Chapter 32: Antenna Arrays Chapter Learning Objectives: After completing this chapter the student will be able to: Determine the radiation pattern for an antenna array consisting of two elements. Determine the radiation pattern for a phased array of arbitrary size. Determine the radiation pattern for a binomial array of arbitrary size. Determine the directivity of an antenna array. You can watch the video associated with this chapter at the following link: Historical Perspective: Jill Tarter (1944-present) is an American astronomer who uses arrays of radio antennas to search for extraterrestrial life. She is the former director of the Center for SETI Research and was the inspiration for the main character in the movie Contact. Photo credit: https://upload.wikimedia.org/wikipedia/commons/3/3b/Jill_Tarter_at_Starmus_IV_Trondheim_2017.jpg, [CC by SA 2.0], via Wikimedia Commons. 1 32.1 Introduction We now understand enough about antennas to know that we will want to carefully design our antenna to have a desirable radiation pattern for our particular application. This can be measured by the half-power beam width, directivity, gain, and effective area. The problem is that we don’t actually have very many things to change on a typical antenna. For a finite dipole, it is just one—the length. But we need more control than that to custom design the radiation pattern for more advanced applications. An antenna array is a group of similar antennas that are spaced in a regular manner, typically in a line or a grid. It turns out that constructive and destructive interference of the radiation from the individual antennas can create quite desirable radiation patterns for the overall array. In addition, we now have several things we can adjust: The number of antennas The placement of the antennas The amplitude of the signal applied to each antenna The phase of the signal applied to each antenna We will consider two main types of antenna arrays: phased arrays and binomial arrays (which might be better called amplitude arrays). But first we will study the simplest case: an array of two identical antennas. 32.2 Two Antennas The radiation coming from an antenna has both an amplitude and a phase angle. When two such signals interact with each other, they can constructively interfere if the signals are “in phase,” or they can destructively interfere if the signals are “out of phase.” Figure 32.1 illustrates a situation with two antennas whose signals are interacting at an observation point P. P r1 r2 r q d Figure 32.1. The Two-Element Array 2 It is clear from this figure that the path length r1 will be greater than the path length r2 for the configuration shown. In general, the path lengths will be different unless q=90°. Our task is to figure out the difference in the path lengths and determine whether the waves will interfere constructively or destructively. If we assume that the observation point P is in the far field (very far away from the antenna array), then the dashed lines in Figure 32.1 will be approximately parallel. The two paths r1 and r2 will come to resemble Figure 32.2. r r 1 2 q q d Figure 32.2. The Two-Element Array with Observation Point in the Far Field Studying the bottom left corner of this figure, we can see that the right triangle that is formed has a hypotenuse of d, meaning that the side adjacent to q is dcosq as shown. Geometrically, this is the difference in path length between r1 and r2. If we then multiply this path length difference by the wave number, we will find the phase difference that is created by the path length difference: (Equation 32.1) Of course, it is also possible that we could intentionally introduce a phase difference between the signal applied to the two antennas. We will call such a phase difference . Combining this intentional phase difference with Equation 32.2 gives the total phase difference between the signals from the two antennas when they reach the observation point in the far field: (Equation 32.2) We will arbitrarily say that the signal from one antenna has a phase angle of +/2 while the other antenna has a phase angle of –/2. (This decision has no impact on the final answer other than an overall phase factor, but it makes the math much easier.) We can then write the sum of the electric fields from these antennas as: (Equation 32.3) 3 If we assume that the two antennas are identical, this equation can be rewritten as: (Equation 32.4) Factoring a 2 out of the parentheses gives: (Equation 32.5) The term in parentheses is now in the form of a cosine, so we can write: (Equation 32.6) Substituting Equation 32.2 into this expression gives: (Equation 32.7) Notice that what has essentially happened is that we have multiplied the electric field of a single antenna (E1(r)) by a new function to obtain the electric field of a two-element array. This is a particular example of the pattern multiplication rule, which says that the radiation pattern of an antenna array will be the product of the radiation pattern of a single antenna, the “element factor” F1(q,), multiplied by an “array factor” Fa(q,) that corresponds to the pattern that constitutes the array: (Equation 32.8) From inspection of Equation 32.7, it is apparent that the array factor for a two-element array spaced a distance d apart with an intentional phase difference of is: (Equation 32.8) Since we typically normalize the radiation pattern to remove any constants multiplying it, we will omit the “2” from this function, giving the final array factor: (Equation 32.9) 4 Example 32.1: What is the radiation pattern for a pair of isotropic antennas with no applied phase difference? The radiation pattern for a two-element array of isotropic antennas, derived in Example 32.1, gives very interesting behavior as the distance d between the antennas increases. The Figure 32.2 shows the radiation pattern for d=10cm with a 1.5GHz wave. Clicking on this figure will also show an animation of this pattern from d=0 to d=50cm. Figure 32.3. Radiation Pattern of the Two-Element Array with f=1.5GHz and d=10cm. Already, we can see that antenna arrays will give us interesting opportunities, even if they only contain two antennas. Let’s look at what happens if we include more than two. 32.3 Phased Arrays A “phased array” is simply a linear array of antennas, all equally spaced at a distance d, with the same amplitude applied to each antenna. Sometimes, the phase is also the same, but sometimes we will introduce an intentional phase difference to each antenna. Essentially, we are generalizing the two-element array to have more than two elements. 5 P r6 r5 r4 r3 r2 r1 q q q q q q 6 5 4 3 2 1 d d d d d Figure 32.4. A Phased Array with Six Elements (N=6) Again, if the observation point P is in the far field, then all of the angles q1-q6 become approximately equal as the lines become approximately parallel: q q q q q q d d d d d Figure 32.5. A Phased Array with Six Elements Observed in the Far Field Notice, then, that we can apply our “dcosq” trick from the two-element array to each of the six antennas. This means that each antenna will have a path length that is dcosq longer than the element to its right. There is also the possibility once again that we will intentionally introduce a phase shift to each antenna signal, meaning that the phase difference at the observation point of each antenna compared to the one to its right is the same as it was for the two-element array: (Equation 32.10) Remember that each element of the array will have an additional phase shift, so they add up as you move from antenna #1 (on the far right) to antenna #6 (on the far left). If each antenna has an element factor of E1(q,), we can write the total electric field at the observation point as: 6 (Equation 32.11) If we generalize to the case of an array with N elements, this becomes: (Equation 32.12) Mathematicians have proven that: (Equation 32.13) Applying this to Equation 32.12 where q=ej, we can find a closed form of this series: (Equation 32.14) If we consider only the magnitude of the electric field, this becomes: (Equation 32.15) By inspection, we can see that the array factor for a phased array is: (Equation 32.16) Bringing in Equation 32.10, this can in turn be written as: (Equation 32.17) We can also determine (using l’Hopital’s rule) that the maximum value of this function will occur when =0. This also makes intuitive sense, because if =0, all of the signals will arrive in phase, and they will add constructively. To achieve this goal, we must introduce a negative 7 phase shift to each antenna that cancels out the positive phase shift that comes from the path length difference: (Equation 32.18) (Equation 32.19) In this case, all the signals will arrive at the observation point perfectly in phase, meaning they simply add magnitudes: (Equation 32.20) Example 32.2: Determine the radiation pattern for a phased array with two half- wavelength electric dipoles separated by a distance d.