DE102014019008B4 - Procedure for calibrating a measurement adaptation with two differential interfaces - Google Patents

Abstract

Method for calibrating a measuring arrangement (100) for measuring electrical components (14), comprising at least one measuring adapter (10) which can be connected to a network analyzer (20) and has four measuring ports (1, 2, 3, 4) which are located in a calibration plane (15 ) are arranged for connecting the electrical component (14) and form two differential interfaces (1/2 and 3/4), wherein the measuring adapter (10) can be described by a linear 8-port error network (C1,2,3,4). ,characterized by the following steps:• Combining 2 test ports (1/2; 3/4) of a differential interface to form a modal port (I, II),• Representation of the error network in modal representation, in which linear errors in the push-pull path for both modal ports are described by a 4x4 matrix CI-II,dd and linear errors in the common-mode path for both modal ports by a 4x4 matrix CI-II,cc, neglecting intermodal coupling between push-pull path and common-mode path,• determining the respective eils 15 error terms of the matrices CI-II,dd and CI-II,cc using a 15-term calibration method,• determining a multiplicative link factor between the determined 15 error terms of the matrix CI-II,cc and the matrix CI-II,dd using a reciprocal Standards (R).

Description

The present invention relates to a method for calibrating a measurement arrangement for measuring electrical components (device under test, DUT) such as RF components. Such a measurement arrangement usually includes a network analyzer, such as a vectorial network analyzer VNA, with which the scattering parameters (such as reflection and transmission factors) can be determined by measuring the wave sizes at ports of the electrical component to be measured as a function of the frequency up to a few GHz . A network analyzer regularly has a calibrated reference level with one, two or more test ports for connecting the electrical component to be measured.

When measuring with a network analyzer, the wave sizes of the outgoing and returning waves are determined according to the real and imaginary parts (coherent measuring points) using a frequency range measuring principle. The measuring method enables a very precise determination of the scattering parameters, since all systematic, linear errors can be removed with the help of a system error correction through a subsequent calculation. On their way to and from the DUT, the measurement signals emitted by the network analyzer pass through various electrical components such as lines and directional couplers with non-ideal electrical properties, which can lead to systematic errors in the network analysis. For this reason, calibration measurements are carried out before the actual DUT measurements in order to determine the electrical properties of the measurement arrangement itself and then to be able to correct/compensate them from the measurement signals. In relation to a freely definable reference level, the influence of cables or measurement adaptations can be fully compensated. For this purpose, a linear error network is defined between the measuring points of the wave quantities in the network analyzer and the reference level ("calibration level"). Calibration methods enable the error coefficients of these error networks to be determined, which can differ significantly depending on the application. The DUT is then connected exactly at the calibration level, where the properties of the measurement setup after calibration are known.

Numerous different calibration methods are known from the prior art. Depending on how many test ports the network analyzer and the DUT have and which properties of the test setup and the DUT can be assumed (symmetry, reciprocity, etc.), certain calibration methods are suitable. For example, a 12-term calibration method or the simpler TSD calibration method can be used to calibrate a two-port VNA to successively measure the four scattering parameters of a two-port DUT. A basic distinction is also made between error models with decoupled and coupled test ports. Coupled error models take into account electrical coupling between the reflectometers of the VNA or crosstalk across the DUT, which is given, for example, by the measurement arrangement. If more than two test ports are considered, partial coupling between any test ports or between all test ports can also be taken into account. A calibration method for two-port VNAs that models the coupled error network works according to the 15-term method (sometimes also called the 16-term method). In this context, for example, reference is made to the publications EP 0 706 059 B1 , DE 433 2273 C2 and DE19736897C2 referred. The error network between the measuring points and the calibration level of a two-port VNA can be described by a 4X4 error matrix with sixteen error terms, of which the determination of 15 error terms is sufficient for ratio measurements such as scatter parameter measurements.

There are basically two types of calibration methods: In the case of direct calibration methods, calibration standards that can be described very precisely electrically must be connected to the reference plane to be calibrated. Knowledge of these standards is achieved through special mechanical structures and high manufacturing accuracy. In contrast to this are the self-calibration methods, which only require limited accuracy with regard to the knowledge of the electrical properties, since these properties are determined automatically in the course of the calibration process through the use of redundancy or special properties of the standards used.

However, some electrical components to be measured do not have the appropriate interface for direct connection to the measurement ports of the VNA in the calibrated reference plane of the VNA (hereinafter: "pre-calibration plane"). Measuring adapters can be used to measure such components, which can be coupled to the pre-calibration level of the VNA on the one hand and to the measurement inputs of the electrical component to be measured on the other. The problem with such measuring adapters, however, is that they can in turn introduce systematic linear errors that require their own correction with regard to measuring results that are as exact as possible. It may therefore be necessary to also add the To calibrate measuring adapter formed measuring adaptation. The measurement ports of the measurement adapter for connecting the measurement object to be measured are arranged in the calibration level.

A test adapter with 4 test ports (i.e. with 4 input ports for connecting to the VNA and 4 output ports for connecting the DUT) can be described by a linear 8-port error network, which basically requires the determination of a total of 64 error terms. However, calibration methods for calibrating such a measurement adapter would be extremely complex and time-consuming, especially if a (partial or full) coupling between the individual test ports is to be taken into account, so that the calibration of test adapters with 4 or more test ports is usually dispensed with and as a result if necessary, systematic measurement errors must be accepted.

In view of the problems described, it is the object of the present invention to provide a simple and reliable calibration method for calibrating a measurement arrangement consisting of a network analyzer and a measurement adapter with four test ports, the four test ports of the measurement adapter being arranged in the calibration plane for connecting the electrical component (DUT). and form two differential interfaces.

• • Zuordnen von jeweils 2 Messtoren, die zu einer differentiellen Schnittstelle gehören, zu einem Modaltor I, II,
• • Darstellung des Fehlernetzwerks in modaler Repräsentation, in der lineare Fehler im Gegentaktpfad für beide Modaltore durch eine 4x4 Matrix C_{I-II, dd} und lineare Fehler im Gleichtaktpfad für beide Modaltore durch eine 4x4 Matrix C_{I-II, cc} beschrieben werden, unter Vernachlässigung von Intermodalverkopplung zwischen Gegentaktpfad und Gleichtaktpfad,
• • Bestimmen von jeweils 15 Fehlertermen der Matrizen C_{I-II, dd} und C_{I-II, cc} unter Verwendung eines 15-Term Kalibrierverfahrens,
• • Bestimmen eines multiplikativen Verknüpfungsfaktors zwischen den bestimmten 15 Fehlertermen der Matrix C_{I-II, cc} und der Matrix C_{I-II, dd} unter Einsatz eines Reziprok-Standards R.

This task is solved by a calibration procedure with the following steps:

• • Assignment of 2 test ports that belong to a differential interface to a modal port I, II,
• • Representation of the error network in modal representation, in which linear errors in the push-pull path for both modal ports are described by a 4x4 matrix C _{I-II, dd} and linear errors in the common-mode path for both modal ports are described by a 4x4 matrix C _{I-II, cc} , with neglect of intermodal coupling between push-pull path and common-mode path,
• • Determination of 15 error terms each of the matrices C _{I-II, dd} and C _{I-II, cc} using a 15-term calibration method,
• • Determine a multiplicative link factor between the determined 15 error terms of the matrix C _{I-II, cc} and the matrix C _{I-II, dd} using a reciprocal standard R.

These error terms of the two matrices C _{I-II, dd} and C _{I-II, cc} fully characterize the error network introduced by the measuring adapter. From the error terms of the matrices C _{I-II, dd} and C _{I-II, cc} determined in this way, correction values can then be calculated, which are taken into account in subsequent measurements of scattering parameters of the electrical component DUT.

On the one hand, the invention is based on the knowledge that, in order to calibrate measurement adaptations with four test ports, it is not always necessary to determine all 64 error terms of the error matrix of the associated 8-port error network. Rather, under certain conditions, there is a representation of the error matrix in which 32 of the 64 error terms are negligible or even disappear completely. This is the case in particular when the four test ports of the measurement adapter form two differential interfaces in the calibration level, and of symmetrical or essentially the same electrical properties of the two test ports that each form a differential interface and the electrical lines and components connected to them on the measuring adapter can be assumed. This simplified representation of the error matrix corresponds to the modal representation of the error network, in which the HF signals passed through the measuring adapter are broken down into differential mode and common mode. In this case, the two test ports assigned to a differential interface are combined to form a modal port, through which the test signals are routed in common mode and differential mode, with intermodal coupling between the differential mode path and the common mode path being intended to be negligible.

These requirements are met in particular when the electrical lines and test ports of the differential interfaces on the test adapter are arranged in a way that corresponds to one another, so that symmetrical electrical properties result for the test ports that belong together. On the other hand, a coupling between the push-pull modes of one modal gate and the push-pull modes of the other modal gate, and between the common-mode modes of one modal gate and the common-mode modes of the other modal gate, should be taken into account. In this context, reference is made to the textbook [Holger Heuermann: "High-frequency technology: components for high-speed and high-frequency circuits", Springer 2009], in which the basics of the modal representation of multiports are described, which are incorporated by reference into the present disclosure .

The invention is now based on the knowledge that the 15-term calibration methods already mentioned at the outset, which were previously used with nodal representation of the error network to be determined, can be transferred to the modal representation. After showing the error network in modal representation, in which linear errors in the push-pull path for both modal ports are described by the 4x4 matrix C _{I-II, dd} and linear errors in the common-mode path for both modal ports are described by the 4x4 matrix C _{I-II, cc} , under neglect of intermodal coupling between push-pull path and common-mode path, it is therefore possible to determine the 15 error terms of the error matrices C _{I-II, dd} and C _{I-II, cc} using a 15-term calibration method adapted to the modal representation. Thus, according to the invention, the determination of the 64 parameters of the nodal 8×8 error matrix can be reduced to the determination of two modal 4×4 error matrices by a modally applied 15-term calibration method. Since only 15 terms per error matrix C _{I-II, dd} and C _{I-II, cc} are determined, a normalization of the two 4x4 error matrices is necessary, in which, for example, one of the error terms of each matrix is set to 1.

However, this does not yet determine the relative ratio between the two error matrices C _{I-II, dd} and C _{I-II, cc} . To know the complete error network, a relative link between the two error matrices C _{I-II, dd} and C _{I-II, cc} or a denormalization of one of the two error matrices C _{I-II, dd} and C _{I-II, cc} is also required .

On the other hand, the invention is based on the surprising finding that such denormalization is possible with comparatively simple means. According to the invention, the multiplicative linkage factor between the determined 15 error terms of the matrix C _{I-II, cc} and the matrix C _{I-II, dd} is determined by using a reciprocal standard R, which can be connected to the four test ports of the measuring adapter. The 4-port reciprocal standard R is necessarily reciprocal, but beyond that may be unknown in its electrical properties. This method is based on the fact that the determinant of the transmission matrix of any reciprocal network is always one and can therefore be assumed to be known in a cascade representation. As a result, the calibration measurement results in the unknown multiplicative linkage factor for denormalization only with a phase ambiguity that can be resolved with approximate knowledge of the geometric length of the network.

It is thus possible according to the invention to characterize the complete fault network, which an intermediate measuring adapter with 4 measuring ports causes, with comparatively simple means and without a great deal of time, if it can be assumed that intermodal coupling between the push-pull path and the common-mode path can be neglected. This is particularly possible when the test ports of the test adapter are constructed in the manner of a double differential interface. A transformation between the error matrices or the wave quantities in modal and in nodal representation is possible using known linear transformations, so that measurements on electrical components can be carried out modally or nodally and then converted accordingly. Reference is made to the relevant textbooks known to the person skilled in the art, such as the textbook “High-frequency technology: Components for high-speed and high-frequency circuits” already mentioned above.

The error terms resulting from the calibration method according to the invention are particularly precise when the adapter is constructed in such a way that the electrical coupling between the two test ports of the first differential interface essentially corresponds to the electrical coupling between the two test ports of the second differential interface, and/or that the electrical coupling between one of the test ports of the first differential interface and one of the test ports of the second differential interface essentially corresponds to the coupling between the two other test ports. The measuring adapter thus has the symmetry properties required for complete disappearance of the error terms, which are negligible according to the method according to the invention. A comparable electrical coupling between two pairs of test ports and the respectively associated pairs of lines results in particular when these are arranged in an analogous arrangement, in particular with a comparable mutual distance and comparable mutual course on the measuring adapter.

The measuring adapter is preferably set up essentially symmetrically with regard to all four measuring ports.

With regard to a simple construction of the measuring adapter, it has proven to be expedient for the measuring adapter to be connected to a calibrated pre-calibration level of the network analyzer via four input ports, preferably set up as coaxial interfaces. Traditional network analysis In the pre-calibration level, ports regularly have coaxial interfaces (e.g. SMA) for connecting test objects. The four coaxial interfaces can protrude in a star shape from a central carrier such as a circuit board of the measuring adapter.

Alternatively or additionally, the two differential interfaces of the measuring adapter have four contact elements, preferably provided in a square arrangement on the measuring adapter, for contacting an electrical component having two differential conductor pairs, such as a 4-port calibration standard or a star quad cable. Such a measuring adapter has on the one hand the symmetry properties preferred according to the invention and on the other hand is particularly suitable for measuring DUTs with two differential interfaces without the need for further adapter elements. The four contact elements can be arranged in the center of a carrier such as a circuit board of the measuring adapter. In this case, preferably two contact elements diagonally opposite each other form one of the differential interfaces.

The four contact elements of the two differential interfaces (the output ports) of the measuring adapter are preferably electrically connected to a coaxial interface of the measuring adapter (the input ports) via electrical lines.

A calibration method is required that performs the calibration of a modal 15-term error model with intermode decoupling (fully coupled error model for two ports per mode). A self-calibration method has proven to be particularly useful for this purpose, since it places fewer demands on the calibration standards to be used.

The TMRG or LRM methods known for the nodal two-port case have proven to be particularly suitable self-calibration methods. The calibration standards required for this can be provided comparatively easily, and a transfer to the four-port case (or modal two-port case) present here is possible with manageable complexity.

In particular with the TMRG method mentioned, it is possible to use partially or even exclusively lumped calibration elements for the determination of the 15 error terms of the matrices C _{I-II, dd} and C _{I-II, cc} . In contrast to extended calibration elements (e.g. line as an impedance standard that can only be used in a finite frequency interval), lumped elements can be electrically short (ie much shorter than the wavelength of the frequency used). Such lumped elements, which in turn represent one-port reflection standards, are, for example, resistors, wave sump/impedance standard (Match M), short circuit (Short S, non-ideal/ largely unknown), open circuit (Open O, non-ideal/ largely unknown). The thru (T) is an ideal or fully known connection line between two ports.

In a particularly preferred embodiment, at least three, preferably four, and in particular five of the following calibration standards are used: Thru, match-open MO, short-match SM, short-open SO, open-short OS. In this case, the respective two-port calibration standard preferably appears in modal representation at the common-mode and differential-mode error networks. In the actual/real nodal representation, two measurement ports are terminated with one and the other two measurement ports of the measurement adapter with the other measurement standard. In particular, the first letter denotes the termination of the test ports of one differential interface and the second letter denotes the termination of the test ports of the other differential interface. The through connection Thru connects two test ports to each other twice, preferably two test ports that do not belong to the same differential interface.

For the step of determining the multiplicative link factor between the 15 error terms of the matrix C _{I-II, cc} and the matrix C _{I-II, dd} , a reciprocal standard R is preferably used, which has a high coupling of common and differential modes at one Has a modal gate or which is set up asymmetrically with respect to the two test gates, each with a differential interface. Such a reciprocal standard can be set up in such a way that it terminates two test ports with a (non-ideal) short circuit and two test ports with a (non-ideal) open circuit. For example, the R standard provides a short S (non-ideal) and an open O (non-ideal) at the two test ports of each differential interface.

According to a further aspect, the present invention relates to a method for calibrating a measurement adaptation with 8 measurement ports. Such a measurement adaptation can be between the network analyzer and two the electrical component to be measured include switched measuring adapters, each with four measuring ports. The two measuring adapters can also be formed integrally. In this case, two associated test ports of a test adapter form a differential interface with the symmetry properties explained above. Each of the two measuring adapters is preferably constructed like the measuring adapter of the first embodiment of the invention described above. It is important that electromagnetic coupling between the two measuring adapters is negligible. This can be achieved particularly well if the two measuring adapters are arranged spatially separate from one another during calibration and during measurement, so that the lines and interfaces of the two measuring adapters cannot influence each other electrically. For example, the lines and interfaces of the two measuring adapters are provided on two separate carriers such as printed circuit boards.

Such a measurement adaptation can be used, for example, to determine the scattering parameters of 8-port test objects, in which two ports each form a differential interface. Such an 8-port device under test is, for example, a cable with two twisted pairs of wires, in particular a cable in a star quad arrangement, which has two differential pairs of wires at each cable end (e.g. high-speed data HSD interfaces).

• • Zuordnen der jeweils 2 Messtore einer differentiellen Schnittstelle der Messadapter zu einem Modaltor,
• • Darstellung der die Messadapter beschreibenden Fehlernetzwerke in modaler Repräsentation, in der je Messadapter lineare Fehler im Gegentaktpfad für beide Modaltore durch eine 4×4 Fehlermatrix (C_{I-II, dd}, C_{III-IV, dd}) und lineare Fehler im Gleichtaktpfad für beide Modaltore durch eine 4x4 Fehlermatrix (C_{I-II, cc}, C_{III-IV, cc}) beschrieben werden, unter Vernachlässigung von Intermodalverkopplung zwischen Gegentaktpfad und Gleichtaktpfad,
• • Bestimmen von jeweils 15 Fehlertermen der Fehlermatrizen C_{I-II, dd}, C_{I-II, cc}, C_{III-IV, dd}, C_{III-IV, cc} unter Verwendung eines 15-Term Kalibrierverfahrens,
• • Bestimmen eines multiplikativen Verknüpfungsfaktors zwischen den bestimmten 15 Fehlertermen der Matrizen C_{I-II, cc} und C_{I-II, dd} und eines zweiten multiplikativen Verknüpfungsfaktors zwischen den bestimmten 15 Fehlertermen der Matrizen C_{III-IV, cc} und C_{III-IV, dd} unter Einsatz eines Reziprok-Standards R,
• • Bestimmen eines dritten multiplikativen Verknüpfungsfaktors zwischen Fehlertermen, die für den ersten Messadapter bestimmt wurden, und Fehlertermen, die für den zweiten Messadapter bestimmt wurden, unter Einsatz eines weiteren Reziprok-Standards R₂.

The calibration method according to this second embodiment of the invention comprises the following steps:

• • Assignment of the 2 test ports of a differential interface of the test adapter to a modal port,
• • Representation of the error networks describing the measuring adapters in modal representation, in which, for each measuring adapter, linear errors in the push-pull path for both modal ports by a 4×4 error matrix (C _{I-II, dd} , C _{III-IV, dd} ) and linear errors in the common-mode path for both Modal gates are described by a 4x4 error matrix (C _{I-II, cc} , C _{III-IV, cc} ), neglecting intermodal coupling between push-pull path and common-mode path,
• • Determination of 15 error terms of the error matrices C _{I-II, dd} , C _{I-II, cc} , C _{III-IV, dd} , C _{III-IV, cc} using a 15-term calibration method,
• • determining a multiplicative link factor between the determined 15 error terms of matrices C _{I-II, cc} and C _{I-II, dd} and a second multiplicative link factor between the determined 15 error terms of matrices C _{III-IV, cc} and C _{III-IV, dd} using a reciprocal standard R,
• • Determining a third multiplicative link factor between error terms determined for the first measurement adapter and error terms determined for the second measurement adapter using a further reciprocal standard R ₂ .

A generalization of the method according to the invention to more than two adapters is possible if a coupling between the individual adapters is negligible.

An 8-port standard with twice two differential interfaces for connection to the two differential interfaces of the two measuring adapters is preferably used as a further reciprocal standard R ₂ .

A 4-wire cable, in particular a star-quad cable, can be used as a further reciprocal standard R ₂ , with the differential wire pairs being connected at both ends of the cable to the four test ports or to the differential interfaces of the test adapters formed thereby for the calibration measurement.

According to a further aspect, the invention relates to a set of calibration standards for use in a calibration method according to the invention, comprising at least three, preferably four or five, concentrated 4-port calibration standards and at least one 4-port reciprocal standard R for calibrating a measurement adaptation with at least four measurement ports , which form two differential interfaces, in particular for calibrating a measurement arrangement consisting of a network analyzer and measurement adapter with four or eight test ports.

The at least three concentrated 4-port standards are preferably selected from the following calibration standards: Thru; MO, SM, SO, OS, where the first letter indicates the termination of two measurement ports of the measurement adaptation and the second letter indicates the termination of the other two measurement ports of the measurement adaptation. In particular, the first letter designates the conclusion of the two measuring gates, the one form the differential interface of the measurement adaptation, and the second letter the termination of the two measurement ports that form the other differential interface of the measurement adaptation.

For this purpose, the individual calibration standards are set up in such a way that they can be connected to a double differential interface, which preferably has four contact elements in a square arrangement, with two diagonally opposite contact elements forming a differential interface.

The set of calibration standards according to the invention can also have an 8-port reciprocal standard for connection to two measurement adapters, each measurement adapter having a double differential interface. The 8-port reciprocal standard can be provided in the form of a cable, preferably a 4-wire cable, in particular a star quad cable.

Furthermore, the invention relates to a measurement assembly consisting of one or two measurement adapters described above and the associated set of calibration standards according to the invention for their calibration.

• 1 einen Messadapter mit vier Messtoren einer Messanordnung, die gemäß dem erfindungsgemäßen Verfahren kalibriert wird, in einer perspektivischen Darstellung,
• 2 ein zu kalibrierendes, nodales Fehlermodell eines an die Vorkalibrierebene eines VNAs angeschlossenen Messadapters,
• 3 ein zu kalibrierendes, modales Fehlermodell des an die Vorkalibrierebene des VNAs angeschlossenen Messadapters,
• 4a eine schematische Darstellung, die die Übertragung eines nodalen Kalibrierverfahrens auf den modalen Fall veranschaulicht, wobei jeweils Reflexions-Kalibrierstandards verwendet werden,
• 4b eine schematische Darstellung, die die Übertragung eines nodalen Kalibrierverfahrens auf den modalen Fall veranschaulicht, wobei jeweils eine Durchverbindung (Thru) verwendet wird,
• 5 eine beispielhafte Zusammenstellung eines erfindungsgemäßen Satzes von Kalibrierstandards (oben), die an die zwei differentiellen Schnittstellen (unten) eines in 1 dargestellten Messadapters anschließbar sind,
• 6 eine perspektivische Darstellung zweier Messadapter mit jeweils vier Messtoren, an denen ein zu vermessendes Messobjekt (DUT) in Form eines Kabels mit zwei differentiellen Aderpaaren angeschlossen ist.
• 7 ein zu kalibrierendes, nodales Fehlermodell zweier an die Vorkalibrierebene eines VNAs angeschlossener Messadapter, und
• 8 ein zu kalibrierendes, modales Fehlermodell der zwei an die Vorkalibrierebene des VNAs angeschlossenen Messadapter.

The invention is explained below with reference to the attached drawings. Furthermore, a calibration process including the mathematical transformations carried out is presented in detail. All scalars and resulting quantities are complex quantities unless otherwise specified, scalars are written in lowercase and italics, vectors are written in lowercase and bold, and matrices are written in uppercase and bold. It is readily apparent to those skilled in the art that the calibration process explicitly presented is merely exemplary, and that alternative notation, representations, orders, transformations, etc. can be used to achieve the same result. The method according to the invention is only limited by the appended claims. It shows:

• 1 a perspective view of a measuring adapter with four measuring ports of a measuring arrangement that is calibrated according to the method according to the invention,
• 2 a nodal error model to be calibrated of a measurement adapter connected to the pre-calibration level of a VNA,
• 3 a modal error model to be calibrated of the measurement adapter connected to the pre-calibration level of the VNA,
• 4a a schematic representation that illustrates the transfer of a nodal calibration method to the modal case, using reflection calibration standards in each case,
• 4b a schematic representation that illustrates the transfer of a nodal calibration method to the modal case, using a thru in each case,
• 5 an exemplary compilation of a set of calibration standards according to the invention (above), which are connected to the two differential interfaces (below) of an in 1 shown measuring adapter can be connected,
• 6 a perspective view of two measurement adapters, each with four test ports, to which a device under test (DUT) to be measured is connected in the form of a cable with two differential wire pairs.
• 7 a nodal error model to be calibrated of two measurement adapters connected to the pre-calibration level of a VNA, and
• 8th a modal error model to be calibrated of the two measurement adapters connected to the pre-calibration level of the VNA.

1 shows a perspective view of a measurement adapter 10 with four measurement ports for a measurement arrangement that is calibrated according to the method according to the invention.

A measuring arrangement 100 with measuring adapter 10 to be calibrated using the method according to the invention is shown schematically in FIG 2 shown. It includes a vector 4-port network analyzer VNA 20 with a calibrated pre-calibration level 12 with four test ports for connecting an electrical component DUT.

However, the measurement adapter 10 is connected to the precalibration level 12 instead of a 4-port DUT connected directly to the precalibration level 12 . The measuring adapter 10 is between the network analyzer 20 and the 4-port DUT 14 to be measured are interposed. The measurement adapter 10 has four input ports 1', 2', 3', 4' for connecting to the precalibration level 12 of the network analyzer 20 and four output ports for connecting the DUT 14 in the calibration level 15, so it includes four test ports 1, 2, 3 , 4. The four test ports 1, 2, 3, 4 form two differential interfaces 1/2 and 3/4. The measuring adapter 10 thus represents an 8-port error network 13 which can be described by an 8×8 error matrix C _1,2,3,4 with 64 error terms. In order to determine the scatter matrix S _SE of the DUT 14 without systematic errors introduced by the measuring adapter 10, the error terms of the error matrix C _1,2,3,4 must be determined. This is possible in a particularly simple and elegant way using the calibration method according to the invention, which is described below:

The calibration method described here develops a new approach for a special type of error network using 4-port calibration standards based on a self-calibration method. This addresses measurements of DUTs with four ports. The error network 13 is said to have the property that four individual measurement ports 1, 2, 3, 4 each form a measurement adapter 10 and may be coupled to one another in front of or on the calibration plane 15 due to the given spatial proximity. Furthermore, the description of a measurement adapter 10 has symmetry with respect to the four individual paths.

When replacing the nodal with the modal view by combining two of the four test ports 1, 2, 3, 4, a decoupling of the common mode and differential mode description is achieved. All error networks are thus intermodally decoupled from each other. However, there is still an intramode coupling for each measurement adapter 10 (i.e. coupling between the modal gates may exist with regard to one mode in each case), which must be taken into account within the system error correction and a suitable self-calibration method. After determining all error networks of common mode and differential mode description, these must be set in relation to each other. This process corresponds to the denormalization of one of the two single error networks. For this purpose, the calibration method described here uses a respective reciprocal calibration element R, which basically connects both modes with one another, but only has to have the property of reciprocity (exchangeability of the directional dependency). A concrete realization is also presented.

The measurement adapter 10 serves accordingly as an application example 1 . For the time being, all wave sizes and the resulting scattering parameters follow a nodal description (also called asymmetrical or unbalanced). According to the 1 and 2 the pairs of gates (I, II) necessary for a modal description (also called mixed mode, symmetrical or balanced) as well as two levels for the measuring station are defined. The DUT 14 has four nodal ports and is connected to the 2 specified wave sizes or the nodal 4 × 4 scattering matrix S _SE at the calibration level 15 described.

The precalibration level 12 is provided coaxially with standard connectors, for example by a commercially available network analyzer 20, and is in a calibrated state, ie conventional calibration and system error correction methods are used. The associated wave sizes of the incoming waves (a) and the returning waves (b) of this measurement are marked with a subscript "m". All systematic, linear errors are combined between the pre-calibration level 12 and the calibration level 15 by the 8-port error network, which is described with the 8×8 error matrix C _1,2,3,4 .

With knowledge of all the coefficients of the error network, the sought-after scattering matrix S _SE of the measurement object 14 can finally be calculated. There are a total of 64 coefficients that must be determined for each relevant frequency.

Assuming that all elements are linear, the common mode description and the differential mode description are selected for a modal description of all wave quantities and the associated scattering parameters. In this connection reference is made to the publication [David E. Bockelman and William R. Eisenstadt. "Combined differential and commonmode scattering parameters: Theory and simulation", Microwave Theory and Techniques, IEEE Transactions on 43.7 (1995), pp. 1530-1539], the content of which is incorporated by reference into the present description.

For this purpose, nodal gates 1 and 2, which form the first differential interface, are combined to form modal gate / and nodal gates 3 and 4, which form the second differential interface, to form modal gate II. In addition, the error network is assumed to have such a symmetry with regard to the nodal properties of port 1 to 2 and 3 to 4 that an intermodal coupling within the modal error learning networks is negligible. In practice, this is guaranteed by a pre-calibrated level for connecting the measurement adaptation and its symmetrical structure. In 3 the resulting total error network is now shown, which also takes into account an intramodal coupling of both modes between gate / and II. Linear errors are combined in the push-pull path with a 4-port error network, which is described with the 4x4 matrix C _I-II,dd (see also eq.(8)). In addition, linear errors in the common-mode path are combined with a 4-port error network, which is described with the 4x4 matrix C _I-II,cc (see also eq.(9)). The total of two 4-port error networks contain 32 error coefficients that can be determined using the self-calibration method described below. In detail--as is usual and permissible in a calibration or system error correction--any error coefficient is standardized and finally the remaining 31 unknowns are determined.

• Die Wellengrößen von 3 in ihrer modalen Darstellung am Messobjekt DUT 14 werden zu Wellen-Vektoren zusammengefasst.

The calibration procedure is explained below:

• The wave sizes of 3 in their modal representation on the test object DUT 14 are combined to form wave vectors.

(1) mit $a_d:={\left[a_{dI},a_{dII}\right]}^T,a_c:={\left[a_{cI},a_{cII}\right]}^T$

(2) Incoming Waves: $a:={\left[a_{\mathrm{i.e}I},a_{\mathrm{i.e}II},a_{cI},a_{cII}\right]}^T={\left[a_{\mathrm{i.e}},a_c\right]}^T$

(1) With $a_{\mathrm{i.e}}:={\left[a_{\mathrm{i.e}I},a_{\mathrm{i.e}II}\right]}^T,a_c:={\left[a_{cI},a_{cII}\right]}^T$

(2)

(3) mit $b_d:={\left[b_{dI},b_{dII}\right]}^T,b_c:={\left[b_{cI},b_{cII}\right]}^T$

(4) Returning Waves: $b:={\left[b_{\mathrm{i.e}I},b_{\mathrm{i.e}II},b_{cI},b_{cII}\right]}^T={\left[b_{\mathrm{i.e}},b_c\right]}^T$

(3) With $b_{\mathrm{i.e}}:={\left[b_{\mathrm{i.e}I},b_{\mathrm{i.e}II}\right]}^T,b_c:={\left[b_{cI},b_{cII}\right]}^T$

(4)

(5) Accordingly, the modal 4×4 scattering matrix S _MM of a measurement object is defined as follows: $b=S_{MM}\cdot a\text{respectively}.\left[\begin{array}{c}{b}_{\mathrm{i.e}}\\ b_c\end{array}\right]=S_{MM}\cdot \left[\begin{array}{c}{a}_{\mathrm{i.e}}\\ a_c\end{array}\right]$

(5)

(6) This scattering matrix can in turn be split into 2×2 individual matrices, which describe the intramode (S _dd , S _cc ) and intermode (S _cd , S _dc ) behavior of the DUT: $S_{MM}=\left[\begin{array}{cc}{S}_{\mathrm{i.e}\mathrm{i.e}}& S_{\mathrm{i.e}c}\\ S_{c\mathrm{i.e}}& S_{cc}\end{array}\right]=\left[\begin{array}{cccc}{S}_{\mathrm{i.e}I\mathrm{i.e}I}& S_{\mathrm{i.e}I\mathrm{i.e}II}& S_{\mathrm{i.e}IcI}& S_{\mathrm{i.e}IcII}\\ S_{\mathrm{i.e}II\mathrm{i.e}I}& S_{\mathrm{i.e}II\mathrm{i.e}II}& S_{\mathrm{i.e}IIcI}& S_{\mathrm{i.e}IIcII}\\ S_{cI\mathrm{i.e}I}& S_{cI\mathrm{i.e}II}& S_{cIcI}& S_{cIcII}\\ S_{cII\mathrm{i.e}I}& S_{cII\mathrm{i.e}II}& S_{cIIcI}& S_{cIIcII}\end{array}\right]$

(6)

(7) definiert. Sie ähnelt in ihrer Form einer Transmissionsmatrix, verknüpft jedoch statt Einzeltoren nun Einzelmoden und eignet sich in dieser Form zur Kaskadierung von Netzwerken unterschiedlicher Moden (ohne Intermodenkopplung). Diese Beschreibungsform ist essenziell für das weitere Vorgehen (siehe z.B. GI.(22)) und darüber hinaus als neuwertig herauszustellen.The 4×4 mode conversion matrix of the DUT T _MM is given as $\left[\begin{array}{c}{b}_{\mathrm{i.e}}\\ a_{\mathrm{i.e}}\end{array}\right]=T_{MM}\cdot \left[\begin{array}{c}{a}_c\\ b_c\end{array}\right]$

(7) Are defined. Its form is similar to that of a transmission matrix, but now it links single modes instead of single ports and is suitable in this form for cascading networks of different modes (without intermode coupling). This form of description is essential for the further procedure (see eg eq. (22)) and should also be emphasized as new.

(8) $\left[\begin{array}{c}{a}_{cI}\\ a_{cII}\\ b_{cI}\\ b_{cII}\end{array}\right]=C_{I-II,cc}\cdot \left[\begin{array}{c}{a}_{m,cI}\\ a_{m,cII}\\ b_{m,cI}\\ b_{m,cII}\end{array}\right]$

(9) The two in 3 The error networks introduced (described by the matrices C _I-II,dd and C _I-II,cc ) are used to link the wave quantities at pre-calibration level 12 and calibration level 15: $\left[\begin{array}{c}{b}_{\mathrm{i.e}I}\\ b_{\mathrm{i.e}II}\\ a_{\mathrm{i.e}I}\\ a_{\mathrm{i.e}II}\end{array}\right]=C_{I-II,\mathrm{i.e}\mathrm{i.e}}\cdot \left[\begin{array}{c}{b}_{m,\mathrm{i.e}I}\\ b_{m,\mathrm{i.e}II}\\ a_{m,\mathrm{i.e}I}\\ a_{m,\mathrm{i.e}II}\end{array}\right]$

(8th) $\left[\begin{array}{c}{a}_{cI}\\ a_{cII}\\ b_{cI}\\ b_{cII}\end{array}\right]=C_{I-II,cc}\cdot \left[\begin{array}{c}{a}_{m,cI}\\ a_{m,cII}\\ b_{m,cI}\\ b_{m,cII}\end{array}\right]$

(9)

(10) The relationship between all ports and all modes at pre-calibration level 12 and calibration level 15 is described while retaining the modal description and combined into an overall equation system with an overall error network (described by an 8x8 matrix C): $\left[\begin{array}{c}b\\ a\end{array}\right]=C\cdot \left[\begin{array}{c}{b}_{\text{m}}\\ a_{\text{m}}\end{array}\right]=\left[\begin{array}{cc}E& f\\ G& H\end{array}\right]\cdot \left[\begin{array}{c}{b}_{\text{m}}\\ a_{\text{m}}\end{array}\right]$

(10)

(11) $b_{\text{m}}:={\left[b_{\text{m},dI},b_{\text{m},dII},b_{\text{m},cI},b_{\text{m},cII}\right]}^T$

(12) The following definition of the measured wave magnitude vectors applies: $a_{\text{m}}:={\left[a_{\text{m},\mathrm{i.e}I},a_{\text{m},\mathrm{i.e}II},a_{\text{m},cI},a_{\text{m},cII}\right]}^T$

(11) $b_{\text{m}}:={\left[b_{\text{m},\mathrm{i.e}I},b_{\text{m},\mathrm{i.e}II},b_{\text{m},cI},b_{\text{m},cII}\right]}^T$

(12)

(13) For a clear representation, the matrix C is broken down into the four 4x4 individual matrices E, F, G, H in accordance with Eq. (10) for the following description. Multiplied out and using eq.(5), this results in: $E{b}_{\text{m}}+f{a}_{\text{m}}=S_{MM}\left(G{b}_{\text{m}}+H{a}_{\text{m}}\right)$

(13)

This equation describes the relationship between all wave quantities at the pre-calibration level and the scattering matrix of a measurement object at the calibration level for the excitation of a mode at a port on the pre-calibration level.

(14) The relationship when all modes are excited at all ports is achieved by replacing the measured wave magnitude vectors (a _m , b _m ) in Eq. (13) with the measured 4x4 wave magnitude matrices (A _m , B _m ), which contain measurement data for all excitations : $E{B}_{\text{m}}+f{A}_{\text{m}}=S_{MM}\left(G{B}_{\text{m}}+H{A}_{\text{m}}\right)$

(14)

(15) $B_{\text{m}}=\left[\begin{array}{cccc}{b}_{\text{m},dI}^{\left(dI\right)}& b_{\text{m},dI}^{\left(dII\right)}& b_{\text{m},dI}^{\left(cI\right)}& b_{\text{m},dI}^{\left(cII\right)}\\ b_{\text{m},dII}^{\left(dI\right)}& b_{\text{m},dII}^{\left(dII\right)}& b_{\text{m},dII}^{\left(cI\right)}& b_{\text{m},dII}^{\left(cII\right)}\\ b_{\text{m},cI}^{\left(dI\right)}& b_{\text{m},cI}^{\left(dII\right)}& b_{\text{m},cI}^{\left(cI\right)}& b_{\text{m},cI}^{\left(cII\right)}\\ b_{\text{m},cII}^{\left(dI\right)}& b_{\text{m},cII}^{\left(dII\right)}& b_{\text{m},cII}^{\left(cI\right)}& b_{\text{m},cII}^{\left(cII\right)}\end{array}\right]$

(16) The measurement matrices are defined as follows (superscript in brackets indicates the respective excitation): $A_{\text{m}}=\left[\begin{array}{cccc}{a}_{\text{m},\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& a_{\text{m},\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& a_{\text{m},\mathrm{i.e}I}^{\left(cI\right)}& a_{\text{m},\mathrm{i.e}I}^{\left(cII\right)}\\ a_{\text{m},\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& a_{\text{m},\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& a_{\text{m},\mathrm{i.e}II}^{\left(cI\right)}& a_{\text{m},\mathrm{i.e}II}^{\left(cII\right)}\\ a_{\text{m},cI}^{\left(\mathrm{i.e}I\right)}& a_{\text{m},cI}^{\left(\mathrm{i.e}II\right)}& a_{\text{m},cI}^{\left(cI\right)}& a_{\text{m},cI}^{\left(cII\right)}\\ a_{\text{m},cII}^{\left(\mathrm{i.e}I\right)}& a_{\text{m},cII}^{\left(\mathrm{i.e}II\right)}& a_{\text{m},cII}^{\left(cI\right)}& a_{\text{m},cII}^{\left(cII\right)}\end{array}\right]$

(15) $B_{\text{m}}=\left[\begin{array}{cccc}{b}_{\text{m},\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& b_{\text{m},\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& b_{\text{m},\mathrm{i.e}I}^{\left(cI\right)}& b_{\text{m},\mathrm{i.e}I}^{\left(cII\right)}\\ b_{\text{m},\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& b_{\text{m},\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& b_{\text{m},\mathrm{i.e}II}^{\left(cI\right)}& b_{\text{m},\mathrm{i.e}II}^{\left(cII\right)}\\ b_{\text{m},cI}^{\left(\mathrm{i.e}I\right)}& b_{\text{m},cI}^{\left(\mathrm{i.e}II\right)}& b_{\text{m},cI}^{\left(cI\right)}& b_{\text{m},cI}^{\left(cII\right)}\\ b_{\text{m},cII}^{\left(\mathrm{i.e}I\right)}& b_{\text{m},cII}^{\left(\mathrm{i.e}II\right)}& b_{\text{m},cII}^{\left(cI\right)}& b_{\text{m},cII}^{\left(cII\right)}\end{array}\right]$

(16)

(17) ${\mathbf{C}}_{I-II,cc}=\left[\begin{array}{cccc}{H}_{33}& H_{34}& G_{33}& G_{34}\\ H_{43}& H_{44}& G_{43}& G_{44}\\ F_{33}& F_{34}& E_{33}& E_{34}\\ F_{43}& F_{44}& E_{43}& E_{44}\end{array}\right]$

(18) With the definition of the partial matrices of C according to Eq. (10) and the linking of the wave quantities through the individual error networks according to Eq. (8), (9), the list of the individual error networks results depending on the error coefficients of the individual matrices of E, F , G, H as follows: $C_{I-II,\mathrm{i.e}\mathrm{i.e}}=\left[\begin{array}{cccc}{E}_{11}& E_{12}& f_{11}& f_{12}\\ E_{21}& E_{22}& f_{21}& f_{22}\\ G_{11}& G_{12}& H_{11}& H_{12}\\ G_{21}& G_{22}& H_{21}& H_{22}\end{array}\right]$

(17) ${\mathbf{C}}_{I-II,cc}=\left[\begin{array}{cccc}{H}_{33}& H_{34}& G_{33}& G_{34}\\ H_{43}& H_{44}& G_{43}& G_{44}\\ f_{33}& f_{34}& E_{33}& E_{34}\\ f_{43}& f_{44}& E_{43}& E_{44}\end{array}\right]$

(18)

All other matrix elements of E, F, G, H are zero due to the mode decoupling assigned to the measurement adaptation.

(19) By modifying known self-calibration methods for the 15-term error model (e.g. a nodal TMRG calibration method, as described in the publication [A. Gronefeld and B. Schiek: "Network-analyzer self-calibration with four or five standards for the 15- term error model", 1997, IEEE MTT-S International Microwave Symposium Digest, pp. 1655-1658], or a nodal LRM calibration method as described in the publication [K. Silvonen: "LMR 16-a self-calibration procedure for a leaky network analyzer", IEEE Transactions on Microwave Theory and Techniques, 45.7 (1997), pp. 1041-1049]), 15 of the 16 error terms of the two error matrices (C _I-II,dd , C _I-II,cc ) can be determined. The respective nodal calibration methods must be adapted to modal wave quantities, modal error networks and modal calibration standards (double nodal TMRG), as will be explained in detail below. A normalization takes place (once per modal error network). For knowledge of the complete error network (C), denormalization of one of the two individual error networks is therefore necessary. Without restricting the generality, the initial normalization should take place in relation to the error network C _I-II,dd . Accordingly, the second error gate C _I-II,cc is determined up to a factor α _I-II,cc , so that: ${\mathbf{C}}_{I-II,cc}=a_{I-II,cc}{\tilde{C}}_{I-II,cc}=a_{I-II,cc}\left[\begin{array}{cccc}{\tilde{H}}_{33}& {\tilde{H}}_{34}& {\tilde{G}}_{33}& {\tilde{G}}_{34}\\ {\tilde{H}}_{43}& {\tilde{H}}_{44}& {\tilde{G}}_{43}& {\tilde{G}}_{44}\\ {\tilde{f}}_{33}& {\tilde{f}}_{34}& {\tilde{E}}_{33}& {\tilde{E}}_{34}\\ {\tilde{f}}_{43}& {\tilde{f}}_{44}& {\tilde{E}}_{43}& 1\end{array}\right]$

(19)

Here, the normalization to the element E ₄₄ took place without loss of generality.

The determination of the unknowns α _I-II,cc is to be carried out using a reciprocal but also unknown calibration standard R. An unknown thru method is used for this. Basically, the method is based on the fact that the determinant of the transmission matrix of any reciprocal network is always one and can therefore be assumed to be known in a cascade representation. In a further step, the unknown quantity for denormalization then results with only one phase ambiguity, which can be resolved with approximate knowledge of the geometric length of the network.

(20) Equations (8) and (9) are used in equation (7) to formulate the cascade equation: ${\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}\left[\begin{array}{c}{b}_{\text{m},\mathrm{i.e}I}\\ b_{\text{m},\mathrm{i.e}II}\\ a_{\text{m},\mathrm{i.e}I}\\ a_{\text{m},\mathrm{i.e}II}\end{array}\right]=T_{I-II,MM}{\mathbf{C}}_{I-II,cc}\left[\begin{array}{c}{a}_{\text{m},cI}\\ a_{\text{m},cII}\\ b_{\text{m},cI}\\ b_{\text{m},cII}\end{array}\right]$

(20)

(21) Analogous to Eq. (13), this equation describes the relationship of all variables for (any) excitation of a mode at a port. If all excitations (excitation of all modes at all ports) are considered, all wave quantities in their respective excitation variants can be lined up in columns to take them into account: ${\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}\left[\begin{array}{cccc}{b}_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}I}^{\left(cI\right)}& b_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}II}^{\left(cI\right)}& b_{m,\mathrm{i.e}II}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}I}^{\left(cI\right)}& a_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}II}^{\left(cI\right)}& a_{m,\mathrm{i.e}II}^{\left(cII\right)}\end{array}\right]=T_{I-II,MM}{\mathbf{C}}_{I-II,cc}\left[\begin{array}{cccc}{a}_{m,cI}^{\left(\mathrm{i.e}I\right)}& a_{m,cI}^{\left(\mathrm{i.e}II\right)}& a_{m,cI}^{\left(cI\right)}& a_{m,cI}^{\left(cII\right)}\\ a_{m,cII}^{\left(\mathrm{i.e}I\right)}& a_{m,cII}^{\left(\mathrm{i.e}II\right)}& a_{m,cII}^{\left(cI\right)}& a_{m,cII}^{\left(cII\right)}\\ b_{m,cI}^{\left(\mathrm{i.e}I\right)}& b_{m,cI}^{\left(\mathrm{i.e}II\right)}& b_{m,cI}^{\left(cI\right)}& b_{m,cI}^{\left(cII\right)}\\ b_{m,cII}^{\left(\mathrm{i.e}I\right)}& b_{m,cII}^{\left(\mathrm{i.e}II\right)}& b_{m,cII}^{\left(cI\right)}& b_{m,cII}^{\left(cII\right)}\end{array}\right]$

(21)

(22) Rearranging the equation causes the measured wave quantities at the pre-calibration level and the cascade of error networks and the measurement object to be separated: $\begin{array}{c}\left[\begin{array}{cccc}{b}_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}I}^{\left(cI\right)}& b_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}II}^{\left(cI\right)}& b_{m,\mathrm{i.e}II}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}I}^{\left(cI\right)}& a_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}II}^{\left(cI\right)}& a_{m,\mathrm{i.e}II}^{\left(cII\right)}\end{array}\right]{\left[\begin{array}{cccc}{a}_{m,cI}^{\left(\mathrm{i.e}I\right)}& a_{m,cI}^{\left(\mathrm{i.e}II\right)}& a_{m,cI}^{\left(cI\right)}& a_{m,cI}^{\left(cII\right)}\\ a_{m,cII}^{\left(\mathrm{i.e}I\right)}& a_{m,cII}^{\left(\mathrm{i.e}II\right)}& a_{m,cII}^{\left(cI\right)}& a_{m,cII}^{\left(cII\right)}\\ b_{m,cI}^{\left(\mathrm{i.e}I\right)}& b_{m,cI}^{\left(\mathrm{i.e}II\right)}& b_{m,cI}^{\left(cI\right)}& b_{m,cI}^{\left(cII\right)}\\ b_{m,cII}^{\left(\mathrm{i.e}I\right)}& b_{m,cII}^{\left(\mathrm{i.e}II\right)}& b_{m,cII}^{\left(cI\right)}& b_{m,cII}^{\left(cII\right)}\end{array}\right]}^{-1}\\ ={\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}^{-1}{T}_{I-II,MM}{\mathbf{C}}_{I-II,cc}\end{array}$

(22)

(23) resultiert die Kaskadendarstellung mit Gl.(19) schließlich zu: $P_{I-II}={\alpha }_{I-II,cc}{\mathbf{C}}_{I-II,dd}^{-1}{T}_{I-II,MM}{\tilde{C}}_{I-II,cc}$

(24) After combining the left side into a measurement matrix $P_{I-II}=\left[\begin{array}{cccc}{b}_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}I}^{\left(cI\right)}& b_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& b_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& b_{m,\mathrm{i.e}II}^{\left(cI\right)}& b_{m,\mathrm{i.e}II}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}I}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}I}^{\left(cI\right)}& a_{m,\mathrm{i.e}I}^{\left(cII\right)}\\ a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}I\right)}& a_{m,\mathrm{i.e}II}^{\left(\mathrm{i.e}II\right)}& a_{m,\mathrm{i.e}II}^{\left(cI\right)}& a_{m,\mathrm{i.e}II}^{\left(cII\right)}\end{array}\right]{\left[\begin{array}{cccc}{a}_{m,cI}^{\left(\mathrm{i.e}I\right)}& a_{m,cI}^{\left(\mathrm{i.e}II\right)}& a_{m,cI}^{\left(cI\right)}& a_{m,cI}^{\left(cII\right)}\\ a_{m,cII}^{\left(\mathrm{i.e}I\right)}& a_{m,cII}^{\left(\mathrm{i.e}II\right)}& a_{m,cII}^{\left(cI\right)}& a_{m,cII}^{\left(cII\right)}\\ b_{m,cI}^{\left(\mathrm{i.e}I\right)}& b_{m,cI}^{\left(\mathrm{i.e}II\right)}& b_{m,cI}^{\left(cI\right)}& b_{m,cI}^{\left(cII\right)}\\ b_{m,cII}^{\left(\mathrm{i.e}I\right)}& b_{m,cII}^{\left(\mathrm{i.e}II\right)}& b_{m,cII}^{\left(cI\right)}& b_{m,cII}^{\left(cII\right)}\end{array}\right]}^{-1}$

(23) the cascade representation with equation (19) finally results in: $P_{I-II}=a_{I-II,cc}{\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}^{-1}{T}_{I-II,MM}{\tilde{C}}_{I-II,cc}$

(24)

(25) Now the determinant of Eq. (24) is to be calculated so that the product of the determinants of all individual matrices and the unknowns α _I-II,cc is in its fourth power on the right-hand side of the equation. The determinant of the mode conversion matrix T _MM expressed by the scattering parameters (see equation (6)) of the test object should be considered in advance: $\text{de}\left(T_{MM}\right)=\frac{S_{\mathrm{i.e}IcI}{S}_{\mathrm{i.e}IIcII}-S_{\mathrm{i.e}IcII}{S}_{\mathrm{i.e}IIcI}}{S_{cI\mathrm{i.e}I}{S}_{cII\mathrm{i.e}II}-S_{cII\mathrm{i.e}I}{S}_{cI\mathrm{i.e}II}}$

(25)

• • S_dlcl=S_cldl
• • S_dllcll=S_cldl
• • S_dlcll=S_cldl
• • S_dllcl=S_cldll

This expression is no longer dependent on the reflection coefficients (S _dldl , S _clcl , ...) so that an almost arbitrarily adapted standard can be used. In the case of a reciprocal standard, the remaining scattering parameters are:

• • _Sdlcl = _Scldl
• • S _dllcll =S _cldl
• • _Sdlcll = _Scldl
• • S _dllcl =S _cldll

The determinant det(T _MM ) thus results in one in the case of a reciprocal connection, where the following restriction applies: The product S _dlcl S _dllcll must be different from the product S _dlcll S _dllcl to avoid a singularity. A calibration standard is proposed for this, which has a high coupling pluna of common and differential modes at one modal port each. Specifically, this is achieved by using non-ideal shorting at one nodal port and non-ideal open-circuiting at the remaining nodal port (see R standard in 5 ). Thus S _dlcl ≈ S _dllcll ≈ ± 1 and S _dlcll ≈ S _dllcl ≈ 0.

(26) The determinant of the cascade representation according to Eq. (24) is considered again, which results in the following expression due to the use of the reciprocal standard: $\text{de}\left(P_{I-II}\right)=a_{I-II,\mathrm{<figure-callout\; id="4"\; label="c\; c"\; filenames="DE102014019008B4\_0029.png,DE102014019008B4\_0030.png"\; state="\{\{state\}\}">c</figure-callout>}c}^4\cdot \text{de}\left({\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}^{-1}\right)\cdot \text{de}\left({\tilde{C}}_{I-II,cc}\right)$

(26)

(27) Solved for the unknown α _I-II,cc we are looking for, this results in the following expression: $a_{I-II,cc}=\sqrt[{}^{{}^{{}^{{}^4}}}]{\frac{\text{de}\left(P_{I-II}\right)\cdot \text{de}\left({\mathbf{C}}_{I-II,\mathrm{i.e}\mathrm{i.e}}\right)}{\text{de}\left({\tilde{C}}_{I-II,cc}\right)}}\cdot \text{ex}\left(\mathrm{<figure-callout\; id="2"\; label="j\; \pi "\; filenames="DE102014019008B4\_0029.png,DE102014019008B4\_0030.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{\pi }{}\frac{}{2}\mu \right)\text{With}\mu =\left\{\mathrm{0,1,2,3}\right\}$

(27)

From the four possible solutions for α _I-II,cc , the correct one is chosen based on approximate knowledge of the standard used (eg phase relationship of both modes). Finally, the calibration process is completed using Eq.(19) for the error networks C _I-II,dd and C _I-II,cc . The use of this calibration method with the reciprocal standard R and the resulting low demands on the calibration element can be seen as particularly advantageous and elegant.

(28) Now all error gates are fully known. A system error correction and thus the corrected determination of the scattering parameters of a measurement object DUT 14 can be achieved by converting Eq. (14) to S _MM : $S_{MM}=\left(E{B}_{\text{m}}+f{A}_{\text{m}}\right){\left(G{B}_{\text{m}}+H{A}_{\text{m}}\right)}^{-1}$

(28)

The nodal scattering matrix can also be calculated from this result by means of linear transformation known to those skilled in the art.

• Es wird ein Kalibrierverfahren benötigt, das die Selbstkalibrierung eines modalen 15-Term-Fehlermodells bei Intermodenentkopplung (vollverkoppeltes Fehlermodell für zwei Tore je Mode) mit konzentrierten Elementen leistet. Im nodalen Fall kann ein vollverkoppeltes Fehlermodell (C_n, siehe dazu 4a, 4b, jeweils links) mit zwei Messtoren mithilfe des TMRG-Verfahrens kalibriert werden. Es wird auf die oben bereits genannte Veröffentlichung von A. Gronefeld und B. Schiek verwiesen. Für das erfindungsgemäße Kalibrierverfahren ist eine Übertragung auf den modalen Fall erforderlich, die im Folgenden beispielhaft aufgezeigt wird:

The following describes how a nodal 15-term calibration method can be applied to the modal case. The nodal TMRG method is to be used as an example, referred to here as a double-nodal TMRG method or as a modal TMRG method:

• A calibration method is needed that performs the self-calibration of a modal 15-term error model in intermode decoupling (fully coupled error model for two ports per mode) with lumped elements. In the nodal case, a fully coupled error model (C _n , see 4a , 4b , each on the left) with two test ports using the TMRG method. Reference is made to the above publication by A. Gronefeld and B. Schiek. For the calibration method according to the invention, a transfer to the modal case is required, which is shown below as an example:

• • Tor 1: Γ_n,a ≈ 1 (realer, unbekannter Leerlauf), Tor 2: Γ_n,b ≈ 0 (realer Wellensumpf)
• • Tor 1: Γ_n,a ≈ 0 (realer Wellensumpf), Tor 2: Γ_n,b ≈ -1 (realer, unbekannter Kurzschluss)
• • Tor 1: Γ_n,a ≈ -1 (realer, unbekannter Kurzschluss), Tor 2: Γ_n,b ≈ 1 (realer, unbekannter Leerlauf)
• • Tor 1: Γ_n,a = 1 (realer, unbekannter Leerlauf), Tor 2: Γ_n,b ≈ -1 (realer, unbekannter Kurzschluss)

Show for the initial case of the nodal TMRG procedure 4a and 4b (each on the left) the error model: the two input ports (m,1 and m,2) are connected to the two accessible test ports 1, 2 in the Calibration plane associated with an error network (described with 4×4 matrix C _n ). The calibration procedure provides for the connection of the following four different two-port reflection standards in the calibration level at the two test ports 1 and 2 (corresponding to 4a Left):

• • Gate 1: Γ _n,a ≈ 1 (real, unknown open circuit), gate 2: Γ _n,b ≈ 0 (real wave sump)
• • Gate 1: Γ _n,a ≈ 0 (real wave sump), Gate 2: Γ _n,b ≈ -1 (real, unknown short circuit)
• • Gate 1: Γ _n,a ≈ -1 (real, unknown short circuit), gate 2: Γ _n,b ≈ 1 (real, unknown open circuit)
• • Gate 1: Γ _n,a = 1 (real, unknown open circuit), Gate 2: Γ _n,b ≈ -1 (real, unknown short circuit)

Furthermore, a through-connection Thru of both ports 1, 2 is provided (corresponding to 4b left: S ₂₁ =1).

• • Tor dl: Γ_dd,a ≈ 1 (realer, unbekannter Leerlauf), Tor dll: Γ_dd,b ≈ 0 (realer Wellensumpf)
• • Tor dl: Γ_dd,a ≈ 0 (realer Wellensumpf), Tor dll: Γ_dd,b ≈ -1 (realer, unbekannter Kurzschluss)
• • Tor dl: Γ_dd,a ≈ -1 (realer, unbekannter Kurzschluss), Tor dll: Γ_dd,b ≈ 1 (realer, unbekannter Leerlauf)
• • Tor dl: Γ_dd,a ≈ 1 (realer, unbekannter Leerlauf), Tor dll: Γ_dd,b ≈ -1 (realer, unbekannter Kurzschluss).

the 4a and 4b (each in the middle) are now exhibiting this 3 known error model of a four-port measurement with modal consideration and the symmetry properties of the measuring adapter assumed above. To use the TMRG method for the differential error network (described with 4x4 matrix Cdd), the following two-port reflection standards (corresponding to middle) can be connected:

• • Gate dl: Γ _dd,a ≈ 1 (real, unknown open circuit), gate dll: Γ _dd,b ≈ 0 (real wave sump)
• • Gate dl: Γ _dd,a ≈ 0 (real wave sump), Gate dll: Γ _dd,b ≈ -1 (real, unknown short circuit)
• • Gate dl: Γ _dd,a ≈ -1 (real, unknown short circuit), Gate dll: Γ _dd,b ≈ 1 (real, unknown open circuit)
• • Gate dl: Γ _dd,a ≈ 1 (real, unknown open circuit), gate dll: Γ _dd,b ≈ -1 (real, unknown short circuit).

• Für die Synthese der tatsächlichen/realen Kalibrierstandards in ihrer nodalen Realisierung wird der Ausdruck für den jeweiligen differentiellen Reflexionsfaktor gemäß dem Lehrwerk [Holger Heuermann, „Hochfrequenztechnik: Komponenten für High-Speed- und Hochfrequenzschaltungen“, Springer 2009] betrachtet. Für den Reflexionsfaktor an Tor dl gilt Γ_dd,a = (S₁₁+S₂₂)/2 = Γ_n,a, d.h. bei Anschluss von zwei identischen Eintor-Reflektionsstandards an die nodalen Tore 1 und 2 von 4a (rechts) wird eben jener Reflexionsfaktor am differentiellen Tor / realisiert. Analog gilt dies auch für das differentieile Tor II in Verbindung mit den nodalen Messtoren 3 und 4, da Γ_dd,b ≈ (S₃₃+S₄₄)/2 = Γ_n,b gilt.

These variables can be determined by connecting calibration standards to the nodal measurement ports 1, 2, 3, 4 that are actually present according to the in 4a (right) shown arrangement. In each case, one calibration standard is connected to both test ports 1, 2 of the first differential interface and the other calibration standard is connected to both test ports 3, 4 of the second differential interface, or vice versa, which is possible for the following reasons:

• For the synthesis of the actual/real calibration standards in their nodal realization, the expression for the respective differential reflection factor according to the textbook [Holger Heuermann, "High-frequency technology: components for high-speed and high-frequency circuits", Springer 2009] is considered. The reflection factor at port dl is Γ _dd,a = (S ₁₁ +S ₂₂ )/2 = Γ _n,a , ie when two identical one-port reflection standards are connected to the nodal ports 1 and 2 of 4a (right) this reflection factor is realized at the differential port /. This also applies analogously to the differential port II in connection with the nodal test ports 3 and 4, since Γ _dd,b ≈ (S ₃₃ +S ₄₄ )/2 = Γ _n,b applies.

In addition, a through-connection between the differential gates dl and dll according to 4b (right) are produced. For this purpose, the calculation of the modal scattering parameter from the modal parameters is also considered: S _dlldl =(S ₃₁ -S ₄₁ -S ₃₂ +S ₄₂ )12. With a double connection (S ₃₁ =S ₄₂ =1, S ₄₁ =S ₃₂ =0, see also 4b , right) this results in S _dlldl= 1. The self-calibration of the common-mode error model can be carried out analogously with the same nodal wiring, since the following applies: Γ _cc,a = (S ₁₁ +S ₂₂ )/2 = Γ _n,a , Γ _{cc ,b} = (S ₃₃ +S ₄₄ )/2 = Γ _n,b , S _clclcl = (S ₃₁ +S ₄₁ +S ₃₂ +S ₄₂ )12.

The nodal four-port calibration standards result as a direct consequence 5 for a double differential interface. The numbering of the measurement ports is that of the 2 corresponds to 5 specified below.

A non-ideal short circuit ("Short") and a non-ideal open circuit ("Open") are used as calibration standards. A wave sump (“match”) serves as an impedance standard. The thru has a known spatial extent. What is new is the use of a double TMRG calibration per adapter with four nodal or two modal gates. For this purpose, the proposed nodal calibration standards Thru, Short-Open (SO), Match-Open (MO), Open-Short (OS) and Short-Match (SM) are duplicated so that the respective two-port calibration standard is now considered modal appears on each of the two mode error networks.

the inside 5 The R standard shown below on the right is not used for the actual TMRG calibration, but is used for the subsequent linking of the modally separated error networks.

The presented modification of the calibration method for determining the error coefficients of a modal error model and the associated four-port calibration standards are particularly advantageous and new.

Calibration procedure:

A pre-calibrated network analyzer 20 and, starting from the given pre-calibration level 12, an existing error network 13 with the symmetry properties described above are required, which in turn achieves an intermodal decoupling of all measurement ports with modal representation. An intramode coupling may continue to exist at the respective measurement adapters 10 . All measurements and subsequent calculations can be based on nodal or modal measurements. Assuming linearity, the modal measured values are subsequently calculated for the nodal measurements.

1. 1. Anwendung der oben erläuterten doppelten TMRG-Kalibrierung bei modaler Betrachtung oder eines anderen 15-Term-Kalibrierverfahrens auf den Messadapter 10 liefert die Fehlerterme der Fehlermatrizen C_I-II,_dd sowie C̃_I-II,cc
2. 2. Einsatz des Reziprok-Standards R mit hoher Modenkonversion an dem Messadapter 10 liefert den Wert des Multiplikationsfaktors α_I-II,_cc; dadurch Entnormierung von C̃_I-II,cc zu C_I-II,cc

The actual course of the calibration method according to the invention can be summarized as follows:

1. 1. Application of the double TMRG calibration explained above with modal consideration or another 15-term calibration method to the measuring adapter 10 supplies the error terms of the error matrices C _I-II , _dd and C _{- I-II,cc}
2. 2. Use of the reciprocal standard R with high mode conversion at the measuring adapter 10 supplies the value of the multiplication factor α _I-II , _cc ; thereby denormalization of C̃ _I-II,cc to C _I-II,cc

• Das dargelegte Kalibrierverfahren und die dadurch mögliche Systemfehlerkorrektur leisten die Bestimmung der vollständigen modalen Streumatrix eines elektrischen Bauteils DUT 10 mit zwei differentiellen Schnittstellen (4 Toren) wie etwa eines HSD-Steckverbinders oder eines HSD-Kabels mit zwei differentiellen Aderpaaren. Zu diesem Zweck wird der Einfluss einer Messadaptierung zwischen den Koaxial-Schnittstellen eines Netzwerkanalysators und den doppelt differentiellen Schnittstellen des Messadapters rechnerisch kompensiert. Das Verfahren ist mit einer beliebigen Anzahl von Messadaptern (mit jeweils vier Messtoren) anwendbar. Begründet durch das eingesetzte Selbstkalibrierprinzip werden nur geringe Ansprüche an die zu verwendenden Kalibrierstandards gestellt. Für die korrekte Funktion sollen folgende Bedingungen erfüllt sein:
• • Die Vorkalibrierung des Netzwerkanalysators 20 an den Koaxial-Schnittstellen (Vorkalibrierebene 12) muss ausreichend gut sein, so dass die Symmetrie der Messadapter ausgenutzt werden kann. Abweichungen führen zu einer reduzierten Messdynamik.
• • Die Messadapter 10 sollen nur eine geringe bzw. keine Modenkonversion aufweisen, da diese innerhalb des Fehlernetzwerks nicht abgebildet wird. Abweichungen führen zu einer reduzierten Messdynamik (in etwa auf Niveau der Modenkonversion des Messadapters) in Bezug auf die Modenkonversions-Streuparameter. Der Einfluss dieser Modellabweichung auf weitere Streuparameter ist als gering zu bewerten.
• • Die für die Viertor-Kalibrierstandards eingesetzten nodalen Reflektionselemente (realer Kurzschluss, realer Leerlauf, Impedanzstandard) sollen sich auf allen Viertor-Standards identisch verhalten. Abweichungen führen zu einer reduzierten Messdynamik und zu Messfehlern in Bezug auf alle Streuparameter. Dies gilt nicht für den Reziprok-Standard.
• • Die Durchverbindung Thru soll elektrisch bekannt sein (Länge und Dämpfung). Die unzutreffende Annahme einer idealen Verbindung führt zur geringfügigen Verschiebung der Referenzebene und zu Messfehlern in Bezug auf sämtliche Transmissions-Streuparameter.

The advantages of the present invention can be summarized as follows:

• The calibration method presented and the system error correction made possible thereby provide the determination of the complete modal scattering matrix of an electrical component DUT 10 with two differential interfaces (4 ports) such as an HSD connector or a HSD cable with two differential pairs of wires. For this purpose, the influence of a measurement adaptation between the coaxial interfaces of a network analyzer and the double differential interfaces of the measurement adapter is compensated by calculation. The method can be used with any number of test adapters (each with four test ports). Due to the self-calibration principle used, only low demands are placed on the calibration standards to be used. For correct function, the following conditions should be met:
• • The pre-calibration of the network analyzer 20 at the coaxial interfaces (pre-calibration level 12) must be sufficiently good so that the symmetry of the measuring adapter can be used. Deviations lead to reduced measurement dynamics.
• • The measuring adapters 10 should have only little or no mode conversion, since this is not mapped within the error network. Deviations lead to reduced measurement dynamics (roughly on the level of the mode conversion of the measurement adapter) in relation to the mode conversion scattering parameters. The influence of this model deviation on other scattering parameters can be rated as low.
• • The nodal reflection elements used for the four-port calibration standards (real short circuit, real open circuit, impedance standard) should behave identically on all four-port standards. Deviations lead to reduced measurement dynamics and measurement errors in relation to all scattering parameters. This does not apply to the reciprocal standard.
• • The through connection Thru should be known electrically (length and attenuation). The incorrect assumption of an ideal connection leads to the slight displacement of the reference plane and to measurement errors with regard to all transmission scattering parameters.

• Von besonderem technischem Interesse ist die Verwendung von zwei Messadaptern 10, 10' mit jeweils zwei differentiellen Schnittstellen (also insgesamt acht Messtoren 1, 2, 3, 4, 5, 6, 7, 8 mit den beiden differentiellen Schnittstellen 1/2, 3/4 des ersten Messadapters 10 und den beiden differentiellen Schnittstellen 5/6, 7/8 des zweiten Messadapters 10'). Auf diese Weise können Messobjekte mit zwei doppeltdifferentiellen Schnittstellen (z.B. Messung des Übertragungsverhaltens eines vieradrigen Kabels, siehe 6) charakterisiert werden.

The following is a brief summary of how the calibration method explained above for the one-adapter case can be extended to two measuring adapters. A corresponding extension to more than two measuring adapters is possible in the same way:

• Of particular technical interest is the use of two measuring adapters 10, 10', each with two differential interfaces (i.e. a total of eight test ports 1, 2, 3, 4, 5, 6, 7, 8 with the two differential interfaces 1/2, 3/ 4 of the first measuring adapter 10 and the two differential interfaces 5/6, 7/8 of the second measuring adapter 10'). In this way, measurement objects with two double diffe profitable interfaces (e.g. measurement of the transmission behavior of a four-wire cable, see 6 ) are characterized.

As in 7 is shown particularly clearly, all systematic, linear errors between the pre-calibration level 12 and the calibration level 15 for each of the two measuring adapters 10, 10' are combined by an 8-port error network, which is combined with the 8x8 matrix C _{1,2,3, 4} and C _5,6,7,8 respectively. Electrical coupling between the individual adapters is not taken into account due to spatial separation. Finally, the DUT 14 described by its nodal 8x8 scattering matrix S _SE is located at the calibration level 15. The number of unknown error coefficients is 128 for the measurement setup with two adapters.

With a procedure analogous to the one-adapter case described above, a modal error model can also be achieved for the error model for two measuring adapters, again assuming symmetry within the individual adapters. The error model results 8th , which considers an intramodal coupling of both modes between gate / and 11 and between gate III and IV. The respective intermode coupling is again negligible. The associated four single 4-port error networks each contain 64 error coefficients and can be determined using the following procedure.

1. 1. Unter Beibehaltung der Moden Gleich- und Gegentakt werden analog zu Gl.(5) sämtliche Wellengrößen definiert, die schließlich die Formulierung einer modalen 8x8-Streumatrix des Messobjekts mit acht Toren zulassen.
2. 2. Der Zusammenhang zwischen Wellengrößen an Vorkalibrierebene und Kalibrierebene wird an dem zweiten Messadapter 10' mit den Modaltoren III und IV analog zu Gl.(8),(9) mit Fehlerviertoren verknüpft, die durch die 8x8-Matrizen C_III-IV,dd und C_III-IV,cc beschrieben werden. Es gilt folglich auch ein Gesamt-Fehlernetzwerk entsprechend Gl.(10), wobei die beschreibende Matrix C nun die Größe 16x16 besitzt. Es erfolgt die Zerlegung in die vier 8x8-Einzelmatrizen E, F, G, H, die bei Betrachtung sämtlicher Anregungen und Verwendung von 8x8-Wellengrößenmatrizen analog zu Gl.(14) den Zusammenhang zwischen Streumatrix des Messobjekts, der Messdaten und den Fehlernetzwerken herstellt.
3. 3. Ein Koeffizientenvergleich ermöglicht analog zu Gl.(17),(18) das Aufstellen der Fehlernetzwerke C_I-II,_dd, C_I-II,_cc, C_III-_IV,_dd und C_III-IV,cc in Abhängigkeit von Elementen der Einzelmatrizen E, F, G, H. 64 der insgesamt 256 Elemente von C sind dabei von Null verschieden.
4. 4. Von den Fehlernetzwerken C_I-II,dd, C_I-II,cc, C_III-IV,dd und C_III-IV,cc werden mit dem oben beschriebenen Modal-TMRG-Selbstkalibrierverfahren oder einem anderen 15-Term-Kalibrierverfahren jeweils 15 der 16 Fehlerterme bestimmt, d.h. es finden vier Normierungen statt, wobei jedoch nur eine Normierung zulässig ist. Entsprechend müssen die drei zugehörigen Entnormierungsfaktoren α_I-II,cc, α_III-IV,dd, α_III-IV,cc bestimmt werden (ohne Beschränkung der Allgemeinheit wurde dabei vorausgesetzt, dass das normierte Element in C_I-II,dd enthalten ist) und eine Entnormierung der Fehlernetzwerke C_I-II,cc, C_III-IV,dd und C_III-IV,cc erfolgen.
5. 5. Die Vorgehensweise zur Bestimmung von α_I-II,cc unter Verwendung eines reziproken Standards R sowie dessen Realisierung ist oben bereits beschrieben. Für den hier betrachteten Fall mit zwei Messadaptern 10, 10' folgt die Bestimmung von α_III-IV,dd, bei der ebenfalls ein reziproker aber sonst unbekannter Standard R₂ zum Einsatz kommen soll. Vorgeschlagen wird eine Kabelverbindung (vieradrig) zwischen beiden Messadaptern 10, 10' mit beliebiger Länge, Leitungsimpedanz und Einfügedämpfung. Die Determinante der Kaskade von C_I-II,dd, reziproker Verbindung R₂ und C_III-IV,dd liefert schließlich nach Umformung und Auswahl von einer von vier möglichen Lösungen α_III-IV,dd. Die korrekte Lösung wird durch Kenntnis der ungefähren Länge der Leitung gewählt. Die Bestimmung von α_III-IV,cc unter Verwendung des reziproken Modenkonverters R erfolgt dann analog dem oben beschriebenen Verfahren an dem zweiten Messadapter 10'.

The procedure for the two-measuring adapter case is only outlined here, since the necessary basics and the mathematical methods are already known from the one-adapter case. The use of lumped calibration elements when using self-calibration methods is again addressed. Specifically, both adapters are calibrated separately using the calibration procedure described above (modal TMRG and reciprocal standard). The set of calibration standards follows suit 5 for use. The error networks of the two measuring adapters 10, 10' must then be related to one another, which in turn corresponds to denormalization. A method with reciprocal standard R ₂ or an unknown thru approach can also be used for this task. In detail, the following measures are necessary to expand the calibration and measurement process for the two-adapter case, with the modal error model already being used 8th should apply:

1. 1. Keeping the modes common and differential mode, all wave quantities are defined in analogy to equation (5), which finally allow the formulation of a modal 8x8 scattering matrix of the test object with eight ports.
2. 2. The relationship between wave sizes at the pre-calibration level and the calibration level is linked to the second measuring adapter 10' with the modal gates III and _IV analogous to Eq and C _III-IV,cc . Consequently, a total error network corresponding to Eq. (10) also applies, with the descriptive matrix C now having the size 16x16. It is broken down into the four 8x8 individual matrices E, F, G, H, which, when considering all excitations and using 8x8 wave size matrices analogous to Eq. (14), establishes the connection between the scattering matrix of the measurement object, the measurement data and the error networks.
3. 3. A coefficient comparison allows analogous to equations (17),(18) to set up the error networks C _I-II , _dd , C _I-II , _cc , C _III - _IV , _dd and C _III-IV,cc as a function of Elements of the individual matrices E, F, G, H. 64 of the 256 elements of C are different from zero.
4. 4. From the error networks C _I-II,dd , C _I-II,cc , C _III-IV,dd and C _III-IV,cc are measured using the modal TMRG self-calibration method described above or another 15-term calibration method 15 of the 16 error terms are determined in each case, ie four normalizations take place, but only one normalization is permissible. Accordingly, the three associated denormalization factors α _I-II,cc , α _III-IV,dd , α _III-IV,cc must be determined (without loss of generality, it was assumed that the normalized element is contained in C _I-II,dd ) and a denormalization of the error networks C _I-II,cc , C _III-IV,dd and C _III-IV,cc take place.
5. 5. The procedure for determining α _I-II,cc using a reciprocal standard R and its implementation has already been described above. For the case considered here with two measuring adapters 10, 10', α _III-IV,dd is determined, in which a reciprocal but otherwise unknown standard R ₂ is also to be used. A cable connection (four-wire) between the two measuring adapters 10, 10' with any length, line impedance and insertion loss is proposed. The determinant of the cascade of C _I-II,dd , reciprocal connection R ₂ and C _III-IV,dd finally gives α _III-IV,dd after transformation and selection of one of four possible solutions. The correct solution is chosen by knowing the approximate length of the line. The determination of α _III-IV,cc using the reciprocal mode converter R then takes place analogously to the method described above on the second measuring adapter 10'.

It is also possible to determine the coefficients α _I-II,cc , α _{III-IV,cc ,} α _III-IV,dd in a different order. In particular, the normalized element is not necessarily included in C _I-II,dd . It is only important that the four error matrices are determined except for a single normalization or a single unknown, so that the relative relationships between them are determined.

Claims (15)

Method for calibrating a measuring arrangement (100) for measuring electrical components (14), comprising at least one measuring adapter (10) which can be connected to a network analyzer (20) and has four measuring ports (1, 2, 3, 4) which are located in a calibration plane (15 ) are arranged for connecting the electrical component (14) and form two differential interfaces (1/2 and 3/4), wherein the measuring adapter (10) can be described by a linear 8-port error network (C _1,2,3,4 ). is characterized by the following steps: • Combining 2 test ports (1/2; 3/4) of a differential interface to form a modal port (I, II), • Representation of the error network in modal representation in which linear errors in the push-pull path for both modal ports are described by a 4x4 matrix C _{I-II, dd} and linear errors in the common-mode path for both modal ports are described by a 4x4 matrix C _{I-II, cc} , neglecting intermodal coupling between push-pull path and common-mode path, • Besti Calculation of 15 error terms each of the matrices C _{I-II, dd} and C _{I-II, cc} using a 15-term calibration method, • determining a multiplicative link factor between the determined 15 error terms of the matrix C _{I-II, cc} and the matrix C _{I-II, dd} using a reciprocal standard (R). procedure after claim 1 , characterized in that correction values are calculated from the determined error terms of the matrices C _{I-II, dd} and C _{I-II, cc} , which are taken into account in subsequent measurements of scattering parameters (S _SE ) of electrical components. procedure after claim 1 or 2 , characterized in that the adapter is constructed in such a way that the electrical coupling between the two test ports of the first differential interface essentially corresponds to the electrical coupling between the two test ports of the second differential interface, and/or that the electrical coupling between one of the test ports of the first differential interface and one of the test ports of the second differential interface essentially corresponds to the coupling between the other two test ports. Method according to one of the preceding claims, characterized in that the measuring adapter (10) is connected to a calibrated pre-calibration level (12) of the network analyzer (20) via four input ports (1', 2', 3', 4'), preferably set up as coaxial interfaces and/or that the two differential interfaces (1, 2; 3, 4) of the measuring adapter (10) preferably have four contact elements (45) provided in a square arrangement on the measuring adapter for contacting an electrical component having two differential conductors, such as a 4- Have gate calibration standards or a star quad cable, wherein preferably each contact element (45) is electrically connected to a coaxial interface. Method according to one of the preceding claims, characterized in that a self-calibration method such as a TMRG or LRM method adapted to modal wave variables is used as the 15-term calibration method. procedure after claim 5 , characterized in that only lumped calibration elements such as bridges, resistors, etc. that can be connected to the two differential interfaces and/or to ground are used to determine the 15 error terms of the matrices C _{I-II, dd} and C _{I-II, cc} . procedure after claim 5 or 6 , characterized in that at least three, preferably four, in particular five of the following calibration standards are used: through connection thru; match-open MO, short-match SM, short-open SO, open-short OS, with the respective two-port calibration standard appearing preferably in modal representation on the common-mode and differential-mode error networks, and in particular the first letter the termination of the test ports ( 1, 2) of one differential interface and the second letter the termination of the test ports (3, 4) of the other differential interface or vice versa. Method according to one of the preceding claims, characterized in that a reciprocal standard (R) is used which has a high coupling of common mode and differential mode at the modal ports (I, II) and/or which is asymmetrical with respect to the two test ports differential interfaces is set up. Method according to one of the preceding claims, characterized in that the reciprocal standard (R) terminates two test ports (1, 3) with (non-ideal) short circuits and the other two test ports (4, 2) with (non-ideal) open circuits. Method according to one of the preceding claims, in which two or more optionally integrally formed measuring adapters (10, 10') each having four measuring ports (1, 2, 3, 4; 5, 6, 7, 8) are connected to the network analyzer (20) be connected, the measurement ports being arranged in the calibration plane (15) for connecting the electrical component (14) and two differential interfaces (1/2, 3/4; 5/6, 7/) per measurement adapter (10, 10'). 8) form, wherein a coupling between individual measuring adapters (10, 10 ') is negligible, characterized by the following steps: • Combining 2 test ports of a measuring adapter to a modal port (I, II; III, IV), • Representation of the Measurement adapter describing error networks in modal representation, in which linear errors in the push-pull path for both modal ports by a 4x4 error matrix (C _{I-II, dd} , C _{III-IV, dd} ) and linear errors in the common-mode path for both modal ports by a 4x4 error matrix ( C _{I-II, cc} , C _{III-IV, cc} ) be are written, neglecting intermodal coupling between push-pull path and common-mode path, • Determination of 15 error terms of the error matrices (C _{I-II, dd} , C _{I-II, cc} , C _{III-IV, dd} , C _{III-IV, cc} ) under Using a 15-term calibration method, • determining first and second multiplicative link factors between the determined 15 error terms of matrices C _{I-II, cc} and C _{I-II, dd} and matrices C _{III-IV, cc} and C _{III-IV, dd} using a reciprocal standard R, • determining a third multiplicative link factor between error terms determined for the first measurement adapter and error terms determined for the second measurement adapter using a further reciprocal standard R ₂ . procedure after claim 10 , characterized in that an 8-port standard with twice two differential interfaces for connecting to the two differential interfaces of two measuring adapters is used as a further reciprocal standard R ₂ . procedure after claim 11 , characterized in that a cable with four wires, in particular a star-quad cable, is used as the further reciprocal standard R ₂ . Set of calibration standards for use in a calibration method according to one of the preceding claims, characterized by at least three, preferably four, in particular five, preferably concentrated 4-port standards and at least one 4-port reciprocal standard R for calibrating a measurement adaptation with at least four measurement ports , which form at least two differential interfaces. set of calibration standards Claim 13 , characterized in that the at least three lumped 4-port standards are selected from the following standards: Thru; MO, SM, SO, OS, where the first letter denotes the termination of two measurement ports and the second letter denotes the termination of the other two measurement ports of the measurement adaptation by the respective standard. set of calibration standards Claim 13 or 14 , further comprising an 8-port reciprocal standard R ₂ for connection to two measurement adapters of the measurement adaptation, each with four measurement ports, preferably in the form of a cable with four wires, in particular in the form of a star quad cable.

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