Chapter 3. Six-Port Network Analyzer Theory
3.3 S-Parameter Measurements with the 6-Port Network Analyzer
3.3 S-Parameter Measurements with the 6-Port Network Analyzer
_____ ..,. Divider ...---,,,
~,p..;;;.;;._~
6-Port #1 6-Port #2
Figure 3.6 A full 5-parameter measurement system using six-port network analyzers.
With equation (3.99), then, a2f a1 can be determined from the observed r's for any measurement. The remaining difficulty is to find the values of the C's.
This can be done by noting that equation (3.98) is linear in the C's. Using at least three known values of r I, r 2 and a2f a1, a system of linear equations is formed which can be solved for the C's.
The values of the r's to plug into this set of equations are, of course, directly available from the measurements. The values of a2f a1 must be derived somewhat indirectly. This is done by measuring a set of calibration standards for which the S-parameters are approximately known. A set of precision transmission lines of approximately known length is a common choice. These lines are reciprocal, and so their complete S-parameters can be found from the knowledge of Sn, S22 and
.6.
which is found as noted above.IS12 I = IS21 I =
v'l.6. -
Sn S22 I (3.100)arg( .6. - S1 i S22) arg(S12) = arg(S12) =
2
+
mr (3.101)The mr in equation (3.101) results from the sign ambiguity of the square root.
Since in this case, the length of each of the calibration standards is approximately known, this sign ambiguity can b_e resolved by calculating the expected value of arg(S12 ) and choosing the sign that gives the value closest to this.
With all the S-parameters for the calibration standards thus measured, equa- tions (3.95) or (3.96) can be used to find a2/ a1 for each measurement, and de- termination of the values of the C's can proceed.
The values of the C's change when the phase shifter or variable attenuators are switched. Thus C's must be calculated and stored for all possible configura- tions of the measurement system.
After all calibration has been completed, measurement of the S-parametcrs of an unknown two-port proceeds as follows: the measurement system is stepped through all combinations of the phase shifter and attenuator positions and the
values of the I''s resulting are stored. These are then summed into a matrix and least-squares estimates of S11 , S22 and 6. for the unknown network are determined through a set of equations like (3.52). Then the C's are used to find az/a1 for each of the measurement configurations, and calculate S12 and S21 by solving equations (3.95) and (3.96), respectively. The resulting values of S12 and S21 , respectively, are averaged to yield the final estimates of these quantities.
This is the general procedure. If it is known that the two-port being mea- sured is reciprocal, then greater accuracy can be achieved by using (3.100-3.101) to find S12 _ S21 . Then the results from (3.95-3.96) and (3.99) can be used only to resolve the sign ambiguity in (3.101).
By changing the positions of the variable attenuators, the value of
az/ a
1can be made arbitrarily large or small. This can be used to advantage in some measurement situations. In measuring an amplifier, for example, the signal on the output side of the amplifier is at a much higher level than that on the input side. Most six-port reflectometers have their best accuracy when measuring values of
r
for which1r1 <
1. By making la2/ a1I
approximately equal to the gain of the amplifier, ratios near unity will be measured by both the input and output reflectometers.In summary, two six-port reflectometers can be combined in a system that can measure the full S-parameters of unknown two-ports. This system has none of the switch-dependency problems that can occur with the four-port-based sys- tems. As long as the various switches (attenuators and phase shifters) in the six-port system are repeatable, their effects are calibrated out.
References
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