Discussion of series and parallel resonance phenomena in the input impedance of antennas

Discussion of series and parallel resonancephenomena in the input impedance of antennas

Khaled A. Obeidat,1 Bryan D. Raines,1 and Roberto G. Rojas1

Received 8 January 2010; revised 12 August 2010; accepted 9 September 2010; published 2 December 2010.

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

Citation: Obeidat, K. A., B. D. Raines, and R. G. Rojas (2010), Discussion of series and parallel resonance phenomenain the input impedance of antennas, Radio Sci., 45, RS6012, doi:10.1029/2010RS004353.

1. Introduction[2] An antenna’s input impedance is one of the most

important parameters to assess its performance. It istherefore crucial to understand its behavior as a functionof frequency, size, materials, etc. This knowledge canbe used to optimize the performance of antennas inmany applications, such as broadening the bandwidth ofan electrically small antenna [Obeidat et al., 2007, 2009,2008]. Although lumped circuit models have generallybeen used to represent the frequency behavior of the inputimpedance, an alternative approach that should lead tomore physical insight into the antenna performance ispresented here based on the theory of characteristic modes.[3] The input impedance of an antenna near series res-

onance can be modeled as a series RLC lumped circuit. Atresonance, the imaginary parts of the series impedancescancel each other:

Z ¼ Rs þ j !sLs � 1

!sCs

� �ð1Þ

where ws =ffiffiffiffiffiffiffiffiffi1�LsCs

q. Consequently, only the real part of the

RLC circuit contributes to the antenna input impedance.Also, note that the derivative of the reactance with respectto frequency is positive at that series resonance. On theother hand, near a parallel resonance, the antenna inputimpedance can be modeled as a parallel RLC lumped

circuit. At resonance, the imaginary parts of the paralleladmittances cancel each other; namely,

Y ¼ 1

Rpþ j !pCp � 1

!pLp

� �ð2Þ

where wp =ffiffiffiffiffiffiffiffiffi1�LpCp

q. Consequently, only the real part of the

RLC circuit contributes to the antenna input impedance. Inthis case, the derivative of the susceptance with respect tofrequency is positive at resonance. For an applied voltageVin at the feed point, the input impedance (Vin/Iin) at theantenna feed port depends mainly on the antenna currentIin flowing into the feed port. Hence, the antenna reso-nance behavior will be studied by analyzing the antennacurrent behavior at the feed port.[4] Series resonance at the antenna input port occurs

when the current at the port reaches its greatest effectivevalue, while the opposite occurs during parallel resonance.However, for some antennas, the behavior of the inputimpedance near parallel resonance is more complicatedthan the model used in equation (2). For example, a probe‐fed microstrip antenna has a nonzero reactance when thereal part is maximum, resulting in a current minimum at afrequency that is slightly shifted from the frequency wherethe real part of the input impedance is maximum. In bothcases, the voltage Vin across the input port is in phase withthe current Iin if resonance is defined as the frequencywhere the reactance or susceptance goes to zero. As will beexplained in this paper, series resonance is a “natural” res-onance; namely, it occurs at the resonances of the charac-teristic modes (CM). On the other hand, parallel resonanceis 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

1Electrical Engineering Department, Ohio State University,Columbus, Ohio, USA.

Copyright 2010 by the American Geophysical Union.0048‐6604/10/2010RS004353

RADIO SCIENCE, VOL. 45, RS6012, doi:10.1029/2010RS004353, 2010

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antenna an edge‐fed Microstrip patch antenna and a loopantenna, will be discussed. Furthermore, since the con-ceptual framework of this discussion relies heavily uponthe theory of characteristic modes, a brief background onthe relevant pieces of characteristicmode theory [Garbacz,1965; Harrington, 1971; Harrington and Mautz, 1971]is provided before discussing the relation between theantenna’s input impedance/admittance resonances observedat the antenna input port and their related characteristicmodes. While other formulations exist [Harrington, 1985;Kabalan et al., 2001], the characteristic mode formulationpresented here defines modes on the surfaces of perfectlyconducting bodies.

2. Theory of Characteristic Modes[6] The surface current density ~J on the body of a

lossless metallic antenna can be written as a superpositionof weighted orthogonal current modes ~J n. Each ~J n cor-responds to an eigenvalue ln, where both components ofan eigenpair are functions of the antenna geometry. Theweighting coefficient of each mode is a function of boththe mode’s corresponding eigenvalue ln and the innerproduct of that mode with the excitation vector ~Ein, asexplained in the equation below [Harrington, 1971]:

~J ~r; !ð Þ ¼X1n

h~Jn ~r; !ð Þ;~Ein ~r; !ð Þi1þ j�n !ð Þð Þ�R ~Jn ~r; !ð Þ� �

;~Jn ~r; !ð Þ~Jn ~r; !ð Þ

ð3Þ

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

�~a;~b ¼ Z

S

~a* �~b dS ð4Þ

where ~J n and ln are the solutions of the generalizedeigenvalue equation shown below [Harrington, 1971;Harrington and Mautz, 1971]:

X ~Jn� � ¼ �nR ~Jn

� � ð5Þ

The operators X and R are the imaginary and real partsof the operator Z, which relates the total surface currentdensity ~J to the tangential applied (or incident) electricfield ~Ein at some radial frequency w: ~Ein(~r, w) = Z(~J (~r,w)). Both X and R are real symmetric operators. Hence,all eigenvalues ln, and current modes ~J n must be real. Itis customary to normalize the eigencurrents such thathR(~J n), ~J ni = 1. This normalization will be assumed inthe rest of this paper. Last, a mode is termed dominantat some frequency w if its modal weighting coefficient

h~Jn;~Eini1þ j�nð Þh~Jn;R ~Jnð Þi is much larger inmagnitude than any of the

other coefficients. This implies that for a dominant moderelative to the other modes, the numerator (associatedwith the feed mechanism) is large in magnitude and thedenominator (associated solely with the geometry andusually determined mostly by ln) is small in magnitude.[8] To perform practical computations, the operator Z

is usually approximated by the matrix [Z] using theMethod of Moments (MoM) and the Galerkin method[Harrington, 1971; Harrington and Mautz, 1971]. Thus,the number of modal currents in equation (3) is reducedfrom infinity to N, assuming that [Z] is an NxN matrix.All equations presented in this section are then appliedto this matrix [Z] to compute the eigenvalue spectrumand modal surface currents shown in the results insections 3–5. The notation in subsequent sections, however,will reflect the fact that modal surface currents derivedfrom MoM are still functions of position and can be inter-polated to an arbitrary position~ro 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 currentdistribution in the frequency band of interest) of a wiredipole antenna are shown in Figure 1. Modes 1 and 3 areeven modes while mode 2 is an odd mode with a null atthe center of the dipole. In the case of a center‐fed dipoleantenna, odd modes will not be excited since the innerproduct h~J n (w), ~Ein(w)i in the numerator of equation (3)will equal zero. Therefore, only the even modes will beconsidered in this example.[10] The input impedance of a center‐fed 1.2 m length

wire dipole antenna simulated using the electromagneticsurface patch code (available at http://electroscience.osu.edu) is shown in Figure 2. It is apparent that two series

Figure 1. First three dominant modes of a dipoleantenna.

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resonances occur in the frequency range 20 MHz to400 MHz. The first resonant frequency occurs at119.5MHz, while the second resonant frequency occurs at367.5 MHz. Between these two series resonances, there isone 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, aplot of the dipole eigenvalue spectrum is shown Figure 3.As can be seen in the spectrum, l1, which corresponds tothe first dipole mode, is equal to zero at 119.5 MHz, whilel3, which corresponds to the third dipole mode, is equal tozero at 367.5 MHz. These two frequencies, for which theeigenvalues are essentially equal to zero, also correspondto the series resonance frequencies of the antenna inputimpedance.[12] The input impedance of the antenna is equal to the

voltage across the input port divided by the current at thesame port, namely, Zin = V in (~ro)/I (~ro). Near the first seriesresonance, the total surface current at the input port ~rocan be written as

~J ~roð Þ ¼ J1* ~roð ÞEin ~roð Þ1þ j�1

~J1 ~roð Þ þ HOT ð6Þ

where “HOT” stands for “higher‐order terms”. It is furtherassumed that mode 1 is dominant because it is near itsresonance (l1 ≈ 0) and the remaining terms have muchlarger eigenvalue magnitudes with ∣~J n (~ro)∣ magnitudes atmost similar relative to mode 1. Following commonpractice, the excitation vector ~Ein is assumed to be highlylocalized such that it can be safely approximated to be zeroalmost everywhere except at the feed port. Assuming the

input feed port is Dl in length and Dw in width, the inputimpedance can be written as follows:

Zin ¼ Vin

Iin¼ Ein ~roð ÞDl

J ~roð ÞDw¼ Ein ~roð Þ 1þ j�1ð ÞDl

J1* ~roð ÞEin ~roð ÞJ1 ~roð ÞDwþ HOT

¼ Dl

~J1 ~roð Þ 2Dwþ j

�1Dl

~J1 ~roð Þ 2Dwþ HOT ð7Þ

[13] Thus, the input impedance will be purely resistivein the case of series resonance when l1 approaches zero(mode 1 resonates). Furthermore, since the magnitudes ofthe remaining eigenvalues are relatively very large, theircorresponding modes will be weakly excited and willhave negligible contribution to the total current and theinput impedance.[14] The same analysis applies to degenerate systems

where two or more eigenvalues approach zero at thesame frequency. Hence, the summation of terms fromequation (3) will still have the same phase as the excitationvector ~Ein. In some cases, more than one higher‐ordermode may be strongly excited along with the dominantmode. If those higher‐order modes have relatively smalleigenvalue magnitudes compared to the dominant mode nwhen ln is equal to zero, then the nth term will not be theonly term in the summation. Therefore, the phase of thetotal current will not be the same as the phase of excitationvector ~Ein. Consequently, the zero‐crossing frequency ofthe input reactance will either shift or the reactance maynot have a zero crossing at all.[15] To explain the parallel resonance of the center‐fed

dipole using the dipole characteristic modes, it is neces-sary to understand what happens to the current at the feedport. At parallel resonance, the current at the feed portbecomes small in magnitude and has zero phase as

Figure 2. Input impedance of a center‐fed dipoleantenna.

Figure 3. Eigenvalue spectrum of the dipole antenna.

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shown in Figure 4. The current distribution in this graphillustrates that the current along the center‐fed dipoleantenna becomes small at the center of the dipole antenna.However, this cannot be due to a specific mode becauseany such current mode with a small magnitude at the feedport will be weakly excited, since the inner product in thenumerator of equation (3) h~J n (w), ~Ein(w)i would be verysmall in magnitude.[16] Thus, in the case of a parallel resonance, there

should be at least two strongly, but equally, excited modeswith opposite phase at the input port in order to cancel eachother at this port (other weakly excited higher‐ordermodesusually prevent the total current magnitude from beingzero at the feed point, although the current magnitude isstill relatively small). We conclude that there is no singlecharacteristic mode called a parallel mode, since this typeof resonance should be an interaction of at least two nearbyseries modes with opposite sign. Therefore, parallel res-onance is a function of two factors: the feed location (innerproduct in numerator of equation (3)), and the antennageometry.[17] Since at parallel resonance the current is composed

of at least two dominant modes, the total current~J alongthe antenna body can be written in the neighborhood of theparallel resonance frequency as follows (assuming twodominant terms):

~J ~rð Þ ¼ 1� j�1ð Þ h~J1;~Eini1þ �2

1

~J1 rð Þ þ 1� j�3ð Þ

� h~J3;~Eini1þ �2

3

~J3 ~rð Þ þ HOT ð8Þ

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

~J ~roð Þ ¼ 1� j�1ð Þ J1* ~roð ÞEin ~roð Þ1þ j�2

1

~J1 ~roð Þ þ 1� j�3ð Þ

� J3* ~roð ÞEin ~roð Þ1þ j�2

3

~J3 ~roð Þ þ HOT ð9Þ

The antenna admittance due to the two strongly excitedmodes can then be written as follows, near the parallelresonance frequency:

Y ¼ I ~roð ÞV in ~roð Þ ¼

J ~roð ÞDw

Ein ~roð ÞDl¼ 1� j�1ð Þ j

~J1 ~roð Þj2Dw

1þ �21

� �Dl

þ 1� j�3ð Þ j~J3 ~roð Þj2Dw

1þ �23

� �Dl

þ HOT ð10Þ

Since both l1 and l3 are much larger than 1 and both haveopposite signs:

Re Yð Þ ¼ Dw

Dl

~J1 ~roð Þ 2�21

þ~J3 ~roð Þ 2�23

!þ Re HOTð Þ

ð11Þ

Im Yð Þj j ¼ Dw

Dl

~J1 ~roð Þ 2�1j j �

~J3 ~roð Þ 2�3j j

!þ Im HOTð Þ

ð12Þ

Hence when

~J1 ~roð Þ 2�1

¼

~J3 ~roð Þ 2�3

ð13Þ

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

4. Edge‐Fed Rectangular MicrostripPatch Antenna[18] A rectangular edge‐fed microstrip patch antenna,

16 mm × 12.448 mm in size, and mounted on a 31 milsubstrate with dielectric constant of 2.2 is shown inFigure 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 at6.1 GHz between two series resonances: one at 2.2 GHzand a second one at 6.9 GHz. As shown in Figure 7, thefirst series resonance is due to mode 1 and occurs at thefrequency at which the eigenvalue of mode 1 approacheszero. Similarly, the second series resonance occurs at thefrequency at which the eigenvalue of mode 2 approacheszero. Note that the first series resonance (mode 1) occurswhen the L‐shaped length of the current path on the patch(see Figure 5b) plus its image (ground plane) is approxi-mately l/2 at 2.2 GHz, where l is the wavelength in a

Figure 4. Current magnitude and phase distributionalong the dipole antenna at parallel resonance. The indexsimply denotes the location along the dipole.

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medium with a dielectric constant equal to the effectivedielectric constant of the structure ("r,eff = 1.8). Using thislength makes sense because when the feed point is nearthe corner, the currents for mode 1 tend to flow in thatdirection at 2.2 GHz. It can be shown that the frequencyat the first series resonance increases if the feed pointis moved toward the center of the edge. This also makessense because the effective length decreases, since thecurrents of mode 1 for this feed point location (near centerof edge) tend to flow parallel to the long edge of the patch.Furthermore, mode 1 does not radiate well because thecurrent on the patch and its image on the ground plane tendto cancel each other.[19] Previous work [Cabedo‐Fabres et al., 2003, 2007]

on determining the characteristic modes of a patch antennawas done by ignoring the feeding structure of the antennaand treating the patch antenna as a plate parallel to theground plane. This treatment yields different characteristicmodes than an actual patch antenna. For an actual patchantenna, the feed structure connects the top plate to the

ground plane physically either through coaxial feed ormicrostrip line, which is a different geometry than just twoparallel plates without feeding structure. There are alsomicrostrip patch antennas that are fed through proximitycoupling [Targonski and Pozar, 1993], but the feedingstructure (e.g., hole in the ground plane) has to be modeledas well. Otherwise, incorrect antenna characteristic modesare obtained.[20] The current magnitudes of modes 1 and 2 as well as

the total current at the input port, which consists mostly ofa summation of the two dominant modes (modes 1 and 2),are shown in Figure 8. Clearly, the current magnitude ofmode 1 approaches its maximum at the frequency whichcorresponds to the first resonant frequency. Similarly,mode 2’s maximum occurs at the frequency which cor-responds to the second series resonance. On the other

Figure 5. Edge‐fed microstrip patch antenna: (a) dimen-sions in mm and (b) dominant current path near the firstseries resonance.

Figure 6. Input impedance of the patch antenna.

Figure 7. Eigenvalue spectrum of the patch antenna.

Figure 8. Current magnitude at the input port of theedge‐fed microstrip patch antenna.

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hand, the parallel resonance at 6.1 GHz is due to currentcancellation at the input port between modes 1 and 2, sinceboth modes have similar magnitude (Figure 8) but oppo-site phase at the input port (Figure 9).

4.1. Current Distribution

[21] The magnitude of the normalized total current dis-tribution of the microstrip patch antenna at the first threeresonant frequencies is shown in Figure 10. As can beobserved, the normalized current distribution at the firstand the second series resonances is very small everywherealong the patch surface except at the input port. In contrast,at parallel resonance, the current distribution along thepatch antenna surface is similar to the well known cavity

mode TM01. Thus, the patch antenna is a poor radiator atits series resonances and a good radiator at its parallelresonance.[22] Figures 11 and 12 show the magnitude of the nor-

malized current distribution of modes 1 and 2, respec-tively, at the first three resonances: 2.2 GHz, 6.1 GHz and6.9 GHz. As can be seen in the graph, at 2.2 GHz, where l1approaches zero, the current distribution of mode 1 issimilar to the normalized patch total current distributionat 2.2 GHz (Figure 10). Also, the normalized currentdistribution of mode 2 (Figure 12) at 6.9 GHz, where l2approaches zero, is similar to the normalized total cur-rent distribution at 6.9 GHz (Figure 10). This behavior is

Figure 9. Phase of currents at the input port of the edge‐fed microstrip patch antenna.

Figure 10. Magnitude of total current distribution of thepatch antenna at its first three resonances: series reso-nances at 2.2 GHz and 6.9 GHz and parallel resonanceat 6.1 GHz.

Figure 11. Magnitude of mode 1 current distribution ofthe patch antenna at its corresponding three resonances:series resonance: 2.2 GHz and 6.9 GHz and parallel res-onance at 6.1 GHz.

Figure 12. Magnitude of mode 2 current distribution ofthe patch antenna at its corresponding three resonances:series resonances at 2.2 GHz and 6.9 GHz and parallelresonance at 6.1 GHz.

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expected since the total current distribution at a seriesresonance is mainly due to the contribution of the soledominant resonant characteristic mode.[23] At 6.1 GHz, since both modes 1 and 2 are dominant

modes and both have similar normalized current magnitudedistributions and phase, the total normalized current distri-bution resembles these modes, except in the vicinity of theinput port where the these two currents are out of phase.

4.2. Cavity Modes Versus Characteristic Modes

[24] Microstrip patch antennas are generally narrow-band antennas operated at frequencies corresponding totheir lowest‐order cavity modes. However, away from theresonant frequencies of the cavity modes, microstrip patchantennas are poor radiators. The microstrip patch antennaunder discussion has its first cavity mode at 6.1 GHz(Figure 10), which corresponds to the first parallel reso-nance of its input impedance. Hence, we conclude thatthe characteristic modes do not correspond to the cavitymodes, but rather cavity modes correspond to the patchparallel resonances, which is an interaction of two nearbycharacteristic 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, whilethe latter is open. In other words, a current can flow in theloop antenna even at zero frequency (DC current), whilethat is physically impossible in the dipole antenna. Thishas an important implication in the behavior of the inputimpedance of the antenna. The imaginary part of the inputimpedance of a 3 cm radius circular loop antenna is shownin Figure 13. The first parallel resonance of this loopantenna occurs around 0.744 GHz. Figure 14 shows theeigenvalue spectrum of the loop antenna. Note thatbecause of the antenna symmetry, degenerate modes willexist. Furthermore, there is a conductive mode (mode 0)that resonates at DC. Hence, the first parallel resonance ofthe loop antenna is due to the interaction of the degeneratemodes 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 ofits modes. We showed that parallel resonance phenomenadoes not correspond to a single characteristic mode, but israther due to an interaction of mainly two nearby modeseach having eigenvalues of opposite sign. This is in con-trast to the series resonance phenomenon, which is mainlydue to a single resonant characteristic mode (when thecorresponding eigenvalue becomes zero). It was also

Figure 13. Imaginary part of the input impedance of the 3 cm radius circular loop antenna.

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shown that the characteristic modes of a microstrip patchantenna do not correspond to its cavity modes. This ex-plains why the microstrip antenna normally operates atparallel resonance since the cavity mode can be expressedas the combination of at least two characteristic modes. Aloop antenna was the third antenna modeled here. Thisantenna has a DCmode which is not the case for the dipoleor microstrip patch antennas. Although we only consid-ered three different types of antennas, we believe for anyantenna, series resonance in the input impedance is due tothe resonance of a characteristic mode while parallel res-onance is due to the interaction of two or more charac-teristic modes.

References

Cabedo‐Fabres, M., E. Antonio‐Daviu, M. Ferrando‐Bataller,and A. Valero‐Nogueira (2003), On the use of characteristicmodes to describe patch antenna performance, IEEE Trans.Antennas Propag., 2, 712–715.

Cabedo‐Fabres, M., E. Antonino‐Daviu, A. Valero‐Nogueira,and M. F. Bataller (2007), The theory of characteristicmodes revisited: A contribution to the design of antennasfor modern applications, IEEE Trans. Antennas Propag.,49, 52–68, doi:10.1109/MAP.2007.4395295.

Garbacz, R. (1965), Modal expansions for resonance scatteringphenomena, Proc. IEEE, 53, 856–864, doi:10.1109/PROC.1965.4064.

Harrington, R. (1971), Computation of characteristic modesfor conducting bodies, IEEE Trans. Antennas Propag., 19,629–639.

Harrington, R. (1985), Characteristic modes for apertureproblems, IEEE Trans. Microwave Theory Tech., 33(6),500–505.

Harrington, R., and J. Mautz (1971), Theory of characteristicmodes for conducting bodies, IEEE Trans. Antennas Propag.,19, 622–628, doi:10.1109/TAP.1971.1139999.

Kabalan, K.Y., A. El‐Hajj, and A. Rayes (2001), A three‐dimensional characteristic mode solution of two perforatedparallel planes separating different dielectric media, RadioSci., 36(2), 183–193, doi:10.1029/1999RS002293.

Obeidat, K., B. D. Raines, and R. G. Rojas (2007), Antennadesign and analysis using characteristic modes, paper pre-sented at Antennas and Propagation Society InternationalSymposium, Inst. Electr. and Electr. Eng., Honolulu, 9–15June.

Obeidat, K., B. D. Raines, and R. G. Rojas (2008), Design andanalysis of a helical spherical antenna using the theory ofcharacteristic modes, paper presented at Antennas and Prop-agation Society International Symposium, Inst. Electr. andElectr. Eng., San Diego, Calif., 5–11 Jul.

Figure 14. Eigenvalue spectrum of the loop antenna.

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Obeidat, K., B. D. Raines, and R. G. Rojas (2009), Design ofantenna conformal to V‐shaped tail of UAV based on themethod of characteristic modes, paper presented at 3rdEur. Conf. on Antennas and Propagation, pp. 23–27, Berlin,23–27 Mar.

Targonski, S. D., and D. M. Pozar (1993), Design of wide‐band circularly polarized aperture coupled microstrip

antennas, IEEE Trans. Antennas Propag., 41, 214–220,doi:10.1109/8.214613.

K. A. Obeidat, B. D. Raines, and R. G. Rojas, ElectricalEngineering Department, Ohio State University, Columbus,OH 43210, USA. ([email protected])

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