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## 24 May 2021

### Mode Conversion

 Figure 1: The lower-left and upper-right quadrants of this matrix show the S-parameters that represent mode conversion from differential to common signal, and vice versa.
As said earlier, mixed-mode S-parameters describe the general case of combinations of differential and common signals. When we speak of mode conversion in mixed-mode S-parameters, we are referring to the change of a differential signal into a common signal, or a common signal into a differential signal, as it travels the transmission line. If we look at the matrix of mixed-mode S-parameters in Figure 1, we see that those mixed mode S-parameters affected by such a mode conversion—with a different type of signal going out than what went in—are in the lower-left and upper-right quadrants.

Let’s take the S-parameters SCD11 and SCD21 to see how the combination of single-ended S-parameters they represent reveal the source of mode conversion. If we look at SCD11, the reflected mode conversion, as a function of its single-ended S-parameters, we see:

SCD11 = 0.5*(S11 – S13 + S31 – S33)

Here, there is a differential signal going in on port 1, and common signal coming out on port 1. This S-parameter represents how much common signal comes back when we send a differential signal into the differential pair on port 1.

What causes the common signal?  The equation above actually tells us.

The differential interconnect obeys reciprocity, and we find that S13 and S31 are going to be the same except for some random noise terms. So, these signals are going to cancel out in the equation, leaving us with:

SCD11 = 0.5*(S11– S33)

This means that the mode conversion term is going to be the difference between S11 and S33. This difference is the asymmetry in the two lines that make up the differential pair. What causes mode conversion is asymmetry between the two lines. Specifically, differential to common conversion in the SCD11 reflected common signal is an asymmetry in the reflection coefficient. If the S11 and S33 are the same, that means that S11 and S33 are symmetrical, and there is no converted signal reflecting back.

The same is true for the insertion loss, the transmitted mode conversion:

SCD21 = 0.5*(S21 – S23 +S41 – S43)

This S-parameter represents a differential signal applied to port 1, and a common signal coming out on port 2. So, we send a differential signal in, and we look at how much of that has been converted to common signal coming out.

Again, the equation tells us what single-ended S-parameters terms contribute to the mode conversion. There's the insertion loss of line 1-2 (the P line) minus the insertion loss of line 3-4 (the N line), then there are these mixed cross talk terms S41 and S23. These are the cross talk terms between the P and the N lines. Generally, these terms are going to be exactly the same and cancel out, as well, so we're going to see that the transmitted mode conversion is strictly related to the asymmetry in the insertion loss of the P and the N lines.

SCD21 = 0.5*(S21 – S43)

If they are perfectly symmetric, their difference is zero and there is no mode conversion. One of the most common sources of mode conversion is difference in the phase delay channel in the P and N lines. Any skew would give us some mode conversion.

When the two lines are symmetric, we do not expect to see mode conversion. We can screw up one line as much as we want, but as long as we screw up the other line the same amount, then we will have the same return losses on the two, and there should not be any mode conversion. So, mode conversion is not about discontinuities along the transmission path, it is about asymmetries between two lines.  The same holds for all the other mode conversion S-parameters.

Watch Dr. Eric Bogatin discusses this further in the on-demand webinar, Mixed-Mode S-parameters and TDR Responses.