Numerical Analysis of a Cylindrical Antenna with Finite Gap

Excitation Based on Realistic Modeling

Di Wu, Takuichi Hirano, Naoki Inagaki, and Nobuyoshi Kikuma

Department of Electrical and Computer Engineering, Nagoya Institute of Technology, Nagoy a, Japan 466-8555

SUMMARY

The delta gap model and the frill magnetic current

model are often used as models for the feed point in numeri-

cal calculations for cylindrical antennas. When the delta gap

model with finite gap width is used, the electrical field

produced by the magnetic current of the feed point may be

included in the excitation function of the integral equation

so that a solution separated into the external and internal

surface currents of a hollow cylindrical antenna can be

derived. Such an analytic solution, however, has been ob-

tained only for the case in which the cylindrical antenna has

infinite length or is placed between two parallel conductive

plates. This paper considers a cylindrical antenna of finite

length using the same integral equation. As the first step, a

numerical calculation of the excitation function for the feed

point is considered. The excitation function is derived nu-

merically by a method based on mode expansion in cylin-

drical coordinates and by a method based on the electric

vector potential of the magnetic current ring. The properties

of the excitation function are examined. As the next step,

the cylindrical antenna is numerically analyzed by the

method of moments. In all of these analytical procedures,

the current is represented by piecewise sinusoidal functions

on the surface of the cylinder, and the Galerkin method is

applied. The convergence of the input admittance thus

obtained is compared to the results of various other

methods, such as point matching without considering the

magnetic current at the gap (approximation by the two-

dimensional surface current and the one-dimensional axial

current), and approximation of the axial current by the frill

magnetic current feed model. ' 2000 Scripta Technica,

Electron Comm Jpn Pt 1, 84(3): 6576, 2001

Key words: Feed model; equivalent magnetic cur-

rent; excitation function; method of moments; internal and

external surface current.

1. Introduction

The delta gap model and the frill magnetic current

model are often used as models for the feed point in numeri-

cal calculations for cylindrical antennas. The frill magnetic

current model is often used in the analysis of coaxial feed

antennas, based on Tsais formula [1], which simply repre-

sents the electric field on the central axis. Sakitani and

colleagues have compared various feed point models and

have investigated the convergence of the method of mo-

ments [2].

Special consideration is required when an integral

equation of the Pocklington type is to be solved by the

method of moments [3, 4] based on the delta gap model.

When using a simple method, such as the axial current

method with a one-dimensional current assumed to be

flowing on the central axis, an unrealistic oscillation of the

calculated current distribution is obtained near the feed

point if the segment length is shorter than the antenna

diameter [5]. When using the surface current method, a

two-dimensional approach assuming current flow on the

surface of the cylinder, the solution is not rigorous if the

' 2000 Scripta Technica

Electronics and Communications in Japan, Part 1, Vol. 84, No. 3, 2001Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J82-B, No. 4, April 1999, pp. 609619

65

gap size is defined as a segment since the antenna feed

structure changes due to reduction of the delta gap size

as the number of segments increases.

One of the authors pointed out this situation in

1969 and presented the correct integral equation for

the cylindrical antenna shown in Fig. 1 with finite gap

excitation [6]. In this method, the electric field pro-

duced by the equivalent magnetic current at the feed

point is included as a part of the excitation function.

The conventional integral equation corresponds to the

case in which the cylindrical antenna has the structure

of a hollow cylinder, and the solution is equal to the

sum of the currents flowing on the external and inter-

nal surfaces. In contrast, the new integral equation can

also be applied to the case in which the cylindrical

antenna is composed of a solid cylinder, in other

words, the external and internal surface currents can

be separated. A problem is that the calculation of the

excitation function produced by the equivalent magnetic

current is not simple, and a solution has been obtained

only for the case of a cylindrical antenna of infinite

length and the case of an antenna placed between two

parallel conductive plates [7].

This paper discusses a cylindrical antenna of finite

length by means of the same integral equation. The feed

gap width is denoted by G. It is assumed that the electricfield Ez has the constant value V/G on the gap and is 0elsewhere, with the antenna conductor being perfectly

conductive. The integral equations considered by the

authors are then as follows [6]:

Here GJe, GM

e , GJi , and GM

i are Greens functions of the

electric and magnetic currents to be used in deriving the

electric fields on the external and internal surfaces, re-

spectively. Jzezc and Jz

izc are the equivalent current den-sities on the external and internal surfaces, respectively.

The left-hand sides of Eqs. (1) and (3) are the integrals

over the region containing the feed gap |z| G /2. Theequivalent current is the same as the actual current on the

surface of the antenna conductor |z| ! G /2. Uez andUiz are called the external and internal excitation func-tions, respectively.

This paper deals with a cylindrical antenna by means

of the integral equation presented by the authors. A proce-

dure for calculating the excitation function for the feed point

is presented, and the excitation function is derived numeri-

cally. The result is applied to a cylindrical antenna of finite

length, and the method of moments is applied.

In Section 2, a procedure for numerical calculation of

the antenna excitation function by mode expansion in cylin-

drical coordinates is presented. We then present a method in

which the excitation function is calculated from the electric

vector potential of the magnetic current ring. Section 3

presents an example of the calculation of the excitation

function and demonstrates the properties of the excitation

function. Section 4 examines the current distributions on

the internal and external surfaces of a half-wave dipole

antenna using the Galerkin method [8] with an expan-

sion in piecewise sinusoidal functions. The conver-

gence of the input admittance is examined, and the

result is compared to the results obtained by a combi-

nation of axial current approximation, pulse expansion

of the frill magnetic current excitation, and point

matching and by a combination of the cylindrical sur-

face current/axial current approximation, pulse expan-

sion of the gap excitation, and point matching.

(4)

(5)

(3)

Fig. 1. Cylindrical antenna.

(1)

(2)

66

2. Calculation of Excitation Function for

Feed Point of Cylindrical Antenna

2.1. Method of mode expansion in cylindrical

coordinates

2.1.1. Calculation of external excitation

function

GMe in Eq. (2) is represented as follows [6]:

Here J1(x) is the first-order Bessel function and H02x is

the zeroth-order Hankel function of the second kind. The

path C of integration is shown in Fig. 2(a), and k ZCCCPH(phase constant).

The first term in Eq. (2) is the contribution from the

magnetic current. Substitution of Eq. (6) yields

where the path of integration Cc is as shown in Fig. 2(b).The above integral is taken over an infinite interval, and

numerical integration cannot be performed. Consequently,

the interval of integration is divided into three parts

0, k, k, x, and x, f, and numerical integration is per-formed on the finite intervals (0, k) and (k, x). When x is set

sufficiently large, an approximate expression can be used

for the integrand on the interval (x, f).In this study, the numerical calculation is performed

by setting x = 50/a, so that the approximate expression for

the integrand given in (iii) is valid in the interval of integra-

tion (x, f).(i) For (0, k),

(ii) For (k, x),

where I1(x) is the first-order modified Bessel function of the

first kind, and K0(x) is the zeroth-order modified Bessel

function of the second kind.

(iii) For (x, f), the approximation

is used, resulting in the following expression:

where Si(x) is the sine integral function.

By taking the sum of Eqs. (9), (10), and (12), the

external excitation function is obtained.

2.1.2. Internal excitation function

GMi in Eq. (4) is given as follows [6]:

where J0(x) is the zeroth-order Bessel function, and

H12x is the first-order Hankel function of the second kind.

(6)

(7)

(8)

(9)

(10)

(11)

(12)

Fig. 2. Integration paths.

(13)

67

The first term in Eq. (4) is the contribution of the

magnetic current. By the same calculation as was used for

the external excitation function, we obtain

The above expression is evaluated in the same way as for

the external excitation function.

(i) For (0, k),

(ii) For (k, x),

where I0(x) is a zeroth-order modified Bessel function of

the first kind, and K1(x) is a first-order modified Bessel

function of the second kind.

(iii) For (x, f), the approximation

is used. Then,

By taking the sum of Eqs. (15), (16), and (18), the

internal excitation function is obtained.

2.2. Method based on electric vector potential

of the magnetic current ring

Below we describe the calculation of the external and

internal excitation functions of the feed point based on the

electric vector potential of the magnetic current ring. The

excitation function in the integral equation for the external

surface current is as follows (the details of the derivation

and the notations are given in Appendix 1):

For the external surface, the radius is set as a+ = a +

. Letting o 0 and M = 0, the excitation function isrewritten as follows by separating the term containing :

2.2.1. The case |z| > G/2

Equation (20) can be integrated directly. The second

term containing tends to 0 when o 0.

2.2.2. The case |z| L G / 2

The denominator R3 of the integrand function be-

comes 0 when zc z and Mc 0. We need a procedure tohandle this singular point. In this study the following steps

are taken in the numerical calculation.

(i) Definition of observation point

z = zs is defined as the point at which the integral is

evaluated.

(ii) Division of range of integration

The range of integration is divided into five rectan-

gular regions as shown in Fig. 3. The singular point occurs

only in the central region containing z = zs. In the other four

regions, the integrals are evaluated by ordinary numerical

integration.

(iii) Evaluation of the integral over an infinitesimal

singular region

The central region containing the singular point is [zs dz, zs + dz], [ dM, dM]. If dz and dM are sufficiently small,R can be approximated as follows:

ejkR

is expanded and is approximated by the three major

terms:

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

68

The integral over the small area containing the singular

point is then

(iii-1) Evaluation of Ug zs

The result is as follows (see Appendix 2):

The contribution of the term containing is 0 outside thegap, but it contributes V/(2G) to the excitation functionwithin the gap. This situation produces discontinuities of

the excitation function at the gap ends rG; their magnitudeis half the value of the discontinuity V/G when the magneticcurrent is not considered.

(iii-2) Evaluation of Uagzs

By the same reasoning as for Ug zs, the following

result is obtained:

The above three steps yield the external excitation

function. In this case the radius is defined as a a forthe internal surface, and we set o 0.

3. Properties of Excitation Function

In Sections 2.1 and 2.2, the calculation of the antenna

excitation function is described in terms of the mode expan-

sion in cylindrical coordinates and the vector potential of

the magnetic current ring. The results yielded by these two

methods agree well. Below we discuss the properties of the

excitation function, based on the result given by the vector

potential of the magnetic current ring.

3.1. Dependence on cylinder radius

Figures 4 and 5 show the calculations for W (W = 2

ln l/a, l = 0.5O), respectively, for a gap size of 0.01O andvarious values of Ue(z) and Ui(z). The results reveal the

following. When W is large (the radius is small), the real

part of the external excitation function has a large absolute

value within the gap while the real part of the internal

excitation function has a small value. When the radius is

very small, the real part of the external excitation function

Fig. 3. Division of the integral area including a singular

point.

(22)

(23)

(24)

(25)

(26)

(27)

Fig. 4. External excitation function with variation of

W(V = 1 V, G = 0.01O).

69

approaches the applied electric field at the feed point with-

out consideration of the magnetic current, and the real part

of the internal excitation function approaches 0. The mag-

nitude of the discontinuity at the gap ends z = rG/2 is V/(2G),which is half as great as the discontinuity when the equiva-

lent magnetic current is ignored. The magnitudes of Ue(z)

and Ui(z) are more than 4 orders of magnitude smaller than

the real parts.

3.2. Dependence on gap width

Figures 6 and 7 show calculation examples for Ue(z)

and Ui(z), respectively, as functions of the gap width for the

case where the radius a is 0.005O. If the radius is kept thesame, the real parts of the external and internal excitation

functions are both decreased when the gap length is de-

creased.

4. Properties of Numerical Solution by

Method of Moments

4.1. Internal and external surface currents

In this study, the current distribution on the internal

surface of an antenna of infinite length is analyzed by mode

expansion in cylindrical coordinates. Using the Galerkin

method with an expansion in piecewise sinusoidal func-

tions as the weight function, the current distributions on the

internal and external surfaces of a half-wave dipole antenna

are analyzed, and the convergence of the input admittance

is examined. As in the calculation of the excitation function,

a problem arises due to the presence of a singular point in

the calculation of the mutual impedance by the Galerkin

method. The calculation is the same as in the case of the

excitation function.

Fig. 5. Internal excitation function with variation of

W(V = 1 V, G = 0.01 O).

Fig. 6. External excitation function with variation of the

gap length (V = 1 V, G = 0.01O).

Fig. 7. Internal excitation function with variation of the

gap length (V = 1 V, G = 0.01O).

70

Figure 8 shows the current distribution on the internal

surface versus the radius for a half-wave dipole antenna

with a feed gap G = 0.01O. It is evident that the current onthe internal surface increases with increasing radius. The

real part is approximately 3 orders of magnitude smaller

than the imaginary part and may be ignored. The current on

the internal surface decreases rapidly toward the antenna

end outside the gap.

The current can be approximated as zero, for exam-

ple, at a distance equal to twice the gap width in an antenna

with radius 0.005O, and at a distance of 10 times the gapwidth in an antenna with radius 0.05O.

The above property can be anticipated by physical

reasoning. If the current on the internal surface has a com-

ponent that is in phase with the voltage, this implies that the

power is emitted toward the inside of the cylindrical an-

tenna. Since the radius of the cylinder is sufficiently small,

however, the inside of the cylinder is in the cutoff state, and

the power cannot be received. For the same reason, the

magnetic field inside the cylinder decreases exponentially.

Figure 9 shows the current distribution on the exter-

nal surface of the half-wavelength dipole for various radii.

Both the real and imaginary parts of the current distribution

on the external surface increase with increasing antenna

radius. When the radius exceeds 0.028O, the imaginary partof the current near the feed point becomes positive.

Figure 10 shows the current distribution on the exter-

nal surface of a half-wavelength dipole antenna with radius

0.05O. The excitation voltage is set as 1 V, and the excitationgap width is G = 0.01O. Curves (1), (2), and (3) respectively

represent the current on the internal surface, the current on

the external surface, and the sum of the currents on the

internal and external surfaces of the hollow antenna. Curve

(4) is the current distribution obtained with the conventional

Fig. 8. Internal current distribution near the feed (V = 1

V, G = 0.01O).

Fig. 9. External current distribution (V = 1 V, G =0.01O).

Fig. 10. Internal, external, and total current distribution

(a = 0.05O, V = 1 V, G = 0.01O).

71

Pocklington integral equation, where the magnetic current

is not considered.

The following observations are made from Fig. 10.

The real parts of (2), (3), and (4) almost entirely overlap.

The current on internal surface (1) is nearly 0. As regards

the imaginary parts, (3) and (4) almost entirely overlap. The

current on the internal surface is approximately 0.45 mA,

which is not negligible when the input admittance is to be

calculated for the solid antenna. It is seen that the sum of

the currents on the external and internal surfaces of the

hollow antenna is the same as the current distribution

derived by the conventional Pocklington integral equation.

4.2. Input admittance

Figures 11 and 12 show the convergence of the input

conductance and the input susceptance, respectively, for a

half-wavelength dipole antenna with a = 0.005O and G =0.01O. The solid line is the result of analysis by Galerkinsmethod based on the surface current, using piecewise

sinusoidal functions, and including the equivalent magnetic

current. The dashed line and the dotted-dashed line are the

results given by the methods based on the surface current

and axial current approximations, respectively, with pulse

expansion and point matching applied, and with the gap size

defined as a segment. The dotted line is the result obtained

by the method based on a combination of the axial current

approximation, pulse expansion, and point matching, with

frill magnetic current excitation.

Comparing Figs. 11 and 12, we see that the conver-

gence is fastest in the case of the solid line. It should be

noted that the dashed line, the dotted line, and the dotted-

dashed line are results for the input admittance of the hollow

antenna since the input admittance is calculated by taking

the sum of the internal and external surface currents. The

above comparison of various numerical methods shows that

Galerkins method based on the surface current and

piecewise sinusoidal functions considering the equivalent

magnetic current is the most reliable in examining the actual

radiation characteristics of the given antenna structure.

4.3. Dependence of input admittance on

antenna length

Figure 13 shows the behavior of the input suscep-

tance (the imaginary part of the feed current when V = 1)

Fig. 11. Conductance convergence of a half-wavelength

dipole (a = 0.005O, G = 0.01O).Fig. 12. Susceptance convergence of a half-wavelength

dipole (a = 0.005O, G = 0.01O).

72

on the antenna length L. It is seen that when L is increased,

the input susceptance approaches the value for an antenna

of infinite length. It is also evident that when L is decreased,

the internal surface current at the feed point decreases. The

reason is apparently the effect of reflection at the end point

when the antenna is short.

It is also evident that the approach to the value of the

antenna of infinite length with increasing antenna length L

is slower when the radius a is increased. The convergence

values of the input susceptance for the cases of finite and

infinite length differ when a = 0.10O, apparently due to theerror of the numerical calculation.

5. Conclusions

This paper has considered models for the feed point

of a cylindrical antenna. The analysis was first performed

by mode expansion in cylindrical coordinates. Next the

finite gap feed model considering the equivalent magnetic

current was discussed. A procedure for numerical integra-

tion of an excitation function containing a singular point

was developed, and the excitation function for the feed

point was numerically derived.

In the conventional Pocklington integral equation,

where it is assumed that the electric field due to the feed

voltage V is uniformly distributed in the gap, the excitation

function is a step function with amplitude V/G, but in theexcitation function considering the equivalent magnetic

current, the discontinuity at the gap end is V/(2G), which ishalf the above value. When the gap width G of the feed pointis kept constant and the radius a of the cylindrical antenna

is decreased, the amplitude of the external excitation func-

tion is increased and that of the internal excitation function

is decreased. When a is kept constant and G is decreased,the amplitudes of both the internal and the external excita-

tion functions increase.

The current distributions on the internal and the ex-

ternal surfaces of the half-wavelength cylindrical antenna

are analyzed by means of Galerkins method based on a

piecewise-sinusoidal function. It is seen that the real part of

the internal surface current is 0 and the imaginary part forms

a peak at the center of the gap, decreasing exponentially

outside the gap. It is verified that the sum of the current

distributions on the internal and external surfaces agrees

with the current distribution obtained by solving the equa-

tion without considering the magnetic current.

The convergence of the input admittance of the half-

wavelength cylindrical antenna is examined, and the result

is compared to the results given by the delta gap model

without considering the equivalent magnetic current, and

by the frill magnetic current model. The convergence of the

input admittance differs between the finite gap feed model

and the frill magnetic current model since the structures are

different. The proposed method provides a strict model for

the antenna structure, and it proves to be the most reliable

in deriving the radiation characteristics.

REFERENCES

1. Tsai LL. A numerical solution for the near and far

fields of an annular ring of magnetic current. IEEE

Trans Antennas Propag 1972;AP-20:569576.

2. Hachiya A, Eto S. A study of the feed point in the

numerical analysis of antennas. Trans IEICE

1984;J67-B:945952.

3. Harrington RF. Matrix methods for field problems.

Proc IEEE 1967;55:136149.

4. Harrington RF. Field computation by moment meth-

ods. Macmillan Co.; 1968

5. Mittra R. Numerical and asymptotic techniques in

electromagnetics. Springer-Verlag; 1975.

6. Inagaki N, Sekiguchi T. A note on the antenna inte-

gral equation. IEEE Trans Antennas Propag

1969;AP-17:223224.

7. Inagaki N, Sekiguchi T. An integral equation for a

cylindrical antenna with a finite-width gap at the feed

point and its solution. Trans IEICE 1969;52-B:292

298.

8. Stutzman WL, Thiele GA, Antenna theory and de-

sign. John Wiley & Sons, Inc.; 1981. p 323332.

Fig. 13. Input susceptance versus antenna length

L (G = 0.01O).

73

APPENDIX

1. Derivation of External and Internal

Excitation Functions

Consider, as in Fig. A.1, a magnetic current ring with

radius a. The electric vector potential is

The M component of the electric vector potential is

where

The following relation exists between the electric

vector potential and the corresponding electric field:

The component of the electric field in the z direction

is

Using integration by parts, the following expression

is derived from Eq. (A.2):

It is also seen that

Substituting Eqs. (A.7) and (A.8) into Eq. (A.6), and using

the relation between wR/wU and wR/wMc, we obtain

On the external surface of the antenna we set

On the gap, we have

Integrating Eq. (A.9) along the gap and applying Eqs.

(A.10) and (A.11), the electric field on the external surface

of the antenna due to the equivalent magnetic current on the

gap is obtained as follows:

The Greens function of the magnetic current in the integral

equation for the external surface current is

where

The corresponding external excitation function is

Fig. A.1. Geometry of magnetic ring.

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

(A.8)

(A.9)

(A.10)

(A.11)

(A.12)

(A.13)

(A.14)

(A.15)

74

Similarly, the Greens function of the magnetic cur-

rent in the integral equation for the internal surface current

is

where

The corresponding internal excitation function is as fol-

lows:

2. Derivation of Ugzs

The second and third terms in the above expression tend to

0 when o 0. Consequently, by integrating the first termand taking the limit as o 0, we obtain

3. Derivation of Uagzs

When Mc is small, we have sin2(Mc/2) | (Mc/2)2. Sub-stituting Eq. (22) into Eq. (24) and rearranging, we obtain

For simplicity, we set adM = dz. Then, integration forA, B, and C results in

Substituting Eqs. (A.23) to (A.25) into Eq. (A.22), we

obtain Eq. (27).

(A.16)

(A.17)

(A.18)

(A.19)

(A.20)

(A.21)

(A.22)

(A.23)

(A.24)

(A.25)

75

AUTHORS (from left to right)

Di Wu (student member) received his B.S. degree from the Department of Communications and System Engineering,

Harbin Institute of Technology, in 1985. He completed his masters program at the Chinese Space Technology Research Institute

in 1988. He was admitted to Xian Space Radio Technology Research Institute and engaged in measurement of satellite repeaters.

He came to Japan as a Foreign Invited Researcher in 1995. He is now a graduate student in the second half of his doctoral

program at Nagoya Institute of Technology, engaged in research on numerical analysis of linear antennas.

Takuichi Hirano received his B.S. degree from the Department of Electronics and Information Science, Nagoya Institute

of Technology, in 1998. He is now a graduate student at Tokyo Institute of Technology.

Naoki Inagaki (member) received his B.S. degree from the Department of Electrical Engineering, Tokyo Institute of

Technology, in 1962 and completed his doctoral program in 1967. He has been a professor at Nagoya Institute of Technology

since 1984. He was a visiting senior research associate at the Electroscience Research Institute, Ohio State University, during

1979 and 1980. He has engaged in research on antennas and electromagnetic theory. He holds a D.Eng. degree. He is the author

of Electromagnetic Engineering for Electrical and Electronic Engineering Students and other books. He received the Inada

Prize in 1964, a Paper Award in 1974, and a Control Award in 1983 from IEICEJ. He is a member of IEEJ, ITEJ, and IEEE.

Nobuyoshi Kikuma (member) received his B.S. degree from the Department of Electronic Engineering, Nagoya Institute

of Technology, in 1982. He completed his doctoral program at Kyoto University in 1987. He has been an associate professor at

Nagoya Institute of Technology since 1992. He holds a D.Eng. degree. His research interests are adaptive arrays, analysis of

multiple wave propagation, in-house radio communication, and electromagnetic theory. He was the recipient of the 4th Award

from the Electronic Communications Foundation. He is a member of IEEE.

76