Chapter 30: Finite Dipole Antennas and Loop Antennas Chapter Learning Objectives: After completing this chapter the student will be able to: Calculate the electric and magnetic fields in the far-field of a finite dipole antenna. Draw the radiation pattern for a finite dipole antenna. Calculate the electric and magnetic fields in the far-field of a small loop antenna. You can watch the video associated with this chapter at the following link: Historical Perspective: Hidetsugu Yagi (1886-1976) and Shintaro Uda (1896- 1976) were Japanese electrical engineers and professors at Tohoku University. They invented the Yagi-Uda antenna, which is often just called the Yagi antenna. Photo credit: https://commons.wikimedia.org/wiki/File:Hidetsugu_Yagi.jpg, [Public domain], via Wikimedia Commons. 1 30.1 Finite Dipole Antennas In the previous chapter, we introduced the infinitesimal dipole antenna. For such an infinitesimal antenna, we know that the length of the antenna is much less than the wavelength being transmitted (L << ). While that is a good place to begin our discussion of antennas, they are actually very inefficient at transmitting the signals applied to them. To obtain good transmission characteristics, we will need much larger antennas. The length of a finite dipole antenna can be a substantial fraction of the wavelength being transmitted, or it can even be greater than the wavelength. Quite often, the dipole will be one-half as long as the wavelength (a “half-wave dipole”), because it has some particularly desirable characteristics. Before we dive into the math, let’s take a moment to develop a conceptual understanding of how it works and why it transmits electromagnetic waves. Figure 30.1 shows a dipole antenna. It is connected to a transmission line that is ultimately being driven by a function generator or other signal source. + - + - + - + - + - + - Function Function Generator Generator - + - + - + - + - + - + Figure 30.1. A Finite Dipole Antenna As this figure illustrates, the antenna is being driven by a sinusoidal wave, which means that half the time the top portion of the antenna is at a higher voltage (i.e. has more positive charges), and the other half of the time, the top portion is at a lower voltage (i.e., has more negative charges). Because the antenna is not infinitesimally small, current is always flowing. In the first half of the cycle, current is flowing upward as positive charges move toward the top half of the antenna. Later, current flows downward as positive charges move toward the bottom half of the antenna. These currents, of course, create corresponding magnetic fields, which reverse direction every half-cycle as the current reverses direction. At the same time, an electric field are always pointing from the positive charges in one half of the antenna toward the negative charges in the other half. Again, this electric field reverses direction every twice per cycle. 2 Figure 30.2 shows an updated version of Figure 30.1 that includes this current, electric field, and magnetic fields. If we calculate the Poynting vector for each case, we will see that it is pointing away from the antenna at all times, meaning that energy is flowing (radiating) away from the antenna during the full cycle of the sinusoidal input. I E + - + H S=E x H - H S=E x H + H S=E x H - H + - S=E x H + H S=E x H - H S=E x H + - Function H S=E x H Function H S=E x H Generator H S=E x H Generator H S=E x H - + H S=E x H + H S=E x H - + - H S=E x H + H S=E x H - + - H S=E x H H S=E x H + - E I Figure 30.2. Radiation Flows Away from a Finite Dipole Antenna Another representation of this conceptual explanation can be seen in Figure 30.3: Figure 30.3. Representation of a Dipole Antenna [Public domain], via Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Dipole_receiving_antenna_animation_6_800x394x150ms.gif, Of course, the mathematical description of the finite dipole is a bit more complicated than this qualitative description. In fact, parts of the derivation can only be solved using computer models, iterative solutions, and integro-differential equations. What we find is that the current on the antenna approaches zero toward the two ends of the antenna, it is symmetric, and it has a maximum value at the center (splitting point) of the antenna. This is shown schematically in Figure 30.3. 3 r’ ’ L z r Flow Current Figure 30.3. Representation of the Current Profile on a Finite Dipole Antenna This figure also shows that we will refer to the total length of the antenna as L, and we can identify any point along the antenna as z, the distance above the midpoint. The value of z will be positive for the top half of the antenna and negative for the bottom half. This figure also shows the observation point, which is a distance r away from the center of the antenna at an angle of . It is also a distance r’ away from an arbitrary point along the antenna and at an angle of ’. Although the current profile shown in Figure 30.3 is actually a very complicated mathematical function, it can be very well approximated by Equation 30.1. (Equation 30.1) We will approximate the finite dipole antenna as a superposition of infinitesimal dipoles, each of which carries the current defined by Equation 30.1. Modifying Equation 29.40 to account for a differential current, we find: (Equation 30.2) A geometric analysis of Figure 30.3 provides the observation that, in the far field (where r >> ): (Equation 30.3) This is not an important distinction for the r’ term in the denominator of Equation 30.2, since in the far field, the difference between r’ and r will be negligible. However, in the e-jkr’ term, even a small difference in phase can lead to dramatic changes in the constructive or destructive 4 interference between the terms. Therefore, we will substitute Equation 30.3 into the exponential term of Equation 30.2, but we will approximate r’ by r in the denominator. We will also substitute Equation 30.1 into the I(z) factor in Equation 30.2. (Equation 30.4) The exponential term can be separated into two parts—one that depends on z and one that does not. We will integrate and separate the exponential term in this way, moving all terms that do not depend on z outside of the integral. (Equation 30.5) This integral turns out to be very difficult to solve, but its solution yields a critically important observation. We will omit the solution (which would require many pages), but we will discuss the observation. It turns out that when an antenna with a known radiation pattern is replicated many times at regular intervals, the radiation pattern of the resulting antenna system has two factors: one that depends on the individual antenna (known as the “element factor”), and one that depends on the structure of the replication (known as the “array factor.”) For an infinitesimal dipole antenna, the element factor is simply a sine function: (Equation 30.6) For a linear array of infinitesimal dipoles (from which we will “construct” a finite dipole), the array factor is: (Equation 30.7) The total electric field in the far field of the finite dipole antenna can be written as the product of the element factor, the array factor, and a few other constants and terms that account for the effect of the distance r between the center of the antenna and the observation point: (Equation 30.8) 5 We can combine these three equations to write: (Equation 30.9) This equation can finally be simplified as follows: (Equation 30.10) This function can be plotted on a polar axis as shown in Figure 30.4. This figure also serves as a link to an animation that shows the radiation pattern for values of L ranging from 0.1 to 3. Lobe Null Null Lobe Figure 30.4. Radiation Pattern of a Finite Dipole Antenna with L=3 Example 30.1: Determine the magnitude of the electric field when a 100MHz wave is transmitted from a one-meter dipole antenna to a receiver 20m away at an angle of 30° from the axis of the antenna. The amplitude of the current entering the antenna is 100mA. 6 The magnetic field that corresponds to the electric field in Equation 30.10 can be calculated from Equation 30.11: (Equation 30.11) Substituting Equation 30.10 into Equation 30.11 gives: (Equation 30.12) Example 30.2: What is the magnetic field that corresponds to the electric field in Example 30.1? As we first learned in chapter 18, we can calculate the average Poynting vector for a sinusoidal system as follows: (Copy of Equation 18.18) Substituting Equations 30.10 and 30.12 into Equation 18.18 and simplifying the result gives: (Equation 30.13) 7 Figure 30.5 and the accompanying animation shows a radiation pattern of the average Poynting vector for a finite dipole antenna. Figure 30.5. Average Poynting Vector for a Finite Dipole antenna [Public domain], via Wikimedia Commons: https://upload.wikimedia.org/wikipedia/commons/a/a6/Dipole_xmting_antenna_animation_4_408x318x150ms.gif Example 30.3: What is the average Poynting vector that corresponds to the electric field in Example 30.1? 30.2 Loop Antennas The second fundamental type of antenna is a loop of wire with current flowing around it. Of course, there will need to be a break somewhere along the loop so that the current can enter and exit the loop, but that break is typically so small that it is negligible.