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

Authors: D. B. Kandić; B. D. Reljin; I. S. Reljin
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

In the paper we presented new results in incremental network modelling of two-wire lines in frequency range [0,3] [GHz], by the uniformRLCG 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.


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## Body

## 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 uniformRLCladders with frequency-dependent parameters, turning thuswith the problem of analysis of real two-wire lines into the analysis problem of high-order passive RLCnetworks 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 (Figure1(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  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 σd≪σc.(a) Two-wire line, (b) coaxial line, and (c) twisted pair.
(a)
(b)
(c)Figure 2
The approximate, incremental, and lumped network model of the uniform transmission line.By neglecting the proximity effect with assumptiond≫2a, edge effect and taking σd≪σc, 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]:
(2.1)C′=π·εdln(d/a),G′=σdεd·C′=π·σdln(d/a),(d≫2a∧σd≪σc).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],
(2.2)Ri′(f)=k2π·a·σc·Re[-bei(k·a)+j·ber(k·a)ber′(k·a)+j·bei′(k·a)]=k2π·a·σc·ber(k·a)·bei′(k·a)-bei(k·a)·ber′(k·a)[ber′(k·a)]2+[bei′(k·a)]2,Li′(f)=k2π·a·σc·ω·Im[-bei(k·a)+j·ber(k·a)ber′(k·a)+j·bei′(k·a)]=k2π·a·σc·ω·ber(k·a)·ber′(k·a)+bei(k·a)·bei′(k·a)[ber′(k·a)]2+[bei′(k·a)]2,
where: j=-1,  ω=2π·f,  k=ω·μc·σc , 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)ber(k·a)=G·cos(k·a2-π8),bei(k·a)=G·sin(k·a2-π8),G=12π·k·a·e(k·a)/2,ber′(k·a)=G·cos(k·a2+π8),bei′(k·a)=G·sin(k·a2+π8).From (2.3) it should be firstly noticed that at high frequencies (i.e., for k·a≫1) it holds
(2.4)-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)cos((k·a)/2+π/8)+j·sin((k·a)/2+π/8)=j·ej·((k·a)/2-π/8)ej·((k·a)/2+π/8)=ej·(π/4),
and then from (2.2) it may be obtained consecutively for k·a≫1,
(2.5)Ri′(f)=k2π·2·a·σc=12π·a·π·f·μcσc,Li′(f)=k2π·2·ω·a·σc=14π·a·μcπ·f·σc.The overall per-unit-length resistanceR′(f) and the inductance L′(f) of two-wire line are
(2.6)R′(f)=2·Ri′(f)|∀f,R′(f)=1π·a·π·f·μcσc=Rs(f)π·a|k·a≫1,Rs(f)=π·f·μcσc,L′(f)=Le′+2·Li′(f)|∀f,L'(f)=μdπ·ln(da)+12π·a·μcπ·f·σc|k·a≫1(d≫2a),
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 μc≈μd≈μ0.Since the following expansions hold for any frequencyf∈[0,∞) [Hz] (i.e., for any k·a) [4],
(2.7)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)1222·42·62·82·102·122±…,bei(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)1022·42·62·82·102∓…,ber′(k·a)=-(k·a)322·4+(k·a)722·42·62·8-(k·a)1122·42·62·82·102·12+(k·a)1522·42·62·82·102·122·142·16∓…,bei′(k·a)=k·a2-(k·a)522·42·6+(k·a)922·42·62·82·10-(k·a)1322·42·62·82·102·122·14±…,
then by using (2.2) and (2.7) when f→0 [Hz], the following consequences are easily obtained: Ri′(f)→1/(σc·π·a2), Li′(f)→μc/(8·π), R′(f)→2/(σc·π·a2), L′(f)→(μd/π)·ln(d/a)+μc/(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·μc·σc∈[0,8]).
(2.8)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)28+ε1(|ε1|<10-9),bei(z)=16·(z8)2-113.77777774·(z8)6+72.81777742·(z8)10-10.56765779·(z8)14+0.52185615·(z8)18-0.01103667·(z8)22+0.00011346·(z8)26+ε2(|ε2|<6·10-9),ber′(z)=z·[-4·(z8)2+14.22222222·(z8)6-6.06814810·(z8)10+0.66047849·(z8)14ssssssss-0.02609253·(z8)18+0.00045957·(z8)22-0.00000394·(z8)26]+ε3(|ε3|<2.1·10-8),bei′(z)=z·[12-10.66666666·(z8)4+11.37777772·(z8)8-2.31167514·(z8)12ssssssss+0.14677204·(z8)16-0.00379386·(z8)20+0.00004609·(z8)24]+ε4(|ε4|<7·10-8).Case  B (z=z(f)=a·k=a·2·π·f·μc·σc∈(8,∞)).Define firstly the following set of auxiliary functions:(2.9)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)6,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)6,β(z)=(0.7071068+0.7071068·j)+(-0.0625001-0.0000001·j)·(8z)+(-0.0013813+0.0013811·j)·(8z)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,β(-z)=(0.7071068+0.7071068·j)+(0.0625001+0.0000001·j)·(8z)+(-0.0013813+0.0013811·j)·(8z)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.
and, also, define another set of auxiliary functions:
(2.10)f(z)=π2·z·exp[-1+j2·z+α(-z)],g(z)=12·π·z·exp[1+j2·z+α(z)],
then, the values of Bessel-Kelvin functions can be efficiently calculated [5] by using the relations:
(2.11)ber(z)=Re[jπ·f(z)+g(z)],bei(z)=Im[jπ·f(z)+g(z)],ber′(z)=Re[-jπ·f(z)·β(-z)+g(z)·β(z)],bei′(z)=Im[-jπ·f(z)·β(-z)+g(z)·β(z)].For the coaxial line with lengthℓ (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)C′=2π·εdln(b/a),G′=σdεd·C′=2π·σdln(b/a),(σd≪σc).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)R′(f)=Rs(f)2π·(1a+1b)=12π·π·f·μcσc·(1a+1b),L′≈Le′=μd2π·ln(ba),(μc≈μd≈μ0).The twisted-pair (Figure1(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 Z_0 [3, 6].To resume our investigation, consider two-wire line with copper conductors and polyethylene dielectric, wherea=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 σc≈5.81·107  [S/m] and magnetic permeability is μc≈μ0. At the temperature T=298 [K] the relative permittivity of polyethylene is εr≈2.26 (for frequencies up to 25 [GHz]) and its specific electric conductivity is σd≈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/(σc·π·a2)=1.0957 [Ω/m] and its per-unit-length inductance is L′(0)=(μd/π)·ln(d/a)+μc/(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·(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 [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
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f  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 Z_0 are given by the following relations [1–3]:
(2.14)c=1L′·C′=1εd·μd=c0εr·μr,c0=1ε0·μ0≈3·108[ms],Z_0=L′C′=1c·C′=μrεr·1π·μ0ε0·ln(da)≈120·μrεr·ln(da)[Ω],
where εr=εd/ε0 is relative permittivity and μr=μd/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 Z_0(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)=|Z_0(j·2π·f)| and ζ(f)=Arg[Z_0(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
Z
0
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) and dZ0(f)/df for the considered two-wire line.Figure 12
ζ
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) and dζ(f)/df for the considered two-wire line.For two-wire line being considered, in Figures13, 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)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).Figure 13
A dimensionless parameterϕ1(f) of two-wire line.Figure 14
A dimensionless parameterϕ2(f) of two-wire line.Figure 15
Parameterϕ3(f) of two-wire line.Figure 16
A dimensionless parameterϕ4(f) of two-wire line.From the numerical data associated with the monotonic functions in Figures13, 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).SinceΓ(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,
(2.16)χ(f)=π2-ϑ(f)=π2-12·atan(2π·f·L′(f)/R′(f)+C′/G′1-4·π2·f2((L′(f)·C′)/(R′(f)·G′)))≈12·atan(R′(f)/L′(f)+G′/C′2π·f)≈R′(f)/L′(f)+G′/C′4π·f.For the transmission line with lengthℓ 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,
(2.17)a(f)=12·L′(f)·C′{[(2π·f)2+R′(f)2L′(f)2]·[((2π·f)2+G'2C'2)]+R′(f)L′(f)·G′C′-(2π·f)2}attenuation“constant”,b(f)=12·L′(f)·C′{[(2π·f)2+R′(f)2L′(f)2]·[((2π·f)2+G'2C'2)]-R′(f)L′(f)·G′C′+(2π·f)2}phase“constant”.
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,
(2.18)aa(f)≈12[R′(f)·C′L′(f)+G′·L′(f)C′],ab(f)≈L′(f)·C′(f){2π·f+116π·f·[R′(f)L′(f)-G′C′]2},and,also,wehave,Z0(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)-G′C′]}.The functionsaa(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 functionθ=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,
(2.19)∂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).
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 ofM(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:
(2.20)M(s,x)=U(s,x)U(s,0)=∏n=1∞{1+[(2Γ·(ℓ-x))/((2n-1)·π)]2}∏n=1∞{1+[(2Γ·ℓ)/((2n-1)·π)]2},N(s,x)=I(s,x)I(s,0)=ℓ-xℓ·∏n=1∞{1+[(Γ·(ℓ-x))/(n·π)]2}∏n=1∞[1+((Γ·ℓ)/(n·π))2],Z(s,x)=U(s,x)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[ZL(s)→∞],
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,
(2.21)sinh(θ)θ=∏n=1∞(1+θ2n2·π2),cosh(θ)=∏n=1∞[1+4·θ2(2n-1)2·π2].For the open-circuited two-wire line with lengthℓ=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 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)].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
(2.22)sinh(θ)θ=∏n=1∞(1+θ2n2·π2)=exp[∑n=1∞ln(1+θ2n2·π2)]=[expθ2π2·∑n=1∞1n2-θ42·π4·∑n=1∞1n4+θ63·π6·∑n=1∞1n6-θ84·π8·∑n=1∞1n8ss+θ105·π10·∑n=1∞1n10±….]=exp(θ26-θ4180+θ62835-θ837800+θ10467775±⋯),cosh(θ)=∏n=1∞[1+4·θ2(2n-1)2·π2]=exp{∑n=1∞ln[1+4·θ2(2n-1)2·π2]}=[exp4·θ2π2·∑n=1∞1(2n-1)2-8·θ4π4·∑n=1∞1(2n-1)4+64·θ63·π6·∑n=1∞1(2n-1)6ss-64·θ8π8·∑n=1∞1(2n-1)8+1024·θ105·π10·∑n=1∞1(2n-1)10±⋯]=exp(θ22-θ412+θ645-17·θ82520+31·θ1014175±⋯).The following finite sums of infinite series [9] have been exploited in (2.22),
(2.23)A2p=∑n=1∞1n2p(p=1,5¯),A2=π26,A4=π490,A6=π6945,A8=π89450,A10=5·28·π1033·10!,B2p=∑n=1∞1(2n-1)2p(p=1,5¯),B2=π28,B4=π496,B6=π6960,B8=17·π8161280,B10=31·π1035·81·210.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
(2.24)sinh(θ)≈sinhA(θ)=θ·exp(θ26-θ4180+θ62835-θ837800+θ10467775),cosh(θ)≈coshA(θ)=exp(θ22-θ412+θ645-17·θ82520+31·θ1014175).For the open-circuited two-wire line withℓ=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(j·2π·f,x)/coshA(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[coshAθ(j·2π·f,x)·cosh(j·2π·f,0)/coshA(j·2π·f,0)·coshθ(j·2π·f,x)];(iii)
the current-transmittance magnitude approximation percentage error:ER3(f,x)={|[sinhAθ(j·2π·f,x)/sinhA(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[sinhAθ(j·2π·f,x)·sinh(j·2π·f,0)/sinhA(j·2π·f,0)·sinhθ(j·2π·f,x)].Figure 27
Voltage-transmittance magnitude approximation percentage errorER1(f,x).Figure 28
Voltage-transmittance phase approximation absolute errorER2(f,x).Figure 29
Current-transmittance magnitude approximation percentage errorER3(f,x).Figure 30
Current-transmittance phase approximation absolute errorER4(f,x).It can be observed in Figures27 ÷ 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)
(2.25)sinh(θ)≈sinhA(θ)=θ·exp(θ26-θ4180+θ62835-θ837800+θ10467775)≈θ·(1+θ26-θ4180+θ62835-θ837800+θ10467775)=θ+θ36-θ5180+θ72835-θ937800+θ11467775≈θ+θ36,which partly resembles to Maclaurin’s expansion of sinh(θ), that is, sinh(θ)=θ+θ3/3!+θ5/5!+…,(ii)
(2.26)cosh(θ)≈coshA(θ)=exp(θ22-θ412+θ645-17·θ82520+31·θ1014175)≈1+θ22-θ412+θ645-17·θ82520+31·θ1014175≈1+θ22,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(2.27)tanh(θ2)≈tanhA(θ2)=θ·(θ2+24)6·(θ2+8).The functionssinh(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 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·fmax)(⇔|θmax|≪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·fmax)|.Figure 31
Magnitude approximation absolute errorER5(f,x).Figure 32
Percentage magnitude approximation errorER6(f,x).Figure 33
Percentage magnitude approximation errorER7(f,x).Figure 34
Percentage magnitude approximation errorER8(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
(2.28)[U(s,0)I(s,0)]=F·[U(s,ℓ0)I(s,ℓ0)],F=[cosh(Γ·ℓ0)Z0·sinh(Γ·ℓ0)sinh(Γ·ℓ0)Z0cosh(Γ·ℓ0)].(a) Linear network model of a short transmission line segment and (b) its linear network structure.
(a)
(b)
(c)DenoteZ(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
(2.29)Z′=Z·{tanh[θ(s,0)/2]θ(s,0)/2},Y′=Y·{sinh[θ(s,0)]θ(s,0)},Z′′=Z·{sinh[θ(s,0)]θ(s,0)},Y′′=Y·{tanh[θ(s,0)/2]θ(s,0)/2}.
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 lumpedRLC 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,
(2.30)ε(ω)=π2-arg[Γ(j·ω)]=π2-12·atan((ω·L′)/R′+(ω·C′)/G′1-(L′·C′)/(R′·G′)·ω2)≈12·atan(R′/L′+G′/C′ω)≈R′/L′+G′/C′2·ω.(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)
Sincex∈[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 forsinh[θ(j·ω,x)]/θ(j·ω,x) the following:
(2.31)sinh{A(ω,x)·sin[ε(ω)]}·cos{A(ω,x)·cos[ε(ω)]}A(ω,x)·{sin[ε(ω)]+j·cos[ε(ω)]}+j·cosh{A(ω,x)·sin[ε(ω)]}·sin{A(ω,x)·cos[ε(ω)]}A(ω,x)·{sin[ε(ω)]+j·cos[ε(ω)]},
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)R(ω,x)=sin2[ε(ω)]·(∑k=0∞A2k(ω,x)·sin2k[ε(ω)](2k+1)!)·(∑m=0∞(-1)mA2m(ω,x)·cos2m[ε(ω)](2m)!)+cos2[ε(ω)]·(∑n=0∞A2n(ω,x)·sin2n[ε(ω)](2n)!)·(∑p=0∞(-1)pA2p(ω,x)·cos2p[ε(ω)](2p+1)!),Q(ω,x)=sin[ε(ω)]·cos[ε(ω)]·{(∑k=0∞A2k(ω,x)·sin2k[ε(ω)](2k)!)·(∑m=0∞(-1)mA2m(ω,x)·cos2m[ε(ω)](2m+1)!)sss-(∑n=0∞A2n(ω,x)·sin2n[ε(ω)](2n+1)!)·(∑p=0∞(-1)pA2p(ω,x)·cos2p[ε(ω)](2p)!)}.
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·fmax)|. 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:
(2.33)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)26,Q(ω,x)≈A2(ω,x)6·sin[2·ε(ω)]≈ω2·L'·C′·(ℓ0-x)26·sin[2·ε(ω)]≈ω2·L′·C′·(ℓ0-x)23·ε(ω)=ω6·L′·C′·(ℓ0-x)2·(R′L′+G′C′)=ω6·(ℓ0-x)2·(R'·C'+G'·L′)≪1.Forx=0, from (2.31) and (2.33) it finally follows:
(2.34)sinh[θ(j·ω,0)]θ(j·ω,0)=R(ω,0)+j·Q(ω,0)≈1+j·ω6·(R′·C′+G′·L′)·ℓ02-ω2·L'·C'·ℓ026,(2.35)sinh[θ(j·ω,0)]θ(j·ω,0)→j·ω→s(-ω2→s2)Byanalyticcontinuationsinh[θ(s,0)]θ(s,0)≈1+s6·(R′·C′+G′·L′)·ℓ02+s2·L′·C′·ℓ026.Now, by using (2.29) and (2.35) we may generate the immitances Y’ and Z’’ (Figures 35(b) and 35(c)),
(2.36)Y′(s)=Y(s)·sinh[θ(s,0)]θ(s,0)=(G′+C′·s)·ℓ0·[1+s6·(R′·C′+G′·L′)·ℓ02+s2·L′·C′·ℓ026],(2.37)Z′′(s)=Z(s)·sinh[θ(s,0)]θ(s,0)=(R′+L′·s)·ℓ0·[1+s6·(R′·C′+G′·L′)·ℓ02+s2·L′·C′·ℓ026],
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)]:
(2.38)Y′(s)=(G′+C′·s)·ℓ0,Z′′(s)=(R′+L′·s)·ℓ0.
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).
(2.39)Z′(s)=(R′+L′·s)·ℓ0,Y′′(s)=(G′+C′·s)·ℓ0.


## 3. Approximation of Two-Wire Line by UniformRLCG Ladder and the Simulation Results

Let us consider a short two-wire line with lengthℓ=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 uniformRLCG ladder purporting to represent the transmission line with length ℓ in frequency range f∈[f0-B/2,f0+B/2].Figure 36
The functionl(f)=ϕ3(f) of the considered two-wire line (determination of ℓ0).Assumeδx=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=GΓ(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 ui be the voltage of the point Ni(i=1,41¯), with respect to the common node 0 (Figure 37).Table 1
Two-wire copper-polyethylene transmission line (central frequencyf0=1 [GHz]).
Electrical parameters ofRLCG ladder sections
R/20 [mΩ]
L/20 [nH]
C/10 [fF]
G/10 [aS]
39.772
2.219
51.123
2.554Figure 37
Electrical network model of two-wire line with lengthℓ=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 Figure37 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 xm=(m-1)·ℓ0/2(m=1,21¯) from the line sending end are presented. To these points correspond the points nodes N2m-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 Figure37.
Node
N
1
N
3
N
5
N
7
N
9
N
11
N
13
N
15
N
17
N
19
Amplitude [V] simulation
1.0000
1.06582
1.12687
1.18792
1.24365
1.29938
1.34929
1.39919
1.44283
1.48647
Phase [mdeg] simulation
0.0000
−15.1412
−27.8853
−39.3196
−48.9810
−57.8137
−65.2500
−72.1558
−77.9040
−83.3147
Amplitude [V] exact analysis
1.0000
1.06455
1.12672
1.18637
1.24337
1.297580
1.348883
1.397165
1.442318
1.484240
Phase [mdeg] exact analysis
0.0000
−14.9246
−27.8250
−39.0532
−48.8790
−57.5121
−65.1181
−71.8286
−77.7501
−82.9687
Node
N
21
N
23
N
25
N
27
N
29
N
31
N
33
N
35
N
37
N
39
Amplitude [V] simulation
1.52345
1.56043
1.59042
1.62041
1.64314
1.66587
1.68114
1.69641
1.70407
1.71174
Phase [mdeg] simulation
−87.7248
−91.9260
−95.2320
−98.4156
−100.775
−103.070
−104.588
−106.079
−106.821
−107.556
Amplitude [V] exact analysis
1.522839
1.558028
1.589727
1.617867
1.642383
1.663221
1.680334
1.693685
1.703242
1.708985#
Phase [mdeg] exact analysis
−87.5547
−91.5659
−95.0498
−98.0454
−100.584
−102.693
−104.391
−105.697
−106.621
−107.172&
The amplitude (#) and the phase (&) of the voltage at the end of transmission line (N41) obtained by exact analysis are 1.710901 [V] and −107.3554 [mdeg], respectively.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 frequencyf0=1 [GHz], amplitude Em=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.The structure of the three types of hierarchical blocks HB1, HB2, and HB3 used in PSPICE simulation.
(a)
(b)
(c)From the data associated with functions depicted in Figures12 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 blockHB2/5 in Figure 38(c) is terminated with impedance, ZL(j·2π·f0)=RL+1/(j·2π·f0·CL)=Z0(j·2π·f0) and find RL=294.6728 [Ω] and CL=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 Em·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 Em·exp[-x·a(f0)], φ(x) and τ(x) at the points on line x=xk=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 ui(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 x600·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)):(3.1)Em=10[V],f0=1[GHz],a(f0)=0.044990161[1m],b(f0)=31.551675282[1m],e=Em·sin(2π·f0·t),ui=Uim·sin(2π·f0·t-φi)=Uim·sin[2π·f0·(t-τi)],Uim=Em·exp[-x600·i·a(f0)]=10·(0.960317668)i[V],φi=φ(x600·i)=300·i·ℓ0·b(f0)=28.39650775·i[rad],τi=τ(x600·i)=φi2π·f0=4.519444575·i[ns],(i=1,5¯).Table 3
The steady-state results of exact analysis and simulation forRLCG lader in Figure 38(c).
Distance @ voltage
Amplitude [V] simulation
Amplitude [V] exact analysis
Phase delayφ [deg] simulation
Phase delayφ [deg] exact analysis
Time delayτ[ns] simulation
Time delayτ [ns] exact analysis
90 cm @u1
9.60377
9.60317
187.60752
187.00004
4.4930
4.51944
180 cm @u2
9.22326
9.22210
15.21326
14.00009
9.0066
9.03888
270 cm @u3
8.85778
8.85614
202.81711
201.00014
13.4584
13.55833
360 cm @u4
8.50664
8.50471
30.41907
28.00018
18.0132
18.07777
450 cm @u5
8.16922
8.16722
218.01932
215.00023
22.4650
22.59722From Table3 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 Figure39 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 ofRLCGladder 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 ofRLCG elements.(ii)
Delayed and slightly attenuated signalsu1÷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 Zc(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 usingRLCG 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 lengthℓ=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/(εr)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):
(3.2)∂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),
we obtain the second-order, hyperbolic, partial differential equation (PDE) equivalent to (3.2):
(3.3)∂2u-(τ,x)∂x2=∂2u-(τ,x)∂τ2+C(f0)·∂u-(τ,x)∂τ+B(f0)·u-(τ,x),u(t,x)=u[τA(f0),x]=u-(τ,x),x∈[0,ℓ],A(f0)=1L'(f0)·C'[ms],B(f0)=R'(f0)·G'[1m2],C(f0)=R'(f0)·C'L'(f0)+G'·L'(f0)C′[1m],
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]}:
(3.4)∂2u--(τ,x)∂x2=∂2u--(τ,x)∂τ2+D(f0)·u--(τ,x),D(f0)=14·(R'(f0)·C′L′(f0)-G′·L′(f0)C′)2=2.0241·10-3[1m2],
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 inτ and x:
(3.5)v--(τ,x)=∂u--(τ,x)∂τ=[12·C(f0)·u(t,x)+1A(f0)·∂u(t,x)∂t]·exp[E(f0)·t]|t=τ/A(f0),E(f0)=12·[R′(f0)L′(f0)+G′C′]=8.959[MHz].If the excitation ise(t)=Em·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:
(3.6)∂u--(τ,x)∂τ=v--(τ,x)∧∂v--(τ,x)∂τ=∂2u--(τ,x)∂x2+D(f0)·u--(τ,x)⟵ThesystemofPDEs,u--(0,x)=∥0ifx>0e(0)otherwise∧v--(0,x)=∥0ifx>0[12·C(f0)·e(0)+2π·f0·Em·cos(φ)A(f0)]otherwise⟵theinitialconditions,u--(τ,0)=exp[12·C(f0)·τ]·e[τA(f0)]∧u--(τ,ℓ)=0⟵theboundaryconditions,u(t,x)=exp[-E(f0)·t]·u--[A(f0)·t,x]⟵Solutionof(3.2)inintervalt∈[0,22.59][ns],x∈[0,4.5][m].In Figure40 the solutions u(t,xk) of (3.2) in the interval t∈[0,T] for xk=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 lengthℓ=4.5 [m] (Em=10 [V] and φ=0).In Figure41 the pulse responses [i.e., the voltages uk(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 uk(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 Figure38(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 RLCGladders with frequency-dependent RLparameters, 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 RLCGnetworks 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 RLCGladders as the approximate incremental models of two-wire lines.



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