Does antenna thermal noise matter if noise power is higher?

Question

I'm working on designing a radio system for CubeSats and I want to make sure that the sensitivity of my receiver is as high as it can be. I'm trying to understand what parameters are going to set the lower bound for the performance I can squeeze out of my design.

One thing I am still confused about is antenna temperature and thermal noise. The closest question I could find is here: https://physics.stackexchange.com/questions/289893/is-antenna-noise-temperature-relevant-if-the-physical-system-temperature-is-high

I don't think the top answer on that post answers the question that the original poster was asking.

Basically, the noise power for a system is given by the following formula for Johnson-Nyquist noise: \$ N = k_bT\Delta B\$. As I understand it, this is present at all points of a system and exists regardless of resistance, material, input loss, or any other parameters. Basically, this sets a lower limit on the amount of noise power you can get your system down to.

I've started learning about antenna noise temperature though and I know that this is a function of what the antenna is actually "seeing" in its radiation pattern. When pointing at the sky, the antenna noise temperature can go as low as 3 K, which is the microwave cosmic background.

For receivers at 300 K, does a low antenna noise temperature matter at all? As I understand it, any antenna at a physical temperature of 300 K or so will have a much higher thermal noise power than anything received as antenna noise.

The advantages of putting the LNA close to the antenna have to deal with avoiding resistive losses over a transmission line and not with minimizing thermal noise because thermal noise is omnipresent in equal amounts in the circuit, correct?

Answer 1

Why does a lossless cable not contribute noise?

Let's look at the circuit, and assume a completely matched system for convenience.

R1 is the source impedance. In the case of a radio telescope, this will be the cold of space, ideally a few K or tens of K. You know how to handle this noise.

R5 is the LNA input impedance. We might be using a cryo-cooled LNA, which would make that noise contribution fairly small as well.

The lossy antenna and cable is modeled by the pi pad of R2,3,4, whose resistors are at 300 K, so are each generating their full KTB available noise. The weird value of 0.87 dB is chosen to get nice round numbers for the attenuator components. 10K and 0.5 Ω resistors would give you 0.087dB (spot a pattern?). There are any number of pi-pad calculators and formulae on the net available with a quick search.

^{simulate this circuit – Schematic created using CircuitLab}

We know that each resistor has Johnson-Nyquist noise, which value is independent of its actual resistance. When such a resistor is loaded with its matched resistance, it will transfer KTB to the load.

I am not going to produce any formulae here. A casual reader of this question will not benefit. Somebody wanting to understand must do the sums, or at least the simulation, themselves, to drive the point home. Therefore, only hand-waving follows.

Consider the noise contribution of R4. We can model the noise, and this works fine in Spice, as a current source in parallel with R4. It also works to model the noise voltage in series though, especially in Spice, using a parallel current source means we don't have to break a wire to add it.

The R4 noise current passing through R4 would produce the normal expected noise voltage in R4. Now, when we put R5 in parallel, R5 is much, much smaller than R4, so reduces the noise voltage to roughly 1/20th of what R4 would produce without being shunted. R5 receives nothing like the noise power you would expect for a fully matched situation.

We can do a similar thing for R3's noise power. The R3 current source in parallel with it produces only a very small voltage across R3, and as a result, little of that noise current flows in R5.

Even though the resistors are at 300 K, and able to produce their full KTB noise power into a matched load, the large mismatch between them and the LNA input impedance means that only a small fraction of that possible noise power is actually manifest in the LNA input resistance.

In the limit, as the line becomes lossless, R2 and R4 tend to infinity, and R3 tends to zero, the power they can produce in R5 tends to zero.

In a very lossy line, R2 to R4 assume values more commensurate with 50 Ω. For instance, a 6dB pad uses R2,4 = 150 Ω, with R3 = 37 Ω. These are a better match to R5, and more of their noise power is transferred to it.

In the limit of a very lossy line, R2 and R4 tend to 50 Ω, R3 to infinity, and a whole KTB gets dissipated in the LNA from R4.