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Keywords:

  • planar arrays;
  • Chebyshev polynomial;
  • array factor;
  • sidelobe level;
  • directivity;
  • half-power beam width

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[1] In this paper, the modified Chebyshev planar arrays are introduced. Like Chebyshev planar arrays, they have the important property of providing a constant sidelobe level in all directions. A formula for determining the current distribution in their elements is presented, and an exact formula for directivity is given for even and odd numbers of elements and any direction of maximum radiation. Compared to Chebyshev planar arrays with the same number of elements, same size, and same sidelobe level, the modified Chebyshev planar arrays provide a higher directivity when the number of elements exceeds a minimum value for the specified sidelobe level and interelement spacing. Numerical examples are given to illustrate the features of this new class of planar arrays and to compare their directivity and beam width with those of conventional Chebyshev planar arrays.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[2] The Chebyshev planar arrays have the important property of providing sidelobes of equal magnitude in their radiation patterns. They are optimum in the sense that for a specified sidelobe level they have the smallest beam width, and for a specified beam width they produce the lowest sidelobe level. However, they suffer from directivity saturation when the number of elements becomes large [Tseng and Cheng, 1968; Lo, 1993].

[3] Like the conventional Chebyshev planar array, the proposed modified planar arrays provide a constant sidelobe level in any cross section of the radiation pattern. In addition, they provide higher directivities than the conventional ones of the same number of elements, same size, and same sidelobe level when the number of elements exceeds a minimum value for the specified sidelobe level and element spacing. This comes at the cost of a slightly larger beam width. The concept of the modified Chebyshev planar arrays stems from the fact that multiplying an equal sidelobe pattern by itself any number of times will also yield an equal sidelobe pattern. An alike observation was used by Safaai-Jazi [1998] for the purpose of designing a new class of linear arrays.

[4] The Chebyshev planar arrays are reviewed briefly in Section 2. In Section 3, the modified Chebyshev planar arrays are formulated and equations of their array patterns and current excitations are developed. A general equation of the directivity of planar arrays as a function of the elements' currents is given in Section 4 for any number of elements and any direction of maximum radiation. In Section 5, a formula for the half-power beam width of modified Chebyshev planar arrays is derived. Examples are given in Section 6 to illustrate the features of the modified Chebyshev planar arrays and to compare them to the conventional Chebyshev planar arrays. Summarizing and concluding remarks are given in Section 7.

2. Chebyshev Planar Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[5] The array factor of a planar array of L × L identical elements, when L = 2N, is given by [Tseng and Cheng, 1968]

  • equation image

In (1), the Imn's denote the current magnitudes,

  • equation image

and

  • equation image

Moreover, dx is the interelement spacing in the x direction, dy is the interelement spacing in the y direction, λ is the wavelength, and θ0 and ϕ0 correspond to the direction of maximum radiation. For odd values of L, L = 2N + 1, the array factor is given by

  • equation image

In (4), εm and εn are equal to 1 for m = n = 1, and equal to 2 for m, n ≠ 1. Furthermore, the normalized array factor of a Chebyshev planar array is given by

  • equation image

In (5), TL−1 denotes a Chebyshev polynomial of the (L − 1)th order, R is the main lobe to sidelobe level ratio, and w is given by

  • equation image

The excitation of a Chebyshev planar array is symmetrical with respect to both the x and y axes, and the line y = x. That is, Imn = Im,n = Im,−n and Imn = Inm. Thus one needs to determine only N(N + 1)/2 excitation currents when L = 2N, and (N + 1)(N + 2)/2 excitation currents for L = 2N + 1 [Lo, 1993].

[6] The exact formula for determining these current amplitudes is given by Tseng and Cheng [1968] for the case by

  • equation image

Likewise, for L = 2N + 1,

  • equation image

3. Modified Chebyshev Planar Arrays

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[7] Let us consider a basis Chebyshev planar array of L0 × L0 elements and 1/R0 sidelobe level. The interelement spacing is dx in the x direction and dy in the y direction. The array factor of this basis array is

  • equation image

We define the array factor of the modified Chebyshev planar array as that of the basis array raised to an integer power b greater than 1; that is

  • equation image

where u and υ are as given in (2) and (3) respectively, and

  • equation image

The resulting modified Chebyshev planar array has L × L elements and a sidelobe level equal to R where

  • equation image
  • equation image

L and b should be chosen such that L0 is an integer.

[8] To find an exact expression for the excitation currents of the modified Chebyshev planar array, a method similar to that given by Tseng and Cheng [1968] is employed. This method is based on transforming the current amplitudes {Imn} into a new set of numbers {apq} by the relation

  • equation image

for L = 2N, and on using equations (1) and (10) to solve for the {apq}. Using this procedure, the following formula for the current magnitudes of modified Chebyshev planar arrays is obtained:

  • equation image

In a similar manner, the current magnitudes of modified Chebyshev planar arrays for L = 2N + 1 turned out to be

  • equation image

Like Chebyshev planar arrays, the excitation of a modified Chebyshev planar array is symmetrical with respect to both the x and y axes, and the line y = x. Equations (15) and (16) respectively give the exact values of the N(N + 1)/2 and (N + 1)(N + 2)/2 excitation currents that need to be determined for the cases of even and odd values of L.

4. Directivity Calculation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[9] The general formula for the directivity of an array is given, as in the work of Balanis [1982], by

  • equation image

where F(θ, ϕ) denotes the array factor. For a 2N × 2N planar array, the array factor is derived from (1) and obtained to be

  • equation image

and

  • equation image

For even values of L and using basic trigonometric identities, the four-cosine product in (19) can be transformed into a constant equal to 1/8 multiplied by the sum of eight cosine terms each of the form cos(1Aiu + 2Bjv), where

  • equation image

Substituting (20) into (19), the following is obtained:

  • equation image

Substituting (18) and (21) into (17), D becomes

  • equation image

where

  • equation image

The next step is to compute the integral in (23). For this purpose, the same procedure that was employed by Hansen [1983] to determine an exact expression for the directivity of a planar array with θ0 = ϕ0 = 0 is used. Consequently, an exact formula for the directivity is obtained for any θ0 and any ϕ0:

  • equation image

where

  • equation image

The same procedure is then used to obtain the directivity for the case when L = 2N + 1. In this case, D is found to be

  • equation image

where S is as given in (25), but

  • equation image

Equations (24) and (26) are applicable to any planar array with equal numbers of elements in the x and y directions for any dx and dy, and any direction of maximum radiation. Thus, they are suitable for both Chebyshev planar arrays and modified Chebyshev planar arrays. These equations can be easily changed to suit planar arrays with unequal numbers of elements in the x and y directions.

5. Half-Power Beam Width

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[10] The half-power beam width is the elevation angle difference between the two directions, in the ϕ = ϕ0 plane, in which the radiation intensity is one-half the maximum value beam. For the modified Chebyshev planar array, whose normalized array factor is given in (10), the two directions of concern correspond to

  • equation image

where

  • equation image

and

  • equation image

Using the definition of Chebyshev polynomials, we can rewrite equation (28) as

  • equation image

From (31), we can deduce

  • equation image

Equation (32) has two solutions, θ1 and θ2. Assuming that θ2 > θ1, the half-power beam width is HPBW = θ2 − θ1.

[11] In the case of ϕ0 = 0 (case a), vh = 0 and (θ1, θ2) are calculated from equations (29) and (32). Solving, we obtain:

  • equation image

In the case of ϕ0 = π/2 (case b), uh = 0 and the HPBW is as given in (33) but with dx replaced by dy.

[12] An analysis of modified Chebyshev planar arrays shows that, like in Chebyshev planar arrays, the beam width decreases with increasing element spacing, for a specified θ0. But increasing dx and dy above a certain value, say dM, will cause grating lobes to appear, and this should be avoided. Thus, dM is optimum in the sense that it gives the narrowest beam width before grating lobes begin to appear.

[13] Evidently, the spacing for a modified Chebyshev planar array is exactly that of the corresponding basis Chebyshev planar array given in (9). It was derived by Tseng and Cheng [1968] as

  • equation image

where w0 is as given in (11).

6. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[14] A planar array with 13 × 13 elements uniformly spaced in the x and y directions is first considered. In this case, dx = dy = 0.5λ are chosen and a sidelobe level equal to –10 dB (R = equation image), and we let θ0 = ϕ0 = 0. The corresponding conventional Chebyshev planar array has a directivity Dc = 19.33 and a half-power beam width BWc = 6.700. The 3-D plot of its array factor power is given in Figure 1 and the current magnitudes are listed in Table 1. Note that, due to their symmetry with respect to the x and y axes, only 49 out of the 169 currents are given. Furthermore, the symmetry around the line y = x is evident in the table. In Table 1, the currents are normalized so that the array center element has a current magnitude of 1.

image

Figure 1. Array factor power of a Chebyshev planar array with 13 × 13 elements, SLL = −10 dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0.

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Table 1. Excitation Currents of a Chebyshev Planar Array With 13 × 13 Elements, and SLL = −10 dB
equation imagen = 1n = 2n = 3n = 4n = 5n = 6n = 7
M = 110.693881.035590.845971.194481.639942.06323
M = 20.693880.9687680.560240.6069940.062660.777121.76848
M = 31.035590.560240.3064140.6420811.154860.6928151.1053
M = 40.845970.6069940.6420811.052090.0254591.180890.491246
M = 51.194480.062661.154860.0254591.093960.7209160.147374
M = 61.639940.777120.6928151.180890.7209160.2167860.026795
M = 72.063231.768481.10530.4912460.1473740.0267950.002233

[15] The modified Chebyshev planar array having 13 × 13 elements, a sidelobe level of –10 dB, dx = dy = 0.5λ, and θ0 = ϕ0 = 0 is next considered. For the case when b = 2 (two basis arrays), the directivity and the half-power beam width are, respectively, found to be Dm = 28.325 and BWm = 8.290. The directivity of the modified Chebyshev planar array is 46.556% more than that of the conventional Chebyshev planar array, and the HPBW is 23.73% larger. The array factor power of the modified Chebyshev planar array is plotted in Figure 2 and its current excitations are given in Table 2.

image

Figure 2. Array factor power of a modified Chebyshev planar array with 13 × 13 elements, SLL = −10 dB, dx = dy = 0.5λ, θ0 = ϕ0 = 0, and b = 2.

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Table 2. Excitation Currents of a Modified Chebyshev Planar Array With 13 × 13 Elements, SLL = −10 dB and b = 2
equation imagen = 1n = 2n = 3n = 4n = 5n = 6n = 7
M = 110.021120.0774230.031750.0836310.078490.131091
M = 20.021120.0741730.014620.0517380.0107770.027960.112364
M = 30.0774230.014620.0351450.0517510.05070.0562590.070228
M = 40.031750.0517380.0517510.045240.0190260.079620.031212
M = 50.0836310.0107770.05070.0190260.0761450.0468250.009364
M = 60.078490.027960.0562590.079620.0468250.0138760.001702
M = 70.1310910.1123640.0702280.0312120.0093640.0017020.000142

[16] One can deduce that, in a modified Chebyshev planar array, it is the center element(s) that has (have) the largest current magnitude, whereas for conventional Chebyshev planar arrays, the largest current value is shared by the elements positioned at the centers of the edge rows and edge columns (4 elements for odd L and 8 elements for even L).

[17] The plots of the array factor powers of the modified Chebyshev array and the conventional one in the plane ϕ = ϕ0 = 0 are given in Figure 3. It is noted that the modified Chebyshev planar array has fewer sidelobes than the conventional one. In fact, it has the same number of sidelobes as the basis conventional Chebyshev planar array with L0 elements in each direction, a sidelobe level equal to 1/R0, and same interelement spacing and direction of maximum radiation.

image

Figure 3. Array factor power of Chebyshev and modified Chebyshev planar arrays with 13 × 13 elements, SLL = −10 dB, dx = dy = 0.5λ, θ0 = ϕ0 = 0, and b = 2 in the plane ϕ = 0.

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[18] For the case of b = 3 (three basis arrays), the resulting modified Chebyshev planar array has a directivity Dm = 30.415, which is 57.37% more than the Chebyshev planar array, and a half-power beam width BWm = 9.650, which is 44.045% larger. As evident, the directivity improvement over the Chebyshev planar arrays comes at the expense of a broader beam width.

[19] Next and in Figure 4, the variations of directivity against L are plotted. For b = 2, L can take only odd integer values and the values of the other parameters are as given before. It is noticed that, for the values of L (larger than 9), the directivity of the conventional Chebyshev planar array saturates, whereas the corresponding modified Chebyshev array shows always-higher saturation directivity. In Figure 4, the dashed line separates two regions of performance: L ≤ 5 where Dm < Dc, and L ≥ 7 where Dm > Dc. The variations of the half-power beam width against L are plotted in Figure 5. For all values of L, the HPBW of the Chebyshev planar array is smaller than that of the modified Chebyshev planar array, but the difference between the two decreases with increasing number of elements. For large L, the improvement in directivity does not occur at the expense of a larger half-power beam width.

image

Figure 4. Directivity versus L for Chebyshev and modified Chebyshev planar arrays with SLL = −10 dB, dx = dy = 0.5λ, θ0 = ϕ0 = 0, and b = 2.

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image

Figure 5. HPBW versus L for Chebyshev and modified Chebyshev planar arrays with SLL = −10 dB, dx = dy = 0.5λ, θ0 = ϕ0 = 0, and b = 2.

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[20] The conventional Chebyshev planar array with 13 × 13 elements and a sidelobe level of –10 dB has a dM value of 0.952λ. For dx = dy = dM, the directivity is 19.235 and the HPBW is 3.5180. The corresponding modified one with b = 2 has dM = 0.9379λ for which the directivity is 24.096 and the HPBW is 4.4170.

[21] For modified Chebyshev planar arrays, the optimum element spacing as defined in Section 5 is always smaller than that of Chebyshev planar arrays of the same number of elements and same sidelobe level. This property is illustrated in Figure 6 where the optimum spacing of both types is plotted versus the number of elements for R = −10 dB, θ0 = ϕ0 = 0, and b = 2. It is concluded that the modified Chebyshev planar array operated at its optimum spacing is always smaller in size than the corresponding Chebyshev planar array having its optimum spacing.

image

Figure 6. dM versus L for Chebyshev and modified Chebyshev planar arrays with SLL = −10 dB, θ0 = ϕ0 = 0, and b = 2.

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7. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information

[22] In this paper, the modified Chebyshev planar arrays, which, like Chebyshev planar arrays, have equal sidelobes in their radiation patterns, are proposed. They are synthesized using the array factor of a smaller well-chosen conventional Chebyshev array. Analytical expressions for their array factor, excitation currents, directivity, and half-power beam width were presented for any number of elements, sidelobe level, element spacing, and any direction of maximum radiation. These arrays result in higher directivities than Chebyshev planar arrays when the number of elements exceeds a minimum value for the specified sidelobe level and interelement spacing. These improvements and other properties of the proposed arrays were illustrated through numerical examples.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information
  • Balanis, C. A., Antenna Theory, John Wiley, New York, 1982.
  • Hansen, R. C., Planar arrays, in The Handbook of Antenna Design, vol. 2, edited by A. W. Rudge, and K. Milne, pp. 151155, Peter Peregrinus, London, 1983.
  • Lo, Y. T., Array theory, in Antenna Handbook, vol. 2, edited by Y. T. Lo, and S. W. Lee, pp. 11-4811-58, Van Nostrand Reinhold, New York, 1993.
  • Safaai-Jazi, A., Modified Chebyshev arrays, Proc. IEE, 145(1), 4548, 1998.
  • Tseng, F. I., and D. K. Cheng, Optimum scannable planar arrays with an invariant sidelobe level, Proc. IEEE, 56(11), 17711778, 1968.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Chebyshev Planar Arrays
  5. 3. Modified Chebyshev Planar Arrays
  6. 4. Directivity Calculation
  7. 5. Half-Power Beam Width
  8. 6. Results
  9. 7. Conclusion
  10. References
  11. Supporting Information
FilenameFormatSizeDescription
rds4830-sup-0001-tab01.txtplain text document1KTab-delimited Table 1.
rds4830-sup-0002-tab02.txtplain text document1KTab-delimited Table 2.

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