Mutual Coupling Effects for Radar Cross Section (RCS) of a Series-Fed Dipole Antenna Array

Abstract

The estimation of RCS of a phased array depends on various parameters, namely, array geometry, operational frequency, feed network, mutual coupling between the antenna elements and so fourth. This paper presents the estimation of RCS of linear dipole array with series-feed network by tracing the signal path from the antenna aperture into the feed network. The effect of mutual coupling exhibited by the dipole antenna is considered for three configurations namely, side by side, collinear, and parallel in echelon. It is shown that the mutual coupling affects the antenna pattern (and hence RCS) significantly for larger scan angles. Further it is inferred that the RCS of phased array can be optimized by (i) reducing the length of the dipole, (ii) termination of the isolation port of the coupler with a suitable load, and (iii) using suitable amplitude distribution.

1. Introduction

Radar Cross Section (RCS) of an aerospace vehicle significantly depends on the scattering of the signals by the antenna array mounted on it [1]. This antenna mode scattering is dependent on (i) geometry of the phased array, (ii) nature of the feed network employed to excite it, and (iii) mutual coupling effect. In general a phased array requires a feed network in order to function as an effective receiver or transmitter. This feed network comprises of radiating elements, phase shifters, couplers, and the terminating resistors. Depending on their mode of arrangement, the feed network may be categorized as series feed, corporate feed and space feed [2, 3]. In order to design a series-fed phased array with specific desired characteristics several factors like the number of elements, their Interelement spacing, and beam scan angle are to be considered. Further if the spacing between the elements is less than half-wavelength, the effects of mutual coupling become predominant. These parameters of the array design are found to affect the RCS of the target considerably [4].

Mutual coupling has its influence on the impedance of the array elements, reflection coefficients, radiation pattern, and thus the RCS of phased array [5]. Every element in the antenna array will be of a definite physical size and shape. The signal incident on each of these elements will be reradiated due to various reasons like manufacturing defects, mismatches in the network, and so forth. The fields radiated from one antenna element interact with the field of the surrounding antennas which give rise to coupling. These reradiated fields when scattered along different directions affect the RCS of the target significantly. Further the effect of mutual coupling on the array performance is influenced by (i) feed network, (ii) array scan angle, and (iii) geometry of the antenna array, namely, side-by-side, collinear, and parallel-in-echelon configuration [5].

The literature available in the open domain presents several methods to estimate the antenna RCS. These include finite difference time domain (FDTD) method [6], shooting and bouncing ray (SBR) approach [7, 8], physical optics (PO) based model [9], scattering matrix approach [10], method of moments [11, 12], analytical formulation approach [13], and method of moments-particle swarm optimization (MoM-PSO) method [14].

Over the years, efforts have been put to reduce or compensate the mutual coupling effect [15]. Gupta and Ksienski [16] presented an open-circuit voltage approach for calculating mutual coupling in transmitting and receiving antenna array. Another approach based on the compensation of receiving mutual impedance was proposed, especially for small and compact receiving monopole array [17, 18]. Lee and Chu [19] studied the mutual coupling effect for a finite nonlinearly loaded antenna array using power series expansion technique. However, the effect of the feed network was neglected. Lu et al. [20] proposed a method to estimate the RCS of antenna array as a product of array factor and element factor, without inclusion of mutual coupling effect. Jenn and Lee [21] estimated the RCS of a series-fed phased array by considering the feed components, but ignoring the mutual coupling effect. Lee and Chu [22] included the mutual coupling factor in RCS estimation of a parallel-fed phased array using iterative scattering matrix approach. The edge effects and multiple reflections within the feed network were included. Najib et al. [23] studied the effect of mutual coupling in a phased array with butler-feed network.

In this paper, the method of estimating the RCS of phased array including mutual coupling effect and the feed network is presented. A step-by-step approach is followed to trace the signal path from the antenna aperture to the feed network, and hence obtaining the RCS of phased array including mutual coupling factor. This involves the derivation of the coupling coefficients in terms of impedance matrix. The parameters like length and radius of the dipole, antenna impedance are included. This work further presents a detailed formulation for the scattered field at each stage of the feed network. The simulation results are presented for side-by-side, collinear, and parallel-in-echelon configuration of linear dipole array. The RCS pattern obtained in the presence of mutual coupling is compared with no mutual coupling case. The authenticity of the results is established by comparing them with the trends, inferences, and results reported in the open domain.

2. Formulation for RCS of Series-Fed Phased Antenna Array

The RCS of an object for a plane wave incidence can be expressed as the ratio of the scattered field to the incident field. The mathematical expression for the antenna mode-scattered field is given by [5] ^→𝐸^𝑠_𝑛𝜃,𝜙=𝑗𝜂4𝜆𝑍_𝑎^→ℎ^→ℎ⋅^→𝐸^𝑖𝑒𝜃,𝜙^{−𝑗→𝑘→𝑅}𝑅Γ^𝑟_𝑛𝜃,𝜙, (1) where ^→𝐸^𝑠_𝑛 is the scattered field of the 𝑛th element, ^→𝐸^𝑖 is the incident field, (θ, ϕ) is direction of the signal, 𝑍_𝑎=𝑅_𝑎+𝑗𝑋_𝑎 is the radiation impedance with 𝑅_𝑎=𝑅_𝑟+𝑅_𝑑, 𝑅_𝑎 is the antenna resistance, 𝑋_𝑎 is the antenna reactance, 𝑅_𝑟 is the radiation resistance, 𝑅_𝑑 is the loss resistance of antenna (including conduction and dielectric losses), 𝜆 is the wavelength, 𝜂 is the impedance of medium surrounding the antenna, 𝑘=2𝜋/𝜆 is the free space wave number, 𝑅 is the distance between the target and the observation point, Γ^𝑟_𝑛 is the total reflected signal returning to aperture element 𝑛, and ^→ℎ is the effective height of the antenna element.

For a lossless antenna placed in free space, 𝜂=𝜂_𝑜=120𝜋 ohms with 𝑅_𝑑=0. Thus 𝑅_𝑎=𝑅_𝑟.

The expression (1) can be thus rewritten as ^→𝐸^𝑠_𝑛𝜃,𝜙=𝑗𝜂_𝑜4𝜆𝑍_𝑎^→ℎ^→ℎ⋅^→𝐸^𝑖𝑒𝜃,𝜙^{−𝑗→𝑘→𝑅}𝑅Γ^𝑟_𝑛𝜃,𝜙. (2) For a unit amplitude incident plane wave [24], we have ^→𝐸^𝑖𝜃,𝜙=𝑒^{−𝑗→𝑘⋅→𝑑𝑛}𝜃, (3) where ^→𝑑_𝑛=𝑥(𝑛−1)𝑑 represents the distance vector, with interelement spacing, 𝑑. The wave vector ^→𝑘 can be expressed in terms of direction cosines and the unit vectors [25] as ^→𝑘=𝑘𝑥sin𝜃cos𝜙+𝑦sin𝜃sin𝜙+𝑧cos𝜃. (4) The dot product in the phase factor of the incident signal can thus be simplified as ^→𝑘⋅^→𝑑_𝑛𝑥()𝑑=()∵=𝑘𝑥sin𝜃cos𝜙+𝑦sin𝜃sin𝜙+𝑧cos𝜃𝑛−1𝑛−1𝑘𝑑sin𝜃cos𝜙,𝑥⋅𝑥=1,𝑥⋅𝑦=𝑥⋅𝑧=0,_→𝑘⋅^→𝑑_𝑛=(𝑛−1)𝛼, (5) where 𝛼=𝑘𝑑sin𝜃cos𝜙 is the Interelement space delay of incident wave along 𝑥 direction.

For a closed body, with the coordinate system’s origin located within it, the scalar or dot product of the outward normal and the incident wave propagation vector must be negative [2]. Mathematically, we have ^→𝑘⋅^→𝑑_𝑛<0. (6) Hence from (3) and (5), the term 𝑒^{−𝑗→𝑘⋅→𝑑𝑛}=𝑒^{𝑗(𝑛−1)𝛼}. (7) For an x-polarized antenna, substitution of (7) into (2) yields ^→𝐸^𝑠_𝑛𝜃,𝜙=𝑗𝜂_𝑜4𝜆𝑍_𝑎ℎ𝑥ℎ𝑥⋅𝑒^{𝑗(𝑛−1)𝛼}𝜃𝑒^{−𝑗→𝑘→𝑅}𝑅Γ^𝑟_𝑛=𝜃,𝜙𝑗𝜂_𝑜4𝜆𝑍_𝑎ℎ²cos𝜃^→𝐸^𝑟_𝑛𝑒𝜃,𝜙^{−𝑗→𝑘→𝑅}𝑅𝑥. (8) This is the scattered field due to a single dipole element in the array. The overall scattered field is obtained by summing (8) over all the array elements. Consider ^→𝐸^𝑠𝜃,𝜙=^𝑁→_𝑛=1𝐸^𝑠_𝑛=𝜃,𝜙_𝑁𝑛=1𝑗𝜂_𝑜4𝜆𝑍_𝑎ℎ²(cos𝜃)^→𝐸^𝑟_𝑛𝑒𝜃,𝜙^{−𝑗→𝑘→𝑅}𝑅𝑥. (9) This yields the overall RCS of the dipole array for a unit magnitude incident plane wave. We have 𝜎𝜃,𝜙=4𝜋^𝑁_𝑛=1𝑗𝜂_𝑜4𝜆𝑍_𝑎ℎ²(cos𝜃)^→𝐸^𝑟_𝑛𝜃,𝜙². (10) The effective height of an x-polarized antenna element is given by [2] ^→ℎ1=ℎ𝑥=𝐼(0)_Δ𝑙𝐼(𝑙)𝑑𝑙𝑥, (11) where 𝐼(0) is the current at the feed terminal of the dipole element, and l is the dipole length.

Assuming the current distribution on the single dipole antenna to be cosine distribution, that is, 𝐼(𝑙)=𝐼(0)cos(𝑘𝑙) [5], the expression for the effective height becomes ^→ℎ=1𝐼(0)_Δ𝑙𝐼(0)()𝑥=cos𝑘𝑙𝑑𝑙_Δ𝑙cos(𝑘𝑙)𝑑𝑙𝑥. (12) Substitution of (12) into (10) yields 𝜎𝜃,𝜙=4𝜋^𝑁_𝑛=1𝑗𝜂_𝑜4𝜆𝑍_𝑎Δ𝑙cos(𝑘𝑙)𝑑𝑙²(cos𝜃)^→𝐸^𝑟_𝑛(𝜃,𝜙)²=4𝜋𝐹^𝑁→_𝑛=1𝐸^𝑟_𝑛𝜃,𝜙², (13) where 𝐹=𝑗𝜂_𝑜4𝜆𝑍_𝑎Δ𝑙cos(𝑘𝑙)𝑑𝑙²cos𝜃 (14)^→𝐸^𝑟_𝑛(𝜃,𝜙) represents the total scattered field returning to the aperture after being reflected from the mismatches prevailing within the feed network. This factor can be computed by traversing the path of the signal as it travels through the different components of the feed network (Figure 1). Thus consider 𝜎𝜃,𝜙=4𝜋𝜎_𝑟𝜃,𝜙²+𝜎_𝑝𝜃,𝜙²+𝜎_𝑐𝜃,𝜙²+𝜎_𝑠𝜃,𝜙². (15) The total RCS of phased array is the summation of individual contribution from radiating elements, phase shifters, and coupling port of the couplers.

Figure 1 

Typical series-fed network of phased array.

3. Impedance at Different Levels of the Feed Network

The signal from the antenna aperture enters into the feed network and then pass through its different levels before reaching the receive port. The feed network is a combination of various components, namely, radiators, phase shifters, couplers, and terminating loads. Each of these components offers finite impedance to the traveling signal (Figure 2). Thus in the feed network, a significant impedance mismatch can occur at the junction. This results in the reflection of the incoming signal depending on the value of reflection coefficient, which in turn affects the total RCS of array.

Figure 2 

Impedances at different stages in a series-fed network.

The following subsections present the calculation of the impedances at each level of the feed network.

3.1. Impedance at the Terminals of the Dipole Antenna

A dipole antenna in general will have finite impedance at its feed terminals. The impedance depends on the location of the antenna, the angle of incident signal, and the mutual coupling. In this paper, the effect of mutual coupling is analyzed for three different configurations of dipole array, namely, side-by-side, collinear, and parallel-in-echelon (Figure 3).

(a)

(b)

(c)

Figure 3 

Schematic of dipole array. (a) Side-by-side configuration, (b) collinear configuration, and (c) parallel-in-echelon configuration.

The antenna impedance is expressed as 𝑍_𝑎𝑛=𝑅_𝑎𝑛+𝑗𝑋_𝑎𝑛. The corresponding impedance matrix can be expressed as [26] 𝑧_{𝑎𝑥,𝑦}=𝑧_𝑎1,1𝑧_𝑎1,2⋯𝑧_𝑎1,𝑁𝑧_𝑎2,1𝑧_𝑎2,2⋯𝑧_𝑎2,𝑁𝑧⋮⋮⋱⋮_𝑎𝑁,1𝑧_𝑎𝑁,2⋯𝑧_{𝑎𝑁,𝑁}, (16) where 𝑧_{𝑎𝑖,𝑖} represents the self-impedance of the 𝑖th element and 𝑧_{𝑎𝑖,𝑗}(𝑖≠𝑗) is the mutual impedance between 𝑖th and 𝑗th element. The expressions for the self- and mutual impedance between the dipoles for different configurations can be referred to Appendix A.

This yields the total antenna impedance at the 𝑛th element as [5] 𝑍_𝑎𝑛=^𝑁_𝑦=1𝑧_{𝑎𝑥,𝑦}𝐼_𝑦𝐼_𝑥, (17) where 𝐼_𝑛 is the current at the feed terminals of 𝑛th antenna element.

3.2. Impedance at the Terminals of the Phase Shifters

The impedance at the terminals of the phase shifters depends on the nature of the phase shifters. In the present work, the phase shifters are modeled as simple lossless delay lines with characteristic impedance 𝑍₀. The length of the delay lines connected to each of the antennas depends on the phase shift required. The length of the delay-lines, 𝐿_𝑛 is obtained from the phase shift as [27] 𝐿_𝑛=𝜆)2𝜋(𝑛−1𝑘𝑑sin𝜃_𝑠cos𝜙_𝑠. (18) These delay lines are connected to the antenna terminals at one end and to the coupling port at the other end. Thus the antenna impedance, which will be the input impedance for the phase shifter, will be translated along its length as per the following formula [28]: 𝑍_𝑝𝑛=𝑍₀𝑍_𝑎𝑛+𝑗𝑍₀tan(2𝜋/𝜆)𝐿_𝑛𝑍₀+𝑗𝑍_𝑎𝑛tan(2𝜋/𝜆)𝐿_𝑛. (19) This yields the impedance at the other end of the phase shifter, that is, 𝑍_𝑝𝑛 (Figure 2).

3.3. Impedance at the Coupler Terminals

The impedances at different ports of the couplers depend on type of the couplers. The feed network is assumed to have lossless four-port couplers (Figure 4).

(a)

(b)

(c)

(d)

Figure 4 

A four-port coupler showing the transmission (𝑡𝑐𝑛) and the coupling (𝑗𝑐𝑛) coefficients.

Port 1 and Port 2 are the input and transmission ports, respectively, and form a part of the main feed line. Port 3 represents the coupling port, which connects to the antenna via phase shifter. Port 4 represents the isolated port, which is terminated with a load.

In general the terminating load is chosen as per the design, and the coupling port impedance is taken to be equal to its complex conjugate, that is, 𝑍_𝑐𝑛=𝑍^∗_𝑙𝑛. This condition ensures that there exists maximum power transfer between the coupler port and the terminating load. This in turn yields the maximum reflection coefficient at the terminating load, and thus high RCS.

4. Scattering Contributions from Different Components of the Feed Network

The total RCS of the target can be expressed as the sum of individual components indicating the scattering at the different levels of the feed network. The expressions for these individual contributions can be obtained by following the path of the signal as it travels from antenna aperture to the feed network. Each of these contributions can be expressed in terms of reflection and transmission coefficients of feed network elements.

4.1. RCS Component due to Scattering from Dipoles

The first source of scattering of the incident signal is the radiating element (e.g., a dipole). A well-designed antenna is expected to radiate/receive the entire energy incident on it. However there exists a mismatch between the antenna impedance, 𝑍_𝑎𝑛, and the characteristic impedance of the delay line, 𝑍₀ connected to it. This can be modeled as the antenna connected to a load representing the remaining part of the feed network (Figure 5). Further the reflection of the incident signal can be expressed in terms of reflection coefficient 𝑟_𝑟𝑛 of the 𝑛th dipole.

Figure 5 

Signal reflection from the radiating element.

Mathematically, we have 𝑟_𝑟𝑛=𝑍_𝑎𝑛−𝑍₀𝑍_𝑎𝑛+𝑍₀. (20) This reradiated portion of the signal from adjacent antenna elements exhibits linear phase variation along the array. In terms of the reflection coefficient 𝑟_𝑟𝑛 and the phase of the 𝑛th element, the scattered electric field from 𝑛th antenna element is given by ^→𝐸^𝑟_𝑟𝑛𝜃,𝜙=𝑒^{𝑗(𝑛−1)𝛼}𝑟_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}=𝑟_𝑟𝑛𝑒^{𝑗2(𝑛−1)𝛼}. (21) Equation (21) represents the scattered field due to a single radiating element. The summation of (21) over all the elements of the array yields the total scattered field due to the dipole array. Thus the corresponding RCS component is given by 𝜎_𝑟𝜃,𝜙=𝐹^𝑁→_𝑛=1𝐸^𝑟_𝑟𝑛𝜃,𝜙=𝐹^𝑁_𝑛=1𝑟_𝑟𝑛𝑒^{𝑗2(𝑛−1)𝛼}. (22)

4.2. RCS Component due to Scattering from the Phase Shifters

The portion of the signal received (not reflected) by the radiating element is transmitted to the next stage of feed network, that is, the input port of the phase shifter (modeled as lossless delay lines). This can be represented in terms of transmission coefficient of the radiating elements, 𝑡_𝑟𝑛, given by 𝑡²_𝑟𝑛=1−𝑟²_𝑟𝑛. The transmitted signal flows along the delay line to reach the coupling port of the coupler. At the end of the delay line a certain portion of the signal may be reflected back as the impedance at that point, 𝑍_𝑝𝑛, differs from the characteristic impedance of the line, 𝑍₀. This is represented in terms of the reflection coefficient of the phase shifter, 𝑟_𝑝𝑛, given by 𝑟_𝑝𝑛=𝑍_𝑝𝑛−𝑍₀𝑍_𝑝𝑛+𝑍₀. (23) This reflected signal propagates back towards the radiating element so as to suffer reflection and transmission for the second time. However, these higher-order reflections can be neglected if their magnitude is negligible. Thus following the path of the signal as shown by Figure 6, the scattered field due to 𝑛th phase shifter can be expressed as ^→𝐸^𝑟_𝑝𝑛𝜃,𝜙=𝑒^{𝑗(𝑛−1)𝛼}𝑡_𝑟𝑛𝑟_𝑝𝑛𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}=𝑡²_𝑟𝑛𝑟_𝑝𝑛𝑒^{𝑗2(𝑛−1)𝛼}. (24) Here the antenna is assumed to be a reciprocal device. Thus, the transmission coefficient for both the signal propagating towards the phase shifter and for the signal propagating back to the radiating element is taken as identical. This results 𝑡²_𝑟𝑛 in (24). Here the load represents the levels of feed network beyond the phase shifters. The exponential term in (24) signifies linear phase variation similar to that in the case of antenna elements.

Figure 6 

Signal reflection at the phase shifter.

Similar to the case of radiating elements, the summation of (24) over all the elements in phased array yields the total scattered field due to phase shifters. The corresponding RCS component is given by 𝜎_𝑝𝜃,𝜙=𝐹^𝑁→_𝑛=1𝐸^𝑟_𝑝𝑛𝜃,𝜙=𝐹^𝑁_𝑛=1𝑡²_𝑟𝑛𝑟_𝑝𝑛𝑒^{𝑗2(𝑛−1)𝛼}. (25)

4.3. RCS Component due to Scattering from the Coupling Port of the Couplers

Figure 7 shows the path of the signal, which passes through the phase shifters towards the coupling port of the coupler. This transmitted signal is determined by the transmission coefficient of the phase shifter, 𝑡²_𝑝𝑛=1−𝑟²_𝑝𝑛. At this junction, significant reflection can occur if the impedance at the end of the phase shifter, 𝑍_𝑝𝑛, and the impedance at the coupler input port, 𝑍_𝑐𝑛, differ. The corresponding reflection coefficient 𝑟_𝑐𝑛 is given by 𝑟_𝑐𝑛=𝑍_𝑐𝑛−𝑍_𝑝𝑛𝑍_𝑐𝑛+𝑍_𝑝𝑛. (26) Thus, the scattered field due to 𝑛th coupler is expressed as ^→𝐸^𝑟_𝑐𝑛=𝑒𝜃,𝜙_{𝑗(𝑛−1)𝛼}𝑡_𝑟𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑟_𝑐𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}=𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑟_𝑐𝑛𝑒^{𝑗2(𝑛−1)𝛼𝑠}𝑒^{𝑗2(𝑛−1)𝛼}=𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑟_𝑐𝑛𝑒^{𝑗2(𝑛−1)(𝛼+𝛼𝑠)}=𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑟_𝑐𝑛𝑒^{𝑗2(𝑛−1)𝜁}, (27) where 𝛼_𝑠 represents the Interelement phase to scan the antenna beam along 𝑥 direction. As both the radiating elements and the phase shifters are considered to be reciprocal devices, the expression (27) has terms 𝑡²_𝑟𝑛 and 𝑡²_𝑝𝑛. The exponential term in (27) represents the linear phase variation. The RCS component due to the scattering of the signal at the coupling port of the couplers is given by 𝜎_𝑐𝜃,𝜙=𝐹^𝑁→_𝑛=1𝐸^𝑟_𝑐𝑛𝜃,𝜙=𝐹^𝑁_𝑛=1𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑟_𝑐𝑛𝑒^{𝑗2(𝑛−1)𝜁}. (28)

Figure 7 

Signal reflection at the coupling port of the coupler.

4.4. RCS Component due to Scattering Beyond the Coupling Port of Couplers

The path of signal not reflected at the coupling port of the coupler, depends on the nature of coupler. Assuming the couplers as lossless four-port devices, the coupling coefficient can be expressed as 𝑐_𝑛=𝑍_𝑎𝑛𝑖²_𝑛∑^𝑁_𝑝=1𝑍_𝑎𝑝𝑖²_𝑝−∑^𝑛−1_𝑞=1𝑍_𝑎𝑞𝑖²_𝑞. (29) The detailed derivation of the coupling coefficient of coupler is given in Appendix B. The transmission coefficient of the coupler will be 𝑡²_𝑐𝑛=1−𝑐²_𝑛.

The signal incident at each of the antenna elements is expected to move towards the receiving port only. However it has the ability to move towards (i) Next antenna element in the array, (ii) Previous antenna element in the array, (iii) Load terminating its own coupler, and (iv) Port (backward direction) through which it was received. Each of these signals gives rise to scattering and hence contributes to the RCS significantly.

The scattered field at the 𝑛th element of the array can be due to the signals which would be incident on it from the next (𝑁−𝑛) elements of the array. The path of such a signal is shown in Figure 8.

Figure 8 

Path of the signal traveling towards the 𝑛th element from the next (𝑁−𝑛) elements.

Here the reflection coefficient of the terminating load, 𝑟_𝑙𝑛 can be expressed as 𝑟_𝑙𝑛=𝑍_𝑙𝑛−𝑍_𝑐𝑛𝑍_𝑙𝑛+𝑍_𝑐𝑛. (30) Following the path of the signals through each of the elements, the scattered electric field is given by ^→𝐸^𝑟_1𝑛=𝑒𝜃,𝜙_{𝑗(𝑁−1)𝛼}𝑡_𝑟𝑁𝑡_𝑝𝑁𝑒^{𝑗(𝑁−1)𝛼𝑠}𝑗𝑐_𝑁𝑒^𝑗𝜓𝑡_𝑐𝑁−1𝑒^𝑗𝜓𝑡_𝑐𝑁−2⋯𝑒^𝑗𝜓𝑗𝑐_𝑛𝑟_𝑙𝑛𝑡_𝑐𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}+𝑒^{𝑗(𝑁−2)𝛼}𝑡_𝑟𝑁−1𝑡_𝑝𝑁−1𝑒^{𝑗(𝑁−2)𝛼𝑠}𝑗𝑐_𝑁−1𝑒^𝑗𝜓𝑡_𝑐𝑁−2⋯𝑒^𝑗𝜓𝑗𝑐_𝑛𝑟_𝑙𝑛𝑡_𝑐𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}+⋯+𝑒^{𝑗(𝑛)𝛼}𝑡_𝑟𝑛+1𝑡_𝑝𝑛+1𝑒^{𝑗(𝑛)𝛼𝑠}𝑗𝑐_𝑛+1𝑒^𝑗𝜓×𝑗𝑐_𝑛𝑟_𝑙𝑛𝑡_𝑐𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}. (31) Regrouping of like factors yields𝐸^𝑟_1𝑛𝜃,𝜙=𝑡_𝑟𝑁𝑡_𝑟𝑛𝑡_𝑝𝑁𝑡_𝑝𝑛𝑗𝑐_𝑁𝑗𝑐_𝑛𝑟_𝑙𝑛𝑒^{𝑗(𝑁−1)𝜁}𝑒^{𝑗(𝑛−1)𝜁}×𝑡_𝑐𝑁−1𝑒^𝑗𝜓𝑡_𝑐𝑁−2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛𝑒^𝑗𝜓+𝑡_𝑟𝑁−1𝑡_𝑟𝑛𝑡_𝑝𝑁−1𝑡_𝑝𝑛𝑗𝑐_𝑁−1𝑗𝑐_𝑛𝑟_𝑙𝑛×𝑒^{𝑗(𝑁−2)𝜁}𝑒^{𝑗(𝑛−1)𝜁}𝑡_𝑐𝑁−2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛+1𝑡_𝑟𝑛𝑡_𝑝𝑛𝑡_𝑝𝑛+1𝑗𝑐_𝑛+1𝑗𝑐_𝑛𝑟_𝑙𝑛×𝑒^{𝑗(𝑛)𝜁}𝑒^{𝑗(𝑛−1)𝜁}𝑡_𝑐𝑛𝑒^𝑗𝜓=𝑡_𝑟𝑁𝑡_𝑟𝑛𝑡_𝑝𝑁𝑡_𝑝𝑛𝑗𝑐_𝑁𝑗𝑐_𝑛𝑟_𝑙𝑛𝑒^{𝑗(𝑁−1)𝜁}𝑒^{𝑗(𝑛−1)𝜁}×^𝑁−1_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓+𝑡_𝑟𝑁−1𝑡_𝑟𝑛𝑡_𝑝𝑁−1𝑡_𝑝𝑛𝑗𝑐_𝑁−1𝑗𝑐_𝑛𝑟_𝑙𝑛×𝑒^{𝑗(𝑁−2)𝜁}𝑒^{𝑗(𝑛−1)𝜁𝑁−2}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛+1𝑡_𝑟𝑛𝑡_𝑝𝑛𝑡_𝑝𝑛+1𝑗𝑐_𝑛+1𝑗𝑐_𝑛𝑟_𝑙𝑛×𝑒^{𝑗(𝑛)𝜁}𝑒^{𝑗(𝑛−1)𝜁𝑛}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓. (32a)(32b)

Removing (𝑡_𝑟𝑛𝑡_𝑝𝑛𝑟_𝑙𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁}) as common yields^→𝐸^𝑟_1𝑛𝜃,𝜙=𝑡_𝑟𝑛𝑡_𝑝𝑛𝑟_𝑙𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁}×𝑡_𝑟𝑁𝑡_𝑝𝑁𝑗𝑐_𝑁𝑒^{𝑗(𝑁−1)𝜁𝑁−1}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓+𝑡_𝑟𝑁−1𝑡_𝑝𝑁−1𝑗𝑐_𝑁−1𝑒^{𝑗(𝑁−2)𝜁𝑁−2}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛+1𝑡_𝑝𝑛+1𝑗𝑐_𝑛+1𝑒^{𝑗(𝑛)𝜁𝑛}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓,_→𝐸^𝑟_1𝑛𝜃,𝜙=𝑡_𝑟𝑛𝑡_𝑝𝑛𝑟_𝑙𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁}×^𝑁_𝑚=𝑛+1𝑡_𝑟𝑚𝑡_𝑝𝑚𝑗𝑐_𝑚𝑒^{𝑗(𝑚−1)𝜁𝑚−1}_𝑖=𝑛𝑡_𝑐𝑖𝑒^𝑗𝜓. (33a)(33b)The scattered field at the same element 𝑛 may also be affected by the signals, which travel from the previous 𝑛−1 elements of the array towards it. The path of such a signal is shown in Figure 9.

Figure 9 

Path of the signal traveling towards the 𝑛th element from the previous (𝑛−1) elements.

Following the path of the signals through each of the antenna elements, one gets ^→𝐸^𝑟_2𝑛=𝑒𝜃,𝜙_𝑗0𝛼𝑡_𝑟1𝑡_𝑝1𝑒^{𝑗0𝛼𝑠}𝑡_𝑐1𝑟_𝑙1𝑗𝑐₁𝑒^𝑗𝜓𝑡_𝑐2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓𝑗𝑐_𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}+𝑒^𝑗1𝛼𝑡_𝑟2𝑡_𝑝2𝑒^{𝑗1𝛼𝑠}𝑡_𝑐2𝑟_𝑙2𝑗𝑐₂𝑒^𝑗𝜓𝑡_𝑐3𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓𝑗𝑐_𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}+⋯+𝑒^{𝑗(𝑛−2)𝛼}𝑡_𝑟𝑛−1𝑡_𝑝𝑛−1𝑒^{𝑗(𝑛−2)𝛼𝑠}𝑡_𝑐𝑛−1×𝑟_𝑙𝑛−1𝑗𝑐_𝑛−1𝑒^𝑗𝜓𝑗𝑐_𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}. (34) Regrouping of like factors yields^→𝐸^𝑟_2𝑛=𝑡𝜃,𝜙_𝑟𝑛𝑡_𝑟1𝑡_𝑝𝑛𝑡_𝑝1𝑗𝑐₁𝑗𝑐_𝑛𝑟_𝑙1𝑒^𝑗0𝜁𝑒^{𝑗(𝑛−1)𝜁}×𝑡_𝑐1𝑒^𝑗𝜓𝑡_𝑐2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓+𝑡_𝑟𝑛𝑡_𝑟2𝑡_𝑝𝑛𝑡_𝑝2𝑗𝑐₂𝑗𝑐_𝑛𝑟_𝑙2𝑒^𝑗1𝜁𝑒^{𝑗(𝑛−1)𝜁}×𝑡_𝑐2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛𝑡_𝑟𝑛−1𝑡_𝑝𝑛𝑡_𝑝𝑛−1𝑗𝑐_𝑛−1𝑗𝑐_𝑛𝑟_𝑙𝑛−1×𝑒^{𝑗(𝑛−2)𝜁}𝑒^{𝑗(𝑛−1)𝜁}𝑡_𝑐𝑛−1𝑒^𝑗𝜓=𝑡_𝑟𝑛𝑡_𝑟1𝑡_𝑝𝑛𝑡_𝑝1𝑗𝑐₁𝑗𝑐_𝑛𝑟_𝑙1𝑒^𝑗0𝜁𝑒^{𝑗(𝑛−1)𝜁𝑛−1}_𝑖=1𝑡_𝑐𝑖𝑒^𝑗𝜓+𝑡_𝑟𝑛𝑡_𝑟2𝑡_𝑝𝑛𝑡_𝑝2𝑗𝑐₂𝑗𝑐_𝑛𝑟_𝑙2𝑒^𝑗1𝜁𝑒^{𝑗(𝑛−1)𝜁𝑛−1}_𝑖=2𝑡_𝑐𝑖𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛𝑡_𝑟𝑛−1𝑡_𝑝𝑛𝑡_𝑝𝑛−1𝑗𝑐_𝑛−1𝑗𝑐_𝑛𝑟_𝑙𝑛−1×𝑒^{𝑗(𝑛−2)𝜁}𝑒^{𝑗(𝑛−1)𝜁𝑛−1}_{𝑖=𝑛−1}𝑡_𝑐𝑖𝑒^𝑗𝜓. (35a)(35b)

Removing (𝑡_𝑟𝑛𝑡_𝑝𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁}) as common yields^→𝐸^𝑟_2𝑛=𝑡𝜃,𝜙_𝑟𝑛𝑡_𝑝𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁}×𝑡_𝑟1𝑡_𝑝1𝑟_𝑙1𝑗𝑐₁𝑒^{𝑗0𝜁𝑛−1}_𝑖=1𝑡_𝑐𝑖𝑒^𝑗𝜓+𝑡_𝑟2𝑡_𝑝2𝑟_𝑙2𝑗𝑐₂𝑒^{𝑗1𝜁𝑛−1}_𝑖=2𝑡_𝑐𝑖𝑒^𝑗𝜓+⋯+𝑡_𝑟𝑛−1𝑡_𝑝𝑛−1𝑟_𝑙𝑛−1𝑗𝑐_𝑛−1𝑒^{𝑗(𝑛−2)𝜁𝑛−1}_{𝑖=𝑛−1}𝑡_𝑐𝑖𝑒^𝑗𝜓=𝑡_𝑟𝑛𝑡_𝑝𝑛𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁𝑛−1}_𝑚=1𝑡_𝑟𝑚𝑡_𝑝𝑚𝑟_𝑙𝑚𝑗𝑐_𝑚𝑒^{𝑗(𝑚−1)𝜁𝑛−1}_𝑖=𝑚𝑡_𝑐𝑖𝑒^𝑗𝜓. (36a)(36b)The scattered field at the 𝑛th element can also be affected due to the signal which moves towards the load terminating its own coupler, as shown in Figure 10. This results in the self-scattering of the incident signal expressed as^→𝐸^𝑟_3𝑛𝜃,𝜙=𝑒^{𝑗(𝑛−1)𝛼}𝑡_𝑟𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑐𝑛×𝑟_𝑙𝑛𝑡_𝑐𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}=𝑟_𝑙𝑛𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑡²_𝑐𝑛𝑒^{𝑗2(𝑛−1)𝛼}𝑒^{𝑗2(𝑛−1)𝛼𝑠},_→𝐸^𝑟_3𝑛𝜃,𝜙=𝑟_𝑙𝑛𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑡²_𝑐𝑛𝑒^{𝑗2(𝑛−1)𝜁}. (37a)(37b)The signal, which succeeds in overcoming all the above-mentioned scattering sources, moves towards the receiving port (Figure 11). However if there exists a mismatch between the receiving port and the input port of the coupler, significant scattering can occur even at this point. The reflection coefficient at this input port can be expressed as 𝑟_𝑖𝑛=𝑍₁₂−𝑍₀𝑍₁₂+𝑍₀, (38) where 𝑍₁₂ is the impedance at the input port of the first coupler in the array, and 𝑍₀ is the characteristic impedance of the coaxial cable connected to the input port of the first coupler.

Figure 10 

Path of the signal traveling towards the load terminating its own coupler.

Figure 11 

Path of the signal travelling towards the load terminating the input port.

The scattered field at the junction of receiving port and input port of coupler is expressed as ^→𝐸^𝑟_4𝑛=𝑒𝜃,𝜙_{𝑗(𝑛−1)𝛼}𝑡_𝑟𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑗𝑐_𝑛𝑒^𝑗𝜓𝑡_𝑐𝑛−1⋯𝑒^𝑗𝜓𝑡_𝑐2𝑒^𝑗𝜓𝑡_𝑐1𝑟_𝑖𝑛×𝑡_𝑐1𝑒^𝑗𝜓𝑡_𝑐2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓𝑗𝑐_𝑛𝑡_𝑝𝑛𝑒^{𝑗(𝑛−1)𝛼𝑠}𝑡_𝑟𝑛𝑒^{𝑗(𝑛−1)𝛼}. (39) Rearrangement of like factors yields^→𝐸^𝑟_4𝑛𝜃,𝜙=𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑒^{𝑗2(𝑛−1)𝛼}𝑒^{𝑗2(𝑛−1)𝛼𝑠}𝑗𝑐_𝑛²×𝑟_𝑖𝑛𝑡_𝑐1𝑒^𝑗𝜓𝑡_𝑐2𝑒^𝑗𝜓⋯𝑡_𝑐𝑛−1𝑒^𝑗𝜓2,_→𝐸^𝑟_4𝑛𝜃,𝜙=𝑟_𝑖𝑛𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑗𝑐_𝑛²𝑒^{𝑗2(𝑛−1)𝜁𝑛−1}_𝑖=1𝑡_𝑐𝑖𝑒^𝑗𝜓2. (40a)(40b)The total field due to the signal scattering beyond the coupling port of couplers is given by ^→𝐸^𝑟_𝑠𝑛𝜃,𝜙=^→𝐸^𝑟_1𝑛𝜃,𝜙+^→𝐸^𝑟_2𝑛𝜃,𝜙+^→𝐸^𝑟_3𝑛𝜃,𝜙+^→𝐸^𝑟_4𝑛𝜃,𝜙. (41) The corresponding RCS component is expressed as 𝜎_𝑠𝜃,𝜙=𝐹^𝑁→_𝑛=1𝐸^𝑟_𝑠𝑛𝜃,𝜙. (42) Substitution of (22), (25), (28), and (42) into (15) gives the total RCS for the phased array due to the mismatches of the feed network. The normalized RCS of phased array is expressed as 𝜎𝜃,𝜙=4𝜋𝜆²𝜎_𝑟𝜃,𝜙²+𝜎_𝑝𝜃,𝜙²+𝜎_𝑐𝜃,𝜙²+𝜎_𝑠𝜃,𝜙². (43)

5. Results and Discussion

In this section, the simulation results for the RCS of a linear series-fed dipole array are presented including mutual coupling factor. The simulations have been done using the Fortran 90 software code (developed by the authors) based on the formulation described above. The graphs are plotted using the graphical subroutine Sigma plot v.10. The variation of the RCS of antenna array due to the changes in the design parameters of the array is analyzed. The RCS pattern of a linear series-fed phased array is validated against the results in open domain. In the earlier work [29], the formulation for RCS was done for a phased array of infinitesimal dipoles (𝑙≪𝜆) fed by a series-feed network. The mutual coupling and edge effects were not taken into account. Although these assumptions simplify the RCS estimation of the array, they are unacceptable in a practical scenario. In this paper, an attempt is made to overcome some of these limitations with acceptable increase in the computational complexity.

The impedance parameter is included into the expression of coupling coefficients of the coupler in feed network. In order to validate with the earlier results (without mutual coupling), the impedance matrix of array is taken as identity matrix. This cancels the mutual coupling effect. Figure 12 shows the RCS pattern of 50-element series-fed array with and without inclusion of impedance in the coupling coefficients formulation. The parameters are chosen as 𝜃_𝑠=0^∘,𝜓=𝜋/2,𝑑=0.4𝜆,𝑙=0.5𝜆, and 𝑟_𝑟=𝑟_𝑝=𝑟_𝑐=𝑟_𝑙=0.2 with uniform unit amplitude distribution. It is evident that both the RCS patterns are in excellent match. This authenticates the RCS estimation of phased array including mutual coupling factor.

Figure 12 

RCS pattern of a 50-element series-fed phased array. 𝜃𝑠=0∘, 𝜓=𝜋/2, 𝑑=0.4𝜆, 𝑙=0.5𝜆, and 𝑟𝑟=𝑟𝑝=𝑟𝑐=𝑟𝑙=0.2; uniform unit amplitude distribution.

Next step includes the effect of effective height of the dipole antenna element on the RCS of phased array. Although an antenna is specified using its physical length (height), its effective aperture differs from the physical one. Thus an accurate computation of the target RCS is expected to include the effective height rather than physical height of antenna element. The effective height of the antenna depends on the nature of the current distribution over its surface [2]. In the present work, the length of dipole antenna is taken as 0.003λ in order to satisfy the assumption of infinitesimal dipole. The simulated RCS pattern (Figure 13) is compared with the earlier result using physical height of dipole [29]. The excellent match between the RCS patterns establishes the authenticity of the formulation.

Figure 13 

RCS pattern of series-fed linear array. 𝑁=50, 𝜃𝑠=0∘, 𝜓=𝜋/2, 𝑑=0.4𝜆, 𝑙=0.003𝜆 and 𝑟𝑟=𝑟𝑝=𝑟𝑐=𝑟𝑙=0.2; uniform unit amplitude distribution.

Next, the RCS of phased array is calculated using antenna impedance instead of antenna resistance. This is done in order to include the finite radius of the dipole antenna element in the expression of antenna impedance, and hence RCS of phased array. In earlier work, radius of dipole antenna was not included, assuming to be infinitesimally small.

It is necessary to choose the radius and length of the dipole element appropriately. Usually the dipole is taken to be a thin wire. In the present work, the radius of the wire is fixed as 10⁻⁵𝜆. Next the antenna is assumed to be resonant. The antenna reactance is expected to vanish for a dipole of resonant length. Hence the length of dipole is taken as 0.488λ. This combination of length and radius of dipole antenna yields a radiation reactance of −7.566 × 10⁻⁵ Ω, which is almost equal to zero. The simulated RCS pattern is shown in Figure 14(a). The RCS results are compared with and without using radiation reactance, keeping all other parameters the same. It can be seen that the level of RCS pattern is different for this choice of dipole length and Interelement spacing. The difference in the RCS patterns can be reduced for smaller Interelement spacing, keeping dipole length constant, as shown in Figure 14(b). This observation drives attention that RCS of phased array can be controlled by multivariant optimization [30–32]. The parameters that can be optimized are the Interelement spacing [4, 13], dipole length, or geometry.

(a) 𝑑=0.4𝜆

(b) 𝑑=0.25𝜆

Figure 14 

RCS of series-fed linear array. 𝑁=50, 𝜃𝑠=0∘, 𝜓=𝜋/2, 𝑙=0.488𝜆 and 𝑟𝑟=𝑟𝑝=𝑟𝑐=𝑟𝑙=0.2; uniform unit amplitude distribution.

In Figures 12 through 14 the electric field scattered due to the termination of the receive port of the array was considered to be [21] ^→𝐸^𝑟_4𝑛𝜃,𝜙=𝑟_𝑙𝑡²_𝑟𝑡²_𝑝^𝑁_𝑛=1𝑗𝑐_𝑛𝑒^{𝑗(𝑛−1)𝜁𝑛−1}_𝑖=1𝑡_𝑐𝑖𝑒^𝑗𝜓2. (44) However if the scattered electric field is calculated by tracing the signal path, one gets ^→𝐸^𝑟_4𝑛𝜃,𝜙=^𝑁_𝑛=1𝑟_𝑖𝑛𝑡²_𝑟𝑛𝑡²_𝑝𝑛𝑗𝑐_𝑛²𝑒^{𝑗2(𝑛−1)𝜁𝑛−1}_𝑖=1𝑡_𝑐𝑖𝑒^𝑗𝜓2. (45) The RCS pattern obtained using (45), that is, through tracing the signal path, is shown in Figure 15. It can be observed that the load reflection lobe is absent in the RCS pattern when compared to the RCS pattern obtained using (44). Lu et al. [20] present the RCS pattern for a series-fed 60 × 60 planar dipole array by considering the effect of feed network. Here the radiation resistance of dipole antenna is taken as 𝑅_𝑎=24.7(𝑘𝑙)^2.4. Based on Lu et al. [20] formulation, the RCS pattern for a 30-element series-fed linear dipole array is shown in Figure 16. The parameters considered are 𝑁=30, 𝜃_𝑠=0^∘, 𝜓=𝜋/2, 𝑑=0.4𝜆, 𝑙=0.5𝜆, and 𝑟_𝑟=𝑟_𝑝=𝑟_𝑐=𝑟_𝑙=0.2, with uniform unit amplitude distribution. It can be observed that the RCS patterns in Figures 15 and 16 are similar.

Figure 15 

RCS of series-fed linear array. 𝑁=30,  𝜃𝑠=0∘,  𝜓=𝜋/2,  𝑑=0.4𝜆,  𝑙=0.5𝜆, and 𝑟𝑟=𝑟𝑝=𝑟𝑐=𝑟𝑙=0.2; uniform unit amplitude distribution.

Figure 16 

RCS of series-fed 30-element linear dipole array. 𝜃𝑠=0∘,  𝑑=0.4𝜆,  𝑙=0.5𝜆, and 𝑟=0.2.

Next, mutual coupling effect is included in the estimation of RCS of dipole array in side-by-side configuration. Figure 17(a) presents the RCS pattern of 10-element dipole array with and without mutual coupling. Other parameters considered are 𝜃_𝑠=0^∘, 𝜓=𝜋/2, 𝑑=0.1𝜆, 𝑙=0.5𝜆, 𝑎=10⁻⁵𝜆, 𝑍₀=150 Ω, and 𝑍_𝑙=225 Ω. The amplitude distribution is taken as uniform distribution with unit amplitude. Similar RCS patterns are shown in Figures 17(b) and 17(c) for 20- and 40-element series-fed linear dipole arrays, respectively. It is apparent that the mutual coupling effect changes the RCS pattern of the dipole array significantly.

(a) 𝑁=10, 𝑍𝑙=225 Ω

(b) 𝑁=20,𝑍𝑙=270 Ω

(c) 𝑁=40,𝑍𝑙=275 Ω

Figure 17 

Effect of mutual coupling on RCS of series-fed linear array. 𝜃𝑠=0∘, 𝜓=𝜋/2, 𝑑=0.1𝜆, 𝑙=0.5𝜆, 𝑎=10−5𝜆, and 𝑍0=150 Ω with unit amplitude uniform distribution for side-by-side configuration.

Figure 18 presents the variation in RCS pattern of dipole array (in collinear configuration) due to beam scanning. The beam scanning includes 0^∘, 45^∘, and 85^∘. The other parameters considered are 𝑁=30, 𝑍₀=75 Ω, and 𝑍_𝑙=150 Ω. It can be seen that the RCS pattern of collinear dipole array with and without mutual coupling is almost identical. However, the difference between the RCS patterns is noticeable with increase in scan angle 𝜃_𝑠 from 0° to 45°, or 85°. This is in accordance with the reported inference [22] that the mutual coupling affects both the array and RCS pattern significantly for large scan angles. The variation of the RCS pattern for different scan angles (0^∘, 45^∘, and 85^∘) in case of a parallel-in-echelon and a side-by-side dipole array is shown by Figures 19 and 20, respectively. The parameters are taken to be same as in Figure 18 except for 𝑍₀ and 𝑍_𝑙. The values of 𝑍₀ and 𝑍_𝑙 are chosen to be 125 Ω, and 235 Ω in case of parallel-in-echelon and 150 Ω and 280 Ω for side-by-side configuration, respectively. It can be observed that the trend observed due to the variation of scan angle is same for all three configurations. However the difference in the RCS pattern with and without mutual coupling is least in collinear configuration of dipole array.

(a) 𝜃𝑠=0∘

(b) 𝜃𝑠=45∘

(c) 𝜃𝑠=85∘

Figure 18 

Effect of mutual coupling on RCS of series-fed linear collinear dipole array of 𝑁=30,𝜓=𝜋/2,𝑑=0.1𝜆,𝑙=0.5𝜆,𝑎=10−5𝜆,𝑍0=75 Ω, and 𝑍𝑙=150 Ω; unit amplitude uniform distribution.

(a) 𝜃𝑠=0∘

(b) 𝜃𝑠=45∘

(c) 𝜃𝑠=85∘

Figure 19 

Effect of mutual coupling on RCS of series-fed linear parallel-in-echelon dipole array of 𝑁=30,𝜓=𝜋/2,𝑑=0.1𝜆,𝑙=0.5𝜆,𝑎=10−5𝜆,𝑍0=125 Ω, and 𝑍𝑙=235 Ω; unit amplitude uniform distribution.

(a) 𝜃𝑠=0∘

(b) 𝜃𝑠=45∘

(c) 𝜃𝑠=85∘

Figure 20 

Effect of mutual coupling on RCS of series-fed linear side-by-side dipole array of 𝑁=30,𝜓=𝜋/2,𝑑=0.1𝜆,𝑙=0.5𝜆,𝑎=10−5𝜆,𝑍0=150 Ω, and 𝑍𝑙=280 Ω; unit amplitude uniform distribution.

The estimation of RCS can provide a profitable insight for reducing the scattered field. Variation of certain design parameters of the feed network can reduce the RCS considerably. RCS of phased array can be reduced significantly by reducing the number of elements in the phased array [29]. However if the number of elements is less, the directivity of the array gets affected adversely. The other feasibility is to reduce the antenna length, which in turn reduces the effective area of the antenna array. Figure 21(a) shows the RCS for dipoles of length λ/2 and λ/3 at a scan angle of 0°. Other parameters are taken to be 𝑁=30, 𝜓=𝜋/2, 𝑑=0.1𝜆, 𝑎=10⁻⁵𝜆, 𝑍₀=125 Ω, and 𝑍_𝑙=235 Ω. The excitation is considered to be unit amplitude uniform distribution. The computed results are for parallel-in-echelon configuration. The plot shows that the RCS reduces as the length of the dipoles reduces. However the trend of the RCS pattern seems to be independent of the dipole length. Figure 21(b) shows the RCS pattern for dipole length variation with the same parameters but for a scan angle of 70°. The observations found are identical to that of Figure 21(a). This indicates that the reduction in the dipole length leads to the reduction in RCS for any scan angle. However the variation in dipole length will affect the resonant frequency of the array. Therefore, reducing the RCS of a phased array just by reducing the dipole length is not recommended. Other factors like load termination or aperture distribution are also required to be chosen suitably.

(a) 𝜃𝑠=0∘

(b) 𝜃𝑠=70∘

Figure 21 

Effect of dipole length on RCS of 30-element series-fed linear parallel-in-echelon dipole array including mutual coupling effect.

Next, the dependency of the array RCS on the amplitude distribution of the current feed exciting the couplers is studied. The variations in the current at the antenna terminals affect the power distribution and hence vary the coupling coefficients of the couplers. Figure 22 shows the RCS pattern for various amplitude distributions, namely, uniform, cosine squared on a pedestal, Doplh-Chebyshev, and Taylor. The parameters chosen are of 𝑁=30,𝜃_𝑠=0^∘,𝜓=𝜋/2,𝑑=0.1𝜆,𝑙=0.5𝜆,𝑎=10⁻⁵𝜆 for 𝑍₀=150 Ω, and 𝑍_𝑙=280 Ω. The mutual coupling factor is considered for side-by-side configuration of dipole array. It can be seen that the specular lobe has the same level irrespective of the amplitude distribution. However the levels of the side lobes vary considerably and are found to be maximum for Dolph-Chebyshev distribution.

Figure 22 

Effect of amplitude distribution on RCS of 30-element series-fed linear side-by-side dipole array including mutual coupling effect.

The termination of the isolated port of couplers varies the reflection coefficient at their respective terminal. This varies the amount of scattered field and hence the RCS of phased array. Figure 23 shows the RCS pattern for terminating impedances of 𝑍_𝑙=0 Ω, 50 Ω, and 280 Ω. Other parameters are kept to be the same as in the previous case (Figure 22). It can be observed that the RCS pattern for the termination of 280 Ω is lower than that for short circuit or for a termination of 50 Ω. However the trend of the RCS pattern remains almost independent of the terminating load value. The result obtained is in accordance with the results obtained in open literature [6]. Figure 24 shows the similar comparison of the short and impedance terminated feed network but for a scan angle of 50°. A similar trend is observed. This indicates that the termination of the coupler port with a suitable load will reduce the RCS irrespective of the scan angle.

Figure 23 

Effect of terminating impedance on RCS of series-fed linear array including mutual coupling effect. 𝑁=30, “𝜃𝑠=0∘”, 𝜓=𝜋/2, 𝑙=0.5𝜆,  𝑑=0.1𝜆, 𝑎=10−5𝜆, and 𝑍0=150 Ω with Dolph-Chebyshev distribution for side-by-side configuration.

Figure 24 

Effect of terminating impedance on RCS of series-fed linear array including mutual coupling effect. 𝑁=30, “𝜃𝑠=50∘”, 𝜓=𝜋/2, 𝑙=0.5𝜆, 𝑑=0.1𝜆, 𝑎=10−5𝜆, and 𝑍0=150 Ω with Dolph-Chebyshev distribution for side-by-side configuration.

6. Summary

This paper presents the study carried out on the estimation of RCS of series-fed linear dipole array in the presence of mutual coupling. Here a step-by-step approach is followed to arrive at the equation of RCS for a phased array with the inclusion of mutual coupling factor. The formulation of the coupling coefficients is first derived in terms of impedance matrix. This is followed by the inclusion of (i) length of the dipole, (ii) effective length of the dipole, and (iii) antenna impedance/radius of the dipole wire into the RCS equation. The summation term of the scattered field due to the terminating load is refined. This work further presents a detailed formulation for the scattered field at each stage of the feed network. The simulation results are presented for side-by-side, collinear, and parallel-in-echelon configurations of linear dipole array. The RCS pattern obtained in the presence of mutual coupling is compared with no mutual coupling case. The authenticity of the results is established by comparing them with the trends, inferences, and results reported in the open domain.

It is shown that the mutual coupling alters the impedance at the terminals of the radiating elements significantly. The observed changes depend upon the configuration of dipole array considered, namely, side-by-side configuration, collinear configuration, or parallel-in-echelon configuration. The simulation results indicate that the mutual coupling affects the RCS pattern pronouncedly for large scan angles. It is seen that the RCS value can be reduced by varying certain feed network parameters. It is inferred that the RCS of the dipole array can be reduced by (i) reducing the length of the dipole, (ii) terminating the isolated port of the coupler with a suitable load, or (iii) choosing a suitable amplitude distribution.

Appendices

A. Self- and Mutual Impedance of Dipole Array

The self-impedance of a dipole is given by [5] the following. 𝑅_self𝑛=𝜂2𝜋𝐶+ln𝑘𝑙_𝑛−𝐶_𝑖𝑘𝑙_𝑛+12sin𝑘𝑙_𝑛𝑆_𝑖2𝑘𝑙_𝑛−2𝑆_𝑖𝑘𝑙_𝑛+12cos𝑘𝑙_𝑛×𝐶+ln𝑘𝑙_𝑛2+𝐶_𝑖2𝑘𝑙_𝑛−2𝐶_𝑖𝑘𝑙_𝑛,𝑋_self𝑛=𝜂4𝜋2𝑆_𝑖𝑘𝑙_𝑛+cos𝑘𝑙_𝑛2𝑆_𝑖𝑘𝑙_𝑛−𝑆_𝑖2𝑘𝑙_𝑛−sin𝑘𝑙_𝑛×2𝐶_𝑖𝑘𝑙_𝑛−𝐶_𝑖2𝑘𝑙_𝑛−𝐶_𝑖2𝑘𝑎²_𝑛𝑙_𝑛, (A.1)(A.2) where 𝐶_𝑖(𝑘𝑙_𝑛) and 𝑆_𝑖(𝑘𝑙_𝑛) are cosine and sine integrals expressed as [5] 𝑆_𝑖(𝑥)=^∞_𝑘=0(−1)^𝑘𝑥^2𝑘+1()!,𝐶2𝑘+1)(2𝑘+1_𝑖(𝑥)=𝐶+ln(𝑥)+^∞_𝑘=1(−1)^𝑘𝑥^2𝑘()!.2𝑘2𝑘 (A.3)

The expressions of the mutual impedances for different configurations are [5] the following.

Side-by-side configuration:
𝑅_𝑠𝑟=𝜂4𝜋2𝐶_𝑖𝑢_0𝑟−𝐶_𝑖𝑢_1𝑟−𝐶_𝑖𝑢_2𝑟,𝑋_𝑠𝑟𝜂=−4𝜋2𝑆_𝑖𝑢_0𝑟−𝑆_𝑖𝑢_1𝑟−𝑆_𝑖𝑢_2𝑟, (A.4)(A.5) where 𝑢_0𝑟=𝑘𝑑_𝑟;𝑢_1𝑟=𝑘(𝑑²_𝑟+𝑙²_𝑛+𝑙_𝑛);𝑢_2𝑟=𝑘(𝑑²_𝑟+𝑙²_𝑛−𝑙_𝑛).

Collinear configuration:
𝑅_𝑐𝑟𝜂=−𝑣8𝜋cos_0𝑟−2𝐶_𝑖2𝑣_0𝑟+𝐶_𝑖𝑣_2𝑟+𝐶_𝑖𝑣_1𝑟−ln𝑣_3𝑟+𝜂𝑣8𝜋sin_0𝑟2𝑆_𝑖2𝑣_0𝑟−𝑆_𝑖𝑣_2𝑟−𝑆_𝑖𝑣_1𝑟,𝑋_𝑐𝑟𝜂=−𝑣8𝜋cos_0𝑟2𝑆_𝑖2𝑣_0𝑟−𝑆_𝑖𝑣_2𝑟−𝑆_𝑖𝑣_1𝑟+𝜂𝑣8𝜋sin_0𝑟2𝐶_𝑖2𝑣_0𝑟−𝐶_𝑖𝑣_2𝑟−𝐶_𝑖𝑣_1𝑟−ln𝑣_3𝑟, (A.6)(A.7) where 𝑣_0𝑟=𝑘(𝑑+𝑙_𝑛); 𝑣_1𝑟=2𝑘(𝑑+2𝑙_𝑛); 𝑣_2𝑟=2𝑘𝑑; 𝑣_3𝑟=(𝑑²+2𝑑𝑙_𝑛)/(𝑑+𝑙_𝑛)².

Parallel-in-echelon configuration:
𝑅_𝑝𝑟𝜂=−𝑤8𝜋cos_0𝑟×−2𝐶_𝑖𝑤_1𝑟−2𝐶_𝑖𝑤^′_1𝑟+𝐶_𝑖𝑤_2𝑟+𝐶_𝑖𝑤^′_2𝑟+𝐶_𝑖𝑤_3𝑟+𝐶_𝑖𝑤^′_3𝑟+𝜂𝑤8𝜋sin_0𝑟2𝑆_𝑖𝑤_1𝑟−2𝑆_𝑖𝑤^′_1𝑟−𝑆_𝑖𝑤_2𝑟+𝑆_𝑖𝑤^′_2𝑟−𝑆_𝑖𝑤_3𝑟+𝑆_𝑖𝑤^′_3𝑟,𝑋_𝑝𝑟𝜂=−𝑤8𝜋cos_0𝑟2𝑆_𝑖𝑤_1𝑟+2𝑆_𝑖𝑤^′_1𝑟−𝑆_𝑖𝑤_2𝑟−𝑆_𝑖𝑤^′_2𝑟−𝑆_𝑖𝑤_3𝑟−𝑆_𝑖𝑤^′_3𝑟+𝜂𝑤8𝜋sin_0𝑟2𝐶_𝑖𝑤_1𝑟−2𝐶_𝑖𝑤^′_1𝑟−𝐶_𝑖𝑤_2𝑟+𝐶_𝑖𝑤^′_2𝑟−𝐶_𝑖𝑤_3𝑟+𝐶_𝑖𝑤^′_3𝑟, (A.8)(A.9) where 𝑤_0𝑟=𝑘ℎ_𝑟;𝑤_1𝑟=𝑘(𝑑²+ℎ²_𝑟+ℎ_𝑟);𝑤^′_1𝑟=𝑘(𝑑²+ℎ²_𝑟−ℎ_𝑟),𝑤_2𝑟=𝑘(𝑑²+(ℎ_𝑟−𝑙_𝑛)²+(ℎ_𝑟−𝑙_𝑛));𝑤^′_2𝑟=𝑘(𝑑²+(ℎ_𝑟−𝑙_𝑛)²−(ℎ_𝑟−𝑙_𝑛)),𝑤_3𝑟=𝑘(𝑑²+(ℎ_𝑟+𝑙_𝑛)²+(ℎ_𝑟+𝑙_𝑛));𝑤^′_3𝑟=𝑘(𝑑²+(ℎ_𝑟+𝑙_𝑛)²−(ℎ_𝑟+𝑙_𝑛)).

The equations (A.1) through (A.9) are valid for the dipoles of equal length l with identical wire radius a. Thus 𝑙_𝑛=𝑙 and 𝑎_𝑛=𝑎 for all 𝑛 elements of the array. However 𝑑_𝑟 represents the relative distance between the pair of elements considered, ℎ_𝑟 represents the offset distance between the elements under consideration, and 𝑟=(𝑥,𝑦) indicates the coordinate pair representation of these elements.

B. Coupling and Transmission Coefficients: Formulation

The coupling coefficient for a coupler is the ratio of amount of power coupled into the coupling port to the amount of power at its input or receiving port. Thus we have couplingcoefficient=PoweratthecouplingportPoweratitsreceivingport. (B.1) For each coupler in the network, the power at its coupling port will be proportional to the product of current and the impedance at its terminals. However the power at the receiving port of the 𝑛th coupler will be the total received power minus the power received due to the elements 1 to 𝑛−1 of the array.

The expression for the coupling coefficient of first element is given by 𝑐₁=𝑃₁𝑃_𝑟𝑥. (B.2) Similarly, for other elements we have 𝑐₂=𝑃₂𝑃_𝑟𝑥−𝑃₁=𝑃₂𝑃_𝑟𝑥−∑¹_𝑞=1𝑃_𝑞couplingcoefficientofthe𝑠𝑒𝑐𝑜𝑛𝑑element𝑐₃=𝑃₃𝑃_𝑟𝑥−𝑃₁+𝑃₂=𝑃₃𝑃_𝑟𝑥−∑²_𝑞=1𝑃_𝑞couplingcoefficientofthe𝑡ℎ𝑖𝑟𝑑element. (B.3)(B.4) In general, 𝑐_𝑛=𝑃_𝑛𝑃_𝑟𝑥−𝑃₁+𝑃₂+⋯+𝑃_𝑛−1=𝑃_𝑛𝑃_𝑟𝑥−∑^𝑛−1_𝑞=1𝑃_𝑞couplingcoefficientofthe𝑛thelement. (B.5) For a lossless network shown in Figure 25, the total power fed into the circuit will be equal to the sum of the powers delivered to the individual antenna elements and to the terminating load. If the same array acts as a receiver, then the total power received should be equal to the sum of the power received by the individual antenna elements (assuming that zero power was delivered to the load while transmission). Thus, we have 𝑃_𝑟𝑥=𝑃₁+𝑃₂+⋯+𝑃_𝑛+⋯+𝑃_𝑁−1+𝑃_𝑁=^𝑁_𝑝=1𝑃_𝑝. (B.6) Expressing power in terms of current and impedance, one gets 𝑃_𝑟𝑥=𝑧₁𝑖²₁+𝑧₂𝑖²₂+⋯+𝑧_𝑛𝑖²_𝑛+⋯+𝑧_𝑁−1𝑖²_𝑁−1+𝑧_𝑁𝑖²_𝑁=^𝑁_𝑝=1𝑧_𝑝𝑖²_𝑝, (B.7) where 𝑧_𝑝 is the net impedance of the 𝑝th antenna element,   𝑖_𝑝 is the current at the terminals of 𝑝th antenna element, and  𝑃_𝑟𝑥 is the total received power.

Figure 25 

Coupling and transmission coefficients of the couplers.

Here 𝑧_𝑝 is given by 𝑧_𝑝=∑^𝑁_𝑞=1𝑧_𝑝,𝑞(𝐼_𝑞/𝐼_𝑝) [5] with p and q being the row and column numbers of the impedance matrix [26] 𝑧_𝑝,𝑞=𝑧_1,1𝑧_1,2⋯𝑧_1,𝑁𝑧_2,1𝑧_2,2⋯𝑧_2,𝑁𝑧⋮⋮⋱⋮_𝑁,1𝑧_𝑁,2⋯𝑧_𝑁,𝑁. (B.8) Thus the coupling coefficient of couplers can be written as 𝑐_𝑛=𝑧_𝑛𝑖²_𝑛∑^𝑁_𝑝=1𝑧_𝑝𝑖²_𝑝−∑^𝑛−1_𝑞=1𝑧_𝑞𝑖²_𝑞, (B.9) and the transmission coefficient is given by 𝑡²_𝑐𝑛=1−𝑐²_𝑛. (B.10)

List of Symbols

𝛼∶	Interelement space delay along 𝑥direction
𝛼𝑠:	Interelement phase to scan antenna beam along 𝑥 direction
Γ𝑟𝑛:	Total reflected signal returning to aperture element 𝑛
𝜂:	Impedance of medium around antenna
𝜂𝑜:	Impedance of free space
(𝜃,𝜙):	Direction of incident signal
(𝜃𝑠,𝜙𝑠):	Beam scan angle
𝜆:	Wavelength
𝜎(𝜃,𝜙):	Overall RCS of phased array
𝜎𝑐(𝜃,𝜙):	RCS due to scattering from coupling port of the couplers
𝜎𝑝(𝜃,𝜙):	RCS due to scattering from phase shifters
𝜎𝑟(𝜃,𝜙):	RCS due to scattering from radiators
𝜎𝑠(𝜃,𝜙):	RCS due to scattering beyond the coupling port of couplers
𝑎:	Radius of dipole wire
𝑐𝑛:	Coupling coefficient of 𝑛th coupler
𝐶:	Euler’s constant
𝐶𝑖(𝑥):	Cosine integral
𝑑:	Interelement spacing
→𝑑𝑛:	Distance vector
𝑑𝑟:	Relative distance between element pair
→𝐸𝑖(𝜃,𝜙):	Incident field
→𝐸𝑟𝑐𝑛(𝜃,𝜙):	Scattered field due to 𝑛th coupler
→𝐸𝑟𝑛(𝜃,𝜙):	Total scattered field
→𝐸𝑟𝑝𝑛(𝜃,𝜙):	Scattered field due to 𝑛th phase shifter
→𝐸𝑟𝑟𝑛(𝜃,𝜙):	Scattered field due to 𝑛th dipole
→𝐸𝑟𝑠𝑛(𝜃,𝜙):	Total scattered field due to the scattering beyond the coupling port of couplers
→𝐸𝑠:	Total scattered field
→𝐸𝑠𝑛:	Total scattered field due to single element
𝐹:	Constant
→ℎ:	Effective height of a 𝑥-polarized antenna element
𝑖𝑛:	Current at the terminals of 𝑛th antenna element
𝐼(0):	Current at the feed terminals of antenna
𝐼𝑛:	Current at the feed terminals of 𝑛th antenna element
𝑘:	Wave number
→𝑘:	Wave vector
𝑙:	Dipole length
𝐿𝑛:	Length of delay lines
𝑁:	Number of elements in the antenna array
𝑃𝑛:	Power received at the 𝑛th coupler
𝑃𝑟𝑥:	Total received power
𝑟𝑐𝑛:	Reflection coefficient at the coupling port of the 𝑛th coupler
𝑟𝑖𝑛:	Reflection coefficient at the receive port of the array
𝑟𝑙𝑛:	Reflection coefficient of the load terminating isolation port of 𝑛th coupler
𝑟𝑝𝑛:	Reflection coefficient of 𝑛th phase  shifter
𝑟𝑟𝑛:	Reflection coefficient of 𝑛th radiating element
𝑅:	Distance between the target and the observation point
𝑅𝑎:	Antenna resistance
𝑅𝑐𝑟:	Radiation resistance due to collinear configuration
𝑅𝑑:	Loss resistance of antenna
𝑅𝑝𝑟:	Radiation resistance due to parallel-in-echelon configuration
𝑅𝑟:	Radiation resistance
𝑅𝑠𝑟:	Radiation resistance due to side-by-side configuration
𝑅self𝑛:	Self-coupling radiation resistance of 𝑛th dipole
𝑆𝑖(𝑥):	Sine integral
𝑡𝑐𝑛:	Transmission coefficient of 𝑛th coupler
𝑡𝑝𝑛:	Transmission coefficient of 𝑛th phase shifter
𝑡𝑟𝑛:	Transmission coefficient of 𝑛th-radiating element
𝑋𝑎:	Antenna reactance
𝑋𝑐𝑟:	Radiation reactance due to collinear configuration
𝑋𝑝𝑟:	Radiation reactance due to parallel-in-echelon configuration
𝑋𝑠𝑟:	Radiation reactance due to side-by-side configuration
𝑋self𝑛:	Radiation reactance of 𝑛th antenna element due to self-coupling
𝑧𝑎𝑥,𝑦:	Mutual impedance between 𝑥th and𝑦th dipoles
𝑧𝑝:	Net impedance at 𝑝th antenna element
𝑍12:	Impedance at the input port of the first coupler in the array
𝑍0:	Characteristic impedance
𝑍𝑎:	Antenna impedance
𝑍𝑎𝑛:	Terminal impedance of 𝑛th antenna element
𝑍𝑝𝑛:	Terminal impedance of 𝑛th phase shifter.

Acknowledgment

This work was funded as a sponsored project at CSIR-National Aerospace Laboratories, Bangalore, India by Aeronautics Research and Development Board (AR&DB), New Delhi, India.