termination techniques for high-speed digital...
TRANSCRIPT
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Crosstalk
Tzong-Lin Wu, Ph.D.
Department of Electrical Engineering
National Taiwan University
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Topics:
Introduction
Transmission lines equations for coupled lines
symmetrical lines and symmetrically driven
homogeneous medium
symmetrical lines in homogeneous medium
symmetrical lines and asymmetrical driven
Qualitative description of crosstalk
Terminations
Measurement of crosstalk parameters
How crosstalk noise can be reduced
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Introduction
Three characteristics: 1. Polarity of noise at near end
is opposite to that at the far end. 2. Noise duration at near end is
larger than that at the far end. 3. Noise magnitude at near end
is smaller than that at the far end.
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Introduction : Physical Geometry of Coupled Lines
Two kind of coupling: 1. Inductive coupling 2. Capacitive coupling
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Introduction: cross-talk mechanism of the inductive coupling
Qualitative understanding of the coupling mechanism
• How about the cross-talk mechanism of the capacitive coupling ??
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Lm Cm Lm Lm Lm Cm Cm
A B
C D
Driving signal
Forward
coupling
Reverse
coupling
Tp
Tr
2Tp
A
B
D
C
Tp
Tr
2Tp
A
B
D
C Rise and fall times
same as input
Derivative of input
signal (negative)
Total coupled areas
are the same
Inductive Capacitive
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Coupled lines equations
V z V z z L zt
I z L zt
I z
I z I z z CtV z z C
tV z z V z z
m
m
1 1 1 1 2
1 1 1 1 1 2
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) [ ( ) ( )]'
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Coupled lines equations
dV z
dzL
tI z
dI z
dzC
tV z
( )( )
( )( )
Ll l
l l
L L
L L
Cc c
c c
C C C
C C C
C C
C C
m
m
m m
m m
m
m
LNM
OQP LNM
OQP
LNM
OQP
LNM
OQP
LNM
OQP
11 12
21 22
1
2
11 21
12 22
1
2
1
2
'
'
• Matrix forms
where
L 1, 2 : self-inductance per unit length of line 1 and 2 when isolated
Lm : mutual inductance between line 1 and 2
C’ 1,2 : capacitance (to GND) of line 1 and 2 when isolated **
Cm : mutual capacitance between line 1 and 2
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Coupled lines equations
11 12
21 22
Capacitance matrix=C C
C C
11 1 12gC C C
11 12
21 22
inductance matrix=L L
L L
11 12 1
21 22
1
Capacitance matrix=
N
N NN
C C C
C C
C C
11 12 1
21 22
1
Inductance matrix=
N
N NN
L L L
L L
L L
For multi-conductor coupled lines
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Coupled lines equations
1 10
2 20
1 10
2 20
( )
( )
( )
( )
j t z
j t z
j t z
j t z
V z V e
V z V e
I z I e
I z I e
0 0
0 0
( ) ( )
( ) ( )
V z j LI z
I z j CV z
2
02( ) 0A V
1 1 2 1
1 2 2 2
[ ]m m m m
m m m m
L C L C L C L CA LC
L C L C L C L C
2
2det( ) 0A
The γcan be solved with two possible solutions
for nontrivial solutions
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Symmetrical lines and symmetrical driven
( )( )L L C Cm m1 1
Conditions:
• L1 = L2 (balanced lines)
C1 = C2
• The balanced lines are driven by
common mode or differential mode only
Characteristics:
• Propagation delay per-unit length:
• Characteristic impedance
1
0 1
1
10 1
1
1
1
1
1
m me
m c
m mo
m c
L L kZ Z
C C k
L L kZ Z
C C k
where
magnetic coupling coefficient
capacitive coupling coefficient
kL
L
kC
C
mm
cm
1
1
:
:
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Symmetrical lines and symmetrical driven
Z e0 ,
Z o0 ,
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Symmetrical lines and symmetrical driven (Microstrip example)
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Symmetrical lines and symmetrical driven (another definition for even
and odd mode)
Note:
Coupling Factor = 20log(Z12)
Z12 = (Ze – Zo)/(Ze + Zo)
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Physical concept for the even-mode and odd mode impedance
Odd mode I1 = -I2 and V1 = -V2
odd 1 112g m mC C C C C 11 11 12odd mL L L L L
odd 11 12odd
odd 11 12
L L LZ
C C C
odd odd odd 11 12 11 12TD L C L L C C
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Physical concept for the even-mode and odd mode impedance
Even mode I1 = I2 and V1 = V2
even 11 mL L L even 0 11 mC C C C
even 11 12even
even 11 12
=L L L
ZC C C
odd even even 11 12 11 12TD L C L L C C
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Impedance trends for the even- and odd-modes
What impedance behaviors do you see ?
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Impedance trends for the even- and odd-modes
1. High-impedance traces exhibit significantly more impedance variation than do the lower-impedance trace because the reference planes are much farther in relation to the signal spacing.
2. At smaller spacing, the single-line impedance is lower than the target. This is because the adjacent traces increase the self-capacitance of the trace and effectively lower its impedance even when they are not active.
3. Higher coupling the traces behave, the larger variation even- and odd- impedances will have.
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Mutual coupling trends for the microstrip and stripline
Mutual parasitic fall off exponentially with trace-trace spacing
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Example by simulation:
Please compare the coupling characteristics for the three cases: • Ze • Zo
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Effect of switching patterns on transmission line
Crosstalk induced flight time and signal integrity variations
Switching pattern
What phenomena do you see ??
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As the switching patterns are in even-mode:
Over-driven behavior is seen (Overshooting).
Longer propagation delay is seen.
As the switching patters are in odd-mode:
Under-driven behavior is seen (Undershooting).
Shorter propagation delay is seen.
Why ??
Effect of switching patterns on transmission line
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Effect of switching patterns on transmission line
• Even-mode pattern impedance changed from Z0 to Ze > Z0 . overdriven when Ze > Zs. Slower propagation velocity.
• Odd-mode pattern impedance changed from Z0 to Zodd < Z0 . under driven when Zodd < Zs. Faster propagation velocity.
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Effect of switching patterns on transmission line
Simulating traces in a multiconductor system using a single-line equivalent model (SLEM)
Target trace is line 2. Please find the equivalent impedance and time delay
22 12 232,eff
22 12 23
L L LZ
C C C
2,eff 22 12 23 22 12 23TD L L L C C C
??
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Effect of switching patterns on transmission line
22 12 232,eff
22 12 23
L L LZ
C C C
2,eff 22 12 23 22 12 23TD L L L C C C
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Example by TDR measurement:
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Example by TDR measurement: question one ??
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Example by TDR measurement: question one??
Differential impedance (D.I.) is 150Ω without GND plane And about 100Ωwith GND plane. Why ??
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First half: two traces refer each other like a twisted line, L↑ and C↓
Second half: two traces refer the GND plane, mutual coupling is quite small (km ≒ kc ≒ 0), so D.I.=2*Zodd ≒ 2Z0=100
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Example by TDR measurement: question two??
What’s the variation of the differential impedance
For the differential pair crossing the slot ?
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Example by TDR measurement: question two??
The gap behaves like a high impedance discontinuity which Will cause reflections.
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Homogeneous Medium
Conditions:
• Real (not Quasi-) TEM mode propagating
on the transmission lines
Characteristics:
• Propagation delay per-unit length:
2
0
1 2
1 2
1
where (Coupling coefficient)m c
mm
mc
k
k k k
Lk
L L
Ck
C C
: For mutual inductance coupling
: For mutual capacitance coupling
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Symmetrical Lines on Homogeneous Medium
Conditions:
• L1 = L2 (balanced lines)
C1 = C2
• The balanced lines are driven by
common mode or differential mode only
•
Characteristics:
• Characteristic impedance
Z Z Z
ZL
C
e o0 0 1
2
11
1
where
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Example: homogeneous coupled line
Please compare the coupling characteristics for the three cases: 1. K 2. Ze 3. Zo
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A Quiz:
Please compare their Ze, Zo, Ve, Vo, and C.
(a) (b)
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Symmetrical Lines and asymmetrically driven
R R Z
RZ Z
Z Z
Z R R
e
e o
e o
o
1 2 0
30 0
0 0
0 1 3
2
1
2
(terminate even mode)
terminate odd mode)(
( ( / / ))
P-Terminations:
V V V
V V V
05
05
1 2
1 2
. ( )
. ( )
Asymmetrical driven
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Symmetrical Lines and asymmetrically driven
R R Z
R Z Z
R Z
o
e o
e
1 2 0
3 0 0
3 0
1
2
2
(terminate odd mode)
(terminate even mode)
( R 2
( )
)
T - terminations
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Imperfect Termination for Differential Transmission Line
Bridge Termination
AC Termination
Single ended termination
1. What are their values of the resistance? 2. What are their advantages and disadvantages ?
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Imperfect Termination for Differential Transmission Line
Bridge Termination
AC Termination
Single ended termination
2Zo
Zo
Zo Zo
Zo
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Imperfect Termination for Differential Transmission Line
Bridge: 1. Odd mode is completely absorbed 2. Even mode is completely reflected 3. Only one resistance is used. Single-ended 1. Odd mode is completely absorbed 2. Even mode has the reflection coefficient T= (Zo–Ze)/(Zo+Ze) 3. The extra damping of the even mode costs an additional resistance. AC • Odd mode is completely absorbed • Even mode is completely reflected at low frequency, but has the reflection T = (Zo–Ze)/(Zo+Ze) at high frequency.
3. Compared to bridge type, even mode is attenuated without increasing static power dissipation. 4. Needing three components.
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Example: Termination for Differential Transmission Line
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Example: Termination for Differential Transmission Line
A simple bridge termination
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Example: Termination for Differential Transmission Line
Differential signal is still good
Common signal on each line is oscillating
A simple bridge termination, but Unbalance driving
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Example: Termination for Differential Transmission Line
Common-mode is damped
A Pi-termination for unbalance driving
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Example: Termination for Differential Transmission Line
Common-mode resonance Differential pair excited by common-mode signal
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Example: Termination for Differential Transmission Line
Single-ended termination for Unbalanced driving
Better than bridge termination
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Example: Termination for Differential Transmission Line
AC T-termination for Unbalanced driving
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Example: Termination for Differential Transmission Line
Bridge termination at the far end and Series termination at the near end
1. Power saving 2. Idea of LVDS
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Example: Termination for Differential Transmission Line
Bridge termination at the far end and Series termination at the near end For LVDS case
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Example: Effects of Termination Networks on Signal-Induced EMI from
the Shields of Fibre Channel Cables Operating in the Gb/s Regime
2000 IEEE EMC Symposium
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Example: Effects of Termination Networks on Signal-Induced EMI from
the Shields of Fibre Channel Cables Operating in the Gb/s Regime
Measurement of S11 for both differential mode and common mode
Termination effect is reduced for high frequency range
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Example: Effects of Termination Networks on Signal-Induced EMI from
the Shields of Fibre Channel Cables Operating in the Gb/s Regime
differential
Why the termination effect is degraded at high frequency range ?
EMI
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Quantitative description of crosstalk
Capacitive coupling Inductive Coupling
V x t K xd
dtV t xf f in( , ) ( )
V x t K V t x V t x Tb b in in d( , ) [ ( ) ( )] 2
KL
ZC Z
KL
ZC Z
fm
m
bm
m
1
2
1
4
1
1
1
1
( )
( )
(ns / cm) : Forward coupling coefficient
(dimensionless) : Backward coupling coefficient
If the coupled lines are symmetrical
(see appendix 7.1)
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Quantitative description of crosstalk
In the loosely coupled system (km, kc << 1)
ZL
CL C1
1
1
1 1 and is accurate enough
KL
Z
C Zk k
KL
ZC Z k k
fm m
m c
bm
m m c
2 2
1
4
1
4
1
1
1
1
( ) ( )
( ) ( )
1. It is clear that the backward coupling can be reduced by decreasing the mutual coupling (Lm and Cm) or by increasing the coupling to ground(L1, L2, C1, and C2).
2. The forward crosstalk is zero for two identical lines in homogeneous medium. (not easy in practical circuits)
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Crosstalk for a ramp step driver
V t V f t f t Tin m r( ) [ ( ) ( )]
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Crosstalk for a ramp step driver
Forward crosstalk at the far end:
V l t K ld
dtV t T
K lV
TT t T T
f f in d
fm
r
d d r
( , ) ( )
= ,
= 0 elsewhere
Backward crosstalk at the near end
• If Tr < 2Td
V t K V t V t Tb b in in d( , ) [ ( ) ( )]0 2
V Kb b,max
• If Tr > 2Td
Vl
TK V
K
Tb
r
b mb
r
,max ( ) 2
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SPICE simulation of crosstalk
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Measurement of coupling coefficient (Frequency domain)
: (example) RAMBUS RIMM Connector
•Test module design
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Measurement of coupling coefficient (Frequency domain)
: RAMBUS RIMM Connector
• Calibration (Open, short, load )
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Measurement of coupling coefficient (Frequency domain)
: (example) RAMBUS RIMM Connector
• Theory (mutual capacitance)
Low freq. assumption
The slope of S21(f) at low frequency is taking to compute the Cm
Open circuit !!
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Measurement of coupling coefficient (Frequency domain)
: (example) RAMBUS RIMM Connector
• Theory (mutual inductance)
Short circuit !!
The slope of S21(f) at low frequency is taking to compute the Lm
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Measurement of coupling coefficient (Frequency domain)
: (example) RAMBUS RIMM Connector
• Measurement setup
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Differential Impedance Measurement by TDR/TDT
They have good EMS (Immunity) and EMI (Interference) Performance.
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Differential Impedance Measurement by TDR/TDT
Equivalent circuits of the differential interconnects: (Type I)
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Differential Impedance Measurement by TDR/TDT
Equivalent circuits of the differential interconnects: (Type II)
Advantage is it is a rigorous equivalent circuit. Disadvantage is only valid in three conductor interconnects.
Negative impedance (not practical)
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Differential Impedance Measurement by TDR/TDT
Equivalent circuits of the differential interconnects: (Type III)
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Differential Impedance Measurement by TDR/TDT
Measurement Principle
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Differential Impedance Measurement by TDR/TDT
Comparison of the equivalent model by the SPICE
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A Final Question:
Can you explain why
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How to reduce crosstalk noise
• Space out signal routing
• Route-void channels adjacent to critical nets
• Increase coupling to ground
• Run orthogonal rather than parallel
• Provide the homogeneous medium
• Using the slowest risetime and falling time
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§10.1 Three conductor lines and crosstalk
Time Domain Crosstalk VS Frequency-domain Crosstalk
a. Two conductors transmission lines has no crosstalk problem. In order to have crosstalk, we need to have three or more conductors.
b.
)(tVs
NERFER
SRLR
_NEV
_FEV
Generator
Receptor
),( tzIG
),( tzIR
z0z Lz
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c. The generator circuit consists of the generator conductor and reference conductor and has current IG(z,t) along the conductors and voltage VG(z,t) between them. The IG and VG will generate EM fields and interact with the receptor circuit.
d. The objective in a crosstalk analysis is to determine the near-end voltage VNE(t) and far-end voltage VFE(t) given the cross-sectional dimension and the termination characteristics VS(t), RS, RL, RNE, RFE.
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e. Two type of crosstalk analysis
(1) Time-domain crosstalk :
(2) Frequency-domain crosstalk :
f. (1) For simplicity, we will assume that the medium surrounding the
conductors for these configurations is homogeneous.
(2) Closed-form expressions for the per-unit-length parameters for the lines in inhomogeneous medium, such as microstrip lines, are difficult to determine.
).( )( )( tVgiventVandtVpredict SNEFE
).cos()( )(ˆ )(ˆ wtVtVgivenjwVandjwVpredict SSNEFE
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Transmission Line Equations
),( tzIG
),( tzIR
),(),( tzItzI RG
zRG
zRR
zRO
zLG
zLRzGm
zGR zCR
zCm zGG zCG
)( zzIG
)( zzIR
_
)( zzVR
_
)( zzVG
z
a. Assume only TEM mode present on the materials, so line voltages
VG(z,t) and VR(z,t), as well as line currents IG(z,t) and IR(z,t) can be
uniquely defined.
b. Equivalent circuit of TEM-mode on 3-conductor material :
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In matrix terms :
t
tzVCC
t
tzVCtzVGGtzVG
z
tzI
t
tzVC
t
tzVCCtzVGtzVGG
z
tzI
t
tzIL
t
tzILtzIRRtzIR
z
tzV
t
tzIL
t
tzILtzIRtzIRR
z
tzV
RmR
GmRmRGm
R
Rm
GmGRmGmG
G
RR
GmRORGO
R
Rm
GGROGOG
G
),(),(),(),(
),(
),(),(),(),(
),(
),(),(),(),(
),(
),(),(),(),(
),(
t
tztz
z
tz
t
tztz
z
tz
),(),(
),(
),(),(
),(
VCGV
I
ILRI
V
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c. Solutions
(1)In time domain, it is a difficult problem.
When R=G=0 (lossless lines), there are exact solutions in SPICE.
(2)In frequency domain, the exact solutions for above material equations are possible, we will show the exact solution for lossless case.
ORO
OOG
R
G
R
G
RRR
RRR
tzI
tzItz
tzV
tzVtz
where
,
),(
),(),( ,
),(
),(),(
RIV
, ,
G m G m m G m m
m R m R m m R m
L L G G G C C C
L L G G G C C C
L G C
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Note: In frequency domain, the equations =>
jwt
jwt
eztz
eztz
where
zdz
zd
zdz
zd
)(ˆRe),(
)(ˆRe),(
jwˆ
jwˆ
)(ˆˆ)(ˆ
)(ˆˆ)(ˆ
II
VV
CGY
LRZ
VYI
IZV
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§ 10.3.1 The per-unit-length parameters
Internal parameters : RG, RR, RO and internal inductance are not dependent on the line configurations.
a. For material in homogeneous medium (μ,ε)
(1)
2
1 1
2
2
2 2
2 2
2 2
1
Example : Three-conductor line
1
( )
( )
( )
( )
G m m R m
m R m m GG R m
mm
G R m
RG m
G R m
GR m
G R m
v
C C C L L
C C C L Lv L L L
LC
v L L L
LC C
v L L L
LC C
v L L L
LC CL 1 C L L
1 G LG GL 1 L(2)
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§ Wide-Separation Approximation for Wires
a. (1) Magnetic flux of a current-carrying wire through a surface
(2) voltage between two points for a charge-carrying wire
1
20 ln2 R
RIμ
1R
2Ra
b
1R
2Rq
baV
1
2
0
ln2 R
RqVba
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b. Parameters of three-conductor transmission lines
(1) Inductance :
00
0
,
RG
GR
IG
R
IR
Gm
IR
RR
oIG
GG
R
G
Rm
mG
R
G
IIL
IL
IL
I
I
LL
LL
LI
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dR
dGR receptorgenerator
GND WOr
WRrWGr
1.
2. Self inductance
WOWR
RR
WOWG
G
WO
G
WG
GG
rr
dL
rr
d
r
d
r
dL
20
2000
ln2
ln2
ln2
ln2
GI
GI
WGr
WOr
G
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3. Mutual inductance
(2) Capacitance
dR
dGR
WOr
WRrWGr
GI
GI
R
WOGR
RG
WO
R
GR
Gm
rd
dd
r
d
d
dL
ln2
ln2
ln2
0
00
R
G
mRm
mG
V
V
CCC
CC
CVq
R
G
Rm
mG
q
q
pp
pP
pqqCV
1
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0000
,
GRRR qR
G
qG
Rm
qR
RR
qG
GG
q
V
q
VP
q
VP
q
VP ,
1. Self capacitance
WOWG
G
WO
G
WG
GG
rr
d
r
d
r
dP
2
0
00
ln2
1
ln2
1ln
2
1
Gq
Gq
WGr
WOr
_V
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86
2. Mutual capacitance
WOGR
RG
WO
R
GR
Gm
rd
dd
r
d
d
dP
ln2
1
ln2
1ln
2
1
0
00
dR
dGR
WOr
WRrWGr
Gq
Gqreference
_RV
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87
2
020
31
3
2010
0
0
00
0
41ln4
ln2
)(;ln2
ln2
2ln
2
2ln
2ln
2
S
hh
S
S
SSS
S
S
S
IL
r
h
h
h
r
h
IL
RG
IG
Rm
WG
G
G
G
WG
G
IG
GG
R
R
c. Example
GI
GI
WGr
WRr
RR
hG
hG
hR
hR
S
S1
S2
S3
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88
§10.1.4 Frequency-Domain (Steady State) Crosstalk
§ General Solution :
a.
b.
CGY LRZ
VYI
I V IZV
jwjwwhere
zdz
zd
zI
zIz
zV
zVzz
dz
zd
R
G
R
G
ˆ,ˆ
)(ˆˆ)(ˆ
)(ˆ
)(ˆ)(ˆ,
)(ˆ
)(ˆ)(ˆ)(ˆˆ
)(ˆ
YZZY
YZII
ZYVV
,
)()(ˆ
)()(ˆ
2
2
2
2
generalin:Note
zdz
zd
zdz
zd
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89
(1)
(2) ZY r
ZY T
rTZYT
T
ˆˆˆ
ˆˆˆ
ˆ0
0ˆˆˆˆˆˆ
:ˆ
2
2
21
ofseigenvaluetheis
ofrseigenvectotheiswhere
eddiagonalizr
r
thatsuchmatrixtiontransformaafindcanweif
R
G
)(ˆ)(ˆ)(ˆ
)(ˆ)(ˆ)(ˆ
)(ˆˆ)(ˆ
)(ˆˆ)(ˆ
)(ˆ
)(ˆ
ˆ0
0ˆ
)(ˆˆˆˆˆ)(ˆ
)(ˆ
)(ˆ)(ˆ)(ˆˆ
)(ˆˆˆˆ)(ˆˆ
ˆˆ
ˆˆ
2
2
2
2
1
2
2
1
11
2
2
zIezIezI
zIezIezI
zIrzI
zIrzI
zI
zI
r
r
zzdz
d
zI
zIentmodal currzzlet
zzdz
d
mRzr
mRzr
mR
mGzr
mGzr
mG
mRRmR
mGGmG
mR
mG
R
G
mm
mR
mG
m
RR
GG
ITZYTI
IIT
IZYTIT
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90
(3)
c.
d.
mR
mG
mzr
zrzr
mzr
mzr
m
I
I
e
ewhere
formmatrixinz
G
G
ˆ
ˆˆ,
0
0
.ˆˆ)(ˆ
ˆ
ˆˆ
ˆˆ
I
e
IeIeI
)IeI(eTTrTZV
rTYrTZ rTZYT
)IeI(erTYIYV
IeIeTITI
mzr
mzr
mzr
mzr
mrz
mrz
m
z
by
zdz
dz
zz
ˆˆˆˆˆˆˆ)(ˆ
ˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆ)(ˆˆ)(ˆ
)ˆˆ(ˆ)(ˆˆ)(ˆ
ˆˆ11
1121
ˆˆ11
1 1 1
1
ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ )
ˆ ˆ ˆ
Z Z Tr T Z( YZ
=Y YZ
c
see below for detail
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91
e. Solving the four unknowns Im+ and Im
- by the
termination condition
(1)
11 1
1 2 1 1 1
1 1 1 1
1 1
11 1
ˆ ˆ ˆ ˆˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ
ˆ ˆ ˆ ˆˆ
ˆ ˆ ˆ ˆˆ
-2
Tr T YZ
T YZT r r T YZTr =1
Tr T YZ Tr T =1
YZ Tr T
Tr T = YZ
上式的
1ˆˆ ˆ ˆ ˆ( )ˆ
similar to the scalar results
zZ Z YZ
y
(0) (0)
( ) ( )
V V Z I
V Z I
S S
LL L
0 0
0 00
V , Z , Z
S LSS S L
NE FE
R RVwhere
R R
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92
(2)
-m m
-m m
- -m m m m
ˆ ˆ -m m
ˆ ˆ -m m
ˆ ˆ ˆ-m m
0
ˆ ˆ ˆ ˆ ˆ(0) ( )
ˆ ˆ ˆ ˆ(0) ( )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )
ˆ ˆ ˆ ˆ ˆ( ) ( )
ˆ ˆ ˆ ˆ( ) ( )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) (
V Z T I I
I T I I
Z T I I V Z T I I
V Z T I I
I T I I
Z T I I Z T I
C
C S S
rL rLC
rL rL
rL rL rLC L
at z
at z L
L e e
L e e
e e e
ˆ -m m
m
-ˆ ˆm
- -m m m m
ˆ )
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
ˆˆ ˆ ˆ ˆ ˆ ˆ 0
ˆ ˆ ˆ ˆ, , .
I
Z Z T Z Z T I V
IZ Z T Z Z T
I I I I
rL
C S C SS
rL rLC L C L
e
e e
and can be solved
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93
§ Exact Solution for Lossless Lines in Homogeneous Media
a. for lossless line :
b. It can be shown that :
2 2 2
ˆ ˆ0 ,
ˆ ˆ
hom .
ˆ ˆ,
R G Z L Y C
YZ LC 1 1
LC 1
T 1 r 1 1
and jw jw
w w diagonal matrix
where for ogeneous transmission lines
wj j
v
2
2
2ˆ ˆ ˆ(0)
1
21 ˆ
1
ˆ ˆ ˆ ˆ( )
NER NE m LG GDC
en NE FE
NE FEm GDC
NE FE LG
NE FEFER FE m GDC m GDC
en NE FE NE FE
j LRS
V V jwL L C S ID R R k
j LR R
jwC L C S VR R k
R RRSV L V jwL LI jwC LV
D R R R R
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94
2 2 2 2
:
1 11
1 1
sin( )cos( ),
( ) , 1
( )
(
SG LR LG SR
en G R G R
SR LR SG LG
m m
G R G m R m
G S LG G m
S L S L
R
where
D C S w k jwCS
LC L S
L
L Ck coupling coefficient k
L L C C C C
L L R Rtime constant C C L
R R R R
time const
) NE FERR m
NE FE NE FE
R RL Lant C C L
R R R R
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95
2
2
ˆˆ ˆ ˆ,
ˆ 1
ˆ 1
, , ,
1 .
1
SLGDC S GDC
S L S L
GCG G
G m
RCR R
R m
S NEL FESG LG SR LR
CG CG CR CR
VRV V I DC excitation at generator circuit
R R R R
LZ vL k
C C
LZ vL k
C C
R RR R
Z Z Z Z
if low impedance load
high
. impedance load
Characteristic impedances
of each circuit in the presence
of the other circuit.
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96
§ Inductive and Capacitive coupling
a. Two reasonable assumptions
(1)
(2)
. . .
cos 1, 1
The line is electrically short i e L
C L
.
1
(1 )(1 )
, 1
en G R
en
The generator and receptor circuits are weakly coupled
k
D jw jw
if w is small then D
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97
b.
(1)
(2)
ˆ ˆ ˆ
ˆ ˆ ˆ
NE NE FENE m GDC m GDC
NE FE NE FE
NE FEFEFE m GDC m GDC
NE FE NE FE
R R RV jwL LI jwC LV
R R R R
R RRV jwL LI jwC LV
R R R R
NERFER
_NEV
_FEV
ˆm GDCjwL LI
ˆm GDCjwC LV
ˆ ˆ( )
ˆ ˆ( ) ( )
.
m GDC m GDC
m GDC m GDC
djwL LI L L I emf
dt
d djwC LV C L V CV
dt dt
which is the independent current source due
to the voltage of generator
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98
(3) Crosstalk is the superposition of two components; One is inductive coupling, the other one is capacitive coupling.
c. Intuitively :
(1) When low-impedance load (high current, low voltage), the
inductive coupling is dominant.
(2) When high-impedance load (low current, high voltage), the
capacitive coupling is dominant.