ECE 546 Lecture 08 Nonideal Conductors and Dielectricsemlab.illinois.edu/ece546/Lect_08.pdf · ECE...

ECE 546 – Jose Schutt‐Aine 1

ECE 546Lecture ‐ 08

Nonideal Conductors and Dielectrics

Spring 2020

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


: conductivity of material medium (‐1m‐1)

HEt

EH Jt

J E

Material Medium

E j H

H J j E

1H E j E E j j Ej

2 2 1E Ej

1

j

or

since then


Wave in Material Medium2 2 21E E E

j

is complex propagation constant

2 2 1j

1j jj

associated with attenuation of wave

associated with propagation of wave


Wave in Material Medium

ˆ ˆz z j zo oE xE e xE e e

1/22

1 12

decaying exponentialSolution:

1/22

1 12

ˆ z j zoEH y e e

jj

Magnetic field

Complex intrinsic impedance



is the loss tangent

2

2 218

Two special cases can be distinguished:

: poor conductor1

2

2 212 8 2

2

2 2

318 2

j



: good conductor1

/4jj j j e

1f j

f f

1f j


Skin Depth

1

The decay of electromagnetic wave propagating into a conductor is measured in terms of the skin depth

2

For good conductors:

ˆ ˆz z j zo oE xE e xE e e

Wave decay

Definition: skin depth is distance over which amplitude of wave drops by 1/e.


Skin Depth

I VCz t

e-1Wave motion

For perfect conductor, = 0 and current only flows on the surface


DC Resistance

dclRwt

l: conductor length: conductivity


AC Resistance

2 /acl l l fRw ww

l: conductor length: conductivityf: frequency


Frequency-Dependent Resistance

1J

0

z JJ e dz J

Approximation is to assume that all the current is flowing uniformly within a skin depth


2 2

2 2

I ICLz t

1

acfR

w

1dcR

wt

Resistance is ~ constant when >t

Resistance changes with f

Frequency-Dependent Resistance


, 6ac groundl fRh

Reference Plane Current


r

H. A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-424,1942

Skin Effect in Microstrip


/ /y jyoJ J e e

/ /

0 1y jy o

oJ wI J we e dy

j

oo o o

JE J E

oo

J DV E D

Current density varies as

Note that the phase of the current density varies as a function of y

The voltage measured over a section of conductor of length D is:



11o

skino

jJ DV DZ j fI J w w

1

skin skinDR X fw

The skin effect impedance is

where


is the bulk resistivity of the conductor

skin skin skinZ R jX

with

Skin effect has reactive (inductive) component


internalac skinR RL

The internal inductance can be calculated directly from the ac resistance

Internal Inductance

Skin effect resistance goes up with frequency

Skin effect inductance goes down with frequency


When the tooth height is comparable to the skin depth, roughness effects cannot be ignored

Surface Roughness

Copper surfaces are rough to facilitate adhesion to dielectric during PCB manufacturing

Surface roughness will increase ohmic losses


Hammerstad Model

whenwhen

H sH

dc

K R f tR fR t

when2

when2

Hexternal

HH t

externalt

R fL t

fL f

R fL t

f


Hammerstad Model

221 arctan 1.4 RMSH

hK

hRMS: root mean square value of surface roughness height

: skin depthf=t: frequency where the skin depth is equal to

the thickness of the conductor


Hemispherical Model

whenwhen

hemi shemi

dc

K R f tR fR t

when2

when2

hemiexternal

hemihemi t

externalt

R fL t

fL f

R fL t

f


Hemispherical Model

1 when 1when 1

shemi

s s

KK

K K

3 1 / 1213 1 / 2 1

r jj krr j

2Re 3 / 4 1 1 / 4/ 4

o tile bases

o tile

k A AK

A

23

2

1 4 / 1/ 1213 1 2 / 1/ 1

j k r jj krj k r j


Huray Model

whenwhen

Huray sHuray

dc

K R f tR fR t

when2

when2

Hurayexternal

HurayHuray t

externalt

R fL t

fL f

R fL t

f


Huray Model_flat N spheres

Hurayflat

P PK

P

3 1 / 1213 1 / 2 1

r jj krr j

/ 4flat o tileP A

23

2

1 4 / 1/ 1213 1 2 / 1/ 1

j k r jj krj k r j

2_ 2

1

1 3Re 1 12

N

N spheres onn

P Hk

/ 'o o

Ho: magnitude of applied H field.


Dielectrics and Polarization

Field causes the formation of dipoles polarization

No field Applied field

Bound surface charge density –qsp on upper surface and +qsp on lower surface of the slab.


1o a o a o e a o e a s aD E P E E E E

s

D

o

eaE: electric flux density

: applied electric field: electric susceptibility: free‐space permittivity: static permittivity

Dielectrics and Polarization

P : polarization vector


Material r

AirStyrofoamParaffinTeflonPlywoodRT/duroid 5880PolyethyleneRT/duroid 5870Glass‐reinforced teflon (microfiber)Teflon quartz (woven)Glass‐reinforced teflon (woven)Cross‐linked polystyrene (unreinforced)Polyphenelene oxide (PPO)Glass‐reinforced polystyreneAmberRubberPlexiglas

1.00061.032.12.12.12.202.262.352.32‐2.402.472.4‐2.622.562.552.62333.4

Dielectric Constant


Material r

LuciteFused silicaNylon (solid)QuartzBakeliteFormicaLead glassMicaBeryllium oxide (BeO)MarbleFlint glassFerrite (FqO,)Silicon (Si)Gallium arsenide (GaAs)Ammonia (liquid)GlycerinWater

3.63.783.83.84.85666.8‐7.081012‐161213225081

Dielectric Constants


When a material is subjected to an applied electric field, the centroids of the positive and negative charges are displaced relative to each other forming a linear dipole.

When the applied fields begin to alternate in polarity, the permittivities are affected and become functions of the frequency of the alternating fields.

AC Variations


Reverses in polarity cause incremental changes in the static conductivity sheating of materials using microwaves (e.g. food cooking)

When an electric field is applied, it is assumed that the positive charge remains stationary and the negative charge moves relative to the positive along a platform that exhibits a friction (damping) coefficient d.

AC Variations


22 2

'2

22 2

1

eo

or

o

N Qm

dm

2"

222 2

er

oo

dN Q m

m dm

Complex Permittivity

eN

o

Q

o

m

d : damping coefficient: mass

: free space permittivity: dipole charge: dipole density

: natural frequency : applied frequency

osm

s : spring (tension) factor


' "i c i sH J J j E J E j j E


" ' 'i s i eH J E j E J E j E

equivalent conductivity "e s s a

alternating field conductivity "a

static field conductivitys

e: total conductivity composed of the static portion s and the alternative part a caused by the rotation of the dipoles


't i ce de i eJ J J J J E j E


iJ

tJ

ceJ

deJ

: total electric current density

: impressed (source) electric current density

: effective electric conduction current density: effective displacement electric current density

' ' 1 ' 1 tan'

et i e i i eJ J E j E J j j E J j j E

tan effective electric loss tangent' ' ' '

e s a s ae


Complex Permittivity"

'

"tan tan tan' '

s ee s a

e

tan static electric loss tangent'

ss

"

'tan alternating electric loss tangent'

aa

' ' 1 ' 1 tan'

ecd ce de e eJ J J E j E j j E j j E


Dielectric Properties

' 1 ''

ecdJ j j E j E

1'

e

' 1'

ecd eJ j j E E

1'

e

Good Dielectrics:

Good Conductors:


Dielectric Properties

' 1 ''

ecdJ j j E j E

1'

e

' 1'

ecd eJ j j E E

1'

e

Good Dielectrics:

Good Conductors:


Kramers-Kronig Relations

"'

2 20

' '21 ''

rr d

There is a relation between the real and imaginary parts of the complex permittivity:

Debye Equation

'"

2 20

1 '2 ''

rr d

' '' " '( ) ( ) ( )

1rs r

r r r re

jj


Kramers-Kronig Relations

'

'

22

rse

r

e is a relaxation time constant:

' '' '

21rs r

r re

' '"

21rs r e

re


Material r’ tanAirAlcohol (ethyl)Aluminum oxideBakeliteCarbon dioxideGermaniumGlassIceMicaNylonPaperPlexiglasPolystyrenePorcelain

1.0006258.84.741.001164*74.25.43.533.452.566

0.16 x 10‐4

22x10‐3

1 x 10‐3

0.16x10‐4

2x10‐2

8 x 10‐3

4 x 10‐2

5x10‐5

14x10‐3

Dielectric Materials


Material r’ tanPyrex glassQuartz (fused)RubberSilica (fused)SiliconSnowSodium chlorideSoil (dry)StyrofoamTeflonTitanium dioxideWater (distilled)Water (sea)Water (dehydrated)Wood (dry)

43.82.5‐33.811.83.35.92.81.032.1100808111.5‐4

6x10‐4

7.5x10‐4

2 x 10‐3

7.5 x 10‐4

0.51x10‐4

7 x 10‐2

1x10‐4

3x10‐4

15 x 10‐4

4x10‐2

4.6401x10‐2

Dielectric Materials


Source: H. Barnes et al, "ATE Interconnect Performance to 43 Gps Using Advanced PCB Materials", DesignCon 2008

PCB Stackup


The skew (time delay) between the two traces of the differential pair should be zero. Any skew between the two traces causes the differential signal to convert into a common signal.

Differential signaling is widely used in the industry today. High-speed serial interfaces such as PCI-E, XAUI, OC768, and CEI use differential signaling for transmitting and receiving data in point-to-point topology between a driver (TX) and receiver (RX) connected by a differential pair.

Differential Signaling


Fiber Weave Effect

Source: S. McMorrow, C. Heard, "The Impact of PCB Laminate Weave on the Electrical Performance of Differential Signaling at Multi-Gigabit Data Rates", DesignCon 2005.

Fiberglass weave pattern causes signals to propagate at different speeds in differential pairs


Source: Lambert Simonovich, "Practical Fiber Weave Effect Modeling",White Paper-Issue 3, March 2, 2012.

Fiber Weave Effect


Source: S. Hall and H. Heck , Advanced Signal Integrity for High-Speed Digital Designs, J. Wiley, IEEE , 2009.

Fiber Weave Effect



Fiber Weave EffectGroup delay variation



Fiber Weave EffectGroup delay variation: effect of angle



Fiber Weave EffectStraight traces



Fiber Weave Effect

45o traces


Source: PCB Dielectric Material Selection and Fiber Weave Effect on High-Speed Channel Routing, Altera Application Note AN-528-1.1, January 2011.

Fiber Weave Effect

straight traces

zig-zagtraces



Fiber Weave Effect

Skew on straight traces



Fiber Weave Effect

Skew on zig-zagtraces


• Mitigation TechniquesUse wider widths to achieve impedance targets. Specify a denser weave (2116, 2113, 7268, 1652) compared to a sparse weave (106, 1080). Move to a better substrate such as Nelco 4000‐13 Perform floor planning such that routing is at an angle rather than orthogonal. Make use of zig‐zag routing

Fiber Weave Effect

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