A fast fully-analytic method for scatter parameter uncertainty evaluation

Abstract

The application discloses a fast full-analytic evaluation method for scattering parameter uncertainty, including the uncertainty of noise and non-ideal calibration standards introduced twice in the calibration and measurement process and propagated in a mixed cross mode, according to the mathematical analysis of the model, combining the vector network calibration method, through dimension increasing, decomposition and calculation of the Jacobian matrix, while fully considering the correlation, the full-analytic solution of the scattering parameter uncertainty is derived, that is, the fast full-analytic evaluation method. The method avoids large-scale random sampling and iterative calculation required by the Monte Carlo method, significantly improves the operation efficiency, and can provide the covariance and correlation coefficient between the real and imaginary parts of different scattering parameters, provides an effective tool for accurately analyzing the uncertainty source, and has a wide application prospect in the field of microwave metrology.

Description

Technical Field

[0001] This invention relates to the field of uncertainty assessment technology for scattering parameter measurement in microwave electronic devices and systems, and particularly to a rapid, fully analytical assessment method for scattering parameter uncertainty. Background Technology

[0002] Scattering parameters are key performance indicators for most microwave electronic components and systems. Their accurate measurement is crucial to ensuring the quality of devices and systems. Currently, the measurement of S-parameters mainly relies on vector network analyzers and their calibration algorithms (such as SOLT, TRL, etc.). Although these algorithms can correct the inherent errors of the system, they cannot completely eliminate measurement errors. Residual errors are often quantified in the form of uncertainty. Therefore, accurately and efficiently evaluating the uncertainty of scattering parameters measured by VNA has important practical significance.

[0003] Among existing uncertainty assessment methods: the Monte Carlo method achieves uncertainty propagation through extensive random sampling, but it is computationally time-consuming and cannot provide analytical solutions; the sensitivity analysis method, while avoiding random sampling, ignores the correlation effects between uncertainty sources; and the method based on the covariance matrix achieves a balance between computational efficiency and accuracy, but it has not yet solved the problem of the joint propagation of uncertainty caused by noise and non-ideal calibration standards, mainly due to the lack of corresponding propagation models and decoupling methods. Therefore, this invention proposes a rapid and fully analytical assessment method for scattering parameter uncertainty, namely, a rapid and fully analytical assessment method, which can accurately characterize the mixed cross-propagation mechanism of the joint propagation of noise and calibration standard uncertainty. This method can improve the assessment efficiency and accuracy of scattering parameter uncertainty, realizing a new rapid and fully analytical assessment method to meet the higher requirements of accuracy and efficiency in modern microwave measurement. Summary of the Invention

[0004] The purpose of this invention is to provide a fast, fully analytical method for evaluating the uncertainty of scattering parameters.

[0005] To achieve the above objectives, the present invention is implemented according to the following technical solution:

[0006] This invention includes the following steps:

[0007] Obtain noise data at each port of the measurement receiver and reference receiver, quantify the influencing factors of non-ideal calibration standards, and obtain calibration standard parameters;

[0008] Construct a first-order measurement Jacobian matrix, input noise data to calculate the measurement uncertainty of the calibration standard at each port and the measurement uncertainty of the device under test, construct a first-order theoretical Jacobian matrix, input calibration standard parameters to calculate the theoretical value uncertainty of the single-port calibration standard;

[0009] Calculate the error terms for each port based on the measured and theoretical values ​​of the scattering parameters of each port calibration standard. Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. Input the uncertainty of the measured and theoretical values ​​of the calibration standards to obtain the uncertainty matrix of the error terms.

[0010] The first-order scattering parameter Jacobian matrix is ​​constructed by taking the partial derivatives of the measured scattering parameter and the error term of the calibrated scattering parameter of the test device. The uncertainty of the measured value and the uncertainty of the error term of the test device are propagated to the calibrated scattering parameter of the test device through the first-order scattering parameter Jacobian matrix. The uncertainty matrix of the scattering parameter of the test device is obtained by using the perturbation method and combining it with the Cauchy-Riemann equation.

[0011] Calculate the covariance matrix and Pearson correlation coefficient between the real and imaginary parts of different scattering parameters of the test device based on the uncertainty matrix of the scattering parameters of the test device;

[0012] The port includes a single port and a dual port; the single port includes port 1 and port 2; port 1 contains a measurement receiver 1 and a reference receiver 1; port 2 contains a measurement receiver 2 and a reference receiver 2;

[0013] The calibration standards include single-port calibration standards and dual-port calibration standards; the single-port calibration standard is required on both port 1 and port 2, and includes Short, Open, and Load; the dual-port calibration standard is connected between port 1 and port 2, specifically Thru.

[0014] The calibration standard parameters include theoretical scattering parameters, amplitude uncertainty, and phase uncertainty;

[0015] The diagonal elements of the error uncertainty matrix represent the variance of each error term, while the off-diagonal elements represent the covariance between different error terms.

[0016] The diagonal elements of the uncertainty matrix of the scattering parameters of the test device are the variances of the real and imaginary parts of each scattering parameter of the test device, and the off-diagonal elements are the covariances of the real and imaginary parts of different scattering parameters of the test device.

[0017] Furthermore, the method for calculating the uncertainty of the calibration standard measurement value at each port includes:

[0018] Construct a first-order measurement Jacobian matrix for the single-port calibration standard scattering parameters, the received level of the measurement receiver, and the received level of the reference receiver. The expression is:

[0019] ;

[0020] in For single-port calibration standards Down port The first-order measurement Jacobian matrix of the port Internally includes a measurement receiver and reference receiver , , Measure receiver index, This also indicates the port index and the reference receiver index. For calibration standards Lower measurement receiver From port Received voltage level For calibration standards Down port Internal reference receiver Received voltage level For calibration standards Down port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined. Indicates the real part For the real part The partial derivative, Indicates the imaginary part For the real part The partial derivative, Indicates the real part imaginary part The partial derivative, Indicates the imaginary part imaginary part The partial derivative;

[0021] Based on the first-order measurement Jacobian matrix of the single-port port, the uncertainty of the single-port calibration standard measurement value is calculated using the input noise data, expressed as follows:

[0022] ;

[0023] in For calibration standards Down port The measurement uncertainty, For the variance of the noise data, , For calibration standards Downlink port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standards Downlink port Time reference receiver The real and imaginary parts of the noise data. This is the transpose of the first-order measurement Jacobian matrix for a single port.

[0024] Based on the measured values ​​of the scattering parameters of the two-port calibration standard, the received level of the measurement receiver, and the received level of the reference receiver, a first-order measurement Jacobian matrix for the two-port system is constructed, expressed as follows:

[0025] ;

[0026] in Measurement receiver under dual-port Thru calibration standard First-order measurement Jacobian matrix of the connection port Measurement receiver under Thru calibration standard From port Received level For Thru calibration standard port Internal reference receiver The received voltage level is measured by the receiver under the Thru calibration standard. Connection port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined;

[0027] Based on the first-order Jacobian matrix of the two-port measurement, the uncertainty of the two-port calibration standard measurement is calculated using the input noise data, expressed as follows:

[0028] ;

[0029] ;

[0030] in The measurement uncertainty under the two-port Thru calibration standard. Measurement receiver under dual-port Thru calibration standard Connection port The measurement uncertainty at that time , For Thru calibration standard connection port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standard Thru connection port Time reference receiver The real and imaginary parts of the noise data. For two-port first-order measurement Jacobian matrix The transpose of .

[0031] Furthermore, the method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes:

[0032] Construct a first-order theoretical Jacobian matrix based on the calibration standard parameters, and calculate the uncertainty of the single-port calibration standard theoretical value by inputting the calibration standard parameters. The expression is:

[0033] ;

[0034] ;

[0035] in For single-port calibration standard The theoretical uncertainty, , For single-port calibration standard Next-order theoretical Jacobian matrix, The first-order theoretical Jacobian matrix The transpose of the matrix, , , , , For single-port calibration standards The theoretical scattering parameters, amplitude, phase, amplitude uncertainty, and phase uncertainty are given below. The real part of the theoretical scattering parameter Partial derivative with respect to amplitude or phase.

[0036] Furthermore, the method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes:

[0037] The error terms for each port are calculated based on the measured and theoretical values ​​of the scattering parameters according to the calibration standards of each port; the error terms include single-port error terms. Two-port error term , , Indexed by category;

[0038] Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. The expression is as follows:

[0039] ;

[0040] in This is a first-order error Jacobian matrix with a size of 20×26. Indicates the error term of port 1 The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the two-port scattering parameter measurements are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The scattering parameters at port 2 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The real and imaginary parts of the measured values ​​of the scattering parameters at port 2 are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 1 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix;

[0041] The uncertainty matrix of the error term is obtained by inputting the uncertainty of the measured value and the uncertainty of the theoretical value of the calibration standard. The expression is as follows:

[0042] ;

[0043] in Here is the uncertainty matrix for the error term. The first-order error Jacobian matrix The transpose of .

[0044] Furthermore, the method for obtaining the uncertainty matrix of the scattering parameters of the test piece includes:

[0045] Acquire the measurement receiver of the device under test Connection port scattering parameter measurement values , Indicates the index of the measurement receiver. Simultaneously representing the port index and reference receiver index, and combining them with the error term to calculate the correction coefficient, the expression is:

[0046] ;

[0047] in For correction factors, , , For port 1 error term, , , For port 2 error term, , , , For the two-port error term, where, , Forward and reverse crosstalk errors can be directly measured. , For the forward and reverse source matching errors, , For positive and negative directional errors, , For forward and backward reflection tracking errors, , For positive and negative load matching errors, , For forward and reverse transmission tracking errors;

[0048] Calculate the correction coefficient based on the error term. The error term refers to the measured values ​​of the scattering parameters of the test piece. The scattering parameter values ​​of the test device were obtained after calibration. The first-order scattering parameter Jacobian matrix is ​​constructed by taking the partial derivatives of the calibrated scattering parameter values ​​of the test piece with the measured scattering parameter values ​​and the error term. The expression is as follows:

[0049] ;

[0050] in This is the first-order scattering parameter Jacobian matrix, with a size of 8×20. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts are respectively measured values ​​of the scattering parameters of the test piece. Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts of the single-port error term are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts respectively correspond to the two-port error term Find the partial derivatives of the real and imaginary parts of the matrix;

[0051] Uncertainty of the measured value of the test piece Uncertainty of the sum of error terms The first-order scattering parameter Jacobian matrix is ​​propagated to the calibrated scattering parameters of the device under test (DUT). The uncertainty matrix of the DUT's scattering parameters is obtained by solving the perturbation method combined with the Cauchy-Riemann equation, and the expression is:

[0052]

[0053] in The uncertainty matrix of the scattering parameters of the test piece is... The first-order scattering parameter Jacobian matrix The transpose of .

[0054] Furthermore, the covariance matrix and the Pearson correlation coefficient are used to quantitatively analyze the strength and direction of the linear correlation between the real and imaginary parts of different scattering parameters.

[0055] The beneficial effects of this invention are:

[0056] This invention provides a rapid, fully analytical method for evaluating the uncertainty of scattering parameters. Compared with existing technologies, this invention offers the following technical advantages:

[0057] This invention significantly improves the efficiency and accuracy of scattering parameter uncertainty assessment. It not only achieves a fast and fully analytical solution for the cross-propagation behavior of uncertainty in both calibration and measurement stages based on the extended and decomposed Jacobian matrix, but also avoids the computational burden of large-scale random sampling in the Monte Carlo method. Experimental comparisons have verified the effectiveness of the proposed method and enhanced the reliability and resolvability of the scattering parameter measurement uncertainty assessment results. Attached Figure Description

[0058] Figure 1 This is a flowchart illustrating the steps of a rapid, fully analytical evaluation method for scattering parameter uncertainty according to the present invention.

[0059] Figure 2 This is a schematic diagram illustrating the evaluation results of the uncertainty introduced by noise and non-ideal calibration standards, which is quantified into the uncertainty of the scattering parameters of the DUT after calibration by the rapid full analysis evaluation method proposed in this embodiment of the invention.

[0060] Figure 3 This is a schematic diagram comparing the quantitative results of the proposed method and the Monte Carlo method in evaluating the uncertainty of scattering parameters in the embodiments of the present invention.

[0061] Figure 4 This is a covariance diagram between the real and imaginary parts of different scattering parameters in an embodiment of the present invention;

[0062] Figure 5This is a Pearson correlation coefficient diagram between the real and imaginary parts of different scattering parameters in an embodiment of the present invention;

[0063] Figure 6 This is a flowchart illustrating the uncertainty propagation process in an embodiment of the present invention. Detailed Implementation

[0064] The present invention will be further described below through specific embodiments. The illustrative embodiments and descriptions herein are used to explain the present invention, but are not intended to limit the present invention.

[0065] The present invention provides a fast, fully analytical method for evaluating the uncertainty of scattering parameters, comprising the following steps:

[0066] like Figure 1 As shown, this embodiment includes the following steps:

[0067] Obtain noise data at each port of the measurement receiver and reference receiver, quantify the influencing factors of non-ideal calibration standards, and obtain calibration standard parameters;

[0068] Construct a first-order measurement Jacobian matrix, input noise data to calculate the measurement uncertainty of the calibration standard at each port and the measurement uncertainty of the device under test, construct a first-order theoretical Jacobian matrix, input calibration standard parameters to calculate the theoretical value uncertainty of the single-port calibration standard;

[0069] Calculate the error terms for each port based on the measured and theoretical values ​​of the scattering parameters of each port calibration standard. Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. Input the uncertainty of the measured and theoretical values ​​of the calibration standards to obtain the uncertainty matrix of the error terms.

[0070] The first-order scattering parameter Jacobian matrix is ​​constructed by taking the partial derivatives of the measured scattering parameter and the error term of the calibrated scattering parameter of the test device. The uncertainty of the measured value and the uncertainty of the error term of the test device are propagated to the calibrated scattering parameter of the test device through the first-order scattering parameter Jacobian matrix. The uncertainty matrix of the scattering parameter of the test device is obtained by using the perturbation method and combining it with the Cauchy-Riemann equation.

[0071] Calculate the covariance matrix and Pearson correlation coefficient between the real and imaginary parts of different scattering parameters of the test device based on the uncertainty matrix of the scattering parameters of the test device;

[0072] The port includes a single port and a dual port; the single port includes port 1 and port 2; port 1 contains a measurement receiver 1 and a reference receiver 1; port 2 contains a measurement receiver 2 and a reference receiver 2;

[0073] The calibration standards include single-port calibration standards and dual-port calibration standards; the single-port calibration standard is required on both port 1 and port 2, and includes Short, Open, and Load; the dual-port calibration standard is connected between port 1 and port 2, specifically Thru.

[0074] The calibration standard parameters include theoretical scattering parameters, amplitude uncertainty, and phase uncertainty;

[0075] The diagonal elements of the error uncertainty matrix represent the variance of each error term, while the off-diagonal elements represent the covariance between different error terms.

[0076] The diagonal elements of the uncertainty matrix of the scattering parameters of the test device are the variances of the real and imaginary parts of each scattering parameter of the test device, and the off-diagonal elements are the covariances of the real and imaginary parts of different scattering parameters of the test device.

[0077] In this embodiment, the method for calculating the uncertainty of the calibration standard measurement value at each port includes:

[0078] Construct a first-order measurement Jacobian matrix for the single-port calibration standard scattering parameters, the received level of the measurement receiver, and the received level of the reference receiver. The expression is:

[0079] ;

[0080] in For single-port calibration standards Down port The first-order measurement Jacobian matrix of the port Internally includes a measurement receiver and reference receiver , , Measure receiver index, This also indicates the port index and the reference receiver index. For calibration standards Lower measurement receiver From port Received voltage level For calibration standards Down port Internal reference receiver Received voltage level For calibration standards Down port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined. Indicates the real part For the real part The partial derivative, Indicates the imaginary part For the real part The partial derivative, Indicates the real part imaginary part The partial derivative, Indicates the imaginary part imaginary part The partial derivative;

[0081] Based on the first-order measurement Jacobian matrix of the single-port port, the uncertainty of the single-port calibration standard measurement value is calculated using the input noise data, expressed as follows:

[0082] ;

[0083] in For calibration standards Down port The measurement uncertainty, For the variance of the noise data, , For calibration standards Downlink port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standards Downlink port Time reference receiver The real and imaginary parts of the noise data. This is the transpose of the first-order measurement Jacobian matrix for a single port.

[0084] Based on the measured values ​​of the scattering parameters of the two-port calibration standard, the received level of the measurement receiver, and the received level of the reference receiver, a first-order measurement Jacobian matrix for the two-port system is constructed, expressed as follows:

[0085] ;

[0086] in Measurement receiver under dual-port Thru calibration standard First-order measurement Jacobian matrix of the connection port Measurement receiver under Thru calibration standard From port Received level For Thru calibration standard port Internal reference receiver The received voltage level is measured by the receiver under the Thru calibration standard. Connection port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined;

[0087] Based on the first-order Jacobian matrix of the two-port measurement, the uncertainty of the two-port calibration standard measurement is calculated using the input noise data, expressed as follows:

[0088] ;

[0089] ;

[0090] in The measurement uncertainty under the two-port Thru calibration standard. Measurement receiver under dual-port Thru calibration standard Connection port The measurement uncertainty at that time , For Thru calibration standard connection port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standard Thru connection port Time reference receiver The real and imaginary parts of the noise data. For two-port first-order measurement Jacobian matrix The transpose of the matrix;

[0091] In practical applications, taking port 1 of a single-port device under the calibration standard Short as an example, a first-order measurement Jacobian matrix is ​​constructed based on the measured values ​​of the scattering parameters of the calibration standard, the received level of the measurement receiver, and the received level of the reference receiver:

[0092] ;

[0093] Calculate the measurement uncertainty caused by port 1 noise of a single-port device under the calibration standard Short:

[0094] ;

[0095] For a two-port device calibrated according to Thru standard, a first-order measurement Jacobian matrix for the two-port device is constructed based on the measured values ​​of the two-port scattering parameters, the received level of the measurement receiver, and the received level of the reference receiver:

[0096] ;

[0097] ;

[0098] ;

[0099] ;

[0100] Calculate the measurement uncertainty caused by two-port device noise under the calibration standard Thru:

[0101] ;

[0102] ;

[0103]

[0104]

[0105]

[0106] This yields the uncertainty of the calibration standard measurement values ​​for each port. , , , , , , .

[0107] In this embodiment, the method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes:

[0108] Construct a first-order theoretical Jacobian matrix based on the calibration standard parameters, and calculate the uncertainty of the single-port calibration standard theoretical value by inputting the calibration standard parameters. The expression is:

[0109] ;

[0110] ;

[0111] in For single-port calibration standard The theoretical uncertainty, , For single-port calibration standard Next-order theoretical Jacobian matrix, The first-order theoretical Jacobian matrix The transpose of the matrix, , , , , For single-port calibration standards The theoretical scattering parameters, amplitude, phase, amplitude uncertainty, and phase uncertainty are given below. The real part of the theoretical scattering parameter Partial derivative with respect to amplitude or phase;

[0112] In practical applications, taking port 1 of a single-port device under the Short calibration standard as an example, a first-order theoretical Jacobian matrix is ​​constructed based on the calibration standard parameters, and the uncertainty of the theoretical value of the single-port calibration standard is calculated by inputting the calibration standard parameters:

[0113] ;

[0114] ;

[0115] Thus, the uncertainty of the theoretical value of the single-port calibration standard is obtained. , , .

[0116] In this embodiment, the method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes:

[0117] The error terms for each port are calculated based on the measured and theoretical values ​​of the scattering parameters according to the calibration standards of each port; the error terms include single-port error terms. Two-port error term , , Indexed by category;

[0118] Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. The expression is as follows:

[0119] ;

[0120] in This is a first-order error Jacobian matrix with a size of 20×26. Indicates the error term of port 1 The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the two-port scattering parameter measurements are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The scattering parameters at port 2 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The real and imaginary parts of the measured values ​​of the scattering parameters at port 2 are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 1 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix;

[0121] The uncertainty matrix of the error term is obtained by inputting the uncertainty of the measured value and the uncertainty of the theoretical value of the calibration standard. The expression is as follows:

[0122] ;

[0123] in Here is the uncertainty matrix for the error term. The first-order error Jacobian matrix The transpose of the matrix;

[0124] In practical applications, the error terms for each port are calculated based on the measured and theoretical values ​​of the scattering parameters according to the calibration standards of each port. First, the error terms for each port are calculated. , The expression is:

[0125] ;

[0126] ;

[0127] ;

[0128] ;

[0129] ;

[0130] ;

[0131] in , , For port 1 error term, , , For port 2 error term, , , , For the two-port error term, where, , Forward and reverse crosstalk errors can be directly measured. , For the forward and reverse source matching errors, , For positive and negative directional errors, , For forward and backward reflection tracking errors, , For positive and negative load matching errors, , For forward and reverse transmission tracking errors, , , The theoretical values ​​of the scattering parameters of receiver 1 at port 1 are obtained by measuring the scattering parameters of receiver 1 under the single-port calibration standard. , , The theoretical value of the scattering parameter of receiver 2 at port 2 is obtained by measuring the scattering parameter of receiver 2 under the single-port calibration standard.

[0132] Based on the single-port error term , Calculate the two-port error term using the measured values ​​of the two-port scattering parameters under the calibration standard Thru. The expression is:

[0133] ;

[0134] ;

[0135] ;

[0136] ;

[0137] in , Forward and reverse crosstalk errors can be directly measured;

[0138] Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. The expression is as follows:

[0139] ;

[0140] In each partial derivative matrix, the elements in the same row are the real / imaginary parts of the same error term, with respect to the real and imaginary parts of all measured / theoretical values. The number of rows is the sum of the number of real and imaginary parts of all error terms at the same port. Specifically:

[0141] ;

[0142] ;

[0143] ;

[0144] ;

[0145] ;

[0146] ;

[0147] ;

[0148] ;

[0149] in , , , , , , , The sizes are 6×6, 8×6, 8×8, 8×6, 6×6, 6×6, 8×6, and 6×6.

[0150] Based on the first-order error Jacobian matrix Uncertainty of calibration standard measurement values ​​at each port ( , , , , , , ) and the theoretical value uncertainty of the single-port calibration standard ( , , Construct the uncertainty matrix of the error term.

[0151] In this embodiment, the method for obtaining the uncertainty matrix of the scattering parameters of the test object includes:

[0152] Acquire the measurement receiver of the device under test Connection port scattering parameter measurement values , Indicates the index of the measurement receiver. Simultaneously representing the port index and reference receiver index, and combining them with the error term to calculate the correction coefficient, the expression is:

[0153] ;

[0154] in For correction factors, , , For port 1 error term, , , For port 2 error term, , , , For the two-port error term, where, , Forward and reverse crosstalk errors can be directly measured. , For the forward and reverse source matching errors, , For positive and negative directional errors, , For forward and backward reflection tracking errors, , For positive and negative load matching errors, , For forward and reverse transmission tracking errors;

[0155] Calculate the correction coefficient based on the error term. The error term refers to the measured values ​​of the scattering parameters of the test piece. The scattering parameter values ​​of the test device were obtained after calibration. The first-order scattering parameter Jacobian matrix is ​​constructed by taking the partial derivatives of the calibrated scattering parameter values ​​of the test piece with the measured scattering parameter values ​​and the error term. The expression is as follows:

[0156] ;

[0157] in This is the first-order scattering parameter Jacobian matrix, with a size of 8×20. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts are respectively measured values ​​of the scattering parameters of the test piece. Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts of the single-port error term are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts respectively correspond to the two-port error term Find the partial derivatives of the real and imaginary parts of the matrix;

[0158] Uncertainty of the measured value of the test piece Uncertainty of the sum of error terms The first-order scattering parameter Jacobian matrix is ​​propagated to the calibrated scattering parameters of the device under test (DUT). The uncertainty matrix of the DUT's scattering parameters is obtained by solving the perturbation method combined with the Cauchy-Riemann equation, and the expression is:

[0159]

[0160] in The uncertainty matrix of the scattering parameters of the test piece is... The first-order scattering parameter Jacobian matrix The transpose of the matrix;

[0161] In practical applications, the scattering parameter values ​​of the calibrated test piece are... The expression is as follows:

[0162] ;

[0163] ;

[0164] ;

[0165] ;

[0166] Using the calibrated scattering parameter values ​​of the test device Measured values ​​of scattering parameters of the test piece Sum of error terms , The partial derivative is used to construct the first-order scattering parameter Jacobian matrix, which is expressed as:

[0167] ;

[0168] ;

[0169] ;

[0170] ;

[0171] in , The sizes are 8×8, 8×12, and 8×8;

[0172] Uncertainty of the measured value of the test piece Uncertainty of the sum of error terms The first-order scattering parameter Jacobian matrix is ​​propagated to the calibrated scattering parameters of the device under test. The uncertainty matrix of the scattering parameters of the device under test is obtained by using the perturbation method combined with the Cauchy-Riemann equation.

[0173] In this embodiment, the covariance matrix and the Pearson correlation coefficient are used to quantitatively analyze the strength and direction of the linear correlation between the real and imaginary parts of different scattering parameters.

[0174] like Figure 2 As shown, the FFEM method proposed in this embodiment quantifies the uncertainty introduced by noise and non-ideal calibration standards into the evaluation result of the uncertainty of the scattering parameters of the calibrated DUT. Figure 2 (a) represents the effect of uncertainty of different non-ideal calibration standards on S. 11 and S 22 The impact; Figure 2 (b) indicates the effect of different non-ideal calibration standard uncertainties on S. 12 and S 21 The impact; Figure 2 (c) indicates the relationship between the non-ideal calibration standard and the overall noise level on S. 11 and S 22 The impact; Figure 2 (d) indicates the relationship between the non-ideal calibration standard and the overall noise level on S. 12 and S 21 The impact;

[0175] The results show that in the SOLT calibration method: 1) the non-ideal load standard is the main source of uncertainty in the reflection coefficient, but has little effect on the transmission coefficient; 2) the uncertainty of the reflection coefficient is generally much higher than that of the transmission coefficient; 3) the influence of noise is more significant in the transmission coefficient.

[0176] To verify the correctness of the proposed scattering parameter uncertainty assessment method, a 40dB attenuator was selected as the device under test and the SOLT method was selected as the calibration method for performance verification. The verification steps were as follows: the uncertainty of noise and non-ideal calibration standards was transferred to the 12 error terms of the SOLT method and then to the scattering parameters of the device under test after calibration. The quantification results of the proposed method for the uncertainty assessment of the scattering parameters after calibration were compared with those of the Monte Carlo method.

[0177] like Figure 3 As shown, a comparison is given between the proposed method in this embodiment and the Monte Carlo method for evaluating the uncertainty of quantized scattering parameters, including reflection and transmission coefficients. Figure 3 (a) represents the reflection coefficient S of port 1 obtained by the two methods. 11 The uncertainty assessment results; Figure 3 (b) represents the two methods for the two-port transmission coefficient S. 12 The uncertainty assessment results; Figure 3 (c) indicates the two methods for the two-port transmission coefficient S 21 The uncertainty assessment results; Figure 3(d) represents the reflection coefficient S at port 2 under the two methods. 22 The uncertainty assessment results;

[0178] Experiments show that the proposed FFEM method, by replacing random sampling with analytical solution, achieves highly consistent quantitative results for the uncertainty assessment of scattering parameters with those obtained by the Monte Carlo method, thus verifying its effectiveness.

[0179] By extracting the off-diagonal elements from the scattering parameter uncertainty matrix obtained in step S4, the covariance values ​​between the real and imaginary parts of different S-parameters can be obtained. The characteristics of their variation with frequency can be observed. At the same time, each covariance value can be divided by the product of the standard deviations of the real and imaginary parts of the corresponding scattering parameters to calculate the Pearson correlation coefficient, thereby quantitatively analyzing the correlation between the real and imaginary parts of different S-parameters.

[0180] like Figure 4-5 As shown, the covariance and Pearson correlation coefficient between the real and imaginary parts of different scattering parameters in embodiments of the present invention are given. The results show that: 1) Except for S 11 and S 22 The real covariance is always positive, while the other covariances fluctuate around zero. This indicates that only two sets of sustained positive correlations exist across the entire frequency band, namely S. 11 With S 22 1) Between the real parts and between their imaginary parts; 2) Only S 11 With S 22 The real part of the Pearson coefficient is close to 1, showing a strong linear positive correlation; 3) S 12 Real part and S 21 The coefficients between the real and imaginary Pearson coefficients vary between 1 and -1, showing alternating positive and negative correlations with frequency; 4) The signs of the coefficients between the other parameters are uncertain, but they are all close to 0, indicating that their linear correlation is weak.

[0181] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A fast, fully analytical method for evaluating the uncertainty of scattering parameters, characterized in that, Includes the following steps: S1. Obtain noise data at each port of the measurement receiver and the reference receiver, quantify the influencing factors of non-ideal calibration standards, and obtain calibration standard parameters. S2. Construct a first-order measurement Jacobian matrix, input noise data to calculate the measurement uncertainty of the calibration standard at each port and the measurement uncertainty of the device under test, construct a first-order theoretical Jacobian matrix, input calibration standard parameters to calculate the theoretical value uncertainty of the single-port calibration standard; S3. Calculate the error terms for each port based on the measured and theoretical values ​​of the scattering parameters of each port calibration standard. Construct a first-order error Jacobian matrix by taking the partial derivatives of the error terms of each port with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. Input the uncertainty of the measured and theoretical values ​​of the calibration standards to obtain the uncertainty matrix of the error terms. S4. Construct a first-order scattering parameter Jacobian matrix by taking partial derivatives of the measured scattering parameter values ​​and error terms of the calibrated scattering parameters of the test device. Propagate the uncertainty of the measured values ​​and the uncertainty of the error terms of the test device to the calibrated scattering parameters of the test device through the first-order scattering parameter Jacobian matrix. Solve the scattering parameter uncertainty matrix of the test device using the perturbation method and in combination with the Cauchy-Riemann equation. S5. Calculate the covariance matrix and Pearson correlation coefficient between the real and imaginary parts of different scattering parameters of the test piece based on the uncertainty matrix of the scattering parameters of the test piece. The port includes a single port and a dual port; the single port includes port 1 and port 2; port 1 contains a measurement receiver 1 and a reference receiver 1; port 2 contains a measurement receiver 2 and a reference receiver 2; The calibration standards include single-port calibration standards and dual-port calibration standards; the single-port calibration standards are required on both port 1 and port 2, and include Short, Open, and Load. The dual-port calibration standard is connected between port 1 and port 2, specifically Thru; The calibration standard parameters include theoretical scattering parameters, amplitude uncertainty, and phase uncertainty; The diagonal elements of the error uncertainty matrix represent the variance of each error term, while the off-diagonal elements represent the covariance between different error terms. The diagonal elements of the uncertainty matrix of the scattering parameters of the test device are the variances of the real and imaginary parts of each scattering parameter of the test device, and the off-diagonal elements are the covariances of the real and imaginary parts of different scattering parameters of the test device.

2. The rapid, fully analytical evaluation method for scattering parameter uncertainty according to claim 1, characterized in that, The method for calculating the uncertainty of the calibration standard measurement value of each port includes: Construct a first-order measurement Jacobian matrix for the single-port calibration standard scattering parameters, the received level of the measurement receiver, and the received level of the reference receiver. The expression is: ； in For single-port calibration standards Down port The first-order measurement Jacobian matrix of the port Internally includes a measurement receiver and reference receiver , , Indicates the index of the measurement receiver. It also represents the port index and the reference receiver index. For calibration standards Lower measurement receiver From port Received voltage level For calibration standards Down port Internal reference receiver Received voltage level For calibration standards Down port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined. Indicates the real part For the real part The partial derivative, Indicates the imaginary part For the real part The partial derivative, Indicates the real part imaginary part The partial derivative, Indicates the imaginary part imaginary part The partial derivative; Based on the first-order measurement Jacobian matrix of the single-port port, the uncertainty of the single-port calibration standard measurement value is calculated using the input noise data, expressed as follows: ； in For calibration standards Down port The measurement uncertainty, For the variance of the noise data, , For calibration standards Downlink port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standards Downlink port Time reference receiver The real and imaginary parts of the noise data. This is the transpose of the first-order measurement Jacobian matrix for a single port. Based on the measured values ​​of the scattering parameters of the two-port calibration standard, the received level of the measurement receiver, and the received level of the reference receiver, a first-order measurement Jacobian matrix for the two-port system is constructed, expressed as follows: ； in Measurement receiver under dual-port Thru calibration standard Connection port The first-order measurement Jacobian matrix at time, Measurement receiver under Thru calibration standard From port Received voltage level For Thru calibration standard port Internal reference receiver Received voltage level Measurement receiver under Thru calibration standard Connection port The scattering parameter measurements are obtained from the measurement receiver. Received level With reference receiver Received level The ratio is determined; Based on the first-order Jacobian matrix of the two-port measurement, the uncertainty of the two-port calibration standard measurement is calculated using the input noise data, expressed as follows: ； ； in The measurement uncertainty under the two-port Thru calibration standard. Measurement receiver under dual-port Thru calibration standard Connection port The measurement uncertainty at that time , For Thru calibration standard connection port Time measurement receiver The real and imaginary parts of the noise data. , For calibration standard Thru connection port Time reference receiver The real and imaginary parts of the noise data. For two-port first-order measurement Jacobian matrix The transpose of .

3. The rapid, fully analytical evaluation method for scattering parameter uncertainty according to claim 1, characterized in that, The method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes: Construct a first-order theoretical Jacobian matrix based on the calibration standard parameters, and calculate the uncertainty of the single-port calibration standard theoretical value by inputting the calibration standard parameters. The expression is: ； ； in For single-port calibration standard The theoretical uncertainty, , For single-port calibration standard Next-order theoretical Jacobian matrix, The first-order theoretical Jacobian matrix The transpose of the matrix, , , , , For single-port calibration standards The theoretical scattering parameters, amplitude, phase, amplitude uncertainty, and phase uncertainty are given below. The real part of the theoretical scattering parameter Partial derivative with respect to amplitude or phase.

4. The rapid, fully analytical evaluation method for scattering parameter uncertainty according to claim 1, characterized in that, The method for calculating the uncertainty of the theoretical value of the single-port calibration standard includes: The error terms for each port are calculated based on the measured and theoretical values ​​of the scattering parameters according to the calibration standards of each port; the error terms include single-port error terms. Two-port error term , , Indexed by category; Construct a first-order error Jacobian matrix by taking the partial derivatives of each port error term with respect to the measured and theoretical values ​​of the scattering parameters of all calibration standards. The expression is as follows: ； in This is a first-order error Jacobian matrix with a size of 20×26. Indicates the error term of port 1 The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the scattering parameters at port 1 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The real and imaginary parts of the two-port scattering parameter measurements are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The scattering parameters at port 2 were measured respectively. Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The real and imaginary parts of the measured values ​​of the scattering parameters at port 2 are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 1 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Represents the two-port error term The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix. Indicates the error term of port 2 The theoretical values ​​of scattering parameters were respectively Find the partial derivatives of the real and imaginary parts of the matrix; The uncertainty matrix of the error term is obtained by inputting the uncertainty of the measured value and the uncertainty of the theoretical value of the calibration standard. The expression is as follows: ； in Here is the uncertainty matrix for the error term. The first-order error Jacobian matrix The transpose of .

5. The rapid, fully analytical evaluation method for scattering parameter uncertainty according to claim 1, characterized in that, The method for obtaining the uncertainty matrix of the scattering parameters of the test object includes: Acquire the measurement receiver of the device under test Connection port scattering parameter measurement values , Indicates the index of the measurement receiver. Simultaneously representing the port index and reference receiver index, and combining them with the error term to calculate the correction coefficient, the expression is: ； in For correction factors, , , For port 1 error term, , , For port 2 error term, , , , For the two-port error term, where, , Forward and reverse crosstalk errors can be directly measured. , For the forward and reverse source matching errors, , For positive and negative directional errors, , For forward and backward reflection tracking errors, , For positive and negative load matching errors, , For forward and reverse transmission tracking errors; Calculate the correction coefficient based on the error term. The error term refers to the measured values ​​of the scattering parameters of the test piece. The scattering parameter values ​​of the test device were obtained after calibration. The first-order scattering parameter Jacobian matrix is ​​constructed by taking the partial derivatives of the calibrated scattering parameter values ​​of the test piece with the measured scattering parameter values ​​and the error term. The expression is as follows: ； in This is the first-order scattering parameter Jacobian matrix, with a size of 8×20. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts are respectively measured values ​​of the scattering parameters of the test piece. Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts of the single-port error term are respectively... Find the partial derivatives of the real and imaginary parts of the matrix. The scattering parameter values ​​of the test piece after calibration The real and imaginary parts respectively correspond to the two-port error term Find the partial derivatives of the real and imaginary parts of the matrix; Uncertainty of the measured value of the test piece Uncertainty of the sum of error terms The first-order scattering parameter Jacobian matrix is ​​propagated to the calibrated scattering parameters of the device under test (DUT). The uncertainty matrix of the DUT's scattering parameters is obtained by solving the perturbation method combined with the Cauchy-Riemann equation, and the expression is: ； in The uncertainty matrix of the scattering parameters of the test piece is... The first-order scattering parameter Jacobian matrix The transpose of .

6. The rapid fully analytical evaluation method for scattering parameter uncertainty according to claim 1, characterized in that: The covariance matrix and the Pearson correlation coefficient are used to quantitatively analyze the strength and direction of the linear correlation between the real and imaginary parts of different scattering parameters.

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