s-parameters

© H. Heck 2008 Section 5.5 1

Module 5: Advanced Transmission LinesTopic 5: 2 Port Networks & S-Parameters

OGI EE564

Howard Heck


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Where Are We?

1. Introduction

2. Transmission Line Basics

3. Analysis Tools

4. Metrics & Methodology

5. Advanced Transmission Lines1. Losses

2. Intersymbol Interference

3. Crosstalk

4. Frequency Domain Analysis

5. 2 Port Networks & S-Parameters

6. Multi-Gb/s Signaling

7. Special Topics


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Acknowledgement

Much of the material in this section has been adapted from material developed by Stephen H. Hall and James A. McCall (the authors of our text).


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Contents

Two Port Networks Z Parameters Y Parameters Vector Network Analyzers S Parameters: 2 port, n ports Return Loss Insertion Loss Transmission (ABCD) Matrix Differential S Parameters (MOVE TO 6.2) Summary References Appendices


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Two Port Networks Linear networks can be completely characterized by

parameters measured at the network ports without knowing the content of the networks.

Networks can have any number of ports. Analysis of a 2-port network is sufficient to explain the theory

and applies to isolated signals (no crosstalk).

The ports can be characterized with many parameters (Z, Y, S, ABDC). Each has a specific advantage.

Each parameter set is related to 4 variables: 2 independent variables for excitation 2 dependent variables for response

2 PortNetworkP

ort

1I1

+

-

V1

Po

rt 2I2

+

-

V2


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Z Parameters

Advantage: Z parameters are intuitive. Relates all ports to an impedance & is easy to calculate.

Disadvantage: Requires open circuit voltage measurements, which are difficult to make. Open circuit reflections inject noise into measurements. Open circuit capacitance is non-trivial at high frequencies.

NNNNN

N

N I

I

I

ZZZ

Z

ZZZ

V

V

V

2

1

21

21

11211

2

1

IZV

0

jkI

j

iij I

VZ (Open circuit impedance)

Impedance Matrix: Z ParametersImpedance Matrix: Z Parameters

or [5.5.1]

where [5.5.2]

2221212

2121111

IZIZV

IZIZV

2 Port example:2 Port example:

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V[5.5.4][5.5.3]


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Y Parameters

NNNNN

N

N V

V

V

YYY

Y

YYY

I

I

I

2

1

21

21

11211

2

1

VYI

0

jkV

j

iij V

IY (Short circuit admittance)

Admittance Matrix: Y ParametersAdmittance Matrix: Y Parameters

or

[5.5.6]

[5.5.5]

where

2221212

2121111

VYVYI

VYVYI

2 Port example:2 Port example:

2

1

2221

1211

2

1

V

V

YY

YY

I

I

Advantage: Y parameters are also somewhat intuitive. Disadvantage: Requires short circuit voltage

measurements, which are difficult to make. Short circuit reflections inject noise into measurements. Short circuit inductance is non-trivial at high frequencies.

[5.5.7] [5.5.8]


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Example

ZC

ZA ZB

+

-

+

-

V1 V2

I1 I2

Por

t 1

Por

t 2

CA

CAI

ZZ

ZZV

V

I

VZ

1

1

1

111

02

CC

I

ZI

ZI

I

VZ

2

2

2

112

01

CC

I

ZI

ZI

I

VZ

1

1

1

221

02

CB

CBI

ZZ

ZZV

V

I

VZ

2

2

2

222

01


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Frequency Domain: Vector Network Analyzer (VNA) VNA offers a means to

characterize circuit elements as a function of frequency.

VNA is a microwave based instrument that provides the ability to understand frequency dependent effects. The input signal is a frequency swept sinusoid.

Characterizes the network by observing transmitted and reflected power waves. Voltage and current are difficult to measure directly. It is also difficult to implement open & short circuit loads at high

frequency. Matched load is a unique, repeatable termination, and is

insensitive to length, making measurement easier. Incident and reflected waves the key measures. We characterize the device under test using S parameters.

2-PortNetwork

V1

+

V2

I1 I2

-

+

-


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S Parameters

We wish to characterize the network by observing transmitted and reflected power waves. ai represents the square root of the power wave injected into port i.

bi represents the square root of the power wave injected into port j.

2 PortNetwork

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1 b2

RVP

2

R

VPai

1

R

Vb jj

use

to get

[5.5.9]

[5.5.10]

[5.5.11]


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S Parameters #2

We can use a set of linear equations to describe the behavior of the network in terms of the injected and reflected power waves.

For the 2 port case:

2 PortNetwork

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1 b2

2221212

2121111

aSaSb

aSaSb

iport at measuredpower

jport at measuredpower

i

jij a

bSwhere

in matrix form:

[5.5.12]

[5.5.13]

2

1

2221

1211

2

1

a

a

SS

SS

b

b

Howard Heck

Fix "S55"


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S Parameters – n Ports

[5.5.14]

[5.5.17]

n

nn Z

Va0

aSb

n

nn Z

Vb0

jkk

jkk

Vj

j

i

i

aj

iij

ZV

ZV

a

bS

,0

,0

0

0

nnnn

N

n a

a

a

SS

S

SSS

b

b

b

2

1

1

21

11211

2

1

nnnnnn

nn

nn

aSaSaSb

aSaSaSb

aSaSaSb

2211

22221212

12121111

or

[5.5.15]

[5.5.16]

[5.5.18]


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64 Scattering Matrix – Return Loss

S11, the return loss, is a measure of the power returned to the source.When there is no

reflection from the load, or the line length is zero, S11 is equal to the reflection coefficient.

50

50

0

00

1

1

0

1

0

1

1

111

02Z

Z

V

V

V

V

ZV

ZV

a

bS

incident

reflected

a

[5.5.19]

RS = 50

RL = Z0Z0

z=0 z=l

0

0,0

jjai

iii a

bSIn general: [5.5.20]


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64 Scattering Matrix – Return Loss #2

When there is a reflection from the load, S11 will be composed of multiple reflections due to standing waves.

Use input impedance to calculate S11 when the line is not perfectly terminated.

)0(1

)0(1)0(

z

zZzZZ oin

If the network is driven with a 50 source, S11 is calculated using equation [5.5.22]

RS = 50

Zin

S11 for a transmission line will exhibit periodic effects due to the standing waves.

In this case S11 will be maximum when Zin is real. An imaginary component implies a phase difference between Vinc and Vref. No phase difference means they are perfectly aligned and will constructively add.

50

5011

in

inv Z

ZS

[5.5.21]

[5.5.22]

ZLZ0

z=lz=0


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Scattering Matrix – Insertion Loss #1

When power is injected into Port 1 and measured at Port 2, the power ratio reduces to a voltage ratio:

incident

dtransmitte

o

o

aV

V

V

V

Z

V

Z

V

a

bS

1

2

1

2

021

221

2 PortNetwork

a1

+

-

V1

Po

rt 2

a2

+

-

V2

Po

rt 1

b1 b2Z0 Z0

S21, the insertion loss, is a measure of the power transmitted from port 1 to port 2.

[5.5.22]


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Comments On “Loss”

True losses come from physical energy losses.Ohmic (i.e. skin effect) Field dampening effects (loss tangent) Radiation (EMI)

Insertion and return losses include other effects, such as impedance discontinuities and resonance, which are not true losses.

Loss free networks can still exhibit significant insertion and return losses due to impedance discontinuities.


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Reflection Coefficients

Reflection coefficient at the load:

0

0

ZZ

ZZ

L

LL

0

0

ZZ

ZZ

S

SS

L

L

L

Lin S

SS

S

SSS

11

212

1122

211211 11

S

Sout S

SSS

11

211222 1

[5.5.23]

[5.5.24]

[5.5.25]

[5.5.26]

Reflection coefficient at the source:

Input reflection coefficient:

Output reflection coefficient:

Assuming S12 = S21 and S11 = S22.


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Transmission Line Velocity Measurements

We can calculate the delay per unit length (or velocity) from S21:

S21 = b2/a1

p

d vlf

S 1

36021

Where (S21 ) is the phase angle of the S21 measurement.f is the frequency at which the measurement was taken.l is the length of the line.

[5.5.27]

Port 1 Port 2

a1

b1

a2

b2

0°180°

+90°

-90°

PositivePhase

NegativePhase

0.8 135°


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Impedance vs. frequency Recall

Zin vs f will be a function of delay () and ZL.

We can use Zin equations for open and short circuited lossy transmission.

Transmission Line Z0 Measurements

lZZ openin tanh0,

lZZ shortin coth0,

lj

lj

in e

eZZ

2

2

0 1

1

openinshortin ZZZ ,,0

[5.5.28]

[5.5.29]

[5.5.30]

ZLZ0,

z=lz=0

Zin

ZLZ0,

z=lz=0

Zin

open &shortUsing the equation for Zin,

in, and Z0, we can find the impedance.


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Transmission Line Z0 Measurement #2

shortin

shortinVNAj

shortin

jshortin

VNAshortin Ze

eZZ

,

,

02,

02,

, 1

1

1

1

openinshortin ZZZ ,,0

[5.5.31]

[5.5.32]

11

212

1111

212

11, 111

1

S

SS

S

SSopenin

Input reflection coefficients for the open and short circuit cases:

11

212

1111

212

11, 111

1

S

SS

S

SSshortin

openin

openinVNAj

openin

jopenin

VNAopenin Ze

eZZ

,

,

02,

02,

, 1

1

1

1

Input impedance for the open and short circuit cases:

Now we can apply equation [5.5.30]:

ZLZ0,

z=lz=0

Zin

open &short

ZVNA


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Scattering Matrix Example

Using the S11 plot shown below, calculate Z0 and estimate r.

01.0 1.5 2.0 2.5 3..0 3.5 4.0 4.5 5.0

Frequency [GHz]

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

S 11 M

agn

itu

de

RS = 50

Z0

z=0 z=5"

RL = 50


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Scattering Matrix Example #21.76GHz 2.94GHz Step 1: Calculate the d

of the transmission line based on the peaks or dips.

dpeaks t

GHzGHzf2

176.194.2

Step 2: Calculate r based on the velocity (prop delay per unit length).

minchinchps

smcv

rrd /37.39/7.84

1/1031 8

Peak=0.384

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45S 11

Mag

nit

ud

e

0.1r

pstd 7.423

inchpsin

psd /7.84

5

7.423


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Example – Scattering Matrix (Cont.)Step 3: Calculate the input impedance to the transmission line

based on the peak S11 at 1.76GHz, assuming a 50 port.

384.050

5011

in

in

Z

ZS

Step 4: Calculate Z0 from Zin at z=0:

LCflj

o

ol eZ

Zex 42

50

500)(

Solution: r = 1.0 and Z0 = 75

33.112inZ

1366.97.84)5(76.144 jpsGHzLCflj eeje

)1(5050

1

)1(5050

1

33.112)5(1

)5(1

o

o

o

o

ooin

ZZZZ

Zz

zZZ

9.74oZ


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Advantages/Disadvantages of S Parameters

Advantages: Ease of measurement: It is much easier to measure

power at high frequencies than open/short current and voltage.

Disadvantages: They are more difficult to understand and it is more

difficult to interpret measurements.


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Transmission (ABCD) Matrix The transmission matrix describes the network in terms of

both voltage and current waves (analagous to a Thévinin Equivalent).

The coefficients can be defined using superposition:

221

221

DICVI

BIAVV

2

2

1

1

I

V

DC

BA

I

V

02

1

2

I

V

IC

2 PortNetwork

I1

+

-

V1

Po

rt 2

I2

+

-

V2

Po

rt 1

02

1

2

V

I

ID

02

1

2

V

I

VB

02

1

2

I

V

VA

[5.5.33]

[5.5.34]

[5.5.35]

[5.5.36]

[5.5.29]

[5.5.31]


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I1

+

-

V1

I2

V2

I1

I3

+

-

V3

Transmission (ABCD) Matrix

Since the ABCD matrix represents the ports in terms of currents and voltages, it is well suited for cascading elements.

The matrices can be mathematically cascaded by multiplication:

3

3

22

2

2

2

11

1

I

V

DC

BA

I

V

I

V

DC

BA

I

V

3

3

211

1

I

V

DC

BA

DC

BA

I

V

This is the best way to cascade elements in the frequency domain. It is accurate, intuitive and simple to use.

2DC

BA

1DC

BA

[5.5.37]


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ABCD Matrix Values for Common Circuits

ZPort 1 Port 2 10

1

DC

ZBA

Port 1 Y Port 2 1

01

DYC

BA

323

3212131

/1/1

//1

ZZDZC

ZZZZZBZZA

Z1

Port 1 Port 2

Z2

Z3

Y1Port 1 Port 2Y2

Y3

3132121

332

/1/

/1/1

YYDYYYYYC

YBYYA

Port 1 Port 2,oZ)cosh()sinh()/1(

)sinh()cosh(

lDlZC

lZBlA

o

o

l

[5.5.38]

[5.5.39]

[5.5.40]

[5.5.41]

[5.5.42]


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Converting to and from the S-Matrix

The S-parameters can be measured with a VNA, and converted back and forth into ABCD the Matrix Allows conversion into a more intuitive matrix Allows conversion to ABCD for cascading ABCD matrix can be directly related to several useful circuit

topologies


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Po

rt 2

Po

rt 1

ABCD Matrix – Example

Create a model of a via from the measured s-parameters.

The model can be extracted as either a Pi or a T network

The inductance values will include the L of the trace and the via barrel

assumes the test setup minimizes the trace length, so that trace capacitance is minimal.

The capacitance represents the via pads.

L1 L1

Cvia


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ABCD Matrix – Example #1 The measured S-parameter matrix at 5 GHz is:

153.0110.0572.0798.0

572.0798.0153.0110.0

2221

1211

jj

jj

SS

SS

Converted to ABCD parameters:

827.00157.0

08.20827.0

2

11

2

112

11

2

11

21

21122211

21

21122211

21

21122211

21

21122211

j

j

S

SSSS

SZ

SSSSS

SSSSZ

S

SSSS

DC

BA

VNA

VNA

Relating the ABCD parameters to the T circuit topology, the capacitance can be extracted from C & inductance from A:

pFC

fCjZ

jC VIA

VIA

5.0

2111

0157.03

nHLLfCj

fLj

Z

ZA

VIA

35.0)2/(1

21827.01 21

3

1

Z1

Port 1 Port 2

Z2

Z3


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Advantages/Disadvantages of ABCD Matrix

Advantages: The ABCD matrix is intuitive: it describes all ports with

voltages and currents. Allows easy cascading of networks. Easy conversion to and from S-parameters. Easy to relate to common circuit topologies.

Disadvantages: Difficult to directly measure: Must convert from

measured scattering matrix.


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Summary

We can characterize interconnect networks using n-Port circuits.

The VNA uses S- parameters.From S- parameters we can characterize

transmission lines and discrete elements.


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References D.M. Posar, Microwave Engineering, John Wiley & Sons,

Inc. (Wiley Interscience), 1998, 2nd edition. B. Young, Digital Signal Integrity, Prentice-Hall PTR, 2001,

1st edition. S. Hall, G. Hall, and J. McCall, High Speed Digital System

Design, John Wiley & Sons, Inc. (Wiley Interscience), 2000, 1st edition.

W. Dally and J. Poulton, Digital Systems Engineering, Chapters 4.3 & 11, Cambridge University Press, 1998.

“Understanding the Fundamental Principles of Vector Network Analysis,” Agilent Technologies application note 1287-1, 2000.

“In-Fixture Measurements Using Vector Network Analyzers,” Agilent Technologies application note 1287-9, 2000.

“De-embedding and Embedding S-Parameter Networks Using A Vector Network Analyzer,” Agilent Technologies application note 1364-1, 2001.


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Appendix

More material on S parameters.


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0Re

any for 0Re

mn

mn

Y

m,nZ

jkkIj

iij I

VZ

,0

jkkVj

iij V

IY

,0

Lossless

Reciprocal jiij ZZ jiij ZZ

1 ZY


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S Parameters

NNNNN

N

N V

V

V

SSS

S

SSS

V

V

V

2

1

21

12

12111

2

1

VSV

jkkV

j

iij V

VS

,0

Scattering Matrix: S ParametersScattering Matrix: S Parameters

or [5.5.1]

where [5.5.2]

nnn VVV

nnnnn VVIII ???? VVVIZIZIZ

VUZVUZ

10

10

001

U

UZUZVVS 11


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S Parameters #2

[5.5.1]

where [5.5.2]

UZUZVVS 11

USZSUZUZS

SUSUZ 1

TSS Reciprocal

N

kkikiSS

1

* 1

N

kkjki jiSS

1

* ,0


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S Parameters – n Ports

[5.5.1]

[5.5.2]

n

nn Z

Va0

aSb

n

nn Z

Vb0

nnnnnn baZVVV 0

nn

nn

nnn ba

ZZ

VVI

00

1

22

2

1

2

1nnn baP

jkkaj

iij a

bS

,0

jkk

jkk

Vj

j

i

i

aj

iij

ZV

ZV

a

bS

,0

,0

0

0

nnnn

N

n a

a

a

SS

S

SSS

b

b

b

2

1

1

21

11211

2

1

nnnnnn

nn

nn

aSaSaSb

aSaSaSb

aSaSaSb

2211

22221212

12121111


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S Parameters #4

[5.5.1]

[5.5.2]

aSb

where

jkkaj

iij a

bS

,0

jkk

jkk

Vj

j

i

i

aj

iij

ZV

ZV

a

bS

,0

,0

0

0

niaSbn

jjiji ,,3,2,1for

Sij = ij is the reflection coefficient of the ith port if i=j with all other ports matched

Sij = Tij is the forward transmission coefficient of the ith port if I>j with all other portsmatched

Sij = Tij is the reverse transmission coefficient of the ith port if I<j with all other portsmatched


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VNA Calibration

Proper calibration is critical!!!There are two basic calibration methods Short, Open, Load and Thru (SOLT)

• Calibrated to known standard( Ex: 50)• Measurement plane at probe tip

Thru, Reflect, Line(TRL)• Calibrated to line Z0

– Helps create matched port condition.

• Measurement plane moved to desired position set by calibration structure design.


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SOLT Calibration Structures

OPEN SHORT

LOAD THRU

Calibration SubstrateCalibration Substrate

G

G

S

S

G

S

Signal

Ground

G

S

G

S


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64 TRL Calibration Structures

TRL PCB Structures Normalized Z0 to line

De-embed’s launch structure parasitics

6mil wide gap

Short

100 mils 100 mils

Open

?

Thru

?

L1

?

L2

Measurement Planes


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Calibration- Verification Always check the calibration prior to taking

measurements. Verify open, load etc..

• Smith Chart: Open & Short should be inside the perimeter.• Ideal response is dot at each location when probing the calibration

structures.

Capacitance

Inductance

NormalizedZo

PerimeterZo = 0+/- j X

Short 1.00.2 20

-j0.5

-j1.0

+j0.5+j1.0

Zo

Open

NormalizedZo = 0.2 - j1

Capacitance

Inductance

NormalizedZo


Short 1.00.2 20

-j0.5

-j1.0

+j0.5+j1.0

Zo

Open


S11(Short) S11(Open)

S11(load)

S21/12(Thru)


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One Port Measurements Practical sub 2 GHz technique for L & C data.

Structure must be electrically shorter than /4 of fmax.

1st order (Low Loss):

• Zin = jwL (Shorted transmission line)

• Zin = 1/jwC (Open transmission line)

• For an electrically short structure V and I to order are ~constant.

At the short, we have Imax and Vmin. Measure L using a shorted transmission line with negligible loss.

At the open you have Vmax and Imin. Measure C using an open transmission line with negligible loss.

V

RS= 50 DUT Short

CurrentZin = jL·I

DUT

Open

V

RS = 50

Zin = V/jC


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64 One Port Measurements – L & C

VNA - Format Use Smith chart

format to read L & C data

Capacitance

Inductance

NormalizedZo


Short 1.00.2 20

-j0.5

-j1.0

+j0.5+j1.0

Zo

Open


Capacitance

Inductance

NormalizedZo


Short 1.00.2 20

-j0.5

-j1.0

+j0.5+j1.0

Zo

Open



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Connector L & C

Use test board to measure connector inductance and capacitance Measure values relevant to pinout Procedure

• Measure test board L & C without connector

• Measure test board with connector

• Difference = connector parasitics

Short Open

s-parameters

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s parameters n ports

parameters z

matrix return loss s

port example

port networks linear

z parameters advantage

port case

network ports