Insights From the Radiation and Energy Storage Modes of Canonical Dipole Antennas

Decomposing the method of moments (MoM) impedance matrix of an antenna into its constituent eigenmodes can provide valuable insight into its behavior. Here, we analyze the radiation and energy storage modal solutions of canonical dipole antennas to establish an intuitive correspondence between an antenna’s mode of operation and its modal solutions. Examining these solutions as a function of frequency reveals clear modal transitions that directly affect radiation behavior. Specifically, we demonstrate that the radiation modes are well suited to identify and quantify changes in radiation behavior, because their solutions are decoupled from the energy storage properties of an antenna. We achieve this through the presentation of several examples, which include the analysis of modal changes that occur when an antenna is electrically small versus electrically resonant, as well as when the geometry of an antenna is parametrically modified.

Abstract-Decomposing the method of moments (MoM) impedance matrix of an antenna into its constituent eigenmodes can provide valuable insight into its behavior.Here, we analyze the radiation and energy storage modal solutions of canonical dipole antennas to establish an intuitive correspondence between an antenna's mode of operation and its modal solutions.Examining these solutions as a function of frequency reveals clear modal transitions that directly affect radiation behavior.Specifically, we demonstrate that the radiation modes are well suited to identify and quantify changes in radiation behavior, because their solutions are decoupled from the energy storage properties of an antenna.We achieve this through the presentation of several examples, which include the analysis of modal changes that occur when an antenna is electrically small versus electrically resonant, as well as when the geometry of an antenna is parametrically modified.

I. INTRODUCTION
A T ITS core, eigendecomposition transforms a complex problem into a set of simpler ones.In the context of antenna analysis, this often involves diagonalizing the method of moments (MoM) impedance matrix, such that it can be projected onto a set of orthogonal basis elements.This is the technique used in characteristic mode analysis (CMA) [1], [2], which is one of the most ubiquitous forms of modal eigendecomposition used for antenna analysis today [3].
The generalized eigenequation that is central to CMA relates radiated power to stored energy.This framework gives intuitive meaning to the eigenvalues-which represent the net stored energy-and ensures that the characteristic modes have predictable behavior over frequency [3], [4].However, the inherent coupling of radiative and reactive properties in CMA can obscure nuanced changes in radiation behavior [5], [6].
An alternative form of modal analysis in which the radiation, electric energy storage, and magnetic energy storage modes are computed is used here.Proposed in [5] and extended in [6], this technique involves decomposing the MoM impedance matrix into three matrix operators, each of which yield a set of eigenmodes.These modes provide specific insight into the radiation and energy storage properties of an antenna.Despite this, an intuitive interpretation of these modal solutions has remained elusive-this is, in part, due to a lack of orthogonality between different sets of modal solutions [6], as well as previously documented concerns about the operators retaining semidefiniteness for electrically large structures [7].The absence of such an interpretation has prevented radiation and energy storage modes from being used to inform design decisions in a similar manner as CMA.Consequently, radiation and energy storage modal solutions have been primarily used in numeric and theoretical contexts, e.g., to define optimization problems and determine physical bounds [7], [8], [9], [10].
In this work, we illuminate the relationship between an antenna's operational state and its radiation and energy storage modes.We achieve this by considering how the contribution of each mode to the total power (or stored energy) changes as a function of frequency.We apply this framework to a variety of canonical dipole antennas as illustrative examples with the intention of building insight.Through this analysis, we show that examining the radiation and energy storage modes of an antenna allows for the influence of these modes to be quantified and connected to observable properties, such as the radiation pattern.
The structure of this article is as follows.Section II provides information about the analytical technique used to perform modal analysis and discusses conventions used herein.Section III presents the results of that modal analysis for four primary cases: a strip dipole (Section III-A), a folded dipole (Section III-B1), a rectangular loop (Section III-B2), and an inductively loaded dipole (Section III-C).An additional case study of a Yagi-Uda antenna is presented in Section III-D as a more complex example.We show here that the radiation modes can be used to quantify changes in behavior, such as the transition of a folded dipole from behaving as an electrically small loop to a resonant antenna.Similar transitions are identified for a resonant loop antenna and an inductively loaded dipole.Conventional metrics, such as excitation current distributions and radiation patterns, are also presented to support the conclusions of the modal analysis.Section IV summarizes the results and identifies potential applications.

A. Radiation and Energy Storage Modes
The radiation and energy storage modes of an antenna are computed by taking the eigendecomposition of three matrix operators, R, X e , and X m , which represent radiation resistance, the electric component of reactance, and the magnetic component of reactance, respectively [6].These modes derive their name from their relationship to power and energy quantities [5], [6].While an expression for radiated power can be obtained directly from the complex Poynting theorem [11], a general solution for the average stored reactive energies W e and W m in terms of the current distribution was not established until 2010 [5].Such expressions and their discretization into a form compatible with the MoM solution procedure [9], [10] enable the results presented here.
To perform this analysis, the radiation and energy storage operators are decomposed into three sets of eigenmodes as follows: where L is used as a general notion for the operators R, X e , and X m .Each operator yields a set of modal eigencurrents, {J L }, that are orthogonal within the set (i.e., (J L i ) H J L j = δ i j ), but are not strictly orthogonal to each other (e.g., (J R i ) H J X e j ̸ = δ i j ).This orthogonality property allows the excitation current to be expressed in terms of the eigencurrents of a single operator, but not as a summation of all operators.
Eigencurrents of the same operator can then be used to expand any of the excited currents on a structure as follows: where a n is the modal weighting coefficient [6].This framework can be extended further to compute the modal power and energy percentage contributions of the total as follows: where each a n J n corresponds to its respective operator L (e.g., in (3), a n J n is a eigensolution of R) and the denominator enforces normalization.Modal power and energy contributions are used in Section III to quantify the relative influence of each mode at a given frequency and define where modal transitions occur.An implication of the normalization in (3)-( 5) is that the modal contributions alone do not indicate anything about the magnitude of the total power radiated or total energy stored by an antenna.This has the benefit of making modal transitions easy to identify when examining the modal contributions as a function of frequency.However, if knowing the exact modal contribution is pertinent to another application, an unnormalized version of (3)-( 5) can be obtained by simply taking the numerator of each expression.
It has been established through numerous works [6], [7], [8], [12] that the Vandenbosch formulation of stored energy can produce negative values, and that this issue is typically encountered when analyzing electrically large structures.While we do not encounter negative-valued W e or W m in this work, we acknowledge that this behavior may occur in the analysis of electrically larger antennas, since it has been reported in the literature.A recent study has suggested that this phenomenon is a result of oversubtracting the radiation energy when computing W e and W m and proposes using a modified version of the Vandenbosch expressions [12].Even so, we choose to use the original Vandenbosch formulation here because of its suitability for surface current analysis and the lack of a universally agreed upon method to calculate the total stored energy in radiating systems [7].For the same reason, we analyze the electric and magnetic energy storage modes separately.
Although the antennas considered here are modeled as perfect electrical conductors (PECs), the computed radiation modes would hold even if the antennas had finite conductivity.This is because the radiation modes are computed using the radiation resistance rather than the total resistance that includes ohmic losses [6].Including loss in the system would change the overall impedance matrix, which would consequently change the excitation current distribution and related parameters, such as input impedance.Thus, the total resistance of the system would have to be decomposed into two terms (R = R rad + R loss ) to compute the radiation modes if material loss is included in the model.The surface current distributions of the radiation modes, therefore, represent the currents that contribute to radiation resistance exclusively.Intuitively, this can be understood by considering that resistance in the PEC case represents what the radiation resistance would be if the antenna had 100% radiation efficiency.

B. Computation
Unlike the radiation operator, R, the electric and magnetic energy storage operators, X e and X m , cannot be extracted from Z alone and must be computed elementwise [6].However, these expressions bear a functional resemblance to the elementwise solution of Z and can be implemented in an existing MoM solver with relative ease [9], [10].
For this study, we used a Python-based MoM solver that utilizes parallel processing [13] to efficiently compute the R, X e , and X m matrix operators.The general workflow is as follows.First, the antenna models are constructed and meshed in Altair FEKO [14].These meshes are then imported into the MoM solver via the .outfile generated by FEKO.After the geometric parameters and feed point location are obtained, all further steps are completed using the Python-based MoM solver.These steps include defining the excitation and frequency sweep parameters and computing the MoM solution.A delta-gap feed with V f eed = 1 V at each frequency step was used to excite all the antennas presented here.Performing the MoM solution procedure yields the Z, X e , and X m matrices at each requested frequency.The R matrices are obtained by simply taking the real part of Z, because the antennas are modeled as PEC.Finally, the eigenanalysis and postprocessing steps are performed-all using Python.
Since the eigenvalue equations are evaluated at each frequency step, each solution must be initially assumed to be independent of the solutions at other frequencies.Thus, it is necessary to implement an eigenmode tracking algorithm to correlate the modal solutions across a frequency band.An approach similar to the surface current correlation method in [15] was used to perform mode tracking.At each sampled frequency, a correlation matrix was computed to compare the modal surface currents at different frequencies as follows: where J n and J m are modal eigenvectors at frequencies f 1 and , respectively, and the denominator enforces normalization.Here, the modal eigenvectors are related to the surface currents by a scaling vector of element lengths that depends only on the mesh.Similar to (1), L is used as a general notation for the radiation and energy storage operators.
The general procedure used for mode tracking is as follows.First, the eigenmodes at the lowest frequency are labeled in order-this initial labeling choice is arbitrary but is used for convenience.A correlation matrix is then computed to compare the eigenmodes of J n ( f 1 ) and J m ( f 2 ).If the indexing of the modes has not changed between frequencies, the correlation matrix will resemble an identity matrix.If the indexing has changed, the maximum value along each column will be determined, and the index of that value will be used to relabel the modes at f 2 .This procedure is repeated over the computed frequency range to correlate the eigenmodes at each frequency step with each other.
An advantage of using eigenmode correlation is that it ensures the surface current distributions of the tracked modes vary negligibly as a function of frequency.This allows us to represent the modal current distributions across a range of frequencies by a single current distribution in Section III.
For the purpose of analyzing the modal solutions, we adopt the following labeling convention: the index of 0 is assigned to the mode that is dominant at the frequency where maximum power radiation occurs.Subsequent modal solutions are labeled in an ascending order according to their modal power or energy contribution across the computed spectrum, e.g., the mode that is responsible for the second largest modal power contribution is labeled mode R1 and so on.

III. MODAL ANALYSIS
Three dipole antennas were selected as representative examples for this study: a strip dipole, a planar folded dipole, and a planar inductively loaded dipole.The geometries of these antennas are shown in Fig. 1, and their nominal dimensions are presented in Table I.These specific dipole antennas were selected for their well-known radiation properties and relatively simple geometries.That being said, we show, in this section, that examining the radiation and energy storage modal results can elucidate interesting radiation behavior, even in these canonical examples.

TABLE I NOMINAL DIMENSIONS OF DIPOLE ANTENNAS
It is also important to note that the electrical lengths shown in Table I are defined at 10 GHz.Any further discussion of electrical length in this article is assumed to be referenced to 10 GHz unless explicitly stated otherwise.

A. Strip Dipole
The first antenna considered is the strip dipole with the geometry shown in Fig. 1 and the dimensions stated in Table I.The total radiated power and stored energy of this antenna were computed from 0.1 to 15 GHz and are shown in Fig. 2. From this figure, it can be observed that maximum power radiation occurs at 9.11 GHz, and the antenna is resonant ( W e = W m ) at 9.34 GHz.The electrical length of the antenna is 0.467λ at its resonant frequency.Since modal analysis is the focus of this study rather than meeting specific design objectives, the dimensions of the dipole antenna were not further optimized to approach the nominal design frequency of 10 GHz-this is also true of the other dipole antennas discussed in this article.
The radiation and energy storage modes of this antenna were computed and tracked over the same frequency range.The mode tracking results along with the normalized surface current distribution of the three most dominant modes are shown in Fig. 3.It can be observed that radiation mode, R0, is responsible for over 99% of the radiated power across the spectrum.Similarly, the electric energy storage mode, X e 0, and the magnetic energy storage mode, X m 0, are dominant across the spectrum.Both dominant energy storage modes display half-wave surface current distributions, which is characteristic of a simple dipole antenna.The modal solutions of the strip dipole are straightforward to interpret, as one mode remains dominant for each operator.To further demonstrate this, the excitation current distribution and radiation pattern of the strip dipole at selected points in frequency are shown in Fig. 4. As discussed previously, Mode R0 is responsible for over 99% of the radiated power across the spectrum.Examining the radiation pattern at selected points in frequency reveals that it remains similarly constant, with the exception of becoming gradually more directive at higher frequencies.This is consistent with one's intuition about the radiation behavior of a strip dipole antenna-i.e., when it is electrically small, the excitation current is modeled as having a triangular profile, and when the electrical length is increased, this distribution broadens into the well-known halfwave current distribution of a resonant dipole.It follows that the current responsible for radiation, and consequently the radiation pattern, does not change significantly over frequency for a strip dipole-this is the behavior captured by radiation mode R0.An interesting nuance of this is that the dominant current distribution responsible for radiated power displays a near-uniform current along the length of the antenna, despite the total excitation current having a half-wave profile, as shown in Fig. 4(a).This arises from the difference between how the excitation current and the modal currents are calculated.
The excitation current is obtained by solving for the current, which satisfies the general matrix equation ZJ = V for a given feeding voltage, and is realizable on an antenna.The modal currents are the result of performing an eigendecomposition on only a component of Z.The radiation modes can, thus, be thought of as the currents that contribute to the real part of the impedance matrix, which ultimately determine the radiated power of an antenna.The electric and magnetic energy storage modes represent currents associated with the electric and magnetic components of reactance, respectively.In the context of this article, the strip dipole serves as an intuitive example of what to expect from the radiation and energy storage modal solutions when the radiation pattern of an antenna does not vary greatly as a function of frequency within the range of interest.

B. Folded Dipole
The next antenna we will consider is the folded dipole.When the folded dipole is electrically small, it behaves as a small loop antenna.When it is electrically resonant, it radiates similar to the strip dipole antenna.The electrically small behavior of the folded dipole and its modal transition to dipole-like radiation is considered in Section I.The spirit of this analysis is continued in Section II, where the parameter D-which defines the spacing between the two arms of the folded dipole-is varied.A case study where the folded dipole is modified to behave as a rectangular loop is considered as well.1) Electrically Small Behavior of the Folded Dipole: We first examine a folded dipole antenna with the geometry shown in Fig. 1 and the dimensions stated in Table I.Similar to the strip dipole case, the total radiated power and stored energy from 0.1 to 15 GHz were calculated and are shown in Fig. 5.This figure shows that maximum power radiation occurs at 8.07 GHz, and that there is a substantial increase in the stored magnetic energy toward the lower frequency end of the spectrum.The antenna is resonant at 9.26 GHz, and the electrical length of L (the physical dipole length) is 0.463λ at this frequency.
The radiation and energy storage modes were computed and tracked over the same frequency range.The tracking results of the three most dominant modes of each operator along with their corresponding normalized surface current distributions are presented in Fig. 6.Unlike the strip dipole discussed in Section III-A, the folded dipole does not display a single dominant mode for each operator and instead exhibits modal transitions.The radiation mode tracking results show that at low frequencies, a circulating current mode (R1) is dominant, but at higher frequencies, a dipole-like mode (R0) is dominant.The electric energy storage mode tracking results display a progression from a lower order (X e 0 with two nulls) to a higher order (X e 1 with two nulls) storage mode.Interestingly, the magnetic energy storage modes resemble the radiation modes.At low frequencies, a circulating current (X m 1) is dominant.When the folded dipole radiates maximum power, a dipole-like current mode (X m 0) is dominant.As the frequency increases beyond this point, a current distribution that resembles a resonant rectangular loop mode becomes dominant.
Examining the modal radiation solutions further reveals that there is a clear transition from the antenna behaving as a small loop to behaving as a dipole.To confirm that these modal solutions are capturing the radiation behavior accurately, the normalized excitation current distribution and total field radiation pattern at select frequencies are presented in Fig. 7. From this figure, it can be seen that between 0.1 and ≈1.5 GHz, the loop mode (R1) is responsible for over 50% of the power radiated.This is further supported by the excitation current distributions in Fig. 7(b), which exhibit the same transition indicated by the radiation modes.Examining the radiation pattern over this same range corroborates this statement, as the antenna radiates as an electrically small loop would.At around 2 GHz, the modal power contributions of the dipole (R0) and loop modes (R1) are equal.At this point, nulls begin to form along the ẑ-and x-axes-resembling a superposition of the small loop and dipole radiation patterns.As the frequency increases above 2 GHz, the dipole radiation mode (R0) becomes dominant, and the radiation pattern more strongly resembles a conventional dipole.
2) Modal Transition to Rectangular Loop: Next, we investigate the behavior of the folded dipole when the spacing between the two longer arms (D in Table I) is varied.Fig. 8 shows the simulated S 11 when the gap size is shrunk, assuming a 50-system.From this figure, it can be observed that the S 11 is minimally affected by decreasing D, and that D = {λ/20, λ/40, λ/60, λ/80, λ/100} converges with the S 11 of the nominal folded dipole.
When D is increased, the behavior of the antenna changes more noticeably, as shown in Fig. 9.As D increases from 0.1λ (the nominal case for the folded dipole) to 0.5λ, the S 11 displays a greater number of resonances (i.e., where imag{Z } = 0).Geometrically, increasing D makes the folded dipole resemble a rectangular loop antenna.The case when D = 0.3λ is analyzed further to demonstrate this; this antenna will be referred to as the rectangular loop herein.
Similar to the treatment of the previously discussed antennas, the radiation and energy storage operators were computed for the rectangular loop antenna from 0.1 to 15 GHz.The total radiated power and stored energy over this same frequency Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.that increasing D from its nominal dimension results in additional maxima in the radiated Specifically, this figure shows that the rectangular loop radiates maximum power at 6.73 GHz and is resonant at 7.10 GHz.At resonance, the perimeter of the rectangular loop has an electrical length of 1.14λ.An additional power maxima occurs at 12.76 GHz, where the electrical length of the perimeter is 2.04λ.
The mode tracking results and corresponding surface current distributions of the three most dominant modes of each operator are shown in Fig. 11.Similar to the modal results  of the folded dipole, multiple modal transitions occur within this frequency range.The radiation modes indicate that the antenna first behaves as a small loop at low frequencies and then transitions to behaving like a dipole-similar to the folded dipole.However, increasing D also has the effect of creating a resonant loop mode at higher frequencies.The electric energy storage modes also resemble the folded dipole case and display a similar trend in their modal energy contributions.Likewise, the magnetic energy storage modes parallel the behavior of the radiation modes.
To confirm that transitions between dominant radiation modes reflect true changes in the radiation behavior, the excitation current distribution and radiation patterns at select frequencies are shown in Fig. 12 (total field) and Fig. 13 (θ and  φ components).From 0.1 to ≈2.5 GHz, the antenna behaves as an electrically small loop.At 3 GHz, there is a modal transition between the small loop mode (R1) and the dipole mode (R0).As was the case for the folded dipole, the radiation pattern resembles a superposition of these two modes with nulls beginning to form along the ẑ-and x-axes.From 4 to 10 GHz, the antenna behaves as a folded dipole antenna; it is within this range that maximum power radiation occurs.It is also notable to mention that there is an increase in φ-polarized components relative to the nominal folded dipole.This is due to the increased length of the arms along the ŷ-direction, which contribute to radiation more significantly for the rectangular loop.At 12 GHz, another modal transition occurs, this time between the dipole mode (R0) and the resonant loop mode (R2).Beyond approximately 12 GHz, the resonant loop mode becomes the dominant radiation mode, with half-wave currents Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.along the ŷ-oriented arms and more uniform currents along the ẑ-oriented arms.Despite the perimeter of the loop having an electrical length of 1λ at approximately 6.25 GHz, the loop mode does not become dominant until its electrical length is closer to 2λ-this is why the pattern takes on a four-lobed shape.

C. Inductively Loaded Dipole
The next antenna considered is an inductively loaded dipole antenna.Geometrically, this antenna represents a transitional case between the strip dipole and the folded dipole (i.e., when L ind = λ/2, the inductively loaded dipole is equivalent to the folded dipole).This correspondence is reinforced by considering the S 11 traces for various values of L ind in Fig. 14.As L ind (physical length) decreases, the resulting S 11 shifts upward and toward the left, relative to the folded dipole.These changes correspond to a general increase in the input reactance and a decrease in the input resistance.Since decreasing L ind causes the antenna to more closely resemble a strip dipole-which has (1/4) of a folded dipole's input resistance at resonance-it is reasonable for the input resistance to decrease.In addition, the inclusion of the central loop imparts an inductive character to the antenna (relative to the strip dipole), because it introduces a path for a circulating current.
These observations about the input impedance are reminiscent of stub-tuning techniques that have been used for electrically small antennas [16].From this perspective, the inductive loop plays an analogous role to a shorted parallel stub and transforms the input impedance of a strip dipole antenna.Modifying other geometric parameters, such as the distance between the base of the strip dipole and the arm of the loop (D), provides an additional degree of freedom for tuning the input impedance.This is shown in Fig. 15, where D is modified to show cases where the length along ŷ is extended (dashed traces) and shrunk (solid traces).Extending the length of the loop has the effect of imparting a capacitive character to the antenna's S 11 , whereas shrinking the loop imparts an inductive character.This analysis further demonstrates the impedance-tuning properties of the inductive loop.
As a specific example, we consider an inductively loaded dipole with the geometry shown in Fig. 1 and dimensions  given by Table I.The total radiated power and stored energy from 0.1 to 15 GHz are shown in Fig. 16; maximum power radiation occurs at 9.26 GHz.The antenna is resonant at      electric energy storage modes show that a dipole mode (R0) is dominant from 0.1 to ≈12 GHz; beyond 12 GHz, a higher order storage mode becomes dominant.The magnetic energy storage modes show that for frequencies lower than 7 GHz, a circulating current isolated to the inductive loop (X m 1) is the dominant mode.At frequencies above 7 GHz, a dipolelike current (X m 0) is the dominant magnetic energy storage mode.
In Fig. 18, the modal power contribution of the radiation modes is compared with the excitation current and radiation pattern at select frequencies.This analysis shows that at low frequencies, the inductively loaded dipole also behaves as a small loop (R1), despite the physical loop area being much smaller than the folded dipole or rectangular loop.At frequencies higher than 2 GHz, the dipole mode (R0) is the dominant radiation mode.From 3 to 15 GHz, the dipole mode is responsible for over 85% of the radiated power.

D. Yagi-Uda
The final antenna considered is a Yagi-Uda, which serves as an example of how the modal solutions can be applied to more complex geometries.Yagi-Uda antennas consist of a driven element arrayed with parasitic elements that function as reflectors and directors.Near the resonance of the driven element, the reflector is electrically longer than (λ/2), and the director(s) are electrically shorter than (λ/2) [17].II.The geometry of a Yagi-Uda with four parasitic elements is shown in Fig. 19.The accompanying dimensions are stated in Table II and were determined using standard design equations [17].Similar to the dipole antennas discussed earlier, the electrical lengths are referenced to 10 GHz.At frequencies near resonance, the interaction between the driven and parasitic elements causes the main lobe of the radiation pattern to be directed along + ŷ, according to the coordinate system depicted in Fig. 19.
While Yagi-Uda antennas are typically analyzed only over their resonant bandwidth, we choose here to examine its modal solutions over a larger frequency range.This is done to emphasize the utility of using radiation and energy storage modal analysis to identify frequency-dependent trends.The S 11 of the Yagi-Uda is shown in Fig. 20, along with the S 11 of the isolated strip and folded dipoles considered earlier.It can be seen that the parasitic elements in the Yagi-Uda modify the input impedance of the driven folded dipole relative to its isolated behavior.In general, these parasitic elements result in a reduction in input resistance and an increase in the input reactance.
The radiation modes of the Yagi-Uda driven with a folded dipole are shown in Fig. 21, along with the excitation current and radiation pattern at select frequencies.The most obvious difference when compared with the modal solutions of an isolated folded dipole is that more modes are required to capture the behavior of the Yagi-Uda antenna over the same frequency range.Below resonance, this antenna has two primary modes of operation: an electrically small loop mode (0.1-2 GHz)  and a dipole mode (3-8 GHz).These results show that the loop mode associated with the electrically small behavior of the folded dipole is dominant at low frequencies, even in the presence of the parasitic elements.This is a nonobvious result, because the mutual coupling that occurs between parallel linear elements is typically nonnegligible.Furthermore, the parasitic elements of the Yagi-Uda are very close spaced in this frequency range, e.g., at 1 GHz, the electrical length of this spacing is only 0.021λ.Despite these factors, the loop mode remains dominant in the electrically small regime.With the exception of this low-frequency behavior, and just above resonance (9-10 GHz), a dipole mode (R0) contributes dominantly to the total radiated power.The influence of this mode is evidenced by the null along the ẑ-direction that occurs in the radiation pattern.Just above resonance, the radiation behavior is described by a combination of two loop modes (R1 and R2) and two dipole modes (R0 and R3).The dominant mode in this frequency range, R3, shows that the currents on the driven element and the reflector are directed opposite to each other; this creates the condition for field cancellation in the − ŷ-direction.At 9 GHz, this effect is especially prominent and yields the familiar directive pattern of a Yagi-Uda antenna.At frequencies further above resonance, the back and side lobes of the radiation pattern increase.This change is attributed to the currents along the reflector and the driven element being in-phase, while the currents on the right-most two directors are out-of-phase with the driven element, causing field cancellation in the + ŷ-direction.In addition, the loop modes (R1 and R2) and mode R4 have higher surface current densities on the short arms of the folded dipole, which promote current along the x-direction rather than the ẑ-direction.
The electric and magnetic energy storage modes are shown in Fig. 22.It can be seen that a dipole-like mode (X e 0) is dominant at low frequencies, and a higher order version of this mode (X e 4) becomes dominant at higher frequenciesthese results mirror the behavior of the isolated folded dipole.
Near resonance, modes X e 1, X e 2, and X e 3 become the dominant electric energy storage mode in succession.Each one of these modes displays surface currents that are dominant on one or two of the parasitic elements.Interestingly, the ordering of when these modes become dominant can be directly related to the electrical lengths of the parasitic elements with significant current densities: the reflector is the largest element (X e 1), followed by the first and third directors, which are the same size (X e 2), and the second director is the smallest (X e 3).
The magnetic energy storage modes show that a small loop mode (X m 3) is dominant at low frequencies and is followed by a dipole mode (X m 0) near resonance.At higher frequencies, a resonant loop-like mode (X m 4) becomes dominant.These trends are similar to the magnetic energy storage behavior of the isolated folded dipole, but differ in that the energy storage modes of the Yagi-Uda interact more significantly with each other.This is attributed to the geometry of the Yagi-Uda antenna consisting of closely spaced parallel linear elements.When the driving antenna is excited, it produces a circulating magnetic field, which, in turn, induces a current on the parasitic elements and vice versa.This interaction is reflected in the modal magnetic energy percentage contributions, as several modes are required to describe the magnetic energy storage behavior at a given frequency.

IV. CONCLUSION
Radiation and energy storage modes can provide valuable insight into the behavior of an antenna over a frequency range much larger than its resonant bandwidth.The resulting modal current distributions can be expressed in terms of the total radiated power (or stored energy), making the identification and quantification of these modes visually intuitive.The canonical dipole examples considered in this study emphasize the relationship between the geometric features of an antenna and its resulting radiation modes.Interestingly, it was shown that even minor changes to the geometry of a simple antenna can introduce new radiation modes and yield appreciably different radiation patterns away from resonance.
Overall, this analysis has shown that the radiation and energy storage modes can be used to gain quantitative insight about the radiative, capacitive, and inductive nature of current distributions on an antenna.Importantly, these modal solutions enable analysis that decouples radiation behavior from energy storage properties, which makes them especially suitable for examining the radiation properties of antennas with substantial energy storage qualities (e.g., electrically small antennas).
Future work includes applying this technique to more complicated antenna geometries and exploring how these modal current distributions can inform design decisions.Electromagnetic compatibility testing and reconfigurable antenna design are two areas where the conclusions of this analysis could be beneficial.In the context of electromagnetic compatibility research, performing an analysis similar to the one presented here could reveal nonobvious coupling paths and inform mitigation strategies.As for the design of reconfigurable antennas, knowledge of radiation and energy storage modal distributions may be especially useful when determining how best to implement reconfigurability (i.e., identifying which currents should be minimally/maximally disturbed to achieve a desired operational state) and aid in the placement of tuning devices.

Fig. 1 .
Fig. 1.Definition of antenna geometries for (a) strip dipole, (b) planar folded dipole, and (c) inductively loaded strip dipole.Each antenna was driven by a delta-gap feed at the point denoted by the blue and red triangles.It should be noted that these models are not drawn to scale.

Fig. 2 .
Fig. 2. Total radiated power [W] and stored energy [J/s] as a function of frequency for the strip dipole.Maximum power radiation occurs at 9.11 GHz, and the antenna is resonant at 9.34 GHz.

Fig. 3 .
Fig. 3. Three most dominant (a) radiation, (b) electric energy storage, and (c) magnetic energy storage modes as a function of frequency for the strip dipole.

Fig. 4 .
Fig. 4. (a) Excitation current distribution of the strip dipole at select frequencies.(b) Normalized total field radiation pattern at selected frequencies-the φ component is negligible.It can be observed that the radiation pattern does not significantly vary across the frequency band.

Fig. 5 .
Fig. 5.Total radiated power [W] and stored energy [J/s] as a function of frequency for the folded dipole.Maximum power radiation occurs at 8.07 GHz, and the antenna is resonant at 9.26 GHz.

Fig. 6 .
Fig. 6.Three most dominant (a) radiation, (b) electric energy storage, and (c) magnetic energy storage modes as a function of frequency for the folded dipole.

Fig. 7 .
Fig. 7. (a) Modal power contribution of the dipole and loop radiation modes as a function of frequency.(b) Normalized excitation current distribution at selected frequencies, scaled according to the color bar in (a).(c) Normalized total field radiation pattern at selected frequencies-the φ component is negligible.It can be observed that the radiation pattern evolves from that of an electrically small loop antenna to a dipole, as predicted by the radiation modes.

Fig. 8 .
Fig. 8. Smith chart representation of the S 11 when the distance between arms (D) of the folded dipole is decreased.Each trace spans 0.1-15 GHz.

Fig. 9 .
Fig. 9. Smith chart representation of the S 11 when the distance between arms (D) of the folded dipole is increased.Each trace spans 0.1-15 GHz.

Fig. 10 .
Fig. 10.Total radiated power [W] and stored energy [J/s] as a function of frequency for the rectangular loop.Maximum power radiation occurs at 6.73 GHz, and a secondary maxima occurs at 12.76 GHz.The antenna is resonant at 7.10 GHz.

Fig. 11 .
Fig. 11.Three most dominant (a) radiation, (b) electric energy storage, and (c) magnetic energy storage modes as a function of frequency for the rectangular loop.

Fig. 12 .
Fig. 12.(a) Three most dominant radiation modes of the rectangular loop.(b) Modal power contribution of the dipole and loop radiation modes as a function of frequency.(c) Normalized excitation current distribution at selected frequencies, scaled according to the color bar in (a).(d) Normalized total field radiation pattern at selected frequencies.

Fig. 14 .Fig. 15 .
Fig. 14.Smith chart representation of the S 11 when the length of the loop along ẑ (L ind ) of the inductively loaded dipole is varied.Each trace spans 0.1-15 GHz.

Fig. 16 .
Fig. 16.Total radiated power [W] and stored energy [J/s] as a function of frequency for the inductively loaded dipole.Maximum power radiation occurs at 9.26 GHz, and the antenna is resonant at 7.85 GHz.

Fig. 17
Fig. 17.Three most dominant (a) radiation, (b) electric energy storage, and (c) magnetic energy storage modes as a function of frequency for the inductively loaded dipole.
Fig. 17.Three most dominant (a) radiation, (b) electric energy storage, and (c) magnetic energy storage modes as a function of frequency for the inductively loaded dipole.

7 .
85 GHz, and the electrical length of L (the physical dipole length) is 0.393λ at this frequency.Compared with the strip dipole, which is resonant at 9.34 GHz, the resonance frequency of the inductively loaded dipole is significantly lowered; this further supports the interpretation that the inductive loop functions as a tuning stub to change the input impedance of the antenna.The mode tracking results and surface current distributions of the three most dominant modes of each operator are shown in Fig.17 . The radiation modes show that a dipole mode (R0) is dominant over the majority of the frequency range, and that a loop mode (R1) is dominant at low frequencies.The

Fig. 18 .
Fig. 18.(a) Modal power contribution of the dipole and loop radiation modes as a function of frequency.(b) Normalized excitation current distribution at selected frequencies, scaled according to the color bar in (a).(c) Normalized total field radiation pattern at selected frequencies-the φ component is negligible.It can be observed that the radiation pattern evolves from that of an electrically small loop antenna to a dipole as predicted by the radiation modes.

Fig. 19 .
Fig. 19.Geometry of Yagi-Uda antenna with four parasitic elements and a folded dipole as a driven element.The associated dimensions are listed in TableII.

Fig. 20 .
Fig. 20.Smith chart representation of the S 11 of a Yagi-Uda antenna driven by a folded dipole (yellow dashed traces), along with the S 11 of isolated strip and folded dipoles for comparison.Each trace spans 0.1-15 GHz.The Yagi-Uda is resonant at 8.3 GHz.

Fig. 21 .
Fig. 21.(a) Modal power contribution of the radiation modes as a function of frequency.(b) Normalized excitation current distribution at selected frequencies, scaled according to the color bar in (a).(c) Normalized total field radiation pattern at selected frequencies.

Fig. 22 .
Fig. 22.Five most dominant (a) electric energy storage and (b) magnetic energy storage modes as a function of frequency for the Yagi-Uda driven with a folded dipole.
Insights From the Radiation and Energy Storage Modes of Canonical Dipole Antennas Sasha S. Y. Cain , Graduate Student Member, IEEE, Elias Wilken-Resman, Member, IEEE, and Jennifer T. Bernhard , Fellow, IEEE

TABLE II NOMINAL
DIMENSIONS OF YAGI-UDA ANTENNA