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### Keywords:

• antenna modeling;
• cavity modes;
• dipole antennas;
• feeding networks;
• patch antennas

### Abstract

[1] The series and parallel resonances of an antenna's input impedance are explained through the theory of characteristic modes (CM). We are showing for the first time that the parallel resonance phenomenon corresponds to the interaction of at least two nearby CM, each having eigenvalues of opposite sign. Additionally, the series resonance phenomenon is shown to be mainly due to the resonance of a single characteristic mode. Analysis of a simple wire dipole antenna, an edge-fed patch antenna, and a loop antenna are used to validate this finding.

### 1. Introduction

[2] An antenna's input impedance is one of the most important parameters to assess its performance. It is therefore crucial to understand its behavior as a function of frequency, size, materials, etc. This knowledge can be used to optimize the performance of antennas in many applications, such as broadening the bandwidth of an electrically small antenna [Obeidat et al., 2007, 2009, 2008]. Although lumped circuit models have generally been used to represent the frequency behavior of the input impedance, an alternative approach that should lead to more physical insight into the antenna performance is presented here based on the theory of characteristic modes.

[3] The input impedance of an antenna near series resonance can be modeled as a series RLC lumped circuit. At resonance, the imaginary parts of the series impedances cancel each other:

where ωs = . Consequently, only the real part of the RLC circuit contributes to the antenna input impedance. Also, note that the derivative of the reactance with respect to frequency is positive at that series resonance. On the other hand, near a parallel resonance, the antenna input impedance can be modeled as a parallel RLC lumped circuit. At resonance, the imaginary parts of the parallel admittances cancel each other; namely,

where ωp = . Consequently, only the real part of the RLC circuit contributes to the antenna input impedance. In this case, the derivative of the susceptance with respect to frequency is positive at resonance. For an applied voltage Vin at the feed point, the input impedance (Vin/Iin) at the antenna feed port depends mainly on the antenna current Iin flowing into the feed port. Hence, the antenna resonance behavior will be studied by analyzing the antenna current behavior at the feed port.

[4] Series resonance at the antenna input port occurs when the current at the port reaches its greatest effective value, while the opposite occurs during parallel resonance. However, for some antennas, the behavior of the input impedance near parallel resonance is more complicated than the model used in equation (2). For example, a probe-fed microstrip antenna has a nonzero reactance when the real part is maximum, resulting in a current minimum at a frequency that is slightly shifted from the frequency where the real part of the input impedance is maximum. In both cases, the voltage Vin across the input port is in phase with the current Iin if resonance is defined as the frequency where the reactance or susceptance goes to zero. As will be explained in this paper, series resonance is a “natural” resonance; namely, it occurs at the resonances of the characteristic modes (CM). On the other hand, parallel resonance is the result of the interaction of two or more CM.

[5] In this paper, the resonances of the input impedance of three antenna structures, namely, a center-fed dipole antenna an edge-fed Microstrip patch antenna and a loop antenna, will be discussed. Furthermore, since the conceptual framework of this discussion relies heavily upon the theory of characteristic modes, a brief background on the relevant pieces of characteristic mode theory [Garbacz, 1965; Harrington, 1971; Harrington and Mautz, 1971] is provided before discussing the relation between the antenna's input impedance/admittance resonances observed at the antenna input port and their related characteristic modes. While other formulations exist [Harrington, 1985; Kabalan et al., 2001], the characteristic mode formulation presented here defines modes on the surfaces of perfectly conducting bodies.

### 2. Theory of Characteristic Modes

[6] The surface current density on the body of a lossless metallic antenna can be written as a superposition of weighted orthogonal current modes n. Each n corresponds to an eigenvalue λn, where both components of an eigenpair are functions of the antenna geometry. The weighting coefficient of each mode is a function of both the mode's corresponding eigenvalue λn and the inner product of that mode with the excitation vector in, as explained in the equation below [Harrington, 1971]:

[7] The inner product is defined on the surface of the antenna as follows:

where n and λn are the solutions of the generalized eigenvalue equation shown below [Harrington, 1971; Harrington and Mautz, 1971]:

The operators X and R are the imaginary and real parts of the operator Z, which relates the total surface current density to the tangential applied (or incident) electric field in at some radial frequency ω: in(, ω) = Z((, ω)). Both X and R are real symmetric operators. Hence, all eigenvalues λn, and current modes n must be real. It is customary to normalize the eigencurrents such that R(n), n〉 = 1. This normalization will be assumed in the rest of this paper. Last, a mode is termed dominant at some frequency ω if its modal weighting coefficient is much larger in magnitude than any of the other coefficients. This implies that for a dominant mode relative to the other modes, the numerator (associated with the feed mechanism) is large in magnitude and the denominator (associated solely with the geometry and usually determined mostly by λn) is small in magnitude.

[8] To perform practical computations, the operator Z is usually approximated by the matrix [Z] using the Method of Moments (MoM) and the Galerkin method [Harrington, 1971; Harrington and Mautz, 1971]. Thus, the number of modal currents in equation (3) is reduced from infinity to N, assuming that [Z] is an NxN matrix. All equations presented in this section are then applied to this matrix [Z] to compute the eigenvalue spectrum and modal surface currents shown in the results in sections 35. The notation in subsequent sections, however, will reflect the fact that modal surface currents derived from MoM are still functions of position and can be interpolated to an arbitrary position o on the antenna's surface.

### 3. Center-Fed Dipole Antenna

[9] The first three dominant characteristic modes (i.e., modes with significant contribution to the total current distribution in the frequency band of interest) of a wire dipole antenna are shown in Figure 1. Modes 1 and 3 are even modes while mode 2 is an odd mode with a null at the center of the dipole. In the case of a center-fed dipole antenna, odd modes will not be excited since the inner product 〈n (ω), in(ω)〉 in the numerator of equation (3) will equal zero. Therefore, only the even modes will be considered in this example.

[10] The input impedance of a center-fed 1.2 m length wire dipole antenna simulated using the electromagnetic surface patch code (available at http://electroscience.osu.edu) is shown in Figure 2. It is apparent that two series resonances occur in the frequency range 20 MHz to 400 MHz. The first resonant frequency occurs at 119.5 MHz, while the second resonant frequency occurs at 367.5 MHz. Between these two series resonances, there is one parallel resonance that occurs at 222 MHz.

[11] To understand the occurrence of the two series resonances in terms of the dipole's characteristic modes, a plot of the dipole eigenvalue spectrum is shown Figure 3. As can be seen in the spectrum, λ1, which corresponds to the first dipole mode, is equal to zero at 119.5 MHz, while λ3, which corresponds to the third dipole mode, is equal to zero at 367.5 MHz. These two frequencies, for which the eigenvalues are essentially equal to zero, also correspond to the series resonance frequencies of the antenna input impedance.

[12] The input impedance of the antenna is equal to the voltage across the input port divided by the current at the same port, namely, Zin = V in (o)/I (o). Near the first series resonance, the total surface current at the input port o can be written as

where “HOT” stands for “higher-order terms”. It is further assumed that mode 1 is dominant because it is near its resonance (λ1 ≈ 0) and the remaining terms have much larger eigenvalue magnitudes with ∣n (o)∣ magnitudes at most similar relative to mode 1. Following common practice, the excitation vector in is assumed to be highly localized such that it can be safely approximated to be zero almost everywhere except at the feed port. Assuming the input feed port is Δl in length and Δw in width, the input impedance can be written as follows:

[13] Thus, the input impedance will be purely resistive in the case of series resonance when λ1 approaches zero (mode 1 resonates). Furthermore, since the magnitudes of the remaining eigenvalues are relatively very large, their corresponding modes will be weakly excited and will have negligible contribution to the total current and the input impedance.

[14] The same analysis applies to degenerate systems where two or more eigenvalues approach zero at the same frequency. Hence, the summation of terms from equation (3) will still have the same phase as the excitation vector in. In some cases, more than one higher-order mode may be strongly excited along with the dominant mode. If those higher-order modes have relatively small eigenvalue magnitudes compared to the dominant mode n when λn is equal to zero, then the nth term will not be the only term in the summation. Therefore, the phase of the total current will not be the same as the phase of excitation vector in. Consequently, the zero-crossing frequency of the input reactance will either shift or the reactance may not have a zero crossing at all.

[15] To explain the parallel resonance of the center-fed dipole using the dipole characteristic modes, it is necessary to understand what happens to the current at the feed port. At parallel resonance, the current at the feed port becomes small in magnitude and has zero phase as shown in Figure 4. The current distribution in this graph illustrates that the current along the center-fed dipole antenna becomes small at the center of the dipole antenna. However, this cannot be due to a specific mode because any such current mode with a small magnitude at the feed port will be weakly excited, since the inner product in the numerator of equation (3)n (ω), in(ω)〉 would be very small in magnitude.

[16] Thus, in the case of a parallel resonance, there should be at least two strongly, but equally, excited modes with opposite phase at the input port in order to cancel each other at this port (other weakly excited higher-order modes usually prevent the total current magnitude from being zero at the feed point, although the current magnitude is still relatively small). We conclude that there is no single characteristic mode called a parallel mode, since this type of resonance should be an interaction of at least two nearby series modes with opposite sign. Therefore, parallel resonance is a function of two factors: the feed location (inner product in numerator of equation (3)), and the antenna geometry.

[17] Since at parallel resonance the current is composed of at least two dominant modes, the total current along the antenna body can be written in the neighborhood of the parallel resonance frequency as follows (assuming two dominant terms):

At the feed port, equation (8) may be written as

The antenna admittance due to the two strongly excited modes can then be written as follows, near the parallel resonance frequency:

Since both λ1 and λ3 are much larger than 1 and both have opposite signs:

Hence when

the imaginary part of the admittance approaches zero while its real part is small because the magnitude of the eigenvalues is large. In other words, the real part of the input impedance becomes large and the imaginary part approaches zero (and changes sign) at parallel resonance.

### 4. Edge-Fed Rectangular Microstrip Patch Antenna

[18] A rectangular edge-fed microstrip patch antenna, 16 mm × 12.448 mm in size, and mounted on a 31 mil substrate with dielectric constant of 2.2 is shown in Figure 5. The input impedance simulated using FEKO (available at http://www.feko.info) is shown in Figure 6. As can be observed, the first parallel resonance occurs at 6.1 GHz between two series resonances: one at 2.2 GHz and a second one at 6.9 GHz. As shown in Figure 7, the first series resonance is due to mode 1 and occurs at the frequency at which the eigenvalue of mode 1 approaches zero. Similarly, the second series resonance occurs at the frequency at which the eigenvalue of mode 2 approaches zero. Note that the first series resonance (mode 1) occurs when the L-shaped length of the current path on the patch (see Figure 5b) plus its image (ground plane) is approximately λ/2 at 2.2 GHz, where λ is the wavelength in a medium with a dielectric constant equal to the effective dielectric constant of the structure (ɛr,eff = 1.8). Using this length makes sense because when the feed point is near the corner, the currents for mode 1 tend to flow in that direction at 2.2 GHz. It can be shown that the frequency at the first series resonance increases if the feed point is moved toward the center of the edge. This also makes sense because the effective length decreases, since the currents of mode 1 for this feed point location (near center of edge) tend to flow parallel to the long edge of the patch. Furthermore, mode 1 does not radiate well because the current on the patch and its image on the ground plane tend to cancel each other.

[19] Previous work [Cabedo-Fabres et al., 2003, 2007] on determining the characteristic modes of a patch antenna was done by ignoring the feeding structure of the antenna and treating the patch antenna as a plate parallel to the ground plane. This treatment yields different characteristic modes than an actual patch antenna. For an actual patch antenna, the feed structure connects the top plate to the ground plane physically either through coaxial feed or microstrip line, which is a different geometry than just two parallel plates without feeding structure. There are also microstrip patch antennas that are fed through proximity coupling [Targonski and Pozar, 1993], but the feeding structure (e.g., hole in the ground plane) has to be modeled as well. Otherwise, incorrect antenna characteristic modes are obtained.

[20] The current magnitudes of modes 1 and 2 as well as the total current at the input port, which consists mostly of a summation of the two dominant modes (modes 1 and 2), are shown in Figure 8. Clearly, the current magnitude of mode 1 approaches its maximum at the frequency which corresponds to the first resonant frequency. Similarly, mode 2's maximum occurs at the frequency which corresponds to the second series resonance. On the other hand, the parallel resonance at 6.1 GHz is due to current cancellation at the input port between modes 1 and 2, since both modes have similar magnitude (Figure 8) but opposite phase at the input port (Figure 9).

#### 4.1. Current Distribution

[21] The magnitude of the normalized total current distribution of the microstrip patch antenna at the first three resonant frequencies is shown in Figure 10. As can be observed, the normalized current distribution at the first and the second series resonances is very small everywhere along the patch surface except at the input port. In contrast, at parallel resonance, the current distribution along the patch antenna surface is similar to the well known cavity mode TM01. Thus, the patch antenna is a poor radiator at its series resonances and a good radiator at its parallel resonance.

[22] Figures 11 and 12 show the magnitude of the normalized current distribution of modes 1 and 2, respectively, at the first three resonances: 2.2 GHz, 6.1 GHz and 6.9 GHz. As can be seen in the graph, at 2.2 GHz, where λ1 approaches zero, the current distribution of mode 1 is similar to the normalized patch total current distribution at 2.2 GHz (Figure 10). Also, the normalized current distribution of mode 2 (Figure 12) at 6.9 GHz, where λ2 approaches zero, is similar to the normalized total current distribution at 6.9 GHz (Figure 10). This behavior is expected since the total current distribution at a series resonance is mainly due to the contribution of the sole dominant resonant characteristic mode.

[23] At 6.1 GHz, since both modes 1 and 2 are dominant modes and both have similar normalized current magnitude distributions and phase, the total normalized current distribution resembles these modes, except in the vicinity of the input port where the these two currents are out of phase.

#### 4.2. Cavity Modes Versus Characteristic Modes

[24] Microstrip patch antennas are generally narrowband antennas operated at frequencies corresponding to their lowest-order cavity modes. However, away from the resonant frequencies of the cavity modes, microstrip patch antennas are poor radiators. The microstrip patch antenna under discussion has its first cavity mode at 6.1 GHz (Figure 10), which corresponds to the first parallel resonance of its input impedance. Hence, we conclude that the characteristic modes do not correspond to the cavity modes, but rather cavity modes correspond to the patch parallel resonances, which is an interaction of two nearby characteristic modes.

### 5. Circular Loop Antenna

[25] This circular loop antenna is considered here because it has a somewhat different behavior compared to the dipole antenna. The former is a closed structure, while the latter is open. In other words, a current can flow in the loop antenna even at zero frequency (DC current), while that is physically impossible in the dipole antenna. This has an important implication in the behavior of the input impedance of the antenna. The imaginary part of the input impedance of a 3 cm radius circular loop antenna is shown in Figure 13. The first parallel resonance of this loop antenna occurs around 0.744 GHz. Figure 14 shows the eigenvalue spectrum of the loop antenna. Note that because of the antenna symmetry, degenerate modes will exist. Furthermore, there is a conductive mode (mode 0) that resonates at DC. Hence, the first parallel resonance of the loop antenna is due to the interaction of the degenerate modes 1 and 2 with mode 0.

### 6. Conclusions

[26] The theory of characteristic modes was used to model the input impedance of various antennas in terms of its modes. We showed that parallel resonance phenomena does not correspond to a single characteristic mode, but is rather due to an interaction of mainly two nearby modes each having eigenvalues of opposite sign. This is in contrast to the series resonance phenomenon, which is mainly due to a single resonant characteristic mode (when the corresponding eigenvalue becomes zero). It was also shown that the characteristic modes of a microstrip patch antenna do not correspond to its cavity modes. This explains why the microstrip antenna normally operates at parallel resonance since the cavity mode can be expressed as the combination of at least two characteristic modes. A loop antenna was the third antenna modeled here. This antenna has a DC mode which is not the case for the dipole or microstrip patch antennas. Although we only considered three different types of antennas, we believe for any antenna, series resonance in the input impedance is due to the resonance of a characteristic mode while parallel resonance is due to the interaction of two or more characteristic modes.

### References

• , , , and (2003), On the use of characteristic modes to describe patch antenna performance, IEEE Trans. Antennas Propag., 2, 712715.
• , , , and (2007), The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications, IEEE Trans. Antennas Propag., 49, 5268, doi:10.1109/MAP.2007.4395295.
• (1965), Modal expansions for resonance scattering phenomena, Proc. IEEE, 53, 856864, doi:10.1109/PROC.1965.4064.
• (1971), Computation of characteristic modes for conducting bodies, IEEE Trans. Antennas Propag., 19, 629639.
• (1985), Characteristic modes for aperture problems, IEEE Trans. Microwave Theory Tech., 33(6), 500505.
• , and (1971), Theory of characteristic modes for conducting bodies, IEEE Trans. Antennas Propag., 19, 622628, doi:10.1109/TAP.1971.1139999.
• , , and (2001), A three-dimensional characteristic mode solution of two perforated parallel planes separating different dielectric media, Radio Sci., 36(2), 183193, doi:10.1029/1999RS002293.
• , , and (2007), Antenna design and analysis using characteristic modes, paper presented at Antennas and Propagation Society International Symposium, Inst. Electr. and Electr. Eng., Honolulu, 9 – 15 June .
• , , and (2008), Design and analysis of a helical spherical antenna using the theory of characteristic modes, paper presented at Antennas and Propagation Society International Symposium, Inst. Electr. and Electr. Eng., San Diego, Calif., 5 – 11 Jul.
• , , and (2009), Design of antenna conformal to V-shaped tail of UAV based on the method of characteristic modes, paper presented at 3rd Eur. Conf. on Antennas and Propagation, pp. 2327, Berlin, 23 – 27 Mar.
• , and (1993), Design of wide-band circularly polarized aperture coupled microstrip antennas, IEEE Trans. Antennas Propag., 41, 214220, doi:10.1109/8.214613.