# Analytical Model for Predesigning Probe-Fed Hybrid Microstrip Antennas

Authors: Nilson R. Rabelo; J. C. da S. Lacava; Alexis F. Tinoco Salazar; P. C. Ribeiro Filho; D. C. Nascimento; Rubén D. León Vásquez; Sidnei J. S. Sant’Anna
Journal: International Journal of Antennas and Propagation
 (2018)

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## Abstract

Based on the equivalent resonant cavity model, an effective analysis methodology of probe-fed hybrid microstrip antennas is carried out in this paper, resulting in a better understanding of the parameter interrelations affecting their behavior. With that, a new design criterion focused on establishing uniform radiation patterns with balanced 3 dB angles is proposed and implemented. Results obtained with the proposed model closely matched HFSS simulations. Measurements made on a prototype antenna, manufactured with substrate integrated waveguide (SIW) technology, also showed excellent agreement, thus validating the use of the cavity model for predesigning hybrid microstrip antennas in a simple, visible, and time- and cost-effective way.


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## Body

## 1. Introduction

Microstrip antennas and arrays can be accurately designed using modern electromagnetic simulators such as CST [1] and HFSS [2]. However, as their focus is on analysis, the development process becomes more simple and time- and cost-effective when the geometry under study is predesigned in the first place. In this context, to predesign means the determination of preliminary antenna dimensions before implementation in the simulators. Once in the software environment—which incorporates significant effects, such as dielectric and ground plane truncation, that are not taken into account in more basic models—the antenna can then be more comprehensively analyzed and its dimensions optimized to meet project specifications. Naturally, the closer the predesigned dimensions are to the optimal ones, the faster the analysis-synthesis process will converge.Although Deschamps [3] proposed the concept of microstrip radiators back in 1953, it was only in the 1970s, with the production of low-loss microwave laminates that this type of antenna started gained popularity [4] and a number of practical applications came about [5]. Nowadays, their peculiar characteristics are established [6–8] and they are found as customary components in modern communication systems [9]. Analytical methods, such as the transmission line [10], resonant cavity [11], and electric surface current [12] models, have been extensively used for predesigning planar, cylindrical, and spherical microstrip antennas [6–8, 13–15].The conventional probe-fed linearly polarized antenna, comprising a metallic rectangular patch printed on top of a grounded planar dielectric layer, is certainly the most popular microstrip radiator [16], but at the cost of high levels of cross-polarization in the H plane, as recently revisited [5]. A convenient way to overcome this limitation consists of using a hybrid microstrip patch, as described in [9, 16–21]. In this publication, hybrid microstrip antennas fed by a coaxial probe are predesigned via the cavity model. Although this model had been previously utilized [9, 16, 18, 22, 23], the systematic determination of adequate design criteria has not been fully carried out yet. Such is the primary goal of this work.To validate our predesigning procedure, HFSS simulations were run, and excellent agreement with our results confirms the effectiveness of the equivalent resonant cavity model for thin hybrid antennas. Since the implementation of vertical electric walls in microstrip structures is not straightforward, a prototype antenna was manufactured using the substrate integrated waveguide (SIW) technique [24]. Here again, an excellent match between predesigned and experimental results was observed.


## 2. Cavity Model

Differently, from their conventional counterpart, hybrid microstrip antennas fed by coaxial probes can exhibit low cross-polarization level in the H plane, as recently reported in [9, 16–19, 25, 26]. That outstanding behavior is obtained by connecting two opposite edges of a rectangular patch to the antenna ground plane. The typical geometry, proposed by Penard and Daniel [23], is shown in Figure 1, where aa and ba denote the patch dimensions and h is the thickness of the substrate, ε of electric permittivity, and μ0 of magnetic permeability. Note the antenna is fed, at coordinates ya and za, by a SMA (subminiature version A) connector whose characteristic impedance is 50 Ω.Figure 1
Hybrid microstrip antenna: (a) top view and (b) lateral view.
(a)
(b)The resonant cavity model, used for the analysis of conventional microstrip radiators [11], is applied here to the hybrid antenna. In this model, the region between the patch and the ground plane is considered equivalent to a cavity made up of electric walls at x=0, x=−h, y=0, and y=b and magnetic walls at z=0 and z=a, as illustrated in Figure 2. The equivalent cavity dimension along the z-axis shall be made greater than the actual antenna dimension (i.e., a > aa) to account for the fringing effect at the edges [10]. On the other hand, since the walls at y=0 and y=b are electrically grounded to the bottom wall, the dimensions along the y-axis of the equivalent cavity and the actual antenna are the same (i.e., ba = b).Figure 2
Equivalent cavity excited by a strip of uniform current density.By modeling the coaxial feeder by a vertical strip of uniform current density,(1)J→f=Jyδz−zcx^,Jy=J0,ifyc−ℓp2≤y≤yc+ℓp2,0,otherwise,located at the point (yc, zc); as illustrated in Figure 2, the electric field amplitude of the resonant mode {m, n} inside the cavity is given by
(2)Emn=i2ωμ0I0ξnabk2−kmn2sinkyyccoskzzcsincmπℓp2b,where
(3)k2=ω2μ0ε,(4)kmn2=ky2+kz2=mπb2+nπa2,with m=1,2,3,…,m≠0 and n=0,1,2,3,…,ξn=1 if n=0 and ξn=2 if n≠0, ω is the angular frequency and I0=J0ℓp is the current on the feeding strip.Therefore, the total electric field inside the resonant cavity excited by a uniform current density strip is given by the following expression:(5)E→=iωμ0∑m∑nTmnk2−kmn2sinmπybcosnπzax^,where
(6)Tmn=2I0ξnabsinmπycbcosnπzcasincmπℓp2b.Consequently, the cavity input impedance becomes(7)Zin=i2hωμ0ab∑m∑nξnkmn2−k2×sin2mπycbcos2nπzcasinc2mπℓp2b.However, this equation does not properly describe the input impedance of a microstrip antenna. According to [11], more accurate results are obtained if the cavity wavenumber k is replaced with the effective wavenumber kefmn, given by
(8)kefmn2=1−itanθefmnk2,where
(9)tanθefmn=tanθ+δmnhb2h+ba2m2+b3n2a2m2+b2n2+m2ωmnμ0hbξn2εraη0Iint,δmn=2ωmnμ0σ,Iint=∫−π/2π/2∫0πcosmπeik0bsinθsinϕ−1k0bsinθsenϕ2−mπ2cosnπeik0acosθ−12×cos2ϕ+sin2ϕcos2θsinθdθdϕ,k0=ωμ0ε0,with tanθefmn denoting the effective loss tangent, tanθ the loss tangent of the substrate, σ the electrical conductivity of the cavity electric walls, δmn their skin depth, calculated at the resonant frequency of the {m, n} mode, k0 the wavenumber, and η0 the intrinsic impedance of vacuum, and the parameter Iint directly proportional to the radiated power of the {m, n} antenna mode is obtained from the far radiation field of the hybrid microstrip antenna. Here, as in [27], equivalent magnetic sources, positioned along the ungrounded patch walls, lead to the following expression [9, 28]:
(10)E→=k02η0Ih0ae−ik0rrgθ,ϕθ^cosϕ−ϕ^sinϕcosθ,where
(11)gθ,ϕ=∑m∑nmξnkefmn2−kmn2cosmπeik0bsinθsinϕk0bsinθsinϕ2−mπ2cosnπeik0acosθ−1×sinmπycbcosnπzcasincmπℓp2b.Thus, the input impedance of the hybrid microstrip antenna is calculated from(12)Zin=i2hωμ0ab∑m∑nξnkmn2−kefmn2×sin2mπycbcos2nπzcasinc2mπℓp2b.


## 3. Antenna Analysis

In this section, the electromagnetic behavior of the hybrid microstrip antenna is analyzed with the purpose of establishing an effective predesigning procedure. As mentioned in [16, 28], TM10x is the first resonant mode. Since its electric field does not vary along the z-axis, the fringing fields are in phase opposition, thus producing a null in the broadside direction of the antenna radiation pattern (perpendicularly to the yz plane of Figure 1). In addition, its input impedance does not vary with zc, what makes impedance matching difficult. Given these undesirable characteristics, this first resonant mode is not adequate for the usual operation of microstrip antennas. On the other hand, the TM11x mode presents a cosinusoidal distribution along the z-axis over the length a of the patch, thus permitting matching the antenna to the coaxial probe feeder. Since its fringing fields are in phase, the radiation pattern maximum occurs in the broadside direction. These characteristics make TM11x the mode of operation to hybrid microstrip antenna.Since the resonant frequency of theTM11x mode is a function of both physical dimensions of the patch, antennas with different values of a and b can be designed for operation on a given frequency. Design criteria are therefore required for determining the patch dimensions a and b and the position (yc, zc) of the coaxial probe feeder, to guarantee the proper operation of the antenna. For this purpose, three different hybrid radiators (HB1, HB2, and HB3), designed to operate at 2.45 GHz, the central frequency of the ISM (industrial, scientific, and medical 2.4–2.5 GHz) band, are compared. The dimensions of their respective equivalent resonant cavities are shown in Table 1, noting that HB1 has a rectangular patch, with b > a; HB2 patch is square, with b = a; and HB3 has also a rectangular patch, but now with b < a. The substrate used for all three radiators is 1.524 mm thick Arlon CuClad 250GX (εr = 2.55 ± 0.04 and tanθ = 0.0022) microwave laminate.Table 1
Dimensions of the equivalent resonant cavities.
Dimension
HB1
HB2
HB3
a (mm)
40.00
54.19
133.30
b (mm)
133.30
54.19
40.00Initially, the resonant frequencies of the modes that are the closest toTM11x are calculated from (4) and shown in Table 2. Thus, in the case of the HB1 antenna, modes {2, 0} and {2, 1} are the closest to TM11x. For HB2, modes {1, 0} and {2, 0} are closest to TM11x, whereas, in the HB3 case, modes {1, 0} and {1, 2} are the closest.Table 2
Resonant modes closest toTM11x (calculated in Mathematica [29]).
Frequency (GHz)
HB1 (b > a)
HB2 (b = a)
HB3 (b < a)
f10
0.7042
1.7322
2.3467
f11
2.4501
2.4497
2.4501
f20
1.4084
3.4644
4.6934
f12
4.7460
3.8734
2.7369
f21
2.7369
3.8734
4.7460Also from Table2, the frequency offset Δf between the mode TM11x and its closest one is promptly determined. It is, for the HB1 antenna, Δf1 = 286.8 MHz; for HB2, Δf2 = 717.5 MHz; and for HB3, Δf3 = 103.4 MHz. The frequency bandwidth of conventional microstrip antennas operating in the fundamental mode is known to be roughly 1% [6–8] or approximately 25 MHz in the ISM band. Hence, in practical terms, the proximity of modes TM10x, TM20x, TM21x, and TM12x to the TM11x one will not significantly affect the operating frequency bandwidth of electrically thin hybrid antennas. Thus, from the perspective of modal interference, any one of the three designed antennas could be used. Nonetheless, a simple way to suppress modes TM20x and TM21x consists of placing the feeder at yc = b/2, where their electric field is minimal [9, 16, 28]. In such case, only modes {1, 0} and {1, 2} need to be controlled in the antenna design.Consequently, the input impedance at the operation modeTM11x can be rewritten from (12) as
(13)Zin=iωα11k112−kef112+iω∑m,m≠1∑n,n≠1amnkmn2−k2,where
(14)αmn=2μ0hξnabsin2mπycbcos2nπzcasinc2mπℓp2b.With that, the next step consisted of determining the value ofzc such that the input impedance matches the 50 Ω characteristic impedance of the feeding probe SMA connector (Figure 1). Plots of Zin and the absolute value of the reflection coefficient, also obtained in Mathematica from (13) and (14) (for ℓp = 1.3 mm), are shown in Figures 3–5 for the three antennas.Figure 3
HB1 input impedance and reflection coefficient module.Figure 4
HB2 input impedance and reflection coefficient module.Figure 5
HB3 input impedance and reflection coefficient module.In addition, the resulting frequency bandwidth (BW) and quality factor (Q) are presented in Table 3, both calculated at 2.45 GHz.Table 3
Electrical characteristics of the antennas under analysis.
HB1 (b > a)
HB2 (b = a)
HB3 (b < a)
Zin (Ω) (2.45 GHz)
50.05 +i10.72
49.94 +i10.61
50.04 +i6.89
zc (mm)
9.72
21.33
52.66
BW (%)
2.29
0.84
0.47
Q
29.19
79.67
141.35As shown in Table3 and in Figures 3–5, the three hybrid antennas exhibit inductive input impedances at the design frequency (2.45 GHz), as expected from a coaxial probe feed. It is also noticed that the bandwidth for the HB1 antenna is larger than that for the HB2 antenna and that for the HB2 antenna is larger than that for the HB3 one. This is directly related to side b being longer than side a—for the larger the b dimension is, the smaller the input impedance will be at the antenna edges, that is, at zc = 0 or at zc = a, resulting in a smoother dependence of Zin with zc. Besides compromising the antenna impedance matching at the design frequency, an inductive Zin, makes for an asymmetrical bandwidth around the center frequency, thus reducing its symmetrical operating bandwidth (i.e., |Γ| < −10 dB for the same frequency spacing both left and right of the design frequency). As shown in Figures 3–5, the bandwidth comes from 56 down to 38 MHz, for HB1; from 21 down to 14 MHz, for HB2; and from 12 to 10 MHz, for HB3. Besides, the larger the antenna bandwidth, the lower its Q factor, as expected from the product BW × Q = 0.6667. Thus, from the point of view of frequency bandwidth, the design of hybrid antennas should achieve a < b. The effect of this criterion on the radiation pattern of hybrid antennas is analyzed next.Since now only modes {1, 0} and {1, 2} need to be controlled, the far electric field can be rewritten from (10) and (11) as [9, 28]
(15)E→=A0fθ,ϕ∑n=02ξnkef1n2−k1n2cosnπzca×cosnπeik0acosθ−1sinϕcosθϕ^−cosϕθ^,where
(16)A0=k02η0Ih0ae−ik0rrsinπycbsincπℓp2b,fθ,ϕ=eik0bsinθsinϕ+1k0bsinθsinϕ2−π2.Since thez-axis of the adopted rectangular coordinate system is normal to the magnetic currents at the ungrounded edges of the patch (Figure 1), the Eθ component in (15) defines the copolarization of the hybrid antenna, whereas the cross-polarization is given by Eϕ. Implementing (15) in Mathematica, radiation patterns for the HB1, HB2, and HB3 antennas were plotted at 2.45 GHz as shown in Figures 6–8. It is noticed that the patterns are asymmetrical in the E (xz) and yz planes.Figure 6
Normalized radiation pattern of the HB1 antenna.Figure 7
Normalized radiation pattern of the HB2 antenna.Figure 8
Normalized radiation pattern of the HB3 antenna.Analysis of Figure1 indicates that the coaxial feeder location introduces an asymmetry in the antenna geometry, by making the field distribution asymmetrical within the equivalent resonant cavity, which what reflects in the radiated field. One also notices, however, that in the H (xy) plane the antenna is perfectly symmetrical. The 3D radiation patterns in Figure 9 permit a clearer visualization.Figure 9
3D radiation patterns of HB1, HB2, and HB3 antennas.
(a)
HB1 antenna
(b)
HB2 antenna
(c)
HB3 antennaShown in Table4 are the results from the Mathematica simulation of the directivity (D) and radiation efficiency (RE) of the antennas under analysis. One notes that HB1 and HB2 are equally directive. Nonetheless, the total field structure in HB2 is more uniform, with better balanced 3 dB angles, circa 69o in the E (xz) plane, and 80.5o in the H (xy) plane, as opposed to the wide discrepancy between those angles for antenna HB1, that is, 99.88° in the E (xz) plane and 53.5° in the H (xy) plane. It is also noticed in Figure 9 that the radiation pattern of HB3 shows considerable secondary lobes, which what makes its directivity less than that of the HB1 and HB2 antennas.Table 4
Directivity and radiation efficiency.
HB1 (b > a)
HB2 (b = a)
HB3 (b < a)
D (dB)
8.33
8.23
7.39
Dcop (dB)
8.98
8.84
8.85
RE (%)
90.4
74.1
50.9Another relevant parameter is the copolarization directivityDcop, promptly calculated in the adopted coordinate system. In this case, only the Eθ component is taken into account. The difference between Dcop and the directivity D is due to the intensity of the antenna cross-field (component Eϕ), which varies substantially from one antenna to another (Figure 9(B)), although it is null on the E and H planes. This means a hybrid microstrip antenna does not exhibit cross-polarization on the main radiation planes xz and xy, differently from the significant cross-polarization level on the H plane of a conventional antenna [5]. Such is a relevant property of hybrid antennas. Also shown in Table 4 are the results obtained for the radiation efficiency; that is, the HB1 antenna outweighs both, whereas HB3 is the worst.From these considerations, if the goal is to provide an operation equivalent to the conventional antenna, the hybrid antenna design should go for a modified square patch, withb ≥ a, for, in this case, the antenna directivity will be in the order of 8 dB, its radiation efficiency close to 80%, frequency bandwidth around 1%, and 3 dB angles balanced in the E and H planes. The complete antenna design will be accomplished in the next section.


## 4. Antenna Design

Given the conditionb ≥ a, a hybrid antenna (HB) was designed for operation in the same frequency (2.45 GHz) of the antennas that were analyzed in the previous section. For the substrate, the 1.524 mm thick Arlon CuClad 250GX (εr = 2.55 and tanθ = 0.0022) was used again. Through Mathematica, the following dimensions were obtained: a=50.00mm, b=59.59mm, yc=b/2, and xc=19.05mm. In this case, the resonant modes closest to TM11x are TM10x at f10 = 1.575 GHz and TM20x at f20 = 3.150 GHz. For a frequency bandwidth in the order of 1%, the antenna design is good enough in this respect.Input impedance and the reflection coefficient module, forℓp = 1.3 mm, are plotted in Figure 10. As expected, the HB input impedance turns out to be inductive at the design frequency (Zin = 50.30 + i13.99 Ω), so the best matching occurs above 2.45 GHz. Consequently, the symmetrical passband of the antenna, in relation to the central operating frequency, goes down from 25 MHz to 14 MHz, as shown in Figure 10.Figure 10
HB input impedance and reflection coefficient module.Results for the radiation patterns in the principal planes, at 2.45 GHz, are shown in Figure11. As expected, the antenna is asymmetrical in the E plane. For a better visualization of this effect, 3D patterns, at the same frequency, are presented in Figure 12.Figure 11
Normalized radiation pattern of the HB antenna.Figure 12
3D pattern for the HB antenna.
(a)
Copolarization
(b)
Cross-polarization
(c)
Total fieldAs intended, the design process produced a uniform radiation pattern, with balanced 3 dB angles: circa 75.63o in the E plane and 78.50o in the H plane. In addition, the antenna shows 7.9 dB directivity, 78.6% radiation efficiency, and 1.02% relative frequency bandwidth, all calculated at 2.45 GHz.As noticed from Figure12(b), no cross field exists in the broadside direction or along the E and H planes. Rather, it is more intense close to the antenna ground plane and on the planes that bisect the quadrants formed by planes (xy) and (xz). Consequently, its most significant effect consists of “beefing up” the total field pattern in the neighborhood of the ground plane, thus lowering the antenna directivity (to circa 7.9 dB) relative to the copolarization (Dcop), calculated as 8.6 dB, in this case.The asymmetrical radiation pattern in theE plane is now analyzed. Since the E plane of a hybrid antenna, as shown in Figure 1, coincides with the xz plane of the adopted coordinate system, its far electric field is given by making ϕ=0o in (15). Thus, the following expression for the normalized Eθ component results, given Eϕ is zero on this plane [28],
(17)eθ=1kef102−k102+2cos2πzc/akef122−k122eik0acosθ−1−2cosπzc/akef112−k112eik0acosθ+1,which can be rewritten as
(18)eθ=2ie10+e12sink0acosθ2−e11cosk0acosθ2eik0acosθ/2,where
(19)e11=2cosπzc/akef112−k112,(20)e10=1kef102−k102,(21)e12=2cos2πzc/akef122−k122.From (13), mode {1, 0} and mode {1, 2} characteristics are seen to be opposite from the primary mode; that is, at the operating frequency, e11 is imaginary, whereas e10 and e12 are real. In the case of the HB antenna at the operating frequency, they are e11 = i0.00696845, e10 = i0.000253281, and e12 = i0.000123797. Thus, in the first quadrant, where θ ranges from 0 to 90 degrees, the terms sin[(k0acosθ/2)] and cos[(k0acosθ/2)] are positive, so their subtraction lowers the amplitude of eθ relative to the primary mode amplitude e11cos[(k0acosθ)/2]. In the second quadrant, on the other hand, where θ ranges from 90 to 180 degrees, a negative cosθ changes the sign of the term sin[(k0acosθ)/2]; consequently, the amplitude of eθ is now larger than the primary mode one e11cos[(k0acosθ)/2], resulting in an asymmetrical radiation pattern in the E plane. In addition, analysis of (19), (20), and (21) shows that e10 is not dependent on zc, but e11 and e12 are. This fact is directly related to the resonant mode field distribution along the plane yc = b/2. In fact,
(22)Exb2,zmn=Emncosnπzca.Therefore, the excitation of mode {1, 0} does not depend on the feeder positionzc (along the yc = b/2 plane), but zc substantially affects the level of {1, 1} and {1, 2} modes, thus becoming one of the causes of the E plane radiation pattern asymmetry of hybrid microstrip antennas. Normalized E plane radiation patterns for the HB antenna are shown in Figure 13 for different feeder positions at 2.45 GHz. They clearly show that pattern asymmetry increases with zc. That is, the lower the antenna input impedance is, the more asymmetrical the E plane pattern is. This fact is noticeable from (18), since the larger zc is, the smaller the contribution from the e11 term, whereas the larger that from e12 will be for a fixed e10.Figure 13
NormalizedE plane radiation pattern of the HB antenna: zc = 5 mm—blue curve; zc = 10 mm—red curve; zc = 15 mm—orange curve; and zc = 20 mm—green curve.


## 5. HFSS Comparison

In order to validate the analysis and design procedures set forth, simulations were run in HFSS. The initial predesigned dimensions of the HB antenna, adjusted according to Hammerstad [30], are presented in Table 5. For the HFSS simulations, the antenna was centered on a ground plane of dimensions Wz × Wy, where the subscripts indicate the ground plane sides parallel to the coordinate axes z and y. It is noticed from Table 5 that the predesigned dimensions are very close to the ones simulated via HFSS, besides being obtained in a significantly reduced processing time.Table 5
HB antenna dimensions (Wz=140mm; Wy=140mm).
Dimension
Hammerstad
HFSS
aa (mm)
48.45
48.28
ba (mm)
59.59
59.59
ya (mm)
29.80
29.80
za (mm)
18.27
18.18Results for the antenna input impedance are shown in Figure14, whereas the comparisons between the predesigned results and those obtained via HFSS are presented in Table 6.Figure 14
Input impedance of the HB antenna.Table 6
Input impedance at 2.45 GHz.
HB
Zin (Ω)—equivalent cavity
50.30 +i13.99
Zin (Ω)—HFSS
50.02 +i10.36The good agreement between these results confirms the effectiveness of the equivalent resonant cavity for predesigning hybrid antennas. Nevertheless, forℓp = 1.3 mm, the predesigned impedance turns out to be more inductive than the HFSS simulation result. One way to reduce the input inductive reactance consists of increasing the effective width of the current strip feeder. The effect of different values of ℓp is also plotted in Figure 14, showing the optimal ℓp value is somewhere between 1.6 and 2.8 mm. Curves for Zin calculated for ℓp = 2.3 mm are presented in Figure 15.Figure 15
Input impedance of the HB antenna:ℓp = 2.3 mm.Last, radiation patterns in theE (xz plane—in blue) and H (xy plane—in red) planes are shown in Figure 16. The HFSS patterns were simulated for an infinite ground plane. The excellent agreement confirms once again the effectiveness of the equivalent resonant cavity model for predesigning hybrid antennas.Figure 16
Normalized radiation pattern in theE and H planes of the HB antenna: solid line—Mathematica; dotted line—HFSS.It is worth mentioning that the HB antenna, although electrically thin at 2.45 GHz, shows an inductive input impedance,Zin = 50.26 + i11.78 Ω (ℓp = 2.3 mm), which shifted up the best matching frequency, causing a significant reduction of its symmetrical operating bandwidth. In the following section, the HB antenna will be optimized at the operating frequency in terms of impedance matching to its SMA connector feeder.


## 6. Project Optimization for Null Reactance

A very effective way to match the antenna to the 50 Ω characteristic impedance of its SMA connector feeder, without any external resource, consists of adjusting the antenna design for the null reactance condition [31]. With that, the following dimensions were obtained for the equivalent cavity: a=50.06mm, b=59.75mm, and yc = b/2 e zc = 18.95 mm. The resulting input impedance and reflection coefficient module are shown in Figure 17.Figure 17
Input impedance and reflection coefficient module.After its redesign for null reactance, the antenna is much better matched to its feeding SMA connector, withZin = 50.19 − i0.15 Ω at 2.45 GHz. It is also noticed the 26 MHz (circa 1.06%) bandwidth is now symmetrical around the design frequency. Other electrical characteristics remain very close to the previous HB design.The antenna dimensions after adjusting per Hammerstad are shown in Table7, whereas the results for the input impedance and the reflection coefficient module are superimposed in Figure 17. Once again, the excellent agreement between predesigned and HFSS results confirms the effectiveness of the equivalent resonant cavity model for predesigning hybrid antennas.Table 7
Antenna dimensions for null reactance design. (Wz=140mm; Wy=140mm).
Dimension
Hammerstad
HFSS
aa (mm)
48.51
48.31
ba (mm)
59.75
59.75
ya (mm)
29.88
29.88
za (mm)
18.17
18.05


## 7. SIW Prototype

To validate further the proposed design approach, a prototype antenna was built and tested, as described in this section. Given the implementation of vertical electric walls through the substrate in microstrip structures is not an easy task, an effective alternative approach is the use of SIW technology. In the present case, the vertical metallic walls are implemented with a sequence of cylindrical pins, as illustrated in Figure18, in which Δp is their center-to-center spacing. The SIW antenna dimensions were determined from the values presented in Table 7.Figure 18
SIW antenna geometry: top view.First, theb dimension was determined in order to make beff equal to 59.75 mm, based on the following relationship,
(23)beff=b−d20.95Δp,set up in [32] for the propagation of the TM01 mode in SIW guiding structures, where d denotes the pin diameter. After further optimization in HFSS, the following dimensions were obtained: aa = 47.63 mm, ba = 60.52 mm, ya = 29.875 mm, and za = 17.64 mm, with d = 0.508 mm and Δp = 4.266 mm. Based on those, radiation patterns in the E (xz plane—in blue) and H (xy plane—in red) planes, simulated in HFSS, are presented in Figure 19.Figure 19
Normalized radiation pattern of the SIW antenna.Results for the input impedance and the reflection coefficient module are presented in Figure20. Other electrical characteristics of the SIW antenna are the following: 49.99 − i0.79 Ω input impedance, 80.5% radiation efficiency, 8.02 dB directivity, and balanced 3 dB angles in the E and H planes: circa 76° on the E plane and 78° on the H plane, consistently with the predesigned values.Figure 20
Input impedance and reflection coefficient module of the SIW antenna, designed under the null reactance condition.With those dimensions established in HFSS, a prototype antenna was manufactured, as shown in Figure21, and tested. Experimental results for the input impedance and reflection coefficient module are shown in Figure 20, overlaid to the simulation results. As noted, the prototype resonant frequency was 16 MHz below requirement (2.45 GHz). Confidence in the simulation results and in the manufacturing process led us to believe this effect could be caused by a printed circuit board (PCB) permittivity shift from its nominal value, specified as εr = 2.55 ± 0.04. To check this hypothesis, a conventional, linearly polarized rectangular microstrip antenna, fed by a 50 Ω SMA connector, was designed to operate at 2.45 GHz and manufactured from the same PCB lot (Figure 22).Figure 21
SIW antenna prototype.Figure 22
Conventional microstrip antenna prototype.This design option was based on ease of construction and numerous previous successful implementations. From HFSS simulation, the following dimensions resulted the following:a=40mm, b=52mm, and p=12.45mm. Simulated and experimental results for the input impedance and the reflection coefficient module of the conventional antenna are presented in Figures 23 and 24.Figure 23
Input impedance of the conventional microstrip antenna.Figure 24
Reflection coefficient module of the conventional microstrip antenna.As noticed in this simple case, the resonant frequency of the prototype antenna is still 16 MHz below the expected HFSS simulation, thus confirming the hypothesis on permittivity variation. Since the resonant frequency shifted down, the actual permittivity of the laminate is greater than 2.55. Further HFSS simulation for a range ofεr values closed on 2.583, as pictured in Figures 23 and 24.Having confirmed the cause of the shift, the SIW antenna was simulated again, but now forεr = 2.583. Results for the input impedance and the reflection coefficient module are presented in Figures 25 and 26. This time, an excellent match between the simulated and experimental results is observed.Figure 25
Input impedance of the SIW antenna.Figure 26
Reflection coefficient module of the SIW antenna.Radiation patterns in theE and H planes at 2.434 GHz are presented in Figures 27 and 28. As noticed, experimental and HFSS co-pol patterns on the E and H planes show a good match. Simulated cross-pol patterns are not plotted since they are below −40 dB. The higher level of the measured cross-polarization patterns relative to their simulation can be traced to the lack of a balun for the antenna under test. Results are good regardless, as expected for hybrid antennas.Figure 27
NormalizedE plane radiation pattern.Figure 28
NormalizedH plane radiation pattern.


## 8. Final Comments

An efficient procedure based on the equivalent resonant cavity model for fast and accurate predesign of probe-fed hybrid microstrip antennas is proposed in this article. This procedure, implemented inMathematica in a straightforward way, has provided a comprehensive understanding of the effect of the electrical and geometrical parameters involved in the antenna analysis and synthesis, thus becoming a powerful tool for educational purposes. The proposed design criteria were focused on establishing an operation equivalent to the conventional antenna, but now with uniform radiation patterns in all planes, that is, balanced 3 dB angles. Besides, as the antenna is fed by a 50 Ω SMA connector, the zero input null reactance condition was used for proper impedance matching, resulting in a symmetrical bandwidth around the design frequency. Moreover, according to the rectangular coordinate system adopted, the Eθ component directly defines the copolarization of the hybrid antenna, whereas the cross-polarization is given by Eϕ, thus facilitating their analysis in 3D patterns. Additionally, the asymmetry of the E plane radiation pattern was addressed, indicating that the lower the antenna input impedance is, the more asymmetrical the E plane pattern will be. Finally, it is important to notice that, differently from their conventional counterparts, hybrid microstrip antennas fed by coaxial probes exhibit low cross-polarization level in the H plane.Predesign results obtained with the proposed model for the hybrid radiator closely matched HFSS simulations, as well as actual measurements in a prototype that was built and tested. The excellent agreement validates the use of the cavity model for predesigning hybrid microstrip antennas in a simple, accurate, and time- and cost-effective way.Since the practical implementation of vertical electric walls in microstrip structures is not an easy task, the SIW technique was used in the manufacturing of the prototype antenna, showing very good results.



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*Source: 1893650-2018-02-28.xml*