1 Introduction

Millimeter wave (mmWave) spectrum ranging from 30-300 GHz has received tremendous importance due to gigabits-per-sec data rate, where the sub-6 GHz bands have reached the bottleneck. However, the attenuations at mmWave bands are severely affected due to high path loss, gaseous loss, and rainfall loss. As a result, these frequencies are suitable for short-distance propagations [1]. This gives the advantage of frequency re-use of the same spectrum over the geographical region, enhancing the number of users and data capacity. Another major limitation of the mmWave spectrum is it undergoes a shadowing effect. At mmWave, the wavelength is smaller, so it experiences significant attenuation by various line-of-sight (LOS) path objects like buildings, vehicles, and humans [2]. However, multipath signals arriving from different angles (with non-line-of-sight (NLOS)) can be combined with the array antennas to obtain a better signal-to-noise ratio at the receiver [3]. Along with the new mmWave bands, today’s electronics are getting compact and lightweight with Internet-of-things (IoT) [4]. Most of these devices, such as smart watches, smart goggles, ear-pods, health monitoring devices, etc., have become flexible and wearable [5]. For seamless connectivity, these devices require a flexible and compact antenna. One possible solution is the flexible planar antennas. These antennas can be developed on various substrates such as polyethylene terephthalate (PET), polyester, jeans, polyimide, and liquid-crystal polymer materials because they are highly conformal [6]. In [7], a liquid crystal polymer is used as a substrate to generate resonance at 28 GHz and 36.5 GHz. The radiator is a H-shaped structure with a co-planar waveguide (CPW) feed. The impedance matching of the antenna is tuned by adding an inverted L-shape stub to CPW. In [8], a flexible antenna with beam-switching ability is developed on the Premixgroup substrate. It operates at 77 GHz, with bandwidth from 75-81 GHz. The beam is tilted with the aid of parametric elements placed around the radiator. The antenna gain can be improved by incorporating metamaterial structures. In [9], it is achieved by carving sigma-shaped structures around the radiating element. This sigma-shaped structure exhibits a band-stop characteristic at 30 GHz, due to which better isolation between elements is achieved. As a result antenna performance is improved along with good gain.

The characteristic mode analysis (CMA) is a rapid antenna development tool currently being used in the design stages of an antenna. The CMA models the surface current \((J)\) for any arbitrarily shaped structure antenna without an excitation. The surface current distribution is studied through weighted eigenvalues from which the modal significance (MS) and its phase characteristic angle (CA) are derived. In [9], a U-slot and E-shaped circularly polarized (CP) antenna with slot mode, TM10 mode, and TM01 mode are developed through CMA. The design is analyzed for five modes, out of which the slot mode, TM10 mode, and TM01 mode resonated at 4.7 GHz, 5.3 GHz, and 6.65 GHz, respectively. In another design, a circularly polarized metasurface is analyzed using CMA [10]. It is a multilayer structure with parasitic elements at the top layer, which are excited by meander line feed at the bottom. The analysis is sorted for four modes: the first two modes resonated at 6 GHz, and the other two modes at 6.5 GHz. In [11], a non-uniform metasurface structure is developed for bandwidth enhancement using CMA. The analysis sorted for five modes: modes 1 and 2 are significant at 5.5 GHz and 6.25 GHz. Further, it is tuned to obtain a wide bandwidth of 5.15-5.825 GHz. In [12], the CMA is applied for a flexible antenna at sub-6 GHz. In this case, the planar CP antenna with parasitic elements is analyzed with six modes, of which Mode 1 resonates at 2.25 GHz. In [13], CMA is used to optimize the flexible antenna dimension at sub-6 GHz. In [14], CMA is applied for bandwidth enhancement through defected ground structure at sub-6 GHz.

The above designs applied CMA to analyze the structure response at sub-6 GHz. However, this article focuses on designing a flexible antenna to operate at mmWave. Therefore, a few more articles are referred to comprehend the CMA study on mmWave structures. In [15], CMA is applied for metasurface structures operating at the mmWave band. In this case, the structure is multilayer with a substrate-integrated waveguide at the bottom and parasitic elements behaving as metasurface at the top layer. The CMA analysis is sorted for 20 modes, of which modes 4, 8, and 9 are significant at 33 GHz, 27 GHz, and 37 GHz. The design achieved 23.5-29.5 GHz bandwidth and 36.5-41 GHz. Another metasurface structure [16] with and without the U-slot effect is studied using CMA. It revealed that without the slot, the structure resonates at 28 GHz, and with the slot, the resonance is shifted to 34 GHz. In [17], a metasurface lens unit cell is studied through CMA. The unit cell is a simple C/inverted C-shaped structure on top and bottom of the substrate. It generated Mode 2 and 4 with high model significance at 28 GHz. The results show that under the equilibrium of the surface and induced current, there is a resonance, with the surface behaving as a lens passing the electromagnetic radiation. The bandwidth enhancement of the antenna is achieved with a metasurface structure, which is analyzed using CMA [18]. In this case, the structure has pairs of patches that generate four modes at 28.86, 32.52, 35.65, and 37.26 GHz. These modes together generate wide bandwidth ranging from 29-39 GHz. The CMA can also be applied to analyze the decoupling structures in multiple-input-multiple-output antennas, as achieved in [19].

From the above literature, it is comprehended that there are numerous wearable antennas exist. However, their dimension is large. Also, the CMA analysis tool has mainly been used to analyze sub-6 GHz flexible and rigid millimeter wave antennas. Thus, it gives us the opportunity to study the flexible millimeter antenna structure using CMA. Therefore, a compact and highly flexible millimeter wave antenna structure is designed using CMA to operate at 30 GHz. The evolution stages of the antenna and the bending analysis are performed through CMA. There are plenty of 5G applications using this band, such as for mobile communication [20, 21], vehicular applications [22], and satellite applications [23]. The antenna has achieved a measured bandwidth of 3.2 GHz (29-32.2 GHz) with a decent gain of 5.6 dBi. The antenna performance for conformal orientation invariably has a stable response over the bandwidth.

2 Design Methodology

The antenna structure presented in this article is inherited from our earlier design [24]. However, the CMA analysis and substrate used are entirely new. Therefore, the antenna is designed on a thin film of polyimide substrate, which has a thickness of 0.1 mm. The polyimide substrate is highly flexible, resulting in constant dielectric properties over conformal orientation. It has a dielectric constant \(({\varepsilon }_{r})\) of 3.5. The chosen substrate dimension is 6 \(\times\) 6 mm2. The design process includes four stages. The analyses of these stages are performed through CMA from the dominant complex characteristic values (an) derived from the phase angle (-αn) between the sets of real characteristic current (In(s)) and its electric field (En(s)) [25]. From the current behavior of the resonant mode, the radiation pattern is predicted without excitation [26]. The modes far from the natural resonance will degrade the antenna performance because these modes store the energy in the form of electric or magnetic energy rather than radiating. The eigenfunction with its eigenvalue constructs the ideal characteristic current based on the structure pattern as in (1) [27].

$$X\left({J}_{n}\right)= {\lambda }_{n}R\left({J}_{n}\right)$$
(1)

where \({\lambda }_{n}\) is eigenvalue, \({J}_{n}\) is surface current, [X, R] is the impedance operator’s \(({Z(J}_{n}))\) imaginary and real parts.

$$Z\left({J}_{n}\right)= R\left({J}_{n}\right)+jX\left({J}_{n}\right)$$
(2)

\(X\left({J}_{n}\right)\) indicates the characteristic modes storing the energy. Therefore, for the modes to be dominant, \(X\left({J}_{n}\right)\) must be made zero such that the impedance Z is real. For this to achieve, the current pattern at maximum/minimum areas in the structure has to be perturbed with slits/ slots such that the mode becomes purely real. The imaginary surface current \(X\left({J}_{n}\right)\) can be derived from the characteristic angle (αn) as a function of an eigenvalue, as given in (3).

$${\alpha }_{n}={180}^{0}- {tan}^{-1}{\lambda }_{n}$$
(3)

The sum of all surface/characteristic current is represented by (J), as shown in (4). The modal significance (MS) can determine the facile and stringent modes computed based on weighted eigenvalues (\({\lambda }_{n})\), as given in (5).

$$J= \sum_{n=1}^{\infty }{\alpha }_{n}{J}_{n}$$
(4)
$$MS= \frac{1}{1+j{\lambda }_{n}}$$
(5)

The first evolution process has a circular ring with two dipole stubs etched on the radiator connected with a microstrip feed line, as displayed in Fig. 1(a). In the second evolution process, two triangular slots are etched on a T-shaped structure, as presented in Fig. 1(b). Further, a variable diameter ring is added to the radiator in the third evolution process, overlaying the T-shape stub and outer circular ring, as in Fig 1(c). Finally, in the last evolution process, the ground plane is etched with a circular slot, making the ground defective, as displayed in Fig. 1(d).

Fig. 1
figure 1

The flexible antenna evolution process. (a) First evolution, (b) second evolution, (c) third evolution, and (d) proposed final design structure with dimension in mm

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3 Design Analysis

3.1 Analysis using CMA

The antenna evolution stages are developed in CST v21 electromagnetic simulation software. The CST software solves the eigenvalue problem of the evolution structures to determine the structure surface current through orthogonal eigen-current along with their eigenvalue. The CST incorporates various eigenvalue solvers, such as integral and multilayer, by which it provides the solution. The mode tracking option in the solver enables tracing the variation in eigenvalue through the surface current for the defined frequency range. The user can define the number of desired modes and the frequency point to analyze. After the analysis, the solver presents the modal significance, characteristic angle, and eigenvalue plots. The solver also displays the surface current based on the structure without excitation. The approximate radiation pattern is plotted from the surface current to provide the antenna's radiation behavior.

In our case, the evolution stages of the antenna are simulated using a CST multilayer solver. For the first evolution stage (Fig. 1a), the CMA is sorted at 30.50 GHz (f0) with five modes in CST simulation software. The MS in Fig. 2a reveals that these modes are significant at different frequencies. Mode 2, with its surface current J2 is substantial at 25 GHz to 33 GHz. At 26.75 GHz, Modes 3 and 5 are significant with surface currents J3 and J5. Mode 1 is noteworthy at 28.4 GHz, and Mode 4 is effective at 28.85 GHz. However, at 30.5 GHz, all the modes are indicated as substantial, with Mode 2 as the fundamental mode and a pair of Modes 3-5 (J3 - J5) as degenerate modes. The characteristic angle in Fig. 2b indicates the two dominant convergence points at 26.85 GHz and 29 GHz, which generate the resonance. The resonance at 26.85 GHz is due to the sum of J2+J3+J5 modes with a bandwidth of 27-28 GHz, as indicated by the eigenvalue in Fig. 2c. Similarly, at 29 GHz, the sum of J1 to J4 modes generates resonance with a narrow bandwidth of 28.8-29.4 GHz. The sum of these surface currents results in a radiation pattern.

Fig. 2
figure 2

First evolution stage results from CMA. (a) MS, (b) CA, (c) Eigenvalue

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However, at 30.5 GHz, only Mode 2 is a fundamental mode, which is completely radiating the energy. The other modes are degenerative, which stores the energy in the form of capacitive and inductive above and below f0 [15, 27]. Due to these degenerative modes, the overall impedance mismatch occurs. Therefore, the structure is modified in the next evolution stage to suppress the effect of degenerative modes [27].

To comprehend the relation between the modal surface current and radiation pattern, look at fundamental Mode 2 of Fig. 3. Here, the current concentration is in the ground plane, in-phase along the horizontal edges. Due to this, it results in bidirectional radiation in the yz-plane. As Modes 1 and 5 are capacitive beyond f0, it resulted in current J1 and J5 along the vertical edges of the ground plane, which resulted in bidirectional radiation in the xz-plane. In the case of Mode 4, the in-phase current in the radiator resulted in bidirectional in the xz and yz-plane. However, Mode 4 is inductive, with the anti-phase current along the edges of the ground plane, resulting in quad/end-fire radiation.

Fig. 3
figure 3

Surface current Jn and radiation pattern of first evolution stage antenna

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Further, in the second evolution process (Fig. 1b), slots are introduced in the dipole structure to perturb the current flow. Due to perturbation, modes become significant at four different frequencies, as indicated in Fig. 4a. The significant modes at 26.75 GHz are Modes 2-3 (J2-J3), at 27.1 GHz, Modes 1-2 (J1-J2), at 29.1 GHz, and at 30.5 GHz Modes 1-4 (J1-J4). However, the characteristics angle (CA) in Fig. 4b indicates only two mode convergence points, 29.1 GHz and 30.5 GHz. The bandwidth at these are 28.7-29.6 GHz and 29.5-32 GHz. The eigenvalue graph of Fig. 4c shows that the radiation at 29.1 GHz is due to the sum of J1-J4 modes and at 30.5 GHz, due to the sum of J1-J3, as shown in Fig 5. However, at 30.5 GHz, Modes 4 and 5 disturb the impedance, resulting in a poor reflection coefficient.

Fig. 4
figure 4

Second evolution stage results from CMA. (a) MS, (b) CA, (c) Eigenvalue

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Fig. 5
figure 5

Surface current Jn and radiation pattern of second evolution stage antenna

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In the previous two design stages, the current concentration is at the edges of the ground plane, indicating that most radiation is from the ground plane and partly by a radiator. In the third evolution stage (Fig. 1c), adding a variable thickness ring structure to the radiator improves impedance matching and maximizes the current on the radiator, as shown in Fig. 6 for Modes 1, 2, 3, and 4.

Fig. 6
figure 6

Surface current Jn and radiation pattern of third evolution stage antenna

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Figure 7a indicates two significant frequency points at 25 GHz and 30.5 GHz. The convergence in Fig. 7b at 25 GHz is because of Modes 1 and 3. At 30.5 GHz, it is due to all five modes. The eigenvalue in Fig. 7c indicates that at 25 GHz, it results in good impedance matching due to perfect convergence of modes J1 and J3 and large suppression of other modes. It has a bandwidth of 24.5-25.5 GHz. However, at 30.5 GHz, impedance mismatch occurs due to the inductive nature of Mode 2 before f0 and Modes 1 and 4 after f0. Due to this, the resonance might not occur.

Fig. 7
figure 7

Third evolution stage results from CMA. (a) MS, (b) CA, (c) Eigenvalue

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In the final evolution stage of Fig. 1(d), the ground plane is defected with a circular slot for bandwidth improvement. As a result, modes significance at three frequencies can be observed in Fig. 8a, that is, at 27 GHz, 28 GHz, and 30.5 GHz. The respective pair of modes at these frequencies are J3/J5, J2/J3, and J1/J2/J3/J4/J5, considering MS as 0.9 value. The characteristic angle in Fig. 8b indicates that the structure modification has resulted in close convergence of significant modes by largely suppressing the unnecessary modes. The modes J3 and J5 converges to zero eigenvalue (\({\lambda }_{n})\) at 27 GHz in Fig. 8c to generate the resonance. However, these two modes are not robust enough to generate resonance. At 28 GHz, the fundamental mode J2 converges with the J3, whereas other modes tend to converge slowly. This indicates the improvement in the impedance matching, which may result in a better reflection coefficient |S11| > 10 dB. The mode J2 eigenvalue (\({\lambda }_{n})\) remain close to zero until 31.5 GHz, converging with modes J3 and J5. It results in a good impedance match over the 28-31.5 GHz frequency range. From Fig. 9, it can be observed that the fundamental mode J2 has an in-phase current in the radiator and in the ground plane. But these are opposite to each other, resulting in broadside radiation. In the case of mode J3, the in-phase current can be seen only in the radiator, which resulted in a generation of bidirectional radiation. The mode J5, from Fig. 8c, indicates it has a capacitance effect, so an anti-phase current can be seen in the ground and the radiator.

Fig. 8
figure 8

Final evolution stage results from CMA. (a) MS, (b) CA, (c) Eigenvalue

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Fig. 9
figure 9

Surface current Jn and radiation pattern of final evolution stage antenna

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It resulted in a generation of end-fire radiation. Overall, it can be concluded that modes J2 and J3 are dominant, with partial contribution of J5 at 30.5 GHz. Also, the summation of these currents results in bidirectional radiation. The antenna of all the evolution stages is excited, and its respective reflection coefficient |S11| plot is shown in Fig. 10.

Fig. 10
figure 10

Reflection coefficient |S11| of all the evolution stages

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3.2 Analysis using Equivalent Circuit

The proposed flexible antenna is modeled using transmission line theory with the aid of an RLC circuit. The equivalent circuit in the Fig. 11 has three series and one parallel RLC circuit. The first series RLC circuit that R1, L1, and C1 represents the feedline of the proposed antenna with its value of 1 Ω, 0.169 nH, and 1 pF results in an impedance of 27.80 Ω. The other two series circuits are to generate the notch below and above the desired resonance band with its RLC values of 113 Ω, 0.181 nH, 0.201 pF (R2, L2, C2), 0.5 Ω, 0.261 nH, and 0.33 pF (R4, L4, C4). These have resulted in impedance Z2 = 113.41 Ω and Z4 = 35.28 Ω. The parallel RLC circuit is to generate the desired resonance |S11| > 10 dB with its resulting impedance of Z3 = 316.015 Ω for the values of R3 = 316 Ω, L3 = 0.11 nH, and C3 = 0.28 pF. Therefore, when the overall impedance is calculated, it results in Z0 = 54.056 Ω, close to an ideal 50 Ω. Consequently, the |S11| curve generated from the equivalent circuit overlaps the curve grom EM simulated software, as shown in Fig. 12.

Fig. 11
figure 11

Equivalent circuit of the proposed antenna

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Fig. 12
figure 12

Reflection coefficient curve from EM simulated software and equivalent circuit

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4 Bending Analysis

The flexible antenna performance is studied for two conformal orientations through CMA. The chosen bending radii are 20 mm and 10 mm. These analyses are performed for antenna orientation along the x and y-axis.

4.1 Bending along x-axis

For the x-axis bends at 20 mm and 10 mm radii, it is observed that modes 4 and 5 are greatly suppressed. When the antenna is in its original state, there are three significant points, as indicated in Fig. 8a. However, the significant points for conformal orientation are reduced to two, at 27 GHz and 30.5 GHz, as shown in Figs. 13a & d. The mode J1 is found to converge between these frequencies with 1800 phase, as indicated by characteristic angle Figs. 13b & e. The eigenvalue in Figs. 13c & f provides clarity on the contributing modes. The sum of modes J1 + J3, generating resonance at 27 GHz. Likewise, modes J1+J2 generate resonance at 30.5 GHz. Therefore, it can be concluded that with J1 dominant mode and support of J3 and J2, the bandwidth is maintained from 27-32 GHz, even with conformal orientation.

Fig. 13
figure 13

The x-axis bend analysis through CMA with MS, CA, and EV graphs. (a-c) At 20 mm bend. (e-f) At 10 mm bend

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The variation of the surface current distribution of modes for x-bend at 30.5 GHz is shown in Fig. 14. As Mode J1 is dominant, it has in-phase current in the vertical edges of the ground plane, resulting in bidirectional radiation in xz-plane, with a maximum gain of 5.35 dBi. The fundamental mode, J2, has an in-phase current in the horizontal edges of the ground plane and along the radiator, resulting in bidirectional radiation in the yz-plane. It predicted a maximum gain of 4.86 dBi. As the other modes are not significant, their mode gain is lower.

Fig. 14
figure 14

Surface current distribution of modes and its radiation pattern at 30.5 GHz for x-axis bend

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4.2 Bending along y-axis

The proposed antenna is also validated for y-axis bend through CMA at 20 mm and 10 mm radii. The structures orientation has resulted in similar results to the x-axis bend, as shown in Figs. 15a-f. The mode J1 is dominant from 27 GHz to 32 GHz for 20 mm and 10 mm radii bend. However, a slight deviation can be seen at these convergence points at different bend radii. For a 20 mm bend, the first convergence point is at 27 GHz, whereas at a 10 mm bend, the first convergence point is at 27.5 GHz. This indicates that the increase of conformal bending along the y-axis decreases the bandwidth. Therefore, with a 20 mm bend, the bandwidth is 27-32 GHz; with a 10 mm bend, the bandwidth is 27.5-32 GHz. The modes J1 and J2 in Fig. 16 have in-phase currents, resulting in bidirectional radiation with a maximum gain of 5.32 dBi and 4.85 dBi, respectively.

Fig. 15
figure 15

The y-axis bend analysis through CMA with MS, CA, and EV graphs. (a-c) At 20 mm bend. (e-f) At 10 mm bend

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Fig. 16
figure 16

Surface current distribution of modes and its radiation pattern at 30.5 GHz for y-axis bend

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5 Results and Discussion

5.1 |S11| and Gain

The proposed antenna is a prototype fabricated, as shown in Fig. 17. Its reflection coefficient |S11| is measured using Anritsu VNA (S820E), having a range from 1 MHz to 40 GHz, and the radiation pattern is an anechoic chamber. We used 2.4 mm end lanch connectors from Johnson (147-0701-261) that operate up to 50 GHz. The |S11| and anechoic chamber measurement setup are displayed in Figs. 18a and b. The proposed antenna aims to achieve resonance at 30.5 GHz with wide bandwidth. This is achieved by tuning the antenna in four evolution stages. Also, to achieve flexibility, a thin polyimide substrate is used. The CMA analysis aided in the rapid development of the antenna design process. From CMA, the structure surface current of the proposed antenna indicated a wide bandwidth of 28-31.2 GHz with resonance at 30.5 GHz. The modes contributing to the resonance are fundamental mode J2 and dominant modes J3 and J5. When the port is excited, the achieved simulated bandwidth is in the range of 28.72-32 GHz, with resonance at 30.7 GHz, as presented in Table 1. The measured bandwidth for reflection coefficient |S11| > 10 dB is in the range of 28.9-31.75 GHz, with resonance at 30.45 GHz, as displayed in Fig. 19. The slight deviation in the measured results is due to antenna bend when it soldered to the SMA connector as the substrate is flexible. However, the variations are in the acceptable range. The maximum simulated and measured realized gain achieved is 5.6 dBi, as shown in Fig. 20. The gain is linearly increasing in the band of interest. The antenna has obtained a decent gain suitable for mobile, IoT, and vehicular applications. Further, the gain can be improved by incorporating the metamaterial superstate of frequency selective surface, as mentioned in [21, 28, 29]. This can be considered for future work. The antenna has obtained an average radiation efficiency of 90 % throughout the bandwidth. The CMA results in terms of resonance and bandwidth are close to the simulated results. Also, the measured results from the prototype fabrication are on good terms with the expected results.

Fig. 17
figure 17

Prototype fabricated antenna

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Fig. 18
figure 18

Fabricated antenna measurement setup. (a) Measurement of |S11| using VNA. (b) Radiation pattern measurement in the anechoic chamber

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Table 1 |S-parameter| and gain analysis of proposed antenna through CMA, port excitation, and measurement.
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Fig. 19
figure 19

Simulated and measured reflection coefficient |S11| of proposed flexible antenna

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Fig. 20
figure 20

Simulated and measured realized gain and radiation efficiency plot of the proposed antenna

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The proposed antenna can work with conformal orientations. For this reason, 20 mm and 10 mm radii bending analysis is performed. First, it is analyzed using the CMA, further with simulated and measured results. Along the x-axis bend, the CMA indicated two convergence points at 27 GHz and 30.5 GHz. It is due to modes J1 + J3 and J1 + J2, respectively. It also shows an increase in bandwidth of 27-32 GHz. Similar results can be seen for the y-axis bend with a bandwidth of 27.5-32 GHz in Fig. 13. However, the simulated results reveal that bending along the x-axis drifts the resonance to a higher frequency, as displayed in Fig. 21a. The simulated bandwidth for 20 mm radii bend is 28.5-31.5 GHz, whereas the measured is 28.5-31.2 GHz, with resonance at 30.2 GHz and 30.4 GHz, respectively. The simulated bandwidth for a 10 mm bend is 29.25-32.25 GHz, whereas the measured bandwidth is 29.8-32 GHz. The resonances are at 31 GHz and 31.4 GHz, respectively. The drift in the band might be due to the stretching of the feed line and radiator. In the case of the y-axis bend, not much deviations are observed. This is because only the edges of the radiator are stretched. The simulated bandwidth for 20 mm and 10 mm bend is 29.4-32.6 GHz. The measured bandwidth at various bending radii along the y-axis is 29.7-32.3 GHz and 29.5-32.6 GHz, as shown in Fig. 21b. Table 2 presents the analyzed results of bending for CMA, simulated and measured .

Fig. 21
figure 21

Simulated and measured reflection coefficient |S11| graphs for bending effects. (a) For x-axis bend, (b) for y-axis bend

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Table 2 Bending analysis |S-parameter| results from CMA, port excitation, and measurement
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5.2 Current Distribution

The flow of current and its oscillation leads to the formation of the radiation pattern in the respective plane. Fig. 22 shows the current path in the proposed antenna structure at various frequencies. It is evident that the bandwidth has a uniform current distribution over the different frequencies. As a result, a similar radiation pattern is achieved in the operational band. It can be observed that the current is at its maximum in the radiator structure at 29, 30.5, and 31.5 GHz. In the ground plane, the current is concentrated over the edges of the slot. Thus, the structure exhibits a stable electromagnetic radiation.

Fig. 22
figure 22

Current distribution in the proposed antenna at (a) 29 GHz, (b) 30.5 GHz, and (c) 31.5 GHz

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5.3 Radiation Pattern

The surface current distribution from CMA in Fig. 9 indicated that the proposed antenna structure resulted in an in-phase current in the radiator as well as in the ground plane for the fundamental mode J2. Due to this, it resulted in bi-directional radiation. The simulated and measured results agree with the CMA analysis. Fig. 23 shows the radiation pattern of the proposed antenna for simulation and measurement at 30.5 GHz along the XZ and YZ planes. It resulted in bidirectional and omnidirectional radiation in the respective planes for co-polarization. The resulting simulated and measured half-power-beamwidth (HPBW) in the XZ plane is 58° and 54°. The proposed antenna has achieved good simulated/measured cross-polarization of -28.5 dB/-26 dB and -19 dB/-16 dB in the XZ and YZ planes.

Fig. 23
figure 23

Radiation pattern of proposed antenna at 30.5 GHz. (a) XZ plane, (b) YZ plane

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5.4 Link Budget

At mmWave, the signal fading is severe, resulting in higher path loss over a short distance. Therefore, the proposed antennas performance must be verified by estimating the link budget in a lossy environment. The setup considered for the analysis is a simple outdoor line-of-sight (LOS) condition (free from obstruction) with a communication link between a base station tower and a mobile phone. The total geographical area assumed is 100 \(\times\) 100 m2. The base station antenna considered in our case is directional, with a narrow beam in an azimuthal plane having a gain (\({{\varvec{G}}}_{{\varvec{T}}{\varvec{X}}})\) of 13 dB, tilted 10° downwards for better coverage. It is placed on a tower at a height of 14 m. The base station antenna (TX) is assumed to have beam steering ability, which can be achieved through passive [30] or active methods [31]. The proposed antenna (RX) is placed inside the mobile phone and carried by a pedestrian along the path, as shown in Fig. 24(a). The actual distance between TX and RX is calculated using the Pythagorean theorem. An ideal phase-shift-keying (PSK) modulation scheme is considered with the symbol-to-noise ratio (Eb/N0) of 9.6 dB as in [32]. The received signal power at RX is calculated by considering the path loss and ground reflection effects (\({{\varvec{L}}}_{{\varvec{m}}{\varvec{i}}{\varvec{s}}})\), which is given by equation (6).

Fig. 24
figure 24

Simple geographical area with LOS condition. (a) 2D representation with TX and RX locations. (b) 3D representation

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$${{\varvec{P}}{\varvec{R}}}_{{\varvec{R}}{\varvec{X}}}=\boldsymbol{ }{{\varvec{P}}{\varvec{R}}}_{{\varvec{T}}{\varvec{X}}}+\boldsymbol{ }{{\varvec{G}}}_{{\varvec{T}}{\varvec{X}}}+\boldsymbol{ }{{\varvec{G}}}_{{\varvec{R}}{\varvec{X}}}-\boldsymbol{ }{{\varvec{L}}}_{{\varvec{m}}{\varvec{i}}{\varvec{s}}}$$
(6)

The link budget is the difference between the received and required power over the distance (d) in meters, as given by equation (7) [33].

$${{\varvec{L}}}_{{\varvec{M}}{\varvec{a}}{\varvec{r}}{\varvec{g}}{\varvec{i}}{\varvec{n}}\boldsymbol{ }}=\boldsymbol{ }{{\varvec{P}}{\varvec{R}}}_{{\varvec{R}}{\varvec{X}}}-\boldsymbol{ }{{\varvec{P}}{\varvec{R}}}_{{\varvec{R}}{\varvec{Q}}}$$
(7)
$${{\varvec{P}}{\varvec{R}}}_{{\varvec{R}}{\varvec{Q}}}({\varvec{d}}{\varvec{B}})=\frac{{{\varvec{E}}}_{{\varvec{b}}}}{{{\varvec{N}}}_{0}}+\boldsymbol{ }{{\varvec{B}}}_{{\varvec{r}}}({\varvec{d}}{\varvec{B}})+{\varvec{K}}{\varvec{T}}$$
(8)

Where K is Boltzmann’s constant, T is temperature in kelvin, and \({{\varvec{B}}}_{{\varvec{r}}}\) is bit rate.

The link margin analysis results in Fig. 25 indicate that with an increase in distance, the received signal at the receiver decreases due to large-scale fading. Also, more transmission power or higher TX gain is required to deliver the higher data rate over a longer distance. From the considered setup with a TX gain of 13 dBi and an RX gain of 5.6 dBi, the proposed antenna can reliably communicate up to 1 Mbps to 100 Mbps data rate over a distance of 100 m for a link margin of 0 dB. However, the proposed structure can support a maximum of 1 Gbps to a distance of 70 m. The ripples in the received signal may be due to the following reasons: (i) it might be due to the reflections from the ground as the RX antenna height is just 1 m above the ground level. (ii) Due to the inaccurate beam steering, which can be improved by adapting a suitable algorithm (beyond the scope of this article).

Fig. 25
figure 25

Link margin estimation of the proposed antenna over distance for varied bit rate

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5.5 Comparative Analysis

The proposed flexible antenna performance is compared with the existing state-of-the-art designs and summarized in Table 3. It is evident that the presented antenna is compact compared to other designs. The chosen substrate is also very thin compared to others, which makes it highly flexible and suitable for conformal applications. The proposed antenna has good bandwidth compared to [34]. This is the only mmWave flexible antenna analyzed using the CMA tool. Therefore, the proposed antenna is very well suited for wearable applications.

Table 3 Comparative analysis of proposed flexible antenna with existing state-of-the-art designs
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6 Conclusion

The article proposed the flexible antenna operating at 30.5 GHz mmWave spectrum to support higher data rates for wearable, IoT, and other wireless applications. The design process of an antenna is performed with the aid of the CMA tool for rapid development. The final structure of the antenna revealed that mode J2 is the fundamental mode generating resonance at 30.5 GHz with the support of the other two dominant modes, J3 and J5. The analysis results from CMA are in close agreement with the simulated and measured results. The antenna resulted in a measured bandwidth of 28.9-31.75 GHz. For conformal orientation along the x-axis, a drift in the resonance is seen. However, for the y-axis, no variations are observed. The antenna has resulted in bidirectional and omnidirectional radiation with a maximum gain of 5.6 dBi. The link budget analysis suggested that the proposed antenna can deliver up to 100 Mbps of data reliability to a distance of 100 m. Further, to a maximum 1 Gbps up to 70 m distance. Therefore, the proposed antenna is suitable for outdoor/indoor 5G wireless applications.