Unknown Thru Calibration Algorithm

Transcription

1 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion Unknown Thru Calibration Algorithm Short-Open-Load-Reciprocal SOLR T Reveyrand NVNA Users Forum - INMMiC 2018 Brive-la-Gaillarde, France July 2018

2 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion 1 Calibration Kit SHORT, OPEN and LOAD standards THRU standard and the need for SOLR calibration 2 SOL 1-Port Error models Error terms calculation 3 SOLT 2-Port from [S] measurements from waves measurements 4 SOLR 2-Port from [S] measurements from waves measurements

3 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion NVNA architecture and 8-terms error model for calibration a m1 b m2 a m2 a 1 DUT a 2 b 2 a m1 X DUT Y b m2 a 1 b 2 e 10 S 21 e 32 e 00 e 11 S 11 S 22 e 22 e 33 e 01 S 12 e 23 a 2 a m2 During SOLx calibration, Γ Short, Γ Open, and Γ Load are assumed to be totally known SOLT : [S T hru ] is assumed to be totally known Model of the Thru may be not accurate enough SOLR : [S T hru ] is unknown but RECIPROCAL valid for passive device [S T hru ] values are identified during the calibration

4 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion Models parameters for standards Open: C0, C1, C2 and C3 Short: L0, L1, L2 and L3 Load: Z0; Line: Delay, Loss and Z0 =? Line STD

5 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SHORT standard calculation Inductance frequency polynomial model Lf = L 0 + L 1 f + L 2 f 2 + L 3 f 3 Offset length model Γ L = jωlf Z 0 jωlf + Z 0 S 11 of the standard T Loss = e Delay Loss f Z GHz 0 T Delay = e j4πfdelay Γ Short = Γ L T Loss T Delay

6 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion OPEN standard calculation Capacitance frequency polynomial model Offset length model S 11 of the standard Cf = C 0 + C 1 f + C 2 f 2 + C 3 f 3 Γ C = 1 jωz 0Cf 1 jωz 0 Cf T Loss = e Delay Loss f Z GHz 0 T Delay = e j4πfdelay Γ Open = Γ C T Loss T Delay

7 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion LOAD standard calculation Fixed impedance Offset length model Z = R + jl T ω Γ Z = Z Z 0 Z + Z 0 S 11 of the standard T Loss = e Delay Loss f Z GHz 0 T Delay = e j4πfdelay Γ Load = Γ Z T Loss T Delay

8 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion THRU standard calculation Ideal Thru [S] = [ ] Modeled Thru [ ] 0 e γl [S] = e γl 0 with γl = τ 2Z 0 Loss f GHz + j2πfτ Issues τ may be unknown Loss may be unknown Are S 11 and S 22 equal to zero? Validity of the model?

9 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion One port relative calibration error models e 10 a m b m a m a 1 a 1 e 00 e 11 b m e 01 LSNA papers [ ] a1 α1 β 1 = γ 1 δ 1 am1 VNA papers [ ] bm1 e00 e 01 = a 1 e 10 e 11 am1 a1 = [ e10 e 01 e 11 e 00 e 01 e 11 e 01 e 00 e 01 1 e 01 ] am1 bm1 a 1 [ γ = δ αδ βγ δ 1 δ β δ ] am1

10 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion One port relative calibration Γ m = bm1 a m1 and Γ = b1 a 1 LSNA papers [ ] a1 α1 β 1 = γ 1 δ 1 am1 VNA papers [ ] bm1 e00 e 01 = a 1 e 10 e 11 am1 a1 [ ] 1 β = α 1 1 γ 1 δ 1 Γ = γ 1 + δ 1Γ m 1 + β 1 Γm am1 Γ = Γm e00 Γ me 11 e with e = e 00e 11 e 10e 01 e 00 + ΓΓ me 11 Γ e = Γ m Γ = β 1ΓΓ m + γ 1 + δ 1Γ m

11 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOL : SHORT-OPEN-LOAD Γ <std> = b1 a 1 and Γ m<std> = bm1 a m1 with < std >= Short ; Open ; Load LSNA papers [ ] a1 1 β = α 1 1 γ 1 δ 1 am1 VNA papers [ ] bm1 e00 e 01 = a 1 e 10 e 11 am1 Γ <std> = β Γ <std>γ m<std> + γ + δ Γ m<std> Γ S β 1 Γ msγ S 1 Γ ms Γ O = γ 1 Γ moγ O 1 Γ mo Γ L δ 1 Γ mlγ L 1 Γ ml Γ msγ S 1 Γ ms Γ moγ O 1 Γ mo Γ mlγ L 1 Γ ml 1 Γ S β 1 Γ O = γ 1 Γ L δ 1 e 00 + Γ <std>γ m<std>e 11 Γ <std> e = Γ m<std> Γ ms e 00 1 Γ msγ S Γ S Γ mo = e 11 1 Γ moγ O Γ O Γ ml e 1 Γ mlγ L Γ L 1 Γ msγ S Γ S 1 Γ moγ O Γ O 1 Γ mlγ L Γ L 1 Γ ms e 00 Γ mo = e 11 Γ ml e

12 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT : general concepts Flow Graph a m1 X a 1 DUT b 2 Y b m2 e 10 S 21 e 32 e 00 e 11 S 11 S 22 e 22 e 33 e 01 S 12 e 23 a 2 a m2 [T ] matrix definition [T ] = 1 [ ] 1 S22 S 21 S 11 S 12 S 21 S 11 S 22 [S] = 1 [ ] T21 T 11 T 22 T 12 T 21 T 11 1 T 12 Cascading and de-embedding properties [T m ] = [T X ][T DUT ][T Y ] [T DUT ] = [T X ] 1 [T m ][T Y ] 1

13 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT from [S] measurements : 7 error terms to identify Port 1 Fwd bm1 a 1 [ ] e00 e 01 = e 10 e 11 am1 Short-Open-Load on port 1 e 00, e 11, and e 10e 01 = e X e 00e 11 Port 2 Rev bm2 a 2 [ ] e33 e 32 = e 23 e 22 am2 b 2 Short-Open-Load on port 2 e 22, e 33, and e 32e 23 = e Y e 22e 33 Transfert [S T hru ] Fwd and Rev e 10e 32 we can use 1 direct solution up to 4 equations least-square method

14 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT : THRU calibration e 10 e 32 and [S] measurements a m1 X a 1 DUT b 2 Y b m2 e 10 S 21 e 32 e 00 e 11 S 11 S 22 e 22 e 33 e 01 S 12 e 23 Finding e 10e 32 from [S T HRU ] a 2 a m2 [T m] = [T m] = [T X][T T HRU ][T Y ] [ ] [ 1 1 e11 1 e33 [T e 10e 32 e 00 e T HRU ] X e 22 e Y Calibrated [S DUT ] measurements [ ] 1 [ ] 1 1 e11 1 e33 [T DUT ] = e 10e 32 [T e 00 e m] X e 22 e Y with e X = e 00e 11 e 10e 01 and e Y = e 22e 33 e 32e 23 ]

15 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT from waves : 7 error terms to identify Port 1 Fwd a1 [ ] 1 β1 = γ 1 δ 1 am1 Short-Open-Load on port 1 β 1, γ 1, δ 1 Port 2 Rev a2 b 2 [ ] 1 β = α 2 2 γ 2 δ 2 am2 b m2 Short-Open-Load on port 2 β 2, γ 2, δ 2 Transfert [S T hru ] Fwd and Rev α 2 we can use 1 direct solution up to 4 equations least-square method

16 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT : THRU transfert α 2 Calibration error matrices [ ] a1 1 β1 a1m = γ 1 δ 1 m and a2 b 2 [ ] 1 β = α 2 2 γ 2 δ 2 a2m b 2m α 2 from b 2 = S 21 a 1 + S 22 a 2 in forward mode α 2 = S 21 a m1 + β 1 γ 2 a m2 + δ 2 b m2 S 22 a m2 + β 2 b m2 With an ideal THRU [S T HRU ] = [ ] α 2 = a m1 + β 1 γ 2 a m2 + δ 2 b m2

17 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLT : [S] from calibrated waves measurements Two independent measurements : Forward mode a m1 b m2 a m2 Reverse mode a 1 DUT b 2 a 2 [ b F wd 1 b Rev 1 b F 2 wd b Rev 2 ] [ ] S11 S 12 = S 21 S 22 [ a F wd 1 a Rev 1 a F 2 wd a Rev 2 ] b1 b 2 [ ] S11 S = 12 S 21 S 22 a1 a 2 [ ] [ S11 S 12 b F wd = S 21 S 22 1 b Rev 1 b F wd 2 b Rev 2 ] [ a F wd ] 1 a Rev 1 1 a F wd 2 a Rev 2

18 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : SHORT-OPEN-LOAD-RECIPROCAL Andrea Ferrero A Ferrero, U Pisani, Two-port network analyzer calibration using an unknown thru IEEE Microwave and Guided Wave Letters, Vol 2, No 12, 1992, pp

19 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : General Concepts Flow Graph a m1 X a 1 DUT b 2 Y b m2 e 10 S 21 e 32 e 00 e 11 S 11 S 22 e 22 e 33 e 01 S 12 e 23 a 2 a m2 [T ] matrix definition [T ] = 1 [ ] 1 S22 S 21 S 11 S 12 S 21 S 11 S 22 Reciprocity assumption S 12 = S 21 det[t ] = 1

20 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Short-Open-Load calibration on port 1 forward < std >=Short ; Open ; Load a m1 X e 10 e 00 e 11 e 01 a 1 STANDARD Standard Γ <std> = and Γ m<std> = a 1 a m1 1 Γ ms Γ S 1 Γ S Γ ms 1 Γ mo Γ O Γ O Γ mo = 1 Γ ml Γ L Γ L Γ ml e 00 e 11 X with X = e 00 e 11 e 10 e 01 e 00, e 11, and e 10 e 01 are known

21 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Short-Open-Load calibration on port 2 reverse < std >=Short ; Open ; Load Γ <std> = b 2 and Γ m<std> = b m2 a 2 a m2 1 1 Γ ms Γ S Γ S Γ ms e 33 1 Γ mo Γ O Γ O Γ mo = e 22 1 Γ ml Γ L Γ L Γ ml Y STANDARD Standard b 2 a 2 e 22 Y e 32 e 23 e 33 b m2 a m2 with Y = e 33 e 22 e 32 e 23 e 33, e 22, and e 32 e 23 are known

22 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Reciprocity on the Unknown Thru a m1 X a 1 DUT b 2 Y b m2 e 10 S 21 e 32 e 00 e 11 S 11 S 22 e 22 e 33 e 01 S 12 e 23 a 2 a m2 [ ] 1 [ 1 e11 1 e33 [T DUT ] = e 10 e 32 [T e 00 X m ] e 22 Y with X = e 00e 11 e 10e 01 and Y = e 33e 22 e 32e 23 Reciprocity det[t DUT ] = 1 e 10e 32 2 = e10e01e32e23 det[t m] e 10e 32 = ± e 10e 01e 32e 23 det[t m] ] 1

23 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR from waves : 7 error terms to identify Port 1 Fwd a1 [ ] 1 β1 = γ 1 δ 1 am1 Short-Open-Load on port 1 β 1, γ 1, δ 1 Port 2 Rev a2 b 2 [ ] 1 β = α 2 2 γ 2 δ 2 am2 b m2 Short-Open-Load on port 2 β 2, γ 2, δ 2 Transfert Reciprocity Fwd and Rev α 2

24 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : SHORT-OPEN-LOAD β 1 ; γ 1 ; δ 1 ; β 2 ; γ 2 ; δ 2 On port i, for < std >=Short ; Open ; Load Γ <std> = b i a i and Γ m<std> = b mi a mi Port 1 Port 2 Γ ms Γ S 1 Γ ms Γ mo Γ O 1 Γ mo Γ ml Γ L 1 Γ ml Γ ms Γ S 1 Γ ms Γ mo Γ O 1 Γ mo Γ ml Γ L 1 Γ ml 1 1 Γ S β 1 Γ O = γ 1 Γ L δ 1 Γ S β 2 Γ O = γ 2 Γ L δ 2

25 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Reciprocity transfert α 2 Uncompleted Relative Calibration [ ] a1 1 β1 am1 = and γ 1 δ 1 [ a 2 1 β b = 2 2 γ 2 δ 2 ] am2 b m2 [X] is the partially calibrated [S thru ] measurement [ ] [ X11 X 12 b F = 1 b R 1 X 21 X 22 b F 2 b R 2 ] [ a F 1 a R ] 1 1 a F 2 a R 2 Finding α 2 from reciprocity assumption S 21 = S 12 [ ] [ ] S11 S [S thru ] = 12 X11 X = 12 /α 2 then α S 21 S 22 X 21 α 2 X 2 = ± 22 X12 X 21

26 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Root solution for α 2 Arg{S 21 } α 2 = ± X12 X 21 S 21 = X 21 α 2 Pick up the solution close to φ est = 2πfτ 2 solutions per frequency Good solution Delay estimation of the unknown Thru f φ est

27 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion SOLR : Complete relative calibration a1 a2 [ ] 1 β1 = γ 1 δ 1 [ ] 1 β = α 2 2 γ 2 δ 2 b 2 am1 am2 b m2 a 1 1 β a m1 a 2 = γ 1 δ α 2 β 2 a m2 b γ 2 δ 2 b m2

28 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion Conclusion Some engineers should pay more attention on the validity of their Thru CalKit model during VNA SOLT calibrations ; Using an Unknown Thru calibration is a must compared to the SOLT method ; NVNA software developers should include SOLR method ; SOLR is easy to include in your code and obvious when the SOLT method already exists ; This presentation includes all you need to know for adding a SOLR calibration method on your NVNA system ;

29 Introduction Calibration Kit SOL 1-Port SOLT 2-Port SOLR 2-Port Conclusion Download this presentation T Reveyrand, «Unknown Thru Calibration Algorithm,» IEEE INMMiC 2018, Brive-la-Gaillarde, France, July 2018 wwwmicrowavefr