Performance analysis of the mutual coupling effect on Phased Array Feeds

1.   INTRODUCTION

A PAF is a type of array antenna placed in the focal plane of a reflector antenna as a feed array. Combined with a beamforming network at the rear stage of the array, it can form multiple continuous beams for expanding the field of view of a radio telescope^[1]. The name, PAF, comes from its ability to achieve amplitude and phase control for each array element, which involves the placement of phased array antennas instead of a traditional single beam feed at the focal point of a radio telescope^[2]. Fig. 1 shows the principle prototype of a 20 cm band (0.7–1.8 GHz)^[3] 110-element dual polarized PAF array, developed by the Microwave Technology Laboratory of the Xinjiang Astronomical Observatory of the Chinese Academy of Sciences, for the QiTai 110 m aperture radio Telescope (QTT)^[4].

Figure  1.  Photograph of the Vivaldi PAF array.

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As a type of array antenna, a PAF cannot avoid the impact of the mutual coupling effect during design. Because the array elements can generate induced currents within other antenna elements besides the excited element through the interaction of electromagnetic fields, termed the mutual coupling effect. Generally, the smaller the spacing between elements, the stronger this coupling effect will be. This leads to changes in the radiation characteristics of the array elements in the array environment, further affecting the performance characteristics of the PAF, such as gain, beamwidth, and input reflection coefficient^[5]. The coupling effect is also related to polarization, which is particularly important for the design of a PAF used for polarization observations. However, this effect is not necessarily entirely detrimental.

2.   ANALYSIS METHOD FOR THE MUTUAL COUPLING EFFECT OF A PAF

2.1   Coupling Path

In transmission array applications, the excitation signals of each antenna element can be set to different amplitudes and phases, so that the radiated beam can be directed in the desired direction with the intended wave shape. In ideal conditions, the total radiation field is the sum of the radiation fields from each array element. However, when coupling exists, the interaction between the elements will be superimposed with the total radiation field, making the beam radiated by the array less than ideal.

For a receiving array like a PAF, the electromagnetic waves received by the array will generate induced currents on each antenna element, generating port voltages. Observers expect that the signal received by the PAF array will depend entirely on the incident signal from the external radio source, but while the PAF array receives the radio frequency signal, each antenna element causes scattering during reception, and its secondary radiation field is also received by other antenna elements. The actual signal received by the PAF array is the sum of the external radio source signal and the secondary radiation field^[6].

Owing to the different coupling behaviors of array antennas in transmission and reception modes, it is necessary to first understand the coupling path when placing the PAF array at the focal plane position of the reflector antenna^[7]. Fig. 2 shows a schematic diagram of the coupling path of an n-element PAF located at the main focal position of the reflector.

Figure  2.  Coupling path of a PAF. Red numbers represent different types of signals throughout the transmission path, while LNA refers to low noise amplifiers.

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Owing to the large number of elements in a PAF, Fig. 2 shows only the coupling paths between two elements, labeled a and b, with the reflector, mainly presenting the signals received by a single element a. Firstly, a plane wave radio frequency signal 1 from the far-field region is incident on an antenna element a via the reflector, generating an induced current. The induced current on this element re-radiates a signal 3 back to the reflector, and the remaining signal 2 is transmitted to the low-noise amplifier at the rear stage of the element. A portion of this re-radiated signal 3, labeled as signal 4, is received by a second element b, where the radiation field generates an additional induced current. In turn, the re-radiative field of this second element generates a further signal 5, which will be received again by the first element a. Simultaneously, if the port of the first element a does not match, a portion of the received signal 2 is reflected back (shown as signal 6), a portion of which can radiate back to the reflector, shown as signal 7. These re-radiated and reflected signals 3, 7 superpose, causing a secondary reflection 8, which is then received again by the first element a. This process cycles infinitely until all energy is consumed. Consequently, the total signal received by the first element a is a composite, containing the incident signal 1, re-radiation from the neighboring element 5, a secondary reflection 8, and an additional contribution from a high-order interaction between the two elements a, b. When many elements are present in the PAF array, mutual coupling causes the overall re-radiation field and higher-order interaction between all elements to contribute to the total received signal at each element.

2.2   Analysis of the Mutual Coupling Effect for a PAF Array

A PAF is placed at the focal plane of the reflector antenna. To improve simulation speed and quickly verify array performance, we select a circularly polarized spiral antenna (line antenna) with a relatively simple structure as the array element^[8], set the operating frequency to 1.25 GHz, and arrange a rectangular PAF array in the focal field area of the reflector. The reflector antenna has a diameter of 25 m and a focal diameter ratio of 0.3. Fig. 3 shows the distribution of the normalized electric field in the focal field at 1.25 GHz and a 25 m reflector, with the focal field analysis area set to 0.84 m× 0.84 m. It is generally believed that when the PAF array covers the edge taper area of −10 dB after the focal field normalization, the beam can be basically restored^[9].

Figure  3.  25 m reflector and focal field distribution.

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Considering the relationship between the mutual coupling effect and element spacing, we adjust the element spacing of a 7 × 7 PAF array and analyze the performance of PAF under different spacing mutual coupling effects. In the focal field of the 25 m reflector, shown in Fig. 3, the circular area illuminated by the −10 dB edge taper at 1.25 GHz has a diameter of approximately 0.25 m. When the 7 × 7 rectangular array is arranged with a spacing of 0.1λ, the size of the PAF array is 0.6λ (approximately 0.144 m), and the entire array with this spacing cannot fully cover the taper area of 0.25 m, meaning that the array cannot restore a complete beam. Similarly, when the array is arranged with 0.2λ spacing, the entire array size is 1.2λ (approximately 0.288 m), which can cover an edge taper area of −10 dB, so it can restore a complete beam. With 0.8λ spacing, the entire array size is 4.8λ (approximately 1.152 m). Although coverage can be achieved, the large spacing between elements means that there is only one central element in the 0.25 m taper area, and all other elements are outside this area, which does not meet the requirements of beamforming. Consequently, we select the spacing to be between 0.2λ and 0.7λ, and analyze the PAF performance under relevant differently spaced mutual coupling effects. Fig. 4 shows the numbering of each element in a 7 × 7 rectangular array and the focal field of the 25 m reflector at different spacings, with an edge taper area of −10 dB. For example, at 0.6λ and 0.7λ spacing, elements 18, 24, 25, 26, and 32 are roughly within this area, and only these five elements are needed for beamforming.

Figure  4.  Element arrangement and corresponding −10 dB edge taper area of the focal field at different spacings.

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3.   PERFORMANCE ANALYSIS OF THE MUTUAL COUPLING EFFECT IN PAF ARRAYS

3.1   The S-parameter

As an important parameter in microwave transmission, S-parameter is used to describe the frequency domain characteristics of the transmission channel. Here, we simulate the S-parameter of a PAF array with 0.2λ–0.7λ spacing under uniform excitation across the entire array. Firstly, we analyze the coupling between the central array element 25 and its neighboring element 24 (as shown in Fig. 4). The result for parameter S_25,24 is shown in Fig. 5A. Within the set bandwidth of 1–1.5 GHz, as the spacing decreases, the coupling gradually increases, so that S_25,24 with a 0.2λ spacing reaches −10 dB. Considering that 0.2λ shows the strongest coupling with minimum spacing, we analyze the coupling between different elements at this spacing. Fig. 5B shows the coupling between the center element 25 and the adjacent elements 24, 23, and 22 in the horizontal direction. The coupling gradually increases with the decreasing distance between the two elements.

Figure  5.  Simulation results of coupling between elements 25 and 24 at different spacing, as well as elements 24, 23, and 22 at 0.2λ spacing.

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We continue to analyze the return loss for the central element of the array. The simulation result for S_{25, 25} is given in Fig. 6A, showing that smaller spacings (0.2λ, 0.3λ, and 0.4λ) cause an enhanced coupling effect, and the return loss at the port is better than with larger element spacings (0.5λ, 0.6λ, and 0.7λ). In particular, a spacing of 0.3λ with four consecutive passbands below −10 dB across the entire bandwidth of 1–1.5 GHz, shows the best return loss performance. This is also in line with Mink's theory, which states that strong coupling dipole antennas can achieve broadband performance^[10], showing the beneficial aspect of the coupling effect on PAF performance. Additionally, large element spacings are only 1 to 2 passbands below −10 dB. Compared with the 0.3λ spacing, S_25,25 with a spacing of 0.2λ shows poorer performance, indicating that this is not entirely dependent on reducing the spacing between elements to improve return loss. Considering that a spacing of 0.3λ shows optimal return loss, Fig. 6B shows the simulation results of the return loss for elements 25, 24, 23, and 22 at this spacing. Although the contours of each curve are relatively close, the return loss of central element 25 is still better than the other three elements, owing to the highest number of peripheral elements and the strongest coupling effect.

Figure  6.  Simulation results of return loss for element 25 at different spacings, as well as elements 22, 23, 24, and 25 at 0.3λ spacings.

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3.2   Input Impedance

To understand the concept of input impedance, we first simplify the n-element PAF array into a 2-element array^[11]. The relationship between the port voltage and port current of the two elements is given by

where Z₁₁ is the self-impedance of Port 1 for element 1, caused by the current at Port 1 when Port 2 of element 2 is open; Z₁₂ is the mutual impedance of Port 1, caused by the current at Port 2 when Port 1 is open; Z₂₁ is the mutual impedance of Port 2, caused by the current at Port 1 when Port 2 is open; and Z₂₂ is the self-impedance of Port 2, caused by the current at Port 2 when Port 1 is open.

When the antenna element is in an independent state, its self-impedance is essentially the input impedance. Mutual impedance Z₁₂ is the ratio of the port voltage of element 1 under open circuit conditions to the port current of element 2 when excited. Similarly, mutual impedance Z₂₁ is the ratio of the port voltage of element 2 under open circuit conditions to the port current of element 1 when excited. At this point, the input impedances of the two element ports are given by

where Z_1d and Z_2d are the input impedances of elements 1 and 2. The actual input impedance of each array element port under the coupling effect depends on the ratio of the self-impedance, mutual impedance, and port current of the elements.

Here, we simulate the input impedance of elements 1, 9, 17, and 25 (diagonal position) at 0.2λ–0.7λ spacing under uniform excitation of the entire PAF array. Fig. 7A shows the real part of the input impedance, which is the resistance distribution, and Fig. 7B shows the imaginary part of the input impedance, which is the reactance distribution. There is still a significant difference between the actual input impedance of each port and the reference impedance of 50 ohms. Therefore, when a PAF is designed to match the antenna elements in the array, the actual input impedance of each antenna element, taking coupling effects into consideration, must be matched.

Figure  7.  Simulation results of input impedance with resistance and reactance for elements 1, 9, 17, and 25 at different spacings.

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3.3   Beamforming Pattern

3.3.1   Axial pattern

We continue by simulating the axial pattern of a PAF with a spacing of 0.2λ–0.7λ, combined with a reflector with an aperture of 25 m and a focal diameter ratio of 0.3 under uniform excitation of the entire array. The results are shown in Fig. 8. To quantify the patterns of each spacing, we summarize the performance results obtained from the simulation in Table 1. As the element spacing gradually decreases, the PAF gain increases, the sidelobe level decreases, and the beam width gradually narrows.

Figure  8.  Simulation results of beamforming patterns using PAFs combined with reflectors, under uniform excitation across the entire array, at different spacings.

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Table  1.  Pattern performance of PAFs under uniform excitation at different spacings

Element spacing	Gain /dB	Beam direction/(°)	Beam width/(°)	Sidelobe level/dB
0.2λ	40.5	0	1.54	−33.65
0.3λ	41.75	0	0.74	−32.81
0.4λ	34.61	0	3.10	−22.90
0.5λ	33.26	0	1.99	−17.86
0.6λ	27.94	0	2.25	−10.54
0.7λ	26.57	0	6.82	−7.36

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A PAF does not use all elements to synthesize a single beam in an analog beamforming network. As shown in Fig. 4, we simulate the PAF axial pattern with spacing of 0.2λ–0.7λ, combined with a reflector which only adopts uniform excitation of the array elements in the edge taper area of the focal field. The simulation results are shown in Fig. 9, and the performance results are summarized in Table 2. Comparing Tables 1 and 2, it can be seen that after uniform excitation by a focal field, the reduction of elements participating in beam forming leads to an overall increase in the gain, as shown in Table 2, with a decrease in sidelobe level, and a slight narrowing of the beam width.

Figure  9.  Simulation results of beamforming patterns using PAFs combined with reflectors, under uniform excitation by the focal field, at different spacings.

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Table  2.  Pattern performance of PAFs under uniform excitation by a focal field at different spacings

Element spacing	Gain /dB	Beam direction/(°)	Beam width/(°)	Sidelobe level/dB
0.2λ	41.71	0	1.04	−37.99
0.3λ	37.56	0	2.08	−33.42
0.4λ	40.34	0	1.41	−36.26
0.5λ	43.46	0	0.93	−20.53
0.6λ	41.56	0	0.81	−34.87
0.7λ	37.29	0	1.46	−25.00

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Considering that conjugate field matching excitation is a more practical method for PAF beamforming, we simulate a PAF axial pattern with spacing of 0.2λ–0.7λ combined with a reflector, adopting conjugate field matching excitation of the array elements in the edge taper area of the focal field. The simulation results are shown in Fig. 10, and the performance results are summarized in Table 3.

Figure  10.  Simulation results of beamforming patterns using PAFs combined with reflectors, under conjugate field matching excitation by the focal field, at different spacings.

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Table  3.  Pattern performance of PAFs under conjugate field excitation at different spacings

Element spacing	Gain /dB	Beam direction/(°)	Beam width/(°)	Sidelobe level /dB
0.2λ	39.99	0	1.01	−38.67
0.3λ	37.05	0	0.59	−3.99
0.4λ	37.46	0	1.67	−34.71
0.5λ	39.92	−1	0.91	−17.07
0.6λ	41.01	0	1.02	−32.85
0.7λ	44.91	0	0.49	−33.27

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Comparison of Tables 2 and 3 shows that after excitation according to the conjugate matching of the focal field, the gain at most intervals remains at a relatively balanced level. The gain at 0.7λ spacing also increases, which is related to the different amplitude excitations carried out according to the focal field. The beam direction at 0.5λ spacing has a slight deviation and the sidelobe level also deteriorates to a certain extent at 0.3λ spacing, which is related to different phase excitations carried out according to the focal field. The overall beam width narrows further compared with the results in Table 2. This is particularly notable at 0.3λ spacing, achieving a beam width of 0.59°, which is the closest match with the theoretical prediction (70λ/D=0.67°).

3.3.2   Cross-polarization

We also combine the axial patterns of the array and PAF under different excitation conditions, at spacings of 0.2–0.7λ, to decompose the main polarization and cross-polarization. The final cross-polarization results of PAFs at different spacings are shown in Fig. 11.

Figure  11.  Simulation results of cross-polarization using PAFs combined with reflectors, with different excitation spacings.

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This PAF model uses circularly polarized spiral antennas as array elements, and Fig. 11 shows that optimal cross polarization does not exceed 16 dB. However, the cross polarization results of a PAF under fully uniform excitation show that compared with larger element spacings, a smaller element spacing does indeed improve the cross polarization of the PAF to a certain extent after enhancing the coupling effect.

3.3.3   Antenna efficiency

As an index of the overall performance of PAFs, antenna efficiency is the most important parameter to consider. According to the principle of reciprocity, when conducting PAF pattern simulation, transmission mode is generally used^[12], giving a radiation efficiency equivalent to the antenna efficiency. Fig. 12 shows the radiation efficiency of the array and PAF at different spacings.

Figure  12.  Simulation results of radiation efficiency using PAFs combined with reflectors under different excitation and spacings.

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Optimal radiation efficiency under three types of excitations, shown in Fig. 12, is consistently found at 0.5λ spacing, and the radiation efficiency can reach 76.3% for PAFs under fully uniform excitation. Meanwhile, the lowest radiation efficiency is found at a spacing of 0.2λ, when a PAF is excited according to conjugate field matching, giving a radiation efficiency of only 6%. Although the radiation efficiency of a PAF with 0.3λ spacing can reach 50%, the overall radiation efficiency of small element spacing is lower than that of larger spacings.

4.   CONCLUSION

In this article, we present a model of a PAF with a 7 × 7 rectangular arrangement of spiral antennas under a 25 m reflector. Based on the distribution of the focal field at a working frequency of 1.25 GHz, the coupling, return loss, input impedance, cross polarization, and radiation efficiency of the PAF array at different element spacings, and their mutual coupling effects, are analyzed by setting the spacing to between 0.2λ to 0.7λ. Based on the current PAF array design, the mutual coupling effect caused by variation of element spacing has a significant impact on the input impedance of each port of the array, and also causes varying degrees of change in the forming pattern, ultimately leading to a decrease in the antenna efficiency of the entire PAF. However, the strengthening of the coupling effect also slightly expands the operating bandwidth of the overall PAF array, improving cross polarization. Consequently, not all of the effects generated are adverse. Through appropriate element selection and array design, combined with various simulation optimizations in the early design stages, the mutual coupling effect has the potential to be reasonably utilized in PAF arrays. Of course, this also requires performance trade-offs between, but low antenna efficiency is a core drawback of the mutual coupling effect in creating PAF arrays. Designs of PAF arrays do not necessarily have to adopt traditional methods that weaken coupling, but can combine specific array element, arrangement, and performance requirements to properly apply coupling effects, improving array performance.