slide 1 network analyzer error models and calibration methods by doug rytting

Network Analyzer

Error Modelsand

Calibration Methods

byDoug Rytting

• Network Analyzer Block Diagram and Error Model• System Error Model for Error-Correction• One-Port Error Model and Calibration• Two-Port Error Models and Calibration

• 12-Term Method• 16-Term Method• 8-Term Method• 8-Term Examples

• Measuring S-parameters• E-Cal• Accuracy of Error-Correction

Presentation Outline

RFSource

LOSource

a0

b0 b3

Port - 1 Port - 2

a3

DUTa2

a1

b1

b2Cable Cable

IF

IF

IF

IF

Network Analyzer Block Diagram

DUT

A/D

DSP

A/D

DSP

A/D

DSP

a2

a1

b1

b2

a0 b0 b3

LO LO

RFSource

Port - 1 Port - 2

IF IF

Block Diagram – Forward Direction

aS

N La0 N Lb0N Lb3

b0 b3

M 2M 1

L2-b3

1+A a0 +N Ha0

a0

Port - 1a1

a2b1

b2

S 11

S 21

S 22

S 12

DUT

Port - 2Cable &

ConnectorCable &

Connector

LC

M C

LC

LCLC

M C M CM C

Branches With No Label = 1

IF IF

M S

La0-b3

LS-1

L1-S

LS-a0 LS-b0 L1-a0

L1-b0

1+A b0 +N Hb0 1+A b3 +N Hb3

Error Model – Forward Direction

a1 = Incident Signal at Port-1b1 = Reflected Signal at Port-1a2 = Incident Signal at Port-2b2 = Transmitted signal at Port-2

aS = Source Porta0 = Measured Incident Portb0 = Measured Reflected Portb3 = Measured Transmitted Port

LS-1 = Loss from Source to Port-1L1-S = Loss from Port-1 to SourceLS-a0 = Loss from Source to a 0LS-b0 = Loss from Source to b 0 (Directivity)L1-a0 = Loss from Port-1 to a 0 (Directivity)L1-b0 = Loss from Port-1 to b 0L2-b3 = Loss from Port-2 to b 3La0-b3 = Loss from a 0 to b 3 (Leakage)LC = Loss of Cables

S 11 = Refl Coef of DUT at Port-1S 21 = Forward Trans Coef of DUTS 12 = Reverse Trans Coef of DUTS 22 = Refl Coef of DUT at Port-2

M 1 = Match at Port-1M 2 = Match at Port-2M S = Match of SourceM C = Match of Cables

N La0 = Low Level Noise at a 0N Lb0 = Low Level Noise at b 0N Lb3 = Low Level Noise at b 3N Ha0 = High Level Noise at a 0N Hb0 = High Level Noise at b 0N Hb3 = High Level Noise at b 3

A a0 = Dynamic Accuracy at a 0 (Linearity)A b0 = Dynamic Accuracy at b 0 (Linearity)A b3 = Dynamic Accuracy at b 3 (Linearity)

Error Model Definitions

Branches With No Label = 1

a0

b0

Port - 1a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b 3

e30

e00 e11

e01

e10

e22

e32

There are also Six Terms in the Reverse Direction

Directivity

ReflectionTracking

Port-1 Match

Leakage

TransmissionTracking

Port-2 Match

There are also Errors caused by the Converter, IF, Cables and Connectors

e e

L L

L

L L L

LS b

S a

S a S b

S a

10 011 1 0

0

1 1 0 0

0

2

e ML L

LS a

S a11 2

1 1 0

0

eL

LS b

S a00

0

0

e L a b30 0 3

e eL L

LS b

S a10 32

1 2 3

0

e M22 2

System Model – Forward Direction

ERRORS REMOVED ERRORS REMAINING

Noise and Residuals

Receiver Linearity

Drift after Error-Correction

Stability after Error-Correction

Repeatability of Connectors, etc

Lower Lever Leakage Paths

Errors of Calibration Standards

Port Match

Directivity

Tracking

Main Leakage Paths

Improvements with Correction

Improvements with Correction

ErrorAdapter

DUT

PerfectReflectometer

a0

b0

a0

b0

b1

a1

3 Error Terms

Port - 1

e00 e11

e10 e01

1

b1

a0

b0

DUT

a1

e00 =

e11 =

(e 10 e01 ) =

Directivity

Port Match

Tracking

One Port: 3-Term Error Model

For ratio measurements there are 3 error termsThe equation can be written in the linear form

Any 3 independent measurements can be used

e00 -e

1 - e 11M =

b0

a0

=M - e 00

Me11 - e

=

Measured Actual

e00 + Me11 - e = M

e = e 00 e11 - (e 10 e01 )

e00 + M1 e11 - e = M1

e00 + M2 e11 - e = M2

e00 + M3 e11 - e= M3

With 3 different known , measure the resultant 3 MThis yields 3 equations to solve for e 00 , e 11 , and e

One Port: 3-Term Error Model

DUTPerfectReflectometer

b0

b3

a0

b0

a3

b3

b1

a1

b2

a2

ErrorAdapter

a0,a 3

Forward

Reverse

12-Term Error Model

ReverseError

Adapter

ForwardError

Adapter

DUT[S]


b' 0

a' 3 b' 3

a' 0

b' 0

a' 3

b' 3

b' 1

a' 1

b' 2

a' 2

6 Error Terms

DUT[S]


a0

b0

b3

a0

b0

a3

b3

b1

a1

b2

a2

6 Error Terms

ForwardModel

ReverseModel

12-Term Error Model

a0

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b3

e30

e00 e11

e10 e01

1

e22

e10 e32

e00 =

e11 =

(e 10 e01 ) =

(e 10 e32 ) =

e22 =

e30 =

Directivity

Port-1 Match

Reflection Tracking

Transmission Tracking

Port-2 Match

Leakage

S 11M =b0

a0

= e 00 + (e 10 e01 )S 11 - e 22 S

1 - e 11 S 11 - e 22 S 22 + e 11 e22 S

S 21M =b3

a0

= e 30 + (e 10 e32)S 21

1 - e 11 S 11 - e 22 S 22 + e 11 e22 S

S = S 11 S 22 - S 21 S 12

FORWARD MODEL

12-Term Error Model

b'0

Port - 1 a'1

a'2b'1

b'2

S 11

S 21

S 22

S 12

DUTPort - 2

b'3

e'11

e'23 e'01

e'22

e'23 e'32

1

e'33

a'3

e'03

REVERSE MODEL

e'33 =

e'11 =

(e' 23e'32) =

(e' 23e'01) =

e'22 =

e'03 =

Directivity

Port-1 Match

Reflection Tracking

Transmission Tracking

Port-2 Match

Leakage

= e' 33 + (e' 23e'32)S 22 - e' 11 S

1 - e' 11 S 11 - e' 22 S 22 + e' 11 e'22 S

S 22M =b'3a'3

= e' 03 + (e' 23e'01)S 12

1 - e' 11 S 11 - e' 22 S 22 + e' 11 e'22 S

S 12M =b'0a'3

S = S 11 S 22 - S 21 S 12

12-Term Error Model

S

S e

e e

S e

e ee e

S e

e e

S e

e e

D

S

S e

e e

S e

e ee

D

S

S e

e e

S e

e e

M M M M

M M

M M

11

11 00

10 01

22 33

23 3222 22

21 30

10 32

12 03

23 01

21

21 30

10 32

22 33

23 3222 22

22

22 33

23 32

11 00

10 01

1

1

1

'

' ''

'

' '

'

' ''

'

' '

e

e eS e

e e

S e

e e

D

S

S e

e e

S e

e ee e

D

DS e

e ee

S e

e ee

S e

e e

M M

M M

M M M

11 1121 30

10 32

12 03

23 01

12

12 03

23 01

11 00

10 0111 11

11 00

10 0111

22 33

23 3222

21 30

10 32

1

1 1

''

' '

'

' ''

'

' ''

S e

e ee eM12 03

23 0122 11

'

' ''

12-Term Error Model

CalibrationSTEP 1: Calibrate Port-1 using One-Port procedure

STEP 2: Connect Z 0 terminations to Ports 1 & 2

STEP 3: Connect Ports 1 & 2 together

Solve for e 11 , e 00 , & (e 10 e01 ), Calculate (e 10 e01 ) from e

Measure S 21M gives e 30 directly

S 11M - e 00

S 11M e11 - e

e22 =

e10 e32 = (S 21M - e 30 )(1 - e 11 e22 )

Use the same process for the reverse model

12-Term Error Model

16-Term Error Model

ErrorAdapter

DUT[S]


ImperfectSwitch

a0

b0

a3 b3

a0

b0

a3

b3

b1

a1

b2

a2

16 Error Terms

To remove the effects of an imperfect switch, use the procedure described later.

DUT

S 11

S 12

S 22

S 21

e20 e13

e10

e01

e00 e11

e30 e03

e23

e21 e12

e22e33

e32

e31 e02

a0

a0 a1

a1

b2

b1

a2

b2b3

b0

a3

b3

b0 b1

a3 a2

One of the 16 error terms can be normalized to yield 15 error terms

e00 , e 33 Directivitye11 , e 22 Port Matche10 , e 01 , e 32 , e 23 Trackinge30 , e03 Primary Leakage

All others are lower levelleakage paths

16-Term Error Model

Measured S-Parameters SM = (T 1S + T 2)(T 3S + T 4)-1

Actual S-Parameters S = (T 1 - S MT3)-1(S MT4 - T 2)

Linear-in-T Form T1S + T 2 - S MT3S - S MT4 = 0

Error Model

With 15 or more independent observations the linear matrixequation can be solved. TRL as well as TOSL calibrationmethods are possible.

b

b

a

a

a

a

b

b

0

3

0

3

1

2

1

2

T T

T T1 2

3 4

16-Term Error Model

8-Term Error Model

DUTPerfect

Reflectometer

ImperfectSwitch

a0

b0

a3 b3

a0

b0

a3

b3

b1

a1

b2

a2

8 Error Terms

XError

Adapter

YError

Adapter

To remove the effects of an imperfect switch, use the procedure described later.

DUT

S 11

S 12

S 22

S 21

e10

e01

e00 e11

e23

e22e33

e32

a0

a0 a1

a1

b2

b1

a2

b2b3

b0

a3

b3

b0 b1

a3 a2

One of the 8 error terms can be normalized to yield 7 error terms

X Error Adapter

Y Error Adapter

8-Term Error Model

233233220110110023

10

22

11

33

00

, ,

0

01

0

0

0

0

0

0

eeeeeeeee

ek

kke

e

ke

e

k

YX

Y

X

43

21

TT

TT

b

b

a

a

a

a

b

b

0

3

0

3

1

2

1

2

T T

T T1 2

3 4

8-Term Error Model

Measured S-Parameters SM = (T 1S + T 2)(T 3S + T 4)-1

Actual S-Parameters S = (T 1 - S MT3)-1 (S MT4 - T 2)

Linear-in-T Form T1S + T 2 - S MT3S - S MT4 = 0

Expanding Yields:

e00 + S 11 S 11M e11 - S 11X + 0 + S 21 S 12M (ke 22 ) + 0 + 0 = S 11M

0 + S 12 S 11M e11 - S 12X + 0 + S 22 S 12M (ke 22 ) + 0 - S 12M k = 0

0 + S 11 S 21M e11 + 0 + 0 + S 21 S 22M (ke 22 ) - S 21 (kY) + 0 = S 21M

0 + S 12 S 21 Me11 + 0 + (ke 33 ) + S 22 S 22M (ke 22 ) - S 22 (kY) - S 22M k = 0

8-Term Error Model

Using the cascade parameters in matrix form yields

MEASURED

TM = T XT T Y

ACTUAL

T = T X-1 TM TY

-1

ATBTT

TT

TT

M

YX

M

321033

22Y

11

00X

3210

23323322Y01101100X

33

22Y

3211

00X

10

M21M12M22M11M21122211S

M22

M11M

M2122

11S

21

ee

1

1e

e

1e

e

ee

1

eeeeeeee

1e

e

e

1

1e

e

e

1

SSSSSSSS

1S

S

S

1

1S

S

S

1

8-Term Error Model

TRL & LRL

TRM & LRM

TraditionalTOSL

(Overdetermined)

LRRM

UXYZ

TXYZ & LXYZ

Thru (T) or Line (L) withknown S-parameters

[4 conditions]

Unknown Line (U) withS 12 = S 21

[1 condition]

Line (L) with knownS 11 and S 22

[2 conditions]

Known Match (M)on port-1 and port-2

[2 conditions]

3 known Reflects (XYZ)on port-1 or port-2

[3 conditions]

3 known Reflects (OSL)on port-1

[3 conditions]

Known match (M)on port-1

[1 condition]

3 known Reflects (XYZ)on port-1

[3 conditions]

2 unknown equal Reflects(RR) on port-1 and port-2

[2 conditions]

3 known Reflect (OSL)on port-2

[3 condition]

Unknown equal Reflect (R)on port-1 and port-2

[1 condition]

Seven or more independent known conditions must be measuredA known impedance (Z 0) and a port-1 to port-2 connection are required

Line (L) with knownS-parameters[4 conditions]


[4 conditions]


[4 conditions]

Thru (T) withknown S-parameters

[4 conditions]

Unknown equal Reflect (R)on port-1 and port-2

[1 condition]

3 known Reflects (XYZ)on port-2

[3 conditions]

8-Term Calibration Examples

a0

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b3

e00 e11

e10 e01

1e' 22

e10 e' 32

a0

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b3

e00 e11

e01

e10

e22

e32

a0

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b3

e00 e11

e01

e10

e22

e32

a3

e23

e33

e23

e33 3

3

33 b

a

333

3232

333

323322222

e1

ee

e1

eeee

8-Term to 12-Term: Forward

12-term model if add 2 crosstalk

terms

0

00 b

a

000

0101

000

001101111

e1

ee

e1

eeee

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

e' 11

e23 e' 01

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

e00 e11

e01

e10

e22

e23 e32

a0

b0

Port - 1 a1

a2b1

b2

S 11

S 21

S 22

S 12

DUTPort - 2

b3

e00 e11

e01

e10

e22

e32

a3

e23

e33

1

e33

b3

a3

e22

e32

e23

e33

b3

a3

8-Term to 12-Term: Reverse

12-term model if add 2 crosstalk

terms

Example: TRL

T T ATB

T ATB

T ATB

T AB T

A B

T

M

M

M

M

M

11 1

1

1

10 32

00

11

22

33 10 32

10 32

10 32

10 32

2

21 12

10 32

e e

e

e

e

e e e

e e

e e

e e S

e e

X Y

det det

det det , since det because S

Therefore

det det

det

,

Example: Unknown T, Known A & B

T T ATB

B T A T D D

M

1 1M

11 1

1

1

1

1

10 32

00

11

22

33 10 32

10 32

10 32

22

10 32

33

10 32 10 32

11 12

21 22

10 3222

2212

2233

21

22

11

22

e e

e

e

e

e e e

e e

e e

e

e ee

e e e e

D D

D D

e eD

eD

De

D

D

D

D

X Y

Y

Y

, and is completely known

Therefore

Example: Unknown B, Known A & T

ErrorAdapter

DUT[S]


a0

b0

a3 b3

a0

b0

a3

b3

b1

a1

b2

a2

Forward

Reverse

Forward

b0 = S 11M a0 + S 12M a3b3 = S 21M a0 + S 22M a3

Reverse

b' 0 = S 11M a' 0 + S 12M a' 3b' 3 = S 21M a' 0 + S 22M a' 3

Measuring S-parametersRemoving Switch Errors

Measuring S-parametersRemoving Switch Errors

Electronic Calibration Standards

Dave Blackham& Ken Wong

Electronic Calibration Standards are Transfer Standards

Definition:

Devices that derived their characteristics from measured data relative to primary or physical standards or other transfer standards.


First Generation Solid State Electronic Calibrator

Transmission Line

PIN Diodes


Second Generation Solid State Electronic Calibrator

Z1

Z4

Z3

Z2

Z1

Z4

Z3

Z2

ATTN

Discrete FET Switches

PORT A PORT B


Second Gen E-Cal Impedance States


Third Generation E-Cal

MMICPORT B

PORT A


Typical E-Cal Calibration Set-Up


1 1 1

2 2 2

1

2

1

1

1

m m

m m

mn

a

b

c

n mnn

H H

H

1 H

A E M

A A E A M

let C A A ;

then E C A M

[C] is also the covariance matrix-1

[A] *[E] = [M]

H = conjugate transpose(31)

1-Port Calibration: E-Cal Case


Differences of Calculated System Error Terms

Delta Directivity

Delta Source Match

|Ref Cal – ECal|


Differences of Calculated System Error Terms

Delta Transmission Tracking

Delta Reflection Tracking

|Ref Cal - ECal|


Noninsertable Calibration Using E-Cal

Noninsertable ECal Module

Insetable ECal Module + Unknown Thru or Adapter Removal

User Characterization


Noninsertable E-Cal Module

SAME GENDER

MIX CONNECTOR


Insertable E-Cal + Unknown Thru

VNA

Cal Port-1



VNA

Cal Port-2



VNA

Connect Unknown Thru


Fixture Embedding

PNA

P1 P2

PNA

P1 P2

Port1Coaxial Access

Point Fixtures

Ecal ModuleConnected to Coaxial

Access Points

Port2Coaxial Access

Point

Full 2-port Cal Characterize E-Cal


Fixture Embedding – alternate set up

SPDT

PNA

P1 P2

SwitchDriver


Insitu Calibration – 2-port E-Cal

VNA

MEASUREMENTPORTS

Requires External Software


Accuracy of Error Correction


Residual Errors

OSL Fixed Load

OSL Sliding Load

TRL TRM

Directivity -40 dB -52 dB -60 dB -40 dB

Match -35 dB -41 dB -60 dB -40 dB

Reflection Tracking

± .1 dB ± .05 dB ± .01 dB ± .01 dB

APC-7 (7 mm Coax) at 18 GHz

Vector Network AnalyzerReferences

slide 1 network analyzer error models and calibration methods by doug rytting

Documents

error model system error

term method

term error modelpageone

term examples

port error models

forward12term model

reverse12term model

crosstalk termspage8term