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Interpretation of the exact results of cross-talk analysis
We are going to consider some particular cases that help understanding
the phenomenon of cross-talk
Assumption 1) electrically short line
When L   , we obtain
C  1 and S  1
(1)
Assumption 2) weakly coupling between generator and receptor circuits
When k  1 ,
Den  (1  j G )(1  j R )
(2)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
As a consequence of approximations 1) and 2) the exact results for the
near-end and far-end voltages become:
1  RNE
RNE RFE
ˆ
ˆ
ˆ

VNE 
j

l
L
I

j

c
L
V
m
G
m
G
Den  RNE  RFE
RNE  RFE
DC



DC
1 
RFE
RNE RFE
ˆ
ˆ
ˆ

VFE 

j

l
L
I

j

c
L
V
m
G
m
G
Den  RNE  RFE
RNE  RFE
DC
DC
(3)



(4)
This results suggests that (principle of superposition of inductive and
capacitive coupling) the cross-talk is a linear combination of the
contributions due to the mutual inductance lm (inductive coupling)
and the mutual capacitance cm (capacitive coupling).
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Assumption 3) sufficiently small frequency
Den  1 , which yields
 RNE
RNE RFE
ˆ
ˆ

VNE  
j lm L I G 
j cm LVˆG
RNE  RFE
 RNE  RFE
DC



DC

RFE
RNE RFE
ˆ
ˆ

VFE   
j lm L I G 
j cm LVˆG
RNE  RFE
 RNE  RFE
DC
DC
(3)



(4)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
This result can be obtained from the following equivalent circuit.
Figure 1
Observe that:
1) The voltage source j  lm L IˆG represents the emf induced in the
receptor circuit, according to Faraday’s law, where lm L is the total
mutual inductance between the generator and receptor circuits.
2) The current source j  cm LVˆGDC represent the charge induced in
the receptor circuit, where cm L is the total mutual capacitance
between the generator and receptor circuits.
DC
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Therefore, the key to understand cross-talk is the notion of
superposition between inductive coupling and capacitive coupling.
Intuitively, one would expect that inductive coupling dominates for
low-impedance loads (high currents, low voltages) and that capacitive
coupling dominates for high-impedance loads (low currents, high
voltages).
Inductive coupling dominates in VˆNE if
RL RFE
1
Z CG Z CR
RNE
j lm L IˆG
RNE  RFE

IND
VˆNE


IND
VˆFE

(7)
DC
and in VˆFE if
RL RNE
1
Z CG Z CR
RFE
j lm L IˆG
RNE  RFE
DC
(8)
(notice that the inductive coupling components are opposite in sign and
different in magnitude)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Capacitive coupling dominates in VˆNE if
RL RFE
1
Z CG Z CR

RNE RFE
CAP
ˆ
VNE 
j cm LVˆG
RNE  RFE
DC
(9)
and in VˆFE if
RL RNE
RNE RFE
CAP
ˆ
1
 VFE 
j cm LVˆG
(10)
Z CG Z CR
RNE  RFE
(notice that the inductive coupling components have the same sign and
are equal in magnitude)
The total coupling is the sum of the individual components, hence
IND
CAP
(11)
VˆNE  VˆNE
 VˆNE
(12)
Vˆ  Vˆ IND  Vˆ CAP
DC
FE
FE
FE
Notice that the termination impedances can be chosen so that the
capacitive and inductive coupling cancel each other. This is at the basis
of the directional coupler in microwave circuits.
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Cross-talk Transfer Functions
Cross-talk can be viewed as transfer function between the input voltage
Vˆs and the output voltages VˆNE and VˆFE . The corresponding transfer
functions are:
VˆNE
IND
CAP
(13)
 j M NE
 M NE
Vˆs
VˆFE
IND
CAP
(14)
 j M FE
 M FE
Vˆs
where
RNE
Lm
IND
M NE 
(15)
RNE  RFE Rs  RL




M
CAP
NE
RNE RFE LmCm

RNE  RFE Rs  RL
(16)
M
IND
FE
RFE
Lm

RNE  RFE Rs  RL
(17)
CAP
CAP
M FE
 M NE
(18)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Effect of losses - Common - impedance coupling
The assumption of lossless line previously considered is valid for
electrically short lines when the frequency is below 1GHz.
When a conductor is not lossless there may be cross-talk at lower
frequencies. Let us refer to the following figure:
Figure 2
Usually the resistance of the reference conductor is such that
(19)
R0  RFE , R0  RL
so most of the current Iˆ comes back through the reference conductor.
G
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Now, the current through the reference conductor produces a voltage
drop V0 given by:
R0
ˆ
(20)
V0  R0 I G 
Vˆs
Rs  RL
where
R0  r0 L
(21)
lumps the total resistance of the reference conductor.
CI
The voltage VˆNE
, FE appears in the receptor circuit producing
contributions to the transfer functions given by:
CI
VˆNE
RNE
R0
CI
 M NE 
(22)
ˆ
RNE  RFE Rs  RL
Vs
CI
R0
VˆFE
RFE
CI
 M FE  
RNE  RFE Rs  RL
Vˆs
(23)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
One should observe that the common-impedance coupling provides a
frequency independent floor as shown in the following diagram.
Figure 3
Finally, the total coupling is approximately the sum the three
contributions considered so far:
VˆNE
IND
CAP
CI

 j M NE
 M NE
 M NE
(24)
ˆ
Vs
VˆFE
IND
CAP
CI

 j M FE
 M FE
 M FE
(25)
ˆ
University of Illinois at Chicago
Vs
ECE 423, Dr. D. Erricolo, Lecture 20