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Mathematical Problems in Engineering
Volume 2012, Article ID 351894, 42 pages
http://dx.doi.org/10.1155/2012/351894
Research Article

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

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

Received 23 March 2012; Accepted 1 May 2012

Academic Editor: Kue-Hong Chen

Copyright © 2012 D. B. Kandić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝜀𝑐, 𝜇𝑐 and 𝜎𝑐 or 𝜀𝑑, 𝜇𝑑, and 𝜎𝑑 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  𝛿𝑥0,  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 𝜎𝑑𝜎𝑐.

fig1
Figure 1: (a) Two-wire line, (b) coaxial line, and (c) twisted pair.
351894.fig.002
Figure 2: The approximate, incremental, and lumped network model of the uniform transmission line.

By neglecting the proximity effect with assumption 𝑑2𝑎, edge effect and taking 𝜎𝑑𝜎𝑐, 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=𝜋𝜀𝑑ln(𝑑/𝑎),𝐺=𝜎𝑑𝜀𝑑𝐶=𝜋𝜎𝑑,ln(𝑑/𝑎)𝑑2𝑎𝜎𝑑𝜎𝑐.(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 𝑅𝑖(𝑓) and the frequency-dependent per-unit-length inner inductance 𝐿𝑖(𝑓)-for 𝑓[0,)[Hz], 𝑅𝑖𝑘(𝑓)=2𝜋𝑎𝜎𝑐Rebei(𝑘𝑎)+𝑗ber(𝑘𝑎)ber(𝑘𝑎)+𝑗bei=𝑘(𝑘𝑎)2𝜋𝑎𝜎𝑐ber(𝑘𝑎)bei(𝑘𝑎)bei(𝑘𝑎)ber(𝑘𝑎)ber(𝑘𝑎)2+bei(𝑘𝑎)2,𝐿𝑖𝑘(𝑓)=2𝜋𝑎𝜎𝑐𝜔Imbei(𝑘𝑎)+𝑗ber(𝑘𝑎)ber(𝑘𝑎)+𝑗bei=𝑘(𝑘𝑎)2𝜋𝑎𝜎𝑐𝜔ber(𝑘𝑎)ber(𝑘𝑎)+bei(𝑘𝑎)bei(𝑘𝑎)ber(𝑘𝑎)2+bei(𝑘𝑎)2,(2.2) where: 𝑗=1,  𝜔=2𝜋𝑓,  𝑘=𝜔𝜇𝑐𝜎𝑐 , and “Bessel real” ber(𝑘𝑎) and “Bessel imaginary” bei(𝑘𝑎) are the Bessel-Kelvin functions with the first-order derivatives ber(𝑘𝑎) and bei(𝑘𝑎), respectively, at the point 𝑧=𝑘𝑎, which can be approximated at high frequencies (i.e., for 𝑘𝑎1) with [4], ber(𝑘𝑎)=𝐺cos𝑘𝑎2𝜋8,bei(𝑘𝑎)=𝐺sin𝑘𝑎2𝜋8,1𝐺=2𝜋𝑘𝑎𝑒(𝑘𝑎)/2,ber(𝑘𝑎)=𝐺cos𝑘𝑎2+𝜋8,bei(𝑘𝑎)=𝐺sin𝑘𝑎2+𝜋8.(2.3)

From (2.3) it should be firstly noticed that at high frequencies (i.e., for 𝑘𝑎1) it holds bei(𝑘𝑎)+𝑗ber(𝑘𝑎)ber(𝑘𝑎)+𝑗bei=(𝑘𝑎)sin(𝑘𝑎)/2𝜋/8+𝑗cos(𝑘𝑎)/2𝜋/8cos(𝑘𝑎)/2+𝜋/8+𝑗sin(𝑘𝑎)/𝑒2+𝜋/8=𝑗𝑗((𝑘𝑎)/2𝜋/8)𝑒𝑗((𝑘𝑎)/2+𝜋/8)=𝑒𝑗(𝜋/4),(2.4) and then from (2.2) it may be obtained consecutively for 𝑘𝑎1, 𝑅𝑖(𝑘𝑓)=2𝜋2𝑎𝜎𝑐=12𝜋𝑎𝜋𝑓𝜇𝑐𝜎𝑐,𝐿𝑖𝑘(𝑓)=2𝜋2𝜔𝑎𝜎𝑐=14𝜋𝑎𝜇𝑐𝜋𝑓𝜎𝑐.(2.5)

The overall per-unit-length resistance 𝑅(𝑓) and the inductance 𝐿(𝑓) of two-wire line are 𝑅(𝑓)=2𝑅𝑖(||𝑓)𝑓,𝑅(1𝑓)=𝜋𝑎𝜋𝑓𝜇𝑐𝜎𝑐=𝑅𝑠(𝑓)||||𝜋𝑎𝑘𝑎1,𝑅𝑠(𝑓)=𝜋𝑓𝜇𝑐𝜎𝑐,𝐿(𝑓)=𝐿𝑒+2𝐿𝑖||(𝑓)𝑓𝜇,𝐿(𝑓)=𝑑𝜋𝑑ln𝑎+12𝜋𝑎𝜇𝑐𝜋𝑓𝜎𝑐||||𝑘𝑎1(𝑑2𝑎),(2.6) where 𝑅𝑠(𝑓) is the surface resistance of line conductors and 𝐿𝑒=(𝜇𝑑/𝜋)ln(𝑑/𝑎) is the external per-unit-length inductance of two-wire line. Throught the paper it will be assumed that 𝜇𝑐𝜇𝑑𝜇0.

Since the following expansions hold for any frequency 𝑓[0,) [Hz] (i.e., for any 𝑘𝑎) [4], ber(𝑘𝑎)=𝑛=0(1)𝑛(𝑘𝑎/2)4𝑛(2𝑛!)2=1(𝑘𝑎)42242+(𝑘𝑎)822426282(𝑘𝑎)1222426282102122±,bei(𝑘𝑎)=𝑛=1(1)𝑛+1(𝑘𝑎/2)4𝑛2[](2𝑛1)!2=(𝑘𝑎)222(𝑘𝑎)6224262+(𝑘𝑎)1022426282102,ber(𝑘𝑎)=(𝑘𝑎)322+4(𝑘𝑎)72242628(𝑘𝑎)1122426282102+12(𝑘𝑎)152242628210212214216,bei(𝑘𝑎)=𝑘𝑎2(𝑘𝑎)52242+6(𝑘𝑎)922426282(10𝑘𝑎)132242628210212214±,(2.7) then by using (2.2) and (2.7) when 𝑓0 [Hz], the following consequences are easily obtained: 𝑅𝑖(𝑓)1/(𝜎𝑐𝜋𝑎2), 𝐿𝑖(𝑓)𝜇𝑐/(8𝜋), 𝑅(𝑓)2/(𝜎𝑐𝜋𝑎2), 𝐿(𝑓)(𝜇𝑑/𝜋)ln(𝑑/𝑎)+𝜇𝑐/(4𝜋). 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=𝑘𝑎[0,) [5],

Case  A  (𝑧=𝑧(𝑓)=𝑎𝑘=𝑎2𝜋𝑓𝜇𝑐𝜎𝑐[0,8]). 𝑧ber(𝑧)=16484𝑧+113.7777777488𝑧32.36345652812𝑧+2.64191397816𝑧0.08349609820𝑧+0.00122552824𝑧0.00000901828+𝜀1||𝜀1||<109,𝑧bei(𝑧)=1682𝑧113.7777777486𝑧+72.81777742810𝑧10.56765779814𝑧+0.52185615818𝑧0.01103667822𝑧+0.00011346826+𝜀2||𝜀2||<6109,ber𝑧(𝑧)=𝑧482𝑧+14.2222222286𝑧6.06814810810𝑧+0.66047849814𝑧0.02609253818𝑧+0.00045957822𝑧0.00000394826+𝜀3||𝜀3||<2.1108,bei1(𝑧)=𝑧2𝑧10.6666666684𝑧+11.3777777288𝑧2.31167514812𝑧+0.14677204816𝑧0.00379386820𝑧+0.00004609824+𝜀4||𝜀4||<7108.(2.8)

Case  B (𝑧=𝑧(𝑓)=𝑎𝑘=𝑎2𝜋𝑓𝜇𝑐𝜎𝑐(8,)).

Define firstly the following set of auxiliary functions: 8𝛼(𝑧)=0.3926991𝑗+(0.01104860.0110485𝑗)𝑧80.0009765𝑗𝑧28+(0.00009060.0000901𝑗)𝑧380.0000252𝑧4+8(0.0000034+0.0000051𝑗)𝑧5+8(0.0000006+0.0000019𝑗)𝑧6,8𝛼(𝑧)=0.3926991𝑗+(0.0110486+0.0110485𝑗)𝑧80.0009765𝑗𝑧28+(0.0000906+0.0000901𝑗)𝑧380.0000252𝑧48+(0.00000340.0000051𝑗)𝑧58+(0.0000006+0.0000019𝑗)𝑧6,𝛽8(𝑧)=(0.7071068+0.7071068𝑗)+(0.06250010.0000001𝑗)𝑧8+(0.0013813+0.0013811𝑗)𝑧28+(0.0000005+0.0002452𝑗)𝑧38+(0.0000346+0.0000338𝑗)𝑧48+(0.00000170.0000024𝑗)𝑧58+(0.00000160.0000032𝑗)𝑧6,8𝛽(𝑧)=(0.7071068+0.7071068𝑗)+(0.0625001+0.0000001𝑗)𝑧8+(0.0013813+0.0013811𝑗)𝑧28+(0.00000050.0002452𝑗)𝑧38+(0.0000346+0.0000338𝑗)𝑧48+(0.0000017+0.0000024𝑗)𝑧5+8(0.00000160.0000032𝑗)𝑧6.(2.9) and, also, define another set of auxiliary functions: 𝑓(𝑧)=𝜋2𝑧exp1+𝑗21𝑧+𝛼(𝑧),𝑔(𝑧)=2𝜋𝑧exp1+𝑗2𝑧+𝛼(𝑧),(2.10) then, the values of Bessel-Kelvin functions can be efficiently calculated [5] by using the relations: 𝑗ber(𝑧)=Re𝜋𝑗𝑓(𝑧)+𝑔(𝑧),bei(𝑧)=Im𝜋,𝑓(𝑧)+𝑔(𝑧)ber𝑗(𝑧)=Re𝜋𝑓(𝑧)𝛽(𝑧)+𝑔(𝑧)𝛽(𝑧),bei𝑗(𝑧)=Im𝜋.𝑓(𝑧)𝛽(𝑧)+𝑔(𝑧)𝛽(𝑧)(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]: 𝐶=2𝜋𝜀𝑑ln(𝑏/𝑎),𝐺=𝜎𝑑𝜀𝑑𝐶=2𝜋𝜎𝑑,𝜎ln(𝑏/𝑎)𝑑𝜎𝑐.(2.12)

It has been shown [13] 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(𝑅𝑓)=𝑠(𝑓)12𝜋𝑎+1𝑏=12𝜋𝜋𝑓𝜇𝑐𝜎𝑐1𝑎+1𝑏,𝐿𝐿𝑒=𝜇𝑑𝑏2𝜋ln𝑎,𝜇𝑐𝜇𝑑𝜇0.(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 𝜎𝑐5.81107  [S/m] and magnetic permeability is 𝜇𝑐𝜇0. At the temperature 𝑇=298 [K] the relative permittivity of polyethylene is 𝜀𝑟2.26 (for frequencies up to 25 [GHz]) and its specific electric conductivity is 𝜎𝑑1015 [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/(𝜎𝑐𝜋𝑎2)=1.0957 [Ω/m] and its per-unit-length inductance is 𝐿(0)=(𝜇𝑑/𝜋)ln(𝑑/𝑎)+𝜇𝑐/(4𝜋)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 𝜆𝑐0/[𝑓(𝜀𝑟)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 [𝑓0𝐵/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 𝑓0𝐵/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[𝑓0𝐵/2,𝑓0+𝐵/2]. And by using Figure 5 we obtain in the most conservative approach that 𝐵37 [kHz]. Similarly, if the integral of function in Figure 9 taken between 𝑓0𝐵/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[𝑓0𝐵/2,𝑓0+𝐵/2]. And by using Figure 9 we obtain in the most conservative approach that 𝐵770 [kHz].

351894.fig.003
Figure 3: 𝑅(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.004
Figure 4: 𝐿(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.005
Figure 5: 𝑑𝑅(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.006
Figure 6: 𝑑𝐿(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.007
Figure 7: 𝑅(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.008
Figure 8: 𝐿(𝑓)  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.009
Figure 9: 𝑑𝑅(𝑓)/𝑑𝑓  for line with 𝑎=0.1 [mm] and 𝑑=4 [mm].
351894.fig.0010
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 [13]: 1𝑐=𝐿𝐶=1𝜀𝑑𝜇𝑑=𝑐0𝜀𝑟𝜇𝑟,𝑐0=1𝜀0𝜇03108ms,𝑍0=𝐿𝐶=1𝑐𝐶=𝜇𝑟𝜀𝑟1𝜋𝜇0𝜀0𝑑ln𝑎120𝜇𝑟𝜀𝑟𝑑ln𝑎[Ω],(2.14) where 𝜀𝑟=𝜀𝑑/𝜀0 is relative permittivity and 𝜇𝑟=𝜇𝑑/𝜇0 relative permeability of dielectric (𝜀0109/36𝜋 [F/m] is permittivity and 𝜇0=4𝜋107 [H/m] permeability of vacuum). The characteristic impedance of a lossy transmission line is generally defined as 𝑍0(𝑗2𝜋𝑓)=[(𝑅+𝑗2𝜋𝑓𝐿)/(𝐺+𝑗2𝜋𝑓𝐶)]1/2. In the case considered, on Figures 11 and 12 the variations of 𝑍0(𝑓)=|𝑍0(𝑗2𝜋𝑓)| and 𝜁(𝑓)=Arg[𝑍0(𝑗2𝜋𝑓)] 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].

351894.fig.0011
Figure 11: 𝑍0(𝑓) and 𝑑𝑍0(𝑓)/𝑑𝑓 for the considered two-wire line.
351894.fig.0012
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)=2𝜋𝑓𝐿(𝑓)𝑅(𝑓),𝜙2(𝑓)=2𝜋𝑓𝐶𝐺,𝜙31(𝑓)=2𝜋𝑓𝐿(𝑓)𝐶,𝜙4𝐶(𝑓)=𝑅(𝑓)𝐺𝐿(𝑓),𝜙5(𝑓)=2𝜋𝑓𝑅(𝑓)/𝐿(𝑓)+𝐺/𝐶=𝜙1(𝑓)1+1/𝜙4.(𝑓)(2.15)

351894.fig.0013
Figure 13: A dimensionless parameter 𝜙1(𝑓) of two-wire line.
351894.fig.0014
Figure 14: A dimensionless parameter 𝜙2(𝑓) of two-wire line.
351894.fig.0015
Figure 15: Parameter 𝜙3(𝑓) of two-wire line.
351894.fig.0016
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)32.6531, 𝜙2(110.9KHz)13.943109 and 𝜙4(0)1.3911010. 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(𝑗2𝜋𝑓)| and 𝜗(𝑓)=Arg[Γ(𝑗2𝜋𝑓)] [“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].

351894.fig.0017
Figure 17: Magnitude of propagation function (two-wire line).
351894.fig.0018
Figure 18: Argument of propagation function (two-wire line).

Since Γ(𝑗2𝜋𝑓)=[𝑅(𝑓)+𝑗2𝜋𝑓𝐿(𝑓)](𝐺+𝑗2𝜋𝑓𝐶)=Λ(𝑓)exp[𝑗𝜗(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)2𝜋𝑓[𝐿(𝑓)𝐶]1/2=1/𝜙3(𝑓),𝜗(𝑓)=𝜋/2𝜒(𝑓)[0<𝜒(𝑓)<𝜋/200] and Γ(𝑗2𝜋𝑓)=Λ(𝑓)exp(𝑗𝜋/2)exp[𝑗𝜒(𝑓)]=𝑗Λ(𝑓){cos[𝜒(𝑓)]𝑗sin[𝜒(𝑓)]}={sin[𝜒(𝑓)]+𝑗cos[𝜒(𝑓)]}/𝜙3(𝑓), where the deviation angle 𝜒(𝑓)=𝜋/2𝜗(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, 𝜋𝜒(𝑓)=2𝜋𝜗(𝑓)=212𝐿𝑎tan2𝜋𝑓(𝑓)/𝑅(𝑓)+𝐶/𝐺14𝜋2𝑓2𝐿(𝑓)𝐶/𝑅(𝑓)𝐺12𝑅𝑎tan(𝑓)/𝐿(𝑓)+𝐺/𝐶𝑅2𝜋𝑓(𝑓)/𝐿(𝑓)+𝐺/𝐶.4𝜋𝑓(2.16)

For the transmission line with length let us define the functions: 𝜀(𝜔)=𝜒(𝜔/2𝜋)=𝜋/2𝜗(𝜔/2𝜋) and 𝜃(𝑗𝜔)=Γ(𝑗𝜔)(𝑥)=𝐴(𝜔,𝑥)exp[𝑗𝜗(𝜔/2𝜋)]=𝐴(𝜔,𝑥){sin[𝜀(𝜔)]+𝑗cos[𝜀(𝜔)]}{𝑥[0,]}, where 𝐴(𝜔,𝑥)=|Γ(𝑗𝜔)|(𝑥)=Λ(𝜔/2𝜋)(𝑥). For this line, in frequency range 𝑓[0.01,3] [GHz] we have 𝐴(𝜔,𝑥)𝜔[𝐿(𝜔/2𝜋)𝐶]1/2(𝑥)=(𝑥)/𝜙3(𝜔/2𝜋) and 0<𝜀(𝜔)<𝜋/200—whereby 𝜃(𝑗𝜔) becomes almost pure imaginary number. We could have obtained this result in a different way. To se that, let us write Γ(𝑗2𝜋𝑓)=[𝑅(𝑓)+𝑗2𝜋𝑓𝐿(𝑓)](𝐺+𝑗2𝜋𝑓𝐶)=𝑎(𝑓)+𝑗𝑏(𝑓), where it holds, 𝑎(𝑓)=12𝐿(𝑓)𝐶(2𝜋𝑓)2+𝑅(𝑓)2𝐿(𝑓)2(2𝜋𝑓)2+𝐺2𝐶2+𝑅(𝑓)𝐿𝐺(𝑓)𝐶(2𝜋𝑓)2attenuationconstant,𝑏(𝑓)=12𝐿(𝑓)𝐶(2𝜋𝑓)2+𝑅(𝑓)2𝐿(𝑓)2(2𝜋𝑓)2+𝐺2𝐶2𝑅(𝑓)𝐿𝐺(𝑓)𝐶+(2𝜋𝑓)2phaseconstant.(2.17) The previous two functions are depicted in Figures 19 and 20 in the frequency range 𝑓[0,3] [GHz].

351894.fig.0019
Figure 19: The attenuation “constant” of two-wire line.
351894.fig.0020
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, 1𝑎𝑎(𝑓)2𝑅(𝑓)𝐶𝐿(𝑓)+𝐺𝐿(𝑓)𝐶,𝑎𝑏(𝑓)𝐿(𝑓)𝐶1(𝑓)2𝜋𝑓+𝑅16𝜋𝑓(𝑓)𝐿𝐺(𝑓)𝐶2,and,also,wehave,𝑍0(𝑗2𝜋𝑓)=𝑅(𝑓)+𝑗2𝜋𝑓𝐿(𝑓)𝐺+𝑗2𝜋𝑓𝐶𝐿(𝑓)𝐶11𝑗𝑅4𝜋𝑓(𝑓)𝐿𝐺(𝑓)𝐶.(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(𝑗𝜔)𝑎𝑎(𝜔/2𝜋)+𝑗𝑎𝑏(𝜔/2𝜋){sin[𝜀(𝜔)]+𝑗cos[𝜀(𝜔)]}/𝜙3(𝜔/2𝜋), as it has been expected.

351894.fig.0021
Figure 21: Approximation of the attenuation “constant”.
351894.fig.0022
Figure 22: Approximation of the phase “constant”.

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 (𝑓0𝐵/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=𝐿𝐶𝜕2𝑢(𝑡,𝑥)𝜕𝑡2+𝐿𝐺+𝐶𝑅𝜕𝑢(𝑡,𝑥)𝜕𝑡+𝑅𝐺𝜕𝑢(𝑡,𝑥),2𝑖(𝑡,𝑥)𝜕𝑥2=𝐿𝐶𝜕2𝑖(𝑡,𝑥)𝜕𝑡2+𝐿𝐺+𝐶𝑅𝜕𝑖(𝑡,𝑥)𝜕𝑡+𝑅𝐺𝑖(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 [𝑍𝐿(𝑠)], in particular, by using expansions: 𝑀(𝑠,𝑥)=𝑈(𝑠,𝑥)=𝑈(𝑠,0)𝑛=1[]1+(2Γ(𝑥))/((2𝑛1)𝜋)2𝑛=1[]1+(2Γ)/((2𝑛1)𝜋)2,𝑁(𝑠,𝑥)=𝐼(𝑠,𝑥)=𝐼(𝑠,0)𝑥𝑛=1[]1+(Γ(𝑥))/(𝑛𝜋)2𝑛=11+((Γ)/(𝑛𝜋))2,𝑍(𝑠,𝑥)=𝑈(𝑠,𝑥)=𝐼(𝑠,𝑥)𝑛=1[]1+(2Γ(𝑥))/((2𝑛1)𝜋)2Γ(𝑥)𝑛=1[]1+(Γ(𝑥))/(𝑛𝜋)2𝑍0𝑍𝐿,(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=1𝜃1+2𝑛2𝜋2,cosh(𝜃)=𝑛=11+4𝜃2(2𝑛1)2𝜋2.(2.21)

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 |𝑀(𝑗2𝜋𝑓,𝑥)|, Arg[𝑀(𝑗2𝜋𝑓,𝑥)] [deg], |𝑁(𝑗2𝜋𝑓,𝑥)| and Arg[𝑁(𝑗2𝜋𝑓,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 𝑍𝐿(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)].

351894.fig.0023
Figure 23: The magnitude of the voltage transmittance for two-wire line (=0.1 [m]) in frequency range 𝑓[0,3] [GHz].
351894.fig.0024
Figure 24: The argument of the voltage transmittance for two-wire line (=0.1 [m]) in frequency range 𝑓[0,3] [GHz].
351894.fig.0025
Figure 25: The magnitude of the current transmittance for two-wire line (=0.1 [m]) in frequency range 𝑓[0,3] [GHz].
351894.fig.0026
Figure 26: The argument of the current transmittance for two-wire line (=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 0<𝜋/{2|[Γ(𝑗2𝜋𝑓)|max}16.6 [mm] {|𝜃|max<𝜋/2} and assume, say, 0=16 [mm]. Recall that =𝜃(𝑗𝜔,𝑥)=Γ(𝑗𝜔)(𝑥)=|𝜃(𝑗𝜔,𝑥)|exp{𝑗arg[Γ(𝑗𝜔)]}{𝑥[0,]}, then take (2.21) and write sinh(𝜃)𝜃=𝑛=1𝜃1+2𝑛2𝜋2=exp𝑛=1𝜃ln1+2𝑛2𝜋2=𝜃exp2𝜋2𝑛=11𝑛2𝜃42𝜋4𝑛=11𝑛4+𝜃63𝜋6𝑛=11𝑛6𝜃84𝜋8𝑛=11𝑛8+𝜃105𝜋10𝑛=11𝑛10𝜃±.=exp26𝜃4+𝜃1806𝜃28358+𝜃3780010,467775±cosh(𝜃)=𝑛=11+4𝜃2(2𝑛1)2𝜋2=exp𝑛=1ln1+4𝜃2(2𝑛1)2𝜋2=exp4𝜃2𝜋2𝑛=11(2𝑛1)28𝜃4𝜋4𝑛=11(2𝑛1)4+64𝜃63𝜋6𝑛=11(2𝑛1)664𝜃8𝜋8𝑛=11(2𝑛1)8+1024𝜃105𝜋10𝑛=11(2𝑛1)10𝜃±=exp22𝜃4+𝜃1264517𝜃8+252031𝜃10.14175±(2.22)

The following finite sums of infinite series [9] have been exploited in (2.22), 𝐴2𝑝=𝑛=11𝑛2𝑝𝑝=1,5,𝐴2=𝜋26,𝐴4=𝜋490,𝐴6=𝜋6945,𝐴8=𝜋8,𝐴945010=528𝜋10,𝐵3310!2𝑝=𝑛=11(2𝑛1)2𝑝𝑝=1,5,𝐵2=𝜋28,𝐵4=𝜋496,𝐵6=𝜋6960,𝐵8=17𝜋8,𝐵16128010=31𝜋103581210.(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 retaining only the first five 𝜃 terms in (2.22) the following small-error approximations are produced sinh(𝜃)sinh𝐴𝜃(𝜃)=𝜃exp26𝜃4+𝜃1806𝜃28358+𝜃3780010,467775cosh(𝜃)cosh𝐴𝜃(𝜃)=exp22𝜃4+𝜃1264517𝜃8+252031𝜃10.14175(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:ER1(𝑓,𝑥)={|[cosh𝐴𝜃(𝑗2𝜋𝑓,𝑥)/cosh𝐴(𝑗2𝜋𝑓,0)]/[cosh𝜃(𝑗2𝜋𝑓,𝑥)/cosh(𝑗2𝜋𝑓,0)]|1}100;(ii) the voltage-transmittance phase approximation absolute error:ER2(𝑓,𝑥)=arg[cosh𝐴𝜃(𝑗2𝜋𝑓,𝑥)cosh(𝑗2𝜋𝑓,0)/cosh𝐴(𝑗2𝜋𝑓,0)cosh𝜃(𝑗2𝜋𝑓,𝑥)];(iii) the current-transmittance magnitude approximation percentage error:ER3(𝑓,𝑥)={|[sinh𝐴𝜃(𝑗2𝜋𝑓,𝑥)/sinh𝐴(𝑗2𝜋𝑓,0)]/[sinh𝜃(𝑗2𝜋𝑓,𝑥)/sinh(𝑗2𝜋𝑓,0)]|1}100;(iv) the current-transmittance phase approximation absolute error:ER4(𝑓,𝑥)=arg[sinh𝐴𝜃(𝑗2𝜋𝑓,𝑥)sinh(𝑗2𝜋𝑓,0)/sinh𝐴(𝑗2𝜋𝑓,0)sinh𝜃(𝑗2𝜋𝑓,𝑥)].

351894.fig.0027
Figure 27: Voltage-transmittance magnitude approximation percentage error ER1(𝑓,𝑥).
351894.fig.0028
Figure 28: Voltage-transmittance phase approximation absolute error ER2(𝑓,𝑥).
351894.fig.0029
Figure 29: Current-transmittance magnitude approximation percentage error ER3(𝑓,𝑥).
351894.fig.0030
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|<105%), whereas the errors ER2 and ER4 are positive (ER2<3.36104 [deg] and ER4<5.75108 [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)sinhA𝜃(𝜃)=𝜃exp26𝜃4+𝜃1806𝜃28358+𝜃3780010𝜃467775𝜃1+26𝜃4+𝜃1806𝜃28358+𝜃3780010𝜃467775=𝜃+36𝜃5+𝜃1807𝜃28359+𝜃3780011𝜃467775𝜃+36,(2.25)which partly resembles to Maclaurin’s expansion of sinh(θ), that is, sinh(𝜃)=𝜃+𝜃3/3!+𝜃5/5!+,(ii)cosh(𝜃)cosh𝐴𝜃(𝜃)=exp22𝜃4+𝜃1264517𝜃8+252031𝜃10𝜃141751+22𝜃4+𝜃1264517𝜃8+252031𝜃10𝜃141751+22,(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 𝜃tanh2tanh𝐴𝜃2=𝜃𝜃2+24𝜃62+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 sinh𝐴(𝜃)/𝜃 and tanh𝐴(𝜃/2)/(𝜃/2), obtained when |𝜃|1. For two-wire line with length =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 0 to be much less than 1/|Γ(𝑗2𝜋𝑓max)(|𝜃max|1). 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(𝑗2𝜋𝑓max)|.

351894.fig.0031
Figure 31: Magnitude approximation absolute error ER5(𝑓,𝑥).
351894.fig.0032
Figure 32: Percentage magnitude approximation error ER6(𝑓,𝑥).
351894.fig.0033
Figure 33: Percentage magnitude approximation error ER7(𝑓,𝑥).
351894.fig.0034
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 0 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,0) and 𝐼(𝑠,0) at the output. The chain-matrix 𝐅 of this network reads 𝑈𝑈(𝑠,0)𝐼(𝑠,0)=𝐅𝑠,0𝐼𝑠,0,𝐅=coshΓ0𝑍0sinhΓ0sinhΓ0𝑍0coshΓ0.(2.28)

fig35
Figure 35: (a) Linear network model of a short transmission line segment and (b) its linear network structure.

Denote 𝑍(𝑠)=(𝑅+𝐿𝑠)0, 𝑌(𝑠)=(𝐺+𝐶𝑠)0 and 𝜃(𝑠,0)=Γ(𝑠)0. 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[]=𝑍tanh𝜃(𝑠,0)/2𝜃(𝑠,0)/2,𝑌[]=𝑌sinh𝜃(𝑠,0),𝑍𝜃(𝑠,0)[]=𝑍sinh𝜃(𝑠,0)𝜃(𝑠,0),𝑌[]=𝑌tanh𝜃(𝑠,0)/2.𝜃(𝑠,0)/2(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)/𝜃1 and tanh(𝜃/2)/(𝜃/2)1 (observe that if 𝜃0, then 𝑍 and 𝑍𝑍 and 𝑌 and 𝑌𝑌). To see that, recall that for Γ(𝑗𝜔)=(𝑅+𝑗𝜔𝐿)(𝐺+𝑗𝜔𝐶)=|Γ(𝑗𝜔)|exp{𝑗arg[Γ(𝑗𝜔)]}(𝜔=2𝜋𝑓), in frequency range 𝑓[0.01,3] [GHz] it holds(a)𝜔𝐿/𝑅𝜔/(𝑅/𝐿+𝐺/𝐶)[32.652,609.312]1, 𝜔𝐶/𝐺[1.2571012,3.7711014]1,(b)|Γ(𝑗𝜔)|𝜔(𝐿𝐶)1/2[0.319,94.598], arg[Γ(𝑗𝜔)][89.122,89.952] [deg],(c)arg[Γ(𝑗𝜔)]=𝜋/2𝜀(𝜔), where, 𝜋𝜀(𝜔)=2[]=𝜋argΓ(𝑗𝜔)212𝑎tan𝜔𝐿/𝑅+𝜔𝐶/𝐺𝐿1𝐶/𝑅𝐺𝜔212𝑅𝑎tan/𝐿+𝐺/𝐶𝜔𝑅/𝐿+𝐺/𝐶.2𝜔(2.30)(d)Γ(𝑗𝜔)=|Γ(𝑗𝜔)|{sin[𝜀(𝜔)]+𝑗cos[𝜀(𝜔)]}, 𝜀(𝜔)(0,𝜋/200) and 𝜃(𝑗𝜔,𝑥)=Γ(𝑗𝜔)(0𝑥)=|𝜃(𝑗𝜔,𝑥)|{sin[𝜀(𝜔)]+𝑗cos[𝜀(𝜔)]}. Observe that 𝜃(𝑗𝜔,𝑥) is produced as almost pure imaginary number.(e) Since 𝑥[0,0] and |𝜃(𝑗𝜔,𝑥)|𝜔(𝐿𝐶)1/2(0𝑥)=𝐴(𝜔,𝑥), 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(𝑗𝜔,𝑥)]/𝜃(𝑗𝜔,𝑥) the following: [][]}sinh{𝐴(𝜔,𝑥)sin𝜀(𝜔)}cos{𝐴(𝜔,𝑥)cos𝜀(𝜔)[][]}+[][]}𝐴(𝜔,𝑥){sin𝜀(𝜔)+𝑗cos𝜀(𝜔)𝑗cosh{𝐴(𝜔,𝑥)sin𝜀(𝜔)}sin{𝐴(𝜔,𝑥)cos𝜀(𝜔)[][]},𝐴(𝜔,𝑥){sin𝜀(𝜔)+𝑗cos𝜀(𝜔)(2.31) that can be rewritten as sinh[𝜃(𝑗𝜔,𝑥)]/𝜃(𝑗𝜔,𝑥)=𝑅(𝜔,𝑥)+𝑗𝑄(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=0𝐴2𝑘(𝜔,𝑥)sin2𝑘[]𝜀(𝜔)(2𝑘+1)!𝑚=0(1)𝑚𝐴2𝑚(𝜔,𝑥)cos2𝑚[]𝜀(𝜔)(2𝑚)!+cos2[]𝜀(𝜔)𝑛=0𝐴2𝑛(𝜔,𝑥)sin2𝑛[]𝜀(𝜔)(2𝑛)!𝑝=0(1)𝑝𝐴2𝑝(𝜔,𝑥)cos2𝑝[]𝜀(𝜔),[][](2𝑝+1)!𝑄(𝜔,𝑥)=sin𝜀(𝜔)cos𝜀(𝜔)𝑘=0𝐴2𝑘(𝜔,𝑥)sin2𝑘[]𝜀(𝜔)(2𝑘)!𝑚=0(1)𝑚𝐴2𝑚(𝜔,𝑥)cos2𝑚[]𝜀(𝜔)(2𝑚+1)!𝑛=0𝐴2𝑛(𝜔,𝑥)sin2𝑛[]𝜀(𝜔)(2𝑛+1)!𝑝=0(1)𝑝𝐴2𝑝(𝜔,𝑥)cos2𝑝[]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(𝐿𝐶)1/2(0𝑥) can be made arbitrarily small by selecting 0 such that 𝐴(𝜔,)|max=[2𝜋𝑓(𝐿𝐶)1/2]|max01 or equivalently 01/|Γ(𝑗2𝜋𝑓max)|. 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: 𝐴𝑅(𝜔,𝑥)12(𝜔,𝑥)6[]𝐴cos2𝜀(𝜔)4(𝜔,𝑥)48sin2[]𝜔2𝜀(𝜔)12𝐿𝐶0𝑥26,𝐴𝑄(𝜔,𝑥)2(𝜔,𝑥)6[]𝜔sin2𝜀(𝜔)2𝐿𝐶0𝑥26[]𝜔sin2𝜀(𝜔)2𝐿𝐶0𝑥23𝜔𝜀(𝜔)=6𝐿𝐶0𝑥2𝑅𝐿+𝐺𝐶=𝜔60𝑥2𝑅𝐶+𝐺𝐿1.(2.33)

For 𝑥=0, from (2.31) and (2.33) it finally follows: []sinh𝜃(𝑗𝜔,0)𝜔𝜃(𝑗𝜔,0)=𝑅(𝜔,0)+