The Role of Reactive Energy in the Radiation by a Dipole Antenna

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 63, NO. 8, AUGUST 2015

analysis and the design of such arrays. This proposed method of analysis is almost as accurate as a full-wave analysis of the entire array, and yet it is very efficient, as it only requires full-wave simulations of a single patch (to determine the coefficients E and γ for each patch) or two patches (to determine the mutual coupling M coefficients). It has also been shown that it is moderately important to account for the mutual coupling between the elements of the array. The results also demonstrate the limitations of the simple transmission line model that is often used to analyze series-fed arrays, where feed reactance is usually neglected. R EFERENCES [1] R. Garg, P. Bhartia, I. Bahl, and A. Ittipiboon, Microstrip Antenna Design Handbook. Norwood, MA, USA: Artech House, 2001. [2] R. B. Waterhouse, Microstrip Patch Antennas: A Designer’s Guide. Norwell, MA, USA: Kluwer, 2003. [3] J. L. Volakis, Ed., Antenna Engineering Handbook. New York, NY, USA: McGraw Hill, 2007. [4] A. G. Derneryd, “Linearly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 24, no. 6, pp. 846–851, Nov. 1976. [5] D. G. Babas and J. N. Sahalos, “On the synthesis of nonuniform linear microstrip arrays: A circuit design approach,” in Proc. 2nd Int. Symp. Trans Black Sea Reg. Appl. Electromagn., Jun. 2000. [6] R. Garg and V. Palanisamy, “Generalized cavity model and its application to series fed microstrip patch arrays,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Philadelphia, PA, USA, Jun. 1986, pp. 829–832. [7] J. Freese, R. Jakoby, H.-L. Blöcher, and J. Wenger, “Synthesis of microstrip series-fed patch arrays for 77 GHz-sensor applications,” in Proc. Asia Pac. Microw. Conf., Dec. 2000, pp. 29–33. [8] T. Metzler, “Microstrip series arrays,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 174–178, Jan. 1981. [9] B. B. Jones, F. Y. M. Chow, and A. W. Seeto, “The synthesis of shaped patterns with series-fed microstrip patch array,” IEEE Trans. Antennas Propag., vol. 30, no. 6, pp. 1206–1212, Nov. 1982. [10] S. Otto, O. Litschke, J. Leib, and K. Solbach, “A unit cell based low side lobe level design for series-fed array antennas,” in Proc. Asia Pac. Microw. Conf. (APMC), Dec. 16–20, 2008, pp. 1–4. [11] S. Otto, A. Rennings, O. Litschke, and K. Solbach, “A dual-frequency series-fed patch array antenna,” in Proc. Eur. Conf. Antennas Propag. (EuCAP), Mar. 23–27, 2009, pp. 1171–1175. [12] D. G. Babas and J. N. Sahalos, “Synthesis method of series-fed microstrip antenna arrays,” Electron. Lett., vol. 43, no. 2, pp. 78–80, Jan. 2007. [13] C. Niu, J. She, and Z. Feng, “Design and simulation of linear series-fed low sidelobe microstrip antenna array,” in Proc. Asia Pac. Microw. Conf. (APMC), Dec. 2007, pp. 1–4. [14] Z. Chen and S. Otto, “A taper optimization for pattern synthesis of microstrip series-fed patch array antennas,” in Proc. 2nd Eur. Wireless Technol. Conf. (EuWIT), Rome, Italy, Sep. 2009, pp. 160–163. [15] T. Yuan, N. Yuan, and L.-W. Li, “A novel series-fed taper antenna array design,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 362–365, Jul. 2008. [16] A. Benalla, K. C. Gupta, and R. Chew, “Computer aided design of linear series-fed microstrip patch arrays with a dielectric cover layer,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. Merg. Technol. 90’s Dig., Dallas, TX, USA, May 1990, vol. 4, pp. 1758–1761. [17] K.-L. Wu, M. Spenuk, J. Litva, and D.-G. Fang, “Theoretical and experimental study of feed network effects on the radiation pattern of series-fed microstrip antenna arrays,” IEE Proc. H Microw. Antennas Propag., vol. 138, no. 3, pp. 238–242, Jun. 1991. [18] P. S. Hall and J. R. James, “Crosspolarisation behaviour of series-fed microstrip linear arrays,” IEE Proc. H Microw. Opt. Antennas, vol. 131, no. 4, pp. 247–257, Aug. 1984. [19] D. R. Jackson and P. Manghnani, “Analysis and design of a linear array of electromagnetically coupled microstrip patches,” IEEE Trans. Antennas Propag., vol. 38, no. 5, pp. 754–759, May 1990. [20] S. Sengupta, D. R. Jackson, and S. A. Long, “Analysis of a linear seriesfed array of rectangular microstrip antennas,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Chicago, IL, USA, Jul. 2012, pp. 1–2. [21] S. Sengupta, “Analysis of a linear series-fed rectangular microstrip antenna array,” M.S. thesis, Univ. Houston, Houston, TX, USA, Dec. 2011.

[22] D. R. Jackson and J. T. Williams, “A comparison of CAD models for radiation from rectangular microstrip patches,” Int. J. Microw. Millimeter Wave Comput. Aided Des., vol. 1, no. 2, pp. 236–248, Apr. 1991. [23] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 30, no. 6, pp. 1191–1196, Nov. 1982.

The Role of Reactive Energy in the Radiation by a Dipole Antenna Constantinos A. Valagiannopoulos and Andrea Alú

Abstract—The spatio-temporal distribution of the near field around an antenna is of fundamental importance in understanding its radiative properties, since it can affect the radiated power flow emitted by the device. In this work, we consider the simplest case of a harmonic point dipole antenna activated at some point in time, and study the distribution of the energy flow velocity and reactive energy in space and time. We discuss how the reactive energy building up around the radiator affects quite strongly the flow of energy in the vicinity of the dipole, while in the far field, the expected radiation features are restored. The reactive energy in the near field decelerates or even inverts the overall energy flow, resulting in partial standing waves near the feed. We also investigate the transient response of the stored reactive energy and identify the convergence to steady state, which is demonstrated through time-averaged quantities. Our results shed interesting observations on the radiation properties of antennas in their near field, as time progresses. Index Terms—Dipole antennas, energy flow velocity, Poynting vector, reactive energy.

I. I NTRODUCTION The reactive (nonradiating) energy stored in the vicinity of any device emitting electromagnetic fields is a significant parameter that affects not only the performance and matching of the radiator but also the coupling with other structures placed in the near field. While the conventional operation of an antenna concerns radiation, the nonradiating energy stored in its vicinity is a quantity whose behavior determines all aspects of the device functionality. In [1], it has been illustrated how reactive energy can affect the signals outside the near-field regions; it is advocated that radio waves interact and combine with each other all the time, generating near fields even arbitrarily far away from the transmitters that create them and the receivers that detect them. Furthermore, reactive energy can influence the localization of electromagnetic power concentration in a transient electromagnetic system as indicated in [2]. Some specific cases of Manuscript received August 12, 2014; revised January 16, 2015; accepted May 12, 2015. Date of publication May 21, 2015; date of current version July 31, 2015. This work was supported in part by the Academy of Finland (postdoctoral project no. 13260996) and in part by the AFOSR under Grant FA9550-13-1-0204. C. A. Valagiannopoulos was with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78701 USA, on leave from the School of Electrical Engineering, Aalto University, Espoo 02150, Finland (e-mail: [email protected]). A. Alú is with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78701 USA (e-mail: alu@mail. utexas.edu). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2015.2436410

0018-926X © 2015 EU


field distributions have been also examined in [3], where the consequences of the space-time localization of the reactive energy have been pointed out. An important quantity related to the reactive energy is the energy flow velocity, which quantifies the velocity with which energy moves, at any given point in space, for an arbitrary field distribution. It is well known that plane waves and all other eigensolutions of the Helmholtz equation are traveling with the speed of light and it would be natural to assume that the same happens for the energy radiated by an antenna. However, this is not the case; in fact, the energy flow velocity is usually smaller than the speed of light, as originally reported by Bateman [4], who has been the first to investigate this quantity as a function of time and not in terms of time averages. The nontrivial features of the time-dependent energy flow have been elaborated in [5] where it is argued that, in the near zone of a radiator, the instantaneous power flow is slowed down by the presence of reactive fields. In a different context, Collin and Rothschild [6] were able to clearly connect the quality factor of spherical and cylindrical antennas with the reactive energy and the respective energy flow velocity; in the same sense, the quality factor has been recently optimized [7] with the use of variable current distributions and proper antenna geometries. It should be stressed that most studies on antennas are based on time-averaged or frequencydomain quantities, while here we study time-domain quantities, which highlight intriguing and, sometimes, unexpected properties. In the same context, many standard antenna textbooks [8], [9], [10], [11] include only few analyses and considerations in the time domain. In this work, we investigate the spatio-temporal variation of the reactive energy Wreac and identify its influence on the energy flow velocity v for the case of harmonic dipole fields activated at some point in time. Our excitation has been selected to combine both transient responses (causal source) and steady-state regimes (harmonic oscillator). In particular, we consider the simplest radiating element, an axially (ˆ z) polarized electric Hertzian dipole (point source), located at the origin of the spherical coordinate system (r, θ, φ) with uniˆ φ). ˆ Since any current distribution can be described tary vectors (ˆr, θ, as a (line, surface or volume) integral of properly weighted Hertzian dipoles, the obtained results for this elementary radiator are useful also when considering antennas of arbitrary shape and texture. While there are several ongoing discussions on the definition of reactive energy, and its correct use in antenna theory [12]–[14], our definition of energy quantities is based on the well-established concept of subtracting the radiated energy from the total energy, as originally introduced by Collin and Rothchild [6]. This is different from, e.g., [15], where the reactive power is defined as a part of the flown power, not the total one. In any case, the small differences among the different definitions of the instantaneous stored energy are not the core of this communication, in which we instead focus on the overall distribution and flow of energy around an antenna as it radiates. These results are not expected to be drastically modified by using one or another definition. It is well known that, if a Hertzian dipole is fed by a timedependent current i(t), the developed electromagnetic field (E, H) in the surrounding vacuum region (ε0 , μ0 ) is expressed in the form η0 h E(t, r, θ) = · 3 2πr ⎫ ⎧ t

⎪ ⎪ ˆr r i t − rc + c −∞ i τ − rc dτ cos θ ⎪ ⎪ ⎬ ⎨ r r i t − ˆ ⎪ 2 c sin θ ⎪ ⎪ ⎪ ⎭ ⎩ +θ + c t i τ − r dτ + r2 i t − r 2 −∞ c 2c c H(t, r, θ) =

h ˆ r

r r + i t− sin θ φ i t− 2 4πr c c c

(1)

(2)

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√ where c = 1/ ε0 μ0 is the speed of light and η0 = μ0 /ε0 is the wave impedance into vacuum. Equations (1) and (2) have been derived by inverse Fourier transforming the frequency-domain radiation from an electric dipole, as found in, e.g., [16]. The prime corresponds to the derivative with respect to the entire argument, while the quantity i(t)h is the dipole moment of the point source. We consider the case of a harmonic current excitation of magnitude i0 and angular frequency ω0 which is turned on at t = 0, namely i(t) = i0 sin(ω0 t)u(t) where u(t) is the unit step function. The use of the unit step function in the expression of the current is a canonical approximation to the actual excitation since there are no circuits with zero raise times. In general, the nonzero time constants of both emitter and receiver will modulate the ideal excitation considered here. As we discuss in the following, the outgoing energy velocity is equal to the speed of light for the precursor of the wave, regardless of its electrical distance from the source, which leads to a primary field distribution. In future time instants, the produced field meets this distribution to create a partially standing wave; as a result, the energy flow velocity can take zero or negative values (incoming direction). Points in space and time that support low or even a negative velocity are investigated in detail. We find that they are more common across the maximal radiation plane, where, on the other hand, the convergence to the far-field radiation happens more abruptly. As far as the reactive energy is concerned, we obtain analytical formulas separating the steady state from the transient term for both time-dependent and time-averaged quantities. By inspection of the graphs, one can observe how the reactive power is accumulated with time and, after several periods, only a time-harmonic distribution survives. Finally, the effect of the initial time point when time-averaging reactive energy over a period T0 = 2π/ω0 is considered and the convergence to the steady-state regime is discussed. The properties acquired by the host medium due to the presence of reactive energy and the ingoing flow velocity may be exploited to control the scattering and the reflection of electromagnetic fields.

II. T IME -D EPENDENT Q UANTITIES It is well known that the instantaneous Poynting vector S = E × H expresses the power density (Watt/m2 ) that flows through a surface at a given instant in time. On the other hand, the stored electromagnetic energy volume density in vacuum, defined by: w = 12 ε0 |E|2 + 1 μ |H|2 , is a volume density (Joule/m3 ) and, when integrated over 2 0 a volume, it evaluates the total electromagnetic energy that exists in it. The vector v = S/w is the electromagnetic energy flow velocity [4], measured in m/s. It should be stressed that this definition differs from that in [17], which may admit group velocities with magnitudes larger than c. With the use of formulas (1), (2), and the necessary definitions, one can readily obtain the analytical expressions for the energy flow velocity associated with the total fields of the infinitesimal dipole: v(t, r, θ) = ˆr vr (t, r, θ) + θˆ vθ (t, r, θ), when the excitation is the aforementioned causal tone i(t). The explicit expressions are not included here for brevity; furthermore, one should always take into account that, with the selected excitation, all the spatio-temporal varying quantities are nonzero only when r < ct = ω0 t/k0 ⇔ t > k0 r/ω0 = r/c, due to causality, i.e., the precursor of the radiated wave travels at velocity c in all directions, as expected. In addition, it is obvious that, due to the spherical form of the produced waves, the most significant component of the velocity is the radial one vr ; the tangential component vθ has a peripheral role balancing the dynamics created due to vr variations. It is also clear that the energy flow velocity is not necessarily equal to the speed of electromagnetic waves into vacuum (speed of light c). In fact, as we discuss in the following, due to the

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already established local field distribution, the waves do not propagate as in the far field, and generally v = ˆrc in the near region. As expected, if we evaluate the analytical expression of vr (t, r, θ) for the wave precursor (r = ω0 t/k0 = ct), we obtain lim vr (t, r, θ) = c

(3)

r→ct

namely the energy flow velocity of the field at the outer surface, which the excitation has just reached, equals that of the far field v = ˆrc, no matter how close to the origin we are (or no matter how small is the quantity k0 r = ω0 t). This finding is physically justified by the Einstein’s causality principle. This implies that, while we are used to assume that the energy flowing from an antenna travels at velocity c, it is only its initial pulse to satisfy this assumption, while, as time evolves, the energy flowing around the antenna can assume very different velocities. When it comes to the evaluation of the reactive energy around a dipole antenna, we should take into account that the density of electromagnetic energy w is not integrable at the origin r = 0 (due to the singularity of the problem at hand). Of course, this nonintegrability of the density w is just a mathematical consequence of our assumption that the source occupies null volume. If we consider an (arbitrary) physical source, the current density as well as the total electromagnetic energy will be finite everywhere. To keep the problem simple, we can consider a sphere of (small) radius r including the source, whatever it is. As long as the fields on this sphere are dipolar, the equivalence principle ensures that from that inner radius r outward, the fields are the same regardless of the selection of the source. In this way, we can study the stored energy outside the radius, and we neglect the (finite in reality, infinite in the point-source model) energy inside the inner sphere, which will depend on the realistic setup of the antenna. Therefore, as our dipole radiator, we effectively consider suitable currents on a spherical surface with radius r producing dipolar fields outside and zero fields inside. The wave at the observation time t extends until the boundary ω0 t/k0 = ct > r, and therefore, we can directly evaluate the total electromagnetic energy W by integrating the density w into the volume included between the spheres at ρ = r and ρ = ω0 t/k0 = ct. In order to find the reactive energy from W , we subtract the portion of energy associated with the radiated fields Wrad stored between the radii r and ω0 t/k0 = ct, which may be computed using the Poynting vector S and the fact that in the far-field region, the flow velocity of the fields equals the speed of light v = ˆrc, in the same way that has been originally suggested in [6]. After performing the necessary algebraic manipulations, the reactive energy existing outside a sphere of radius r at the time t > k0 r/ω0 is given by Wreac (t, r)

⎧ ⎨ η0 i20 2 (k0 h) = ⎩ ω0

1−cos(ω0 t−k0 r)+k0 r sin(ω r)+(k0 r)2 0 t−k0 ω t−k r

0 0 6π(k0 r)3 csc2 2 ω0 t sin(ω0 t−ξ) 1 − 12π k0 r dξ ξ

⎫ ⎬ ⎭

. (4)

The radiated energy is Wrad = W − Wreac . In the derivation of the aforementioned quantities, we used the same rationale followed in [18] and [19] where the excitation is purely harmonic. The energy density in the time domain has also been split into irreversible and reversible portions [20]. The time-dependent reactive energy Wreac can now be written in terms of the instantaneous radial velocity vr = (ˆr · S)/w, since in the far field, the radiation happens only radially Wreac (t, r) ct = 2π r

π 0

vr (t, ρ, θ) w(t, ρ, θ) sin2 θ dθ dρ. ρ2 1 − c

(5)

Fig. 1. Normalized radial energy flow velocity vr /c in (azimuthally independent) spherical contour plots expressed in the coordinate system (x, y, z) ⇔ (r, θ, φ) for: (a) t = T0 /4; (b) t = T0 /2; (c) t = 3T0 /4; and (d) t = T0 .

From this relation, it is clear that even if a spatio-temporal snapshot of the velocity v(t, r, θ) possesses the standard value ˆrc, it does not mean that the reactive energy Wreac (t, r) is zero. In fact, (5) indicates that the presence of reactive energy around the dipole necessarily corresponds to a spatio-temporal distribution of radial velocities sufficiently different from c. This is because the reactive energy accumulated outside a sphere of radius r at a time instant t is associated with all values of vr (t, ρ, θ) within the whole spherical mantle r < ρ < ct, not simply the instantaneous one vr (t, r, θ). In Fig. 1, we show the spatial variation in the radial component of vr /c for various time instants t = T0 /4, T0 /2, 3T0 /4, T0 within the first period since we turned on the dipole. The first two snapshots [Fig. 1(a) and 1(b)] are almost identical and differ only in the size of the region in which the field has been developed; the velocity is positive anywhere, as we are exploring the precursor dynamics and the reactive energy is not yet established around the dipole. However, in the next two snapshots [Fig. 1(c) and 1(d)], the radiated field is now affected by the reactive energy already distributed right around the dipole during the previous half-period and that forms a quasi-standing wave [3], yielding small and even negative values of vr /c in the vicinity of the origin, and accordingly points with vr = 0, which we name in the following as velocity nodes. In Fig. 1, we notice that, as the reactive fields around the antenna get established, they slow down the energy flow around the antenna, and can indeed sustain vortices and local inward power flow. As a result, the energy transport velocity is definitely not directed only radially outward unless we are referring to the far zone. Equation (3) is seen to clearly hold for all time instants of Fig. 1. To observe the spatial variation in the reactive energy for the corresponding time instants t of Fig. 1, we represent the normalized quantity ω0 2 Wreac as function of k0 r in Fig. 2. The vertical dashed lines η0 i2 0 (k0 h) correspond to the radii of the respective precursors k0 r = ω0 t. The curves are rapidly decreasing with k0 r, which is obvious from the definition of Wreac . Of course, the reactive energy is zero for k0 r > ω0 t. It is remarkable that the magnitude of Wreac is quite substantial close to the dipole for t = T0 /4 and in the next snapshot, (t = T0 /2) gets further increased. On the contrary, during the second half of the period


Fig. 2. Normalized reactive energy

ω0 2 Wreac η0 i2 0 (k0 h)

as function of the elec-

trical distance k0 r for various time snapshots t within the first period 0 < t < T0 . The dashed lines correspond to the radial surfaces k0 r = ω0 t that the wave has just reached for each time snapshot t (wave precursor).


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as function of the elec-


on the map of normal-

trical distance k0 r for various time snapshots t within: (a) T0 < t < 2T0 and (b) 100T0 < t < 101T0 (steady state).


ized time and electrical radius (t/T0 , k0 r): (a) for 2T0 < t < 5T0 and (b) for 102T0 < t < 105T0 (steady state).

Fig. 3. Normalized radial energy flow velocity vr /c in (azimuthally independent) spherical contour plots expressed in the coordinate system (x, y, z) ⇔ (r, θ, φ) for: (a) t = T0 + T0 /4; (b) t = T0 + T0 /2; (c) t = T0 + 3T0 /4; and (d) t = T0 + T0 .

T0 /2 < t < T0 , we observe a gradual deflation of Wreac to assume to low values at t = T0 . By comparing the behavior of Wreac with the variations in radial velocity in Fig. 1, it appears that during the first half period, a large amount of reactive energy is accumulated around the dipole location, which affects the energy velocity at future instants, to slowing down or even inverting the direction of energy flow (vr < 0) during the second half of the period. Kaiser in [3], using a relativistic analogy, compared the reactive energy to an effective inertial mass for the energy flow, and these findings are consistent with his analysis. In Fig. 3, we show the variation of vr /c for the second period after we switched on the dipole but at the respective time instants as in Fig. 1. By juxtaposing the corresponding snapshots, we realize that, due to the periodic variation of v, the distribution of the velocity at an instant of the first period is similar to that of the corresponding instant, no matter how many (integer in number) periods T0 have passed (this is restricted of course by the obvious inequality r < ω0 t/k0 = ct). This is quite an interesting finding on its own, since it shows that the time evolution of the energy velocity is very regular, even in the very beginning of the transient response. It is seen indeed that Fig. 1(a) is identical to Fig. 3(a) for k0 r < π/2, and similarly, for the other pairs

of figures; obviously, Figs. 1(d) and 3(d) are actually the same because, in both cases, the excitation has been propagated across the entire region 0 < k0 r < 2π. Again, one can notice interesting regions in space with negative radial velocity vr < 0 and accordingly the velocity nodes along which vr = 0. Similar to Fig. 2, we present in Fig. 4(a) the reactive energy as function of k0 r for the time snapshots of Fig. 3. The values of Wreac in the near field are much lower during the second period; however, the decaying trend with k0 r is much smoother compared to the first period, as the energy is spread over larger volumes. If we exclude the area with k0 r < 1, the stored reactive energy outside the sphere of radius r gets, on average, larger with time. This is further demonstrated in Fig. 4(b) where the steady-state Wreac distributions (100T0 < t < 101T0 ) are shown; it is clear that, on average, the accumulated reactive energy has an increasing trend, while the curve shapes stabilize over time. The spatio-temporal variation in Wreac can be better understood in Fig. 5, where the instantaneous reactive energy Wreac stored outside of the sphere of radius r is represented in contour plot on the (t/T0 , k0 r) map for two different time intervals of length 3T0 : one very close to the ignition time t = 0 and one very far from it. In Fig. 5(a), we see again how the reactive energy is gradually accumulated (increasing, on average, as a function of t) in the corresponding complementary sphere, since the steady-state regime has not been reached yet. On the contrary, in Fig. 5(b), what we observe is a periodic pattern with respect to t (steady state). As mentioned above, the most interesting scenario concerns the case in which the radial component of the energy velocity has a negative sign (incoming energy flow), which is quite counter-intuitive, given that the dipole is expected to radiate power outward. The nodes, namely the spatio-temporal paths along which vr = 0, delimit these

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the (natural) v = ˆrc correspond to the isolated nodes (7) lying close to straight lines on the (t/T0 , k0 r) map, i.e., we already reached the region in which (7) holds. This is reflected in the significantly lower values of reactive energy as we move away from the dipole location. As indicated in Figs. 1 and 3, the main occurrence of negative radial velocities happens during the second half of the periods (starting at t = 0), which is also verified by inspection of Fig. 6. In particular, we obtain vr < 0 mainly in the near field (indicatively k0 r < 3) and for 5T0 /4 < t < 2T0 . These regions widen when moving from small θ to θ = 90◦ . On the contrary, the larger is the angle 0 < θ < 90◦ , the more rapid convergence to the isolated nodes (7) is found once the observation point gets more distant from the near region.

III. T IME -AVERAGED Q UANTITIES

Fig. 6. Normalized radial energy flow velocity vr /c on the map of normalized time and electrical radius (t/T0 , k0 r) for: (a) θ = 30◦ ; (b) θ = 60◦ ; (c) θ = 75◦ ; and (d) θ = 90◦ .

Since we have a harmonic excitation, it is reasonable to analyze also the time averages over one period T0 = 2π/ω0 , by acting on the time-dependent quantity of interest with the operator: J { } = τ +T0 1 dt (we assume that at t = τ , the wave has reached the T0 τ point at observation radius r, namely r < cτ ). If this is done for the Poynting vector S and the electromagnetic energy density w, the energy velocity becomes ¯ (r, θ) = v

interesting regions and can be explicitly derived as the roots of the following transcendental equations: ⎧ ⎫ F ≡ k0 r cos(ω0 t − k0 r) ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ + sin(ω0 t − k0 r) = 0 or vr (t, r, θ) = 0 ⇒ . (6) 2 G ≡ (k0 r) − 1 cos(ω0 t + k0 r) ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ −k0 r sin(ω0 t + k0 r) + 1 = 0 The functions F, G are defined above and they are not dependent on θ, namely a single node constitutes a spherical surface. When k0 r is not small, the velocity nodes (6) are analytically determined as follows: k0 r1 F (ω0 t, k0 r) = 0 == ==⇒ G(ω0 t, k0 r) = 0 1 π (7) cos(ω0 t − k0 r) = 0 ⇒ k0 r = ω0 t + n + 2 for each integer number n ∈ N. They are present in the entire volume in which fields exist, and in this far-field range, they correspond to a simultaneous nullification of electric and magnetic fields; they are very localized in space time, and therefore are not that interesting, being just isolated “breaks” at which the velocity ceases instantly to possess the natural far-field value v = ˆrc. Most importantly, they do not correspond to cases with negative energy velocities vr < 0, since the value of v gets immediately restored to ˆrc, in the vicinity of these “breaks.” In the near field, on the contrary, the nodes of (6) represent the boundaries in which an inward energy flow can be found. In Fig. 6, in order to visualize this evolution from near field to far field, we represent the radial velocity vr /c in contour plot with respect to the time t/T0 and radius k0 r for various elevation angles θ; thus, what we are observing is the spatio-temporal evolution of the energy flow velocity across a cone of fixed θ in spherical coordinates. As stated above, the time dependence of vr is harmonic (with angular frequency ω0 ); therefore, all the variation in the energy flow velocity is included in a single period. In Fig. 6, we consider the second period after switching on the antenna; in this way, every single observation point under investigation 0 < k0 r < 2π has been excited. Regardless of the angle θ, it is clear that for k0 r > 3, the only combinations of time t and space r leading to velocities different from

J {S(t, r, θ)} = ˆrv¯r (r, θ) J {w(t, r, θ)}

=ˆ rc

(k0 r)4 sin2 θ . 2 [(k0 r)2 + 3] cos2 θ + (k0 r)4 + 32 sin2 θ

(8)

Note that this expression does not equal the time average of the instantaneous velocity defined above, since the ratio of two averages is not the average of the ratio [3]. We note that this averaged definition is the most commonly used in studying the energy velocity, yet it cannot capture the dynamics outlined in the previous section. It is interesting ¯ is not dependent on the initial time that the time-averaged quantity v point of integration τ (provided, of course, that τ > k0 r/ω0 , as mentioned above). This feature is due to the fact that the expressions of the transient fields (E, H) contain either harmonic (with period T0 ) or dc terms, whose integrals are not affected by the initial integration point τ . One can also show that, when the time average operator J { } is applied, only the radial component vr survives: the tangential velocity vθ has a neutral effect within one period T0 , as expected, since it is associated with a circulation of energy around the origin. Finally, it is worth noting that the time-averaged velocities are inherently positive for every point (r, θ) under investigation. Applying the operator J { } to the reactive energy Wreac to obtain its time average, we find that the result is dependent on the initial time point τ due to the presence of the transient integral term in (4). In particular, the time-averaged reactive energy is given by 2 2 ¯ reac (r) = η0 i0 (k0 h)2 3 + 2(k0 r) + 1 W ω0 24π(k0 r)3 24π 2 ω0 τ +2π ω0 τ cos(ω0 t − ξ) dξ + ln · . (9) ξ ω0 τ + 2π ω0 τ As expected, if we assume that a significant amount of time has gone by since we turned on the source and we perform the time averaging at a distant time interval (ω0 τ → +∞), the term between the brackets in (9) vanishes and we obtain the same result found in the steady-state case [6] lim

ω0 τ →+∞

2 2 ¯ reac (r) = η0 i0 (k0 h)2 3 + 2(k0 r) . W ω0 24π(k0 r)3

(10)


Fig. 7. Time-averaged normalized radial energy flow velocity v¯r /c in (azimuthally independent) spherical contour plot expressed in the coordinate system (x, y, z) ⇔ (r, θ, φ).

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velocity, the reactive energy, and their time averages. The representation of these quantities with respect to time, the electric radius, and the elevation angle yield several interesting conclusions regarding the energy flow velocity as time progresses, the combinations of space and time providing negative velocity, and the evolution toward the steady-state regime. This work may be expanded to analyze arrays of radiators which may form layers or surfaces with unconventional properties to control the energy flow velocity in space and time. For instance, a synchronized selection of current excitations could lead to maximization of the reactive energy and corresponding regions in which the energy flow velocity is largely slowed down or negative. On the other hand, with proper design of source placement and excitation, the near-field deceleration effect may be substantially mitigated and improve the radiation process in the transient and in the steady state.

R EFERENCES

Fig. 8. Normalized time-averaged reactive energy

ω0 ¯ 2 Wreac : η0 i2 0 (k0 h)

(a) on

the map of normalized initial time point and electrical radius (τ /T0 , k0 r) and (b) as function of normalized initial time point τ /T0 for various electrical observation radii k0 r. The dashed lines correspond to the steady-state regime.

Fig. 7 shows the polar contour plot of the time-averaged velocity component (8). One can clearly observe that the velocity in the vicinity of the origin is small, an indication that reactive energy is heavily affecting the energy flow, while its value gradually increases when moving toward the outer radial direction. The velocity converges to the speed of light c as the observer gets closer to the plane θ = 90◦ , while for points closer to z axis, its convergence is slower. Note that, despite its variation, the time-averaged energy flow velocity remains positive everywhere. ¯ reac In Fig. 8, we evaluate the time-averaged reactive energy W ¯ reac using (9). In particular, in Fig. 8(a), we show the variation of W with respect to the initial time point τ and the radius r. It is worth noting that the inclination of the iso-contour levels with respect to τ gets vanishing for larger initial time-averaging points, which means that when ω0 τ → +∞, the quantity becomes independent from τ as indicated by (9). More specifically, the k0 r-varying pattern at the rightmost vertical axis of the plot is very close to that of the steady-state case given by (10). In Fig. 8(b), this convergence to the steady-state ¯ reac is represented response is shown more clearly since the quantity W as function of τ for three different sphere radii k0 r. With dashed lines, we denote the steady-state values and one directly notices how the integral term in (9) becomes insignificant when we perform the time averaging at larger times and how the curves tend to coincide with the dashed lines for ω0 τ → +∞. IV. C ONCLUSION By considering the case of a monochromatic dipole source turned on at a specific time point, we derived and analyzed the analytical expressions for the spatio-temporal electromagnetic energy flow

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