Hl-l-<30 ii II ll ll O D a: EE JQ. ii < 0. Cl w w ,— ,— II s: T— r— NIZHhDO II Z _3 c_ z -J cl < _j (flE<ZE mm ZCOCOCOCOOCOOIfi CO Z Z < Z E 1- H ii II II II II r — 7LT. ' CO Z < Z H _J DOLJD •—

_J D K H h- h ii UJ 2 DOOUJODUJ — CM co ^r •- cm MM -J o h- ll UJ Z D O UJ O UJ < WOKW JZQ CO CO CO CO X 0_ LJ COOOQ-OCONXXXX-J-J-JZa CO X CC UJ CO O 0. O N O

CQcOcQcOCQcOcQcOcQcQCQcOcDcQcQcOcOcOcQcOcQcO OOOOC.(DON^OO(Dr-^NW10MnC\l(DO OOOOOO-'-r-OOO^^'JlOlfllDwNNO OOOOOOOOOOOOOOOOOOOOO^- oooooooooooooooooooooo

oooooooooooooooooooooo

C0cOCQCOCQcQcOcQcOa3COCQCOCOcOc0cOCQC0C0cOCQCQC0CQCQCQc0 00000000(0000"-COIfiNN1lOK)C)W70lD'-lfi OOOOOOWWNOIIOIDNNNNOO-WIDCJOT^NN 00000000000000000'-«-«-'-<-CMCMCMCMCMCM oooooooooooooooooooooooooooo oooooooooooooooooooooooooooo ooooooooooooooooOOOOOOOOOOOO ooooooooooooooo

CO CO CO CO CO CO o o o o o o

o o o o o «- o o o o o o

o o o o o o

o o o o o o

o o o o o o

«-N«TIO(DNCOO>0 — CMCOTlOCDNCOCnO'-CM

M-M-M-M-M-M-M-M-M-M-CMCMCM WO^K)tDMOOIO-NO^«(DN(OfflO- CMC0TT1OCDNC0

^^----^-^^^-r-T-CMCMCMCMCMCMCMCMCM — CM CO T ID CO

38
> CO H .- _l ~ s^

UJ O ^

OH - •—* — X UJ - X -3 - o

X _l X i -< a cm o

UJ > i X CO D - CM O

_l CM X ~ o" < X «-* O CD > a o s_ < r—

- > _i z:

u. «- I

< - ^—

-X 1 UJ < o

a. o X or w

Ul - u - CO o »-> h s: w CO CO

CO CO 3 \ co < UJ

Q. Z CO *-. < + _l

Ixl -. -. ^^ + ~

- CO 1- ^-. s— ct

-> OL - w -> CM ct a.

CO *— UJ — a CM X o o /^ z: o

E <r- CM ^x -^ o i x a CO \ > -> o en I CO < CO z o CO *^ a i ^-* H CD o -

< X H -o a. CO *— 1 ^ O CD _l < CO - o

Z - r~ UJ < u. o «— UJ \ + -^ — H < UJ Z a. a: *- I Q h- N w -, ^-. ,-v 1 -, o o ~ «— w £ CK — UJ o *-

- X co S. a: to CO CO CM ^ .-. en w o w O O - X X X - - < o * CM CD z ft. 0. \ — X CM ., a m a - CD X o o o

I * o H UJ o UJ Ul X w z —

* o ., ~ CD -, tr J" CO • - •-

CL ~ ^ CO H- u. -. -* X H o X ^ CO + o _i < -> _l x < a. o o ^

_l Z * * z - < CO ^ *-> < u_ z £ z: -. .—

in < £ ~ o < w uj UJ i - < - •— ~> - < Z Z CD O X -. o z> 3 o r- -, UJ — — - UJ »- a: x > UJ ~ ^

~ CO 0- CD z o o z Z _1 X < u. co CO o z o + o ct < - o or x x o t- _i ^: £ \ X Z UJ o Ll u. o o * u. x X + UJ X X T— -. •—

UJ 01 O w ^. w - X < - • _l - i -- H

D — O X Z U_ CD ~ - u. 1 *— X .— w 5: => x x CO -1 \r W => • CO CO o o ^ zxcouj<o_ r-_j

CO < X ~ < CO + o II O X + + ^ II 1 * o z -~* O CO CO — UJ — o < - oc a. ui — o - - s \ o s H X _l o o w w < o ~m + co ~ o 1- CM -> <— X —

CO II < < CO x rv ii >— XXIhW-N — OQ. x o co x z D UJ < u. u. z z o CM < ii h- w 1 o + u. ^> t- 1 > ii ii co a: < O > X ii ii D CO X ~ x ii Ul * I x CC CO _J O O O < o X CM w o II ii _l 2: Z CM II z o z II —

zzoi-_icD<oro t-<<irozoox-owH _l II II < D a: a. — ii II 1 _i _i u. T— X \ II CM — < < 3 CD \ *-v w 1 ^ II X w D w _J < < o — <i-o_ii:q-Q- — w ii b z + i-sw

_] .- C\J _l (- Q CO £ CD - CM CO _l _J v^ II — CM X Z. X X II —

u. o O UJ n uuj<j:j: -cmiiz jz < h- H ii UJ z 3 O O X X ii < < II X II II II 3 o II w U. X u_ II a:o:-itii.ctoct:a: u OOU.U_XXXliD<n

o co co >- a: w WOJOOIOOhZZ^IWD H O "TM U. Q _ X X X X o — XCDXXXOO oozzoo

o o

o O o o o o o o o o —

<— CM PJ ^ in CO N CO 0) r- ^

(0 CO CO CO CO CO t o in .- cm io

<- CM CM CO ^ T

o o o o o o

o o o o o o

o o o o o o

o o o o o o

cQcOcoajiococQajcocOcocDcococDcOiOcocDiococoajcocOflJcocoaDcocDcoco OOOOfflONNC0O--(\l^l0NW^C!)Wl0ONI0^WI0©O-O^lf)ini0N(\inNOOONW1NN1N-(\IOlD OOOOLOCDCDCOI^OOOOOO^^^CMCMCOCO'q-ininininCDI^r^NNNO^CMCMCMCOCOOCOTi'q-TTiniOinCDCOCOCO OOOOOOOOO^r-^^^r-r-r-^r-^r-^^-^^^^^,-^^>-r-NNC\ICVlNNWWWWWWWC\IWWCMCVlCVI oooooooooooooooooooooooooooooooooooooooooooooooooooo

OOOCJOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

N CO 0> O <- CM ^cvicoTincoNcocno CMCO'TinCDNCOCnO — CMCOTinCDNCOCnO — CMC0Tl0CDNC00>O^-CMC0^inC0I^COCJ»O^CM

.-.- — r-^^-^^CMCMCMCMCMCMCVICMCMCMCOCOCOOCOCOCOCOCOCOTTirT^T^T^r^T'T'T^rmiOlO

39
<7

<x> o

CO

00

CO

0)

0)

CM

en

•sr o

CO + z

0TH

Q

UJ Z

or uj

CO CO O CO

CM CM o o o o o o

CO

T

in id

oo3D

_Jo

to<

UJ

0T< o o o o

CO o o o o o z o o o o o o o o o o o — o o o o o

CO o o o o o Z - - - - CM

Ul ^ >- -- ^ -

UJQ

CD

UJXH

II II II II — CM «- CM II << CO CO CO

CO3o

-J-Io

u.

CO<

UJ<

CO oooo

ID

Ul 00 £ -

UJ

COo

0T

n II

L±J I w3D

_l_lD

U.

CO<

UJ

01<

CO o

U) CO O CM Z ID CJ) CM CM < ^r en co cm Q CO CO «- «-

uj o en cm • 0- T CO CD i s co cm cn

CM CM CM

CD CD CO z - — CO

n h- i —

_1

N 3 II II ii _J

COO 0T

— UJ

UlNNNQ

01

hiX

0)

G>

CO0)

CO

CM

CM

UJOz<Q

LU

Q_<HO

CO CO 3 CO D N

CO _l <^r CO 3 _l ^r 3 o O O _l u_ CM _l _J

_l o CO D u. <

U_ CO Ul

CO < or ii <

UJ < N

UJ 01 CO CD UJ

or < Ul CM

O < o <— z o o o o CO z in CD CM O <

CO o o o o o z O

O < \r CO r- COQ z o o o o o o o o Q CO CO CD <— Ul o o o o o o ~- O O UJ o N N 0. — o o o o o CO LO <— 0. ^T T ^, 1 z (0OOO o o z N o z CD ST t— <~ z - - - CM UJ CO o ~

Ul

— <- •- <— z CM CM 0) _l z ~~ CD CD <tf <

-~

Q Z O

H

Q

UJ 1-

CO1

II

D

h- Ul CQ _l N Q II II II II o D II 11 II _l Ul — .- CM »- CM ii or II II co o or — UJ X D < < CO CO CD 0- D H Ul N N N Q \-

CD

Ul

or

UJ i UJ X h- i

co3DO

U.

CO<

UJor< o o o o

CO o o o o o z o o o o o o o o o o o — o o o o o

CO o o o o o

Ul

UlQ II II II II — .- CM <- CM II 3 < < CO CO CD

CD

UlX

40
CD o

CO

CO

S> CD => s- 1— rs D_ D1 UJ •i

CO U_

41
>>

CD -QO

S_Q.

CDcI

03Xou

C_3

CUSCJ5

42
DC

Figure 3. Radiation resistance for a

] = a 2 = b, = b~ = 1 m,

t = 1 mm, and f = 1 MHz

43
COO

X

X

Figure 4. Reactance for a., = a

2 = b. = b~ = 1 m, t = 1 mm, and f = 1 MHz

44
-.04 -

X

-.12 -

Figure 5. Perturbation reactance for a^ = a 2 = b-j = b^ = 1 m,

t = 1 mm, and f = 1 MHz

45
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBS TN-1054

2. Performing Organ. Report No 3. Publ ication Date

May 1982

4. TITLE AND SUBTITLE

INPUT IMPEDANCE OF A PROBE ANTENNA EXCITING A TEM CELL

5. AUTHOR(S)

P.F. Wilson, D.C. Chang, and M, T. Ma

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

10. SUPPLEMENTARY NOTES

]] Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The input impedance of a probe antenna exciting

magnetic (TEM) transmission line cell is formulated

The formulation also utilizes the results from a pre

distribution inside a TEM cell excited by a vertical

dipole. The final result of impedance is shown to c terms, which are respectively contributed by the ord

waveguide and the gap perturbation. Numerical resul

and imaginary parts of the impedance are given. The

to be proportional to the square of the probe length

largely capacitive.

a transverse electroby a variational approach,

vious work on the field

electrical Hertzian

onsist of two distinct

inary rectangular

ts for both the real

resistive part is found

, and the reactive part

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Green's function, input impedance; probe antenna; radiation resistance,

rectangular coaxial transmission line; TEM cell; variational method.

13. AVAILABILITY

[jCl Unlimited

| | For Official Distribution. Do Not Release to NTIS

~\ Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[~X1 Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

52

15. Price

$9.00

USCOMM-DC 6043-P80
NBS TECHNICAL PUBLICATIONS

PERIODICALS

JOURNAL OF RESEARCH—The Journal of Research of the National Bureau of Standards reports NBS research and develop- ment in those disciplines of the physical and engineering sciences in which the Bureau is active. These include physics, chemistry,

engineering, mathematics, and computer sciences. Papers cover a broad range of subjects, with major emphasis on measurement

methodology and the basic technology underlying standardization. Also included from time to lime are survey articles on topics

closely related to the Bureau's technical and scientific programs. As a special service to subscribers each issue contains complete

citations to all recent Bureau publications in both NBS and non- NBS media. Issued six times a year. Annual subscription: domestic

$18: foreign $22.50. Single copy, $4.25 domestic; $5.35 foreign.

NONPERIODICALS

Monographs—Major contributions to the technical literature on

various subjects related to the Bureau's scientific and technical activities. Handbooks—Recommended codes of engineering and industrial practice (including safety codes) developed in cooperation with in- terested industries, professional organizations, and regulatory

bodies. Special Publications—Include proceedings of conferences sponsored by NBS, NBS annual reports, and other special publications

appropriate to this grouping such as wall charts, pocket cards, and

bibliographies. Applied Mathematics Series—Mathematical tables, manuals, and

studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE: The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at N BS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on N BS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's tech- nological marketplace. Order the above NBS publications from: Superintendent of Document.',, Government Printing Office, Washington, DC 20402. Order the following NBS publications—FIPS and NBSIR's—from

the National Technical Information Services, Springfield, VA 22161

.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May II, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC. 20234

OFFICIAL BUSINESS

Penalty for Private Use. $300

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE COM-215

SPECIAL FOURTH-CLASS RATE BOOK