Crosstalk between two terminated lines (RS=RT=Zg)

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Active line-1 and passive line-2 are terminated by a common resistance R at both ends.

The voltages at both ends of active line are:

The voltages at both ends of passive line are:

where

Especially if R is equal to the geometric average of Z+ and Z|:

then the voltages at both ends of active line are:

and the voltages at both ends of passive line are:

Therefore no crosstalk pulse induced at near end of active line and at far end of passive line.

An example waveform for a rising step input v01(t) with rise time tr=/2:

Focusing on the first wave (n=0), the voltage on passive line is:

where is the crosstalk coefficient:

We will calculate the crosstalk between two identical lines, terminated by a common resistance R at both ends.

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The boundary conditions are:

(1) The termination resistances are equal at both ends : S+=T+=+ , S|=T|=|

(2) The propagation speeds are equal between even and odd mode : +=|= (TEM mode)

Then the voltages on line-1,2 results in more simple form.

Active line:

Passive line:

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Here we assume the special condition that R is equal to the geometric average of Z+ and Z|:

This means that the reflection coefficients + and | are equal in magnitude but opposite in sign:

Note that + can never be equal to | because Z| is always larger than Z+ i.e. +<|.

Substituting +=|| into the equations above:

Active line:

Passive line:

Therefore no crosstalk pulse induced at near end of active line and at far end of passive line.

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We will compute an waveform using these equations. For convenience, we express the reflection coefficient

with the crosstalk coefficient and the normalized resistance r:

Reminding that the R is chosen to equal to the geometric average of Z+ and Z|:

Substituting this resistance, we obtain the reflection coefficients:

Assuming the input voltage v01(t) is a rising step with rise time tr=/2, and rS=1, =0.4, the waveform is:

Note that no crosstalk pulse induced at near end of active line and at far end of passive line.

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For more intuitive understanding, we will simplify the equation above. We assume that the input voltage v01(t) is a rising step.

Using the identity

, the voltages on line-1,2 can be expressed in terms of the amplitude in n< t <m: v01(t-n)-v01(t-m),

Active line:

Passive line:

or, explicitly

Active line:

Passive line:

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Noting ||<1, the amplitude attenuates exponentially with time. Then we will focus on the first wave (n=0) on passive line:

Noting 01, its polarity is the same as the input voltage. And the amplitude is almost proportional to the crosstalk coefficient .


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