Excitation of multiconductor shielded cables by an incident electromagnetic field

The transmission line equations for multiconductor shielded cables with perfect, infinite conducting and excited by an incident EM field, as shown in Fig. 1, can be written as follows:

Fig. 1. Coaxial shielded cable over a perfectly conducting ground.

Outer system

(1)$$\left\{\matrix{{\delta V_{shd}\over \delta z} + jwL_{shd}I_{shd} + R_{shd}I_{shd} = V_{FD}\cr\cr {\delta I_{shd}\over \delta z} + jwC_{shd}V_{shd} + G_{shd}V_{shd} = I_{FD}}\right.$$

Inner system

(2)$$\left\{\matrix{{\delta V_{wr}\over \delta z} + jwL_{wr}I_{wr} + R_{wr}I_{wr} & = Z_{t}I_{shd}\cr\cr {\delta I_{wr}\over \delta z} + jwC_{wr}V_{wr} + G_{wr}V_{wr} & = 0}\right.$$

where V _shd and I _shd are the shield-to-ground voltage and the current flowing between the external shield and the ground, respectively. G _shd, R _shd, L _shd, and C _shd represent the per-unit-length (PUL) conductance, resistance, inductance, and capacitance of the outer system, respectively.

V _wr = [V _wr1,  V _wr2,  …V _wrN] ^T and

I _wr = [I _wr1,  I _wr2,  …I _wrN] ^T are the line voltage and current vectors,respectively.

N×N matrices G _wr, R _wr, I _wr, and C _wr are the PUL conductance, resistance, inductance, and capacitance matrices of the inner system, respectively.

Z _t = [Z _t1,  Z _t2,  …Z _tN] ^T represents the transfer impedance vector. For the braided shield, the transfer impedance is given by the following equation [Reference Saih, Rouijaa and Ghammaz22]:

(3)$$Z_{tk} = Z_{dk} + jwL_{tk}$$

where Z _dk and L _tk represent the diffusion term and the inductance, respectively, which indicates the effect of the field penetrating through the braid apertures, respectively. The expression of these two parameters is derived from[Reference Wang, Yin, Fang and Liu21]. For simplicity, the transfer impedance Z _tk is expressed as [Reference Saih, Rouijaa and Ghammaz23]:

(4)$$Z_{tk} = R_{dk} + jwL_{tk}$$

where R _dk is the constant PUL transfer resistance of the shield relative to the k ^th wire.

V _FD and I _FD are the distributed voltage and current sources; they represent the effects of the incident fields and are computed based on the components of the incident magnetic and electric fields (for further details, see [Reference Caniggia and Maradei24]).

Excitation fields of multiconductor shielded cables in a complex structure

To analyze multiconductor shielded cables within a complex structure, such as an automobile or a cavity excited by a nonuniform incident wave, as shown in Fig. 2(a), the line must be discretized into cells; the length of each cell is ΔZ = λ/10. A currents recovery method is used to compute the current of the shield in each segment I _shdi. Currents are needed to compute the terminal voltages and currents of the wires, as expressed in equation (2).

Fig. 2. (a) Multiconductor shielded cables inside a cavity with slot excited by an external plane wave field. (b) The configuration composed of a cavity with slot and shield only used to compute the current of the shield in each segment by MoM. (c) The configuration composed by inner wires used to compute the terminal voltages and currents after incorporating currents of the shield for each cell.

Circuit models of the inner wires

Equation (2) consists of coupled sets of partial differential equations. Here, a similar transformation is needed to decouple them into modal voltages V _wrm and currents I _wrm[Reference Ferrieres, Parmantier, Bertuol and Ruddle20].

(5)$$\left\{\matrix{{ V_{wr}} = T_{V}V_{wrm}\cr {I_{wr}} = T_{I}I_{wrm}}\right.$$

Substituting (5) in (2), yields

(6)$$\left\lbrace \matrix{{\delta\over \delta z}V_{wrm} + jwL_{wrm}I_{wrm} + R_{wrm}I_{wrm} = T^{-1}_{v}Z_{t}I_{shd} \cr {\delta\over \delta z}I_{wrm} + jwC_{wrm}V_{wrm} + G_{wrm}V_{wrm} = 0\hfill\cr }\right.$$

where

(7a)$$L_{wrm} = T^{-1}_{V}L_{wr}T_{I}$$

(7b)$$C_{wrm} = T^{-1}_{I}C_{wr}T_{V}$$

(7c)$$R_{wrm} = T^{-1}_{V}R_{wr}T_{I}$$

(7d)$$G_{wrm} = T^{-1}_{I}G_{wr}T_{V}$$

and where T _I and T _V represent the voltage and current transformation matrices, respectively. T _I and T _V are chosen in order to simultaneously diagonalize both L _wr and C _wr. From (6), the relation between the terminal mode voltages and currents for each cell can be written as follows [Reference Saih, Rouijaa and Ghammaz22]:

(8)$$\left\lbrace \matrix{V_{rm} = V_{rwm}( z_{0}) -Z_{cm}I_{wrm}( z_{0}) \hfill\cr V_{im} = V_{m}( z_{0} + \Delta z) + Z_{cm}I_{wrm}( z_{0} + \Delta z) }\right.$$

(9)$$\left\lbrace \matrix{V_{rm} = e^{-\gamma_m\Delta z}[ V_{rwm}( z_{0} + \Delta z) -Z_{cm}I_{wrm}( z_{0} + \Delta z) \cr -T^{-1}_V Z_t I_{sdh}( z_0 + \Delta z) ] \cr { V_{im}} = e^{-\gamma_m\Delta z}[ V_{rwm}( z_{0}) + Z_{cm}I_{wrm}( z_{0}) ] \cr \quad + T^{-1}_V Z_t I_{sdh}( z_0 + \Delta z) }\right.$$

where Z _cm and γ_m are the mode characteristic impedances and propagation constants, respectively. They are given by the following equations:

(10)$$Z_{cw} = \sqrt{{R_{wrm} + jwL_{wrm}\over G_{wrm} + jwC_{wrm}}}$$

γ_m = α_w + jβ_w

(11)$$\gamma_{m} = \sqrt{( R_{wrm} + jwL_{wrm}) ( G_{wrm} + jwC_{wrm}) }$$

Ignoring the PUL conductance G _wrm and using the first term of the Taylor series expansion, we obtain the following equations:

(12)$$Z_{cw} = \sqrt{{R_{wrm} + jwL_{wm}\over jwC_{wm}}} = R_{cm} + {1\over jwC_{cm}}$$

where $R_{cm} = \sqrt {{L_{wrm}\over C_{wrm}}}$ and $C_{cm} = {2L_{wrm}\over R_{wrm}R_{cm}}$

(13)$$\gamma_{m} = \alpha_w + j\beta_w = {R_{wrm}\over 2R_{cm}} + jw\sqrt{L_{wrm}C_{wrm}}$$

In this case, the characteristic impedance can be viewed as resistance and capacity connected in series. From (8), (9), and (12), we can deduce the equivalent circuit presented in Fig. 3. From (5), the terms of the generators controlled voltage and current placed in each cell can be obtained by the following equations:

(14)$$V_{wr}( z,\; t) _i = \sum_{k = 1}^N[ T_V] _{ik}V_{wm}( z,\; t) _k$$

(15)$$I_{wrm}( z,\; t) _i = \sum_{k = 1}^N[ T^{-1}_I] _{ik}I_{wr}( z,\; t) _k$$

Fig. 3. Equivalent circuit of inner wires obtained from the method of characteristics (Branin's method) [Reference Bayram and Volakis18] as implemented in ESACAP [Reference Saih, Rouijaa and Ghammaz22].

Analysis of the radiated susceptibility of coaxial cables

Figure 4 shows the configuration of the radiated susceptibility analyzed to study the validity of the hybrid MoM circuit models method. The configuration consists of a coaxial shielded cable, which consists of an inner solid wire (R _w = 5 mm) and a shield with an interior radius R _S of 2.5 mm, over a perfectly conducting ground. The line length is 1 m, and the conductors are suspended 5 m above a ground plane. The relative permittivity ɛ_r of the inner insulation equals 2.375.

Fig. 4. Coaxial shielded cable over a perfectly conducting ground.

The constant PUL transfer resistance R _T and the transfer inductance L _T are 100 mΩ/m and 0.5 nH/m, respectively. The shield is grounded at both ends with terminations impedances R _sh1 = 1 MΩ and R _sh2 = 154.365Ω, while the inner terminations are matched: R _w1 = R _w2 = 50Ω. The coaxial cable is excited by an incident plane wave oriented along the x axis of amplitude E = 1V/m and propagating along z axis (E _x − k _z).

Figure 5 illustrates the amplitude evolution of the wire-to-shield voltage at the far end as a function of frequency, which is obtained by the proposed hybrid method, together with the outcomes determined by the MoM. The results obtained by our method are in excellent agreement with those obtained by the MoM method, which confirms the validity of the proposed hybrid method approach.

Fig. 5. Wire to shield voltage of the coaxial cable at the far end.

The two curves depicted in this figure have two characteristic parts corresponding to the low frequency domain (assumption of short lines: length < <λ wavelength) and high frequencies (long line). In the first part of the curves, the voltages are proportional to the transfer impedance Z _t [Reference Degauque and Hamelin28]. The second part is characterized by anti-resonances specific to the propagation phenomenon.

The shielding is short-circuited on the right (Z ₂ = 0.5Ω). Figure 6 illustrates the induced current in dBA on shielded cables excited by uniform Ex-Kz, which is obtained by the hybrid model. Comparing these with the analytical results published by Smith [Reference Smith27] shown in Fig. 7 shows exact agreement between the results of the hybrid and analytic methods.

Fig. 6. Result of the induced current in dBA, on shielded cable excited by uniform Ex-Kz, computed with ESACAP.

Fig. 7. Solution for the induced current on shielded cable excited by uniform Ex-Kz [Reference Smith27].

Analysis of the radiated susceptibility of coaxial cable in the presence of a trough

To validate the proposed hybrid method in complex structures, we consider a coaxial shielded cable similar to that depicted in Fig. 4 in the presence of a trough. The length L _g, width W _g, and height H _g of the trough are 2, 0.2, and 0.2 m, respectively, as shown in Fig. 8. The entire structure is excited by an incident plane wave oriented along the x-axis and propagated along the z-axis (E _x − k _z) with an amplitude of E = 1 V/m.

Fig. 8. Coaxial shielded cable in the vicinity of a trough excited by an external plane wave field.

The wire-to-shield voltage at the far end of the coaxial cable obtained by the proposed approach is given in Fig. 9. The results obtained by our method are in excellent accord with those obtained by the MoM method, which validates the use of the hybrid method in complex structures.

Fig. 9. Wire-to-shield voltage of the coaxial cable at the far end.

Table 1 compares the computing time needed in the two methods. This shows that the MoM method requires significantly more computing time, than the hybrid method. Because the inner wires must not mesh while computing the shields currents, they are not meshed when using the hybrid method.

Table 1. Needed time by the hybrid method and MoM

Analysis of the radiated susceptibility of coaxial cables in the presence of an automobile

Figure 10 shows that the complex wiring inside a car, illuminated by an incident plane wave oriented along the x-axis and propagated along the z-axis (E _x − k _z), has an amplitude of E = 1 V/m. The wiring consists of three shielded cables: C ₁, a coaxial cable; C ₂, shielded cables with two parallel wires (i.e. twinax cable); and C ₃, shielded cables with three parallel wires (i.e. triaxial cable).

Fig. 10. Three shielded cables within a car excited by an external plane wave field.

The lengths of shielded cables C ₁, C ₂, and C ₃ are L _C1 = 0.5 m, L _C2 = 0.4 m, and L _C3 = 0.5 m respectively. All the cables have the same height, h = 5 mm. The characteristics of the shielded cables can be defined in terms of the shield radius R _s, the inner wire radius R _w, and the interval between wires d (the values of which are reported in Table 2). The transfer impedances and load networks with the adopted values are given in Table 3.

Table 2. Parameters of shielded cables

Table 3. Electrical parameters of lines configuration shown in Fig. 8

The voltages calculated using the proposed method at the far end in the frequency analysis are illustrated in Fig. 11. It can be observed that the lines resonate at f = n × (3 × 10⁸ 4/λ), n = 1,  3,  5 (f1 = 75 MHz, f2 = 225 MHz, and f3 = 375 MHz). The magnitudes of these induced voltages can reach − 70 V.

Fig. 11. Three shielded cables within a car excited by an external plane wave field.