Scattering parameters

Scattering parameters, abbreviated S- parameters are used to describe the behavior of linear electrical components and networks in the small signal behavior by means of shaft sizes. Application, see the S-parameters for the dimensioning and calculations in the field of high frequency technology, such as communication systems and the systems of telecommunications.

The importance of S- parameter is in particular in the field of metrology, as opposed to other parametric representations, such as the Z-, Y-, and H- parameters, the detection of the S- parameters of the characteristic impedance is carried out, which in normal operation at the terminals are available. This will in the measurements of S- parameters to the inputs and outputs of a network, due avoided through the necessary test leads and their spatial extent, unwanted impedance transformations.

The number of required S- parameter depends on the number of goals of the network, and results from the square of its goals is. For describing a Eintors ( dipole ), a single S- parameter is sufficient to a two-port will be fully described by means of four S- parameters of a three-port with nine, a four-port ( octopole ) sixteen S- parameters, and so on ( multiport ). The representation of a general linear Mehrtors means of S- parameters is always possible.

In addition to the dispersion parameter representation, there are linear networks with any number of signal ports, other network parameter representations, such as admittance parameters ( Y parameters, complex conductances ) or impedance parameters ( Z- parameters, complex resistance ). S-, Y - and Z- parameters can convert into each other. In this way, obtained by measuring the S- parameters for use in circuit simulations can be processed (for example, SPICE). This feature is in many simulation programs already in place. In contrast to the S- parameter representation of the existence of the Y- and Z- parameter representations of general linear More Goals can not be guaranteed universal, since the Y- matrix or the Z- matrix special More Goals is singular.

General

The S- parameters, the conditions at the gate of a network not directly by the electrical voltage applied Uν there currently are, and the electric current Iν flowing into the gate, described, but the description is this equivalent to a running into the gate wave and a reflected wave from the gate.

Based on the impedance Z0 of the measurement system at the gate to let the two representations from the following equations, which is also called the Heaviside transformation, set in relation ( we assume in the following that is positive real ):

And by reversing the relations:

The voltage Uν and the current Iν at the gate of the outwardly acting impedance Zν can interact:

Which can be described rν with the impedance Z0 of the measuring system, the reflection coefficient as:

In the simplest case of a scalar Eintors the reflection factor r is equal to the one and only S- parameter S11. In electrical networks with more than one gate, this relationship is expressed using a matrix equation in the form of a linear equation system. Generally, the S- parameters of a n-port network as a n x n matrix S, and the two elements each comprising n vectors a and b are expressed as a matrix equation:

Or in the element syntax:

Linear networks with any number of gates

More generally, one goes from a complex reference impedance of the target. Each port can be assigned an individual reference impedance and these need not necessarily be real-valued. The Heaviside transform ( We assume in the following that the real part of is positive ):

The squares of these expressions have the physical dimension of a performance. The air flowing into the gate active power is given by:

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

Z0 is the reference impedance, ie, the impedance of the test ports of the vector network analyzer used, usually 50 ohms.

E is the unit matrix.

Two goals

In the high-frequency technology, particularly Two goals play a significant role. Examples of two-ports are amplifiers or filters which have an input and an output. The S-parameters then comprise the elements S11, S12, S21 and S22:

A1 is the incoming wave at Gate 1, a2 the incoming wave at port 2. b1 describes the entrance (Gate 1) outgoing wave, b2 describes the output ( port 2 ) outgoing wave.

The S- parameters have the following meaning:

Represents the reflection at the entrance without Welleneinprägung at the gate 2 is:

Represents the reflection at port 2 without excitation at port 1 is:

Represents the forward transmission without excitation at port 2 is:

Represents the backward transmission without excitation at port 1 is:

Measurement

In practice, the S- parameters with the help of network analyzers are measured as a function of frequency. S- parameters are dimensionless complex numbers, referred to as phasor that are specified in terms of magnitude in decibels (dB ) and phase in degrees ( °). The wave impedance is typically set at 50 Ω.

Based on the measured amount of data practically all network analyzers have the ability to store data of the S - parameters of data carriers. A common data format is the Touchstone file format with the file extension " s2p ". This data format is a text file with the following structure is:

! Created Fri July 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 After the header data is in each data line S- parameter set. A data line begins with the indication of the frequency, in this case, 50 MHz, 51 MHz, etc. in the line following eight numerical values ​​represent the S- parameters S11, S21, S12 and S22, each with two values ​​in the form of the amount in decibels and phase angle dar. in degrees in the above extract, for example, has the parameter S11 at 50 MHz -15.4 dB the amount at a phase angle of 100.2 °. Those records can be processed with programs such as Advanced Design System (ADS ) in the area of ​​circuit simulation.

In the illustration, the transmission coefficients are usually represented in a cartesian diagram, for the reflection display in the Smith chart is often preferred. So can also directly read the impedance of the measuring object and optimize the fit.

In practice, the S- parameters have the following meaning:

S11 adjustment of the input. How good (or bad) the input on my reference system (50 ohms or 75 ohms ) is adapted. A low ( absolute amount: as high as possible ) value indicates that an input signal is barely reflected.

S21 gain / attenuation of the input signal. When an amplifier is S21 the gain. In a passive element, the insertion loss.

S12 should correspond with passive bidirectional elements S21.

S22 adaptation of the output. How good (or bad) the output is adjusted to my reference system (50 ohms or 75 ohms). In poor matching the output power is already reflected at the output.

Manufacturer of network analyzers also offer devices with four test ports for measuring the M parameter.

T parameter

While the S-parameters of the outgoing wave variables of a 2 - describe or n-port device as a function of the incoming wave sizes, represent the T parameter is an alternative spelling available, which the incoming and outgoing wave sizes at a gate as a function of wave parameters at the other gate describes:

This wording can be easily cascaded components calculated by matrix multiplication.

Conversions between S and T parameters:

From S to T:

From T to S:

Depending on the sources used in the following formulation of the T- parameter is used.

M parameter

Two goals for differential line systems to describe the so-called M- parameters, which are also referred to as mixed-mode parameters. These are in close relationship to the S- parameters and can be converted directly with linear components. Such a differential two port in which the influences of the common mode sizes to be considered is referred to as Zweitorpaar.

Differential lines and components found in the high-frequency technology and high-speed digital and computer technology use.

X - parameters and S-functions

The expansion of the S- parameters for X- parameters allows to determine the properties of non-linear components in the high frequency technology. The component to be measured will not be excited by a stimulus at a frequency but by multiple frequencies in contrast to the conventional S- parameters. In this way it is also possible to describe components in the large signal range. Furthermore, this can be frequency-converting scattering parameter vector Intermodulation Measurements. These measurements allow detection of IM interferers.