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An Introduction to S-Parameters
What are S-Parameters?
The wave equation in one dimension (Telegrapher’s Equation) allows for the
propagation of RF waves in both directions simultaneously within a transmission line.
This makes intuitive sense if you were to imagine two signal generators launching
signals, one on each end of the transmission line. The waves from each end propagate
toward each other, overlap, interfere, and perhaps terminate in the source impedance of
the generator on the other side.
Figure 1 - Two Waves on a Transmission Line
Because it is difficult to annotate two signals on a single line that are going in different
directions, they are depicted using a network diagram like the one in Figure 2.
Figure 2 - Network Diagram
The arrow from a0 to b1 represents a voltage wave moving from left to right, and the
arrow from a1 to b0 represents a voltage wave moving from right to left. Although there
are four nodes shown here, there are only two connections, and these two waves are
traveling on the same line. We separate them in the network diagram for clarity and
convenience. There are two interfaces shown also shown in this diagram – a0:b0 and
a1:b1. Reflections may occur at these interfaces, and they are represented by the two
vertical arrows.
A part of the voltage wave entering from the left of a0 may reflect at the a0-b0 interface
and add to the existing wave from a1. Similarly, a reverse voltage wave entering the
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An Introduction to S-Parameters
diagram from the right of a1 may reflect at the a1:b1 interface and add to the forward
wave from a0. Waves at the top of the diagram always flow from left to right, while
waves at the bottom of the diagram always flow from right to left.
These wave reflections are analogous to what occurs in optical systems where a light
wave incident on a glass surface results in a reflected wave and a transmitted wave
which passes into the glass.
Figure 3 - Optical Reflection
The reflection is proportional to the ratio of the dielectric constant of glass to that of air.
If the incident power is normalized to a value of 1, then the reflected wave would be 0.2
and the transmitted wave would be 0.8.
S-parameters are normalized in this fashion as well, and the reflection is related to the
impedance encountered by the wave vs the source impedance of the generator. S-
parameters are complex numbers with magnitude and phase, also called reflection
coefficients. The reflection, Γ from an impedance Z, fed by a generator with source
impedance Zo is:
𝑍−𝑍
Γ = 𝑍+𝑍0
0
This reflection always has a magnitude less than or equal to 1 and may have any
phase. If Z is 0 – a short – then Γ = -1, if Z is infinite – an open – then Γ = +1.
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An Introduction to S-Parameters
A Vector Network Analyzer (VNA) is capable of separating and measuring the voltage
waves moving in each direction on a transmission line and can determine their
magnitudes and phases. An example is shown in Figure 4, where a 2-port VNA
produces an outgoing incident signal on port 1, which travels to a DUT. Some of the
incident signal may be reflected, and some may proceed through the DUT to arrive at
port 2.
Figure 4 - VNA Measurement
Based on the port voltages and currents, we can write the equations for the normalized
incident and arbitrary reflected voltage waves from Figure 5 like this:
𝑉1 +𝐼1 𝑍0 𝑉2 +𝐼2 𝑍0 𝑉1 −𝐼1 𝑍0 𝑉2 −𝐼2 𝑍0
𝑎1 = 𝑎2 = 𝑏1 = 𝑏2 =
2√𝑍0 2√𝑍0 2√𝑍0 2√𝑍0
Where the “a” variables are independent and “b” are dependent.
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An Introduction to S-Parameters
Figure 5 - Port Voltages and Currents
The linear equations describing the network with S-parameter coefficients are then:
𝑏1 = 𝑠11 𝑎1 + 𝑠12 𝑎2 and 𝑏2 = 𝑠21 𝑎1 + 𝑠22 𝑎2
And the S-parameters are defined thusly:
𝑏 𝑏 𝑏 𝑏
𝑠11 = 𝑎1 | 𝑠22 = 𝑎2 | 𝑠21 = 𝑎2 | 𝑠12 = 𝑎1 |
1 𝑎2 =0 2 𝑎1 =0 1 𝑎2 =0 2 𝑎1 =0
Since power is proportional to the square of the voltage:
𝑃𝑜𝑤𝑒𝑟 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑝𝑜𝑟𝑡 1 𝑖𝑛𝑝𝑢𝑡
|𝑠11 |2 =
𝑃𝑜𝑤𝑒𝑟 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑜𝑛 𝑝𝑜𝑟𝑡 1 𝑖𝑛𝑝𝑢𝑡
𝑃𝑜𝑤𝑒𝑟 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑝𝑜𝑟𝑡 2 𝑖𝑛𝑝𝑢𝑡
|𝑠22 |2 =
𝑃𝑜𝑤𝑒𝑟 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑜𝑛 𝑝𝑜𝑟𝑡 2 𝑖𝑛𝑝𝑢𝑡
𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑 𝑡𝑜 𝑎 𝑍 𝑙𝑜𝑎𝑑
|𝑠21 |2 = 0
(𝑇𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟 𝑃𝑜𝑤𝑒𝑟 𝐺𝑎𝑖𝑛)
𝑃𝑜𝑤𝑒𝑟 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑓𝑟𝑜𝑚 𝑍 𝑠𝑜𝑢𝑟𝑐𝑒 0
|𝑠12 |2 = 𝑅𝑒𝑣𝑒𝑟𝑠𝑒 𝑇𝑟𝑎𝑛𝑠𝑑𝑢𝑐𝑒𝑟 𝑃𝑜𝑤𝑒𝑟 𝐺𝑎𝑖𝑛
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An Introduction to S-Parameters
S-parameters displayed in log format on a vector network analyzer portray them as
20*Log(S), which is equivalent to these power ratios.
Why are S-Parameters Useful?
Using a Vector Network Analyzer (VNA), S-parameters may be directly measured
without the need for elaborate fixturing. For instance, a 50Ω connectorized amplifier
might be connected to the VNA, and the directional bridges will separate the incident
and reflected waves. The return loss (S11) and transducer power gain (S21) would then
be displayed on the screen.
The measurement of all four S-parameters over a frequency represents a full
characterization of any linear device. S-parameter data saved as a touchstone file may
be imported into most linear simulators to demonstrate the device performance within a
larger system.
Amplifiers
S-Parameters may also be used to evaluate the stability of an amplifier – see reference
[1]. We know that the input reflection coefficient, Γin, of a linear system characterized by
S-parameters and loaded with output reflection coefficient ΓL is given by:
𝑠 𝑠 Γ
21 12 𝐿
Γ𝑖𝑛 = 𝑠11 + 1−𝑠 Γ
22 𝐿
The load, ΓL, has an effect on the input reflection coefficient as long as neither S21 nor
S12 are zero. An amplifier will have S21 > 0 (gain) and non-zero S12 (reverse isolation). If
there exists a load, |Γ𝐿 | < 1 (passive load), which causes |Γ𝑖𝑛 | > 1 (reflection greater
than the incident wave), then the amplifier will likely oscillate with that load impedance
applied. Rollet’s stability factor, K, indicates whether an amplifier is potentially unstable
for some input source impedance or output load.
1−|𝑠11 |2 −|𝑠22 |2+Δ2
𝐾= where Δ = 𝑠11 𝑠22 − 𝑠12 𝑠21
2|𝑠21 𝑠12 |
An amplifier with stability factor K>1 is said to be unconditionally stable. A conditionally
stable amplifier with K<1 may still be used, as long as the input and output impedances
are controlled and not allowed to venture into a region where the amplifier becomes
unstable.
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An Introduction to S-Parameters
Gain (S21), return loss (S11), 1dB compression, AM-AM conversion, and AM-PM
conversion are all characteristics of an amplifier that may be evaluated with S-
parameters.
Filters
S-parameters may be used to characterize an RF filter response. A filter will have an
insertion loss in the passband and one or more stopbands with required rejection. S21 is
the S-parameter evaluated for these, and S11 is the reflection from the filter or return
loss. See reference [6].
Figure 6 - 100 MHz Bandpass Filter
Transmission Lines and Antennas
Coaxial cables and balanced transmission lines connect RF devices to each other and
to antennas. Measuring S21 (insertion loss) of these lines may be important to determine
how much RF energy is being transferred to the load at the end of the line. Poor return
loss could indicate a damaged cable or failing connector. The measurement of the
return loss looking into a cable that feeds an antenna is an effective way to evaluate the
health of both. Physical damage to the antenna or ingress of moisture into the feed-line
cable increases reflection and degrades the return loss. See references [3] and [4].
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An Introduction to S-Parameters
Material Properties and Moisture Content
mmWave S-Parameter measurements of materials may be used to derive the
permittivity of unknown materials. This information is critical for the development of
radomes for aerospace and automotive applications. See reference [5].
Figure 7 - Material Measurement from Compass Technology Group
Because the relative permittivity of water is very high – 80 – it raises the overall relative
permittivity of any material containing it. Dry soil might have a relative permittivity
between 3 and 5, but added moisture drives this number up very quickly, so it is
possible to estimate moisture content. Soil moisture estimation is extremely important in
the agricultural industry. See reference [2].
Conclusion
S-parameters are very useful for a great number of RF measurements. With the advent
of the modern VNA, there is simply no reason to use the older characterization
parameters such as Z, Y, or h. These older parameters require special fixturing and are
not suitable for high-frequency measurement. S-parameters can characterize devices
up to hundreds of GHz.
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An Introduction to S-Parameters
References:
1) Copper Mountain Technologies, “Amplifier Measurements”,
https://coppermountaintech.com/amplifier-measurements/
2) Arkadiusz Levandowski et al., “One-Port Vector Network Analyzer
Characterization of Soil Dielectric Spectrum”, IEEE Transactions on Geoscience
and Remote Sensing, Vol 57, No 6, June 2019.
3) Copper Mountain Technologies, “Testing and Matching PCB Antennas”,
https://coppermountaintech.com/testing-matching-pcb-antennas/
4) Copper Mountain Technologies, “Near and Far Field Measurement”,
https://coppermountaintech.com/near-and-far-field-measurement/
5) Copper Mountain Technologies, “Focused Beam Materials Measurement System
with Compass Technology Group”, https://coppermountaintech.com/video-focus-
beam-materials-measurement-system-with-compass-technology-group/
6) Copper Mountain Technologies, “Measuring a Filter with a 2-Port VNA”,
https://coppermountaintech.com/video-measuring-a-filter-with-a-2-port-vna/
