The Reactive and Radiation Electromagnetic Energies of Antennas: a New Formulation

It is required to calculate the stored reactive energy of an antenna in order to evaluate its Q factor. Although it has been investigated for a long time, some issues still need further clarifying. The main difficulty involved lies in that the reactive energy of an antenna tends to become infinitely large when integrating the conventional defined energy density in the whole space outside a small sphere containing the antenna. The reactive energy can be made bounded by subtracting an additional term associated with the radiation fields. However, there exists no well-founded accurate definition for this term that is valid for all cases. By re-checking the definition of energy densities, a new formulation is proposed in this paper to separate the reactive energy and the radiation energy explicitly. The clearly defined reactive energy is bounded without necessary to subtract the additional term, and the formulae are easy to implement.


I. INTRODUCTION
The stored reactive electromagnetic energy of an antenna is a very important parameter, which can be used to evaluate the Q factor of the antenna and predict its bandwidth.The reactive energy has been investigated by many researchers, and the calculation methods proposed so far can be roughly divided into two categories: (1) methods in early stage that based on spherical mode expansion technique [1]- [3], and (2) methods of determining the fields using computational electromagnetic methods or simulators [4]- [7].In 1948, Chu in his paper [8] discussed the radiation problem associated with electrically small antennas, and derived ladder type equivalent circuits for 0 n TM / 0 n TE spherical waves, based on which the upper bound on their Q factors can be predicted.The reactive energy only includes those stored in the reactive elements in the equivalent circuit, hence is bounded and can be accurately evaluated.Collin [1] calculated the reactive energies strictly with fields obtained using mode decomposition method [9] [10], where the reactive energies of spherical modes and cylindrical modes are obtained by directly integrating the term     in the whole space outside a sphere with a small radius.Since the integration is infinite, the energy density associated with the radiation fields has to be subtracted from the integrand.Fante [2] extended the results of Collin, and McLean re-examined the case of small antennas and calculated the Q factor of 10 TM and 10 TE mode [3].For small antennas, spherical mode expansion solution for reactive energy is a good approximation, and can provide satisfactory upper bound for Q factors.It has been extended to analyzing larger antennas [11], where the radiation fields by a current distribution are expanded with spherical modes.However, it is not efficient because many modes may be needed for antennas with large size and complicated structures.Furthermore, the fields inside the sphere enclosing the antenna cannot be addressed accurately.Therefore, it is more natural to use numerical methods to calculate the reactive energies, as has been investigated by many researchers [5][12]- [14].Basically, a typical numerical procedure to calculate the reactive energies can be carried out in two steps.First, determine all currents involved in the system, such as the induced currents on metal surfaces, the polarization currents in dielectrics, the magnetization currents in magnetic materials, and the currents in lossy media.This can be done using common methods that have been developed in the computational electromagnetic community, such as methods based on electric field integral equations (EFIEs) [15]- [18], volume integral equations (VIEs) [19] [20], and methods based on equivalence principles [21]- [24].Second, calculate the radiation electromagnetic fields from these current sources, and then the stored reactive energies by integrating the energy densities in the whole space.However, as pointed out by many researchers, the stored reactive energy obtained in this way is infinitely large if the conventionally defined electric energy density and magnetic energy density are used.Subtracting a special term of energy density associated with the radiation fields from the integrand has become a common strategy.However, this additional energy density is ultimately an ambiguous concept that has no rigorous definition.It might be acceptable for small antennas, but is quite inaccurate for large antennas, in which the propagation pattern is largely different from spherical waves, especially in the region near the radiators.
Many efforts have been made to overcome this difficulty.Vandenbosch [25] proposed a set of formulae for calculating the reactive energies, which are expressed in closed form of integrations with respect to the current densities in the antenna structure.
The Key Laboratory of Ministry of Education of Design and Electromagnetic Compatibility of High-Speed Electronic Systems, the Department of Electronic Engineering, Shanghai Jiaotong University, Shanghai, 200240, China (e-mail: gaobiaoxiao@sjtu.edu.cn).

The Reactive and Radiation Electromagnetic
Energies of Antennas: a New Formulation

Gaobiao Xiao
The formulation has been extended to time domain [26][27].Some special techniques have been introduced in determining the stored reactive energies.The method has attracted much attention from many researchers [28]- [33], and has been successfully applied for analysis and optimization of small antennas [34]- [40].However, some issues might need further clarifying.For example, those expressions are derived by assuming that the current is frequency-independent.This assumption may be reasonable for excitation currents that can be adjusted at the feeding port.However, it needs to be proved that the other types of currents, such as the induced currents, polarization currents and magnetization currents are frequency-independent.It is reported that the proposed expressions can produce negative values of stored energy for electrically large structures [38].Furthermore, the formulation in time domain may give results that are a little bit different from those obtained with the formulation in frequency domain [27].
A careful re-examination on this issue reveals that the difficulty involved in reactive energy can be traced back to an old classical problem: for a given time-varying current distribution   are respectively the total electric energy and magnetic energy that have the same form as their counterparts for static fields.The electromagnetic energy calculated with this formula is generally bounded for finite source densities.For point sources, it is also bounded if a small spherical region containing the point source is excluded.Intuitively speaking, bounded reactive energy seems more acceptable, because it looks strange for an antenna to store infinitely large amount of reactive energies in the surrounding background.
Based on this consideration, the basic concept of the reactive energy is revisited and a new formulation for calculating the reactive energies of radiators is derived, in which the reactive energy of a radiator can be explicitly separated from its radiation energy.The detailed deduction is described in Section II, and validated with simple examples in Section III.The expressions for calculation of reactive energies and Q factors of antennas are in shown Section IV, with a summary in Section V.

II. SEPARATING THE REACTIVE ENERGY FROM THE RADIATION ENERGY
In order to give a clear description on the formulation, the introduction of the energy densities for static fields are re-examined at first and the derivations are described in detail even though some of them are quite fundamental.
To begin with, consider the simple case that a static charge density  , is located in the infinitely large free space.A popular method to obtain the total energy generated by the charge is to assume that all charges are moved piece by piece from infinite to their current positions.Based on energy conservation law, it can be deduced that the total electrostatic energy of the whole system is equal to the work done to the charges, which is derived to be where   r   is the scalar electric potential and E     for static fields.Applying Gauss' Law, D      , eq. ( 1) can be cast into For the sake of simplicity, hereafter, S  and V  are used to denote the spherical surface and the space when r   .In the case of static fields, , the first term at the RHS of (2) approaches zero.The electric energy density is defined as Similarly, the magnetic energy in the free space associated steady current distribution   , can be expressed by where   A r   is the vector magnetic potential relating to the magnetic flux density . Applying Ampere's Law in static case, J H     , eq.( 4) can be transformed to , the surface integration in (5) also vanishes.The conventional magnetic energy density is then defined as 1 2 The procedure for deriving (1) and ( 4) is basically valid for time-varying sources，as discussed in [41].However, it is obvious that the total energies associated with time-varying sources cannot be correctly calculated by integrating the energy densities defined in (3) and (6).It can be seen more clearly in the following derivation that the unbounded integration results discussed by many researchers are actually not the correct reactive energy of time varying fields.
In time-varying situations, E A t , the electric energy associated with charge distribution can be expressed as The upper script '~' is used to indicate time varying variables.Although the asymptotic behavior of the amplitudes of the potentials when approaching infinity is similar to those of static fields, the asymptotic behavior of time varying electromagnetic fields is quite different from the static fields.Both electric fields and magnetic fields have a transverse components of order of   O r .The surface integral in (7) does not vanish if judging from the asymptotic behavior of the amplitude of its integrand.Fortunately, only the radial component of the electric flux contributes to the surface integral in (7), the amplitude of which approaches infinity at speed of   . Therefore, instead of (3), for time varying fields, it is reasonable to introduce a reactive electric energy density as defined below, However, the introduction of reactive magnetic energy density has to be more careful.In time varying case, the Ampere's Law must include the displacement current.
We have to take care that the surface integral in ( 9) is not zero.However, it can be proven that it is bounded making use of the asymptotic behavior of the magnetic fields and the vector potential.This term can be considered as the energy stored at the infinity beyond surface S  , or equivalently considered as absorbed by the radiation resistor at infinity.Apparently, it is reasonable to consider the second term in the RHS of (9) as the reactive energy associated with the current source, from which the reactive magnetic energy density for time varying magnetic fields can be defined as Integrating m w  over the whole region gives the total reactive magnetic energy For time varying fields, the reactive energy densities are defined with ( 8) and (10).For static charges and steady currents, 0 t    , these formulae return to their conventional forms for static fields.Integration of part of the energy densities cannot provide meaning information for the reactive energies of the time varying sources.
The electric energy and the magnetic energy are coupled with each other for time varying fields, hence, it might be more proper to consider them as a whole reactive electromagnetic energy defined as The energies can be computed with integrations either over the region where sources exist, or over the region where fields exist.For radiation problems in open environment, it is obviously more efficient to calculate the reactive electric energy in terms of sources, Since the reactive magnetic energy is basically calculated with (9).In some situations, it is possibly more convenient to calculate the reactive magnetic energy in terms of sources and the radiation energy in terms of surface integral, We will show in the following that, for time harmonic fields, it is very convenient to calculate the reactive magnetic energy with (14) because the surface integral is zero.Apparently, the new formulation only concerns with the energy balance at time t , the reactive energy and radiation energy are clearly separated.Both the reactive electric energy and the reactive magnetic energy are bounded.The total radiation energy depends on the radiation time.In practical applications, the radiation power is of much more significance than the radiation energy.In order to evaluate the Q factors of antennas, it is efficient to calculate the stored reactive energies with ( 13) and ( 14), meanwhile, calculate the radiation power based on Poynting theorem,

III. FOR TIME HARMONIC FIELDS
For sinusoidal time varying electromagnetic fields with time dependence of   exp j t  , the complex energy densities can de expressed with phasors, For the sake of simplicity, the same symbols are used for phasors in the expressions.It can be verified that the Poynting theorem has the form as usual It can be justified from ( 16) and ( 17) that both energy densities for sinusoidal time varying fields differ from the static ones by an identical term * 4 j A D  

 
, which cancel out in the Poynting theorem.The time-averaged stored reactive energies of an antenna in free space can then be calculated in terms of sources The real part of the surface integral in (21) can be proved to be zero, so it can be simplified as As has mentioned in previous discussion, the radiation energy depends on how long the antenna has worked.It may be unbounded for sinusoidal time varying fields since the time integration of the radiation power should be performed over period of

 
,t  .The reactive energies can be calculated with integrations over sources area, and are bounded consequently.
The radiation power is calculated either in terms of sources or in terms of fields,

IV. VALIDATION WITH HERTZIAN DIPOLE
In order to compare the proposed formulation with conventional methods, the stored energy and Q factor of a Hertzian dipole is analyzed, which has been calculated by Mclean in [3].It's known that a Hertzian dipole generates 10 TM spherical mode in free space (denoted with 01 TM in [3]).The fields of a Hertzian dipole are symmetrical about the z-axis, the components of which can be readily derived as below 2 1 sin The amplitude of the Hertzian dipole is assumed to be    so as that these expressions are exactly the same as those shown in [3].However, the vector potential given in [3] is not appropriate because we cannot find a scalar potential to satisfy the Lorentz Gauge.A proper vector potential can be obtained directly from the dipole current as follows, As can be readily checked that the integration of e w and m w in the space outside a small sphere with radius a is infinite due to the contribution of the   2 1 r terms.These terms are canceled in [3] by subtracting the energy density associated with the radiation fields, which is With the new definition, the stored reactive electric energy density and magnetic energy density calculated with ( 16) and ( 17) are Obviously, there is no   2 1 r term.Integrating them in the space outside a sphere with radius a, the total reactive energies can be obtained The radiation power is found to be, 4 3 Hence, the Q factor for the 10 TM mode is It can be further verified that, by multiplying a proper scale factor, the reactive electric energy and the reactive magnetic energy are exactly the same of that stored in the capacitor and the inductor in the equivalent circuit proposed by Chu [8].The Q factor is also exactly in agreement with that obtained by Mclean [3].Note that the propagating wave of this point source behaves much like a spherical one, therefore, subtracting the energy density associated with the radiation power happens to give good results for the total reactive energy in this case.However, the reactive energy densities are different from those obtained with the new formulation.
V. EXPRESSIONS FOR Q FACTOR OF ANTENNA Consider a typical radiation problem shown in Fig. 1(a).An excitation current ex J  exists on the antenna port p S .There is a dielectric with permittivity 1 The radiation problem can be solved based on equivalence principle.Assume that there is an induced surface current s J  on C S , a polarization current pol J  in region d V .These equivalent sources are then all placed in free space and are used to account for the effect of the conductor and the dielectric, as shown in Fig. 1(b).
Denote the electric field generated by the excitation current as input field to the conductor and dielectric, which is denoted by The operator where is the scalar Green's function and k is the wave number, in free space.The tangential component of the electric field vanishes on the PEC surface, so we have the electric field integral equation (EFIE), In the dielectric, the total electric field includes two parts, where the polarization current relates to the total electric field with Inserting ( 41) into (40) results in a volume integral equation (VIE) with respect to the polarization current.Given an excitation current ex J  , the current S J  and pol J  can be obtained by solving the electric field integral equation of (39) and (40) with method of moment (MoM).The stored energies are then computed with c o s Re 4 16 Here the integration region is denoted by The Poynting theorem in this case can still be expressed as where the electric field includes contribution from all sources, From (44), we have the power balance equation, in rad lc ld Therefore, the Q factor of the antenna can be denoted as All the Q factors are defined with (36), with the power there being replaced by in P , rad P , lc P and ld P , respectively.For PEC conductors, 0 lc P  since the tangential component of the total electric field on the surface is zero.The current on the PEC surface contributes to the storage of reactive energies, but does not contribute to the radiation power directly.However, the surface current will generate electric field on the antenna, hence, indirectly affect the radiation power.For dielectrics, The power loss is represents reactive energy stored in the dielectric, which has been already addressed in the reactive energies related to the corresponding polarization current.The polarization currents also influence the radiation power through affecting the electric field distribution at the antenna port.

VI. NUMERICAL EXAMPLES
Three examples with simple structures are analyzed just to show the difference between the new formulation and the method proposed in [25].
A plate dipole is analyzed at first.It consists of two PEC plates with size of 50mm 0.5mm  , and a feeding patch between them with size of 0.5mm 0.5mm  .The two plates are meshed to generate 326 RWGs.The surface current on the antenna is calculated by solving the corresponding EFIF with Galerkin method.The results of the reactive energies and Q factors calculated with the new formulation and the expressions provided in [25] are plotted in Fig. 2, where the label 'Ref.' in the figures indicates that the results are obtained using formulae in [25].The second example is a square loop antenna with edge length of 30mm, as shown in Fig. 3(a), excluding the PEC ground plate.The width of the PEC strip is 0.5mm.A 0.5mm 0.5mm  feeding patch is put at the center of one segment of the square.The mesh structure results in 318 RWGs.The results of Q factors calculated with the new formulation and the expressions provided in [25] are plotted in Fig. 3  In the third example, a PEC plate with size of 35mm 35mm  is put 2mm away under the loop antenna, as shown with dot lines in Fig. 3(a).Totally 684 RWGs are used for numerical calculation.The results of the reactive energies and Q factors calculated with the new formulation and the expressions provided in [25] are plotted in Fig. 4.These examples show that the reactive energies computed using the new formulation are bounded.For dipole and loop antennas with narrow strips, the results using the expressions in [25] are close to those with the new formulation.However, for antenna with more complex structures, like the one in the third example, the results are quite different.

VII. CONCLUSION
By integrating the radiation power over the past period, the radiation energy of an antenna is explicitly expressed.As a consequence, the stored reactive energy of the antenna is also clearly defined, which has clarified a long existing ambiguous understanding of the stored reactive energy.The new formulation can provide an insight into the composition of the energies radiated by an antenna.The expressions for the reactive energies are quite simple and easy to implement.

Fig. 1
and permeability  in region d V , together with a PEC conductor with surface c S .Antenna scattering problem.(a) Antenna near a PEC conductor and a dielectric obstacle; (b) Equivalent problem with all equivalence currents located in free space.

Fig. 4 A
PEC square loop antenna on a PEC square plate.(a) Reactive electric energy.(b)Reactive magnetic energy.(c) Q factor.
the total current denoted by * E E