1 interconnect and packaging lecture 3: skin effect chung-kuan cheng uc san diego
TRANSCRIPT
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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)
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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III. Skin EffectBoundary condition:•Charges at the boundary shield the electric field.•Thus, capacitance C does not change with frequency.
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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
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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
Skin Effect
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