Crosstalk between two terminated lines (R_T=∞,R_S= Z₀)

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An active line-1 and a passive line-2 are terminated by R_S at near end, and are open at far end.

The voltages at both ends of active line are:

The voltages at both ends of passive line are:

where

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

The pulse induced on passive line is called backward crosstalk or near-end crosstalk (NEXT).

Focusing on the first wave (n=1), the voltage on passive line is:

where ξ is the crosstalk coefficient and r_s is the normalized source resistance:

The amplitude of backward crosstalk saturates at the line length:

In the case of a parallel data transmission or a source synchronous clocking, transmission lines are identical. We will calculate the crosstalk between two identical lines.

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The voltages of line-1,2 will be derived from the voltages of even and odd mode. (→Multiple reflection in terminated coupled-lines)

where

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For simplicity, we assume the conditions below:

(1) The far-end impedance is HiZ: R_T=∞ i.e. Γ_T+=Γ_T−=1

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

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

Active line:

Passive line:

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

with the crosstalk coefficient ξ and the normalized resistance r_S:

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

The pulse induced on passive line is called backward crosstalk or near-end crosstalk (NEXT).

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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 identities

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

Active line:

Passive line:

or, explicitly

Active line:

Passive line:

Note that the amplitude v₂(l,t) is twice as large as v₂(0,t) because of the total reflection (Γ_T+=Γ_T−=1).

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

Because of 0≦ξ＜1, its polarity is the same as the input voltage. And the amplitude is almost proportional to the crosstalk coefficient ξ.

Especially, when the source resistance matches the characteristic impedance:

, the first wave on passive line is:

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If the propagation time 2τ is smaller than the rising time t_r, the amplitude of crosstalk becomes small.

Conversely, the amplitude of backward crosstalk saturates at the line length:

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