i: . APPENDIX TO AN ANALYSIS OF VECTOR MEASUREMENT ACCURACY ENHANCEMENT TECHNIQUES DOUG RYTTING NETWORK MEASUREMENTS DIVISION 1400 FOUNTAIN GROVE PARKWAY SANTA ROSA, CALIFORNIA 95401 RF L Microwave Measurement Symposium And Exhibition FIJP'I HEWLETT PACKARD www.HPARCHIVE.com
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VECTOR MEASUREMENT ACCURACY …. APPENDIX TO AN ANALYSIS OF VECTOR MEASUREMENT ACCURACY ENHANCEMENT TECHNIQUES DOUG RYTTING NETWORK MEASUREMENTS DIVISION 1400 FOUNTAIN GROVE PARKWAY
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i:.APPENDIX
TOAN ANALYSIS
OFVECTOR MEASUREMENT
ACCURACY ENHANCEMENTTECHNIQUES
DOUG RYTTINGNETWORK MEASUREMENTS DIVISION
1400 FOUNTAIN GROVE PARKWAYSANTA ROSA, CALIFORNIA 95401
To solve for ell and e22 let us use the standard one-port calibration
procedure. With a termination fA on port-l of error adapter x, step
3 of Fig. 3.
(32)
Solving for fA and using equations (22) and (23)
(33)1 b - f mx
f = ---A ell a - f mx
~ith the same termination fA on port-2 of error adaptor y, see
Fig. 3, step 3.
( 34)
Solving for fA and using equations (30) and (31)
(35)1 d + f my
f = - -.,.----:--"'-A e22 C + rmy
Eliminating fA from (33) and (35)
( 36 ) _1_ = _1_ (b - ~::) (' ~ : f my)e22 ell a - fmy
During the thru connection, step of Fig. 3, we know that we can
measure e22 with the port-l reflectometer
( 37 )
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Solving (37) for ell
b - f ml(38) e =_l_11 e22 a - f ml
We can now substitute the value of ~ from (36) into (38) to obtain22
(39)
also from (38)
from (23)
(41) I(elDeOl ) = (b - a) ell Iand from (31 )
(42) I (e23e32 ) = (c - d) e221
We still need the two transmission tracking terms (e,Oe32 ) and
(e23eOl ). This can be obtained from the thru connection, step 1
of Fig. 3, since
and
1(44) S12m = (e23eOl ) 1 e e
- 11 22
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We know ell and e12t therefore
(45) I(e10e32 ) = S21m (1 - ell e22 )
(46) [(e23e01 ) = S12m (1 - ell e22 ) I
Notice that we solved for the e-parameters instead of [T ] or [T ].x y
We chose the e-parameters so that this calibration technique would
be compatible with the other error correction procedures developed
and used earlier.
A1so, the swi tch repea tabil ity errors can be removed by the procedure
in Appendix IX if we use four measurement ports with four mixers or
samplers connected at all times.
The value of fA and 6£ were not needed but can be calculated by
1 b - fr =- mxA ell a - r mx
And~.£. by taking the ratio of (17) to (14) which yields
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Appendix IX: Source and Load Match Error Removal
If we have a system block diagram as shown in Fig. 1 the
characteristics of the switch can be removed by assuming that
the a3 1 0 (non Zo term,) in the forward configuration and
aO' 1 0 in the reverse configuration. This approach is a
generalized method of measuring S-parameters where Zo terminations
are not assumed.
~lbo ~ ao b,X X ---. ---.
PERFECT4- ERROR. -- UNKNOWNbo b;1
Zo REFLECTOMETER a3 ADAPTER TWO-PORT-. ---.SWEEP REVERSE X X 4- --
OSCILLATOR. ~~b3 82
Figure'
Measurement System Block Diagram
In the forward configuration
bo = S'lm aO + S12m a3(1)
b3 = S21m aO + S22m a3
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And in the reverse configuration
b I = Sllm aOI
+ S12m a3'0(2)
b ' = S21m aOI
+ S22m a3I
3
Combining the forward and reverse configurations
( 3)
or
(4) [b] = [Sm] [a]
Since [a] and [b] are square and non-singular
Expanding (5) gives
( 6)
bO'ao - bOaO,
(7) S12m = t; , reverse
b3a3I - b3'a 3(8) S21m = , forward
/:;
b3'aO - b3aOI
(9) S22m = , reverset;
Where
(10) t;t:, aOa3, - a3aO
I
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The typical network analyzer, which measures phase, needs to make a
ratio measurement. Equations (6) through (9) can be factored into
form as follows. Where the incident signals are aO and a3'
(11 )
(12) S12m =
b'b a Io 0 0-, - ---,a3 aO a3
d
, forward
, reverse
( 13)
(14)
Where
a I aaO 3 0S21 m = --...;=----d~---:- , forward
, reverse
(15) d ~
The leakage, missmatch, and repeatability of the switch are
removed by this procedure.
If a3 = 0
( 16)
(zO termination) for the forward configuration
bO _ b3Sllm = a
Oand S21m - a
O
And if aO' = 0 (zO termination) for the reverse configuration
( 17)b '
S = ---L and S22m a3' 12m
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Acknowledgments
The following individuals at HewlettPackard, Santa Rosa, have contributedto error correction algorithms andmeasurement techniques:
John BarrJim FitzpatrickSy Ramey
I would like to thank them for allowingme to present many of their ideas inthis seminar. Thanks are also due toMargie Brown for many hours of assistance.
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References
The following papers have been usefulin stimulating our thinking in errorcorrection techniques. Some of theapproaches used in this seminar originatedin these references.
(1) 1. Kasa, "A Circle Fit t ing Procedureand its Error Analysis," IEEETransactions on Instrumentation andMeasurement, pp 7-14, March 1976.
(2) 1. Kasa, "Closed-Form MathematicalSolutions to Some Network AnalyzerCalibration Equations," IEEETransactions on Instruments andMeasurement, Vol. IM-23, No.4,pp 309-402, December 1974.
(3) Glenn F. Engen, "Calibration Techniquefor Automated Network Analyzers withApplication to Adapter Evaluation,"IEEE Transactions on Microwave Theoryand Technique, Vol. MTT-22 , No. 12,pp 1255-1260, December 1974.
(4) Ross A. Speciale, "A Generalization ofthe TSO Network Analyzer CalibrationProcedure, Covering n-Port ScatteringParameter Measurements, Affected byLeakage Errors," IEEE Transactionson Microwave Theory and Techniques,Vol. MTT-25, No.12, pp 1100-1115,December 1977.
~5) w. Kruppa and Kenneth F. Sodomsky,"An Explicit Solution for the ScatteringParameters of a Linear Two-Port Measuredwith an Imperfect Test Set," IEEETransactions on Microwave Theory andTechniques, Vol. 19, No.1, pp 122-123,January, 1971.
(6) Stig Rehnmark, "On the CalibrationProcess of Automatic Network AnalyzerSystems," IEEE Transactions on MicrowaveTheory and Techniques, pp 457-458April,1974.
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MARCH 1112
(7) Jim Fi tzpa trick. "Error Models forSystems Measurement." MicrowaveJournal. pp 63-66. ~~y. 1978.
(8) C. McKay Allred. "The Calibration andUse of Directional Couplers WithoutStandards." IEEE Transactions onInstrumentation and Measurement. Vol. 25No.1. pp 84-89. March, 1976.
(9) Glenn F. Engen and C1etus A. Hoer,'~n Improved Technique for Calibratingthe Dual Six-Port Automatic NetworkAnalyzer," IEEE Transactions onMicrowave Theory and Techniques, Vol.MTT-27, No. 12, pp 987-993, December 1979