Lecture 13 Vector Network Analyzers and Signal Flow...
Transcript of Lecture 13 Vector Network Analyzers and Signal Flow...
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