Calibration Method for Active Element Patterns corrupted by the Parasitic Effects of Feeding Networks

Article information

J. Electromagn. Eng. Sci. 2025;25(2):167-174

Publication date (electronic) : 2025 March 31

doi : https://doi.org/10.26866/jees.2025.2.r.288

Seongjung Kim

Department of System Large Scale Integration, Samsung Electronics, Hwaseong, South Korea

^*Corresponding Author: Seongjung Kim (e-mail: sjkim@ael.snu.ac.kr)

Received 2024 March 6; Revised 2024 May 9; Accepted 2024 July 7.

Abstract

This paper presents a novel method for calibrating the active element patterns (AEPs) of array antennas distorted by feeding networks. These distortions include signal loss, multiple reflection, and parasitic radiation. This methodology is particularly relevant for millimeter-wave frequencies, at which feeding networks prominently impact antenna performance. The proposed approach utilizes the inherent consistency of vertical launch connectors, overcoming the issue of distortion and focusing on enhancing the accuracy and reliability of antenna performance measurements. Using an innovative calibration process, we demonstrate the ability to effectively correct distorted AEPs while simultaneously ensuring the fidelity of the antenna’s radiation pattern and reflection characteristics. The method’s effectiveness is validated through comparison with ideal simulated AEPs, showing significant improvement in radiation pattern characteristics, particularly at higher frequencies.

Keywords: Active Element Pattern; Calibration; Connector; Impedance Matching; mmWave; Radiation Leakage; Realized Array Gain; RF Feeding

I. Introduction

Millimeter-wave (mmWave) frequencies offer the advantage of higher bandwidths but come with the trade-off of increased propagation loss and sensitivity to environmental factors [1–3]. This unique interplay of benefits and challenges at mmWave frequencies necessitates innovative and efficient antenna designs to harness their full potential.

A critical aspect of mmWave antenna design is the feeding mechanism—a component that significantly influences overall antenna performance—which is fraught with challenges. One of the primary concerns in feeding mmWave antennas is the considerable ohmic loss occurring at feeding lines or connectors, which is particularly problematic because it can significantly diminish antenna efficiency and effectiveness. Furthermore, the use of connectors in the feeding process can introduce undesirable signal reflections. These reflections not only reduce the antenna’s efficiency but also complicate the impedance matching process, which is a crucial factor for securing optimal antenna performance [4]. Another issue is the parasitic radiation emanating from large reflections or diffractions at the feeding structure [4], which can distort the antenna’s radiation pattern and reflection characteristics. Such distortions can be deceptive—for instance, ohmic losses at the feed structure may appear to enhance the antenna’s impedance matching bandwidth. In other words, the reflection coefficient, defined as the reflected voltage to the incident voltage (Γ = V⁻/V⁺), could be small, since V⁻→0 due to the losses. However, this is a misleading indication because it does not truly represent the antenna’s performance capabilities [5–10]. In addition, the realized array gain pattern loss caused by the feeding structure has so far been compensated for only by transmission loss (that is, by simply scaling up the magnitude) [8, 9]. This leads to inaccuracies in the radiation pattern, which becomes more prominent with increased reflection, refraction, and loss.

With regard to the feeding method, consistency is a key concern. Although various feeding approaches, such as mini-SMP or soldering-based connectors, have been employed over time [5–7], these methods often suffer from consistency issues, primarily due to the inherent variability of the soldering process. This inconsistency may result in performance variations among different antenna units, ultimately diminishing the reliability of the obtained results. In this context, vertical launch connectors offer better consistency [8–10], but they are not without drawbacks. If the recommended printed circuit board (PCB) pattern, thickness, and material are not used, it can lead to substantial reflection or loss.

In this paper, we propose an innovative approach that utilizes the consistency offered by vertical launch connectors while also addressing their associated challenges. The focus of this study is on mmWave array antennas, in which issues related to reflection characteristics and distorted active element patterns (AEPs) are particularly pronounced. We introduce a novel calibration method that effectively corrects distorted AEPs, thus enhancing the accuracy of antenna performance measurements and the reliability of these results.

II. Calibration Process

1. Array Antenna Structure

Fig. 1(a) depicts a 2 × 2 array antenna structure operating within the 24–40 GHz range that requires calibration. Notably, this structure is adapted from [11], where its dimensions and operational principles are thoroughly detailed. Fig. 1(b) presents the backside of the array antenna, where the antenna element on the top side is connected through the feeding point originating from the discrete port illustrated in Fig. 1(c). Furthermore, the recommended PCB pattern, designed for connection to a 2.92-mm vertical launch connector operating up to 40 GHz, was employed [12]. Notably, the discontinuity pattern was intentionally designed to enhance reflection and parasitic backward radiation in the simulation environment. The discontinuity, as a simple model, mimics incomplete connections to reproduce certain situations that feature parasitic backward radiation in the feeding network. This deliberate design choice was made to demonstrate the impact of the proposed calibration method more conclusively. It was anticipated that reflection and parasitic backward radiation would primarily occur in the part depicted in Fig. 1(c), while significant line loss may occur through the bent microstrip line. The entire structure extending from the discrete port to the front of the feeding point is referred to as the feeding network, which can be characterized by an ABCD matrix or a 2 × 2 scattering matrix S^C. Notably, throughout this paper, bold characters are used to denote the matrices.

Fig. 1

(a) Front view of the array antenna configuration. (b) Back view, where the feeding network (black dashed box) is characterized as ABCD or the scattering matrix S^C. (c) Enlarged view of the blue dashed box. (d) Feeding networks terminated by 50 Ω.

2. Scattering Parameter of the Array Antenna

Fig. 2(a) presents the N-port network of the array antenna, where N is the number of antenna elements (in this case, N = 4). The measured scattering matrix, including the feeding network, can be represented as N × N S^dis. Meanwhile, the pure array antenna structure, ranging from the feeding point to the array aperture, can be modeled as the N × N impedance matrix Z_array or the N × N scattering matrix S. Furthermore, S^C can be calculated by thru-reflect-line (TRL) calibration [13], as depicted in Fig. 2(b), where T and L are the scattering parameters of the calibration kits. In addition, Z₀ is the characteristic impedance of the extended transmission line, β is the propagation constant along the transmission line, and l is the physical length of the transmission line, which can be arbitrarily chosen to be shorter than half the guided wavelength at the highest operating frequency. In this study, it was assumed that Z₀, β, and l of the fabricated kits are identical to the estimated values. This assumption was considered reasonable due to the simple configuration of the transmission line. The calculated S^C is shown in Fig. 3(a) and the simulated S^dis at the four discrete ports is presented in Fig. 3(b). As anticipated, the results show significant reflection and a notably low transmission coefficient. To correct the detrimental distortion, Z_array was determined by simple manipulation [14] using Eq. (1), as noted below:

Fig. 2

(a) N-port network of the array antenna. (b) Calibration kits for characterizing the feeding network.

Fig. 3

(a) S^C. (b) S^dis. (c) S, where two identical lines for each type are drawn—one represents the calibrated result obtained using Eq. (1), and the

(1) Zarray=(AIN-CZdis)-1(DZdis-BIN),

where I_N is the N × N identity matrix, and Z_dis is the N × N impedance matrix converted from S^dis. Finally, S was converted from Z_array, as demonstrated in Fig. 3(c). It is observed that S is significantly different from S^dis, and the calibration process seems to be imperative. Moreover, the lines of the same type exhibit excellent agreement with each other, thus validating the calibration process.

3. Active Element Pattern

AEP refers to the radiation pattern generated when one port is excited and the others are matched-terminated. Fig. 4(a) illustrates the signal flow of an array antenna on excluding the feeding networks. When port 1 is excited with a voltage of 1, it produces scattering at all ports. This can be expressed as the scattering matrix S, which corresponds to the scattering matrix shown in Fig. 2(a). Notably, the total voltage (V) at the antenna port is the summation of the incident (V⁺) and reflected voltages (V⁻). Eq. (2) represents the voltages by vector quantity (^→) for all port elements, where B is a N × N transformation matrix that maps the incident voltage vector to the total voltage vector:

Fig. 4

Signal flow in the N-port network of the array antenna when port 1 is excited: (a) without the feeding networks, (b) with the feeding networks.

(2) V→=V→+V→-=BV→+,V→+=[V1+⋮VN+],

(3) B=IN+S.

The value for matrix B can be acquired from the S of the array antenna. Furthermore, it is well known that a complex array gain pattern (G⃗_array) can be calculated by a linear summation of complex AEPs (g⃗), as described in Eq. (4) [15], where the dimensions of these patterns are (observation samples at the far field) by 1. In other words, a complex array gain pattern can be linearly transformed by the total or incident voltage vectors, as shown in Eq. (5). Here, A is a kernel determined by the geometry of the radiating aperture, mainly contributed by the top of the structure (Fig. 1(a)), enabling the transformation of the total voltage vector into a complex array gain pattern.

(4) G→array=∑n=1Ng→nVn+

(5) G→array=AV→=ABV→+

By implementing Eqs. (4) and (5), we found that AB is a matrix composed of the complex AEPs of all ports, as shown below:

(6) AB=[g→1∣⋯∣g→N]=[g1(θ=0°,φ)⋯gN(θ=0°,φ)⋮⋱⋮g1(θ=359°,φ)⋯gN(θ=359°,φ)]

Furthermore, if we sample the radiation pattern in the φ-plane with a 1° spacing, g⃗ will assume a size of 360 × 1 and AB will have a size of 360 × N.

Fig. 4(b) illustrates a scenario involving feeding networks, which introduce distortions in the scattering behavior. First, the excited incident voltage is attenuated by S21C through the feeding networks. Subsequently, the transmitted voltage wave is subjected to multiple scattering events. In other words, the first scattered voltage wave reaching another port undergoes partial reflection at that feeding network ( S22C) and is then scattered again. This process repeats infinitely. In Fig. 4(b), the total voltage vector is the voltage at the node where it passes through the feeding network, while the incident voltage vector corresponds to the voltage at the front node of the feeding network. Notably, this definition is identical to that in the case of Fig. 4(a). The voltage vectors in Fig. 4(b) can be expressed as Eq. (7):

(7) V→=CV→+,

(8) C=[c→1⋯c→N].

Here, C is an N × N transformation matrix that maps the incident voltage vector to the total voltage vector, and c⃗₁ is the total voltage vector when port 1 is excited by V⁺ = 1.

(9) c→1=S21C[1+S11(1+S22C)+(1+S22C)S22C∑n=1NS1nSn1+(1+S22C)S22C2∑nj=1NSn1SjnS1j+⋯⋮SN1(1+S22C)+(1+S22C)S22C∑n=1NSNnSn1+(1+S22C)S22C2∑nj=1NSn1SjnSNj+⋯]]=S21C{[10⋮0]+(1+S22C)([S11S21⋮SN1]+S22CS[S11S21⋮SN1]+(S22CS)2[S11S21⋮SN1]+⋯)}.

Eq. (9) represents c⃗₁ by accounting for the multiple scatterings illustrated in Fig. 4(b).

Here, the S21C outside the brackets indicates that the first excited voltage at port 1 is attenuated. The first term of the first row, which is 1, indicates that only port 1 is excited, and there are no incident voltages to the feeding networks from the other ports. Whenever a voltage wave travels to another port, it experiences S22C due to reflection at the feeding networks, undergoing repetitive multiplication until it dissipates completely. Ultimately, once c⃗₁ is decomposed into column vectors, as shown in Eq. (9), a clear rule can be identified. To generalize C for all port excitations, it can be expressed as Eq. (10):

(10) C=S21C{IN+(1+S22C)(IN+S22CS+(S22CS)2+⋯)S}=S21C{IN+(1+S22C)(IN-S22CS)-1S}.

For the case depicted in Fig. 4(b), the complex array gain pattern and the complex AEP matrix can be expressed as Eqs. (11) and (12), respectively:

(11) G→arraydis-ACV→+,

(12) AC=[g→1dis⋯∣g→Ndis].

As mentioned before, A is a kernel determined by the geometry of the radiating aperture, which is mainly contributed by the top side of the structure (Fig. 1(a)). Since the feeding networks were positioned at the back of the array antenna structure (Fig. 1(b)), they did not interfere with the broadside radiation. Therefore, formulating the same A in Eq. (11) as identical to that in Eq. (5) is a reasonable assumption. Ultimately, Eq. (12) was obtained by measuring the array structure (Fig. 1(b)) in the case of negligible parasitic backward radiation. The goal of this study is to recover the ideal complex AEP (Eq. 6) from the measured distorted radiation pattern (Eq. 12). This was achieved using the following equations:

(13) [g→1∣⋯∣g→N]=[g→1dis∣⋯∣g→Ndis]C-1B,

(14) G→array=(g→1dis∣⋯∣g→Ndis]C-1B)V→+.

Once the distorted complex AEPs of the array antenna (AC) were measured, the complex AEPs of the pure array antenna (AB) were obtained using Eqs. (3), (10), and (13).

4. Parasitic Backward Radiation

In Section II-3, we did not account for any parasitic radiation from the feeding network. Furthermore, in Eq. (11), we assumed that the complex array gain pattern is primarily determined by kernel A, considering the array antenna’s geometry. However, if parasitic backward radiation significantly contributes to the total radiation pattern, Eqs. (11) and (12) should be modified into Eqs. (15) and (16), respectively, where Aback is a kernel determined by the geometry of the feeding network. Fig. 1(d) shows the feeding network terminated by 50 Ω resistors, which cannot excite the four antenna elements but can excite the discontinuities. Eq. (17) accounts for the pure active backward radiation patterns of the feeding network obtained by measuring the structure in Fig. 1(d). Finally, the modified calibrated AEP (Eq. 18) was obtained if Eq. (16) was subtracted by the contribution of Eq. (17).

(15) G→arraydis=(A+Aback)CV→+,

(16) (A+Aback)C=[g→1dis∣⋯∣g→Ndis],

(17) AbackC=[g→1back∣⋯∣g→Nback],

(18) [g→1∣⋯∣g→N]=([g→1dis∣⋯∣g→Ndis]-[g→1back∣⋯∣g→Nback])C-1B.

In this context, it should be noted that Eqs. (15) and (16) will be valid only when the feeding network does not interfere with the radiation pattern emanating from the array antenna aperture and, simultaneously, the parasitic backward radiation is not influenced by the array antenna aperture, meaning that they are independent of each other.

The calibration procedure is summarized in Fig. 5. First, using a vector network analyzer (VNA), the two-port device under test (DUT), consisting of one of the array antennas and two calibration kits, was measured, and the scattering parameters (S^dis and S^C) were obtained. Next, the parameters were converted into Z_dis and ABCD. Eq. (1) was calculated to obtain Z_array, which was then converted into S. Furthermore, B (Eq. 3) and C (Eq. 10), which are necessary for calibrating AEPs, were calculated. Second, the AEPs of the array antenna were measured in the anechoic chamber. The measured AEP was found to be equal to that of Eq. (16), which was distorted by the feeding network. Third, if the parasitic backward radiation was not significant, A^back was neglected, and Eq. (13) was calculated. However, if the feeding network showed potential for strong parasitic radiation at the feeding connection, as seen in Fig. 1(d), the AEP of the array terminated by 50 Ω (= Eq. 17) was measured and calculated using Eq. (18).

Fig. 5

Flow diagram of the calibration procedure.

Fig. 6 presents the AEPs (|g⃗|²) for four cases: without the feeding networks (direct excitation), without calibration (with the feeding networks and no calibration process), calibration by Eq. (13) (not accounting for backward radiation), and calibration by Eq. (18) (accounting for backward radiation). As expected, the uncalibrated AEP displayed significant degradation. Furthermore, the AEP derived using Eq. (13) exhibited a notably different back lobe pattern compared to the first case, since it neglects parasitic backward radiation even though it definitely exists. It is particularly dominant at high frequencies owing to the resonance of the discontinuity behaving like a dipole at such frequencies. Finally, the calibrated result obtained when accounting for backward radiation demonstrated remarkable agreement with the first (ideal) case.

Fig. 6

AEP in the E-plane (a) and the H-plane (b).

For port 1 in Fig. 6, the root mean square (RMS) of deviation from the AEP (Eq. 6; without the feeding network) was calculated at 25, 30, 35, and 40 GHz using Eq. (19), the results of which are presented in Fig. 7.

Fig. 7

RMS of deviation from AEP (Eq. 6) for port 1 in Fig. 6: (a) upper (270° ≤ θ ≤ 90°) and (b) lower (90° ≤ θ ≤ 270°) sides of the AEPs.

(19) RMS={1181∑θ=270°90°(∣g1(θ,φ)∣-∣g1case(θ,φ)∣)21181∑θ=90°270°(∣g1(θ,φ)∣-∣g1case(θ,φ)∣)2,

where the upper and lower equations relate to the upper (270° ≤ θ ≤ 90°) and lower (90° ≤ θ ≤ 270°) sides of the AEPs, respectively. Notably, every term is considered to be in the linear scale, while 181 indicates the number of sampled θ. Furthermore, the superscripted term “case” indicates one of the three considered cases (Eqs. 16, 13, and 18). Without calibration (Eq. 16), the RMS was found to be significantly large for both AEP sides since the injected power was significantly decreased and distorted at the lower side. Furthermore, the RMS obtained based on Eq. (13) was fairly small for the upper side but very large for the lower side owing to the scaling up and persistence of parasitic backward radiation. As a result, the RMS was the largest for the lower side. Meanwhile, the RMS based on Eq. (18) was almost zero for all directions, thus calibrating every distortion.

III. Conclusion

In conclusion, this study successfully addresses the challenges posed by the feeding network in mmWave antenna measurement. The proposed calibration method effectively compensates for the distortions in AEPs caused by feeding network issues, such as ohmic loss, signal reflection, and parasitic backward radiation. Our findings underscore the importance of accounting for the entire antenna structure, including the feeding network, to accurately characterize ideal antenna performance. Furthermore, the calibration technique developed in this study offers ample scope for advancements in the field of antenna engineering, particularly for mmWave applications, since it enhances the accuracy of antenna performance assessments while also paving the way for more efficient and reliable antenna designs in the future.

Notes

This work was supported by the Institute of New Media Communication (INMC), School of Electrical and Computer Engineering, Seoul National University, South Korea.

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Biography

Seongjung Kim, https://orcid.org/0000-0001-5189-8023 received his B.S. degree in electronic and electrical engineering from Hongik University, Seoul, South Korea, in 2017. In 2023, he received his M.S. and Ph.D. degrees in electrical engineering and computer science from Seoul National University, Seoul, South Korea. Since 2023, he has been a staff engineer in the Department of System Large Scale Integration at Samsung Electronics, Hwaseong, South Korea. His main research interests are phased array antenna theory and design.

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