ece 451 coupled lines - emlab.uiuc.eduemlab.uiuc.edu/ece451/notes/coupled.pdf · ece 451 –jose...

ECE 451 – Jose Schutt‐Aine 1

ECE 451Coupled Lines

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


Signal Integrity

Crosstalk Dispersion Attenuation

Reflection Distortion Loss

Delta I Noise Ground Bounce Radiation

Sense Line

Drive Line

Drive Line

Crosstalk Noise


TEM PROPAGATION

L

z

C

I

V

+

-

z

Zo Z1 Z2

Vs


Telegrapher’s Equations

L: Inductance per unit length.

C: Capacitance per unit length.

L

z

C

I

V

+

-

V ILz t

I VCz t


50 line 1

line 2

50

line 1

line 2

50 line 1

line 2

line 1

line 2

Crosstalk noise depends on termination


50 line 1

line 2

50

line 1

line 2

line 1

line 2

tr = 1 ns tr = 7 ns

Crosstalk depends on signal rise time


50 line 1

line 2

50

line 1

line 2

line 1

line 2

tr = 1 ns tr = 7 ns



tr = 1 ns tr = 7 ns


50 line 1

line 2

line 1

line 2

line 1

line 2


Coupled Transmission Lines

r

w s

h

Cs

V1

V2

I1

I2

Cs

Cm Lm


1 1 211 12

V I IL Lz t t

1 1 211 12

I V VC Cz t t

2 1 221 22

I V VC Cz t t

Telegraphers Equations for Coupled Transmission Lines

Maxwellian Form

2 1 221 22

V I IL Lz t t


1 1 2s m

V I IL Lz t t

2 1 2m s

V I IL Lz t t

1 1 1 2s m m

I V V VC C Cz t t t

2 1 2 2m m s

I V V VC C Cz t t t

Telegraphers Equations for Coupled Transmission Lines

Physical form


Relations Between Physical and Maxwellian Parameters

(symmetric lines)

L11 = L22 = Ls

L12 = L21 = Lm

C11 = C22 = Cs+Cm

C12 = C21 = - Cm


11 12e eV IL L

z t

11 12e eI IC Cz t

Add voltageand currentequations

Ze = L11 + L12C11 + C12

= Ls + LmCs

ve = 1(L11 + L12 )(C11 + C12 )

= 1(Ls + Lm )Cs

Ve : Even mode voltage

Ie : Even mode current

Ve = 12

V1 + V2( )

Ie = 12

I1 + I2( )

Impedance

velocity

Even Mode


Subtract voltageand currentequations

Vd : Odd mode voltage

Id : Odd mode current

Impedance

velocity

Odd Mode

11 12d dV IL L

z t

11 12d dI IC Cz t

d 1 21V2

V V

d 1 21I2

I I

11 12d

11 12 2Z = = s m

s m

L L L LC C C C

d11 12 11 12

1 1v = = ( )( ) ( )( 2 )s m s mL L C C L L C C


+1 +1

EVEN

+1 -1

ODD

Mode Excitation


PHYSICAL SIGNIFICANCE OF EVEN- ANDODD-MODE IMPEDANCES

* Ze and Zd are the wave resistance seen by the even and odd mode travelling signals respectively.

V1 = Z11 I1 + Z12 I2

V2 = Z21 I1 + Z22 I2

* The impedance of each line is no longer describedby a single characteristic impedance; instead, we have


Even-Mode Impedance: ZeImpedance seen by wave propagating through the coupled-line system when excitation is symmetric (1, 1).

Odd-Mode Impedance: ZdImpedance seen by wave propagating through the coupled-line system when excitation is anti-symmetric (1, -1).

Common-Mode Impedance: Zc = 0.5ZeImpedance seen by a pair of line and a common return by a common signal.

Differential Impedance: Zdiff = 2ZdImpedance seen across a pair of lines by differential mode signal.

Definitions


EVEN AND ODD-MODE IMPEDANCES

Z11, Z22 : Self Impedances

Z12, Z21 : Mutual Impedances

For symmetrical lines,

Z11 = Z22 and Z12 = Z21


EXAMPLE(Microstrip)

r = 4.3Zs = 56.4

Single LineDielectric height = 6 milsWidth = 8 mils

r = 4.3

Coupled LinesHeight = 6 milsWidth = 8 milsSpacing = 12 mils

Ze = 68.1 Zd = 40.8 Z11 = 54.4 Z12 = 13.6


( ,0) ( ,0) ( ,0) ( ,0)e d e dtdr

e d e d

a t a t a t a tIZ Z Z Z

( ,0) ( ,0)tdr e dV a t a t da ( ,0) 0t

= 2

tdr e

tdr

V ZI

Even Mode

coaxial line

Zgline 1

line 2

Zg

stepgenerator

Vb

Vf IT

+

-VT I2

I1

reference plane tied to ground

e g1Z = 2( ) Z1

e

e

e 2v =

e

l


coaxial line

Zg

Zg

stepgenerator

Vb

Vf line 1

line 2

IT

VTI2 reference plane floating

+-

I1

tdr e d e dV = a (t,0)+a (t,0)- a (t,0)-a (t,0) f bV V

tdr ( ,0) ( ,0)I = e d

e d

a t a tZ Z

tdr

( ,0) ( ,0)I =- e d

e d

a t a tZ Z

ae(t,0) = 0, VtdrItdr

= 2Zd

dd

1 1 2l, v = 2 1

dd g

d

Z Z

Odd Mode


EXTRACT INDUCTANCE AND CAPACITANCE COEFFICIENTS

s1L = + 2

e d

e d

Z Zv v

1s

e e

CZ v

m 1L = - 2

e d

e d

Z Zv v

m 1 1 1C = - 2 e e d dZ v Z v

d s eZ < Z < Z


20

40

60

80

100

120

4 6 8 10 12 14 16 18Spacing (mils)

Even-Mode Impedanceh=3 milsh=5 milsh=7 mils

h=10 milsh=14 mils

h=21 mils

Ze(

)

Measured even-mode impedance


20

25

30

35

40

45

50

4 6 8 10 12 14 16 18Spacing (mils)

Odd-Mode Impedanceh=3 mils

h=5 mils

h=7 mils

h=10 mils

h=14 mils

h=21 mils

Zd

(

)

Measured odd-mode impedance


0.158

0.16

0.162

0.164

0.166

0.168

0.17

4 6 8 10 12 14 16 18Spacing (mils)

Even-Mode velocityh=3 mils

h=5 mils

h=7 mils

h=10 mils

h=14 mils

h=21 mils

v e(m/n

s)

Measured even-mode velocity


0.175

0.18

0.185

0.19

0.195

0.2

0.205

0.21

4 6 8 10 12 14 16 18Spacing (mils)

Odd-Mode Velocity

h=3 milsh=5 mils

h=7 mils

h=10 milsh=14 milsh=21 mils

vd(m

/ns)

Measured odd-mode velocity


0

50

100

150

200

250

4 6 8 10 12 14 16 18Spacing (mils)

Mutual Inductance h=3 mils

h=5 mils

h=7 mils

h=10 mils

h=14 mils

h=21 mils

Lm

(nH

/m)

Measured mutual inductance


10

15

20

25

30

35

40

4 6 8 10 12 14 16 18Spacing (mils)

Mutual Capacitance h=3 mils

h=5 mils

h=7 milsh=10 mils

h=14 mils

h=21 mils

Cm

(pF/

m)

Measured mutual capacitance


40

50

60

70

80

90

100

110

10 20 30 40 50

Typical Even & Odd Mode Impedances

ZevenZodd

Zeve

n, Z

odd

(Ohm

s)

Distance (mils)

Even & Odd Mode Impedances


Microstrip : Inhomogeneous structure, odd and even-mode velocities must have different values.

Stripline : Homogeneous configuration, odd and even-mode velocities have approximately the same values.

Modal Velocities in Stripline and Microstrip


Microstrip vs Stripline

Microstrip (h =8 mils)w = 8 milsr = 4.32Ls = 377 nH/mCs = 82 pF/mLm = 131 nH/mCm = 23 pF/mve = 0.155 m/nsvd = 0.178 m/ns

Stripline (h =16 mils)w = 8 milsr = 4.32Ls = 466 nH/mCs = 86 pF/mLm = 109 nH/mCm = 26 pF/mve = 0.142 m/nsvd = 0.142 m/ns

50

50


-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Volts

microstrip

C

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Vol

ts

stripline

C

50

50

Microstrip vs Stripline

Sense line response at near end

Probe


1( ) e e d d

j z j z j z j zv v v v

e e d d

even odd

V z A e B e A e B e

2 ( ) e e d d


e e d d

even odd

V z A e B e A e B e

Zs1

Zs2 ZL2

ZL1line 1

line 2

General Solution for Voltages


11 1( ) e e d d


e e d de d

even odd

I z A e B e A e B eZ Z

21 1( ) e e d d


e e d de d

even odd

I z A e B e A e B eZ Z

Zs1

Zs2 ZL2

ZL1line 1

line 2

General Solution for Currents


odd

even

First reflection

even

even

odd

odd

Second reflection

odd

odd

odd

oddeven

even

even

even

Zs1

Zs2 ZL2

ZL1line 1

line 2

Coupling of Modes (asymmetric load)


Coupling of Modes(symmetric load)

Zs

Zs ZL

ZLline 1

line 2

odd

even

First reflection

even

odd

Second reflection

odd

even


-0.2

0

0.2

0.4

0.6

0.8

1

Vol

ts

0 5 10 15 20 25 30

Time (ns)

Drive Line at Near End

35 40

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Vol

ts

0 5 10 15 20 25 30

Time (ns)

Sense Line at Near End

35 40


1 2 3

31 1 211 12 13

VI V VC C Cz t t t

32 1 221 22 23

VI V VC C Cz t t t

3 31 231 32 33

I VV VC C Cz t t t

31 1 211 12 13

IV I IL L Lz t t t

32 1 221 22 23

IV I IL L Lz t t t

3 31 231 32 33

V II IL L Lz t t t

Three‐Line Microstrip


11 13( )V IL Lz t

11 13( )I VC Cz t

Subtract (1c) from (1a) and (2c) from (2a), we get

This defines the Alpha mode with:

1 3V V V 1 3I I I and

The wave impedance of that mode is:

11 13

11 13

L LZC C

and the velocity is 11 13 11 13

1uL L C C

Three‐Line – Alpha Mode


In order to determine the next mode, assume that

1 2 3V V V V

1 2 3I I I I

31 211 21 31 12 22 32 13 23 33

V II IL L L L L L L L Lz t t t

31 211 21 31 12 22 32 13 23 33

I VV VC C C C C C C C Cz t t t

By reciprocity L32 = L23, L21 = L12, L13 = L31

By symmetry, L12 = L23

Also by approximation, L22 L11, L11+L13 L11

Three‐Line – Modal Decomposition


31 211 12 13 12 112

V II IL L L L Lz t t t

In order to balance the right-hand side into I, we need to have

12 11 2 11 12 13 2 11 12 22L L I L L L I L L I

212 122L L

or 2

Therefore the other two modes are defined as

The Beta mode with

Three‐Line – Modal Decomposition


The Beta mode with

1 2 32V V V V

1 2 32I I I I

The characteristic impedance of the Beta mode is:

11 12 13

11 12 13

22

L L LZC C C

and propagation velocity of the Beta mode is

11 12 13 11 12 13

1

2 2u

L L L C C C

Three‐Line – Beta Mode


The Delta mode is defined such that

1 2 32V V V V

1 2 32I I I I

The characteristic impedance of the Delta mode is

11 12 13

11 12 13

22

L L LZC C C

The propagation velocity of the Delta mode is:

11 12 13 11 12 13

1

2 2u

L L L C C C

Three‐Line – Delta Mode


1 0 1

1 2 1

1 2 1

E

1 2 3

Alpha modeBeta mode*

Delta mode*

Symmetric 3‐Line Microstrip

In summary: we have 3 modes for the 3-line system

*neglecting coupling between nonadjacent lines


1 2 3

r = 4.3

113 17 5( / ) 16 53 16

5 17 113C pF m

346 162 67( / ) 152 683 152

67 162 346L nH m

0.45 0.12 0.450.5 0 0.50.45 0.87 0.45

E

0.44 0.49 0.440.5 0 0.50.10 0.88 0.10

H

Coplanar Waveguide


73 0 0( ) 0 48 0

0 0 94mZ

56 23 8( ) 22 119 22

8 23 56cZ

0.15 0 0( / ) 0 0.17 0

0 0 0.18pv m ns

1 2 3

r = 4.3

Coplanar Waveguide


WS' S'WS

rh

2Sk

S W

( ) : Complete Elliptic Integral of the first kindK k

'( ) ( ')K k K k

2 1/ 2' (1 )k k

30 '( ) (ohm)( )1

2

ocpr

K kZK k

1/ 22

1cpr

v c

Coplanar Waveguide


S WW

rh

120 '( ) (ohm)( )1

2

ocsr

K kZK k

Coplanar Strips


Characteristic Microstrip Coplanar Wguide Coplanar strips

eff* ~6.5 ~5 ~5

Power handling High Medium Medium

Radiation loss Low Medium Medium

Unloaded Q High Medium Low or High

Dispersion Small Medium Medium

Mounting (shunt) Hard Easy Easy

Mounting (series) Easy Easy Easy

* Assuming r=10 and h=0.025 inch

Qualitative Comparison


2p

h

w

dielectric

ground

signal strip

V Transmission Line


0

50

100

150

200

250

0 0.1 0.2 0.3 0.4 0.5 0.6w/h

CHARACTERISTIC IMPEDANCE

30º37.5º45º52.5º

60ºMicrostrip

0.7 0.8 0.9 1

Zo (o

hms)

Calculated values of the characteristic impedance for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is r = 2.55.

V‐Line Characteristic Impedance


1.8

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

0 0.1 0.2 0.3 0.4 0.5 0.6

w/h

EFFECTIVE PERMITTIVITY

30º

37.5º

45º

52.5º

60º

Microstrip

0.7 0.8 0.9 1

Effe

ctiv

e R

elat

ive

Die

lect

ric C

onst

ant

Calculated values of the effective relative dielectric constant for a single-line v-strip structure as a function of width-to-height ratio w/h. The relative dielectric constant is r = 2.55

V‐Line – Effective Permittivity


h

y

x

Region 2

Region 3

Region 4

Region 6

Region 1

Region 5

2pw

dielectricground surface

signal strip

s

Three‐Line – V Transmission Line


74.00 -0.97 -0.23C (pF/m) = -0.97 74.07 -0.97

-0.23 -0.97 74.00

425.84 20.35 7.00L (nH/m) = 20.36 422.85 20.36

7.00 20.35 425.84

52.49 -5.90 -0.57C (pF/m) = -5.90 53.27 -5.90

-0.57 -5.90 52.49

609.74 113.46 41.79L (nH/m) = 113.54 607.67 113.54

41.79 113.46 609.74

Microstrip V-line

Comparison of the inductance and capacitance matricesbetween a three-line v-line and microstrip structures. The parameters are p/h = 0.8 mils, w/h = 0.6 and r = 4.0.

h

y

x

Region 2

Region 3

Region 4

Region 6

Region 1

Region 5

2pw

dielectricground surface

signal strip

s1 2 3

V‐Line and Microstrip


0

50

100

150

200

250

300

350

0 0.2 0.4 0.6 0.8 1 1.2s/h

Mutual Inductance

V-linemicrostrip

Lm

(nH

/m)

0

5

10

15

20

25

30

35

40

0 0.2 0.4 0.6 0.8 1 1.2s/h

Mutual Capacitance

V-linemicrostrip

C m(p

F/m

)

Plot of mutual inductance (top) and mutual capacitance (bottom) versus spacing-to-height ratiofor v-line and microstrip configurations. The parameters are w/h = 0.24, r = 4.0.

V‐Line: Coupling Coefficients


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2s/h

Coupling Coefficient

V-linemicrostrip

Kv

Plot of the coupling coefficient versus spacing-to-height ratio for v-line and microstrip configurations. The parameters are w/h = 0.24, r = 4.0.

V‐Line vs Microstrip: Coupling Coefficients


0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17

microstrip

V-60

V-30

Frequency (GHz)

Linea

r Atte

nua ti

on

Propagation Function

2p

h

w

dielectric

ground

signal strip

Advantages of V Line* Higher bandwidth

* Lower crosstalk

* Better transition

ground reference

ground reference

dielectric

substrate

signal strip

Advantages of V‐Line


-8

-7

-6

-5

-4

-3

-2

-1

0

0 5 10 15 20 25

Insertion Loss

VlineMicrostrip20

*log1

0*|S

21|

Frequency (GHz)

V‐Line vs Microstrip: Insertion Loss

ece 451 coupled lines - emlab.uiuc.eduemlab.uiuc.edu/ece451/notes/coupled.pdf · ece 451 –jose...

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