Research Article | Open Access
D. B. Kandić, B. D. Reljin, I. S. Reljin, "On Modelling of Two-Wire Transmission Lines with Uniform Passive Ladders", Mathematical Problems in Engineering, vol. 2012, Article ID 351894, 42 pages, 2012. https://doi.org/10.1155/2012/351894
On Modelling of Two-Wire Transmission Lines with Uniform Passive Ladders
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.
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 , and or , , and are denoted: the electric permittivity, the magnetic permeability, and the specific electric conductivity of conductor (“”) or dielectric (“”), respectively. Nevertheless, no matter what type of transmission line is considered, each line segment (section) with physical length , 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 , , , and : 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 , 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 . Dielectrics are assumed isotropic, linear, and homogeneous and, if imperfect, linear in ohmic sense, with constant specific electric conductivity .
By neglecting the proximity effect with assumption , edge effect and taking , the per-unit-length capacitance and the per-unit-length dielectric conductance of two-wire line in Figure 1(a) are calculated according to the following relations :
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 and the frequency-dependent per-unit-length inner inductance -for , where: , , , and “Bessel real” ber() and “Bessel imaginary” bei() are the Bessel-Kelvin functions with the first-order derivatives and , respectively, at the point , which can be approximated at high frequencies (i.e., for ) with ,
The overall per-unit-length resistance and the inductance of two-wire line are where is the surface resistance of line conductors and is the external per-unit-length inductance of two-wire line. Throught the paper it will be assumed that .
Since the following expansions hold for any frequency [Hz] (i.e., for any ) , then by using (2.2) and (2.7) when [Hz], the following consequences are easily obtained: , , , . Automatic, fast, and accurate numerical calculation of Bessel-Kelvin functions (2.7) imposes a real need to distinguish between the following two cases of approximation, depending on magnitude of ,
Case A .
Case B ().
Define firstly the following set of auxiliary functions: and, also, define another set of auxiliary functions: then, the values of Bessel-Kelvin functions can be efficiently calculated  by using the relations:
To resume our investigation, consider two-wire line with copper conductors and polyethylene dielectric, where and (Figure 1(a)). Let the operating frequency range of this line be [GHz]. The specific electric conductivity of copper is [S/m] and magnetic permeability is . At the temperature [K] the relative permittivity of polyethylene is (for frequencies up to 25 [GHz]) and its specific electric conductivity is [S/m]. From (2.1) it is calculated [pF/m] and [fS/m]. At frequency [Hz] the per-unit-length resistance of this line is [Ω/m] and its per-unit-length inductance is [μH/m]. At frequency [GHz] it is calculated from (2.2) and (2.9)–(2.11): [Ω/m] ≫ and [μH/m], and for the current wavelength it is obtained [cm]. In Figures 3, 4, 5, and 6 the variations of , and are depicted, respectively, in the frequency range [MHz], whereas the variations of these quantities in the frequency range [GHz] are depicted in Figures 7, 8, 9, and 10, respectively. Let the frequency spectrum of the signal being transmitted is ( is the central frequency of the signal spectrum and the signal bandwith). If the integral of function in Figure 5 taken between and is less than 0.01, then we see from Figure 3 that it may be taken , for . And by using Figure 5 we obtain in the most conservative approach that [kHz]. Similarly, if the integral of function in Figure 9 taken between and is less than 0.1, then we see from Figure 5 that it, also, holds , for . And by using Figure 9 we obtain in the most conservative approach that [kHz].
For a lossless transmission line ( and [S/m]) with linear and homogeneous dielectric the phase-velocity of electromagnetic perturbation (i.e., the propagation speed of current wave in the line) and characteristic impedance are given by the following relations [1–3]: where is relative permittivity and relative permeability of dielectric ( [F/m] is permittivity and [H/m] permeability of vacuum). The characteristic impedance of a lossy transmission line is generally defined as . In the case considered, on Figures 11 and 12 the variations of and in the frequency range [GHz] are respectively depicted. In older telephony applications at lower frequencies, 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 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 [Ω] and the propagation speed approximately . For the coaxial lines the standard is 50 or 75 [Ω] and their propagation speed is approximately . For the printed transmission lines, is in the range [Ω], and their propagation speed is approximately .
From the numerical data associated with the monotonic functions in Figures 13, 14, and 16 it is obtained , , , and . Herefrom and from (2.15) it follows . Since in this case the Heaviside’s condition  is not satisfied, distortionless transmission is not possible. Another two functions, and [“” 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 [GHz]. From data associated with these functions it is obtained: (10 MHz) ≈ 0.319, Λ (3 GHz) ≈ 94.598, [deg] and (3 GHz) ≈ 89.953 [deg].
Since and since in the range [GHz] it holds: , , and [deg], then: and , where the deviation angle can be approximated (The percentage error of this approximation is positive and <0.032% in the entire frequency range [GHz].) with,
For the transmission line with length ℓ let us define the functions: and , where . For this line, in frequency range [GHz] we have and —whereby becomes almost pure imaginary number. We could have obtained this result in a different way. To se that, let us write , where it holds, The previous two functions are depicted in Figures 19 and 20 in the frequency range [GHz].
From relations (2.17) we obtain the approximations and of and , respectively, in the frequency range [GHz], since there it holds and ,
We will now emphasize the importance of function in the following.
Constituting of functions and that play fundamental role in producing uniform three-terminal networks nominally equivalent to short-line segments  and in realization of these networks in the specified frequency range by approximately equivalent three-terminal lumped 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, 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 , , , and from (A.17) and from (A.18), in general, and for the finite length open-circuited line , in particular, by using expansions: thus placing into evidence the pole-zero location of , , and .
For the open-circuited two-wire line with length [m] in Figures 23 ÷ 26 are depicted for [m] and [GHz] the variations of , [deg], and [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 the diagrams analogous to those in Figures 23 ÷ 26 could also be drawn easily, provided that the impedance is taken into account [see relation (A.17)].
When the line is sufficiently short, some approximations can be made leading to satisfactory results without need to cope with the cumulative products (2.21). To see that, suppose that line length is [mm] and assume, say, [mm]. Recall that , then take (2.21) and write
The infinite complex series (2.22) in almost pure imaginary are convergent for and [GHz]. If is sufficiently less than and , then by retaining only the first five terms in (2.22) the following small-error approximations are produced
For the open-circuited two-wire line with [m], in Figures 27, 28, 29, and 30 they are depicted on grid(x) × grid(f) = 40 × 60 in the range [GHz] and range [m], respectively:(i) the voltage-transmittance magnitude approximation percentage error:;(ii) the voltage-transmittance phase approximation absolute error:;(iii) the current-transmittance magnitude approximation percentage error:;(iv) the current-transmittance phase approximation absolute error:.
It can be observed in Figures 27 ÷ 30 that in the given range of and , the errors ER1 and ER3 are negative ( and ), whereas the errors ER2 and ER4 are positive ( [deg] and [deg]). The upper limit of is lower than of and the upper limit of ER4 is lower than of ER2.
If is sufficiently small (), from (2.24) further approximations are obtained:(i)which partly resembles to Maclaurin’s expansion of sinh(θ), that is, ,(ii)which, partly resembles to Maclaurin’s expansion of cosh(θ), that is, .(iii) combining (i) and (ii) it follows that
The functions and play fundamental role in effort to transform short transmission line segments into equivalent lumped three-terminal RLC networks . The same role is played their respective approximating functions and , obtained when . For two-wire line with length [mm], in Figures 31, 32, 33, and 34, the magnitude approximation absolute error and three percentage magnitude approximation errors: , and , are depicted on grid(x) × grid(f) = 40 × 60 in the frequency range [GHz] and range of [mm]. Obviously, all these errors can be kept arbitrarily small in magnitude in the entire frequency range [GHz] if sufficiently small step of uniform line segmentation is applied. The key action in achieving the previous goals is providing the maximum of to be much less than . For example, from the numerical data associated with Figure 17, it can be calculated that 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 , which is at least ten times less than .
The equations (A.15) and (A.16) offer an opportunity to view on a transmission line segment with length as on a linear two-port network (Figure 35(a)) with boundary conditions and at the input and and at the output. The chain-matrix of this network reads
Denote , and . In study of short transmission lines it is found convenient to replace them, either with nominally equivalent networks (Figure 35(b)) or with nominally equivalent networks (Figure 35(c)) , whose immitances are given as follows The aforementioned criterion for selection of relies on attempt to find convenient that provides physical realizability of immitances , , , and 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 . Observe that the immitances and are realizable by trivial two-element-kind RLC networks. Nevertheless, we will show now that, in general, the imitances , , , and are not realizable by lumped RLC networks, except in the limiting case when is as small, so that the complex approximations hold: and (observe that if , then and and and ). To see that, recall that for , in frequency range [GHz] it holds(a), ,(b), [deg],(c), where,