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1

Differential Signaling

Introduction to Frequency Domain Analysis (3 Classes)Many thanks to Steve Hall, Intel for the use of his slidesReference Reading: Posar Ch 4.5

http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

Slide content from Stephen Hall

Instructor: Richard Mellitz

2


Outline

Motivation: Why Use Frequency Domain Analysis2-Port Network Analysis TheoryImpedance and Admittance MatrixScattering MatrixTransmission (ABCD) MatrixMason’s RuleCascading S-Matrices and Voltage Transfer FunctionDifferential (4-port) Scattering Matrix

3


Motivation: Why Frequency Domain Analysis?Time Domain signals on T-lines lines are hard to analyze

Many properties, which can dominate performance, are frequency dependent, and difficult to directly observe in the time domain• Skin effect, Dielectric losses, dispersion, resonance

Frequency Domain Analysis allows discrete characterization of a linear network at each frequencyCharacterization at a single frequency is much easier

Frequency Analysis is beneficial for Three reasonsEase and accuracy of measurement at high frequenciesSimplified mathematicsAllows separation of electrical phenomena (loss, resonance … etc)

4


Key Concepts

Here are the key concepts that you should retain from this class

The input impedance & the input reflection coefficient of a transmission line is dependent on:Termination and characteristic impedanceDelayFrequency

S-Parameters are used to extract electrical parametersTransmission line parameters (R,L,C,G, TD and Zo) can be

extracted from S parametersVias, connectors, socket s-parameters can be used to create

equivalent circuits=

The behavior of S-parameters can be used to gain intuition of signal integrity problems

5


Review – Important Concepts

The impedance looking into a terminated transmission line changes with frequency and line length

The input reflection coefficient looking into a terminated transmission line also changes with frequency and line length

If the input reflection of a transmission line is known, then the line length can be determined by observing the periodicity of the reflection

The peak of the input reflection can be used to determine line and load impedance values

6


Two Port Network Theory

Network theory is based on the property that a linear system can be completely characterized by parameters measured ONLY at the input & output ports without regard to the content of the system

Networks can have any number of ports, however, consideration of a 2-port network is sufficient to explain the theoryA 2-port network has 1 input and 1 output port.The ports can be characterized with many parameters, each

parameter has a specific advantage

Each Parameter set is related to 4 variables2 independent variables for excitation2 dependent variables for response

7


Network characterized with Port ImpedanceMeasuring the port impedance is network is the most

simplistic and intuitive method of characterizing a network

Port 1 Port 2

Case 1Case 1: Inject current I1 into port 1 and measure the open circuit voltage at port 2 and calculate the resultant impedance from port 1 to port 2

1

2,21

port

portopen

I

VZ

Case 2Case 2: Inject current I1 into port 1 and measure the voltage at port 1 and calculate the resultant input impedance

1

1,11

port

portopen

I

VZ

2-port Network

I1 I2

+-

V1 V2+-

2- port Network

I1 I2

+-

V1 V2+-

8


Impedance MatrixA set of linear equations can be written to describe the network in terms of its port impedances

Where:

If the impedance matrix is known, the response of the system can be predicted for any input

2221212

2121111

IZIZV

IZIZV

j

iij I

VZ Open Circuit Voltage measured at Port i

Current Injected at Port j

2

1

2221

1211

2

1

I

I

ZZ

ZZ

V

V

Zii the impedance looking into port iZij the impedance between port i and j

Or

9


Impedance Matrix: Example #2

Calculate the impedance matrix for the following circuit:

Port 1 Port 2

R1R2

R3

10



Step 1: Calculate the input impedance

R1R2

R3I1 V1

+

-

Step 2: Calculate the impedance across the network

R1 R2

R3I1 V2

+

-

311

111

3111 )(

RRI

VZ

RRIV

31

221

3131

3131

31

312

)(

RI

VZ

RIRR

RRRI

RR

RVV

11



Step 3: Calculate the Impedance matrix

Assume: R1 = R2 = 30 ohmsR3=150 ohms

18030

30180MatrixZ

1803111 RRZ

3021Z

12


Measuring the impedance matrix

Question: What obstacles are expected when measuring the impedance

matrix of the following transmission line structure assuming that the micro-probes have the following parasitics? Lprobe=0.1nH Cprobe=0.3pF

Assume F=5 GHz

T-line0.1nHPort 1

Port 20.3pF

0.1nH

0.3pF

Zo=50 ohms, length=5 in

13


Measuring the impedance matrix

1062

1_ fC

Z Cprobe

32_ fLZ Lprobe

T-line

Port 2

Answer: Open circuit voltages are very hard to measure at high frequencies

because they generally do not exist for small dimensions Open circuit capacitance = impedance at high frequencies Probe and via impedance not insignificant

0.1nH

106 ohms106 ohms

Zo = 50

Without Probe Capacitance

Zo = 50

With Probe Capacitance @ 5 GHz

Z21 = 50 ohms

Z21 = 63 ohms

Port 1 Port 2

Port 1 Port 2

T-line0.1nHPort 1

Port 20.3pF

0.1nH

0.3pF

Zo=50 ohms, length=5 in

14


Advantages/Disadvantages of Impedance Matrix

Advantages:The impedance matrix is very intuitive

Relates all ports to an impedanceEasy to calculate

Disadvantages:Requires open circuit voltage measurements

Difficult to measureOpen circuit reflections cause measurement noiseOpen circuit capacitance not trivial at high frequencies

Note: The Admittance Matrix is very similar, however, it is characterized with short circuit currents instead of open circuit voltages

15


Scattering Matrix (S-parameters)Measuring the “power” at each port across a well

characterized impedance circumvents the problems measuring high frequency “opens” & “shorts”

The scattering matrix, or (S-parameters), characterizes the network by observing transmitted & reflected power waves

2-port Network2-port

NetworkPort 1 Port 2

ai represents the square root of the power wave injected into port i

R

VaP

R

VP i

i

2

a1a2

b2b1

bj represents the power wave coming out of port jR

Vb j

j

RR

16


Scattering MatrixA set of linear equations can be written to describe the network in terms of injected and transmitted power waves

Where:

Sii = the ratio of the reflected power to the injected power at port i

Sij = the ratio of the power measured at port j to the power injected at port i

2221212

2121111

aSaSb

aSaSb

jportatinjectedPower

iportatmeasuredPower

a

bS

j

iij

2

1

2221

1211

2

1

a

a

SS

SS

b

b

17


Making sense of S-Parameters – Return LossWhen there is no reflection from the load, or the line length

is zero, S11 = Reflection coefficient

50

50

1

1

1

1

021

111

o

oo

incident

reflected

aZ

Z

V

V

V

V

R

VR

V

a

bS

S11 is measure of the power returned to the source,

and is called the “Return Loss”

R=Zo

Z=-l Z=0

Zo

R=50

18


Making sense of S-Parameters – Return LossWhen there is a reflection from the load, S11 will be

composed of multiple reflections due to the standing waves

)(1

)(1)(

l

lZlZZ oin

RL

Z=-l Z=0

Zo

inZ

If the network is driven with a 50 ohm source, then S11 is calculated using the input impedance instead of Zo

50 ohms

50

50

11

in

in

Z

ZSS11 of a transmission line will exhibit periodic effects due to the standing waves

19


Example #3 – Interpreting the return loss

Based on the S11 plot shown below, calculate both the impedance and dielectric constant

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

1.0 1.5 2.0 2.5 3..0 3.5 4.0 4.5 5.0

Frequency, GHz

S11

, Mag

nit

ud

eR=50

L=5 inchesZoR=50

20


Example – Interpreting the return loss

Step 1: Calculate the time delay of the t-line using the peaks

inchpspsinchTD

inchTDpsTDTD

GHzGHzf peaks

/7.84"5/7.423/

/7.4232

176.194.2

Step 2: Calculate Er using the velocity

0.1

)/37.39/7.84

1

/1031 8

Er

minchinchps

Er

sm

Er

cv

TD

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Frequency, GHz

S11

, Mag

nit

ud

e

1.76GHz 2.94GHz

Peak=0.384

21


Example – Interpreting the return loss

Step 3: Calculate the input impedance to the transmission line based on the peak S11 at 1.76GHz

Note: The phase of the reflection should be either +1 or -1 at 1.76 GHz because it is aligned with the incident

33.112

384.050

5011

in

in

in

Z

Z

ZS

Step 4: Calculate the characteristic impedance based on the input impedance for x=-5 inches

9.74

)1(5050

1

)1(5050

1

33.112)5(1

)5(1

1

50

50)(

366.97.84)5(76.144

42

o

o

o

o

o

ooin

jpsGHzjLCflj

LCflj

o

olo

Z

ZZZZ

Zx

xZZ

eee

eZ

Zex

Er=1.0 and Zo=75 ohms

22


Making sense of S-Parameters – Insertion Loss

When power is injected into Port 1 with source impedance Z0 and measured at Port 2 with measurement load impedance Z0, 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


NetworkV1

a1a2=0

b2b1

V2ZoZo

S21 is measure of the power transmitted from

port 1 to port 2, and is called the “Insertion Loss”

23


Loss free networks For a loss free network, the total power exiting the N ports must

equal the total incident power

exitincident PP If there is no loss in the network, the total power leaving the

network must be accounted for in the power reflected from the incident port and the power transmitted through network

121_1_

incident

portportdtransmitte

incident

portreflected

P

P

P

P

Since s-parameters are the square root of power ratios, the following is true for loss-free networks

1221

211 SS

If the above relationship does not equal 1, then there is loss in the network, and the difference is proportional to the power dissipated by the network

24


Insertion loss exampleQuestion: What percentage of the total power is dissipated by the

transmission line? Estimate the magnitude of Zo (bound it)

S-parameters; 5 inch microstrip

0

0.2

0.4

0.6

0.8

1

1.2

0.E+00 2.E+09 4.E+09 6.E+09 8.E+09 1.E+10 1.E+10

Frequency, Hz

Mag

nit

ud

e S(1,1)S(1,2)

25


Insertion loss example What percentage of the total power is dissipated by the transmission line ? What is the approximate Zo? How much amplitude degradation will this t-line contribute to a 8 GT/s signal? If the transmission line is placed in a 28 ohm system (such as Rambus), will

the amplitude degradation estimated above remain constant? Estimate alpha for 8 GT/s signal

S-parameters; 5 inch microstrip;

0

0.2

0.4

0.6

0.8

1

1.2

0.E+00 2.E+09 4.E+09 6.E+09 8.E+09 1.E+10

Frequency, Hz

Ma

gn

itu

de

S(1,1)

S(1,2)

26


Insertion loss exampleAnswer: Since there are minimal reflections on this line, alpha can be

estimated directly from the insertion loss S21~0.75 at 4 GHz (8 GT/s)

057.075.0 )5(21 eeS l

When the reflections are minimal, alpha can be estimated

If the reflections are NOT small, alpha must be extracted with ABCD parameters (which are reviewed later)

The loss parameter is “1/A” for ABCD parameters ABCE will be discussed later.

If S11 < ~ 0.2 (-14 dB), then the above approximation is valid

27


Important concepts demonstrated

The impedance can be determined by the magnitude of S11

The electrical delay can be determined by the phase, or periodicity of S11

The magnitude of the signal degradation can be determined by observing S21

The total power dissipated by the network can be determined by adding the square of the insertion and return losses

28


A note about the term “Loss”

True losses come from physical energy lossesOhmic (I.e., skin effect)Field dampening effects (Loss Tangent)Radiation (EMI)

Insertion and Return losses include effects such as impedance discontinuities and resonance effects, which are not true losses

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

29


Advantages/Disadvantages of S-parametersAdvantages:Ease of measurement

Much easier to measure power at high frequencies than open/short current and voltage

S-parameters can be used to extract the transmission line parametersn parameters and n Unknowns

Disadvantages:Most digital circuit operate using voltage thresholds. This suggest

that analysis should ultimately be related to the time domain.Many silicon loads are non-linear which make the job of

converting s-parameters back into time domain non-trivial. Conversion between time and frequency domain introduces errors

30


Cascading S parameter

While it is possible to cascade s-parameters, it gets messy.

Graphically we just flip every other matrix. Mathematically there is a better way… ABCD parameters We will analyzed this later with signal flow graphs

a1a111

b1b111

s11

s21

s12

s22

a2a211

b2b211 a1a122

b1b122

a2a222

b2b222 a1a133

b1b133

a1a133

b1b133

s111 s121

s211 s221 s112 s122

s212 s222

s113 s123

s213 s223

3 cascaded s parameter blocks3 cascaded s parameter blocks

31


ABCD Parameters

The transmission matrix describes the network in terms of both voltage and current waves


NetworkV1

I1I2

V2

The coefficients can be defined using superposition

221

221

DICVI

BIAVV

2

2

1

1

I

V

DC

BA

I

V

02

1

02

1

02

1

02

1

2222

VIVI

I

ID

V

IC

I

VB

V

VA

32


Transmission (ABCD) Matrix

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

V1

I1I2

V2

The matrices can be cascaded by multiplication

3

3

22

2

2

2

11

1

I

V

DC

BA

I

V

I

V

DC

BA

I

V

I3

V3

1DC

BA

2DC

BA

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 simplistic.

33


Relating the ABCD Matrix to Common Circuits

ZPort 1 Port 2 10

1

DC

ZBA

Port 1 Y Port 21

01

DYC

BA

Z1

Port 1 Port 2323

3212131

/1/1

//1

ZZDZC

ZZZZZBZZA

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

Assignment 6:

Convert these to s-parameters

34


Converting to and from the S-Matrix

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

DCZZBA

DCZZBAS

DCZZBAS

DCZZBA

BCADS

DCZZBA

DCZZBAS

oo

oo

oo

oo

oo

oo

/

/

/

2

/

)(2

/

/

11

21

12

11

21

21122211

21

21122211

21

21122211

21

21122211

2

)1)(1(

2

)1)(1(1

2

)1)(1(

2

)1)(1(

S

SSSSD

S

SSSS

ZC

S

SSSSZB

S

SSSSA

o

o

35


ABCD Matrix – Example #1

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

Port 1

Port 2

153.0110.0572.0798.0

572.0798.0153.0110.0

2221

1211

jj

jj

SS

SS

36


ABCD Matrix – Example #1The model can be extracted as either a Pi or a T network

L1

CVIA

L2

The inductance values will include the L of the trace and the via barrel (it is assumed that the test setup minimizes the trace length, and subsequently the trace capacitance is minimal

The capacitance represents the via pads

Port 1

Port 2

37



Assume the following s-matrix measured at 5 GHz

153.0110.0572.0798.0

572.0798.0153.0110.0

2221

1211

jj

jj

SS

SS

38



Assume the following s-matrix measured at 5 GHz

153.0110.0572.0798.0

572.0798.0153.0110.0

2221

1211

jj

jj

SS

SS

Convert to ABCD parameters

827.00157.0

08.20827.0

j

j

DC

BA

39


ABCD Matrix – Example #1Assume the following s-matrix measured at 5 GHz

153.0110.0572.0798.0

572.0798.0153.0110.0

2221

1211

jj

jj

SS

SS

Convert to ABCD parameters

827.00157.0

08.20827.0

j

j

DC

BA

Relating the ABCD parameters to the T circuit topology, the capacitance and inductance is extracted from C & A

Z1

Port 1 Port 2

Z2

Z3

pFC

fCjZ

jC VIA

VIA

5.0

2111

0157.03

nHLLfCj

fLj

Z

ZA

VIA

35.0)2/(1

21827.01 21

3

1

40


ABCD Matrix – Example #2Calculate the resulting s-parameter matrix if the two

circuits shown below are cascaded

2-port Network

Network X 5050

Port 1 Port 2

2221

1211

XX

XXX SS

SSS

2-port Network

Network Y 5050

Port 1 Port 2

2-port Network

Network X50

Port 1

2-port Network

Network Y 50

Port 2

?XYS

2221

1211

YY

YYY SS

SSS

41



Step 1: Convert each measured S-Matrix to ABCD Parameters using the conversions presented earlier

XX

XXXX DC

BATS

YY

YYYY DC

BATS

Step 2: Multiply the converted T-matrices

XYXY

XYXY

YY

YY

XX

XXYXXY DC

BA

DC

BA

DC

BATTT

Step 3: Convert the resulting Matrix back into S-parameters using thee conversions presented earlier

2221

1211

XX

XXXYXY SS

SSST

42


Advantages/Disadvantages of ABCD Matrix

Advantages:The ABCD matrix is very intuitive

Describes all ports with voltages and currents

Allows easy cascading of networksEasy conversion to and from S-parametersEasy to relate to common circuit topologies

Disadvantages:Difficult to directly measure

Must convert from measured scattering matrix

43


Signal flow graphs – Start with 2 port first

The wave functions (a,b) used to define s-parameters for a two-port network are shown below. The incident waves is a1, a2 on port 1 and port 2 respectively. The reflected waves b1 and b2 are on port 1 and port 2. We will use a’s and b’s in the s-parameter follow slides

44


Signal Flow Graphs of S Parameters

“In a signal flow graph, each port is represented by two nodes. Node an represents the wave coming into the device from another device at port n, and node bn represents the wave leaving the device at port n. The complex scattering coefficients are then represented as multipliers (gains) on branches connecting the nodes within the network and in adjacent networks.”*

a1

b1

b2

a2

S L

s21

s12

s11 s22

Example

Measurement equipment strives to be match i.e. reflection coefficient is 0

See: http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf

0

011 21

1 LS

abas

45


Mason’s Rule ~ Non-Touching Loop Rule

T is the transfer function (often called gain) Tk is the transfer function of the kth forward path

L(mk) is the product of non touching loop gains on path k taken mk at time.

L(mk)|(k) is the product of non touching loop gains on path k taken mk at a time but not touching path k.

mk=1 means all individual loops

mk

mkk mk

kmkk

mkL

mkLT

))()1(1(

))()1(1(T

)(

46


Voltage Transfer function

What is really of most relevance to time domain analysis is the voltage transfer function.

It includes the effect of non-perfect loads. We will show how the voltage transfer functions for a 2 port

network is given by the following equation.

Notice it is not S21

47


Forward Wave Path

a1

b1

b2

a2

Vs

S L

s21

s12

s11 s22

Z0

ZS Z0( )

48


Reflected Wave Path

a1

b1

b2

a2

Vs

S L

s21

s12

s11 s22

Z0

ZS Z0( )

49


Combine b2 and a2

50


Convert Wave to Voltage - Multiply by sqrt(Z0)

51


Voltage transfer function using ABCD

ABCD_CHANNEL

1 s11( ) 1 s22( ) s12 s212 s21

1 s11( ) 1 s22( ) s12 s212 s21 Z0

1 s11( ) 1 s22( ) s12 s21[ ] Z02 s21

1 s11( ) 1 s22( ) s12 s212 s21

ABCD_SOURCE1

0

Zs

1

ABCD_LOAD

1

1

ZL

0

1

ZL Z01 L

1 L Zs Z0

1 s

1 s

Let’s see if we can get this results another way

52


Cascade [ABCD] to determine system [ABCD]

VOLTAGE_TRANSFER_FUNCTION ABCD_SOURCEABCD_CHANNEL ABCD_LOAD

1

0

Z01 s

1 s

1

1 s11( ) 1 s22( ) s12 s212 s21

1 s11( ) 1 s22( ) s12 s212 s21 Z0

1 s11( ) 1 s22( ) s12 s21[ ] Z02 s21

1 s11( ) 1 s22( ) s12 s212 s21

1

1

Z01 L

1 L

0

1

Simplify

21 s11 s s22 L s11 s22 s L s12 s21 s L

1 s s21 1 L

1 s22 L s11 s11 s22 L s12 s21 L

s21 Z0 1 L

Z01 s22 s11 s s11 s22 s s12 s21 s

1 s s21

12

1 s22 s11 s11 s22 s12 s21s21

53


Extract the voltage transfer function

"A" parameter which input over output transfer. We are looking for "1/A" which is output over input

21 s11 s s22 L s11 s22 s L s12 s21 s L

1 s s21 1 L

1

Simplify and re-arange

s21 1 L 1 s

2

1 s11 s s22 L s12 s21 s L s11 s22 s L

Same as with flow graph analysis

54


Cascading S-Parameter

As promised we will now look at how to cascade s-parameters and solve with Mason’s rule

The problem we will use is what was presented earlier The assertion is that the loss of cascade channel can be

determine just by adding up the losses in dB. We will show how we can gain insight about this

assertion from the equation and graphic form of a solution.

a1a111

b1b111

s11

s21

s12

s22

a2a211

b2b211 a1a122

b1b122

a2a222

b2b222 a1a133

b1b133

a1a133

b1b133

s111 s121

s211 s221

s112 s122

s212 s222

s113 s123

s213 s223

55


Creating the signal flow graph

We map output a to input b and visa versa. Next we define all the loops Loop “A” and “B” do not touch each other

11

11 11

11A1A111 B2B211 A1A122 B2B222 A1A133 B2B233

B1B111 A2A211 B1B122 A2A222 B1B133 A2A233

s12s1211

s22s2211

s21s2111

s12s1233

s21s2133

s11s1133 s22s2233

s21s2122

s11s1122 s22s2222

s12s1222

a1a111

b1b111

s11

s21

s12

s22

a2a211

b2b211 a1a122

b1b122

a2a222

b2b222 a1a133

b1b133

a1a133

b1b133

s111 s121

s211 s221

s112 s122

s212 s222

s113 s123

s213 s223

56


Use Mason’s rule

There is only one forward path a11 to b23.

There are 2 non touching looks

Mason’s RuleMason’s Ruleb6

a1

s211

s212

s213

1 s222

s111

s223

s112

s113

s221

s122

s212

s221

s112

s222

s113

11

11 11

11A1A111 B2B211 A1A122 B2B222 A1A133 B2B233

B1B111 A2A211 B1B122 A2A222 B1B133 A2A233

s12s1211

s22s2211

s21s2111

s12s1233

s21s2133

s11s1133 s22s2233

s21s2122

s11s1122 s22s2222

s12s1222

57


Evaluate the nature of the transfer function

• If response is relatively flat and reflection is relatively low– Response through a channel is s211*s212*213…

Assumption is that these are ~ 0Assumption is that these are ~ 0

b6

a1

s211

s212

s213

1 s222

s111

s223

s112

s113

s221

s122

s212

s222

s111

s223

s114

58


Jitter and dB Budgeting Change s21 into a phasor

Insertion loss in db

S Smag ej

=

i.e. For a budget just add up the db’s and i.e. For a budget just add up the db’s and jitterjitter

S211 ej 211

S212

ej 211

S213 ej 213

S211

S212

s213

ej 211 212 213

20 log S211

S212

s213 =

20 log s211 20 log s21

2 20 log s213

dbsys

n

dbi i 1 delay

n

i i 1

59


Differential S-Parameters are derived from a 4-port measurement

Traditional 4-port measurements are taken by driving each port, and recording the response at all other ports while terminated in 50 ohms

Although, it is perfectly adequate to describe a differential pair with 4-port single ended s-parameters, it is more useful to convert to a multi-mode port

Differential S-Parameters

4-port

a1a2

b1b2

S11

S22S21

S12

S33

S44S43

S34S31

S42S41

S32

S13

S24S23

S14b1

b2

b3

b4

a1

a2

a3

a4

=a3

b3

a4

b4

60


Differential S-Parameters

Matrix assumes differential and common mode stimulus

Multi-ModePort

Mu

lti-

Mo

de

Po

rt 1

adm1adm2

acm2acm1

bdm1 bdm2

bcm1 bcm2

Mu

lti-Mo

de P

ort 2

It is useful to specify the differential S-parameters in terms of differential and common mode responsesDifferential stimulus, differential responseCommon mode stimulus, Common mode responseDifferential stimulus, common mode response (aka ACCM Noise)Common mode stimulus, differential response

This can be done either by driving the network with differential and common mode stimulus, or by converting the traditional 4-port s-matrix

DS11

DS22DS21

DS12

CS11

CS22CS21

CS12CDS11

CDS22CDS21

CDS12

DCS11

DCS22DCS21

DCS12bdm1

bdm2

bcm1

bcm2

adm1

adm2

acm1

acm2

=

61


Explanation of the Multi-Mode Port

Differential Matrix:Differential Stimulus, differential responsei.e., DS21 = differential signal [(D+)-(D-)] inserted at port 1 and diff signal measured at port 2

Common mode Matrix:Common mode stimulus, common mode Response. i.e., CS21 = Com. mode signal [(D+)+(D-)] inserted at port 1 and Com. mode signal measured at port 2

Common mode conversion Matrix:Differential Stimulus, Common mode response. i.e., DCS21 = differential signal [(D+)-(D-)] inserted at port 1 and common mode signal [(D+)+(D-)] measured at port 2

differential mode conversion Matrix:Common mode Stimulus, differential mode response. i.e., DCS21 = common mode signal [(D+)+(D-)] inserted at port 1 and differential mode signal [(D+)-(D-)] measured at port 2

DS11

DS22DS21

DS12

CS11

CS22CS21

CS12CDS11

CDS22CDS21

CDS12

DCS11

DCS22DCS21

DCS12bdm1

bdm2

bcm1

bcm2

adm1

adm2

acm1

acm2

=

62


)()()()( 3414433133321223111131

4343332321313

4143132121111

031

31

0;01

111

422

SSaSSaSSaSSabb

aSaSaSaSb

aSaSaSaSb

aa

bb

a

bDS

aaaadm

dm

cmdm

Differential S-ParametersConverting the S-parameters into the multi-mode requires just a little algebra

Example Calculation, Differential Return LossThe stimulus is equal, but opposite, therefore:

2413 ; aaaa


Network

21

3 4

Assume a symmetrical network and substitute 2413 ; aaaa

3313311111 2

1SSSSDS

14323412 ; SSSS

Other conversions that are useful for a differential bus are shown

4323412121 2

1SSSSDS 4123432121 2

1SSSSCDS

Differential Insertion Loss: Differential to Common Mode Conversion (ACCM):

Similar techniques can be used for all multi-mode Parameters

63


Next class we will develop more differential concepts

64


backup review

65


Advantages/Disadvantages of Multi-Mode Matrix over Traditional 4-portAdvantages:Describes 4-port network in terms of 4 two port matrices

DifferentialCommon modeDifferential to common modeCommon mode to differential

Easier to relate to system specificationsACCM noise, differential impedance

Disadvantages:Must convert from measured 4-port scattering matrix

66


High Frequency Electromagnetic WavesIn order to understand the frequency domain analysis, it is

necessary to explore how high frequency sinusoid signals behave on transmission lines

The equations that govern signals propagating on a transmission line can be derived from Amperes and Faradays laws assumimng a uniform plane waveThe fields are constrained so that there is no variation in the X and Y

axis and the propagation is in the Z direction

Wf

min

sm

r

1

4.39103 8

Z

X

Y

Direction of propagation

This assumption holds true for transmission lines as long as the wavelength of the signal is much greater than the trace width

For typical PCBs at 10 GHz with 5 mil traces (W=0.005”)

"005.0"59.0

67


High Frequency Electromagnetic Waves For sinusoidal time varying uniform plane waves,

Amperes and Faradays laws reduce to:

xy Ej

z

B

Amperes Law:

A magnetic Field will be induced by an electric currentor a time varying electric field

Faradays Law: An electric field will be generated by a time varying magnetic flux

yx Bj

z

E

Note that the electric (Ex) field and the magnetic (By) are orthogonal

68


High Frequency Electromagnetic Waves

If Amperes and Faradays laws are differentiated with respect to z and the equations are written in terms of the E field, the transmission line wave equation is derived

0

1

222

2

2

2

2

2

xx

xxyyx

Ejz

E

Ejjz

E

z

B

z

Bj

z

E

This differential equation is easily solvable for Ex:

zjzjx eCeCzE )(

2)(

1)(

69


High Frequency Electromagnetic WavesThe equation describes the sinusoidal E field for a plane

wave in free space

zjM

zjMx eEeEzE )()()(

Portion of wave traveling In the +z direction

Portion of wave traveling In the -z direction

Note the positive exponent is because the wave is traveling in the opposite

direction

= permittivity in Farads/meter (8.85 pF/m for free space) (determines the speed of light in a material)

= permeability in Henries/meter (1.256 uH/m for free space and non-magnetic materials)

Since inductance is proportional to & capacitance is proportional

to , then is analogous to in a transmission line, whichis the propagation delay

LC

70


High Frequency Voltage and Current Waves

The same equation applies to voltage and current waves on a transmission line

RL

z=-l z=0

If a sinusoid is injected onto a transmission line, the resulting voltageis a function of time and distance from the load (z). It is the sum of the incident and reflected values

tjzref

tjzin eeVeeVtzV ),(

Voltage wave traveling towards the load

Voltage wave reflecting off the Load and traveling

towards the source

Incident sinusoidReflected sinusoid

tje Note: is added to specifically represent the time varying Sinusoid, which was impliedin the previous derivation

71



))(( CjGLjR LCj

= Attenuation Constant (attenuation of the signal due to transmission line losses)

= Phase Constant (related to the propagation delay across the transmission line)

j = Complex propagation constant – includes all the transmission line parameters (R, L C and G)

(For the loss free case) (lossy case)

C

LG

L

CR

2

1 (For good conductors)

LC (For good conductors and good dielectrics)

The parameters in this equation completely describe the voltage on a typical transmission line

tjzref

tjzin eeVeeVtzV ),(

72



ze

)sin()cos( je j

LC

Subsequently:

)(sin)(cos

)(sin)(cos

),( )()(

ztj

ztVe

ztj

ztVe

eeVeeVtzV

refz

inz

tjzjref

tjzjin

The voltage wave equation can be put into more intuitive terms by applying the following identity:

The amplitude is degraded byThe waveform is dependent on the driving function

( ) & the delay of the linetjt sincos

73


Interaction: transmission line and a load

(Assume a line length of l (z=-l))

)(1)( leVeeVeVeVlV lin

lo

lin

lref

lin

This is the reflection coefficient looking into a t-line of length l

Zl

Z=-l Z=0

l

ol

ollol

in

lref e

ZZ

ZZe

eV

eVl

22)(

The reflection coefficient is now a function of the Zo discontinuities AND line lengthInfluenced by constructive & destructive combinations of the forward & reverse waveforms

Zo

)(l

74


)(111

)(

)(1)(

leVZ

eVeVZ

lI

leVeeVeVeVlV

lin

o

lref

lin

o

lin

lo

lin

lref

lin

This is the input impedance looking into a t-line of length l

RL

Z=-l Z=0

)(1

)(1

)(11

)(1

)(

)()(

l

lZ

leVZ

leV

lI

lVlZZ o

lin

o

lin

in

Interaction: transmission line and a loadIf the reflection coefficient is a function of line length, then the

input impedance must also be a function of length

Note: is dependent on and

)(l

l

Zin

75


Line & load interactions In chapter 2, you learned how to calculate waveforms

in a multi-reflective system using lattice diagrams Period of transmission line “ringing” proportional to the line delay Remember, the line delay is proportional to the phase constant

In frequency domain analysis, the same principles apply, however, it is more useful to calculate the frequency when the reflection coefficient is either maximum or minimum This will become more evident as the class progresses

LCjLCjj

GR

CjGLjRj

22

00

To demonstrate, lets assume a loss free transmission line

76


Line & load interactions

o

loel 2)(

The frequency where the values of the real & imaginaryreflections are zero can be calculated based on the line length

LCfljLCfleee oLClj

olj

ol

o 4sin4cos2)(2)(2

Term 1 Term 2

...5,3,18

24

nLCl

nf

nLCfl

Term 1=0Term 2 =

...3,2,14

4

nLCl

nf

nLCfl

Term 2=0Term 1 =

Note that when the imaginary portion is zero, it means the phase of the incident & reflected waveforms at the input are aligned. Also notice that value of “8” and “4” in the terms.

Remember, the input reflection takes the form

o

77


Example #1: Periodic ReflectionsCalculate:1. Line length2. RL

(assume a very low loss line)

RL

Z=-l Z=0

)(l

-2.5E-01

-2.0E-01

-1.5E-01

-1.0E-01

-5.0E-02

0.0E+00

5.0E-02

1.0E-01

1.5E-01

2.0E-01

2.5E-01

0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 3.0E+09Frequency

Ref

lect

ion

Coe

ff.

Real

Imaginary

Zo=75

Er_eff=1.0

78


Example #1: SolutionStep 1: Determine the periodicity zero crossings or peaks & use the

relationships on page 15 to calculate the electrical length

psGHzff

LClTD

GHzMHzGHzLClLClLCl

ff

nn

nn

42535.2

1

)(2

1

176.158876.12

1

4

1

4

3

13

13

Imaginary

79


Example #1: Solution (cont.)

Since TD and the effective Er is known, the line length can be calculated as in chapter 2

inm

ms

ps

c

effEr

TDlength 5127.0

1031

425

_8

Note the relationship between the peaks and the electrical length

This leads to a very useful equation for transmission lines)(2

1

13

nn ffLClTD

TDFpeaks 2

1

80


Example #1: Solution (cont.) The load impedance can be calculated by observing the peak values of the reflection

When the imaginary term is zero, the real term will peak, and the maximum reflection will occur If the imaginary term is zero, the reflected wave is aligned with the incident wave and the phase term = 1

Important Concepts demonstrated The impedance can be determined by the magnitude of the reflection The line length can be determined by the phase, or periodicity of the reflection

50

2.0)1(75

75)( 2

L

L

Ll

oL

oL

R

R

Re

ZR

ZRl

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