Lecture 10 Vector Network Analyzers and Signal Flow Graphs
Transcript of Lecture 10 Vector Network Analyzers and Signal Flow Graphs
HP8510
Lecture 10
Vector Network Analyzers and Signal Flow Graphs
Sections: 6.7 and 6.11
Homework: From Section 6.13 Exercises: 4, 5, 6, 7, 9, 10, 22
Acknowledgement: Some diagrams and photos are from M. Steer’s book “Microwave and RF Design”
Vector Network Analyzers
port 1 port 2
DUT
HP8510
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R&S®ZVA67 VNA2 ports, 67 GHz
Agilent N5247A PNA-X VNA, 4 ports, 67 GHz
Agilent 8719ES
Test
Instruments
MICROPROBERS
UnderDevice
Vector Network Analyzer and IC Probes
HP8510
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measurements of circuits with non-coaxial connectors (HMIC, MMIC)
NETWORKANALYZER
S PARAMETERTEST SET
SYNTHESIZER
CONTROLLER
PROBES
RF
HF
GPIB
GSG Probe
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2-Port Vector Network Analyzer: Schematic
1b1a
2a
2b
1a 1b
2b2a
1a
2aLO1
LO2
[Pozar, Microwave Engineering] 4
N-Port Vector Network Analyzer: Schematic
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[Hiebel, Fundamentals of Vector Network Analysis]
1a 1a
1b1b
Vector Network Analyzer: Directional Element
reversed directional coupler
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port 3 terminated with a matched load (power is absorbed, not used)
measure scattered wave bb
a
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
nodeseach port has two nodes, ak and bk
branches• a branch shows the
dependency between pairs of nodes
• it has a direction – from input to output
example: 2-port network
1 11 1 12 2
2 21 1 22 2
b S a S ab S a S a
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Signal Flow Graphs of Two Basic 1-port Networks
11l S lb a
1bsV
0Z
0
0( )s
ZZ Z
8
a
b
0
0( )s ss
Zb a VZ Z
0
0 0
0
0
, ( )s
s
ss
s
Z VV V bZ Z Z
Z ZZ Z
(a) load
0Z
0
0
ll
l
Z ZZ Z
(b) source
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Decomposition Rules of Signal Flow Graphs
(1) series rule
(2) parallel rule
(3) self-loop rule
(4) splitting rule
9
Nikolova 2012 LECTURE 10: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
Signal Flow Graphs: Example
Express the input reflection coefficient Γ of a 2-port network in terms of the reflection at the load ΓL and its S-parameters.
21
221 L
ss
2
rule #4at a
2rule #3 at b
rule #1
10
1 12 2111
1 221L
L
b s ssa s
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 = S11 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
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3-term Error Model: Signal-flow Graph
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error box
S-parameters of the error box contain 3 unknowns
00
10 01 11
1E
ee e e
S
10e
01e
00 01
10 11E
e ee e
S
M
[Rytting, Network Analyzer Error Models and Calibration Methods]
Note: SFG branches without a coefficient have a default coefficient of 1.
3-term Error Model: Error-term Equations
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Using the result from the example on sl. 10 and the signal flow graph in sl. 12, prove the formula
00M
111ee
e
error de-embedding formula(see sl. 10)
3-term Error Model
• for accurate results, one has to know the exact values of Γo, Γs and Γm – use manufacturer’s cal kits!Nikolova 2012 14LECTURE 10: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• ideally, in the OSM calibration,
1 o
2 s
3 m
11
0
• the 3 calibration measurements with the 3 standard known loads (Γ1, Γ2, Γ3) produce 3 equations for the 3 unknown error terms
00 11( , , )ee e M 00
M 11 e
ee
error de-embedding
linear system for 00 11[ , , ]Tee e x
2-port Calibration: Classical 12-term Error Model
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consists of two models:• forward (excitation at port 1): models errors in S11M and S21M• reverse (excitation at port 2): models errors in S22M and S12M
port 1
port 2
[Rytting, Network Analyzer Error Models and Calibration Methods]
12-term Error Model: Reverse Model
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port 1
port 2
12-term Error Model: Forward-model SFG
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( )
12-term Error Model: Forward-model SFG
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Using signal-flow graph transformations derive the formulas for S11Mand S21M in the previous slide.
( )
12-term Error Model: Reverse-model SFG
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( )
12-term Calibration Method
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Step 1: Port 1 Calibration using the OSM 1-port procedure.Obtain e11, e00, and Δe, from which (e10e01) is obtained.
Step 2: Connect matched loads (Z0) to both ports (isolation). (S21 = 0)The measured S21M yields e30 directly.
Step 3: Connect ports 1 and 2 directly (thru). (S21=S12=1, S11=S22=0)
Obtain e22 and e10 e32 fromeqns. (*) usingS21 = S12 = 1, S11 = S22 = 0.
• All 6 error terms of the forward model are now known.
• Same procedure is repeated for port 2.
12-term Calibration Method: Error De-embedding
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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) calibration structures
thru: the 2 ports must be connected directly, sets the reference planesreflect: load on each port identical; must have large reflectionline (or delay): 2 ports connected with a matched (Z0) transmission line (TL must represent the IC interconnect for the measured DUT)
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Thru, Reflect, and Line Calibration Connections
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(a) thru
(b) reflect
(c) line
Thru-Reflect-Line Calibration Fixtures
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DC needle probes
GSG probes
2-port Calibration: 8-term Error Model
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[Rytting, Network Analyzer Error Models and Calibration Methods]
port 1
port 2
Signal-flow Graph of 8-term Error Model
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port 1
port 2
XT
YT
T
Signal-flow Graphs of the 3 TRL Calibration Measurements
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Signal-flow Graphs of the 3 TRL Calibration Measurements (2)
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Scattering Transfer (or Cascade) Parameters
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• when a network is a cascade of 2-port networks, often the scattering transfer (T-parameters) are used
1 11 12 221 22 21
V VT TT TV V
• relation to S-parameters
11 12 11111 22 12 2121
21 22 22, 1
SS
T T SS S S S ST T S
A BT T TAT BT
1 11 12 2
1 21 22 2
b T T aa T T b
or
8-term Error Model in Terms of T-parameters for TRL Calibration
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error de-embedding
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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)
1 110 32 M( )e e T A T B
• TRL measurement procedure
M X Y
M1 X C1 Y
M2 X C2 Y
M3 X C3 Y
(1) measured with DUT(2) measured with 2-port cal standard #1(3) measured with 2-port cal standard #2(4) measured with 2-port cal standard #3
T T TTT T T TT T T TT T T T
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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
• 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 TC1 must be completely known – thru cal standard #2 TC2 can have 2 unknown transmission terms – line cal standard #3 TC3 can have 3 unknowns; if its reflection
coefficients satisfy S11 = S22 (it is best if S11 = S22 are large!) then its 3 coefficients can be unknown – reflect
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• errors are introduced when measuring a device due to parasitic coupling, leakage and imperfect connections
• 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• there exists also a 16-term error model, many other cal techniques
VNA Calibration – Summary