NPSEC-93-011

NAVAL POSTGRADUATE SCHOOL

Monterey, California

Design of a Continuous ResistivelyLoaded Monopole Antenna

by

Darwish Abd El Aziz Mohamed

Ramakrishna Janaswamy

May 1, 1993

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE

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3. REPORT TYPE AND DATES COVERED

Interim Report 1 Jan 93-1 May 93

4. TITLE AND SUBTITLE

Design of a Continuous Resistively Loaded Monopole

Antenna

5. FUNDING NUMBERS

6. AUTHOR(S)

Darwish Abd El Aziz Mohamed and Ramakrishna Janaswamy

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Naval Postgraduate School

Monterey, CA 93943-5000

8. PERFORMING ORGANIZATIONREPORT NUMBER

NPSEC-93-011

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U.S. Army CECOMCenter for C3 Systems

Fort Monmouth, NJ 07703

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official policy or position of the Department of Defense or the United States Government.12a. DISTRIBUTION /AVAILABILITY STATEMENT

Approved for public release;

distribution unlimited.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

Hallen's integral equation with continuous resistive loading is used to design a broadband

monopole antenna. The resistive profile is different from that of the well known Wu-King profile

and results in a higher efficiency compared to the latter. An optimization function for determining

the optimal continuous resistive load of the monopole antenna in order to satisfy the broadband

property is constructed. Simplex method is used to perform the optimization. The integral equa-

tion is solved by the moment method to determine the current distribution, the efficiency, and the

radiation pattern of the monople.

14. SUBJECT TERMS 15. NUMBER OF PAGES

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sndard Form 298 (Rev 2-89)scribed by ANSI Std Z39-'8

4 3S3

r*

ABSTRACT

Hallen's integral equation with continuous resistive loading is

used to design a broadband monopole antenna. The resistiveprofile is different from that of the well known Vu - King profileand results in a higher efficiency compared to the latter. An

optimization function for determining the optimal continuousresistive load of the monopole antenna in order to satisfy thebroadband property is constructed. Simplex method is used toperform the optimization. The integral equation is solved by the

the moment method to determine the current distribution, theefficiency and the radiation pattern of the monopole.

- 1 -

1- INTRODUCTION

A straight wire antenna of fixed length is narrowband due to the

rapid variations in its input impedance, and to a lesser degree,

in its radiation pattern. A wire antenna is however, light weight,

easy to fabricate and simple to mount on any platform. Travelingwave antennas are desirable for purposes of broadband and

directional communication. A traveling wave antenna may be

obtained by introducing dissipative elements into the antennasystem. The introduction of lossy elements will cause a decrease

in the efficiency. Altshuler [1] found that a traveling wavecould be maintained along a part of a thin cylindrical monopole,

if a resistive loading of appropriate magnitude is inserted 1/4

wavelength from the monopole end. Wu. and King [23 found that if

the antenna is made of resistive material so that the internalimpedance per unit length, is a particular function of the

position along the antenna, a pure outward traveling wave can

exist on an antenna of finite length. An attempt had been made by

Janaswamy [3] to design a broadband monopole antenna by using thecontinuous resistive profile proposed by Wu-King. However theWu-King profile results in a low radiation efficiency. In orderto design a monopole wire antenna that has a broadband propertyand has at the same time higher radiation efficiency than thatobtained using the Wu-King profile, we consider a modified Wu-Kingprofile and incorporate it into Hallen's integral equation [43.

The solution of the integral equation will depend upon the loadingimpedance. We construct a real function which assumes a minimumvalue when the broadband property is reached. FORTRAN programsare developed for solving the integral equation of the antenna as

well as for determining the optimized load impedance. A comparisonis made between the performance of the monopole lo»i»tfo<:l with

Wu-King profile and that of the monopole loaded with the optimumprofile.

2- FORMULATION OF THE PROBLEM

For thin monopole wire antenna, the current distribution wasassumed to be of sinusoidal form. For a finite diameter wire,

the sinusoidal current distribution is not exact. To find thecurrent distribution for a cylindrical antenna, an integralequation can be derived [53. All other antenna properties can be

deduced once the current distribution along the wire antenna has

been determined. We present a brief outline of the integralequation below.

- 2 -

Consider a thin, cylindrical monopole of radius a and length h,

center driven by a delta -function generator of voltage V^ at the

angular frequency o. Let the monopole be situated in vacuum, have

an internal impedance per unit length Z , which could be a

function of position along the monopole. The notation used is as

follows

E Cz) : the tangential component of E at a point z on thez

antenna surface.

I Cz) the current intensity at the same point.

Z Cz) : the internal impedance per unit length of the antenna.

Ve then have

E Cz) = Z Cz) I Cz) c 1 3Z I

Also, E Cz) can be expressed in terms of the vector magnetic

potential A* which has only, a z - component A_ , as

E Cz) = -jo 1 + — A Cz) c 2 :

L k2 dz 2

J

Equating CI) and C2), we obtain on the antenna surface that

^2 • .2

A Cz) + k 2 A Cz) = Z Cz) I Cz) c 3 3

dz 2 r 2gj

where, k = o _/ u co o

The particular solution of equation C3), is

kAip Cz> = J —

JZ Cz ) I Cz ) sin kCz—z ) dz c a 3

The homogeneous solution of equation C3) and corresponding to thecondition A^C-z) = A Cz) and to a driving voltage V, is

A u Cz) =Z n

-J k

oD cos kz + —— sin k|z| c 5 3

where D is a constant to be determined. Combining the particularand homogeneous solutions, the solution of the differentialequation C3) is obtained as

- 3 -

A Cz) = A u Cz) + A Cz)Z Z h 2 p

^[° cos kz + si n k|z|j

kr—I

z,Cz ) I Cz) sin kCz-SD dz

C 6 3

If the antenna considered is thin, i.e. if h » a, the vectorpotential, A Cz) can be expressed in terms of I Cz) as [5]

A Cz) =4 n

I Cz)

-jkR -jkR.

dz C 7 3

where

.-[

R =

K2

=

C z - z )2

C z + z )2 + a

2

1/2

1/2C 8 D

Combining equations C6) and C7), an integral equation for the

current I Cz) is obtained, where

I Cz)

-jkR -jkR.

dz

J r Cz ) I Cz ) sin kCz-z ) dz

o o

+ C cos kz = - J2 r

sin k |z|

C 9 3

where, K, = / fJQ / c is the intrinsic impedance of vacuum and C

is a new constant. equation C9) is the Hallen's integral equationfor the current distribution of the loaded monopole antenna.

- 4 -

3- METHOD OF SOLUTION

Moment method techniques [4] will be used to solve the integral

equation C9). The current I_ is expanded in terms of N piecewiselinear subdomain basis functions, as

N

2 +- n nn = l

- Z n-1

G Cz')n

Z Zn n-1

Z — Zn + i

Z — Zn + 1 n

z < z < zn-1 n

< z < zn + 1

C lOtO

elsewhere

Also, the internal impedance per unit length Z is assumed to be

of the form

A MC 1 1 DZ Cz) =

h - z

2 m-1Z C z • h )

m = l

In comparison the Wu-King profile is of the formwk R

Z Cz) = 2

h - z

where R depends on ka and kh. Substituting CIO) and Cll) intoC9) yields

N

I I.

n = l

G (Z )n

L 4 » Ri

-jkR.

4 rc R.dz

N 2 M

"JX p X

o n = 1 O m=

1

2 J-1

CZ ) Cz /h) G (z ) sin kCz-z 5 dz

+ C cos kz2 <

sin k |z|

(12)

which can be written in a simple form as

- 5 -

N N M

T I A Cz) +2 1 I, z™ B Cz) + c cos kz = -j —£- sink|z|

n = l n = 1 m = 1 ^o

where,

C 13)

A Cz) =n

G (z )n

-jkR

4 n R

-,kR.

4 tt R.dz C 14}

B Cz)r J h - zo O

m-1Cz /h) G (z ) sin kCz-z ) dz

c is:

To determine the coefficients I , n = 1.2,...,N and the unknown C,

we will use the point matching method which will result in CN+1)

linear equations of the form

N M

5 I A Cz ) +5 Z B Cz )fc rilnp *- mnmpn = 1 m= 1

+ C cos kz = -j —— sin klzplp 2 t;

P = 1.2, 3, , N+lC 16)

Solution of this system of equations can be considered as functionof the M parameters Z .

4- THE OPTIMIZATION FUNCTION

A broadband monopole antenna in admittance can be synthesized by

minimizing a convenient optimization function. An optimization

function is constructed which includes the current distributionparameters I , and which is minimized by varying the values of

the impedance parameters Z . The proposed optimization functionincludes values of both the admittance as well as of the radiationefficiency weighted differently. Ve construct the optimizationfunction as:

F< Zl ,Z2,.. Zm>

= V.Yi n Sir c f ) - Y

c r efV, I I

«f > - K cr el

C 17 j

where,

n. is a number of spotfrequency range

frequencies in the desired

- 6 -

n

y Cf ) is the input, admit,Lance of the monopole antenna atl n n

frequency f , and it is equal to

Y Cf ) = 2 I (f ) / V, cisDt n n In o

^Cf^) is the radiation efficiency at frequency f^, and

it is equal to

£Cf ) = 1 - P, / P .where^ n Loes l n

P. (f ) = -4- II (f )|2 R „Cf ), (19)

in n ^Z in t n n

P, Cf ) = -i- Cz ) lICz )|2 dz

lose ) = -1- Z (z ) lICz )

I

n ^ J v I

C 20J

Y is the reference admittance in the desiredcrsl

frequency range, which we take as the average of

Y Cf ) over f .

i. n n n

? , is the reference efficiency in the desiredcrel

frequency range, which we take as the average of

^ Cf ) over f .

n n

V and W* are the weights for the admittance and efficiencyi v n C,

respectively.

The Simplex method of optimization C5] is used for determining the

coefficients C Z ) of the internal impedance by minimizing theIT)

optimization function given in C17). The Simplex Cbody in multi-

dimentional space) optimization method computes the values of the

optimization function at the vertices of a simplex in the

parameter space, and on the bases of these values chooses a new,

presumably smaller simplex within which an optimum should be

situated. The number of the vertices of the simplex is equal to

the number of parameters plus 1. Here we chose M = 4 thus,

resulting in five vertices CX„, X,, X . X' , X ), where X. is a

vector of the impedance coefficients CZ , Z , Z , Z ). We haveused coefficients corresponding to the Vu - King profile as one of

the starting vectors needed to accomplish the optimization.

5- DESIGN OF THE MATCHING NETWORK

Once a resistive profile has been determined, we design a matchingnetwork to make the antenna even more broadband. We have designeda matching network using the EESOF C ESYN / TOUCHSTONE ) circuitdesign code [73. The matching network is a 2-port device insertedbetween the source with 50 ohm source resistance and the designedmonopole antenna. First a matching network was synthesized and

- 7 -

optimized using the program ESYN with lossless lumped elements.Then the synthesized matching network was simulated and furtheroptimized using TOUCHSTONE program but with lossy elements with Q-

factor equal to 75. Two matching networks are then synthesized.

Two matching networks have also been designed for the monopoleloaded with Vu - King profile.

d- RESULTS

For a monopole of length h = 1.5 m and radius a = 0.5 cm operating

over the frequency range 30 - 90 MHz, a FORTRAN program was

developed in order to solve the integral equation C16) subjectto minimize the optimization function C17) with respect to the

coefficients Z of the resistive loading impedance. The number of

the coefficients Z was taken as M =4 and the current distributionwas expanded using N = 10 basis functions. Different optimalloadings were determined for different permissible values of the

optimization function C17), where the broadband property wassatisfied. A flat input impedance can be obtained at the expenseof the radiation efficiency and vice versa. the optimizedresistive load is the one which satisfies the good broadbandproperty and at the same time gives a high antenna radiationefficiency. The optimum load was found with equal weights

V = Vx = 1 ) to be

42 m-i

Z Cz) = J Z C z / h ) C2ijh — z m=i

where,

Zx

= 6.41

Z3

= -13.49y Z

2= 2. 95,

y 2. = 6 81

h = 1

.

5 m, a =0.5 cm ), Vu -

the geometric mean frequencyFor the considered monopole C

profile was calculated at the geometric mean frequency of

51.96 MHz. It is

vk 564.64 -j 148.67

Z Cz) = ohm/m C22Ji. h - z

For implementation we use only the resistive part. Fig. 1 showsthe variation of the resistive loading of the optimum load

compared with that for Vu - King profile. It is clear from the

figure that the internal resistance R per unit length for the

Vu-King profile is higher compared to the optimum loading.

Table CI) shows the variation of the input impedance of the

- 8 -

monopole loaded with the optimum load compared with that of

the monopole loaded with Vu and King profile [33. It can be seen

from Fig. 2 that the input resistance R for the monopole with the

Vu - King profile is quite flat whereas the one using the optimumload varies from 54.4 ohm at 30 MHz to 343.9 ohm at 90 MHz. Also

from Fig. 3, it is clear that the input reactance is capacitive

for the Vu-King profile over the entire frequency range and has a

little change while for the case of the optimum load there is a

resonsance at 52.7 MHz. Fig. 4 shows the main disadvantage of

using Vu - King profile : lower value of the radiation efficiency

£. It can be seen that the maximum value of £; for the monopoleloaded with Vu - King profile is 11.85 fc C at 90 MHz ) which is

less than the minimum value of < for the monopole loaded with

optimum load, where < changes from 15.83 v. at 30 MHz to 39.74v. at 90 MHz. Table C2> shows the variation of the efficiencywith the frequency for the optimum load compared with that for Vu

- King load. The VIRE software program 161 has been used to

determine the radiation pattern of the designed monopole using

lumped equivalent loading. Fig. 5 shows the radiation pattern at

frequency 70 MHz for the case of the optimum load. Table.

3

summarizes the scattering parameters of the designed matchingnetworks using lossless lumped elements for the two impedanceprofiles. Fig. 6 shows the S parameter variation across the

frequency range and Fig. 7 shows that the VSVR for the optimumprofile is less than 2 all over the operating frequency. Also it

is clear from Fig. 8 that the gain CS21> is greater than 0.94

across the frequency band. The overall efficiency of the monopoleC < x 1^21 I

^ ^or ^ne t'wo cases OI " loading is shown in Fig. 9.

It can be seen from this figure that the overall efficiencyof our design has improved significantly compared to the Vu -Kingprofile. To be the matching networks realizable, the previouslydesigned networks are further optimized using TOUCHSTONE code bymaking the inductors and capacitors lossy with a Q factor valueof 75. Table. 4 summarizes the scattering parameters of thedesigned matching networks obtained using lossy lumped elements.Figs. 10. 11 show the variation of the S - parameters across thefrequency range for the lossy matching network case. Figs. 12 and13 show the VSVR and the overall efficiency for the lossy matchingnetworks. It is seen that even with lossy elements the presentdesign yields an input VSVR < 2 while still maintaining a highoverall efficiency compared to the Vu - King case. Fig. 14 showsthe topology and element values of the matching network for theoptimum loading, while Fig. 15 shows the circuit diagram for the

- 9 -

matching network which corresponds to The Wu - King profile.

7- CONCLUSION

It was shown that efficient optimization of wire-antenna

admittance can be performed by varying distributed loadings alongits length. Depending upon specific requirements on the wire -

antennas, it is possible to construct many optimization functions

by changing the values of the weights V and W~. The choice

of the values of V and W* depends upon the specificrequirements of whether a broadband input admittance or overall

efficiency is desired. The decision on terminating the optim-ization procedure depends on the goal of the antenna synthesisprocess. This report has introduced a new loading to improve the

efficiency of a broadband monopole. A matching network was

synthesized in order to maintain good impedance matching such thatthe VSVR is less than 2 across the required frequency band.

REFERENCES

[1] Altshuler,E. E., "The Traveling-Vave Linear Antenna, ' IRE. Trans.

Ant. Prop. AP-9, No. 4, pp. 324-329, 1961.

[2] Vu,T.T and R. V.P.King, '"The Cylindrical Antenna Vith Non

Reflecting Resistive Loading, " IEEE Trans. Ant. Prop. AP-13,

No. 3, pp. 369-373, 1965.

[3] Janaswamy , R. , "Videbanding Techniques For VHF Antennas-I,"Naval Postgraduate School, Monterey, Report # NPS-EC 010,1992.

[4] Balanis, C. A. , Antenna Theory Analysis And Design, Harper and

Row, Inc. , 1982.

[53 B.D.Popovic, M.B. Dragovic and A. R. Djordjevic, Analysis and

Synthesisof Wire Antennas, Research Studies Press, England ,

1982.

[6D Davis, V. A. , VIRE Program, Virginia Polytechnic Institute and

State Univ. , 1980.

E7] EESOF, Inc., Touchstone User's Manual, 1986.

- 10 -

TABLE 1. THE INPUT IMPEDANCE OF LOADED MONOPOLE

FREQUENCY INPUT IMPEDANCE INPUT IMPEDANCE INPUT IMPEDANCE

MHz UNLOADED OPTIMUM LOAD VU-KING LOAD

30 9. 47-J196. 03 54. 65-

j

203. 37 252.85-j 305.60

40 20.16-j 80.47 75.98-j 96.28 254.51-j 241.49

50 40.76+j 17.53 112. 19-j 18.23 253.09-j 209.85

60 84.49+J120.62 171. 58+

j

36. 96 248. 56-j 188.14

70 193.44+J242.89 256. 88+

j

51.89 241.91-j 174.81

80 502.85+J310.63 336. 08-

j

02.04 234.43-j 164.34

90 789.49-jl96.58 343.98-j 98.20 227.13-j 155.28

h = 15 id a = 0.5 cm

TABLE 2. SCATTERING PARAMETERS OF THE LOADED MONOPOLE

WITH LOSSLESS MATCHING NETWORK

FREQUENCY

MHz

OPTIMUM LOAD VU - KING LOAD

|S11| |S21| |S11| |S21|

30 0. 335 0.942 0.036 0.99940 0.299 0.954 0.019 1.00050 0. 273 0.962 0.019 1.00060 0.217 0. 976 0.029 1.00070 0.210 0. 978 0.014 1.00080 0.169 0.986 0.034 0.99990 0.184 0. 983 0.033 0.999

h = 1.5 m . a = 0.5 c m

- 11 -

TABLE 3. SCATTERING PARAMETERS OF THE LOADED MONOPOLE

WITH LOSSY MATCHING NETWORK

FREQUENCY

MHz

OPTIMUM LOAD VU - KING LOAD

|S11| |S21| |S11| |S21|

30 0.259 0.882 0.133 0.918

40 0.265 0.918 0.116 0.93950 0.243 0.930 0.058 0.94760 0.193 0. 941 0.038 0.94670 0.187 0.937 0.066 0.93880 0.153 0.935 0.058 0.932

90 0.201 0.913 0.071 0.922

h = 1.5 m a = 0. 5 c m

- 12 -

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Fig. 15 Matching Network For The Monopole Loaded

The Vu - King Profile

26

INITIAL DISTRIBUTION LIST

No. Copies

1. Defense Technical Information Center 2

Cameron Station

Alexandria, VA 22304-6145

2. Dudley Knox Library, Code 52 2

Naval Postgraduate School

Monterey, CA 93943-5002

3. Chairman, Code EC 1

Department of Electrical and Computer Engineering

Naval Postgraduate School

Monterey, CA 93943-5000

4. Professor Ramakrishna Janaswamy, Code EC/Js 10

Department of Electrical and Computer Engineering

Naval Postgraduate School

Monterey, CA 93943-5000

5. Professor Richard Adler, Code EC/Ab 1

Department of Electrical and Computer Engineering

Naval Postgraduate School

Monterey, CA 93943-5000

6. Dr. Darwish Abdel Aziz Mohamed 5

Officers Building Home 2BFlat 44

Toson- Abokir- Alexandria, Egypt

27

DUDLEY KNOX LIBRARY

3 2768 00342562