Finite Dipole Antennas and Loop Antennas

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.

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.

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.

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)

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.

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)

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. You may recall that we have already studied loops of current and the magnetic fields they create back in section 12.4, leading to the definition of the magnetic dipole moment:

(Copy of Equation 12.12)

Thus, although we call it a loop, we could also call it a magnetic dipole, which is very similar in some ways to the electric dipole we have been discussing thus far in this chapter.

Figure 30.6 shows the geometry associated with a loop antenna.

H S z E

r r’

 y f a Idl’

x Figure 30.6. Geometry of a Loop Antenna

Loop antennas can be classified as small loops (similar to infinitesimal dipoles) in which the circumference of the loop is much less than the wavelength being transmitted:

(Equation 30.14)

If the loop is small and the observation point is in the far field, then the magnetic field can be determined to be:

(Equation 30.15)

Similarly, the electric field can be shown to be:

(Equation 30.16)

Following Equation 18.18, we can calculate the average Poynting vector for a small loop in the far field to be:

(Equation 30.17)

This can be simplified to:

(Equation 30.18)

It is worth noting two things about this equation: 1. It is very, very similar to Equation 29.42 for the infinitesimal dipole antenna. Both the small loop and the infinitesimal dipole have a 1/r dependence for both E and H, leading to a 1/r2 dependence for S. Both also have a sin dependence for E and H, leading to a sin2 dependence for S. 2. While the infinitesimal dipole has a lobe (maximum strength) in the direction perpendicular to the antenna, the small loop has a maximum in the plane of the antenna. (Notice that the maximum of Equation 30.18 will occur when q=90°, and Figure 30.6 shows this will be in the antenna’s plane.) The small loop actually has a null (zero signal strength) perpendicular to the plane of the loop.

Example 30.4: A loop antenna with a radius of 6 cm has a sinusoidal current of amplitude 200mA and frequency of 10MHz applied to it. What is the average Poynting vector at a distance of 10m at an angle of 45° away from the normal vector of the loop?

30.3 Yagi Antennas

A Yagi-Uda antenna, more commonly just known as a Yagi antenna, consists of multiple parallel linear antenna elements (finite dipole antennas). Often, these are each one-half wavelength, but sometimes they are of variable length. Yagi antennas are widely used as both transmitting antennas and receiving antennas. In addition to the primary dipole elements, Yagi antennas also include reflectors and directors, which are designed to modify the antenna’s radiation pattern, increasing its directivity and gain.

Figure 30.7. Drawing of a Yagi Antenna [Public domain], via Wikimedia Commons: https://upload.wikimedia.org/wikipedia/commons/7/74/Yagi_TV_antenna_1954.png

The operation of a Yagi antenna is more complicated than a simple dipole antenna, but the gain and directivity are correspondingly better. Its operation relies on constructive interference of the signals from the dipoles, the reflector, and the director elements, as shown in Figure 30.8.

Figure 30.8. Constructive Interference in a Yagi Antenna [Public domain], via Wikimedia Commons: https://upload.wikimedia.org/wikipedia/commons/3/36/Yagi_antenna_animation_16_frame_1.6s.gif

30.4 Complementarity of Antennas

Before wrapping up this chapter, we should briefly discuss the concept of complementarity. Many electrical components are complementary to each other, meaning that they can serve one of two roles. For example, an electric motor converts electrical energy into rotational energy. But if you took that same motor and, rather than applying a voltage to it, you forced its rotor to turn, it would generate a voltage at its terminals. Thus, a motor and a generator are really the same physical structure. (There are some engineering optimizations we would make if we know that a particular device was going to be used as a motor or as a generator, but fundamentally, the physics is the same.) Similarly, capacitors and inductors are complementary of each other—one converts a time-dependent voltage into a current, and the other converts a time-dependent current into a voltage.

It turns out that antennas are probably the best example of complementarity in all of electrical and computer engineering. If you design an antenna to be an excellent transmitter, it will also serve very well as an excellent receiver as well. Yes, there are some optimizations we would make for one or the other, but the physics is the same. We have been thinking and talking about antennas as transmitters until now, but the symmetry of Maxwell’s equations makes it entirely possible for those transmitters to also serve as receivers. For example, the loop antenna, which we have just discussed, uses variable current along the wire to generate magnetic fields and the accompanying electric fields, but if that same loop antenna is subject to time-varying magnetic flux passing through its loop, Faraday’s Law says that it will generate a voltage across its terminals.

As we conclude our discussion of antennas over the next two chapters, this complementarity of transmitters and receivers will become more evident. In fact, we will find that several other characteristics of antennas, such as gain and directivity, will also be complementary.

30.5 Summary

 When an antenna with a known radiation pattern is duplicated in a regular array, the radiation pattern of the resulting antenna system will be the product of an element factor, caused by the single antenna, and an array factor, caused by the pattern. For a linear array of infinitesimal dipoles that are used to model a finite dipole, the element factor and array factor are:

 Taking the product of these terms along with a handful of constants that represent the dependence on distance from the source, the finite dipole has the following far-field electric field, magnetic field, and average Poynting vector:

 The finite dipole antenna is a much better transmitter (and receiver) than the infinitesimal dipole. It also has a relatively complicated radiation pattern depending on the length of the antenna relative to the wavelength of the signal being transmitted.  A small loop antenna has the following electric field, magnetic field, and average Poynting vector:

 The small loop antenna has a radiation pattern identical to that of the infinitesimal dipole, although with different constants. Also, the small loop has maximum radiation emitted within the plane of the loop, whereas the infinitesimal dipole has maximum transmission perpendicular to the antenna.  The Yagi antenna uses multiple dipoles, reflectors, and directors to create constructive interference, which significantly improves the performance of these antennas.  Antennas are highly complementary, meaning that an antenna designed to operate as a transmitter will also work well as a receiver, and vice versa.