Explicit Definitions for the Electromagnetic Energies in Electromagnetic Radiation and Mutual Coupling

Abstract

It is still difficult to accurately evaluate the reactive electromagnetic energy and the radiative electromagnetic energy of a radiator, because there are no explicit expressions for them. This paper proposes to borrow the energy concept in the charged particle theory and separate the total electromagnetic energy of a radiator into three parts: a Coulomb–velocity energy, a radiative energy and a macroscopic Schott energy. Consequently, the Poynting vector is considered to include a real radiative power flow by the radiative energy and a pseudo power flow caused by the fluctuation of the reactive energy. The energies involved in the electromagnetic mutual coupling are separated in a similar way. All energies are defined with explicit expressions in which the vector potential plays an important role. The time domain formulation and the frequency domain formulation of the theory are consistent with each other. The theory is verified with the Hertzian dipole. Numerical examples demonstrate that this theory may provide proper interpretations for electromagnetic radiation and mutual coupling problems.

1. Introduction

Electromagnetic radiation and coupling problems have been intensively investigated for more than a hundred years. It is a little strange that there is still no widely accepted explicit expression for macroscopic electromagnetic reactive energy and the radiative energy of a radiator [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. It is commonly known in classical charged particle theory that the fields associated with charged particles can be divided into Coulomb fields, velocity fields and radiative fields [16,17]. The energy carried by Coulomb fields and velocity fields is referred to as Coulomb–velocity energy in this paper. The radiative fields are generated by the acceleration of charged particles, emitting radiative energy to the surrounding space. The Coulomb–velocity fields and energy are considered to be attached to the charged particles. They appear and disappear with the charged particles. On the contrary, after being radiated by the charged particles, the radiative fields and energy depart from the sources and propagate to remote infinity. They exist after their generation sources have disappeared and can couple with other sources they have encountered during their journey. Schott energy was first introduced in 1912 by Schott [18,19,20]. It is reversible and is responsible for energy exchanges between Coulomb–velocity energy and radiative energy. Although it is natural to consider that the reactive energy in macroscopic electromagnetics is similar to the Coulomb–velocity energy and the Schott energy, no successful attempt has been found or well accepted to handle the reactive energy in this manner.

For harmonic fields over the time interval $\left(-\infty <t<\infty \right)$, the radiative energy occupies the whole space and is infinitely large [14], as is the total electromagnetic energy. The reactive energy can be evaluated by subtracting an additional term associated with the radiative power from the total energy density. However, the additional term is usually obtained by an approximate way, and it is not easy to give a general expression for that term, because the propagation patterns are quite different for different radiators [1,5].

A better strategy Is to examine a pulse radiator, because the radiative energy and the total electromagnetic energy are all finite. Consider a radiator in free space with a charge density $\rho \left(r_1,t\right)$ and $J\left(r_1,t\right)$, $r_1\in V_s$. From Maxwell equations, the electric energy density and the magnetic energy density can be transformed into

$$\frac{1}{2}D•E=\frac{1}{2}\rho \varphi -\frac{1}{2}D•\frac{\partial A}{\partial t}-\frac{1}{2}\nabla •\left(D\varphi \right)$$

(1)

$$\frac{1}{2}B•H=\frac{1}{2}J•A+\frac{1}{2}\frac{\partial D}{\partial t}•A-\frac{1}{2}\nabla •\left(H\times A\right)$$

(2)

where $E$ and $H$ are the electromagnetic fields, and $D$ and $B$ are the flux densities. The scalar potential $\varphi$ and the vector potential $A$ are subject to the Lorentz Gauge, and their reference zero points are put at infinity. For a pulse radiator in $\left[0,T\right]$, the total electric energy and the total magnetic energy can be written as

$$W_{tot}^e\left(t\right)={\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}D•E\right)d{r}_1}={\displaystyle {\int }_{V_s}\left(\frac{1}{2}\rho \varphi \right)d{r}_1}+{\displaystyle {\int }_{V_{\infty }}\left(-\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}$$

(3)

$$W_{tot}^m\left(t\right)={\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}B•H\right)d{r}_1}={\displaystyle {\int }_{V_s}\left(\frac{1}{2}J•A\right)d{r}_1}+{\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}\frac{\partial D}{\partial t}•A\right)d{r}_1}$$

(4)

Note that the integrals of the divergence terms on the right-hand side of Equations (1) and (2) are zeros because they can be transformed to the surface integrals at $S_{\infty }$, where the fields of the pulse radiator will never reach.

For the sake of convenience, we introduce two notations as follows:

$$W_{\rho }\left(t\right)={\displaystyle {\int }_{V_s}\frac{1}{2}\rho \left(r_1,t\right)\varphi \left(r_1,t\right)}d{r}_1$$

(5)

$$W_J\left(t\right)={\displaystyle {\int }_{V_s}\frac{1}{2}J\left(r_1,t\right)•A\left(r_1,t\right)}d{r}_1$$

(6)

Obviously, for static sources, $W_{\rho }$ describes the static electric energy, and $W_J$ describes the static magnetic energy. When the charge $\rho \left(r_1,t\right)$ reduces to a single charge, $W_{\rho }$ is exactly its Coulomb energy. When the single charge moves uniformly, $W_J$ becomes its velocity energy. We just borrowed the notations from the charged particle theory. Their properties will be investigated later from their behaviors revealed from the explicit expressions. Hereafter, $W_{\rho J}=W_{\rho }+W_J$ is referred to as the Coulomb–velocity energy. As can be clearly checked from Equations (5) and (6), the Coulomb–velocity energy appears and disappears simultaneously with its sources. It seems natural to choose the Coulomb–velocity energy alone as the reactive energy. However, if we make this choice, we have to define the second integral at the RHS of Equation (3) as the electric radiative energy and that in Equation (4) as the magnetic radiative energy. They are not equal. Therefore, the choice is not reasonable. For far fields of a radiator in free space, we use $\widehat{k}\times E\left(r,t\right)\approx {\eta }_{0}H\left(r,t\right)$, so the electric energy density and magnetic energy density are approximately equal. Since, at the far field zone in free space, the reactive energy is much smaller than the radiative energy, it is natural to consider that the radiative electric energy should equal the radiative magnetic energy. We further check in Section 2 that it is not proper to define $W_J$ in Equation (6) as the magnetic reactive energy, but it may be proper to define $W_{\rho }$ in Equation (5) as the electric reactive energy. Therefore, in order to overcome the inconsistency, we explicitly define the electric and the magnetic radiative energy as

$$W_{rad}^m\left(t\right)=W_{rad}^e\left(t\right)=\frac{1}{2}{W}_{rad}\left(t\right)=-{\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}$$

(7)

Consequently, the total electric energy in Equation (3) can be expressed as

$$W_{tot}^e\left(t\right)=W_{\rho }\left(t\right)+W_{rad}^e\left(t\right)$$

(8)

and a special term has to be introduced in Equation (4):

$$W_{tot}^m\left(t\right)=W_J\left(t\right)+W_{rad}^m\left(t\right)+{\displaystyle {\int }_{V_{\infty }}\frac{1}{2}\frac{\partial }{\partial t}\left(D•A\right)d{r}_1}$$

(9)

It can be further demonstrated that the last term in the RHS of Equation (9) corresponds to the Schott energy in the charged particle theory [18,20] by applying the Lienard–Wiechert potentials [21] to a moving charge [22]. We denote it as the macroscopic Schott energy:

$$W_S\left(t\right)={\displaystyle {\int }_{V_{\infty }}\frac{1}{2}\frac{\partial }{\partial t}\left(D•A\right)d{r}_1}$$

(10)

The explicit expression for the macroscopic energy derived in Section 2 clearly shows that it will disappear soon after its sources disappear.

The total electromagnetic energy is the sum of Equations (3) and (4). Substituting Equations (7)–(10) into it gives the energy separation equation:

$$W_{tot}\left(t\right)=W_{\rho J}\left(t\right)+W_S\left(t\right)+W_{rad}\left(t\right)$$

(11)

The reactive energy is accordingly expressed by

$$W_{react}\left(t\right)=W_{\rho J}\left(t\right)+W_S\left(t\right)$$

(12)

This is the energy separation for a pulse radiator proposed in this paper. We want to emphasize three points first.

(1).

The energy separation in Equation (11) for a pulse radiator is directly derived from Maxwell equations with no approximation.

(2).

The electric radiative energy equals the magnetic radiative energy, which is consistent with the practical situation of a radiator in free space.

(3).

The macroscopic Schott energy is related to the Schott energy in the charged particle theory. Both are full-time derivatives.

We are to further justify the separation in the following sections.

(1).

In Section 2, explicit expressions for the energies are derived, which show that the Coulomb–velocity energy and the macroscopic Schott energy are attached to the sources, while the radiative energy defined in the proposed theory keeps constant after the sources have disappeared. The temporal evolution property of the energies supports the reasonability of the energy separation.

(2).

In Section 4, by applying the energy separation to harmonic waves, it is verified that the time domain formulation of the theory is consistent with its frequency domain formulation, which was discussed in [14] and was comprehensively compared with other frequency domain formulations with several examples.

(3).

The Hertzian dipole with harmonic excitation is a standard validation example for these situations, because the exact solutions for its fields and potentials are available both in the time domain and in the frequency domain, together with a well-established equivalent circuit model. In Section 5, all the expressions of the electromagnetic energies and powers corresponding to the Hertzian dipole are derived. They are exactly in agreement with those obtained with the equivalent circuit model.

A closely related issue is electromagnetic mutual coupling, which plays a very important role in many systems. Efficient and accurate analysis of electromagnetic mutual coupling is still a challenging issue [23]. In Section 3, the theory is extended for handling multiple radiators. We can aggregate all radiators together and treat them as a single larger radiator, similar to an antenna array. The mutual electromagnetic coupling energies are defined in the same way. One radiator may exert electromagnetic coupling to other sources through its potentials instead of fields. The key issue involved in electromagnetic mutual coupling is the same as that in the electromagnetic radiation problem.

On the other hand, the Poynting vector is widely considered as the electromagnetic power flux density [24]. The Poynting theorem describes the relationship among the Poynting vector, the varying rate of the total electromagnetic energy densities, and the work rate done by the exciting source. It provides an intuitive description of the propagation of electromagnetic energy. However, the interpretation of the Poynting vector has always been controversial [21,25,26,27,28,29,30,31,32,33,34], and some researchers have pointed out that the Poynting theorem may have not been used in the correct way in some situations [35,36]. This difficulty is largely due to the fact that it is not easy to separate the real radiative power flux from the Poynting vector. As the Poynting vector is related to the total electromagnetic energy, it should include a real radiative power flow from the contribution of the radiative energy and a pseudo power flow due to the fluctuation of the reactive energy. A new energy–power balance equation at a certain instant time is proposed in Section 2. It is based on the Poynting relation, only with some substitution and reorganization that can be derived from Maxwell equations.

Two kinds of time domain formulations for this issue can be found in the published literature. One was proposed by Shlivinski and Heyman [2,3], and the other was proposed by Vandenbosch [6,7]. The first one is an approximate method, while the second one is sometimes not consistent with its counterpart formulation in the frequency domain. In Section 6, a loop pulse radiator and a Yagi antenna are analyzed with the proposed theory. They are not for comparison with the other time domain formulations but for the purpose of showing what we can do with the proposed expressions. Numerical examples for comparison among various formulations in the frequency domain can be found in [14,15].

The theory is briefly discussed in the Section 7, where it is concluded that the theory is neither a static limit formulation nor a kind of updated version of the Carpenter formulation [27].

2. Energies and Powers of a Pulse Radiator

For a pulse radiator existing in the time period of $\left[0,T\right]$, the scalar potential $\varphi \left(r,t\right)$ and the vector potential $A\left(r,t\right)$ evaluated at the observation point $r$ and the time $t$ are defined in their usual way:

$$\varphi \left(r,t\right)={\displaystyle {\int }_{V_s}\frac{\rho \left(r_1,\tau \right)}{4\pi {\epsilon }_0{R}_1}}d{r}_1$$

(13)

$$A\left(r,t\right)={\mu }_0{\displaystyle {\int }_{V_s}\frac{J\left(r_1,\tau \right)}{4\pi R_1}}d{r}_1$$

(14)

In the above equations, $\tau =t-R/c$ is the retarded time, c is the light velocity in vacuum and $R_1=\left|r-r_1\right|$ is the distance between the two positions. ${\mu }_0$ and ${\epsilon }_0$ are, respectively, the permeability and permittivity in free space.

Integrating Equation (1) over a domain $V_a\supset V_s$ and rearranging the terms gives

$${\displaystyle {\int }_{V_a}\left(\frac{1}{2}\rho \varphi \right)d{r}_1}={\displaystyle {\int }_{V_a}\left(\frac{1}{2}D•E+\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}+{\displaystyle {\oint }_{S_a}\frac{1}{2}\varphi D•\widehat{n}dS}$$

(15)

where $S_a$ is the surface enclosing $V_a$ with outward normal units $\widehat{n}$. In order to investigate the property of $W_{\rho }\left(t\right)$ and $W_J\left(t\right)$, we assume $T\to +\infty$, so that the fields can spread over the whole space. Let $r\to \infty$; then, we obtain $V_a\to V_{\infty }$ and $S_a\to S_{\infty }$. Recalling $\underset{r\to \infty }{\lim }\left(D\cdot \widehat{r}\right)\sim O\left(1/r^2\right)$ and $\underset{r\to \infty }{\lim }\varphi \sim O\left(1/r\right)$, where $\widehat{r}$ is the unit radial vector, the surface integral at the RHS of Equation (15) approaches zero at $S_{\infty }$. The energy at the LHS of Equation (15) becomes $W_{\rho }\left(t\right)$. It really has the meaning of being stored in the space with no energy leaking to infinity. Therefore, we may define $W_{\rho }\left(t\right)$ as the electric reactive energy and define the electric radiative energy by Equation (7).

Following the same procedure, integrating Equation (2) over the domain $V_a\supset V_s$ and rearranging the terms gives

$${\displaystyle {\int }_{V_a}\left(\frac{1}{2}J•A\right)d{r}_1}={\displaystyle {\int }_{V_a}\left(\frac{1}{2}B•H-\frac{1}{2}\frac{\partial D}{\partial t}•A\right)d{r}_1}+{\displaystyle {\oint }_{S_a}\left(\frac{1}{2}H\times A\right)•\widehat{n}dS}$$

(16)

when $T\to +\infty$, we obtain $\underset{r\to \infty }{\lim }\left(H\times A\right)•\widehat{r}\sim O\left(1/r^2\right)$, and the surface integral in Equation (16) at $S_{\infty }$ is usually a bounded but nonzero value. The LHS of Equation (16) is the velocity $W_J\left(t\right)$. Obviously, the velocity energy is not an energy being purely stored in the whole space $V_{\infty }$, because it contains a part of the energy leaking to or coming back from infinity. Therefore, it is not proper to define $W_J\left(t\right)$ as the reactive magnetic energy. This has also been verified in our previous work [37]. In the case of the Hertzian dipole, the electric reactive energy defined by $W_{\rho }\left(t\right)$ is exactly in agreement with the electric energy stored in the capacitor in its equivalent circuit model proposed by Chu [38]. However, the energy calculated with $W_J\left(t\right)$ does not exactly equal the magnetic energy stored in the equivalent inductor.

This inconsistency makes it necessary to introduce the macroscopic Schott energy. The electric reactive energy and the magnetic reactive energy are then changed to

$$W_{react}^e\left(t\right)={\displaystyle {\int }_{V_s}\left(\frac{1}{2}\rho \varphi \right)d{r}_1}={\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}D•E+\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}$$

(17)

$$W_{react}^m\left(t\right)={\displaystyle {\int }_{V_s}\left(\frac{1}{2}J•A\right)d{r}_1}+{\displaystyle {\int }_{V_{\infty }}\frac{1}{2}\frac{\partial }{\partial t}\left(D•A\right)d{r}_1}={\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{2}B•H+\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}$$

(18)

In order to reveal the property of the macroscopic Schott energy, we are to derive its explicit expression from the vector potential and the electric flux density:

$$D\left(r,t\right)=-{\epsilon }_0\nabla \varphi \left(r,t\right)-{\epsilon }_0\frac{\partial }{\partial t}A\left(r,t\right)=-{\displaystyle {\int }_{V_s}{\displaystyle {\int }_{-\infty }^{\infty }\rho \left(r_1,t_1\right)\nabla G\left(t-t_1-R_1/c\right)d{t}_1}d{r}_1}-\frac{1}{c^2}{\displaystyle {\int }_{V_s}{\displaystyle {\int }_{-\infty }^{\infty }J\left(r_1,t_1\right)\dot{G}\left(t-t_1-R_1/c\right)d{t}_1}d{r}_1}$$

(19)

where the “˙” on the top of the variables means a derivative with respect to time. The time domain Green’s function can be expressed with the Dirac delta function:

$$G_1\left(r,r_1;t-R_1/c\right)=\frac{\delta \left(t-R_1/c\right)}{4\pi R_1}$$

(20)

Substituting Equations (14) and (19) into Equation (10) yields

$$W_S\left(t\right)={\displaystyle {\int }_{V_{\infty }}\frac{1}{2}\frac{\partial }{\partial t}\left(D•A\right)dr}=-{\mu }_0{\displaystyle {\int }_{V_{\infty }}{\displaystyle {\int }_{V_s}{\displaystyle {\int }_{V_s}\left\{\left[{\displaystyle {\int }_{-\infty }^{\infty }\rho \left(r_1,t_1\right)\nabla G_{1}d{t}_1}+c^{-2}{\displaystyle {\int }_{-\infty }^{\infty }J\left(r_1,t_1\right){\dot{G}}_{1}d{t}_1}\right]•{\displaystyle {\int }_{-\infty }^{\infty }J\left(r_2,t_2\right)G_{2}d{t}_2}\right\}}}d{r}_{1}d{r}_2}dr$$

(21)

where $G_{1,2}=G\left(t-t_{1,2}-R_{1,2}/c\right)$ and $R_{1,2}=\left|r-r_{1,2}\right|$. With the derivation detailed in the Appendix A, the integral can be explicitly expressed by an integration over the source region:

$$W_S\left(t\right)=-\frac{1}{8\pi {\epsilon }_0}{\displaystyle {\int }_{V_s}{\displaystyle {\int }_{V_s}\frac{1}{r_{21}}{\displaystyle {\int }_{t-r_{21}/c}^t\times }}}\left[\rho \left(r_1,\tau \right)\dot{\rho }\left(r_2,2t-\tau -\frac{r_{21}}{c}\right)+c^{-2}\dot{J}\left(r_1,2t-\tau -\frac{r_{21}}{c}\right)•J\left(r_2,\tau \right)\right]d\tau d{r}_{2}d{r}_1$$

(22)

where $r_{21}=\left|r_2-r_1\right|$. Note that $\rho \left(r_1,t_1\right)$, $J\left(r_1,t_1\right)$, and $\rho \left(r_2,t_2\right)$, $J\left(r_2,t_2\right)$ represent the sources at $\left(r_1,t_1\right)$ and $\left(r_2,t_2\right)$, respectively. They are the same functions related to the same radiator. For a pulse source in $\left[0,T\right]$, as checked in Appendix A, the integral becomes zero when $t\ge T+0.5t_{\max }$, where $t_{\max }=r_{21,\max }/c$ is the largest traveling time between two source points. This means that, when the sources have disappeared, $W_S\left(t\right)$ does not become zero simultaneously. Instead, it will continue to be nonzero for a short time. At $t\ge T+0.5t_{\max }$, although $\partial \left(D•A\right)/\partial t$ is not zero everywhere in the space, its volume integral over the whole space becomes zero.

We introduce a principal radiative energy as

$$W_{rad}^{pri}\left(t\right)={\displaystyle {\int }_{V_{\infty }}\frac{1}{2}\left(\frac{\partial D}{\partial t}•A-D•\frac{\partial A}{\partial t}\right)d{r}_1}$$

(23)

The radiative energy can then be expressed by

$$W_{rad}\left(t\right)=W_{rad}^{pri}\left(t\right)-W_S\left(t\right)$$

(24)

As shown in Appendix A, the principal radiative energy can be evaluated with the integration over the source region:

$$W_{rad}^{pri}\left(t\right)=\frac{1}{8\pi {\epsilon }_0}{\displaystyle {\int }_{V_s}{\displaystyle {\int }_{V_s}\frac{1}{r_{21}}{\displaystyle {\int }_{r_{21}/c}^t}}}\left\{\begin{array}{l}\left[\dot{\rho }\left(r_1,\tau \right)\rho \left(r_2,\tau -r_{21}/c\right)-\dot{\rho }\left(r_1,\tau -r_{21}/c\right)\rho \left(r_2,\tau \right)\right]\\ +c^{-2}\left[J\left(r_1,\tau \right)\dot{J}\left(r_2,\tau -r_{21}/c\right)-J\left(r_1,\tau -r_{21}/c\right)\dot{J}\left(r_2,\tau \right)\right]\end{array}\right\}d\tau d{r}_{2}d{r}_1$$

(25)

It can be checked that, for a pulse source over $\left[0,T\right]$, $W_{rad}^{pri}\left(t\right)=W_{rad}^{pri}\left(T\right)$ for $t\ge T$. However, as seen from Equation (24), the total radiative energy continues to vary in a small time period $\left[T,T+0.5t_{\max }\right]$ due to the effect of $W_S\left(t\right)$.

For a pulse radiator with sources existing in $\left[0,T\right]$, its total energy can be divided into $W_{tot}\left(t\right)=W_{\rho J}\left(t\right)+W_{rad}^{pri}\left(t\right)$, as shown in Equation (11). This is what we proposed in our previous work [37]. However, careful examination shows that $W_{rad}^{pri}\left(t\right)$ does not exactly equal the radiative energy. As will be shown in Section 4, the principal radiative energy exactly equals the time-averaged value of the radiative energy for harmonic waves. At the same time, $W_J\left(t\right)$ does not exactly reflect the stored magnetic energy for a Hertzian dipole. We revisited the issue of the electromagnetic radiation of a moving charged particle [16,17] and finally realized that radiative fields can interact with other sources before they completely leave the source area. The interaction causes an oscillatory energy exchange. Therefore, we introduce macroscopic Schott energy $W_S\left(t\right)$ into our formulation and divide the total energy of a radiator into a reactive energy $\left(W_{\rho J}+W_S\right)$ and a radiative energy $\left(W_{rad}^{pri}-W_S\right)$, where macroscopic Schott energy plays the role of energy exchanging. It has been verified that the Coulomb–velocity energy is strictly attached to the sources. It appears and disappears simultaneously with the sources. The macroscopic Schott energy does not disappear simultaneously with its sources. It continues to remain nonzero within the period of $\left[T,T+0.5t_{\max }\right]$ and then disappears. On the other hand, the radiative energy keeps constant for $t\ge T+0.5t_{\max }$. Therefore, the temporal evolution property of the three energies supports the energy separation scheme in Equation (11). For static electromagnetic fields, the radiative energy is zero, and the reactive electric energy is exactly the electric energy associated with the static charges, while the reactive magnetic energy is exactly the energy associated with the static currents.

The Poynting theorem correctly describes the relationship among the work rate done by the source, the total electromagnetic energy in region $V_a\supseteq V_s$ containing the source and the total electromagnetic power flux crossing the boundary $S_a$ of the region:

$$-{\displaystyle {\int }_{V_s}J•Ed{r}_1}=\frac{\partial }{\partial t}{\displaystyle {\int }_{V_a}\left(\frac{1}{2}D•E+\frac{1}{2}B•H\right)d{r}_1}+{\displaystyle {\oint }_{S_a}S•\widehat{n}dS}$$

(26)

where the Poynting vector $S=E\times H$ is the electromagnetic power flux density.

Now, we will show that $W_{rad}^{pri}$ associated with a bounded volume is a convenient quantity for engineering applications. Substituting Equations (1) and (2) into Equation (26) and reorganizing it gives

$$-{\displaystyle {\int }_{V_s}J•E}d{r}_1-\frac{\partial }{\partial t}{W}_{\rho J}\left(t\right)=\frac{\partial }{\partial t}{\displaystyle {\int }_{V_a}\frac{1}{2}\left(\frac{\partial D}{\partial t}•A-D•\frac{\partial A}{\partial t}\right)d{r}_1}+{\displaystyle {\oint }_{S_a}\left[E\times H-\frac{1}{2}\frac{\partial }{\partial t}\left(H\times A+D\varphi \right)\right]•\widehat{n}dS}$$

(27)

For the sake of convenience, a new vector is introduced for the integrand of the surface integral in Equation (27):

$$S_{rad}\left(r,t\right)=E\times H-\frac{\partial }{\partial t}\left(\frac{1}{2}H\times A+\frac{1}{2}D\varphi \right)$$

(28)

It has to be noted that $S_{rad}$ is not the radiative power density. We denote its surface integral as

$$P_{rad}^{pri}\left(t\right)={\displaystyle {\oint }_{S_a}{S}_{rad}•\widehat{n}dS}$$

(29)

Meanwhile, the total work done by the source is expressed by

$$W_{exc}\left(t\right)=-{\displaystyle {\int }_{-\infty }^t{\displaystyle {\int }_{V_s}J\left(r_1,\tau \right)•E\left(r_1,\tau \right)}d{r}_{1}d\tau }$$

(30)

Integrating both sides of Equation (27) gives

$$W_{exc}\left(t\right)-W_{\rho J}\left(t\right)=W_{rad}^{pri}\left(t\right)+{\displaystyle {\int }_0^t{P}_{rad}^{pri}\left(\tau \right)d\tau }$$

(31)

where $W_{rad}^{pri}\left(t\right)$ is defined using Equation (23) but with the integration domain replaced by $V_a$. Accordingly, $P_{rad}^{pri}\left(t\right)$ can be interpreted as the principal radiative power passing through the observation surface $S_a$. It is associated with the principal radiative energy $W_{rad}^{pri}\left(t\right)$. Since it is not easy to find a general explicit expression for the radiative power passing through the observation surface, the principal radiative power $P_{rad}^{pri}\left(t\right)$ can provide a good measurement. As shown in the Hertzian dipole and the other examples, the principal radiative power $P_{rad}^{pri}\left(t\right)$ gives a kind of time-averaged value of the total radiative power passing through the observation surface.

For $t\ge T+0.5t_{\max }$, we obtain $W_{\rho J}\left(t\right)=W_S\left(t\right)=0$. The total radiative energy crossing the observation surface $S_a$ enclosing the radiator can be evaluated with the temporal integration of $P_{rad}^{pri}\left(t\right)$ on $S_a$:

$$W_{rad}\left(t\right)=W_{rad}^{pri}\left(t\right)={\displaystyle {\int }_{t_{\min }}^{t_{\max }}{P}_{rad}^{pri}\left(t\right)}dt=W_{exc}\left(T\right)$$

(32)

For pulse sources, $P_{rad}^{pri}\left(t\right)$ has nonzero values only over the period $\left(t_{a,\min }<t<t_{a,\max }\right)$, in which $t_{a,\min }$ and $t_{a,\max }$ are, respectively, the earliest and the latest times that the fields pass through the observation surface $S_a$. As a special case, we may choose $V_a=V_s$, and put the observation surface $S_a$ close to the surface of the sources. Ignoring the radiative energy stored in $V_s$, we may obtain the power coming out of the surface of the sources as

$$P_{rad0}^{pri}\left(t\right)=-{\displaystyle {\int }_{V_s}\left[J•E+\frac{\partial }{\partial t}\left(\frac{1}{2}\rho \varphi +\frac{1}{2}J•A\right)\right]d{r}_1}$$

(33)

Apparently, $P_{rad}^{pri}\left(t\right)$ at different observation surfaces are not expected to be equal, but their integrations over the time interval $\left(t_{a,\min }<t<t_{a,\max }\right)$ are equal and are approximately equal to that of $P_{rad0}^{pri}\left(t\right)$, since all the radiative energy of the pulse source in vacuum will eventually pass through the observation surface and propagate to infinity.

As was discussed in [20], the radiative energy is always nonnegative, and it describes an irreversible loss of energy, while the Schott energy can change reversibly. For $t\ge T+0.5t_{\max }$, although $W_S\left(t\right)=0$, it is not necessary for its integrand to be zero everywhere. It induces an energy oscillation and contributes to the Poynting vector. The integration of the normal component of the Poynting vector on the observation surface $S_a$ can be separated into a real radiative power and a pseudo radiative power:

$$P_{pv}\left(t\right)={\displaystyle {\oint }_{S_a}S\left(r_1,t\right)\cdot \widehat{n}d{r}_1}=P_{rad}^{real}\left(t\right)+P_{rad}^{pseudo}\left(t\right)$$

(34)

It will be shown in Section 4 that, for harmonic waves, the time-averaged value of the principal radiative power defined in Equation (29) equals the time-averaged value of the real radiative power.

3. Mutual Couplings

Consider a group of N radiators in vacuum. The i-th radiator has a source $\left(J_i,{\rho }_i\right)$ in the region $V_{si}$. The reactive electromagnetic energy coupled from source-j to source-i is expressed as

$$\begin{array}{l}{W}_{ij}^{react}\left(t\right)={\displaystyle {\int }_{V_{si}}\left(\frac{1}{2}{\rho }_i{\varphi }_j+\frac{1}{2}{J}_i•A_j+\frac{1}{2}\frac{\partial }{\partial t}\left(D_i•A_j\right)\right)d{r}_i}\\ =\frac{1}{8\pi {\epsilon }_0}{\displaystyle {\int }_{V_{si}}{\displaystyle {\int }_{V_{sj}}\frac{1}{r_{ij}}}}\times \left\{\begin{array}{l}{\rho }_i\left(r_i,t\right){\rho }_j\left(r_j,t-r_{ij}/c\right)+c^{-2}{J}_i\left(r_i,t\right)•J_j\left(r_j,t-r_{ij}/c\right)-\\ {\displaystyle {\int }_{t-r_{ij}/c}^t\left[{\rho }_i\left(r_i,\tau \right){\dot{\rho }}_j\left(r_j,2t-\tau -\frac{r_{ij}}{c}\right)+c^{-2}{\dot{J}}_i\left(r_i,2t-\tau -\frac{r_{ij}}{c}\right)•J_j\left(r_j,\tau \right)\right]d\tau }\end{array}\right\}d{r}_{j}d{r}_i\end{array}$$

(35)

where $A_{i,j}$ and ${\varphi }_{i,j}$ are, respectively, the vector potential and the scalar potential generated by the source $\left({\rho }_{i,j},J_{i,j}\right)$.

The mutual coupled reactive energies also include the macroscopic Schott energies, which may be denoted by $W_{Sij}\left(t\right)$. It is straightforward to check that the total reactive energy of the system is

$$W_{tot}^{react}\left(t\right)={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N{W}_{ij}^{react}\left(t\right)}}$$

(36)

It contains the self-reactive energies ($i=j$) and the mutual coupled reactive energies ($i\ne j$). Note that conventional formulations may be difficult to be extended to contain multiple radiators, because it is difficult to determine the coordinate origin and the subtraction term.

The mutual radiative energy from source-j to source-i is

$$W_{ij}^{rad}\left(t\right)={\displaystyle {\int }_{V_{si}}\left(-D_i•\frac{\partial A_j}{\partial t}\right)d{r}_i}$$

(37)

which is the energy radiated by radiator-i when it is excited by radiator-j.

In circuit theory, mutual coupling usually refers to the coupling between the energies stored in inductors or capacitors, not including the losses dissipated by resistors. As the mutual radiative energy is a kind of radiation loss to the radiator, it is reasonable to consider that electromagnetic mutual coupling energies only include the mutual reactive energies and use them to determine the mutual coupling coefficients.

The electromagnetic radiation and coupling problem of two-pulse radiators are illustrated in Figure 1. We only analyze the radiation of radiator-1 in the region $V_{s1}$. It generates a Coulomb–velocity field carrying Coulomb–velocity energy and emits a radiation field carrying the radiative energy to the surrounding space. Specifically, we consider a small part of the source in the internal region of $V_{s1}$ denoted by the red star in Figure 1. The radiative fields generated by the red star source interact with other sources in the source region $V_{s1}$ when they propagate through the source region to the outside space. Part of the radiative energy is transferred to the sources they have encountered. A nonzero macroscopic Schott energy appears in this period corresponding to the energy exchange. Only after they have completely left the source region can the radiative fields by the red star source propagate to far regions with a constant radiative energy until they reach radiator-2 and interact with the sources there. The radiative fields from other sources in $V_{s1}$ experience a similar journey and carry the radiative energy of radiator-1, inducing a real electromagnetic radiative power flow.

On the other hand, the reactive energy of radiator-1 affects radiator-2 through mutual coupling. When the reactive energy of radiator-1 varies with time, the fluctuation of the reactive energy of radiator-1 causes a pseudo power flow and travels with the real power flow to radiator-2.

The Poynting vector represents the total power flow, including the real radiative power flow $P_{rad}^{real}\left(t\right)$ and the pseudo power flow $P_{rad}^{pseudo}\left(t\right)$. As shown in Figure 2, the real power flow always propagates from its source, crossing the observation surface from left to right. However, the direction of the pseudo power flow is reversible. It crosses the observation surface from left to right when the reactive energy of the source increases and from right to left when the reactive energy decreases.

In the radiation process of a radiator, mutual coupling occurs between different parts of sources in the same radiator. Before leaving the radiator, the fields generated by one part of the sources in the radiator will interact with the other sources in the radiator. The electromagnetic energy carried by the fields may be absorbed and will not propagate to outside the radiator. The total reactive energy and the total radiative energy of the radiator are the overall results including all these mutual coupling effects. If the radiator contains several separate parts with different sources, the radiation process can be analyzed in the same way. A good example is an antenna array. It can be handled as a single radiation problem if we take the whole array as a single radiator. It is a radiation and mutual coupling problem if we want to reveal the mutual couplings among different units of the array.

Because the mutual coupling energies are expressed in terms of potentials, it is possible that the mutual coupling can exist with no electromagnetic fields. The theory can provide a reasonable interpretation for the Aharonov–Bohm effect [39].

4. Radiation of Harmonic Sources

For harmonic fields with a time convention of $e^{j\omega t}$, the radiation is assumed to last temporally from $-\infty$ to $+\infty$, so the radiative energy is infinitely large. The Poynting theorem can be applied to describe the balance between the time averaged powers and the varying rate of the energies:

$$-\frac{1}{2}{\displaystyle {\int }_{V_s}{J}^{\ast }•Ed{r}_1}=2j\omega {\displaystyle {\int }_{V_a}\left[\frac{1}{4}B•H^{\ast }-\frac{1}{4}E•D^{\ast }\right]d{r}_1}+\frac{1}{2}{\displaystyle {\oint }_{S_a}E\times H^{\ast }•\widehat{n}dS}$$

(38)

from which the time-averaged radiative power at infinity can be evaluated with source distributions:

$${\left(P_{rad}\right)}_{av}=\mathrm{Re}\left\{\frac{1}{2}{\displaystyle {\oint }_{S_{\infty }}E\times H^{\ast }•\widehat{n}dS}\right\}=-\mathrm{Re}\left\{\frac{1}{2}{\displaystyle {\int }_{V_s}E•J^{\ast }d{r}_1}\right\}$$

(39)

The same symbols are used for the corresponding phasors for the sake of simplicity.

However, the evaluation of the reactive energies in conventional formulations requires subtracting the radiative energy from the total energy. Since both the energies are unbounded, all those formulations based on energy subtraction are not always satisfactory.

With the theory proposed here, the power balance can be evaluated within any domain enclosed by an observation surface $S_a$ containing the source region $V_s$:

$$-{\displaystyle {\int }_{V_s}{J}^{\ast }•E}d{r}_1=2j\omega {\displaystyle {\int }_{V_s}\left(\frac{1}{4}{\rho }^{*}\varphi +\frac{1}{4}{J}^{\ast }•A\right)d{r}_1}+{\displaystyle {\oint }_{S_a}\left[\frac{1}{2}E\times H^{\ast }-j\omega \left(\frac{1}{4}{H}^{\ast }\times A+\frac{1}{4}{D}^{\ast }\varphi \right)\right]•\widehat{n}dS}$$

(40)

The time-averaged radiative power crossing the observation surface can be obtained using the radiative power flux vector $S_{rad}$ or the source distributions:

$${\left(P_{rad}\right)}_{av}=\mathrm{Re}\left\{{\displaystyle {\oint }_{S_a}\left[\frac{1}{2}E\times H^{\ast }-\frac{j\omega }{4}\left(H^{\ast }\times A+D^{\ast }\varphi \right)\right]•\widehat{n}dS}\right\}={\displaystyle {\oint }_{S_a}{\left(S_{rad}\right)}_{av}•\widehat{n}dS}=-\mathrm{Re}\left\{{\displaystyle {\int }_{V_s}\frac{1}{2}{J}^{\ast }•Ed{r}_1}\right\}$$

(41)

Note that, with Equation (41), the observation surface is not required to approach infinity for evaluating the radiative power. It can be checked that the result is consistent with that obtained using the Poynting vector, since it has been proven in [14] that

$$\mathrm{Re}\left\{{\displaystyle {\oint }_{S_{\infty }}j\omega \left(\frac{1}{4}{H}^{\ast }\times A+\frac{1}{4}{D}^{\ast }\varphi \right)•\widehat{n}dS}\right\}=0$$

(42)

The time-averaged reactive energy can be calculated with the fields and the vector potential:

$${\left(W_{react}\right)}_{av}=\mathrm{Re}\left\{{\displaystyle {\int }_{V_{\infty }}\left(\frac{1}{4}E•D^{*}+\frac{1}{4}B•H^{*}+\frac{1}{2}j\omega D^{*}•A\right)}\right\}$$

(43)

It is easy to verify that ${\left(W_S\right)}_{av}=0$, so the time-averaged reactive energy can be alternatively calculated using the source potential products as follows:

$${\left(W_{react}\right)}_{av}=\mathrm{Re}\left\{{\displaystyle {\int }_{V_s}\left(\frac{1}{4}{\varphi }^{*}\rho +\frac{1}{4}{J}^{*}•A\right)d{r}_1}\right\}$$

(44)

Note that the reactive energy calculated with Equations (43) and (44) are equal. We may choose to use one of them based on practical situations.

It can be also checked that ${\left(W_{Sij}\right)}_{av}=0$. Therefore, the time-averaged mutual coupled reactive electromagnetic energies are found to be

$${\left(W_{ij}^{react}\right)}_{av}=\mathrm{Re}{\displaystyle {\int }_{V_{si}}\left(\frac{1}{4}{\rho }_i{\varphi }_j^{*}+\frac{1}{4}{J}_i^{*}•A_j\right)d{r}_i}$$

(45)

As expected, ${\left(W_{ij}^{react}\right)}_{av}={\left(W_{ji}^{react}\right)}_{av}$ holds for mutual coupling in the free space.

5. Hertzian Dipole

A Hertzian dipole located at the origin is analyzed to show the energy–power balance relationship. The moment of the dipole is assumed to be $ql\cos \omega t$, the scalar potential and the vector potential of which can be readily derived from the Hertzian potential $\Pi =\left(ql/4\pi r\right)\cos \left(\omega t-kr\right)$ [37,40,41]:

$$A=-\frac{\omega {\mu }_{0}ql}{4\pi r}\sin \left(\omega t-kr\right)\left(\widehat{r}\cos \theta -\widehat{\theta }\sin \theta \right)$$

(46)

$$\phi =\frac{{\omega }^2{\mu }_{0}ql}{4\pi }\cos \theta \left[\frac{1}{k^2{r}^2}\cos \left(\omega t-kr\right)-\frac{1}{kr}\sin \left(\omega t-kr\right)\right]$$

(47)

from which the fields are found to be

$$E=\frac{k^{2}ql}{4\pi {\epsilon }_{0}r}\left\{\widehat{r}2\cos \theta \frac{1}{kr}\left[\begin{array}{l}\frac{1}{kr}\cos \left(\omega t-kr\right)\\ -\sin \left(\omega t-kr\right)\end{array}\right]+\widehat{\theta }\sin \theta \left[\begin{array}{l}\left(\frac{1}{k^2{r}^2}-1\right)\cos \left(\omega t-kr\right)\\ -\frac{1}{kr}\sin \left(\omega t-kr\right)\end{array}\right]\right\}$$

(48)

$$H=-\frac{\omega kql}{4\pi r}\sin \theta \left[\frac{1}{kr}\sin \left(\omega t-kr\right)+\cos \left(\omega t-kr\right)\right]\widehat{\phi }$$

(49)

As is known, the Hertzian dipole is a point source, and its total reactive energy is infinite if the integration region contains the source point. A common strategy is to evaluate the second integrals Equations (17) and (18) in the whole space, excluding a small sphere with radius a. The results are listed below.

$$W_{react}^e\left(t\right)={\displaystyle {\int }_{V_{\infty }-V_a}\left(\frac{1}{2}D•E+\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}={\alpha }_0\left[\begin{array}{l}\frac{1}{k^3{a}^3}+\frac{1}{ka}+\left(\frac{1}{k^3{a}^3}-\frac{1}{ka}\right)\cos 2\left(\omega t-ka\right)\\ -\frac{2}{k^2{a}^2}\sin 2\left(\omega t-ka\right)\end{array}\right]$$

(50)

$$W_{react}^m\left(t\right)={\displaystyle {\int }_{V_{\infty }-V_a}\left(\frac{1}{2}B•H+\frac{1}{2}D•\frac{\partial A}{\partial t}\right)d{r}_1}=\frac{2{\alpha }_0}{ka}{\sin }^2\left(\omega t-ka\right)$$

(51)

The principal radiative power evaluated at a spherical observation surface is

$$P_{rad}^{pri}\left(t\right)={\displaystyle {\oint }_{S_a}{S}_{rad}•\widehat{n}dS}=2\omega {\alpha }_0$$

(52)

It is a constant value independent of the radius of the sphere, clearly indicating that the total radiative power associated with $W_{rad}^{pri}$ crossing any concentric spherical surface is the same.

The surface integral of the Poynting vector on the spherical surface $S_a$ is calculated to be

$$\begin{array}{c}{P}_{pv}\left(t\right)={\displaystyle {\oint }_{S_a}S•\widehat{n}dS}=2\omega {\alpha }_0\left[1+\cos 2\left(\omega t-ka\right)\right]\hfill \\ \hfill +2\omega {\alpha }_0\left[\left(\frac{2}{ka}-\frac{1}{k^3{a}^3}\right)\sin 2\left(\omega t-ka\right)-\frac{2}{k^2{a}^2}\cos 2\left(\omega t-ka\right)\right]\end{array}$$

(53)

which varies with the radius of the surface due to the effect of the reactive energy. As expected, the time average of $P_{pv}\left(t\right)$ equals that of $P_{rad}^{pri}\left(t\right)$.

Since the Hertzian dipole is a point source, its fields propagate radially and across all concentric spherical observation surfaces with light velocity. Therefore, the radiative energy per unit time near the spherical surface $S_a$ can be considered as the real radiative power crossing $S_a$:

$$P_{rad}^{real}\left(t\right)dt={\displaystyle {\int }_S{\displaystyle {\int }_a^{a+cdt}\left(-D•\frac{\partial A}{\partial t}\right)}}drdS=cdt{\displaystyle {\int }_{S_a}\left(-D•\frac{\partial A}{\partial t}\right)dS}$$

(54)

from which the real radiative power is found to be

$$P_{rad}^{real}\left(t\right)=2\omega {\alpha }_0\left[1+\cos 2\left(\omega t-ka\right)\right]$$

(55)

the amplitude of which is not dependent on the radius of the observation surface. It is readily recognizable from Equation (53) that it is the first term in the Poynting power $P_{pv}\left(t\right)$. The other terms of $P_{pv}\left(t\right)$ in Equation (53) comprise the pseudo power flow, which decreases with the distance to the dipole.

The time-averaged energies are listed below for the readers’ reference:

$$\{\begin{array}{l}{\left(W_m\right)}_{av}={\alpha }_0\left(\frac{1}{ka}\right)\\ {\left(W_e\right)}_{av}={\alpha }_0\left(\frac{1}{k^3{a}^3}+\frac{1}{ka}\right)\end{array}$$

(56)

The Q factor of the dipole is then calculated to be

$$Q=\frac{2\omega {\left(W_e\right)}_{av}}{{\left(P_{rad}\right)}_{av}}=\frac{1}{k^3{a}^3}+\frac{1}{ka}$$

(57)

which is exactly in agreement with the results shown in [42].

The well-established equivalent circuit model proposed by Chu [38] for the Hertzian dipole is shown in Figure 3. Assume that the current in the radiation resistor at the interface of $r=a$ is $i_R=I_0\cos \left(\omega t-ka\right)$. The energies stored in the capacitor and the inductor can be derived to be

$$\{\begin{array}{c}{W}_C\left(t\right)=\frac{I_0^2}{4\omega }\left[\frac{1}{ka}+\frac{1}{k^3{a}^3}+\left(\frac{1}{k^3{a}^3}-\frac{1}{ka}\right)\cos 2\left(\omega t-ka\right)-\frac{2}{{\left(ka\right)}^2}\sin 2\left(\omega t-ka\right)\right]\hfill \\ W_L\left(t\right)=\frac{I_0^2}{2\omega }\left(\frac{1}{ka}{\sin }^2\left(\omega t-ka\right)\right)\hfill \end{array}$$

(58)

If we choose $I_0^2=4\omega {\alpha }_0$, it can be readily verified that $W_C\left(t\right)=W_e\left(t\right)$ and $W_L\left(t\right)=W_m\left(t\right)$. This exact agreement gives good support to the energy separation in Equation (11).

The integration regions for $W_{rad}\left(t\right)$, $W_{rad}^{pri}\left(t\right)$ and $W_S\left(t\right)$ are all modified in a similar way. They are found to be

$$W_{rad}\left(t\right)=-{\displaystyle {\int }_{V_{\infty }-V_a}D•\frac{\partial A}{\partial t}}dV=2k{\alpha }_0\underset{r\to \infty }{\lim }\left(r-a\right)+{\alpha }_0\left[\sin 2\left(\omega t-ka\right)-\underset{r\to \infty }{\lim }\sin 2\left(\omega t-kr\right)\right]$$

(59)

$$W_{rad0}\left(t\right)={\displaystyle {\int }_{V_{\infty }-V_a}\frac{1}{2}\left(\frac{\partial D}{\partial t}•A-D•\frac{\partial A}{\partial t}\right)}dV=2{\alpha }_{0}k\underset{r\to \infty }{\lim }\left(r-a\right)$$

(60)

$$W_S\left(t\right)={\displaystyle {\int }_{V_{\infty }-V_a}\frac{\partial }{\partial t}\left(\frac{1}{2}D•A\right)dV}=-{\alpha }_0\left[\sin 2\left(\omega t-ka\right)-\underset{r\to \infty }{\lim }\sin 2\left(\omega t-kr\right)\right]$$

(61)

When the wave travels to infinity, the principal radiative energy $W_{rad}^{pri}\left(t\right)$ monotonically increases with the radius, revealing that the radiative rate is always positive. The macroscopic Schott energy $W_S\left(t\right)$ oscillates in the propagation with a zero average value. Its amplitude remains constant in this case.

6. Numerical Results

We provide two examples to demonstrate the temporal evolution properties of the electromagnetic energies in radiation problems:

A. 

Solenoidal Loop

The radiation of a solenoidal loop current is analyzed. The solenoidal surface current on a ring is described by $J_s\left(r,t\right)=f\left(r\right)I\left(t\right)$ [A/m], as shown in Figure 4. Here, we choose

$$f\left(r\right)=1.0\widehat{\phi }$$

(62)

The inner and outer radii of the ring are 0.08 m and 0.1 m, respectively. The temporal function is a modulated Gaussian pulse:

$$I\left(t\right)=\{\begin{array}{l}{e}^{-{\gamma }^2}\sin \omega t,0\le t\le T\\ 0, \mathrm{else}\end{array}$$

(63)

with $\omega =2\pi \times {10}^{10}$, $\gamma =2\sqrt{5}\left(t-0.5T\right)/T$ and $T=1\mathrm{ns}$. Therefore, both its initial and final reactive energies are zero. Two spherical surfaces with radii of 0.2 m and 10 m are chosen as the observation surfaces, with their centers coinciding with that of the source. They are labeled as sphere-1 and sphere-2, respectively. The principal radiative energies passing through sphere-1 and sphere-2 are calculated with the integration of $P_{rad}^{pri}\left(t\right)$, as expressed in Equation (32). $W_{pv}\left(t\right)$ is the integration of the Poynting vector power passing through the observation surface:

$$W_{pv}\left(t\right)={\displaystyle {\int }_0^t{P}_{pv}\left(\tau \right)d\tau }$$

(64)

The excitation energy, the principal radiative energy and the energy evaluated with the Poynting vector are shown in Figure 5a. In this case, the current is solenoidal, and its corresponding charge is zero, so the reactive energy includes the contribution from the current alone and is labeled as $W_J$ in the figures. It can be seen that $W_J$ oscillates with the source and admits negative values periodically. It is acceptable, because the reactive energy is dependent on the potentials, which are values relative to their reference zero points. When the current varies and changes its direction periodically, the retarded vector potential in the source region lags behind and may point in the direction opposite to that of the current, causing negative values. The macroscopic Schott energy is plotted in Figure 5a as well and is zoomed in for Figure 5b, together with $W_J$. It can be seen that the macroscopic Schott energy oscillates like $W_J$ but continues to exist for about 0.33 ns after the source has disappeared at 1 ns. Note that the Schott energy in the charged particle theory may also be negative [20,43].

The energies passing through sphere-1 are shown in Figure 5c. The smallest and the largest distances between the source and sphere-1 are, respectively, 0.1 m and 0.3 m, so the earliest and latest times for the fields passing sphere-1 are, respectively, 0.33 ns and 2 ns. Therefore, the fields have all passed sphere-1 at 2 ns. The total radiative energy passing through sphere-1 at t = 2 ns is equal to that evaluated at the source region.

The excitation power, the principal radiative power and the time-varying rate of the reactive energy are shown in Figure 6a. The powers crossing sphere-1 and sphere-2 are shown in Figure 6b,c, respectively. $P_{rad}^{pri}\left(t\right)$ varies smoothly and remains positive. The Poynting power contains ripples coming from $P_{react}\left(t\right)$, which gradually decrease with the propagation distance.

B. 

Thin Plate Yagi Antenna

The geometrical structure and parameters of the Yagi antenna are shown in Figure 7. It consists of three PEC plates with zero thickness: a dipole in the middle, a reflector in the left and a director in the right. The width of the plate is 2 mm. The dipole is fed at its center with a Delta gap voltage source of

$$V_{feed}\left(t\right)=1.0\sin \left(\omega t\right)\left[\mathrm{V}\right],t\ge 0$$

(65)

where $\omega =2\pi \times 1.5\times {10}^8$, corresponding to 150 MHz.

The first step is to calculate the surface current by solving the surface electric field integral equation (EFIE) with the marching-on-in-time scheme (MOT) [44,45]. The plates are triangularly meshed, and the surface current is expanded with RWG basis functions [46]. There are 53, 59 and 50 RWGs on the left, the middle and the right plates, respectively. The Delta gap voltage feeding is put on the common edge of the RWG in the middle of the dipole. The time step of the MOT is 2.67 ps. The second step is to evaluate the energies and powers using the obtained surface currents and the expressions we have proposed. When calculating the macroscopic Schott energy and the principal radiative energy, the integration interval of the innermost integral is dependent on the distance between two points. It may become much smaller than the time step and should be handled carefully to obtain satisfactory numerical accuracy. For r_ij = 0, we can use the L’Hospital rule to find the limit of the innermost integral.

The principal radiative powers of the three plates are calculated separately with Equation (33), with the integration domain respectively replaced by those of the three plates. The results are shown in Figure 8. The Coulomb–velocity energies are shown in Figure 9. As shown in the figures, the principal radiative power of the dipole is always positive. It radiates electromagnetic energy to the space from the beginning. However, the principal radiative powers of the reflector and the director are negative at the beginning, which means that they absorb energy from the dipole to generate the reactive energy of themselves at the transition stage. When the radiation enters the steady state, the reactive energies tend to become steadily stable. Since the PEC reflector and the PEC director are passive elements, they do not radiate by themselves but only scatter away the electromagnetic energies they receive. Consequently, their principal radiative powers are zero at their steady state.

The total energies are plotted in Figure 10. After a short transition stage, the radiation approaches a steady state. It can be seen that the macroscopic Schott energy gradually becomes an oscillation with approximately uniform amplitude and the same period of excitation. It brings equi-ripples to the total reactive energy and the total radiative energy. The total reactive energy tends to become steady and bounded, while the radiative energy increases approximately in a linear way.

The principal radiative power and the electromagnetic power calculated with the Poynting vector are evaluated on an observation spherical surface with a radius of 2 m, the center of which is located at the feeding point of the Yagi antenna. The results are illustrated in Figure 11.

The directivity of the antenna inevitably varies in the transition stage. The directivity patterns in the E-plane and the H-plane at 10 ns, 20 ns and 40 ns are depicted in Figure 12. As can be seen, in the beginning, the passive reflector and the director absorb more electromagnetic energies than the energies they receive. The radiation is mainly determined by the center dipole, and the pattern is much like that of a single dipole. The antenna performs like a Yagi antenna only after the two passive plates have achieved a balance between their absorbed and scattered powers.

In the steady state, we can evaluate the Q factor of the radiator with a formula as follows:

$$Q\left(t\right)\approx \frac{2\pi }{T}\frac{W_{\rho J}\left(t\right)}{P_{rad}^{pri}\left(t\right)}$$

(66)

For comparison, the Q factor for $t>$ 0.15 μs is calculated to be 18.5, while the Q factor obtained with the frequency domain formulation described in [15] is 17.5 at 150 MHz.

7. Conclusions and Discussions

Some issues concerning the electromagnetic radiation and mutual couplings have remained ambiguous or even controversial for decades long, especially the definitions for the reactive energy. This theory proposed clear definitions and explicit expressions for the reactive energy and the radiative energy of a radiator. The introduction of the macroscopic Schott energy makes it possible to separate the radiative energy and the reactive energy in a consistent formulation. Consequently, a new form of power balance equation is given based on the Poynting relation. The Poynting vector includes the contributions from the radiative energy propagation and the fluctuation of the reactive energy. The newly defined principal reactive energy $W_{rad}^{pri}\left(t\right)$ and its flux, the principal radiative power $P_{rad}^{pri}\left(t\right)$, can characterize the main property of the radiative energy. Furthermore, they can be numerically evaluated more efficiently, so the mutual electromagnetic coupling energies are defined with potentials.

As is pointed out in the Introduction, the main problem of the issue is how to separate the radiative energy and the reactive energy. We separated the electromagnetic energy into three parts: the Coulomb–velocity energy $W_{\rho J}\left(t\right)$, the macroscopic Schott energy $W_S\left(t\right)$ and the radiative energy $W_{rad}\left(t\right)$. Then, we derived the explicit and accurate expressions for them and verified the separation with a pulse radiator judging from their temporal evolution properties. All the expressions are strictly derived from Maxwell equations, with no approximations. The retarded time is included in them, and there are no static limits, although they may look alike at a first glance.

The basic theory is for time-varying pulse radiators. We also provided formulae for harmonic waves. Unlike the Vandenbosch formulation, in which the results obtained with the time domain formulation sometimes may not agree with the results obtained with the frequency domain formulation, in the theory proposed here, the results in the time domain and frequency domain are completely consistent, because they are, respectively, directly derived from the time domain Maxwell equations and the frequency domain Maxwell equations. The electromagnetic fields in the frequency domain and time domain can be converted with Fourier transform, which is a common sense in computational electromagnetics.

It has been discussed in previous sections that the electromagnetic radiation and mutual coupling issues are closely related and can be handled in the same manner. If we let $i=j$ in Equations (35) and (37), we obtain Equations (12) and (7), which are expressions for a single radiator. The terms in these expressions are all dot products of two vectors: those at the right side are potentials, and those at the left side are source densities or their fields. In systems with electromagnetic mutual couplings, the potentials are associated with the radiators that impose the couplings, while the left quantities are associated with the radiators affected by the couplings.

The theory is completely different from the Carpenter formulation [27]. In the Carpenter formulation, it was proposed to use the Coulomb–velocity energy alone as the total electromagnetic energy and to replace the Poynting theorem with a new equation. The formulation, as well as the power flow vector $\varphi J$ proposed by Slepian [47], were pointed out to be mathematically flawed by Dr. Endean [48]. In our theory, we combine the Coulomb–velocity energy and the macroscopic Schott energy together to form the reactive energy, which is only part of the total electromagnetic energy of the radiator. This theory does not suffer from the mathematical flaw, since expressions in the formulation are all derived directly from the Maxwell equations, and there is no modification to the total electromagnetic energy and the Poynting theorem.

Basically, the formulation is only valid for evaluating the nonrelativistic electromagnetic energies of a radiator in free space. Although the role of the macroscopic Schott energy needs to be further investigated, the energy separation strategy proposed here can perhaps provide a more consistent formulation than the other conventional formulations related to this issue.