System and method for calibrating a measuring arrangement and characterizing a measurement mount

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First Claim
1. A method for characterizing a measurement holder connected to a network analyzer based on an error model, wherein the method comprises:
 determining a matrix with measured scattering parameters from different calibration standards in the measurement holder and with associated actual scattering parameters of the calibration standards;
determining linearinT system errors of the error model by solving a linear equation system with the matrix; and
transforming the linearinT system errors into corresponding system errors of a systemerror matrix of the error model which characterizes the measurement holder,wherein in order to solve the linear equation system, a first and second linearinT system error are each selected and the given system error of the systemerror matrix is weighted with a correct first linearinT system error or with a correct second linearinT system error,a value of the correct first and correct second linearinT system error is determined by a reciprocity condition between two reciprocal transmission coefficients of a transmission path of the measurement holder in the systemerror error matrix of a transmissive reciprocal calibration standard in the measurement holder, andwherein system errors previously caused by the network analyzer and by measurement lines are determined by calibration, and the previously determined system errors are removed from the measured scattering parameters by the different calibration standards in the measurement holder.
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Abstract
A method and a system for calibrating a measuring arrangement on the basis of a 16term error model determines a matrix (A) with measured scattering parameters (S_{m}) from different calibration standards (3) and with associated actual scattering parameters (S_{a}) of the calibration standards (3) and determines linearinT system errors (T_{i}) for the calibration of a network analyzer (1) by solving a linear equation system with the determined matrix (A). To solve the linear equation system, a first and a second linearinT system error (k, p) are freely selected in each case. With use of reciprocal calibration standards, the determined linearinT system errors are weighted with the freely selected first linearinT system error (T_{i}) or with a correct second linearinT system error p_{kor}(k)) dependent upon the first linearinT system error (k).
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9 Claims
 1. A method for characterizing a measurement holder connected to a network analyzer based on an error model, wherein the method comprises:
determining a matrix with measured scattering parameters from different calibration standards in the measurement holder and with associated actual scattering parameters of the calibration standards; determining linearinT system errors of the error model by solving a linear equation system with the matrix; and transforming the linearinT system errors into corresponding system errors of a systemerror matrix of the error model which characterizes the measurement holder, wherein in order to solve the linear equation system, a first and second linearinT system error are each selected and the given system error of the systemerror matrix is weighted with a correct first linearinT system error or with a correct second linearinT system error, a value of the correct first and correct second linearinT system error is determined by a reciprocity condition between two reciprocal transmission coefficients of a transmission path of the measurement holder in the systemerror error matrix of a transmissive reciprocal calibration standard in the measurement holder, and wherein system errors previously caused by the network analyzer and by measurement lines are determined by calibration, and the previously determined system errors are removed from the measured scattering parameters by the different calibration standards in the measurement holder.  View Dependent Claims (2, 3, 4)
 5. A method for characterizing a measurement holder connected to a network analyzer based on an error model, wherein the method comprises:
determining a matrix with measured scattering parameters from different calibration standards in the measurement holder and with associated actual scattering parameters of the calibration standards; determining linearinT system errors of the error model by solving a linear equation system with the matrix; and transforming the linearinT system errors into corresponding system errors of a systemerror matrix of the error model which characterizes the measurement holder, wherein in order to solve the linear equation system, a first and second linearinT system error are each selected and the given system error of the systemerror matrix is weighted with a correct first linearinT system error or with a correct second linearinT system error, wherein a value of a relationship between the first and second linearinT system error is determined by a reciprocity condition from transformed transmission factors of a transmissive and reciprocal calibration standard in the measurement holder, and wherein the transformed transmission factors of the transmissive and reciprocal calibration standard in the measurement holder is implemented by transformation of measured transmission factors of the transmissive and reciprocal calibration standard in the measurement holder with the given linearinT system errors.  View Dependent Claims (6)
 7. A method for characterizing a measurement holder connected to a network analyzer based on an error model, wherein the method comprises:
determining a matrix with measured scattering parameters from different calibration standards in the measurement holder and with associated actual scattering parameters of the calibration standards; determining linearinT system errors of the error model by solving a linear equation system with the matrix; and transforming the linearinT system errors into corresponding system errors of a systemerror matrix of the error model which characterizes the measurement holder, wherein in order to solve the linear equation system, a first and second linearinT system error are each selected and the given system error of the systemerror matrix is weighted with a correct first linearinT system error or with a correct second linearinT system error, wherein a value of a relationship between the first and second linearinT system error is determined by a reciprocity condition from transformed transmission factors of a transmissive and reciprocal calibration standard in the measurement holder, and wherein the value of either the correct first or the correct second linearinT system error is determined by a reciprocity condition between two reciprocal transmission coefficients of a transmission path of the measurement holder in the system error matrix with the transmissive and reciprocal calibration standard in the measurement holder.  View Dependent Claims (8, 9)
1 Specification
The present application is a national phase application of PCT Application No. PCT/EP2013/053337, filed Feb. 20, 2013, and claims priority to German Application No. DE 10 2012 003 285.4, filed on Feb. 20, 2012, and German Application No. DE 10 2013 201 914.9, filed on Feb. 6, 2013, the entire contents of which are herein incorporated by reference.
The invention, according to the various embodiments described herein, relates to a system and a method for calibrating a measuring arrangement and characterizing a measurement holder.
If the scattering parameters of a device under test are measured using a network analyzer, the measured scattering parameters of the device under test are influenced by nonidealities of the network analyzer, the measurement lines and the measurement holder in which the device under test is fixed and accordingly do not correspond to the actual scattering parameters of the device under test. The system errors caused by the nonidealities of the network analyzer, the measurement lines and the measurement holder must be determined by means of a calibration. With the system errors determined, the scattering parameters measured by the network analyzer are converted into the actual scattering parameters during the measuring operation.
If a crosstalk in the test setup must be considered when determining the system errors, it is advantageous to use the 16term error model.
Equation (1) shows in each case the associated mathematical interrelation between the individual elements of the systemerror matrix E and the individual propagated wave values a_{0}, a_{1}, a_{2 }and a_{3 }respectively the reflected wave values b_{0}, b_{1}, b_{2 }and b_{3}. It contains a total of 16 elements, from which the name is derived, and is illustrated with its 16 elements in the signalflow diagram of
The systemerror matrix E describes the influence caused by the measuring device, the test setup and the measurement holder on the wave values a_{1}, a_{2}, b_{1 }and b_{2 }actually present in the measuring device, which leads to the test values a_{0}, a_{3}, b_{0 }and b_{3 }accessible to the measuring device. The secondary diagonal elements of the submatrices E_{1 }to E_{4 }describe, in this form, the crosstalk contained in the measurement system. In practice, the crosstalk is caused, for example, by inadequate insulation of the signal paths of the vectorial network analyzer, inadequate shielding of the highfrequency measurement lines and inadequate shielding of the measurement holder, as indicated schematically in
Equation (2), in which error terms T occur in the place of error terms E, shows an presentation for the interrelation between the individual propagated wave values a_{0}, a_{1}, a_{2 }and a_{3 }respectively the reflected wave values b_{0}, b_{1}, b_{2 }and b_{3}, which is equivalent with equation (1).
With reference to Silvonen K. “New FiveStandard Calibration Procedures for Network Analyzers and Wafer Probes”, ISBN 9512219891, 24 Jul. 1998, the content of which is accordingly included in the disclosed content of the patent application, equation (3) shows an interrelation between the measured scattering parameters s_{m }of the device under test, the actual scattering parameters s_{a }of the device under test and the individual submatrices of the systemerror matrix E, and equation (4) shows a corresponding interrelation between the measured scattering parameters s_{m }of the device under test, the actual scattering parameters s_{a }of the device under test and the individual submatrices of the systemerror matrix T. While equation (3) leads to a nonlinear equation system for determining the system errors E, a linear equation system can be derived from equation (4) for the determination of the system errors T. This presentation of the system errors with the matrix T is therefore designated as a linearinT system error—as also in the following—and, in practice, is a conventional form for presenting system errors with the use of a 16term error model.
S_{m}=E_{1}+E_{2}·(I−S_{a}·E_{4})^{−1}·S_{a}·E_{3} (3)
S_{m}·(T_{3}·S_{a}+T_{4})=(T_{1}·S_{a}+T_{2}) (4)
Schramm M. et al. “MOS16: New Method for InFixture Calibration and Fixture Characterization” in Proceedings of the ARFTG, Baltimore, USA, June 2010, describes a 16term calibration method. Alongside the reciprocity of the measurement holder, this 16term calibration method disadvantageously requires identical reflection coefficients at the input and output side of the measurement holder in each case.
Another 16term calibration method is known from Silvonen K. “LMR 16—A SelfCalibration Procedure for a Leaky Network Analyzer” in IEEE Transactions on Microwave Theory and Techniques, Volume 45, Number 7, July 1997, pages 1041 to 1049. Alongside four nontransmissive calibration standards, the calibration method additionally uses a fifth transmissive calibration standard. The transmission behavior of the fifth calibration standard must therefore be known exactly over the entire measurementfrequency range. With the use of a measurement holder, this requirement is, under some circumstances, either technically very difficult or completely impossible.
Embodiments of a method for calibrating and a method for characterizing and in each case an associated system which overcomes the disadvantages of the publications named above are provided.
In one embodiment, for the case of calibrating a measuring arrangement with the use of calibration standards and also for the case of characterizing a measurement holder in which a calibration standard or device under test is fixed, the transmission and reflection coefficients measured with the network analyzer of, preferably four, nontransmissive calibration standards and one transmissive calibration standard are inserted into an equation system together with the actual properties of the calibration standards taken as known, which allow a determination of 15 linearinT parameters, in the case of the calibration, and the calculation of all scattering parameters of a measurement holder, in the case of a characterization.
A maximal rank of 15 is determined from the structure of the equation system obtained for the 16term error model. Accordingly, the value of one of the 16 linearinT system errors can be freely selected and the accordingly selected 16th linearinT system error can be used as the scaling value for scaling the other 15 linearinT system errors. If only the known nontransmissive calibration standards are used, the resulting equation system provides a maximal rank of 14. With an initially free selection of one further parameter, a solution of the equation system can be found without a use of the transmissive calibration standard. Conventional numerical methods, such as, the NewtonRaphson or the GaussJordan method can be used to solve the linear equation system.
In order to implement a calibration of the measuring arrangement, a total of 15 linearinT system errors are required, which provide a dependence on only one 16^{th }linearinT system error. In the case of the characterization of the measurement holder, all 16 linearinT system errors are determined.
As will be explained in greater detail below, in one embodiment, with the use of a transmissive, reciprocal calibration standard, one of the two freely specified linearinT system errors—designated in the following as second linearinT system errors—can advantageously be determined according to one embodiment of the invention dependent upon the respectively other freely specified linearinT system parameter—referred to in the following as first linearinT system errors. The remaining 14 linearinT system errors determined by solving the linear equation system are finally weighted either with the freely specified first linearinT system error or with a correct second linearinT system error dependent upon the first linearinT system error.
In one embodiment, for the case of the calibration of a measuring arrangement, the fifth transmissive and reciprocal calibration standard is preferably used to determine the dependence of the correct, second linearinT system error upon the first linearinT system error.
The value of the correct second linearinT system error in this context is preferably obtained from exploitation of the reciprocity condition of the transmissive and reciprocal calibration standard.
For this purpose, the measured transmission coefficients of the fifth transmissive calibration standard are converted with the 14 determined linearinT system errors, the first and second freely specified linearinT system errors into transformed transmission coefficients of the fifth transmissive calibration standard.
A sign to be additionally considered in the value of the correct, second linearinT system error can preferably be determined with knowledge of the known electrical properties, by further preference, of the known electrical length, of the fifth transmissive and reciprocal calibration standard by means of plausibility considerations between the electrical length and the transformed transmission coefficients of the fifth calibration standard. In view of the plausibility considerations, advantageously by contrast with the prior art, the actual transmission coefficients of the fifth transmissive and reciprocal calibration standard need not be exactly known.
A preferred characterization according to one embodiment of the invention of a measurement holder connected to a network analyzer, exploits the fact that the parameters characterizing the transmission, reflection and crosstalk behavior of the measurement holder correspond to individual system errors of a systemerror matrix E which result from transformation of corresponding linearinT system errors, which are derivable from the actual and measured scattering parameters of different calibration standards fixed successively in the measurement holder.
For this purpose, the measured and actual scattering parameters of the calibration standards fixed in the measurement holder are first inserted into a matrix, which, together with the linearinT system errors to be determined, forms a linear equation system. The remaining 14 linearinT system errors can be determined dependent upon the first and second linearinT system error through free selection of a first and second linearinT system error, by solving the linear equation system. The determined linearinT system errors can be transformed in a conventional manner into corresponding system errors of a systemerror matrix E of the 16term error model.
In the case of the characterization according to one embodiment the invention of a measurement holder connected to a network analyzer, a correct first linearinT system error and a correct second linearinT system error can be determined. In this manner, the scaling to the first linearinT system error can be cancelled, so that the scattering parameters of the measurement holder according to a magnitude and phase are present as elements of the systemerror matrix E.
For this purpose, the measurement holder must be reciprocal only with regard to the transmission in its two transmission paths. Advantageously, by contrast with the measuring arrangement in Schramm M. et al. “MOS16: A New Method for InFixture Calibration and Fixture Characterization” in Proceedings of the ARFTG, Baltimore, USA, June 2010, it need not provide an identical reflection behavior at the two inputs or at the two outputs.
In a first preferred embodiment of the characterization according to one or embodiment the invention of a measurement holder connected to a network analyzer, the value of the correct, first linearinT system error and the value of the correct second linearinT system error is determined in the systemerror matrix E, in each case with exploitation of the reciprocity condition of the reciprocal measurement holder and of the reciprocal and transmissive calibration standard fixed in the reciprocal measurement holder, on the basis of two mutually reciprocal transmission coefficients associated respectively with one transmission path of the measurement holder. Since these transmission coefficients of the two transmission paths of the measurement holder in the systemerror matrix E provide either a dependence upon the first or the second linearinT system error, the first and second linearinT system errors can be determined independently of one another.
A sign to be additionally considered in the value of the correct, first linearinT system error and a sign to be additionally considered in the value of the correct, second linearinT system error can be determined in a first preferred embodiment of the characterization according to one embodiment of the invention of a measurement holder connected to a network analyzer, in each case with knowledge of the known electrical properties, preferably the known electrical length, from one of the transmission paths of the measurement holder by means of plausibility considerations between the electrical length and the mutually reciprocal transmission coefficients in the determined systemerror matrix E of one of the two transmission paths of the measurement holder. Advantageously, in this context, the length of the measurement holder need not be known exactly, respectively, only with a knowledge of the phase of ±90°.
In a second preferred embodiment of the characterization according to one embodiment of the invention of a measurement holder connected to a network analyzer, the value of the relation between first and second linearinT system error and the sign to be additionally considered in the value of the relation between first and second linearinT system error are initially determined by means of a fifth transmissive and reciprocal calibration standard fixed in the measurement holder.
The value of the relationship between first and second linearinT system error is preferably obtained in an equivalent manner to the calibration of a measuring arrangement with exploitation of the reciprocity condition of a fifth transmissive and reciprocal calibration standard. For this purpose, transformed transmission coefficients of the fifth transmissive and reciprocal calibration standard are determined by transformation of the measured transmission coefficients of the fifth transmissive calibration standard fixed in the measurement holder with the 14 determined linearinT system errors and the first and second linearinT system errors.
The sign to be considered in the value of the relationship between first and second linearinT system error can be determined in an equivalent manner to the calibration of the measuring arrangement with knowledge of the known electrical properties, preferably the known electrical length, of the fifth transmissive and reciprocal calibration standard, by means of plausibility observation between the electrical length and the transformed transmission coefficients of the fifth transmissive and reciprocal calibration standard which is fixed in the measurement holder.
Following this, either the value of the correct, first linearinT system error or the value of the correct, second linearinT system error is preferably determined. This is obtained in an equivalent manner to the first embodiment of the characterization according to one embodiment the invention of a measurement holder connected to a network analyzer with exploitation of the reciprocity condition of the reciprocal measurement holder and of the reciprocal and transmissive calibration standard fixed in the reciprocal measurement holder on the basis of two mutually reciprocal transmission coefficients associated with one transmission path of the measurement holder in the systemerror matrix E with fixing of a fifth transmissive and reciprocal calibration standard in the measurement holder.
Following this, the sign to be additionally considered in the value of that correct linearinT system error—correct first or correct second linearinT system error, of which the value has been previously calculated is determined. The determination of the sign to be additionally considered in the value of the correct first or correct second linearinT system error is obtained in an equivalent manner to the first embodiment of the characterization according to one embodiment the invention of a measurement holder connected to a network analyzer with a knowledge of the known electrical properties, preferably the known electrical length, of one of the transmission paths of the measurement holder by means of plausibility considerations between the electrical length and the two mutually reciprocal transmission coefficients of the respective transmission path of the measurement holder in the systemerror matrix E with fixing of the fifth transmissive and reciprocal calibration standard in the measurement holder. Advantageously, in this context, the electrical length of the transmission path need not be exactly known.
The value and the sign to be additionally considered in the value of the respectively other correct, linearinT system error—this means the correct first or correct second linearinT system error—is obtained from the initially determined correct linearinT system error and the determined relation between the first and second linearinT system error.
Finally, in the method according to one embodiment of the invention for the characterization according to one embodiment of the invention of a measurement holder connected to a network analyzer, the transmission, reflection and crosstalk parameters of the measurement holder are preferably determined from the associated system errors of the systemerror matrix E by weighting the linearinT system errors determined from the linear equation system of the systemerror matrix T in each case with the first or second linearinT system error determined correctly with regard to value and sign to be additionally considered in the value, and then transferring into the systemerror matrix E.
The individual embodiments of the method according to one embodiment of the invention and the system according to one embodiment of the invention for calibrating a measuring arrangement and of the method according to one embodiment of the invention and the system according to one embodiment the invention for characterizing a measurement holder connected to a network analyzer are explained in detail in the following, by way of example with reference to the drawings, on the basis of a 16term error model. The Figs. of the drawings show:
Before the technical realizations of various embodiments of the invention are explained in detail with reference to the block diagram in
If the mathematical relationship in equation (4) between the actual scattering parameters S_{a }of the device under test described mathematically in equation (5A), the measured scattering parameters S_{m }of the device under test described mathematically in equation (5B) and the coefficients of the linearinT systemerror matrix T described mathematically in equation (2) are resolved, the relationship in equation (6A) respectively the abbreviated version in equation (6B) is obtained.
In order to determine the linearinT system errors of the vector t according to equation (6B), a total of five calibration standards are used. These are, for example, the four nontransmissive calibration standards OM, SM, MS and SO according to equation (7A), (7B), (7C) and (7D) and the transmissive calibration standard R according to equation (7E). In this context, o denotes an open (English: open) termination, s a short (English: short) termination and m a matched (English: matched) termination of the respective nontransmissive calibration standard and a respectively c a reflection coefficient in each case, and b a transmission coefficient of the transmissive calibration standard.
The table in
The 4×16dimensional matrix A^{4×16 }is expanded through the use of five calibration standards instead of a single calibration standard to form a 20×16 matrix A^{20×16 }according to equation (8A). Because of its internal structure, the rank of the 20×16 dimensional matrix A^{20×16 }is 15. Accordingly, 15 linearinT system errors can be scaled by a 16^{th }linearinT system error. The 16^{th }linearinT system error is selected freely and independently from zero. The remaining 15 linearinT system errors are therefore dependent upon the 16^{th }linearinT system error in a linear manner.
A^{20+16}·t^{16×1}=0 (8A)
With the four nontransmissive calibration standards, according to equation (7A) to (7D), a matrix A^{16×16 }is obtained of which the maximum rank assumes the value 14 because of the internal structure of the equation system. Consequently, in total, 14 linearinT system errors of the vector t can be determined. Accordingly, one further linearinT system error must still be freely selected. In the following, the first freely selected linearinT system error is designated k, the second freely selected linearinT system error is designated p. The linear equation system (8A) is therefore transferred into the linear equation system (8B) with the 16×14dimensional matrix A^{16×14}, the 14 linearinT system errors {tilde over (t)}^{14×1 }to be determined, the freely selected first and second linearinT system errors k and p, and the associated vectors V_{1}^{16×1 }and V_{2}^{16×1}.
A^{16×14}·{tilde over (t)}^{14×1}=−p·V_{1}^{16×1}−k·V_{2}^{16×1} (8B)
The values for k and p are preferably each selected to be 1. Since the secondary diagonal elements of the submatrices T_{1},T_{2},T_{3},T_{4 }disappear, if no crosstalk is present, the system errors k and p are positioned in primary diagonal elements of the submatrices T_{1},T_{2},T_{3},T_{4}. Furthermore, the first freely selected linearinT system error k is a linearinT system error in a righthand column of one of the submatrices T_{1},T_{2},T_{3},T_{4}—for example t_{15}—and the second freely selected linearinT system error p is a linearinT system error in a lefthand column of one of the submatrices T_{1},T_{2},T_{3},T_{4}—for example t_{12} or vice versa.
Since the values k and p for the first and respectively second linearinT system error represent merely assumptions which can deviate from the correct values for the first respectively second linearinT system error, the remaining 14 linearinT system errors obtained from the solution of the linear equation system according to equation (8) represent only intermediate results which may still deviate from the correct linearinT system errors. In the following, these intermediate results are supplied, according to equation (9), to a matrix T, (i=English: intermediate=intermediate result).
With regard to the elements t_{ji }of the individual submatrices of the matrices T_{i }according to equation (10A) to (10D), the following property can be derived:
If the individual matrix multiplications in equation (4) are implemented, a 4×4 matrix is obtained on the righthand and lefthand side of the equals sign, of which the elements 1,1 and 1,2 originate, in each case, only from the two elements in the left column of each of the submatrices T_{1 }to T_{4 }and of which elements 2,1 and 2,2 originate in each case only from the two elements in the right column of each of the submatrices T_{1 }to T_{4}. Since the freely selected value p is assigned, for example, to the linearinT system error t_{12 }disposed in the left column of the submatrix T_{4i}, and the freely selected value k is assigned, for example, to the linearinT system error t_{15 }disposed in the right column of the submatrix T_{4i}, all linearinT system errors in a left column of the submatrices T_{1 }to T_{4 }must provide a dependence only upon k, and all linearinT system errors in a right column of the submatrices T_{1i }to T_{4i }must provide a dependence only upon p, so that equation (4) continues to provide its validity with the intermediate results for the linearinT system errors disposed in the submatrices T_{1i }to T_{4i}.
In order to separate the effects of p respectively k from the individual linearinT system errors t_{ji }determined by solving the linear equation system in each case, a further intermediate value t_{ji}′ must be introduced, which is obtained for an element in the left column of the submatrices T_{1i }to T_{4i }from the weighting of the relation between the respective correct linearinT system error t_{j }and the correct linearinT system error t_{12 }according to equation (11A) with the freely selected value p, and for an element in the right column of the submatrices T_{1i }to T_{4i }from the weighting of the relation between the respective correct linearinT system error t_{j }and the correct linearinT system error t_{15 }according to equation (11B) with the freely selected value k. Equation (11A) and (11B) explain the scaling of the remaining 14 linearinT system errors through the freely selected first and second linearinT system errors k and p.
Accordingly, starting from equations (10A) to (10D) for the submatrices T_{1i }to T_{4i}, the following relationships are obtained in equation (12A) to (12B):
If the elements in the left column of the submatrices T_{1i }to T_{4i }are weighted with the relation
between the correct linearinT system error k still to be determined and dependent upon the freely selected first linearinT system error p_{kor}(k), and the originally freely selected value p, with reference to equation (11A) respectively (11B), the respective correct linearinT system error t_{j }is obtained, all of which still provide a dependence upon the freely selected first linearinT system error k. Equation (13A) illustrates this weighting, which is also named a renormalizing, which leads to submatrices T_{1c }to T_{4c}. According to equation (13B), the individual submatrices T_{1c }to T_{4c }result in the linearinT system error matrix T_{c}.
According to one embodiment of the invention, the second linearinT system error—for example, the linearinT system error t_{12 }with the originally freely selected value p—is placed in dependence upon the first linearinT system error—for example, the linearinT system error t_{15 }with the originally freely selected value k. In order to determine the dependence of the second linearinT system error upon the first linearinT system error, the reciprocity of the fifth reciprocal and transmissive calibration standard R is used according to equation (7E).
For this purpose, starting from a mathematical relationship, also presented in Silvonen K. “New FiveStandard Calibration Procedures for Network Analyzers and Wafer Probes”, ISBN 9512219891, 24 Jul. 1998, between the measured scattering parameters S_{m }and the actual scattering parameters S_{a }of a device under test and the individual submatrices of the systemerror matrix T according to equation (14), a mathematical relationship for scattering parameters S_{aRez }Of the fifth reciprocal and transmissive calibration standard according to equation (15) is introduced with a factorized relationship for the individual submatrices T_{1 }to T_{4 }from equation (12A) to (12D). System errors have still not been removed from the scattering parameter S_{aRez }Of the fifth reciprocal and transmissive calibration standard determined according to equation (15), since the submatrices T_{1 }to T_{4 }used in the transformation still provide a dependence upon the originally freely selected first and second linearinT system errors k and p, which have not yet been accurately determined. Accordingly, the scattering parameters S_{aRez }represent intermediate values and are designated in the following merely as transformed scattering parameters S_{aRez}.
A mathematical transformation of equation (15) leads to a mathematical relationship for the transformed scattering parameters S_{aRez }of the fifth reciprocal and transmissive calibration standard according to equation (16):
From equation (16), the mathematical relationships for the two transformed and mutually reciprocal transmission coefficients {tilde over (S)}_{arez}(1,2) and {tilde over (S)}_{arez}(2,1) of the fifth reciprocal and transmissive calibration standard to the freely selected parameters k and p of the first and second linearinT system error can be determined according to equation (17A) respectively (17B):
The reciprocity condition {tilde over (S)}_{arez}(1,2)={tilde over (S)}_{arez}(2,1) between the two transformed and mutually reciprocal transmission coefficients {tilde over (S)}_{arez}(1,2) and {tilde over (S)}_{arez}(2,1) of the fifth transmissive and reciprocal calibration standard applies only for the case of a correct choice of the parameters p and k. If the parameters p and k are not selected correctly, because the still unknown dependence relationship between the parameters p and k in the free selection of the parameters p and k has typically not been considered, a still unknown relationship factor M accordingly exists between the two transformed transmission coefficients {tilde over (S)}_{arez}(1,2) and {tilde over (S)}_{arez}(2,1) according to equation (18):
The relationship factor M therefore follows directly from the transformed transmission coefficients {tilde over (S)}_{arez}(1,2) and {tilde over (S)}_{arez}(2,1) and provides the dependence illustrated in equation (18) upon the freely selected parameters p and k of the first and second linearinT system error.
To ensure that the identity of the transmission coefficients following from the reciprocity of the fifth transmissive calibration standard also applies for the transformed transmission coefficients {tilde over (S)}_{arez}(1,2) and {tilde over (S)}_{arez}(2,1), the relation of the freely selected parameters must be configured in such a manner that the relationship factor M in equation (18) results in the value one.
Accordingly, a correct second linearinT system error p_{kor}(k) must be found, which provides a dependence upon the freely selected first linearinT system error k and corrects the relation between the originally freely selected first and second linearinT system error k and p. Starting from equation (18), an equation (19) is therefore obtained, which delivers a determination equation for the correct second linearinT system error p_{kor}(k).
Accordingly, also with reference to equation (18), a mathematical relationship for the relationship factor M follows from equation (19) according to equation (20).
The correct second linearinT system error p_{kor}(k) dependent upon the freely selected first linearinT system error k is therefore obtained according to equation (21).
p_{kor}(k)=±√{square root over (M·p^{2})} (21)
After the renormalization according to equation (13A), the 15 linearinT system errors t_{ji }now provide only a dependence upon the freely selected first linearinT system error k. With the 16 linearinT system errors t_{ji }determined in this manner, a correct error correction is possible.
The correct sign to be additionally considered in the correct value of the correct second linearinT system error p_{kor}—the sign which results from the root formation according to equation (21)—can be derived with known correct value of the correct second linearinT system error p_{kor }with plausibility considerations between the known electrical length and the scattering parameters {tilde over (S)}_{arez }of the transmissive and reciprocal calibration standard determinable according to equation (13A) for both possible signs and transformed according to equation (14). The electrical length of the transmissive and reciprocal calibration standard need not be known exactly. An accuracy of the electrical length at the level of ±90° is adequate for the determination.
The linearinT system errors of the network analyzer, the highfrequency lines and the measurement holder can be determined in this manner. With system errors determined in this manner, the system errors of the network analyzer, the highfrequency lines and the measurement holder can be removed from the measured values of unknown devices under test to be characterized according to equation (14).
According to the idea of one embodiment of the invention, a measurement holder, which is connected to the network analyzer via the highfrequency measurement lines, and in which calibration standards or a device under test to be characterized is fixed, can be characterized. In this context, the reflection and transmission coefficients of the measurement holder and the individual terms which describe a crosstalk within the measurement holder are regarded as system errors to be determined, which occur between device under test respectively calibration standard and network analyzer or between the reference planes 1 and 2 as shown in
After the system errors of the network analyzer and the highfrequency lines have been determined in advance within the framework of a preliminary calibration, scattering parameters of reciprocal and nontransmissive calibration standards, which are fixed in succession in the measurement holder in each case, are measured for this purpose in reference plane 1. 14 linearinT system errors can be determined from the measured scattering parameters and the actual scattering parameters of the individual calibration standards, as in the calibration method described above, by solving a linear equation system, if a first linearinT system error k and a second linearinT system error p are freely selected in each case.
In order to determine the reflection and transmission coefficients of the measurement holder and the individual terms which describe a crosstalk within the measurement holder, the individual linearT system errors are transformed into corresponding system errors of a systemerror matrix E of the 16term error model.
The submatrices E_{1i }to E_{4i }of the systemerror matrix E_{i }of the 16term error model can be determined, as also presented in Silvonen K. “New FiveStandard Calibration Procedures for Network Analyzers and Wafer Probes”, ISBN 9512219891, 24 Jul. 1998, by transformation of the submatrices T_{1i }to T_{4i }of the linearinT system error matrix T_{i }according to equations (22A) to (22D).
E_{1i}=T_{2i}T_{4i}^{−1} (22A)
E_{3i}=T_{4i}^{−1} (22B)
E_{2i}=T_{1i}−T_{2i}T_{4i}^{−1}T_{3i} (22C)
E_{4i}=−T_{4i}^{−1}T_{3i} (22D)
The inverse submatrix T_{4i}^{−1}, required for this purpose can be calculated according to equation (23).
After the implementation of the matrix multiplications, the submatrices E_{1i }to E_{4i }of the system matrix E_{i }are obtained according to equations (24A) to (24D):
Equations (24A) to (24D) explain the effects of the two freely selected parameters k and p for the first and respectively second linearinT system error on the calculation of the matrices E_{ji}.
The transmission, reflection and crosstalk parameters of a 4port unit shown in the signalflow graph in
From the two general reciprocity conditions e_{10}=e_{01 }and e_{23}=e_{32}, in each case between two mutually reciprocal transmission coefficients of the two transmission paths of the measurement holder according to the signal flow graph in
E_{2i}(1,1)=E_{3i}(1,1) (26A)
E_{2i}(2,2)=E_{3i}(2,2) (26B)
Since the first linearinT system error k and the second linearinT system error p are not generally correctly selected, the quotient between the matrix elements E_{2i}(1,1) and E_{3i}(1,1) from equation (26A) and the quotient between the matrix elements E_{2i}(2,2) and E_{3i}(2,2) from equation (26B) results in a relationship factor M different from one. This relationship factor M is obtained in each case starting from (26A) respectively (26B) taking into consideration equation (25A) and (25B), as shown in equation (27A) respectively (27B) in each case.
If the correct second linearinT system error p_{kor }is selected in equation (27A) instead of the freely selected second linearinT system error p, a relationship factor M=1 is obtained according to equation (28A). In an equivalent manner, if the correct first linearinT system error k_{kor }is selected in equation (27B) instead of the freely selected first linearinT system error k, a relationship factor M=1 is also obtained according to equation (28B).
Accordingly, from equation (28A) respectively (28B), a mathematical relationship for the relationship factor M according to equation (29A) respectively (29B) therefore follows in a similar manner.
Accordingly, starting from equation (29A), a mathematical relationship for the correct second linearinT system error p_{kor }and starting from equation (29B) a mathematical relationship for the correct first linearinT system error k_{kor }is obtained.
p_{kor}=±√{square root over (M·p^{2})} (30A)
k_{kor}=±√{square root over (M·k^{2})} (30B)
The sign of the correct first and second linearinT system error p_{kor }and k_{kor }to be additionally considered in the value can be derived in the case of a known correct value of the correct first and second linearinT system error p_{kor }and k_{kor }on the basis of plausibility considerations between the known electrical length and the transformed and mutually reciprocal transmission coefficients E_{2i}(1,1) and E_{3i}(1,1), E_{2i}(2,2) and E_{3i}(2,2) of the two transmission paths of the measurement holder. Here also, an accurate knowledge of the electrical length is not necessary. An estimate accuracy of ±90° is sufficient for the correct determination of the correct sign.
The individual submatrices E_{1i }to E_{4i }of the systemerror matrix E_{i }partially dependent upon the originally freely selected first and second system errors k and p are transferred into the submatrices E_{1c }to E_{4c }of the systemerror matrix E_{c }with the originally freely selected first and second system errors k and p removed, in that the individual elements of the linearinT system error submatrices T_{ni }originally determined from the linear equation system are weighted with the relationship between the correct first linearinT system error k_{kor }and the originally freely selected first linearinT system error k or with the relationship between the correct second linearinT system error p_{kor }and the originally freely selected second linearinT system error p within the framework of the renormalization. With reference to equations (22A) to (22D), the associated systemerror submatrices E_{ci }of the systemerror matrix E_{c }are determined from the renormalized systemerror submatrices T_{ci }according to equation (31B).
In the following, the method according to one embodiment of the invention for calibrating a measuring arrangement on the basis of a 16term error model is explained in detail with reference to the flow diagram in
In the first method step S10 of the method according to one embodiment of the invention, the two test ports P_{1 }and P_{2 }of the vectorial network analyzer 1 are connected in succession via the two highfrequency measurement lines 2_{1 }and 2_{2 }to five calibration standards 3, and the scattering parameters S_{m }of the five calibration standards are measured. The five calibration standards preferably comprise four nontransmissive calibration standards according to equations (7A) to (7D) in one of the exemplary combinations which are presented in the table of
In the next method step S20, a linear equation system according to equation (9B) is prepared. The measured scattering parameters S_{m }and the actual scattering parameters S_{a }of the four nontransmissive calibration standards are entered into the 16×14 dimensional matrix A^{16×14 }and into the 16dimensional vectors V_{1}^{16×1 }and V_{2}^{16×1 }in a manner equivalent to equation (6A) for the case of a single calibration standard.
For the freely selected first and second linearinT system errors k and p, arbitrary values other than k=0 and p=0 can be used. By preference, a value 1 is selected in each case because the scaling of the 14 remaining linearinT system errors at the conclusion of the method according to one embodiment of the invention is simplified in this case.
The 14 linearinT system errors to be determined are determined in the next method step S30 dependent upon the freely selected first and second linearinT system errors k and p by solving the linear equation system. For this purpose, conventional numerical methods for solving linear equation systems, such as the method for Eigenvalue analysis, the GaussJordan method or the NewtonRaphson method can be used.
In the next two method steps S40 and S50, the dependence of the second linearinT system error p, which was originally freely selected and is no longer independent from now on, is determined for the freely selected and still independent first linearinT system error k, in order to find a total of 15 dependent linearinT system errors and one independent linearinT system error k for the 16dimensional solution.
In method step S40, the value of the second linearinT system error p is determined dependent upon the freely selected value of the first linearinT system error k. For this purpose, the transformed transmission coefficients S_{arez}(1,2) and S_{arez}(2,1) of the fifth transmissive and reciprocal calibration standard R from equation (7E) used as calibration standard 3 are determined according to equation (15) by transforming the measured scattering parameters S_{mrez }of this fifth transmissive and reciprocal calibration standard with the submatrices T_{1i }to T_{4i }of the linearinT systemerror matrix T_{i }determined in the last method step S30.
The correct value p_{kor}(k) for the second linearinT system error dependent upon the freely selected parameter k for the first linearinT system error can be determined according to equation (21).
In method step S50, the sign to be additionally considered on the basis of the root formation in the correct value of the second linearinT system error p_{kor}(k) is determined on the basis of plausibilities between the known electrical length and the transformed scattering parameters S_{arez }of the fifth transmissive and reciprocal calibration standard determined in the preceding method step S40. For this purpose, for every sign of the roots in equation (21), the associated “signed” correct value of the second linearinT system error p_{kor}(k) is used in each case to determine 15linearinT system errors, which are dependent only upon the freely selected first linearinT system error k, by “renormalization” of the 14 linearinT system errors determined in method step S30 according to equation (13A). With the accordingly determined 15 linearinT system errors, the measured scattering parameters of the fifth transmissive and reciprocal calibration standard are transformed according to equation (14) in order to determine actual scattering parameters (socalled deembedding). By comparing the phase of the two accordingly determined and mutually reciprocal transmission coefficients with the known electrical length of the fifth transmissive and reciprocal calibration standard, the correct sign to be additionally considered for the correct value of the second linearinT system error can be determined by means of plausibility considerations.
In the final method step S60, the correct linearinT system errors for the individual submatrices T_{1c }to T_{4c }of the linearinT systemerror matrix T_{c}, which now provide only a dependence upon the first freely selected linearinT system error k, are stored for further use, whereas the solution vector with use of the incorrect sign is rejected.
The system errors of the network analyzer, the measurement lines and the measurement holder can be removed from scattering parameters of a device under test measured with the vectorial network analyzer with the elements of the linearinT systemerror matrix T. For this purpose, for example, the mathematical relationship for systemerror adjustment in equation (14) must be used.
In the following, the first embodiment of the method according to one embodiment of the invention for characterizing a measurement holder connected to a network analyzer is explained in detail with reference to the flow diagram in
In the first method step S100 of the method according to one embodiment of the invention, the system errors caused by the vectorial network analyzer and the highfrequency measurement lines are determined by means of a calibration. For this purpose, conventional calibration methods according to the prior art can be used, which determine parameters for error correction by measuring the scattering parameters of calibration standards.
In the next method step S110, in an equivalent manner to method step S10 in the case of the method according to one embodiment of the invention for calibrating a measuring arrangement, for example, on the basis of a 16term error model, the scattering parameters, for example, of a total of four nontransmissive and reciprocal calibration standards according to equation (7A) to (7D) and of one fifth transmissive and reciprocal calibration standard according to equation (7E), which are fixed in succession in a measurement holder 4 according to
The determined system errors of the network analyzer 1 and the measurement lines 2_{1 }and 2_{2 }are removed from the scattering parameters S_{m }measured in each case in the reference plane 1 with the parameters determined in the preceding method step S10 according to an appropriate method.
In the next method step S120, in an equivalent manner to method step S20 in the method according to one embodiment of the invention for calibrating a network analyzer, for example, on the basis of a 16term error model, a linear equation system according to equation (8B) is prepared. In each case the matrix A^{16×15 }and the vectors V_{1}^{16×1 }and V_{2}^{16×1 }contain the actual and the measured scattering parameters of the four nontransmissive and reciprocal calibration standards. The first and second linearinT system errors k and p are specified appropriately in each case, preferably with the value 1.
The determination of the 14 unknown linearinT system errors dependent upon the specified first and second linearinT system error k and p by solving the linear equation system in the next method step S130 corresponds to method step S30 in the method according to one embodiment of the invention for calibrating a measuring arrangement on the basis of a 16term error model.
The linearinT system errors determined in this manner or freely specified are supplied to the four submatrices T_{1i }to T_{4i }according to equation (12A) to (12D). From the elements of these linearinT system error submatrices T_{1i }to T_{4i}, the corresponding systemerror submatrices E_{1i }to E_{4i }of the 16term error model are determined in the next method step S140 using equations (23) and (24A) to (24D).
The system errors determined in this manner in the systemerror submatrices E_{1i }to E_{4i }of the 16term error model correspond to the parameters illustrated in the signalflow graph of
Since the individual elements of the systemerror matrices E_{1i }to E_{4i }of the 16term error model in some cases also provide dependences upon the first and/or second linearinT system errors k and p, in the following method steps S150 and S160, both the first linearinT system error k and also the second linearinT system error p are determined in each case with regard to the value and the sign to be additionally considered in the value.
In method step S150, the values of the first and second linearinT system errors k and p are determined. For this purpose, the fact is exploited that the transmission behavior in both transmission paths of the measurement holder 4 is reciprocal in each case. Accordingly, e_{10}=e_{01 }applies in the one transmission path and e_{23}=e_{32 }in the other transmission path of the measurement holder.
Since both the transmission coefficient e_{01 }in the forward direction of the one transmission path in the element E_{2i}(1,1) of the systemerror matrix E_{2i }and also the transmission coefficient e_{10 }in the backward direction of the same transmission path in the element E_{3i}(1,1) of the systemerror matrix E_{3i }is weighted with the second linearinT system error p and are additionally reciprocal to one another (e_{01}=e_{10}), the elements E_{2i}(1,1) and E_{3i}(1,1) determined in the last method step S140 for determining the value of the correct second linearinT system error p_{kor }according to equation (30A) exploiting the relationship factor M between the elements E_{2i}(1,1) and E_{3i}(1,1) according to equation (27A) are used.
Since, in an equivalent manner, the transmission coefficient e_{23 }in the forward direction of the other transmission path of the measurement holder in the element E_{2i}(2,2) of the systemerror matrix E_{2i }and also the transmission coefficient e_{32 }in the reverse direction of the same transmission path in the element E_{3i}(2,2) of the systemerror matrix E_{3i }with the first linearinT system error are reciprocal with one another (e_{23}=e_{32}) and are additionally weighted in each case with k, the elements E_{2i}(2,2) and E_{3i}(2,2) determined in the last method step S140 for determining the value of the correct first linearinT system error k_{kor }according to equation (30B) can be used by exploiting the relationship factor M between the elements E_{2i}(2,2) and E_{3i}(2,2) according to equation (27B).
For the determination of the sign to be additionally considered in the value of the correct first and correct second linearinT system error k_{kor }and p_{kor }in the next method step S160, in an equivalent manner to the method according to one embodiment of the invention for calibrating a measuring arrangement on the basis of a 16term error model, plausibilities, that is, interrelations between the known electrical length and the associated transmission elements in the systemerror matrices E_{2i }and E_{3i }of the respective transmission path of the measurement holder 4 are used. Here also, the electrical length need only be known to ±90°. For this purpose, taking into consideration all possible signs, the determined systemerror matrices T_{1i }to T_{4i }are renormalized according to equation (31A) with the correct first and correct second linearinT system error k_{kor }and p_{kor }and, from the accordingly obtained systemerror matrices T_{1c }to T_{4c}, the associated systemerror matrices E_{1c }to E_{4c }are determined with reference to equations (22A) to (22D). On the basis of the phases of the transmission coefficients determined in this manner, the correct sign is selected through a consideration of plausibility.
In the final method step S170, the matrices determined with incorrect sign are rejected, so that only adjusted submatrices E_{1c }to E_{4c }remain. These form the correct system error matrix E_{c }according to equation (31B) which contain the elements for characterizing the transmission, reflection and crosstalk behavior of the measurement holder without a remaining normalization factor.
In the following, the second embodiment of the method according to one embodiment of the invention for characterizing a measurement holder connected to a network analyzer is explained in detail on the basis of the flow diagram in
Method steps S200 to S230 of the second embodiment correspond to method steps S100 to S130 of the first embodiment, and the explanation will therefore not be repeated at this point. In the next method steps S240 and S250, the relationship between the first and second linearinT system error k and p is determined.
In method step S240, the value of the relationship between the first and second linearinT system error k and p is determined by using the measured scattering parameters of the fifth transmissive and reciprocal calibration standard R from equation (7E) fixed in the measurement holder 4. In an equivalent manner to method step S40 of the method according to one embodiment of the invention for calibrating a measuring arrangement on the basis of a 16term error model, the transformed transmission coefficients S_{arez}(1,2) and S_{arez}(2,1) of the transmissive and reciprocal calibration standard R according to equation (16) fixed in the measurement holder 4 are determined from the measured scattering parameters S_{mrez }and the linearinT submatrices T_{1i }to T_{4i }determined in method step S230 and dependent upon p and k. The value for the relationship between the second linearinT system error p and the first linearinT system error k can be determined by analogy with the calculation of the correct second linearinT system error p_{kor}(k) dependent upon the freely selected first linearinT system error k according to equation (21). Since the measurement holder 4 provides a reciprocal transmission behavior in both transmission paths, the reciprocity condition of the fifth transmissive and reciprocal calibration standard in the reference plane 2 also applies for the measurement in reference plane 1.
In the next method step S250, the sign to be additionally considered in the value of the relationship between the first and second linearinT system error k and p is determined in an equivalent manner to the method according to one embodiment of the invention for calibrating a measuring arrangement on the basis of a 16term error model by means of plausibility considerations, that is, on the basis of plausible interrelations, between the electrical length and the mutually reciprocal transmission coefficients S_{arez}(1,2) and S_{arez}(2,1) of the fifth calibration standard fixed in the measurement holder transformed according to equation (16).
In the next method step S250, the linearinT system errors in the submatrices T_{1i }to T_{4i }of the linearinT systemerror matrix T_{i }determined in method step S230 are transformed by means of the equations (22A) to (22D) into corresponding system errors in the submatrices E_{1i }to E_{4i }of the systemerror matrix E_{i }in an equivalent manner to method step S140 of the first embodiment.
In the next method step S270, in an equivalent manner to method step S150 of the first embodiment, the value of the correct first or correct second linearinT system error k_{kor }respectively p_{kor }is determined on the basis of equation (30A) respectively (30B) by using for this purpose the transmission elements E_{2i}(1,1) and E_{3i}(1,1) respectively E_{2i}(2,2) and E_{3i}(2,2) of the submatrices E_{2i }and E_{3i }of the systemerror matrix E_{i }determined in the preceding method step S260.
In the next method step S280 also, in an equivalent manner to method step S160 of the first embodiment, the sign to be additionally considered in the value of the correct first linearinT system error k_{kor }or the correct second linearinT system error p_{kor }is determined on the basis of plausibility considerations between the electrical length and the two mutually reciprocal transmission coefficients of one transmission path of the measurement holder determined in method step S260 as elements in a submatrix of the systemerror matrix E_{i}.
In the next method step S290, the respectively other correct—first or second—linearinT system error k_{kor }or p_{kor }is determined on the basis of the value established in each case in method steps S240 and S250 and the sign to be additionally considered in the value of the relation between first and second linearinT system error k and p and of the value established in the preceding method steps S270 and S280 in each case and the sign to be additionally considered in the value of the first correct linearinT system error k_{kor }or the second correct linearinT system error p_{kor}. The final method step S295 corresponds to the method step S170 in the first embodiment, and the explanation will therefore not be repeated at this point.
If the individual elements of the systemerror matrix E_{c }have been determined according to the first and second embodiment, a device under test fixed in the measurement holder 4 can be characterized by removing the system errors of the network analyzer 1 and the highfrequency measurement lines 2_{1 }and 2_{2 }and the transmission, reflection and crosstalk influences of the measurement holder 4 from the scattering parameters S_{m }measured with the vectorial network analyzer 1 in the reference plane 1 with the elements of the determined systemerror matrix E_{c }transformed into linearinT system errors. This corresponds to a measurement of the scattering parameters of the device under test in the reference plane 2.
For an adjusted crosstalk at the level of −30 dB,
At this point, reference is made to the fact that additional crosstalk can erroneously occur through numerical inaccuracies of the method used for solving the linear equation system in the case of measured data with superposed noise. This disadvantageous restriction of the dynamic range of the measurement with the use of a 16term error model is significantly limited in the method according to one embodiment of the invention by comparison with other methods of the prior art.
The method according to one embodiment of the invention and the systems according to one embodiment of the invention are not restricted to the embodiments presented. In particular, measurements on devices under test and measurement holders with more than two ports are also covered by one embodiment of the invention by using the method according to one embodiment of the invention sequentially, in each case for two ports of the device under test, respectively measurement holder. In particular, the combination of all of the features claimed in the patent claims, all of the features disclosed in the description and all of the features presented in the Figs. of the drawings are also covered by the one embodiment of invention.