Finding the optimum input impedance for an LNA on 1296 MHz.
(May 5 2013)

#### The impedance transformers.

By use of three sections of coaxial cable in different orders one can arrange for an impedance transformation that converts a 50 ohm source to SWR 1.5 with four orthogonal phases that well span the Smith chart.

One 50 ohm 0.25 wl section, one 50 ohm 0.125 wl section and one 0.25 wl 70 ohm section would do the job. While trivial on 144 MHz it is not so easy to make a 0.125 wl cable on 1296. Connectors are too long. A set of three cables with SMA connectors, one male and one female on each one was arranged and evaluated in a previous study Far from perfect, but good enough for the present study.

The performance of the impedance transformers is shown in table 1.

 ```Device Zre Zim Phase Ideal Diff VSWR DUT138 73.82 5.71 11 0 11 1.49257 DUT183 45.26 -18.94 -93 -90 -3 1.50320 DUT381 34.18 2.49 169 180 -11 1.46963 DUT831 57.36 19.41 59 90 -31 1.46996```
 Table 1.Impedance transformers.

The dissipative losses of the impedance transformers is 0.0807 dB (With an unknown uncertainty that might be in the order of 0.001 dB.

#### Theory for the NF dependence of source impedance.

Combining equations (2) and (3) from this theoretical investigation: Shallowness of NF minimum and absolute NF. gives this expression:

S/N = Es2 / [ En2 + Ea2/T + T * Ea2] .....................(1)

S/N is the signal to noise power ratio at the output of an amplifier that is fed with a weak signal from a room temperature source at an arbitrary but fixed bandwidth.

Es is the signal voltage.

En is the RMS noise voltage of the room temperature signal source.

Ea is the RMS noise voltage added by the amplifier.

T is the impedance ratio between the actual feed impedance and the feed impedance that gives the optimum NF. (Obviously T=1 when the feed impedance is optimum.)

Equation (1) is valid if the voltage noise and current noise on the amplifier are totally uncorrelated. In case they are fully correlated one obtains the result shown in equation (2).

S/N = Es2 / [ En2 + Ea2/T + 2 * Ea2 + T * Ea2] ....................... (2)

Equations (1) and (2) are closely related to the definition of NF, the noise figure: With a signal coming from a room temperature source, the noise figure is the actual S/N of a signal at the output of an amplifier divided by the S/N that would have been present if the amplifier were ideal and did not add any noise of its own. In other words, S/N from equations (1) or (2) with the actual value of Ea divided by S/N from the same equation with Ea set to zero.

NF= En2 / [ En2 + Ea2/T + T * Ea2] ...................(3)

NF= En2 / [ En2 + Ea2/T + 2 * Ea2 + T * Ea2] ...................(4)

(3) and (4) are derived from (1) and (2) respectively. By setting P=Ea2/En2 we can write (3) and (4) as:

NF=1 / [ 1 + P * T + P / T ].........(5)

NF=1 / [ 1 + P * T + 2 * P + P / T ].........(6)

Equations 5 and 6 represent the upper and lower limit for the NF depending on to what extent the equivalent noise voltage and current is correlated. The NF has a minimum for T=1.

It must be noted that the conventional formula for NF vs source impedance is like this: The equation is derived here: http://mobiledevdesign.com/images/archive/302Harter20.pdf
I do not know whether it is equivalent to the simple formula I have deduced......

There are two reliable methods for relative NF evaluation. One can use S/N measurements with stable signals or one could use conventional NF meters such as the HP 8970A with circulators. Both methods provide very accurate results on a relative scale, but both methods require a stable temperature. One can compare two amplifiers with good accuracy even if the zero point drifts with the temperature.

The measurements on this page were performed after midnight on April 29 2013 at a time with no winds outside. (In daytime with varying sunshine through the windows the temperature is always unstable.) Measurements were done with a HP8970A and circulators. For details look here: Using the 8970A with circulators.

The temperature on the noise head was between 23.90 and 24.05 degrees on the noise head and between 23.77 and 23.86 on the attenuator between the circulator and the LNA. The error in Te due to the temperature variations is in the order of 0.7 times the temperature difference so the this error is about 0.1 K and it is not compensated for in the tables.

There are 5 measurements for each amplifier and the temperature differences within each group of five measurements is below 0.05 degrees Centigrade.

#### Application of the NF vs impedance function on the G4DDK amplifier.

A G4DDK amplifier (this particular amplifier, SM0ERR, showed NF=0.187 dB at the EME meeting Orebro 2012.) was measured directly and with the impedance transformers listed in table 1. The results are shown in table 2.

 ```Device Zre Zim Te NF NFCorr Gain (Ohms) (Ohms) (K) (dB) (dB) (dB) Direct 50.00 0.00 16.3 0.2375 0.2375 40.59 DUT138 73.82 5.71 21.0 0.3036 0.2229 39.38 DUT183 45.26 -18.94 26.7 0.3825 0.3018 40.91 DUT381 34.18 2.49 26.2 0.3756 0.2949 40.83 DUT831 57.36 19.41 20.8 0.3008 0.2201 39.35 ```
 Table 2.The G4DDK amplifier. The NFCorr column is with 0.0807 dB subtracted for the impedance tuner dissipative losses.
 The G4DDK design has only one degree of freedom on the input tuning. Table 2 shows that the NF is not minimum for a 50 ohm source impedance. The amplifier is however tuned for optimum NF on a 50 ohm source. We may make a guess on the optimum source impedance, the associated NF and the offset of the measured NF values. (Remember they are on a relative NF scale.) We can then compute the impedance transformation ratios T for the 5 measurements and apply equation (5) to compute the expected NF at 5 points in the Smith chart. With five data points and four unknowns a least squares fit should give the optimum feed impedances together with a reasonable estimate of the errors. From (5) we would get a lower limit for the absolute NF and from (6) an upper limit. Table 3 is generated from the data in table 2 by use of this fortran program lna-1.0.tbz (5219 bytes)
 ```DUT000 (50.00, 0.00) 0.2375 0.2373 -0.0002 DUT138 (73.82, 5.71) 0.2229 0.2217 -0.0012 DUT183 (45.26,-18.94) 0.3018 0.3024 0.0006 DUT381 (34.18, 2.49) 0.2949 0.2943 -0.0006 DUT831 (57.36, 19.41) 0.2201 0.2216 0.0015 Optimum source impedance (65.1, i 12.9) Ohms Optimum NF (experimental scale) 0.2174 dB (Te= 14.9 K) Shallowness parameter P=0.0410 Optimum NF computed from shallowness assuming uncorrelated Rn and In 0.3426 dB ```
 Table 3.Output from lna-1.0 with the data of table 2 for the G4DDK amplifier as input. The columns are impedance, measured NF, computed NF from equation (5) and the difference between computed and measured.

The optimum NF at (65.1, i12.9) ohms is 0.02 dB below the NF with a 50 ohm feed impedance. The difference is in the order of 2.4 K in system noise temperature and significant only with very quiet antennas. Near 50 ohms, the NF with impedance variation is not negligible however so the G4DDK amplifier is not suitable for measurement of small losses of adapters, relays and other stuff by their influence on the NF.

#### Application of the NF vs impedance function on the L LNA amplifier by AD6IW.

Table 4 gives source impedances and associated NF values for the L LNA amplifier.

 ```Device Zre Zim Te NF NFCorr Gain (Ohms) (Ohms) (K) (dB) (dB) (dB) Direct 50.00 0.00 17.8 0.2587 0.2587 22.88 DUT138 73.82 5.71 25.9 0.3715 0.2908 22.15 DUT183 45.26 -18.94 25.8 0.3701 0.2894 22.60 DUT381 34.18 2.49 25.3 0.3633 0.2826 23.08 DUT831 57.36 19.41 25.7 0.3688 0.2881 22.42 ```
 Table 4.The L LNA amplifier. The NFCorr column is with 0.0807 dB subtracted for the impedance tuner dissipative losses.
 Table 5 is generated from the data in table 4 by use of this fortran program lna-1.0.tbz (5219 bytes)
 ```DUT000 (50.00, 0.00) 0.2587 0.2587 0.0000 DUT138 (73.82, 5.71) 0.2908 0.2909 0.0001 DUT183 (45.26,-18.94) 0.2874 0.2873 -0.0001 DUT381 (34.18, 2.49) 0.2826 0.2827 0.0001 DUT831 (57.36, 19.41) 0.2881 0.2880 -0.0001 Optimum source impedance (48.9, i -0.4) Ohms Optimum NF (experimental scale) 0.2586 dB (Te= 17.8 K) Shallowness parameter P=0.0448 Optimum NF computed from shallowness assuming uncorrelated Rn and In 0.3726 dB ```
 Table 5.Output from lna-1.0 with the data of table 4 for the L LNA as input. The columns are impedance, measured NF, computed NF from equation (5) and the difference between computed and measured.

The optimum NF is at an impedance very close to 50 ohms. This amplifier is well suited for the measurement of dissipative losses in adapters, relays and other things we may want between the antenna and the LNA.

#### Application of the NF vs impedance function on an experimental LNA for 50 ohm feed impedance.

For details look here: An experimental amplifier optimized for 50 ohm feed impedance. Table 6 gives source impedances and associated NF values for the experimental amplifier.

 ```Device Zre Zim Te NF NFCorr Gain (Ohms) (Ohms) (K) (dB) (dB) (dB) Direct 50.00 0.00 12.6 0.1847 0.1847 25.93 DUT138 73.82 5.71 20.2 0.2924 0.2117 25.29 DUT183 45.26 -18.94 19.6 0.2840 0.2033 24.54 DUT381 34.18 2.49 20.0 0.2896 0.2089 26.57 DUT831 57.36 19.41 20.4 0.2952 0.2145 26.69 ```
 Table 6.The experimental amplifier. The NFCorr column is with 0.0807 dB subtracted for the impedance tuner dissipative losses.
 Table 7 is generated from the data in table 6 by use of this fortran program lna-1.0.tbz (5219 bytes)
 ```DUT000 (50.00, 0.00) 0.1847 0.1847 0.0000 DUT138 (73.82, 5.71) 0.2117 0.2117 0.0000 DUT183 (45.26,-18.94) 0.2033 0.2033 -0.0000 DUT381 (34.18, 2.49) 0.2089 0.2089 0.0000 DUT831 (57.36, 19.41) 0.2145 0.2145 -0.0000 Optimum source impedance (49.8, i -2.7) Ohms Optimum NF (experimental scale) 0.1842 dB (Te= 12.6 K) Shallowness parameter P=0.0385 Optimum NF computed from shallowness assuming uncorrelated Rn and In 0.3222 dB ```
 Table 7.Output from lna-1.0 with the data of table 6 for the experimental amplifier as input. The columns are impedance, measured NF, computed NF from equation (5) and the difference between computed and measured.

The optimum NF is at an impedance close to 50 ohms. It would be possible to tune it much closer, but it is close enough for this amplifier to be useful for loss measurements. The NF is significantly lower than the NF of the L LNA

#### Verifying loss measurements.

DUT31 which gives a very small impedance transformation was inserted in front of the L LNA and the experimental amplifier. In both cases the noise temperature increased by 4.0 K which corresponds to 0.0568 dB. The previously determined value was 0.05533 dB. The measurements on this page have an uncertainty of about 0.1 K (0.0014 dB) so there is no disagreement.

Further studies will be made when the HP8970A measurements are automated. Then there will be one more decimal on noise temperatures.