S-Parameters: Insertion Loss, Return Loss, and Mixed Mode

Why not just measure voltage and current?

Think of testing a pipe. You could try to measure the pressure at every point inside it, but it is far easier to push a known flow in one end and see what comes out the other. The ratio of what got through to what you sent is a complete description of what the pipe does — and if some of the flow bounced back at you, that tells you something about obstructions inside.

S-parameters are exactly that idea, applied to electrical waves. Instead of pressure and flow, you send a known electromagnetic wave into a port and measure two things: what fraction comes back (the reflection, S₁₁) and what fraction arrives at the other end (the transmission, S₂₁). Do this at every frequency of interest and you have a complete frequency-domain description of the interconnect — its "report card." That is why every VNA measurement, every channel simulation, and every Touchstone file you will encounter in SI work is built on S-parameters.

At low frequency, describing a two-terminal network is easy: apply a voltage, measure the current, and you have an impedance. Everything in circuit theory is built that way.

Two things break that at high frequency. First, you have to load the node to measure it, and a probe that loads a 50 Ω node meaningfully has changed the thing it was measuring. Second, voltage and current require a defined port geometry and mode once fields are no longer well represented by a single lumped node. Different reference conductors or propagating modes can describe different quantities at the same physical structure.

But there is something that stays well defined and is directly measurable: send a known wave in, and see what comes back out. Not how much voltage is at a node, but what fraction of an incident wave is reflected and what fraction is transmitted. You need to agree a reference impedance to define what "a wave" means, and after that everything is a ratio of two measurable quantities.

That is an S-parameter, and the naming is simply S_out,in — S₂₁ is what comes out of port 2 when you drive port 1. Port order is a convention that has to be stated, which is the first thing to check on any file somebody hands you.

Ratios of waves

At high frequencies you can still define voltage and current at a terminal, but they stop being the convenient description: they depend on the field distribution across the port, and a measurement that loads the node changes them. What is well defined and directly measurable instead is how much of an incident wave is reflected and how much is transmitted, once a reference impedance and a port mode are agreed. That is all an S-parameter is.

Insertion loss (S₂₁) — how much survives the trip. Reported in dB and at or below 0 dB for an individual power-normalised path through a passive network, which is why “12 dB of loss” and “S₂₁ = −12 dB” mean the same thing. The number at half the data rate — the Nyquist frequency — appears in many serial-channel budgets, but the full magnitude and phase response determine the pulse.
Return loss (S₁₁) — how much bounces back. Impedance mismatches, via stubs and connector footprints all live here. Poor return loss is doubly expensive: the energy is missing from S₂₁ and it comes back to cause reflections.
Reference impedance — every S-parameter is relative to one, almost always 50 Ω single-ended. A file measured against a different reference is not comparable until it is renormalised.

Mixed mode: four blocks, not sixteen numbers

A differential pair is a 4-port, and the raw 4×4 matrix is nearly unreadable. Converting to mixed-mode reorganises it into four meaningful blocks, and the naming is mechanical: the first subscript letter is the output mode, the second is the input mode.

Term	Means	What it tells you
SDD21	differential in → differential out	the real insertion loss of the pair
SDD11	differential reflected	differential impedance discontinuities
SCD21	differential in → common out	mode conversion — asymmetry, skew, weave
SCC21	common in → common out	how common-mode noise propagates and radiates

SDD21 is usually the headline. SCD21 can explain failures that a differential-only review misses, because it is where a perfectly good-looking differential channel is quietly leaking into common mode.

Check the file before you trust it A Touchstone file from a solver or a VNA can be physically impossible, and simulators will happily run it. Three checks catch nearly everything, and they test three separate properties: passivity — the network may not deliver more power than it absorbs, which for a power-normalised matrix means every singular value of S stays at or below one at every frequency; causality — no energy may arrive before it was sent, which shows up as a non-zero response at negative time after transformation; and reciprocity — S₂₁ = S₁₂, which holds for ordinary linear isotropic interconnect but is not implied by passivity. A ferrite isolator or circulator is passive and deliberately non-reciprocal. Expect reciprocity from a trace, a via or a connector; do not treat a violation as automatic proof the file is broken until you know what the structure contains. A non-causal file makes a time-domain simulation produce eye openings that do not exist.

A practical file-intake checklist Record the port map and mode order, reference impedance, reference planes, frequency start, stop and spacing, units and Touchstone convention, fixture or de-embedding history, and the expected reciprocity. Then run passivity and causality checks over the band used by the simulation. This small record prevents a correct matrix from being connected incorrectly.

One more habit: check the frequency extent and the DC point. A file that stops at 20 GHz cannot support a time-domain simulation with a 20 ps edge, and a file with no DC point forces the simulator to extrapolate one — badly. Both produce results that look plausible and are not.

Why this decides how your board behaves

S₂₁ is your loss budget, and its shape matters more than its value. Two channels with the same loss at Nyquist behave very differently if one is flat and the other is falling hard, because it is the slope that creates ISI.
S₁₁ reveals mismatch and delayed energy. Reflected energy can reduce forward delivery and return later through additional discontinuities. Its time-domain impact depends on where the mismatch sits and what it reflects from next.
A null in S₂₁ is not a budget item. Loss you can equalise; a resonance removes a frequency entirely, and nothing puts it back. See vias.
SCD21 identifies a path into common mode. Mode conversion can turn a good differential channel into common-mode excitation that may radiate when the structure provides an antenna path. Use it with the common-mode current path and enclosure geometry.
A bad file affects every downstream result. A non-causal or non-passive Touchstone file can produce unrealistic eye openings; confirm that the selected simulator reports or corrects these conditions as expected.

What to do about it

Check the file before you trust the result. Passivity, causality and reciprocity are three separate properties and all three are worth testing — see the note above for what each one actually asserts. Every serious simulator has these checks; most people run them only after something looks wrong.

Check the frequency extent against your edge, not your data rate. A file that stops at 20 GHz cannot support a time-domain simulation with a 20 ps edge. The rule from edge rate applies directly: you need the model to be accurate up to roughly the knee.

Check the DC point exists. A file with no DC point forces the simulator to extrapolate one, and extrapolating to DC from a measurement that starts at 10 MHz is guesswork. The symptom is a baseline wander in the time domain that no physical mechanism explains.

Always state the reference plane. An S-parameter file describes whatever was between the two calibration planes. Measured at the connector, it includes the launch; de-embedded to the board, it does not. Comparing two files calibrated to different planes is the most common way simulation and measurement "disagree".

Using this to find a fault

Smooth S₂₁ falling steadily. Ordinary loss. Compare against α·length — if it is higher than the trace explains, look at roughness or the laminate.
Ripple on S₂₁ with a regular period. A reflection between two discontinuities. The period in frequency gives you the delay between them: Δf = 1/(2T_d), which tells you how far apart they are.
A deep narrow null. A resonance — usually a via stub. Invert f = c/(4L√D_k) to get its length.
S₁₁ good at low frequency and bad above a few GHz. The trace impedance is fine and a discontinuity is not. Transform to the time domain and look at where it is.
SCD21 rising with frequency. Asymmetry that gets worse with frequency — skew, most likely. Compare against the intra-pair length mismatch you know about; if it is larger than that explains, suspect glass weave or an asymmetric via pair.

Go deeper — what the matrix means, and the transform to the time domain

Formally, each port has an incident wave a and a reflected wave b, normalised so that |a|² and |b|² are powers. The matrix relates them:

b_i = Σ_j S_ij a_j S_ij is measured by driving port j with all other ports terminated in the reference impedance

The phrase “all other ports terminated” is doing real work there. It is why S-parameters are easy to measure — a matched termination is easy to build, where the open and short circuits that Z- and Y-parameters need are not, at these frequencies — and it is why a file is only valid at its stated reference impedance. Renormalising to a different one is a defined operation, not a relabelling.

The three checks, stated precisely.

Passivity: the network cannot output more power than it takes in. For a power-normalised matrix that means every singular value of S is at or below one, at every frequency. A solver that has extrapolated badly, or a measurement with a calibration error, routinely violates this.
Causality: no energy can arrive before it was sent. It shows up as non-zero response at negative time after an inverse transform, and it is usually caused by a truncated or badly extrapolated frequency range rather than by a bad measurement.
Reciprocity: S₂₁ = S₁₂. This holds for ordinary linear isotropic interconnect but is not implied by passivity — a ferrite isolator is passive and deliberately non-reciprocal. Expect it from a trace, a via or a connector; do not treat a violation as automatic proof of a broken file until you know what the structure contains.

Frequency and time are the same information. The impulse response is the inverse Fourier transform of S₂₁, and the TDR trace is the step response of S₁₁ read back as an impedance. Within a consistent bandwidth, frequency grid, window, and reference plane, they are two views of the same linear network — see Lab B, which draws both from one network, exposes the discrete impulse weights, and lets you watch them move together.

Two practical consequences of that equivalence. The bandwidth of your file sets the time resolution you can recover: a 20 GHz file cannot resolve features closer than about 25 ps of round trip, or roughly 0.07 inch. And the frequency step sets the time window before the response wraps around on itself — a 10 MHz step gives you 100 ns, which is plenty for a board and marginal for a long cable.

In the real world

S-parameters are the interface between organisations, which is why the discipline around them matters more than the mathematics. A connector vendor's model, a cable assembly's measured file and your own board simulation all have to compose, and they will only compose correctly if everyone agreed on port order, reference impedance and reference plane.

Most of the time lost to S-parameter work is not spent understanding the physics. It is spent discovering that two files have their ports numbered differently, or that one includes a launch the other assumed was de-embedded. Ask those three questions of every file before you use it, and most of that time comes back.

Matched lossless channel reference — unity transmission magnitude with group delay exported in seconds and shown in picoseconds
PCB Transmission-Line Loss Mechanisms
Differential Signaling and Mode Conversion
High-Speed Vias: Stubs, Resonance, and Backdrilling
S-Parameter De-Embedding and Fixture Removal