Post on 16-Jun-2020
HP8510
Lecture 13
Vector Network Analyzers and Signal Flow Graphs
ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS 1
Vector Network Analyzers
port 1 port 2
DUT
HP8510
ElecEng4FJ4 2LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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
ElecEng4FJ4 3
measurements of circuits with non-coaxial connectors (HMIC, MMIC)
NETWORKANALYZER
S PARAMETERTEST SET
SYNTHESIZER
CONTROLLER
PROBES
RF
HF
GPIB
GSG Probe
[M. Steer, Microwave and RF Design]LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
2-Port Vector Network Analyzer: Schematic
1b1a
2a
2b
1a 1b
2b2a
1a
2aLO1
LO2
[Pozar, Microwave Engineering] 4ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
N-Port Vector Network Analyzer: Schematic
ElecEng4FJ4 5LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
[Hiebel, Fundamentals of Vector Network Analysis]
1a 1a
1b1b
Vector Network Analyzer: Directional Element
• reversed directional coupler enables the measurement of reflection coefficients
ElecEng4FJ4 6LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
port 3 terminated with a matched load (power incident from port 1 is absorbed, not used)
measure scattered wave b
ba
• power dividers must ensure good output-port isolation
generator & power divider
Terminology related to VNA calibration:
21 42
21 42
22
41
- test port match- reflection tracking
/( ) - directivitys s
s
s
ss
(called isolation in L12)
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, ak
and bk
branches• a branch shows the
dependency between pairs of nodes
• it has a direction – from input (ai) to output (bj)
example: 2-port network
1 11 1 12 2
2 21 1 22 2
b S a S ab S a S a
7ElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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) sourceElecEng4FJ4 LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
ElecEng4FJ4
Decomposition Rules of Signal Flow Graphs
(1) series rule
(2) parallel rule
(3) self-loop rule
(4) splitting rule
9LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
ElecEng4FJ4 LECTURE 13: 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
ElecEng4FJ4 11LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
3-term Error Model: Signal-flow Graph
ElecEng4FJ4 12LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
error box
• the S-matrix of the error box contains in effect 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.
equivalent
11e
00eport 0 port 1
3-term Error Model: Error-term Equations
ElecEng4FJ4 13LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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
Prove that the S-matrices of the error box in sl. 12, SE and SꞌE, result in the same expression for ΓM.
compare with 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!ElecEng4FJ4 14LECTURE 13: 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
ElecEng4FJ4 15LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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
ElecEng4FJ4 16LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
port 1
port 2
12-term Error Model: Forward-model SFG
ElecEng4FJ4 17LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
( )
excitation
response
response
same as 3-term error model
12-term Error Model: Forward-model SFG
ElecEng4FJ4 18LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
Using signal-flow graph transformations derive the formulas for S11Mand S21M in the previous slide.
( )
12-term Error Model: Reverse-model SFG
ElecEng4FJ4 19LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
( )
excitation
response
response
12-term Calibration Method
ElecEng4FJ4 20LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
Step 1: (Port 1 Calibration) using the OSM 1-port procedure,obtain e11, e00, and Δe, from which (e10e01) is obtained.
Step 2: (Isolation) Connect matched loads (Z0) to both ports. (S21 = 0)The measured S21M yields e30 directly. (S12M = eꞌ03)
Step 3: (Thru) Connect ports 1 and 2 directly. (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.
transmission tracking
port 2 match
(sl. 12)
(sl. 17)
12-term Calibration Method: Error De-embedding
ElecEng4FJ4 21LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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 planesreflect: same load on each port (preferred); must have large reflectionline (or delay): 2 ports connected with system interconnect (represents the IC interconnect for the measured DUT and sets Z0)
ElecEng4FJ4 22LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
Thru, Reflect, and Line Calibration Connections
ElecEng4FJ4 23LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
(a) thru
(b) reflect
(c) line
Thru-Reflect-Line Calibration Fixtures
ElecEng4FJ4 24LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
DC needle probes
GSG probes
[Steer, Microwave and RF Design]
2-port Calibration: 8-term Error Model
ElecEng4FJ4 25LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
[Rytting, Network Analyzer Error Models and Calibration Methods]
port 1
port 2
Signal-flow Graph of 8-term Error Model
ElecEng4FJ4 26LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
port 1
port 2
XT
YT
T
TRL Calibration: SFG with DUT
ElecEng4FJ4 27LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• unfolded SFG of the DUT measurement
TRL Calibration: SFG of Thru Measurement
ElecEng4FJ4 28LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• we must know all 4 Thru S-parameters
• if Thru is assumed of zero length, then reference plane for all ports is set in its middle:
• if Thru assumed perfectly matched, then (then it must be made with the same line as that in the Line standard, which determines Z0)
thru thru21 12 1S S
thru thru11 22 0S S
TRL Calibration: SFG of Line Measurements
ElecEng4FJ4 29LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• we need to know only 2 Line S-parameters
• Line determines Z0 and is, therefore, assumed perfectly matched to Z0: (2 known parameters)
• must have different physical length compared to Thru
unknown (determined from calibration)
line line11 22 0S S
0a
0b
3b
00e 11e
01e
22e32e
Line
1
1
1
1
1
1
1
port 1
10e
port 2
23e
33e
3a
line21S
line12S
ElecEng4FJ4 30LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
TRL Calibration: SFG of Reflect Measurements
• must have high reflection on both ports!
• only one piece of information is needed, for example
0a
0b
3b
00e 11e
01e
22e32e
Reflect
1
1
1
1
1
1
1
port 1
10e
port 2
23e
33e
3a
refl11S
refl22S
refl21S
refl12S
o (most common)o
refl refl11 22S Srefl refl21 12S S
refl refl11 22( , )S S
Scattering Transfer (or Cascade) Parameters
ElecEng4FJ4 31LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• 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
ElecEng4FJ4 32LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
error de-embedding
T in terms of S
T matrices of error boxes
ElecEng4FJ4 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)
1 110 32 M( )e e T A T B
• TRL measurement procedure
M1 X C1 Y
M2 X C2 Y
M3 X C3
M X
Y
Y
(1) measured with 2-port cal standard #1(2) measured with 2-port cal standard #2(3) measured with 2-por(
t cal standard #34) measured with DUT
T T T TT T T T
TT
TT
TTT
T
• we need to find the 7 error terms from (1), (2) and (3)
A
B
LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
ElecEng4FJ4 34LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
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) TC1 must be completely known(common choice: ref. plane in the middle, perfect match)
cal standard #2 (line) TC2 can have 2 unknowns cal standard #3 (reflect) TC3 can have 3 unknowns
(sl. 28)
(sl. 29)
(sl. 30)
ElecEng4FJ4 35LECTURE 13: VECTOR NETWORK ANALYZERS AND SIGNAL FLOW GRAPHS
• 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
VNA Calibration – Summary