# Modal Resonant Frequencies and Radiation Quality Factors of Microstrip Antennas

Authors: Jan Eichler; Pavel Hazdra; Miloslav Capek; Milos Mazanek
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

The chosen rectangular and fractal microstrip patch antennas above an infinite ground plane are analyzed by the theory of characteristic modes. The resonant frequencies and radiationQ are evaluated. A novel method by Vandenbosch for rigorous evaluation of the radiation Q is employed for modal currents on a Rao-Wilton-Glisson (RWG) mesh. It is found that the resonant frequency of a rectangular patch antenna with a dominant mode presents quite complicated behaviour including having a minimum at a specific height. Similarly, as predicted from the simple wire model, the radiation Q exhibits a minimum too. It is observed that the presence of out-of-phase currents flowing along the patch antenna leads to a significant increase of the Q factor.


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## Body

## 1. Introduction

Evaluation of the basic properties of microstrip patch antennas (MPA) has been numerously discussed in literature, see, for example, [1–3]. The two main MPA attributes are resonant frequency (or frequencies but we will deal mostly with the dominant mode) and the radiation Q factor. So far, only approximate results and semianalytic equations have been published. To our knowledge, this is the first time that these important characteristics have been studied in a rigorous way. The antennas are treated by using a modal approach (hence we do not a priori consider any feeding to be connected), namely, by the theory of characteristic modes (TCM), [4, 5]. Evaluation of the radiation Q is performed both by the TCM from the eigenvalues slope and by novel rigorous equations derived by Vandenbosch [6] and Vandenbosch and Volski [7].


## 2. The Theory of Characteristic Modes

For completeness, let us formulate the basics of the characteristic modes for perfectly conducting bodies of areaS. The scattered field Es is related to the electric surface currents J by the electric field integral equation (EFIE) [8](1)[L(J)-Ei]tan⁡=0.
Equation (1) is usually treated within the method of moments (MoMs) [8] framework and, due to the structure discretization, the L operator is known as the “complex impedance matrix” [Z]=[R]+j[X].Then the associated Euler’s equation to be solved is(2)XJn=λnRJn.
Equation (2) is a standard weighted eigenvalue equation leading to a set of real characteristic eigencurrents Jn and associated eigenvalues λn. Properties of eigenvalues are described in [9], at this moment it is important to note that λn  reflects the amount of net reactive power (thus λn=0 means resonance). Instead of eigenvalues, the so-called characteristic angles αn are introduced to show more visible behavior with frequency [9]. Characteristic currents form a complete orthogonal set, and hence the total current on a conducting body may be expressed as a linear combination of these mode currents [10].


### 2.1. Implementation of the Characteristic Modes Theory

Implementation of the modal decomposition process has been done in the MATLAB [11] environment using Makarov EFIE codes [12] with the RWG basis functions [13]. This usage is restricted to arbitrary 3D PEC structures with air dielectrics. Our developed TCM tool [14] has the following main advantages:(i)
Comsol Multiphysics [15]/MATLAB’s PDE TooLbox mesh import,(ii)
Optional Green’s function for infinite ground plane simulations,(iii)
Single solver/multicore solver/distributed solver (within a computer network with installed MATLAB).


## 2.1. Implementation of the Characteristic Modes Theory

Implementation of the modal decomposition process has been done in the MATLAB [11] environment using Makarov EFIE codes [12] with the RWG basis functions [13]. This usage is restricted to arbitrary 3D PEC structures with air dielectrics. Our developed TCM tool [14] has the following main advantages:(i)
Comsol Multiphysics [15]/MATLAB’s PDE TooLbox mesh import,(ii)
Optional Green’s function for infinite ground plane simulations,(iii)
Single solver/multicore solver/distributed solver (within a computer network with installed MATLAB).


## 3. The RadiationQ Factor

In [6] a novel theory able to rigorously calculate radiated power and stored energies directly from currents flowing along the antenna has been presented. The radiation Q factor is then readily evaluated by the definition [16]:
(3)Q=2ωmax⁡⁡(W̃m,W̃e)Pr.
The equations for radiated power Pr and stored electric and magnetic energies W̃e, W̃m are
(4)Pr=(18πωε0)∫Ω1∫Ω2[k2J(r1)J(r2)-∇⋅J(r1)∇⋅J(r2)]×sin⁡(kr21)r21dΩ1dΩ2,(5)W̃e=116πω2ε0(Ie-IR),(6)W̃m=116πω2ε0(Im-IR),
where(7)IR=k2∫Ω1∫Ω2[k2J(r1)J(r2)-∇⋅J(r1)∇⋅J(r2)]×sin⁡(kr21)dΩ1dΩ2,(8)Ie=∫Ω1∫Ω2∇⋅J(r1)∇⋅J(r2)cos⁡⁡(kr21)r21dΩ1dΩ2,(9)Im=k2∫Ω1∫Ω2J(r1)J(r2)cos⁡(kr21)r21dΩ1dΩ2,
where k is a free-space wavenumber, J is the surface current density, and r21 is the distance between interacting current elements. The tilde denotes that the radiation contribution IR has been subtracted from the stored energies at every point in space [17]. It is assumed that the currents are flowing in a vacuum.


### 3.1. The Modal RadiationQ Factor

The modal radiationQ factor may be evaluated from the slope of modal eigenvalues [18]:(10)Qeig=ω02dλdω.
In [18], (10) is supposed to be an approximation of the radiation Q, but in resonance it is actually exact.Since characteristic modes are normalized to radiate unit powerPr=1 [4], (3) reduces to(11)Q=2ωmax⁡⁡(W̃e,W̃m).
For parallel or series RLC circuit (hence, for one mode), the “impedance QZ” equals the exact “current Q” [6]:(12)Q=QZ=ω02|∂Z∂ω|=ω02|∂R∂ω+∂X∂ω|.
Inserting(13)Z=R+jX=1|I2|[Pr+j2ω(W̃m-W̃e)]
valid for lossless antennas [19] and using the fact that Pr=1, (12) results in(14)Q=QX=ω02|∂X∂ω|=ω02∂∂ω[2ω(W̃m-W̃e)]=ω02∂λ∂ω=Qeig,
providing that(15)λ=2ω(W̃m-W̃e).
It is therefore concluded that the modal Qeig equals the QX by definition, and it can be proven (using the reactance theorem [20, 21]) that in resonance it also equals the radiation Q defined from energies by (11).


### 3.2. Software Implementation

The above equations were implemented in MATLAB for the RWG triangular mesh where two different interaction situations occur:(a) Distant Elements
When the triangular elements are not overlapping, current density on triangles may be simply approximated as point sources located at the centre of triangles [22], see Figure 1. No actual integration is then needed. This centroid approach is very fast with satisfactory accuracy as will be shown later (however it may fail for patches located very close to the ground plane).Figure 1
Distance between nonoverlapping current elements [23].(b) Overlapping (Self) Elements
As known from the method of moments, the so-called “self” contributions are of great importance when dealing with calculations on discrete elements (meshes).
Here, the self-interaction occurs when two triangles are overlapping each other. Due to the behavior of integral kernels, only rapidly varying termcos(kr21)/r21 has to be carefully treated. Since k0R21→0 (R21 being the longest side of the triangle T) is satisfied, one needs only to use the first term in the Taylor series expansion. The dominant singular static part is 1/r21 and the integral to be worked out is
(16)I=∫T∫T′1(x-x′)2+(y-y′)2dxdydx′dy′,
where T=T′ is a triangular area. Using simplex coordinates transformation (Figure 2), the result is [23, 24]
(17)I=-43A2[ln⁡(1-2h12/L)h12+ln⁡(1-2h13/L)h13+ln⁡(1-2h23/L)h23],
where A is the triangle area, hij are the edge lengths (see Figure 2), and L is the perimeter of the triangle.Figure 2
Self-term evaluation. (a) Original problem, (b) simplex coordinates transformation [23].


## 3.1. The Modal RadiationQ Factor

The modal radiationQ factor may be evaluated from the slope of modal eigenvalues [18]:(10)Qeig=ω02dλdω.
In [18], (10) is supposed to be an approximation of the radiation Q, but in resonance it is actually exact.Since characteristic modes are normalized to radiate unit powerPr=1 [4], (3) reduces to(11)Q=2ωmax⁡⁡(W̃e,W̃m).
For parallel or series RLC circuit (hence, for one mode), the “impedance QZ” equals the exact “current Q” [6]:(12)Q=QZ=ω02|∂Z∂ω|=ω02|∂R∂ω+∂X∂ω|.
Inserting(13)Z=R+jX=1|I2|[Pr+j2ω(W̃m-W̃e)]
valid for lossless antennas [19] and using the fact that Pr=1, (12) results in(14)Q=QX=ω02|∂X∂ω|=ω02∂∂ω[2ω(W̃m-W̃e)]=ω02∂λ∂ω=Qeig,
providing that(15)λ=2ω(W̃m-W̃e).
It is therefore concluded that the modal Qeig equals the QX by definition, and it can be proven (using the reactance theorem [20, 21]) that in resonance it also equals the radiation Q defined from energies by (11).


## 3.2. Software Implementation

The above equations were implemented in MATLAB for the RWG triangular mesh where two different interaction situations occur:(a) Distant Elements
When the triangular elements are not overlapping, current density on triangles may be simply approximated as point sources located at the centre of triangles [22], see Figure 1. No actual integration is then needed. This centroid approach is very fast with satisfactory accuracy as will be shown later (however it may fail for patches located very close to the ground plane).Figure 1
Distance between nonoverlapping current elements [23].(b) Overlapping (Self) Elements
As known from the method of moments, the so-called “self” contributions are of great importance when dealing with calculations on discrete elements (meshes).
Here, the self-interaction occurs when two triangles are overlapping each other. Due to the behavior of integral kernels, only rapidly varying termcos(kr21)/r21 has to be carefully treated. Since k0R21→0 (R21 being the longest side of the triangle T) is satisfied, one needs only to use the first term in the Taylor series expansion. The dominant singular static part is 1/r21 and the integral to be worked out is
(16)I=∫T∫T′1(x-x′)2+(y-y′)2dxdydx′dy′,
where T=T′ is a triangular area. Using simplex coordinates transformation (Figure 2), the result is [23, 24]
(17)I=-43A2[ln⁡(1-2h12/L)h12+ln⁡(1-2h13/L)h13+ln⁡(1-2h23/L)h23],
where A is the triangle area, hij are the edge lengths (see Figure 2), and L is the perimeter of the triangle.Figure 2
Self-term evaluation. (a) Original problem, (b) simplex coordinates transformation [23].


## 4. Applications: Rectangular Patch Antenna

Let us first concentrate on a rectangular patch antenna of dimensionsL=50 mm and W=30 mm (further noted as R50 × 30) placed in air at a height H above an infinite ground plane. Only the dominant TM01 mode will be studied. The reason for choosing a patch with L/W≠1 is that we do not have to deal with degenerated modes.Using the image theory, the radiator in theXY plane at height z=H above an infinite electric ground plane is modelled as two patches separated by 2H. The total number of triangular elements is 676. In the TCM analyser, a proper out-of-phase mode is selected (Figure 3).Figure 3
Model of MPA above infinite ground plane forH=10 mm, dominant mode TM01 shown.The resonant frequency of the dominant mode is shown as a function of heightH, see Figure 4. It has been evaluated from a modal resonant condition for eigenvalues λ=2ω(W̃m-W̃e)=0 employing an adaptive frequency sweep for each height. The behaviour is quite peculiar, especially for greater heights. For low heights (H<10 mm or H/λres<0.08), the resonant frequency decreases “regularly,” and quasianalytical formulas (see, e.g., [1, 3]) based on the fringing field concept are valid below this range. For H≅25mm (H/λres≅0.188) there is absolute minimum of the TM01 resonant frequency. Further on, the resonant frequency rises to reach its maximum for H≅40mm (H/λres≅0.51). Around this specific height the patch also shows the minimum of the radiation Q. The above described process repeats periodically. It is yet unclear to the authors as what is the physical background to the resonant frequency discontinuity around H/λres≅0.5.Figure 4
R50 × 30 resonant frequency of the dominant TM01 mode. The dashed red curve is a quasianalytical equation from [1].The terms2ωW̃m,2ωW̃e, and 2ω(W̃m-W̃e) obtained from (5)–(9) and eigenvalues λ are plotted at Figure 5 for H=25 mm as a function of frequency. There is excellent agreement between the difference in stored energies and the eigenvalues, both obtained in a completely different manner.Figure 5
Reactive energies and their differences for an R50 × 30 patch at height of 25 mm.There is also very good agreement between the exactQJ and Qeig confirming the validity of the proposed algorithm via (14), see Figure 6. Note that Qeig in (10) does not require the currents to be calculated on the structure while QJ is evaluated in a rigorous way from modal currents (11).Figure 6
The radiationQ for dominant mode of an R50 × 30 patch as a function of height H.From Figure6 it is seen that the radiation Q has a minimum for a specific height. It is deduced that the reason lies in the cancelling of the radiated power between the two out-of-phase currents. Similar behaviour has been observed in the case of two half-wave thin-wire dipoles with opposite sinusoidal currents, separated by d=2H, see [25] for details. Actually these two out-of-phase dipoles may serve as a very simple model for a patch antenna with a dominant mode. When the dipoles are reduced to elementary (Hertzian) ones, an approximate analytical solution is available and in [25] we showed that the Q is led by the function(18)fQ(H)≅2Hk2H-sin⁡⁡(k2H).
After deriving (18), the condition is worked-out(19)tan⁡(k2H)=k2H,
and the first nontrivial root of (19) could be approximated as [25](20)(Hλ)min⁡≅38-16π2=0.358.
For sinusoidal currents on dipoles the minimum (evaluated numerically) occurs for H=0.36λ.The minimum of the patch under study is obtained atH≅0.4λ, a value that is remarkably close to the simple dipole model.


### 4.1. Algorithm Convergence

Since no other methods for calculating modalQ are available, Qeig is taken as a reference, and the relative error percentage is defined as:(21)relativeerror=|QJ-Qeig|Qeig⋅100,
where QJ is calculated from the currents using (11). Four different heights H were chosen, H=1 mm (0.01λ), H=2 mm (0.0185λ), H=10 mm (0.0803λ), and H=20 mm (0.151λ), and the relative error was evaluated as a function of total triangular elements (including the mirror), see Figure 7. All quality factors were evaluated at the resonant frequency of the dominant mode for the R50 × 30 patch. As discussed earlier, the centroid approximation became more inaccurate with low heights H. However, even for the lowest analyzed value H=0.01λ, the relative error is in the order of a few percent for reasonable mesh density (hundreds of elements). Further improvements to the integration routine are considered for the future.Figure 7
Relative error of theQ factor as a function of triangular elements (mesh density).


### 4.2. Fractional Bandwidth of the R50 × 30 Patch Antenna

It is known that the fractional bandwidth (FBW) is related to the unloadedQ factor and the desired matching VSWR level. For VSWR < s we have [26](22)FBW≅s-1Qs[%].
Using a full-wave simulator CST-MWS [27], an R50 × 30 patch has been simulated and the FBWCST for VSWR < 2 was calculated as:(23)FBWCST=f2-f1f0,
where f2 and f1 are margins for VSWR < 2 and f0 is the centre frequency. Only very low heights were studied since we used a simple probe feed which introduces an inductance component to the total input impedance. The comparison in Figure 8 shows good agreement of both fractional bandwidths.Figure 8
Fractional bandwidth FBW (VSWR < 2) for a R50 × 30 patch.


## 4.1. Algorithm Convergence

Since no other methods for calculating modalQ are available, Qeig is taken as a reference, and the relative error percentage is defined as:(21)relativeerror=|QJ-Qeig|Qeig⋅100,
where QJ is calculated from the currents using (11). Four different heights H were chosen, H=1 mm (0.01λ), H=2 mm (0.0185λ), H=10 mm (0.0803λ), and H=20 mm (0.151λ), and the relative error was evaluated as a function of total triangular elements (including the mirror), see Figure 7. All quality factors were evaluated at the resonant frequency of the dominant mode for the R50 × 30 patch. As discussed earlier, the centroid approximation became more inaccurate with low heights H. However, even for the lowest analyzed value H=0.01λ, the relative error is in the order of a few percent for reasonable mesh density (hundreds of elements). Further improvements to the integration routine are considered for the future.Figure 7
Relative error of theQ factor as a function of triangular elements (mesh density).


## 4.2. Fractional Bandwidth of the R50 × 30 Patch Antenna

It is known that the fractional bandwidth (FBW) is related to the unloadedQ factor and the desired matching VSWR level. For VSWR < s we have [26](22)FBW≅s-1Qs[%].
Using a full-wave simulator CST-MWS [27], an R50 × 30 patch has been simulated and the FBWCST for VSWR < 2 was calculated as:(23)FBWCST=f2-f1f0,
where f2 and f1 are margins for VSWR < 2 and f0 is the centre frequency. Only very low heights were studied since we used a simple probe feed which introduces an inductance component to the total input impedance. The comparison in Figure 8 shows good agreement of both fractional bandwidths.Figure 8
Fractional bandwidth FBW (VSWR < 2) for a R50 × 30 patch.


## 5. Applications: Fractal Antennas

In this section, a bit more complex structures will be studied. The first one (the “Self Affine U” fractal, SAU), has been described in [28] and further analyzed in [29]. This kind of radiating motif is employed as a dual-band radiator with mutually orthogonal radiation patterns at both bands. Therefore we are analyzing the first two modes, where the currents are orthogonal. These are depicted in Figures 9 and 10 for first (SAU1) and second (SAU2) fractal iteration, respectively. The current of the first (lower) mode J1 has two out-of-phase components (see Figure 11 for schematic current paths) while the second mode comprises inphase currents only. As we know from previous studies, opposite currents contribute to a rapid increase of the radiation Q, and it is expected that J1 will have a much higher Q than J2.Figure 9
The first two characteristic modes (currents and charges) for the SAU1 structure.Figure 10
The first two characteristic modes (currents and charges) for the SAU2 structure.Figure 11
The main current paths for the first two modes of the SAU1/2 structure.Figure11 presents a very simple concept showing the main current paths for the J1 and J2 modes discussed above including the mirroring effect of the infinite ground plane. It could be simply stated that more opposing current paths lead to significant increase in Q.We show detailed behaviour only for SAU2 (the situation is similar for SAU1)—see Figure12 that confirms high Q for the J1 mode. Characteristic angles are calculated for H=29 mm, the actual height for which the dual-band antenna was designed [29].Characteristic angles (left) and radiationQ for the SAU2.
(a)(b)


### 5.1. The FCL-2 Fractal Antenna

The second presented structure is the so-called fractal clover leaf (FCL) of the second iteration, [14]. The antenna is fed by an L-probe [30] that excites its dominant mode and is located at height H=36 mm. Actually, the dominant mode is composed of two degenerated modes J1 and J2 (Figure 13). The second higher mode J3 is shown at Figure 14 for completeness.Degenerated dominant modeJ1, and  J2 of the FCL2 antenna (currents and charges).
(a)(b)(c)(d)Second higher modeJ3 (currents and charges).
(a)(b)Figure15 shows the main current paths of these modes, and we can again deduce that the dominant mode will exhibit lower Q compared to J3. This is confirmed by Figure 16—J3 has more than 200x higher radiation Q.Figure 15
Schematic depiction of the dominant current paths for the dominant (J1+J2) and the second higher J3 modes together with their modal radiation patterns.Figure 16
Characteristic angles for the FCL2 structure atH=29 mm.


## 5.1. The FCL-2 Fractal Antenna

The second presented structure is the so-called fractal clover leaf (FCL) of the second iteration, [14]. The antenna is fed by an L-probe [30] that excites its dominant mode and is located at height H=36 mm. Actually, the dominant mode is composed of two degenerated modes J1 and J2 (Figure 13). The second higher mode J3 is shown at Figure 14 for completeness.Degenerated dominant modeJ1, and  J2 of the FCL2 antenna (currents and charges).
(a)(b)(c)(d)Second higher modeJ3 (currents and charges).
(a)(b)Figure15 shows the main current paths of these modes, and we can again deduce that the dominant mode will exhibit lower Q compared to J3. This is confirmed by Figure 16—J3 has more than 200x higher radiation Q.Figure 15
Schematic depiction of the dominant current paths for the dominant (J1+J2) and the second higher J3 modes together with their modal radiation patterns.Figure 16
Characteristic angles for the FCL2 structure atH=29 mm.


## 6. Resonant Properties of Studied Antennas

The properties of studied antennas are summarized in this section. At first we observed that microstrip antenna could support different kinds of modes regarding theirQ factors (see Figure 17):(a)
lowQ modes with the current flowing in one direction and not changing its phase (dominant modes of simple shapes like rectangular, circular patch, and so forth.)(b)
highQ modes with part of the currents flowing in the opposite direction. These modes exist even on simple “U” shaped patch (Figure 9 left) and on complex (fractal) geometries.Figure 17
RadiationQs for different antennas/modes.Secondly, it has been observed that resonant frequency is quite a complicated function of height. Unfortunately we do not yet have any physical explanation as to why some modes present minimum values offr.Looking at Figure18, it is clear (and interesting) that the resonant frequency behaves quite differently for low-Q and high-Q modes. The resonant frequency of low-Q modes is much more sensitive to the height, whereas high-Q modes exhibit almost constant fr when the height is varied. The proposed explanation is that the opposite currents (responsible for high Q) keep reactive fields very close to the radiating structure so the effect of a fringing field coupled to the ground plane becomes almost negligible.Figure 18
Resonant frequencies for different antennas/modes.


## 7. Conclusions

Modal resonant properties of selected microstrip patch antennas have been studied with the help of characteristic modes and the novel theory published by Vandenbosch. It has been found that the resonant frequency of a simple rectangular patch antenna is quite a complicated function of its height above the infinite ground. Moreover, the dependency of resonant frequency is also found to be a function of the radiationQ factor (which is now possible to calculate in a rigorous way). Due to the complexity of the problem, no physical explanation for the resonant frequency behaviour has yet been found.It is observed that the radiationQ factor decreases for “standard” heights (<~0.1λ), however there exists an absolute minimum value of Q that has already been predicted by simple modeling of two elementary out-of-phase dipoles. Using proper feeding techniques (like the L-probe) allows us to design wideband compact antennas.The theory now puts current distribution and the radiationQ factor into objective context. Whenever the current mode exhibits opposite components, high Q may appear.Future work is needed to connect the presented theory with parameter sweeps or even optimization, so we will be able to design novel wideband/multimode compact antennas.



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