ridged waveguide phased array elements

46 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL AP-24, NO. 1, JANUARY 1976

Ridged Waveguide Phased Array Elements JAMES P. MONTGOMERY, MEMBER, IEEE

Abstract-The mutnal coupling solution of dual and quad ridged waveguide is described for an in6nite array emironment. The method of analysis is the numerical solution of integral equations for both the eigen solution and the boundary matching problem. In addition to numerical results, comparison of theory with waveguide simulator measnrements is also given.

INTRODUCTION

R IDGED WAVEGUIDE is well known for its bandwidth properties. This is particularly true for dual

ridged waveguide where it is realized that the TE,o-TE20 modal bandwidth is increased with ridge insertion. Mont- gomery [l] has reported on the complete solution of ridged waveguide using the numerical solution of integral equations. This study also included the calculation of the fields of the dominant and higher order modes. The form of these modal fields were Fourier series. This allowed a convenient basis for further numerical manipulations. A preliminary discussion of these results used in phased array analysis has already been given [Z]. The primary purpose of this paper is to more fully discuss the solution of dual ridged waveguide in an array, give the array solution of quad ridged waveguide, and present comparison with experiment. In addition, limited parametric results are given.

Theoretical investigation of ridged waveguide has recently been the topic of several authors. M. H. Chen et al. [3], have performed detailed modal studies of quad ridged waveguides using Silvestor's finite element program [4]. C. C. Chen [SI has recently reported on the use of quad ridged waveguide in an array. However, the ridges are a . part of the feed with the junction characteristics being measured. The actual radiating element was a circular dielectric-loaded waveguide element. Also, Wang has recently finished work on a Ph.D. dissertation [6] dealing with dual ridged waveguide radiators. His work contains detailed parametric studies. In contrast to this recent work, much of the work for this paper was performed in the time frame of [l] and [a].

THEORY There are two separate problems in the characterization

of ridged waveguide in an array environment. The first is a numerical solution giving the eigenvalues and modal fields of the waveguide. The second problem is to solve the boundary value problem of the element in an infinite array environment.

The modal solution of ridged waveguide (ref. Fig. 1) is described in [l]. Using the notation of this reference, we

The author is with Texas Instruments Incorporated, Dallas, TX Manuscript received December 22, 1974; revised August 19, 1975.

75222.

Y

t a2 _I

S

(MAGNETIC OR ' a3 ELECTRIC X

2 a 1 = s

2 a = a 2 a q + a 4 = b I - a + a ' = d 3 3 I

Fig. 1. Geometry of dual ridged waveguide.

have the following expression for the electric basis field (for a particular eigenvalue) :

where

e+.(rT) = (el(rT); 0 s x s a,, -a3' I y I a3 e2(rT); a, I x I u2, -a,' < y s a,.

Series representations for el(rT) and e2(rT) are given in [I]. e-(rT) is found from e+(r,) using appropriate symmetry conditions. The coefficients of these series are normalized such that

s et(vT) e,(rT) dr, = 1. (2) cross section

From general cylindrical waveguide theory [7] we have

s et(rT,kTn) e t ( r T , k T d drT = O, n # (3) cross section

where the eigenvalue is given as an argument to distinguish basis fields. In general, (3) is satisfied only approximately due to the numericaUy determined fields; however, all array equations are derived assuming orthogonality holds exactly.

A similar modal solution for quad ridged waveguide [SI may be accomplished (ref. Fig. 2) by solving for the aperture fields at both x = a l , and a6. This solution also appears in series form, except that quarter symmetry is assumed and three geometrical regions (instead of two) are involved.

The second part of the problem is the infinite array problem shown in Fig. 3. This is accomplished using a Galerkin solution for the aperture fields in a similar manner to that used by Amitay et al. [9]. However, the modal functions are now computed using (1). In fact, one need only modify the Fourier transform subroutine of any existing program setup to solve rectangular or circular waveguide. The Fourier

MONTGOMERY: PHASED ARRAY E L E M E N S 47

Fig. 2. Quad ridged waveguide geometry.

2

Fig. 3. Inlinite array geometry.

transforms may be found in closed form using the detailed series expressions in [ 1). Due to the series representation of the modal functions, the computational time will, of course, be much greater than rectangular or circular waveguide computer programs (nominally by a factor of 10 times).

EXPERIMENTAL RESULTS

Due to the complexity of the numerical tasks, one must necessarily place a large emphasis on comparison between waveguide simulator measurements and theory to assess the solution. Fig. 4 illustrates the 60” H-plane simulator used in measurements of dual ridged waveguide. Transi- tions to standard X band waveguide (and, hence, a standard hardware) were used for both the simulator and ridged waveguide. Fig. 5 illustrates measured data and comparison with theory for the case: a , = 0.127 cm, a, = 0.635 cm, a3 = a3’ = 0.1397 cm,a4 = u4‘ = 0.508cm, dl , = 1.8161 cm, dZx = 0.908 cm, d2,, = 1.5748 cm. The agreement is excellent adding credibility to the solution. Fig. 6 illustrates this same case except a dielectric sheath flush with the aperture has been added. The parameters of the sheath are: t = 0.0381 cm, E, = 4.3. Again, the agreement is excellent.

Simulator measurements have also been made using quad ridged waveguide near 7.5 GHz. Due to hardware difficulties? the agreement of theory and experiment was not as good as the dual ridged results. However, the results do tend to enforce the validity of the quad ridged analysis. Obtaining dual ridged waveguide results with the quad ridged waveguide program lends further credibility to the

INTER m DETECTOR 0

\ 4 SI h\ULATOR PART BLOWUP OF RIDGED lVAVEGUl DE

;- X-BAND LOAD

RIDGEDGUIDE -RECTANGULAR SIMULATOR

Fig. 4. Simulator setup.

1 - 1 0.8 -

- 0.6 - E -

0.4 - - -10

OTHEORY OEXP VJlO SHEATH

0 ‘ I 9. E 9.35 9.45 f K H z )

Fig. 5. Dual ridged waveguide simulator measurements without dielectric sheath.

0.0 I I I 9.25 9.35 9.5

f GHz)

Fig. 6. Dual ridged waveguide simulator measurements with dielectric sheath.

quad ridged solution. Also, quad ridged waveguide modal solutions were compared with Silvestor’s results [4] and less than a 0.5 percent difference was observed in the eigenvalues.

NUMERICAL DISCUSSION This section addresses the rather complex problem of the

numerical convergence of the solution. There are basically three convergence problems: 1) the number of terms in-

_ - .~

48 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, JANUARY 1976

cluded in the modal representations, 2) the number of waveguide modes included in the boundary value problem, and 3) the number of Floquet modes.

The number of terms in the modal representations of dual ridged waveguide must be chosen consistent with the edge condition at x = a,, y = a3, -a3'. Hence, we chose the ratio of the gap to trough indices to be the same as the ratio of the geometrical heights [lo]. Normally, five to six terms in the gap are sufficient. Of course, if we limit our discussion to dual ridged waveguide, only the even (or odd) terms of the modal series contribute.

The number and type of waveguide modes included in the boundary value problem are chosen consistent with the eigenvalue spectrum of ridged waveguide. It is well known that the dominant TE,, mode is lowered in cutoff by a large percentage. Hence, generally only the dominant TE,, mode and the first few higher order modes are required. Higher order mode classification is, in general, accomplished using the cutoff characteristics and the boundary conditions at x = 0. Discussion of ridged waveguide modes using rectangular waveguide classification is interesting but not necessary for the analysis in this paper. In this analysis, the TM modes have generally been omitted since the ridge insertion increases the cutoff frequency of the dominant TM modes.

For a given modal function approximation and number of waveguide modes, one must also decide on the number of Floquet modes. One approach might be to use an excessive number of spatial modes. However, this is time consuming. A more economical approach is to consider that a lower bound on the modes is obtained from edge condition considerations using only the outer rectangular dimensions of the ridged waveguide and the periodic cell dimensions. Of course, for ridged waveguide, the fields are more confined in the ridge region and, hence, an upper bound might be obtained using the dimensions of the ridge gap rather than the overall height. For most of the results in this paper, the number of Floquet modes are near the lower bound discussed in the preceding.

The convergence considerations for quad ridged waveguide are similar to those discussed here for dual ridged waveguide. The primary difference is that the TE,, mode is present and the TE, , eigenvalue is lowered, thus reducing the modal bandwidth.

DUAL RIDGED WAVEGUIDE NUMEFUCAL RESULTS

Wider bandwidths in phased arrays must generally be achieved using aperture matching techniques, since no excessive distance lies between the array and the matching junction. In circular and rectangular waveguides, aperture matching can be accomplished using any combination of dielectric aperture sheaths, dielectric plugs, aperture irises, and waveguide transformers in close proximity to the aperture junction. Ridge waveguide offers the possibility of optimizing the ridge design for matching purposes. An example of such matching is now given.

Fig. 7 illustrates the frequency and scan behavior of the element considered in the simulator measurements (ref.

4 1 Fig. 7. Aperture impedance versus scan angle and frequency.

Fig. 8. Effect of changing element size and ridge design.

Fig. 6). We note that the aperture mismatch is large, however, not excessively so for 3 percent operation with internal matching. In this case, the matching was incorporated in the microstrip to ridged waveguide excitation mechanism. The array considered in Fig. 8 uses the same lattice dimensions and dielectric sheath as in the preceding example, except the waveguide now has the parameters: a, = 0.1740 cm, a2 = 0.8496 cm, u3 = a3' = 0.4346 cm, a, = a,' = 0.7366 cm. Here the outer dimensions of the waveguide have been made larger, the ridge gap has increased, and the ridge width has increased. We note that the aperture mismatch has improved. Changing only the gap and the sheath to have the parameters: a3 = a3' = 0.5893 cm, t = 0.0381 cm, E = 2; we arrive at the result given in Fig. 9. This last stage of matching was clearly accomplished by the reduction of the aperture capacitance using ridge gap adjustment and dielectric constant adjustment. Fig. 10

MONTGOMERY: PHASED ARRAY ELEMENTS 49

.I/ Fig. 9. Final matched element.

Fig. 10. Frequency variation of final matched element.

illustrates the frequency variation of a few scan angles for this same design. We note that for 60" conical coverage, the YSWR is less than 4: 1 up to 9.5 GHz. The lower operational frequency, in this case, is determined by the ratio above cutoff one desires. We note that if the scan volume is limited in the H-plane to 40°, the useful bandwidth of this array is greatly extended. This scan limitation is practical in many radar applications.

In the preceding example, we note that the E-plane element spacing is 0.5)., at f = 9.518 GHz, while the element dominant mode cutoff occurs at f = 8.157 GHz (9.518/8.157 = 1.167). In contrast, consider the array with the following parameters: a, = 0.8400 cm, u2 = 4.1999 cm, u3 = u3' = 1.8102 cm, u4 = u4' = 3.6204 cm, d l , = 8.6538 cm, d,, = 4.3269 cm, d,, = 7.4948 cm (with no dielectric sheath). The E-plane element spacing of this array is o.%, at 2.0 GHz and the dominant mode cutoff is

44 Fig. 11. Broadband array.

1.324 GHz (2/1.324 = 1.51 1). Fig. 11 illustrates the frequency variation of a few scan angles for this array. We note that the impedance behavior is "flat" in contrast to the previous case. We also note that the mismatch is more balanced between the E-plane at low frequencies and the H-plane at the high frequencies. This array exhibits a 60" conical coverage with a VSWR less than 4: 1 from 1.35 GHz to 1.6 GHz. Note that we are operating close to cutoff at the lower end of the frequency band. Again, note that if the scan is limited in the H-plane to 40", the upper operating frequency of the array is extended to 2.1 GHz.

We note that both of the latter examples avoid excessive ridge loading. The interested reader is referred to Wang [6] for more detailed parametric studies.

Having given typical dual ridged waveguide array data, let us briefly present additional data illustrating the convergence of the presented data. Table I gives specific reflec- tion coefficients for the data in Fig. 9 for various mode choices. Data (a) and Data (b) are identical except for the number of waveguide modes. We note that significant errors can be made with only a dominant mode solution. This discrepancy is amplified as grating lobes move inside real space. Data (c) illustrate this error. Data (a), (b), and (c) are for 35 Floquet modes. Data (d) increase the number of Floquet modes to 117. We note little change in the solution of the critical E-plane data point.

QUAD RIDGED WAVEGUIDE NU~MERICAL RESULTS Quad ridged waveguide solutions are similar to dual

ridged waveguide in their general impedance characteristics. However, when optimizing quad ridged waveguide, its circular polarization characteristics, such as axial ratio and polarization coupling [ll], [12], are important design parameters. These latter parameters are only meaningful if we examine a matched waveguide element.

Figs. 12 and 13 illustrate data for an array with parameters: a, = u5 = 0.1295 cm, a, = a4 = 0.6477 cm, a3 =

50 IEEE TRANSACTIONS ON A N T E N N A S AND PROPAGATION, JANUARY 1976

TABLE I CONVERGENCE OF SOLUTION GIVEN IN FIG. 9

(a) 9.35 GHz OD O0 1 0.132 e~p(j145.7~)

9.35 60 0 1 0.532 q(-j64.7")

9.35 60 90 1 0.286 exp(-j150.8")

9.35 60 45 1 0.045 exp(-j128.lo)

(b) 5.35 O0 O0 6 0.118 exp(j145.5')

9.35 60 0 6 0.583 exp(-j68.3")

5.35 60 90 6 0.226 exp(-jl21.jo)

9.35 60 45 6 0.087 exp(j27.1')

(c) 9.8 60 90 1 0.427 exp(-j131.So)

9.8 60 90 6 0.731 exp(-j119.Z0)

(d) 9.35 60 90 6'f 0.227 exp(-j122.4')

5.8 60 90 6' 0.720 emI-ill9.lo'1

t All above data k i t h 35 Floquet modes, this data h-ith 117 Floquet modes.

Fig. 12. Quad' ridged waveguide with dielectric sheath matching.

a6 = 0.2591 cm, d l , = 1.7907 cm, d,, = 0.8954 cm, and d,,, = 1 S494 cm. Two dielectric sheaths are present: the sheath nearest the array having parameters E,, = 1.0, t , = 0.254 cm; the outer sheath having parameters E , ~ = 9.0, t , = 0.1016 cm. This matching sheath was designed by minimizing the 60" H-plane, 60" E-plane, and broadside impedances for various parameters of the two sheath combinations. The impedance behavior of this array is quite acceptable, having a VSWR less than 3.8: 1 over a 60" conical scan volume. The axial ratio for this array is given in Fig. 14. This is very poor performance. We note that in this example the TE,, mode is propagating, having a

1 4 Fig. 13. Quad ridged waveguide with dielectric sheath matching.

cutoff frequency 1.06 times the TEl0 and TEol cutoff frequencies. However, omitting the TE,, mode from the calculations did not significantly affect the axial ratio. The high axial ratio is due to a phase imbalance in the 8 and 4 component fields.

Fig. 15 illustrates the axial ratio of the same quad ridged waveguide array without dielectric sheaths. Matching is accomplished using an internal match, each having a dominant mode normalized generator impedance of Zg = (2.2193, 2.1090). This axial ratio is quite good out to 60" in all planes. The impedance characteristics of this array were slightly better than the sheath matched example.

MOSVTGOMERY: PHASED ARRAY ELEMEhTS 51

10

8

2

9 x 4 d

2

0

Fig. 14. Axial ratio of quad ridged waveguide array.

/-

The preceding examples have illustrated two important points: 1) the TE,, mode does not necessarily have a detrimental effect on quad ridged waveguide performance and 2) polarization and impedance parameters must be optimized simultaneously.

OTHER PHASED ARRAY ELEMENTS As a basis for comparison, this section will present data

for circular dielectrically loaded waveguide in the same lattice as that considered for the quad ridged waveguide. We also note that this lattice is quite close to that considered for the X band dual ridged waveguide arrays.

Figs. 16 and 17 illustrate the two diagonal components of the aperture scattering matrix of an array of dielectrically loaded circular waveguide elements with the parameters : waveguide radius = 0.6477 cm, waveguide dielectric loading ( E , =) 2.7, d l , = 1.7907 cm, d2x = 0.8954 cm, d2y = 1.5494 cm: dielectric sheath near array with t , = 0.254 cm and E , ~ = 1.0, outer sheath with t , = 0.0508 cm and E , ~ = 9.0. The VSWR at this frequency is less than 2: 1 over a 60" scan volume. Fig. 18 illustrates the axial ratio versus scan for this element design. Note that the axial ratio has a worst case value of -3.5 dB at 6 = 60'.

At f = 9.9 GHz, a grating lobe occurs in real space at 8 = 72.5' for 6 = 30" and 90' for the infinite array. Figs.

Fig. 16. Circular waveguide impedance.

Fig. 15.

x) 40 M Ea 0

Internally matched quad ridged waveguide axial ratio.


10 - r1 - C=O 1 1

, I 90 I 1 -_-

8 - ...... 45 j I A . A - 1 : j

1 1 : - 0 ' 1 - m - 2

-5 5 4 -

0 x) 4 0 6 0 Ea 8 IDEGREESl

Fig. 18. Circular waveguide axial ratio versus scan.

52 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGA~ON, JANUARY 1976



Fig. 21. Circular waveguide axial ratio versus scan.

19-21 illustrate pertinent data for the preceding array at f = 9.3 GHz. Due to the absence of the grating anomaly, the characteristics are quite good even beyond 0 = 60”.

CONCLUSIONS AND FINAL COhlhlENTS

This paper has endeavored to illustrate the array solutions for dual and quad ridged waveguide elements. The solution is accomplished by using accurate Fourier series representations of the modal functions obtained via integral equation solutions [I]. The modal functions so derived can be easily Fourier transformed, allowing one to conveniently solve the array problem. Hence, the first conclusion of this paper is that ridged waveguide elements can be accurately analyzed with some economy when compared to experiment. This is particularly important to the designer of a large phased array where a failure to predict possible modal resonances can be extremely expensive.

The second conclusion is that ridged waveguide offers a convenient matching mechanism by variation of the ridge width and gap size. Indeed, the arrays illustrated by Figs. 9-11 were both matched by ridge parameter variation. Additionally, increased bandwidth is available using the modal characteristics of ridged waveguide. Indeed, the array described in Fig. 11 has less than a 4: 1 VSWR in a scan volume defined by a 60” E-plane-40” H-plane scan over a 43 percent bandwidth. This appears to be an improve- ment over other designs using rectangular waveguide [13]. Even broader bandwidths may be achievable with more investigation than was allowed for the foundation of this paper. In order to achieve this performance, other forms of aperture matching might be used in addition to ridge optimization.

Although no ridge optimization was illustrated for quad ridged waveguide, clearly the ridge can be used to improve the match. However, the quad ridged analysis has established insight into the answers to questions about any degradation due to the TE, mode. However, aperture matching has not yet yielded performance comparable to that obtained by the circular waveguide examined in the latter section of this paper. This should not be construed to imply that improved aperture matched element designs are not achievable. Instead, it should indicate the need for additional research.

ACKNOWLEDGMENT

The author wishes to express appreciation to the reviewer of this paper for careful reading and his many suggestions which lead to a considerably better paper than the original draft submitted.

REFERENCES

[ l ] J. P. Montgomery, “On the complete eigenvalue solution of ridged waveguide,” ZEEE Trans. Microware Theory Tech., vol. MTT-19, pp. 547-555, June 1971.

[2] -, “On the characterization of ridged waveguide in an array environment,” in 1970 GAP Znt. Symp. Digest, pp. 236-241.

[3] M. H. Chen et a/., “Modal characteristics of quadruple-ridged circular and square waveguides,” ZEEE Tram. Microwace Theory

[4] P. Silvester, “High order finite-element waveguide analysis,” ZEEE f i -ms. Microwace Theory Tech., vol. MIT-17, p. 651, Aug. 1969.

Tech., VOI. IMTT-22, pp. 801-804, Aug. 1974.

IEEE TRANSACTIOSS OX AhTENNAS AND PROPAGATION, VOL. AP-24, NO. 1, JANUARY 1976 53

[j] C. C. Chen, “Quadruple ridge-loaded circular waveguide phased

483, May 1974. arrays,” lEEE Trans. Antennas Propagat., vol. AP-22, pp. 481-

[6] S. S. Wang, “Wide angle wide band elements for phased arrays,“

[7] R. E. Collin, Field TIleoryoofCuided Wares. New York: McGraw- Ph.D. dissertation. Polytechnic Institute of New York, 1975.

[8] J. P. Montgomery, ”Numerical eigenvalue solution of quad Hill, 1960, ch. 5, sect. 2.

ridged waveguide,” unpublished Texas Instruments Rep. Antenna Systems Branch, June 1971.

[9] X. Amitay, V. Galindo, and C. P. \Vu. Theory and Analysis of

ch. 7. Phased Arra? Antennas. New York: Wiley-Interscience. 1972,

[lo] R. Mittra, T. Itoh, and T. S. Li, “Analytical and nu.merical studies of the relative convergence phenomenon arlsmg In the solutlon of an integral equation by the moment method,” IEEE Trans. Xlicrowace Theory Teelz., vol. MTT-20, pp. 96-104, Feb. 1972.

[ l l ] C. C. Chen, “Wideband wide-angle impedance matching and polarization characteristics of circular waveguide phased arrays,” IEEE Trans. Antennas Propagat., vol. AP-22, pp. 414-418, May 1974.

[12] G. N. Tsandoulas and G. N. Knittel, “The analysis and design of dual-polarization square-waveguide phased arrays,” IEEE

[13] G. N. Tsandoulas, “Wideband limitations of waveguide arrays,” Trans. Antennas Propagar., vol. AP-21, pp. 796-808, Nov. 1973.

d.licrowaL.e J., pp. 49-56, Sept. 1972.

Analysis of the Radiation Patterns of Reflector Antennas

Abstract-The development and application of a numerical technique for the rapid calculation of the far-field radiation patterns of a reflector antenna from either a measured or computed feed pattern are reported. The reflector is dehed hy the intersection of a cone with any surface of revolution or an offset sector of any surface of revolution. The feed is assumed to be linearly polarized and can have an arbitrary location. Both the copolarized and the cross polarized reflector radiation patterns are computed. Calculations using the technique compare closely with measured radiation patterns of a waveguide-fed offset parabolic reflector. The unique features of this technique are the freedom from restrictive feed assumptions and the numerical methods used in preparing the aperture plane electric field data for integration.

INTRODUCTIOS

T HE MOTIVATION for this work was the development of a technique for the rapid computation of the radia-

tion patterns of reflector antennas from the measured radiation patterns of multiple feeds. For communications satellite applications the feeds could be connected and phased to produce the required shaped beams [l]; in radiometric applications each feed, or set of feeds, might be operated at widely separated frequencies for the remote sensing of the physical properties of the earth’s surface. Due to the blockage and scattering from such multiple feed systems for the conventional parabola, there is strong interest in the offset reflector geometry. The feeds may or may not be on focus depending upon the particular configuration. For the case of the feed on focus, Chu and

J. F. Kauffman is with the Department of Electrical Engineering, Manuscript received November 21,1974; revised September 4,1975.

W. F. Croswell is with the kASA Langley Research Center, Hamp-

L. J. Jowers is with LTV Aerospace Corporation, Hampton,

North Carolina State Lniversijy, Raleigh. NC 17607.

ton, VA 23663.

VA 23665.

Turrin [2] have published measurements of the beam displacement and cross polarization of the offset reflector along with calculations assuming a symmetrical feed pattern.

Briefly, the calculation reported here is formulated in the following manner. The equations of geometrical optics are used to calculate the reflected electric field using the radiation patterns of the feed and the parameters defining the reflector surface. Also obtained using geometrical optics are the direction of the reflected ray, and the point of intersection of the reflected ray with the aperture plane. These fields comprise the aperture distribution which is integrated over the aperture plane to yield the far-field radiation pattern.

The case of the on-axis-fed full paraboloid is the only one where the projected rays defined on 4 = constant cuts of the feed radiation pattern, strike the aperture plane at points defining radial lines emanating from the origin. For other feed positions these data points will not lie on constant coordinate lines in any convenient coordinate system. Moreover, the point configuration changes each time the feed position and/or reflector geometry is changed. This is illustrated by comparing Fig. 5 later in the paper. In this paper an algorithm is presented for ordering these aperture data points in a rectangular coordinate system prior to the numerical integration. This coordinate system is used for all antenna configurations.

The most commonly used alternative formulation is the current distribution method [3], where the surface current J, on the reflector is taken to be 2(8 x Hi). Hi is the incident magnetic field calculated from the feed pattern using geometrical optics, and 8 is the unit normal to the reflector surface. This current is then integrated over the reflector

ridged waveguide phased array elements

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