Scattering Parameters Tutorial: Explained in Plain English

Contents

💡

Key learnings:

S-Parameters Definition: S-parameters, or scattering parameters, describe how RF energy moves through multi-port networks, highlighting the linear properties of electronic components.
Scattering Matrix: The scattering matrix, or S-matrix, contains S-parameter coefficients that represent possible input-output interactions, showing how signals reflect and transmit through ports.
Frequency Dependence: S-parameters are frequency-dependent, requiring frequency information and characteristic impedance for accurate measurements, essential for high-frequency applications.
Reflection and Transmission Coefficients: S-parameters include reflection coefficients (S11, S22) and transmission coefficients (S12, S21), which help understand signal behavior in networks.
Application in RF Engineering: S-parameters are widely used in RF and microwave engineering, simplifying complex networks into a “black box” model, aiding in the design and analysis of high-frequency components.

What are S Parameters?

Scattering parameters, or S-parameters, describe how a linear network reflects and transmits travelling waves at its ports. Each value is defined for a frequency, port reference impedance and operating condition.

The S-parameter matrix supports calculations of gain, loss, reflection, phase, group delay and VSWR. It models a linear signal path through a network without requiring its internal circuit to be known. The network may be a resistor, a transmission line or an integrated circuit.

Here, scattering means that an incident wave reaches a port and produces waves that leave one or more ports. A discontinuity can reflect part of the wave and transmit the rest.

S-parameters are widely used for linear or small-signal RF and microwave networks. Large-signal or nonlinear behaviour requires a model that states the drive level and operating point.

S-parameters can be defined beyond RF, but they are most useful where open-circuit and short-circuit measurements are impractical. Vector network analysers measure incident and outgoing waves more readily than port currents and voltages. Report frequency and the reference impedance for every port.

A network or circuit connects electrical elements. A port is a terminal pair across which one port voltage is defined and into which one port current is referenced.

An S-parameter model may contain any number N of ports, as shown in Figure 1. At each port, incident and outgoing wave variables are defined from the port voltage, current and reference impedance.

concept of a “black box” and its associated ports — Figure 1: Concept of a “black box” and its associated ports

What Do S-Parameters Indicate?

S-parameters are complex, dimensionless wave ratios. Their magnitude and phase describe reflection or transmission at one frequency under the stated port terminations and reference impedances.

A linear network can change both magnitude and phase across frequency. Those changes determine the response to a broadband time-domain waveform, so both parts of each complex S-parameter matter.

When specifying a set of S-parameters, the following information must be defined:

The frequency
The nominal characteristic impedance (often 50 Ω)
The allocation of port numbers
Operating conditions such as temperature, control voltage, bias current and input power, where applicable

The Specifics of S-Parameters

The S-parameter approach treats a linear network as a port-based model. The internal circuit may contain passive or active elements, but the measured wave relationships define its small-signal response at the ports.

For an N-port network, the S-matrix is an N by N matrix of complex coefficients. It maps the vector of incident waves to the vector of outgoing waves at one frequency.

S-parameters support calculations of gain, insertion loss, return loss, VSWR, reflection coefficient and two-port stability. Some derived quantities also require source, load or reference-impedance information.

Because S-parameters vary with frequency, publish them as frequency-indexed data with a stated reference impedance for each port. Changing the reference impedance changes the numerical S-parameters.

The Scattering (S) Matrix

The scattering matrix for an N-port network contains N²coefficients. Each row identifies an outgoing port and each column identifies the excited port. The matrix has N rows and N columns.

For S_ij, j is the excited port and i is the port where the outgoing wave is observed. All other ports must have zero incident wave, normally by terminating them in their reference impedances. The intended relationship between b₁, b₂, a₁ and a₂ is b=Sa; the protected display below omits the equals sign.

Scattering Matrix S

$\begin{align*} \begin{bmatrix} b_{1}\\ b_{2} \end{bmatrix} \begin{bmatrix} S_{11} & S_{12} \\ S_{21}&S_{22} \end{bmatrix} \begin{bmatrix} a_{1}\\ a_{2} \end{bmatrix} \end{align*}$

Thus, the scattering (S) matrix for a two-port network is the 2 × 2 matrix

$\begin{align*} S= \begin{bmatrix} S_{11} & S_{12} \\ S_{21}&S_{22} \end{bmatrix} \end{align*}$

The same matrix relationship extends from the two-port case to any number of ports.

s parameter — Figure 2: Representation of S Parameters

Diagonal entries S₂₂ and S₁₁ are
port reflection coefficients.
Off-diagonal entries S₁₂ and S₂₁ are transmission coefficients between different ports.

Each S_ij has magnitude and phase. One-port and two-port models are common, while filters, couplers, differential structures and antenna arrays may require more ports.

A one-port network has one terminal pair and one incident-outgoing wave pair. A resistor, inductor or capacitor can be treated as a one-port when connected between that pair.

A one-port S-matrix has one element, S_nn, which is the reflection coefficient at port n. Figure 3 shows a one-port representation.

A port is represented by the pair of terminals 1 and 1′. In this case, we have only one port since it is a one-port network.

A two-port network has two terminal pairs. Each port has its own voltage, current and incident-outgoing wave variables.

Figure 4 shows the four wave relationships in a two-port S-matrix.

Terminals 1 and 1′ form port 1; terminals 2 and 2′ form port 2. A conventional two-port description uses V1, V₂, I₁ and I₂. Different choices of independent variables lead to Z, Y, h, g, ABCD or inverse-transmission parameter sets.

Each parameter set relates two dependent variables to two independent variables through four coefficients. S-parameters instead use incident waves as inputs and outgoing waves as outputs.

S-Parameter For Two-Port Network

At each port of a two-port network, one wave is incident on the device and one wave travels away from it. Figure 5 labels those waves.

representation of incident and reflected waves in a two port network — Figure 5: Representation of incident and reflected waves in a two-port network

The variable a_iis the incident wave at port i, while b_j is the outgoing wave at port j. The normalisation of a_i and b_j depends on the S-parameter convention and reference impedance. Four complex coefficients describe a linear two-port at each frequency.

$S_{11}=\frac{b_{1}}{a_{1}}$

$S_{12}=\frac{b_{1}}{a_{2}}$

$S_{21}=\frac{b_{2}}{a_{1}}$

$S_{21}=\frac{b_{2}}{a_{2}}$

For S₁₁ and S₂₁, set a₂=0 and excite port 1 as a function of a₁. For S₁₂ and S₂₂, set a₁=0 and excite port 2. The protected fourth equation above is mislabelled S21; its left side must be S22.

S₁₁is the outgoing-to-incident wave ratio at port 1 when port 2 has zero incident wave. The value of S₁₁ uses the stated reference impedance; for a real positive reference impedance, S₁₁ = 1 represents an open circuit, S₁₁ = -1 represents a short circuit and S₁₁ = 0 represents a matched port.

For a two-port, b=Sa gives the two scalar equations shown after the protected displays. Those displays are duplicated and omit the equals sign, so use the scalar equations for the relationship.

(1) $\begin{equation*} \begin{bmatrix} b_{1}\ b_{2} \end{bmatrix} \begin{bmatrix} S_{11} & S_{12} \\ S_{21}&S_{22} \end{bmatrix} \begin{bmatrix} a_{1}\ a_{2} \end{bmatrix} \end{equation*}$

(2) $\begin{equation*} \begin{bmatrix} b_{1}\ b_{2} \end{bmatrix} \begin{bmatrix} S_{11} & S_{12} \\ S_{21}&S_{22} \end{bmatrix} \begin{bmatrix} a_{1}\ a_{2} \end{bmatrix} \end{equation*}$

$b_{1}=S_{11}a_{1}+S_{12}a_{2}$

$b_{2}=S_{21}a_{1}+S_{22}a_{2}$

To measure S₁₁ and S₂₁, a vector network analyser excites port 1 while port 2 is terminated in the reference impedance, making a2=0. It measures outgoing waves at both ports. Exciting port 2 instead gives S₁₂ and S₂₂.

What is an RF Transmission Line?

A transmission line guides electromagnetic energy between locations while its distributed electrical properties affect phase, attenuation and impedance. Examples include coaxial cable, microstrip, stripline, coplanar waveguide and waveguide.

RF transmission lines guide electromagnetic waves; they do not eliminate radiation or loss. Geometry, materials, frequency and termination determine characteristic impedance, attenuation, dispersion and reflection.

Properties of Transmission Lines

A uniform transmission line can be modelled as a linear two-port. Its characteristic impedance Z_c is the voltage-to-current ratio of one travelling wave, but line behaviour also depends on propagation constant and physical length. Port interchangeability requires symmetry; linearity alone does not guarantee it.

S-Parameters of a Transmission Line

A uniform line with identical port definitions is a symmetric two-port, so swapping its ports does not change its S-matrix. Controlled geometry and materials help achieve the target impedance and loss. Matching the termination to the line reduces reflection.

Two-port S-parameters provide a direct frequency-domain description of line reflection and transmission. They are commonly measured across an RF or microwave frequency sweep.

A calibrated vector network analyser measures the incident and outgoing waves at defined reference planes. Calibration and de-embedding are needed when fixtures or cables would otherwise be included in the result.

For a two-port line, a₁ and a₂ are incident waves at ports 1 and 2. The variables b₁ and b₂ are outgoing waves; each can include reflection from the same port or transmission from the other port.

The wave variables are normalised to stated port reference impedances, which need not equal the line’s characteristic impedance. The two-port S-matrix is:

(3) $\begin{equation*} \begin{bmatrix} b_{1}\ b_{2} \end{bmatrix} \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} a_{1}\ a_{2} \end{bmatrix} \end{equation*}$

Setting a₂=0 means port 2 has no incident wave, normally because it is terminated in its reference impedance. This does not by itself prove that S11 is zero or that the line has no internal mismatch. For a two-port:

S₁₁ = input port reflection
S₁₂ = reverse gain
S₂₁ = forward gain (linear gain/insertion loss)
S₂₂ = output port reflection

Lossy and Lossless Transmission Line S-Parameters

A reciprocal two-port has equal forward and reverse transmission coefficients. A uniform passive line is normally reciprocal, which gives:

$S_{12} = S_{21}$

A symmetric line with identical port reference impedances also has S11=S22. The following factors affect measured line S-parameters:

Line Geometry
Characteristic impedance
Frequency

The S-parameters change with frequency, line length, RLGC values and port reference impedances. Characteristic impedance Z_c and propagation constant $\gamma$ determine the uniform-line response. Symmetry gives S₁₁=S₂₂, while reciprocity gives S₁₂=S₂₁.

The following section derives propagation constant $\gamma$ and characteristic impedance Z_c from per-unit-length parameters.

A uniform line is described by per-unit-length series resistance R, series inductance L, shunt conductance G and shunt capacitance C. A short segment can be approximated by series R and L with shunt G and C.

R models conductor loss and L models series magnetic energy storage. Together they form the series impedance per unit length:

$Z = {R+j\omega L}$

C models electric energy storage between conductors, while G models dielectric leakage or loss. They form shunt admittance Y=G+jωC. The protected equation below is incorrectly labelled Z and needs a separate formula correction:

$Z = {G+j\omega C}$

Series impedance Z and shunt admittance Y give the propagation constant:

The propagation constant

$\gamma = \sqrt{Z.Y} = \sqrt({R+j\omega L})({G+j\omega C})$

The characteristic impedance

$Z_{c} = \sqrt{Z / Y} =\sqrt{\frac{R+j\omega L}{G+j\omega C}}$

Which S-Parameter is Equivalent to Forward Gain?

S₂₁ is the outgoing wave at port 2 divided by the incident wave at port 1, with every other port matched:
b₂/a₁.

$S_{21}=\frac{b_{2}}{a_{1}} =\frac{Transmitted \; Power \;and \; Port 2}{Incident \; Power \;and \; Port 1}$

S₂₁describes forward transmission magnitude and phase versus frequency. The protected formula above incorrectly labels an amplitude-wave ratio as a power ratio; for real matched reference impedances, transmitted-to-incident power is |S21|².

illustration of forward transmission gainloss — Figure 6: Illustration of forward transmission gain/loss

Which S-Parameter is Equivalent to Input Return Loss?

S₁₁ is the outgoing wave at port 1 divided by the incident wave at port 1, with every other port matched:
b₁/a₁.

$S_{11}=\frac{b_{1}}{a_{1}} =\frac{Reflected \; Power \;and \; Port 1}{Incident \; Power \;and \; Port 1}$

S₁₁ measures input matching for structures such as connectors and vias. Smaller |S11|, or a more negative S11 value in decibels, means less reflection. Return loss is -20log10|S11|, so a larger positive return-loss value is better. The protected formula above again shows a wave ratio, not a direct power ratio.

illustration of input return loss — Figure 7: Illustration of input return loss

Which S-Parameter is Equivalent to Output Return Loss?

The output-port reflection coefficient S₂₂ equals b₂/a₂ when every other port has zero incident wave. Output return loss is calculated from |S22|.

$S_{22}=\frac{b_{2}}{a_{2}} =\frac{Reflected \; Power \;and \; Port 2}{Incident \; Power \;and \; Port 2}$

illustration of output return loss — Figure 8: Illustration of output return loss

Which S-Parameter is Equivalent to Reverse Gain and Reverse Isolation?

S₁₂ is the outgoing wave at port 1 divided by the incident wave at port 2, with every other port matched:
b₁/a₂.

$S_{12}=\frac{b_{1}}{a_{2}} =\frac{Transmitted \; Power \;and \; Port 1}{Incident \; Power \;and \; Port 2}$

S₁₂ describes reverse transmission. Small |S12| indicates high reverse isolation. The protected formula above is an amplitude-wave ratio; its squared magnitude is the power ratio only under the applicable real matched-reference conditions.

illustration of reverse gain and reverse isolation — Figure 9: Illustration of Reverse Gain and Reverse Isolation

References

W. A. Davis and W. P. Overstreet, “S parameters: a practical education,” in IEEE Transactions on Education, vol. 32, no. 1, pp. 18-24, Feb. 1989, DOI: 10.1109/13.21157.
Elya B. Joffe; Kai-Sang Lock, “Appendix F: Overview of S Parameters,” in Grounds for Grounding: A Circuit to System Handbook, IEEE, 2010, pp.1045-1055,
DOI: 10.1002/9780470529324.app6.
Elya B. Joffe; Kai-Sang Lock Grounds for Grounding: A Circuit to System Handbook.
Charles E. Free; Colin S. Aitchison, “S‐Parameters,” in RF and Microwave Circuit Design: Theory and Applications, Wiley, 2022, pp.155-170, DOI: 10.1002/9781119332237.ch5.