On Modelling of Two-Wire Transmission Lines with Uniform Passive Ladders

1 Department of Physics & Electrical Engineering, School of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia 2 Department of General Electrical Engineering, School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia 3 Department of Telecommunications, School of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia


Introduction
A comprehensive theory of linear, time-invariant, lumped-parameter networks is presented in many references, where the physical dimensions of network elements are assumed small, compared to the wavelength associated with the highest frequency in the spectrum of the signal being processed.In those networks, two-terminal passive elements, such as resistors, capacitors, and inductors, are specified by single, spatially independent parameters.In passive electrical networks there may be also four-terminal elements, such as transformers and gyrators.The equilibrium equations of linear, time-invariant, lumped-parameter

The Incremental Network Model of Two-Wire Line
In engineering practice the most widely and frequently used types of transmission lines are: a two-wire line Figure 1 a , coaxial line Figure 1 b and twisted pair Figure 1 c .In Figure 1 with ε c , μ c and σ c or ε d , μ d , and σ d are denoted: the electric permittivity, the magnetic permeability, and the specific electric conductivity of conductor "c" or dielectric "d" , respectively.Nevertheless, no matter what type of transmission line is considered, each line segment section with physical length δx, which is sufficiently small compared to the wavelength associated with highest frequency in the spectrum of the signal being transmitted, can be represented with approximate, incremental, lumped network model depicted in Figure 2. Thereon are denoted with R Ω/m , L H/m , C F/m , and G S/m : the resistance, inductance, capacitance, and conductance, respectively, of transmission line, in per-unit-length form.The lumped network model in Figure 2 becomes more and more accurate as δx → 0, and it is proved to be an adequate representation of any transmission line, since it is in good agreement with the experimental observations.Throughout the paper the length of the line will be denoted by and the considered frequency range will be f ∈ 0, 3 GHz .Dielectrics are assumed isotropic, linear, and homogeneous and, if imperfect, linear in ohmic sense, with constant specific electric conductivity σ d σ c .By neglecting the proximity effect with assumption d 2a, edge effect and taking σ d σ c , the per-unit-length capacitance C and the per-unit-length dielectric conductance G of two-wire line in Figure 1 a are calculated according to the following relations 1 : It has been proved 2, 3 that due to the influence of the skin-effect each conductor of the two-wire line should be characterized by the frequency-dependent per-unit-length  resistance R i f and the frequency-dependent per-unit-length inner inductance L i f -for f ∈ 0, ∞ Hz , , and "Bessel real" ber k • a and "Bessel imaginary" bei k • a are the Bessel-Kelvin functions with the first-order derivatives ber k • a and bei k • a , respectively, at the point z k • a, which can be approximated at high frequencies i.e., for k • a 1 with 4 ,

2.3
From 2.3 it should be firstly noticed that at high frequencies i.e., for k • a 1 it holds e j• π/4 ,

2.4
and then from 2.2 it may be obtained consecutively for k • a 1,

2.5
The overall per-unit-length resistance R f and the inductance L f of two-wire line are where R s f is the surface resistance of line conductors and L e μ d /π • ln d/a is the external per-unit-length inductance of two-wire line.Throught the paper it will be assumed that μ c ≈ μ d ≈ μ 0 .
Since the following expansions hold for any frequency f ∈ 0, ∞ Hz i.e., for any

2.7
then by using 2.2 and 2.7 when f → 0 Hz , the following consequences are easily obtained: Automatic, fast, and accurate numerical calculation of Bessel-Kelvin functions 2.7 imposes a real need to distinguish between the following two cases of approximation, depending on magnitude of z k

2.8
Case Define firstly the following set of auxiliary functions: 2.9 and, also, define another set of auxiliary functions: then, the values of Bessel-Kelvin functions can be efficiently calculated 5 by using the relations:

2.11
For the coaxial line with length Figure 1 b , the per-unit-length capacitance C and the per-unit-length conductance G of dielectric are calculated according to the following relations 1 :

2.12
It has been shown 1-3 that at high frequencies the coaxial line is characterized by the per-unit-length resistance R f and the per-unit-length inductance L f given with,

2.13
The twisted-pair Figure 1 c has characteristics similar to those of the two-wire line, except for the smaller inductivity and the smaller modulus Z 0 of its characteristic impedance Z 0 3, 6 .
To resume our investigation, consider two-wire line with copper conductors and polyethylene dielectric, where a 0.1 mm and d 4 mm Figure 1 Frequency (Hz) ×10 6    frequency of the signal spectrum and B the signal bandwith .If the integral of function in Figure 5 taken between f 0 − B/2 and f 0 B/2 is less than 0.01, then we see from Figure 3 that it may be taken R f ≈ R f 0 , for f ∈ f 0 − B/2, f 0 B/2 .And by using Figure 5 we obtain in the most conservative approach that B ≤ 37 kHz .Similarly, if the integral of function in Figure 9 taken between f 0 − B/2 and f 0 B/2 is less than 0.1, then we see from Figure 5 that it, also, holds R f ≈ R f 0 , for f ∈ f 0 − B/2, f 0 B/2 .And by using Figure 9 we obtain in the most conservative approach that B ≤ 770 kHz .
For a lossless transmission line ⇔ R 0 Ω/m and G 0 S/m with linear and homogeneous dielectric the phase-velocity c of electromagnetic perturbation i.e., the propagation speed of current wave in the line and characteristic impedance Z 0 are given by the following relations 1-3 :  where ε r ε d /ε 0 is relative permittivity and μ r μ d /μ 0 relative permeability of dielectric ε 0 ≈ 10 −9 /36π F/m is permittivity and μ 0 4π • 10 −7 H/m permeability of vacuum .The characteristic impedance of a lossy transmission line is generally defined as In the case considered, on Figures 11 and 12 the variations of • f in the frequency range f ∈ 0.01, 3 GHz are respectively depicted.In older telephony applications at lower frequencies, Z 0 was typically 600 Ω for air two-wire lines.For symmetric antenna feeding at frequencies up to 500 MHz , sometimes the two-wire lines with standard characteristic impedances Z 0 240 or 300 Ω are used.At shorter distances in telephony and local computer networks, nowdays are used the twisted-pairs two-wire lines with reduced inductance with standard Z 0 100 Ω and the propagation speed approximately c 0 /2.For the coaxial lines the standard Z 0 is 50 or 75 Ω and their propagation speed is approximately 2c 0 /3.For the printed transmission lines, Z 0 is in the range 100 ÷ 150 Ω , and their propagation speed is approximately c 0 /2 6 .For two-wire line being considered, in Figures 13, 14, 15, and 16 variations of several functions in the frequency range f ∈ 0, 3 GHz are depicted, which will be used in later consideration,

2.15
From the numerical data associated with the monotonic functions in Figures 13, 14, and 16 it is obtained φ 1 110.9KHz ≈ 1, φ 1 1289.9KHz ≈ 10, φ 1 are the propagation function, see A.10 in Appendix , also, play important role in analysis and they are depicted in Figures 17 and 18, respectively, in range f ∈ 0.01, 3 GHz .From data associated with these functions it is obtained: Λ 10 MHz ≈ 0.319, Λ 3 GHz ≈ 94.598, ϑ 10 MHz ≈ 89.123 deg and ϑ 3 GHz ≈ 89.953 deg . Since , where the deviation angle χ f π/2 − ϑ f can be approximated The percentage error of this  approximation is positive and <0.032% in the entire frequency range f ∈ 0.01, 3 GHz .with,

2.16
For the transmission line with length let us define the functions:  write 2.17  From relations 2.17 we obtain the approximations aa f and ab f of a f and b f , respectively, in the frequency range f ∈ 0.01, 3 GHz , since there it holds φ 1 f 1 and and, also, we have,

2.18
The functions aa f and ab f are depicted in Figures 21 and 22, respectively, in the frequency range f ∈ 0.01, 3 GHz , where we have as previously that it holds Γ j • ω ≈ aa ω/2π j • ab ω/2π ≈ {sin ε ω j • cos ε ω }/φ 3 ω/2π , as it has been expected.We will now emphasize the importance of function θ θ s, x Γ s • −x {x ∈ 0, } in the following.
a Constituting of functions sinh θ /θ and tanh θ/2 / θ/2 that play fundamental role in producing uniform three-terminal networks nominally equivalent to short-line segments 7 and in realization of these networks in the specified frequency range f 0 − B/2, f 0 B/2 by approximately equivalent three-terminal lumped RLC networks.The purpose of this approach is to involve the application of PSPICE, so as to facilitate the steady-state analysis of transmission lines with arbitrary terminations and band-limited signals, instead of solving the pair of so-called telegraph equations, hyperbolic, linear, partial diffrential equations obtained from relations A.4 in the Appendix,

2.19
To alternatively determine the voltage and current variations in time at any place on the finite length line, we may firstly perform the Fourier analysis of excitation signal and retain a reasonable number of its spectral components, then determine their transfer one at a time  to the specified place on the transmission line by using A.17 from the Appendix and finally synthesize the overall response by superposition of the obtained single-frequency responses.
b Calculation of M s, x , N s, x , U s, x , and I s, x from A.17 and Z s, x from A.18 , in general, and for the finite length open-circuited line Z L s → ∞ , in particular, by using expansions: thus placing into evidence the pole-zero location of M s, x , N s, x , and Z s, x .The relations 2.20 are produced by using Weierstass's factor expansions 8 of transcendental functions appearing in A.17 and A.18 into infinite product forms, For the open-circuited two-wire line with length 0.1 m in Figures 23 ÷ 26 are depicted for x ∈ 0, 0.1 m and f ∈ 0, 3 GHz the variations of  at six discrete frequencies altogether, in the two disjoint sets.Also, we may notice in Figures 24 and 26 that variations of Argfunctions are very complex with abrupt transitions.For the line terminated in Z L s the diagrams analogous to those in Figures 23 ÷ 26 could also be drawn easily, provided that the impedance Z 0 s is taken into account see relation A.17 .
When the line is sufficiently short, some approximations can be made leading to satisfactory results without need to cope with the cumulative products 2.21 .To see that, suppose that line length is  The following finite sums of infinite series 9 have been exploited in 2.22 ,

2.23
The infinite complex series 2.22 in almost pure imaginary θ are convergent for ≤ 0 , x ∈ 0, and f ∈ 0, 3 GHz .If is sufficiently less than 0 and x ∈ 0, , then by

2.24
For the open-circuited two-wire line with 0.01 m , in Figures 27, 28, 29, and 30 they are depicted on grid x × grid f 40 × 60 in the range f ∈ 0, 3 GHz and range x ∈ 0, 0.01 m , respectively: i the voltage-transmittance magnitude approximation percentage error: ii the voltage-transmittance phase approximation absolute error: iii the current-transmittance magnitude approximation percentage error: iv the current-transmittance phase approximation absolute error:     which, partly resembles to Maclaurin's expansion of cosh θ , that is, cosh θ

2.27
The functions sinh θ /θ and tanh θ/2 / θ/2 play fundamental role in effort to transform short transmission line segments into equivalent lumped three-terminal RLC networks 7 .The same role is played their respective approximating functions sinh A θ /θ Mathematical Problems in Engineering    in achieving the previous goals is providing the maximum of 0 to be much less than For example, from the numerical data associated with Figure 17, it can be calculated that 0 should be at most 1 cm on the upper limit of VHF and at most 1 mm on the upper limit of UHF band.Therein it has been tacitly assumed that transmission line is uniformly partitioned in segments of length 0 , which is at least ten times less than 1/|Γ j • 2π • f max |.
The equations A.15 and A.16 offer an opportunity to view on a transmission line segment with length 0 as on a linear two-port network Figure 35   conditions U s, 0 and I s, 0 at the input and U s, 0 and I s, 0 at the output.The chainmatrix F of this network reads In study of short transmission lines it is found convenient to replace them, either with nominally equivalent T

2.29
The aforementioned criterion for selection of 0 relies on attempt to find convenient θ s, 0 Γ s • 0 that provides physical realizability of immitances Z , Y , Z , and Y by lumped, transformerless RLC networks.The necessary and sufficient condition for existence and realizability of these immitances is that they must be rational, positive real functions in complex frequency s 10 .Observe that the immitances Z s and Y s are realizable by trivial two-element-kind RLC networks.Nevertheless, we will show now that, in general, the imitances Z , Y , Z , and Y are not realizable by lumped RLC networks, except in the limiting case when 0 is as small, so that the complex approximations hold: sinh θ /θ ≈ 1 and tanh θ/2 / θ/2 ≈ 1 observe that if θ → 0, then Z and Z → Z and Y and Y → Y .
To see that, recall that for , then by selecting 0 sufficiently small, A ω, x can always be produced arbitrarily small.
Bearing in mind the properties a ÷ e , we obtain for sinh θ j • ω, x /θ j • ω, x the following:

2.31
Mathematical Problems in Engineering that can be rewritten as sinh θ j • ω, x /θ j • ω, x R ω, x j • Q ω, x .R ω, x and Q ω, x are even and odd functions in ω, respectively, which are represented with the following expanded forms:

2.32
We must always bear in mind that ε ω is small c and 2.30 and that A ω, x ω Then, retaining only the first two terms in each of the convergent infinite sums in 2.32 , the approximations of R ω, x and Q ω, x are obtained which hold for all x ∈ 0, 0 and for all ω corresponding to f from the specified frequency range:

2.33
For x 0, from 2.31 and 2.33 it finally follows:

2.37
which are not realizable by lumped RLC networks, since they are not the positive real functions in complex frequency s 10 , except when 0 → 0. Putting it in other words we may say that if in the whole operating frequency range practically hold the three conditions: Relations 2.38 may be considered as a consequence of approximation sinh θ s, 0 ≈ θ s, 0 applied to 2.36 and 2.37 when θ s, 0 → 0. Similarly, under the conditions: 1, the complex approximation tanh θ s, 0 /2 ≈ θ s, 0 /2 applied to 2.29 when θ s, 0 → 0 gives the other two immitances from 2.29 , which are necessary to accomplish forming of linear networks in Figures 35 b and 35 c , which are nominally equivalent to the network in Figure 35 a .

Approximation of Two-Wire Line by Uniform RLCG Ladder and the Simulation Results
Let us consider a short two-wire line with length 30 mm or a longone partitioned in sections of length .Assume that the bandwith of the signal being transmitted is B < 770 kHz and that its central frequency is f 0 1 GHz .Then, make the graph of function 36 and from its associated numerical data find that l 10 9 ≈ 31.7 mm .Let the maximum length 0 of line segments be selected to satisfy the condition 0 ≤ l 10 9 /10 ≈ 3.17 mm .Finally, assume 0 3 mm and calculate the number of cells N / 0 10 in uniform RLCG ladder purporting to represent the transmission line with length in frequency range Assume δx 0 and calculate the parameters of uniform lumped RLCG network representing the line segments of length 0 see Figure 2 .If the overall short-line parameters  line with length is depicted in Figure 37, whereon the conductance elements G/10 present in Figure 2 are omitted only for the simplicity of drawing but are included in the PSPICE simulation network.Also, observe that at frequency f 0 1 GHz the quantity 2π • f 0 • C /G takes on extremely high value.Let u i be the voltage of the point N i i 1, 41 , with respect to the common node 0 Figure 37 .Now we will present the results obtained by PSPICE simulation of the open-circuited network in Figure 37 and compare them to the results obtained by exact analysis of considered open-circuited line.The amplitude of excitation voltage e in simulation was 1 V , its frequency was f 0 1 GHz and the initial phase 0 deg .The steady-state, odd numbered point voltages, and their phase angles obtained through PSPICE analysis of open-circuited network in Figure 37, are summarized in Table 2. Also, in this table the exactly obtained voltages at points on the line with distance x m m − 1 • 0 /2 m 1, 21 from the line sending end are presented.To these points correspond the points nodes N 2m−1 m 1, 21 in the simulation RLCG ladder on Figure 37.
When analysis of long lines is considered in the time domain it is useful to resort to forming of multilevel hierarchical blocks in PSPICE.To see that, suppose that we are to consider transmission of a signal with frequency f 0 1 GHz , amplitude E m 10 V , and zero initial phase in two-wire line with length 4.5 m and parameters as in Table 1.Let us designate the ladder on Figure 37, which represents a two-wire line with length 3 cm , as level 1 hierarchical block HB 1 on Figure 38 a .By cascading, say, 30 level 1 blocks: HB 1 /1, HB 1 /2, . .., and HB 1 /30, we produce a level 2 hierarchical block HB 2 on Figure 38 b , which represents a two-wire line with length 90 cm .By cascading 5 level 2 blocks: HB 2 /1, HB 2 /2, . .., and HB 2 /5, we make a level 3 hierarchical block HB 3 on Figure 38 c , which represents a two-wire line with length 450 cm , and so on.Hierarchical blocks of arbitrary length may be considered as independent entities or sophisticated parts in PSPICE.

Table 2:
The steady-state results of exact analysis and simulation for the open-circuited two-wire line in Figure 37.

Mathematical Problems in Engineering
From the data associated with functions depicted in Figures 12 and 13 we can easily: i obtain Z 0 j •2π •f 0 294.67280−j•0.42017 Ω and ii notice that variations of |Z 0 j •2π •f | and Arg Z 0 j • 2π • f are very small in the frequency range f 0 − B/2, f 0 B/2 .Now, suppose that the block HB 2 /5 in Figure 38 c is terminated with impedance, 6728 Ω and C L 378.7781 pF .From A.17 it follows that M j , and then at any place x on the line we obtain the exact values of signal amplitude 20 we see that function b f is practically linear in f, so that τ x should be linear in x, too.To verify the ladder model of two-wire line with length 4.5 m Figure 38 c , we will compare the exact values of E m • exp −x • a f 0 , ϕ x and τ x at the points on line x x k k • 0 /2 k 1, 3000 0 3 mm , with values obtained by PSPICE simulation.Bearing in mind the topological uniformity of the considered ladder, it is felt that for estimation of the proposed model it will suffice to check the ladder response only at the selected set of points whose voltages are u i i 1, 5 with respect to the common node 0 i.e., ground Figure 38 c .The distances of these five points from the line sending end are x 600•i 300 • i • 0 90 • i cm i 1, 5 , respectively.For these voltages in Table 3 are given the steady-state results of both the simulation and the exact analysis, where the following notation has been used Figures 19, 20, and 38 c :

3.1
From Table 3 it is evident the good agreement between the simulation results and those obtained by exact analysis.The selection of less segmentation step 0 will improve this agreement at the expense of rising the complexity of RLCG network, as a consequence of proliferation in number of network elements.
In Figure 39 the results of transient PSPICE analysis of RLCG ladder with zero initial conditions Figure 38 c representing the approximate network model of two-wire line with 4.5 m are depicted.Now we can summarize our obtained results i A transmission line is physically dispersive system with respect to frequency, having the infinite number of poles and zeros and complex transient dynamics, which cannot be represented perfectly with common-ground ladder with possibly great, but finite number of RLCG elements.37 and Table 1 .At frequency f 0 the characteristic impedance 8 of ladder is • f 0 it is close to the characteristic impedance of transmission line .Then, it can be shown that: i for n 1, 1500, the complex voltage at the end of the nth cell is is the complex representative of e t , and ii the time-delay at that place with respect to excitation is, Im{n of the line at frequency f 0 Figure 38 c and not by real two-wire line.As will be seen, the transient response of real two-wire line, even with the same initial conditions and termination, is more complex.
iii A deeper insight into transient phenomena in real lines can be acquired, either by applying the numeric inverse Laplace transform of A.17 with restricted number of poles/zeros or by numerical solving of linear, second-order, hyperbolic partial differential equations 2.19 , telegraph equations, with specified initial and boundary conditions depending on line termination and excitation voltages in consecutive time intervals determined according to the line length.The method of lines seems to be the most appropriate for solving of hyperbolic and parabolic partial differential equations 11 .
iv The PSPICE simulation method is applied herein only to facilitate the approximate steady-state analysis of two-wire lines with arbitrary loads and limited frequencyband signals, by using RLCG ladders as approximate network models of these Mathematical Problems in Engineering lines, instead of resorting to numerical solving of partial differential equations or application of complex analytic methods.
To illustrate the complexity of transient phenomena in transmission lines let we consider two-wire line with length 4.5 m and excitation e t as in Figure 38 c see, also, 3 , terminated with its characteristic impedance at frequency f 0 .At the moment of appearing of excitation at t 0, the line did not have any initial energy.Let us determine the solution u t, x of the telegraph equation 2.19 in the interval t ∈ 0, T , where T / 2π • f 0 /b f 0 ≈ 22.5972 ns is the perturbation propagation time from the line sending end to its receiving end, and 2π we obtain the second-order, hyperbolic, partial differential equation PDE equivalent to 3.2 : , then from 3.3 the following PDE of Klein-Gordon's type 12 is obtained {τ, x ∈ 0, 4.5 m }: which would be a pure wave equation if "diffusion" term D f 0 • u τ, x was not present, or in other words, if the line parameters satisfy the Heaviside's condition of distortionless at frequency f 0 .
Let we define, also, the following auxiliary function in τ and x: v τ, x ∂u τ, x ∂τ

3.5
If the excitation is e t E m • sin 2π • f 0 • t ϕ , then from 3.4 and 3.5 the system of two PDEs is produced to be solved in the interval τ, x ∈ 0, 4.5 m , by using of MATHCAD "Pdesolve" block: 2 in interval t ∈ 0, 22.59 ns , x∈ 0, 4.5 m .

3.6
In Figure 40 the solutions u t, x k of 3.2 in the interval t ∈ 0, T for x k k • /5 m k 1, 5 are depicted.Certainly the solutions of 3.2 for any x ∈ 0, can be produced easily, also by using the relations 3.4 -3.6 .
In Figure 41 the pulse responses i.e., the voltages u k k 1, 5 of the ladder in Figure 38 c terminated with the characteristic impedance of two-wire line are depicted.The ladder is excited by pulsed emf e t with amplitude 10 V , frequency f 10 MHz , dutycycle 0.2 and rise and fall times equal 1 ps .The voltages u k k 1, 5 have overshoots, undershoots, and delay times close to those of the network in Figure 38 c with continuous excitation of frequency f 0 1 GHz , in spite of the fact that frequency spectrum of the periodic, pulsed signal e t has components 10 • k MHz k ∈ N and that energetically significant part of spectrum is concentrated in the frequency range f ∈ 0, 150 MHz .

Conclusions
In the paper new results are presented in incremental network modelling of Two-wire lines in the frequency range 0, 3 GHz , by uniform RLCG ladders with frequency-dependent RL parameters, which are analyzed by using of the PSPICE.Some important frequency limitations of the proposed approach have been pinpointed, restricting the application Mathematical Problems in Engineering   of the developed models to the steady-state analysis of RLCG networks processing the limited-frequency-band signals.The basic intention of the approach considered herein is to circumvent solving of telegraph equations and application of the complex, numerically demanding procedures in determining two-wire line responses at selected set of equidistant points.The key to the modelling method applied is partition of the two-wire line in sufficiently short segments having defined maximum length, whereby couple of new polynomial approximations of line transcedental functions is introduced.It is proved that the strict equivalency between the short-line segments and their uniform ladder counterparts does not exist, but if some conditions are met, satisfactory approximations could be produced.This is illustrated by several examples of short and moderately long two-wire lines with different terminations, proving the good agreement between the exactly obtained steadystate results and those obtained by PSPICE simulation of RLCG ladders as the approximate incremental models of two-wire lines.

Appendix
By using of Kirchoff's voltage and current laws the following equlibrium equations can be written for the uniform transmission line depicted in Figure 2, no matter what its length or type is:  If Z L s is the load impedance termination of the uniform, finite length line, then from A.12 -A.14 we finally produce voltage-and current-transmittances M s, x and N s, x , respectively, M s, x U s, x U s, 0 A.17 Since U s, Z L s • I s, , then from A.12 -A.14 it follows, Z s, 0 U s, 0 I s, 0 where Z s, 0 is the input impedance of line and Z s, x is the impedance at place x seen towards the line end.From A.17 and A.18 we see that analysis of transmission line is equivalent to analysis of its segments terminated with impedances given with A.18 .The characteristic impedance and the propagation function of a distortionless line are Z 0 s L /C 1/2 and Γ s R • G 1/2 s • L • C 1/2 , respectively.And further if the line load impedance is Z L s Z 0 s , then for all x ∈ 0, it holds: Z s, x Z 0 s L • C 1/2 and M s, x N s, x exp −Γ s When the line parameters are constant, we will have u t, x that is, voltage u t, x and current i t, x at place x on distortionless line with constant parameters and load Z L s L /C 1/2 are as those on the line sending end, except for the time delay τ x • L • C 1/2 and the attenuation exp R • G • x .By using of lossless, constant parameter line with resistive load L /C 1/2 , it cannot be produced realistically even a relatively small signal delay.For example, pulse delay τ 5 ms can be obtained from a lossless, constant parameter line with parameters L 1.5 μH/m , C 18 pF/m , and the load resistance L /C 1/2 500/ √ 3 Ω at distance x τ • L • C −1/2 ≈ 962.25 km from the line sending end.But, if lossy, distortionless line is terminated with its characteristic impedance Z 0 s L /C 1/2 , the attenuation factor exp x • R • G 1/2 must, also, be taken into account.

cFigure 1 :
Figure 1: a Two-wire line, b coaxial line, and c twisted pair.

Figure 2 :
Figure 2: The approximate, incremental, and lumped network model of the uniform transmission line.

Figure 3 :
Figure 3: R f for line with a 0.1 mm and d 4 mm .

Figure 5 :
Figure 5: dR f /df for line with a 0.1 mm and d 4 mm .

Figure 6 :
Figure 6: dL f /df for line with a 0.1 mm and d 4 mm .

Figure 8 :
Figure 8: L f for line with a 0.1 mm and d 4 mm .

Figure 10 :
Figure 10: dL f /df for line with a 0.1 mm and d 4 mm .

Figure 12 :
Figure 12: ζ f and dζ f /df for the considered two-wire line.

Figure 13 :
Figure 13: A dimensionless parameter φ 1 f of two-wire line.

Figure 14 :
Figure 14: A dimensionless parameter φ 2 f of two-wire line.

Figure 18 :
Figure 18: Argument of propagation function two-wire line .

Figure 20 :
Figure 20: The phase "constant" of two-wire line.

Figure 24 :Figure 25 :
Figure 24: The argument of the voltage transmittance for two-wire line 0.1 m in frequency range f ∈ 0, 3 GHz .

Figure 26 :
Figure 26: The argument of the current transmittance for two-wire line 0.1 m in frequency range f ∈ 0, 3 GHz .

Figure 35 :
Figure 35: a Linear network model of a short transmission line segment and b its linear network structure.

networks
Figure 35 b or with nominally equivalent Π networks Figure 35 c 7 , whose immitances are given as follows

Figure 36 :
Figure 36: The function l f φ 3 f of the considered two-wire line determination of 0 .
& of the voltage at the end of transmission line N 41 obtained by exact analysis are 1.710901 V and −107.3554mdeg , respectively.

Figure 37 :
Figure 37: Electrical network model of two-wire line with length 3 cm in the frequency range f ∈ f 0 − B/2, f 0 B/2 .

Figure 40 :
Figure 40: Transient voltages at five equidistant places on the two-wire line with length 4.5 m E m 10 V and ϕ 0 .

Figure 41 :
Figure 41: Pulsed responses in selected points on the uniform ladder in Figure 38 c .
where Γ σ R L • s / G C • s is propagation function of the line.The important parameter of any line is, also, its generalized characteristic impedanceZ 0 s R L • s / G C • s .The line is distortionless if R /L G /C 7 ,and it is lossless when R 0 Ω/m and G 0 S/m .The general solution to the set of linear, homogeneous differential equations A.10 readsU s, x A 1 • cosh Γ • x A 2 • sinh Γ • x , I s, x B 1 • cosh Γ • x B 2 • sinh Γ • x , A.11where the terms A 1 , A 2 , B 1 , and B 2 are not the functions of x and are determined from the boundary conditions.From A.11 for x 0 we obtain, A 1 U s, 0 and B 1 I s, 0 and from A.9 and A.11 ; after differentiation in x; it follows, A 2 −Z 0 s • I s, 0 and B 2 −U s, 0 /Z 0 s .Then A.11 takes on the following form:U s, x U s, 0 • cosh Γ • x − Z 0 s • I s, 0 • sinh Γ • x , A.12 I s, x I s, 0 • cosh Γ • x − U s, 0 Z 0 s • sinh Γ • x .0 • cosh Γ • − Z 0 s • I s, 0 • sinh Γ • , I s, I s, 0 • cosh Γ • − U s, 0 Z 0 s • sinh Γ • , A.14 then from A.14 we obtain, U s, 0 U s, • cosh Γ • Z 0 s • I s, • sinh Γ • , A.15 I s, 0 I s, • cosh Γ • U s, Z 0 s • sinh Γ • .A.16 a .Let the operating GHz .The specific electric conductivity of copper is σ c ≈ 5.81 • 10 7 S/m and magnetic permeability is μ c ≈ μ 0 .At the temperature T 298 K the relative permittivity of polyethylene is ε r ≈ 2.26 for frequencies up to 25 GHz and its specific electric conductivity is σ d ≈ 10 −15 S/m .From 2.1 it is calculated C ′ (f) (H/m)Figure 4: L f for line with a 0.1 mm and d 4 mm .frequency range of this line be f ∈ 0, 3 10 MHz ≈ 32.653 1, φ 2 110.9 KHz ≈ 13.943 • 10 9 and φ 4 0 ≈ 1.391 • 10 10 .Herefrom and from 2.15 it followsFigure 11: Z 0 f and dZ 0 f /df for the considered two-wire line.
Magnitude of propagation function two-wire line .
respectively, on the grid x × grid f 50 × 60.In Figures 23 and 25 we observe the presence of voltage and current resonances at different places on the line, as is it might be expected from 2.20 , Problems in Engineering 29 Now, by using 2.29 and 2.35 we may generate the immitances Y' and Z" Figures 35 b and 35 c , the immitances Y s and Z s Figures 35 b and 35 c are produced in the following simple form and are realizable by two-element-kind RLC networks see 2.36 and 2.37 :

Table 1 :
GHz according to 2.1 , 2.2 , 2.8 -2.11 , are given in Table1.The RLCG network representing the short Two-wire copper-polyethylene transmission line central frequency f 0 1 GHz .

Table 3 :
The steady-state results of exact analysis and simulation for RLCG 1.9913• 10 8 m/s is the propagation velocity.If we introduce substitution t τ/A f 0 {A f 0 1/ L f 0 • C 1/21.9913 • 10 8 m/s } into the telegraph equation in u t, x obtained from 2.19 : If s is the complex frequency, let we suppose that the following conditions hold.Both u t, x and i t, x have continuous derivatives with respect to x. c The following two integrals are uniformly convergent with respect to x:Taking into account all conditions a ÷ d and A.1 -A.6 , it follows: wherefrom the equations describing voltage and current distribution in uniform line are readily produced, regardless to its length and/or terminal conditions i.e., the generator and load impedances : a Both u t, x and i t, x possess Laplace transform with respect to time, d The initial conditions u 0, x and i 0, x are assumed for convenience to be zero for all x.
and generally,