These calculations are good but they assume TRM increasing without any compromises and changes. So your 2x TRM scenario is basically 2x power, 2x collection area and 2x gain scenario. Without platform-level changes you won't be able to grow your radar and its total power output much. TRM increase would still result in a gain increase but that wouldn't be fully realized since a higher gain also means a slower scan at the same dwell time. A dwell time decrease would be needed. Range gain would likely be around 12%.

Thank you and you know what, You can also add dwell time too as it is also a variable in radar range equation. Also in AESA you can at least in theory changes the antenna tapering e.g from Taylor -40 dB to something like Cos or if you want a full gain an untapered one. This will increase or decrease your beamwidth and sacrifice or put some more gain (within limits of course) and you can optimize your dwell time for desired coverage.

Shall we revisit our 1200 TRM AESA and 2400 one with 10 Watt emitted power ? to demonstrate the effect of dwell time. So you dont have to guesstimates or even use word "Likely" at all. Let's say we want to introduce the beam dwell time into the problem. This will introduce us into two terminology Dwell time and Frame time. The dwell time is how long your beam sticks in one area or one beam, this can usually be 0.025, 0.01 or 0.1 or even longer in seconds depending on requirement. the Frame time is your total time required to scan your area of interest.

As for the addition it would look like this :

Rfac = ((2400/1200)^3*(10/10)*(Dwellx/Dwellr))^(1/4)

Where dwell time is added

Now let's assume both AESA have pencil beamwidth which is the beamwidth where vertical and horizontal one is the same or close enough for not to be a Fan beam. We can find the beamwidth of each antenna by simple equation :

Beam in Deg = 100/SQRT(N)

for AESA with 2400 TRM

Beam=100/SQRT(2400)

Beam = 2 Degrees

AESA with 1200 TRM

Beam=100/SQRT(1200)

Beam = 2.9 Degrees

After that we can assume a scan sector.. let's say a 7 bar scans in an arc of 120 degrees and dwell time of say 0.025 seconds which typical for fighter aircraft with an X-band operating frequency. if we divided those area with beamwidth it will result with following :

Radar1200 TRM will complete the scan with 158 Beam Thus the total frame time required would be 158 * 0.025 = 3.96 Seconds

Radar 2400 TRM will complete the scan with 316 Beam Thus the total frame time required would be 316 * 0.025 = 7.92 Seconds

or twice the amount of the 1200 TRM Radar. So yeah the scan is slowed. However Since the dwell time is the same the factor would be 1, thus the growth is the same.

Rfac = ((2400/1200)^3*(10/10)*(.025/.025))^(1/4)

Rfac =1.68

But now, what if we want the frame time or your total scan time the same ? let's say we want to complete the scan in 3.96 seconds same as the 1200 TRM AESA ? We have to reduce the beam dwell time of our 2400 TRM Radar by half which is 0.0125 seconds to get the same frame time. Does our 2400 TRM AESA still have advantage ?

Rfac = ((2400/1200)^3*(10/10)*(.0125/.025))^(1/4)

Rfac = 1.41

So as you see, our 2400 TRM AESA Still have 41% advantage over the other with smaller TRM counts. although yes not as extreme as one with slower scan. and now we increased the power of the 1200TRM AESA by twice.. see how much it can offset the advantage of having more TRM's

Rfac =((2400/1200)^3*(10/20)*(.0125/.025))^(1/4)

Rfac = 1.18

The AESA with larger TRM counts still trump over the other although with markedly 23% "loss" compared to the one where the power of the modules are equal. If the 1200 TRM AESA also increase the dwell time then there would be point where it can compensate the smaller number of TRM's

Shouldn’t two radars with different transmit powers have two different P-x and P-ref? If that’s the case shouldn’t those differences also be included in the N^3 term, not just module count?

It's included already. if you properly read my writings you tried to quote you will see how the N^3 factor arise. I honestly confused on what you are trying to criticize here. Considering that people has been using 4th root laws the way i did for long time to compare radar.

Or maybe you mean i should do the calculations twice. So each radar will have its own factor then compared the result. Let's say.. now we calculate 2 AESA separately. One with 1200 TRM vs another with 1200 TRM but twice the power in this manner ?

So i have to write them separately like that ?

Saying that signal processing power and complex waveform management matter is “bringing something to the table.”

I am expecting something that is in Radar range equation. Let's say what it can do to reduce the required SNR. Something like pulse integration.. like when you integrate the pulse in coherent manner vs one with noncoherent. There are calculation methods for it and there is simplification too to allow introduction to Radar Range Equation.

and if you are into processing. Let's say you wish to introduce a complex algorithm like say a STAP to cancel clutter, you can show how much load it could have in computing maybe in terms of MIPS or maybe how a faster clock processor can handle that. Then we can actually have something to compare with.

And i will close this post with remarks from the book "Radar Techniques using Array Antenna" by Wulf-Dieter Wirth on AESA. You can find it in page 76 chapter 4 of the book.

It is very revealing. The most effective means of increasing performance for AESA is having more TRM's.