We’ve been doing a blog series focused on the new Phaser (CN0566) Phased Array exploration kit from Analog Devices.
And in the last blog post, we finally generated an array pattern
So, we did see the maximum gain response in the direction that our RF source was pointed (30 deg). That was good and confirmed that our simple beam steering equation was working. But what that equation didn’t predict was all the other peaks and nulls in that pattern. Where did they come from, and what can we do about them?
Antenna Gain, Array Factor, and Element Factor
To understand those peaks and nulls, we have to first understand that the total antenna gain is a function of two parts: the element factor and the array factor.
The element factor is the radiating pattern from a single element of the array. This is defined by the construction of the element, and it’s not something we can change electrically – so we’ll just leave it as a constant in our analysis. The Array Factor is the portion of the antenna gain that we can influence by beamforming. Therefore let’s focus on understanding that.
Each element of our linear array receives a signal that is delayed relative to the element next to it. And the array factor is the summation of all those signals:
If we do some math to rearrange this, and assuming a linear array with uniform spacing and all elements at the same gain, then we can arrive at a normalized array factor in this form:
Arik Brown's book, "Active Electrically Scanned Array", has an excellent derivation of this equation in Appendix A. And at the end, we have this sin(N) over N * sin typeofequation. It’s not intuitively obvious what this equation means, so let’s plot out a few cases.
This is the array factor plotted for different numbers of elements (N). The Xaxis is the steering angle – which is just a function of the phase delta. In this plot, you can see that as the number of elements (N) increases from 2 to 4 to 8, the main lobe beamwidth narrows and the number of lobes and nulls increases.
From this plot, there are a couple of key measurements to consider:
 Half Power Beam Width (HPBW): this is the width of the main lobe measured at half power (i.e. 3dB) down from the peak
 First Null Beam Width (FNBW): this is the width of the main lobe measured at its null locations
 First sidelobe amplitude: This is how far down from the main lobe the first sidelobe is. For a large array of uniform element gain, this will be 13 dB down from the main lobe.
If we simplify the array factor equation to just mechanical boresight (i.e. sin(theta) = 0), we can quickly solve for the desired HPBW (GA=1/sqrt(2)) and FNBW (GA=0). Doing so gives us the following results:

HPBW 
FNBW 
N=8 
13° 
30° 
N=4 
27° 
62° 
N=2 
62° 
180° 
d = 14 mm, and f=10.3 GHz 
Putting Array Math to the Test with the Phaser
Now we know where the sidelobes are coming from – and how many we should expect for various array sizes. And we also calculated what our ideal beamwidth should be. So let’s try it out with the Phaser, and compare results. As we did in the last blog post, we’ll use the “beamsteer.py” file found here.
Setting the RF signal directly in front of the Phaser (mechanical boresight), and using a frequency of 10.3 GHz, gives us these patterns:
HPBW (Calc/Meas) 
FNBW (Calc/Meas) 

N=8 
13° / 14° 
30° / 29° 
N=4 
27° / 28° 
62° / 62° 
N=2 
62° / 58° 
180° / 180° 
d = 14 mm, and f=10.3 GHz 
Hey, that’s not too bad! Our measurements are only off by a degree or so. This math stuff works! But what about the element factor? Where does that come in?
Antenna Gain, Array Factor, and Element Factor
I’ve been careful not to call these plots antenna plots – instead, I call them array factor plots. And to be more accurate, it's an array factor plot at one particular element gain. But if you want the true antenna pattern, you would use an antenna chamber to measure the gain at different angles. But that can be time consuming and expensive. So for our phased array “exploration” purposes, how far off are we?
Here's a comparison of what we’re doing (aka “Electrical Scan”) vs. an antenna chamber (“Mechanical Scan”) vs an HFSS simulation of the array (thank you to Dr. Laila Fighera Marzall at CU Boulder for the simulation!):
We’re not far off on the peaks and nulls locations. But where we differ is on the gain that is influenced by the element gain – so you see some variation at the peak levels of the sidelobes as we move away from the main lobe. But for our exploration into tapering, grating lobes, monopulse tracking, and radar, this is not going to be a problem. Just know that it’s there!
Summary
We’ve explained where all those peaks and nulls come from. And even the sizes and shapes of them. But unfortunately, sidelobes are never desirable. So how do we get rid of them? That is what we’ll tackle next in a blog about array tapering!