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
1 TRL Standards THRU and LINE Measurements T OUT parameters : T12 and T IN parameters : The THRU equality : T 11 T 12 and T11 and T21 T 22 T 22 T21 T 22 The REFLECT equality : Extracting 2 Reciprocity 3 Scilab Code Presentation Example Insights
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.
Some definitions : [S parameters a 1 a 2 A b 1 b 2 b1 b 2 [ S11 S = 12 S 21 S 22 a1 a 2 1
Some definitions : [T parameters a 1 a 2 A b 1 b 2 a1 b 1 [ T11 T = 12 T 22 b2 a 2 2
Some definitions : [T parameters a 1 a 2 A B C b 1 b 2 [T Total = [T A [T B [T C 3
Conversions between [S and [T [S to [T [T = [ 1 S 21 S 11 S 22 S 21 S 21 S 12 S 11 S 22 S 21 S 21 4 [T to [S [S = [ T 22 T 12 1 T 12 5
Standards for the TRL algorithm THRU : totally known [ TTHRU Std = [ 1 0 0 1 6 REFLECT : unknown Sgn { } R Γ Std REFLECT = ±1 7 LINE : partially known [ TLINE Std = [ e γ l 0 0 e +γ l 8
Measuring the THRU [ [ TTHRU Meas = [T IN TTHRU Std [T OUT 9 [ [ [T IN 1 TTHRU Meas = TTHRU Std [T OUT 10 [ [T IN 1 TTHRU Meas = [T OUT 11 [T IN 1 = [ T IN [ TIN [T THRU Meas = [T OUT 12
Measuring the LINE [ T Meas LINE [ [T IN 1 [ = [T IN TLINE Std [T OUT 13 T Meas LINE = [ [ TIN [T LINE Meas e γ l 0 = 0 e +γ l [ TLINE Std [T OUT 14 [T OUT 15
OUTPUT : Defining the [M matrix Equation 15 is : [ T 12 T 22 12 in 16 give : [ T11 T 12 or T 22 [ T11 T 12 [ T Meas LINE [ 1 [ TTHRU Meas T 22 [ T11 e = γ l T 12 e γ l e +γ l T 22 e +γ l T Meas LINE 16 [ T11 e = γ l T 12 e γ l e +γ l T 22 e +γ l [ T11 e [M = γ l T 12 e γ l e +γ l T 22 e +γ l 17 18 with [M = [ 1 [ TTHRU Meas T Meas LINE
OUTPUT : TRL Equations Equations given by 18 are : M 11 + T 12 M 21 = e γ l 19 M 12 + T 12 M 22 = T 12 e γ l 20 M 11 + T 22 M 21 = e +γ l 21 M 12 + T 22 M 22 = T 22 e +γ l 22
OUTPUT : Solving T 12 20 gives : 23 in 19 gives : e γ l = T11 T 12 M 11 + T 12 M 21 = T12 M 11 + T12 2 M 21 + M 21 = T12 M 12 + M 22 23 [ T11 T 12 T11 T 12 M 12 + M 22 24 M 12 + M 22 25 M 11 M 22 M 12 = 0 26
OUTPUT : Solving T 22 21 gives : 27 in 22 gives : e +γ l = M 11 + M 12 + T 22 M 22 = T 22 T22 M 22 + T21 2 M 21 + M 12 = T 22 T22 T22 [ T22 T22 M 21 27 M 21 + M 11 28 M 21 + M 11 29 M 11 M 22 M 12 = 0 30
OUTPUT : T 12 and T 22 X 2 M 21 + X [M 11 M22 M 12 31 This polynom has 2 solutions : T12 and T22 T 21 Usually, T 12 < T 22 If we consider the following polynom : X 2 M 12 + X [M 22 M11 M 21 32 Then the 2 solutions are T11 T 12 and T21 T 22
INPUT : Defining the [N matrix Equation 15 is : [ T 12 T 22 [ 12 in 33 gives : [ T 12 or T 22 [ [ T 12 T 22 T Meas LINE T Meas LINE = = [ e γ l 0 0 e +γ l [ e γ l 0 0 e +γ l [ T11 T 12 T 22 [ T 11 T 12 T 22 [ T [N = 11 e γ l T 12 e γ l e +γ l T 22 e +γ l 33 [ TTHRU Meas 34 35 with [ [N = T Meas LINE [ 1 TTHRU Meas
INPUT : T 12 and T 22 Equation 35 is similar to 18. Thus we can consider : X 2 N 21 + X [N 11 N 22 N 12 36 This polynom has 2 solutions : T 12 and T 22 Usually, T 12 T < T 22 11 If we consider the following polynom : X 2 N 12 + X [N 22 N 11 N 21 37 Then the 2 solutions are and T 12 T 22
The THRU equality a 1 a 2 a 3 T IN T OUT b 1 b 2 b 3 b2 a 2 [ T = 11 T 12 T 22 a1 b 1 b2 a 2 [ T11 T = 12 T 22 b3 a 3
Forward mode : b 2 equality to extract The b 2 equality leads us to : a 1 + T 12 b 1 = b 3 + T 12 a 3 38 And by definition, about the THRU measurement, we know : S21 Meas = b 3 a3 a 1 and S Meas 11 = b 1 a3 =0 a 1 =0 Thus, a 1 + T 12 b1 1 + T11 T 12 = a 1 S11 Meas 1 + T 12 S Meas 21 = b 3 39 = S Meas 21 40 S11 Meas 41
Reverse mode : a 2 equality to extract The a 1 equality leads us to : a 1 + T 22 b 1 = b 3 + T 22 a 3 42 For the THRU measurement, we know : S12 Meas = b 1 a1 a 3 and S Meas 22 = b 3 a1 =0 a 3 =0 Thus, leads us to : b 1 T 22 = b 3 + a 3 43 T21 T 22 = S12 Meas S Meas 22 + T22 44
REFLECT Measurement a 1 a 2 Γ Std REF LECT Γ Std REF LECT b 1 b 2 Γ Std REFLECT = b 1 = + T 22 S11 Meas a 1 + T 12 S Meas Γ Std REFLECT = b 2 = T 12 + S22 Meas a 2 T 22 + S Meas 11 22 45 46
REFLECT Equality + T 22 S Meas 11 + T 12 S Meas 11 = T 12 + S22 Meas T 22 + S22 Meas 47 1 + T 22 1 + T 12 2 T21 T 22 S Meas 11 S Meas 11 T 22 = 2 = S Meas T11 22 + T12 S Meas 22 + T22 48 S22 Meas T12 + T 11 S22 Meas T22 + T 21 1+ T 22 S Meas 21 T 1+ 12 S Meas 11 49
REFLECT Equality = ± S Meas T11 T21 T 22 T 22 T12 22 + T 11 S22 Meas T22 + T 21 1+ T 22 S Meas 21 T 1+ 12 S Meas 11 50 There are 2 solutions. We select the good one thanks to the knowledge of Sgn R { Γ Std REFLECT } = ±1 in 45 : T Γ Std REFLECT = 21 1 + T 22 1 + T 12 S Meas 11 S Meas 11 51
TRL Completed The TRL algorithm is completed. We got 7 parameters from : The [M matrix : The [N matrix : The THRU equality : The REFLECT equality : T12 and T22 T 12 and T 22 T 21 T11 from 31 ; from 36 ; from 41 and from 50 ; T21 from 44 ; T 22 It is suffisent for [S parameters de-embedding but not for power measurement. We need to normalize correctly the system of equation finding the absolute value of. For that purpose we will consider a reciprocity assumption : S21 IN = S 12 IN.
Reciprocity assumption We should have : [ T IN = [ S21 S 12 S 11 S 22 S 12 S 22 S 12 S 11 S 12 1 S 12 Thus the reciprocity assumption S 21 = S 12 leds to : 52 T 22 T 12 = 1 53
Reciprocity assumption We can obtain from TRL the complete [ T IN and [TOUT matrix from an arbitrary value of. Those matrix has to be multiplied by K in order to fullfill equation 53 such as : K 2 1 = T 22 T 12 54 K = ± 1 T 22 T 12 55 There are 2 solutions. The good one is selected such as the extrapolated phase of S 21 on DC is as close as possible of zero.
This code is now available in Scilab http://www.microwave.fr/uw.html
Example with an OPEN reflect S2P Measurements : Thru black, Line blue and Open red.
Example with an OPEN reflect S2P Extracted : Port 1 black, Port 2 blue and De-embedded open red.
Example with an SHORT reflect S2P Measurements : Thru black, Line blue and Short red.
Example with an SHORT reflect S2P Extracted : Port 1 black, Port 2 blue and De-embedded short red.
Source Code : uw_trl_calc.sci
Source Code : uw_trl_calc.sci