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Analysis of Linear Array Modules

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CHAPTER 6

ANALYSIS OF LINEAR ARRAY MODULES

In the foregoing chapter, the efficacy of the shunt-slot feeding technique has been

validated through two alternate analysis approaches with regard to a single-element

radiator. In this chapter, we further develop the investigations of this feeding

technique by utilizing the proposed element as a unit-cell for a linear array. Already

referred in Chapter 1, such a module has been termed a “stick” array by authors [35].

This module may be used as a building block for a full-fledged planar array. Since the

other end of the feeding waveguide is available, a cascade of radiating elements may

be placed ahead of the single element naturally resulting in a series-fed linear array.

The fraction of power coupled out at the first element is relatively small but the

remaining elements together with the first may couple out most of the energy

propagating down the guide resulting in high radiation efficiency. Further, the

amplitude excitation of the array elements may be controlled for a variety of

requirements like reduced sidelobe level, null placement, etc. In the proposed

configuration, variation of the longitudinal slot offset offers a convenient way of

modifying the coupled power to each element. The parametric studies presented in

Chapter 4 may be effectively utilized for this purpose.

In this chapter, analysis and simulated results of linear array modules based on the

proposed radiator configuration are presented. It is perceived that the linear array

modules may be designed for any arbitrary amplitude distribution. To illustrate this

versatility, two well-known variants of the linear array module have been selected for

investigation as part of this doctoral work. These are a: a) Uniformly Excited Linear

Array; and b) Dolph-Chebyshev Reduced Sidelobe Linear Array. Both of these will


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be described. Considerations for the design of a linear array using waveguide shunt-

slots are addressed in the next section. The number of elements and the design

parameters chosen based on the design criteria. An array factor formula that best suits

the configuration of the linear array modules is chosen. The MOM-computed

radiation fields for the single element (of Chapter 2) are used in conjunction with this

to obtain the field patterns of the modules. This is followed by simulation on Ansoft®

HFSS® and the computed performance is compared to the MOM results for

validation. The second case of sidelobe reduction using tapered excitation is chosen

as an example of how the analysis developed in this work may be used for the rapid

design of a linear array where the element amplitudes are tailored for a desired

radiation pattern. Simulated results for this module using both the MOM + array

factor formula and HFSS® are also presented in the succeeding section. It is certainly

possible to modify the excitation phase to obtain a squinted beam or frequency

scanning also but this has not been attempted presently.

6.1 A Linear Array Module of Shunt-Slot Fed Microstrip Patch Elements with Uniform Excitations

As mentioned above, a linear array of shunt-slot excited microstrip patch elements

can be realized in a manner similar to an array of broadwall, longitudinal slots. Elliott

has detailed a design procedure based on the Schelkunoff unit circle method and the

pattern prediction for a 5-element uniformly excited broadside array (see pp. 130 of

[49].) The array factor for such an array should provide maximum directivity for the

given number of elements i.e. maximum radiation efficiency; a number of sidelobes

and prominent nulls. The relative power levels for the two principal sidelobes should

be -13.5 & -17.9dB respectively for λ/2 element spacing. As seen in the following, it

is not possible to enforce this spacing on a series-fed linear array designed with


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waveguide. Elliott also outlines a design procedure for a 4-element linear array of

longitudinal broadwall slots (see pp. 412 of [49].) The change in resonant length of

slots due to transverse offset calls for a length adjustment. Further, he describes the

stacking of “stick” arrays as transverse unit cells for obtaining a planar array. More

information regarding this last may be referred in [49] and is not discussed in greater

detail here.

The design considerations for a module using the proposed single radiator for a

uniformly-excited linear array are discussed in the following.

6.1.1 Design of the Linear Array Module

For a broadside radiation pattern, all the radiating elements need to be excited in-

phase. Owing to the peculiar feeding medium, i.e. a continuous waveguide, the linear

array becomes a series-fed structure with the intervening waveguide lengths deciding

the inserted phase between the elements. This may be forced equal to 360; in which

case the element spacing has to be one guide wavelength leading to undue grating

lobes. Fortunately, there is a more elegant solution. The coupling slots may be placed

λg/2 apart resulting in out-of-phase excitation. However, by placing the slots to

alternate sides of the waveguide longitudinal axis, the remaining 180 phase may be

imparted resulting in broadside reinforcement without grating lobes. This approach

was followed for designing a linear array module using the proposed C-band

prototype shunt-slot fed patch element (see Fig. 6.1).

With the resonant frequency selected according to the Case-1 analysis of the previous

chapter, the inter-element spacing of λg/2 = 35.91mm was chosen for the linear array.

A nominal number five radiating elements were selected to obtain a reasonably long


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linear array while keeping the dimensions handy. A transverse offset of + 10mm for

the feeding slot was chosen on either side of the median line of waveguide. This

choice is not critical since the elements are to be excited in equal amplitudes in this

variant. However, xs = 10mm ensures a good fraction of power coupling through the

feeding slots to the patches. This should yield a high radiation efficiency and a good

radiating structure from this design.

By specifying these dimensions, the design of the uniformly-excited linear array

module is completely defined and we next address its pattern computation from array

factor considerations along with the element pattern already computed in Chapter 2.

Exploded Linear

Array PCB Ground

Slots

Patches

Fig. 6.1: Configuration of 5-Element Linear Array (Exploded View)


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6.1.2 Array Factor for Arbitrary Element Positions and Pattern Computation

Due to the staggering of the adjacent elements on either side of the waveguide

median, the resulting array is not exactly linear although the approximation is still

good. For exactness, we select the formula for an array factor with arbitrary element

positions given as follows (pp. 116 of [49]).

,

(6.1)

This formula allows any ith element of the array to be positioned at its particular

location (xi, yi, zi). In our case, zi = constant in the plane of the coupling slots and the

two lateral offsets vary per element.

Fig. 6.2: Array Factor for 5-Element Linear Array with Uniform

Excitations (after Eq. 6.1)

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The computation was implemented as a Microsoft® Excel® sheet. For uniform

excitations, a unity value of the coefficients Ii/I0 was fixed and array factor computed

for N = 5 elements (Fig. 6.2). We observe the first sidelobe close to the expected level

of -13 dB. The element pattern computed from Eq. 2.119 was used with the principle

of pattern multiplication to (6.1) to obtain the net array pattern in Fig. 6.3 below.

Fig. 6.3: Computed Principal Plane Patterns for 5-Element Linear Array with Uniform Excitations using MOM X Array Factor Formulation

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A comparison of Figs. 6.2, 6.3 and Fig. 4.39 reveals that the element gain roll-off

causes the array sidelobes to be lower in the array factor. Even the E-plane pattern of

the array rolls-off faster to about -15 dB in the plane of the substrate compared to

about -7 dB in Fig. 4.39. Hence the effect of the element staggering on the radiation

pattern is also accounted although mutual-coupling between the elements is neglected.

Next, as for the case of the single element, in order to validate the analysis using

array factor formula, we utilized HFSS® for the simulation of this linear array module

– features of the analysis model are discussed next.

6.1.3 The HFSS®-Simulation Model for 5-Element Linear Array Module with

Uniform Excitations

The substrate parameters and patch/slot dimensions are retained identical to the single

element cases analyzed in Chapters 4 & 5. The ground plane size is taken as 180 X

Fig. 6.4: Ansoft® HFSS® Simulation Model of 5-Element Linear Array Module with Uniform Excitations


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100, with the longer dimension along the waveguide axis (see Fig. 6.4.) The ABC is

extended 5mm to either side beyond this and its height is 40mm which is optimum

from previous simulations. Meshing operations and boundary conditions are as for the

earlier cases in the previous chapter.

An examination of the top-wall of the feeding waveguide indicates the correct

alignment of the exciting slots with the overlying patch radiators (Fig. 6.5.) Also, it

assures that the order of surface assignment in HFSS® is correct. Otherwise the slots

may be shorted out in the problem definition resulting in a trivial solution.

To reduce the execution time, the frequency sweep is carried out for a coarser step of

0.02GHz resulting in 51 steps from 5.0 to 6.0GHz. A total CPU-time of 7h 11m was

needed for obtaining convergence. The final number of tetrahedra in the adaptive

mesh was 149,656. The simulated results obtained for this problem are presented.

Fig. 6.5: Waveguide Top-Wall Boundary Display from Ansoft® HFSS® showing Patch & Slot Alignment


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6.1.4 The HFSS®-Simulation Results for 5-Element Linear Array Module with

Uniform Excitations

The computed VSWR for the uniformly excited linear array module shows a very

good impedance match (see Fig. 6.6.) The trend is different from the earlier responses

(see for instance, Fig. 5.4) but is understandable due to low value of VSWR.

5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]

1.0000

1.0005

1.0010

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Ansoft Corporation HFSSDesign1VSWR Quick Report

Curve Info

VSWR(WavePort1)Setup1 : Sw eep1


VSWR(WavePort1)_1Setup1 : Sw eep2


Fig. 6.6: HFSS®-Computed VSWR of 5-Element Linear Array Module with Uniform Excitations

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3.00E-006

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Po

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Ansoft Corporation HFSSDesign1Power Coupled Out

m1

Curve Info

PoutSetup1 : Sw eep1Name X Y

m1 5.6000E+000 7.7244E-006

Fig. 6.7: HFSS®-Computed Coupled-Power, Pout for 5-Element Linear Array Module with Uniform Excitations


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The maximum power coupling is 7.7244 X 10-6 and takes place at 5.59 GHz although

it appears shifted to 5.60GHz due to coarser analysis steps (see Fig. 6.7.) The contour

plots at the resonant frequency show an in-phase reinforcement of the fields radiating

from the individual elements in the space towards the front of the substrate (see Fig.

6.8.) The E-plane contours are similar to the Case-1 analysis presented earlier.

The computed radiation patterns show behaviour close to that expected for the 5-

element linear array configuration (see Fig. 6.9) and compare very well to the MOM-

predictions for the forward half-space also (Figs. 6.10 & 11). The predicted peak

directivity is 11.509 dBi and the half-power beamwidths in the principal planes are

16 X 72. The H-plane beamwidth shows the expected beam narrowing due to the

presence of five radiating elements.

The two principal sidelobe levels are -14.43 dB and -18.18 dB respectively. These are

seen to differ slightly from the values expected from the array factor. A possible

reason is that the element spacing of λg/2 = 35.91mm actually corresponds to 0.67 λ0.

This results in a different sidelobe level from the case of λ0/2. Also, this is a constraint

on a broadside array using this feeding technique. The element spacing is governed by

the requirement of in-phase excitation of elements based on the guide wavelength at

the operating frequency. Null-filling is seen in the H-Plane near in-plane angles (Fig.

6.10) apparently due to surface-wave scatter off the substrate edge. Even though

mutual coupling is neglected, the E-Plane pattern matches MOM well yet shows

slight undulations in the forward space pattern predictions (see Fig. 6.11).

The predicted backlobe is -22.29 dB which is better than Case-1 earlier. This is on

account of the focusing behaviour due to the reinforcement in the broadside direction.


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Fig. 6.8: Electric Field Contour Plots through the HFSS®-Solved

Problem Geometry (Uniformly Excited Linear Array Module) a) parallel to waveguide longitudinal section; b) parallel to waveguide cross-section

(a)

(b)


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Fig. 6.9: HFSS®-Computed Radiation Pattern Cuts for 5-Element Uniformly Excited Linear Array Module

red H-Plane; and brown E-Plane

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m1

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Curve Info

dB10normalize(DirL3Y)Setup1 : Sw eep2Freq='5.59GHz' Phi='0deg'


Name Theta Ang Mag

m1 -26.00 -26.00 -14.43

m2 -48.00 -48.00 -18.18

m3 -180.00 -180.00 -22.29


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Fig. 6.10: Comparison of Radiation Patterns for 5-Element Uniformly-Excited Linear Array using MOM X Array Factor with HFSS Analysis (H-Plane)

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Fig. 6.11: Comparison of Radiation Patterns for 5-Element Uniformly-Excited Linear Array using MOM X Array Factor with HFSS Analysis (E-Plane)

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6.2 A Linear Array Module of Shunt-Slot Fed Microstrip Patch Elements with Dolph-Chebyshev Excitations

The relatively large first sidelobe level of ~ -13 dB is a characteristic of a uniformly-

excited linear array. It is possible to reduce the sidelobe level at the expense of

boresight gain by applying a taper to the excitation amplitudes. The sidelobes may be

thought of as a secondary reinforcement away from boresight as the path differences

from the elements results in in-phase signal arrival in those directions. With a tapered

amplitude, the side elements contribute lesser to both the main beam as well as the

sidelobes resulting in lower levels of both. In terms of the Schelkunoff unit circle

method, the relative power level in any direction is proportional to the product of the

distances to all the roots. As a result, clustering the roots closer to = will result in

a reduction of the sidelobe levels in general. A graphical solution for a five-element

linear array shows that placing the roots at + 87 and + 147 gives a -20 dB sidelobe

level (see pp. 134 of [49].) A procedure due to Dolph entails placing array factor roots

at the appropriate position to obtain a symmetrical pattern and reduced sidelobes

exhibiting a behaviour as per Chebyshev polynomials (see pp. 134 of [49].) This

results in equalized sidelobes at the specified relative level. With a more exact

calculation, the correct root positions for a 5-element case are obtained as + 88.82

and + 145.16 for a -20 dB sidelobe level. This requires the excitation of the elements

in the following ratio:

1 : 1.60 : 1.93 : 1.60 : 1

As an example of the use of tapered element excitations to obtain a reduced sidelobe

level, it was proposed to design and analyze this 5-element linear array of the

proposed shunt-slot coupled C-band elements with a Dolph-Chebyshev distribution.


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In the next subsection, we discuss the use of parametric variations presented earlier

for the proposed radiating element to design this variant of the linear array module.

6.2.1 Design of the Linear Array Module with Tapered Excitations

The design of the 5-element linear array module with Dolph-Chebyshev distributions

of amplitude excitations requires the determination of the transverse offset of each

coupling slot from the waveguide axes in order to obtain the ratios mentioned above.

For this purpose, the parametric behaviour of the power coupling vs. slot offset xs

presented in Section 4.7 (Fig. 4.25) is utilized to obtain the corresponding slot

positions.

The offset for the central element that requires the maximum power coupling is fixed

at 10mm since in Section 4.7.11, we observe that this is close to the maximum

coupling level that may be achieved by the slot. The other two coupling levels are

used to obtain the offsets for the end elements and the next inner elements

respectively as:

6.64mm and 8.71mm.

A simple linear interpolation is used to obtain these values using the data of Fig. 4.25.

This is justified since the trend of coupling in dB vs. distance is approximately linear.

A higher-order interpolation may be used for a more accurate estimate or the

developed program may be executed to compute the exact power coupling with these

distances. A reverse interpolation was used to verify the correctness of these numbers.

With these positions of the individual elements, an analysis was carried out.


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6.2.2 Array Factor Calculation and Pattern Computation

The amplitude distribution presented in the beginning of this section may be inserted

into the array factor calculation of Eq. (6.1) as Ii/I0 along with the new element

positions in the preceding section. The resulting H-Plane array factor plot (Fig. 6.12)

clearly shows the sidelobe suppression targeted by the amplitude tapering. All

sidelobes, including the one in the plane of the array are equalized at -20 dB. A

main-beam dilation is also apparent as a result of the reduction in aperture efficiency

– the penalty of sidelobe reduction as radiated power is “pulled out” of the sidelobes.

To compute the actual array pattern, we again take the element response of Eq. 2.119

and obtain the plot of Fig. 6.13 through pattern multiplication.

Fig. 6.12: Comparison of H-Plane Array Factors for the 5-Element Linear Array with Uniform & Dolph-Chebyshev Excitations

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We observe that the element roll-off causes the far-out sidelobes to be even lower

than -20 dB. The E-plane also shows an additional roll-off of ~ 0.7 dB.

We proceed to analyze this array variant using HFSS® as a confirmatory exercise.

Fig. 6.13: Computed Principal Plane Patterns for 5-Element Linear Array with Dolph-Chebyshev Excitations using MOM X Array Factor Formulation

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6.2.3 The HFSS®-Simulation Model for 5-Element Linear Array Module with

Dolph-Chebyshev Excitations for Reduced Sidelobes

Apart from the excitation slot locations, the simulation model for the present case is

identical to the uniform excitation case. The ground plane size is retained as 180 X

100 (see Fig. 6.14.) Similar pre-analysis operations are carried out on the analysis

model before invoking the solver.

Program execution takes place using a total CPU-time of 5h 2m. The optimized mesh

consists of a total of 130,426 tetrahedral elements. Convergence occurs in a total of

five passes with a final s = 0.000054. The initial mesh refinements have clearly

helped in reducing the number of passes as well as the execution time.

Fig. 6.14: Ansoft® HFSS® Simulation Model of 5-Element Linear Array Module with Dolph-Chebyshev


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6.2.4 The HFSS®-Simulation Results for 5-Element Linear Array Module with

Dolph-Chebyshev Excitations

The VSWR for the analyzed linear array module with Dolph-Chebyshev excitation is

very good (see Fig. 6.15.) The trend is similar to the case of uniform excitation in the

previous section. The maximum power coupling is 7.1473 X 10-6 and takes place at

5.60 GHz (Fig. 6.16) but is taken to be at the original value of 5.59GHz due to

reasons mentioned in the previous section. The radiation pattern calculations

presented later are again performed at this original resonant frequency.

The contour plots at the resonant frequency show a resemblance to the earlier contour

plots for the uniformly excited linear array (see Fig. 6.17.) The computed radiation

patterns show the expected sidelobe reduction due to the tapered amplitude weights

(see Fig. 6.18.) The predicted peak directivity is 10.383 dBi which is about 1.1 dB

lower than the uniformly-excited linear array module. The half-power beamwidths in

the principal planes are 18 X 86 which represents a mainlobe enlargement vs. the

previous case. The two principal sidelobe levels are -18.62 dB and -19.38 dB

respectively. These are close to the intended equalized level of -20 dB. The results

also closely match the computations from MOM + array factor calculation (Figs. 6.19

& 20). Reasons for the differences are: 1) approximation in the excitation amplitudes

actually realized by the given transverse offsets; and 2) surface-wave scatter from

ground-plane edges (causing an increase in sidelobe levels and radiation to the rear. It

is possible to carry out a more accurate estimate of the offsets and a few design

iterations to obtain a 20-dB suppression if desired. This is not attempted here but is

clearly within the scope of the developed program. The predicted backlobe level is -

21.89 dB which is not significantly different from the uniform excitation case.


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5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]

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Ansoft Corporation HFSSDesign1VSWR Plot

Curve Info





Fig. 6.15: HFSS®-Computed VSWR of 5-Element Linear Array Module with Dolph-Chebyshev Excitations

5.00 5.20 5.40 5.60 5.80 6.00FREQUENCY [GHz]

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Po

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Ansoft Corporation HFSSDesign1Power Coupling

m1

Curve Info

PoutSetup1 : Sw eep1

Name X Y

m1 5.6000E+000 7.1473E-006

Fig. 6.16: HFSS®-Computed Coupled-Power, Pout for 5-Element Linear Array Module with Dolph-Chebyshev Excitations


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Fig. 6.17: Electric Field Contour Plots through Problem Geometry

(Linear Array Module with Dolph-Chebyshev Excitations) c) parallel to waveguide longitudinal section; d) parallel to waveguide cross-section

(a)

(b)


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Fig. 6.18: HFSS®-Computed Radiation Pattern Cuts for 5-Element Linear Array Module with Dolph-Chebyshev Excitations red H-Plane; and brown E-Plane

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m1

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Curve Info



Name Theta Ang Mag

m1 -26.0000 -26.0000 -18.6174

m2 -48.0000 -48.0000 -19.3795

m3 -180.0000 -180.0000 -21.8857


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Fig. 6.19: Comparison of Radiation Patterns for 5-Element Dolph-Chebyshev Linear Array using MOM X Array Factor with HFSS Analysis (H-Plane)

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Fig. 6.20: Comparison of Radiation Patterns for 5-Element Dolph-Chebyshev Linear Array using MOM X Array Factor with HFSS Analysis (E-Plane)

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6.3 Summary

This chapter has presented the results of investigations on an important application of

the proposed WGMPA – that of cascading several such elements to realize a linear

array or “stick” module. It is also proposed that the WGMPA can be used for

obtaining any prescribed amplitude distribution across the array by varying the

transverse offset of each coupling slot. Two cases have been selected for illustrating

the concept, both using a set of five radiating elements each. The first case provides

uniform excitations to all the array elements. In the second case, a tapered distribution

conforming to Dolph-Chebyshev distribution was applied that targets an equalized

sidelobe envelope of -20 dB w.r.t. beam peak. The expected patterns for both these

cases are well-known from fundamental array factor considerations. For the

uniformly-excited case, the design of the array in terms of the inter-element spacing is

described. Alternate elements are disposed on either side of the waveguide centre-line

to allow closer spacing between the elements, yet retain an in-phase relationship

between the radiated signals. An array factor formulation for arbitrary element

positions is utilized for the pattern computation since the elements are staggered to

alternate sides of the waveguide median. In conjunction with this, expressions for the

radiated field of the element obtained in Chapter 2 are used to obtain the array

response through pattern multiplication. Excitation amplitudes and element positions

were accordingly varied for the two example cases. Subsequently, validation of these

analyses is carried out using Ansoft® HFSS® and the simulated results for both these

cases have been presented. The simulation model is described that uses a seeded

mesh as employed earlier for the single element along with material and boundary

definitions. The results of the analysis are presented and discussed. A good match is

observed between them and the expected sidelobe levels for a uniformly-excited


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array. Next, the reduced sidelobe array using Dolph-Chebyshev excitations is

described. The procedure for obtaining the amplitude excitation ratios for the targeted

-20-dB sidelobe envelope is described. In the WGMPA, these excitations are

implemented using the offsets obtained from the parametric study carried out with the

developed M-o-M program (summarized in Chapter 4). The central element has the

highest offset, hence the maximum amplitude. The excitation reduces till the end

elements are closest to the waveguide centre-line, thus receiving minimum amplitude

excitation. The simulated results of this array module also show the sidelobes nearly

equalized at the target level. Finally, radiation patterns obtained from the MOM and

the confirmatory HFSS® analysis are compared and are found to be in close

agreement. In this manner, by simulation, the utility of the proposed element to

realize linear array modules in a simple, robust configuration has been illustrated.

These may be excited with any desired aperture excitation function using the

parametric results derived in this thesis earlier. The linear array modules have the

potential to be used for realizing a full planar array by stacking a number of stick

modules in the transverse direction.

analysis of linear array modules - shodhganga...

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