Junming Diao
Editor’s Note
This issue’s “Education Corner†column presents a useful technique to help students and practitioners understand the receiving characteristics of antennas using a graphical method illustrating how the Poynting vector of an incident field interacts with the antenna. This is facilitated by employing streamlines visualized in the vicinity of the antenna. The technique is particularly useful in helping one to visualize the concept of the effective area of an antenna, which can be particularly challenging with certain antenna types, such as dipoles. Such methods can aid the understanding of how antennas interact with incident fields and provide a complementary tool for educators in the classroom.
Classical antenna theory mainly focuses on the transmitting mode and gives less consideration to the receiving mode. However, the streamlines of the Poynting vector field can directly model and help us understand the behavior of receiving antennas. By analyzing the distribution of Poynting streamlines, we can visualize the flow of field energy near the receiving antennas and obtain a geometry that represents the shape of the antenna’s effective area. In this article, I adopt the Poynting streamline method to model and analyze typical antennas. This method serves as an effective teaching tool in antenna courses, providing intuitive pictures to help students understand the receiving mechanism and physics of antennas. According to survey data, students highly value the Poynting streamline method and recommend incorporating it into future courses.
Since antennas are often modeled and understood as transmitters, the receiving characteristics are not well employed in the antenna community. When a plane wave is incident on a receiving antenna, the field energy flow from the far-field region to the antenna can be represented by the streamlines of the real part of Poynting vectors, which are called the Poynting streamlines. In the far-field region around the receiving antenna, the incident field is a uniform plane wave, and the Poynting streamlines are straight and evenly distributed. In the near-field region of the receiving antenna, some of the Poynting streamlines are bent, concentrated, and absorbed by the receiving antenna load, while the other Poynting streamlines are scattered by the receiving antenna. Therefore, this method can visualize the absorbed field energy by tracing the trajectories of Poynting streamlines from the far-field region to the receiving antenna load. The effective area shape can be obtained from the Poynting streamlines absorbed by the antenna load.
The Poynting streamline method was first introduced in the 1970s [1], but it has not been widely recognized and applied to the antenna field. In the past decades, only limited literature has reported the use of this method for antenna analysis and design, including the analysis of dipole antennas [2], [3], the design of superdirective antennas [4], [5], [6], the analysis of mutual coupling and element-gain paradox for array antennas [7], [8], the improvement of antenna gain [9], and the design of energy-harvesting rectennas [10].
To the best of my knowledge, almost all textbooks used to teach antenna courses focus on transmitting properties for many types of antennas and simply use equivalent circuits to explain the high level of receiving characteristics. This is a good way to understand the general properties of receiving antennas but lacks detailed information and clear physical pictures of how receiving antennas absorb the field energy from the incident wave. In this article, the presented results based on the Poynting streamline analysis are of great value for students to intuitively and quickly understand the receiving mechanism between various typical antennas. This would make antenna theory less mysterious by showing intuitive physical pictures of the effective area shape to indicate the field energy absorbed by the antenna load and the distribution of Poynting streamlines to track the flow of field energy near the antennas. It would help to enrich and speed up the learning process without introducing additional potentially burdensome mathematics, advanced equivalent circuit models, and abstract physical concepts. This is especially true given that the Poynting streamline method is a graphical tool for teaching antenna theory in upper-level undergraduate and graduate classes. Survey data indicate that the students found the Poynting streamline method to be highly valuable and worthy of incorporation into future coursework.
The average Poynting vector over a period can be expressed as \[{\bf{S}} = \frac{1}{2}{\text{Re}}{\{}{\bf{E}}\,{\times}\,{\bf{H}}^{\ast}{\}} \tag{1} \] where E and H are the steady electric and magnetic field vectors, and ${\ast}$ denotes the complex conjugate. The Poynting vector provides information about the direction and magnitude of energy flow at a given point in space. The local tangent of a streamline is the direction of the Poynting vector, which represents the flow of energy at that point in space. A streamline is governed by a differential equation by [11] \[{\frac{{d}{\vec{p}}{(}{a}{)}}{da}} = {\bf{S}}{(}{\vec{p}}{(}{a}{)}{)} \tag{2} \] where ${\vec{p}}$ is the position in three dimensions, a is the parameter along the streamline, and ${\bf{S}}{(}{\vec{p}}{)}$ is the Poynting vector at position ${\vec{p}}$. The contours of ${\vec{p}}$ at the curve level are streamlines [12].
The Poynting streamlines provide a visual representation of the energy flow in the vicinity of a receiving antenna and can be determined based on the total field ${E}_{\text{total}}$. The total field comprises incident fields ${E}_{\text{inc}}$ and scattered fields ${E}_{\text{sca}}$ generated by the antenna. A minimum scattering antenna, such as an open-circuited receiving dipole antenna, is highly transparent to the incident field energy and exhibits minimal ${E}_{\text{sca}}$ [13]. Consequently, as depicted in Figure 1(a), the Poynting streamlines above an open-circuit half-wave receiving dipole antenna appear as almost straight lines, with no visible bending. The absence of any Poynting streamlines terminating at the dipole antenna load indicates that no energy is being absorbed by the load.
Figure 1. The bending effect of the Poynting streamline near a half-wave receiving dipole antenna. (a) An open circuit. (b) An impedance match.
In contrast, a matching half-wave receiving dipole antenna, shown in Figure 1(b), exhibits bent Poynting streamlines that terminate at the antenna load owing to the scattered fields generated by the dipole arm current. This behavior indicates that the energy of the incident plane wave can be absorbed by the impedance-matched receiving antenna load.
A full-wave model based on the finite-element method is used to calculate the Poynting streamlines near the receiving antenna. The analysis assumes that the antenna is lossless and that a uniform plane wave with an electric field strength of 1 V/m is incident in the −z direction, with polarization matching that of the receiving antenna. The load impedance of the receiving antenna is complex conjugate matched to the antenna input impedance.
The effective area ${A}_{\text{load}}$ can be determined by the Poynting streamlines terminated by the antenna load by \[{A}_{\text{load}} = {\sum}{\alpha}{(}{i}{)}{\Delta}{x}{\Delta}{y} \tag{3} \] where ${\Delta}{x}$ and ${\Delta}{y}$ represent the sample intervals of the Poynting streamlines in an x-y plane in the far-field region above the antenna. In Figure 1, this plane is situated at ${z} = {0.7}{\lambda}$. The ith Poynting streamline in this plane is denoted by ${\alpha}{(}{i}{)}$. If the minimum distance between the ith Poynting streamline and the antenna load is fewer than ${0.005}{\lambda}$, ${\alpha}{(}{i}{)}$ is set to one; otherwise, it is zero. Therefore, the effective area shape can be determined by identifying the Poynting streamlines in the x-y plane where ${\alpha}{(}{i}{)}$ is equal to one.
The absorbed field power ${P}_{\text{load}}$ of a receiving antenna can be determined by ${A}_{\text{load}}$ using the equation \[{P}_{\text{load}} = {A}_{\text{load}}{P}_{\text{inc}} \tag{4} \] where ${P}_{\text{inc}}$ is the incident field power density. ${A}_{\text{load}}$ is equal to the effective area ${A}_{\text{eff}}$ when the antenna input impedance and the antenna load impedance are complex conjugate matched [14].
The Poynting streamline distribution and effective area shape for a short dipole antenna are shown in Figure 2. The antenna has a length of ${0.054}{\lambda}$, an arm width of ${0.017}{\lambda}$, and a gap distance between the arms of ${0.01}{\lambda}$. The effective area shape is similar to an ellipse, with the major axis along the dipole antenna. However, compared to the half-wave dipole antenna, the short dipole antenna has a shorter major axis and a longer minor axis. The Poynting streamlines absorbed by the antenna load are much larger than the physical size of the short dipole antenna, which can be attributed to the bending and concentration of the Poynting streamline caused by the near-field expansion of the antenna.
Figure 2. An X-polarized short dipole antenna with a length of ${0.1}{\lambda}$. (a) The 3D Poynting streamline distribution. (b) The E-plane. (c) The effective area shape.
Figure 3 shows the Poynting streamline distribution and the effective area shape for a monopole antenna. The length of the monopole antenna is ${0.215}{\lambda}$, and the width of the antenna arm is ${0.017}{\lambda}$. The length of the square ground plane is changed from ${0.6}{\lambda}$ to ${0.1}{\lambda}$.
Figure 3. The Poynting streamline distribution and effective area shape for a y-polarized monopole antenna. From (a) to (c), the length of the ground plane is ${0.6}{\lambda}$, ${0.4}{\lambda}$, ${0.2}{\lambda}$, and ${0.1}{\lambda}$, respectively.
When the length of the ground plane is ${0.6}{\lambda}$, the effective area shape of the monopole antenna is similar to the left half of the effective area shape for the receiving short dipole antenna in Figure 2(c), which is due to the imaging effect of the ground plane to the vertically polarized monopole antenna. Decreasing the length of the ground plane helps to mitigate the imaging effect, and the effective area shape is expanded in the +y-direction. When the length of the ground plane is ${0.1}{\lambda}$, the effective area shape for the monopole antenna is similar to that for the short dipole antenna in Figure 2(d).
It is important to note that while the radiation properties, such as the radiation pattern, for a transmitting monopole antenna with a large ground plane are similar to those of a transmitting half-wave dipole antenna, the receiving properties, such as the effective area shape and energy flow distribution, for the receiving monopole antenna are different from those of the receiving half-wave dipole antenna. Due to the imaging effect produced by the large size of the ground plane, the field energy absorbed by the antenna on the right side of the ground plane is close to zero, thus forming an asymmetric shape of the effective area. When the size of the ground plane is close to zero, the imaging effect can be ignored, and the effective area shape becomes nearly symmetrical.
Figure 4 depicts the distribution of Poynting streamlines and effective area shape for a loop antenna with a circumference of ${1}{\lambda}$. Figure 4(b) shows the Poynting streamline distribution in a 2D cut plane when ${y} = {0.1}{\lambda}$, where some of the Poynting streamlines flow clockwise along the loop antenna and get absorbed by the antenna load. The loop antenna structure helps in bending and driving the field energy into a semicircle around the loop antenna structure, resulting in an almost circular effective area.
Figure 4. A Y-polarized loop antenna with ${1}{\lambda}$ circumference. (a) A 3D Poynting streamline distribution. (b) A 2D cut plane at ${y} = {0.1}{\lambda}$. (c) The effective area shape.
To gain a better understanding of the energy flow near the loop antennas, Figure 5 illustrates the effective area shape and Poynting streamlines in a 2D cut plane when ${y} = {-}{0.1}{\lambda}$ for loop antennas with different circumferences. When the circumference of the loop antenna is ${1.5}{\lambda}$, a hollow shape is found toward the left center of the effective area. The presence of this hollow shape indicates that the aperture efficiency is relatively low when the circumference of the loop antenna is sufficiently large. The Poynting streamlines absorbed by the antenna load can be attributed to two parts; one part of the Poynting streamlines is directly absorbed by the antenna load from free space, while the other part of the Poynting streamlines gets absorbed by the antenna load after rotating along the loop wire. The wire of the loop antenna helps to conduct the field energy along the wire to the antenna load.
Figure 5. The effective area shape for y-polarized loop antennas with ${0.5}{\lambda}$, ${1}{\lambda}$, and ${1.5}{\lambda}$ circumferences, respectively.
Figure 6 illustrates the Poynting streamline distribution and the effective area shape for a circular-polarized helix antenna. The helix radius and wire radius are ${0.125}{\lambda}$ and ${0.00667}{\lambda}$, respectively, with a pitch of ${0.133}{\lambda}$ and five turns. The ground plane measures ${1}{\lambda}\,{\times}\,{1}{\lambda}$. Figure 6(a) displays the Poynting streamline distribution in the x-z plane. Figure 6(b) shows the trajectory of a single Poynting streamline rotating along the helix antenna wire and then absorbed by the antenna load. The helix antenna rotates and bends the field energy from top to bottom along the wire, which is absorbed by the antenna load and demonstrates the working principle of the circularly polarized receiving antenna. This observation is similar to the Poynting streamline distribution near other wire antennas, where the field energy is conducted along the antenna wire to the load. The effective area shape in Figure 6(c) is approximately the shape of the square ground plane of the helix antenna with an angle of rotation.
Figure 6. A circular-polarized helix antenna. (a) The E-plane. (b) A single Poynting streamline. (c) The effective area shape.
After analyzing the dipole, monopole, loop, and helix antennas, a common characteristic of receiving wire antennas has been identified; wire antennas tend to make the Poynting streamline tangent to the wire and conduct it to the antenna load. This behavior can be observed in Figures 1(b) and 3, where the outer red Poynting streamlines are tangent to the antenna arms and conducted to the antenna load. A similar behavior is described in Figures 5 and 6(b) for the rotated Poynting streamlines along the loop wire. Therefore, it can be concluded that the wire used in receiving wire antennas helps to conduct the field energy along the wire to the antenna load.
The Poynting streamline distribution and the effective area shape of a slot dipole antenna are presented in Figure 7. The slot dimensions are ${0.525}{\lambda}$ in length and ${0.01}{\lambda}$ in width, while the feed point is located at ${(}{0},{0.166}{\lambda},{0}{)}$, and the square ground plane length is ${0.667}{\lambda}$. The effective area shape of the slot dipole antenna differs from that of the half-wave dipole antenna as it is more rectangular in shape. The concentrated red Poynting streamlines in both the E- and H-planes are shown in Figure 7(a) and (b), respectively, indicating that the energy is absorbed by the antenna load. The energy distribution of the slot antenna conforms to the complementary theorem; the Poynting streamline distribution in the E-plane is similar to that in the H-plane of the half-wave dipole antenna, and the Poynting streamline distribution in the H-plane is similar to that in the E-plane of the half-wave dipole antenna. The larger effective area shape of the slot dipole antenna is primarily due to the larger ground plane compared to the half-wave dipole antenna.
Figure 7. An X-polarized half-wave slot dipole antenna. (a) The 3D Poynting streamline distribution. (b) The E-plane. (c) The effective area shape.
Figure 8 displays the distribution of Poynting streamlines and the effective area shape of a ${\text{TE}}_{10}$-mode horn antenna, with an aperture size of ${2}{\lambda}\,{\times}\,{1}{\lambda}$. The effective area shape, as shown in Figure 8(b), is contained within the physical aperture of the horn antenna, indicating that the aperture efficiency of the horn antenna is less than one. The red Poynting streamlines in Figure 8(b) illustrate that energy is absorbed by the aperture of the horn antenna in the E-plane. The shape of the effective area of the horn antenna is similar to a rectangle, matching the geometry of the horn antenna aperture.
Figure 8. A Y-polarized horn antenna. (a) The 3D Poynting streamline distribution. (b) The effective area shape. (c) The E-plane. (d) The H-plane.
Figure 9 illustrates the Poynting streamline distribution and the effective area shape for a dielectric dipole antenna, where a dipole antenna is positioned at the center of a dielectric sphere. The dipole antenna has a length of ${0.247}{\lambda}$ and an arm width of ${0.017}{\lambda}$. The radius of the dielectric sphere is ${0.183}{\lambda}$, and the relative permittivity is four. The effective area shape in Figure 9(b) is close to an ellipse. Figure 9(a) and (b) depicts the distribution of Poynting streamlines in the E- and H-planes, respectively. In the E-plane, the Poynting streamlines are discontinuous at the boundary between the dielectric sphere and air, while they are continuous in the H-plane. Due to the high dielectric constant of the sphere, the effective wavelength is shortened by half, resulting in the length of the dielectric dipole antenna being approximately half that of a half-wave dipole antenna in free space.
Figure 9. A half-wave dipole antenna in a dielectric sphere with ${\epsilon}_{r} = {4}$. (a) The E-plane. (b) The H-plane. (c) The effective area shape.
Figure 10 illustrates the effective area shape and the Poynting streamline distribution of a square patch antenna. The square patch and ground plane have lengths of ${0.43}{\lambda}$ and ${0.86}{\lambda}$, respectively, and the substrate’s ${\epsilon}_{r}$ is one. Figure 10(a) and (b) displays the distribution of Poynting streamlines in the E- and H-planes, respectively. The red Poynting streamlines are concentrated and absorbed by the antenna feed via the left (+y) and right (−y) slots of the patch antenna, which agrees with the transmitting patch antenna theory that the same slots are the antenna’s effective radiating portion. The effective area shape in Figure 10(c) is similar to a square shape, matching the ground plane’s geometry, which suggests high aperture efficiency for the patch antenna.
Figure 10. A Y-polarized patch antenna. (a) The H-plane. (b) The E-plane. (c) The effective area shape.
Figure 11 illustrates the Poynting streamline distribution in the E-plane for the patch antenna with different ground planes. Solid and dashed red lines indicate the Poynting streamlines absorbed by the left and right slots, respectively. The effective receiving portion of the patch antenna is independent of the ground plane changes. Compared to Figure 10, the effective area shape reduces as the ground plane decreases. Figure 11(a) shows that when the ground plane’s dimension equals the patch’s dimension, the effective area decreases from ${0.61}\,{\lambda}^{2}$ in Figure 10 to ${0.36}{\lambda}^{2}$. In this case, more Poynting streamlines are absorbed by the left slot than the right slot. To increase the effective area, a possible way is to extend the ground plane on the right side to enhance the Poynting streamlines absorbed by the right slot. Figure 11(b) and (c) illustrates that the patch antenna’s effective area is more significant when the ground plane is extended on the right than on the left. Hence, the Poynting streamline method provides an intuitive guide to optimize the ground plane’s design for the patch antenna when the antenna’s dimension is limited.
Figure 11. A Y-polarized patch antenna over the different ground planes.
Figure 12 shows the Poynting streamline distribution and the effective area shape for a ${2}\,{\times}\,{2}$ array using the patch antenna elements in Figure 10 with ${0.86}{\lambda}$ element spacing. Figure 12(a) and (b) shows the Poynting streamline distribution in the E- and H-planes, where the red Poynting streamlines are absorbed by the antenna load through the left (+y) and right (−y) slots of the patch antenna. The effective area shape in Figure 12(c) is close to the size of the array aperture, indicating that a large array antenna can potentially achieve high aperture efficiency.
Figure 12. A Y-polarized 2 × 2 patch array antenna. (a) The E-plane. (b) The H-plane. (c) The effective area shape.
Figure 13(a) shows the Poynting streamline distribution and the effective area shape for a receiving Yagi-Uda antenna. The lengths of the reflector element, driven element, and four identical director elements are ${0.475}{\lambda}$, ${0.466}{\lambda}$, and${0.424}{\lambda}$, respectively. The distance between the reflector element and the driven element is ${0.2}{\lambda}$, and the distance between the director elements is ${0.308}{\lambda}$. The Poynting streamline distribution in the E-plane indicates that the reflector and director elements help to concentrate and guide the Poynting streamlines absorbed by the antenna load. The effective area shape in Figure 13 is close to an ellipse, and the area is much larger than that of a half-wave dipole antenna. Compared to the half-wave dipole antenna in Figure 13(b), the Yagi-Uda antenna absorbs more field energy, resulting in a larger effective area shape.
Figure 13. (a) A Y-polarized Yagi-Uda antenna. (b) A half-wave dipole antenna.
Figure 14 illustrates the Poynting streamline distribution and effective area shape of a spiral antenna. The length of the spiral antenna is ${2.21}{\lambda}$, and the number of turns is 1.5. The spiral antenna rotates and bends the field energy as it propagates toward the antenna. The effective area shape is similar to an ellipse, where the length of the effective area shape in the E-plane is greater than that in the H-plane. This pattern is similar to that of a y-polarized receiving dipole antenna with a length close to the effective diameter of the two arms of the spiral antenna. This observation is consistent with the theory of transmitting spiral antenna, which states that the antenna radiates the field energy using half the wavelength of the diameter of the two arms.
Figure 14. A Y-polarized spiral antenna. (a) The E-plane. (b) The H-plane. (c) The effective area shape.
During the spring semester of 2023, the author taught the Department of Electrical and Computer Engineering 4313/6313 Antennas course to senior undergraduate and graduate students at Mississippi State University. The course used Balanis’s Antenna Theory book [15] as the primary textbook. To enhance the comprehension of students, the Poynting streamline method was introduced and utilized as supplementary teaching materials toward the end of the course. Anonymous questionnaires were distributed to 20 students. The questionnaires utilized a one-to-five Likert scale for students to express their opinions, and the survey questions are available in Table 1. Based on the questionnaire results, it was found that more than 80% of the students strongly agreed that the Poynting streamline method improves their understanding of antenna theory and should be included in future antenna textbooks and recommended to other students taking antenna classes in the future. This indicates that the students found the Poynting streamline method to be highly valuable and worthy of incorporation into future courses.
Table 1. Survey questions and results.
This article employs the Poynting streamline method to investigate various receiving antennas, such as dipole antennas, loop antennas, helix antennas, small antennas, slot antennas, horn antennas, patch antennas, reflector antennas, dielectric antennas, bow-tie antennas, array antennas, log-periodic antennas, and spiral antennas. The method serves as a useful pedagogical tool by facilitating an intuitive visualization of the energy flow near the receiving antennas and the geometry of the antenna effective areas, thereby enhancing the comprehension of the receiving characteristics of these antennas.
Junming Diao (jdiao@ece.msstate.edu) is with Mississippi State University, Starkville, MI 39762 USA. He is a Member of IEEE.
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Digital Object Identifier 10.1109/MAP.2023.3304558
