Lects 451 TRL 07 - University Of Illinoisemlab.uiuc.edu/ece451/appnotes/TRL.pdfrac ca 1b 1 a β ρα β γ α γ α ... Microsoft PowerPoint - Lects_451_TRL_07 Author: Jose Created

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Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 451 1

ECE 451Automated Microwave Measurements

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]

TRL Calibration


Board with traces

L-shaped support

Center pin

screw

Flange-mountconnector

Coaxial‐Microstrip Transition


L1CL1 CR2

L2CL2 CR1

TL1SMA SMA

In Out

Coaxial‐Microstrip Transition

TDR Plot

Equivalent Circuit


With parasiticsNo parasitics


DUT

microstrip microstrip

coaxial connector coaxial connector

TRL CALIBRATION SCHEME

Want to measure DUT only and need to remove the effect of coax-to-microstrip transitions. Use TRL calibration


Error BoxA

Error BoxB

Port1

Port2Γ1 Γ2

1 2

W1 W2

MeasurementPlanes

Error boxes A and B account for the transition parasitics and the electrical lengths of the microstrip.

Make three standards: Thru, Line and Reflect

A model for the different error boxes can be implemented

TRL Error Box Modeling


connect thru

A B

RaRb

Rt

t a bR R R=

Step 1 - THRU Calibration


LINE

connect line (Note: difference in length between thru and line)

A B

Ra Rb

Rd

LINE

RL

Step 2 - LINE Calibration


REFLECT

connect reflect

Step 3 - REFLECT Calibration

A B

Ra Rb


-5

0

5

1 1.5 2 2.5 3 3.5 4 4.5 5

Measured |S11| of Microstrip Unknown Relative to TOUCHSTONE Models

PORT EXT. calibrationTRL calibration

Rel

ativ

e M

agni

tude

, dB

Frequency, GHz

PORT EXT. data compared to L=.808 nH modelTRL data compared to L=.948 nH model

TRL – Measurement Comparison


-20

-15

-10

-5

0

1 1.5 2 2.5 3 3.5 4 4.5 5

Measured Data for Microstrip Unknown

with TRL calibrationwith 722 ps port ext. (inc. barrel)

|S11

| (d

B)

Frequency, GHz

Measured 10/18/94

TRL – Measurement Comparison


TRL Derivation

- Obtain network parameters of error boxes A and B- Remove their effects in subsequent measurements

TRL Objectives


S11A

a1R

b1R

S22A

S12A

S21A

ΓR

S12B

S21B

S22BS11

B

b2R

a2R

ΓR

R2

R1R1 a 0

ba

= R2

R2R2 a 0

ba

=

Model for Reflect

2 Measurements


S11A

1

1

a1T

b1T

S22A

S12A

S21A

S12B

S21B

S22BS11

B

b2T

a2T

T2

T1T1 a 0

ba

= T2

T2T1 a 0

ba

=T1

T2T2 a 0

ba

=T1

T1T2 a 0

ba

=

Model for Thru

4 Measurements


S11A

e-γLa1L

b1L

S22A

S12A

S21A

S12B

S21B

S22BS11

B

b2L

a2L

e-γL

L2

L1R1 a 0

ba

= L2

L2L1 a 0

ba

=L1

L2L1 a 0

ba

= L1

L1L2 a 0

ba

=

Model for Line

4 Measurements


Using T parameters (transfer parameters), we can show that if

1 11 2

1 22 221

11

b S ba S aS

−Δ⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠

1 11 1 12 2b S a S a= +2 21 1 22 2b S a S a= +

11 22 12 21Δ = −S S S S

11

2221

11

−Δ⎛ ⎞= ⎜ ⎟−⎝ ⎠

SR

SS

Use R (or T) Parameters


M A BR R RR=

A B

1 1MR R R R− −=

11 12A 22

21 22

r r a bR r

r r c 1⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

11 12B 22

21 22

R1

ρ ρ α βρ

ρ ρ γ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

A

1

22

1 b1 1Rc ar a bc

− −⎡ ⎤= ⎢ ⎥−− ⎣ ⎦

The measurement matrix RM is just the product of the matrices of the error boxes and the unknown DUT

or

Let RA be written as

The inverse of RA is

RB is similarly written as

TRL Derivation


B

1

22

11 1Rβ

γ αρ α βγ− −⎡ ⎤

= ⎢ ⎥−− ⎣ ⎦

M22 22

1 b 11 1 1 1R Rc c ar a 1 b 1a

ββ γ αρ α γα

− −⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦− −

T A BR R R=

And the inverse of RB is

The matrix of the DUT is then found from

Note that although there are eight terms in the error boxes, only seven quantities are needed to find R. They are a, b, c, α, β, γ, and r22ρ22

From the measurement of the through and of the line, seven quantities will be found. They are b, c/a, β/ α, γ, r22ρ22, αa and e2γl

In addition to the seven quantities, if a were found, the solution would be complete. Let us first find the above seven quantities.

The ideal through has an R matrix which is the 2 x 2 unit matrix. The measured Rmatrix with the through connected will be denoted by RT and is given by

Where RA and RB are the R matrices of the error box A and B respectively. With the line connected, the measured R matrix will be denoted by RD and is equal to

TRL Derivation


A

1D A L TR R R R R−=

T

1D A A LR R R R R− =

T

1DT R R−=

A A LTR R R=

11 12

21 22

t tT

t t⎡ ⎤

= ⎢ ⎥⎣ ⎦

l

L l

e 0R

0 e

γ

γ

−

+

⎡ ⎤= ⎢ ⎥

⎣ ⎦

D A L BR R R R=

A

1B TR R R−=

where RL is the R matrix of the line

Now

so that

Define Which when substituted into the above equations results in

The matrix T is known from measurements and will be written as

, since the line is non-reflecting

TRL Derivation


11 12A 22

21 22

r r a bR r

r r c 1⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

11 12B 22

21 22

R1

ρ ρ α βρ

ρ ρ γ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

A A LTR R R=

l11 12

l21 22

t t a b a b e 0t t c 1 c 1 0 e

γ

γ

−

+

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

RA is unknown and was written as

RB similarly was written as

Recalling and writing the matrices results in

Next, writing out the four equations gives:

TRL Derivation


l11 12t a t c ae γ−+ =

l21 22t a t c ce γ−+ =

l11 12t b t be γ++ =

γ++ = l21 22t b t e

++= =

+ +

11 1211 12

21 2221 22

at tt a t c a cat a t c c t tc

( )2

21 22 11 12a at t t t 0c c

⎛ ⎞ + − − =⎜ ⎟⎝ ⎠

Dividing the first of the above equation by the second results in

which gives a quadratic equation for a/c

Dividing the third equation in the group by the fourth results in

TRL Derivation


11 12

21 22

t b t bt b t

+=

+

( )221 22 11 12t b t t b t 0+ − − =

2 L 21 22 21 22

21 2221 22

t b t t b te c at a t c t tc

γ + += =

+ +

11 12 2111

21 22

r S Sa Sc r S

= = −

which gives the analogous quadratic equation for b as

Dividing the fourth equation in the group by the second results in

Since e2γL is not equal to 1, b and c/a are distinct roots of the quadratic equation. The following discussion will enable the choice of the root. Now b=r12/r22=S11 and

TRL Derivation


abc

A A LTR R R=

( )( ) ( )( )A A Ldet T det R det R det R=

( ) ( )Ldet T det R 1= =

11 22 12 21t t t t 1− =

For a well designed transition between coax and the non-coax |S22|, |S11| <<1 which yields |b|<< 1and |a/c|>>1. Therefore,

which determines the choice of the root

Recalling

or

so that

which implies that there are only three independent Tij. Then there are only three independent results, e.g. b, a/c, and e2γL.

TRL Derivation


22 22 A B T

a b d er R R R g

c 1 1 f 1α β

ργ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1a b 1 b1c 1 c aa bc

− −⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥−−⎣ ⎦ ⎣ ⎦

22 22

1 b d egr1 c a f 1a bc

α βρ

γ−⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

22 22

d bf e bgr1 af cd a cea bc

α βρ

γ− −⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥− −−⎣ ⎦ ⎣ ⎦

Now let us find four more quantities

Now

So that

or

TRL Derivation


ρ−−

= =− −

22 22

c1 ea ce ar g g ca bc 1 ba

d bf e b11 af cd a cea ce

α βγ

− −⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥− −−⎣ ⎦ ⎣ ⎦

cf dac1 ea

γ−

=−

e bd bf

βα

−=

−

TRL Derivation

and

We also have

from which we can extract

from which we obtain


d bfa c1 ea

α −=

−

R1

R

a bwc 1ΓΓ

+=

+

1

R 1

w bac1 wa

Γ

−=

⎛ ⎞−⎜ ⎟⎝ ⎠

12 21 R 22 R2 22

11 R 11 R

S S Sw S1 S 1 S

Γ ΔΓΓ Γ

−= + =

− −

The additional four quantities found are β/α, γ, r22ρ22 and αa. To complete the solution, one needs to find a. Let the reflection measurement through error box A be w1. Then

which may be solved for a in terms of the known b and a/c as

We need a method to determine a. Use the measurement for the reflect from through the error box B. Let w2 denote the measurement

TRL Derivationand


21 11R

22 222

12R

22

w1

ρ ρ Γρ ρ

ρ Γρ

− +=

−

R2

R

w1

αΓ γβΓ

−= −

−

2

R 2

w

1 w

γαβΓα

+=

⎛ ⎞+⎜ ⎟⎝ ⎠

1

R 1

w bac1 wa

Γ

−=

⎛ ⎞−⎜ ⎟⎝ ⎠

or

α may be found in terms of γ and β/α as

Recall

TRL Derivation


21

21

1 ww bacw 1 wa

βα

α γ

+−=

+ −

d bfa c1 ea

α −=

−

22 1

21

1 ww b d bfa c cw 1 w 1 ea a

βα

γ

+− −=

+ − −

12

21

21

1 ww b d bfa c cw 1 w 1 ea a

βα

γ

⎛ ⎞+⎜ ⎟− −= ±⎜ ⎟+⎜ ⎟− −

⎝ ⎠

so that

From earlier

so that

which determines a to within a ± sign.

TRL Derivation

or


1R

1

w bca 1 wa

Γ −=

⎛ ⎞−⎜ ⎟⎝ ⎠

So if ΓR is known to within ± then a may be determined as well. Calibration is complete and we can now proceed to the measurement of the DUT.

TRL Derivation

M22 22

1 b 11 1 1 1R Rc c ar a 1 b 1a

ββ γ αρ α γα

− −⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦− −

From earlier, the matrix of the DUT is found from

in which all the terms have now been determined.

Documents

Lects 451 TRL 07 - University Of Illinoisemlab.uiuc.edu/ece451/appnotes/TRL.pdfrac ca 1b 1 a β ρα β γ α γ α ... Microsoft PowerPoint - Lects_451_TRL_07 Author: Jose Created