Parent Category: 2019 HFE

*By Kenneth S. Schneider*

**Ed. Note: Due to the length of this article, it will be published in segments. This is part one, with the remainder to follow in an upcoming issue.**

**1. INTRODUCTION**

A complete model for a Quad Cable used for broadband transmission is presented. This model is accurate and useable at the high frequencies used for new broadband technologies such as Gfast, GIGAWIRE, G. mgfast etc which extend well into the microwave region. The frequency range for these new broadband technologies currently extends to 212 MHz. However, in the next few years it is expected to extend to 500 MHz and then shortly thereafter to 1 GHz.

Transmission lines-basically telephone network local loops- used by these technologies for the transmission media, present many different and complex characteristics at these microwave frequencies-characteristics which were either unknown or ignored until relatively experimental measurements uncovered them. These characteristics may well affect the performance of the new broadband technologies. The model presented captures these characteristics analytically and can be used as a guide in developing simulation equipment for the testing of these technologies. The model presented includes the direct path attenuation, phase (Group Delay), input impedance and Characteristic Impedance of the direct path-single line loop. The model includes the characterization of Far End Crosstalk, FEXT and especially the “normalized” FEXT (ELFEXT) both within a single Quad (Intra-Quad) - -which is often alternatively referred to as “In-Quad”- - and between Quads (Inter-Quad). After presentation, this model is compared to experimental measurements made with an actual Quad Cable in current use. The model presented is derived as a further development of the approach to this problem presented by Youla [1]. This is based essentially upon Maxwell’s equations and the electro-magnetic theory associated with transmission lines- but applied to multiple transmission lines-- essentially a matrix-vector extension of the “traditional” transmission line equations. This development also makes use of significant mathematical techniques such as the work of Brillouin [2] to characterize the twists in these cables. It employs a deterministic geometric/topological modelling of the relative placement of Quads-rather than the statistical modelling used in [3]. Both Youla’s presentation and the additional mathematical analysis are quite involved and for conciseness the mathematical details have been omitted in this presentation. Note that the presence of a shield with a Quad cable allows the direct application of Youla’s formalism. The application of this formalism to Unshielded Twisted pair requires additional development and this will be the subject of future work. It must be emphasized that the model presented is not based upon computer simulation. It is based strictly upon theoretical analyses with resulting computation applied to specific cases. Yet, it does employ the assumptions used by van den Brink [4] in describing Quad crosstalk modelling based upon computer simulation-but enhancing these with other assumptions with regards both to the frequency behavior of the complex dielectric constant and of the conductivity between transmission lines. In fact, the model presented provides results which, for the most part, are generally, but not totally, consistent with the main results of van den Brink. Furthermore, while the approach here is that of a strict mathematical analysis this does follow the general development used by van den Brink [4]. It differs, somewhat, from the results of [3] in the dependence of ELFEXT on loop exposure length, the phase (Group Delay) of the FEXT, characterization of slope discontinuities, the oscillatory behavior of the ELFEXT and the characterization of ELFEXT between Quads. The characterization of ELFEXT by the model presented is very general. It does include the slope discontinuity usually referred to as the “Dual Slope Effect” which has been observed experimentally and included in other models. The characterization also includes frequency and loop exposure length behavior associated with cable twist properties and capacitive coupling imbalance.

The primary cable properties used by the model presented are based upon the characteristics of PE4D-ALT and an example of PE4D-ALT which is commercially available. The “completeness” of the model presented, its foundation on an analytical approach to characterizing FEXT and its use of the characteristics of PE4D-ALT recommend it as the basis for a loop simulation system for repeatable testing of the new broadband technologies mentioned.

With respect to computational examples provided in the modelling in the sequel the length of the direct paths formed from individual Quad Cables will be taken as varying from 50 m to 400 m- lengths which are of interest to the emerging broadband communication technologies. While the current presentation is focused on Quad Cables the analytical and computational techniques can also be used for carrying a similar characterization of Unshielded Twisted Pair Cables.

**2. CHARACTERIZING THE UNDERLYING CABLE**

For computational results, this entire model will use the complex transmission line exponent, γ and the Characteristic Impedance, Z_{0}, for PE4D-ALT Quad Cable. While there a variety of different Quad Cable types this is a reasonable representative. The model will be directed at a multiplicity of these Quad Cables within a shielded binder which is grounded. Each of the individual Quad Cables in the binder is composed of 4 conductors, 2 pairs. Each pair of conductors constitutes essentially a single line, direct path, that can be used for communications. There can be crosstalk between the 2 pairs in an individual Quad Cable and this is termed “Intra-Quad” crosstalk. There can be crosstalk between pairs in different Quad Cables in the Binder and this is termed “Inter-Quad” crosstalk. Interest here is only focused on the Far End Crosstalk (FEXT). Near End Crosstalk (NEXT) is considered less of an issue in the emerging broadband communication technologies as it is not even present when these technologies have Half Duplex implementations. For reference purposes, γ and Z_{0} are “usually” principally related to the distributed transmission line series impedance and shunt admittance, Z_{s} (f) and Y_{p} (f). However, there has also been anecdotal evidence based upon experiment measurements that for multiple Quad Cables in a Binder that γ is also a function of the number of other Quad pairs in the Binder, “Nu,” -with the influence of “Nu” being more prominent at the higher frequencies of interest by the new broadband technologies. This is recognized and accounted for in the analytical formula for γ given herein. Specifically, γ and Z_{0} are in turn, these are defined in terms of the distributed line parameters which are:

Zs (f) = R (f) + j2πf L(f),

Yp (f) = G (f) + j2πf C(f). (1)

R(f), L(f), C(f), G(f) are defined respectfully as the distributed line series resistance and inductance and the shunt capacitance and conductance. With this the complex transmission line exponent, γ, is defined as:

γ= [ ((1 + B (Nu, f)) Zs (f)) Yp (f)]^{0.5} = ά+ j β.(2)

where ά is identified as the real part of γ and β is identified as the imaginary part of γ—the “Phase.” The convention is adopted that exp(-γ) is associated with the forward wave propagated. ά is considered positive- resulting in an attenuation. β is considered as being positive to allow a positive Group Delay required by causality. At this point it is worthwhile to have some discussion of the physical origin underlying the function B (Nu, f). Basically, consider the situation where there is a Binder with one active Quad Cable pair and Nu others which are not active- no generators driving them. There will be fields radiated outward from the single active Quad Cable pair which will induce currents in the Nu inactive pairs. The induced currents will almost certainly radiate energy out as heat- essentially “sucking” energy out of the original single active Quad Cable. However, the dominant effect will be to radiate back and affect the original active pair. The radiation becomes more pronounced as frequency increases. The induced radiation back and its effect on the original active pair will become more noticeable as frequency increases because of the skin effect. It most likely will be represented by increased transmission losses. Ultimately, as Nu increases, this has the appearance of a radiated field being reflected from a conductive ground plane. The overall effect is that of a perturbation on Zs(f) (not Yp(f))- which is why it is has be “grouped as such” in eq. (2). In turn, this will affect γ and ά. It is conjecture d that B (Nu, f) increases with “Nu” and with “f” and follows an asymptote with increasing “f.” It must be noted that the overall effect of B (Nu, f) has been observed in actual experimental measurements [12]. But, in the sequel no further statements about this functional dependence can be made- and it will be “ignored” in the further specification.

The Characteristic Impedance, Z_{0}, is defined as:

Z_{0} = ((1 + B (Nu, f)) Zs (f))/ Yp (f)] 0.5, where the units are in Ohms.(3)

Figure 1 • a. ά in Nepers per meter vs. frequency, f, in MHz for the model, b. Insertion Loss per meter vs. frequency, f, in MHz for the model.

For the Quad Cable model presented ά was obtained indirectly from measurements [12]. It is well approximated by the following formula which is also shown in Figure 1a.

ά = (-1.1723(10^{-19})f^{6} +6.806(10-16)f^{5} -1.5247(10^{-12})f^{4} + 1.6525(10^{-9})f^{3} - 9.3339(10^{-7})f^{2} +

3.9040(10^{-4})f-1.969910^{-3} (4)

where f is in MHz.

In applications of the model ά will often be used to compute the corresponding Insertion Loss per meter for a corresponding direct loop. This is illustrated in Figure 1b. Rather than specifying β, for the Quad Cable model presented the Group Delay is specified. This is more useful for purposes of simulation than β itself. Group Delay is defined as – (1/2π) ∂β/∂f. For the Quad Cable model presented Group Delay was obtained indirectly from measurements [12]. It is will approximated by:

Group Delay = 4.5 nsec per meter for f ≥ 1 MHz.(5)

This frequency range for “f” covers the range of interest for the emerging broadband communication technologies. The Characteristic Impedance for the model presented (ignoring B (Nu, f)) is already at its asymptotic value of 135 Ohms -resistive- at 1 MHz. Accordingly, for the frequency range of interest- corresponding to the emerging broadband communications technologies, it is specified as:

Z∞ =135 Ohm, resistive(6)

The subscript “∞” indicates that this is the asymptotic value with frequency.

**3. CHARACTERIZING THE INDIVIDUAL QUAD CABLE PAIR**

An individual Quad Cable Pair in a Binder- a direct path used for a local loop- is characterized by its Transfer Function and by its Input Impedance. These are provided below.

**Transfer Function**

The transfer function for the individual Quad Cable Pair will be based upon the complex transmission line exponent, γ. The following nomenclature is used: Multipair Cable Path of length = “d” :Source Impedance= Z_{s}o which for computational purposes in the sequel is taken as 100 Ohms-resistive- a “typical value.” Load impedance = “Z_{L}“ which is taken as 100 Ohms which for computational purposes in the sequel is taken as 100 Ohms-resistive- a “typical value.”(7)

The Individual Quad Cable Pair transfer function,” H_{P},” is then given by the following formula

H_{P} (f, d) = [Z_{inP} / (Zin_{P} +Z_{so} ] T_{P}(8)

It should be noted that T_{P} is the same as the scattering parameter S_{12}.

Here:

T_{P}= [cosh (γd) + (Z_{0} /Z_{L}) sinh (γd)] ^{-1}(9)

Z_{inP} = The Input Impedance of the individual Quad Cable pair– which follows below.

In the sequel, for simplification purposes, the functional dependence of H_{P} (· , ·) on “f” and “d” will be suppressed in the notation unless needed for clarity.

The magnitude of “T_{P}” given by |T_{P}| gives the” attenuation” of the individual Quad Cable direct path. The phase of “T_{P}” gives the “phase” of the individual Quad Cable direct path. The negative derivative of the phase with respect to frequency (divided by 2 π) gives the “Group Delay” of the individual Quad Cable direct path.

**Input Impedance: **This will be based upon the complex transmission line exponent, γ.

The input impedance of the individual Quad Cable direct path, “Z_{in},” is then given by the following formula:

Z_{inP} = Z_{∞} [(Z_{L}+ Z_{0} tanh (γd)) / [(Z_{∞}+ Z_{L} tanh (γd))](10)

**FEXT- Crosstalk**

This FEXT transfer function between two Quad Cable paths “i” and “k” where the signal generating the FEXT comes from path “i” and the FEXT “output” is at the load on path “k” will be taken as:

H_{F} (f, d) = ELFEXT (f, X) [10 ^{θik /20}] │H_{P} (f, d) │ [ e ^{–j Ψik}](11)

Relative to the formula of (11)

“ELFEXT (f, X)” is the “normalized” FEXT.” This can be represented as:

ELFEXT (f, X) = |ELFEXT (f, X) | e^{ j Φik (f, X)}(12)

It is important to emphasize that “θik = 0” corresponds only to the FEXT within a single Quad Cable- which will be referred to as “Intra-Quad” FEXT. Equation (7) is generalized also to represent FEXT between Quads- which will be referred to as “Inter-Quad” FEXT- through the θik term when θik ≠0 . With respect to (11) the new nomenclature is defined as follows: “X” is the exposure length, the minimum of the 2 path lengths corresponding to paths “i” and “k” - as FEXT is only generated by coupling over the two paths- it is only this overlap that enters the formula. θik is called the “Amplitude Offset Factor” and is given in dB. This accounts for the effect of varying distances between different Quads on the amplitude of the FEXT. Ψik is the phase of the transfer function. The term, │|H_{P} (f, d)|│, provides the attenuation that the FEXT signal encounters as it traverses its path which begins at the generating end of path “i” and terminates at the load of path “k.”

“ELFEXT (f, X)”, “θik “and “Ψik” are given explicitly below along with the rational supporting the specifications.

**“ELFEXT (f, X)”**

The characterization ELFEXT (f, X) is now considered. While an exact mathematical development allows this characterization, this development does not allow for a precise, explicit, formula to be given. However, formulas are provided which are reasonable approximations for both the magnitude and phase of ELFEXT (f, X). As mentioned previously ELFEXT (f, X) corresponds to Intra-Quad of a single Quad Cable. This single Quad Cable is shown “geometrically” in Figure 2.

Figure 2 • Geometry of a single Quad Cable

This consists of 2 twisted pairs. Each pair consists of 2 conductors. These are clearly labelled in Figure 2. In a “perfect Quad” the geometry is symmetric and there is “total balance” between the capacitance between corresponding conductors. That is, the capacitance between Conductors “1” of each pair is the same as the capacitance between Conductors “2” of each pair. This perfect balance results in no crosstalk. This is a point to be emphasized. It is the presence of imperfect balance here, along with other effects- such as the twist- and other imbalances- which result in FEXT. The approach that has been used to derive formulas representing |ELFEXT (f, X) | and Φik (f, X) has been to follow the theoretical developments given by Youla [1] and consider a segment of the Quad Cable consisting of a single twist. A matrix version of the “Telegrapher’s Equation” of transmission line theory is then set up and solved resulting in the Network Theory { A, B, C ,D}parameters for this single twist. These are summarized in the single { A, B, C ,D} matrix M_{twist.} The entire details of this mathematical development are not given here to simplify the discussion. The mathematical procedure does very much follow the same development carried out by van den Brink [4] for Quad Cable modelling and the interesting work of Strobel in [5] and [6] as reported in Annex B of [7]. The development also employs many of the same parameters used by van den Brink [4]. The parameters that are required and the assumed values are given, in Table 1. It is worth noting that the “Capacitances” in Table 1—Cc, Cp and Cq—are not actual “physical capacitances” but rather mathematical constructs which are related to the real “physical capacitances.”While these are not “physical capacitances” it must be noted that they are related to the actual physical capacitances between the Quad Cable conductors and between those conductors and the shield. It is also worth noting that sensitivity analyses have been performed for the model with respect to the specific values provided in Table 1. No great variation in the results has been observed-and these will be brought up in the sequel as needed. Consequently, a model based upon Table 1 can be considered “Robust” and applicable to Quad Cables other than those based upon PE4D-ALT. The value assumed for the Imbalance has the greatest effect. This generally tends to move the |ELFEXT (f, X)| -either as a function of “f” or “X”- uniformly up or down. Once M_{twist} is determined the techniques provided by Brillouin [2] for Floquet Theory are used to handle the periodic repetition of the twists- very similar to the way this was handled by van den Brink [4] with a “twist amplitude” of 0.55. With this a new “A, B, C, D” matrix is obtained, M_{total} which is [M_{twist}]^{K} that is where M_{twist} raised to the “K^{th}” power—where K = X/(Twist Length). From M_{total }can be obtained all the transmission line characteristics for the case of the 2 coupled transmission lines constituting a single Quad Cable. This includes both |ELFEXT (f, X) | and Φik (f, X). It must be noted that the actual computations carried out in really results from the model noted above require careful attention to the algorithms for compute matrix inverses. Variable precision computation must be used in order to get accurate results due to matrices being raised to large powers and matrices which have very small determinants. Numerical analysis is very much a part of the development of this model.

*Table 1 • Parameters used in Model and assumed values*

In addition to the parameters of Table 1 it is assumed that the dielectric constant of all capacitances is complex and that the relationship between the complex dielectric constant, ε and the DC dielectric constant, εDC is given by:

ε = εDC (1 - j (aτ f ))(13)

where “τ” is the relaxation time of the dipoles in the dielectric material and is taken with a representative value of 10^{-11}. “a” is related to the polarizability of the dielectric material and the density of dipoles per unit volume and is taken with a representative value = 30 which allows alignment of the model with experimentally measured results.

The DC dielectric constant, εDC , of course depends upon the type of insulation. It is different for paper and polyethylene. The values assumed here are considered representative. Of course, these can be readily changed in any extension of the model presented. It is noted that the general functional behavior of the imaginary part of (13) with respect to “f” is that of a “Bell Shaped”- unimodular- Gaussian type density having its mean value and mode at f = τ^{-1} with the peak value at τ^{-1} being a small fraction of εDC . But, τ^{-1} = 100 GHz and this is >> than the frequencies of operation for the emerging broadband communication technologies. The functional dependence on “f” in (13) along with the values of “a” approximate the left tail of this “Bell Shaped” function. Sensitivity studies have been performed in developing the model and there are very minor changes to the results in considering other forms than (13). It is also assumed that the twist amplitude is = 0.55. and that the frequency dependence of the distributed resistance of the direct loop is given by the formula:

Rs0 + (Rs1) sqrt(2πf/w0) + (Rs2)2πf/w0

with Rs0=.0863, Rs1= 0.13, Rs2=.018, w0 = 2(106 ) π,(14)

It is assumed that there is a conductivity between all conductors and that this is used to compute the distributed admittance and that this is frequency dependent. This frequency dependence is conjectured to be caused by the non-uniformity of the conductive material. The frequency dependence given by Annex B.1.2 of [13] has been used as a guide to formulate this in the model. However, this has been adjusted to make results consistent with experimental measurements. In addition, for physical reasons this frequency dependence is expected to asymptote above a threshold frequency and this has been considered. The corresponding conductivity matrix would have all zeros on the diagonal and this term on all off-diagonal elements. Accordingly, for the model the following formulation of this conductivity term, G(f) is used:

For f ≤ 200 MHz. G(f) = g0f^{Nge} where g0 = 0.2 x 10 ^{-11}, Nge = 1.10

For f > 200 MHz. G(f) = g0 (200000000) ^{1.10} (15)

This inclusion of the complex dielectric constant- a “leaky” dielectric- and the leakage due to the conductivity is very important in the model. The complex dielectric constant affects the variation of the of the derivative of ELFEXT with “f” of at very high frequencies- typically above 100 MHz. The conductivity is the origin of the Dual Slope effect. Finally, it is assumed that the capacitive imbalance is 0.95.

ELFEXT (f, X) is of course a complex function having a magnitude and phase. The variation this with respect to “X” and “f” is considered now and discussed and in that order. This will be done by considering the variation of illustrate |ELFEXT (f, X) | dB- the logarithmic representation = 20 Log |ELFEXT(f, X)| . After this variation is discussed formulae providing approximations illustrating the variation are given for the possible use in simulations.

Before continuing it is important to emphasize the following caveat-- The discussion of the characteristics in the sequel strictly only applies to PE4D-ALT and under the assumed values provided in Table 1 and the assumptions regarding, complex dielectric constant, frequency dependence of resistance. However, it is reasonable to assume that other Quad Cable types with other assumed values will have similar characteristics- though the quantified values of various threshold frequencies, slopes, periodic resonances etc may differ. The model presented herein provides a guide to the characteristics of other Quad Cable types.

First consider the variation with respect to “f.” Figures 3a, b, c and d illustrate|ELFEXT(f, X )| dB versus “f” as derived from the analyses for over a range of 5 values of “X.” This range extending from a “low” of 66 m to a “high” of 400 m This encompasses the range of “X” of interest in the emerging broadband communication technologies. Figures 3a-d allow conclusions to be drawn concerning the overall characteristics with respect to “f. “ As will be shown in a later section the characteristics of these figures follow that of experimentally measured data obtained from [12].

Ed. Note: End part 1 of this article. Part 2 to follow in an upcoming issue.

**About the Author:**

Mr. Schneider is Chairman and CEO of Telebyte, Inc.

### July 2020

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