1142 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, KO. 10, OCTOBER 198*

ACKTYOWLEDGMENT The author wishes to express his gratitude to Mr. William

C. Cummings, Dr. Alan J . Simmons, and Dr. Leon J . Ricardi for their encouragement and technical discussions throughout this study. The computer programming of the gradient search by Mr. David S. Besse and Mr. Leon Niro is appreciated.

REFERENCES [ I ] W. F. Gabriel, “Adaptive arrays-An introduction.’’ Proc. IEEE,

[2] J. T. Mayhan, A. J . Simmons. and W. C. Cummings, “Wide-band adaptive antenna nulling using tapped delay lines,” IEEE Tram. Antennas Propagar., vol. AP-29. no. 6, pp. 923-936, 1981.

131 J. T. Mayhan, “Some techniques for evaluating the bandwidth characteristics of adaptive nulling systems.’’ IEEE Trans. Antennas Propagat., vol. AP-27, no. 3. pp. 363-373, 1979.

VOI. 6 4 , pp. 239-271, 1976.

[4] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York: Wiley. 1980.

[ j ] G. Strang, Linear Algebra and Its Applications. New York: Academic, 1976.

[6] B. Friedman. Principles and Techniques of Applied Mathematics. New York: Wiley, 1956, pp. 28-33.

[7] B. Noble and J . W. Daniel, Applied Linear Algebra. New York: Rentice-Hall, 1977. pp. 323-330.

[8] R. L. Zahradnik: Theory and Techniqum of Optimization for Q Practicing Engineers. New York: Barnes and Noble, 1971, pp. 118- 124.

[9] F. W . Byron. Jr. and R. W. Fuller, Mathematics of Classical and Quantum Physics. vol. I , Reading, MA: Addison-Wesley, 1969.

[ IO] D. A. Pierre, Optimization Theory with Applications. New York: Wiley, 1969. pp, 296-299.

Alan J. Fenn (S’74-M’78), for a photograph and biography please see page e 564 of the July 1982 issue of this TRANSACTIONS.

Wide-Band Matching of a Small Disk-Loaded Monopole

Abstract-A wide-band, electrically small, disk-loaded antenna, com- prising of a disk, 0.26 wavelength (at midband) in diameter, located 0.097 wavelength above a ground plane, has been designed and tested. A unique experimental procedure was used to determine the parameters of the impedance matching network, which consists of a conductive biconical center post and two structural side posts localed in the space under the disk. The resulting antenna has a maximum voltage standing-wave ratio (VSWR) of 2:l over a frequency bandwidth ratio of approximately 2:l. A second model, designed using the same technique, has a maximum VSWR of 3:l over a frequency bandwidth ratio of 3:l. This antenna was compared to a multi-element disk-loaded antenna (with the same size profile) designed by Dr. Georg Goubau. This multi-element antenna also has a maximum VSWR of 2:l over a frequency band of approximately 2:l. The comparison shows that the simple disk-loaded antenna, with fixed double tuning, achieves the same low VSWR as the multi-element disk-loaded antenna with fixed triple tuning. Therefore, an increase in bandwidth could be achieved in the simple disk-loaded antenna by applying higher order tuning.

E INTRODUCTION

LECTIUCALLY SMALL antennas pose a major problem vith respect to their electrical performance. These types

of antennas have radiation resistances which decrease rapidly with size. As a consequence, tuning and matching become very difficult to compute. With these concepts in mind, the antenna efficiency deteriorates and performance parameters, such as a bandwidth, tend to decrease to unacceptable levels.

Manuscript received November 9, 1984; revised June 10. 1985. This work

The author is with the Hazeltine Corporation, Greenlawn, NY 11740. was supported under Contract DAAK80-81-C-0124.

q Therefore, designing compact antennas which are efficient in spite of their electrical size is a very difficult task. For a more complete discussion of fundamentals of small antennas, refer

One small antenna design concept that was developed was that of a multi-element sectored-disk-loaded antenna [4], [ 5 ] . This design concept was developed by Dr. G. Goubau. This 4 multi-element antenna has a maximum voltage standing-wave ratio (VSWR) of 2: 1 over a frequency band of approximately 2: 1. Fig. 1 illustrates the multi-element sectored-disk-loaded antenna along with its performance.

The problem with the multi-element antenna is how to describe the proper field interaction between each of the conductive structure elements so that one could use these rules 4 to design other small wide-band antennas.

Presented in this paper is a procedure for the development of a wide-band matching method to achieve a low VSWR over a desired bandwidth for a small antenna. Then to adapt this method, by a unique experimental procedure, to tune the disk- loaded antenna via special structural features. Next, a compar- ison of performances of the multi-element disk-loaded antenna and the simple disk-loaded antenna will be performed and i conclusions and future goals will be discussed.

to ~ 1 , [21, [31.

CIRCUIT REPRESENTATION OF THE RADIATION FROM A

DISK-LOADED MONOPOLE

The development of an equivalent circuit for a simple disk-

0018-926X/85/1000-1142$01.00 @ 1985 IEEE

c FRIEDMAN: DISK-LOADED MONOPOLE 1143

I

VSWR

3

2

1

400 500 600 700 800 900 1000

MHz

(b) Fig, 1. (a) Broad-band multi-element monopole antenna. (b) Measured

VSWR of the antenna of Fig. l(a).

loaded antenna is presented in this section. This network is useful as a dummy antenna for use in wide-band matching circuit synthesis.

When the disk-loaded monopole in Fig. 2(a) has a radius which is both comparable with the radian length (W2a) and its height, then its behavior is more complicated than that of the usual “small” antenna [I]. Under these conditions, the radiation loading outside the cylindrical space under the disk shown in Fig. 2(a) can be approximated by a simple R , L , C network with constant coefficients determined by the dimen- sions. Figs. 2(b), 2(c), and 2(d) illustrate the different disk- loaded monopole circuit representations, all of which yield the same input impedance.

The following expressions for the internal and external capacitance of the simple disk-loaded monopole were derived from [6 ] :

C, = co7rr2/ h (1)

ce=c,,t-[8+j 2 In [ 1 +0.8(r/h)2+ (0.31r/h)4 1 +0.9(r/h)

It can be shown that the “natural logarithm” term is a small fraction of the total capacitance, C, + C, (and happens to be negative if r = h). Therefore, it may be ignored for purposes of estimating the capacitance.

In the limit of a “small” antenna, the radiation resistance (R,), referred to the current in the axial wire, and is given in (3) :

(a) EXTERIOR

Ra

( a

Fig. 2. Circuit representation of a disk-loaded monopole.

The effective height (hefi) is defined as the actual height (h) of the thin disk; this ignores the extra capacitance of the vertical wire. The frequency dependent R,, in series with the total capacitance (C, + C,), is transformed to an equivalent frequency dependent parallel conductance. This is given by (4) and is represented in Figs. 2(b) and 2(c) as a parallel R, and L, circuit.

We should note that the conductance has the fourth power frequency variation that one might expect. Fig. 3 illustrates the frequency variation of the conductance and is typical of the range of shape and size for which the above formulation is intended. The dashed curve is the continuation of the w4 variation from the limiting case of low frequency or “small“ antenna.

The conductance variation is primarily dependent on the radius of the disk, so the points of a quarter- and half-wave diameter are indicated. Note how the half-wave diameter is near the peak of conductance. So, for small heights, the conductance very nearly tracks the limiting w 4 variation.

In the limit of “high” frequency, the aperture loading approaches the “plane wave” radiation R of the cylindrical

I

1144 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 10, OCTOBER 1985 &

FORMULATION OF DOUBLE-TUNING OF THE DISK-LOADED MONOPOLE

P E A K In this section, the development of a double-tuned impe- dance matching network will be presented. This technique will then be applied to a simple disk-loaded monopole to achieve a maximum VSWR of 2: 1 over a 2: 1 frequency bandwidth ratio. z 2 1.00: *

4 The first step in the formulation of wide-band double-tuning 9 of the disk-loaded monopole is to determine the constant

coefficients of the equivalent circuit representation for the ’ simple disk-loaded monopole (C,, C,, L,, R,), which is

2rl h- mainly capacitive in this case. I

0 ’ I The remaining task is to design a network of lossless 1 w,2 2 reactors for double tuning to impedance match a line resistance (I

NORMALIZED ANGULAR FREQUENCY of R. The particular matching network configuration is chosen Fig. 3. Normalized radiation conductance for a r / h = 1.42 (disk radius/ for the following features:

height ratio).

interface. This resistance is represented in Fig. 2(c) and is given by (5):

Roh h R,=-=60 - ohms.

27rr r

The value of inductance, which is across the resistance to give the proper frequency variation, is derived from the equivalent parallel conductance from (4) and is obtained by (6):

In terms of dimensions,

Another circuit representation for the disk-loaded monopole is illustrated in Fig. 2(d). The transformation from one model to another is given as follows:

c,‘ =(C,+C,)

C’ = (C, - C,)/(C, + C,)

L‘ =(C,/(C,+C,))~L,

R,‘ = (C,/(C,+ C,))’R,. (8)

The latter circuit of Fig. 2(d) is used to calculate the radiation power factor (RPF), which, in turn, is used to calculate the bandwidth. Therefore, it might be advantageous to use the model of Fig. 2(d).

Two expressions are given for the W F of a disk-loaded monopole, including its image. The first formula is based on a volume expression; the other is based on the circuit represen- tation of Fig. 2(d).

(volume)

(circuit) RPF=w~(C,’)(L’)~/R,‘ (10)

RPF CE - a bandwidth. 1

Q

to integrate shunt capacitance at the line terminals; to match the antenna to a lower impedance line; to achieve double-peaked response to miximize the bandwidth for a defined VSWR.

To obtain the maximum bandwidth for a defined VSWR, it is necessary that the reflection coefficient be unity everywhere outside the band of interest and constant and equal to the defined VSWR in the band of interest. A better match anywhere in the band, or a reflection coefficient less than unity anywhere outside the band, is a waste of “matching area” and results in less bandwidth [2], [7], [8], [93.

The second step is preliminary series tuning of the mainly capacitive antenna. This is achieved by adding a series * inductor (Lz) to the network described in Fig. 2(c). By adding this series inductance, it will give equal conductance (G1 = G3) at the lower and upper cutoff frequencies (w,, w3). This is illustrated in Fig. 4.

At this point we note the corresponding values of suscep- tance (B, and B3) at the lower and upper frequencies.

The third step is to compute the correct amount of 4 susceptance (AB) to explicitly cancel the susceptances at the lower and upper frequencies (B, = B3 = 0). This is illustrated in Fig. 5(a). This results in the magnitude of impedance having equal peaks of resistance at the upper and lower frequencies and a valley of resistance near the midband frequency (double tuning) (see Fig. 3 3 ) ) . On the hemisphere chart (Fig. 5(c)), the ends of the impedance locus would coincide at a single 4 point on the chart as pure resistances (R1 = R3) , and the point of minimum resistance (Rz) corresponds to the midband frequency.

The circuit that will yield this proper amount of susceptance (AB) is a parallel capacitance and inductance, as illustrated in Fig. 5(d). These circuit values can be determined from (11) and (12).

NOTE

riF

The realization with positive values of L and C can only be achieved if the difference between B3 and Bl is nonnegative.

(

c3 = B3W3 - Blwl

w: - w ; (1 1)

FRIEDMAN: DISK-LOADED MONOPOLE 1145

R c4 Cj L3

EQUAL VSWR AT a,, w2’

0

Fig. 6 . Series capacitor for reducing asymmetry.

Fig. 4. Midband series tuning of dummy antenna.

Z = 0

0

(C)

R % S L 2

R, = R3

R ==

(dl Fig. 5. Edgeband parallel tuning.

Since this antenna is not really “electrically small” because it fits into a Wheeler radian sphere only within the lower portion of the frequency band. Therefore the tuned impedance locus departs from symmetry, such that the VSWR is greater above the midband frequency. This condition may be compensated for by adding a series reactor to Fig. 5(d) and is illustrated in Fig. 6 .

The final step is to transform the tuned impedance to the desired line impedance. This can be achieved by the design of an integral inductive transformer made up of L2 and L3 (see Fig. 6) .

In Fig. 7(a), the parallel inductance of L3 and the series inductance of L2 are illustrated as an equivalent transformer with the coefficient of coupling given in (13):

(1 3)

In Fig. 7(b) the coefficients are labeled to reflect the new values of capacitance caused by the transformer which is shown in Fig. 7(a). The transformer is represented by a T- configuration of uncoupled inductors. The values of the inductors are given in (14)-(17):

L; =L2+L3-nL3=L2-(n- l )L3 (1 4)

L j =nL3 (1 5 )

L; = n2L3 - nL3 = n(n - l)L3 ( 1 4)

n = J R ‘ / R , .

Referring to Fig. 7(c), the T-configuration is to be made up of two coupled inductors, one in series and one in shunt with the circuit representation of the antenna. The purpose of configuring the transformer this way was to adapt it to the antenna structure so it could be realized. The values are given in (18)-[21):

L , , = L i + L j (18)

K ’ = L i /

1146 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-33, NO. 10, OCTOBER 1985 &

(C)

Fig. 7, Transformer designed to be adapted to a simple disk-loaded monopole.

EXPERIMENTAL PROCEDURE TO STRUCTURALLY REALIZE A

DOUBLE TUNED MATCHING NETWORK Fig. 8 illustrates the structure of the simple disk-loaded

monopole that can be designed to realize the computed network. This simple antenna structure has three radii, which can be proportioned to yield the desired result of a 2: 1 or 3: 1 frequency bandwidth ratio with a 2: 1 or 3: 1 VSWR, respec- tively.

Structural realization of the design is complicated by the mixed fields under the disk, so C or L cannot be measured individually. Placing the end of the line in a gap halfway up the center post has the effect of minimizing the interation of fields.

(refer to Fig. 7(c)), the series L of the post and wires with their coupling. It can be measured on the L-meter at about 1 MHz. It reflects the coupling therebetween.

First the radius ( rJ of the center post can be adjusted approximately by this procedure:

The one quantity that can be measured individually is

remove the wires (L23) and omit C;; compute the series-resonance frequency ( w ~ ~ ) of LZ4 with the dummy; measure this frequency of minimum Z or zero R . The separation between these two conditions indicates the uncertainty of defining this frequency; adjust the radius to give a measured frequency close to the computed frequency.

From rl and L; can be computed the required radius (r3) to the wires:

L; =poh(1/2x) In ( r 3 / r l )

r3 = r, exp (2xL; /hpO). (22)

From these radii and L34, the required radius (r2) of each wire can be computed. In the examples studied to date, the center-

BlCONlCAL LINE R ' SERIESPOST 1124) PARALLEL POSTS IL231

D = 7.26 INCHES

Fig. 8. Simple disk-loaded monopole with double-tuning for a 2:l or 3:l frequency band.

post radius is large enough to shield the wires from each other, so each wire can be computed individually. Also each wire is small enough to use the small-wire approximation. Then this formula can be used for each wire:

2 L 3 4 = poh (1/2n) In [2(r3 - rl)(l/r2 + I/rl)]

1 l/r2=-

2(r3 - r1) exp ( 4 x L 3 , / h ~ ) - l/r,. (23)

A structure with these dimensions can be measured before providing parallel and series C . Its locus on the chart would indicate what should be added to C; so that the insertion of computed C,' will yield the desired closed loop. It should be centered near the specified R ' and should give the defined VSWR.

To the extent that the structure may fall short of prediction, some trimming of the dimensions under the disk may be required. The effects of the parts are not simple, so no formula for the trimming process is apparent. So an interpolation procedure to logically trim the antenna dimensions to come up with a desired result was performed. The procedure was as follows.

1) The construction of four models including two different values of each radius to be designed. One model is taken as a reference, from which each radius has one modified value in one or more of the other modes.

2) The measurement of each model at the three identified frequencies.

3) The solution of a set of three simultaneous linear equations to obtain interpolation factors.

4 ) The computation of a new set of radius dimensions intended to realize this result in a practical design.

5) The extension of the interpolation to evaluate the r

! FRIEDMAN: DISK-LOADED MONOPOLE

IYPEOANCE OR CCMTTANCE COOROINATES

1147

IHPEOAWE OR IDMmAHCE WOROIHATES

Fig. 9. Measured input impedance of simple disk-loaded monopole, double tuned for a 2:l frequency band.

expected performance, such as the VSWR intended to be minimized.

Figs. 9 and 10 show the impedance versus frequency of the 2: 1 and 3: 1 frequency bandwidth ratio models, respectively. The final dimensions for the 2:l bandwidth model are given below:

disk diameter 7.25 in disk height 2.55 in center post radius (rl) = 1.18 in side post radius (rz) = 0.19 in separation radius (r3) = 1.64 in bionical angle (0,) = 45" capacitive overlap (C,) = 0.20 in.

The final dimensions for the 3:l bandwidth model are given below :

disk diameter 7.25 in disk height 2.55 in center post radius ( r ] ) = 1.45 in side post radius (r2) = 0.0625 in separation radius (r3) = 1.9 in biconical angle (19,) = 45" capacitive overlap (C3) = 0.10 in.

COMPARISONS AND CONCLUSION

The multi-element disk-loaded monopole was a design concept of Dr. Georg Goubau. This design is characterized by a disk divided into four sectors on four posts, with adjacent sectors being interconnected by wires. Two of the posts are connected to the ground plane, while two other posts are for feed connections. Fig. l(a) depicts the essential features of the Goubau model.

Fig. 10. Measured impedance of simple disk-loaded monopole, double tuned for a 3: 1 frequency band.

Dr. Goubau indicated that the sectored-disk or "multi- element" monopole, which consists of a system of closely coupled radiating elements could achieve a bandwidth which is greater than that achievable by a simple monopole of the same overall dimensions in wavelengths [4]. The performance of the multi-element disk-loaded monopole is remarkable for an integrated structure. The simple disk-loaded monopole with the same overall dimensions in wavelengths, however, does match the performance of the multi-element disk-loaded monopole. The results are shown in Fig. 11 as a plot of VSWR versus frequency. These results are based on measurements of the actual antenna models. The VSWR of the simple disk tracks very closely to the multi-element disk-loaded mono- pole. The graph of the actual measured impedance of the Goubau sectored disk monopole is shown in Fig. 12. Note that the impedance plot of the sectored disk monopole forms three circles on the chart, while the simple disk-loaded monopole (see Fig. 9) forms only two circles. This may be interpreted that the multi-element disk monopole is triple tuned, while the simple disk monopole is double tuned, therefore leaving the possibility of greater bandwidth by triple tuning of the simple disk monopole.

In summary, an impedance matching approach has been developed, which is a general approach and can be used to double tune various types of antennas. This approach was used to match a small antenna comprising of a 0.27 wavelength diameter disk, located 0.097 wavelength above a ground plane. A unique experimental procedure is given to determine the parameters of the impedance matching structural realiza- tion, which consisted of a conductive biconical center post and two structural posts. These structural features achieved, by double tuning, a 2:l or 3:l frequency bandwidth ratio with a 2: 1 or 3: 1 VSWR, respectively.

1148

KEY

h = 097 hm

h = 11.39

SIMPLE-DISC LOADED ANTENNA , = .,39 ~m i Og7 h- L NOTE 1 THE TOP FREQUENCY SCALE IS FOR GOUBAU’S A m = X.185

ANTENNA. THE BOTTOM FREQUENCY SCALE IS FOR THE SIMPLE-DISC-LOADED ANTENNA

+ SANDWIDTH = 1.9 1 FREDUENCY lYHzl -

Fig. 1 1 . Comparison of VSWR versus frequency of multi-element disk- loaded monopole to simple disk-loaded monopole.

lMPED&YCE OR IDMITTAICE C00fiDINATES

G Frequency-dependent shunt conductance of radiation

h Height of the disk above the ground plane. heK Effective height of the disk above the ground plane. L Inductance. L, Constant inductance in the circuit representation of the

R Resistance. r Radius of the thin disk. R, Constant resistance in the circuit representation of the

Ro Free space resistance = 120 T ohms = m. R, Radiation resistance of the disk-loaded monopole.

loading G a: w4.

solid disk-loaded monopole. e

solid disk-loaded monopole.

ACKNOWLEDGMENT

IEEE TRANSAmIONS ON ANTENNAS AhTD PROPAGATION, VOL. AP-33, NO. 10, OCTOBER 1985 *

Fig. 12. Measured input impedance of multi-element disk-loaded monopole.

Finally, this simple disk-loaded antenna has the promise of a.ider bandwidth by the implementation of higher order tuning.

NOMENCLATURE

C Capacitance. C, Interior capacitance under the area of the disk. C, Exterior capacitance.

The author wishes to express his deep gratitude to Dr. Harold A . Wheeler, Life Fellow, IEEE, for his valuable guidance and technical support. The author is also grateful to Dr. Donn Campbell of CECOM, Ft. Monmouth, N J , for making available the actual Goubau multielement monopole for measurements. kL:

REFERENCES

H. A. Wheeler, “Fundamental limitations of small antennas,” Proc.

-, “The wide-band matching area for a small antenna,” ZEEE Trans. Antennas Propagat., vol. AP-31, pp. 364-367, Mar. 1983. -, “Small antennas,” ZEEE Trans. Antennas Propagat., vol. AP-23, pp. 462-469, July 1975. G. Goubau, “Multi-element monopole antennas,” (the first published description of the sectored disk monopole) Proc. ECOM-ARO d Workshop on Electrically Small Antennas, Fort Monmouth. NJ, May 6 and 7, 1976, G . Goubau and F. K. Schwering, Eds. pp. 63-67. G . Goubau, N. N. Puri. and F. K. Schwering, “Diakoptic theory for multi-element antennas” (includes a description of the sectored disk- loaded monopole, but. it is not used as an example of diakoptic theory), ZEEE Trans. Antennas Propagat., vol. AP-30, pp. 15-26. 1982. H. A. Wheeler, “A simple formula for the capacitance of a disk on a dielectric on a plane.” IEEE Trans. Microwave Theory Tech., vol.

R. M. Fano. “Theoretical limitations on the broadband matching of arbitrary impedances.” Tech. Rep. 41. M a s s . Inst. Technol.. Cam- bridge, MA. Jan. 1918. L. J. Chu, “Physical limitations of omnidirectional antennas.” J . Appl. Phys., vol. 19. pp. 1163-1175. Dec. 1948. H. W. Bode. Network Analysis and Feedback Amplifier Design. New York: D. Van Nostrand, 1945.

IRE, VOI. 35, pp. 1479-1484.

MTT-30, pp. 2050-2053, 1982. 4

Clifford H. Friedman (S’80-M’81) received the B.S. degree in electrical engineering and computer science and the M.S. degree in electrophysics both from the Polytechnic Institute of New York, Brook- lyn, NY, in 1981 and 1982, respectively.

He is currently a Senior Engineer at the Wheeler Laboratory, which is a part of Hazeltine Corpora- tion Research Laboratories. At Wheeler Laboratory he has performed studies investigating special wide- band antennas; in-depth computer modeling of antenna structures; analysis and predictions for the I

performance of electrically small antennas; studies using fiber-optic links for microwave systems: designed UHFlVHF dual bond sonobuoy antennas, high power VHF wide-baod vehicular antennas and active receiving antennas.

Mr. Friedman is the Treasurer of IEEE Antennas and Propagation Society on Long Island and is a Visiting Adjunct Professor at the State University of New York at Stony Brook.