University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at...

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





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





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)





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





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)





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.





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)





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





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





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

University of Illinois at Chicago ECE 423, Dr. D. Erricolo, Lecture 20 University of Illinois at...

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