Phased Array Adjustment for Ham Radio

Grant Bingeman, P.E. KM5KG

__Introduction__

This article presents a lumped-parameter tee-network model of the coupling between radiating elements in a phased array in order to provide a physical and
intuitive understanding of phased array operation and adjustment interactions. It then analyzes the performance of a practical phased array antenna which
appeared in a QST magazine construction article.^{1 } The technique of connecting mismatched feeders to the driven elements in a phased array is an expedient
method for obtaining relative current phases in the elements that approximate the desired currents. A simple technique to provide instant pattern adjustability
with a variable capacitor is discussed later in the article. This practical technique helps to compensate for the coarse method of using mismatched transmission
lines to provide phasing.

__Basic Theory__

A phased array is an antenna having more than one driven element. The desired radiation pattern is obtained by adjusting the current magnitude and phase in each
element. Ideally, we want to be able to *independently* adjust the current magnitude and phase in each element and also obtain a perfect impedance match to the
transmission lines carrying power to each element. It is nearly impossible to do this by simply fanning out different length transmission lines from the common
feed point to the driven elements. However, a compromise can be reached, which is the main thrust of this article.

An antenna design is begun by defining a set of desired performance features. This set may include gain, beamwidth, departure angle, polarization, front-to-back ratio, front-to-side ratio, impedance and pattern bandwidths, component stress (current, voltage, compression, tension, shear), overall size and weight, feed-point impedance, etc. Since many of the desired features interact and some may be improved only at the expense of others, priorities must also be assigned to these performance criteria. In other words you must weight your criteria, and set upper and lower boundaries. Like most things in life, antenna design is a set of compromises.

In order to efficiently optimize your design, it certainly helps to understand how the antenna performance features interact on both the quantitative and qualitative levels. For example, it is generally true that there is a trade-off between gain and bandwidth. Sometimes a slight improvement in gain is paid for by a major degradation in bandwidth. Therefore you have to find a balance, and this depends on the relative importance of each of the antenna's performance criteria. Obviously if we are operating CW at a single frequency, we don't care too much about bandwidth. If we are only operating in the SSB portion of the 20 meter band, then we only care about antenna performance in the upper half of the band. And so on.

Optimizing algorithms have been around for a long time, and can readily be applied to antenna design problems. A formula is written to create an error sum based on each of the weighted antenna performance criteria. Various parameters in the antenna are adjusted until a minimum error is found. This iterative process is both an art and a science, and has a jargon all its own. There are brute-force methods that may take a long time and never reach the best solution, and there are "intelligent" algorithms that employ advanced mathematical concepts, and converge quickly. Digital computers love these kinds of tasks.

Since the outcome of such optimization is dependent on the number of parameters that can be varied, the more the better (usually). Too many knobs can get a human operator into deep water, whereas a computer program seldom complains about these iterative exercises. Typical high-gain amateur radio antennas consist of one driven element and many parasitic elements. The input parameters to your typical parasitic antenna design would therefore include element length, diameter and spacing, height above ground, and perhaps some transmission lines and other discrete components. If you also introduce multiple RF sources, such that both the source current magnitudes and phases are added to the adjustable parameter list, then you increase the possibility of finding a better compromise between all the desired performance features. In other words, for a given field intensity maybe you can get twice the bandwidth from an array of two driven elements compared to an array of one driven and one parasitic element. Or perhaps you can get a certain gain from a driven array that requires only half the volume of a conventional parasitic array with the same gain.

And so this article will explore the physical inter-actions and rules-of-thumb that apply when you are designing an antenna having more than one driven element. Admittedly the complexity of the problem increases exponentially as we add more knobs to the "box," but a human operator will eventually develop a feel for the problem by simply turning the cranks and observing what happens. Since the 1960's, the "box" has often consisted simply of a method-of-moments antenna analysis program. Some general rules will appear as you play with the box, and if you read this article carefully, you will find some useful short-cuts to the antenna design process. Rule number one: write everything down in a notebook so you don't end up repeating yourself, and enter enough detail so you can readily verify your results later. Define a simple set of goals and try to stick by them.

__A Starting Point__

The behavior of a single isolated antenna element, such as a dipole in free space, is a simplified or reduced model of an actual antenna. In reality the behavior of a dipole is influenced by the presence of the earth, weather, support structures, transmission line, insulators, etc. Often these influences are minor, but when an additional radiating element is deliberately added to an antenna, major changes occur in the first element. When both elements are physically aligned and within a wavelength or so of each other, they are said to be strongly coupled.

We can loosely tie everything together by observing the current in the antenna elements, and remembering Ohm's Law. The electric field intensity produced by an
antenna is directly proportional to the currents in that antenna. For a given power input to an antenna, high-gain multi-element wire antennas have relatively high
currents compared to single-element antennas of comparable dimensions. This higher gain is manifested as a decrease in the input resistance to the antenna. For
example, assume you have a 144 MHz half-wave dipole fed at its center. You measure the input impedance and find it to be about 70 + j0 ohms. Then you add a
second antenna element consisting simply of a half-wavelength of wire in parallel with your driven element. As you move this wire closer to your dipole, you will
see the input resistance to the dipole drop. For a spacing of 30 cm, the input impedance becomes about 25 + j25 ohms and the field intensity increases by about
3.6 dB in one direction compared to the dipole alone, and decreases in the opposite direction. The parasitic wire acts as a reflector at this spacing, but at a much
closer spacing it can act as a director and produce even more gain, albeit in the opposite direction. For a spacing of 10 cm, the input impedance is about 4 - j20
ohms, and the gain is about 4.0 dB. However, since the currents are much higher and the input resistance much lower for this close spacing, you can lose quite a
bit of your input power in the form of i^{2}R losses, depending on the resistance of the wires. I assumed copper for the gain figures cited; aluminum would yield a
gain of perhaps 3.9 dB, and steel would just about eat up all of our gain for the 10 cm spacing.

You must never assume that maximum current means maximum gain. If we were to narrow the spacing between the two antenna elements such that the second element shorted out the driven element, then the input resistance would approach zero, the current would get very high, but the field would approach a theoretical minimum. And remember that real wires and real insulators have losses, and the higher the currents in these wires, and the higher the voltages across these insulators, the more RF power will be lost as heat. Or worse, you may encounter voltage breakdown, especially under wet conditions during modulation peaks.

Let us also remember that the field produced by a wire depends on the current distribution along that wire, and it is not just the maximum current magnitude that defines the electric field intensity. In fact if you have equal magnitude but opposite phase currents on a wire, your total field at some locations can approach zero no matter how large your currents are. The field at any point in space is in fact the sum of the fields from the currents in all the antenna elements delayed by the individual times each field takes to reach that point. Because the current is not uniform over the length of each wire element, it is useful to apply a little integral calculus to determine the area under the current distribution curve in order to get an accurate handle on the electric field produced by an antenna. This is one reason why computer programs are so useful in the arena of antenna analysis. They can integrate by brute force. Antennas may have very complex shapes, and operate in very cluttered environments, so it is nice to have a computer program that can explicitly incorporate many of the physical details, thereby reducing the number of assumptions and simplifications one has to make in order to create a practical model.

In the world of antenna design, gain is traded off for just about every other performance feature. Higher gain typically is paid for with an increase in one or more of the following: antenna size, weight, windage, current, voltage, bandwidth, feed complexity, difficulty of impedance matching, importance of manufacturing tolerances, etc. At one extreme, if you want your antenna to present a resonant 50 ohms across several octaves, you may be better off building a dummy load out of non-inductive resistors! At the other extreme, you may end up with an antenna Q so high that when you try amplitude modulating, the reflected power at the sidebands knocks you off the air. Thus SSB may be more tolerant of bandwidth than AM.

It is possible to compensate for a narrow impedance bandwidth by placing a special impedance transformation network between the antenna and the transmission line, or between the common feed-point of a phased antenna array and the transmitter. Power losses and voltage stresses in such a broad-banding network become important considerations, so the components tend to be large. However at maximum legal ham radio power, such a network is sometimes warranted. Just keep in mind that improving the impedance bandwidth of a phased array does not necessarily mean that the pattern bandwidth has also been improved. This article addresses adjustment of a phased array at one frequency only, and does not investigate effects on bandwidth such adjustments might cause.

Getting back to our analysis of the QST construction article^{1},
consider the case of two quarter-wave elements cut for the
two-meter band and spaced a quarter wavelength, each
having four ground radials bent downward at a 45 degree
angle (Figure 1). If the phase of the current in one element is
90 degrees out of phase with the other, then the fields
radiated from these elements are in phase in one direction,
and out of phase in the opposite direction. That is, the signal
reinforces in the beam direction and cancels out back. If the
current magnitudes are equal, the null out back goes to zero.
In Figure 1, if the left element's current phase is -90 degrees
relative to the right element's phase, then by the time the left
element's electro-magnetic wave arrives at the right element a
quarter wavelength distant, its overall delay is 180 degrees.
So the signal from the left element cancels that from the right
element, and the pattern develops a null to the right. Just the
opposite occurs to the left bearing (Figure 2A). For other
relative phases the pattern may have higher gain, but a
secondary lobe may appear out back, and the front-to-back
ratio would be compromised. Phasing of 135 degrees
maximizes forward gain for the given quarter wavelength
spacing (Figure 2B).

__Physical Particulars__

The two vertical antenna elements are 19.2 inches long, and
all calculations in this article were done at 146.5 MHz. The
elements are spaced 19.2 inches, and the image-plane radials are 18 inches long.^{1} The ground is 30 feet below the feed-points of the elements. The ground
conductivity is 5 mS/m and the relative dielectric constant is 15. All wire diameters are 1/4 inch. The transmission lines are lossless RG8, which has a surge
impedance of 52 ohms and a velocity factor of 66 percent. At 146.5 MHz a wavelength in free space is about 80 inches, but it is only 53 inches in RG8. By
convention we assign 360 degrees of phase shift to a wavelength, so a quarter wavelength is 90 degrees. A piece of RG8 that is physically a quarter wavelength
long (in this case, 20 inches) actually has a phase shift of 90/.66 = 136 degrees, but only if the VSWR is 1.0 (more about this later). The outer conductors of the
RG8 were included in the moment-method antenna model, and formed a vee shape beneath the ground radials. This affected the pattern only slightly, perhaps 0.2
dB. The gain figures cited in this article are for an elevation angle of one degree above the horizon. See appendix for a typical vertical pattern.

__Phase Shift Sign Conventions__

There is some confusion about the sign of the phase shift across a transmission line -- do we call this 20 inches of RG8 minus 136 degrees or plus 136 degrees? In the commercial broadcast industry most engineers assign a minus sign to the phase shift across a transmission line, and this is in keeping with the concept of "delay." That is, the arrival of the RF energy at the end of the line is delayed by 136 degrees in this case. And if that same energy were traveling through 20 inches of space, it would only be delayed by 90 degrees. If we were to model this 20 inch length of transmission line as a tee network, it would have a capacitor in its shunt leg and identical input and output inductors in its series arms. If we chose a pi network model, it would have identical capacitors in the shunt input and output arms, and a coil in the series arm.

I think some of the confusion about the sign convention of phase shift stems from Circuits 101, where a current entering an inductor is said to lag the voltage applied to that inductor by 90 degrees, and the current entering a capacitor is said to lead the voltage across that capacitor by 90 degrees. The current in the inductor is defined as V/jX, so its phase shift is -90 degrees relative to the applied voltage, or -jV/X. I suppose it all goes back to the way we plot waveforms against a time axis. If time increases from left to right on the graph, then the inductor current sinusoid starts 90 degrees later, which is to the right of the voltage sinusoid. And in a Cartesian coordinate system we typically assign positive values to the right of the vertical axis, and negative values to the left, which tends to contradict the value of -90 degrees we just defined on the time axis. Again, this is just convention, and if you are consistent with whatever convention you choose, then your results will be okay. Above all, remember that we are interested in comparing one current with another current, not with a voltage.

On the outer ring of a Smith chart is a legend stating which direction is "towards the generator." When you move in this direction (clockwise) on the Smith chart, you are moving away from the load, which in the case of our feeder lines is away from the base of the antenna elements. Keep this in mind, and everything should work out fine. To check your math, make a rough plot of your impedance results on the Smith chart to see if everything makes sense. Note that the phase shift around the complete circumference of a Smith chart is 180 degrees, not 360 degrees.

So the practical question becomes, how does one obtain equal current magnitudes and a relative phase of 90 to 135 degrees in the two antenna elements?
Perhaps the first approach that comes to mind is to feed the two elements with different lengths of transmission line, one that is a quarter wavelength longer than
the other (Figure 5). However, on closer inspection, this technique is flawed because it assumes that the transmission lines are terminated in their characteristic
impedance. You see, the current phase shift through a transmission line is dependent on its load impedance. And the feed-point "operating" impedances of the
two elements in our array are not equal to each other, nor are they equal to their individual or "self" impedances. This is caused by the electro-magnetic coupling
between the two elements. They are close enough physically that they influence each other very significantly. So every time we change the length of one
transmission line, the mismatch on *both* lines changes, the relative element currents change, etc. In fact, just about everything interacts in a multi-element
antenna. You can't make a change in one element without influencing the others.

We can get a handle on the amount of coupling and its exact impedance effects by using the following tee-network model (Figure 3). Think of this as a
transformer model, where the self-inductance of the primary and the secondary are the same in this case (the two antenna elements are the same), and the mutual
inductance between transformer windings is an analog of the coupling between the two elements in our array. Except in the world of antennas the resistance is
more significant, so we refer to self and mutual *impedance*, rather than inductance. Of course, the mutual impedance between antenna elements can also be
capacitive.

Where Z11 = self impedance of element 1

Z22 = self impedance of element 2

Z12 = mutual impedance between both elements

Z1 = operating input impedance to element 1

Z2 = operating input impedance to element 2

In this case Z11 = Z22, since the array is bilaterally symmetrical

If you prefer, you can model the coupling between elements as a pi network equivalent (Figure 4) of the tee network we just defined. This might give us some additional insight into the workings of a phased array. In this case, Z1 = 1/Y1, Z11 = 1/Y11, etc. The port input voltages and currents are the same in Figures 3 and 4. The pi network has the advantage of having one less "node" than the tee network, which makes for a more efficient computer analysis model. Personally I tend to think in series R and X, rather than shunt G and B (even though those are my initials). Most of us are right-handed, some are left-handed, and a lucky few are ambidextrous.

Keep in mind that these tee and pi networks are not the same as the standard impedance matching networks we use in antenna tuners, since those have only one driven port. Also the impedances seen looking into these antenna models are not resonant, whereas matching networks typically are. So you can't make too many safe inferences about antenna behavior based on normal antenna matcher behavior.

We can measure the *self* impedance of an antenna element by connecting our impedance meter to the feed-point of one of the elements when the other element's
feed-point has nothing connected to it. Yes, you must do this in the presence of the other element, since the self impedance of a single element alone in free space
will not be the same. We can then determine the *mutual* impedance by measuring the feed-point impedance when the second element's feed-point is
short-circuited (we will call Z1 in this special case Zsc), and plugging that value into Equation 1 (does this remind you a bit of the short-circuit/open-circuit test
for measuring transmission line characteristic impedance?):

_____________

Z12 = sqrt[ Z22 (Z11 - Zsc) ] . . . . . . . . . Equation 1

An alternative approach is to terminate the base of element 2 in an open circuit, and measure the complex voltage across this termination, V2, and also measure the complex input current at the base of element 1, I1. Then we could use Equation 2 to determine the mutual impedance. But since most of us don't have access to a vector voltmeter, this isn't too practical. However, it is something we can easily model with a moment-method antenna analysis program.

Z12 = V2 / I1 . . . . . . . . . Equation 2

It should be intuitively obvious that the closer the antenna elements are to each other, the higher the value of mutual impedance, and the more Zsc will differ from Z11. If the elements were very far apart, Z12 would be zero and Zsc would be equal to Z11. Now the real deal is portrayed by the following two equations, where we can determine what the actual base operating impedances are. In this case, they are very different from the self-impedances of the two elements, and the total RF power input to the antenna only rarely splits evenly between the two elements in a phased array. Remember that the operating resistance and power of an individual element can be zero, and can also be negative. However, this is never true at the overall input to the array.

Z1 = Z11 + Z12 (i2/i1) . . . . . . . . . . Equation 3

Z2 = Z22 + Z12 (i1/i2) . . . . . . . . . . Equation 4

Where: i1 = complex base current in element 1

and i2 = complex base current in element 2

The general form of the operating impedance equation for n antenna elements is

Zi = (Ij/Ii) Zij, for j = 1 to n

__Math Example:__

The self impedance of these elements is 49.2 + j10.0 ohms, per NEC2 when all wire diameters are 1/4 inch, and we are 30 feet above 5 mS/m earth. The self impedance is determined by placing a current source at the feed-point of the first element, and placing an open-circuit load at the feed-point of the second element (I used a 10k resistor, which is close enough for our purposes). The pattern is essentially omni-directional and the gain is 1.4 dBi when the second element is left "floating" this way. However, it is important to realize that self-impedance is not determined by de-tuning the other elements in the array in order to obtain an omni-directional pattern. With electrically taller elements, say a half wavelength, you may have to short-circuit the parasitic element's feed-point in order to obtain an omni pattern, and this would yield very wrong self-impedance values for the driven element. So just ignore the pattern shape you get when "floating" the other elements in the array. All you care about is the impedance of the driven element obtained with this test, which we define as self impedance (Z1 becomes Z11 when there is no current in the output arm of our Figure 3 tee-network model).

Using Equation 2, the mutual impedance, Z12, is then equal to the voltage across this 10k resistor, V2, divided by the input current, I1. If you use a source current of one ampere at zero degrees and NEC to determine what the current is through the 10k resistor in order to find V2, be careful you don't get the sign wrong, or your value for Z12 will be off by 180 degrees. This can occur because in our coupling model of Figure 3, I2 is going into the port, but in the NEC open-circuit model I2 is coming out of the port. A vector voltmeter would verify this, but as I said, most of us don't have one. It is always a good idea to compare your measured or calculated mutual impedance with textbook values to be sure you haven't made a gross error somewhere.

However, most of us can get our hands on an inexpensive impedance measuring device, several of which are advertised in the various amateur radio magazines. When the open circuit at the base of element 2 is replaced with a short circuit, Z1 = Zsc = 55 + j36.2 ohms. The mutual impedance is then calculated from equation 1, thus Z12 = 36.7 / -45.6 = 25.7 + j26.2 ohms. By the way, shorting out the second element makes it act as a reflector, which produces a gain of 4.7 dBi, not bad compared to the 1.4 dBi value we obtained earlier for the omni-directional pattern.

Now if we drive both elements, we can obtain a maximum gain of 5.2 dBi, which occurs when the currents are equal in magnitude and phased 135 degrees. If we phase them 90 degrees, we get a better front-to-back ratio, but less forward gain (3.9 dBi). So you might ask the question, why should I bother with a complicated phasing system when I can get pretty good gain simply by feeding one element and tuning the other as a parasitic reflector? This is a good point, but if you want the sharpest possible null out the back, you can't do it unless you independently control the phase and magnitude of the currents. And when the QRM originates from the back lobe of your antenna's pattern, you might wish for a better front-to-back ratio, regardless of forward gain. A second and perhaps more important reason for having a simple method to adjust the relative current magnitudes and phases is to compensate for stray capacitance and inductance within the feeder system and the antenna, undesired radiation from the feeder lines, construction irregularities, etc.

Table 1 describes the differences for various element currents when 100 watts of power is delivered to the array. Note that dBi is the maximum gain of the pattern relative to an isotropic radiator, and dBf is the front-to-back ratio of the pattern. The parasitic case looks attractive (where the second element is not driven and has no insulator between it and its counterpoise radials). How can we tell when an element is parasitic? Answer: when it has current but no power (i.e., when its operating resistance is zero, or it is short-circuited).

__Table 1 Effects of Adjusting Element Currents__

__input impedance at bases complex current at bases__ __ (watts) __ __gain__

__Z1 (ohms). . . Z2 (ohms) . . . i1 (amps) i2 (amps) . . . P1 P2 . . . dBi dBf__

75.5 + j35.7 23.1 - j15.8 . . . 1.00 / -90 1.00 / 0 . . . . 77 23 . . . 8.8 22.8

49.6 + j46.7 12.5 + j10.3 . . 1.27 / -135 1.27 / 0 . . . . 80 20 . . . 10.1 7.7

55.1 + j36.2 0.0 + j0.0 . . . . 1.35 / -123 0.98 / 0 . . . 100 0 . . . . 9.6 8.7

53.0 + j26.8 -27.5 - j5.7 . . . 1.46 / -123 0.69 / 0 . . . 113 -13 . . . 8.9 5.6

__Negative Resistance__

You might be wondering if it is possible to create a situation where one of the operating resistances is actually negative. In this case the power in one element
would be negative, and the power in the other element would actually be greater than the total input power to the antenna. This may sound strange, but it does
happen in practice. The sum of all the element powers must equal the total input power to the antenna. The negative element is passing power back down its
transmission line, and the positive element is passing some of its power to the negative element. To see it in our simple array, all we have to do is change the
relative current. We know that the parasitic case has zero resistance at the base of the reflector, so we can start from there by reducing the magnitude of the
element 2 current while maintaining the same phase relative to element 1. The pattern obtained with this particular negative resistance case isn't that great, and
the bandwidth is likely to be narrower compared to a comparable pattern obtained from an array having all positive operating resistances. The negative resistance
implies an increased circulating current. But there are some instances where a very sharp null is desired in the pattern, and a negative resistance element is the
only way to get it. The correct method of returning the negative power to the common feed-point in phase is widely misunderstood, but is not something we
need to discuss in detail at this time.^{3}

__Feed Lines__

So how can we feed this array with the desired complex currents (i1 = 1.0 / -90 and i2 = 1.0 / 0)? Let's start with a pair of transmission lines per Figure 5, and look at two cases. In both cases we connect 23 inches of RG8 to element 1. In the first case line 2 (connected to element 2) is 17 inches long, but in the second case it is 3 inches long. In both cases the RF current takes longer to get from the common feed point of the two transmission lines to element 1 than it takes to get to element 2. We would expect the relative phase in the first case to be less than it is in the second case, since the difference in physical line lengths is 6 inches in the first case and 20 inches in the second. A wavelength at 146.5 MHz in RG8 is 53 inches, so the phase difference in the first case is (6 / 53) 360 = 41 degrees if the VSWR is 1.0, and in the second case it is 135 degrees. In practice, the actual phase shifts turn out to be much different because the lines are not matched. In fact the relative phase in the first case is 91 degrees, and in the second case is 150 degrees. So we missed the boat by 50 degrees in the first case, and by 15 degrees in the second case (Table 2). But since we actually wanted a relative phase of 90 degrees, rather than 41 degrees, our first case looks pretty good as a practical feeder solution. When I say relative phase I mean that i2 leads i1 by 90 degrees, or i1 lags i2 by 90 degrees.

__Table 2 Effects of Feeding with Mis-matched Lines__

__input impedance at bases . . . . . current at bases . . . . . power (watts) __

__case Z1 (ohms) Z2 (ohms) . . . i1 (amps) i2 (amps) . . . P1 P2 . . . . . dBi dBf__

**1** 73.0 + j41.4 30.4 - j18.3 . . . 0.91 / -91 1.11 / 0 . . . 61W 39W . . . 9.2 22.3

**2** 32.1 + j41.7 15.6 + j8.8 . . . 1.40 / -150 1.54 / 0 . . . 63W 37W . . . 10.9 5.5

Keep in mind that the sign of the electrical length of the transmission line is normally construed as positive, but the current phase shift is normally considered negative to connote the time lag associated with travel through the line. So in Figure 6, if we have a quarter wavelength line, we say that theta is 90 degrees. However, the phase shift phi associated with this line is - 90 degrees, for the matched condition. The formulas listed in this article assume that the value for theta is positive. So think electrical length when you see theta, and think time-lag phase shift when you see phi.

The transmission lines are terminated in the impedances above, so they are mismatched. The lines transform their load impedances (Equation 5) to the Z1' and Z2' values in Table 3 below. The combined input impedance Zin at their common junction is simply Z1' in parallel with Z2'. The current phase shift through the mismatched transmission lines can be determined with Equation 6. If we change the line lengths in an attempt to produce higher gain, comparing case 2 to case 1 above our impedance match becomes even worse, as the element currents increase and the operating resistances decrease farther below the 52 ohm characteristic impedance of the transmission line. Increased currents for a given power is the price you pay for higher gain.

We don't have to use a moment-method antenna analysis program to determine the currents in the individual array elements. Instead we can simply plug into an AC network analysis program the tee network model of the coupled antenna elements per Figure 3, and the two feeder lines. If the network analysis program does not have a transmission line model, you can make your own pi or tee network equivalent.

__Table 3 Mis-matched Feeder Lines__

. . . . . . . Input 1. . . . . . . . . . . . . . . . . . . . Input 2 . . . . . . . . . . . . . . . . . .Combined

__line1 . . Z1' (ohms) . . VSWR . . . . line 2 . . Z2' (ohms) . . VSWR . . . . Zin (ohms) . . VSWR__

23 in 39 + j31 ohms . . 2.08 . . . . . 17 in . 104 + j4 ohms . . 2.00 . . . . . 31.6 + j15.2 . . 1.85

23 in 20 + j18 ohms . . 2.96 . . . . . . 3 in . 108 + j6 ohms . . 2.09 . . . . . .18.5 + j12.7 . . 3.00

Z' = (Z - jZo tan Theta) / (1 - j [Z / Zo] tan Theta) . . . . . . . . . . . Equation 5

Phi = tan-1 [ R / (X + Zo / tan Theta)] . . . . . . . . . . . . . .Equation 6

Note that the current phase shift through the transmission lines given by Equation 5 is only part of the story. There is an additional phase shift that occurs at the common end of the transmission lines because the input current does not split evenly between the two lines. That is, the input impedance to each line is different (Z1' and Z2'). Finding this additional phase shift requires a simple exercise of Ohm's Law, where the relative phase of the two input currents can be determined by Equation 7.

i2' / i1' = Z1' / Z2' . . . . . . . . Equation 7

So what we have done by decreasing the length of line 2 is to compromise our impedance match in order to obtain maximum gain, but in neither case do we have a sharp null out the back because the element current magnitudes are never equal. Is there some other combination of transmission line lengths and types that would offer a better impedance match and also produce the desired pattern? Maybe, but finding it could be a painfully iterative process. It is not something you would want to do while standing on a ladder, that's for sure (does this sound familiar?).

__Practical Adjustment Techniques__

Before we get into a full-blown phasing and coupling network design, let's look at what a single variable capacitor installed at the input to element 1 buys us in the way of adjustability. We know from Table 1 that the inductive input reactance to this element varies from 27 to 47 ohms for the range of radiation patterns that interest us. If we tune this out with a capacitive reactance, we reduce the VSWR on line 1 and we also afford some adjustability of the pattern (Table 4). However, the pattern degrades with a capacitor, but gives us what we want with an inductor. So intuition doesn't always work in a mis-matched feed system, does it? Apparently the design is relying heavily on mismatch to produce additional phase shift in the transmission line. The following is for the case where we have 23 inches of RG8 connected to element 1 and 17 inches of RG8 connected to element 2. Note how touchy the pattern is; this should tell you to be very careful with your transmission line wiring because even a little inductance has a big effect at the base of the driven elements. It should also tell you that you need a practical adjustment handle if you want the pattern to arrive when and where you expect it. Take a while to appreciate the range of forward gain and front-to-back values in Table 4 that can be had with a simple turn of a variable capacitor in series with a fixed inductor.

__Table 4 Pattern Adjustment with Series Coil and Capacitor at Element 1__

#1 tuning. . . . . . . . . . . . . . __ current at bases __

__reactance . . . Zin (ohms) . . . . i1 (amps) i2 (amps) . . dBi dBf__

-30 ohms . . 35 + j11 ohms . . 0.62 / -56 1.10 / 0 . . . . 7.9 4.5

-20 ohms . . 34 + j12 ohms . . 0.65 / -65 1.08 / 0 . . . . 8.2 6.7

-10 ohms . . 33 + j13 ohms . . 0.80 / -76 1.08 / 0 . . . . 8.6 10.9

0 ohms . . . .32 + j15 ohms . . 0.91 / -91 1.11 / 0 . . . . 9.2 22.3

10 ohms . . . 31 + j18 ohms . . 1.02 / -107 1.21 / 0 . . . 9.9 19.6

20 ohms . . . 31 + j21 ohms . . 1.08 / -123 1.36 / 0 . . 10.4 12.0

30 ohms . . . 33 + j25 ohms . . 1.08 / -137 1.53 / 0 . . 10.6 8.1

40 ohms . . . 36 + j27 ohms . . 1.02 / -146 1.65 / 0 . . 10.4 5.7

Intuition suggests that if an inductor in series with element 1 gives us the adjustment range we want, then perhaps a capacitor in series with element 2 would do the same. A small variable capacitor seems better suited to our application than a variable inductor, so to cover the reactance range of -40 to +40 ohms we could use a variable capacitor in series with a fixed coil. In the 2 meter band, a 5 to 30 pF variable capacitor in series with a 90 nH air-core coil would work fine (about three turns on a half inch diameter).

Table 5 tells the story of what happens when we tune the base of element 2, and it should be no surprise that again the results are not as expected. Specifically we just don't get the big forward gain that we get by tuning at the base of element 1. So it looks like we have to use an inductive reactance in series with element 1 in order to get the maximum gain from our array. Perhaps adjustment at the base of element 2 is less effective because there is less power in this element?

Perhaps a variable capacitor at the input to one of the transmission lines will do the trick. Or perhaps a variable capacitor at the input and output of both transmission lines would give us enough adjustment latitude. Or would it just serve to get us lost in the woods? I leave this as an exercise for the reader, because I don't believe in taking all the fun out of a project and leaving no room for experimentation. Remember that at these relatively low power levels, using mis-matched feeder lines is not a bad practice, so take advantage of it.

__Table 5 Pattern Adjustment with Series Coil and Capacitor at Element 2__

#2 tuning . . . . . . . . . . . . . __ current at bases __

__reactance Zin (ohms) . . . i1 (amps) i2 (amps) . . . dBi dBf__

-40 ohms 24 + j22 . . . . . 1.02 / -85 0.84 / 0 . . . . . 8.9 13.2

-30 ohms 26 + j21 . . . . . 1.08 / -86 0.90 / 0 . . . . . 9.0 15.6

-20 ohms 28 + j20 . . . . . 1.04 / -87 0.96 / 0 . . . . . 9.0 19.1

-10 ohms 30 + j18 . . . . . 0.98 / -89 1.03 / 0 . . . . . 9.1 23.4

0 ohms 32 + j15 . . . . . . . 0.91 / -91 1.11 / 0 . . . . . 9.2 22.3

10 ohms 33 + j11 . . . . . . 0.83 / -87 1.20 / 0 . . . . . 9.2 16.9

20 ohms 32 + j6 . . . . . . . 0.73 / -98 1.29 / 0 . . . . . 9.2 12.6

30 ohms 30 + j1 . . . . . . . 0.62 / -111 1.38 / 0 . . . . 9.1 9.3

There is nothing egregiously wrong with having a high VSWR on the feeder lines. The extra losses and higher voltages and currents in short feeder lines caused by the mismatch are not too significant at ham radio power levels, in most cases. But remember, the higher the gain, the higher the currents in those feeder lines. If you are in doubt, be sure to calculate all component voltages and currents, and be aware that the wet voltage ratings at the ends of the lines are a lot lower than the ideal dry sea-level ratings (yes, you have to de-rate for altitude).

Of course, some effort should be made to match the main transmission line at the common junction of the feeder lines, since the main line is a lot longer than the feeders and its overall power loss could be significantly higher in the mis-matched condition. A couple of stubs would do the trick, as long as we are careful that they don't become additional parasitic radiating elements in our array.

The alternative to the mis-matched feeder approach is to match the operating base impedance of each element to the transmission line surge impedance, and to provide a controlled-phase-shift power divider at the common input to the array. But we need to keep track of the phase shift across the element matching networks, too. And it is not that easy to make an impedance matching network that works well in the two-meter band, and has independent phase and impedance transformation adjustability. We could make a tee network phasing and coupling unit out of three stubs. A quarter wavelength ladder-line stub with a sliding short would work well as an inductor, and an adjustable length open-circuited stub would work as a capacitor, but then you have to deal with the problem of parasitic re-radiation from the stubs. That is, the stubs start to act as undesired antenna elements. So a lumped parameter tee network may be the best approach, since these components would be considerably smaller than the stubs. An adjustable co-axial capacitor is easy to fashion from a couple inches of half-inch copper pipe stuffed with a plastic dowel bored and tapped to accept a 3/8 inch diameter screw. This gives you a surge impedance of about 10 ohms, which yields about 14 pF per inch. A 5/16 inch screw would yield about 9 pF per inch, and a 1/4-20 screw about 6 pF per inch. A DC capacitance meter will get you in the ball park, but it is safer to use an impedance meter at VHF, if you use it properly. The time-honored method of playing an air-core solenoidal coil like an accordion provides a practical form of adjustable inductor. Or you could try an adjustable ferrite slug, if you can find one that is not too lossy at 146 MHz. In any case you need to make a good estimate of the initial component values, which means you need to utilize the formulas and measurement techniques in this article or run a moment-method analysis to determine what your driving-point impedances are likely to be.

__Phasing and Coupling Network Example__

Assume we really want equal current magnitudes in each element, and a relative phase of 90 degrees. We already know the driving point impedances and powers
to expect. Now we want to match those impedances to 52 ohms, and we also want 52 ohms at the common feed point to the array. Let's start at the power
divider, using what is called an Ohm's Law configuration. We know that element 1 radiates 77 watts and element 2 radiates 23 watts. In order for more power to
be transferred to line 1, we need to present a lower resistance to the common input point looking towards line 1 as compared to line 2 (V^{2}/R). We also know that
the paralleled resistances looking toward the two lines is 52 ohms. So for 100 watts input to the array the math looks like this, where V is the input voltage
across the power divider:

V^{2} = PR = 100 (52) = 5200

V^{2} / R1 = 77 watts, so R1 = 68 ohms

V^{2} / R2 = 23 watts, so R2 = 222 ohms

Checking our work, R1R2 / (R1 + R2) = 52 ohms

So if we connect an L network between the common point and each transmission line, we might have a network that looks like Figure 7. Note that the shunt elements of the two high-pass L networks combine to form one coil. I chose "leading" phase shift networks, as opposed to "lagging" phase shift, low-pass networks simply because it is cleaner and easier to adjust a co-axial screw capacitor than it is to adjust a coil. So I end up with two capacitors and one coil, instead of two coils and one capacitor. A co-axial screw inductor would be physically too long for this application, in my opinion, because it would begin to act as another radiating element in the array. However, if you are willing to add more-or-less a quarter wavelength of line to your capacitor, you can make it look like an inductor. Note that the two capacitors in the power divider are not connected to ground, so you have to be careful how you mount them. If you make them from ½ inch copper pipe, you could align the pipes in parallel and solder them together, and this would form the common side of the power divider.

Now that we have taken care of the front end of our phasing and coupling design, we need to determine the phase shift needed in the impedance matching units at the other ends of the feeders. Note that the phase shift across the two legs of the power divider are not equal -- leg 1 lags 31 degrees behind leg 2 (62 - 31 = 31). In order to simplify our design, let's use equal length feeder lines. This has the benefit of creating a symmetrical perturbation of the antenna structure and will tend to minimize pattern distortion and other side-effects. We want the relative current in element 1 to lag that of element 2 by 90 degrees. We already have 31 degrees of lag provided by the power divider, so we need an additional -59 degrees. We can get this by selecting a phase shift across matching network 1 of 50 degrees, and across matching network 2 of 109 degrees per Figure 8. The component values shown are for 146.5 MHz.

During adjustment it may also be useful to have a dual-trace, wide-band oscilloscope to monitor the currents and phases in each element using a small sample loop located near, but not connected to, each element. However, you would have to be very careful with the placement and de-tuning of your sample lines so as not to disturb the array. This gets complicated at VHF. Unless you are very patient, this may be more trouble than it is worth. However, it is quite practical at lower frequencies.

It would be nice to have two helpers with hand-held 2-meter transceivers, one aligned with the main bang of the antenna, and the other on the other side. Then you could communicate with them using your transceiver plugged into the antenna under test. As you adjusted your phasing and coupling components, your helpers would relay their S-meter readings back to you. Or you could use a single helper, and rotate the antenna 180 degrees of azimuth to measure the forward and reverse fields. Be sure to maintain a constant input power to the antenna when you do this; an auto-tuner would be ideal for these circumstances. If you don't have a helper, you could key down the hand-held and leave it radiating in your neighbor's pasture while you adjust your antenna for maximum received signal in the forward direction, and minimum signal in the reverse direction. Keep a clip-board and pencil at your side to keep track of component settings and the resulting field intensities. Otherwise you may lose your bearings and end up repeating the same tests.

__Summary__

In summary, the technique of connecting mismatched feeders to the driven elements in a phased array is an expedient method for obtaining relative current phases that approximate the desired values. A variable capacitor in series with one or more of the lines will often provide a good degree of adjustability. However, it won't always work, in which case phasing and coupling networks may have to be inserted between the elements and the feeder lines, and a power divider will have to be constructed at the input to the feeders. In order for this approach to be successful, it implies a knowledge of the operating feed impedances of the driven elements, and the way they interact with one another. In actual practice it can be a real "can of worms." A simplified approach that will allow easy adjustability of the pattern is therefor suggested where a single series coil and/or capacitor is inserted at the feed-point of one or more of the driven elements. In all cases where lumped parameters are added to the design, impedance and pattern bandwidth may suffer, but this needs to be addressed on a case-by-case basis. If your moment-method program does not allow you to enter all the network components in your phasing and coupling equipment, a network analysis program can do the job if you model the antenna self and mutual impedances using the multi-port network models discussed in this article.

You don't need an antenna analysis program to measure and model the interaction between radiating elements in an antenna; all you need is an impedance measuring device, some math skills, and some network analysis software. However, a moment-method program is certainly helpful, and should be part of your arsenal of design tools. Sometimes the stray reactances and pattern "scattering" within and around an antenna can mask what is really going on, and if you can approach an antenna problem simultaneously from different angles, you will converge on a design solution in less time.

A note about the mutual coupling model -- I used a tee network in this article, but a special pi network requiring some fancy matrix math is the practical form
when more than two antenna elements are used.^{2} Topologically a pi network has one less voltage node than a tee network, and this has advantages over a tee
network. However a detailed discussion of the generalized antenna impedance model is beyond the scope of this article, so I have attached an appendix giving
just the basics.

__References:__

1) Harold Thomas, "A 2-Meter Phased-Array Antenna," January 1998, *QST*.

2) Dane Jubera, "Toward Improved Control of Medium Wave Directional Arrays," December 1981, *IEEE Transactions on Broadcasting*.

3) Grant Bingeman, "Negative Towers," November 1980, *Broadcast Management/Engineering*.

____________________________________________________________________

author's biography:

Grant Bingeman, P.E., KM5KG (ex-WN6AIW, WB6MBX, WB4AOI) has been a Senior Engineer at Continental Electronics in Dallas, Texas for 17 years, where he designs high power antennas and transmitters. Call him at 214-381-7161 and he'll give you a special deal on a 500 kw auto-tuned short-wave transmitter. His e-mail address is DrBingo@compuserve.com.

______________________________________________________________________

Appendix

A rigorous multi-port network model of a phased array containing n radiating elements can be developed using the following procedure. First form a matrix of
the mutual impedances between all the elements. This matrix will contain n^{2} complex values, and look like this:

Z11 Z12 ... Z1n

Z21 Z22 ... Z2n

. . .

. . .

Zn1 Zn2 ... Znn

Then you must invert the matrix, keeping in mind that matrix inversion is a tedious math process that you will probably not want to program yourself, but simply call as a function from a math library. In other words, an individual matrix element Yij does not equal 1/Zij.

Y11 Y12 ... Y1n

Y21 Y22 ... Y2n

. . .

. . .

Yn1 Yn2 ... Ynn

Next you must calculate the lumped-parameter values for the network shown in Figure 9, using these equations:

Zi' = 1 / [Yi1 + Yi2 + ... + Yin]

Zij' = -1 / Yij

Figure 10 shows the vertical or elevation pattern of the array described in Figure 1 when the element currents are equal in magnitude and phased 135 degrees. The forward gains and front-to-back ratios will vary depending on the elevation angle of interest. The values throughout this article are for an elevation angle of one degree above the horizon plane. Since communication at these frequencies is almost always line-of-sight, this is a valid angle. Note that the far-field goes to zero when the elevation angle is zero, but the near field does not. For 100 watts input to the array, you can expect a vertically polarized field intensity at one kilometer two meters above ground of about 1.5 mV/m for the pattern shown in Figure 10 on a bearing of 270 degrees true over 5 mS/m earth, assuming no reflections. This value would be slightly higher if the ground conductivity were better. Over perfect ground the field would be about 12 mV/m.

file: array.wpd 3-10-98