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

Port - 1 Port - 2

b2Cable Cable

Network Analyzer Block Diagram

a0 b0 b3

RFSource

Port - 1 Port - 2

Block Diagram – Forward Direction

N La0 N Lb0N Lb3

M 2M 1

1+A a0 +N Ha0

Port - 1a1

Port - 2Cable &

ConnectorCable &

Connector

M C M CM C

Branches With No Label = 1

La0-b3

LS-a0 LS-b0 L1-a0

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

Port - 1a1

DUTPort - 2

e00 e11

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

S a S b

10 011 1 0

1 1 0 0

e ML L

S a11 2

e L a b30 0 3

e eL L

S a10 32

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

ErrorAdapter

PerfectReflectometer

3 Error Terms

Port - 1

e00 e11

e10 e01

(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 =

=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

ErrorAdapter

a0,a 3

Forward

Reverse

12-Term Error Model

ReverseError

Adapter

ForwardError

Adapter

DUT[S]

a' 3 b' 3

6 Error Terms

DUT[S]

6 Error Terms

ForwardModel

ReverseModel

12-Term Error Model

Port - 1 a1

DUTPort - 2

e00 e11

e10 e01

e10 e32

(e 10 e01 ) =

(e 10 e32 ) =

Directivity

Port-1 Match

Reflection Tracking

Transmission Tracking

Port-2 Match

Leakage

S 11M =b0

= 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

= 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

Port - 1 a'1

a'2b'1

DUTPort - 2

e'23 e'01

e'23 e'32

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

e ee e

M M M M

23 3222 22

e eS e

e ee e

11 1121 30

10 0111 11

10 0111

23 3222

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

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

16 Error Terms

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

e20 e13

e00 e11

e30 e03

e21 e12

e22e33

e31 e02

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.

T T1 2

16-Term Error Model

8-Term Error Model

DUTPerfect

Reflectometer

ImperfectSwitch

8 Error Terms

XError

Adapter

YError

Adapter

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

e00 e11

e22e33

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

eeeeeeeee

T T1 2

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

321033

23323322Y01101100X

M21M12M22M11M21122211S

eeeeeeee

SSSSSSSS

8-Term Error Model

TRL & LRL

TRM & LRM

TraditionalTOSL

(Overdetermined)

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]

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

Port - 1 a1

DUTPort - 2

e00 e11

e10 e01

1e' 22

e10 e' 32

Port - 1 a1

DUTPort - 2

e00 e11

Port - 1 a1

DUTPort - 2

e00 e11

323322222

8-Term to 12-Term: Forward

12-term model if add 2 crosstalk

001101111

Port - 1 a1

DUTPort - 2

e23 e' 01

Port - 1 a1

DUTPort - 2

e00 e11

e23 e32

Port - 1 a1

DUTPort - 2

e00 e11

8-Term to 12-Term: Reverse

12-term model if add 2 crosstalk

Example: TRL

T T ATB

T AB T

33 10 32

det det

det det , since det because S

Therefore

det det

Example: Unknown T, Known A & B

T T ATB

B T A T D D

33 10 32

10 32 10 32

10 3222

e e e e

, and is completely known

Therefore

Example: Unknown B, Known A & T

ErrorAdapter

DUT[S]

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

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

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

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

Cal Port-1

Cal Port-2

Connect Unknown Thru

Fixture Embedding

Port1Coaxial Access

Point Fixtures

Ecal ModuleConnected to Coaxial

Access Points

Port2Coaxial Access

Full 2-port Cal Characterize E-Cal

Fixture Embedding – alternate set up

SwitchDriver

Insitu Calibration – 2-port E-Cal

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

Accuracy of Error Correction

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