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

Abstract

In the paper we presented new results in incremental network modelling of two-wire lines in frequency range [0,3] [GHz], by the uniform RLCG ladders with frequency dependent RL parameters, which are analyzed by using PSPICE. Some important frequency limitations of the proposed approach have been pinpointed, restricting the application of developed models to steady-state analysis of RLCG networks transmitting the limited-frequency-band signals. The basic intention of this approach is to circumvent solving of telegraph equations or application of other complex, numerically demanding procedures in determining line steady-state responses at selected equidistant points. The key to the modelling method applied is partition of the two-wire line in segments with defined maximum length, whereby a couple of new polynomial approximations of line transcendental 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 steady-state results and those obtained by PSPICE simulation.

1. 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 networks are ordinary, linear differential equations with constant coefficients. Unfortunately, all physical components cannot be treated as lumped, since their spatial configuration plays important role in understanding their physical behaviour at high frequencies. The systems such as electrical transmission lines, passive integrated circuits, as well as many physical processes: thermal conduction in rods, carrier motion in transistors, vibration of strings, and so forth, are characterized by partial differential equations, and distributed parameters must be introduced for correct mathematical description of their physical behaviour. Equilibrium equations of those distributed-parameter systems (i.e., partial differential equations) have solutions which are more difficult to find than the solutions of ordinary differential equations with constant coefficients. Since differential equations of transmission lines are analogous to those of many other systems and one-dimensional physical processes (e.g., of the heat flow in solids), in this paper we will: (a) make brief overview of partial differential equations describing voltage and current distributions in finite length lines, (b) develop the appropriate physical model of two-wire line with frequency dependent per-unit-length parameters by taking into account all the physically relevant parameters (geometry, dielectric, magnetic, and conductivity properties of media, the operating frequency range, and skin-effect), and (c) propose an approximate representation of real two-wire lines by uniform RLC ladders with frequency-dependent parameters, turning thuswith the problem of analysis of real two-wire lines into the analysis problem of high-order passive RLC networks by extensive use of PSPICE.

2. 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  ℓ  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.

(a)

(b)

(c)

Figure 1 

(a) Two-wire line, (b) coaxial line, and (c) twisted pair.

Figure 2 

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

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]: 𝐶^=𝜋⋅𝜀^𝑑ln(𝑑/𝑎),𝐺^=𝜎^𝑑𝜀_𝑑⋅𝐶^=𝜋⋅𝜎^𝑑ln(𝑑/𝑎),𝑑≫2𝑎∧𝜎^𝑑≪𝜎_𝑐.(2.1)

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], 𝑅^𝑖(𝑓)=𝑘2𝜋⋅𝑎⋅𝜎_𝑐⋅Re−bei(𝑘⋅𝑎)+𝑗⋅ber(𝑘⋅𝑎)=𝑘ber^(𝑘⋅𝑎)+𝑗⋅bei^(𝑘⋅𝑎)2𝜋⋅𝑎⋅𝜎_𝑐⋅ber(𝑘⋅𝑎)⋅bei^(𝑘⋅𝑎)−bei(𝑘⋅𝑎)⋅ber^(𝑘⋅𝑎)ber^(𝑘⋅𝑎)²+bei^(𝑘⋅𝑎)²𝐿_𝑖,(𝑓)=𝑘=2𝜋⋅𝑎𝑘⋅𝜎_𝑐⋅𝜔⋅Im−bebre^i((𝑘𝑘⋅⋅𝑎𝑎))++𝑗𝑗⋅⋅bbeei^r((𝑘𝑘⋅⋅𝑎𝑎))2𝜋⋅𝑎⋅𝜎_𝑐⋅𝜔⋅ber(𝑘⋅𝑎)⋅ber^(𝑘⋅𝑎)+bei(𝑘⋅𝑎)⋅bei^(𝑘⋅𝑎)ber^(𝑘⋅𝑎)²+bei^(𝑘⋅𝑎)²,(2.2) where: 𝑗=√−1,  𝜔=2𝜋⋅𝑓,  𝑘=√𝜔⋅𝜇_𝑐⋅𝜎_𝑐 , 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], ber(𝑘⋅𝑎)=𝐺⋅cos𝑘⋅𝑎√𝐺2=−𝜋81,bei(𝑘⋅𝑎)=𝐺⋅sin𝑘√⋅2𝑎−𝜋8,√2𝜋⋅𝑘⋅𝑎⋅𝑒^{√(𝑘⋅𝑎)/2}ber_,(𝑘⋅𝑎)=𝐺⋅cos𝑘⋅𝑎√2+𝜋8,bei^(𝑘⋅𝑎)=𝐺⋅sin𝑘⋅𝑎√2+𝜋8.(2.3)

From (2.3) it should be firstly noticed that at high frequencies (i.e., for 𝑘⋅𝑎≫1) it holds −bei(𝑘⋅𝑎)+𝑗⋅ber(𝑘⋅𝑎)ber^(𝑘⋅𝑎)+𝑗⋅bei^(𝑘⋅𝑎)=−sin(𝑘⋅𝑎)/√2−𝜋/8+𝑗⋅cos(𝑘⋅𝑎)/√2−𝜋/8=𝑗c⋅o𝑒s(𝑘⋅𝑎)/√_{√𝑗⋅((𝑘⋅𝑎)/2−𝜋/8)}2+𝜋/8+𝑗⋅sin(𝑘⋅𝑎)/√2+𝜋/8𝑒^{√𝑗⋅((𝑘⋅𝑎)/2+𝜋/8)}=𝑒^{𝑗⋅(𝜋/4)},(2.4) and then from (2.2) it may be obtained consecutively for 𝑘⋅𝑎≫1, 𝑅^𝑖(𝑓)=𝑘2𝜋⋅√2⋅𝑎⋅𝜎_𝑐=12𝜋⋅𝑎⋅𝜋⋅𝑓⋅𝜇_𝑐𝜎_𝑐𝐿_𝑖,(𝑓)=𝑘2𝜋⋅√2⋅𝜔⋅𝑎⋅𝜎_𝑐=14𝜋⋅𝑎⋅𝜇_𝑐𝜋⋅𝑓⋅𝜎_𝑐.(2.5)

The overall per-unit-length resistance 𝑅^(𝑓) and the inductance 𝐿^(𝑓) of two-wire line are 𝑅^(𝑓)=2⋅𝑅^𝑖(𝑓)||_∀𝑓,𝑅^(𝑓)=1𝜋⋅𝑎⋅𝜋⋅𝑓⋅𝜇_𝑐𝜎_𝑐=𝑅^𝑠(𝑓)𝜋⋅𝑎||||_{𝑘⋅𝑎≫1},𝑅_𝑠(𝑓)=𝜋⋅𝑓⋅𝜇_𝑐𝜎_𝑐𝐿_,(𝑓)=𝐿^𝑒+2⋅𝐿^𝑖(𝑓)||_∀𝑓,𝐿′(𝑓)=𝜇^𝑑𝜋⋅ln𝑑𝑎+2𝜋1⋅𝑎⋅𝜇_𝑐𝜋⋅𝑓⋅𝜎_𝑐||||_{𝑘⋅𝑎≫1}(𝑑≫2𝑎),(2.6) 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≈𝜇₀.

Since the following expansions hold for any frequency 𝑓∈[0,∞) [Hz] (i.e., for any 𝑘⋅𝑎) [4], ber(𝑘⋅𝑎)=^∞_𝑛=0(−1)^𝑛⋅(𝑘⋅𝑎/2)^4𝑛(2𝑛!)²=1−(𝑘⋅𝑎)⁴2²⋅4²+(𝑘⋅𝑎)⁸2²⋅4²⋅6²⋅8²−(𝑘⋅𝑎)¹²2²⋅4²⋅6²⋅8²⋅10²⋅12²bei(𝑘⋅𝑎)=_∞±…,_𝑛=1(−1)^𝑛+1⋅(𝑘⋅𝑎/2)^4𝑛−2[(2𝑛−1)!]²=(𝑘⋅𝑎)²2²−(𝑘⋅𝑎)⁶2²⋅4²⋅6²+(𝑘⋅𝑎)¹⁰2²⋅4²⋅6²⋅8²⋅10²ber_∓…,(𝑘⋅𝑎)=−(𝑘⋅𝑎)³+2²⋅4+2²⋅((𝑘𝑘4²⋅⋅⋅𝑎𝑎6))⁷²₁₅⋅8−2²⋅4²⋅(6𝑘²⋅⋅𝑎8)²⋅¹¹10²⋅12bei_2²⋅4²⋅6²⋅8²⋅10²⋅12²⋅14²⋅16∓…,(𝑘⋅𝑎)=𝑘⋅𝑎−2−2(²𝑘⋅⋅4𝑎²()𝑘⋅⁵6⋅𝑎+)2²₁₃⋅4²(𝑘⋅6⋅²𝑎⋅)⁹8²⋅102²⋅4²⋅6²⋅8²⋅10²⋅12²⋅14±…,(2.7) then by using (2.2) and (2.7) when 𝑓→0 [Hz], the following consequences are easily obtained: 𝑅^𝑖(𝑓)→1/(𝜎_𝑐⋅𝜋⋅𝑎²), 𝐿^𝑖(𝑓)→𝜇_𝑐/(8⋅𝜋), 𝑅^(𝑓)→2/(𝜎_𝑐⋅𝜋⋅𝑎²), 𝐿^(𝑓)→(𝜇_𝑑/𝜋)⋅ln(𝑑/𝑎)+𝜇_𝑐/(4⋅𝜋). 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⋅𝑎∈[0,∞) [5],

Case  A  (𝑧=𝑧(𝑓)=𝑎⋅𝑘=𝑎⋅√2⋅𝜋⋅𝑓⋅𝜇_𝑐⋅𝜎_𝑐∈[0,8]). ber(𝑧)=1−64⋅𝑧8⁴+113.77777774⋅𝑧8⁸−32.36345652⋅𝑧8¹²+2.64191397⋅𝑧8¹⁶−0.08349609⋅𝑧8²⁰+0.00122552⋅𝑧8²⁴−0.00000901⋅𝑧8²⁸bei(𝑧)=16⋅𝑧+𝜀₁||𝜀₁||<10⁻⁹,8²−113.77777774⋅𝑧8⁶+72.81777742⋅𝑧8¹⁰−10.56765779⋅𝑧8¹⁴+0.52185615⋅𝑧8¹⁸−0.01103667⋅𝑧8²²+0.00011346⋅𝑧8²⁶ber_+𝜀₂||𝜀₂||<6⋅10⁻⁹,(𝑧)=𝑧⋅−4⋅𝑧8²+14.22222222⋅𝑧8⁶−6.06814810⋅𝑧8¹⁰+0.66047849⋅𝑧8¹⁴−0.02609253⋅𝑧8¹⁸+0.00045957⋅𝑧8²²−0.00000394⋅𝑧+𝜀₃8²⁶bei_||𝜀₃||<2.1⋅10⁻⁸,(𝑧)=𝑧⋅12−10.66666666⋅8𝑧⁴+11.37777772⋅𝑧8⁸−2.31167514⋅𝑧8¹²+0.14677204⋅𝑧8¹⁶−0.00379386⋅𝑧8²⁰+0.00004609⋅𝑧+𝜀₄8²⁴||𝜀₄||<7⋅10⁻⁸.(2.8)

Case  B (𝑧=𝑧(𝑓)=𝑎⋅𝑘=𝑎⋅√2⋅𝜋⋅𝑓⋅𝜇_𝑐⋅𝜎_𝑐∈(8,∞)).

Define firstly the following set of auxiliary functions: 𝛼(𝑧)=−0.3926991⋅𝑗+(0.0110486−0.0110485⋅𝑗)⋅8𝑧−0.0009765⋅𝑗⋅8𝑧²+(−0.0000906−0.0000901⋅𝑗)⋅8𝑧³−0.0000252⋅8𝑧⁴+(−0.0000034+0.0000051⋅𝑗)⋅8𝑧⁵+(0.0000006+0.0000019⋅𝑗)⋅8𝑧⁶𝛼(−𝑧)=−0.3926991⋅𝑗+(−0.0110486+0.0110485⋅𝑗)⋅8,𝑧−0.0009765⋅𝑗⋅8𝑧²+(0.0000906+0.0000901⋅𝑗)⋅8𝑧³−0.0000252⋅8𝑧⁴+(0.0000034−0.0000051⋅𝑗)⋅8𝑧⁵+(0.0000006+0.0000019⋅𝑗)⋅8𝑧⁶𝛽(𝑧)=(0.7071068+0.7071068⋅𝑗)+(−0.0625001−0.0000001⋅𝑗)⋅8,+(−0.0013813+0.0013811⋅𝑗)⋅8𝑧𝑧²+(0.0000005+0.0002452⋅𝑗)⋅8𝑧³+(0.0000346+0.0000338⋅𝑗)⋅8𝑧⁴+(0.0000017−0.0000024⋅𝑗)⋅8𝑧⁵+(0.0000016−0.0000032⋅𝑗)⋅8𝑧⁶𝛽(−𝑧)=(0.7071068+0.7071068⋅𝑗)+(0.0625,001+0.0000001⋅𝑗)⋅8+(−0.0013813+0.0013811⋅𝑗)⋅8𝑧𝑧²+(−0.0000005−0.0002452⋅𝑗)⋅8𝑧³+(0.0000346+0.0000338⋅𝑗)⋅8𝑧⁴+(−0.0000017+0.0000024⋅𝑗)⋅8𝑧⁵+(0.0000016−0.0000032⋅𝑗)⋅8𝑧⁶.(2.9) and, also, define another set of auxiliary functions: 𝑓(𝑧)=𝜋2⋅𝑧⋅exp−1√+2𝑗⋅𝑧+𝛼(−𝑧),𝑔(𝑧)=√2⋅1𝜋⋅𝑧⋅exp1√+2𝑗⋅𝑧+𝛼(𝑧),(2.10) then, the values of Bessel-Kelvin functions can be efficiently calculated [5] by using the relations: ber(𝑧)=Re𝑗ber_𝜋⋅𝑓(𝑧)+𝑔(𝑧),bei(𝑧)=Im𝜋𝑗⋅𝑓(𝑧)+𝑔(𝑧),(𝑧)=Re−𝑗𝜋⋅𝑓(𝑧)⋅𝛽(−𝑧)+𝑔(𝑧)⋅𝛽(𝑧),bei^(𝑧)=Im−𝑗𝜋⋅𝑓(𝑧)⋅𝛽(−𝑧)+𝑔(𝑧)⋅𝛽(𝑧)(2..11)

For the coaxial line with length ℓ (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]: 𝐶^=2𝜋⋅𝜀^𝑑ln(𝑏/𝑎),𝐺^=𝜎^𝑑𝜀_𝑑⋅𝐶^=2𝜋⋅𝜎^𝑑ln(𝑏/𝑎),𝜎^𝑑≪𝜎_𝑐.(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, 𝑅^(𝑓)=𝑅^𝑠(𝑓)2𝜋⋅𝑎1+1𝑏=21𝜋⋅𝜋⋅𝑓⋅𝜇_𝑐𝜎_𝑐⋅1𝐿_𝑎+1𝑏,≈𝐿^𝑒=𝜇^𝑑2𝜋⋅ln𝑎𝑏,𝜇^𝑐≈𝜇_𝑑≈𝜇₀.(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 𝑍₀ of its characteristic impedance 𝑍₀ [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(a)). Let the operating frequency range of this line be 𝑓∈[0,3]  [GHz]. The specific electric conductivity of copper is 𝜎_𝑐≈5.81⋅10⁷  [S/m] and magnetic permeability is 𝜇_𝑐≈𝜇₀. 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^=17.04 [pF/m] and 𝐺^=0.85 [fS/m]. At frequency 𝑓=0 [Hz] the per-unit-length resistance of this line is 𝑅^(0)=2/(𝜎_𝑐⋅𝜋⋅𝑎²)=1.0957 [Ω/m] and its per-unit-length inductance is 𝐿^(0)=(𝜇_𝑑/𝜋)⋅ln(𝑑/𝑎)+𝜇_𝑐/(4⋅𝜋)≈1.575 [μH/m]. At frequency 𝑓=1 [GHz] it is calculated from (2.2) and (2.9)–(2.11): 𝑅^(𝑓)=26.514 [Ω/m] ≫ 𝑅^(0) and 𝐿^(𝑓)≈1.479 [μH/m], and for the current wavelength it is obtained 𝜆≈𝑐₀/[𝑓⋅(𝜀_𝑟)^1/2]≈20 [cm]. In Figures 3, 4, 5, and 6 the variations of 𝑅^(𝑓),𝐿^(𝑓),𝑑𝑅^(𝑓)/𝑑𝑓, and 𝑑𝐿^(𝑓)/𝑑𝑓 are depicted, respectively, in the frequency range 𝑓∈[0,10] [MHz], whereas the variations of these quantities in the frequency range 𝑓∈[0.01,3] [GHz] are depicted in Figures 7, 8, 9, and 10, respectively. Let the frequency spectrum of the signal being transmitted is [𝑓₀−𝐵/2,𝑓₀+𝐵/2] (𝑓₀ is the central frequency of the signal spectrum and 𝐵 the signal bandwith). If the integral of function in Figure 5 taken between 𝑓₀−𝐵/2 and 𝑓₀+𝐵/2 is less than 0.01, then we see from Figure 3 that it may be taken 𝑅^(𝑓)≈𝑅^(𝑓₀), for 𝑓∈[𝑓₀−𝐵/2,𝑓₀+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 𝑓₀−𝐵/2 and 𝑓₀+𝐵/2 is less than 0.1, then we see from Figure 5 that it, also, holds 𝑅^(𝑓)≈𝑅^(𝑓₀), for 𝑓∈[𝑓₀−𝐵/2,𝑓₀+𝐵/2]. And by using Figure 9 we obtain in the most conservative approach that 𝐵≤770 [kHz].

Figure 3 

𝑅(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 4 

𝐿(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 5 

𝑑𝑅(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 6 

𝑑𝐿(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 7 

𝑅(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 8 

𝐿(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 9 

𝑑𝑅(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

Figure 10 

𝑑𝐿(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].

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 𝑍₀ are given by the following relations [1–3]: 𝑐=1√𝐿^⋅𝐶^=1√𝜀_𝑑⋅𝜇_𝑑=𝑐⁰√𝜀_𝑟⋅𝜇_𝑟,𝑐₀=1√𝜀₀⋅𝜇₀≈3⋅10⁸m𝑍₀s,=𝐿^𝐶^=1𝑐⋅𝐶^=𝜇_𝑟𝜀_𝑟⋅1𝜋⋅𝜇₀𝜀₀⋅ln𝑑𝑎≈120⋅𝜇_𝑟𝜀_𝑟⋅ln𝑑𝑎[Ω],(2.14) where 𝜀_𝑟=𝜀_𝑑/𝜀₀ is relative permittivity and 𝜇_𝑟=𝜇_𝑑/𝜇₀ relative permeability of dielectric (𝜀₀≈10⁻⁹/36𝜋 [F/m] is permittivity and 𝜇₀=4𝜋⋅10⁻⁷ [H/m] permeability of vacuum). The characteristic impedance of a lossy transmission line is generally defined as 𝑍₀(𝑗⋅2𝜋⋅𝑓)=[(𝑅^+𝑗⋅2𝜋⋅𝑓⋅𝐿^)/(𝐺^+𝑗⋅2𝜋⋅𝑓⋅𝐶^)]^1/2. In the case considered, on Figures 11 and 12 the variations of 𝑍₀(𝑓)=|𝑍₀(𝑗⋅2𝜋⋅𝑓)| and 𝜁(𝑓)=Arg[𝑍₀(𝑗⋅2𝜋⋅𝑓)] in the frequency range 𝑓∈[0.01,3] [GHz] are respectively depicted. In older telephony applications at lower frequencies, 𝑍₀ 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 𝑍₀=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 𝑍₀=100 [Ω] and the propagation speed approximately 𝑐₀/2. For the coaxial lines the standard 𝑍₀ is 50 or 75 [Ω] and their propagation speed is approximately 2𝑐₀/3. For the printed transmission lines, 𝑍₀ is in the range 100÷150 [Ω], and their propagation speed is approximately 𝑐₀/2 [6].

Figure 11 

𝑍0(𝑓) and 𝑑𝑍0(𝑓)/𝑑𝑓 for the considered two-wire line.

Figure 12 

𝜁(𝑓) and 𝑑𝜁(𝑓)/𝑑𝑓 for the considered two-wire line.

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, 𝜙₁(𝑓)=2⋅𝜋⋅𝑓⋅𝐿^(𝑓)𝑅^(𝑓),𝜙²(𝑓)=2⋅𝜋⋅𝑓⋅𝐶^𝐺^,𝜙₃(𝑓)=12⋅𝜋⋅𝑓⋅√𝐿^(𝑓)⋅𝐶^𝜙₄,(𝑓)=𝐶^⋅𝑅^(𝑓)𝐺^⋅𝐿^(𝑓),𝜙⁵(𝑓)=2⋅𝜋⋅𝑓𝑅^(𝑓)/𝐿^(𝑓)+𝐺^/𝐶^=𝜙¹(𝑓)1+1/𝜙₄(𝑓).(2.15)

Figure 13 

A dimensionless parameter 𝜙1(𝑓) of two-wire line.

Figure 14 

A dimensionless parameter 𝜙2(𝑓) of two-wire line.

Figure 15 

Parameter 𝜙3(𝑓) of two-wire line.

Figure 16 

A dimensionless parameter 𝜙4(𝑓) of two-wire line.

From the numerical data associated with the monotonic functions in Figures 13, 14, and 16 it is obtained 𝜙₁(110.9KHz)≈1, 𝜙₁(1289.9KHz)≈10, 𝜙₁(10MHz)≈32.653≫1, 𝜙₂(110.9KHz)≈13.943⋅10⁹ and 𝜙₄(0)≈1.391⋅10¹⁰. Herefrom and from (2.15) it follows 𝜙₁(𝑓)≈𝜙₅(𝑓). Since in this case the Heaviside’s condition [7] [⇔𝜙₄(𝑓)=1] is not satisfied, distortionless transmission is not possible. Another two functions, Λ(𝑓)=|Γ(𝑗⋅2𝜋⋅𝑓)| and 𝜗(𝑓)=Arg[Γ(𝑗⋅2𝜋⋅𝑓)] [“Γ” 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(10MHz)≈89.123 [deg] and 𝜗 (3 GHz) ≈ 89.953 [deg].

Figure 17 

Magnitude of propagation function (two-wire line).

Figure 18 

Argument of propagation function (two-wire line).

Since Γ(𝑗⋅2𝜋⋅𝑓)=√[𝑅^(𝑓)+𝑗⋅2𝜋⋅𝑓⋅𝐿^(𝑓)]⋅(𝐺^+𝑗⋅2𝜋⋅𝑓⋅𝐶^)=Λ(𝑓)⋅exp[𝑗⋅𝜗(𝑓)] and since in the range 𝑓∈[0.01,3] [GHz] it holds: 𝜙₁(𝑓)≈𝜙₅(𝑓)≫1, 𝜙₂(𝑓)≫1, Λ(𝑓)∈[0.319,94.598] and 𝜗(𝑓)∈[89.123,89.953] [deg], then: Λ(𝑓)≈2𝜋⋅𝑓⋅[𝐿^(𝑓)⋅𝐶^]^1/2=1/𝜙₃(𝑓),𝜗(𝑓)=𝜋/2−𝜒(𝑓)[0<𝜒(𝑓)<𝜋/200] and Γ(𝑗⋅2𝜋⋅𝑓)=Λ(𝑓)⋅exp(𝑗⋅𝜋/2)⋅exp[−𝑗⋅𝜒(𝑓)]=𝑗⋅Λ(𝑓)⋅{cos[𝜒(𝑓)]−𝑗⋅sin[𝜒(𝑓)]}={sin[𝜒(𝑓)]+𝑗⋅cos[𝜒(𝑓)]}/𝜙₃(𝑓), 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, 𝜒(𝑓)=𝜋2−𝜗(𝑓)=𝜋2−21⋅𝑎tan2𝜋⋅𝑓⋅𝐿^(𝑓)/𝑅^(𝑓)+𝐶^/𝐺^≈11−4⋅𝜋²⋅𝑓²𝐿^(𝑓)⋅𝐶^/𝑅^(𝑓)⋅𝐺^2⋅𝑎tan𝑅^(𝑓)/𝐿^(𝑓)+𝐺^/𝐶^2𝜋⋅𝑓≈𝑅^(𝑓)/𝐿^(𝑓)+𝐺^/𝐶^4𝜋⋅𝑓.(2.16)

For the transmission line with length ℓ let us define the functions: 𝜀(𝜔)=𝜒(𝜔/2𝜋)=𝜋/2−𝜗(𝜔/2𝜋) and 𝜃(𝑗⋅𝜔)=Γ(𝑗⋅𝜔)⋅(ℓ−𝑥)=𝐴(𝜔,𝑥)⋅exp[𝑗⋅𝜗(𝜔/2𝜋)]=𝐴(𝜔,𝑥)⋅{sin[𝜀(𝜔)]+𝑗⋅cos[𝜀(𝜔)]}{𝑥∈[0,ℓ]}, where 𝐴(𝜔,𝑥)=|Γ(𝑗⋅𝜔)|⋅(ℓ−𝑥)=Λ(𝜔/2𝜋)⋅(ℓ−𝑥). For this line, in frequency range 𝑓∈[0.01,3] [GHz] we have 𝐴(𝜔,𝑥)≈𝜔⋅[𝐿^(𝜔/2𝜋)⋅𝐶^]^1/2⋅(ℓ−𝑥)=(ℓ−𝑥)/𝜙₃(𝜔/2𝜋) and 0<𝜀(𝜔)<𝜋/200—whereby 𝜃(𝑗⋅𝜔) becomes almost pure imaginary number. We could have obtained this result in a different way. To se that, let us write Γ(𝑗⋅2𝜋⋅𝑓)=√[𝑅^(𝑓)+𝑗⋅2𝜋⋅𝑓⋅𝐿^(𝑓)]⋅(𝐺^+𝑗⋅2𝜋⋅𝑓⋅𝐶^)=𝑎(𝑓)+𝑗⋅𝑏(𝑓), where it holds, 𝑎(𝑓)=⎷12⋅𝐿^(𝑓)⋅𝐶^⎧⎪⎨⎪⎩(2𝜋⋅𝑓)²+𝑅^(𝑓)²𝐿^(𝑓)²⋅(2𝜋⋅𝑓)²+𝐺^2𝐶^2+𝑅^(𝑓)𝐿^(𝑓)⋅𝐺𝐶^−(2𝜋⋅𝑓)²attenuation⎫⎪⎬⎪⎭𝑏(𝑓)=⎷”constant”,12⋅𝐿^(𝑓)⋅𝐶^⎧⎪⎨⎪⎩(2𝜋⋅𝑓)²+𝑅^(𝑓)²𝐿^(𝑓)²⋅(2𝜋⋅𝑓)²+𝐺^2𝐶^2−𝑅^(𝑓)𝐿^(𝑓)⋅𝐺𝐶^+(2𝜋⋅𝑓)²phase⎫⎪⎬⎪⎭”constant”.(2.17) The previous two functions are depicted in Figures 19 and 20 in the frequency range 𝑓∈[0,3] [GHz].

Figure 19 

The attenuation “constant” of two-wire line.

Figure 20 

The phase “constant” of two-wire line.

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 𝜙₁(𝑓)≫1 and 𝜙₂(𝑓)≫1, 𝑎𝑎(𝑓)≈12⎡⎢⎢⎣𝑅^(𝑓)⋅𝐶^𝐿^(𝑓)+𝐺^⋅𝐿^(𝑓)𝐶^𝑎𝑏(𝑓)≈√⎤⎥⎥⎦,𝐿^(𝑓)⋅𝐶^(𝑓)2𝜋⋅𝑓+116𝜋⋅𝑓⋅𝑅^(𝑓)and,also,wehave,𝑍₀𝐿^(𝑓)−𝐺𝐶^²,(𝑗⋅2𝜋⋅𝑓)=𝑅^(𝑓)+𝑗⋅2𝜋⋅𝑓⋅𝐿^(𝑓)𝐺^+𝑗⋅2𝜋⋅𝑓⋅𝐶^≈𝐿^(𝑓)𝐶^⋅1−𝑗⋅14𝜋⋅𝑓𝑅^(𝑓)𝐿^(𝑓)−𝐺𝐶^.(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)]}/𝜙₃(𝜔/2𝜋), as it has been expected.

Figure 21 

Approximation of the attenuation “constant”.

Figure 22 

Approximation of the phase “constant”.

We will now emphasize the importance of function 𝜃=𝜃(𝑠,𝑥)=Γ(𝑠)⋅(ℓ−𝑥){𝑥∈[0,ℓ]} 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 (𝑓₀−𝐵/2,𝑓₀+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, 𝜕²𝑢(𝑡,𝑥)𝜕𝑥²=𝐿^⋅𝐶^⋅𝜕²𝑢(𝑡,𝑥)𝜕𝑡²+𝐿^⋅𝐺^+𝐶^⋅𝑅^⋅𝜕𝑢(𝑡,𝑥)𝜕𝑡+𝑅^𝜕⋅𝐺^⋅𝑢(𝑡,𝑥),₂𝑖(𝑡,𝑥)𝜕𝑥²=𝐿^⋅𝐶^⋅𝜕²𝑖(𝑡,𝑥)𝜕𝑡²+𝐿^⋅𝐺^+𝐶^⋅𝑅^⋅𝜕𝑖(𝑡,𝑥)𝜕𝑡+𝑅^⋅𝐺^⋅𝑖(𝑡,𝑥).(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: 𝑀(𝑠,𝑥)=𝑈(𝑠,𝑥)𝑈(𝑠,0)=∏^∞𝑛=11+[(2Γ⋅(ℓ−𝑥))/((2𝑛−1)⋅𝜋)]²𝑁(𝑠,𝑥)=𝐼(𝑠,𝑥)∏^∞𝑛=11+[(2Γ⋅ℓ)/((2𝑛−1)⋅𝜋)]²,𝐼(𝑠,0)=ℓ−ℓ𝑥⋅∏^∞𝑛=11+[(Γ⋅(ℓ−𝑥))/(𝑛⋅𝜋)]²𝑍(𝑠,𝑥)=𝑈(𝑠,𝑥)∏^∞𝑛=11+((Γ⋅ℓ)/(𝑛⋅𝜋))²,𝐼(𝑠,𝑥)=∏^∞𝑛=11+[(2Γ⋅(ℓ−𝑥))/((2𝑛−1)⋅𝜋)]²Γ⋅(ℓ−𝑥)⋅∏^∞𝑛=11+[(Γ⋅(ℓ−𝑥))/(𝑛⋅𝜋)]²⋅𝑍⁰𝑍_𝐿(𝑠)→∞,(2.20) 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, sinh(𝜃)𝜃=^∞_𝑛=11+𝑛²𝜃⋅²𝜋²,cosh(𝜃)=^∞_𝑛=11+(2𝑛−4⋅1𝜃)²²⋅𝜋².(2.21)

For the open-circuited two-wire line with length ℓ=0.1 [m] in Figures 23 ÷ 26 are depicted for 𝑥∈[0,0.1] [m] and 𝑓∈[0,3] [GHz] the variations of |𝑀(𝑗⋅2𝜋⋅𝑓,𝑥)|, Arg[𝑀(𝑗⋅2𝜋⋅𝑓,𝑥)] [deg], |𝑁(𝑗⋅2𝜋⋅𝑓,𝑥)| and Arg[𝑁(𝑗⋅2𝜋⋅𝑓,𝑥)] [deg], 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), 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 𝑍₀(𝑠) is taken into account [see relation (A.17)].

Figure 23 

The magnitude of the voltage transmittance for two-wire line (ℓ=0.1 [m]) in frequency range 𝑓∈[0,3] [GHz].

Figure 24 

The argument of the voltage transmittance for two-wire line (ℓ=0.1 [m]) in frequency range 𝑓∈[0,3] [GHz].

Figure 25 

The magnitude of the current transmittance for two-wire line (ℓ=0.1 [m]) in frequency range 𝑓∈[0,3] [GHz].

Figure 26 

The argument of the current transmittance for two-wire line (ℓ=0.1 [m]) in frequency range 𝑓∈[0,3] [GHz].

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 ℓ≤ℓ₀<𝜋/{2⋅|[Γ(𝑗⋅2𝜋⋅𝑓)|_max}≈16.6 [mm] {⇔|𝜃|_max<𝜋/2} and assume, say, ℓ₀=16 [mm]. Recall that =𝜃(𝑗⋅𝜔,𝑥)=Γ(𝑗⋅𝜔)⋅(ℓ−𝑥)=|𝜃(𝑗⋅𝜔,𝑥)|⋅exp{𝑗⋅arg[Γ(𝑗⋅𝜔)]}{𝑥∈[0,ℓ]}, then take (2.21) and write sinh(𝜃)𝜃=^∞_𝑛=11+𝑛²𝜃⋅²𝜋²=exp^∞_𝑛=1ln1+𝜃²=exp𝜃₂𝑛²⋅𝜋²𝜋²⋅^∞_𝑛=11𝑛²−𝜃⁴2⋅𝜋⁴⋅^∞_𝑛=11𝑛⁴+𝜃⁶3⋅𝜋⁶⋅^∞_𝑛=11𝑛⁶−𝜃⁸4⋅𝜋⁸⋅^∞_𝑛=11𝑛⁸+𝜃¹⁰5⋅𝜋¹⁰⋅^∞_𝑛=11𝑛¹⁰=exp𝜃₂±….cosh(𝜃)=_∞6−1𝜃8⁴0+28𝜃3⁶5−37𝜃8⁸00+46𝜃77¹⁰75±⋯,_𝑛=11+(2𝑛−4⋅1𝜃)²²⋅𝜋²=exp^∞_𝑛=1ln1+4⋅𝜃²=exp4⋅𝜃₂(2𝑛−1)²⋅𝜋²𝜋²⋅^∞_𝑛=11(2𝑛−1)²−8⋅𝜃⁴𝜋⁴⋅^∞_𝑛=11(2𝑛−1)⁴+64⋅𝜃⁶3⋅𝜋⁶⋅^∞_𝑛=11(2𝑛−1)⁶−64⋅𝜃⁸𝜋⁸⋅^∞_𝑛=11(2𝑛−1)⁸+1024⋅𝜃¹⁰5⋅𝜋¹⁰⋅^∞_𝑛=11(2𝑛−1)¹⁰=exp𝜃₂±⋯2−1𝜃2⁴+4𝜃5⁶−1275⋅2𝜃0⁸+3114⋅1𝜃75¹⁰±⋯.(2.22)

The following finite sums of infinite series [9] have been exploited in (2.22), 𝐴_2𝑝=^∞_𝑛=11𝑛^2𝑝𝑝=1,5,𝐴₂=𝜋²6,𝐴⁴=𝜋⁴90,𝐴⁶=𝜋⁶945,𝐴⁸=𝜋⁸𝐴9450₁₀,=5⋅2⁸⋅𝜋¹⁰𝐵33⋅10!_2𝑝,=^∞_𝑛=11(2𝑛−1)^2𝑝𝑝=1,5,𝐵₂=𝜋²8,𝐵⁴=𝜋⁴96,𝐵⁶=𝜋⁶960,𝐵⁸=17⋅𝜋⁸𝐵161280₁₀,=31⋅𝜋¹⁰35⋅81⋅2¹⁰.(2.23)

The infinite complex series (2.22) in almost pure imaginary 𝜃 are convergent for ℓ≤ℓ₀,𝑥∈[0,ℓ] and 𝑓∈[0,3] [GHz]. If ℓ is sufficiently less than ℓ₀ and 𝑥∈[0,ℓ], then by retaining only the first five 𝜃 terms in (2.22) the following small-error approximations are produced sinh(𝜃)≈sinh_𝐴(𝜃)=𝜃⋅exp𝜃²cosh(𝜃)≈cosh_𝐴6−1𝜃8⁴0+28𝜃3⁶5−37𝜃8⁸00+46𝜃77¹⁰75,(𝜃)=exp𝜃²2−1𝜃2⁴+4𝜃5⁶−1275⋅2𝜃0⁸+3114⋅1𝜃75¹⁰.(2.24)

For the open-circuited two-wire line with ℓ=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:ER₁(𝑓,𝑥)={|[cosh_𝐴𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)/cosh_𝐴(𝑗⋅2𝜋⋅𝑓,0)]/[cosh𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)/cosh(𝑗⋅2𝜋⋅𝑓,0)]|−1}⋅100;(ii) the voltage-transmittance phase approximation absolute error:ER₂(𝑓,𝑥)=arg[cosh_𝐴𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)⋅cosh(𝑗⋅2𝜋⋅𝑓,0)/cosh_𝐴(𝑗⋅2𝜋⋅𝑓,0)⋅cosh𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)];(iii) the current-transmittance magnitude approximation percentage error:ER₃(𝑓,𝑥)={|[sinh_𝐴𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)/sinh_𝐴(𝑗⋅2𝜋⋅𝑓,0)]/[sinh𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)/sinh(𝑗⋅2𝜋⋅𝑓,0)]|−1}⋅100;(iv) the current-transmittance phase approximation absolute error:ER₄(𝑓,𝑥)=arg[sinh_𝐴𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)⋅sinh(𝑗⋅2𝜋⋅𝑓,0)/sinh_𝐴(𝑗⋅2𝜋⋅𝑓,0)⋅sinh𝜃(𝑗⋅2𝜋⋅𝑓,𝑥)].

Figure 27 

Voltage-transmittance magnitude approximation percentage error ER1(𝑓,𝑥).

Figure 28 

Voltage-transmittance phase approximation absolute error ER2(𝑓,𝑥).

Figure 29 

Current-transmittance magnitude approximation percentage error ER3(𝑓,𝑥).

Figure 30 

Current-transmittance phase approximation absolute error ER4(𝑓,𝑥).

It can be observed in Figures 27 ÷ 30 that in the given range of 𝑓 and 𝑥, the errors ER₁ and ER₃ are negative (|ER₁|<0.06% and |ER₃|<10⁻⁵%), whereas the errors ER₂ and ER₄ are positive (ER₂<3.36⋅10⁻⁴ [deg] and ER₄<5.75⋅10⁻⁸ [deg]). The upper limit of |ER₃| is lower than of |ER₁| and the upper limit of ER₄ is lower than of ER₂.

If |𝜃| is sufficiently small (|𝜃|≪1), from (2.24) further approximations are obtained:(i)sinh(𝜃)≈=s𝜃in⋅hex_Ap(𝜃)𝜃₂≈𝜃⋅1+𝜃6₂−1𝜃8⁴0+28𝜃3⁶5−37𝜃8⁸00+46𝜃77¹⁰75=𝜃+𝜃₃6−1𝜃8⁴0+28𝜃3⁶5−37𝜃8⁸00+46𝜃77¹⁰75≈𝜃+𝜃6₃−1𝜃8⁵0+28𝜃3⁷5−37𝜃8⁹00+46𝜃77¹¹756,(2.25)which partly resembles to Maclaurin’s expansion of sinh(θ), that is, sinh(𝜃)=𝜃+𝜃³/3!+𝜃⁵/5!+…,(ii)cosh(𝜃)≈cosh_𝐴(𝜃)=exp𝜃²≈1+𝜃₂2−1𝜃2⁴+4𝜃5⁶−1275⋅2𝜃0⁸+3114⋅1𝜃75¹⁰2−1𝜃2⁴+4𝜃5⁶−1275⋅2𝜃0⁸+3114⋅1𝜃75¹⁰≈1+𝜃2²,(2.26)which, partly resembles to Maclaurin’s expansion of cosh(θ), that is, cosh(𝜃)=1+𝜃²/2!+𝜃⁴/4!+….(iii) combining (i) and (ii) it follows that tanh𝜃2≈tanh^𝐴𝜃2=𝜃⋅𝜃²+246⋅𝜃²+8.(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 and tanh_𝐴(𝜃/2)/(𝜃/2), obtained when |𝜃|≪1. For two-wire line with length ℓ=1 [mm], in Figures 31, 32, 33, and 34, the magnitude approximation absolute error ER₅(𝑓,𝑥)=|(𝜃+𝜃³/6)|−|sinh(𝜃)| and three percentage magnitude approximation errors: ER₆(𝑓,𝑥)={|{𝜃⋅(𝜃²+24)/[6⋅(𝜃²+8)]}/tanh(𝜃/2)|−1}⋅100, ER₇(𝑓,𝑥)=[|𝜃/sinh(𝜃)|−1]⋅100 and ER₈(𝑓,𝑥)={|(𝜃/2)/tanh(𝜃/2)|−1}⋅100, are depicted on grid(x) × grid(f) = 40 × 60 in the frequency range 𝑓∈[0,3] [GHz] and range of 𝑥∈[0,1] [mm]. Obviously, all these errors can be kept arbitrarily small in magnitude in the entire frequency range 𝑓∈[0,3] [GHz] if sufficiently small step of uniform line segmentation is applied. The key action in achieving the previous goals is providing the maximum of ℓ₀ to be much less than 1/|Γ(𝑗⋅2𝜋⋅𝑓_max)(⇔|𝜃_max|≪1). For example, from the numerical data associated with Figure 17, it can be calculated that ℓ₀ 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 ℓ₀, which is at least ten times less than 1/|Γ(𝑗⋅2𝜋⋅𝑓_max)|.

Figure 31 

Magnitude approximation absolute error ER5(𝑓,𝑥).

Figure 32 

Percentage magnitude approximation error ER6(𝑓,𝑥).

Figure 33 

Percentage magnitude approximation error ER7(𝑓,𝑥).

Figure 34 

Percentage magnitude approximation error ER8(𝑓,𝑥).

The equations (A.15) and (A.16) offer an opportunity to view on a transmission line segment with length ℓ₀ as on a linear two-port network (Figure 35(a)) with boundary conditions 𝑈(𝑠,0) and 𝐼(𝑠,0) at the input and 𝑈(𝑠,ℓ₀) and 𝐼(𝑠,ℓ₀) at the output. The chain-matrix 𝐅 of this network reads ⎡⎢⎢⎣𝑈𝐼((𝑠𝑠,,00))⎤⎥⎥⎦=𝐅⋅⎡⎢⎢⎣𝑈𝐼𝑠𝑠,,ℓℓ₀₀⎤⎥⎥⎦,𝐅=⎡⎢⎢⎢⎣coshΓ⋅ℓ⁰sinhΓ⋅ℓ₀𝑍₀⋅sinhΓ⋅ℓ₀𝑍₀coshΓ⋅ℓ₀⎤⎥⎥⎥⎦.(2.28)

(a)

(b)

(c)

Figure 35 

(a) Linear network model of a short transmission line segment and (b) its linear network structure.

Denote 𝑍(𝑠)=(𝑅^+𝐿^⋅𝑠)⋅ℓ₀, 𝑌(𝑠)=(𝐺^+𝐶^⋅𝑠)⋅ℓ₀ and 𝜃(𝑠,0)=Γ(𝑠)⋅ℓ₀. In study of short transmission lines it is found convenient to replace them, either with nominally equivalent 𝑇 networks (Figure 35(b)) or with nominally equivalent Π networks (Figure 35(c)) [7], whose immitances are given as follows 𝑍^=𝑍⋅tanh[𝜃(𝑠,0)/2]𝜃(𝑠,0)/2,𝑌^=𝑌⋅sinh[𝜃(𝑠,0)]𝑍𝜃(𝑠,0)_,=𝑍⋅sinh[𝜃(𝑠,0)]𝜃(𝑠,0),𝑌^=𝑌⋅tanh[𝜃(𝑠,0)/2]𝜃(𝑠,0)/2.(2.29) The aforementioned criterion for selection of ℓ₀ relies on attempt to find convenient 𝜃(𝑠,0)=Γ(𝑠)⋅ℓ₀ 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 ℓ₀ 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 Γ(𝑗⋅𝜔)=√(𝑅^+𝑗⋅𝜔⋅𝐿^)⋅(𝐺^+𝑗⋅𝜔⋅𝐶^)=|Γ(𝑗⋅𝜔)|⋅exp{𝑗⋅arg[Γ(𝑗⋅𝜔)]}(𝜔=2𝜋⋅𝑓), in frequency range 𝑓∈[0.01,3] [GHz] it holds(a)𝜔⋅𝐿^/𝑅^∧𝜔/(𝑅^/𝐿^+𝐺^/𝐶^)∈[32.652,609.312]≫1, 𝜔⋅𝐶^/𝐺^∈[1.257⋅10¹²,3.771⋅10¹⁴]≫1,(b)|Γ(𝑗⋅𝜔)|≈𝜔⋅(𝐿^⋅𝐶^)^1/2∈[0.319,94.598], arg[Γ(𝑗⋅𝜔)]∈[89.122,89.952] [deg],(c)arg[Γ(𝑗⋅𝜔)]=𝜋/2−𝜀(𝜔), where, 𝜀(𝜔)=𝜋2−arg[Γ(𝑗⋅𝜔)]=𝜋2−21⋅𝑎tan𝜔⋅𝐿^/𝑅^+𝜔⋅𝐶^/𝐺^≈11−𝐿^⋅𝐶^/𝑅^⋅𝐺^⋅𝜔²2⋅𝑎tan𝑅^/𝐿^+𝐺^/𝐶^𝜔≈𝑅^/𝐿^+𝐺^/𝐶^2⋅𝜔.(2.30)(d)Γ(𝑗⋅𝜔)=|Γ(𝑗⋅𝜔)|⋅{sin[𝜀(𝜔)]+𝑗⋅cos[𝜀(𝜔)]}, 𝜀(𝜔)∈(0,𝜋/200) and 𝜃(𝑗⋅𝜔,𝑥)=Γ(𝑗⋅𝜔)⋅(ℓ₀−𝑥)=|𝜃(𝑗⋅𝜔,𝑥)|⋅{sin[𝜀(𝜔)]+𝑗⋅cos[𝜀(𝜔)]}. Observe that 𝜃(𝑗⋅𝜔,𝑥) is produced as almost pure imaginary number.(e) Since 𝑥∈[0,ℓ₀] and |𝜃(𝑗⋅𝜔,𝑥)|≈𝜔⋅(𝐿^⋅𝐶^)^1/2⋅(ℓ₀−𝑥)=𝐴(𝜔,𝑥), then by selecting ℓ₀ 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: sinh{𝐴(𝜔,𝑥)⋅sin[𝜀(𝜔)]}⋅cos{𝐴(𝜔,𝑥)⋅cos[𝜀(𝜔)]}+𝑗⋅𝐴co(s𝜔h,𝑥{)𝐴⋅({𝜔si,n𝑥)[⋅𝜀s(i𝜔n)][𝜀+(𝑗𝜔⋅)]c}os⋅s[in𝜀({𝜔𝐴)](}𝜔,𝑥)⋅cos[𝜀(𝜔)]}𝐴(𝜔,𝑥)⋅{sin[𝜀(𝜔)]+𝑗⋅cos[𝜀(𝜔)]},(2.31) 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: 𝑅(𝜔,𝑥)=sin²[𝜀(𝜔)]⋅^∞_𝑘=0𝐴^2𝑘(𝜔,𝑥)⋅sin^2𝑘[𝜀(𝜔)](2𝑘+1)!⋅^∞_𝑚=0(−1)^𝑚𝐴^2𝑚(𝜔,𝑥)⋅cos^2𝑚[𝜀(𝜔)]+cos₂(2𝑚)![𝜀(𝜔)]⋅^∞_𝑛=0𝐴^2𝑛(𝜔,𝑥)⋅sin^2𝑛[𝜀(𝜔)](2𝑛)!⋅^∞_𝑝=0(−1)^𝑝𝐴^2𝑝(𝜔,𝑥)⋅cos^2𝑝[𝜀(𝜔)]𝑄(𝜔,𝑥)=s⋅in[𝜀(_∞𝜔)]⋅cos[𝜀(𝜔)](2𝑝+1)!,_𝑘=0𝐴^2𝑘(𝜔,𝑥)⋅sin^2𝑘[𝜀(𝜔)](2𝑘)!⋅^∞_𝑚=0(−1)^𝑚𝐴^2𝑚(𝜔,𝑥)⋅cos^2𝑚[𝜀(𝜔)]−(2𝑚+1)!_∞_𝑛=0𝐴^2𝑛(𝜔,𝑥)⋅sin^2𝑛[𝜀(𝜔)](2𝑛+1)!⋅^∞_𝑝=0(−1)^𝑝𝐴^2𝑝(𝜔,𝑥)⋅cos^2𝑝[𝜀(𝜔)](2𝑝)!.(2.32) We must always bear in mind that 𝜀(𝜔) is small [(c) and (2.30)] and that 𝐴(𝜔,𝑥)=𝜔⋅(𝐿^⋅𝐶^)^1/2⋅(ℓ₀−𝑥) can be made arbitrarily small by selecting ℓ₀ such that 𝐴(𝜔,ℓ)|_max=[2𝜋⋅𝑓⋅(𝐿^⋅𝐶^)^1/2]|_max⋅ℓ₀≪1 or equivalently ℓ₀≪1/|Γ(𝑗⋅2𝜋⋅𝑓_max)|. 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,ℓ₀] and for all 𝜔 corresponding to 𝑓 from the specified frequency range: 𝑅(𝜔,𝑥)≈1−𝐴²(𝜔,𝑥)6⋅cos[2⋅𝜀(𝜔)]−𝐴⁴(𝜔,𝑥)48⋅sin²[2⋅𝜀(𝜔)]≈1−𝜔²⋅𝐿′⋅𝐶′⋅ℓ₀−𝑥²𝑄(𝜔,𝑥)≈𝐴₂6,(𝜔,𝑥)6⋅sin[2⋅𝜀(𝜔)]≈𝜔²⋅𝐿′⋅𝐶^⋅ℓ₀−𝑥²≈𝜔6⋅sin[2⋅𝜀(𝜔)₂]⋅𝐿^⋅𝐶^⋅ℓ₀−𝑥²3⋅𝜀(𝜔)=𝜔6⋅𝐿^⋅𝐶^⋅ℓ₀−𝑥²⋅𝑅^𝐿^+𝐺^=𝜔𝐶^6⋅ℓ⁰−𝑥²⋅𝑅′⋅𝐶′+𝐺′⋅𝐿^≪1.(2.33)

For 𝑥=0, from (2.31) and (2.33) it finally follows: sinh[𝜃(𝑗⋅𝜔,0)]𝜃(𝑗⋅𝜔,0)=𝑅(𝜔,0)+𝑗⋅𝑄(𝜔,0)≈1+𝑗⋅𝜔6⋅𝑅^⋅𝐶^+𝐺^⋅𝐿^⋅ℓ²⁰−𝜔²⋅𝐿′⋅𝐶′⋅ℓ²⁰sinh[𝜃(𝑗⋅𝜔,0)]6,(2.34)𝜃(𝑗⋅𝜔,0)^{𝑗⋅𝜔→𝑠(−𝜔2→𝑠2)}−−−−−−−−−−−→_{Byanalyticcontinuation}sinh[𝜃(𝑠,0)]𝜃(𝑠,0)≈1+6𝑠⋅𝑅^⋅𝐶^+𝐺^⋅𝐿^⋅ℓ²⁰+𝑠²⋅𝐿^⋅𝐶^⋅ℓ²⁰6(2..35)

Now, by using (2.29) and (2.35) we may generate the immitances Y’ and Z’’ (Figures 35(b) and 35(c)), 𝑌^(𝑠)=𝑌(𝑠)⋅sinh[𝜃(𝑠,0)]𝜃(𝑠,0)=𝐺^+𝐶^⋅𝑠⋅ℓ₀⋅1+𝑠6⋅𝑅^⋅𝐶^+𝐺^⋅𝐿^⋅ℓ²⁰+𝑠²⋅𝐿^⋅𝐶^⋅ℓ²⁰𝑍6(2.,36)_(𝑠)=𝑍(𝑠)⋅sinh[𝜃(𝑠,0)]𝜃(𝑠,0)=𝑅^+𝐿^⋅𝑠⋅ℓ₀⋅1+𝑠6⋅𝑅^⋅𝐶^+𝐺^⋅𝐿^⋅ℓ²⁰+𝑠²⋅𝐿^⋅𝐶^⋅ℓ²⁰6(2.3,7) which are not realizable by lumped RLC networks, since they are not the positive real functions in complex frequency 𝑠 [10], except when ℓ₀→0. Putting it in other words we may say that if in the whole operating frequency range practically hold the three conditions: 𝜔⋅𝐿^/𝑅^≫1, 𝜔⋅𝐶^/𝐺^≫1, and ℓ₀⋅𝜔⋅√𝐿^⋅𝐶^≪1, 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)]: 𝑌^(𝑠)=𝐺^+𝐶^⋅𝑠⋅ℓ₀,𝑍^(𝑠)=𝑅^+𝐿^⋅𝑠⋅ℓ₀.(2.38) 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: 𝜔⋅𝐿^/𝑅^≫1, 𝜔⋅𝐶^/𝐺^≫1 and ℓ₀⋅𝜔⋅√𝐿^⋅𝐶^≪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). 𝑍^(𝑠)=𝑅^+𝐿^⋅𝑠⋅ℓ₀,𝑌^(𝑠)=𝐺^+𝐶^⋅𝑠⋅ℓ₀.(2.39)

3. Approximation of Two-Wire Line by Uniform RLCG Ladder and the Simulation Results

Let us consider a short two-wire line with length ℓ=30 [mm] (or a longone partitioned in sections of length ℓ). Assume that the bandwith of the signal being transmitted is 𝐵<770 [kHz] and that its central frequency is 𝑓₀=1 [GHz]. Then, make the graph of function 𝑙(𝑓)=𝜙₃(𝑓)=1/2𝜋⋅𝑓⋅[𝐿^(𝑓)⋅𝐶^)^1/2]≈1/|Γ(𝑗⋅2𝜋⋅𝑓)| (Figure 36) and from its associated numerical data find that 𝑙(10⁹)≈31.7 [mm]. Let the maximum length ℓ₀ of line segments be selected to satisfy the condition ℓ₀≤𝑙(10⁹)/10≈3.17 [mm]. Finally, assume ℓ₀=3 [mm] and calculate the number of cells 𝑁=ℓ/ℓ₀=10 in uniform RLCG ladder purporting to represent the transmission line with length ℓ in frequency range 𝑓∈[𝑓₀−𝐵/2,𝑓₀+𝐵/2].

Figure 36 

The function 𝑙(𝑓)=𝜙3(𝑓) of the considered two-wire line (determination of ℓ0).

Assume 𝛿𝑥=ℓ₀ and calculate the parameters of uniform lumped RLCG network representing the line segments of length ℓ₀ (see Figure 2). If the overall short-line parameters are 𝑅=𝑅^⋅ℓ-resistance, 𝐿=𝐿^⋅ℓ-inductance, 𝐶=𝐶^⋅ℓ-capacitance, and 𝐺=𝐺^{Γ(𝑗⋅2𝜋⋅𝑓)}⋅ℓ-conductance, then the lumped RLCG network parameters: 𝑅^⋅𝛿𝑥/2=𝑅/20, 𝐿^⋅𝛿𝑥/2=𝐿/20, 𝐶^⋅𝛿𝑥=𝐶/10, and 𝐺^⋅𝛿𝑥=𝐺/10, calculated at frequency 𝑓₀=1 [GHz] according to (2.1), (2.2), ((2.8)–(2.11)), are given in Table 1. The RLCG network representing the short line with length ℓ 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 𝑓₀=1 [GHz] the quantity 2𝜋⋅𝑓₀⋅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).

Table 1 

Two-wire copper-polyethylene transmission line (central frequency 𝑓₀=1 [GHz]).

Figure 37 

Electrical network model of two-wire line with length ℓ=3 [cm] in the frequency range 𝑓∈[𝑓0−𝐵/2,𝑓0+𝐵/2].

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 𝑓₀=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)⋅ℓ₀/2(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.

Table 2 

The steady-state results of exact analysis and simulation for the open-circuited two-wire line in 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 𝑓₀=1 [GHz], amplitude 𝐸_𝑚=10 [V], and zero initial phase in two-wire line with length ℓ=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 HB₁ on Figure 38(a). By cascading, say, 30 level 1 blocks: HB₁/1, HB₁/2,…, and HB₁/30, we produce a level 2 hierarchical block HB₂ on Figure 38(b), which represents a two-wire line with length 90 [cm]. By cascading 5 level 2 blocks: HB₂/1, HB₂/2,…, and HB₂/5, we make a level 3 hierarchical block HB₃ 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.

(a)

(b)

(c)

Figure 38 

The structure of the three types of hierarchical blocks HB1, HB2, and HB3 used in PSPICE simulation.

From the data associated with functions depicted in Figures 12 and 13 we can easily: (i) obtain 𝑍₀(𝑗⋅2𝜋⋅𝑓₀)=294.67280−𝑗⋅0.42017 [Ω] and (ii) notice that variations of |𝑍₀(𝑗⋅2𝜋⋅𝑓)|] and Arg[𝑍₀(𝑗⋅2𝜋⋅𝑓)] are very small in the frequency range (𝑓₀−𝐵/2,𝑓₀+𝐵/2).

Now, suppose that the block HB₂/5 in Figure 38(c) is terminated with impedance, 𝑍_𝐿(𝑗⋅2𝜋⋅𝑓₀)=𝑅_𝐿+1/(𝑗⋅2𝜋⋅𝑓₀⋅𝐶_𝐿)=𝑍₀(𝑗⋅2𝜋⋅𝑓₀) and find 𝑅_𝐿=294.6728 [Ω] and 𝐶_𝐿=378.7781 [pF]. From (A.17) it follows that 𝑀(𝑗⋅2𝜋⋅𝑓₀,𝑥)=𝑁(𝑗⋅2𝜋⋅𝑓₀,𝑥)=exp[−𝑥⋅Γ(𝑗⋅2𝜋⋅𝑓₀)] = exp[−𝑥⋅𝑎(𝑓₀)]⋅{cos[𝑥⋅𝑏(𝑓₀)]−𝑗⋅sin[𝑥⋅𝑏(𝑓₀)]}, and then at any place 𝑥 on the line we obtain the exact values of signal amplitude 𝐸_𝑚⋅exp[−𝑥⋅𝑎(𝑓₀)] [V], phase delay 𝜑(𝑥)=𝑥⋅𝑏(𝑓₀) [rad] and time delay 𝜏(𝑥)=𝑥⋅𝑏(𝑓₀)/(2𝜋⋅𝑓₀) [s]. In Figure 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 ℓ=4.5 [m] (Figure 38(c)), we will compare the exact values of 𝐸_𝑚⋅exp[−𝑥⋅𝑎(𝑓₀)], 𝜑(𝑥) and 𝜏(𝑥) at the points on line 𝑥=𝑥_𝑘=𝑘⋅ℓ₀/2  (𝑘=1,3000) (ℓ₀=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⋅ℓ₀=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)):𝐸_𝑚=10[V],𝑓₀=1[GHz],𝑎𝑓₀=0.0449901611𝑒=𝐸_𝑚m,𝑏𝑓⁰=31.551675282m1,⋅sin2𝜋𝑈⋅𝑓₀⋅𝑡,𝑢_𝑖=𝑈_𝑖𝑚⋅sin2𝜋⋅𝑓₀⋅𝑡−𝜑_𝑖=𝑈_𝑖𝑚⋅sin2𝜋⋅𝑓₀⋅𝑡−𝜏_𝑖_𝑖𝑚,𝜑_𝑖=𝐸_𝑚⋅exp−𝑥_600⋅𝑖⋅𝑎𝑓₀=10⋅(0.960317668)^𝑖[𝑉],𝜏=𝜑𝑥_600⋅𝑖=300⋅𝑖⋅ℓ₀⋅𝑏𝑓₀=28.39650775⋅𝑖[rad_𝑖],=𝜏𝑥_600⋅𝑖=𝜑^𝑖2𝜋⋅𝑓₀=4.519444575⋅𝑖[ns],𝑖=1,5.(3.1)

Table 3 

The steady-state results of exact analysis and simulation for RLCG lader in Figure 38(c).

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 ℓ₀ 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 ℓ=4.5 [m] are depicted.

Figure 39 

Transient analysis of RLCG ladder in Figure 38(c) and “measuring” of delay times 𝜏1÷𝜏5 [see, also, (3.1)].

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.(ii)Delayed and slightly attenuated signals 𝑢₁÷𝑢₅ with frequency 𝑓₀ and smooth transient intervals (Figure 39) are produced by uniform RLCG ladder terminated with characteristic impedance (The uniform ladder in Figure 38(c) consists of 1500 identical cells with impedances 𝑍₁(𝑗⋅2𝜋⋅𝑓₀)=𝑅/20+𝑗⋅2𝜋⋅𝑓₀⋅𝐿/20 and 𝑍₂(𝑗⋅2𝜋⋅𝑓₀)=[𝑌₂(2𝜋⋅𝑓₀)]⁻¹, where 𝑌₂(2𝜋⋅𝑓₀) = 𝐺/10+𝑗⋅2𝜋⋅𝑓₀⋅𝐶/10 (see Figure 37 and Table 1). At frequency 𝑓₀ the characteristic impedance [8] of ladder is 𝑍_𝑐(𝑗⋅2𝜋⋅𝑓) = {𝑍₁(𝑗⋅2𝜋⋅𝑓₀)⋅[𝑍₁(𝑗⋅2𝜋⋅𝑓₀)+2⋅𝑍₂(𝑗⋅2𝜋⋅𝑓₀)]}^1/2≈𝑍₀(𝑗·2𝜋·𝑓₀) (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 ≈ exp{−𝑛⋅acosh[1+𝑍₁(𝑗⋅2𝜋⋅𝑓₀)/𝑍₂(𝑗⋅2𝜋⋅𝑓₀)]}⋅𝐸(𝑗⋅2𝜋⋅𝑓₀)[𝐸(𝑗⋅2𝜋⋅𝑓₀) is the complex representative of 𝑒(𝑡)], and (ii) the time-delay at that place with respect to excitation is, Im{𝑛⋅acosh[1+𝑍₁(𝑗⋅2𝜋⋅𝑓₀)/𝑍₂(𝑗⋅2𝜋⋅𝑓₀)]/(2𝜋⋅𝑓₀)}≈15.07⋅𝑛 [ps].) of the line at frequency 𝑓₀ (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 frequency-band signals, by using RLCG ladders as approximate network models of these 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 ℓ=4.5 [m] and excitation 𝑒(𝑡) as in Figure 38(c) [see, also, (3.1)], terminated with its characteristic impedance at frequency 𝑓₀. 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=ℓ/[2𝜋⋅𝑓₀/𝑏(𝑓₀)]≈22.5972 [ns] is the perturbation propagation time from the line sending end to its receiving end, and 2𝜋⋅𝑓₀/𝑏(𝑓₀)≈𝑐₀/(𝜀_𝑟)^1/2≈1.9913⋅10⁸ [m/s] is the propagation velocity. If we introduce substitution 𝑡=𝜏/𝐴(𝑓₀){𝐴(𝑓₀)=1/[𝐿^(𝑓₀)⋅𝐶^]^1/2 = 1.9913⋅10⁸[m/s]} into the telegraph equation in 𝑢(𝑡,𝑥) obtained from (2.19): 𝜕²𝑢(𝑡,𝑥)𝜕𝑥²=𝐿^𝑓₀⋅𝐶^⋅𝜕²𝑢(𝑡,𝑥)𝜕𝑡²+𝐿^𝑓₀⋅𝐺^+𝐶^⋅𝑅^𝑓₀⋅𝜕𝑢(𝑡,𝑥)𝜕𝑡+𝑅^𝑓₀⋅𝐺^⋅𝑢(𝑡,𝑥),(3.2) we obtain the second-order, hyperbolic, partial differential equation (PDE) equivalent to (3.2): 𝜕²𝑢(𝜏,𝑥)𝜕𝑥²=𝜕²𝑢(𝜏,𝑥)𝜕𝜏²+𝐶𝑓₀⋅𝜕𝑢(𝜏,𝑥)𝜕𝜏+𝐵𝑓⁰⋅𝑢(𝑡,𝑥)=𝑢𝜏𝑢(𝜏,𝑥),𝐴𝑓₀,𝑥=𝐴𝑓𝑢(𝜏,𝑥),𝑥∈[0,ℓ₀],=1𝐿′𝑓₀⋅𝐶′m𝐶𝑓₀s,𝐵𝑓⁰=𝑅′𝑓₀⋅𝐺′m1₂,=𝑅′𝑓₀⋅𝐶′𝐿′𝑓₀+𝐺′⋅𝐿′𝑓₀𝐶^m1,(3.3) where 𝜏∈[0,𝐴(𝑓₀)⋅𝑇] [m], that is, 𝜏∈[0,4.5] [m]. If 𝑢(𝜏,𝑥)={exp[−(1/2)⋅𝐶(𝑓₀)⋅𝜏]}⋅𝑢(𝜏,𝑥), then from (3.3) the following PDE of Klein-Gordon’s type [12] is obtained {𝜏,𝑥∈[0,4.5] [m]}: 𝜕²𝑢(𝜏,𝑥)𝜕𝑥²=𝜕²𝑢(𝜏,𝑥)𝜕𝜏²𝐷𝑓+𝐷𝑓₀⋅𝑢(𝜏,𝑥),₀=14⋅⎛⎜⎜⎝𝑅′(𝑓⁰)⋅𝐶^𝐿^𝑓₀−𝐺^⋅𝐿^(𝑓₀)𝐶^⎞⎟⎟⎠²=2.0241⋅10⁻³1m²,(3.4) which would be a pure wave equation if “diffusion” term 𝐷(𝑓₀)⋅𝑢(𝜏,𝑥) was not present, or in other words, if the line parameters satisfy the Heaviside’s condition of distortionless at frequency 𝑓₀.

Let we define, also, the following auxiliary function in 𝜏 and 𝑥: 𝑣(𝜏,𝑥)=𝜕𝑢(𝜏,𝑥)𝜕𝜏=12⋅𝐶𝑓⁰⋅𝑢(𝑡,𝑥)+𝐴1𝑓₀⋅𝜕𝑢𝜕(𝑡𝑡,𝑥)⋅exp𝐸𝑓⁰⋅𝑡|||||_{𝑡=𝜏/𝐴(𝑓0)}𝐸𝑓₀,=12⋅𝑅^𝑓₀𝐿^𝑓₀+𝐺𝐶^=8.959[MHz].(3.5)

If the excitation is 𝑒(𝑡)=𝐸_𝑚⋅sin(2𝜋⋅𝑓₀⋅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: 𝜕𝑢(𝜏,𝑥)𝜕𝜏=𝑣(𝜏,𝑥)∧𝜕𝑣(𝜏,𝑥)𝜕𝜏=𝜕²𝑢(𝜏,𝑥)𝜕𝑥²+𝐷𝑓₀⋅𝑢(𝜏,𝑥)⟵ThesystemofPDEs,𝑢(0,𝑥)==‖‖‖‖‖‖‖‖‖‖‖𝑒(010)oifth𝑥er>w0ise∧𝑣0(0,𝑥)if𝑥>02⋅𝐶𝑓⁰⋅𝑒(0)+2𝜋⋅𝑓⁰⋅𝐸_𝑚⋅cos(𝜑)𝐴𝑓₀otherwise⟵theinitialconditions,𝑢(𝜏,0)=exp12⋅𝐶𝑓⁰⋅𝜏⋅𝑒𝐴𝜏𝑓₀∧𝑢(𝑡,𝑥)=exp−𝐸𝑓₀𝑢(𝜏,ℓ)=0⟵theboundaryconditions,⋅⋅𝑡𝑢𝐴𝑓₀⋅𝑡,𝑥⟵Solutionof(3.2)ininterval𝑡∈[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⋅ℓ/5 [m] (𝑘=1,5) are depicted. Certainly the solutions of (3.2) for any 𝑥∈[0,ℓ] can be produced easily, also by using the relations (3.4)–(3.6).

Figure 40 

Transient voltages at five equidistant places on the two-wire line with length ℓ=4.5 [m] (𝐸𝑚=10 [V] and 𝜑=0).

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], duty-cycle 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 𝑓₀=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].

Figure 41 

Pulsed responses in selected points on the uniform ladder in Figure 38(c).

4. 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 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 steady-state 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 ℓ or type is: 𝑢𝑡,𝑥+𝛿𝑥2−𝑢(𝑡,𝑥)=−𝛿2𝑥⋅𝐿^⋅𝜕𝑖(𝑡,𝑥)𝜕𝑡+𝑅^𝑖(𝑡,𝑥+𝛿𝑥)−𝑖(𝑡,𝑥)=−𝛿𝑥⋅𝐶_⋅𝑖(𝑡,𝑥),(A.1)⋅𝜕𝑢(𝑡,𝑥+𝛿𝑥/2)𝜕𝑡+𝐺^⋅𝑢𝑡,𝑥+𝛿𝑥𝑢(𝑡,𝑥+𝛿𝑥)−𝑢𝑡,𝑥+𝛿𝑥2,(A.2)2=−𝛿2𝑥⋅𝐿^⋅𝜕𝑖(𝑡,𝑥+𝛿𝑥)𝜕𝑡+𝑅^⋅𝑖(𝑡,𝑥+𝛿𝑥).(A.3)

If 𝛿𝑥→0, from (A.1)–(A.3) it immediately follows: 𝜕𝑢(𝑡,𝑥)𝜕𝑥=−𝐿^⋅𝜕𝑖(𝑡,𝑥)𝜕𝑡−𝑅^⋅𝑖(𝑡,𝑥),𝜕𝑖(𝑡,𝑥)𝜕𝑥=−𝐶^⋅𝜕𝑢(𝑡,𝑥)𝜕𝑡−𝐺^⋅𝑢(𝑡,𝑥).(A.4)

If 𝑠 is the complex frequency, let we suppose that the following conditions hold.(a) Both 𝑢(𝑡,𝑥) and 𝑖(𝑡,𝑥) possess Laplace transform with respect to time, 𝑈𝐼((𝑠𝑠,,𝑥𝑥))==ℓ[[𝑖𝑢(𝑡(,𝑡,𝑥𝑥)])]==^∞0_−∞0−𝑢(𝑡,𝑥)⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡,𝑖(𝑡,𝑥)⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡.(A.5) (b)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: ^∞0₋𝜕𝑢(𝑡,𝑥)𝜕𝑥⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡,^∞0₋𝜕𝑖(𝑡,𝑥)𝜕𝑥⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡.(A.6) (d) The initial conditions 𝑢(0,𝑥) and 𝑖(0,𝑥) are assumed for convenience to be zero for all 𝑥.

Taking into account all conditions (a) ÷ (d) and (A.1)–(A.6), it follows: ℓ𝜕𝑢(𝑡,𝑥)𝜕𝑥=^∞0−𝜕𝑢(𝑡,𝑥)𝜕𝑥⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡=𝜕𝜕𝑥^∞0−𝑢(𝑡,𝑥)⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡=𝜕𝑈(𝑠,𝑥)ℓ𝜕𝑖(𝑡,𝑥)𝜕𝑥,(A.7)𝜕𝑥=^∞0−𝜕𝑖(𝑡,𝑥)𝜕𝑥⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡=𝜕𝜕𝑥^∞0−𝑖(𝑡,𝑥)⋅𝑒^{−𝑠⋅𝑡}⋅d𝑡=𝜕𝐼(𝑠,𝑥)𝜕𝑈(𝑠,𝑥)𝜕𝑥,(A.8)𝜕𝑥=−(𝑅′+𝐿′⋅𝑠)⋅𝐼(𝑠,𝑥),𝜕𝐼𝜕(𝑠𝑥,𝑥)=−(𝐺′+𝐶′⋅𝑠)⋅𝑈(𝑠,𝑥),(A.9) wherefrom the equations describing voltage and current distribution in uniform line are readily produced, regardless to its length ℓ and/or terminal conditions (i.e., the generator and load impedances): 𝜕²𝑈(𝑠,𝑥)𝜕𝑥²+Γ²(𝑠)⋅𝑈(𝑠,𝑥)=0,𝜕²𝐼(𝑠,𝑥)𝜕𝑥²+Γ²(𝑠)⋅𝐼(𝑠,𝑥)=0,(A.10) 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 impedance 𝑍₀(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) reads 𝑈(𝑠,𝑥)=𝐴₁⋅cosh(Γ⋅𝑥)+𝐴₂⋅sinh(Γ⋅𝑥),𝐼(𝑠,𝑥)=𝐵₁⋅cosh(Γ⋅𝑥)+𝐵₂⋅sinh(Γ⋅𝑥),(A.11) where the terms 𝐴₁, 𝐴₂, 𝐵₁, and 𝐵₂ are not the functions of 𝑥 and are determined from the boundary conditions. From (A.11) for 𝑥=0 we obtain, 𝐴₁=𝑈(𝑠,0) and 𝐵₁=𝐼(𝑠,0) and from (A.9) and (A.11); after differentiation in 𝑥; it follows, 𝐴₂=−𝑍₀(𝑠)⋅𝐼(𝑠,0) and 𝐵₂=−𝑈(𝑠,0)/𝑍₀(𝑠). Then (A.11) takes on the following form: 𝑈(𝐼𝑠,(𝑥𝑠,)𝑥=)=𝑈𝐼(𝑠(,𝑠0,)0⋅)c⋅ocsohsh(Γ(Γ⋅𝑥⋅)𝑥−)−𝑍₀𝑈(𝑠)(𝑠⋅,𝐼0)(𝑠,0)⋅sinh(Γ⋅𝑥),(A.12)𝑍₀(𝑠)⋅sinh(Γ⋅𝑥).(A.13)

Since for 𝑥=ℓ it holds: 𝑈(𝐼𝑠,(𝑠ℓ,)ℓ=)=𝑈𝐼(𝑠(,𝑠0,)0⋅)c⋅ocsohsh(Γ(Γ⋅ℓ⋅)ℓ−)−𝑍₀𝑈(𝑠)(𝑠⋅,𝐼0)(𝑠,0)⋅sinh(Γ⋅ℓ),𝑍₀(𝑠)⋅sinh(Γ⋅ℓ),(A.14)

then from (A.14) we obtain, 𝑈(𝐼𝑠,(𝑠0,)0=)=𝑈𝐼(𝑠(,𝑠ℓ,)ℓ⋅)c⋅ocsohsh(Γ(Γ⋅ℓ⋅)ℓ+)+𝑍₀𝑈(𝑠)(𝑠⋅,𝐼ℓ()𝑠,ℓ)⋅sinh(Γ⋅ℓ),(A.15)𝑍₀(𝑠)⋅sinh(Γ⋅ℓ).(A.16)

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)=𝑍^𝐿(𝑠)⋅cosh[Γ⋅(ℓ−𝑥)]+𝑍₀(𝑠)⋅sinh[Γ⋅(ℓ−𝑥)]𝑁(𝑠,𝑥)=𝐼(𝑠,𝑥)𝑍_𝐿(𝑠)⋅cosh(Γ⋅ℓ)+𝑍₀(𝑠)⋅sinh(Γ⋅ℓ),𝐼(𝑠,0)=𝑍^𝐿(𝑠)⋅sinh[Γ⋅(ℓ−𝑥)]+𝑍₀(𝑠)⋅cosh[Γ⋅(ℓ−𝑥)]𝑍_𝐿(𝑠)⋅sinh(Γ⋅ℓ)+𝑍₀(𝑠)⋅cosh(Γ⋅ℓ).(A.17)

Since 𝑈(𝑠,ℓ)=𝑍_𝐿(𝑠)⋅𝐼(𝑠,ℓ), then from (A.12)–(A.14) it follows, 𝑍(𝑠,0)=𝑈(𝑠,0)𝐼(𝑠,0)=𝑍^𝐿(𝑠)⋅cosh(Γ⋅ℓ)+𝑍₀(𝑠)⋅sinh(Γ⋅ℓ)𝑍_𝐿(𝑠)⋅sinh(Γ⋅ℓ)+𝑍₀(𝑠)⋅cosh(Γ⋅ℓ)⋅𝑍⁰𝑍(𝑠,𝑥)=𝑈(𝑠,𝑥)(𝑠),andgenerally,𝐼(𝑠,𝑥)=𝑍^𝐿(𝑠)⋅cosh[Γ⋅(ℓ−𝑥)]+𝑍₀(𝑠)⋅sinh[Γ⋅(ℓ−𝑥)]𝑍_𝐿(𝑠)⋅sinh[Γ⋅(ℓ−𝑥)]+𝑍₀(𝑠)⋅cosh[Γ⋅(ℓ−𝑥)]⋅𝑍⁰(𝑠),𝑥∈[0,ℓ],(A.18) 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 𝑍₀(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)=𝑍₀(𝑠), then for all 𝑥∈[0,ℓ] it holds: 𝑍(𝑠,𝑥)=𝑍₀(𝑠)=(𝐿^⋅𝐶^)^1/2 and 𝑀(𝑠,𝑥)=𝑁(𝑠,𝑥)=exp[−Γ(𝑠)⋅𝑥)]=exp(−𝑅^⋅𝐺^⋅𝑥)⋅exp[−𝑥⋅(𝐿^⋅𝐶^)^1/2⋅𝑠]. When the line parameters are constant, we will have 𝑢(𝑡,𝑥)=ℒ⁻¹[𝑈(𝑠,𝑥)] = ℒ⁻¹[𝑀(𝑠,𝑥)⋅𝑈(𝑠,0)]=ℒ⁻¹{exp(−𝑅^⋅𝐺^⋅𝑥)⋅exp[−𝑥⋅(𝐿^⋅𝐶^)^1/2⋅𝑠]⋅𝑈(𝑠,0)}=exp(−𝑅^⋅𝐺^⋅𝑥)⋅𝑢[𝑡−𝑥⋅(𝐿^⋅𝐶^)^1/2,0] and, also, 𝑖(𝑡,𝑥)=ℒ⁻¹[𝐼(𝑠,𝑥)]=ℒ⁻¹[𝑁(𝑠,𝑥)⋅𝐼(𝑠,0)]=ℒ⁻¹{exp(−𝑅^⋅𝐺^⋅𝑥)⋅exp[−𝑥⋅(𝐿^⋅𝐶^)^1/2⋅𝑠]⋅𝐼(𝑠,0)}=exp(−𝑅^⋅𝐺^⋅𝑥) · 𝑖[𝑡−𝑥⋅(𝐿^⋅𝐶^)^1/2], 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 𝑍₀(𝑠)=(𝐿^/𝐶^)^1/2, the attenuation factor exp[𝑥⋅(𝑅^⋅𝐺^)^1/2] must, also, be taken into account.

Acknowledgment

This work is supported in part by the Serbian Ministry of Science and Technological Development through Projects TR 32048 and III 41006.