Crossfire Phasing 
Most of our phased amateur antenna arrays are copies of systems or ideas used to build single frequency systems, like AM BC stations. Most of my work has been in HF systems in and around amateur bands. Generally these systems require wide bandwidth, rather than a certain pattern on one single frequency. The proper way to extend bandwidth is to use crossfire phasing, where one element's phase is rotated 180degrees and the delay is set approximately at the spacing. Here's how it works: We have two elements with spacing "s" at X and Y. The signals come from direction 1 or 2.
Let's assume antennas X and Y above are 90 degrees apart on 80 meters. A traditional array would delay phase of the feedline signal to either X or Y by 180s = 90 degrees. In other words we normally think with 1/4 wl spacing we need a 1/4 wl delay line. In this example, let's assume the feed to X lags the feed to Y by 90 degrees. A signal arriving at X from direction 1 will be our reference. We will call this 0 degrees for the reference. The same signal from direction 1 continues on to arrive at Y. Since the signal traveled 90 degrees of extra distance, it arrives at Y 90 degrees later. This delay occurs because of the physical distance it traveled over the space between the two antennas, X and Y, and the finite speed of the wave (the speed of light). Remember X lags Y in the array phasing system. This means the two signals from X and Y arrive at the common point of the feed system with additional changes. X's reference phase of zero is added to the 90 delay. The result is the signal from X arrives at the common point with a 90 degree delay. Now let's look at Y. The additional signal delay in space to Y is also 90 degrees. X arrives at the common point delayed by only 90 degrees. Signals from X and Y are perfectly in phase, and the resulting signal is the inphase sum of the two signals. Our receiving level has doubled over one element. Now let's look at this from direction 2.
For a signal coming from 2, the phase of element Y becomes our zero degree reference. Element X receives the signal 90degrees later for 90 phase. This element is the one fed through a 90 degree delay line. The 90 degree phase delay in element X from additional time to travel from Y to X through space adds to the 90 delay in the delay line, and the result is a 180 degree phase shift at the common point. Since Y is at zero and added to the 180 delay of the signal from X at the common point, the signals subtract exactly to zero. There is no response in direction 2 as long as the signal levels at X and Y are equal. Now assume we move the array to 160 meters, where our fixed length delay line and spacing are both 45degrees long.
Spacing s is now 45 degrees. We did not change the physical spacing. The inphase direction of 1 is still the direct sum of the two signals, but the outofphase direction becomes (45) + (45) degrees or 90 degrees at the common point. The signal vector addition (sin of 90 deg = 1) is now unity. This is a very poor null. Now let's go back to 80 meters and invert one element 180degrees someplace in the system. In this case, we have:
From 1 we have X=0 degrees plus 90 though air is 90 at Y. Assume Y is inverted (it also works if we invert only X. Doesn't matter which element is phase flipped or inverted). The phase at Y is 90 but rotated to the opposite side of a "phase circle" by the mirror flip of 180 in element Y's inverted driving system. A 90 with a 180 flip is now +90. This combines with the 90 delay from element X through it's phasing line delay, and the sum of two equal signals of 90 and +90 is zero. 90 is exactly 180 degrees from +90, and the array now has no response at all towards 1. When the signal arrives from 2, the phase is now inverted the same way and the result is perfect addition. We have Y = 0 degrees (it is the 0 degree reference point now) plus inversion of 180 is 180 degrees. Spatial delay is 90 plus 90 in the delay line for a net delay of 180. The result is signals from 2 are now in phase from both elements. The only change by adding a 180degree element flip is the array inverts direction!
Here is where it gets interesting, and I am amazed so many amateur antenna designers miss this. Y is still inverted 180 degrees by the phase flip.
When we move to 160 meters, the delay from direction 1 is: X=0 degrees, our reference point. The delay line adds 45 degrees delay, the same as the distance the wave is delayed when it travels from through space from X to Y. Since the delay line and space are the same (in this case 45 degrees), the delays in the delay line and space effectively cancel. They are both 45 degrees. Since element Y is inverted 180 degrees by the phase flip in the feed system, the combined phase at the common point is now exactly 180degrees out of phase. The result is zero response from direction 1!!! More amazing, this is true for any frequency! There is always a perfect null from direction 1 so long as the signal levels are even and impedances are matched from the two antennas, and we use a physical delay line rather than lumped components. From direction 2 we have: Y=0 degrees (the signal arrives there first, so it is the reference) plus a 180 flip at the feedpoint for 180 phase into the delay system. At X we have 45 spatial delay as the wave continues through space from Y to X, plus the delay line of 45 degrees for 90 degrees total delay. We have 90 degree phase difference. The signal is the same as the signal from one element. The only effect, as frequency is reduced, is sensitivity of the array drops. We have less signal, but we would have that anyway even if we used the narrow band phasing systems commonly used! Since there is mutual coupling in a low loss transmitting array, the antenna would have gain in that application. It would be exactly like a perfect backside null. In a lossy receiving array where mutual coupling is swamped out by losses, the receive signal level would be the same as a single element by itself. Why do amateurs, who almost always move around in frequency, use singlefrequency 180s ( where s is the element spacing in degrees) phasing systems? Probably for the same reason we use 90 and 180degree shift in a four square. We started out wrong and just kept doing the same thing. While this won't provide octaves of bandwidth on transmitting, it does reduce phase errors across a single band substantially. It also provides a phasing method that allows us to build receiving arrays covering octaves without any loss of null direction or depth!
