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![Page 1: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/1.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Interpretation of the exact results of cross-talk analysisWe are going to consider some particular cases that help understandingthe phenomenon of cross-talk
Assumption 1) electrically short line
Assumption 2) weakly coupling between generator and receptor circuits
1kWhen ,
When , we obtain L
)1)(1(Den RG jj
(1)
(2)
1C and 1S
![Page 2: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/2.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
As a consequence of approximations 1) and 2) the exact results for thenear-end and far-end voltages become:
DCDC Gm
FENE
FENEGm
FENE
NENE VLcj
RR
RRILlj
RR
RV ˆˆ
Den
1ˆ
DCDC Gm
FENE
FENEGm
FENE
FEFE VLcj
RR
RRILlj
RR
RV ˆˆ
Den
1ˆ
(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 (inductive coupling)and the mutual capacitance (capacitive coupling).
ml
mc
![Page 3: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/3.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Assumption 3) sufficiently small frequency
1Den , which yields
DCDC Gm
FENE
FENEGm
FENE
NENE VLcj
RR
RRILlj
RR
RV ˆˆˆ
DCDC Gm
FENE
FENEGm
FENE
FEFE VLcj
RR
RRILlj
RR
RV ˆˆˆ
(3)
(4)
![Page 4: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/4.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
This result can be obtained from the following equivalent circuit.
Observe that: 1) The voltage source represents the emf induced in the receptor circuit, according to Faraday’s law, where is the total mutual inductance between the generator and receptor circuits. 2) The current source represent the charge induced in the receptor circuit, where is the total mutual capacitance between the generator and receptor circuits.
DCGm ILlj ˆ
Llm
DCGm VLcj ˆLcm
Figure 1
![Page 5: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/5.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
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, highvoltages).
Inductive coupling dominates in ifNEV
DCGmFENE
NEINDNE
CR
FE
CG
L ILljRR
RV
Z
R
Z
R ˆˆ1
and in ifFEV
DCGmFENE
FEINDFE
CR
NE
CG
L ILljRR
RV
Z
R
Z
R ˆˆ1
(notice that the inductive coupling components are opposite in sign anddifferent in magnitude)
(7)
(8)
![Page 6: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/6.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Capacitive coupling dominates in ifNEV
DCGmFENE
FENECAPNE
CR
FE
CG
L VLcjRR
RRV
Z
R
Z
R ˆˆ1
and in ifFEV
DCGmFENE
FENECAPFE
CR
NE
CG
L VLcjRR
RRV
Z
R
Z
R ˆˆ1
(notice that the inductive coupling components have the same sign andare equal in magnitude)
(9)
(10)
The total coupling is the sum of the individual components, hence
CAPFE
INDFEFE
CAPNE
INDNENE
VVV
VVV
ˆˆˆ
ˆˆˆ
(11)
(12)
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.
![Page 7: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/7.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Cross-talk Transfer FunctionsCross-talk can be viewed as transfer function between the input voltage and the output voltages and . The corresponding transferfunctions are:sV NEV FEV
CAPNE
INDNE
s
NE MMjV
V
ˆ
ˆ
where
Ls
m
FENE
NEINDNE RR
L
RR
RM
Ls
mm
FENE
FENECAPNE RR
CL
RR
RRM
Ls
m
FENE
FEINDFE RR
L
RR
RM
CAPFE
INDFE
s
FE MMjV
V
ˆ
ˆ
CAPNE
CAPFE MM
(13)
(14)
(15)
(16)
(17)
(18)
![Page 8: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/8.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Effect of losses - Common - impedance couplingThe 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 lowerfrequencies. Let us refer to the following figure:
Usually the resistance of the reference conductor is such that
so most of the current comes back through the reference conductor.LFE RRRR 00 ,
GI
(19)
Figure 2
![Page 9: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/9.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
Now, the current through the reference conductor produces a voltage drop given by:
0V
sLs
G VRR
RIRV ˆˆ 0
00
where
LrR 00
lumps the total resistance of the reference conductor. The voltage appears in the receptor circuit producing contributions to the transfer functions given by:
LsFENE
NECINE
s
CINE
RR
R
RR
RM
V
V
0
ˆ
ˆ
LsFENE
FECIFE
s
CIFE
RR
R
RR
RM
V
V
0
ˆ
ˆ
(20)
(21)
(22)
(23)
CIFENEV ,
ˆ
![Page 10: University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 Interpretation.](https://reader036.fdocuments.us/reader036/viewer/2022082418/5697bfd61a28abf838cadd2f/html5/thumbnails/10.jpg)
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
University of Illinois at Chicago
ECE 423, Dr. D. Erricolo, Lecture 20
One should observe that the common-impedance coupling provides afrequency independent floor as shown in the following diagram.
Finally, the total coupling is approximately the sum the three contributions considered so far:
CINE
CAPNE
INDNE
s
NE MMMjV
V
ˆ
ˆ
CIFE
CAPFE
INDFE
s
FE MMMjV
V
ˆ
ˆ
(24)
(25)
Figure 3