An antenna should transfer power efficiently. That means that its impedance should match that of its connecting transmission line. The transmission line should transfer all of its power to the antenna and not radiate energy itself. This means that the mode of the transmission line should be matched to mode of the antenna. Often one wants the antenna to radiate in a specified direction or directions. This is accomplished by designing it to have the proper radiation pattern. Closely related to this is the antenna polarization. Many times antennas are arranged in arrays in order to achieve the desired pattern. These arrays may then be electronically steered. A passive antenna, that is one with no amplifiers attached, will have the same characteristics whether it is transmitting or receiving.

This thin antenna will have an impedance of about 72 ohms at its first series resonance and tens of thousands of ohms at its parallel resonances. It is a high Q antenna with a very narrow bandwidth. As the antenna increases in diameter, the values of the reactances decrease, and the radiation resistance rises for the series resonance and decreases for the parallel resonance. As the Q becomes lower and the bandwidth becomes wider. A typical dipole antenna will now have a resonate impedance of closer to 100 ohms. In most cases, one wants as low of a Q and as broad a bandwidth as possible. That means that it is best to make the dipole as fat as is possible. Another advantage to the fat dipole is that it resonates at a somewhat lower frequency and can be cut shorter. So far, only the center fed dipole has been considered. It is possible to change the impedance by adjusting the location of the feed point and this is sometimes done to achieve a good match to the transmission line.

Transmission lines have currents that flows in two directions the vector sum of which should be zero. If the sum is not zero, there exists what is known as a common mode and this common mode will radiate and distort the pattern of the antenna connected to it. This common mode is usually introduced in two ways, one by coupling to the energy radiating from the antenna and secondly from a mismatch of modes which occurs when a balanced antenna is connected to a unbalanced transmission line or visa versa. A dipole antenna is usually a balanced antenna where as a whip is an unbalanced antenna. Twin lead transmission line is balanced while coaxial cable is unbalanced. If you connect a coaxial transmission line to a balanced dipole, the following will happen. The current on the center conductor of the coax will flow out to one arm of the dipole. The current on the outer conductor will flow out to the other arm of the dipole, and it will also flow down the outside of the shield exciting a common mode on the cable which will also radiate. This problem can often be relieved by the use of an interface called a balun (BAlance-UNbalance).

So far, consideration has been given only to transmission lines that are matched at both ends. While this is desirable in most cases, there are some situations where transmission lines that are not matched are useful. As one moves away from a mismatched termination the impedance varies as one moves along the transmission line, however in no case will it ever be equal to the characteristic impedance of the transmission line. The impedance will vary in value from that of the termination to that given by

Let us now look at these concepts in detail. As mentioned above, the larger the antenna the more structured the pattern and the following is usually a good rule of thumb. For each wavelength in dimension that the antenna spans there will be 4 lobes in one 360 degree pattern cut. As an example, the 10 wavelength antenna has about 40 lobes in its pattern. Many times two additional nulls are added to the pattern due the nulls off the end of a dipole. At the antiresonance frequencies, however, some of these lobes may lay on top of each other and there may appear to be only half the number of expected lobes. These lobes will be closest together in a direction broadside to the longest dimension of the antenna where the separation between lobes, in degrees, will be about 60/D, where D is the dimension of the antenna in wavelengths. At an angle of about 45 degrees from the broadside the lobes are about 90/D degrees apart. Do not try to use these algorithms near the endfire of an array. The beamwidth of an antenna is a measure of the directivity of an antenna and is usually defined by the angles where the pattern drops to one half of its peak value or known as the 3db points. If the antenna is uniformly excited, so that there is a uniform distribution of current over it, this beam width is about 50/D degrees. The next lobe in the pattern, usually called the first sidelobe, will be about 1/20 of the value of the main lobe and any others will be of even lesser value. Antennas that are designed to suppress the sidelobes will have a beam width about twice as wide as that of the uniformly excited array. When an antenna is designed to focus its energy in a given direction, the energy radiated in that direction is more intense than if it were an omnidirectional antenna. The ration between these values is called the antenna gain. This can be approximated by the following formula

where G is the power gain of the antenna, BWh is the horizontal beamwidth of the antenna and BWv is the vertical beamwidth of the antenna. As an example, consider an antenna that had a vertical beam width of 27 degrees and a horizontal beam width of 10 degrees; it will have a power gain of 100 or 20 db. This would also have a vertical dimension of about 2 wavelengths and a horizontal dimension of about 5 wavelengths if the antenna is uniformly excited.

A closely related parameter is the impedance of a wave; this is the ratio of E/H and for free space is close to 377 ohms. This is not to be confused with the radiation resistance of an antenna; it’s just that they have the same units. If a propagating radio wave encounters a medium of a different impedance, part of the wave is reflected, much like the reflections at a discontinuity in a transmission line. The remaining energy of the wave that passes through the discontinuity is refracted in a different direction of propagation, just like the distortion one sees as a light beam passes through water. The reflection and refraction properties often depend upon the polarization of the EM wave.

Lets look at how the pattern of an array of antennas can be steered electronically and at the same time we can see how we can change the shape of the array to remain within given physical constraints and still have a pattern reasonably close to what we want. Figure 6 shows two small dipole antennas that are one wavelength apart. We will call the boresite direction as being broadside to a line that connects the two antennas. If each antenna is excited with an identical signal, the waves from each will be in phase along the boresite. The two waves add and the energy is summed in the direction of the boresite. See figure 7. If we look thirty degrees to either side of the boresite, the wave coming from one antenna is delayed behind the wave coming from the other antenna because it has to travel one half wavelength further. This means that the two waves will subtract in this direction giving us a null in the pattern. If we continue around until we are looking down the endfire direction of the array, we see that one wave has to travel a full wavelength further than the other and they will again be in phase. Here the waves will add again and give us another peak in the pattern. The pattern resulting from such an array of omnidirectional elements is referred to as the array factor. It is also important to note that the null is closer to the broadside than to the endfire. This results in narrower lobes which will be closer together in the broadside direction than in the endfire direction. This is because as one rotates the array, the radial distances to the elements changes faster when you are observing from the broadside than from the endfire. Now, lets excite these two elements 180 degrees out of phase or rotating the array by 30 degrees. This will be the same as having one element displaced by one half wavelength. Now the waves will be out of phase along the boresite and the endfire directions producing nulls in the patterns in those directions and they will be in phase at 30 degrees off of the boresite producing peaks in those directions. See figure 8. You can steer the beams by changing the relative phase of the signals exciting the array elements. The above example demonstrates that the pattern of an array of antennas is the product of the pattern of a single element, called the element factor, times the pattern generated by having an array of elements called the array factor. This example also introduces another concept called grading lobes. If the array elements are spaced more than 1/2 wavelength apart other lobes, identical to the main lobe will start to appear in the array factor. In the above example they would be the lobes off the ends of the array.

Let’s look at some arrays that we may encounter. The most straight forward arrays are a linear array, made up by a straight row of elements. Let's look at some examples of this. figure 9 shows a half wavelength dipole vertically orientated. The colors represent the magnitude of current distribution red for the maximum current levels graded to blue for lesser values. The radiation pattern is shown in Figure 10 with the colors representing the magnitude of the radiated field as shown by the color bar, Figure 11 shows an array of two half wavelength dipoles stacked vertically on one wavelength centers and the resulting pattern is shown in Figure 12. The pattern has a narrower beamwidth in the vertical direction. In Figure 13 this array is extended to 4 elements with the pattern shown in Figure 14 and the pattern is narrower still. Now, lets suppose that our array is on a mountain top and we want to scan the beam down. Figure 14a illustrates an extreme example where the elements are progressively phased 90 degrees down the array which scans the main lobe down 15 degrees. Now we see another problem creeping in, grading lobes, because the elements are spaced further than a half wavelength apart. Previously they presented no problem because they were in the null of the element pattern of the individual dipoles. Now they too have scanned and you can see one starting to scan into view about 45 degrees from the zenith, but the element factor is still suppressing it to a level less than the main beam. If we continue to scan the main beam down by phasing each element progressively by 180 degrees, the main beam is now down to 30 degrees below the horizontal as shown in Figure 14b, but look what happened to that grading lobe. It is now down well into the element factor and has an amplitude comparable to the main beam. Another rule of thumb is also demonstrated here. For elements spaced one wavelength apart, the beam is scanned roughly one degree for every 6 degrees of phase shift in adjacent elements. This number is also roughly proportional to the spacing of the elements.

The Yagi antenna is an example of a fairly high gain array where most of the elements are fed parasitically from one or more driven elements. This is a relatively inexpensive antenna as the feed network is fairly simple but dimensional adjustments may be critical. The phase in the parasitic elements, which is what controls the array factor, is controlled by adjusting the element length and spacing. It is this combination of adjustment parameters is rather critical and the bandwidth of a Yagi antenna is usually only a few percent but it does have a fairly high gain considering its electrical size. Figure 15 show a three element Yagi antenna and its relative current distribution. It has a driven element which is half a wavelength long, a parasitic reflecting element which is a little longer and a parasitic director which is shorter than the driven element. The radiation pattern is shown in figure 16.

Now lets's develop an array with thin wires that are endfed. Figure 17 shows the current distribution along a wire that is two wavelengths long and end fed and Figure 18 shows its radiation pattern with large lobe about 30 degrees of the end of the wires and the larger one in a direction away from the fed end of the wire. Now, lets build an array using four wire connected to form a rhombic with a vertex of 60 degrees so that the lobes of the individual wires add up along the long axis of the rhombic. This is shown in Figure 19; note that the current distribution shows moding due to reflections off the end of the array just as with an electrically long dipole. The pattern is shown in Figure 20; note the large lobes to the back end of the array. If we terminate the end of the antenna with a matching impedance, just like on a transmission line, the current reflections and modes will be eliminated as well as the large lobes to the rear. A resistive load of 300 ohms was use in this case. Figure 21 shows the current distribution without reflections and Figure 22 shows the radiation pattern with the backlobes removed. A rhombic antenna is usually supported on poles above the ground and the resulting reflections can cause a serious multipath problem. If the ground is considered a perfect conductor, the reflections can be accounted for by creating an identical image at a distance below the ground equal to the distance the antenna is above the ground as shown in Figure 23. The radiation pattern is shown in Figure 24 which is ideal for launching a sky wave at HF frequencies. The lobe depicted going into the ground really does not exist because array theory is only valid for the half space above the reflecting surface.

An example of a broadband array is the log periodic antenna. This is an array of closely spaced elements, each 180 degrees out of phase with the next and the length of each and the spacing changing proportional to its distance from an apex This forms a pattern by subtraction rather than by addition. Figure 25 is a typical log periodic antenna. Only a few of the elements, which are near resonance, will radiate at a given frequency, and this can be seen in the computed model of this antenna in figure 26. The alternate phasing of the elements is necessary to force the array to radiate through the electrically shorter elements which will perturb the pattern less than the longer ones would. The feed is usually a parallel transmission line which also serves as the antenna boom. The combination of element size, element spacing and transmission line geometry is rather complex. A log periodic array has a typical gain of 7-9 db and patterns that are about 60 degrees by 120 degrees wide. The computed 3D pattern is shown in figure 27.

Suppose now that we design an antenna array that is on a flat surface, but we must be physically constrained to follow a curved surface such as on an aircraft. Figure 28 shows 5 elements as if they were mounted on flat surface and also as if they were on a cylindrical surface with a radius of 3 meters. In order to correct the radiation pattern for the effects of the curved surface, we advance the phase of the excitation in direct proportion to the elements distance from the strait line. Figure 29 shows the pattern of the array with the elements mounted on a flat surface and the same voltage applied to each element at a frequency of 300 MHz. Figure 30 shows the resultant distortion caused by mounting them on the cylindrical surface with the same excitation. If the phase is delayed to account for the displacement by the cylindrical surface we get the pattern in Figure 31 which is reasonably close to the original pattern, in the foward direction.

Extending this further, a planer array that is a two dimensional array of elements lying in a plane. This is a way of achieving a relatively high gain using fairly simple elements. These arrays have the advantage that you can steer the beam along its axis and also have control of the shape of the beam and its side lobes. The two dimensional array can scan and shape the beam in two directions. Actually we can extend this concept to three dimensions by making each element of a planer array an array but care must be taken that the element pattern is wide enough to cover the range of azimuth and elevation that the beam is to be steered over. These arrays usually have a rather complex and costly feed network.

The examples of array theory have been greatly simplified here, but the purpose was to develop the concept of array factor and the fact that you can steer and shape an antenna radiation pattern by varying the excitation on the array elements. We can also broadband an array, change its shape somewhat and even feed elements parasitically.

Be sure to bookmark this page as its development will probably continue for some time. As time becomes available, I will expand on these concepts and create missing and additional figures. If you see any errors, have any comments or if there are additional topics that you would like to see covered, be sure to drop me a line. If there is anything on this page that is not explained clearly enough, please drop me a line with the specifics. If you would be interested in helping me author this page, I would also like to hear from you. Please be patient with me, but give me a kick once in a while to make sure I get this task completed.

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