Interpretation in the Case of the Screened Pair

As both wires are electromagnetically coupled to each other, launching a signal in a wire of a screened pair is exactly like exciting one pendulum in the state described above. The signal will propagate along the pair length and its energy will be transferred back and forth between both wires (see Figure 11).

Figure 11. Signal propagation in a screened single pair when only one wire is excited.

Any excitation mode can be described as a linear combination of the eigenmodes of the system. Assuming a perfect cable, a single wire excitation can be constructed by the linear superposition of the common and differential modes with identical amplitudes. Hence, assuming a perfect connector, by injecting a signal into a single wire, both propagation modes occur with equal amplitudes.

These two modes will then propagate through the pair. Measuring a single wire at some place along the pair will reveal the superposition of the state of these two modes at that position (see Figure 12).

Figure 12. Decomposition of a single wire excitation into common and differential mode signals and their propagation. The propagation speed of both modes is different, so they will not reach the cable end simultaneously.

This can be easily simulated by the sum of two sine waves with different propagation speeds and attenuation, as it is the case for common and differential modes. Doing so, one can perfectly fit the measured curves (see Figure 13).

Figure 13. Comparison between measured S parameters and calculated ones using common and differential mode propagation with different speeds and attenuations.  

Note that using auto mixed mode theory, S_dd21 and S_cc21 (the differential and common mode signals, respectively,) can be calculated from the single wire measurements. These two signals show the expected monotonic power decrease with frequency (see Figure 14).

Figure 14. S_dd21 and S_cc21.

Time Domain Curves: Propagation Speeds

When looking at the calculated TDR of S₂₁, one gets two distinct reflections (see Figure 15). The propagation speeds of differential and common mode can be computed from measured S_cc21 and S_dd21, respectively. The product of propagation speed and cable length results in the exact time where both peaks appear. This corroborates our interpretation of wave propagation based on the superposition of common and differential modes.

Figure 15. Time domain representation of S₂₁ showing dual propagation times of the signal corresponding to common and differential modes. 

Notes

The theoretical description given above corresponds to a situation using perfect connectors and cable. In the real world, the transfer of the signal from the connector to the wire may not be perfect and may thus result in different amplitudes of the differential and common modes. This will lead to S₂₁ and S₄₃ parameter that do not completely reach “zero” amplitude at dips. If the cable is perfect, this property might be used to characterize the connector.

In the real world also, the cable is not ideal and the eigenmodes may deviate slightly from the differential and common modes shown in Figure 8. These new eigenmodes are called “quasi-differential” and “quasi-common modes” by J.Poltz et al.⁴ This will not significantly affect our analysis, except that both modes will not be populated with equal amplitude using a perfect connector.

Eigenmodes are by definition orthogonal and hence propagate without interaction through the cable.

Irregularities along the cable length will however lead to alteration of the quasi-eigenmodes and hence to mode coupling.

For short cable length or low frequencies, when the transfer of power on the second wire remains low, the “usual” description of in-pair skew may still have some relevance. 

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