Much is made in phased array literature of the effects of errors in the phase (and amplitude) for each element. This page discusses the results of some analysis that shows that for transmit arrays with small numbers of elements, the phase control requirements are remarkably loose. There's an enormous amount of literature showing the effects of errors for arrays with large numbers of elements: Mailloux's and Hansen's books on phased arrays are good examples. Going back a bit in time, Lo published a paper back in the 1960s with some analysis. This page deals with arrays having few elements, typical of small HF phased arrays, and also, as it happens, with the use of power combiners and multiple amplifiers.
Excitation errors in a phased array result in two effects of interest: 1) a reduction in directivity and forward gain; and 2) sidelobes. In the case of transmit arrays one generally is not concerned about sidelobes, except insofar as the power in the sidelobe isn't going in the direction you want it to. A notable exception would be if you are trying to reduce power in a particular direction, as for a low probability of intercept/detection application, or where you are trying to "protect" the coverage area from another transmitter. In any case, for this analysis, what we are interested in is squirting the maximum power in the direction of interest, and for that purpose, the usual metric is the Equivalent Isotropic Radiated Power (EIRP). The EIRP is the power which would have to be radiated from an isotropic antenna (radiating in all directions equally) to produce the same power radiated in the desired direction. EIRP rolls together both the antenna gain (which raises EIRP) and system losses (which lower EIRP).
The analysis here applies to simple phased arrays, and doesn't necessarily hold true for superdirective arrays.
The analysis is quite simple. We consider the power from each antenna as being a vector (a phasor) and the power we radiate in a given direction is determined by the magnitude vector sum of the contributions from each element. In the ideal case, the vectors all line up in a straight line, so the magnitude of the sum is the sum of the magnitudes of the contributors. However, in the real world, there are errors, so the magnitude of the sum isn't quite as big. The picture below shows the difference between Pcombined and the Pideal as Delta EIRP. In most transmit phased arrays, we don't really care about the absolute phase error (Delta Theta). In the analysis below, the delta EIRP is expressed as dB degradation.
One thing to keep in mind is that this analysis doesn't depend on how the elements are arranged, or what the desired phasing is, or what direction the beam is steered. The assumption is: In the desired direction, all the signals add up in phase. It's when you start to look at directions other than the desired direction (i.e. to look at sidelobes) that you start worrying about physical displacements of the elements and relative phases. The positions of the elements are also important when trying to design a feed network to produce those desired phases, because of the mutual coupling, but, for the analysis of error effects, all we care is that the optimum is when they all line up.
The data presented below was produced by a Monte Carlo type analysis. For each of ten thousand tries, 4 vectors were created, with random errors in phase, uniformly distributed over a specified range. For instance, if the error is 90 degrees, the angle might be anywhere between +45 and 45 degrees, with equal probability. This is a good model for a system where you don't have apriori knowledge of the phases required, you might be required to steer in any arbitrary direction, and your phase control system has discrete steps. Simulations were run with angular errors (total) of 5,10,15,20,30,45,60, and 90 degrees. The 90 degree position would be the case where you can feed each element with the transmitted signal either unshifted, inverted (180 degrees), or shifted by 90 or 270 degrees. This is typical of hybrid phasing boxes like those sold by ComTek or DX Engineering, or the plethora of quadrature phased schemes in the ARRL Antenna Handbook or Devoldevere's "Low Band DXing". The finer resolution cases are nice submultiples from there (i.e. 60 degrees is one of 6 possible phases, etc.).
The 10,000 tries were statistically processed to determine the degradation in forward gain (EIRP, really) that would would result a certain percentage of the time. For example, we can say that 99% of the time, the gain degradation will be no worse than 0.5 dB. With this sort of statistical information, you can make an intelligent system design. Why spend countless hours and dollars trying to cover the worst case, which might occur only 0.1% of the time. The following graph shows an example (for a completely different system, with a different distribution of errors, but it's illustrative). You can see that for the vast majority of cases, the degradation is <0.02dB, but there's a few outliers (aka pathological cases) where there's 45 times more degradation. It's up to the designer to decide whether it's worth it for those cases.
( Click the picture or here for
the full size (60kB) drawing )
dB degradation

Percent power lost

3

50%

2

37%

1.5

29%

1

21%

0.5

11%

0.4

9%

0.3

7%

0.2

4.5%

0.1

2.3%

0.05

1.1%

0.025

0.6%

So here we are with the data. There are 3 cases, with 4, 6, and 8 elements (referred to as "feeds" on the plots. The curves are for percentiles of 50, 80, 90, 95, 99 and 99.7%, and show the degradation (in dB) that will occur for the specified phase errors. Looking at the first graph (for 4 elements), you can see that with 45 degrees of phase error, the degradation will be no worse than about 0.2dB, 50% of the time (the top blue line), and no worse than a half dB, 99.7% percent of the time (the bottom red line). Even huge errors like 90 degrees aren't all that bad, with about 0.6 dB degradation half the time (but the pathological cases bite you more often: 1% of the time, the degradation will be almost 2 dB).
In general (and shown in the data below), the more elements in the array, the less degradation you get from a certain amount of random errors. With very large numbers of elements, the errors sort of average out. The tabular data used to generate the graphs can be found in the accompanying text file.
And the data for 6 elements
And the data for 8 elements
The "take home" message here is that if you are willing to accept a 1 dB degradation only 1% of the time(which is, after all, a fraction of an Sunit on the other end), you can have pretty big phasing errors (like 70 degrees). A system that generates 45 degree phasing steps (a 3bit phase shifter) will have less than 0.5 dB degradation, basically all the time. If you are designing a higher powered phased array, this also means that one can use fairly simple (and low loss) networks to accomplish the phasing.
This also means that extreme measures to match impedances or trim phasing lines are probably not worthwhile, especially if all those matching networks add loss. A typical phased array element might have a driving point impedance of 50+20j ohms, compared to 50 +0j ohms for the unphased, isolated element. The phase shift from the extra 20j introduces about a 20 degree phase shift. We can see that 20 degree errors result in an almost imperceptible change in the power in the desired direction.
WARNING: This analysis ONLY applies to transmitting. Small errors in phase can have a huge effect on the depth of a null or the size of back and side lobes. (In numerical terms, a phase error of 5 degrees will turn a 20dB null into a 10 dB null, so will a 10% variation in loss or excitation on one element.) The null and sidelobe performance of a phased array (one of the big potential virtues) is so sensitive to small excitation errors that I think it's unlikely that simple switched LC networks will be adequate, and one should, instead, rely on more sophisticated signal processing techniques, such as digitizing all receive channels and doing the beamforming in software.
The analysis in this page is derived from work done for the High EIRP Cluster at NASA's Jet Propulsion Laboratory. In that work, one large dish antenna and power amplifier is replaced by a small number of smaller antennas and multiple amplifiers, with the phases of each amplifier adjusted to steer the beam over small angular deflections as well as compensate for changes in their physical properties.
Readers interested in obtaining a copy of the modeling codes used to produce this data should contact the author at:
james.p.lux@jpl.nasa.gov
James Lux
Jet Propulsion Laboratory
Mail Stop 161213
4800 Oak Grove Rd
Pasadena CA 91109