Vivaldi antenna

The design of the Vivaldi antenna with two directors for enhanced gain is shown in Fig. 1. The antenna is designed on an FR-4 substrate of thickness 1.6 mm with a relative permittivity of 4.3 and loss tangent 0.025 (Table 1). The inner edge and outer edge of both the arms of the antenna are exponential and each arm is metallized on either side of the substrate, flaring in the opposite direction to form a tapered slot. EM waves travel along the inner edges of the flared aperture and couple with each other to produce radiation. According to conventional theory [Reference Gazit21], the lower cutoff frequency of a Vivaldi antenna is determined by the width of antenna aperture, which can be expressed as λ _cutoff = 2W. The exponential curve for the inner edge C1 is defined as:

(1)$$X( t) = {\,fw} - 0.5 \, {\,fw}\, e^{( a1( t-5) } + {W1\over 2} ,\; \quad Y( t) = t$$

where t ∈ [5, 148], fw is the width of feeding microstrip, and a1 = 0.027 is the rate of exponential curve for C1, and for the outer edge C2 is defined as:

(2)$$X( t) = - 0.5 \, {\,fw}\, e^{a2( t-5) } + {W1\over 2},\; \quad Y( t) = t$$

where t ∈ [5, 29] and a2 = 0.16 is the rate of exponential curve for C2.

Figure 1.

Schematic diagram of a Vivaldi antenna.

Table 1.

List of design parameters for Vivaldi antenna

The antenna is designed to be fed through a 50 Ω SMA (SubMiniature version A) connector followed by a gradual transition consisting of an exponentially contoured ground plane and a microstrip line, shown in Fig. 1. Figure 2(a) shows the S ₁₁ plots for the designed Vivaldi antenna. As shown in Fig. 2(b), with 0 directors, the gain of the antenna varies from 6.8 to 8.8 dBi. To enhance the gain of the Vivaldi antenna, two directors are added between the inner edges. Consequently, the gain increases by 1 dB. Table 2 lists the peak memory requirement for matrix calculation, and total solver run for the simulation in CST for Vivaldi antenna.

Figure 2.

(a) Frequency variation of |S ₁₁| for the Vivaldi antenna design in Fig. 1, and (b) frequency variation of peak gain for the Vivaldi antenna for different number of directors.

Table 2.

Computational burden for simulations of Vivaldi antennas (in CST)

The radiation pattern plots for the E- and H-planes are shown in Fig. 3. The antenna has a directivity of 10.44 dBi and front-to-back ratio of 14.44 dB at 2.4 GHz. The fabricated antenna and the measurement setup are shown in Fig. 4. The measured result is obtained using a Keysight N5230A PNA series network analyzer, and the influence of the coaxial line is eliminated by calibration. It is observed that simulation results match the experimental results reasonably well.

Figure 3.

Radiation pattern (a) E-plane and (b) H-plane for the proposed Vivaldi antenna at 2.4 GHz.

Figure 4.

Fabricated prototype of the Vivaldi antenna and test setup in the anechoic chamber.

Proposed quasi-Yagi antenna

This section describes the design procedure for the proposed compact quasi-Yagi antenna for the TWR on-chip (TWRoC). Figure 5 illustrates the evolution of the designed antenna structure at various stages. A microstrip-fed printed directive dipole antenna operating at 2.34 GHz with an impedance bandwidth (IBW) of 13.24% (S ₁₁ < −10 dB) is chosen as the initial design (Fig. 5(a)). The directive dipole consists of an x-directed printed dipole antenna with a partial ground plane. The printed dipole is excited by broadside-coupled co-planar strip-lines (BC-CPS). The partial ground plane acts as the reflector, which is essential for the directive operation of the antenna. The antenna is designed on a 1.6 mm thick FR-4 epoxy substrate with a relative permittivity ($\epsilon _{r}$) of 4.3 and loss tangent (tan δ) of 0.025 (Table 3). Table 4 lists the computational burden for the simulation of the quasi-Yagi antenna at different design stages.

Figure 5.

Design process for the proposed quasi-Yagi antenna. (a) Stage-I: initial design, (b) stage-II: initial design with a rectangular slot introduced in the ground, (c) stage-III: two-element array for the design in stage-II, (d) stage-IV: design in stage-III with flared dipoles, and (e) stage-V: final design of the two-element array for the quasi-Yagi antenna with modified ground, flared dipoles and 12 directors.

Table 3.

List of design parameters for quasi-Yagi antenna

Table 4.

Computational burden for simulations of quasi-Yagi antennas (in CST)

For a length of 1.25 λ (46 mm) the dipole radiates at 2.34 GHz. To reduce the size of the dipole a rectangular slot is introduced near the BC-CPS (Fig. 5(b)). This modification of the ground plane shifts the center frequency. The introduced slot has two parameters (glc and gwc) which can be tuned to obtain matching at the desired frequency. A parametric study to demonstrate the impact of glc and gwc variation is shown in Fig. 6.

Figure 6.

Frequency variation of |S ₁₁| with slot parameters: (a) lgc (mm) and (b) wgc (mm).

The dimensions of the slot were optimized (glc = gwc = 10 mm) to achieve a smaller dipole length (=0.97 λ) for the same resonant frequency. The design was then used to obtain a two-element array (Fig. 5(c)) and to increase the bandwidth, the dipoles were flared at an angle of θ = 15° (Fig. 5(d)). Flaring in the printed microstrip dipole antenna has been shown to improve the IBW without adversely affecting the efficiency [Reference Dey, Venugopalan, Jose, Aanandan, Mohanan and Nair22]. Upon flaring the dipoles, the IBW increased from 13.24 to 27.35%.

Next, to improve the gain of the antenna, directors were added (Fig. 5(d)). Starting with no directors, the number of directors was increased gradually. The gain of the antenna at 2.4 GHz for 0, 2, 6, and 12 directors was observed to be 3.09, 4.90, 6.80, and 8.62 dBi, respectively. Due to the trade-off between the gain and the size of the antenna, the number of directors was chosen to be 12. The final design offers a high IBW of 44.62%. The |S ₁₁| versus frequency plots for the antenna at each design stage are shown in Fig. 7(a) and the gain plot for increasing the number of directors is shown in Fig. 7(b). The radiation patterns for the E- and H-planes at 2.4 GHz are shown in Fig. 8. The antenna has a measured gain of 8.7 dBi and front-to-back ratio of 25.76 dB at 2.4 GHz.

Figure 7.

(a) Frequency variation of |S ₁₁| for the proposed quasi-Yagi antenna design in Fig. 5, and (b) frequency variation of peak gain for the proposed quasi-Yagi antenna for different number of directors.

Figure 8.

Radiation pattern (a) E-plane and (b) H-plane for the proposed quasi-Yagi antenna at 2.4 GHz.

A comparison between the proposed and previously reported compact planar quasi-Yagi antennas is shown in Table 5. The gain bandwidth product for the antennas is defined as the figure of merit. It can be seen in Table 5 that our final design in Fig. 5(e) has the maximum gain bandwidth product. The proposed quasi-Yagi antenna has a modified ground structure for size reduction and offers a 22.4% reduction in dipole length and 36% reduction in the area of a single element antenna. Figure 5 shows the structure of the antenna at different design stages. The fabricated antenna and measurement setup is shown in Fig. 9. The measured results agree with the simulated results, and the fabrication tolerance can account for the slight discrepancies.

Figure 9.

Fabricated prototype of the proposed quasi-Yagi antenna and test setup in the anechoic chamber.

Table 5.

Comparison with previously reported compact planar quasi-Yagi antennas

Working principle of FMCW radar

Figure 11 shows the block diagram of an FMCW radar system. For an FMCW radar with start frequency f _c, chirp bandwidth B, and modulation time T with a target located at a distance d from the radar, the transmitted LFM chirp can be expressed as:

(3)$$S_{Tx} ( t) = a_{0} \cos \left[2 \pi \left(\,f_{c} t + {st^{2}\over 2} \right) + \phi_{0} \right]$$

where s is slope of the chirp defined as, s = B/T and ϕ ₀ is the initial phase. After reflecting off a target situated at a distance d from the radar, the received signal is a time delayed replica of the transmitted chirp, given by:

(4)$$S_{Rx} ( t) = a_{1} \cos \left[2 \pi \left(\,f_{c} \left(t- \tau \right) + {s\left(t- \tau \right)^{2}\over 2} \right) + \phi_0 \right]$$

The change in the amplitude incorporates the path loss and the radar cross section of the target. The time delay (τ) is the time taken by the EM wave to travel the round-trip distance, 2d given as τ = 2d/c.

Figure 11.

Block diagram of an FMCW radar.

The received signal is then mixed with an exact replica of the transmitted signal which is applied to the LO port of the mixer. The resulting waveform at the output port of the mixer is called the intermediate frequency (IF) signal expressed as

(5)$$S_{IF} ( t) = b \cos \left[2\pi \left(s\tau t + f_c\tau -{s\tau^{2}\over 2} \right)\right]$$

This process is termed de-chirping [Reference Amin8] as illustrated in Fig. 12. For a single stationary target, the IF signal is a single tone with its frequency linearly proportional to the delay τ of the echo, and thus related to the beat frequency f _b as

(6)$$f_b = s\tau = {2sd\over c}$$

Figure 12.

Graphical representation of de-chirping.

The target range can thus be determined from the beat frequency using (6). The maximum detected range depends on the transmit power and the sampling frequency of the analog-to-digital converter, used to digitize the de-chirped signal. The range resolution of a radar is its ability to resolve two closely spaced objects. For two objects separated by a distance of Δd, the corresponding difference in their beat frequencies using (6) can be written as

(7)$$\Delta f_b = {2\, s \Delta d\over c}$$

Also, an observation window of T can resolve two frequencies that are separated by more than 1/T Hz, i.e. Δf _b ≥ 1/T. Using (7), we get

(8)$$\Delta d \geq {c\over 2B}$$

Thus, the range resolution of the FMCW radar depends directly on the bandwidth of the radar.

Spectrogram

The short-time Fourier transform (STFT) is a standard method to study time-varying signals [Reference Cohen37]. The STFT of a signal x(t) is defined as

(9)$$X( t,\; w) = {1\over \sqrt{2\pi}} \int_{-\infty}^{\infty} x( \tau) w^\ast ( t-\tau) e^{-jw\tau}\, {d}\tau$$

where w(t) is a suitably chosen analysis window. The spectrogram corresponds to the squared magnitude of the STFT and is widely used in radar applications. The spectrogram can visualize the strength of the signals reflected from the targets over time at various frequencies [Reference Almeida38]. For all the spectrogram plots in this paper, the x-axis represents frequencies between 1–5 MHz in steps of 5 kHz, and the resolution is 15 kHz with 12 801 frequency points. The y-axis represents the time (in s) and the color-map represents the power (−66 to −15 dBm) of the received echoes. The power level of the color-map is selected to minimize clutter from the environment and to make the signatures of interest clearly visible.

Distance calculation from beat frequency

Since we have used a long coaxial cable (l ₁ = 40 m, $\epsilon _{r1}$) and other RF cables (l ₂ = 7.45 m, $\epsilon _{r2}$), the target is no longer located at a distance of d from the radar. We need to modify (6) to compensate for the additional length which will be reflected in the observed beat frequencies. The total distance of the target from the radar is D given by:

(10)$$D = 2d + l_1 + l_2$$

Incorporating the different lengths and dielectric constants of the cables, we modify (6) as:

(11)$$f_b = s \left[{2d\over c} + {l_1\over v_1} + {l_2\over v_2} \right]$$

where v ₁ and v ₂ are the velocities of wave propagation inside the cables with lengths l ₁ and l ₂. Hence,

(12)$$d = {s\over c} \left[2d + d_0 \right]$$

where d ₀ is a constant representing the effective length of both the cables given by $d_0 = l_1\epsilon _{r1} + l_2\epsilon _{r2}$:

(13)$$d = {1\over 2}\left({cf_b\over s} - d_0 \right)$$

We have used the modified equation (12) for all our calculations instead of (6). Once we know the value of the constant d ₀, we can calculate the distance d of the target from the antennas using (13). The constant d ₀ for both the environments as well as for the through-wall setup is 55.97 m.