TRL algorithm to de-embed a RF test fixture T. Reveyrand University - PowerPoint PPT Presentation

Introduction TRL Reciprocity Scilab Code TRL algorithm to de-embed a RF test fixture T. Reveyrand University of Colorado - Boulder ECEE department 425 UCB Boulder, Colorado 80309 USA July 2013

Introduction TRL Reciprocity Scilab Code TRL 1 Standards THRU and LINE Measurements � � � � T 12 T 21 T OUT parameters : and T 11 T 22 � � � � T 12 T 21 T IN parameters : and T 11 T 22 � � � � T 11 T 21 The THRU equality : and T 11 T 22 � � T 21 The REFLECT equality : Extracting T 11 Reciprocity 2 Scilab Code 3 Presentation Example Insights

Introduction TRL Reciprocity Scilab Code Motivations for this talk De-embedding of a test-fixture measured in the coaxial reference planes ; Perform a TRL calibration when the VNA provides some ill conditioned solutions ("Reflect" checking not correct) ; Offer a open, complete and ready-to-use source code for educational purpose.

Introduction TRL Reciprocity Scilab Code Some definitions : [S] parameters a 1 a 2 A b 1 b 2 � b 1 � S 11 � a 1 � � � S 12 = · (1) b 2 S 21 S 22 a 2

Introduction TRL Reciprocity Scilab Code Some definitions : [T] parameters a 1 a 2 A b 1 b 2 � a 1 � T 11 � b 2 � � � T 12 = · (2) b 1 T 21 T 22 a 2

Introduction TRL Reciprocity Scilab Code Some definitions : [T] parameters a 1 a 2 A B C b 1 b 2 [ T Total ] = [ T A ] · [ T B ] · [ T C ] (3)

Introduction TRL Reciprocity Scilab Code Conversions between [S] and [T] [ S ] to [ T ] � � − 1 − S 22 S 21 S 21 [ T ] = (4) S 11 S 21 · S 12 − S 11 · S 22 S 21 S 21 [ T ] to [ S ] � � − T 21 T 11 · T 22 − T 12 · T 21 T 11 T 11 [ S ] = (5) 1 − T 12 T 11 T 11

Introduction TRL Reciprocity Scilab Code Standards for the TRL algorithm THRU : totally known � 1 � 0 � � T Std = (6) THRU 0 1 REFLECT : unknown � � �� Γ Std Sgn = ± 1 (7) ℜ REFLECT LINE : partially known � e − γ · l � 0 � � T Std = (8) LINE e + γ · l 0

Introduction TRL Reciprocity Scilab Code Measuring the THRU � � � � T Meas T Std = [ T IN ] · · [ T OUT ] (9) THRU THRU [ T IN ] − 1 · � � � � T Meas T Std = · [ T OUT ] (10) THRU THRU [ T IN ] − 1 · � � T Meas = [ T OUT ] (11) THRU [ T IN ] − 1 = � � T IN � � T Meas � � T IN · = [ T OUT ] (12) THRU

Introduction TRL Reciprocity Scilab Code Measuring the LINE � � � � T Meas T Std = [ T IN ] · · [ T OUT ] (13) LINE LINE [ T IN ] − 1 · � � � � T Meas T Std = · [ T OUT ] (14) LINE LINE � e − γ · l � 0 � � T Meas � � = · [ T OUT ] (15) T IN · LINE e + γ · l 0

Introduction TRL Reciprocity Scilab Code OUTPUT : Defining the [M] matrix Equation (15) is : � T 11 � T 11 · e − γ · l � T 12 · e − γ · l � T 12 � � T Meas · = (16) LINE T 21 · e + γ · l T 22 · e + γ · l T 21 T 22 (12) in (16) give : � T 11 � T 11 · e − γ · l � T 12 · e − γ · l � � − 1 T 12 � � � T Meas T Meas · · = THRU LINE T 21 · e + γ · l T 22 · e + γ · l T 21 T 22 (17) or � T 11 � T 11 · e − γ · l � T 12 · e − γ · l � T 12 · [ M ] = (18) T 21 · e + γ · l T 22 · e + γ · l T 21 T 22 with � − 1 � � � T Meas T Meas [ M ] = · THRU LINE

Introduction TRL Reciprocity Scilab Code OUTPUT : TRL Equations Equations given by (18) are : T 11 · M 11 + T 12 · M 21 = T 11 · e − γ · l (19) T 11 · M 12 + T 12 · M 22 = T 12 · e − γ · l (20) T 21 · M 11 + T 22 · M 21 = T 21 · e + γ · l (21) T 21 · M 12 + T 22 · M 22 = T 22 · e + γ · l (22)

Introduction TRL Reciprocity Scilab Code � � T 12 OUTPUT : Solving T 11 (20) gives : � T 11 � e − γ · l = · M 12 + M 22 (23) T 12 (23) in (19) gives : �� T 11 � � T 11 · M 11 + T 12 · M 21 = T 11 · · M 12 + M 22 (24) T 12 � T 12 � � T 11 � M 11 + · M 21 = · M 12 + M 22 (25) T 11 T 12 � 2 � T 12 � T 12 � · M 21 + · ( M 11 − M 22 ) − M 12 = 0 (26) T 11 T 11

Introduction TRL Reciprocity Scilab Code � � T 22 OUTPUT : Solving T 21 (21) gives : � T 22 � e + γ · l = M 11 + · M 21 (27) T 21 (27) in (22) gives : �� T 22 � � T 21 · M 12 + T 22 · M 22 = T 22 · · M 21 + M 11 (28) T 21 � T 21 � � T 22 � M 22 + · M 12 = · M 21 + M 11 (29) T 22 T 21 � 2 � T 22 � T 22 � · M 21 + · ( M 11 − M 22 ) − M 12 = 0 (30) T 21 T 21

Introduction TRL Reciprocity Scilab Code � � � � T 12 T 22 OUTPUT : and T 11 T 21 X 2 · M 21 + X · [ M 11 − M 22 ] − M 12 (31) � � � � T 12 T 22 This polynom has 2 solutions : and T 11 T 21 � � � � � T 12 � T 22 Usually, � < � � � � T 11 T 21 � If we consider the following polynom : X 2 · M 12 + X · [ M 22 − M 11 ] − M 21 (32) � � � � T 11 T 21 Then the 2 solutions are and T 12 T 22

Introduction TRL Reciprocity Scilab Code INPUT : Defining the [N] matrix Equation (15) is : � T 11 � e − γ · l � T 11 � � � T 12 0 T 12 � � T Meas · = · (33) LINE e + γ · l 0 T 21 T 22 T 21 T 22 (12) in (33) gives : � T 11 � e − γ · l � T 11 � � � 0 T 12 T 12 � � � � T Meas T Meas · = · · LINE e + γ · l THRU T 21 T 22 0 T 21 T 22 (34) or � T 11 � T 11 · e − γ · l � T 12 · e − γ · l � T 12 · [ N ] = (35) T 21 · e + γ · l T 22 · e + γ · l T 21 T 22 with � − 1 � � � T Meas T Meas [ N ] = · LINE THRU

Introduction TRL Reciprocity Scilab Code � � � � T 12 T 22 INPUT : and T 11 T 21 Equation (35) is similar to (18). Thus we can consider : X 2 · N 21 + X · [ N 11 − N 22 ] − N 12 (36) � � � � T 12 T 22 This polynom has 2 solutions : and T 11 T 21 � � � � � T 12 � T 22 Usually, � < � � � � T 11 T 21 � If we consider the following polynom : X 2 · N 12 + X · [ N 22 − N 11 ] − N 21 (37) � � � � T 11 T 21 Then the 2 solutions are and T 12 T 22

Introduction TRL Reciprocity Scilab Code The THRU equality a 1 a 2 a 3 T IN T OUT b 1 b 2 b 3 � b 2 � T 11 � a 1 � � � T 12 = · a 2 T 21 T 22 b 1 � b 2 � T 11 � b 3 � � � T 12 = · a 2 T 21 T 22 a 3

Introduction TRL Reciprocity Scilab Code � � T 11 Forward mode : b 2 equality to extract T 11 The b 2 equality leads us to : T 11 · a 1 + T 12 · b 1 = T 11 · b 3 + T 12 · a 3 (38) And by definition, about the THRU measurement, we know : � � S Meas = b 3 a 3 = 0 and S Meas = b 1 � � 21 11 a 1 a 1 � � a 3 = 0 Thus, � � T 11 + T 12 · b 1 a 1 · = T 11 · b 3 (39) a 1 � � T 12 � � · S Meas = T 11 · S Meas T 11 · 1 + (40) 11 21 T 11 � T 11 � � � � T 12 · S Meas 1 + � 11 T 11 = (41) S Meas T 11 21

Introduction TRL Reciprocity Scilab Code � � T 21 Reverse mode : a 2 equality to extract T 21 The a 1 equality leads us to : T 21 · a 1 + T 22 · b 1 = T 21 · b 3 + T 22 · a 3 (42) For the THRU measurement, we know : � � = b 1 = b 3 S Meas a 1 = 0 and S Meas � � 12 22 a 3 a 3 � � a 1 = 0 Thus, b 1 · T 22 = T 21 · b 3 + T 21 · a 3 (43) leads us to : � T 21 S Meas � 12 = (44) � � T 22 T 22 S Meas + 22 T 21

Introduction TRL Reciprocity Scilab Code REFLECT Measurement a 1 a 2 Γ Std Γ Std REFLECT REFLECT b 1 b 2 = T 21 + T 22 · S Meas REFLECT = b 1 Γ Std 11 (45) T 11 + T 12 · S Meas a 1 11 = T 12 + T 11 · S Meas REFLECT = b 2 Γ Std 22 (46) T 22 + T 21 · S Meas a 2 22

Introduction TRL Reciprocity Scilab Code REFLECT Equality T 21 + T 22 · S Meas = T 12 + T 11 · S Meas 11 22 (47) T 22 + T 21 · S Meas T 11 + T 12 · S Meas 22 11 � � � �  T 22   T 12  · S Meas S Meas 1 + + T 21  = T 11 11 22 T 11 T 21 · · (48)    � � � � T 21 T 11 T 12 T 22 · S Meas S Meas 1 + + 11 22 T 21 T 11 � T 12 � � � S Meas + 22 T 11 � T 22 � T 21 � T 11 � S Meas � � T 22 � � + � 2 · � 2 · 22 T 21 � � = T 21 · T 11 ·  � �  T 22 T 21 T 11 T 22 · S Meas 1 + 11 T 21   � � T 12 · S Meas 1 + 11 T 11 (49)

Introduction TRL Reciprocity Scilab Code REFLECT Equality � � T 12 � � � � S Meas � � + T 11 22 T 11 � · � T 22 � � T 11 S Meas � T 21 � + � 22 T 21 = ± � (50) �  � �  T 11 T 22 · S Meas 1 + � � � � � 11 T 21 T 21 T 22 � · ·   � � � T 22 T 21 T 12 · S Meas 1 + 11 T 11 There are 2 solutions. We select the good one thanks to the � � Γ Std �� knowledge of Sgn = ± 1 in (45) : ℜ REFLECT � �   T 22 · S Meas 1 + � T 21 � 11 T 21 Γ Std REFLECT = · (51)   � � T 11 T 12 · S Meas 1 + 11 T 11