Influence of Twist-Pitch Random Non-Uniformity on the Radiated Immunity of Twisted-Wire Pairs

Sergio A. Pignari and Giordano Spadacini

Politecnico di Milano, Dept. of Electrical Engineering, P.za Leonardo Da Vinci 32, 20133, Milano, Italy,

sergio.pignari@polimi.it, giordano.spadacini@polimi.it

Abstract Random non-uniformity of the twist pitch play an important role in radiated immunity of twisted-wire pairs (TWPs). In this work, a numerical prediction model based on transmission-line theory is used to investigate the response of a TWP with non-uniform twisting. The TWP is connected to balanced loads, is placed above an ideal ground plane, and illuminated by a plane-wave electromagnetic field. Unlike in previous works, where non-uniform TWPs were simply described as a cascade of individual twists of helical shape with different pitch, here the TWP is represented as a bifilar helix with continuous variation of the twist pitch. The modeling approach is therefore more consistent with the physical structure of a real TWP, and exploits a random function which relates the local value of the twist-pitch with the position along the TWP. It is shown that twist-pitch non uniformity does not affect the common-mode component of the response to the external field, whereas it influences the differential-mode voltage, whose sensitivity to twist-pitch non uniformity can be conveniently analyzed by resorting to a statistical approach.

1. Introduction Balanced twisted-wire pairs (TWPs) are interconnecting structures able to increase the system robustness to electromagnetic disturbances. Such a beneficial feature has an impact both on crosstalk (i.e., unintended electromagnetic coupling among wiring structures operating in close proximity) and on radiated susceptibility (i.e., coupling with an external electromagnetic field). From the theoretical standpoint, it is well known that interference-reduction in ideal TWPs is strictly related to (a) balancing of the loads connected at the terminations, and (b) symmetry due to wire twisting, which allows for induced effects to cancel out within each twist (whose axial length is called “twist pitch”). Nevertheless, it turns out that imperfections in the geometry of real TWPs may play a prominent role in reducing the overall immunity of TWPs to electromagnetic disturbances. Despite a careful manufacturing process, these imperfections are unavoidable, and show random and/or uncontrolled nature. Typical deviations from the ideal TWP geometry include: a) small twist non-uniformities, i.e., variations of the twist pitch along the TWP, b) residual portions of twists at the TWP terminations, and c) local deformations of the bifilar helix formed by the wire pair. Some of these phenomena have been experimentally and theoretically investigated in recent years. Particularly, in [1-3] their effect on crosstalk was investigated, whereas [4] contains an analysis of their impact on radiated susceptibility. In the aforementioned works, twist non-uniformity was modeled by describing the TWP as a cascade of perfect bifilar-helix structures, corresponding either to half twists [3] or full twists [4], with different pitch. This modeling approach, involving the partitioning of a TWP into a sequence of ideal sections with slightly different characteristics, leads to a very simplified representation of real TWPs. In order to refine the modeling, in this work we assume the twist non-uniformity to be continuously distributed in space, so that the bifilar helix shows continuous variation of the twist-pitch along the TWP. In this connection, the present work proposes a refined method to investigate the radiated immunity of a TWP with continuous non-uniform twisting, running above a ground plane, connected to balanced loads and illuminated by a plane-wave electromagnetic field. In particular, a random function relating the local value of the twist-pitch with the position along the TWP is used to construct the geometry of a bifilar helix with continuous deformation. As a result, the obtained geometry is more consistent with the physical structure of a real TWP. In passing, it is worth noting that this approach leads to a unified description of the distributed non-uniformity along the TWP and the possible presence of a final portion of incomplete twist. Conversely, these aspects were addressed separately in previous works [1-4]. Field-to-wire coupling with the obtained wiring structure is modeled by resorting to a computationally-efficient multiconductor-transmission-line (MTL) model, which is obtained via a modified numerical implementation of the analytical model developed for uniform TWPs in [5]. It is shown that twist non-uniformity does not affect the induced common-mode (CM) voltage, whereas it influences the induced differential-mode (DM) voltage, whose sensitivity is conveniently analyzed by means of a

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statistical approach. Namely, a repeated-run (i.e., Monte Carlo) analysis is exploited in this work to obtain the distribution and statistical estimates of the induced DM voltage.

2. Generation of a Random Non-Uniform Bifilar Helix For an ideal TWP described as a uniform bifilar helix running above ground at height h with wire separation ,s the curves describing the coordinates of wire #1 and #2 can be parameterized with respect to the arc length (the so-called “natural parameterization”) as [5]

zyx apasashrπ

ααα2

)sin(2

)cos(2

)(21 +±⎥⎦

⎤⎢⎣⎡ ±= , ],0[ L∈ , (1)

where xa , ya , and za are Cartesian unit vectors, p is the twist pitch, L is the wire length and α is a normalization parameter given by

2

122

22

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

πα ps . (2)

In order to introduce non-uniformity in the twists, it is assumed that the twist-pitch is not constant along the TWP, and that it can be described as a function )(pp = . Consequently, also α in (2) becomes a function of , i.e.,

)()]([ ααα == p , and the natural parameterization of the non-uniform bifilar helix takes the following form:

zyx aasashrπ

ξϕϕ2

)()](sin[2

)](cos[2

)(21 +±

⎭⎬⎫

⎩⎨⎧ ±= , ],0[ L∈ , (3)

where

∫ ′′=0

)()( dαϕ , ∫ ′′′=0

)()()( dpαξ . (4)

The function )(p provides all the information about twist deformation, and its definition should be consistent with the actual characteristics of the TWP. Here, it is assumed that non-uniformity has an inherent probabilistic nature, and )(p is generated by starting from a set of twist pitches randomly selected, i.e., Nkpp knomk ...,,2,1, =+= δ , (5) where nomp is the nominal twist pitch, and kδ are independent random variables (RVs) having uniform distribution in a symmetric range ],[ nomnom pkpk− , where k is a constant. Each RV kp is associated with position Δ−= )1(kk along the arc length, where Δ is a constant step which determines the rate of non-uniformity, i.e., the smaller is Δ , the larger is the deformation of the bifilar helix. Finally, the continuous function )(p is obtained via a spline-interpolation algorithm of the random data set )( kkp . As a specific example, an outcome for the random function )(p is exemplified in Fig. 1(a) with cm 1=nomp , 1.0=k , cm 25.0=Δ (for the sake of graphical clearness, the abscissa in Fig. 1 is limited to 5 cm only). The resulting non-uniform TWP is shown in Fig. 1(b).

3. Field-to-wire Coupling Model

In [5], an analytical model based on MTL theory was recently developed to predict the response of a TWP with uniform twist pitch, running above ground, and illuminated by a plane-wave electromagnetic field. According to this model, a TWP exposed to an external plane-wave field can be replaced by the Thevenin equivalent circuit (at the terminal ports) shown in Fig. 2.

0 0.01 0.02 0.03 0.04 0.050.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Random twist pitchesSpline Interpolation

y

x

z

(a) (b)

Fig. 1 (a) Function )(pp = generated via spline interpolation of a set of random twist pitches; (b) Non-uniform TWP

Fig. 2 Thevenin equivalent circuit of the TWP at the terminal ports. The center block is the passive part of the model, which is represented by the chain-parameter matrix )(LΦ of an unexcited three-conductor transmission line (TL). In particular, as a consequence of balancing and symmetry due to twisting the wires in the TWP, this TL can be approximately represented as an equivalent uniform balanced TL with averaged per-unit-length (p.u.l.) parameters [5]. In Fig. 2, the equivalent voltage sources at the terminal ports account for the field-to-wire coupling effects and can be expressed in closed-form [5]. This approximate MTL model can be modified to analyze the non-uniform TWP structure defined in Section 2. Namely, as concerns the passive part of the model, it can be assumed that slight deformation of twisting do not jeopardize the substantial symmetry and balancing of the TWP, so that an equivalent uniform balanced TL with averaged per-unit-length (p.u.l.) parameters (i.e., averaged along a uniform twist) can still be adopted for the circuit model in Fig. 2. Conversely, the equivalent voltage sources in Fig. 2 have to be evaluated by means of numerical-integration algorithms, as the solution of the field-to-wire coupling equations requires the integration of the longitudinal component of the electric field along a non-uniform line [5].

4. Simulations and Conclusion For the sake of exemplification, a TWP with the following parameters is considered in this Section: height

cm 2=h above ground, wire separation mm 1=s , wire radius mm 0.25=wr , nominal twist pitch cm 1=nomp , wire length m 1=L . The TWP is connected to balanced loads at both terminations, which are characterized by DM resistance Ω= 50DMR and CM resistance Ω= 502CMR (see [5]). The structure is illuminated by an incident plane-wave field having elevation angle o50=ϑ , azimuth angle o20=ψ and polarization angle o60=η (see [5] for the definition of the reference system).

Δ

p(ℓ)

, cm

ℓ, m

The CM and DM voltage induced at the left termination in the case of uniform twisting (i.e., in the case of constant pitch nompp = ) are plotted in Fig. 1(a) versus frequency, as evaluated via the closed-form TL model in [5]. Similarly, the CM and DM voltage induced at the left termination are plotted in Fig. 2(b) for a TWP with non-uniform twisting characterized by parameters 1.0=k , cm 25.0=Δ . This repeated-run simulation was based on 1000 different samples of the random function ( ).p

106

107

108

109

-80

-60

-40

-20

0

20

40

60

80

100

frequency, Hz

VL, d

BμV

CM

DM

106

107

108

109

-80

-60

-40

-20

0

20

40

60

80

100

frequency, HzV

L, dB

μV

DM

CM

(a) (b)

Fig. 3 CM and DM Induced Voltage: (a) Uniform TWP; (b) Non-uniform TWP (1000 repeated runs)

One can note that the induced CM voltage in Fig. 3(b) does not depend on twist-pitch non-uniformity (all the 1000 curves are perfectly superimposed), and results to be equal to the CM voltage reported in Fig. 3(a), corresponding to a TWP with uniform twisting. Conversely, the induced DM voltage in Fig. 3(b) shows large sensitivity (on the order of 40 decibels) to the small random fluctuations of the involved twists. Particularly, it is observed that the induced DM voltage may result largely lower than the corresponding response for a TWP with ideal (uniform) twisting The repeated-run simulations exemplified in Fig. 3(b) are the basis to perform a statistical analysis aimed at evaluating statistical estimates (e.g., mean, standard deviation, confidence intervals, etc.) of the induced DM voltage. In a similar fashion, influence of the non-uniform helix parameters defined in Section 2, i.e., k and Δ , on the induced DM noise voltage can be readily quantified.

5. References 1. C .R. Paul and M. B. Jolly, “Sensitivity of crosstalk in twisted-pair circuits to line twist,” IEEE Trans. Electromagn. Compat., vol. 24, no. 3, pp. 359-364, Aug. 1982. 2 D. Bellan, G. Spadacini and S. Pignari, “Prediction of twist non-uniformity and twist-residual effects on crosstalk in twisted-wire pairs”, in Proc. EMC Zurich 2003 Int. Symp. Electromagn. Compat., Zurich, Switzerland, Feb. 18-20 2003, pp. 181-186. 3. G. Spadacini and S. A. Pignari, “Impact of common-to-differential mode conversion on crosstalk in balanced twisted pairs,” in Proc. 2006 IEEE Int. Symp. on Electromagn. Compat., Portland, OR, USA, Aug. 14-18, 2006, pp. 145-150. 4. R. B. Armenta and C. D. Sarris, “Modeling the terminal response of a bundle of twisted-wire pairs excited by a plane wave,” IEEE Trans. Electromagn. Compat., vol. 49, no. 4, pp. 901-913, Nov. 2007. 5. S. A. Pignari and G. Spadacini, “Plane-wave coupling to a twisted-wire pair above ground,” IEEE Trans. Electromagn. Compat., pp. 1-16, in press (currently available on-line in the early-acces papers, http://www.ieeexplore.ieee.org).