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In this paper we consider the complexity problem in electronics production process. Particularly, we investigate the ways to reduce sensitivity of transmission line characteristics to their parameter variations. The reduction is shown for the per-unit-length delay and characteristic impedance of several modifications of microstrip transmission lines. It can be obtained by means of making an optimal choice of parameter values, enabling proper electric field redistribution in the air and the substrate. To achieve this aim we used an effective simulation technique and software tools. Taken together, for the first time, they have allowed formulating general approach which is relevant to solve a wide range of similar tasks.

Electronics is more and more widely used in human life. Complexity of electronics is continuously increasing. It results in increasing the number of electronics parameters. Unfortunately, the parameters undergo undesirable variations caused by the manufacturing process. As undesirable results, we obtain the reduced yield ratio in mass production and the reduced quality of a product or the increased production cycle due to necessity of a product to be redesigned or reproduced.

To treat the problem we can consider two main parts of the problem: concerning electronic components and supporting structures for their packaging. As for components, their nomenclature is very wide. Nevertheless, a great attention is paid to ensure their tolerance. As for supporting structures, their number is very small (a printed circuit board (PCB), an integrated circuit (IC), and a chip). The most popular structure is a PCB, while an IC and a chip can be considered as dedicated to electronic components, especially due to the recent trends of system-in-package (SiP) and system-on-chip (SoC) designs. Thus, a PCB is becoming the main concern of a designer, while a microstrip-like-lines are the main paths for signal propagation in PCBs and even ICs and chips as well. Therefore, to assure stable characteristics of the lines are an important task, the importance is illustrated by a representative example in Figure

Comparison of measured (_{M}) and calculated (_{C}) impedance values for 186 test line samples of various cross-sections [

With increasing requirements to electronics characteristics it is necessary to have transmission lines with more stable per-unit-length delay (

One of the most widely used high-speed signal transmission lines is a microstrip line (MSL) [

However the possibilities to minimize the sensitivity of MSL characteristics are limited by the simplicity of the classical MSL construction. Therefore, various modifications of an MSL, such as suspended and inverted microstrip lines, allowing zero sensitivity of

To obtain necessary characteristics of a structure, it is required to carry out a multivariate analysis in the range of parameter variations. However, for a structure of an arbitrary cross-section, the quick and accurate expressions are not available. Therefore, it is necessary to use numerical methods, wherein the main costs fall on a multiple linear algebraic systems solution. In this case, depending on the type of the problem, only the matrix or the matrix and the right-hand side can be varied. In the matrix case we can write

The first approach to solving sequence (^{T}-decomposition of matrix

Thus, it is useful to analyze the results obtained as well as the techniques and the tools used for these studies. The aim of this paper is to propose a general approach to solving the complexity problem in electronics production process based on the summary of the recent studies devoted to minimizing the sensitivity of microstrip-like lines characteristics to variations of their parameters.

Various modifications of the MSL under study are presented in Figures

Cross-sections of covered (a) and shielded (b) MSLs.

Cross-sections of MSLs with side grounded conductors in air (a) and dipped in substrate (b).

Cross-sections of MSLs with side grounded conductors above (a), among (b), and under (c) air–substrate boundary.

The values of the characteristics (_{0} is the speed of light in a vacuum;

The investigation of a microstrip structure of various characteristics, especially in the first stage should be performed through simulation, as it is less costly and may be more accurate than measurements. In this regard, in this work, the construction of cross-sections and calculations are performed in the TALGAT system available to the authors [

Before using any software, one must validate it properly by using simulation and measurements. However, in case of coupled transmission lines, the off-diagonal coefficients of a per-unit-length matrix have large (up to 25%) error [

Results of simulation by Green’s Function Method (GFM), Method of Moments (MoM) and Variational Method (VM) and measurement (pF/cm).

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Size (mills) for | Results | _{11} | –_{12} |

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| GFM | 5.61 | 0.77 |

MoM | 5.62 | 0.76 | |

VM | 5.64 | 0.68 | |

Measured | 5.59±0.06 | 0.62±0.15 | |

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| GFM | 3.78 | 0.70 |

MoM | 3.78 | 0.70 | |

VM | 3.78 | 0.63 | |

Measured | 3.69±0.04 | 0.38±0.10 | |

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| GFM | 2.66 | 0.29 |

MoM | 2.65 | 0.30 | |

VM | 2.67 | 0.24 | |

Measured | 2.64±0.03 | 0.20±0.05 | |

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| GFM | 2.77 | 0.59 |

MoM | 2.76 | 0.60 | |

VM | 2.77 | 0.53 | |

Measured | 2.75±0.30 | 0.48±0.12 | |

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| GFM | 3.00 | 0.97 |

MoM | 2.99 | 0.97 | |

VM | 2.96 | 0.94 | |

Measured | 2.95±0.03 | 0.92±0.23 |

Firstly, we compared results for a simple case of 2 coupled strips on a substrate [

Comparison of computed entries of matrix

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| Results | _{11} | _{12} | ||

Without walls | With walls | Without walls | With walls | ||

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6.0 | Result [ | 92.36 | 92.05 | 8.494 | 8.473 |

Our result, pF/m | 91.11 | 91.11 | 9.162 | 9.162 | |

Error, % | –1.3 | –1.0 | 7.8 | 8.1 | |

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4.0 | Result [ | 92.44 | 92.14 | 8.506 | 8.485 |

Our result, pF/m | 91.15 | 91.14 | 9.167 | 9.168 | |

Error, % | –1.4 | –1.1 | 7.7 | 8.0 | |

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2.0 | Result [ | 92.40 | 92.10 | 8.539 | 8.517 |

Our result, pF/m | 91.15 | 91.08 | 9.179 | 9.182 | |

Error, % | –1.3 | –1.1 | 7.5 | 7.8 | |

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0.5 | Result [ | 91.44 | 90.50 | 8.595 | 8.565 |

Our result, pF/m | 90.73 | 89.88 | 9.185 | 9.184 | |

Error, % | –0.7 | –0.7 | 6.8 | 7.2 | |

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0 | Result [ | 89.68 | 87.97 | 8.603 | 8.569 |

Our result, pF/m | 89.34 | 87.68 | 9.146 | 9.146 | |

Error, % | –0.3 | 0.3 | 6.2 | 6.7 |

Then, we considered the same 3 strips of various positions in a two-layer dielectric medium [

Comparison of computed

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Results | _{11} | _{21} | _{31} | _{22} | _{32} | _{33} |

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Result [ | 142.1 | 21.7 | 0.9 | 93.5 | 18.1 | 88.0 |

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Our result | 143.6 | 19.8 | 0.9 | 88.6 | 17.7 | 83.1 |

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Error, % | 1.1 | –8.8 | 0 | –5.2 | –2.2 | –5.6 |

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Results | _{11} | _{21} | _{31} | _{22} | _{32} | _{33} |

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Result [ | 277.7 | 87.8 | 36.8 | 328.6 | 115.8 | 338.0 |

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Our result | 279.4 | 87.6 | 36.5 | 330.7 | 115.5 | 339.0 |

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Error, % | 0.6 | –0.2 | –0.8 | 0.6 | –0.3 | 0.3 |

Thus, the performed comparisons showed satisfactory coincidence of the results and the relevance of the TALGAT system for computing per-unit-length matrices for structures of various complexities. Meanwhile, for final testing, one can compare a time response of a structure. There exist indicative and commonly available examples comparing the TALGAT system results with the measurement [

We are using several approaches to reduce the computational complexity of the analysis. The main feature of these approaches is the use of iterative methods. So, when solving the first linear system, we calculated a proper preconditioner. This preconditioner (frozen) is used to solve subsequent linear systems. At the same time, the use of the previous system solution as the initial guess is more preferable than the fixed initial guess [

Constant parameters for the line in Figures _{r}=4.5 (fiberglass).

For Figure

Dependencies of

For Figure

Dependencies of

Dependencies of

Consider the influence of the side walls on the calculated characteristics. Quantitative estimates can be done from the comparison of the relevant dependencies from Figures

Consider the results for various modifications of MSLs with side grounded conductors (Figures _{r}=4.5.

In the TALGAT software we built the geometric models of the line cross-section and calculated (using the method of moments) the matrices (3×3) of per-unit-length coefficients of electrostatic induction, taking into account the dielectric as well as ignoring it. Calculations were performed, changing the distance (2

Dependencies of

Dependencies of

For Figure

Dependencies of

Dependencies of

As

Dependencies of

Dependencies of

Dependencies of

Dependencies of

We have presented the systematic results of our study into the values of

The presented results have been obtained for particular values of line parameters. However, it is easy to obtain similar dependencies for other values of the parameters and even other parameters. Besides, transmission lines with arbitrary number and shapes of conductors and dielectrics can be studied easily [

For multiple calculations, a lot of described accelerations of linear system solutions can be effectively used. In this paper, for quick estimations, we have used calculations of transmission line parameters only. However, for more comprehensive analysis, you can calculate the frequency or time response and use optimization by genetic algorithms similarly to [

Thus, taken together, for the first time, the above-mentioned techniques form a general approach to solving complexity problem in the electronics production process through the reduction of sensitivity of transmission line characteristics to their parameter variations. Versatility of the approach allows solving a wide range of tasks similar to considered examples.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by The Ministry of Education and Science of the Russian Federation (project 8.9562.2017/8.9).