Scattering parameters

Scattering parameters , abbreviated as S parameters, are used to describe the behavior of linear electrical components and networks in small-signal behavior using wave quantities . The S parameters are used for dimensioning and calculations in the field of high-frequency technology , such as communication systems and telecommunications systems .

The importance of the S-parameters lies above all in the metrological area, since in contrast to other parameter representations such as the Z , Y and H parameters, the S parameters are recorded with the wave impedance , which is also carried out at the connections during normal operation is available. This avoids undesired impedance transformations when measuring the S-parameters at the inputs and outputs of a network, due to the necessary measuring lines and their spatial extension .

The number of S parameters required depends on the number of gates in the network and results from the square of its number of gates. A single S-parameter is sufficient to describe a one- port (two- port ), a two-port is fully described with the help of four S-parameters, a three-port with nine, a four-port (eight-port) with sixteen S-parameters and so on ( multi-port ). It is always possible to represent a general linear multi-port using S parameters.

In addition to the scatter parameter display, there are also other network parameter displays for linear networks with any number of signal ports, such as B. Admittance parameters (Y parameters, complex conductance) or impedance parameters (Z parameters, complex resistance). S, Y and Z parameters can be converted into one another. In this way, S-parameters obtained through measurements can be prepared for use in circuit simulations (e.g. SPICE ). This function is already available in many simulation programs. In contrast to the S-parametric representation, the existence of the Y- and Z-parametric representation of general linear multi-ports cannot be universally guaranteed, since the Y-matrix or the Z-matrix of special multi-ports is singular .

General

Equivalent diagram at the terminals of an electric gate with the voltage  U _ν and current  I _ν plus equivalent forward and backward, each with a traveling wave a _ν and  b _ν

With the S parameters, the relationships at the gate of a network are not described directly by the electrical voltage  U _ν currently present there or the electric current  I _ν flowing into the gate, but the description is equivalent to that in the gate incoming wave and the gate reflected wave  . ${\displaystyle \nu }$ ${\displaystyle a_{\nu }}$ ${\displaystyle \nu }$ ${\displaystyle b_{\nu }}$

Based on the impedance Z _{0 of} the measuring system at the gate , the two representations can be related to the following equations, which are also referred to as Heaviside transformation (we assume in the following that positive is real): ${\displaystyle \nu }$ ${\displaystyle Z_{0}}$

${\displaystyle a_{\nu }={\frac {1}{2}}\cdot \left({\frac {U_{\nu }}{\sqrt {Z_{0}}}}+I_{\nu }{\sqrt {Z_{0}}}\right)}$

${\displaystyle b_{\nu }={\frac {1}{2}}\cdot \left({\frac {U_{\nu }}{\sqrt {Z_{0}}}}-I_{\nu }{\sqrt {Z_{0}}}\right)}$

and by reversing the relationships:

${\displaystyle U_{\nu }={\sqrt {Z_{0}}}\cdot (a_{\nu }+b_{\nu })}$

${\displaystyle I_{\nu }={\frac {1}{\sqrt {Z_{0}}}}\cdot (a_{\nu }-b_{\nu })}$

The voltage U _ν and the current I _ν at the gate are related to one another via the outwardly acting impedance Z _ν : ${\displaystyle \nu }$

${\displaystyle Z_{\nu }={\frac {U_{\nu }}{I_{\nu }}}}$

with which, with the impedance Z _{0 of} the measuring system, the reflection factor r _ν can be described as:

${\displaystyle r_{\nu }={\frac {b_{\nu }}{a_{\nu }}}={\frac {Z_{\nu }-Z_{0}}{Z_{\nu }+Z_{0}}},\qquad b_{\nu }=r_{\nu }\cdot a_{\nu }}$

In the simplest case of a one-port, the scalar reflection factor r is equal to the one and only S-parameter S ₁₁ . In electrical networks with more than one port, this relationship is expressed using a matrix equation in the form of a linear system of equations . In general, the S parameters of an n-gate are expressed as an n × n matrix S and the two vectors a and b, each comprising n elements, are expressed as a matrix equation:

${\displaystyle \mathbf {b} =\mathbf {S} \cdot \mathbf {a} }$

or in the element notation:

${\displaystyle {\begin{pmatrix}b_{1}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}S_{11}&\dots &S_{1n}\\\vdots &\ddots &\vdots \\S_{n1}&\dots &S_{nn}\end{pmatrix}}\cdot {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}}$

Linear networks with any number of gates

A more general assumption is a complex reference impedance at the gate . An individual reference impedance can be assigned to each gate and this does not necessarily have to be real. The Heaviside transformation becomes (in the following we assume that the real part of is positive): ${\displaystyle Z_{0\nu }}$ ${\displaystyle \nu }$ ${\displaystyle Z_{0\nu }}$

${\displaystyle a_{\nu }={\frac {U_{\nu }+Z_{0\nu }I_{\nu }}{2{\sqrt {\operatorname {Re} (Z_{0\nu })}}}}}$

${\displaystyle b_{\nu }={\frac {U_{\nu }-Z_{0\nu }^{*}I_{\nu }}{2{\sqrt {\operatorname {Re} (Z_{0\nu })}}}}}$

The squares of these expressions have the physical dimension of a power. The effective power flowing into the door results from: ${\displaystyle \nu }$

${\displaystyle P_{\nu }={\frac {1}{2}}\operatorname {Re} (U_{\nu }I_{\nu }^{*})={\frac {1}{2}}|a_{\nu }|^{2}-{\frac {1}{2}}|b_{\nu }|^{2}}$

S , Y and Z are the matrices of the network parameters.

Z ₀ is the reference impedance, i.e. the impedance of the test ports of the vector network analyzer used , usually 50 ohms.

E is the identity matrix .

Y parameters as a function of the S parameters:

${\displaystyle \mathbf {Y} ={\frac {1}{Z_{0}}}\cdot (\mathbf {E} +\mathbf {S} )^{-1}\cdot (\mathbf {E} -\mathbf {S} )}$

Z parameters as a function of the S parameters:

${\displaystyle \mathbf {Z} =Z_{0}\cdot (\mathbf {E} -\mathbf {S} )^{-1}\cdot (\mathbf {E} +\mathbf {S} )}$

S parameters as a function of the Y parameters:

${\displaystyle \mathbf {S} =(\mathbf {E} -Z_{0}\cdot \mathbf {Y} )\cdot (\mathbf {E} +Z_{0}\cdot \mathbf {Y} )^{-1}}$

S parameters as a function of the Z parameters:

${\displaystyle \mathbf {S} =\left({\frac {1}{Z_{0}}}\cdot \mathbf {Z} -\mathbf {E} \right)\cdot \left({\frac {1}{Z_{0}}}\cdot \mathbf {Z} +\mathbf {E} \right)^{-1}}$

Two goals

Schematic representation of the S-parameters on a two-port

Two ports in particular play an important role in high frequency technology . Examples of two ports are amplifiers or filters which have an input and an output. The S parameters then comprise the elements S ₁₁ , S ₁₂ , S ₂₁ and S ₂₂ :

${\displaystyle {\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}={\begin{pmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{pmatrix}}{\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}}$

a ₁ is the wave entering gate 1, a _{2 is} the wave entering gate 2. b ₁ describes the wave exiting from the entrance (gate 1), b ₂ describes the wave exiting from the exit (gate 2).

The S parameters have the following meaning:

Input reflection factor S ₁₁

represents the reflection at the entrance without excitation at gate 2: ${\displaystyle S_{11}=\left.{\frac {b_{1}}{a_{1}}}\right|_{a_{2}=0}}$

Output reflection _factor S ₂₂

represents the reflection at gate 2 without excitation at gate 1: ${\displaystyle S_{22}=\left.{\frac {b_{2}}{a_{2}}}\right|_{a_{1}=0}}$

Forward transmission factor S ₂₁

represents the forward transmission without excitation at gate 2: ${\displaystyle S_{21}=\left.{\frac {b_{2}}{a_{1}}}\right|_{a_{2}=0}}$

Backward transmission factor S ₁₂

represents the reverse transmission without excitation at gate 1: ${\displaystyle S_{12}=\left.{\frac {b_{1}}{a_{2}}}\right|_{a_{1}=0}}$

Measurement

Measurement of the S-parameters of a bandpass filter with a network analyzer

In practice, the S parameters are measured as a function of frequency with the aid of network analyzers. The S-parameters are dimensionless complex numbers , also known as phasors , which are specified by amount in decibels (dB) and phase in degrees (°). The wave impedance is typically set at 50 Ω.

Due to the amount of data measured, practically all network analyzers have the option of being able to save data sets of the S parameters on data carriers. A common data format is the Touchstone file format with the filename extension “s2p” for a two-port . This data format represents a text file with the following typical structure:

! Created Fri Jul 21 14:28:50 2005
# MHZ S DB R 50
! SP1.SP
50 -15.4 100.2 10.2 173.5 -30.1 9.6 -13.4 57.2
51 -15.8 103.2 10.7 177.4 -33.1 9.6 -12.4 63.4
52 -15.9 105.5 11.2 179.1 -35.7 9.6 -14.4 66.9
53 -16.4 107.0 10.5 183.1 -36.6 9.6 -14.7 70.3
54 -16.6 109.3 10.6 187.8 -38.1 9.6 -15.3 71.4

The header data (introduced with #) describe the frequency range (MHZ), the type of parameter (S), the type of display (DB = amplitude in dB and phase in degrees), and the normalization (R = 50 Ohm).

After the header data, there is an S parameter set in each data line. A data line begins with the specification of the frequency (in the order of magnitude as in the header), in this case 50 MHz, 51 MHz, etc., sorted by increasing frequency. The eight numerical values ​​that follow in the line, separated by whitespace , represent the S parameters S ₁₁ , S ₂₁ , S ₁₂ and S ₂₂ each with two values ​​in the form of the amount in decibels and phase angle in degrees Parameter S ₁₁ at 50 MHz is −15.4 dB at a phase angle of 100.2 °. Those data sets can be processed with programs such as Advanced Design System (ADS) in the field of circuit simulation .

The transmission coefficients are usually shown in a Cartesian diagram; for the reflection, a display in a Smith diagram is often preferred. In this way, the impedance of the measurement object can also be read directly and the adaptation can be optimized.

In practice, the S parameters have the following meaning:

S ₁₁ Input reflection Adjustment of the input. How well (or badly) is the input adapted to my reference system (50 Ohm or 75 Ohm). A low value indicates that an input signal is hardly reflected.

S ₂₁ Forward transmission Gain / attenuation of the input signal. In the case of an amplifier, S ₂₁ indicates the gain. In the case of a passive element, the insertion loss.

S ₁₂ backward transmission should correspond to S ₂₁ for passive bidirectional elements .

S ₂₂ Output reflection factor Adjustment of the output. How well (or badly) is the output adapted to my reference system (50 ohms or 75 ohms). If the match is poor, the output power is already reflected at the output.

Network analyzer manufacturers also offer devices with four test ports for measuring the M parameters.

T parameters

While the S-parameters describe the outgoing wave sizes of a 2- or n-port as a function of the incoming wave sizes, the T-parameters provide an alternative notation that shows the incoming and outgoing wave sizes at one gate as a function of the wave sizes at the other gate describes:

${\displaystyle {\begin{pmatrix}b_{1}\\a_{1}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{pmatrix}}{\begin{pmatrix}a_{2}\\b_{2}\end{pmatrix}}\,}$

This formulation allows components connected in series to be easily calculated by matrix multiplication.

${\displaystyle {\begin{pmatrix}T_{\text{Gesamt}}\end{pmatrix}}={\begin{pmatrix}T_{1}\end{pmatrix}}{\begin{pmatrix}T_{2}\end{pmatrix}}...{\begin{pmatrix}T_{n}\end{pmatrix}}\,}$

That is why the T parameters are also called wave chain parameters and the T matrix is ​​also called wave chain matrix in the literature.

Conversions between S and T parameters:

From S to T:

${\displaystyle T_{11}={\frac {-\det {\begin{pmatrix}S\end{pmatrix}}}{S_{21}}}=S_{12}-{\frac {S_{11}S_{22}}{S_{21}}}\,}$

${\displaystyle T_{12}={\frac {S_{11}}{S_{21}}}\,}$

${\displaystyle T_{21}={\frac {-S_{22}}{S_{21}}}\,}$

${\displaystyle T_{22}={\frac {1}{S_{21}}}\,}$

From T to S:

${\displaystyle S_{11}={\frac {T_{12}}{T_{22}}}\,}$

${\displaystyle S_{12}={\frac {\det {\begin{pmatrix}T\end{pmatrix}}}{T_{22}}}=T_{11}-{\frac {T_{12}T_{21}}{T_{22}}}\,}$

${\displaystyle S_{21}={\frac {1}{T_{22}}}\,}$

${\displaystyle S_{22}={\frac {-T_{21}}{T_{22}}}\,}$

Depending on the sources used, the following definition of the T parameters, which differs from the above and cannot be equated, is also used:

${\displaystyle {\begin{pmatrix}a_{1}\\b_{1}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{pmatrix}}{\begin{pmatrix}b_{2}\\a_{2}\end{pmatrix}}\,}$

M parameters

Two ports for differential line systems are described with the so-called M parameters, which are also referred to as mixed-mode parameters . These are closely related to the S parameters and can also be converted directly in the case of linear components. Such a differential two-port, in which the influences of the common-mode quantities are also taken into account, is called a two-port pair.

${\displaystyle {\begin{pmatrix}b_{1}^{-}\\b_{2}^{-}\\--\\b_{1}^{+}\\b_{2}^{+}\end{pmatrix}}={\begin{pmatrix}M_{11}^{-}&M_{12}^{-}&\!\mid \!&M_{11}^{-+}&M_{12}^{-+}\\M_{21}^{-}&M_{22}^{-}&\!\mid \!&M_{21}^{-+}&M_{22}^{-+}\\----&----&\!\!&----&----\\M_{11}^{+-}&M_{12}^{+-}&\!\mid \!&M_{11}^{+}&M_{12}^{+}\\M_{21}^{+-}&M_{22}^{+-}&\!\mid \!&M_{21}^{+}&M_{22}^{+}\end{pmatrix}}{\begin{pmatrix}a_{1}^{-}\\a_{2}^{-}\\--\\a_{1}^{+}\\a_{2}^{+}\end{pmatrix}}}$

Differential lines and components are used in high-frequency technology and high-speed digital and computer technology.

X parameters and S functions

The extension of the S-parameters to X-parameters makes it possible to determine the properties of non-linear components in high-frequency technology. In contrast to the classic S-parameters, the component to be measured is not excited by a single frequency stimulus, but by several frequencies. This also makes it possible to describe components in the large-signal range. In addition, these frequency-converting scatter parameters can be used to carry out vectorial intermodulation measurements. These measurements allow the IM interferers to be located.

literature

• Holger Heuer Men: high-frequency technology: components for high-speed and high frequency circuits . 2nd Edition. Vieweg + Teubner-Verlag, Wiesbaden 2009, ISBN 978-3-8348-0769-4 .
• David M. Pozar: Microwave Engineering . 3. Edition. Wiley, 2004, ISBN 978-0-471-44878-5 .

Web links

Touchstone® File Format Specification Rev 1.1. Retrieved October 9, 2019 . 

Touchstone® File Format Specification Version 2.0. Retrieved October 9, 2019 . 

swell

1. ↑ Peter Vielhauer : Linear networks . Verlag Technik, Berlin 1982.
2. ↑ S-parameter design ; Application Note AN 154; Agilent Technologies; Page 14ff (PDF; 851 kB)