Comparison of the measured  and computed dominant mode resonant frequency of an RMPA.
In modern microwave systems, rectangular microstrip patch antennas (RMPAs) are probably the most investigated topics among the planar antennas. There are several methods available in literature, for designing and analyzing such antennas, but most of them are very complex and give only approximate results. In this chapter, we have discussed the most accurate and updated computer-aided design (CAD) formulations related to probe-fed RMPA for computing its fundamental input characteristics (resonant frequency and input impedance) and improving radiation characteristics, i.e. gain and polarization purity (the parameter that signifies how much an RMPA is free from spurious modes). These formulations have evolved in the last decades and have been validated against numerous simulations and measurements. The present CAD formulas for resonant frequency and input impedance can accurately address a wide range of RMPA with patch width to patch length ratio (W/L) from 0.5 to 2.0, a substrate having thickness up to 0.23 λg where λg is the guide wavelength and relative permittivity (εr) ranging over 2.2–10.8. The role of a finite air gap on resonant frequency and gain of an RMPA have also been presented. The chapter will be surely useful to antenna designers to achieve a concrete understanding of the RMPA theory.
- rectangular microstrip antenna
- resonant frequency
- input impedance
- polarization purity
- grounded microstrip patch
‘Microstrip antennas’, the class of antennas which has been capturing the attention of the antenna research community for the last 63 years, starting from the 3rd Symposium on the US Air Force Antenna Research and Development Program, was proposed firstly by Deschamp and Sichak  in 1953. In , they proposed a microstrip feeding network for a waveguide system which comprised of 300 waveguide horn antennas. But, the ‘microstrip patch’, to which all the researchers associated to the antenna theory are familiar with, was theoretically analyzed by Howell in  and applied into practical applications by Munson in  for the first time. Still, the credit of the authors of the work  was to foster a very much new discipline of antenna engineering, and in addition to this, they highlighted the related performance superiority of this new antenna over the other commercially available conventional antenna. From 1972 onwards, researchers started to understand a microstrip patch in two approaches: (i) treating the patch as a lossy and open resonator cavity  and (ii) as an extended section of a microstrip line .
In the last four decades, several books [4–11] and collection of research papers [12, 13] have been published unfolding rigorous analytical and numerical techniques dealing with microstrip antennas. The computational methods like FDTD, FEM and MOM are very much versatile in nature particularly in analyzing irregular-shaped patch geometries with huge various types of substrates, but any of them give neither any physical insights into the radiation mechanism of the antenna nor any closed-form design formulations which are utmost necessary to any practicing antenna engineer, researcher, academician or scientist. In this context, cavity resonator model appears to be more effective than the other available methods to estimate the fundamental input characteristics (i.e. resonant frequency and input impedance) and to improve the crucial radiation characteristics (i.e. gain and polarization purity) for commonly available microstrip antennas of regular geometries with thin substrate. This method not only improvises the design steps of antenna design but also aids in achieving better performance.
In this chapter, the authors have presented comprehensive electromagnetic analyses on the fundamental input characteristics (i.e. resonant frequency and input impedance) and radiation characteristics (i.e. gain and polarization purity) of rectangular microstrip antennas with conventional and suspended geometries in light of the versatile cavity model method and discussed some improved and closed-form computer-aided design (CAD) formulas. Unlike other theories and work, the present CAD formulas can accurately address a wide range of aspect ratio or patch width to patch length ratio (
Air gap over the ground plane is shown in Figure 1. The variable air-gap height
2. Input characteristics
2.1. Resonant frequency
After the work of Howell , Hammerstad  proposed comprehensive CAD formulations on resonance characteristics of an RMPA using the cavity resonator model (CRM) method. Till now, several theoretical analyses employing CRM method [6, 16–18], transmission-line method [19, 20], method of moments [21, 22] and integral equation technique  are available in literature. But the CAD formulas presented in [6, 15] are found to be the most popular for the design purpose.
Nevertheless, a close inspections into these works show that the formulas available in [6, 15] can provide a reasonably good approximation only when the patches have aspect ratio (W/L) near to 1.5 and the substrate thickness is lower than 0.02 λg, where λg is the guide wavelength at the resonant frequency
An empirical relation is used to determine Δ
Figure 2 shows the computed as obtained from the previous work of Chattopadhyay in [14, 35] and measured  resonant frequencies of an RMPA as a function of substrate thickness
In , Chattopadhyay et al. have also shown the close agreement between the values obtained using MOM and their theory.
The simulated and measured values ranging from no air gap to an air gap of 4 mm show good agreement with the present formulations. The tunability of the RMPA as a function of the air gap height has also been studied. Table 1 compares the resonant frequencies as computed using the presented formulations (as given in the previous work of Chattopadhyay in [14, 35]) with that of Hammerstad , James et al.  and Chew and Liu  for different sets of RMPAs with
|Length, ||Normalized thickness (||Measured ||Computed ||Computed ||Computed ||Computed |
|38||1.5||0.037||2.31||2.38 (3.03%)||2.30 (0.4%)||2.37 (2.6%)||2.32 (0.4%)|
|30.5||1.49||0.047||2.89||2.90 (0.3%)||2.79 (3.4%)||2.90 (0.3%)||2.83 (2%)|
|19.5||1.51||0.068||4.24||4.34 (2.35%)||4.11 (3.06%)||4.32 (1.88%)||4.18 (1.4%)|
|13||1.5||0.094||5.84||6.12 (4.79%)||5.70 (2.39%)||6.07 (3.93%)||5.86 (0.3%)|
|11||1.54||0.110||6.80||7.01 (3.08%)||6.47 (4.85%)||6.90 (1.5%)||6.65 (2.2%)|
|9||1.55||0.125||7.70||8.19 (6.36%)||7.46 (3.11%)||7.87 (2.2%)||7.73 (0.38%)|
|8||1.50||0.141||8.27||9.01 (8.94%)||8.13 (1.7%)||8.39 (1.45%)||8.50 (2.7%)|
|7||1.50||0.148||9.14||9.97 (9.01%)||8.89 (2.73%)||8.69 (4.92%)||9.3 (1.75%)|
|6||1.50||0.166||10.25||11.18 (9.07%)||9.82 (4.19%)||10.4 (1.46%)|
|Average error w.r.t measurement |
Moreover, in  Chattopadhyay et al. showed the versatility of these formulations in accurately predicting the higher-order modes of an RMPA for W/L = 1. In , Chattopadhyay has predicted the higher-order modes of an RMPA for W/L = 0.7, 1.2, 1.5, 1.7 extending the work in . One can refer to Table 1 of one of the previous works of Chattopadhyay et al. in  for a closer look into the topic. It is seen that the significant higher-order modes of an RMPA are TM01, TM02, TM12, TM20, TM30, TM03, etc. When W/L = 1, TM10 and TM01 become degenerate modes. The separation between resonant frequency of dominant TM10 mode and that of net higher-order mode TM02 is from 2 to 1.25
2.2. Input impedance
An RMPA can be represented as an equivalent R-L-C parallel resonant circuit in order to find out its input impedance . Near resonance of the dominant mode and its input impedance can be expressed as [39, 40]
In this context, another parameter
The same approach is also valid for circular patches, and a detailed discussion on the resonant frequency and input impedance of a circular patch can be found in . From , one can find that the dominant mode of a circular patch is TM11. The immediate higher-order modes are TM21, TM01, TM31, etc. The formulas are found to be very accurate in case of substrate with thin and moderate height.
Cavity model analysis of the resonant frequency and input impedance for a 60°–60°–60° equilateral triangular patch is found in . The dominant mode of a triangular patch is TM10 . The immediate higher-order modes are TM11, TM20, TM21, etc.
(a) The resonant frequency of the patch can be obtained from Eq. (3.1) as
As the dominant mode is TM10,
The expression for effective relative permittivity of the medium below the patch is
As air gap height
The fringing length
Therefore, the resonant frequency
(b) Now, if
Therefore, the resonant frequency
(a) The input impedance of a patch can be expressed as
=0.001 as W = 30 mm which is smaller than 0.35
Therefore, the resonant resistance at edge (
(b) To obtain the optimum feed point, we need to find the point where input impedance of the patch becomes 50 Ω. From part (a), we get
λg = 95.84 mm
Putting these in Eq. (3.3), we may write
4. Radiation characteristics
4.1. Gain enhancement
Any RMPA has a strong influence of substrate permittivity (
The gain of an RMPA loaded with air substrate is directly related to its effective radiating area
Now, the standard formula of gain (G) of any rectangular aperture is (as given in one of the earlier works of the authors in )
It is seen that when air substrate is used in lieu of PTFE substrate, the electric field lines along the patch edges becomes more relaxed or loosely bound resulting in an increase in
The formulations presented in this section are well validated against simulations and measurements [36, 41]. These formulations are found to be very much accurate for L-Ku band and for wide range of aspect ratios. Figure 6 shows increase in gain when PTFE substrate is replaced by air substrate for
Theoretically computed, simulated and measurement results show very close agreement among themselves.
4.2. Polarization purity
In general, a conventional RMPA radiates in the fundamental TM10 mode along the broadside of the element, and the field is primarily linearly polarized, called co-polarized (CO) radiation. However, some orthogonally polarized, called cross polarized (XP), radiations take place due to weak oscillations of higher-order modes inside an RMPA. The XP radiation becomes considerably prominent for probe-fed designs particularly when the thicknesses as well as the dielectric constant of the substrate increase. Thus, the XP radiation becomes an important issue for investigation for microstrip antenna research. The (XP) fields are more significant in H plane than in E plane as obtained in our earlier work in . Therefore, the polarization purity (CO-XP isolation) deteriorates in H plane (only 9 dB), and the suppression of XP radiation performance of an RMPA to improve its polarization purity is the challenging issue for antenna research community. Lower polarization purity also limits the use of RMPA in different array applications There are several techniques to improve polarization purity of an RMPA such as the use of defected ground structure (DGS) , grounding the non-radiating edges of a patch [43, 45] and defected patch surface . A thorough discussion on DGS-integrated RMPAs can be found in [11, 42]. However, DGS-integrated RMPAs always possess high back radiation, and only 15–20 dB of CO-XP isolation in H plane can be obtained from those [11, 42]. The two later techniques can address the limitations of DGS and minimum 25 dB of CO-XP isolation from those, and these are discussed clearly in this section. The two techniques are very simple to understand and very effective to implement over a wide microwave frequency range (L-Ku band).
An RMPA with three pairs of shorting plates placed at the non-radiating edges is shown in Figure 7. If the non-radiating edges are grounded using pairs of thin strips, the EM boundary conditions get altered and result in a significant change in field structure with in the cavity. Hence, the restructuring of field structure within the patch inevitably modifies the radiation properties of the RMPA. Some recent work in [43, 44] show the XP radiations are typically from the non-radiating edges of the RMPA. In fact, the oscillations of electric field beneath the patch in a direction, orthogonal to E plane, produce higher-order orthogonal resonance (higher-order orthogonal modes). The XP radiations are typically due to those higher-order orthogonal modes and the fields of those modes, located near the non-radiating edges of the patch. The electric field vectors for a grounded patch in TM
The electric surface current over the patch surface can be obtained from our earlier work in  as
The co-sinusoidal variation in Eq. (4.5) shows the variation of
The use of grounded strip loading in a conventional RMPA not only modifies the radiation property but also regulates the input characteristics of the RMPA . The length (
which comes parallel to patch input impedance (
Therefore, from 
The expression for reactive impedance can be written as 
The input impedance of conventional probe-fed RMPA can be written as
Putting the values of
From our earlier work in , it is observed that when grounded strips are placed along the non-radiating edges, the structure becomes thick dipole loaded (dipole length to diameter ratio ~ 4.2 around resonant frequency), and it prevents the usual sharp variation of input reactance over the operating bandwidth. It is found from our previous work in  that the reactance of thick dipole slowly varies with the frequency, and as it is in parallel to patch reactance, the resultant reactance of the proposed patch varies slowly with frequency (Figure 8).
The formulations presented in this section were validated in case of an RMPA with length
Three pairs of thin copper strips of thickness 0.1 mm with height of
In E plane, CO-XP isolation is more than 35 dB (Figure 10).
Eqs. (4.20 and 4.21) show that at z = 0 and at
Hence, the null occurs in between these limits, i.e. 
From Eq. (4.22) one can write that
Along the middle section of the patch, i.e. when
Hence, from our previous work in , we can write
A defect can be incorporated within this region, i.e. from
The electric surface current (
In case of an RMPA with length
In this chapter, some recent developments in the CAD techniques have been presented lucidly but thoroughly for rectangular microstrip antennas. The presented formulations are very much accurate and are valid for wide range of aspect ratios and substrate thickness compared to other formulations. It is hoped that the work would be helpful for researchers and engineers working in the field of microstrip antennas and will help them to gain an insight into the physics of any RMPA.
Authors would like to express their deep sense of gratitude to Prof. L. Lolit Kumar Singh of Mizoram University, Mizoram; Prof. Gautam Das of Siliguri Institute of Technology, West Bengal; Prof. Debatosh Guha and Dr. Jawad Y. Siddiqui of the Institute of Radio Physics and Electronics, Calcutta University and Prof. B. N. Basu of Sir. J. C. Bose School of Engineering, Mankundu, West Bengal, India, for fruitful discussions during the preparation of the manuscript.
Subhradeep Chakraborty thanks Prof. Santanu Chaudhury, Director, CSIR-CEERI, Pilani; Dr. S. N. Joshi, Ex-Emeritus Scientist; Chirag P. Mistry, Scientist, TWT Group; Dr. Amitavo Roy Choudhury, Senior Scientist, TWT Group; Dr. Sanjay Kumar Ghosh, Principal Scientist and Head of TWT Group and Dr. R. K. Sharma, Principal Scientist and Head of MWT Division, CSIR-CEERI, Pilani, for always encouraging research endeavours and their support.
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