Figure 1: Single-ended vs. differential signal "world views" of S-parameters |

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:

- A differential signal can be applied to a port and be output as a differential signal.
- A common signal can be applied to a port and be output as a common signal.
- A differential signal can be applied to a port be output as a common signal.
- A common signal can be applied to a port and be output as a common signal.

Figure 2: Mixed-mode S-parameter nomenclature. |

**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. |

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.

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