ecen689: special topics in high-speed links …spalermo/ecen689/lecture3_ee689_tlines.pdfsam palermo...
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Sam PalermoAnalog & Mixed-Signal Center
Texas A&M University
ECEN689: Special Topics in High-Speed Links Circuits and Systems
Spring 2011
Lecture 3: Transmission Lines
Announcements
• HW1 due 1/28• One page summary of recent link design paper
• Lecture Reference Material• Dally, Chapter 3.1 – 3.4
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Agenda
• Transmission Lines• Propagation constant• Characteristic impedance• Loss• Reflections• Termination examples• Differential transmission lines
3
Wire Models
• Model Types• Ideal• Lumped C, R, L• RC transmission line• LC transmission line• RLGC transmission line
• Condition for LC or RLGC model (vs RC)
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LRfπ20 ≥
Wire R L C >f (LC wire)
AWG24 Twisted Pair 0.08Ω/m 400nH/m 40pF/m 32kHz
PCB Trace 5Ω/m 300nH/m 100pF/m 2.7MHz
On-Chip Min. Width M6 (0.18µm CMOS node) 40kΩ/m 4µH/m 300pF/m 1.6GHz
RLGC Transmission Line Model
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( ) ( ) ( )t
txILtxRIx
txV∂
∂−−=
∂∂ ,,,
( ) ( ) ( )t
txVCtxGVx
txI∂
∂−−=
∂∂ ,,,
0 dx As →(1)
(2)
General Transmission Line Equations
Time-Harmonic Transmission Line Eqs.
• Assuming a traveling sinusoidal wave with angular frequency, ω
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( ) ( ) ( )xILjRdx
xdV ω+−=
( ) ( ) ( )xVCjGdx
xdI ω+−=
• Differentiating (3) and plugging in (4) (and vice versa)
( ) ( )xVdx
xVd 22
2
γ=
( ) ( )xIdx
xId 22
2
γ=
• where γ is the propagation constant
( )( ) ( )-1m CjGLjRj ωωβαγ ++=+=
(5)
(6)
Time-Harmonic Transmission Line Equations
(3)
(4)
Transmission Line Propagation Constant
• Solutions to the Time-Harmonic Line Equations:
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( ) ( ) ( ) xr
xfrf eVeVxVxVxV γγ
00 +=+= −
• What does the propagation constant tell us?• Real part (α) determines attenuation/distance (Np/m)• Imaginary part (β) determines phase shift/distance (rad/m)• Signal phase velocity
( ) ( ) ( ) xr
xfrf eIeIxIxIxI γγ
00 +=+= −
where ( )( ) ( )-1m CjGLjRj ωωβαγ ++=+=
(m/s) βωυ =
Transmission Line Impedance, Z0
• For an infinitely long line, the voltage/current ratio is Z0
• From time-harmonic transmission line eqs. (3) and (4)
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( )( ) ( )Ω
++
== 0 CjGLjR
xIxVZ
ωω
• Driving a line terminated by Z0 is the same as driving an infinitely long line
[Dally]
Lossless LC Transmission Lines
• If Rdx=Gdx=0
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LC
LCjj
ωβ
αωβαγ
=
==+=
0
CLZ
LC
=
==
0
1βωυ
No Loss!
• Waves propagate w/o distortion• Velocity and impedance
independent of frequency• Impedance is purely real
[Johnson]
Low-Loss LRC Transmission Lines
• If R/ωL and G/ωC << 1
• Behave similar to ideal LC transmission line, but …• Experience resistive and
dielectric loss• Frequency dependent
propagation velocity results in dispersion
• Fast step, followed by slow DC tail
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( )( )
βαα
ωωω
ωω
ωωβαγ
j
CG
LRLCjGZ
ZR
LCGLRCjLCj
CjGLjRj
DR ++=
+
+++≅
+−≅
++=+=
220
0
21
81
811
22
1
2
2
0
0
GZZR
D
R
≅
≅
α
α
+
+≅
22
81
811
CG
LRLC
ωωωβ
122
81
811
−
+
+≅
CG
LRLC
ωωυ
Resistive Loss
Dielectric Loss
Skin Effect (Resistive Loss)
• High-frequency current density falls off exponentially from conductor surface
• Skin depth, δ, is where current falls by e-1 relative to full conductor• Decreases proportional to
sqrt(frequency)
• Relevant at critical frequency fswhere skin depth equals half conductor height (or radius)• Above fs resistance/loss increases
proportional to sqrt(frequency)
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δd
eJ−
= ( ) 21
−= µσπδ f
2
2
=h
fs
πµ
ρ
( )21
=
sDC f
fRfR
21
02
=
s
DCR f
fZ
Rα
For rectangular conductor:
[Dally]
Skin Effect (Resistive Loss)
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[Dally]
MHzfmR sDC 43 ,7 =Ω=
5-mil Stripguide
kHzfmR sDC 67 ,08.0 =Ω=
30 AWG Pair
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02
=
s
DCR f
fZ
Rα
Dielectric Absorption (Loss)
• An alternating electric field causes dielectric atoms to rotate and absorb signal energy in the form of heat
• Dielectric loss is expressed in terms of the loss tangent
• Loss increases directly proportional to frequency
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CG
D ωδ =tan
LCf
CLfCGZ
D
DD
δπ
δπα
tan2
tan22
0
=
==
[Dally]
Total Wire Loss
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[Dally]
Reflections & Telegrapher’s Eq.
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T
iT ZZ
VI+
=0
2
+−
=
+−=
−=
0
0
0
00
2
ZZZZ
ZVI
ZZV
ZVI
III
T
Tir
T
iir
Tfr
0
0
ZZZZ
VV
IIk
T
T
i
r
i
rr +
−===
Termination Current:
• With a Thevenin-equivalent mode of the line:
• KCL at Termination:Telegrapher’s Equation or Reflection Coefficient
[Dally]
Termination Examples - Ideal
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RS = 50ΩZ0 = 50Ω, td = 1nsRT = 50Ω
050505050
050505050
5.05050
501
=+−
=
=+−
=
=
+=
rS
rT
i
k
k
VVV
in (step begins at 1ns)
source
termination
Termination Examples - Open
17
RS = 50ΩZ0 = 50Ω, td = 1nsRT ~ ∞ (1MΩ)
050505050
15050
5.05050
501
=+−
=
+=+∞−∞
=
=
+=
rS
rT
i
k
k
VVV
in (step begins at 1ns)
source
termination
Termination Examples - Short
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RS = 50ΩZ0 = 50Ω, td = 1nsRT = 0Ω
050505050
1500500
5.05050
501
=+−
=
−=+−
=
=
+=
rS
rT
i
k
k
VVV
in (step begins at 1ns)
source
termination
Arbitrary Termination Example
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RS = 400ΩZ0 = 50Ω, td = 1nsRT = 600Ω
778.05040050400
846.05060050600
111.050400
501
=+−
=
=+−
=
=
+=
rS
rT
i
k
k
VVV
in (step begins at 1ns)
sourcetermination
0.111V0.205V
0.278V0.340
Lattice Diagram
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RS = 400Ω
RT = 600ΩZ0 = 50Ω, td = 1ns
in (step begins at 1ns)
Rings up to 0.6V(DC voltage division)
Termination Reflection Patterns
21
RS = 25Ω, RT = 25Ω
krS & krT < 0
Voltages Converge
RS = 25Ω, RT = 100Ω
krS < 0 & krT > 0
Voltages Oscillate
RS = 100Ω, RT = 25Ω
krS > 0 & krT < 0
Voltages Oscillate
RS = 100Ω, RT = 100Ω
krS > 0 & krT > 0
Voltages Ring Up
source
termination
sourcetermination
source
termination
source
termination
Termination Schemes
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• No Termination• Little to absorb line energy• Can generate oscillating
waveform• Line must be very short
relative to signal transition time• n = 4 - 6
• Limited off-chip use
• Source Termination• Source output takes 2 steps up• Used in moderate speed point-
to-point connections
LCnlnTt triproundr 2=> −
LClt porch 2≅
Termination Schemes
23
• Receiver Termination• No reflection from receiver• Watch out for intermediate
impedance discontinuities• Little to absorb reflections at driver
• Double Termination• Best configuration for min
reflections• Reflections absorbed at both driver
and receiver
• Get half the swing relative to single termination
• Most common termination scheme for high performance serial links
Differential Transmission Lines
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• Differential signaling advantages• Self-referenced • Common-mode noise rejection• Increased signal swing• Reduced self-induced power-
supply noise
• Requires 2x the number of signaling pins relative to single-ended signaling• But, smaller ratio of
supply/signal (return) pins• Total pin overhead is typically
1.3-1.8x (vs 2x)
[Hall]
• Even mode• When equal voltages drive both
lines, only one mode propagates called even more
• Odd mode• When equal in magnitude, but out
of phase, voltages drive both lines, only one mode propagates called odd mode
Balanced Transmission Lines
• Even (common) mode excitation• Effective C = CC
• Effective L = L + M
• Odd (differential) mode excitation• Effective C = CC + 2Cd
• Effective L = L – M
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21
21
2
+−
=
+=
dcodd
ceven
CCMLZ
CMLZ
[Dally]
2 ,2 even
CModdDIFFZZZZ ==
PI-Termination
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1RZeven =
2||2|| 221 RZRRZ evenodd ==
−
=oddeven
evenodd
ZZZZR 22
T-Termination
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12 2RRZeven +=
( )oddeven
odd
ZZR
RZ
−=
=
21
1
2
Next Time
• Channel modeling• Time domain reflectometer (TDR)• Network analysis
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