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ECE 124a/256cTransmission Lines as Interconnect
Forrest BrewerDisplays from Bakoglu, Addison-Wesley
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Interconnection
Circuit rise/fall times approach or exceed speed of light delay through interconnect
Can no longer model wires as C or RC circuits Wire Inductance plays a substantial part Speed of light: 1ft/nS = 300um/pS
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Fundamentals
L dI/dt = V is significant to other effects Inductance: limit on the rate of change of the current E.g. Larger driver will not cause larger current to flow initially
R LC G
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Lossless Tranmission R=G=0
Step of V volts propagating with velocity v Initially no current flows – after step passes, current of +I After Step Voltage V exists between the wires
dx
-I
+I
vV
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Lossless Tranmission R=G=0
Maxwell’s equation:
B is field, dS is the normal vector to the surface is the flux For closed surface Flux and current are proportional
S
dSB
IL
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Lossless Tranmission R=G=0
For Transmission Line I and are defined per unit length At front of wave: Faraday’s Law: V = d/dt = IL dx/dt = ILv Voltage in line is across a capacitance Q=CV
I must be:
Combining, we get:
Also:
ILdxILdd )(
CVvdt
dxCV
dt
CVd
dt
dQI
)(
LCv
1
C
L
I
VZ
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Units
The previous derivation assumed = = 1
In MKS units:
c is the speed of light = 29.972cm/nS
For typical IC materials r = 1
So:
rr
cv
1
cCZ r0
CcL r
2
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Typical Lines
We can characterize a lossless line by its Capacitance per unit length and its dielectric constant.
Material Dielectric Constant
Propagation Velocity
Polymide 2.5-3.5 16-19 cm/nSSiO2 3.9 15
Epoxy PC 5.0 13Alumina 9.5 10
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Circuit Models
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Circuit Models II Driver End
TM modeled by resistor of value Z Input voltage is function of driver and line impedance:
Inside Line Drive modeled by Step of 2Vi with source resistance Z Remaining TM as above (resistor)
Load End Drive modeled by Step of 2Vi with source resistance Z
Voltage on load of impedance ZL:
SS
line VZR
ZV
iL
LL V
ZZ
ZV 2
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Discontinuity in the line (Impedance)
Abrupt interface of 2 TM-lines Incident wave: Vi = Ii Z1 Reflected Wave: Vr = IrZ1
Transmitted Wave: Vt = ItZ2
Conservation of charge: Ii = Ir + It
Voltages across interface: Vi + Vr = Vt
We have: 211 Z
V
Z
V
Z
V tri 12
12
ZZ
ZZVV ir
12
22
ZZ
ZVV it
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Reflection/Transmission Coefficients
The Coefficient of Reflection = Vr/Vi
Vi incident from Z1 into Z2 has a reflection amplitude vi
Similarly, the Transmitted Amplitude = 1+
12
12
ZZ
ZZ
21
22
ZZ
Z
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Inductive and Capacitive Discontinuities
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Typical Package Pins
Package Capacitance (pF) Inductance (nH)40 pin DIP (plastic) 3.5 2840 pin DIP (Ceramic) 7 2068 pin PLCC 2 768 pin PGA 2 7256 pin PGA (with gnd plane) 5 15Wire bond (per mm) 1 1Solder Bump 0.5 0.5
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Discontinuity Amplitude
The Amplitude of discontinuity
Strength of discontinuity
Rise/Fall time of Impinging Wave
To first order Magnitude is:
Inductive: Capacitive:r
iDpeak Zt
VLV
2
r
iDpeak t
ZVCV
2
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Critical Length (TM-analysis?)
TM-line effects significant if: tr < 2.5 tf
Flight time tf = d/v c
d r
Rise Time (pS)
Critical Length (15 cm/nS)
25 150m
75 0.45
200 1.3
500 3.0
1000 6.0
2000 12.0
Technology On-ChipRise time
Off-ChipRise time
CMOS (0.1) 18-70pS 200-2000pS
GaAs/SiGe/ (ECL)
2-50pS 8-300pS
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Un-terminated Line Rs = 10Z
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Un-terminated Line Rs = Z
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Un-terminated Line Rs = 0.1Z
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Unterminated Line (finite rise time)
Rise Time Never Zero For Rs>Zo, tr>tf
Exponential Rise Rs<Z0,
“Ringing” Settling time can be
much longer than tf.
vltt fr /
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Line Termination (None)
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Line Termination (End)
V R = Z
A B
VVA
VVB
tF
Z = 0
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Line Termination: (Source)
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Piece-Wise Modeling Create a circuit model for short section of line
Length < rise-time/3 at local propagation velocity E.G. 50m for 25pS on chip, 150nm wide, 350nm tall
Assume sea of dielectric and perfect ground plane (this time) C = 2.4pF/cm = 240fF/mm = 12fF/50m L = 3.9/c2C = 1.81nH/cm = 0.181nH/mm = 9.1pH/50m R = L/(W H) = 0.005cm*2.67cm/(0.000015cm*0.000035cm) = 25/50m
25
12f
9.1p
200m:
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Lossy Transmission
Attenuation of Signal Resistive Loss, Skin-Effect Loss, Dielectric Loss
For uniform line with constant R, L, C, G per length:
lexV
lxV )0(
)(
22220
0
GZ
Z
R
C
LG
L
CRGR
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Conductor Loss (Resistance)
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Conductor Loss (step input) Initial step declines
exponentially as Rl /2Z Closely approximates RC
dominated line when Rl >> 2Z
Beyond this point, line is diffusive
For large resistance, we cannot ignore the backward distrbitued reflection
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Conductor Loss (Skin Effect) An ideal conductor would exclude any electric or magnetic field
change – most have finite resistance The depth of the field penetration is mediated by the frequency of the
wave – at higher frequencies, less of the conductor is available for conducting the current
For resistivity (cm), frequency f (Hz) the depth is:
A conductor thickness t > 2 will not have significantly lower loss For Al at 1GHz skin depth is 2.8m
f
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Skin Effect in stripline (circuit board)
Resistive attenuation:
At high frequencies:
WHZZR
22/
WZ
f
ZWZ
Rskin
2
2
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Dielectric Loss
Material Loss tangent:
Attenuation:
r
DD C
G
tan
Dr
D
D
D
c
LCf
CLfC
GZ
tan/
tan
/tan
2