# How to feed an array of dipoles for an end-fire pattern

An array of dipoles can be fed to achieve an end-fire pattern.

If the phase of the current on each element is equal, the array will have maximum radiation broadside to the array. If the phase of the currents is equal to the physical spacing of the elements, then the array radiates from its end.

An attempt to show this is shown here:

The three circles represent the dipoles, and the three waves are the electric fields radiated by them. To the right, the waves all add in phase, this is the direction of maximum radiation. To the left, they do not add in phase, and the pattern is smaller. (in fact, with three dipoles, this isn't the optimum phasing for best front-to-back, but for larger arrays this is less important)

How should such an array be fed, to ensure the currents are as required for end-fire radiation?

• If I understand your graph: to the right, the phases compliment eachother, and theoretically would have a gain of about 4dBD, this is the "end-fire" effect. To the left, two of the three waves are cancelling eachother out, leaving only 1 wave to continue, so that would be a 0dBD similar as a single dipole. I don't really understand your question; it seems all in order to me (but I am not an expert, only a beginner experimenter with end-fire arrays) – Edwin van Mierlo May 18 '16 at 19:58
• Green 180° out of phase to Red & Blue. – Optionparty May 18 '16 at 20:25
• Creating a network to produce the required currents from a single source is a huge subject. The best treatment I have seen is Chapter 11 of *Low Band DXing" by ON4UN. – Brian K1LI Jan 29 '19 at 20:52

## 1 Answer

Short answer:

For an array of dipoles with equal spacing $$d$$, the direction of maximum gain with respect to the end-fire direction (call this $$\theta_0$$) can be obtained using the following equation:

$$\delta = k d\cos(\theta_0)$$ where $$\delta$$ is the incremental phase difference between adjacent elements, $$kd$$ is in radians. ($$k$$ is the wave number defined as $$2\pi/\lambda$$). See figure below. NOTE: the angle $$\delta$$ in the figure below should be a $$\theta_0$$ !!! This is an errata in the edition I took this figure from...

Longer Answer:

To plot the gain, we first will first need the array factor given by $$\gamma' = k d(\cos\theta-\cos\theta_0)$$ where $$\theta$$ is the clockwise angle from the end-fire direction ($$\theta_0$$ is the direction we want to steer the beam). The array factor is found as $$F_{an}(\gamma')=\frac{\sin^2(N\gamma'/2)}{N^2\sin^2(\gamma'/2)}$$. True gain is found by multiplying the gain of a dipole with the array factor. An example of various $$\theta_0$$ for a normalized 10-element array with $$\lambda/2$$ spacing is shown below:

I can post more info if there are more questions. The full derivation takes over 10 pages so there is a lot of info I glossed over.

Source: Fundamentals of Applied Electromagnetics, 6th edition, Section 9-9.

• The questioner wants to know how to feed the elements to produce the desired currents, not what the pattern looks like. – Brian K1LI Jan 29 '19 at 20:44
• The answer I provided is the more general solution but has all of information on how to feed the elements. When theta_0 = 0 the please difference delta = kd = 2*pi*d/lambda. Spacing of d = lambda/2 would mean you simply feed them with each with a phase shift of pi radians. Phase delays can physically be created in plenty of ways. Longer feeds between elements (coax/micro strip), unity gain filters, etc. – Andrew W. Jan 30 '19 at 21:16