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Interconnect and PackagingLecture 3: Skin Effect

Chung-Kuan ChengUC San Diego

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Outlines

I. Transmission Line Model

II. Spectrum of Configurations

III. Skin Effect

IV. Coaxial Cable

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I. Transmission Line Model

RΔl LΔlCΔl

RΔl LΔl

CΔl …

i(z,t)

RΔl LΔl RΔl LΔl

Voltage drops through serial resistance and inductance

Current reduces through shunt capacitance

Resistance increases due to skin effect

Shunt conductance is caused by loss tangent

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I. Interconnect Model

• Telegrapher’s equation:

),(),(),(

),(),(

),(

tzGVdt

tzdVC

dz

tzdIdt

tzdILtzRI

dz

tzdV

• Propagation Constant:

jCjGLjR ))((

• Wave Propagation:

//,),( 0 tzveVtzV tjzjz

• Characteristic Impedance

)/()( CjGLjRZ

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I. Interconnect Model

LCRGRGRCLGLCj

LCRGRGRCLGLCj

2222224

2222224

)(])()[()(2

1

)(])()[()(2

1

• Propagation Constant:

• Wave Propagation:

//,),( 0 tzveVtzV tjzjz

• Characteristic Impedance

)/()( CjGLjRZ

jCjGLjR ))((

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I. Interconnect Model (Constants)

• AWG (American Wire Gauge• Wire Diameter = 2.54x10-(AWG+10)/20

• Copper p= 2.2uohm-cm• Copper thickness 1oz(/sqft)= 36um• Electric Permittivity of Air 8.85x10-12F/m• Magnetic Permeability of Air• Characteristic Impedance of Air 8.376/

mH /104 7

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II. Spectrum of Configurations jCjGLjR ))((

RLGC R L G C

0001 ( jwC ) Capacitance

0010 ( G ) Shunt

0011 ( G + jwC ) Leaky Capacitance

0100 ( jwL ) Inductance

0101 ( jwL )( jwC ) Lossless LC Line

0110 ( jwL )( G ) Skin Effect Derivation

0111 ( jwL )( G + jwC ) Skin Effect + Permitivity

1000 ( R ) Resistance

1001 ( R )( jwC ) RC Line

1010 ( R )( G ) Leaky Resistance

1011 ( R )( G + jwC ) Leaky RC Line

1100 ( R + jwL ) Lossy Inductance

1101 ( R + jwL )( jwC ) Lossy LC Line

1110 ( R + jwL )( G ) Lossy and Leaky Inductance

1111 ( R + jwL )( G + jwC ) Transmission Line

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III. Skin Effect

2

22

))((

LGj

LG

LGj

CjGLjR

Skin Depth

(Equivalent Depth of Uniform Current)

Assuming that resistance and capacitanceare negligible.

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IV. Coaxial Cable

02,

2),

11(

2

1

Z

R

baR

776.3/

/1/ln0

0Zab

baabda

d

)/ln(60

2

)/ln(0 ab

abZ

r

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IV. Coaxial Cable: Inductance

abL

rdra

IbrLI

a

/ln28

2),(0 2

2

abI

raa

I

drr

Idr

a

Ir

rbr

b

a

a

r

/ln2

)(4

22),(

222

2

2

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IV. Coaxial Cable: Inductance

iLR

2

11,

2

1

2

20

ppR

ppL

p

Irdr

p

IIL

i

i

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IV. Coaxial Cable: Inductance

)/()(

/)(

)(

2

)(

1

0

jj

mjaI

jaI

aZi

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IV. Coaxial Cable: Impedance

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IV. Coaxial Cable: Impedance22ACDC RRR

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HW1: Remarks on z900

• Chip (Processor)• 10x17 sqmm, 38W, 918MHz, 250MIPS• 128bits to each L2 cache chip• 280um pitch chip to MCM

• MCM• 127x127sqmm, 5xTerabits/s, 459MHz• 20 Processors, 8x4MB Memory• 11Knets, 95mm max length on critical path• 1ns on MCM, 1.4ns off MCM• 33um think film pitch, 396um ceramic substrate• 101Kpins, 35.3pin/sqcm• 4224pins/1735pg to PCB

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HW1: Remarks

• Board• 553x447sqmm, 1 MCM, 64GB memory, 24 STI• 7 mils pitch (3.3width/4sep)• 24GB/s IO, 1GB/s bus, 240lines/MBA (6GB/s), • 10pairs/STI, 9 signals/1 clock• 3516nets, 19,788pins (signals + pg), 8pins/sqcm