1 Multi Port Measurements Slides from Dave Blackham and Ken Wong At Agilent Technologies With some...

1

Multi Port Measurements

Slides from Dave Blackham and Ken WongAt Agilent Technologies

With some additions by Doug Rytting

Dave Blackham& Ken Wong

2

Agenda

Two Port Network AnalysisMultiport Network AnalysisMultiport Network Analysis Port Match Correction

Single Reference Receiver ExampleMultiport Using a 2-port VNA Example

Multiport Calibration ApproachHow Many Connections Are Needed

Examples

3

RFSource

LOSource

a0

b0 b3

Port - 1 Port - 2

a3

DUTa2

a1

b1

b2Cable Cable

IF

IF

IF

IF

Network Analyzer Block Diagram

4

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.

5

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

6

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

7

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

0 + S 12 S 11M e11 - S 12 X + 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

8

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

9

ErrorAdapter

DUT[S]

PerfectReflectometer

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 Port Match Changes Caused by Switch

10

Measuring S-parameters

213

0

0

3

13

0

0

3

3

3

M22

20

3

3

3

0

3

M21

13

0

0

0

3

0

M12

20

3

3

0

0

0

M11

'a

'b

a

b1d

d

'a'b

ab

'a'b

Sd

ab

'a'b

ab

S

d

'a'b

ab

'a'b

Sd

ab

'a'b

ab

S

By defining

3

32

0

01 b

a and

b

a

11

Agenda




Examples

12

Multiport error correction

Is multiport error correction hard?


13


Is multiport error correction hard?No, multiport error correction with constant match is as easy as single port error correction.


14

Multiport error box diagram

IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

mi i mi i i

i i mi i i

b e a e b

a e a e b


15


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

00 01

10 11

mi i mi i i

i i mi i i

mi mii i

i ii i

b e a e b

a e a e b

b ae e

a be e


16


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

00 01

10 11

00 01

10 11

mi i mi i i

i i mi i i

mi mii i

i ii i

i ii

i i

b e a e b

a e a e b

b ae e

a be e

e e

e e

E


17


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

00 01

10 11

00 01

10 11

10 01

00

111

mi i mi i i

i i mi i i

mi mii i

i ii i

i ii

i i

ii i

mi ii

imii

i

b e a e b

a e a e b

b ae e

a be e

e e

e e

be e

b ae

ba ea

E


18


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

1 1

1 1

00 01

10 11

; ;

; ;

mi i mi i i

i i mi i i

i i

n n

m m

m mi m mi

mn mn

m m

b e a e b

a e a e b

a b

a b

a b

a b

a b

a b

a b

a b

b a

a b


19


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 01

10 11

1

1

100 01 11 10

1 101 00 10

111

111

ˆ

ˆ

ˆ ˆ

m m

m m m

a

m a a

n m

n a a

a n n

b a

a b

S b a

S b a

S S I S

S S

S S I S

S S I S


20


IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00 10 01 00

1

2

1 101 00 10

00111 1 12

10 01 10 01 10 011 1 1 2 1

2110 02 1

For the non-leaky model

, , , and are

each diagonal matricies

0 0

0

0

0 0

ˆ

ˆ

ik

ikik

ikn

n m

n

n

n

e

e

e

SS e S

e e e e e e

S

e e

S S

S

0022 2

1 10 012 2

001

10 01 10 011

n nn n

n n n

S e

e e

S S e

e e e e


21

Multiport error box diagramwith “12 term” crosstalk

IdealVNA

E2

En

DUT

E1

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

00

00 00 001 1:2 1:00 00

00 2:1 2

00 00:1

101 00 10

For the multiport equivalent to two-port

12 term model fills out to include

additional isolation terms

ˆ

n

n n

n m

e e e

e e

e e

S S

1

00 000012 1:2 1 1:11 1

10 01 10 01 10 011 1 1 2 1

00 0021 2:1 22 2

10 01 10 012 1 2 2

00 001 :110 01 10 01

1

ˆ

n n

n

n

n n nn n

n n n

S e S eS e

e e e e e e

S e S e

e e e e

S e S e

e e e e

S


22

Multiport error box diagramwith full leaky model

00 10 01 11

1 1:2 1:

2:1 2

:1

1

00 01 11 10

For the multiport full leaky model

, , , and all fill out to

include additional crosstalk termsij ij ij

nij ij

ij

ij ijn n

m a a

e e e

e e

e e

S I S S

1 101 11 01

1 111 01 00 10 01 00

1

0

0

m a m

a

m a m a

a m m

S S S

S

K S S L S S H M

S K S M L S H

2n

n+2

n+11

2

n

IdealVNA

DUT

1ma 1a

2ma 2a

mna na

nbmnb

2mb 2b

1mb 1b

E


23

Agenda




Examples

24


Models presented thus far assume a constant port match

similar to 8 term two-port model for non-leaky casesimilar to 16 term two-port model for leaky case

Due to switching, port match is not constant similar to 12 term two-port model


25

What Is Switch Correction?

TRL and unknown thru algorithms belong to a class that assumes a constant match at each test port.In reality, the match at each test port will vary as the source is switched from port to port.Switch correction is the process of characterizing the match difference then factoring it out of the calibration process

Generalized s-parameters factor out match differences during raw measurements for receivers that have dual couplers at each port (reference receiver at each port).Two-tier calibration approaches characterize match differences with a first tier calibration using SOLT. This allows the use of generalized s-parameters approach for systems that have a single reference receiver.


26

Ideal S-Parameters

N port DUT

iia

iib

ii

îib

îia

jia

jib

ji

ˆjib

nia nibni

n̂ib

kia

kib

ki

ˆkib

Ideal s-parametersNon-source ports terminated in perfect match—incident signal only from source port

111 12 111 12 1 11

21 22 2 2221 22 2

1 21 2

111 12

11 22

221 22

11 22

ˆ ˆ ˆ ˆ 0 0ˆ ˆ ˆ ˆ0 0

ˆ0 0ˆ ˆ ˆ

ˆˆ ˆ

ˆ ˆ ˆ

ˆˆ ˆ

ˆ ˆ ˆ

ˆ

nn

n n

n n nn nnn n nn

n

nn

n

nn

b b bS S S a

S S S ab b b

S S S ab b b

bb b

a a a

bb b

a a a

1 2

11 22

ˆ ˆ

ˆ ˆ ˆn n nn

nn

b b b

a a a


27

Use Generalized S-Parameters

N port DUT

iia

iib

ii

îib

îia

jia

jib

ji

ˆjib

nia nibni

n̂ib

kia

kib

ki

ˆkib

Ideal s-parametersNon-source ports terminated in perfect match—incident signal only from source port

Generalized s-parametersUses incident signals from all ports & removes port match error

111 12

11 22

11 12 1

221 2221 22 2

11 22

1 2

1 2

11 22

ˆˆ ˆ

ˆ ˆ ˆ

ˆˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

n

nn

n

nn

nn

n n nn

n n nn

nn

bb b

a a aS S S

bb bS S S

a a a

S S Sb b b

a a a

1

11 12 1 11 12 1 11 12 1

21 22 2 21 22 2 21 22 2

1 2 1 2 1 2

n n n

n n n

n n nn n n nn n n nn

S S S b b b a a a

S S S b b b a a a

S S S b b b a a a


28

Agenda




Examples

29

Single Reference Receiver

Rcvr ARcvr B Rcvr C

Rcvr D

Recr R

RF

LO

Port-A Port-B Port-C Port-D


30

S-parameter measurement (two-port, ideal)

11 121

21 22 22

1 212 22

2 2

0

;

r

rr

r r

r r

S Sb

S S ab

b bS S

a a

11 121 1

21 222

1 211 21

1 1

0

;

f f

f

f f

f f

S Sb a

S Sb

b bS S

a a

Forward s-parameters Source at port 1

Reverse s-parametersSource at port 2


31

11 121

21 22 22

1 212 22

2 2

0

;

r

rr

r r

r r

S Sb

S S ab

b bS S

a a

11 121 1

21 222

1 211 21

1 1

0

;

f f

f

f f

f f

S Sb a

S Sb

b bS S

a a

11 121 1 1

21 222 2 2

1

11 12 1 1 1

21 22 2 2 2

0

0

0

0

f r f

f r r

f r f

f r r

S Sb b a

S Sb b a

S S b b a

S S b b a

S-parameter measurement (two-port, ideal)


32

S-parameter measurement (two-port, non-ideal)

1

11 12 1 1 1 1

21 22 2 2 2 2

f r f r

f r f r

S S b b a a

S S b b a a

Generalized s-parameters

• Dual reflectometers at each testport allow measurement of all signals required to determine s-parameters.

• Using this method will correct for the changing port match caused by the switch.


33


1

11 12 1 1 1 1

21 22 2 2 2 2

f r f r

f r f r

S S b b a a

S S b b a a

Generalized s-parameters

• Dual reflectometers at each testport allow measurement of all signals required to determine s-parameters.

• Benefit allows constant match to be assumed for error correction (eight term model)

• Match variations tracked by incident wave measurements


34


• Non dual reflectometer analyzers can’t measure signals reflected from switch in off position.

• Requires mathematical equivalent computed from difference between source and load match at each port (delta match)

• Generalized S-parameter in ratio form:

1

1 1 1

1 2 211 12

21 22 2 2 2

1 2 1

1

1

f r r

f r r

f r f

f r f

b b a

a a aS S

S S b b a

a a a


35


1 1 1

1 1 1 1

2 2 1 2

2 2 2 2

1 1 2 1

1 1 1

1 1 1

r r r r

rr r r r

f f f f

ff f f f

a b a b

a a b a

a b a b

a a b a

2 1Need to replace and terms.f ra a


36

Calculate F and RFor Single Reference Receiver

Error terms were measured during the first tier calibration using SOLT.With F and R determined the generalized s-parameters can be used to remove the port match variations.Also TRL or unknown thru, etc. can be used in a second tier calibration.

ELF

EDR

ESR

ERR/

F

ETF

a1 b2

( ) ( )

LF SR LR SF

F RRR DR LF SR RF DF LR SF

E E E E

E E E E E E E E

2 2 1 1andf f r r

F Ra b a b


37

Agenda




Examples

38

Multiport Using a 2-port VNA Example

Rcvr ARcvr B

Recr R

RF

LO

Recr R

Switches Terminatedin off state


39

Multiport Using a 2-port VNA

i ijii:j 11 12

i:jji jj21 22

0S SSm ; ; i 1 N, j i

0S S

Let:

Smi:j = measured S-parameters between ports i and j.

Rmi:j = Port impedance normalized Scattering Matrix

i:j = Diagonal matrix of reflection coefficient of imperfect

port terminations at ports i and j. [i..N values must not

change when signal paths are changed.]

i:i i:j

1i:j * i:j i:ji:j i:j j:i j:j

R RRm Sm I Sm

R R


40


Let:Rn = Composite port impedance normalized N-port Scattering Matrix

R1:1 R1:2

R2:1 R2:2

•Fill Rn matrix with calculated Rm sub-matricesi=1, j=2

[Rn] matrix


41



R1:1 R1:2

R2:1 R2:2 R2:3 R3:2 R3:3


[Rn] matrix


42



R1:1 R1:2 R1:3

R2:1 R2:2 R2:3 R3:1 R3:2 R3:3


[Rn] matrix


43



R1:1 R1:2 R1:3 R1:N

R2:1 R2:2 R2:3 R2:N

R3:1 R3:2 R3:3 R3:N

Ri:1 Ri:i Ri:j Ri:N

Rj:1 Rj:i Rj:j Rj:N

RN:1 RN:N

•Fill Rn matrix with calculated Rm sub-matricesDo N(N-1)/2 2-port measurements to fill

[Rn] matrix


44


Let:Rn = Composite port impedance normalized N-port Scattering Matrixn = Diagonal matrix of reflection coefficient of imperfect port terminations at ports 1 to N.Sn = S-parameters of corrected N-port

•Normalize Result back to System Impedance

1

1 2*

N

0 0

0 0 0Sn I Rn n Rn n ; n

0 0


45

Agenda




Examples

46

Multiport calibration Approach

Use all of the same calibration standards used by two port calibrations.Brute force method: calibrate all possible two-port pairs

This will get tedious very quickly as the number of ports increases

:

:2

!

! !

1

2 2

n r

n

n nC

r r n r

n n nC

n 2 3 4 6 8 12

Cn:2 1 3 6 15 28 66


47

Multiport calibration error terms

00ie

10ie

01ie

11ie

00je

01je

10je

11je

00ke

01ke

10ke

11ke

Port terms Path terms

00

10 01

11

11

sets of terms

Directivity

Reflection tracking

Source match

Load match

i

i i

Si

Li

n

e

e e

e

e

10 01

00

* 1 sets of path terms

Transmission tracking

Crosstalk

i j

ij

n n

e e

e


48

Minimizing Connections During Multiport calibration

Characterize each set of port terms once (n).Characterize (n-1) thru standards to characterize (n) load match terms and 2x(n-1) sets of transmission tracking terms. Compute the other (n-1)x(n-2) transmission tracking terms.If desired, connect loads to each port then characterize n x (n-1) sets of crosstalk terms.Full leaky model would connect multiple permutations of one port reflection standards to the ports and measure n x (n-1) paths for each permutation.


49

Required Number of ThrusConnect (n-1) thru connections and characterize 2x(n-1) transmission tracking terms. The other (n-1)x(n-2) terms can be calculated.


Port 1

Port N

Port 2

Port 3

Required Thrus

50

Compute Transmission Tracking Characterize transmission tracking between ports i and j Characterize transmission tracking between ports i and k Compute transmission tracking between ports j and k Accuracy of computed transmission tracking terms less than characterized transmission tracking terms. Actual equation includes compensation for varying port match (source match not equal to load match at port i).

transmission tracking transmission tracking

port to port port to port

10 10 0110 01 10 01

10 01

reflection trackingport

j i i k

j i ij i i k

i i

i

e e ee e e e

e e

01

10 01

k

i i

e

e e 10 01

transmission trackingport to port

j k

j k

e e


51

Agenda




Examples

52

Port 1

Port 4

Port 2

Port 3

Precision Mechanical 2-port Cal (SOLT or TRL)

Multiport Mechanical Cal

Mechanical Cal Method


Port 1

Port 4

Port 2

Port 3Unknown Thrus (adapters)

AND

Finish 4-port CalUsing Unknown Thru.

Only Transmission trackingneeds to be determined.

53

Port 1

Port 4

Port 2

Port 3

EC

al

1E

Cal

2

Port 1

Port 4

Port 2

Port 3Unknown Thrus (adapters)

AND

Multiport ECal Cal

ECal Method




54

Port 1

Port N

Port 2

Port 3

Port 1

Port N

Port 2

Port 3

Unknown Thrus (Adapters)

1-Port Calibrations, ECal or Mech

Multiport Unknown Thru Cal

Can have differentconnector on

Each Port




55

Port 1

Port 4

Port 2

Port 3

Port 1

Port 4

Port 2

Port 3

AND

TRL on Wafer Cal

Multiport On-Wafer Cals

Straight Thrus

Imperfect Unknown Thrus




56

Advantages of Unknown Thru Calibration in Multiport

Systems

Unknown Thru is very convenient for right-angle or not-in-line thru calibrations.S-parameters of the thru standard need not to be characterized.Eliminates the need to move test ports and cables or probes.Passive DUTs may be used as the unknown thru.Noninsertable cal (mix connectors, transitions, F-F or M-M combinations) is just as easy as an insertable cal


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1 Multi Port Measurements Slides from Dave Blackham and Ken Wong At Agilent Technologies With some...

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