Circuit models of Lossy Coaxial Shielded cables to Analyze Radiated and Conducted Susceptibilities with unmatched line loads

Brazilian Microwave and Optoelectronics Society-SBMO received 09 Aug 2016; for review 09 Aug 2016; accepted 04 Feb 2017 Brazilian Society of Electromagnetism-SBMag © 2017 SBMO/SBMag ISSN 2179-1074 Abstract— This paper presents a circuit models to analyze the variation effects of incident plane wave on shielded coaxial cables, using Branin’s method, which is called the method of characteristics. The model can be directly used for the time-domain and frequency-domain analyses and for all arbitrarily loaded. This makes it easy to insert in circuit simulators, such as SPICE, SABER, and ESACAP. The obtained results are in good agreement with those from others methods. Finally, we will discuss the effects of the variation of the incident plane wave.

Nevertheless, there exists coupling between the exterior and interior of the shield for the reason that the imperfect nature of the shield.Consequently, electromagnetic interference (EMI) and electromagnetic compatibility (EMC) problems associated with cables [6] connecting these devices should be taken into consideration.
Circuit models for multiconductor transmission lines (MTLs) with and without shields [7], [8] have been a subject of great interest in recent years.Spice models to analyze the conducted immunity of coaxial cables have been presented in [9], then some spice models have presented for the analyses of the conducted and radiated immunity of lossless shielded cables [10].These models are not inherently capable to analyze directly the time-domain, the inverse Fourier transform (IFT) is needed for the models to obtain the time-domain results.Recently, some Spice models have been proposed to analyze the conducted and the radiated susceptibilities of lossless shielded coaxial cables [8], [11].The principle point of interest of these models comprises in the likelihood of utilizing them as a part of frequency and time domains, with linear and non linear loads individually, and the discretization of shielded cables is not needed.However, these models cannot be used to analyze lossy shielded cables.
After Then, some lossy models have been presented to analyze the conducted and radiated  [12].However, a similarity transformation is needed to decouple the inner transmission-line equations.
In this paper, an equivalent circuit models to analyze the radiated susceptibility of uniform shielded coaxial shielded cables is presented.These models can be used to analyze both the time-domain and frequency domain and for all arbitrarily loaded.There is a good correlation with those from other methods.

A. Model of shielded cables
For a coaxial cable over an infinite and perfectly conducting, as illustrated in Fig. 1, the coupling with external fields can be described by [8]: Where V s is the shield-to-ground voltage, I s is the current flowing between the external shield and the ground, V w is the wire-to-shield voltage and I w is the current of the wire.
L s , C s , R s , and G s are the per-unit-lenght (p.u.l) inductance, capacitance, resistance and conductance of the outer system, respectively, while L w , C w , R w and G w are the p.u.l inductance, capacitance, resistance, and conductance matrices of the inner system.
Z t is the transfer impedance, In case of braided shield, the transfer impedance is given by the complex expression [13][14] Where Z d is the diffusion term, and L t is the inductance which accounts for the field penetrating through the braid apertures.The expression of both Z d and L t in terms of the braid weave parameters can be found in [13][14].In our application, we used a simplified expression [8] , where R t is the constant p.u.l.transfer resistance of the shield.
V f (z,t) and I f (z,t) are distributed sources that represent external excitation of the transmission line.These source terms can be written solely in terms of the incident electric field using Faraday's law.For coaxial shielded cable, shown in Fig. 2, we have: 0 ( , ) ( , , ) (0, , ) ( , , ) where h is the height of the line, and ( , , ) Where e x , e y and e z are the components of the incident electric field vector along the x, y , and z axes, and are given by: sin sin sin cos cos cos sin sin cos sin cos cos   zero and vertical if it is equivalent to 90°.The angle θ p decides the rise with respect to the ground.
This one is generally called the incident angle.The angle ϕ p gives the propagation direction relative to the axis Oz.
The components of the phase constant along those coordinate axes are: cos sin cos sin sin The phase constant is related to the frequency and properties of the medium as:

B. Equivalent Circuit Model for conducted Immunity: Outer system
In order To solve the equations ( 1) and ( 2) we use the 'discrete line' model.For this reason, the cable is discretized in the form of cell; the length of each cell is 10 z   .Using Branin's method, each cell can be written in the case of conducted mode as [12]: s cs s br In (10), Z cs represents the characteristic impedance of the outer system.Using the first term of the Taylor series expansion, we obtain [12] 1 Where Eq. ( 10) becomes Where The characteristic impedance in this case, is presented as a characteristic resistance R c and capacit f C as shown in Fig. 3.Where T s is the one-way delay of the shield, and is denoted by s s s

T z L C 
With the same estimation, the constant of propagation gets to be:

C. Equivalent circuit model for conducted immunity: Inner system
Using the same procedure, the inner system can be written for each cell as Where

D. Equivalent Circuit Model for Radiated Immunity of coaxial shielded cable
This is the same representation as the conducted immunity by adding generators 'forced' of voltage E z0 and E z0+ ∆z , which are representing the coupling between the shield and the incident wave, as shown in Fig. 5.   E z0 and E z0+ ∆z modeling the influence of the incident field in the time domain.For a perfect ground plane, their expressions are defined by [7] 00 00 00 00 Where ε0(t) represents the amplitude of the electric field in the time domain, α z0 et α z0+ ∆z are the coefficients dependent on the parameters of the line defined by: 00 00 ( )( )

z e x e y T T e T z e x e y T T
, if a component wave that propagates along the axis T z =T s , in the oposite case T z =0.

A. Conducted susceptibility and validation
The configuration used for the conducted susceptibility is shown in Fig. 6.The length L and the height h of the cable are 1m and 1cm, respectively.The shield and the inner wire radius are r s =2.5mm and r w = 0.25mm, respectively.The relative perttivity is ε r =2.The wire-to-shield voltage at the cable ends acquired by the proposed model is appeared in Fig. 7 together with the outcomes determined by the FDTD [15], where the "FDTD" implies the finite difference time domain solution to the transmission-line equations of the cable, and by the compact circuit model proposed in [10].They are in very good agreement with each other.
The lumped current source is set to 1A for the frequency-domain analysis.Fig. 8 demonstrates the magnitude of the frequency responses of the inner terminators acquired by the proposed model.The outcomes got by the ESACAP test system are in great concurrence with the analytical solution [16].
As shown in Fig. 8.The coupling into end side load is clearly stronger than in near side load, because the injection is asymmetrically located on near side of external shield, the total coupling is under a flat envelop and the anti-resonance frequencies are located as by the following formula at

B. Radiated Susceptibility Analysis of Coaxial Cable
The analysis of the radiated immunity is carried out on the coaxial cable as shown in Fig. 9.The shield radius and the inner wire radius are 0.25mm and 0.108mm, respectively.The cable's characteristic impedance is  The shield is short circuited on the right (Z 2 =0.5).At the output of the coaxial we recover the current into dBA, which is matched with the canonic results published by Smith [16], as shown in Fig. 10.
When the shield is open on the left (Z 1 =0.5) a towering resonances at about /4, as shown in Fig. 10.To drastically diminish the coupling to internal wire, a two-side grounded configuration for the shield must be utilized.Fig. 11 demonstrates the voltage reactions at the inner loads of the cable in the time-domain analysis acquired by the diverse methods, when Z 1 = Z 2 equivalent to the characteristic impedance of the shield-to-ground (Z 1 =Z 2 =244.5).The arrangements of the distinctive techniques concur exceptionally well.

C. Variation effect of incident plane wave on Coaxial cable with unmatched line loads, over ground plane
Fig. 12 shows a coaxial cable of 1m length at 5.25mm above a perfectly conducting ground plane, the shield radius R sh and the inner wire radius r w are 0.25mm and 0.0716mm, respectively, with dielectric constant ε r = 2.25.The loads R 1 and R 2 between the inner wire and the shield at the two terminations are R 1 =10Ω and R 2 =1000Ω.The value of the transfer impedance is set to R T =0.01Ω/m and L T =1nH/m.The incident electromagnetic field is a plane wave, while the incident field E = 1V/m.The Analysis performed for three reference field directions as described in Fig. 7   The two different curves for each case, with similar field illuminations, correspond to typical applications where the shield is connected to the ground on both sides, or only on one side, mainly at the receiving end.The internal wire is always loaded, as typical for real signal bus, with low resistance on the transmitting side and high resistance on the receiving side.
In Figs. 13, we compare the voltages at the far-end for three reference field directions.For all cases, it is seen that the line resonates at /(3x4) f =225 MHz when the shield load is open at the near-end.
The value of the open circuit resistance chosen here is 500MΩ.Also the high impedance resistance values located at opposite ends, for internal line and external shield, provides some phase compensation, practically eliminating the resonance /4. f =75 MHz.However, with short circuit at the ends, it is seen that for cases 1 and 2, eliminates practically all resonance, and the internal immunity is improved more than 20dB.For the case 3, the side vertical illumination excites the shield with maximum efficiency.This is the reason why the line resonates at /2f=150MHz and /(3x2)f=450MHz, even under short circuit conditions.

Fig. 1 .
Fig. 1.A Shielded coaxial cable over an infinite and perfectly conducting ground velocity in the space and the medium is characterized by the permeability

Fig. 3 .
Fig. 3. Circuit model of each cell of the outer system: shield

Fig. 4 .
Fig. 4. Circuit model of each cell of the inner system: wire

Fig. 5 .
Fig. 5. Equivalent Circuit Model for Radiated Immunity of coaxial shielded cable s

Fig. 6 .
Fig. 6.(a) Geometrical cross-section of the coaxial cable.(b) Configuration of the simulation for conducted analysis

50 cZFig. 7 .Fig. 9 .Fig. 10 .
Fig.7.Voltage responses of the inner loads in the transient analysis obtained by different methods Fig. 8. Magnitude of the frequency responses in decibels of the inner terminations

Fig. 11 .
Fig. 11.Voltages induced at the cable ends excited by an incident plane wave Ex-Kz, obtained by different methods

Fig. 13a. Case 1 -
Fig. 12a.Case 1-Coaxial cable over an infinite and perfectly conducting ground excited by an incident plane with 90 , 90 and 90 p p E          