Grasping Y-factor and antenna testing

Article By : John Dunn

A technique for testing an RF or microwave amplifier's properties may also be applied to antennas.

An amplifier's properties can be tested along the lines of the following sketch. Typically, this technique would be applied to an RF or microwave amplifier, but this technique's additional applicability to antenna testing will be addressed shortly.

[EDNAOL 2016JUN08 TA 01Fig1]

A resistor provides Johnson noise at an RMS value of Erms = sqrt (4 k T B R) where "k" is Boltzmann's constant, "T" is temperature in degrees Kelvin, "B" is bandwidth in Hz, and "R" is the resistance value in Ohms. If the temperature of the resistor is varied, the magnitude of the Johnson noise varies too. What we do is use the Johnson noise of a resistor as a test signal to the amplifier's input. We get a noise level from the resistor that varies with the resistor's temperature and in response to that, we look at what happens at the amplifier's output.
A resistor can be used as a variable noise source if we run it at two different operating temperatures. The output noise power of the amplifier is measured as Phot for resistor temperature Thot and as Pcoldfor resistor temperature Tcold. The difference between the two output power levels is taken in terms of their equivalent temperatures which is divided by the difference between the two temperatures of the resistor. That quotient is the measurement of the amplifier's power gain.
If the amplifier were ideal and noiseless, output versus input would look like this:

[EDNAOL 2016JUN08 TA 01Fig2]

With a totally quiet amplifier making no noise contribution at all, the straight line trace in the above graph would go to the origin for a resistor temperature of absolute zero. However, the amplifier does have some noise of its own so the output versus input trace gets shifted upward as follows:

[EDNAOL 2016JUN08 TA 01Fig3]

The amplifier still has some power output even for a resistor that has been cooled to absolute zero.

[EDNAOL 2016JUN08 TA 01Fig4]

When we extrapolate our upper straight line back to the negative intercept on the horizontal axis, we find what we may call the amplifier's own noise temperature, Tamp. As a general statement, the higher the power gain and the lower the value of Tamp, the better.
Apart from the obvious pun of this being a really cool way to test a hot amplifier, this test philosophy can be extended to antenna testing as well.
Mounting the antenna
We mount our antenna of interest in a setting where it can be either exposed to the night time sky or it can be completely covered up by a black-box enclosure. A clear and open sky behaves like a black-box radiator into the antenna at a sky temperature of approximately 35°K. However, when a black box is placed over the antenna, the black box enclosure radiates into the antenna as well and does so at the ambient temperature of the test site, say 300°K.

[EDNAOL 2016JUN08 TA 01Fig5]

This makes the antenna look like that resistor that was being operated at two different temperatures. We will get two different noise voltages coming from the antenna, one for the open sky and one for the box. Putting the cover over the antenna and removing the cover changes the level of noise excitation being delivered by black-box radiation into the antenna under test. Think of that black box as if it were another "sky".
Just to mention as two aside comments, this open sky is not the sky of a busy metropolitan area. It is the sky of a remote and isolated location. Also, I've read some recommendations to aim the antenna at the sun for the hot test condition, but we won't get into that idea here. You might want to look into it though.
In any case, we are setting up two antenna test conditions that are analogous to those two resistor test conditions. Just as before, the output of the antenna will be different when exposed to these two different radiation stimuli, one at the "cold" temperature of the open, night time sky and the other one at the "hot" temperature of the enclosure.

Of course, no antenna is ever used all by itself. There also has to be a feedline connecting the antenna to the first amplifier in the signal path. With that, we can now model our two antenna test conditions as follows:

[EDNAOL 2016JUN08 TA 01Fig6]

We look at the antenna itself, at the loss properties of the feed line and at the properties of a low-noise amplifier (LNA), all of which will be involved if the antenna being tested is ever going to be of any real world use other than as some kind of decorative roof ornament.
First, we define the mean sky noise temperature, TMS. This number will be either Tcold or Thot depending on whether the antenna is exposed to the sky (cold) or is under the black-box cover (hot).
We choose that Thot is the ambient temperature because we are not using the sun in any of this. That hot temperature, the ambient temperature, is also the temperature of the feedline. Just to draw a conceptual distinction, we call the temperature of the feedline by the designator, To, which always stays put at the ambient temperature.
We let TAS be the "sky" (open or covered up) antenna temperature as seen at the output of the feed cable at which location we defined a G/T reference plane (see the preceding sketch) and for which we may write the following:
TAS = alpha * TMS + (1—alpha) * TO whereTMS is either Thot or Tcold and where To is always Thot = Tambient.
If the feedline were lossless
If the feedline were lossless with alpha = 1, the TAS would arise only from incoming radiation from the sky or from the cover and would depend solely on the pickup performance of the antenna.
Conversely, if the feedline were totally lossy with alpha = 0, the TAS would arise only from the feedline itself at the ambient temperature. In that case, whatever was going on with the antenna would be completely masked by the cable's immense loss which would utterly negate the antenna's reason for existence except perhaps as that roof ornament.
At the low noise amplifier's input at the G/T plane, we have a signal coming from the cable which we may represent as TAS. The output of the LNA in response to that input may be represented as (TAS + TLNA.) * GLNA.
We pause for a moment at that G/T plane (metaphorically speaking) to consider the noise factor, F, of the low noise amplifier. Its effect will be the equivalent of having more cable loss. There will be a diminished "signal to noise ratio" and a larger number for TAS. I know that everything we're doing is with noise, but the sky noise is our "signal" for our test purposes.
Recalling now that we had previously assigned the ambient temperature at the measurement site as TO, the "hot" condition, usually around 300°K, let us now call that temperature our Thot of 300°K. Similarly, we have the sky temperature as our Tcold, typically at that 35°K. (Brrrrr.)
We define Phot as the power measured out of the LNA with the antenna exposed to the "hot" source and Pcold as the power measured out of the LNA with the antenna exposed to the "cold" source. We next define that y = Phot / Pcold for which we write Y = Phot dB—Pcold dB = 10 * log10 (y).
We may refer to all of these hardware goodies as an antenna module consisting of the antenna itself, the feedline with its ohmic loss and the LNA with its gain and its linear noise factor. For all of this stuff taken together which we may now call the module, we come to the following:

[EDNAOL 2016JUN08 TA 01Fig7]

Starting with the expression: y = Phot / Pcold = (Thot + TAS) / (Tcold + TAS).
We rearrange the equation to get: y * Tcold + y * TAS = Thot + TAS
By further rearrangement, we have: (y—1) * TAS = Thot—y * Tcold
By still further rearrangement, we have: TAS = (Thot—y * Tcold) / (y—1)
What does this all mean? It means that we want the slope of the output versus the input to be as steep as possible and we want the TAS intercept to the left of the origin to be as close to the origin as possible. The better the antenna, the more these two statements will be so.
As Karen Carpenter once sang: "We've only just begun." If you would like to go deeper into the use of the Y-Factor and into using that to find noise figures, please see:

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