# Time-Domain Analytical Expression for Near Fields of Arbitrarily Oriented Electric Dipole and Its Application

Authors: Qian Yang; Bing Wei; Xinbo He; Minghao Gong
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

The near fields of electric dipole are commonly used in wide-band analysis of complex electromagnetic problems. In this paper, we propose new near field time-domain expressions for electric dipole. The analytical expressions for the frequency-domain of arbitrarily oriented electric dipole are given at first; next we give the time-domain expressions by time-frequency transformation. The proposed expressions are used in hybrid TDIE/DGTD method for analysis of circular antenna with radome. The accuracy of the proposed algorithm is verified by numerical examples.


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## Body

## 1. Introduction

The electric dipole is an important unit in electromagnetism; its analytical expressions of the radiation field have been described in a number of works, for example, the expressions of special oriented electric dipole [1–3] and the expressions of arbitrarily oriented electric dipole by using the far-field approximation ‎[4]. But the time-domain expressions of arbitrarily oriented electric dipole are necessary to analyze the time-domain near fields problems. The near field is governed by several type fields, the relationship between E and H becomes very complex, and special care must be taken when dealing with near fields problems.The hybrid method takes advantage of several methods and is often used to analyze multiscale electromagnetic problems [5–7], like the shortwave antenna in complex environment ‎[8]. In this study, the analytical time-domain expressions of arbitrarily oriented electric dipole are proposed and used in hybrid TDIE/DGTD method for analysis of circular antenna with radome. The numerical results verify the validity of our algorithm.


## 2. The Frequency-Domain Near Fields for an Arbitrarily Oriented Electric Dipole

Suppose an electric dipole which alongr^0 is located at original point; the polar angle and the azimuthal angle of r^0 are θ0 and φ0, as Figure 1(a) shows. The observation point is located at P, and the coordinates of point P are r, θ, and φ, respectively, as Figure 1(b) shows.Figure 1
The arbitrarily oriented electric dipole.
(a)
An electric dipole at the original point
(b)
The observation point(r,θ,φ)
(c)
Integrating point and observation pointThe magnetic vector potential produced by the electric current in an infinite medium can be written as(1)Αr=μ∭vJr′expikr-r′4πRdυ′,where μ, r′, r, R, and k are the permeability, the position vector of the integrating point, the position vector of the observation point P, the distance from the integrating point to the observation point, and the wave number, respectively, as Figure 1(c) shows.A short current sourcer^0Il can be seen as an electric dipole; considering the characteristic of the δ(r′) function, we have(2)Ar=μ∭vr^0Ilδr′expikr-r′4πRdυ′=r^0Ilμexpikr4πr,where r^0 is defined as(3)r^0=x^sin⁡θ0cos⁡φ0+y^sin⁡θ0sin⁡φ0+z^cos⁡θ0,where the x^, y^, and z^ are the unit vector of the rectangular coordinate system. Magnetic field can be evaluated as(4)H=1μ∇×A=1μ∇×r^0Ilμexpikr4πr=1μ∇×x^sin⁡θ0cos⁡φ0+y^sin⁡θ0sin⁡φ0+z^cos⁡θ0Ilμexpikr4πr=Il4πsin⁡θ0cos⁡φ0∇×x^expikrr+sin⁡θ0sin⁡φ0∇×y^expikrr+cos⁡θ0∇×z^expikrr,(5)∇×A=x^∂Az∂y-∂Ay∂z+y^∂Ax∂z-∂Az∂x+z^∂Ay∂x-∂Ax∂ywhich implies curl elements of (4) can be rewritten as(6)∇×x^expikrr=y^cos⁡θ-z^sin⁡θsin⁡φikr-1r2expikr,∇×y^expikrr=z^sin⁡θcos⁡φ-x^cos⁡θikr-1r2expikr,∇×z^expikrr=x^sin⁡θsin⁡φ-y^sin⁡θcos⁡φikr-1r2expikr.Substituting (6) into (2), we get expression of magnetic field for the frequency-domain of an arbitrarily oriented electric dipole(7)H=ikIlexp⁡ikr4πrx^cos⁡θ0sin⁡θsin⁡φ-sin⁡θ0sin⁡φ0cos⁡θ+y^sin⁡θ0cos⁡φ0cos⁡θ-cos⁡θ0sin⁡θcos⁡φ+z^sin⁡θ0sin⁡φ0sin⁡θcos⁡φ-sin⁡θ0cos⁡φ0sin⁡θsin⁡φ1+ikr.The expression of electric field is(8)E=-iωε∇×H=-iωμε∇×∇×A=-iωμε∇2A-∇∇·A=iωμεk2A+∇∇·A;substituting (2) into (8), we have(9)E=iωμεk2r^0Ilμexpikr4πr+∇∇·r^0Ilμexpikr4πr=iIlμ4πωμεk2r^0expikrr+∇∇·r^0expikrr,where(10)∇∇·r^0expikrr=sin⁡θ0cos⁡φ0∇∇·x^expikrr+sin⁡θ0sin⁡φ0∇∇·y^expikrr+cos⁡θ0∇∇·z^expikrr.


## 3. The Time-Domain Near Fields for an Arbitrarily Oriented Electric Dipole

An electric dipole consists of a positive charge+q and an negative charge -q, distance l apart. The dipole moment p=ql, so that electric current can be represented as(11)Il=-iωp.Substituting (11) into (7),(12)Hr,ω=ω2pexp⁡ikr4πcr1+icωrx^cos⁡θ0sin⁡θsin⁡φ-sin⁡θ0sin⁡φ0cos⁡θ+y^sin⁡θ0cos⁡φ0cos⁡θ-cos⁡θ0sin⁡θcos⁡φ+z^sin⁡θ0sin⁡φ0sin⁡θcos⁡φ-sin⁡θ0cos⁡φ0sin⁡θsin⁡φ.The time-frequency conversion relation and the flourier transformation are(13)-iω⟶∂∂t,pt=12π∫pωexp⁡-iωtdω,pt-rc=12π∫pωexp⁡ikrexp⁡-iωtdω.The time-domain expression of (12) can be obtained by (13)(14)Hr,t=-14πcr∂2∂t2+cr∂∂tpt-rcx^cos⁡θ0sin⁡θsin⁡φ-sin⁡θ0sin⁡φ0cos⁡θ+y^sin⁡θ0cos⁡φ0cos⁡θ-cos⁡θ0sin⁡θcos⁡φ+z^sin⁡θ0sin⁡φ0sin⁡θcos⁡φ-sin⁡θ0cos⁡φ0sin⁡θsin⁡φ.The time-domain expression of electric field can be obtained in the same way(15)Ex=sin⁡θ0cos⁡φ0A1·sin2θcos2φ-A2·cos2θcos2φ+sin⁡θsin2φ-∂2∂t2+sin⁡θ0sin⁡φ0A1·sin2θsin⁡φcos⁡φ-A2·sin⁡φcos⁡φcos2θ-sin⁡θ+cos⁡θ0A1·sin⁡θcos⁡θcos⁡φ+A2·sin⁡θcos⁡θcos⁡φμ4πrpt-rc,Ey=sin⁡θ0cos⁡φ0A1·sin2θcos⁡φsin⁡φ-A2·sin⁡φcos⁡φcos2θ-sin⁡θ+sin⁡θ0sin⁡φ0A1·sin2θsin2φ-A2·cos2θsin2φ+sin⁡θcos2φ-∂2∂t2+cos⁡θ0A1·sin⁡θcos⁡θsin⁡φ+A2·sin⁡θcos⁡θsin⁡φμ4πrpt-rc,Ez=sin⁡θ0cos⁡φ0A1·sin⁡θcos⁡θcos⁡φ+A2·sin⁡θcos⁡θcos⁡φ+sin⁡θ0sin⁡φ0A1·sin⁡θcos⁡θsin⁡φ+A2·sin⁡θcos⁡θsin⁡φ+cos⁡θ0A1·cos2θ-A2·sin2θ-∂2∂t2μ4πrpt-rc,A1=2c2r2+2cr∂∂t+∂2∂t2A2=c∂∂t+c2r.


## 4. The Hybrid Method Combining DGTD and TDIE for Wire Antenna-Dielectric Interaction

TDIE is widely used for analyzing thin-wire antenna (radiation on scattering) problems, but it is difficult to deal with interaction of the antenna and complexity media. The TDIE/FDTD method and the FDTD/FETD/TDIE method have been proposed for complex electromagnetic problems ‎[9, 10]. Discontinuous Galerkin Time-Domain (DGTD) method [11–15] inherits from Finite Element Time-Domain (FETD) the advantage of unstructured grids without solving large linear systems. DGTD is more flexible than Finite Difference Time-Domain (FDTD) method in geometry modeling, but like FDTD, it also becomes resource-consuming dealing with thin-wire structures.Gao et al. proposed the TDIE/DGTD method ‎[16]; in our study, the currents on the antenna can be easily obtained by TDIE. These currents are used to calculate E-H fields on Huygens surface which yields the same radiating fields. The E-H fields are deduced from our new expressions, as shown in Figure 2.Figure 2
The hybrid TDIE/DGTD method.
(a)
Hybrid method setup
(b)
Computing procedures


### 4.1. A Circular Antenna

The antenna is parallel to theXOY plane, the radius of the antenna is 0.5 meter, the excitation wavelength is 0.5 meter, and the time harmonic current is excited at every line element (Figure 3). The normalized results of TDIE/DGTD with our new expressions are shown in Figures 4(a) and 4(b); a good agreement is observed between our algorithm (circle) and analytical solution (solid line) ‎[17].Figure 3
Circular antenna (divided into 35 straight lines).Figure 4
The radiation pattern of a circular antenna.
(a)
The radiation pattern ofXOZ plane
(b)
Three-dimensional radiation pattern


### 4.2. A Right Angle Antenna with Radome

We analyzed a right angle antenna located in the center of a dielectric ellipsoid-shell radome (semiprincipal axesa=1 meter, b=1 meter, and c=0.5 meters); the thickness of the radome is 0.25 meters, the relative electric permittivity is 3.0, and the relative permeability is 1.0. The antenna is parallel to the planeYOZ (Figure 5); each arm of the antenna is 0.2 meters long. Antenna is excited by time harmonic current whose wavelength is 0.5 meters. The results of TDIE/DGTD and DGTD are shown for comparison in Figure 6; a good agreement is observed between two results.Figure 5
Sectional view grid.Figure 6
The radiation pattern ofYOZ plane.


### 4.3. A Circular Antenna with Radome

In this example, the antenna in the example of Section4.2 is replaced with the circular antenna in the example of Section 4.1; excitation parameters of antenna are the same as the example of Section 4.2 (Figure 7).Figure 7
Sectional view grid.Figure8(a) is the normalized radiation pattern of circular antenna with radome. It is obvious that the main lobe of the pattern with radome is more intensive than without radome, as Figure 8(a) shows.Figure 8
The radiation pattern of circular antenna with radome.
(a)
The radiation pattern ofXOZ plane
(b)
Three-dimensional radiation patternNext, the antenna is excited by a voltage pulse source, as shown in Figure9, the distribution of currents on antenna is derived from TDIE. The voltage source is defined as(16)V=expt-t0τ2τ=2ns,t0=8ns.Figure 9
Voltage source.Follow the steps in Figure2(b); the snapshots in Figure 11 of the absolute value of electric field E are obtained from the hybrid TDIE/DGTD method. As shown in Figure 11(a), the electric field density is higher in the half-space x>0, because the voltage source is located at the cross point of antenna and +x axis. The reflection radiation caused by radome can be seen in Figures 11(b), 11(c), and 11(d). The current waveform of source point is shown in Figure 10; the current tends to a value that is not zero.Figure 10
Current of source point.Figure 11
Snapshots ofE.
(a)
1000th time-step
(b)
1600th time-step
(c)
2200th time-step
(d)
2800th time-step


## 4.1. A Circular Antenna

The antenna is parallel to theXOY plane, the radius of the antenna is 0.5 meter, the excitation wavelength is 0.5 meter, and the time harmonic current is excited at every line element (Figure 3). The normalized results of TDIE/DGTD with our new expressions are shown in Figures 4(a) and 4(b); a good agreement is observed between our algorithm (circle) and analytical solution (solid line) ‎[17].Figure 3
Circular antenna (divided into 35 straight lines).Figure 4
The radiation pattern of a circular antenna.
(a)
The radiation pattern ofXOZ plane
(b)
Three-dimensional radiation pattern


## 4.2. A Right Angle Antenna with Radome

We analyzed a right angle antenna located in the center of a dielectric ellipsoid-shell radome (semiprincipal axesa=1 meter, b=1 meter, and c=0.5 meters); the thickness of the radome is 0.25 meters, the relative electric permittivity is 3.0, and the relative permeability is 1.0. The antenna is parallel to the planeYOZ (Figure 5); each arm of the antenna is 0.2 meters long. Antenna is excited by time harmonic current whose wavelength is 0.5 meters. The results of TDIE/DGTD and DGTD are shown for comparison in Figure 6; a good agreement is observed between two results.Figure 5
Sectional view grid.Figure 6
The radiation pattern ofYOZ plane.


## 4.3. A Circular Antenna with Radome

In this example, the antenna in the example of Section4.2 is replaced with the circular antenna in the example of Section 4.1; excitation parameters of antenna are the same as the example of Section 4.2 (Figure 7).Figure 7
Sectional view grid.Figure8(a) is the normalized radiation pattern of circular antenna with radome. It is obvious that the main lobe of the pattern with radome is more intensive than without radome, as Figure 8(a) shows.Figure 8
The radiation pattern of circular antenna with radome.
(a)
The radiation pattern ofXOZ plane
(b)
Three-dimensional radiation patternNext, the antenna is excited by a voltage pulse source, as shown in Figure9, the distribution of currents on antenna is derived from TDIE. The voltage source is defined as(16)V=expt-t0τ2τ=2ns,t0=8ns.Figure 9
Voltage source.Follow the steps in Figure2(b); the snapshots in Figure 11 of the absolute value of electric field E are obtained from the hybrid TDIE/DGTD method. As shown in Figure 11(a), the electric field density is higher in the half-space x>0, because the voltage source is located at the cross point of antenna and +x axis. The reflection radiation caused by radome can be seen in Figures 11(b), 11(c), and 11(d). The current waveform of source point is shown in Figure 10; the current tends to a value that is not zero.Figure 10
Current of source point.Figure 11
Snapshots ofE.
(a)
1000th time-step
(b)
1600th time-step
(c)
2200th time-step
(d)
2800th time-step


## 5. Conclusion

The analytical expressions of time-domain near fields of an arbitrarily oriented electric dipole are derived in this paper; then the expressions are used in hybrid TDIE/DGTD method for analysis of circular antenna with radome. Our study provides a new way to study the time-domain radiation problems of complex medium and structure.



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