S-Parameters Explained
S-parameters — scattering parameters — are the standard mathematical language for describing how an RF circuit or component behaves when driven by a signal. Every VNA measurement result, whether displayed as a number, a trace on a log-magnitude plot, or a curve on a Smith chart, is fundamentally the measurement of one or more S-parameters. Once you understand what S11, S21, S12, and S22 represent, you can read any VNA output correctly and understand exactly what the measurement is telling you about your circuit.
S-parameters were originally developed by Kaneyuki Kurokawa in the 1960s as a way to characterize high-frequency circuits without needing to short or open-circuit the ports — something dangerous and impractical at RF because parasitic reactances immediately distort the measurement. Instead, S-parameters use matched terminations (50 Ω) at each port and measure the ratio of incident to reflected (or transmitted) wave amplitudes. This makes them directly applicable to real-world RF circuits operating in 50 Ω systems.
S-Parameter Notation
S-parameters are written with two subscripts: Soutput portinput port. The second subscript is the port where the signal is injected (the stimulus); the first subscript is the port where the response is measured. So S21 means the signal is injected at Port 2 — no, wait, let us be precise: the convention is Sresponse portstimulus port.
For a two-port network:
| Parameter | Stimulus At | Response At | What It Describes | Common Name |
|---|---|---|---|---|
| S11 | Port 1 | Port 1 | Ratio of reflected wave to incident wave at Port 1 | Input reflection coefficient, input return loss |
| S21 | Port 1 | Port 2 | Ratio of forward wave at Port 2 to incident wave at Port 1 | Forward transmission, insertion gain or loss |
| S12 | Port 2 | Port 1 | Ratio of reverse wave at Port 1 to incident wave at Port 2 | Reverse transmission (reverse isolation if small) |
| S22 | Port 2 | Port 2 | Ratio of reflected wave to incident wave at Port 2 | Output reflection coefficient, output return loss |
S-parameters are complex numbers — they have both magnitude and phase. The VNA measures them at every frequency in the sweep. When you look at a VNA trace in log-magnitude format, you are seeing 20·log₁₀(|S|) in dB as a function of frequency.
The four S-parameters of a two-port RF device. S11 and S22 describe how much signal reflects at each port; S21 and S12 describe how much signal passes through in each direction. For passive reciprocal devices (filters, cables, attenuators), S21 = S12. For amplifiers, S21 is positive in dB (gain) while S12 is highly negative (isolation).
View LargerS11 — Input Reflection Coefficient
S11 is the parameter most familiar to amateur radio operators, because it is equivalent to what an antenna analyzer measures. It describes how much of the signal applied to Port 1 reflects back into Port 1 rather than entering the device under test.
In magnitude terms, S11 is the reflection coefficient Γ (gamma):
|S11| = |Γ| = (ZDUT − Z0) / (ZDUT + Z0)
Where ZDUT is the impedance of the device under test and Z0 is the reference impedance (50 Ω). When ZDUT = 50 Ω exactly, the numerator is zero, |S11| = 0, return loss is infinite, and no signal reflects. When ZDUT deviates from 50 Ω, |S11| rises and return loss decreases.
Displaying S11: The VNA shows S11 in dB as a negative number. S11 = −20 dB means |Γ| = 0.1, return loss = 20 dB (1% of power reflected). S11 = −6 dB means |Γ| = 0.5, return loss = 6 dB (25% reflected). S11 = 0 dB means the circuit is an open or short circuit — all power reflects.
The VNA sweeps the 40m band (7.0–7.3 MHz) with Port 1 connected to the dipole feedpoint. At 7.150 MHz, S11 = −18.4 dB. What does this mean?
Return loss = 18.4 dB → |Γ| = 10^(−18.4/20) = 10^(−0.92) = 0.120
SWR = (1 + 0.120) / (1 − 0.120) = 1.120 / 0.880 = 1.27:1
Power reflected = 0.120² = 0.0144 = 1.44%
The dipole is an excellent match at 7.150 MHz. SWR 1.27:1 is well within any practical antenna tuner's range, and 1.44% reflected power is insignificant.
S21 — Forward Transmission
S21 describes how much signal passes through the device from Port 1 to Port 2. For a passive device like a filter or attenuator, |S21| is always less than or equal to 1 (0 dB or less), because passive devices cannot add energy. For an active device like an amplifier, |S21| can be greater than 1 (positive dB) — S21 = +20 dB means the amplifier provides 20 dB of power gain.
Insertion Loss is −S21 for passive devices. A filter with S21 = −3 dB in the passband has 3 dB of insertion loss — it loses half the signal power passing through it. A well-designed LC bandpass filter might achieve 0.5 dB insertion loss in the passband and 60 dB rejection (S21 = −60 dB) at two octaves above the passband.
S21 for a straight through connection (the VNA's two ports connected directly with a cable) should be 0 dB — the "through" calibration standard is precisely this measurement used during calibration to establish the 0 dB reference point for transmission measurements.
A VNA measures S21 across 1–30 MHz on a 7 MHz Chebyshev bandpass filter. The results:
- At 7.15 MHz (passband center): S21 = −1.2 dB (1.2 dB insertion loss in the passband)
- At 3.5 MHz (one octave below): S21 = −32 dB (32 dB rejection)
- At 14.0 MHz (second harmonic): S21 = −48 dB (48 dB rejection)
- At 21.0 MHz (third harmonic): S21 = −58 dB (58 dB rejection)
This filter would reduce second-harmonic power by 48 dB — far exceeding FCC Part 97 requirements — while accepting the 40m signal with only 1.2 dB loss.
S12 and S22 — Reverse Parameters
S12 is reverse transmission — the signal injected at Port 2 and measured at Port 1. For amplifiers, this is the reverse isolation: a high S12 means signals can leak backward from the amplifier output to the input (or the antenna), which can cause instability and spurious radiation. A well-designed amplifier might achieve S12 = −40 dB or better.
For passive reciprocal devices (filters, attenuators, cables), S12 = S21 — the device behaves identically in both directions. Most passive RF components are reciprocal. Non-reciprocal devices (circulators, isolators, ferrite-based components) deliberately have different S12 and S21.
S22 is the output reflection coefficient — the same concept as S11 but measured at Port 2. For an amplifier, S22 describes how well the amplifier output is matched to 50 Ω. A poorly-matched output causes the effective output power to depend on the load impedance and can cause filter-amplifier interactions. For an antenna, measuring from both directions should give the same S11/S22 result (because the antenna is a passive reciprocal device).
More Worked Examples
Amplifier Characterization
A 40m preamplifier connected between VNA Port 1 and Port 2 shows: S11 = −10 dB, S21 = +18 dB, S12 = −35 dB, S22 = −12 dB.
- S21 = +18 dB: The amplifier provides 18 dB of power gain
- S11 = −10 dB: The amplifier input has 10 dB return loss, SWR ≈ 1.92:1 (acceptable but not ideal)
- S22 = −12 dB: The amplifier output has 12 dB return loss, SWR ≈ 1.67:1
- S12 = −35 dB: Reverse isolation is 35 dB — signals from the antenna cannot leak significantly back through the amplifier in reverse
Coaxial Cable Loss
A 30-foot length of RG-58 coaxial cable is connected between the two VNA ports. At 30 MHz, S21 = −1.8 dB. This means the cable has 1.8 dB of insertion loss at 30 MHz. S11 = −26 dB — the cable input is well-matched to 50 Ω as expected. The combination of low S11 and moderate S21 is the signature of a lossy but well-matched transmission line.
VNA Display Formats for S-Parameters
| Display Format | What It Shows | Best Used For |
|---|---|---|
| Log Magnitude | |S| in dB vs frequency | Filter response, insertion loss, return loss, amplifier gain |
| Phase | Angle of S in degrees vs frequency | Phase response, group delay calculation |
| Smith Chart | Complex S11 on normalized impedance chart | Impedance matching, antenna impedance, component characterization |
| Polar | |S| and angle as a vector on a circular plot | Complex reflection analysis |
| Real / Imaginary | Real and imaginary parts of S separately vs frequency | Component modeling, simulation comparison |
| Group Delay | Rate of change of S21 phase (ns) vs frequency | Filter phase distortion, cable delay measurement |
S11 to SWR and Impedance Calculator
This calculator converts between S11 magnitude (in dB as shown on a VNA), SWR, reflection coefficient magnitude, and — for purely resistive terminations — resistance. For complex impedances, a full Smith chart analysis is needed, but this calculator is useful for quick antenna and transmission line checks.
S11 ↔ SWR ↔ Impedance Converter
Enter S11 in dB (negative value, as shown on VNA display) to calculate SWR, return loss, reflection coefficient, and the equivalent purely-resistive impedance (50 Ω reference).
Frequently Asked Questions
Why does the VNA show S11 as a negative dB number?
S11 represents reflected power divided by incident power. Since a matched load reflects less power than arrives, the ratio is always less than 1 (for passive devices), which is a negative number when expressed in dB. A perfect 50 Ω match reflects nothing — S11 approaches negative infinity dB. An open circuit reflects everything — S11 = 0 dB. The more negative the S11 reading, the better the match.
What is a good S11 value for an antenna?
For most amateur HF antennas, S11 = −10 dB (SWR ≈ 2:1) or better is acceptable with a tuner. S11 = −14 dB (SWR ≈ 1.5:1) is good. S11 = −20 dB (SWR ≈ 1.2:1) is excellent and usually achievable with a resonant antenna cut to the correct frequency. For VHF/UHF antennas driving power amplifiers, stricter matching (S11 ≤ −20 dB) reduces stress on the final stage.
Is S21 the same as gain?
For amplifiers, yes — S21 in dB is the transducer gain (the ratio of power delivered to a 50 Ω load at Port 2 to available power from a 50 Ω source at Port 1). For passive devices, S21 is negative in dB (representing loss). S21 = 0 dB means all of the input power transfers to the output, which is the ideal case for a through connection or lossless attenuator (though an actual attenuator has loss). S21 does not account for mismatch at the ports, so it is strictly valid only when the ports are properly terminated in 50 Ω.
Test Your Knowledge
Answer the questions below to check your understanding. Every answer can be found in the lesson above.