Lecture 13 Vector Network Analyzers and Signal Flow Graphsael.chungbuk.ac.kr/.../L02a-VNA-and-signal-flow-graph.pdf · Vector Network Analyzers and Signal Flow Graphs ElecEng4FJ4

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


[Hiebel, Fundamentals of Vector Network Analysis]

1a 1a

1b1b

Vector Network Analyzer: Directional Element

• reversed directional coupler enables the measurement of reflection coefficients


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


3-term Error Model: Signal-flow Graph


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


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


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


12-term Error Model: Reverse Model


port 1

port 2

12-term Error Model: Forward-model SFG


( )

excitation

response

response

same as 3-term error model

12-term Error Model: Forward-model SFG


Using signal-flow graph transformations derive the formulas for S11Mand S21M in the previous slide.

( )

12-term Error Model: Reverse-model SFG


( )

excitation

response

response

12-term Calibration Method


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


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)


Thru, Reflect, and Line Calibration Connections


(a) thru

(b) reflect

(c) line

Thru-Reflect-Line Calibration Fixtures


DC needle probes

GSG probes

[Steer, Microwave and RF Design]

2-port Calibration: 8-term Error Model



port 1

port 2

Signal-flow Graph of 8-term Error Model


port 1

port 2

XT

YT

T

TRL Calibration: SFG with DUT


• unfolded SFG of the DUT measurement

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 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


• 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


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


• 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


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


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)


• 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

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Lecture 13 Vector Network Analyzers and Signal Flow Graphsael.chungbuk.ac.kr/.../L02a-VNA-and-signal-flow-graph.pdf · Vector Network Analyzers and Signal Flow Graphs ElecEng4FJ4