Introduction
Multiple-input–multiple-output (MIMO) systems with frequency diversity are the optimal choice for improved coverage and ultra-fast transmission rates [Reference Zhang, Cheng, Wang, Lou, Gao, Wu and Ng1–Reference Wen, Huang, Peng, Wu, Zheng and Zhang3]. The authors of reference [Reference Singh, Kumar, Kanaujia and Pandey4] presented a MIMO antenna with a square patch and a modified complementary electric inductive capacitive resonator designed on the FR4 substrate, resonating at 2.45, 3.5, and 4.4 GHz. This design features a peak isolation of 47 dB due to hexagonal-shaped split ring resonators (SRRs) in the ground plane. In reference [Reference Singh, Kumar, Kanaujia and Pandey5], the authors proposed a 20 × 20 mm MIMO antenna on FR4 substrate that produces triple bands at 1.8, 2.4, and 3.5 GHz with the help of meta-resonator to achieve gain of 1.9, 1.75, and 1.52 dBi, respectively. The references [Reference Singh, Kumar, Kanaujia and Pandey4, Reference Singh, Kumar, Kanaujia and Pandey5] present the different techniques used for multiband operation, reduced antenna size, and frequency diversity. The MIMO antenna has a challenge of coupling among the resonating elements, which can be encountered with the help of defected ground surface, parasitic reflectors and neutralization lines. In reference [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6], a single band MIMO antenna of 26.43 × 38.25 mm, with four ports designed on FR4 substrate, is used for 4.4–5 GHz band, wherein the radiators are orthogonally placed to produce an isolation of 25 dB, reducing mutual coupling and obtain a peak gain of 2.8 dBi used for 5G applications. MIMO antenna system that covers 3.4–3.6 GHz with H-shaped resonators, of size of 20 × 20 mm, in orthogonal arrangement used for 5G applications is reported in reference [Reference Kiani, Altaf, Anjum, Afridi, Arain, Anwar, Khan, Alibakhshikenari, Lalbakhsh, Khan, Abd-Alhameed and Limiti7]. This offers a bandwidth of 200 MHz and isolation of >12 dB with FR4 substrate of 0.8 mm thickness. A quad-port MIMO antenna of 45 × 45 mm with FR4 as a substrate is proposed in reference [Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8], which consists of a circular stub in the ground to achieve ultra-wideband range of 3.1–11 GHz for Wi-MAX, WLAN, and C-band applications. A four-port antenna of 60 × 60 mm with FR4 substrate is presented in reference [Reference Saxena, Kanaujia, Dwari, Kumar, Choi and Kim9], in which two arms with an open slot ground plane is used for bandwidth enhancement that operates at 3.4–3.8 GHz. The single band and low-level isolation limit its practical application.
In reference [Reference Ameen, Ahmad and Chaudhary10], the authors proposed a MIMO antenna of 46 × 21 mm, designed on the FR4 substrate of 1.6 mm thickness, with a circular-shaped SRR on top and a defected ground at the bottom to produce dual bands that provide linear polarization and circular polarization in each band. The gain of the antenna is 3.25 and 3.4 dBi in 3.4–3.6 GHz and 4–8 GHz bands, respectively, with more than 15 dB isolation used for 5G and C band applications. In reference [Reference Sarkar and Srivastava11], the authors proposed a four-port SRR induced inverted L-monopole antenna on the FR4 substrate of 40 × 40 mm. The lower frequency mode merges with SRR to obtain a wider bandwidth of 35.21% and gain of 4 dBi at 2.9 GHz for 5G applications. It also presents MIMO parameters like envelope correlation coefficient (ECC), total active reflection coefficient (TARC), channel capacity loss (CCL) with acceptable value limits. A MIMO antenna of 35 × 28 mm designed on FR4 substrate is presented in reference [Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12] to produce dual-band operation, with the help of closely spaced U-slots, at 2.6 and 3.6 GHz. It has parasitic structures of C-shape to reduce mutual coupling in the operating bands, where the isolation obtained is of 13 and 10 dB at resonant frequencies. Antenna configurations proposed in references [Reference Ameen, Ahmad and Chaudhary10–Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12] have larger substrate area, complicated design, and poor performance.
In reference [Reference Chen, Liu, Yuan and Wong13], a dual-band MIMO antenna with FR4 substrate is reported with dimensions of 7 × 6.2 mm for 5G mobile applications, and use a parasitic stub method to cover 3.4–3.93 GHz and 4.5–5.3 GHz with isolation of 10 dB, efficiency > 50%, and ECC < 0.23. In reference [Reference Hu, Li, Wu, Chen, Wen, Jiang and Gao14], a T-shaped monopole MIMO antenna with FR4 substrate is presented for 5G mobile applications. The antenna has dimensions of 21.8 × 7 mm, produces an isolation of 12.8 dB, ECC lower than 0.18 and efficiencies of 71% and 68% in 3.3–3.6 GHz and 4.4–5 GHz bands, respectively. In reference [Reference Tian and Du15], a four-port low profile MIMO antenna is presented for 5G mobile applications mounted on FR4 substrate with wideband shared-radiator. The antenna with dimensions of 19 × 19 mm has four corner-cut patches with grounded square analyzed through characteristic mode analysis, and it produces isolation of 10 dB and efficiencies of 40.3%–48.5% in the frequency range of 4.4–5 GHz. Another eight-element MIMO antenna for 5G mobile applications is presented in reference [Reference Kiani, Savci, Parchin, Rimli and Hakim16], which resonates at 3.1–3.7 GHz, 4.47–4.91 GHz, and 5.5–6.0 GHz. The antenna dimensions are 19 × 7 mm and produces spatial diversity characteristics with efficiencies of 62% and 78%, isolation greater than 16 dB and 5.8 dBi of peak gain. In reference [Reference Tian and Du17], a mobile phone antenna is designed of dimensions of 46 × 1 mm with dual feed shared radiator on FR4 substrate to cover Global Positioning System (GPS)and Long Term Evolution (LTE) bands at 1172–1205 MHz and 1410–2790 MHz with improved isolation of 14.8 dB and efficiencies from 52.5%–63.9%. In reference [Reference Li, Sim, Luo and Yang18], an 8 × 8 MIMO antenna with dimensions of 3 × 21.5 mm is presented, which has isolation greater than 17.5 dB with the help of balanced open-slot design between the ports. The antenna is designed for 5G mobile applications, and it has an efficiency of >62% and ECC < 0.05. The designs in the above references are used in miniaturizing MIMO antennas in smartphones. However, these designs cannot reduce the size of antenna further with a greater number of antenna elements. Therefore, designing a miniaturized antenna with larger number of antenna elements, which can be used to develop a MIMO antenna for smartphones, is still a challenge.
In this paper, the designed antenna is a compact multiband MIMO configuration for 5G smartphones with improved isolation. The proposed antenna resonates at 2.4, 3.5, and 5.5 GHz with omnidirectional radiation patterns. The objectives of the proposed eight-port multiband MIMO antenna are:
• To achieve the frequency bands that resonates at 2.4, 3.5, and 5.5 GHz without any parasitic elements.
• To obtain isolation of >17.5 dB between the antennas.
• To study the antenna characteristics on a hand phantom model to check its performance such as S11, total efficiency, and ECC.
This work is organized as follows: The “Design of an eight-port multiband MIMO antenna” section elaborates the design of an eight-port multiband MIMO antenna, the “Analysis of the eight-port multiband MIMO antenna” section describes the analysis of the proposed MIMO antenna, the “Performance analysis using hand-phantom mode” section analyzes the efficiency, ECC, and S-parameters using hand phantom model, and the “Conclusion” section describes the conclusion of the work.
Design of an eight-port multiband MIMO antenna
The proposed antenna has meandering elements that are etched on 150 × 80 mm FR4 substrate (${\varepsilon _r}$ = 4.4 and tan δ = 0.02) with 0.8 mm of thickness, and the single antenna has a size of 16.5 × 8.5 mm (0.132λ 0 × 0.068λ 0, where λ 0 is the free space wavelength calculated at the lowest frequency). The antenna has eight elements (ANT 1–8), in which, antenna 1 and antenna 5 are placed in horizontal direction along the short edges, and antennas 2–4 and 6–8 are placed in vertical direction along the long edges.
The expanded layout of the proposed antenna is indicated in Fig. 1. The proposed eight-port multiband MIMO antenna is simulated using CST Microwave Studio Suite software, and Table 1 shows the dimensions of the proposed eight-port multiband MIMO antenna. It has an inverted E-shaped slot radiator on the ground plane, which has two open ends. In the radiator, there is a branch slot on the either of the open-ended vertical slot, and a microstrip line is used in the slot radiator to feed it.
Every antenna element is designed to resonate at 2.4, 3.5, and 5.5 GHz using equations (1–3), respectively.
Design novelty
The proposed MIMO antenna composed of eight resonators with an ordinary ground plane and no decoupling element among the radiators is used to improve isolation.
• The antenna elements 1 and 5, 2 and 8, 3 and 7, and 4 and 6 are mirror images of each other so the coupling current vectors are in the opposite direction. Due to this mirror-image arrangement of the resonating elements, the proposed MIMO antenna achieves polarization diversity thereby providing high isolation without the need for an additional decoupling mechanism.
• The proposed antenna design achieves multiband by using an E-shaped slot in the ground plane, and the ground plane disrupts unwanted current paths, reducing coupling between antenna elements.
• The meandering of the conductive patch reduces the physical dimensions of the antenna while maintaining the desired electrical length. This allows for a more compact design, which is beneficial in space-constrained applications like mobile devices.
• Despite the compact size, the proposed antenna elements can achieve good radiation efficiency ensuring strong signal transmission and reception, particularly in low-power applications.
• The antenna element can be tailored to meet specific design requirements, such as desired resonant frequency, bandwidth, and radiation pattern. This flexibility makes it adaptable to integrate into printed circuit boards (PCBs) and other electronic devices, facilitating the design of compact and integrated wireless gadgets/devices.
• Mirrored antenna elements in the proposed antenna can help to achieve better isolation between individual antenna elements, improving the performance of MIMO algorithms that rely on separating the multiple transmitted and received signals.
• With reduced correlation, the proposed MIMO antenna can achieve higher diversity and multiplexing gains, resulting in better performance metrics such as signal-to-noise ratio.
• The planar structure of the proposed meandered line antenna contributes to a low-profile design, which is aesthetically pleasing and practical for embedding in thin devices like smartphones and tablets.
Evolution stages of the proposed multiband single antenna
The evolution stages of the proposed antenna are shown in Fig. 2. The antenna has a meandering element over the feed lines to resonate at 3.5 GHz, as shown in Fig. 2(a). To cover the 5.5 GHz frequency for Wi-MAX application, an additional meandering element is introduced internally, as shown in Fig. 2(b). Finally, to obtain the WLAN operating frequency at 2.4 GHz, external meandering element is incorporated in the antenna, as shown in Fig. 2(c). The S11 of the three stages are given in Fig. 3.
Eight-port multiband MIMO antenna placement
The placement of the radiating elements for an eight-port antenna is decided by placing them in different positions and analyzing their S-parameters.
Case 1
In this case, the position of the radiating elements 1 and 5 is reversed, as shown in Fig. 4(a), so that the open ends of the two antennas (ANT 1 and ANT 5) are very close to the PCB corners. Due to this, the eight-port arrangement for the multiband MIMO antenna does not resonate at 3.5 GHz frequency band. Also, the isolation between antennas 1 and 2 and antennas 4 and 5 (S21 and S54) is 13.5–17.5 dB, as shown in Fig. 4(b).
Case 2
In this case, the position of the radiating elements 2, 4, 6, and 8 is reversed, as shown in Fig. 5(a). The isolation between the antennas is reduced to 12 dB, as shown in Fig. 5(b). As the open ends of the radiating elements are facing each other, due to the lesser distance between both the antennas, an undesirable isolation is obtained.
It is observed that the two antennas that are adjacently placed causes poor isolation than the antennas placed away from each other. Therefore, the two face to face adjacent radiating elements should be placed away from each other (as in antennas 1 and 6 and antennas 5 and 8) to obtain enhanced isolation.
Case 3
This case shows the orthogonal arrangements of the radiating elements in which they are placed perpendicularly, as shown in Fig. 6(a). The orthogonal antenna polarization does not cover the resonating frequencies (3.5 and 5.5 GHz) and provides deteriorated isolation, as shown in Fig. 6(b). This case cannot yield good antenna performance.
Case 4
To improve the isolation between the antenna elements, the antennas 1 and 5 are placed away from the nearby antennas 2 and 4, as shown in Fig. 1. This produces the bandwidth of 2%, 6.28%, and 2.53% at 2.4, 3.5, and 5.5 GHz, respectively. From Fig. 7, it is observed that the proposed eight-port multiband MIMO antenna operates at 2.4, 3.5, and 5.5 GHz with an isolation of 17.5 dB.
Current distribution
The two antennas connecting at the common null are in reverse direction. As there is a less current distribution in the ground, a high isolation level is achieved. Also, as stipulated by the current distribution in Fig. 8, if the antennas are in perpendicular direction, the x-axis and y-axis generate orthogonal polarizations, resulting in low isolation.
The antenna proposed in this work resonates at 2.4, 3.5, and 5.5 GHz, making it suitable for 5G smartphones while having smaller dimensions. The fabrication process is simple because the substrate used in this work, FR4 with a thickness of 0.8 mm, is widely available. FR4 is inexpensive while comparing with other high-performance substrates, making it a cost-effective choice for production of PCBs. It is a mechanically robust material, providing good support and stability to the antenna design that helps the PCB to withstand physical stress during manufacturing. Also, FR4 is easy to process and handle during PCB manufacturing, including drilling, cutting, and etching.
Figure 9 shows the prototype of the proposed eight-port multiband MIMO antenna whose S-parameter and far-field characteristics are tested using vector network analyzer and anechoic chamber. The measured S-parameters and transmission coefficients are shown in Fig. 10(a) and (b).
Radiation characteristics
The radiation patterns for antennas 1–4 operating in the three bands are given in Fig. 11, which that shows the patterns of the antennas are omnidirectional. The radiation patterns of antennas 5–8 are not presented, as they are nearly the mirror images of antennas 1–4. Figure 12 shows the gain and efficiency of the proposed eight-port multiband MIMO antenna. The obtained gain values are 2.78, 3.014, and 3.438 dBi and efficiencies are 78%, 71%, and 58% at 2.4, 3.5, and 5.5 GHz, respectively.
Analysis of the eight-port multiband MIMO antenna
To verify the proposed multiband MIMO characteristics, several metrics such as CCL, ECC, diversity gain (DG), total efficiency, and TARC are studied in this section.
The ECC explains the correlation between the radiating elements. The actual value of ECC is zero but in practical, 0.5 is the acceptable limit. The ECC calculated using equation (4) [Reference Li, Sim, Luo and Yang18] is less than 0.04, as shown in Fig. 13.
where Fi denotes far-field radiated by the antenna. $\left( {\theta ,\varphi } \right)$ are the angles (elevation, azimuth) and $\Omega $ is the solid angle. The measured DG shown in Fig. 14 is mathematically related to ECC as given in equation (5) [Reference Hu, Li, Wu, Chen, Wen, Jiang and Gao14] whose value is >9.98 for the proposed MIMO antenna.
TARC describes the effective operating bandwidth of the antenna system, whose obtained value is less than –10 dB, is shown in Fig. 15, and can be given by equation (6) [Reference Tian and Du17].
CCL, shown in Fig. 16, compares the performance of the MIMO system with a single antenna system which produces CCL less than 0.25 bits/Hz/s that can be calculated using equation (7) [Reference Kiani, Savci, Parchin, Rimli and Hakim16, Reference Xiao, Wang, Chen, Wang, Wen and Li19]. In Fig. 17, it is observed that the total efficiency of the antennas 1–4 is 55%–78%, and the antennas 5–8 are nearly the mirror image of antennas 1–4.
where ${\xi _{11}} = 1 - \left[ {{{\left| {{S_{11}}} \right|}^2} + {{\left| {{S_{12}}} \right|}^2}} \right]$; ${\xi _{11}} = - \left[ {S_{11}^{\text{*}}{S_{12}} + S_{21}^{\text{*}}{S_{12}}} \right]$; ${\xi _{11}} = - \left[ {S_{22}^{\text{*}}{S_{21}} + S_{12}^{\text{*}}{S_{21}}} \right]$; ${\xi _{11}} = 1 - \left[ {{{\left| {{S_{22}}} \right|}^2} + {{\left| {{S_{21}}} \right|}^2}} \right]$.
Performance analysis using hand-phantom model
In 5G smartphones, where data communication plays a dominant role compared to voice calls, understanding the impact of user hands on antenna performance becomes crucial [Reference Syrytsin, Pedersen and Zhang20, Reference Zhou, Sheng, Bao, Hao, Ji and Li21]. This section explains the influence of user’s single and double hands on the proposed eight-port multiband MIMO antenna. Through simulations employing a standard hand phantom model, the antenna’s behavior is analyzed under single hand mode (SHM) and dual hand mode (DHM) scenarios, as shown in Fig. 18.
Figure 19 shows the S-parameters for SHM operation. From the figure, it is observed that, in SHM, shifting of frequencies affect the antennas 2, 7, and 8, which have direct contact with the fingers. The efficiency values of antennas 2, 7, and 8 are below 30% due to the absorption of energy in the user’s hand, as shown in Fig. 20.
Similarly, the scattering parameters for DHM operations are shown in Fig. 21, which shows that there is a shifting of resonant frequencies in antennas 1 and 5, that have direct contact with the thumb. The scattering parameters of antennas 3 and 6–8 are almost the same, as they are distant from the hand phantom. From Fig. 22, in antennas 1 and 5, the total efficiencies are reduced to 25%. Figure 23 shows that the ECC values of the affected antennas 2, 7, and 8 in SHM and antennas 1 and 5 in DHM gets shifted.
From the analysis of S-parameters, total efficiencies, and ECCs, the proposed MIMO antenna has good performance under SHM and DHM hand-phantom conditions. Table 2 provides the comparison between the hand-held and non-hand-held states of the proposed antenna design.
Table 3 shows the comparison of the proposed antenna with the related published works, which can be explained as,
• The proposed eight-port multiband MIMO antenna has a compact size of 0.132λ 0 × 0.068λ 0 compared to the designs reported in references [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6–Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8, Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12, Reference Hu, Li, Wu, Chen, Wen, Jiang and Gao14, Reference Tian and Du17, Reference Li, Sim, Luo and Yang18].
• The proposed eight-port multiband MIMO antenna covers three operating frequencies at 2.4, 3.5, and 5.5 GHz, unlike reported in references [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6–Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8, Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12–Reference Tian and Du15, Reference Tian and Du17, Reference Li, Sim, Luo and Yang18].
• The proposed eight-port multiband MIMO antenna produces improved ECC value of 0.04 than reported in references [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6, Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8, Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12–Reference Li, Sim, Luo and Yang18].
• The proposed eight-port multiband MIMO antenna produces improved DG value of 9.98 dB than reported in references [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6, Reference Kiani, Altaf, Anjum, Afridi, Arain, Anwar, Khan, Alibakhshikenari, Lalbakhsh, Khan, Abd-Alhameed and Limiti7, Reference Tian and Du15, Reference Kiani, Savci, Parchin, Rimli and Hakim16].
• The proposed eight-port multiband MIMO antenna produces improved CCL of 0.25 bits/Hz/s than reported in references [Reference Kiani, Altaf, Anjum, Afridi, Arain, Anwar, Khan, Alibakhshikenari, Lalbakhsh, Khan, Abd-Alhameed and Limiti7, Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8, Reference Tian and Du15, Reference Kiani, Savci, Parchin, Rimli and Hakim16].
• The proposed eight-port multiband MIMO antenna produces isolation of 17.5 dB unlike in references [Reference Ali, Ren, Bari, Bashir, Hashmi, Khan, Majid, Jan, Tareen and Anjum6–Reference Kumar, Lee, Kim, Mohyuddin, Choi and Kim8, Reference Ojaroudi Parchin, Al-Yasir, Basherlou and Abd-Alhameed12–Reference Li, Sim, Luo and Yang18].
Conclusion
In this work, an eight-port antenna is proposed with small size, which is easy to fabricate and resonates at 2.4, 3.5, and 5.5 GHz. It provides an isolation of >17.5 dB, antenna efficiency (58%–78%) and its parameters such as low ECC < 0.04, high DG > 9.98 dB, improved TARC < –10 dB and CCL < 0.25 bits/Hz/s are observed. The proposed antenna also exhibits good characteristics with hand-phantom. Based on the simulated and measured results, the proposed multiband MIMO antenna is a suitable design for 5G applications.
Competing interests
The author(s) declare no competing interests.
A. Vincy Lumina has received her Bachelors degree in Electronics and Communication Engineering from PTR college of Engineering, Madurai and Master’s degree in St. Joseph’s College of Engineering, Chennai, India in 2015 and 2018 respectively. Currently, she is pursuing her PhD degree in SRM Institute of Science and Technology, Chennai. Her current research interests include Wireless Communications, Antenna Design, Sub–6 GHz, MM-wave applications, 5G applications.
Dr. M. Sangeetha is a senior IEEE member, who received her PhD degree from S.R.M. University, Kattankulathur, and Chennai, India in 2014. She was a gold medalist in her M.Tech degree. Her research interests include Wireless Chaotic Communications, Signal Processing application for Wireless Communication Systems, Internet of Things (IoT) and Machine Learning (ML) algorithms for IoT. Currently she is working as an Associate Professor in the Department of Electronics and Communication Engineering at S.R.M. Institute of Science and Technology (formerly known as SRM University), Kattankulathur, India. She has published 35 International Journal papers, presented 19 International Conference papers and presented 5 papers in National Level Conferences.
Sachin Kumar received the B.Tech. degree in Electronics and Communication Engineering from Uttar Pradesh Technical University, Lucknow, India, in 2009, and the M.Tech. and Ph.D. degrees in Digital Communication and RF & Microwave Engineering from Netaji Subhas University of Technology (East Campus), Delhi, India, in 2011 and 2016, respectively. He was a Post-Doctoral Research Fellow at the College of IT Engineering, Kyungpook National University, South Korea, from 2018 to 2021. He is currently working as Deputy Dean (R&D) and Associate Professor (Research) at Galgotias College of Engineering and Technology, Greater Noida, India. Dr. Kumar has published two books, twelve book chapters, five patents, one hundred and fifty research papers in SCI journals, and over fifty articles in international conferences, and his articles have been cited over 3000 times with an h-index of 30. He is an Associate Editor for the Journal Frontiers in Antennas and Propagation and Franklin Open, Advisory Board Member for the Journal Measurement, Editorial Board Member for the Journal Current Chinese Science, Journal of High-Frequency Communication Technologies, Transactions on Electromagnetic Spectrum, Wearable Technology, and Guest Editor for the International Journal of Antennas and Propagation, Electronics, and Micromachines. He is also a frequent reviewer for more than sixty scientific journals and book publishers. He has given several invited talks at prestigious institutions, and serves as the session chair, organizer, and member of the technical program committee for various national/international conferences, summits, and workshops. His name was featured in the list of “World’s Top 2% Scientists” in the 2022, 2021, and 2020 database released by Stanford University, USA and Elsevier. He was a recipient of the Teaching-cum-Research Fellowship from the Government of NCT of Delhi, India, and the Brain Korea 21 Plus Research Fellowship from the National Research Foundation of South Korea. He is a Fellow of the Institute of Electronics and Telecommunication Engineers, India, a Life Member of the Indian Society for Technical Education, and a Member of the Korean Institute of Electromagnetic Engineering and Science.