S-parameters

The formula

S-parameters describe a network in terms of traveling waves instead of voltages and currents. At each port you define an incident wave a going in and a reflected wave b coming out, both normalized to a reference impedance Z0 (almost always 50 Ω):

a_i = (V_i + Z0·I_i) / (2·√Z0)      incident wave at port i
b_i = (V_i − Z0·I_i) / (2·√Z0)      reflected wave at port i

The whole network is one matrix equation:

b = S · a

Reads: every outgoing wave is a weighted sum of every incoming wave, and the weights are the S-parameters. For a two-port that expands to:

b1 = S11·a1 + S12·a2
b2 = S21·a1 + S22·a2

Each parameter is defined by driving one port and terminating the others in Z0 so their incident wave is zero:

S11 = b1/a1  with a2 = 0    input reflection  — what bounces back off port 1
S21 = b2/a1  with a2 = 0    forward transmission — what makes it through, port 1 to 2
S12 = b1/a2  with a1 = 0    reverse transmission — leakage backward
S22 = b2/a2  with a1 = 0    output reflection — what bounces back off port 2

Reads: S11 is the reflection coefficient looking into port 1 when port 2 is matched; S21 is the gain (or loss) from port 1 to port 2. The "terminate the other port in Z0" condition is not a formality — it is why you need matched loads on every unused port when you measure.

The two dB numbers you quote on a datasheet fall straight out of the magnitudes:

Return loss  = −20·log10|S11|   (dB, how well the input is matched)
Insertion loss = −20·log10|S21|  (dB, how much signal is lost passing through)

S11 also maps directly to VSWR, the older way of stating a match:

VSWR = (1 + |S11|) / (1 − |S11|)

A quick check: S11 = −10 dB gives |S11| = 0.316, so VSWR = (1.316)/(0.684) = 1.92, and the reflected power is |S11|² = 0.10, exactly 10% of the incident power. Ten dB of return loss means one-tenth of your power bounces back.

Where you meet it

On the bench with a VNA. You calibrate a vector network analyzer to the ends of your cables, connect a filter or amplifier, and the screen shows S11, S21, S12, S22 swept across frequency. Every RF component acceptance test in the cluster — filters, isolators, LNAs, attenuators — comes down to reading these four traces against a spec mask.
Cable and connector qualification. A run of coax gets characterized by S21 (insertion loss versus frequency) and S11 (return loss from connector mismatches and kinks). A bad crimp shows up as a return-loss spike long before it kills the link.
Amplifier design review. S21 is the small-signal gain, S11 and S22 tell you how the input and output present to the rest of the chain, and S12 — the reverse isolation — tells you whether the output can talk back and make the thing oscillate. Stability circles are drawn from these four numbers.
Antenna and matching-network work. A single-port antenna is characterized by S11 alone. You tune the match until the return loss is deep enough across the band, then hand off the touchstone file.

How it works

The reason RF engineers use waves instead of Z, Y, or ABCD parameters is that at high frequency you cannot build a true open or short circuit to make the classical measurements — parasitic inductance and capacitance turn any "open" into something else by a few gigahertz. But you can build a good matched load. S-parameters are defined against matched terminations, which are the one boundary condition you can actually realize on a bench above a gigahertz. That is the whole trick.

A few things that bite people:

S-parameters are complex. Each one has a magnitude and a phase. Datasheets often print only the magnitude in dB, but the phase carries the delay and is what you need to cascade parts correctly or to de-embed a fixture. Throw away the phase and you cannot rebuild the network.
They are tied to a reference impedance. A 50 Ω S11 describes behavior in a 50 Ω system. Drop the same part into a 75 Ω CATV system and the numbers change. You can re-reference them mathematically, but you have to actually do it — you cannot read 50 Ω numbers as if they were 75 Ω.
Matched terminations are mandatory. S21 is only the clean forward transmission if port 2 is terminated in Z0. Leave a port open or hang a mismatched load on it and reflections come back through S22 and S12 and corrupt the reading. This is the number-one bench mistake: an unused port left open.
Reciprocity and losslessness are checkable. A passive network with no ferrites is reciprocal, so S12 = S21. If your measurement disagrees, suspect the calibration. A lossless network has a unitary S-matrix — the columns are orthonormal, meaning |S11|² + |S21|² = 1 for a lossless two-port. Power in equals power reflected plus power transmitted. If those squared magnitudes sum to more than one, something is wrong with the data or the part is active.
Calibration is the measurement. The VNA measures raw ratios; the S-parameters you trust exist only after calibration moves the reference plane to the ends of your cables (SOLT, TRL, and friends). An uncalibrated S11 is mostly your cables, not your part.
They are linear, small-signal descriptors. Standard S-parameters assume the device is linear. Drive an amplifier into compression and the "gain" S21 is no longer a fixed number — it depends on drive level, and you have left the regime where a single S-matrix describes the part.

History

The idea of describing a network by what it does to traveling waves rather than to terminal voltages and currents shows up first in the 1945 doctoral thesis of the Belgian engineer Vitold Belevitch, who called his construction the repartition matrix and worked it out for lumped-element circuits ^[1]^[2]. Two years later, working the other side of the problem, the American physicist Robert Henry Dicke — better known later for cosmology — used the term scattering matrix while doing wartime microwave and radar work at MIT's Radiation Laboratory, borrowing the language of how waves scatter off an obstacle ^[1]^[2].

The version RF engineers use every day was pinned down by Kaneyuki Kurokawa in a March 1965 paper, "Power Waves and the Scattering Matrix," in the IEEE Transactions on Microwave Theory and Techniques (vol. 13, pp. 194–202) ^[3]^[4]. Kurokawa defined the power-wave variables carefully enough that the squared magnitudes correspond to real, exchangeable power and the definitions hold for complex reference impedances — the rigorous footing under the "50 Ω" numbers everyone quotes. The timing was not an accident: commercial network analyzers were arriving in the same decade, and an instrument that measures wave ratios needed a wave-based parameter set to report. The measurement hardware and the math grew up together.

Related tools

/tools/vswr-return-loss
/tools/db-converter
/tools/coax-loss
/tools/microstrip-impedance
/tools/coax-impedance-dimensions
/tools/l-network-match
/tools/series-rlc-impedance
/tools/dbm-dbuv
/tools/frequency-wavelength

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →