Interactive Tutorial on
Fundamentals of Signal Integrity
for High-Speed/High-Density Design
Andreas Cangellaris, Jose Schutt-Aine
and Umberto Ravaioli
ECE Department, University of Illinois at Urbana-Champaign
cangella@uiuc.edu, jose@decwa.ece.uiuc.edu, ravaioli@uiuc.edu
Alina Deutsch
IBM Corporation, T.J. Watson Research Center
deutsch@us.ibm.com
DAC 2001
© SEMCHIP
1
Outline
• Introduction (A. Deutsch)
– The interconnect bottleneck in high-speed systems
• Interconnect Modeling Fundamentals
(A.Cangellaris/U. Ravaioli)
– Time-domain & frequency-domain transmission line analysis
– Lossy lines and signal dispersion
– Crosstalk for short lengths of coupled interconnects
• On-Chip Interconnects (A. Deutsch)
– Modeling of on-chip interconnects
– Interconnect impact on system performance
– Future trends
DAC 2001
© SEMCHIP
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Outline (cont.)
• Interconnects at the Package and Board Level
(J.Schutt-Aine/U. Ravaioli)
– Multiconductor transmission line theory
– Crosstalk modeling and measurement
– Lumped vs. distributed modeling of interconnects
• Concluding Remarks
DAC 2001
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Fundamentals of
Transmission Line Theory
DAC 2001
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Transmission-line theory quantifies signal propagation
on a system of two parallel conductors with crosssectional dimensions much smaller than their length
Coaxial
Stipline
Microstrip
Familiar
Cross sections
I(z,t)
Generic symbol for
a two-conductor line
V(z,t)
z (direction along interconnect axis)
© SEMCHIP
DAC 2001
5
For a uniform transmission line, the electric and magnetic fields
are transverse to the direction of wave propagation (and hence, to
the axis of the line). Thus, transmission line fields are called
Transverse Electromagnetic (TEM) Waves
y
Electric field lines
The electric field behaves like an electrostatic field.
x
Over the cross section, the potential difference between
any two points A and B on the two conductors is constant:
∫
E( x, y, z, t ) ⋅ dl = V ( z, t )
A→B
Magnetic field lines
Over the cross section, the magnetic field looks like a
magnetostatic field. Its line integral around one of the
conductors equals the total current in the conductor.
H ( x, y , z , t ) ⋅ d l = I ( z , t )
∫
center
conductor
DAC 2001
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In a transmission line configuration as much charge
moves down the “active” wire that much charge of
negative polarity moves down the “return path”
Active wire
Active wire
(+++++)
Return path (- - - - -)
Return path
Direction along interconnect
Direction along interconnect
At every cross section of a transmission line the
currents on the active and return wires
balance each other.
This balance leads to field confinement and
reduced interference.
Electric field
between wires
© SEMCHIP
DAC 2001
7
Signal propagation is quantified in terms of the
solution of the so-called Telegrapher’s equations
Time-Domain Form
∂v ( z , t )
∂z
∂i ( z , t )
∂z
= − Ri ( z , t ) − L
= − Gv ( z , t ) − C
Frequency-Domain Form
∂i ( z , t )
dV ( z , ω )
∂t
dz
∂v ( z , t )
dI ( z , ω )
∂t
dz
= − GV ( z , ω ) − jω CV ( z , ω )
I(z,t) R(∆z)
I(z,t)
V(z,t)
V(z,t)
z
DAC 2001
= − RI ( z , ω ) − jω LI ( z , ω )
L(∆z)
C(∆z)
I(z+∆z,t)
G(∆z)
V(z+∆z,t)
∆z
8
Transmission-Line Parameters
• Per-unit-length capacitance C
• Per-unit-length conductance G
• Per-unit-length inductance L
– Loop inductance
– Frequency dependence due to
Current Loop
skin effect
• Per-unit-length resistance R
∆z
– Strong frequency dependence
due to skin effect
© SEMCHIP
DAC 2001
9
Time-Domain Solution of Telegrapher’s Equations
Neglecting losses for simplicity:
∂v ( z , t )
∂i ( z , t ) 
= −L
∂ 2v 1 ∂ 2v
∂z
∂t 
⇒
−
=0

∂i ( z , t )
∂v ( z , t ) 
∂z 2 v 2p ∂t 2
= −C
∂z
∂t 
where v p = 1
is the wave velocity on the line.
LC
General solution: v ( z , t ) = f + ( z − v p t ) + f − ( z + v p t )
forward wave
Current wave: i ( z , t ) =
1 +
1 −
f ( z − v pt ) −
f ( z + v pt )
Z0
Z0
forward wave
where Z 0 =
DAC 2001
backward wave
backward wave
L
is the characteristic impedance of the line.
C
© SEMCHIP
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A voltage signal f (t) launched on a lossless line
propagates unaltered with speed vp dependent on the
transmission-line properties.
I+(z0,t)
1
vp =
V+(z0,t)
Vs
z
V+(z0,t)
LC
V + ( z, t )
L
=
Z
0
I + ( z, t )
C
Vs(t) = f (t)
t
Characteristic
Impedance
I+(z0,t)
t
Time of
flight
t0 = z0/vp
t
11
DAC 2001
The characteristic impedance dictates the
amplitude of the voltage waveform launched
on the line
+
RS I (0,t)
Vs
V+(0,t)
To infinity (or matched)
VS (t ) = V + (0, t ) + I + (0, t ) RS 
Z0
+
⇒
=
(0,
)
(
)
V
t
V
t

S
Z 0 + RS
V + (0, t ) = Z 0 I + (0, t )

DAC 2001
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Discontinuities in the characteristic impedance
of a transmission line give rise to reflections
At the junction it is:
Z01
V1 = V + + V − = V2 = V ++
Z02
(V ++, I ++)
1
V ++
+
−
I1 =
V −V = I2 =
Z 01
Z 02
V − = ΓV + 
V ++ = TV + 


V −  and ++ V ++ 
−
I =− 
I =
Z01 
Z02 
Z02 − Z01 
Z02 + Z01 

2Z02 
Transmission Coefficient: T =
Z02 + Z01 
(V +, I +)
(
(V −, I −)
Reflection Coefficient:
)
Γ=
Maintaining a fairly constant value of the characteristic impedance
along an interconnect path is essential for reflection suppression.
© SEMCHIP
DAC 2001
13
Source and load impedances impact transmission line
performance of the interconnect
Vs
ZS(f)
Z0
Load reflection coefficient:
Source reflection coefficient
ΓS ( f ) =
ZL(f)
Z S ( f ) − Z0
Z S ( f ) + Z0
ΓL ( f ) =
ZL ( f ) − Z0
Z L ( f ) + Z0
Load transmission coefficient:
2Z L ( f )
TL ( f ) = 1 + Γ L ( f ) =
Z L ( f ) + Z0
DAC 2001
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Example: Unterminated interconnect (ZL=∞) driven by high
source impedance driver with ZS>>Z0 (e.g. unbuffered CMOS)
Source (ΓS ≈ 1)
Load (ΓL = 1)
Excitation: Step Pulse of amplitude V0
V+
V + = V0
T=d / v
(One-way
Delay)
ΓLV+
ΓL =
Z0
Z − Z0
<< V0 , Γ S = S
≈1
Z S + Z0
ZS + Z0
Z L − Z0
= 1, TL = 2
Z L + Z0
2T
ΓSΓLV+
VL(t )
3T
ΓL ΓSΓL
t
Slow response
V+
4T
~6V+
~4V+
t
2V+
Bounce diagram
Steady-state value V0
T
3T
5T
t
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DAC 2001
Example: Unterminated interconnect (ZL=∞) driven by low
source impedance driver with ZS<Z0 (e.g. ECL or strong TTL)
Source (ΓS ≈ –1)
Load (ΓL = 1)
V + = V0
V+
ΓLV+
Excitation: Step Pulse of amplitude V0 ; Z0 = 7ZS
T=d / v
(One-way
Delay)
ΓL =
7
Z0
Z −Z
≈ V0 , ΓS = S 0 = −0.75
ZS + Z0 8
ZS + Z0
ZL − Z0
= 1, TL = 2
ZL + Z0
VL(t )
2T
Overshoot & Ringing
2V+
ΓSΓLV+ = - 0.75 V+
1.625V+
3T
ΓL ΓSΓLV+ LV+ = -0.75 V+
SteadyState: V0
0.5V+
4T
t
Bounce diagram
DAC 2001
T
t
© SEMCHIP
3T
5T
t
16
A capacitor CL represents the load at the gate
input of the receiver. Its presence adds delay.
VL = V + + V −
(V+, I+)
IL =
(V-, I-) C
L
1
V + −V − )
(
Z0
I L = CL
Interconnect delay = T
dVL
, VL (t = T ) = 0.
dt
VL (t ) = V + (1 − exp( − (t − T ) / τ ) ) , t > T
where τ =Z 0 C .
Let Td be the time at which VL (t = Td ) = 0.9V + .
Td = T + 2.3τ ⇒
Extra delay due to the capacitor is 2.3Z 0 C
© SEMCHIP
DAC 2001
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Delay is introduced by all capacitive and inductive
discontinuities present in a signal path
Wire bonds
(primarily inductive)
Z0
Interconnect bends
(primarily capacitive)
Z0
Z0
Via
(primarily inductive)
Z0
Equivalent circuit for SPICE-based transient simulation
DAC 2001
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The delay due to a capacitive o an inductive
discontinuity depends on the values C or L and Z0
Inductive Discontinuity
Capacitive Discontinuity
(V+, I+)
Z0
C
(V++, I++)
(V+, I+)
Z0
Z0
(V++, I++)
L
Z0
(V-, I-)
(V-, I-)
V ++ (t ) = V + (1 − exp( −t / τ C ) )
V ++ (t ) = V + (1 − exp( −t / τ L ) )
CZ 0
2
++
V (t ) reaches 0.9V + at t = 2.3τ C
where τ L =
V ++ (t ) reaches 0.9V + at t = 2.3τ L
Hence, the capacitor adds a delay of
Hence, the inductor adds a delay of
Td = 1.15CZ 0 .
Td = 1.15 L / Z 0
(C=1 pF, Z 0 = 50 Ohm; Td = 57.5 ps)
( L = 2.5 nH, Z 0 = 50 Ohm; Td = 57.5 ps)
where τ C =
L
2Z0
19
DAC 2001
Slots in ground planes increase interconnect delay and
enhance noise generation and interference
The slot in the ground plane
acts as a slot antenna.
interconnect
The return current in the ground plane
flows around the slot. Hence,
• Extra L ⇒ extra delay
• Unbalanced currents lead to
enhanced emissions
• Interference (crosstalk) with
other wires beyond immediate
neighbors
DAC 2001
© SEMCHIP
Ground plane
Interference occurs on
either side of the plane
20
Transmission line models of interconnects
predict only “differential-mode” currents
Displacement
current (radiation)
I1
+
-
Transmission-Line
Model
Conduction (return)
current
I2
ID
+
-
- ID
Radiated emissions calculation
can be grossly incorrect if the
common-mode current is not
taken into account
DAC 2001
I1 − I 2
2
I +I
Common-mode current: I C = 1 2
2
I1 = I C + I D
Differential-mode current: I D =
I 2 = IC − I D
© SEMCHIP
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The input impedance of a match-terminated interconnect with a
continuous return path remains essentially contact over a broad
frequency range
'(
)(
!"#$%&
Plots generated using UIUC’s fast time-domain solvers (Prof. E. Michielssen)
DAC 2001
© SEMCHIP
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The disruption of the return path caused by the slot manifests itself
as an added inductance at lower frequencies and radiated emissions
(radiation resistance) at higher frequencies
*
'(
*
)(
*
+,)
+,",)
!"#$%&
Plots generated using UIUC’s fast time-domain EM solvers (Prof. E. Michielssen)
© SEMCHIP
DAC 2001
23
Mesh (Grid) Planes in PCBs increase the
characteristic impedance of the lines
Return current
Per-unit-length inductance, L, increases.
Per-unit-length capacitance, C , decreases.
Z0 =
DAC 2001
L
increases
C
© SEMCHIP
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In the case of mesh (grid) planes, high-speed lines
should be routed right above the plane metallization
interconnects
Grid Plane
(cross section)
Advantages
-Better impedance control
-Reduced cross-plane interference
DAC 2001
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Lossy Transmission Lines
• Ohmic loss in the metallization
– Frequency-dependent R and L (skin effect)
• Insulating substrate loss
– Frequency-dependent G
• Semiconductor substrates
– Frequency-dependent R and L
– Frequency-dependent G and C
DAC 2001
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Frequency-Domain Solution of Telegrapher’s Equations
In the frequency domain, interconnect loss can be accounted for easily.
dV ( z , ω )

= [ R (ω ) + jω L (ω )] I ( z , ω ) 
d 2V ( z , ω )

dz
⇒
− Z (ω )Y (ω )V ( z , ω ) = 0

2
dI ( z , ω )
dz
−
= [G (ω ) + jω C (ω )]V ( z , ω ) 

dz
where Z (ω ) = R (ω ) + jω L (ω ) is the per-unit-length impedance of the line
−
and Y (ω ) = G (ω ) + jω C (ω ) is the per-unit-length admittance of the line.
V ( z , ω ) = V + (ω ) exp( −γ z ) + V − (ω ) exp(γ z )

General solution : 
1
+
−
 I ( z , ω ) = Z (ω ) V (ω ) exp( −γ z ) − V (ω ) exp(γ z ) 
0

γ (ω ) = [ R (ω ) + jω L (ω )][G (ω ) + jω C (ω )] is the complex propagation constant ,
and Z 0 (ω ) =
R (ω ) + jω L (ω )
is the characteristic impedance.
G (ω ) + jω C (ω )
© SEMCHIP
DAC 2001
27
The characteristic impedance of a lossy line
is a complex number!
When G ≈ 0 it is the per-unit-length ohmic loss in the wires that
dominates the loss; hence,
Z 0 (ω ) =
R (ω ) + jL (ω )
=
jω C
L (ω )
R (ω )
1− j
ω L (ω )
C
For the interconnect structures of interest , L is in the order of nH/cm;
hence, for f < a few tens of MHz, ω L << R (especially for thin-film wire).
1 − j R (ω )
ωC
2
Notice that the real and imaginary parts are of the same magnitude.
On the other hand , for high frequ e ncie s such that R << ω L ,
Thus, for low frequenc ies: Z 0 (ω ) ≈
Z 0 (ω ) =
1 R (ω ) 
L 
1 − j

2 ω L (ω ) 
C 
Notice that, since at high frequencies R (ω ) ∝ ω , the characteristic
impedance is predominantly real.
DAC 2001
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The presence of loss is responsible for signal attenuation
and distortion
The propagation constant becomes frequency dependent:
γ (ω )= [ R (ω ) + jω L (ω )][G (ω ) + jω C ] = α (ω ) + j β (ω ).
α (ω ) is the attenuation constant; β (ω ) is the phase constant.
V + (ω , z ) = V0+ exp( −α (ω ) z ) exp( − j β (ω ) z )
attenuation
phase shift
The characteristic impedance and the phase velocity are frequency dependent:
Z 0 (ω ) =
ω
R (ω ) + jω L(ω )
, v p (ω ) =
G (ω ) + jω C
β (ω )
Different frequencies in the spectrum of a pulse propagate at
different speeds and suffer different attenuation.
This results in pulse distortion often referred to as dispersion
Input
Pulse
Output
Pulse
Lossy line
29
DAC 2001
Input Impedance of a Transmission Line
(or, how long wires “transform” load impedances)
Z in ( d ) =
V (d )
Z + Z 0 tanh γ d
= Z0 L
I (d )
Z 0 + Z L tanh γ d
Neglecting losses, γ = j β , Z 0 is real, and it is:
Z in ( d ) = Z 0
Z L + jZ 0 tan β d
Z 0 + jZ L tan β d
I(d)
− Periodic function with period λ / 2
1+
− max( Z in ) = Z 0
Z L − Z0
Z L + Z0
Z − Z0
1− L
Z L + Z0
ZL
Z0,γ
V(d)
d
1−
Z L − Z0
Z L + Z0
1+
Z L − Z0
Z L + Z0
; min( Z in ) = Z 0
• Matched Load: Z L = Z 0 ⇒ Z in ( d ) = Z 0
• Shorted Line: Z L = 0 ⇒ Z in ( d ) = jZ 0 tan β d
- A shorted line of length equal to an odd multiple of λ / 4 has infinite input
impedance and th us appears as an open circuit.
DAC 2001
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Grounding wires running some distance along a
ground plane or chassis exhibit transmission-line
behavior at RF frequencies.
d
Zg
Z0
• Safety earth is not an RF ground.
• At high frequencies, the claim that
“everything is connected to earth”
through safety earth is
meaningless
• At high frequencies, “single point
ground” is meaningless
DAC 2001
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Skin-Effect Resistance
Skin effect
At frequencies such that the skin depth is
larger or comparable with the conductor
thickness, the current distributes uniformly
over the conductor cross section.
Skin depth: δ =
At high frequencies, where the skin
depth is smaller than the conductor
thickness, current crowding around
the perimeter occurs.
1
π f µσ
• At f = 1 GHz, for aluminum with conductivity σ = 4×107 S/m and
permeability µ = 4π × 10-7 H/m, the skin depth is 2.5 µm.
• For high enough frequencies, the p.u.l. resistance increases as √f
DAC 2001
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Frequency dependence
of the p.u.l. resistance
(top) and inductance
(bottom) of the single
stripline configuration
with w=50 µm, t=10 µm,
g=10 µm, and h=100 µm.
R per-unit-length (Ohms/cm)
The contribution of the return path to interconnect
resistance may need to be taken into account
4
3
2
PEC reference planes
Copper reference plane
1
0
-2
10
-1
0
10
1
10
10
L per-unit-length (Ohms/cm)
Frequency (GHz)
3.4
3.2
3
2.8
-2
10
-1
0
10
1
10
10
Frequency (GHz)
DAC 2001
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Extraction of the frequency-dependent p.u.l. interconnect
resistance must take into account the presence of
adjacent conductors (proximity effect)
DAC 2001
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The per-unit-length resistance matrix has non-zero off-diagonal
elements. Taking these off-diagonal elements into account is
important, especially for the tightly coupled wires
5
Per-unit-length Resistance (Ohms/cm)
4.5
g
w
1
t
h
s
2
4
3.5
3
2.5
2
1.5
R12
R11
1
0.5
g
0
-4
10
10
-3
-2
-1
10
10
Frequency (GHz)
0
10
10
1
3.5
w= 50 µm, t=10 µm, s= 30 µm
h= 110 µm, g= 10 µm, εr=4
Aluminum: σ=3.3E7 S/m
Copper: σ =5.8E7 S/m
L11
Per-unit-length Inductance (nH/cm)
3
2.5
2
1.5
L12
1
0.5
-4
10
DAC 2001
10
-3
-2
-1
10
10
Frequency (GHz)
10
3510
0
1
Impact of different metallization on delay
–
Vs
+
g
length=20 cm
w
Zo
1
1 pF
h
s
Zo
2
Risetime=50 ps
1 pF
g
Far-end Voltage of Victim line
Far-end Voltage of Driven line
1
t
0.2
90%
0.15
Voltage (V)
Voltage (V)
0.8
0.6
0.1
0.05
0.4
~ 1 ns
0.2
0
0
0.2
0.4
0.6
Time (sec)
DAC 2001
0
0.8
1
x 10
-0.05
-8
© SEMCHIP
0
0.2
0.4
0.6
Time (sec)
0.8
1
x 10
36
-8
Impact of frequency-dependent loss on
interconnect transient response
Cross-section of a stripline
geometry.
s=50 µm, w=50 µm,
t=10 µm, g=10 µm,
h=200 µm and εr=4.
Copper metallization.
Constant Model (Rdc, L , C)
Frequency dependent model: PEC reference planes (R(f), L(f), C)
Frequency dependent model: Copper reference plane (R(f), L(f), C)
37
DAC 2001
The effect of frequency-dependent loss is
particularly apparent in the cross-talk levels
Constant Model (Rdc, L , C)
Frequency dependent model: PEC reference planes (R(f), L(f), C)
Frequency dependent model: Copper reference plane (R(f), L(f), C)
DAC 2001
38
Insulating Substrate Loss
• Characterized in terms of the substrate material
conductivity or loss tangent

ε ′′(ω ) 
ε (ω ) = ε ′(ω ) − jε ′′(ω ) = ε ′(ω ) 1 − j
 = ε ′(ω ) (1 − j tan δ (ω ) )
ε ′(ω ) 



σ (ω )  
σ (ω ) + jωε ′(ω ) = jω ε ′(ω ) 1 − j

ωε ′(ω )  


ε ′′(ω ) σ (ω )
tan δ (ω ) =
=
ε ′(ω ) ωε ′(ω )
• Transverse electric field between conductors results in a
transverse leakage current and, thus, ohmic loss
E
J
ε
ε,σ
C
G
© SEMCHIP
DAC 2001
39
The assumption of constant loss tangent leads to
physically inconsistent models for G
Coaxial
2a
2πε
2πσ
, G=
⇒
ln(b / a )
ln(b / a)
G σ
G
= ⇒ = ω tan δ
C ε
C
C=
2b
• Assuming tanδ is constant yields G(ω) ∝ ω
– Such a behavior violates causality!
– For a causal circuit
Re{Y (ω )} is an even function of frequency
Im{Y (ω )} is an odd function of frequency
DAC 2001
© SEMCHIP
Y(ω)
G
C
40
Simply assuming the loss tangent to remain constant
over a broad (multi-GHz) frequency range leads to
a non-physical behavior of G(ω)
• A physically correct model needs to start with a physicallycorrect description of the frequency dependence of the complex
permittivity.
– Use measured data for the complex permittivity to synthesize a Debye
model for it
– Use the synthesized Debye model for the extraction of C(ω) and G(ω)
K
εk
⇒
ε (ω ) = ε ′(ω ) − jε ′′(ω ) = ε ∞ + ∑
k =1 1 + jω τ k
K
tan δ (ω ) =
ε ′′(ω )
=
ε ′(ω )
ω ε kτ k
∑ 1+ ω τ
k =1
2 2
k
εk
2 2
k =1 1 + ω τ k
K
ε∞ + ∑
G (ω ) ∝ ω tan δ ⇒ even function of frequency
41
DAC 2001
Capacitive and Inductive Crosstalk
in Short Interconnects
y
V1
V2
V3
VN
Reference
conductor
x
V1   R11
  
d V2   R21
−  =
dz   
VN  RN1
 I1   G11
  
d  I2  G21
−  =
dz  
IN  GN1
DAC 2001
R12 R1N  I1 
 L11 L12 L1N  I1 


 
R22 R2N 
  I2  + jω  L21 L22 L2N   I2 
 

 
 

 
RN 2 RNN 
 IN 
 IN 
LN1 LN 2 LNN 
G12 G1N V1 
 C11 C12 C1N V1 


 
G22 G2N 
 V2  + jω C21 C22 C2N  V2 
 

 



GN2 GNN VN 
CN1 CN2 CNN VN 
42
Crosstalk in Coupled Lines
• For interconnects with more than two (active) conductors,
crosstalk analysis is most effectively performed in terms of a
circuit simulator that can support MTL models (*).
– Most common (and computationally efficient) SPICE
equivalent circuits for MTL assume lossless transmission
lines.
– Models for MTLs with losses (including frequency-dependent
losses associated with skin effect) are available also. They are
essential for accurate analysis of interconnect-induced delay,
dispersion, and crosstalk at the board level for signals of GHz
bandwidths.
– It is assumed that the interconnect structure is uniform
enough for its description in terms of per-unit-length L,C,R,
and G matrices to make sense.
(*) V.K. Tripathi and J.B. Rettig, “A SPICE Model for Multiple Coupled Microstrips and Other
Transmission Lines,” IEEE Trans. Microwave Theory Tech., vol. 33(12), pp. 1513-1518,
Dec. 1985.
© SEMCHIP
DAC 2001
43
For the case of a three-conductor, lossless line in
homogeneous dielectric, with resistive terminations,
an exact solution is possible.
• Exact solutions are useful because:
– they help provide insight into the crosstalk mechanism;
– they can be used to validate computer-based simulations.
• The following results were first published by C.R. Paul (C.R.
Paul, “Solution of transmission line equations for threeconductor lines in homogeneous media,” IEEE Trans. On
Electromagnetic Compatibility, vol. 20, pp. 216-222, 1978.
Rs
Vs ~
IG (0)
RNE
IR (0)
DAC 2001
IG (d)
l
VG (0)
VG (d ) RFE
VNE
VFE
© SEMCHIP
RL
IR (d)
44
Exact solution for crosstalk in a lossless, threeconductor line with resistive terminations
L M 
 CG
=
Per unit length parameters: L =  G
,
C

−C
 M LR 
 M
Near - end and far - end crosstalk voltages :
VNE =



S  RNE
j 2π l / λ
α LG S  IGDC  +
jω Ml  C +

D  RNE + RFE
1− k 2




S  RNE RFE
j 2π l / λ 1
jω CM l  C +

D  RNE + RFE
1 − k 2 α LG

VFE =
−CM 
CR 


S VGDC 



R R
S
RFE
jω MlI GDC + NE FE jω CM lVGDC 
−
D  RNE + RFE
RNE + RFE

where, C = cos β l , S =
Inductive &
Capacitive
Coupling
Coefficients
sin β l
M
CM
, k=
=
≤ 1, and
βl
LG LR
CG CR
45
DAC 2001
Exact solution for crosstalk in a lossless, threeconductor line with resistive terminations

(1 − αSGαLR )(1 − αLGαSR )  + ω τ + τ ;
D = C 2 − S 2ω 2τ Gτ R 1 − k 2
 j CS ( G R )
(1 + αSRαLR )(1 + αSGαLG ) 

R
R
R
R
αSG = S , α LG = L , αSR = NE , α LR = FE ;
ZCG
ZCG
ZCR
ZCR
ZCG = LG / CG , ZCR = LR / CR are the characteristic impedances of each
line in the presence of the other one;
τG =
LGl
RR
LRl
R R
+ CGl S L , τ R =
+ CRl NE FE ,
RS + RL
RS + RL
RNE + RFE
RNE + RFE
are the time constants of the coupled lines;
VGDC =
RL
VS
, are the voltage and current of the
VS , IGDC =
RS + RL
RS + RL
generator circuit under dc excitation (no coupling to the receptor circuit).
DAC 2001
46
Under the assumptions of electrically short lines, and weak
coupling, the crosstalk equations simplify considerably
• A line is said to be electrically short if its length is a small fraction of
the wavelength at the highest frequency of interest.
Package interconnects fall in this category
• Two lines are said to be weakly coupled if the coupling coefficient,
k , is sufficiently smaller than 1.
Under these assumptions the equations for the near-end and
far-end crosstalk voltages become:
VNE =

1
RNE
RNE RFE RL
jω Ml +
jω CM l 

( RNE + RFE )( RS + RL )
D  ( RNE + RFE )( RS + RL )

VFE =

1
RNE RFE RL
RFE
jω Ml +
jω CM l 
−
( RNE + RFE )( RS + RL )
D  ( RNE + RFE )( RS + RL )

DAC 2001
© SEMCHIP
47
For weakly coupled, electrically short wires, the crosstalk is a
linear combination of contributions due to the mutual
inductance between the lines (inductive coupling) and the
mutual capacitance between the lines (capacitive coupling).
IND
CAP
+ VNE
=
VNE = VNE
jω
RNE
( Ml + (RFE RL )CM l )
D ( RNE + RFE )( RS + RL )
CAP
=
VFE = VFINE D + VFE
jω
RNE
( −Ml + (RNE RL )CM l )
D ( RNE + RFE )( RS + RL )
Notice that:
• The higher the frequency the larger the crosstalk
• Inductive coupling dominates for low-impedance loads
• Capacitive coupling dominates for high-impedance loads
DAC 2001
© SEMCHIP
48
Lumped versus Distributed Modeling
• When the interconnect length is much smaller than
the wavelength of interest, lumped models provide
sufficient accuracy and can be used
– Typical case for package interconnects at RF
frequencies
– Inaccurate for interconnects at the MCM and PCB level
• What does “interconnect length is much smaller
than the wavelength of interest” really mean?
– Typical rule of thumb: d < λ/10
– …but one can take a closer look at this rule of thumb as
shown next
© SEMCHIP
DAC 2001
49
There are four possible implementations of lumped
models for a two-conductor interconnect of length d
L/2
C
C = C p.u .l .d
C
Right L - model
T- model
L
L
C/2 C/2
C
π - model
Left L - model
DAC 2001
L = Lp.u .l .d
L
L/2
© SEMCHIP
50
The accuracy of a lumped model can be examined by considering
the input impedance obtained when the model is terminated at the
characteristic impedance of the line
L/2
L/2
(see B. Young, Digital Signal Integrity, Prentice Hall, 2000)
For the model to exhibit "transmission line" behavior,
C
Zin
Z0
its input impedance should equal the load impedance Zˆ0 :
Zin = jω L / 2 +
2
1
jωC +
1
ˆ
Z0 + jω L / 2
ω 
2
L
= Zˆ0 ⇒ Zˆ0 =
1 −   , where ωT =
C
LC
 ωT 
• For ω <<ωT , Zˆ0 ≈ Z0
• A bandwidth of validity of the T-model can be obtained by finding ωmax such that
Z − Zˆ ≤ aZ for ω ≤ ω .
0
0
0
max
For a = 0.05, ωmax =
0.62
0.62
=
.
LC d Lp.u.l C p.u.l .
Application: d = 2 cm, Lp.u.l . = 4 nH/cm, C p.u.l = 1 pF/cm; ωmax = 4.9 GHz
© SEMCHIP
DAC 2001
51
5% accuracy bandwidth for the four lumped models
π - model
T- model
L/2
L/2
C
2
 ω LC 
L
1− 
Zˆ0 =

C
 2 
0.62
ωmax =
LC
−1/2
Right L - model
L
C
DAC 2001
L
2
L   ω LC  
ˆ
1−
Z0 =

C   2  


0.62
ωmax =
LC
C/2 C/2
Left L - model
2
 ω LC 
L
1 − 
Zˆ0 = jω L / 2 +

C
 2 
0.1
ωmax =
LC
2
 ω LC 
L
1−
Zˆ0 =− jωL /2 +

C
 2 
0.1
ωmax =
LC
© SEMCHIP
L
C
52
The number of segments is dictated by the
maximum frequency of interest that must be
represented accurately in the simulation
• The effective bandwidth criteria described earlier can be used to
select the segment size.
Example : Interconnect lead with total capacitance
C = 20 pF, and (loop) inductance L = 50 nH.
Let f max = 10 GHz be the maximum frequency of interest.
Find the minimum number n of lumped segments required
for accurate modeling.
Lseg = L / n and C seg = C / n.
2π f max ≤
2π f max LC
0.62
0.62 n
d
or n ≥ 10
=
⇒n≥
0.62
λ min
Lseg C seg
LC
For the given numbers, nmin = 101
DAC 2001
© SEMCHIP
53
Use of an insufficient number of segments leads to
artificial filtering and phase distortion of the
transmission-line response
The load response for
a source and load
match-terminated, lossless
transmission line is:
 2π 
VL = 0.5exp  − j

λ 

DAC 2001
© SEMCHIP
54
If done properly, distributed RLCG circuit
modeling of MTLs works
Such an approach is preferable when:
•
•
•
•
the wire resistance must be taken into account; (as
already mentioned, some SPICE vendors provide lossy
line modeling through extensions of the exact model
mentioned earlier);
the line is electrically short, and a few lumped-circuit
segments are sufficient for its modeling;
the MTL exhibits non-uniformity (variable cross section)
along its axis;
modeling of radiation coupling to the MTL is desired.
DAC 2001
© SEMCHIP
55