# The need for infinite preamplifier gain?

A recent posting on social media kicked off some discussion about infinite gain preamplifiers, mostly in the context of an unloaded very short vertical.

Over the past couple of years I’ve had a number of comments and questions about active antennas, instigated by my ARRL book, Receiving Antennas for the Radio Amateur.

The “main ingredient” of an active antenna (in this discussion, we’ll center on the very short WHIP), is the preamplifier, which generally takes the form of an FET source follower.

A true source follower (or ideal cathode follower) is theoretically capable of INFINITE power gain). In practice, modern FET input op-amps have an input resistance on the order of a teraohm or so, and an input capacitance of about a picofarad.

Although we can’t QUITE get to infinite power gain with a real FET (or FET input op amp), we can get EXTREMELY high power gains. Assuming an output (source) resistance of 1Kohm and an input resistance of 1 teraohm, a voltage follower will have a power gain of 10^21:1…..not too shabby. (This is assuming essentially a DC signal, where the input parallel capacitance can be ignored).

At this point in time, there has been no mention of noise… but it is key to the problem.

Let’s consider the case of a low noise preamplifier (assume Noise Figure 1dB) used ahead of a receiver with Noise Figure 6dB. We can calculate the system Noise Figure, ie the cascade of the two elements, vs preamplifier gain.

It is easier to solve this problem by working in equivalent Noise Temperatures, $$T_s=T_1+T_2/G_1$$. So converting the two Noise Figures to Noise Temperatures, we can calculate system Noise Temperature vs preamplifier Gain and plot it. Above is a plot of system Noise Temperature vs preamplifier Gain, also shown on the right hand axis is the system Noise Figure.

For this scenario, there is a significant improvement in Ts and NFs as Gain increases… up to about 20dB, after which further increase in Gain makes very little difference. Above, zooming in on the results, the behavior is very clear. Note that it is dependent on the scenario, and the plot will be different for different scenarios.

We do not need to consider the antenna gain to find this point of diminishing returns of Ts and NFs vs preamplifier gain, and to show that for this scenario, a Gain of 20dB or 100 is sufficient to have achieved most of the benefit of a preamplifier with NF=1dB.

The bigger question is whether that system Noise Figure or Noise Temperature is ‘sufficient’ for a specific application.

Again, it is easier to work in equivalent Noise Temperature.

Consider a scenario of an antenna system, with gain -30dB (or 1/1000) at 3.5MHz. ITU P.372 suggests the expected external or ambient noise is 47350000K, and with an antenna system gain of 1/1000, the noise power presented to the preamplifier is $${T_e}^{\prime}=\frac{47350000}{1000}=47350 \text{ K}$$.

A useful metric in system design is the extent to which the external S/N is degraded by the receiver system, I will call it Signal to Noise Degradation (SND).

$$SND=10 log\frac{\frac{S_{ext}}{N_{ext}}}{\frac{S_{ext}}{N_{int}+N_{ext}}}$$

Simplifying this by dividing top and bottom by $$S_{ext}$$ we get

$$SND=10 log\frac{N_{int}+N_{ext}}{N_{ext}}$$.

So, SND gives us a metric that simply depends on the external noise and the receiver internal noise, a quantitative measure of the system in an application context.

Let’s take the system Noise Temperature with the above scenario and 20dB of preamplifier gain to be 84K, we can calculate SND due to receiver system internal noise to be $$SND=10 log\frac{N_{int}+N_{ext}}{N_{ext}}=10 log\frac{84+47350}{47350}=0.08 \text{dB}$$.

So, in this scenario, the receive system equivalent noise temperature is so low that there is only a 0.08dB degradation in the off-air S/N ratio.

You might see that a system Noise Temperature of 5000K (NFs=12dB) is not going to degrade S/N much (<0.5dB).

## Caveats

But here is the problem…

The methods presented here apply to linear systems,  they do not capture the effects of non-linear behavior such as IMD noise.

It is easy to build a preamplifier with high gain, harder to build one with low noise, and even harder to build a broadband one with very load IMD noise.