Figure 1 illustrates a model of n-conductor bounded in a braid shielding structure and irradiated by an incident EM field. The radiation field excites the induced current along each conductor. For the immunity problems, the huge complexity of the whole cable bundles seems to be unreasonable to model when considering the required computer resources. Though previous “ECBM” puts forward a solution to the immunity case that all the equivalent unshielded conductors are located above the infinite ideal conducting plane, it cannot be directly used to simplify the modeling of the coupling prediction of the conductors within the shielding screen. Thus, this paper focuses on a multiconductor reduction technique for modeling immunity problem on shielding cable bundles. The presented technique overcomes the limit to the MTL formalism at high frequency by using three-dimensional (3D) computation codes.

Fig. 1.

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	Fig. 1. (color online) Illustration of the electromagnetic immunity on braid shielding multiconductor cable bundles model.

The generation of the equivalent cross-section geometrical characteristic parameters in the braid shielding structure is quite different from that in the half-free space. As the analytical expressions of self and mutual inductance of the braid shielding multiconductors are difficult to obtain, an approximate model of multiconductors within an ideal conducting cylindrical shield is adopted to compute the electrical parameters. In this paper, we define a central distance d and central angle θ to determine the relative position parameters of the reduced bundles. It should be noted that the shield transfer impedance associated with inductance is not discussed in the simplification technique because the approximate method is used in computing the inductance of braid shielding multiconductors. We do not need to analyze the coupling mechanism by using the shield transfer impedance, because it is considered in the CST simulation.

The purpose of the approach is to lower the computation time and simplify the modeling process by reducing the multiconductor cable bundles model into four groups shown in Fig. 2. For the immunity problems, the provided modeling technique presents a modified six-phase procedure to generate the electrical and geometrical characteristic parameters of the reduced cable bundles, which contains conductor grouping, cable bundle matrices reduction, cross-section geometry modeling and equivalent loads determination. The creation of groups of conductors is based on the knowledge of Z-Smith chart, which provides a convenient way to describe the relationship between load impedance and reflection coefficient. The whole problem is calculated over a large frequency range. Based on the assumption in Ref. [17], the proposed simplification technique provides a way of modeling the CM current induced at the extremity of inner conductors.

Fig. 2.

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	Fig. 2. (color online) Cross-section geometries of the complete and reduced cable bundle.

a) Step 1: Creation of groups of conductors based on Z-Smith chart

In the first step, all the conductors of the complete cable bundle are classified as four groups with respect to the location of normalized terminal load |Z_ij| in Z-Smith chart, where represents the terminal impedance and Z_C the equivalent CM characteristic impedance of all conductors, i corresponds to the terminal number (“1” represents the near end terminal and “2” the far end terminal) and j the conductor number as illustrated in Table 1. It should be noted that the management of the cable harness grouping is just to simplify the coupling CM current calculation and the classification method will not affect the overall performance.

Table 1.

Table 1.

Table 1. Conductor classification table. . Extremity 1 Extremity 2 Group 1 Z1j ∈ I1 Z2j ∈ II1 Group 2 Z1j ∈ I1 Z2j ∈ II2 Group 3 Z1j ∈ I2 Z2j ∈ II1 Group 4 Z1j ∈ I2 Z2j ∈ II2

Table 1. Conductor classification table. .

Figure 3 demonstrates the classification of groups of 9-conductor model. The dots in the Z-Smith chart represent near end normalized loads of each conductor, while the asterisks represent far end normalized loads. All the dots and asterisks are included in four sections denoted by I₁, I₂, II₁, and II₂. From the characteristic of Z-Smith chart, the terminal load has similar reflective properties in an adjacent area. Here, let , , where and are the arbitrary terminal impedances connected to the near ends. Then we can obtain the following inequality:

Fig. 3.

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	Fig. 3. (color online) Criterion of cable bundle grouping based on the Z-Smith chart.

b) Step 2: Reduced cable bundle matrices

The second step aims to determine the reduced cable bundle inductance [L_eq] and capacitance matrix [C_eq]. According to the general MTLN theory, the equations of N-conductor cable bundle can be written as

On the assumption that the same group CM current I_gci equals the sum of the currents induced on each conductor of the group and all the conductors in group i have the same group voltage V_gci, the reduced cable bundle containing four conductors is established and the equivalent MTLN equations can be modified as

From Eqs. (1)–(4), the reduced inductance matrix [L_eq] and capacitance matrix [C_eq] can be obtained as follows:^[23]

c) Step 3: Reduced cable bundle cross-section geometry

The third step is to generate the cross-section of the reduced cable bundle, which consists of six phases. As the inductance parameters of the shielding multiconductor are associated with the structural factors, such as the distance between the analytical conductor and screen, the relative distance among conductors, the calculation of the cross-sectional geometry parameters of equivalent model are obtained thanks to study made in Ref. [24] and the knowledge of the [L_eq] and [C_eq] matrices.

1) Phase 1: Estimate the central distance d_i between each equivalent conductor and the central axis of the braid shielding structure. Distance d_i of each equivalent conductor corresponds to the average of the distance of all the conductors of the group.

2) Phase 2: Estimate the radius r_i of the braid shielding structure. The analytical formula for self-inductance l_ii of a wire in an ideal perfectly conducting shield is equal to . Thus, the radius r_i of each equivalent conductor can be approximated by

3) Phase 3: The analytical formula of mutual inductance l_ij between two conductors in an ideal perfectly conducting shield can be written as

Thus, the central angle between two conductors can be approximated by

4) Phase 4: Optimize the reduced cross-section geometrical parameter d_i, r_i, and θ_ij based on the dichotomic optimization and exact electrostatic calculations.

5) Phase 5: Determine the thickness of the dielectric coating surrounding each equivalent conductor while avoiding dielectric coating overlapping.

6) Phase 6: Calculate and optimize the relative permittivity ε_r of each wire dielectric coating, which is in accordance with the [C_eq] matrix using an electrostatic calculation.

Figure 4 shows the corresponding computational process of the proposed simplification technique used to build the cross-section geometry. Computational process for modeling the cross-section geometry of a reduced cable bundle in braid shielding structure.

Fig. 4.

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	Fig. 4. Computational process for modeling the cross-section geometry of a reduced cable bundle in braid shielding structure. (a)–(f) Phases 1–6.

d) Step 4: Reduced cable bundle equivalent terminal loads

The fourth and the last step consists in computing the terminal loads of the reduced model connected to each equivalent conductor end. Here, the equivalent CM loads are defined as the loads that connect conductor ends to the shield screen. The terminal load connected to an equivalent conductor end is equal to the loads at the same end of all the conductors of the group connected in parallel.

It should be noted that the previous four-step procedure is used for a simple point-to-point connected shielding cable bundle. Nevertheless, due to the calculation methods of the self and mutual inductance of shielding multiconductor, there are two important restrictions on their applicability. These are that the wires must be widely separated and the dielectric medium surrounding the wires must be homogeneous. The charge distributions around closely spaced wires will be nonuniform around their peripheries but will be approximately uniform for ratios of wire separation to wire-radius 4 and higher.

As illustrated in Fig. 2, a 14-conductor point-to-point connected cable bundle, 1-meter long, located in a copper braid shielding structure with a radius of 12 mm and a thickness of 0.5 mm is modeled. Each conductor has a radius of 0.5 mm and is surrounded by dielectric coating with a thickness of 0.2 mm and dielectric constant of ε_r = 2.5 and μ_r = 1. The conductors with a serial number “2, 3, 4, 5, 6” are evenly distributed on the circle with a radius of 2 mm, while the conductors with a serial number “6, 7, 8, 9, 10, 11, 12, 13, 14” on the circle with a radius of 4 mm. The central coordinate of each conductor is presented in Table 2. By using modal analysis,^[19] the analytical 14-conductor cable bundle characteristic impedance Z_C equals 244.4 Ω. The terminal loads connected to the ends of each conductor are described in Table 3. According to the classification method proposed in the first equivalent step, the 14-conductor can be sorted into the following four groups: 1)

group 1: conductors 1, 2, 12;

Table 2.

Table 2.

Table 2. Coordinates of each conductor of the 14-conductor shielding cable bundle (unit: mm). . Conductor 1 2 3 4 5 6 7 (x, y) (0,0) (0,2) (1.9,0.62) (1.18, −1.62) (−1.18, −1.62) (−1.9,0.62) (0,4) Conductor 8 9 10 11 12 13 14 (x, y) (2.83,2.83) (4,0) (2.83, −2.83) (0, −4) (−2.83, −2.83) (−4,0) (−2.83, 2.83)

Table 2. Coordinates of each conductor of the 14-conductor shielding cable bundle (unit: mm). .

Table 3.

Table 3.

Table 3. Terminal loads of the 14-conductor shielding cable bundle. . Conductor 1 2 3 4 5 6 7 Near end Z1(Ω, nH, pF) (50, 10, 0) (100, 5, 0) (1.0×104, 0, 0.5) (50, 0, 0) (20, 0, 0) (5.0×103, 0, 0.2) (100, 0, 0) Far end Z2(Ω, nH, pF) (10, 0, 100) (60, 0, 50) (1.0×103, 500, 0) (1.0×103, 0, 0) (2.0×103, 0, 0) (1.5×104, 2.0×103, 0) (1.0×104, 400, 0) Conductor 8 9 10 11 12 13 14 Near end Z1(Ω, nH, pF) (15, 0, 0) (1.0×106, 0, 0.01) (1.0×104, 0, 0.1) (1.0×103, 300, 0) (80, 6, 0) (1.5×104, 1.0×103, 0) (5.0×103, 0, 0) Far end Z2(Ω, nH, pF) (3.0×103, 0, 0) (50, 1, 0) (20, 3, 0) (100, 0, 100) (40, 10, 0) (15, 0, 50) (8.0×103, 1.0×103, 0)

Table 3. Terminal loads of the 14-conductor shielding cable bundle. .

The p.u.1. inductance matrix [L] and capacitance matrix [C] of the complete model in Fig. 5 are

Fig. 5.

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	Fig. 5. (a) Cross-sectional geometry of the 14-conductor complete shielding cable bundle model and (b) the corresponding 4-equivalent-conductor reduced shielding cable bundle model.

After applying the six-phase procedure described in the third step in Section 2, the cross-section geometry of 4-equivalent-conductor model is obtained and illustrated in Fig. 5. The equivalent CM terminal loads are calculated and the results are listed in Table 4. In addition, the relevant central distance, radius and central coordinate are listed in Table 5.

Table 4.

Table 4.

Table 4. Terminal loads of each conductor of the reduced shielding bundle. . Conductor 1 2 3 4 Near end Z1(Ω, nH, pF) (27.4, 2.5, 0) (1.6×103, 0, 1.3) (6, 0, 0) (1.6×103, 214, 0) Far end Z2(Ω, nH, pF) (7.5, 0, 131) (1.2×103, 303, 0) (0.49, 0, 0) (14, 0, 1.2×103)

Table 4. Terminal loads of each conductor of the reduced shielding bundle. .

Table 5.

Table 5.

Table 5. Central coordinate of each conductor of the reduced shielding cable bundle (unit: mm). . Conductor 1 2 3 4 Conductor radius 1.5 1.6 2.4 2.5 Central distance 1.5 2 4.2 4 (x, y) (0, 1.5) (1, −1.73) (3.99, 1.3) (−3.46, −2)

Table 5. Central coordinate of each conductor of the reduced shielding cable bundle (unit: mm). .

Cross-sectional geometry of the 14-conductor complete shielding cable bundle model and the corresponding 4-equivalent-conductor reduced shielding cable bundle model.

The induced CM current at the near and far ends of each cable in frequency domain are calculated by using CST Cable Studio. Based on the co-simulation technique, the adopted numerical approach uses the TLM technique to analyze the electric field around the conductors and then use AC result solver to compute the coupling to terminal loads. These computations correspond to the case of a plane wave illumination with a 1 V/m magnitude, the propagation direction 45° with respect to the reference conducting ground plane and the electric field direction parallel to the conductor. Figures 6–9 illustrate the comparison between the CM currents obtained at the ends of the complete cable bundle and the reduced cable bundle.

Fig. 6.

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	Fig. 6. (color online) Comparison of the CM current in frequency domain on cable 1 between the complete (conductors 1, 2, 12) and reduced cable bundle model: (a) near end; (b) far end.

Fig. 7.

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	Fig. 7. (color online) Comparison of the CM current in frequency domain on cable 2 between the complete (conductors 3, 6, 14) and reduced cable bundle model: (a) near end; (b) far end.

Fig. 8.

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	Fig. 8. (color online) Comparison of the CM current in frequency domain on cable 3 between the complete (conductor 4, 5, 7, 8) and reduced cable bundle model: (a) near end; (b) far end.

Fig. 9.

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	Fig. 9. (color online) Comparison of the CM current in frequency domain on cable 4 between the complete (conductor 9, 10, 11, 13) and reduced cable bundle model: (a) near end; (b) far end.

Using the feature selective validation (FSV) technique, it provides a way to validate computational electro magnetics (CEM) and predict the assessment of EMC data.^[25,26] The FSV technique is used to evaluate the results between the complete model and the simplified model. The FSV technique shows a measure of the “quality” of the correlation between the two sets of data according to specific criteria. Among those measure indicators, we mainly focus on the total amplitude difference measure (ADM_tot), the total feature difference measure (FDM_tot), and the total global difference measure (GDM_tot). These are available as numerical values and can be converted into a natural language descriptor on a six-level scale: excellent (0–0.1), very good (0.1–0.2), good (0.2–0.4), fair (0.4–0.8), poor (0.8–1.6), and very poor (> 1.6).

The FSV evaluation results of Figs. 6–9 are listed in Table 6. From the visual evaluation, it is obviously found that the agreement between the reduced and complete model results is still satisfactory. Thus, it is demonstrated that the proposed simplification technique can be successfully used to model the CM currents on braid shielding multiconductor cable bundles.

Table 6.

Table 6.

Table 6. FSV evaluation of results in Figs. 6–9. . Terminals of the TLs FSV ADMtot FDMtot GDMtot Near end 1 0.239/good 0.351/good 0.465/fair Far end 1 0.353/good 0.386/good 0.581/fair Near end 2 0.323/good 0.395/good 0.564/fair Far end 2 0.356/good 0.428/fair 0.616/good Near end 3 0.198/very good 0.278/good 0.343/good Far end 3 0.368/good 0.418/fair 0.612/fair Near end 4 0.220/good 0.365/good 0.425/fair Far end 4 0.283/good 0.298/good 0.457/fair

Table 6. FSV evaluation of results in Figs. 6–9. .

Several reasons are identified to explain the degradation of the agreement at some frequency points. In particular, we suspect the modeling of the cross-section geometry, which is determined by an approximated method. As the terminal loads involve inductance and capacitance, the impedance value is associated with frequency. In this paper, we use the mean value over frequency (0–3 GHz) of the terminal loads of the complete model to represent the equivalent loads.

Table 7 shows the computation times for both types of cable bundle models, calculated by CST Cable Studio. The CST simulation configuration is set to satisfy a convergence criterion and all of the computations are completed by a desktop computer with Intel(R) Xeon(R) CPU E3-1231 v3 @3.40 GHz and 32 G RAM under Windows 7. From the table, it is obviously found that the reduced model behaves better in efficiency. Furthermore, it greatly simplifies the modeling process at the expense of certain calculation accuracy.

Table 7.

Table 7.

Table 7. Analysis time of the simplified and complete model. . Complete cable bundle Reduced cable bundle CST 3208 s 1784 s

Table 7. Analysis time of the simplified and complete model. .