LOW NOISE AMPLIFIERS FOR MEASUREMENT OF SMALL LOSSES.
(Jan 29 2013)

The DUTs

Combinations of three cables is the set of DUTs used to present different input impedances to low noise amplifiers.
 ```device Zre Zim |Zdiff| Phase (ohms) (ohms) (degrees) None 49.41 -0.05 - DUT312 70.02 -12.29 23.97 -31 (2) DUT132 57.64 19.68 21.38 67(100) DUT231 34.03 5.81 16.46 159(192) DUT213 38.09 -12.63 16.92 -132(-99)```
 Table 1. The four DUTs used for presenting different impedances to a LNA. The summed dissipative losses for the three cables is 0.1084 dB (With an unknown uncertainty.) The phase is followed by the phase plus 33 degrees to present values close to the 0,90,180 and -90 that we are used to see.

Measurements on existing amplifiers.

The problem of measuring noise figures with good accuracy is closely related to the problem of measuring small losses accurately. Some amplifiers that are accurately known from here PRECISION MEASUREMENTS OF NOISE FIGURES are measured with the different feed impedances listed in table 1.

 ``` LNA NONE DUT312 DUT132 DUT231 DUT213 Approx phase 0 90 180 270 PSA4-5043 0.6191 0.6434 0.6657 0.6662 0.6383 AD6IW 0.4731 0.4762 0.3871 0.5321 0.5638 MGF1801 0.2874 0.2850 0.2830 0.3205 0.3294 ATF33143 0.2525 0.2074 0.2343 0.3324 0.2981 ATF33143nov 0.1620 0.1486 0.1608 0.1917 0.1812 NE334-S01 0.1420 0.1429 0.1308 0.1607 0.1698```
 Table 2. The NF in dB in the different cases.
 It is obvious from Table 2 that only the PSA4-5043 has its best NF near 50 ohms. The sensitivity of the NF vs the source impedance can be estimated from the second order differences in two dimensions. (DUT312-NONE)-(NONE-DUT231)=DUT312+DUT231-2*NONE =d2re DUT132+DUT213-2*NONE =d2im and
 ``` LNA -d2re -d2im NF PSA4-5043 0.0714 0.0658 0.6191 AD6IW 0.0621 0.0047 0.4731 MGF1801 0.0307 0.0376 0.2874 ATF33143 0.0348 0.0274 0.2525 ATF33143nov 0.0235 0.0180 0.1620 NE334-S01 0.0196 0.0166 0.1420```
 Table 3. Second order differences.
 The measurements can be seen as stepping the impedance in two orthogonal directions. If the optimum NF is close to 50 ohms one should observe parabolas while doing such stepping and the curvature, the second derivative should be proportional to the second order differences given in table 3. Without doing any propher math one can immediately see from table 3 that the PSA4-5043 is optimized for 50 ohms but that the minimum is sharp. This LNA is not well suited for loss measurements. The AD6IW has optimum NF too far from 50 ohms to be useful. None of the remaining amplifiers has its best NF for 50 ohm feed impedance, but they are presumably not far away from having that. The best candidate for use for attenuation measurement is the NE334-S01. It has the most shallow minimum. Unfortunately my attempt to adjust the input coil of the NE334-S01 amplifier failed because I destroyed the FET and have no replacement. For that reason the amplifier for loss measurements by NF degradation is the previous ATF33143nov which has been converted to ATF33143feb. It is very difficult to get the optimum tuning by moving a tap on a coil so I have added a second tunable capacitor. For that reason and because the L/C ratio is now smaller the NF is not quite as good as before. To evaluate the NF this amplifier was compared to a couple of the known amplifiers, see table 4.
 ``` LNA NF (dB) PSA4-5043 0.1847 MGF1801 0.1766 FHX05FAnov 0.1777 ATF33143 0.1813 Average 0.1801 ```
 Table 4. The NF of ATF33143feb evaluated from a comparison with known amplifiers.
 The changes that allowed fine tuning of the input network has degraded the NF by about 0.03 dB. There should be ways to do it better, but for the purpose of this study it is good enough. The effect on S/N when inserting the cables that change the source impedance is shown in table 5.
 ``` LNA NONE DUT312 DUT132 DUT231 DUT213 Approx phase 0 90 180 270 ATF33143feb 0.1800 0.1914 0.1847 0.1903 0.1901 Differences 0 0.0114 0.047 0.0103 0.0101```
 Table 5. The NF in dB in the different cases.

As is obvious from table 5 it would be possible to tune this amplifier a little better, but for the purpose of studying dissipative losses it is not necessary.

The loss in NF when the source impedance has WSWR=1.5 is 0.01 dB. Maybe it could be 0.009 dB with optimum tuning.

Shallowness of NF minimum and absolute NF.

It is obvious that the shape of the NF minimum contains information about the absolute NF of an amplifier. An ideal amplifier would add no noise at all regardless of the feed impedance so it would have a flat NF surface.

It is also clear that an amplifier preceded by a large attenuator would show NF values that are degraded by the mismatch losses.

If one makes a model of the amplifier as a black box and attributes all the noise it produces to two noise sources at the input as illustrated in figure 1 one can draw some interesting conclusions.

 Figure 1. An LNA with a noise EMF and a noise current on the input and a signal source with a signal EMF and a noise EMF.
 Es is the electromotoric force of the signal. En is the electromotoric force of the thermal noise in the source R. Ea is the equivalent noise electromotoric force of the amplifier. Ia is the equivalent noise current of the amplifier. The S/N in U, the output voltage from the LNA is determined only by the the electromotoric forces, currents and impedances on the source side. The impedance of the amplifier, whether it is matched or not does not affect the output S/N. For simplicity of arguments, assume the impedance is infinitely high so no current would flow into the amplifier.. Assume also (for now) that all the noise sources are uncorrelated. If a lossless impedance transformation were applied between the source R and the LNA that would lift the source impedance by a factor of T, the signal voltage and the thermal noise of the source would increase by a factor of the square root of T. The noise voltage due to Ea would remain unchanged but the noise voltage due to Ia would increase by a factor of T. Uncorrelated signals add by power. The effect on S/N (power ratio) of a lossles feed impedance transformation with a factor of T is thus: S/N = T * Es2 / [ T * En2 + Ea2 + T2 * 502 * Ia2] ....................... (1) This simplifies to: S/N = Es2 / [ En2 + Ea2/T + T * 2500 * Ia2] ........................ (2) For the amplifier to have optimum NF for a 50 ohm source impedance, Ea2/T + T * 2500 * Ia2 must have a minimum for T=1. That means that the derivative of this expression -Ea2/T2 + 2500 * Ia2 has to be zero for T=1 which leads to: Ea2 = 2500 * Ia2 ......................(3) Now, we know that for input impedances of 75 and 33 ohms, T = 1.5 as well as for T = 0.667 the NF degrades by about 0.01 dB for the ATF33143feb amplifier. (SWR=1.5) In linear power scale that is a factor of 0.9977. From that we get: 0.9977 * Es2 / [ En2 + Ea2 + 2500 * Ia2] = Es2 / [ En2 + Ea2/1.5 + 1.5 * 2500 * Ia2] = .......................(4) Es2 / [ En2 + Ea2/0.667 + 0.667 * 2500 * Ia2] ....................... (5) Knowing that Ea2 = 2500 * Ia2 we can simplify to: 0.9977 / [ En2 + 2 * Ea2] = 1 / [ En2 + (1.5 + 1 / 1.5) * Ea2] = ....................... (6) 1 / [ En2 + (0.667 + 1 / 0.667 ) * Ea2] ....................... (7) This is two equations which both give: 0.9977 * (En2 + 2.167 * Ea2) = En2 + 2 * Ea2 ....................... (8) from which we can compute: -0.0023 * En2 + 0.162 * Ea2 = 0 ....................... (9) or Ea2 = 0.0142 * En2 ....................... (10) The noise figure is by definition the actual S/N divided by the S/N that would have been obtained with an ideal amplifier having Ea as well as Ia equal to zero. By use of equation (1) twice with T=1 we get the ratio of the result without and with the losses included as: NF = {Es2 / En2} / {Es2 / [En2 + Ea2 + 502 * Ia2]} ..........................(11) By use of equations (3) and (10) we get: NF = {Es2 / En2} / {Es2 / [En2 + 2* 0.0142 * En2]} ..........................(12) This means that NF=1.0284 in linear power scale or 0.12 dB. This is a lower limit for the NF based on the assumption that Ea and Ia are uncorrelated. If we assume that Ea and Ia are 100% correlated equation 1 would become: S/N = T * Es2 / [ T * En2 + ( Ea + T * 50 * I)2] ....................... (13) This leads to: S/N = Es2 / [ En2 + Ea2/T + 100 * Ia * Ea + 2500 * T * Ia2] ....................... (14) Equation (14) is similar to (2). It has one more term in the denominator but the derivative is unchanged so equation (3) is still valid but equation (4) becomes different: 0.9977 * Es2 / [ En2 + Ea2 + 100 * Ia * Ea + 2500 * Ia2] = Es2 / [ En2 + Ea2/1.5 + 100 * Ia * Ea + 2500 * 1.5 * Ia2] .......................(15) By use of equation (3) this becomes: 0.9977 * Es2 / [ En2 + 4 * Ea2 ] = Es2 / [ En2 + ( 1.5 + 2 +1/1.5 ) * Ea2] .......................(16) From (16) we find: Ea2 = 0.0146 * En2 ................(17) Now we can compute the NF as the ratio of (S/N) values computed without and with the amplifier noise included: NF = {Es2 / [ En2 } / {Es2 / [ En2 + Ea2 + 100 * Ia * Ea + 2500 * Ia2]} ................. (18) By use of (3) this becomes: NF = {Es2 / [ En2 } / {Es2 / [ En2 + 4* Ea2]} ................. (18) Inserting (17) we find: NF=1.0584 as power ratio or 0.264 dB. By a comparison with other amplifiers the NF was determined to 0.18 dB. That value is traceable to measurements with boiling water and ice that probably are correct within 0.04 dB. Now, by a completely different method, we have found that the NF has to be between 0.12 and 0.26 dB. (There is a small uncertainty from adopting 0.010 dB NF change for VSWR=1.5 as the differences from table 5. By measuring more accurately and by fitting a paraboloid surface to the five NF points at the five source impedances this error can be made very small.) The NF contour map tells us that the NF of the ATF33143feb amplifier is 0.19 +- 0.07 dB (still assuming that the 0.010 dB change for VSWR=1.5 is correct) The interesting aspect of this result is that with access to a better amplifier with a true NF of a little less than 0.1 dB one would get the absolute NF to within something like 0.04 dB from measurements at a single temperature, the room temperature. Maybe a cryogenic amplifier could be used to calibrate the noise head in a measurement with a conventional NF meter and a circulator in front of the DUT on higher frequencies where it is difficult to know the temperature distribution over the different losses in a hot/cold measurement.