Sensitivity inversion-based scattering parameter traceability uncertainty analysis method
By combining sensitivity inversion and Monte Carlo simulation, the scattering parameter uncertainty of microwave electronic components and systems is evaluated, which solves the problem that measurement errors cannot be eliminated in the prior art. It realizes the traceable uncertainty transfer from the geometry of the air medium transmission line to the S-parameter, and improves the reliability and analytical quality of the evaluation results.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NATIONAL INSTITUTE OF METROLOGY CHINA
- Filing Date
- 2026-01-26
- Publication Date
- 2026-05-05
AI Technical Summary
Existing technologies cannot effectively assess the uncertainty of scattering parameters of microwave electronic components and systems, especially cannot completely eliminate measurement errors. Furthermore, existing methods struggle to balance computational efficiency and accuracy, and lack corresponding propagation models and decoupling methods.
By combining sensitivity inversion and Monte Carlo simulation, the characteristic impedance and its uncertainty are calculated by obtaining the geometric uncertainty of a standard air-medium transmission line. The TRL calibration model is then used for linearization, and Monte Carlo method is used for joint sampling to evaluate the scattering parameter uncertainty of the device under test.
This enables traceable assessment of the uncertainty propagation path from the geometry of the air medium transport line to the final calibrated S-parameters, enhancing the reliability and resolvability of the scattering parameter measurement uncertainty assessment.
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Figure CN121980796A_ABST
Abstract
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 in particular to a traceable uncertainty analysis method for scattering parameters based on sensitivity inversion. 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; sensitivity analysis, while avoiding random sampling, ignores the correlation effects between uncertainty sources; methods based on the covariance matrix achieve a balance between computational efficiency and accuracy, but have 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 novel uncertainty analysis method combining sensitivity inversion and Monte Carlo simulation. The sensitivity inversion technique decouples the uncertainty of the known transmission line S-parameters from the uncertainty of the error coefficients, and then uses joint Monte Carlo simulation to analyze the propagation of uncertainty from the error coefficients and measurement noise to the S-parameters of the device under test. Summary of the Invention
[0004] The purpose of this invention is to provide a method for analyzing traceable uncertainties in scattering parameters by combining sensitivity inversion and Monte Carlo simulation.
[0005] To achieve the above objectives, the present invention is implemented according to the following technical solution: This invention includes the following steps: Obtain the measured values of the inner diameter of the outer conductor and the outer diameter of the inner conductor of a standard air-medium transmission line, and evaluate the standard uncertainty of the inner diameter of the outer conductor and the outer diameter of the inner conductor, respectively. Based on the inner diameter of the outer conductor and the outer diameter of the inner conductor, calculate the characteristic impedance and its uncertainty of the standard air-dielectric transmission line; Based on the characteristic impedance and its uncertainty, and in combination with the influence of measurement noise, the uncertainty matrices of the scattering parameters and transmission parameters of the standard air medium transmission line are calculated respectively. Based on the uncertainty matrix of the transmission parameters, the uncertainty vector of the error coefficients is obtained by linearizing the eight-term error model calibrated by TRL and solving it through sensitivity inversion. The uncertainty vector of the error coefficients and the measurement noise are jointly sampled using the Monte Carlo method. The original scattering parameters of the device under test are repeatedly corrected by TRL calibration and the discreteness is statistically output to obtain the uncertainty matrix of the scattering parameters of the device under test after TRL calibration. Based on the known values and uncertainties of the calibration standard obtained by TRL calibration, the known value uncertainty of the calibration standard is introduced into SOLT calibration and jointly propagated with the measurement noise to obtain the uncertainty matrix of the scattering parameters of the device under test after SOLT calibration. Furthermore, the steps for evaluating the standard uncertainty of the inner diameter of the outer conductor and the outer diameter of the inner conductor include: for the measurement process of the inner and outer diameters of the air transmission line, identifying uncertainty components such as temperature correction, gauge calibration, and repeatability measurement, and combining each uncertainty component using the root sum of squares method to obtain the combined standard uncertainty of the inner and outer diameter measurements.
[0006] Furthermore, the step of calculating the uncertainty of the characteristic impedance includes propagating the uncertainty of the inner diameter of the outer conductor and the uncertainty of the outer diameter of the inner conductor to the characteristic impedance based on the error propagation law.
[0007] Furthermore, in the step of calculating the uncertainty of the characteristic impedance, the following formula is used:
[0008] Where μ and ε are the magnetic permeability and dielectric constant of air, respectively.
[0009] Furthermore, the method for obtaining the uncertainty vector of the error coefficients through sensitivity inversion specifically includes: An error model equation based on TRL calibration is established, which is associated with the known transmission matrix of the standard air line, the transmission matrix of the through standard, the corresponding measurement transmission matrix, and the error coefficients a, b, c, α, β, ε, r, ρ of the eight error models. The error model equations are linearized by total differential to construct a system of linear equations with the uncertainty of the error coefficients as unknowns. Solve the system of linear equations to obtain the uncertainty vector of the error coefficients.
[0010] Furthermore, the method of obtaining the uncertainty matrix of the scattering parameters of the device under test after TRL calibration using the Monte Carlo method specifically includes: Based on the uncertainty vector of the error coefficients and the statistical characteristics of the measurement noise, the error coefficients and measurement noise are subjected to multiple joint random samplings. For each sample, a complete TRL calibration procedure is performed to correct the original measured scattering parameters of the device under test; The discreteness of the output results after multiple calibration corrections is statistically analyzed to determine the uncertainty matrix of the scattering parameters of the device under test after TRL calibration.
[0011] Furthermore, the uncertainty matrix of the scattering parameters of the device under test after SOLT calibration specifically includes: Obtain standard values for short-circuit, open-circuit, and load standards for SOLT calibration. The standard values and their uncertainties are obtained through TRL calibration and measurement techniques. The Monte Carlo method is used to perform joint propagation analysis on the standard value uncertainty of the calibration standard, the uncertainty introduced by its original measurement noise, and the uncertainty introduced by the measurement noise of the device under test. The uncertainty matrix of the scattering parameters of the device under test after SOLT calibration is obtained from the statistical propagation results.
[0012] Furthermore, the uncertainty matrix of the S-parameters of the device under test (DUT) after SOLT calibration is solved. ,include: Uncertainty vector associated with known values of SOLT calibration standard The uncertainty can be determined using the uncertainty calculation method derived from the aforementioned five-step TRL calibration technique. When the known values of the calibration standard are treated as random variables, this does not introduce additional complexity to solving the 12-term error model or calibrating the S-parameters of the device under test. Therefore, the Monte Carlo method can be used to synthesize the uncertainty vector. Initial measurement uncertainty vector of S-parameters caused by noise and matrix The measurement uncertainty matrix of the S-parameters of the device under test is obtained by transferring the data. .
[0013] The beneficial effects of this invention are as follows: Compared with the prior art, the method for analyzing the traceable uncertainty of scattering parameters by combining sensitivity inversion and Monte Carlo simulation has the following technical advantages: This invention establishes a traceability chain for S-parameter measurements integrating TRL and SOLT calibration technologies, tracing coaxial S-parameter measurements back to the inner and outer diameter geometry of a standard air-medium transmission line. Based on this, a traceable uncertainty propagation model considering the influence of non-ideal calibration standards and noise is proposed, clarifying the uncertainty propagation path from the air line geometry and measurement noise to the final calibrated S-parameters. A novel uncertainty analysis method combining sensitivity inversion and Monte Carlo simulation is introduced. The sensitivity inversion technique decouples the uncertainty of the known S-parameters of the air line from the uncertainty of the error coefficients. Subsequently, a joint Monte Carlo simulation is used to analyze the process by which the error coefficients and measurement noise propagate together to the uncertainty of the device under test (DUT) S-parameters. Experimental comparisons verify the effectiveness of the proposed method, enhancing the reliability and resolvability of the scattering parameter measurement uncertainty assessment results. Attached Figure Description
[0014] Figure 1 This is a flowchart of the steps in the method for traceable uncertainty analysis of scattering parameters based on the joint sensitivity inversion and Monte Carlo simulation of the present invention. Figure 2 This is a schematic diagram of the S-parameter measurement traceability chain that integrates TRL and SOLT calibration technologies in this invention; Figure 3 This is a schematic diagram of the traceability uncertainty propagation model considering non-ideal calibration standards and measurement noise in this invention; Figure 4 This is a flowchart of the uncertainty analysis method combining sensitivity inversion and Monte Carlo simulation in this invention; Figure 5 This is an example of the S-parameter measurement device for the Agilent coaxial 1.0 mm mismatch verification component and the 2.4 mm attenuator in this invention. Figure 6 The corrected S-parameters of the mismatch verification device after TRL calibration and the attenuator after SOLT calibration in the example of this invention; Figure 7 This refers to the traceable uncertainty of the S-parameters of the mismatched device caused by the uncertainty of the inner and outer diameters of the standard air line and measurement noise in the examples of this invention. Figure 7 (a) is the reflection coefficient S11 of port 1. Figure 7 (b) is the transmission coefficient S12 of port 1. Figure 7 (c) is the transmission coefficient S21 of port 2. Figure 7 (d) is the reflection coefficient S22 of port 2; Figure 8 The traceable uncertainty of the attenuator S-parameters caused by non-ideal calibration standard components and measurement noise in this invention example is as follows: Figure 7 (a) is the reflection coefficient S11 of port 1. Figure 7 (b) is the transmission coefficient S12 of port 1. Figure 7 (c) is the transmission coefficient S21 of port 2. Figure 7 (d) is the reflection coefficient S22 of port 2. Detailed Implementation
[0015] 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.
[0016] The present invention provides a method for traceable uncertainty analysis of scattering parameters based on sensitivity inversion, comprising the following steps: like Figure 1 As shown, this embodiment includes the following steps: S1. Evaluation of the uncertainty of the outer conductor inner diameter and the inner conductor outer diameter of the standard air wire: Considering the effects of instrument traceability / calibration, resolution, temperature and repeatability, the standard uncertainties of the two geometric quantities are synthesized. S2. Calculate the characteristic impedance and uncertainty of the air line: Calculate the characteristic impedance from the structural dimensions of the air line, and propagate the geometric uncertainty of S1 to the impedance to obtain the standard uncertainty of the characteristic impedance, which will be used as the input for subsequent source tracing. S3. Calculate the uncertainty matrix of S-parameters and T-parameters of standard air line: Calculate the scattering parameters and transmission parameters of air line, and propagate the impedance uncertainty and measurement noise effects into the electrical parameters to form the uncertainty matrix of scattering parameters and transmission parameters of standard part. S4. Solve for the uncertainty vector of the eight error coefficients: Establish the solution relationship based on the TRL calibration error model, linearize the model, and obtain the uncertainty vector of the error coefficients from the uncertainty of the electrical parameters of the standard parts through sensitivity inversion. S5. Calculate the uncertainty matrix of the TRL correction S-parameters of the device under test using the Monte Carlo method: By combining the Monte Carlo sampling error coefficients and measurement noise, the TRL correction is repeatedly performed and the output discreteness is statistically analyzed to obtain the uncertainty matrix of the scattering parameters of the device under test after TRL correction. S6. Solve for the uncertainty matrix of the S-parameters of the measured DUT after SOLT calibration: In SOLT calibration, the known uncertainty of the calibration standard and the measurement noise are introduced, and the joint propagation and statistical output are performed to obtain the uncertainty matrix of the scattering parameters of the DUT after SOLT calibration. In practical evaluation, a traceability chain for S-parameter measurements was established, integrating TRL and SOLT calibration technologies. This chain traces the coaxial S-parameter measurements back to the inner and outer diameter geometry of a standard air-medium transmission line. Figure 2As shown in the figure. Based on this, a traceable uncertainty propagation model considering the effects of non-ideal calibration standards and noise is proposed, clarifying the uncertainty propagation path from air line geometry and measurement noise to the final calibrated S-parameters.
[0017] The traceability of S-parameters is crucial for achieving measurement consistency and the flattening of measurement values. The measurement of S-parameters can ultimately be traced back to the inner and outer diameters of a standard air-dielectric transmission line, as well as the permeability and dielectric constant of air. These parameters collectively determine the characteristic impedance of the standard air-dielectric transmission line, which in turn determines the scattering matrix (S-matrix) and transmission matrix (T-matrix) of the standard air-dielectric transmission line during calibration. Based on these matrix parameters, the error coefficients of an eight-term error model can be solved, and these error terms can be used to correct the measured S-parameters of the device under test (DUT), thereby realizing the TRL calibration process.
[0018] In SOLT calibration, the parameters of short-circuit, open-circuit, and loaded standard components are measured and extracted using TRL calibration technology. These known standard values are then used to solve a twelve-term error model, and finally, error coefficients are applied to correct the measured S-parameters of the device under test. Therefore, the SOLT calibration process is essentially traceable to the geometry of a standard air-dielectric transmission line and the dielectric constant and permeability of air.
[0019] like Figure 2 and 3 As shown, the measurement uncertainty of the inner and outer diameters of a standard air-medium transmission line. u d and u D The uncertainty propagated first to the characteristic impedance of the standard transmission line u Zn This then propagates further to the uncertainty matrices of the S-parameters and T-parameters of the standard transmission line. and In addition, there is noise at the port of the Vector Network Analyzer (VNA). n 1 and n 2 The uncertainty matrices of the original measured S-parameters of the standard transmission line and the device under test (DUT) will be introduced separately. and When solving the eight-term error model, the uncertainty matrix... and The uncertainty vector propagated to the error coefficients Ultimately, when the error model is used to correct the measured values of the device under test, the uncertainty vector... With matrix The uncertainty matrix of the S-parameters after calibration of the device under test is finally propagated. .
[0020] In SOLT calibration, the uncertainty vectors related to open-circuit, short-circuit, and load standards are used. Derived from TRL calibration and measurement technology. Similarly, measurement noise. n 1 and n 2 This will introduce an uncertainty vector into the original measurement S-parameters of these standard parts. When solving the twelve-term error model, the uncertainty vector... and The uncertainty vector propagated to the error coefficients Similarly, in the process of calibrating the measurement S-parameters of the device under test, the uncertainty vector... With matrix This will further propagate to the uncertainty matrix of the final calibration result. .
[0021] Based on the traceable uncertainty propagation principle of the TRL and SOLT calibration techniques described in this invention, this example evaluates the uncertainty of the S-parameters of an Agilent coaxial 1.0 mm mismatch verification component and a 2.4 mm attenuator. Figure 5 As shown.
[0022] Inner diameter of outer conductor of air wire D The measurements were performed using a pneumatic measuring instrument in conjunction with a standard ring gauge. The standard uncertainty of the ring gauge temperature correction value was then calculated. u DT Next, the uncertainty introduced by the correction of readings after calibration of the pneumatic measuring instrument is analyzed. u ΔD This mainly includes the resolution and drift error of the pneumatic measuring instrument. Finally, for three air transmission lines of different lengths, the Type A standard uncertainty was evaluated based on the pooled sample standard deviation by repeatedly measuring at multiple locations. u a The combined standard uncertainty of the inner diameter of the outer conductors of the three air wires was calculated. u Da This example demonstrates the measurement model and Type B uncertainty. u D It can be expressed as follows:
[0023]
[0024] Outer diameter of the inner conductor of the air transmission line d The diameter was obtained by measuring with a laser diameter gauge and a standard needle gauge. The standard uncertainty of the inner diameter of the outer conductor was used. uD Using a similar calculation method, the standard uncertainty components of the outer diameter of the inner conductor of the air wire are obtained. The measurement model and uncertainty calculation in this example can be expressed as follows:
[0025]
[0026] in, d The outer diameter of the inner conductor of the air wire being measured. ds and u dS These are the outer diameter values of standard needle gauges and their uncertainties. D T and u DT These are the temperature correction values and their uncertainties for needle gauges, Δ. D and u ΔD These are the correction values and uncertainties of the readings after calibration of the pneumatic needle gauge. u Da It is also a Type A standard uncertainty.
[0027] According to transmission line theory, the characteristic impedance Z n The calculation formula is
[0028] The standard uncertainty of the characteristic impedance of the three air transmission lines is further calculated using the following formula. u Zn .
[0029]
[0030] To simplify subsequent calculations and ensure the accuracy of uncertainty assessment, this example selects the maximum value among the uncertainties of the characteristic impedances of the three air lines as the representative value.
[0031] S-parameter uncertainty matrix of standard air line and T-parameter uncertainty matrix Perform an evaluation. When a matched load is connected to port 2 of the air line, the input impedance of port 1 is... It can be represented as
[0032] in Z L The value is 50Ω, representing the impedance of the matched load. γ and l Let be the propagation constant and length of the air line, respectively. Therefore, considering the reciprocity of the air line, its S-parameters can be expressed as:
[0033]
[0034] in Z 0 The value is 50Ω, representing the reference impedance. Based on sensitivity analysis, the uncertainty matrix... and The expressions for each parameter are given by the following formula.
[0035]
[0036]
[0037]
[0038]
[0039]
[0040] Subsequently, the uncertainty matrix of the S-parameters of the device under test (DUT) after calibration was determined by using the uncertainty assessment method combining sensitivity inversion and Monte Carlo simulation proposed in the invention.
[0041] When applying the known transmission matrix of airline and pass-through standards and With measurement transfer matrix and When solving for the error coefficients, the system of equations is as follows:
[0042]
[0043] in a, b, c, α, β, ε, r and ρ These are the error coefficients of the eight-item error model. In traditional sensitivity analysis and Monte Carlo methods, it is necessary to... and The elements of the matrix are treated as deterministic variables and random numbers, respectively, under this condition:
[0044]
[0045] therefore and The system of equations becomes highly complex, making it difficult to obtain analytical expressions for the eight error coefficients. Therefore, this example adopts a sensitivity inversion method: through total differential... and The system of equations is transformed into a linear system of equations:
[0046]
[0047] The expressions for some of the parameters are given by the following formula.
[0048] Subsequently, by inverting the system of equations, the uncertainty vector of the error term can be obtained. .
[0049] Furthermore, the uncertainty matrix of the S-parameters of the device under test (DUT) after TRL calibration is calculated using the Monte Carlo method. The following equation gives the uncertainty vector from the error term. The noise of the T-parameters of the device under test introduces into the original measurement uncertainty matrix. Propagation to the T-parameter uncertainty matrix after calibration The model.
[0050]
[0051] Based on this result, the following equation characterizes the uncertainty matrix from the T parameter. The S-parameter uncertainty matrix after calibration The process of its spread.
[0052]
[0053] Uncertainty vector associated with known values of SOLT calibration standard The uncertainty can be determined using the uncertainty calculation method derived from the aforementioned five-step TRL calibration technique. When the known values of the calibration standard are treated as random variables, this does not introduce additional complexity to solving the 12-term error model or calibrating the S-parameters of the device under test. Therefore, the Monte Carlo method can be used to synthesize the uncertainty vector. Initial measurement uncertainty vector of S-parameters caused by noise and matrix The measurement uncertainty matrix of the S-parameters of the device under test is obtained by transferring the data. .
[0054] like Figure 6 As shown, the mismatch verification device after TRL calibration and the corrected S-parameters of the attenuator after SOLT calibration are displayed.
[0055] like Figure 7 As shown, the traceable uncertainty of the S-parameters of the mismatched device caused by the uncertainty of the standard air wire inner / outer diameter and measurement noise is illustrated. Figure 7(a) is the reflection coefficient S11 of port 1. Figure 7 (b) is the transmission coefficient S12 of port 1. Figure 7 (c) is the transmission coefficient S21 of port 2. Figure 7 (d) represents the reflection coefficient S22 at port 2. The combined uncertainty of the S-parameters includes the combined contributions of the uncertainties of the inner and outer diameters of the air line and measurement noise. As shown in the figure, the trends of the combined uncertainty of the reflection coefficients at the two ports are highly similar: when the real uncertainty increases or decreases, the imaginary uncertainty shows the opposite trend. Furthermore, for the transmission coefficient and the reflection coefficient, the trends of the real and imaginary uncertainty components caused solely by noise are basically consistent. The combined uncertainty of the S-parameters is mainly dominated by the propagation of the uncertainty of the air line geometry, which has a dominant contribution, so much so that its uncertainty curve almost completely overlaps with the combined uncertainty curve. The above-mentioned variation also indicates that, during the propagation process, different uncertainty sources affect the real and imaginary uncertainties of the S-parameters in different ways.
[0056] like Figure 8 As shown, the traceable uncertainty of the attenuator S-parameters caused by non-ideal calibration standards and measurement noise is... Figure 8 (a) is the reflection coefficient S11 of port 1. Figure 8 (b) is the transmission coefficient S12 of port 1. Figure 8 (c) is the transmission coefficient S21 of port 2. Figure 8 (d) represents the reflection coefficient S22 at port 2. It can be seen that for the reflection coefficients of both ports, the real and imaginary parts of the combined uncertainty exhibit similar trends and are of comparable magnitude. The main contribution to the uncertainty originates from the non-ideal characteristics of the calibration standard, while the uncertainty caused by measurement noise is negligible. Furthermore, both measurement noise and the non-ideal calibration standard contribute significantly to the combined uncertainty of the transmission coefficient.
[0057] 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 method for analyzing traceable uncertainties in scattering parameters based on sensitivity inversion, characterized in that, Includes the following steps: Obtain the measured values of the inner diameter of the outer conductor and the outer diameter of the inner conductor of a standard air-medium transmission line, and evaluate the standard uncertainty of the inner diameter of the outer conductor and the outer diameter of the inner conductor, respectively. Based on the inner diameter of the outer conductor and the outer diameter of the inner conductor, calculate the characteristic impedance and its uncertainty of the standard air-dielectric transmission line; Based on the characteristic impedance and its uncertainty, and in combination with the influence of measurement noise, the uncertainty matrices of the scattering parameters and transmission parameters of the standard air medium transmission line are calculated respectively. Based on the uncertainty matrix of the transmission parameters, the uncertainty vector of the error coefficients is obtained by linearizing the eight-term error model calibrated by TRL and solving it through sensitivity inversion. The uncertainty vector of the error coefficients and the measurement noise are jointly sampled using the Monte Carlo method. The original scattering parameters of the device under test are repeatedly corrected by TRL calibration and the discreteness is statistically output to obtain the uncertainty matrix of the scattering parameters of the device under test after TRL calibration. Based on the known values and uncertainties of the calibration standard obtained from TRL calibration, the known value uncertainty of the calibration standard is introduced into SOLT calibration and jointly propagated with the measurement noise to obtain the uncertainty matrix of the scattering parameters of the device under test after SOLT calibration.
2. The method for analyzing traceable uncertainty of scattering parameters based on sensitivity inversion according to claim 1, characterized in that: The steps for evaluating the standard uncertainty of the inner diameter of the outer conductor and the outer diameter of the inner conductor include: for the measurement process of the inner and outer diameters of the air transmission line, identifying the uncertainty components of temperature correction, gauge calibration, and repeatability measurement, and combining each uncertainty component using the root sum of squares method to obtain the combined standard uncertainty of the inner and outer diameter measurements.
3. The method for traceable uncertainty analysis of scattering parameters based on sensitivity inversion according to claim 1, characterized in that: The step of calculating the uncertainty of the characteristic impedance includes propagating the uncertainty of the inner diameter of the outer conductor and the uncertainty of the outer diameter of the inner conductor to the characteristic impedance based on the error propagation law.
4. The method for analyzing traceable uncertainty of scattering parameters based on sensitivity inversion according to claim 3, characterized in that: The following formula is used in the step of calculating the uncertainty of the characteristic impedance: ； Where μ and ε are the magnetic permeability and dielectric constant of air, respectively.
5. The method for traceable uncertainty analysis of scattering parameters based on sensitivity inversion according to claim 1, characterized in that: The methods for obtaining the uncertainty vector of the error coefficients through sensitivity inversion include: An error model equation based on TRL calibration is established, which is associated with the known transmission matrix of the standard air line, the transmission matrix of the through standard, the corresponding measurement transmission matrix, and the error coefficients a, b, c, α, β, ε, r, ρ of the eight error models. The error model equations are linearized by total differential to construct a system of linear equations with the uncertainty of the error coefficients as unknowns. Solve the system of linear equations to obtain the uncertainty vector of the error coefficients.
6. The method for traceable uncertainty analysis of scattering parameters based on sensitivity inversion according to claim 1, characterized in that, The uncertainty matrix of the scattering parameters of the device under test after TRL calibration, obtained by using the Monte Carlo method, specifically includes: Based on the uncertainty vector of the error coefficients and the statistical characteristics of the measurement noise, the error coefficients and measurement noise are subjected to multiple joint random samplings. For each sample, a complete TRL calibration procedure is performed to correct the original measured scattering parameters of the device under test; The discreteness of the output results after multiple calibration corrections is statistically analyzed to determine the uncertainty matrix of the scattering parameters of the device under test after TRL calibration.
7. The method for analyzing traceable uncertainty of scattering parameters based on sensitivity inversion according to claim 1, characterized in that, The uncertainty matrix of the scattering parameters of the device under test obtained after SOLT calibration specifically includes: Obtain standard values for short-circuit, open-circuit, and load standards for SOLT calibration. The standard values and their uncertainties are obtained through TRL calibration and measurement techniques. The Monte Carlo method is used to perform joint propagation analysis on the standard value uncertainty of the calibration standard, the uncertainty introduced by its original measurement noise, and the uncertainty introduced by the measurement noise of the device under test. The uncertainty matrix of the scattering parameters of the device under test after SOLT calibration is obtained from the statistical propagation results.