Analysis Example on the Electronic-Steering 1D FP LWA

The method proposed can be applied to many types of electronically tunable LWAs with continuous scanning at a fixed operation frequency f0, for instance [Lim 2004-I, II], [Archbold 2010], [Sievenpiper 2005]. According to Eq.(2.2.4), the leaky-mode normalized phase constant

/k0 must be properly modified to cover a wide range of scanning angles R at the design frequency f0. In this section, we study the SPBW for the reconfigurable Fabry-Pérot (FP) LWA proposed in this chapter. Previous works of other authors have treated the bandwidth of FP LWAs which are normally narrow-band ([Boutayeb 2007], [Feresidis 2006], [Hosseini 2011], [Lovat 2006-II], [Mateo-Segura 2014], [Yuehe 2012], [Zhao 2005]). However, these studies were focused on fixed (not reconfigurable) two-dimensional (2D) FP LWAs radiating at broadside. However, the study proposed here is totally different since we deal with reconfigurable scanning 1D LWAs.

For the reconfigurable beam-scanning 1D FP LWA designed in this chapter, Fig. 2.9.2(a) shows the leaky-mode dispersion curves as a function of frequency Eq.(2.9.6) for different control voltages (VR). From these dispersion curves and applying equations Eq.(2.9.7) and Eq.(2.9.3), the scanning angle R(f) and the scanning ratio SR(f) responses of the antenna are respectively obtained and plotted in Fig. 2.9.2(b). Note that Eq.(2.9.3) has been readily obtained as a numerical finite difference from the dispersion curves. It is evident that the control signal VR

modifies the frequency dispersion response of the leaky mode, increasing the cutoff frequency as the voltage VR is increased.

(b)

(c)

Fig. 2.9.2 (a) Normalized leaky-mode dispersion curves (β/k0 and /k0) obtained from TEN for the half-space electronic scanning 1D FP LWA and (b) Scanning angle (θR) and scanning ratio (SR) as a function of frequency for different VR cases. (c) Scanning response and frequency

scanning ratio as a function of VR at the fixed design frequency f0=5.6GHz.

Again, it can be observed that if the operating frequency is fixed to f0=5.6GHz, this mechanism provides a continuous electronic control over the scanning angle, as observed in Fig. 2.9.2(c), where theoretical, simulated and measured scanning angle and scanning ratio is represented as a function of VR. The dependence of the scanning ratio SR (Eq.(2.9.3)) on the antenna operating state can be attributed to the non-linear dispersion of the R(f) curves, which experiment an increase of their slope close to the lower cutoff frequency and especially to the upper stopband region. This type of response is common to structures loaded with High Impedance Surfaces

(HIS) ([Sievenpiper 2003, 2005], [García-Vigueras 2012-II]). As a result, the scanning ratio SR for the design frequency of f0=5.6GHz shows a dependence with the control signal, as shown by the red curve in Fig. 2.9.2(c). Particularly, when the antenna is scanning at angles around

These theoretical curves shown in Fig. 2.9.2 are the same obtained from the TRM in previous sections. Simulated and measured results presented in Fig. 2.9.2(c) validate the leaky-mode dispersive response as a function of the control signal (VR). Remembering from previous sections, the prototype presented an aperture length LA=5.2λ0 (280mm) and an approximated usable scanning range from R=+9º to R=+30º, limited by the effect of losses and assuming a minimum realized gain G≥5dBi.

Fig. 2.9.3 Fractional pattern bandwidth (FPBW) as a function of scanning angle (θR) and antenna aperture length (LA) (theoretical lossless case).

Once the dispersion curves have been obtained for different operating states of the reconfigurable antenna, the SPBW can be directly estimated from Eq.(2.9.9) and for different scanning angles. Fig. 2.9.3 shows the relative Fractional SPBW (FSPBW=SPBW/f0) at f0=5.6GHz as a function of the antenna operating scanning angle (θR) and for three different antenna aperture lengths (LA). For a fixed antenna length, it is observed that the SPBW depends on the scanning state, observing higher bandwidth for intermediate scanning angles (θR20º). As previously discussed, this is mainly due to the increase of the SR for scanning angles close to broadside (θR<10º where the leaky-mode lower band cutoff occurs) and angles tending to endfire with θR>30º (where the HIS upper stop band arises). This is an important effect which limits the overall BW of a reconfigurable LWA, and which has not been reported until now. For

10 15 20 25 30 35

instance, the case of LA=5.2λ0 shows a theoretical SPBW=2.88% when scanned at R=20º, which is reduced to SPBW=2.02% for R=10º and to SPBW=2.26% for R=30º.

In addition, it is also appreciated the strong influence of LA, evidencing an inverse relationship with the SPBW as stated by Eq.(2.9.9). This is reasonable since longer LWAs provide reduced half-power beamwidth  (Eq.(2.9.8)), which results in a stronger effect of the frequency beam steering and thus lower SPBW. This can be clearly observed if we fix θR=20º in Fig. 2.9.3, observing a theoretical maximum bandwidth of SPBW=6.65% for LA=2.6λ0, whereas a lower bandwidth of FSPBW=1.66% is achieved for LA=10.4λ0.

Theoretical results of Fig. 2.9.3 are compared with full-wave simulations (plotted with circles), obtained for the ideal (lossless and perfectly matched) reconfigurable antenna prototype. Good agreement with theoretical predictions is obtained despite the simplicity of our formula, which only takes into account the dispersion of the antenna, neglecting directivity or beamwidth variability. In addition, the derived expressions are more accurate for longer antennas which are narrowband and present lower beamwidths, as discussed in the approximations made to derive the formulation (Eq.(2.9.4)-Eq.(2.9.5)). This is observed for the longer studied case (LA=10.4λ0), which simulations almost fit the theoretical curve.

Fig. 2.9.4 Comparison of Fractional PBW as a function of operating scanning point (θR) obtained from theory, ideal lossless simulations and lossy simulations.

All previous results have treated with ideal lossless antennas and neglected mismatching.

However, in the real world these effects must be considered to obtain the real usable bandwidth.

Fig. 2.9.4 compares the FSPBW presented in Fig. 2.9.3 for an ideal antenna with LA=5.2λ0, with the FSPBW obtained from simulations of the final prototype, which have taken into account losses and a real excitation source. From this graph it is observed how our model still remains in

good agreement. Table 2.9.1 compares theory, lossless simulations and lossy simulations of the final prototype for three representative antenna operating states (R=10º, R=20º, R=30º). It is demonstrated good SPBW theoretical prediction with absolute errors lower than 0.5%, whereas lower error is expected for longer antennas, given the results presented in Fig. 2.9.3. In addition, it is demonstrated that the maximum usable bandwidth for this antenna is approximately 2.22%, given by the most restrictive operating states (in terms of bandwidth ) which are observed in this case (R=10º , 30º), which are located at the extremes of the considered scanning range, as

Table 2.9.1 Calculated Scanning Ratio for three scanning angles (θRAD) of the reconfigurable FP LWA (Case LA=5.2λ0).

Finally, to give a better insight about the influence of the gain variation on the SPBW, Fig.

2.9.5(a) shows the measured gain radiation patterns at the three frequencies which define the FSPBW f0, f1 and f2 for each case of Table 2.9.1. In the first state, the LWA operates close to broadside (R=10º), and the gain decreases rapidly when frequency is reduced due to the proximity to the lower cutoff regime of the leaky mode. This gain fluctuation can be verified in Fig. 2.9.5(b) (blue curve), which represents the relative variation of the measured gain at R with frequency normalized w.r.t the maximum gain measured at f0 for each case. The second state (R=20º) presents a softer decay of the gain when frequency is varied, thus obtaining higher bandwidth (green curve in Fig. 2.9.5(b)). The third state (R=30º) presents lower bandwidth as a consequence of the stronger sensitivity of R with frequency (high scanning ratio) and especially the abrupt gain pattern variation as frequency is increased due to the proximity of this operating state to the upper stopband of the leaky mode (red curve of Fig. 2.9.5(b)), where also dissipative losses and mismatching take more effect.

(a)

(b)

Fig. 2.9.5 a) Simulated Realized Gain patterns at f1, f0 and f2 for different scanning configurations b) Variation of the realized gain (simulated from final prototype) for each case as

a function of frequency (values normalized w.r.t. maximum gain measured at central design frequency f0=5.6GHz for each respective case).

0 20 40