Statistical Characterization of Line-Imbalance in Differential Lines

Xinglong Wu1, Yuehong Yang1, Flavia Grassi2, Giordano Spadacini2, and Sergio A. Pignari2

1State Key Lab. of Electrical Insul. and Power Equip., Xi’An Jiaotong Univ., 28 Xianning W. Rd., Xi’An, P. R. China {xjtuwuxinglong; xjtuyyh}@gmail.com

2Dept. of Electronics, Information and Bioengineering, Politecnico di Milano, P.za L. da Vinci 32, Milan, Italy {flavia.grassi; giordano.spadacini; sergio.pignari}@polimi.it

Abstract

In this work, multi-conductor transmission line modeling and modal analysis are combined to develop a probabilistic model for the characterization of the unwanted common-mode (CM) voltages across the terminations of a differential line by possible imbalance of the wiring structure. Estimates for the probability density function of the transfer ratio between the differential-mode signal source and the theoretically null CM voltages induced at line terminations are cast in closed form as a function of the line geometrical parameters. The obtained predictions, validated versus Monte-Carlo simulations, are then used for a quantitative comparison with unwanted CM voltages due to imbalance of the line terminal networks.

1. Introduction

Differential signaling schemes are nowadays widely used, as they ideally allow the transmission of high- frequency signals without interference. However, in order for this goal to be achieved, the interconnection must be perfectly balanced. Indeed, in the presence of imbalance of the wiring structure or of the terminal networks, mode conversion takes place with detrimental consequences in terms of signal integrity and electromagnetic compatibility (EMC). More specifically, unwanted conversion of the desired differential-mode (DM) signal into common-mode (CM) is at the basis of undesired radiated emission. Conversely, conversion into DM of the CM noise picked-up from an external electromagnetic (EM) field may seriously degrade line immunity, thus compromising the integrity of the received signal [1, 2]. For this reason, ad hoc parameters and related measurement setups are currently foreseen by International Standards on information technology equipment, e.g., [3], to experimentally characterize the amount of mode-conversion in telecommunication cables and devices. Among these, the concepts of Longitudinal and Transverse Conversion Loss (i.e., LCL and TCL in [2-4]) represent a measure of system ability to convert the CM into DM and vice versa, respectively. However, for EMC-oriented design of telecommunication cables and devices, the information provided by these parameters (experimentally measured in well-defined test setups with specific terminal networks) need to be supplemented by prediction models, able to show the role of each parameter involved in mode conversion.

In a previous work [1], mode conversion due terminations imbalance was studied, and the correlation between the unwanted CM voltages and the common mode rejection ratio (CMRR) of each termination was pointed out. Conversely, in this paper, effects due to possible imbalance affecting the wiring structure are investigated, with the objective to identify the parameters that play a significant role on mode conversion. Particularly, since line imbalance stems from uncertainties in cable manufacture, that is deviation of some geometrical/electrical parameters from their nominal values, the main quantities responsible for mode conversion result to be inherently unknown and uncontrolled. Hence, instead of a deterministic model (quite inadequate to properly represent the phenomenon, and scarcely helpful to the designer), the model here proposed provides a statistical characterization of mode conversion, where (a) the unbalanced line is treated as a perturbation of the corresponding balanced line; and (b) uncontrolled/unknown geometric parameters are treated as random variables.

2. TL Model and Modal Analysis

To evidence the effect of line imbalance, the canonical differential-line structure shown in Fig. 1(a) is considered. Line terminal networks are modeled by lumped T-circuits, with impedances ZS1,X, ZS2,X in series with each wire, and ground impedances ZG,X (where X = L, R is used to denote the left and right termination, respectively). Such networks are assumed to be ideally matched to the line DM impedance, ZDM, and –at least at this point of the analysis– perfectly balanced (i.e., ZS1,X = ZS2,X = ZD/2, X = L, R). The wiring structure is assumed to be composed of two identical

ZG,L ZG,R V/2 h S   Z CM 0 , L  Z LCM VLE RCM L (a) (b) (c) Fig. 1. Differential-line interconnection under analysis: (a) Circuit model; (b) Cross-section view of the wiring structure; (c) Equivalent circuit model in the modal domain (upper: DM; lower: CM). bare wires of radius rw ideally running at the same height hμ above an ideal ground plane. To include possible line imbalance due to uncertainties in wires’ positioning, we will let the two wires move with respect to the nominal position, so that the actual wire height above ground (i.e., h1, h2) may slightly differ from hμ. Particularly, in order to keep wire separation, s, constant (as, for example, in an appliance cord) we will hereinafter constrain the wires to rotate along a circle with diameter s as shown in Fig. 1(b). In line with this assumption, their actual heights h1, h2 can be expressed as function of the rotation angle α as: h1(α) = hμ + sin(α) s/2; h2(α) = hμ – sin(α) s/2.

The above described wiring structure is modeled as a uniform and lossless three-conductor transmission line 21 (TL), characterized by 2×2 p.u.l. inductance, L , and capacitance, CL c0 (c0 being the speed of light in free space), matrices with entries evaluated by the analytical expressions in [5, Sec. 9.3] under the simplifying assumption s  5rw. Since prediction of the CM voltages induced at line terminals is the target, CM and DM quantities are introduced as in [6], and the whole structure is transformed in the modal domain. This yields the circuit model in Fig.

1(c), which clearly puts in evidence that although the voltage source VS excites the DM only, non-null voltages can be measured across the terminations of the CM circuit due to mode conversion and mixing along the wires. Particularly, as long as condition hμ >> s is satisfied (as in typical EMC test setups, where hμ  50mm ), the modal characteristic unb impedance matrix ZC of the unbalanced structure in Fig. 1(b) can be interpreted as a perturbation of the modal  characteristic impedance matrix ZC of the corresponding balanced structure (i.e., the reference TL with h1 = h2 = hμ). Indeed, it can be written as

Z 0 0()Z  unb  CM  CD  ZZZCC ()    , (1) 0 ZDM ZDC () 0  where: Z  csr1 ln( / ), Z  chrsr(2 )1 ln 2 / /  denote DM and CM characteristic impedances, DM00 w CM00    w w  respectively, [7], whereas impedances ZDC(α), ZCD(α) account for mode conversion (analytical expressions for these impedances are here omitted for brevity). These results are the rationale for the choice to keep wire separation constant. Namely, under such a condition, CM and DM line impedances are not affected by line imbalance, and coincide with those of the reference (i.e., balanced) configuration. Hence, line imbalance does not affect the matching of the DM circuit, which can be anyway enforced by ZD=ZDM. Furthermore, coefficients βL, βR in Fig. 1(c) can be readily defined as βX = (ZG,X + ZD/4)/ ZCM, X = L, R.

3. Statistical Analysis of Line Imbalance

CM As target variable, the voltage transfer ratio FLE of the CM voltage VLE induced at the left termination of the modal circuit in Fig. 1(c) over the voltage source VS is considered, since this quantity can be easily correlated to TCL as long as ZG,L = ZG,R = 0, [3, 4]. For statistical analysis, the rotation angle α is treated as a random variable (RV), with 2 normal distribution N(;  ) around the nominal value α = 0 (i.e., μα = 0). Additionally, the transfer function FLE is written as the product of the following two functions:

CM FVVMLE LE/(,,)(,,,,,,). S LE 0 L L R  G LE  R ZZhsr DM CM w  (2) -50 -10 Analytical x 10 2 Numerical Exact -60 Approximate -70 1 -80  +  F F

) Z = Z = 0  , [dB] -90

 G,L G,R (

LE  0 F LE F

G -100

-110 -1  -  F F -120 Z = Z = 1 k G,L G,R

-2 -130 6 7 8 9 -1.5 -1 -0.5 0 0.5 1 1.5 10 10 10 10 , [rad] Frequency, [Hz] (a) (b)

Fig. 2. Statistical analysis of line imbalance: (a) Exact dependence of GLE in (2)-(4) on the rotation angle α (solid curves) versus linear approximation by Taylor expansion (dashed curves); (b) Mean value and standard deviation of FLE (σα = π/18; ZG,L = ZG,R = 1 kΩ): Theoretical prediction (solid curves) versus numerical results obtained by 1000 repeated-run simulations (dashed curves).

The transfer function MLE retains the frequency dependence of FLE , and takes the expression:

LcL00sinh( ) MLLE(,,)0 L R  , (3) 2 sinh(00LL )(1LR ) cosh( )( L  R ) where L denotes the line length and γ0 = jω/c0 is the common propagation constant of DM and CM quantities (degenerate mode). Conversely, effects due to line imbalance are accounted for by the transfer function GLE. The exact analytical expression of GLE is quite complex as it involves the difference between the p.u.l. self-inductance and self- capacitance parameters in [5, Sec. 9.3]. As explicative examples, the solid curves in Fig. 2(a) show the dependence of GLE as a function of the rotation angle  for a differential line with hμ = 50 mm, rw = 0.5 mm, s = 2.5 mm, and for two different values of the ground impedances ZG,L, ZG,R. The observed almost-linear behavior of GLE around α = 0 suggests to approximate the exact analytical expression by resorting to a Taylor series expansion in a suitably-small interval around α = 0. This leads to the approximate expression [see dashed lines in Fig. 2(a)]:

 21  sZZ0004 DM CM R GZZhsrKLE(, R D ,,,,) C w LE  , KLE 1, (4) 4hZDM ln(/)ln(4/) h s h s ln(/ h r w )ln(4/ h r w )

The analytical expressions in (2)-(4) can be readily used to evaluate the probability density function (pdf) of FLE 2 in (2), by recalling that if  follows a normal distribution N(0; ) , the pdf of its magnitude, |α|, is an half-normal 22 distribution [8], with || 2/ , and ||(1 2 / ). Hence, also FLE in (2) is half-normally distributed, with:

2/ M K and 2212/ M 22K . (5) FLE  LE LE FLE    LE LE

To validate these analytical results, a repeated-run analysis is applied to a specific differential line with geometrical parameters: hμ = 50 mm, rw = 0.5 mm, s = 2.5 mm, L = 1 m. At each step, the solution process requires (a) to evaluate the line p.u.l. parameters, (b) to find suitable transformation matrices which allow to decouple the multi- conductor TL equations, (c) to solve the system equations with respect to the physical voltages V1, V2, and (d) evaluate CM voltages as VCM = (V1 + V2)/2. As a specific example, the simulation results obtained for 1000 randomly selected 2 values of the RV  N(0; ), with σα = π/18, and ground impedances ZG,L = ZG,R = 1 kΩ [worst case in Fig. 2(a)], are here considered, that exhibit a spread in FLE levels larger than 60 dB. The good agreement (the maximum discrepancies are less than 1 dB) between the mean value and standard deviation of FLE predicted by the analytic model (solid line) and obtained by post-processing numerical results (dashed lines) is shown in Fig. 2(b), and proves the accuracy of the statistical estimates in (5).

4. Comparison versus Termination Imbalance and Conclusions

The statistical model developed in the previous Section is here exploited for a quantitative comparison versus effects due to possible imbalance affecting the terminal networks. To this end, the series impedances in Fig. 1(a) are let to slightly differ from their nominal value by the imbalance coefficients δL, δR as [1]: -40 -40

-60 -60

-80 -80 , [dB] , , [dB] , LE LE F -100 F -100

 ,  +  due to  -120 -120 F F F  ,  +  due to  F F F

-140 5 6 7 8 9 -140 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 Frequency, [Hz] Frequency, [Hz] (a) (b) Fig. 3. Comparison of line imbalance (statistical estimates of FLE) versus termination imbalance (1000 repeated-run simulations) for two different values of the ground impedances ZG,L, ZG,R: (a) ZG,L, ZG,R = 0 Ω; and (b) ZG,L, ZG,R = 1 kΩ.

ZZSL1, D(1 L )/2, ZZ SR 1, D (1 R )/2, (7) SL2, SR 2, and the obtained transfer function FLE is evaluated by repeated-runs for 1000 randomly selected values of 2 2 RR N(0; ), LL N(0; ). Simulation results obtained by assuming a 2% degree of uncertainty with respect to the nominal impedance are shown in Fig. 3 for two different values of ZG,L, ZG,R that is: ZG,L= ZG,R = 0 Ω in Fig. 3(a), and ZG,L= ZG,R = 1 kΩ in Fig. 3(b). In the same figures, the black lines represent the predictions obtained by (5) for the mean value (solid curve) and standard deviation (dashed curves) of the transfer function FLE due to line imbalance (σα = π/18). Apart from the low-frequency range, where imbalance due to line terminations is undoubtedly prevailing, the comparison shows that in the standing-wave region of the line the two contributions may become comparable depending on the CM impedance of the terminal networks. As a matter of fact, while FLE due to termination imbalance results to be almost unaffected by ZG,L, ZG,R, the corresponding transfer function due to line imbalance exhibits a considerable sensitivity to these impedances. Indeed, an increase of more than 15 dB can be observed by comparing the black curves in Fig. 3(a) with those in Fig. 3(b). Therefore, although line imbalance is usually considered smaller than imbalance due to line terminations, the analysis here proposed shows that such an effect may not-negligibly contribute to the overall CM voltage in the presence of high-valued CM impedances. This finding takes a significant meaning considering that LCL/TCL measurement setups usually involve low-valued CM impedances (e.g., [3] foresees ZG,L, ZG,R =0 Ω), and further stresses the need for supplementing the information provided by such measurement-based parameters with a statistical characterization of the possible effects due to this inherently random and un-controlled phenomenon.

5. References

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