Accurate Formulation of the Skin and Proximity Effects in High-Speed Cable System

An efficient transmission line model of high-speed cables is established, to predict its response at a high-frequency region (up to 20 GHz). Especially, an appropriate conformal mapping technique is applied to formulate the per-unit-length parameters of high-speed cables. Moreover, the skin and proximity effects of the conducting wires and the outer shield are derived in the closed form. As a result, the AC resistance/ inductance of all the conducting materials due to the skin and proximity effects are accurately incorporated for both layered and unlayered conductors. Furthermore, the mixed-mode S-parameters are precisely predicted for both balanced and unbalanced transmission line cables in the high-frequency region. In addition, the effects of the nonuniformities and cable geometry deformation on the mode conversions are also investigated. The proposed model is firstly validated with commercial software tools, COMSOL, FEKO, and HFSS; then, we further performed physical measurements to verify our new algorithm.

thick coaxial cable was utilized. After that, this standard was 22 The associate editor coordinating the review of this manuscript and approving it for publication was Flavia Grassi . revised by the Institute of Electrical and Electronics Engi-23 neers (IEEE) in 1983. Hence, the first IEEE standard, which 24 has the name IEEE 802.3, was released [1], and the thick 25 coaxial cable was standardized as 10BASE5. Due to the limi- 26 tations of the thick coaxial cable, the twisted-wire pair (TWP) 27 Ethernet was applied and standardized as 10BASE-T, then 28 10BASE-CX was released for shielded TWP cables [2]. In the 29 1990s, the speed rate increased, and the Gigabit Ethernet was 30 developed, where the shielded copper was used and called 31 1000BASE-CX and 10GBASE-CX for the speed rate 1 Gbps 32 and 10 Gbps, respectively [3], [4]. Moreover, the develop-33 ment of digital technology requires high-speed data rates in 34 order to transfer data reliably and safely between electronic 35 characteristic impedance of planar transmission lines on lossy 92 substrates was extracted using the method of series resistance. 93 However, in all the overmentioned papers about high-94 speed cables, the analysis was restricted only using full-95 wave solutions provided by commercial software tools or 96 direct measurement testing because there are no accurate 97 theoretical models were developed for the high-speed trans-98 mission line cables. Consequently, in this paper, our attention 99 will be focused on developing a transmission line model for 100 the high-speed cables, which is applicable at high-frequency 101 ranges, which help us to understand its physical operation, 102 hence, make the analysis and design of high-speed cables 103 straightforward. To do that, all the side effects of the dielec-104 tric and conducting materials should be modeled accurately. 105 It means that the skin and proximity effects of both the inner 106 conductors and the outer shield should be included in the 107 modeling process because their impact on the high-speed 108 cable operation cannot be neglected at the high-frequency 109 range. 110 This paper is organized as follows. In Section II, the 111 high-speed twinax cable is described, where its per-unit-112 length (p.u.l) parameters are derived, and also the skin 113 and proximity effect of inner conductors and the shield 114 are formulated analytically. In Section III, a generalized 115 transmission line model is developed for the noncoaxial 116 cable as an intermediate step toward the high-speed cable 117 modeling, where the effect of layered conductors in the 118 AC resistance is discussed. In Section IV, the transmission 119 line model is generalized to the high-speed twinax cables, 120 where the conversions between the common-mode (CM) and 121 differential-mode (DM) signals are studied using the mixed-122 mode S-parameters. In Section V, the proposed model is 123 validated by comparing its results with one generated by the 124 commercial software tools FEKO and HFSS. In Section VI, 125 experimental testing of actual cables is also conducted to fur-126 ther validate the proposed model. Finally, some conclusions 127 are drawn in Section VII.

129
The cross-section of a twinax cable is shown in Fig. 1. It con-130 sists of two inner conductors with a radius r w , they are usually 131 VOLUME 10,2022 frequency, accurate derivation of the p.u.l parameters, which cable are given as [20], [21], and [22].

154
where V and I are the voltage between the inner conductor 155 and the shield, and the current passes through the inner con-156 ductor and returns along with the shield, respectively. The 157 constant D is derived for the noncoaxial cable by using a 158 conformal mapping as [20] 159 The vectors e 0 and h 0 are the normalized transverse electric 163 and magnetic fields of the TEM mode, and they are given for 164 the noncoaxial cable in the polar coordinate as [20] 165 e 0 = −D r − αr sh cos φ r 2 + α 2 r 2 sh − 2αr sh r cos φ 166 − α 2 r − αr sh cos φ α 2 r 2 + r 2 sh − 2αr sh r cos φ It should be noted that (r, φ) are the polar coordinates and 171 e r , e φ , and e z are the unit vectors of the cylindrical coordinate. 172 The self-inductance and capacitance of the inner conductor 173 with respect to the shield are defined as [23] in which a n and a n are the normal unit vectors to the curves c 177 and c , respectively, as shown in Fig. 2.

178
It is worth indicating that the paths c and c can be chosen 179 arbitrarily; however, for simplifying the integration, they are 180 selected to follow the magnetic and electric fields curvatures, 181 respectively. For more details about the magnetic and electric 182 fields distributions in the noncoaxial cable, the reader can 183 refer to [20] 184 Without losing any generality, the path c is selected to be 185 the circle, which overlaps the inner surface of the shield, and 186 c is selected as a straight line started at the right edge of the 187 inner conductor and ended at the inner surface of the shield 188 along the x-axis, therefore, Along the path c the normalized electric and magnetic 192 fields of the TEM mode are given by whereas along path c, they are given by tance and capacitance are obtained as is considered in (8a) and (8b). Furthermore, by taking the fact 208 and the p.u.l. inductance and capacitance can be simplified as The solution of equation (12) can be expressed by It should be noted that H φ,r sh and H φ,R sh are the magnetic 237 fields at the inner and outer surfaces of the tubular shield, 238 respectively. Therefore, H φ,r sh can be obtained from (1b) 239 and (3b) by taking r = r sh , hence Noting that I in is the current passing through the inner 244 conductor and return through the inner surface of the tubular 245 shield. Generally, let us consider another current I out , which 246 uniformly flows along the outer surface of the tubular shield 247 and returns through an external return, and it could be a 248 ground plane or another covering shield, therefore The distribution of the current density within the tubular 251 shield can be obtained from ∇ × H = J as whereas the electric field can be obtained from E z (r) = 257 J z (r) /σ sh . Note that I 0 and K 0 are the modified Bessel func-258 tions of order zero.

259
The electric field along the inner and outer surfaces of the 260 tubular shield are given, respectively, as It is indicated that the property I 1 (Z ) K 0 (Z ) + 268 K 1 (Z ) I 0 (Z ) = 1/Z is used, where Z could be any complex 269 number.

270
In order to obtain the inner/outer surface impedances p.u.l 271 of the tubular shield, it is better to obtain the p.u.l complex 272 power flowing through the conductor at the inner and outer 273 surface of the shield, hence, Therefore, By considering the definition of the p.u.l complex power 285 P = 1 2 Z |I | 2 , hence, (21a) and (21b) can be written as In which Z in and Z out are the inner and outer p.u.l surface 289 impedances of the tubular shield, respectively, and Z t is the 290 p.u.l transfer impedance of the shield. They can be expressed As compared to the results obtained by Schelkunoff in r w 1 and r w 2 as shown in Fig. 3. By applying the same analysis 317 done above on the current density J in the cross-section of the 318 double-layered conductor, the following results are obtained 319 as [25]: Therefore, the electric and magnetic fields within the 323 double-layered conductor, respectively, are described as By applying the electric and magnetic fields boundary 329 conditions at the layer's boundaries [26], i.e., E z | r=r w 1 − = 330 E z | r=r w 1 + and H φ r=r w 1 − = H φ r=r w 1 + , respectively, where 331 the magnetic permeability is considered the same in both 332 conductors. hence, the constants d 1 and d 2 are obtained in 333 terms of c 1 as To determine the constant c 1 , the magnetic along 339 the outer layer of the inner conductor should be 340 used. From (3a), the magnetic field along the layered 341 Since d 1 = c 1 m 1 and d 2 = c 2 m 2 , therefore Once again, by applying the definition of the p.u.l complex 348 power, the p.u.l inner impedance of the layered conductor is 349 obtained as means that the inner conductor is centered at the origin of 356 the tubular shield, hence, Z cond is reduced to the expression 357 of an isolated/coaxial wire [25]. This indicated that the fac-  For the case of a non-layered conductor, i.e., σ 1 = σ 2 = σ , 361 and by taking r w 2 = r w , Z cond is simplified as where V (z, f ) and I (z, f ) are the voltage and currents of the 376 noncoaxial cable at a given operating frequency f , Z cable and 377 Y cable are the p.u.l impedance and admittance, and they are 378 expressed as The term tan δ Loss represents the loss tangent, due to the 382 bound charge and dipole relaxation in the dielectric mate-383 rial [26].
where L is the length of the noncoaxial cable, Z c its charac-391 teristic impedance, γ T is the propagation constant, and Consequently, the S-parameters of the noncoaxial cable are 395 given as [ where Z 0 is the S-parameter's reference impedance.

404
In order to validate the expressions of the skin and proximity 405 effects, a noncoaxial cable is considered, where the inner 406 wire is a double-layered conductor. The parameters of the 407 noncoaxial cable are taken as: the inner wire is copper with 408 a radius r w 1 = 0.2675 mm, and covered with tin layer has 409 thickness t tin = 5 µm, the outer shield is aluminum with a 410 radius r sh = 1.56 mm and thickness t AL = 0.13 mm. At first, 411 the dielectric material is considered to be air, i.e., lossless and 412 homogeneous, so as to make the validation straightforward. 413 It is seen that the transmission line model is valid under 414 the condition λ > 2 (r sh − r w ), which ensures only the 415 propagation of the TEM mode [22]. It is indicated that the 416 modified Bessel functions do not give correct results for large 417 function arguments. To overcome this problem, the method 418 proposed in [28] is used in this paper. The results of the skin 419 and proximity effect for different inner cable positions are 420 shown in Fig. 4. It can be observed that the current density 421 tends to accumulate at the edge side located at the closest 422 return path in both the inner conductor and the shield.    (27) and (16), respectively.

441
Therefore, the AC resistance of the inner wire and the 442 shield increase due to the skin and proximity effects as pre-443 sented in Eqs. (23a), (29), and (30), and the results are shown 444 in Fig. 7. Also, it can be observed that the position of the inner 445 conductor has a considerable effect on the AC inductance of 446 both inner conductor and shield. Moreover, it can be deduced 447 that the Tinned layered conductor has higher AC resistance 448 and inductance at high frequency region as compared to the 449 untinned conductor because the Tin has lower conductivity 450 than the copper, and most of the current flow in Tin layers 451 at high frequency. Whereas, at low frequency region the AC 452 resistance of the Tinned and untinned conductors are identical 453

465
The transmission line theory is generalized to be applicable 466 to high-speed twinax cables with an arbitrary cross-section.

467
The primary purpose is to investigate the effect of the cable 468 geometry on its performance and also investigate the longi-  dz where Z(f ), Y (f ) are the p.u.l impedance and admittance 477 matrices of the twinax cable and they are given as In which L and C are the p.u.l inductance and capacitance 481 of the twinax cable, and Z ac_Imp is the p.u.l ac impedance of 482 the conductors and the shield and it can be formulated as When conductors 1 and 2 of twinax cable are located in the 485 x-axis at Z 1 and Z 2 , respectively, then It is worth noticing that for layered conductors, the expres-490 sion (29) should be used instead of (30) in (40a).

492
In this section, the modal domain decomposition is per-493 formed in order to investigate the CM and DM modes and 494 VOLUME 10, 2022 (41)

500
The transmission line equations of the twinax cable (37a) 501 and (37b) can be written in the modal domain as Therefore, (42a) and (42b) can be written as It is indicated that V CM,DM or V DM,CM and I CM,DM or 547 I DM,CM are the total voltage and current due to DM-to-CM 548 or CM-to-DM conversion:

561
For the DM excitation, the input/output terminals are taken 562 as shown in Fig.10 (a). Therefore, Since the input/output terminals of the highspeed twinax Note that the closed expressions of

591
The induced CM currents at the input and output termi-592 nals due to DM-to-CM conversions are given as (56a)-(56c), 593 shown at the bottom of the page.

594
For CM excitation, the input/output terminals are taken as 595 By repeating the previous analysis, the total voltage and 600 currents due to CM-to-DM conversion are given as   After performing some algebra simplifications, it can be 658 deduced that s cd11 = s dc11 and s cd21 = s dc21 .

660
In this section, the proposed model of high-speed cable is 661 validated by making a comparison with FEKO/HFSS soft-662 ware. Three different cable geometries have been considered, 663 as shown in Fig. 11. The parameters of the cable are taken 664  It is seen that for the arbitrary shape as shown in slot can be obtained as [20] and [21] 711 where N = 1 4π 2 D 2 and d is the slot width. Hence, the 714 p.u.l inductance and capacitance of the high-speed cable are 715 modified as [20] 716 L a = L + diag (L slot (α, 0) , L slot (α, π)) (65a) in which the ''diag'' operation represents the diagonal matrix.

719
In the second case, the positions of the inner coated con-   Fig. 11).

768
In this section, an experiment is carried out to further validate 769 the proposed model. The measurement setup is shown in 770     slightly unbalanced cable, the level of the TCL parameters 802 increases, and good agreement is obtained because, in fact, 803 the actual fabricated cables are not perfectly balanced, and 804 higher conversions between modes have been occurred. This 805 due to the fact that the real fabricated highspeed cables have 806 many deformations that cannot be avoided due to the uncer-807 tainties during the fabrication process. We have plotted two 808 circles with exactly the same radius as shown in Fig. 18. It can 809 be observed that there are many different deformations in the 810 fabricated cable. a) deformation in the shield b) deformation 811 in the dielectric. c) deformation in the location of the wires. 812 To model this deformation we have, considered 5% shift of 813 one of the inner conductors.

815
In this paper, an efficient transmission line model of high-816 speed cables has been developed to predict its response at 817 high-frequency ranges up to 20 GHz. To do that, accurate 818 modeling of the p.u.l parameters has been established, and 819 the skin and proximity effect of the conductors and shield 820 have been formulated in the closed-form. Consequently, the 821 mixed-mode S-parameters of the highspeed cables have been 822 generated, and the conversions between the CM and DM 823 signals have been investigated in detail. It has been shown that 824 any slight unbalances in the highspeed cable can cause high 825 conversions between the modes, which can deteriorate the 826 transmitted signal. The obtained results have been compared 827 with measurement one, and very good agreement has been 828 obtained, and it has been shown that the S-parameters of 829 the cable are critical and can be affected by the measuring 830 instruments.