# Simulation Based Poynting Vector Description of the Field Regions for Simple Radiating Structures

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals and Theory

#### 2.1. Poynting Vector

#### 2.2. Field Regions

#### 2.2.1. Boundary between the Radiating Near-Field and the Far-Field

#### 2.2.2. Boundary between the Reactive and Radiating Near-Field

#### 2.3. Hertzian Dipole

## 3. Dipole Antennas and Dipole-Based Array

#### 3.1. Dipole Antennas

#### 3.2. Linear Antenna Array

## 4. Loop Antennas and the Coupling Behavior of Large Loop Antennas

#### 4.1. Loop Antennas

#### 4.2. Coupling Behavior of an Electrically Large Loop Antenna

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NF | Near-field |

FF | Far-field |

PEEC | Partial element equivalent circuit |

EMI | Electromagnetic interference |

## References

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**Figure 1.**Schematic view of an antenna and the surrounding field regions, described by the radii ${r}_{far}$ and ${r}_{near}$.

**Figure 3.**(

**a**) Real part of the Poynting vector around an HD in the E-plane, respectively, in the x/z plane. (

**b**) Imaginary part of the Poynting vector around an HD in the E-plane.

**Figure 4.**The error-estimator 9, showing the reactive NF region around the HD in the E-plane.

**Figure 5.**(

**a**) The error-estimator 5, describing the radial and real portion of the power flow applied on an HD in the E-plane. (

**b**) The error-estimator 6 applied on an HD in the E-plane, describing the error-estimator weighted with the directivity $D(\theta ,\varphi )$. The FF boundary ${r}_{far}$ for very small antennas is displayed as a black circle.

**Figure 6.**(

**a**) The real part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The imaginary part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$.

**Figure 7.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole. (

**b**) The error-estimator 6, describing a qualitative measure of the FF quality, applied on a $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole in the E-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ given in 4 is displayed as a black circle.

**Figure 8.**(

**a**) The real part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The imaginary part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$.

**Figure 9.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole. (

**b**) The error-estimator 6, describing a qualitative measure of the FF quality, applied on a $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole in the E-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ given in 4 is displayed as a black circle.

**Figure 10.**(

**a**) The real part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The imaginary part of the Poynting vector in the E-plane of a dipole antenna with an overall length of $1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$.

**Figure 11.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a $1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole. (

**b**) The error-estimator 6, describing a qualitative measure of the FF quality, applied on a $1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole in the E-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ given in 4 is displayed as a black circle.

**Figure 12.**The reactive NF region, described by the angular-dependent radius ${r}_{near}(\theta ,\varphi )$, when applying the error estimator 11 with $er{r}_{NR2}({r}_{near}(\theta ,\varphi ),\theta ,\varphi )=1$ of a (

**a**) $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ (

**b**) $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, and (

**c**) $1.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ dipole. The corresponding dipole can be seen inside the region.

**Figure 13.**The three investigated linear antenna arrays, each consisting of five $\lambda /2$-dipoles. The distance between these array elements is (

**a**) $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, (

**b**) $0.3\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, and (

**c**) $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. The reactive NF region, proposed in Figure 12b, is shown around each dipole.

**Figure 14.**The current distribution along the different dipoles for the array with (

**a**) $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ spacing, (

**b**) $0.3\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ spacing, and (

**c**) $0.5\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ spacing between each element. For the blue curve, each element has been computed separately, hence mutual coupling effects were not considered. For the red curve, the elements have been computed as an array; consequently, mutual coupling effects have been considered.

**Figure 15.**The difference of the impedance between the two computation methods for every dipole as a function of the distance between the array elements in x-direction.

**Figure 16.**The difference of the impedance between the two computation methods for every dipole as a function of the distance between the array elements in z-direction.

**Figure 17.**(

**a**) The real and imaginary part of the Poynting vector in the xy-plane of a loop antenna with an overall circumference of $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The real and imaginary part of the Poynting vector in the yz-plane of a loop antenna with an overall circumference of $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**c**) The real and imaginary part of the Poynting vector in the zx-plane of a loop antenna with an overall circumference of $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$.

**Figure 18.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a loop antenna with the circumference of $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. The error-estimator 6, describing a qualitative measure of the FF quality, applied on a loop antenna with the circumference of $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ in the (

**b**) xy-plane, (

**c**) yz-plane, and (

**d**) zx-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=\frac{C}{\pi}$ given in 4 is displayed as a black circle.

**Figure 19.**(

**a**) The real and imaginary part of the Poynting vector in the xy-plane of a loop antenna with an overall circumference of $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The real and imaginary part of the Poynting vector in the yz-plane of a loop antenna with an overall circumference of $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**c**) The real and imaginary part of the Poynting vector in the zx-plane of a loop antenna with an overall circumference of $0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. In each sub-figure, the loop antenna is indicated as a white circle.

**Figure 20.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a loop antenna with the circumference of $C=0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. The error-estimator 6, describing a qualitative measure of the FF quality, applied on a loop antenna with the circumference of $C=0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ in the (

**b**) xy-plane, (

**c**) yz-plane, and (

**d**) zx-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=\frac{C}{\pi}$ given in 4 is displayed as a black circle.

**Figure 21.**(

**a**) The real and imaginary part of the Poynting vector in the xy-plane of a loop antenna with an overall circumference of $0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**b**) The real and imaginary part of the Poynting vector in the yz-plane of a loop antenna with an overall circumference of $0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. (

**c**) The real and imaginary part of the Poynting vector in the zx-plane of a loop antenna with an overall circumference of $0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. In each sub-figure, the loop antenna is indicated as a white circle.

**Figure 22.**(

**a**) A 3D plot of the directivity $D(\theta ,\varphi )$ of a loop antenna with the circumference of $C=0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. The error-estimator 6, describing a qualitative measure of the FF quality, applied on a loop antenna with the circumference of $C=0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ in the (

**b**) xy-plane, (

**c**) yz-plane, and (

**d**) zx-plane. The FF boundary ${r}_{far}$ for the given antenna dimension $l=\frac{C}{\pi}$ given in 4 is displayed as a black circle. The loop itself is given as a white circle.

**Figure 23.**The reactive NF region, described by the angular-dependent radius ${r}_{near}(\theta ,\varphi )$, when applying the error estimator 11 with $er{r}_{NR2}({r}_{near}(\theta ,\varphi ),\theta ,\varphi )=1$ of a loop antenna with the circumference of (

**a**) $C=0.01\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, (

**b**) $C=0.1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, (

**c**) $C=0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, and (

**d**) $C=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$.

**Figure 24.**The reactive NF region of a loop antenna with the circumference of $C=0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$, determined by applying the error-estimator in 11, is given. The red surface shows the region where the imaginary part of the normal component of the Poynting vector is positive, and the blue surface shows the region with a negative imaginary part.

**Figure 25.**Simulation setup for the coupling investigations. For both setups, a $C=0.45\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ loop antenna (red) is placed in the xy-plane. The feed-gap is placed in the $\varphi =0$-direction. In (

**a**), an electrically small loop antenna is placed parallel to the big loop with a z-distance of $0.05\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. In (

**b**), a y-orientated electrically small dipole antenna is placed above the big loop antenna, with a z-distance of $0.05\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$. The small loop (

**a**) and small dipole (

**b**) are placed along the green line to analyze the coupling behavior for different points.

**Figure 26.**The induced voltage of a loop probe (

**a**) and a dipole probe (

**b**) when moved along the green line denoted in Figure 25 above a $0.05\phantom{\rule{3.33333pt}{0ex}}\mathsf{\lambda}$ loop antenna.

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**MDPI and ACS Style**

Baumgartner, P.; Masiero, A.; Riener, C.; Bauernfeind, T.
Simulation Based Poynting Vector Description of the Field Regions for Simple Radiating Structures. *Electronics* **2022**, *11*, 1967.
https://doi.org/10.3390/electronics11131967

**AMA Style**

Baumgartner P, Masiero A, Riener C, Bauernfeind T.
Simulation Based Poynting Vector Description of the Field Regions for Simple Radiating Structures. *Electronics*. 2022; 11(13):1967.
https://doi.org/10.3390/electronics11131967

**Chicago/Turabian Style**

Baumgartner, Paul, Anna Masiero, Christian Riener, and Thomas Bauernfeind.
2022. "Simulation Based Poynting Vector Description of the Field Regions for Simple Radiating Structures" *Electronics* 11, no. 13: 1967.
https://doi.org/10.3390/electronics11131967