Interconnections and Signal
Integrity
José Schutt-Ainé
SemChip
DAC Tutorial
22, June 2001
DAC 2001
© SEMCHIP
1
Future System Needs and Functions
Auto
Digital Wireless
MEMS
Analog, RF
Computer
Consumer
2.5
Limits of Optical
Log (Capacity Gb/s)
2
1.5
1
0.5
0
1980
1985
1990
1995
2000
2005
2010
A
DAC 2001
High bandwidth
© SEMCHIP
High-speed Digital
2
The Interconnect Bottleneck
Al
3.0 µΩ -cm
Cu
1.7 µΩ -cm
SiO2
κ = 4.0
Low κ
κ = 2.0
Al & Cu .8µ Thick
Al & Cu Line 43µ Long
SPEED/PERFORMANCE ISSUE
45
40
Gate Delay
35
Sum of Delays, Cu & Low K
30
Delay (ps)
Gate wi Al & SiO2
Sum of Delays, Al & SiO2
Interconnect Delay, Al & SiO2
25
Interconnect Delay, Cu & Low K
20
15
10
Gate
5
0
650
595
540
485
430
375
320
265
210
155
100
Generation (nm)
DAC 2001
© SEMCHIP
3
Semiconductor Technology Trends
1997
2003
2006
2012
Chip size
(mm2)
300
430
520
750
Number of transistors
(million)
11
76
200
1400
200
100
70
35
2.16
2.84
5.14
24
Interconnect width
(nm)
Total interconnect length
(km)
DAC 2001
© SEMCHIP
4
The Interconnect Bottleneck
Technology
Generation
MOSFET Intrinsic
Switching Delay
Response
Time
1.0 um
~ 10 ps
~ 1 ps
0.01 um
~ 1 ps
~ 100 ps
DAC 2001
© SEMCHIP
5
Chip-Level Interconnect Delay
Line
Pulse Characteristics:
Line Characteristics
rise time: 100 ps
fall time: 100 ps
pulse width: 4ns
length : 3 mm
near end termination: 50 Ω
far end termination 65 Ω
Near End Response
Far End Response
1
0.7
0.6
Volts
0.45
Logic
threshold
0.175
Board
VLSI
Submicron
Deep Submicron
0.5
Volts
Board
VLSI
Submicron
Deep Submicron
0.725
0.4
0.3
Logic
threshold
0.2
0.1
0
-0.1
-0.1
0
0.4
0.8
1.2
1.6
2
0
Time (ns)
DAC 2001
0.4
0.8
1.2
1.6
2
Time (ns)
© SEMCHIP
6
Interconnect Bottleneck
Signal Integrity
Crosstalk
Dispersion
Attenuation
Reflection
Distortion
Loss
Delta I Noise
Ground Bounce
Radiation
Drive Line
Sense Line
Drive Line
DAC 2001
© SEMCHIP
7
Mixed Signal Noise
Analog
Digital
Power bus
Power bus
Interconnect
Interconnect
coupled
noise
Substrate
Chip-package
interconnect
bond Inductance
GND
DAC 2001
•
Simultaneous switching and inductance (Leff)
•
Leff is f( current magnitude and direction)
•
Interactions between noise generated by power/ground and signal paths
© SEMCHIP
8
INDUCTANCE
z
r
H
R
Idl
Inductance =
DAC 2001
Total flux linked
Current
© SEMCHIP
9
INDUCTANCE
NΦ
L= I
N : number of turns
Φ : flux per turn
DAC 2001
© SEMCHIP
10
CAPACITANCE
Capacitance =
DAC 2001
Total charge
Voltage
© SEMCHIP
11
CAPACITANCE
d
+
V
εο
ε oA
C= d
A : area
εo : permittivity
DAC 2001
© SEMCHIP
12
Package Inductance & Capacitance
Component
Capacitance
Inductance
(pF)
(nH)
68 pin plastic DIP pin†
4
35
68 pin ceramic DIP pin ††
7
20
68 pin SMT chip carrier†
2
7
† No ground plane; capacitance is dominated by wire to wire component.
†† With ground plane; capacitance and inductance are determined by the distance
between the lead frame and the ground plane, and the lead length.
DAC 2001
© SEMCHIP
13
Package Inductance & Capacitance
Component
Capacitance
Inductance
(pF)
(nH)
68 pin PGA pin††
4
35
256 pin PGA pin ††
7
20
Wire bond
1
1
0.5
0.1
Solder bump
† No ground plane; capacitance is dominated by wire to wire component.
†† With ground plane; capacitance and inductance are determined by the distance
between the lead frame and the ground plane, and the lead length.
DAC 2001
© SEMCHIP
14
CONDUCTIVITY OF DIELECTRIC
MATERIALS
Material
Conductivity (Ω
Ω -1 m-1)
Germanium
2.2
Silicon
0.0016
Glass
10-10 - 10-14
Quartz
0.5 x 10-17
Loss TANGENT : tanδδ = ωσε
DAC 2001
© SEMCHIP
15
Propagation Speeds
εr
v
c
r
εr
v (cm/ns)
Polymide
2.5-3.5
16-19
Silicon Dioxide
3.9
15
Epoxy Glass (PC board)
5
13
Alumina (ceramic)
9.5
10
Dielectric
DAC 2001
© SEMCHIP
16
Signal Integrity
Ideal
Transmission
Channel
Common
Transmission
Channel
Noisy
Transmission
Channel
DAC 2001
© SEMCHIP
17
Signal Degradation
DAC 2001
© SEMCHIP
18
90%
magnitude
10%
width
rise time
fall
time
DAC 2001
© SEMCHIP
19
Frequency Components
of Digital Signal
C0
+
C1
=
+
C2
+
C3
+
C4
DAC 2001
© SEMCHIP
20
RC Network
R
+
+
C
vi
vo
-
-
vo (f)
A is in the steady state gain of the network; A =
v (f)
i
A=
1
f2 =
1+ (f / f2 )2
1
2πRC
The gain falls to 0.707 of its low-frequency value at
the frequency f2. f2 is the upper 3-dB frequency or
the 3-dB bandwidth of the RC network.
DAC 2001
© SEMCHIP
21
RC Network
R
+
+
V
C
vi
vo
-
-
−t / RC
vo = V (1 − e
)
v/V
vi/V = 1
1.0
0.9
vo/V
0.1
0
t10%
DAC 2001
tr
t
t90%
© SEMCHIP
22
RC Network
R
+
+
V
C
vi
-
-
Rise time :
vo
tr = t90% - t10%
t r = 2.2RC =
2.2 0.35
=
2 πf 2 f 2
Rule of thumb : A 1-ns pulse
requires a circuit with a 3-dB
bandwidth of the order of 2 GHz.
DAC 2001
© SEMCHIP
23
WAVE PROPAGATION
λ
λ
z
Wavelength : λ
λ=
DAC 2001
propagation velocity
frequency
© SEMCHIP
24
Why Transmission Lines ?
In Free Space
At 10 KHz : λ = 30 km
At 10 GHz : λ = 3 cm
Transmission line behavior is prevalent when the
structural dimensions of the circuits are comparable
to the wavelength.
DAC 2001
© SEMCHIP
25
Transmission Line Model
Let d be the largest dimension of a circuit
circuit
z
λ
If d << λ, a lumped model for the circuit can be used
DAC 2001
© SEMCHIP
26
Transmission Line Model
circuit
z
λ
If d ≈ λ, or d > λ then use transmission line model
DAC 2001
© SEMCHIP
27
Frequency Dependence of Lumped
Circuit Models
At higher frequencies, a lumped circuit model is no
longer accurate for interconnects and one must use
a distributed model. Transition frequency depends
on the dimensions and relative magnitude of the
interconnect parameters.
0.3 ×109
f ≈
10d εr
DAC 2001
tr ≈
© SEMCHIP
0.35
f
28
Lumped Circuit or Transmission Line?
A) Determine frequency or bandwidth of the signal
-Microwave: f = operating frequency
0.35
-Digital: f = rise time
B) Determine propagation velocity in medium, v,
v
next calculate wavelength λ =
f
DAC 2001
© SEMCHIP
29
Lumped Circuit or Transmission Line?
C) Compare wavelength with dimensions (feature size) d.
Case 1: If λ >> d use lumped circuit equivalent
Total inductance = L x length
Total capacitance = C x length
Case 2: If λ ≈ 10d or λ < 10d, use transmission-line
model
DAC 2001
© SEMCHIP
30
Frequency Dependence of Lumped
Circuit Models
Dimension Frequency Rise time
Printed circuit line
10 in
>55 MHz <7 ns
(epoxy, glass)
Package lead frame 1 in
>400 MHz <0.9 ns
(ceramic)
VLSI
interconnection*
100 µm
>8 GHz
<50 ps
(silicon)
* Using RC criterion for distributed effect
DAC 2001
© SEMCHIP
31
Types of Transmission Lines
εr
Coaxial line
εerr
Microstrip
DAC 2001
Air
Waveguide
eεrr
Stripline
© SEMCHIP
εerr
Coplanar line
εerr
Slot line
32
Parallel-plate Transmission Line
w
µ, ε
a
µa
L= w
εw
C= a
DAC 2001
© SEMCHIP
33
Coaxial Transmission Line
µ
ε
b
a
b
L = µ ln a
2πε
C = ln(b/a)
DAC 2001
© SEMCHIP
34
Microstrip
ε
DAC 2001
© SEMCHIP
35
Microstrip
M icrostrip C haracteristic Im pedance
100
90
h = 21 m ils
h = 14 m ils
h = 7 m ils
Zo (ohm s)
80
70
60
50
40
30
0
1
2
3
W /h
dielectric constant : 4.3.
DAC 2001
© SEMCHIP
36
Electric Field Configuration
Microstrip
Stripline
Consequence: Wave propagation in stripline is
closer to the TEM mode of propagation and the
propagation of velocity is approximately c/ εr .
DAC 2001
© SEMCHIP
37
TEM PROPAGATION
L
I
+
V
C
∆z
z
Z1
Zo
β
Z2
Vs
DAC 2001
© SEMCHIP
38
Telegrapher’s Equations
L
I
+
V
C
∆z
−
∂V
∂I
=L
∂z
∂t
−
∂I
∂V
=C
∂z
∂t
L: Inductance per unit length.
DAC 2001
C: Capacitance per unit length.
© SEMCHIP
39
Transmission Line Solutions
z
Zo
β
forward wave
backward wave
DAC 2001
© SEMCHIP
40
Reflection in Transmission Lines
1.
2.
3.
DAC 2001
© SEMCHIP
41
Time Domain Reflectometry
Zg
Vf
microstrip line
Zg
step
generator
DAC 2001
Vb
coaxial line
© SEMCHIP
42
Metallic Conductors
σ
Area
th
Leng
Re s ist an ce : R
R=
Le ng th
σ Are a
Submicron level:
W=0.25 microns
R=422 Ω/mm
Package level:
W=3 mils
R=0.0045 Ω/mm
DAC 2001
© SEMCHIP
43
Metallic Conductors
σ (Ω-1 m −1 × 10-7)
Metal
Silver
6.1
Copper
5.8
Gold
3.5
Aluminum
1.8
Tungsten
1.8
Brass
1.5
Solder
0.7
DAC 2001
© SEMCHIP
44
Loss in Transmission Lines
RF
SOURCE
DAC 2001
© SEMCHIP
45
Skin Effect in Transmission Lines
δ
Low Frequency
DAC 2001
High Frequency
© SEMCHIP
Very High Frequency
46
Skin Effect in Microstrip
..
D
y
w
d
- y/
J oe
J=
/
- jy
d
e
ty
ensi
nt d
e
r
r
f cu
de o
nitu
g
a
V
M
σ
t
d
εe
DAC 2001
© SEMCHIP
47
Skin Effect
The electric field in a material medium propagates as
E o e −γz = E oe −αz e− jβz
z
where γ = α + jβ
β. We also have
γ=ω
σ
µε(1+j ) .
ωε
Wint
Hint
δs
CURRENT AREAS
δs
DAC 2001
© SEMCHIP
48
Lossy Transmission Line
I
L
R
+
V
C
G
∆z
Telegraphers Equation
−
∂V
= (R+ jωL)I = ZI
∂z
−
∂I
= (G + jωC)V = YV
∂z
DAC 2001
© SEMCHIP
49
Lossy Transmission Line
z
R, L, G, C,
forward wave
backward wave
DAC 2001
© SEMCHIP
50
Network Analyzer
DAC 2001
© SEMCHIP
51
S Parameters of Transmission Lines
Short line
Category 5/ 1-meter
200
0.5
150
100
0.4
S11 (phase)
S11 (magnitude)
Category 5/ 1-meter
0.6
0.3
0.2
50
0
-50
0.1
-100
0
-150
0
0.05
0.1
Frequency (GHz)
0.15
0
0.2
0.15
0.2
200
150
0.95
100
0.9
S21 (phase)
S21 (magnitude)
0.1
Frequency (GHz)
Category 5/ 1-meter
Category 5/ 1-meter
1
0.85
0.8
50
0
-50
0.75
-100
0.7
-150
-200
0.65
0
DAC 2001
0.05
0.05
0.1
Frequency (GHz)
0.15
0
0.2
© SEMCHIP
0.05
0.1
Frequency (GHz)
0.15
0.2
52
Two-Port Characterization
a1
Zo
a2
Two-Port
Network
Zo
Zo
b1
Zo
b2
Port 1
Port 2
b1 = S11 a1 + S12 a2
b2 = S21 a1 + S22 a 2
b1
a1 a = 0
2
b1
S12 =
a2 a1 =0
S11 =
DAC 2001
S21 =
b2
a1 a2 =0
S22 =
b2
a2 a1 = 0
© SEMCHIP
53
Microstrip Characterization
- Network analyzer measurement of S parameters
- Use de-embedding scheme
- Use extraction algorithm
copper mi cr os t r i p
w =5 mi l
l =20 .2 cm
f i ber gl ass
subs t r at e
Microstrip
1
Microstrip
30
0.99
SMA connect or
25
0.98
20
0.97
10
R
e
In
s
15
0.96
0.95
0.94
5
0
DAC 2001
0.02
0.04
0.06
Frequency (GHz)
0.08
0.1
0
0.02
© SEMCHIP
0.04
0.06
Frequency (GHz)
0.08
0.1
54
Package Level Characterization
v13.s21
0
-10
-20
-30
-40
-50
-60
0
2
4
6
8
10
12
14
Frequency (GHz)
Zo
a1
Zo
L1
Lb1
4
C1L
b1
S11m
Zo
Cm12L L2
Lb2
Zo
10
1
5
R1
7
C1R
Lm12
11
R2
Cm12R
8
2
C2L
0
3
Lb3
Cm23L L
3
6
C2R
Lm23
12
R3
Cm23R
9
Lbg
C3L
C3R
99
DAC 2001
© SEMCHIP
55
Crosstalk Noise and Coupled
Transmission Lines
José Schutt-Ainé
SemChip
DAC Tutorial
22, June 2001
DAC 2001
© SEMCHIP
56
TEM PROPAGATION
L
I
+
V
C
∆z
z
Z1
β
Zo
Z2
Vs
DAC 2001
© SEMCHIP
57
Telegrapher’s Equations
L
I
+
V
C
∆z
−
∂V
∂I
=L
∂z
∂t
−
∂I
∂V
=C
∂z
∂t
L: Inductance per unit length.
DAC 2001
C: Capacitance per unit length.
© SEMCHIP
58
Crosstalk and Coupled Line
Analysis
RF
SOURCE
DAC 2001
© SEMCHIP
59
Crosstalk Noise
Signal Integrity
Crosstalk
Dispersion
Attenuation
Reflection
Distortion
Loss
Delta I Noise
Ground Bounce
Radiation
Drive Line
Sense Line
Drive Line
DAC 2001
© SEMCHIP
60
Crosstalk noise depends on termination
50 Ω
50 Ω
line 1
line 1
line 2
line 2
50 Ω
line 1
line 1
line 2
line 2
DAC 2001
© SEMCHIP
61
Crosstalk depends on signal rise time
50 Ω
line 1
line 2
50 Ω
tr = 1 ns
tr = 7 ns
line 1
line 1
line 2
DAC 2001
line 2
© SEMCHIP
62
Crosstalk depends on signal rise time
50 Ω
line 1
line 2
tr = 1 ns
tr = 7 ns
line 1
line 1
line 2
line 2
DAC 2001
© SEMCHIP
63
Coupled Transmission Lines
w
εr
h
V1
I1
Cm
I2
s
Cs
Lm
V2
Cs
DAC 2001
© SEMCHIP
64
Telegraphers Equations for Coupled Transmission Lines
Maxwellian Form
−
∂ V1
∂I
∂I
= L11 1 + L12 2
∂z
∂t
∂t
−
∂ V2
∂I
∂I
= L21 1 + L22 2
∂z
∂t
∂t
−
∂ I1
∂V
∂V
= C11 1 + C12 2
∂z
∂t
∂t
−
∂ I2
∂V
∂V
= C21 1 + C22 2
∂z
∂t
∂t
DAC 2001
© SEMCHIP
65
Telegraphers Equations for Coupled Transmission Lines
Physical form
−
∂ V1
∂I
∂I
= Ls 1 + Lm 2
∂z
∂t
∂t
−
∂ V2
∂I
∂I
= Lm 1 + Ls 2
∂z
∂t
∂t
−
∂ I1
∂V
∂V
∂V
= C s 1 + C m 1 − Cm 2
∂z
∂t
∂t
∂t
−
DAC 2001
∂ I2
∂V
∂V
∂V
= −C m 1 + C m 2 + C s 2
∂z
∂t
∂t
∂t
© SEMCHIP
66
Relations Between Physical
and Maxwellian Parameters
L11 = L22 = Ls
L12 = L21 = Lm
C11 = C22 = Cs+Cm
C12 = C21 = - Cm
DAC 2001
© SEMCHIP
67
Even Mode
−
∂Ve
∂I
= ( L11 + L12 ) e
∂z
∂t
−
∂I e
∂I
= (C11 + C12 ) e
∂z
∂t
Add voltage
and current
equations
1
(V + V )
2 1 2
1
Ie : Even mode current Ie = (I1 + I2 )
2
Ve : Even mode voltage Ve =
Ze =
L11 + L12
=
C11 + C12
ve =
1
=
(L11 + L12 )(C11 + C12 )
DAC 2001
Ls + Lm
Cs
Impedance
1
(Ls + Lm )Cs
© SEMCHIP
velocity
68
Odd Mode
∂Vd
∂I
= ( L11 − L12 ) d
∂z
∂t
∂I
∂I
− d = (C11 − C12 ) d
∂z
∂t
−
Subtract voltage
and current
equations
Vd : Odd mode voltage
Id : Odd mode current
Zd =
vd =
L11 − L12
Ls − Lm
=
C11 − C12
Cs + 2Cm
1
=
( L11 − L12 )(C11 − C12 )
DAC 2001
1
(V1 − V2 )
2
1
Id = ( I1 − I 2 )
2
Vd =
Impedance
1
( Ls − Lm )(Cs + 2Cm )
© SEMCHIP
velocity
69
Even Mode
+1
+1
Ze =
ve =
DAC 2001
Ls + Lm
Cs
1
(Ls + Lm )Cs
© SEMCHIP
Impedance
velocity
70
Odd Mode
-1
+1
Zd =
vd =
DAC 2001
Ls − Lm
Cs + 2Cm
Impedance
1
(Ls − Lm )(Cs + 2Cm )
velocity
© SEMCHIP
71
PHYSICAL SIGNIFICANCE OF EVEN- AND
ODD-MODE IMPEDANCES
* Ze and Zd are the wave resistance seen by the even
and odd mode travelling signals respectively.
* The impedance of each line is no longer described
by a single characteristic impedance; instead, we have
V1 = Z11 I1 + Z12 I 2
V2 = Z21 I1 + Z22 I2
DAC 2001
© SEMCHIP
72
EVEN AND ODD-MODE IMPEDANCES
Z11, Z22 : Self Impedances
Z12, Z21 : Mutual Impedances
For symmetrical lines,
Z11 = Z22 and Z12 = Z21
DAC 2001
© SEMCHIP
73
EXAMPLE
(Microstrip)
Single Line
Dielectric height = 6 mils
Width = 8 mils
εr = 4.3
Zs = 56.4 Ω
Coupled Lines
Height = 6 mils
Width = 8 mils
Spacing = 12 mils
εr = 4.3
Ze = 68.1 Ω Zd = 40.8 Ω
Z11 = 54.4 Ω Z12 = 13.6 Ω
DAC 2001
© SEMCHIP
74
Even Mode
Vf
Zg
I1
IT
line 1
Zg
+
Vb VT
step
- I2
generator
coaxial line
line 2
reference plane tied to ground
 a (t ,0) ad (t ,0)   ae (t ,0) ad (t ,0) 
I tdr =  e
+
−
+ Z
Z
Z
Z d 
e
d
e

 
Vtdr = ae (t ,0) − ad (t ,0)
Vtdr Z e
=
2
I tdr
a d (t ,0) = 0
1 + ρe
) Zg
1 − ρe
Ze = 2(
DAC 2001
ve =
2l
τe
© SEMCHIP
75
Odd Mode
Zg
Vf
Zg
step
generator
Vb
IT
+
V
-T
I1
line 1
line 2
I2
reference plane floating
coaxial line
Vtdr = a e (t,0)+a d (t,0)- [a e (t,0)-a d (t,0) ] = V f + Vb
 a (t ,0) ad (t ,0) 
I tdr =  e
+
Z
Z d 
e

ae (t,0) = 0,
1  1 + ρd
Zd = 
2  1 − ρd
DAC 2001
 a (t ,0) ad (t ,0) 
I tdr =-  e
−
Z d 
 Ze
Vtdr
= 2Zd
Itdr

 Zg ,

© SEMCHIP
vd =
2l
τd
76
EXTRACT INDUCTANCE AND
CAPACITANCE COEFFICIENTS
Ls =
1  Ze Zd 
+
2  ve
vd 
Cs =
1
Z e ve
Lm =
1  Ze Zd 
2  ve vd 
Cm =
1 1
1 
2  Z e ve Z d vd 
Z d < Zs < Z e
DAC 2001
© SEMCHIP
77
Measured even-mode impedance
Even-Mode Impedance
120
h=3 mils
h=5 mils
h=7 mils
100
h=10 mils
Z ( Ω)
e
80
h=14 mils
h=21 mils
60
40
20
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
78
Measured odd-mode impedance
h=3 mils
Odd-Mode Impedance
h=5 mils
50
h=7 mils
45
h=10 mils
h=14 mils
40
h=21 mils
Z ( Ω)
d
35
30
25
20
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
79
Measured even-mode velocity
Even-Mode velocity
0.17
h=3 mils
h=5 mils
0.168
h=7 mils
h=10 mils
0.166
v e(m/ns)
h=14 mils
0.164
h=21 mils
0.162
0.16
0.158
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
80
Measured odd-mode velocity
h=3 mils
h=5 mils
Odd-Mode Velocity
0.21
h=7 mils
h=10 mils
0.205
h=14 mils
0.2
h=21 mils
0.195
v (m/ns)
d
0.19
0.185
0.18
0.175
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
81
Measured mutual inductance
Mutual Inductance
h=3 mils
h=5 mils
250
h=7 mils
h=10 mils
200
h=14 mils
h=21 mils
100
L
m
(nH/m)
150
50
0
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
82
Measured mutual capacitance
h=3 mils
Mutual Capacitance
h=5 mils
40
h=7 mils
h=10 mils
35
h=14 mils
h=21 mils
25
20
m
C (pF/m)
30
15
10
4
6
8
10
12
14
16
18
Spacing (mils)
DAC 2001
© SEMCHIP
83
Modal Velocities in Stripline and Microstrip
Microstrip : Inhomogeneous
structure, odd and even-mode velocities
must have different values.
DAC 2001
© SEMCHIP
Stripline : Homogeneous configuration,
odd and even-mode velocities have
approximately the same values.
84
Microstrip vs Stripline
50 Ω
50 Ω
Microstrip (h =8 mils)
w = 8 mils
εr = 4.32
Ls = 377 nH/m
Cs = 82 pF/m
Lm = 131 nH/m
Cm = 23 pF/m
ve = 0.155 m/ns
vd = 0.178 m/ns
Stripline (h =16 mils)
w = 8 mils
εr = 4.32
Ls = 466 nH/m
Cs = 86 pF/m
Lm = 109 nH/m
Cm = 26 pF/m
ve = 0.142 m/ns
vd = 0.142 m/ns
DAC 2001
© SEMCHIP
85
Microstrip vs Stripline
50 Ω
50 Ω
Probe
Sense line response at near end
C
C
stripline
microstrip
0.3
0.4
0.3
0.2
0.2
0.1
Volts
Volts
0.1
0
-0.1
0
-0.1
-0.2
-0.2
-0.3
-0.4
DAC 2001
-0.3
© SEMCHIP
86
General Solution for Voltages
ZL1
Zs1
line 1
line 2
Zs2
ZL2
−
jω z
ve
+
jω z
ve
−
jω z
vd
+
jω z
vd
+
jω z
vd
V1 ( z ) = Ae e
+ Be e
+ Ad e
+ Bd e
even
−
jω z
ve
odd
+
jω z
ve
−
jω z
vd
V2 ( z ) = Ae e
+ Be e
− Ad e
− Bd e
even
DAC 2001
odd
© SEMCHIP
87
General Solution for Currents
ZL1
Zs1
line 1
line 2
Zs2
ZL2
jω z
jω z
jω z
jω z
+
−
+
 1 

1  − ve
ve
vd
vd
I1 ( z ) =  Ae e
A
e
B
e
− Be e
+
−

 d

d
Z e 
 Z d 


even
odd
jω z
jω z
jω z
jω z
+
−
+
 1 

1  − ve
ve
vd
I 2 ( z ) =  Ae e
− Be e
− Bd e vd 
−
 Ad e
Z e 
 Z d 


even
DAC 2001
odd
© SEMCHIP
88
Coupling of Modes
(asymmetric load)
ZL1
Zs1
line 1
line 2
Zs2
ZL2
even
odd
even
odd
even
even
odd
even
odd
even
odd
even
odd
odd
First reflection
DAC 2001
Second reflection
© SEMCHIP
89
Coupling of Modes
(symmetric load)
ZL
Zs
line 1
line 2
Zs
ZL
even
even
even
odd
odd
odd
First reflection
DAC 2001
Second reflection
© SEMCHIP
90
Sense Line at Near End
Drive Line at Near End
0.2
1
0.15
0.8
0.1
Volts
Volts
0.6
0.05
0.4
0
-0.05
0.2
-0.1
0
-0.15
-0.2
0
0
5
10
15
20
25
30
35
5
10
15
20
25
30
35
40
40
Time (ns)
Time (ns)
DAC 2001
© SEMCHIP
91
TELGRAPHER’S EQUATION FOR N
COUPLED TRANSMISSION LINES
z=0
z=l
V1(z)
V2(z)
...
V3(z)
L, C
−
∂V
∂I
=L
∂z
∂t
−
∂I
∂V
=C
∂z
∂t
V and I are the line voltage and line current VECTORS
respectively (dimension n).
DAC 2001
© SEMCHIP
92
N-LINE SYSTEM
V1 
V 
 2
V = ⋅ 
 
⋅
Vn 
z=0
z=l
V1(z)
V2(z)
...
V3(z)
L, C
 I1 
I 
 2
I = ⋅ 
 
⋅
 I n 
L and C are the inductance and capacitance MATRICES respectively
 L11
L
L =  21
 ⋅

 ⋅
L12 ⋅ ⋅ 
L22 ⋅ ⋅ 

⋅ ⋅ ⋅ 

⋅ ⋅ Lnn 
DAC 2001
 C11 C12
C
C22
C =  21
⋅
 ⋅

⋅
 ⋅
⋅ ⋅ 
⋅ ⋅ 

⋅ ⋅ 

⋅ Cnn 
© SEMCHIP
93
COUPLED LOSSY TRANSMISSION LINES
z=0
z=l
V1(z)
V2(z)
.
...
V3(z)
R, L, G, C
Time Domain
−
∂V
∂I
= RI + L
∂z
∂t
−
DAC 2001
∂I
∂V
= GV + C
∂z
∂t
© SEMCHIP
94
COUPLED LOSSY TRANSMISSION LINES
z=0
z=l
V1(z)
V2(z)
.
...
V3(z)
R, L, G, C
Frequency Domain
∂V
= ZI
∂z
∂I
− = YV
∂z
−
Y = G + jω C
Z = R + jω L
DAC 2001
© SEMCHIP
95
7-Line Coupled-Microstrip System
ALS04
ALS240
Drive Line 1
ALS04
Drive Line 2
ALS04
Drive Line 3
ALS240
ALS240
Sense Line 4
ALS04
Drive Line 5
ALS04
Drive Line 6
ALS04
ALS240
ALS240
ALS240
Drive Line 7
z=l
z=0
Ls = 312 nH/m; Cs = 100 pF/m;
Lm = 85 nH/m;
DAC 2001
© SEMCHIP
Cm = 12 pF/m.
96
Drive Line 3
Drive line 3 at Near End
Drive Line 3 at Far End
5
5
4
4
3
3
2
2
1
1
0
0
-1
0
100
Time (ns)
200
0
100
200
Time (ns)
DAC 2001
© SEMCHIP
97
Sense Line
Sense Line at Near End
2
2
Sense Line at Far End
1
1
0
0
-1
-1
0
100
200
Time (ns)
DAC 2001
0
100
200
Time (ns)
© SEMCHIP
98