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## 12 May 2021

### Introduction to Mixed-Mode S-parameters

 Figure 1: Single-ended vs. differential signal "world views" of S-parameters
We’ve treated single-ended S-parameters quite extensively in this blog. Links to several entries are listed at the bottom of this post. Now, we’re going to look at how we go from single-ended to mixed-mode S-parameters and what new information we can find in them. This will come in handy when we start looking at some of the MDI S-parameter tests that are performed for Automotive Ethernet compliance a bit down the road.

With single-ended S-parameters, we look at every combination of ‘going in signals’ and ‘coming out signals’. For example, two single-ended transmission lines and their return paths would yield a four-port S-parameter file. We take the complex ratios of each port combination to obtain the S-parameter value in the form of:

S_(OUT,IN) =  V_OUT/V_IN

The bold typeface indicates complex quantities.

But what happens if we drive two transmission lines with a differential source? Figure 1 compares the single-ended and differential signal world views.

In the single-ended world view, we have four ports. The driving and termination impedances for all the ports are all 50 Ohms. The only things that can change between any to two ports are the magnitude and phase of the signal determined by the characteristics of the path.

For single-ended S-parameters, we used index numbers to identify these inputs and outputs. So S11 indicates the signal going in on port 1 and coming out on port 1, while S21 indicates the signal going in on port 1 and coming out on port 2 (remember, the “out” port is always listed first).

In the differential world view, there are only two ports. The ports are labeled to reflect that we have a single differential port in and a single differential port out. Any waveform applied to either of the differential ports can be described by a combination of differential and/or common signals.  The differential signals are composed of two, out-of-phase components on each element of the differential pair. The common signals have components on each of the differential elements, which are in phase.

We use the term mixed-mode S-parameters to describe the general case of combinations of differential and common signals. Sometimes, in the special case when we have only differential signals going in and out, we can use the term differential S-parameters.

A signal applied to a differential pair can interact with the differential pair in any of four possible ways:

1.  A differential signal can be applied to a port and be output as a differential signal.
2.  A common signal can be applied to a port and be output as a common signal.
3.  A differential signal can be applied to a port be output as a common signal.
4.  A common signal can be applied to a port and be output as a common signal.

 Figure 2: Mixed-mode S-parameter nomenclature.
Therefore, for mixed mode S-parameters, we need a nomenclature that represents the input and output port designations, as well as the differential or common signal type. Figure 2 shows that format, SxxOI, where:

• xx represents the signal type, using D for differential signal and C for common signal
• O represents the output port, and I represents the input port

Matrix math is often used to perform calculations on S-parameters, so mixed-mode S-parameters are usually organized in a way that simplifies the jump to matrix manipulation, shown in Figure 3. This table is a convenient way of keeping track of mixed mode S-parameters in a format that describes them in detail, both the input and output ports as well as the signal types.

 Figure 3: Matrix organization for mixed-mode S-parameters.
Note that there is a common pattern. The differential signals in and differential signals out (SDDxx) are all located in the upper left quadrant. The common signals in and common signals out (SCCxx) are all in the bottom right quadrant. Then when we have differential signal in and common signal out (SCDxx) in the bottom left, and finally the common signal in and differential signal out (SDCxx) in the upper right quadrant. This is the universally accepted organization for all the mixed-mode S-parameters.

It is pretty much an open industry secret that the way we get mixed-mode S-parameters is actually by measuring the single-ended S-parameters, then doing some matrix math on them—assuming the interconnect in question is a linear, passive, time invariant system. Linear meaning if you send one frequency in, you get that same frequency out. Passive because there is no energy conversion except some possible loss of energy at most. Time invariant because it's stable, it doesn’t change over time.

We’ll outline the single-ended to mixed-mode conversion process in our next post.

Dr. Eric Bogatin discusses this at more length in his webinar on Mixed Mode S-parameters and TDR Responses.