Lecture 13. Vector Network Analyzers and Signal Flow Graphs

Transcription

1 HP8510 Lecture 13 Vector Network Analyzers and Signal Flow Graphs 1

2 Vector Network Analyzers HP8510 Agilent 8719ES R&S ZVA67 VNA 2 ports, 67 GHz port 1 port 2 DUT Agilent N5247A PNA-X VNA, 4 ports, 67 GHz 2

3 Vector Network Analyzer and IC Probes measurements of circuits with non-coaxial connectors (HMIC, MMIC) HP8510 HF NETWORK ANALYZER S PARAMETER TEST SET SYNTHESIZER RF GPIB PROBES CONTROLLER Instruments GSG Probe MICROPROBERS Device Under Test ElecEng4FJ4 [M. Steer, Microwave and RF Design] LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS 3

4 2-Port Vector Network Analyzer: Schematic a 1 a 1 a1 b1 b 1 a 2 b 2 b 2 a 2 a 2 LO1 LO2 ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS [Pozar, Microwave Engineering] 4

5 N-Port Vector Network Analyzer: Schematic b 1 b 1 a1 a1 5 [Hiebel, Fundamentals of Vector Network Analysis]

6 Vector Network Analyzer: Directional Element reversed directional coupler enables the measurement of reflection coefficients generator & power divider a b Terminology related to VNA calibration: s22 - test port match s21s42 - reflection tracking s /( s s ) - directivity (called isolation in L12) measure scattered wave b port 3 terminated with a matched load (power incident from port 1 is absorbed, not used) power dividers must ensure good output-port isolation 6

7 Signal Flow Graphs used to analyze microwave circuits in terms of incident and scattered waves used to devise calibration techniques for VNA measurements components of a signal flow graph nodes a node represents a state variable (root-power wave) each port has two nodes, a k and b k branches a branch shows the dependency between pairs of nodes it has a direction from input (a i ) to output (b j ) example: 2-port network b1 S11a1 S12a2 b S a S a ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS 7

8 Signal Flow Graphs of Two Basic 1-port Networks a Z 0 (a) load b l b a l l S 11 Z Z l l Z Z 0 0 V s Z0 ( Z Z ) 0 s b 1 Z 0 Z0 V Z0 V Vs, b bsa V ( Z0 Zs ) Z0 ( Z0 Zs ) Zs Z0 (b) source s Z Z LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS ElecEng4FJ4 s 0 s 8

9 Decomposition Rules of Signal Flow Graphs (1) series rule (2) parallel rule (3) self-loop rule (4) splitting rule ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS 9

10 Express the input reflection coefficient Γ of a 2-port network in terms of the reflection at the load Γ L and its S-parameters. Signal Flow Graphs: Example rule #4 at a 2 b s s a s s22 L L rule #1 s 21 1 s22l rule #3 at b 2 ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS 10

11 VNA Calibration for 1-port Measurements (3-term Error Model) the 3-term error model is known as the OSM (Open-Short-Matched) cal technique (aka OSL or SOL, Open-Short-Load) the cal procedure includes 3 measurements performed before the DUT is measured: 1) open circuit, 2) short circuit, 3) matched load used when Γ = S 11 of a single-port device is measured actual measurements include losses and phase delays in connectors and cables, leakage and parasitics inside the instrument these are viewed as a 2-port error box calibration aims at de-embedding these errors from the total measured S-parameters 11

12 3-term Error Model: Signal-flow Graph [Rytting, Network Analyzer Error Models and Calibration Methods] M error box the S-matrix of the error box contains in effect 3 unknowns S E e 00 e 01 e00 e e e 10 port 0 port 1 e 11 Note: SFG branches without a coefficient have a default coefficient of 1. 1 e equivalent e S E e e e

13 3-term Error Model: Error-term Equations compare with sl. 10 error deembedding formula Using the result from the example on sl. 10 and the signal flow graph in sl. 12, prove the formula e00 e M 1 e11 Prove that the S-matrices of the error box in sl. 12, S E and Sꞌ E, result in the same expression for Γ M. 13

14 3-term Error Model the 3 calibration measurements with the 3 standard known loads (Γ 1, Γ 2, Γ 3 ) produce 3 equations for the 3 unknown error terms T linear system for x [ e00, e11, e ] M e00 ( e00, e11, e ) Me11 e error de-embedding ideally, in the OSM calibration, 1 o 1 2 s m for accurate results, one has to know the exact values of Γ o, Γ s and Γ m use manufacturer s cal kits! 14

15 2-port Calibration: Classical 12-term Error Model [Rytting, Network Analyzer Error Models and Calibration Methods] consists of two models: forward (excitation at port 1): models errors in S 11M and S 21M reverse (excitation at port 2): models errors in S 22M and S 12M port 1 port 2 15

16 12-term Error Model: Reverse Model port 1 port 2 16

17 12-term Error Model: Forward-model SFG excitation response response same as 3-term error model () 17

18 12-term Error Model: Forward-model SFG Using signal-flow graph transformations derive the formulas for S 11M and S 21M in the previous slide. () 18

19 12-term Error Model: Reverse-model SFG response response excitation ( ) 19

20 12-term Calibration Method Step 1: (Port 1 Calibration) using the OSM 1-port procedure, obtain e 11, e 00, and Δ e, from which (e 10 e 01 ) is obtained. (sl. 12) Step 2: (Isolation) Connect matched loads (Z 0 ) to both ports. (S 21 = 0) The measured S 21M yields e 30 directly. (S 12M =eꞌ 03 ) Step 3: (Thru) Connect ports 1 and 2 directly. (S 21 =S 12 =1, S 11 =S 22 =0) Obtain e 22 and e 10 e 32 from eqns. (*) using S 21 = S 12 = 1, S 11 = S 22 = 0. All 6 error terms of the forward model are now known. transmission tracking Same procedure is repeated for port 2. (sl. 17) port 2 match 20

21 12-term Calibration Method: Error De-embedding 21

22 2-port Thru-Reflect-Line Calibration TRL (Thru-Reflect-Line) calibration is used when classical standards such as open, short and matched load cannot be realized TRL is the calibration used when measuring devices with non-coaxial terminations (HMIC and MMIC) TRL calibration is based on an 8-term error model TRL calibration requires three (2-port) custom calibration structures thru: the 2 ports must be connected directly, sets the reference planes reflect: same load on each port (preferred); must have large reflection line (or delay): 2 ports connected with system interconnect (represents the IC interconnect for the measured DUT and sets Z 0 ) 22

23 (a) thru Thru, Reflect, and Line Calibration Connections (b) reflect (c) line 23

24 Thru-Reflect-Line Calibration Fixtures GSG probes DC needle probes [Steer, Microwave and RF Design] 24

25 2-port Calibration: 8-term Error Model [Rytting, Network Analyzer Error Models and Calibration Methods] port 1 port 2 25

26 Signal-flow Graph of 8-term Error Model port 1 T X T Y port 2 T 26

27 TRL Calibration: SFG with DUT unfolded SFG of the DUT measurement 27

28 TRL Calibration: SFG of Thru Measurement we must know all 4 Thru S-parameters if Thru is assumed of zero length, then reference plane for all ports thru thru is set in its middle: S S thru thru if Thru assumed perfectly matched, then S11 S22 0 (then it must be made with the same line as that in the Line standard, which determines Z 0 ) 28

29 TRL Calibration: SFG of Line Measurements we need to know only 2 Line S-parameters Line determines Z 0 and is, therefore, assumed perfectly matched to line line Z 0 : S S (2 known parameters) must have different physical length compared to Thru unknown (determined from calibration) a 0 port 1 1 e 10 e00 e11 1 line S 21 1 e 22 e 32 e 33 1 b 3 port 2 b 0 1 e 01 1 Line line S 12 1 e 23 a 3 29

30 TRL Calibration: SFG of Reflect Measurements refl refl must have high reflection ( S, S ) on both ports! only one piece of information is needed, for example o o S S S refl refl refl refl 21 S12 (most common) a 0 port 1 b e 10 e00 e11 e refl S 21 refl S 11 refl S 22 refl S 12 Reflect 1 1 e 32 e 22 e 33 e 23 1 b 3 port 2 a 3 30

31 Scattering Transfer (or Cascade) Parameters when a network is a cascade of 2-port networks, often the scattering transfer (T-parameters) are used V 1 T11 T12 V 2 b1 T11 T12 a2 V 1 T21 T22 V 2 or a1 T21 T22 b2 relation to S-parameters T11 T12 1 S S11 S 21, S S S S S T21 T22 S TA T T T A B TB 31

32 8-term Error Model in Terms of T-parameters for TRL Calibration error de-embedding T in terms of S T matrices of error boxes 32

33 8-term Error Model for TRL Calibration the number of unknown error terms is actually 7 in the simple cascaded TRL network (see sl. 26) TRL measurement procedure M X Y A B T( e e ) A T B M (1) TM1 TXTC1 TY measured with 2-port cal standard #1 (2) TM2 TXTC2 TY measured with 2-port cal standard #2 (3) TM3 TXT C3TY measured with 2-port cal standard #3 ( 4) T T TT measured with DUT we need to find the 7 error terms from (1), (2) and (3) 33

34 8-term Error Model for TRL Calibration measuring the 3 two-port cal standards yields 12 independent equations while we have only 7 error terms THRU (4) + LINE(4) + REFLECT(4) thus 5 parameters of the 3 cal standards need not be known and can be determined from the calibration measurements which 5 parameters are chosen for which cal standards is important in order to reduce errors and avoid singular matrices cal standard #1 (thru) T C1 must be completely known (common choice: ref. plane in the middle, perfect match) cal standard #2 (line) T C2 can have 2 unknowns cal standard #3 (reflect) T C3 can have 3 unknowns (sl. 28) (sl. 29) (sl. 30) 34

35 VNA Calibration Summary errors are introduced when measuring a device due to parasitic coupling, leakage, reflections and imperfect directivity these errors must be de-embedded from the overall measured S- parameters the de-embedding relies on the measurement of known or partially known cal standards calibration measurements, which precede the measurement of the DUT 1-port calibration uses the 3-term error model and the OSM method 2-port calibration may use 12-term or 8-term error models the 12-term error model requires OSM at each port, isolation, and thru measurements the 8-term error model with the TRL technique is widely used for non-coaxial devices requires custom fixtures for thru, reflect & line there exists also a 16-term error model, many other cal techniques 35