Rapid Analysis of Bent Cable Crosstalk Based on Numerical Calculation

In this paper, a fast prediction method of cable crosstalk is introduced, which can simulate the crosstalk problem more accurately in the actual cable situation. The traditional flexible cable model considers few factors to truly reflect the spatial pose of the cable. In this paper, constraints of the assembly space are considered to truly reflect the structural characteristics of the cable, providing a theoretical basis for cable bundle layout in the virtual environment. Based on the element method and the theory of multi-conductor transmission lines, the crosstalk coupling model of curved cable is established. In this model, the traditional multi-conductor transmission line theory is improved, and the bending cable bundle model is divided into several equivalent subsegments by using the element method. In the subsegment, the multi-conductor transmission line theory is used for modeling, and the chain parameter method is used for crosstalk coupling analysis. The accuracy of the model introduced in this paper is verified by MATLAB and CST Studio simulation.


Introduction
Electronic information technology has been used in many fields.As an indispensable signal transmission way for electronic and electrical equipment, cable plays an important role in the field of information.With the increasing complexity of the electromagnetic environment of equipment, the electromagnetic interference of signal equipment is becoming more and more serious, and the electromagnetic interference of cable is gradually attracting people's attention, among which cable crosstalk is an important electromagnetic interference problem.
Cable crosstalk is greatly affected by position changes.At present, many scholars have studied crosstalk prediction in cable crosstalk.The main methods are multi-conductor transmission line theory, finite difference time domain method, moment method, etc.Although these methods can accurately predict the crosstalk problem of random cable bundles, they are regarded as the ideal cable model [1].In actual cases, the position of equipment or cable bundle and the influence of such factors as the geometry of cable arrangement often have certain randomness.Due to cable crosstalk under the influence of disturbance, the error could be as high as 20 dB.In this case, if you use the traditional crosstalk of multiple conductor transmission line theory, ideal accuracy will fall sharply [2].Therefore, a new method is needed to predict the crosstalk of random cable bundles.In this paper, a fast prediction method of cross-talk in multi-conductor cable bundles is proposed.Based on the Hermite interpolation method, the cross-talk characteristics of multi-conductor cable bundles are calculated by using multi-

Hermite interpolation model
The interpolation method is the main method of function approximation.The traditional Hermite interpolation method not only needs to know the function value at the node but also needs to know the first derivative value at the node.However, in the actual situation, the information of some nodes may not be complete.The Hermite interpolation method is now extended to a more general situation.We suppose that the node is  , then the interpolation polynomial is: . ( In this formula: The Hermite interpolation method can effectively solve the problem of insufficient known node information, but the interpolation polynomial calculated by using the Hermite interpolation method directly has a high number of times.The function curve is easy to oscillate violently at two endpoints, that is, the Runge phenomenon occurs.The higher the interpolation times are, the more obvious the Runge phenomenon will be.To avoid this situation and ensure a smooth curve, piecewise multiple interpolations are used in this paper.

Cable bundle crosstalk model based on Hermite interpolation method
To facilitate cable installation and maintenance, cable before installation usually carries on a reasonable layout, and part of the cable will be together, which is bound to form a larger cable bundle and be installed to the device.Although there are fixed banding points and endpoint cable bundles, the position of the conductor in the cable bundles is not fixed, which is not entirely parallel between cables.To sum up, the geometry of the cable bundle is usually random, and the spacing between cables may change continuously along the length of the line.
The spacing of the cable will affect the self-inductance and mutual inductance coefficients of the distribution parameters, which indicates that the crosstalk characteristics of the cable bundle will also change along the line length [3].If the prior conditions are insufficient, it can be assumed that the distribution parameters of the cable bundle change randomly within a certain range.Similarly, the selfinductance and mutual inductance coefficients of cable bundles also obey a probability distribution [4].

Spatial cable bundles model
The geometric information of spatial cable can be described by the flexible cable model.To represent the spatial position and shape of the cable, the coordinate system shown in Figure 1 is established.

Figure 1. Example of spatial geometry model
In Figure 1, r is the position vector of the cable center line.In a fixed coordinate system, it can be considered that the position information of the cable is only expressed by the single-valued function of r.If the local coordinates are shifted to the fixed point O, the local coordinates can be transformed into fixed coordinates, and the transformation relationship is related to Euler Angle a, b, and c, as follows: is the initial value of the local coordinate origin concerning the fixed coordinates.Assuming that the cross-sectional area of the cable remains unchanged during bending, the coordinates of any point on the cable and the changing trend of the cable can be expressed in Formula 3.This model can accurately describe the spatial pose of the cable.
Although the spatial pose of the cable can be clearly expressed, the cable shown in Figure 1 may not appear in actual situations.Cable assembly is often subject to many constraints.Considering that the spatial pose of the cable does not change freely, as shown in Figure 1 when the cable is in a narrow space or bundled into a cable bundle [5].
To sum up, the traditional flexible cable modeling method cannot accurately describe the actual cable geometry information, and it is necessary to analyze various constraint types of assembly space.Based on this model and combined with the physical characteristics of cables, this paper provides a new idea for analyzing the binding of cable bundles.To facilitate discussion, the cable bundle model studied in this paper needs to meet the following conditions:  This paper only considers the crosstalk between signal lines and ignores external electromagnetic interference;  The transmission line is loss-free and the charge distribution on the cable is uniform;  The total current in the section of the cable bundle is 0, i.e., the current needs to return along another section of the conductor.

Utilization of the Hermite interpolation method to the cable bundle model
The layout of the bundle usually has the following characteristics: Restricted by the device, the two ends of the bundle often have fixed endpoints, that is, a definite starting and ending point; The cable elements at the binding point have the same trend, that is, the cable segments at the binding point are in the same direction.Constrained by factors, such as cable slots, cable changes often have certain boundaries, that is, there are maximum points and minimum points.Cable spacing is an important factor in cable crosstalk analysis.The position relation of cables is often random.To obtain the distance between cables in the cable bundle, the relative distance between two points on the line should be analyzed first.For located on two cables, the fixed coordinates mentioned above can be used to represent the distance between two points.
Although the values in Formula 8 are uncertain, they can be combined into a new variable for analysis.Therefore, assuming that the value of cable spacing follows a random distribution, the relationship between cable spacing and cable length can be further obtained.Through the content of the first chapter, we know the conditions of the Hermite interpolation method for the known node function value or a derivative.This characteristic can correspond to the binding point of the cable bundles model.If we assume the binding point for a random node and this variable is normal distribution, there will be a new generation of random variables into the interpolation polynomial curve expression.
where the coefficient can be calculated by Formula 2, which 0 y is a modified constant.The conductor spacing of any node can be obtained by this expression, and then the distribution parameters of the cable bundle can be calculated.In this paper, some examples are given to illustrate the geometric descriptions of cable bundles generated by the above models in some typical cases.Table 1 is the function value and first-order derivative value of each node under normal distribution, corresponding to the position information of the binding point of the cable bundle.

Distribution parameters of the cable bundle crosstalk model
The establishment of the cable bundle crosstalk model is based on the multi-conductor transmission line theory.For the convenience of discussion, this paper takes the three-conductor transmission line as an example, in which one cable is the reference ground cable [6].Figure 3 shows the transmission line coupling equivalent circuit.

Figure 3. Transmission line coupling equivalent circuit
The calculation of distributed parameters is based on the mirror image theory, and the distributed inductance can be obtained by the ampere law.The inductance matrix and capacitance matrix of unit length cable bundle can be formed.Since the distance between the conductors in the cable bundle is small, it can be considered that the distance between the two cables and the reference ground cable is approximately equal [7].

Two-port network theory
Two-port network theory is a common analysis model for transmission lines.Its advantage is that transmission lines can be divided into independent small networks for analysis [8].Small networks are connected through input and output ports.In the two-port network shown in Figure 4, the voltage and current of the ports can be expressed in the form of a matrix.
When the transmission line is a lossless cable, the impedance and admittance coefficient in the twoport network is defined by Formulas 14 and 15.
In Formula 6, four second-order matrices constitute the chain parameter matrix, and the calculation of matrix elements is as follows: where the T matrix satisfies , where  is regarded as the diagonal matrix of order 2 and the boundary condition can be obtained by the Davinen theorem [10].By substituting boundary conditions into Formula 6, the relation between terminal current and source voltage is obtained.
The crosstalk value can be obtained from the terminal current.Therefore, as long as the probability distribution of a random node is known, the cable bundle crosstalk model under this node can be obtained.

The simulation results
For the crosstalk model, corresponding examples are given in this paper, which are simulated by MATLAB and CST Studio respectively.The calculation example follows the three-conductor transmission line model in Chapter 2, with the same conductor specifications of the radius r = 0.003 m and cable length l = 1 m.The relationship between characteristic impedance and terminal impedance is The multi-conductor transmission line model should obtain a three-dimensional image with the xaxis and y-axis as plane coordinates and the z-axis as the current size.Since the distance between the conductor and the reference ground cable in the cable bundle is the same, the spatial coordinates of the cable bundle are simplified in this paper and transformed into a two-dimensional image for a more intuitive representation.In the figure, when the frequency is below 10 MHz, the simulation results of MATLAB and CST Studio are compared, and the curves are in good agreement.It can be considered that the two are consistent at low frequencies, which proves that the above algorithm is effective at low frequencies.At high frequencies, the equivalent circuit is no longer suitable because of radiation effects.The establishment of a bending cable is based on the cable geometric modeling method mentioned above.MATLAB and CST are also used to simulate it, and the results are shown in Figure 6.In the results shown in Figure 6, under the condition of low frequency, there is an error of less than 5 dB in the simulation results of the two software, but in general, the trend of the two software is the same.The prediction error is mainly related to the mesh division of the model.As mentioned above, the crosstalk calculation of MATLAB needs to go through several iterations.According to the principle of numerical calculation, this kind of model is greatly affected by truncation error.

Conclusion
In this paper, a method is proposed to predict the crosstalk characteristics of cable bundles quickly based on Hermite interpolation theory.The crosstalk characteristics of the cable bundle under the uncertain geometric arrangement obey a certain probability distribution, which is related to the cable installation mode.Due to the randomness of the binding points of the cable bundle, the spatial pose of the cable bundle changes randomly, and the spacing of the conductors in the cable bundle also changes along the line length, thus making the crosstalk of the cable bundle appear random.A cable bundle model based on the Hermite interpolation method is proposed, and the crosstalk model of the cable bundle is solved in the frequency domain combined with transmission line theory.The results show that Hermite interpolation can accurately and efficiently simulate the geometric uncertainty of bundling cables.As the modern electronic power system becomes more and more complex, this method has a certain guiding significance for the prediction of cable bundle crosstalk.

Figure 2
Figure2respectively represents the possible geometric arrangement of cable bundles in some typical cases.This model can accurately predict the geometric structure of cable bundles in complex cases, to facilitate the calculation of cable bundle distribution parameters.

Figure 4 .
Figure 4. Two port network equivalent circuit.

Figure 5 .
Figure 5. Crosstalk simulation of straight cable Figure 6.Crosstalk simulation of straight cableAs an equivalent sub-segment model of the element method, crosstalk simulation of straight cable is necessary, and the simulation results are shown in Figure5.Since the classical transmission line theory is based on the equivalent circuit method, it is only applicable to the crosstalk calculation under the

Table 1 .
A more complex table with a narrow caption