There are applications where one might wish to adjust the relative phase of the signals radiated by two antennas without needing an external phasing network. One way to do this is to connect the two antennas together, and deliberately tune them off resonance, one high and the other low. The reactive component of the feedpoint impedance will result in a phase shift, but the combination of the two still presents a resistive impedance to the feedline.

While this is a is pretty standard technique for quadrifilar helixes and turnstile antennas for VHF and UHF, where the interaction between the "crossed dipoles" is low, it can also be used in other situations with more coupling. In fact, it's not much different than the design of a Yagi-Uda, where the lengths of the elements are adjusted to get the phases of the currents in the elements to be as desired for the design radiation pattern. These applications are all aiming at having a fixed phase relationship.

There are antennas available, such as the SteppIR, which have adjustable length elements which can be adjusted by remote control. This brings up the interesting possibility of doing things like adjusting the relative phase of stacked Yagi-Uda's in real time. It could also be used to create a "virtual rotator" with two crossed antennas.

### A bit of the theory behind this

Let's assume that the two antennas have complex feed point impedances Ra+jXa and Rb+jXb, and that they are connected in parallel. Further, let's assume that there is no coupling between the antennas. The equivalent circuit looks like the following:

<diagram here>

The radiated fields from the antennas are proportional to the current in the antenna, which can be calculated by I=E/Z. If Xa=Ra, then the phase of the current will be 45 degees lagging the voltage phase. If Xa=-Ra, then the phase of the current will be 45 degrees leading the voltage phase. The net effect is that the currents are 90 degrees apart.

There is a limit. The best you can do is 180 degrees apart (and at that point, the reactances will be huge, so losses from circulating currents might be limiting). However there might be lots of applications where a maximum range of 90-130 degrees might be enough.

### A practical computational example

How much do you need to change the lengths to make it work? Let's take a look at some actual impedance data. Here's a subset of some modeled data for a 25 ft vertical over ground which I happened to have handy.

`F	R	X6.3	32.7342	-40.88776.4	33.4842	-35.20726.5	34.251	-29.66.6	35.0351	-24.06146.7	35.8367	-18.58666.8	36.6563	-13.17126.9	37. 	 -7.811467	38.3523	 -2.503337.1	39.2297	  2.756987.2	40.1272	  7.973127.3	41.0458	 13.14857.4	41.9859	 18.28647.5	42.9481	 23.397.6	43.9332	 28.46257.7	44.942	 33.50667.8	45.9751	 38.52547.9	47.0335	 43.5213`

If we wanted the relative 90 degee phase shift (X=R or -R), you can see this is like tuning the antenna to 7.8 MHz or 6.45 MHz. Since, to a first order, the length is proportional to resonant frequency, we're looking at changes of about 10% in length, which is certainly reasonable.
Neglecting the mutual coupling (which is important in a real system), if we tune one to 7.8 and the other to 6.45 (one will lead by 45 and the other will lag by 45 degrees) we'll have our 90 shift.  The parallel combination will look like about 19 or 20 ohms with some reactance, so a tuner or network would definitely be needed.

This isn't much different than a Yagi, by the way. Instead of having a piece of coax to one of the elements, it's done by mutual coupling, but just as in this case, the feedpoint impedance is low.  The reactive term is dealt with by making both elements shorter or longer as required. One could use an optimizing antenna modeling program (like 4nec2) to determine the element lengths, because you can build the model with the feedlines, and then optimize for zero reactance AND foward gain, and it should converge to the appropriate lengths.

Do this for a series of frequencies, and you have numbers to program into your SteppIR controller for a vertical array.

More to come...

radio/antenna/phased/phbylen.htm - 24 Jan 2009 - Jim Lux