Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2012, Article ID 351894, 42 pages
doi:10.1155/2012/351894
Research Article
On Modelling of Two-Wire Transmission Lines
with Uniform Passive Ladders
D. B. Kandić,1 B. D. Reljin,2 and I. S. Reljin3
1
Department of Physics & Electrical Engineering, School of Mechanical Engineering,
University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia
2
Department of General Electrical Engineering, School of Electrical Engineering, University of Belgrade,
Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
3
Department of Telecommunications, School of Electrical Engineering, University of Belgrade,
Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
Correspondence should be addressed to D. B. Kandić, dbkandic@gmail.com
Received 23 March 2012; Accepted 1 May 2012
Academic Editor: Kue-Hong Chen
Copyright q 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.
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
2
Mathematical Problems in Engineering
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 twowire 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 1a, coaxial line Figure 1b and twisted pair Figure 1c.
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 and the considered frequency range will be
f ∈ 0, 3GHz. Dielectrics are assumed isotropic, linear, and homogeneous and, if imperfect,
linear in ohmic sense, with constant specific electric conductivity σd σc .
By neglecting the proximity effect with assumption d 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 1a are calculated according to the following relations 1:
C π · εd
,
lnd/a
G σd
π · σd
,
· C εd
lnd/a
d 2a ∧ σd σc .
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
Mathematical Problems in Engineering
3
Dielectric
en
t
x-axis
2
ur
r
en
t
C
ur
r
i(t, x)
on
du
ct
or
C
C
on
du
ct
or
1
εd , μd , σd
C
ε c , μc , σ c
ℓ
2a
2a
0
d
a
or
uct
ond
2
C
r1
cto
du
n
Co
Dielectric
εd , μd , σd
εc , μc , σc
a
b
ℓ
ℓ
b
c
Figure 1: a Two-wire line, b coaxial line, and c twisted pair.
i(t, x + δx)
i(t, x)
···
R′ · δx/2
u(t, x)
L′ · δx/2
G′ ·δx
R′ · δx/2
C′ ·δx
···
L′ · δx/2
u(t, x + δx/2)
u(t, x + δx)
e
···
0
···
x
x + δx/2
x + δx
x-axis
Figure 2: The approximate, incremental, and lumped network model of the uniform transmission line.
resistance Ri f and the frequency-dependent per-unit-length inner inductance Li f-for
f ∈ 0, ∞Hz,
−beik · a j · berk · a
k
· Re
Ri f 2π · a · σc
ber k · a j · bei k · a
berk · a · bei k · a − beik · a · ber k · a
k
·
,
2 2
2π · a · σc
ber k · a bei k · a
4
Li f Mathematical Problems in Engineering
−beik · a j · berk · a
k
· Im
2π · a · σc · ω
ber k · a j · bei k · a
berk · a · ber k · a beik · a · bei k · a
k
·
,
2 2
2π · a · σc · ω
ber k · a bei k · a
2.2
√
√
where: j −1, ω 2π · f, k ω · μc · σc , and “Bessel real” berk · a and “Bessel
imaginary” beik · 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,
k·a π
k·a π
berk · a G · cos √ −
,
beik · a G · sin √ −
,
8
8
2
2
1
G √
√
· ek·a/ 2 ,
2π · k · a
k·a π
k·a π
ber k · a G · cos √ ,
bei k · a G · sin √ .
8
8
2
2
2.3
From 2.3 it should be firstly noticed that at high frequencies i.e., for k · a 1 it
holds
√
√
− sin k · a/ 2 − π/8 j · cos k · a/ 2 − π/8
√
√
ber k · a j · bei k · a
cos k · a/ 2 π/8 j · sin k · a/ 2 π/8
−beik · a j · berk · a
j·
√
ej·k·a/ 2−π/8
√
ej·k·a/ 2
π/8
2.4
ej·π/4 ,
and then from 2.2 it may be obtained consecutively for k · a 1,
π · f · μc
1
k
·
Ri f ,
√
σc
2π · 2 · a · σc 2π · a
μc
1
k
·
.
Li f √
π · f · σc
2π · 2 · ω · a · σc 4π · a
2.5
The overall per-unit-length resistance R f and the inductance L f of two-wire line
are
R f 2 · Ri f ∀f ,
L f Le 2 · Li f∀f ,
π · f · μc
π · f · μc
Rs f ,
Rs f ,
σc
π · a k·a1
σc
μd
μc
d
1
· ln
·
L f d 2a,
π
a
2π · a
π · f · σc R f 1
·
π ·a
k·a1
2.6
Mathematical Problems in Engineering
5
where Rs f is the surface resistance of line conductors and Le μd /π · lnd/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 frequency f ∈ 0, ∞ Hz i.e., for any
k · a 4,
∞
k · a/24n
k · a4
k · a8
1− 2 2 2 2 2 2
berk · a −1n ·
2
2 ·4
2 ·4 ·6 ·8
2n!
n0
−
beik · a ∞
k · a/24n−2 k · a2
k · a6
−
−1n
1 ·
22
22 · 42 · 62
2n − 1!2
n1
ber k · a −
bei k · a k · a12
± ...,
22 · 42 · 62 · 82 · 102 · 122
k · a10
∓ ...,
22 · 42 · 62 · 82 · 102
3
7
2.7
11
k · a
k · a
k · a
2 2 2
− 2 2 2 2
2
2 ·4
2 · 4 · 6 · 8 2 · 4 · 6 · 8 · 102 · 12
k · a15
∓ ...,
22 · 42 · 62 · 82 · 102 · 122 · 142 · 16
k·a
k · a5
k · a9
− 2 2
2 2 2 2
2
2 · 4 · 6 2 · 4 · 6 · 8 · 10
−
k · a13
± ...,
22 · 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 · lnd/a μc /4 · π. Automatic, fast, and accurate numerical calculation of BesselKelvin 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 zf a · k a · 2 · π · f · μc · σc ∈ 0, 8.
4
8
12
z
z
z
berz 1 − 64 ·
113.77777774 ·
− 32.36345652 ·
8
8
8
16
20
24
z
z
z
2.64191397 ·
− 0.08349609 ·
0.00122552 ·
8
8
8
28
z
− 0.00000901 ·
ε1
|ε1 | < 10−9 ,
8
2
6
10
14
z
z
z
z
beiz 16 ·
− 113.77777774 ·
72.81777742 ·
− 10.56765779 ·
8
8
8
8
18
22
z
z
0.52185615 ·
− 0.01103667 ·
8
8
26
z
0.00011346 ·
ε2
|ε2 | < 6 · 10−9 ,
8
6
Mathematical Problems in Engineering
6
10
14
2
z
z
z
z
14.22222222 ·
− 6.06814810 ·
0.66047849 ·
ber z z · −4 ·
8
8
8
8
18
22
26 z
z
z
−0.02609253 ·
0.00045957 ·
− 0.00000394 ·
8
8
8
ε3
|ε3 | < 2.1 · 10−8 ,
4
8
12
z
z
z
11.37777772 ·
− 2.31167514 ·
8
8
8
16
20
24 z
z
z
0.14677204 ·
− 0.00379386 ·
0.00004609 ·
8
8
8
ε4
|ε4 | < 7 · 10−8 .
bei z z ·
1
− 10.66666666 ·
2
2.8
Case B z zf a · k a · 2 · π · f · μc · σc ∈ 8, ∞.
Define firstly the following set of auxiliary functions:
2
8
8
− 0.0009765 · j ·
z
z
3
4
8
8
−0.0000906 − 0.0000901 · j ·
− 0.0000252 ·
z
z
5
6
8
8
−0.0000034 0.0000051 · j ·
0.0000006 0.0000019 · j ·
,
z
z
2
8
8
− 0.0009765 · j ·
α−z − 0.3926991 · j −0.0110486 0.0110485 · j ·
z
z
3
4
8
8
0.0000906 0.0000901 · j ·
− 0.0000252 ·
z
z
5
6
8
8
0.0000034 − 0.0000051 · j ·
0.0000006 0.0000019 · j ·
,
z
z
8
βz 0.7071068 0.7071068 · j −0.0625001 − 0.0000001 · j ·
z
3
2
8
8
0.0000005 0.0002452 · j ·
−0.0013813 0.0013811 · j ·
z
z
4
5
8
8
0.0000346 0.0000338 · j ·
0.0000017 − 0.0000024 · j ·
z
z
6
8
0.0000016 − 0.0000032 · j ·
,
z
αz − 0.3926991 · j 0.0110486 − 0.0110485 · j ·
Mathematical Problems in Engineering
7
β−z 0.7071068 0.7071068 · j 0.0625001 0.0000001 · j ·
8
z
3
2
8
8
−0.0000005 − 0.0002452 · j ·
z
z
4
5
8
8
0.0000346 0.0000338 · j ·
−0.0000017 0.0000024 · j ·
z
z
6
8
0.0000016 − 0.0000032 · j ·
.
z
2.9
−0.0013813 0.0013811 · j ·
and, also, define another set of auxiliary functions:
fz 1
j
π
· exp − √ · z α−z ,
2·z
2
1
j
1
· exp √ · z αz ,
gz √
2·π ·z
2
2.10
then, the values of Bessel-Kelvin functions can be efficiently calculated 5 by using the
relations:
j
· fz gz ,
π
j
ber z Re − · fz · β−z gz · βz ,
π
berz Re
j
· fz gz ,
beiz Im
π
j
bei z Im − · fz · β−z gz · βz .
π
2.11
For the coaxial line with length Figure 1b, the per-unit-length capacitance C and
the per-unit-length conductance G of dielectric are calculated according to the following
relations 1:
C 2π · εd
,
lnb/a
G σd
2π · σd
,
· C εd
lnb/a
σd σc .
2.12
It has been shown 1–3 that at high frequencies the coaxial line is characterized by the
per-unit-length resistance R f and the per-unit-length inductance L f given with,
R
f
π · f · μc
1
1
1
1 1
s
R f ·
·
,
·
2π
a b
2π
σc
a b
2.13
μ
b
d
· ln
L ≈ Le ,
μc ≈ μd ≈ μ0 .
2π
a
The twisted-pair Figure 1c 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 1a. Let the operating
8
Mathematical Problems in Engineering
2.92
R′ (f) (Ω/m)
2.55
2.19
1.82
1.46
1.1
0
2
4
6
8
Frequency (Hz)
10
×106
Figure 3: R f for line with a 0.1 mm and d 4 mm.
×10−6
1.576
L′ (f) (H/m)
1.564
1.552
1.54
1.529
1.517
0
2
4
6
Frequency (Hz)
8
10
×106
Figure 4: L f for line with a 0.1 mm and d 4 mm.
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·lnd/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
Mathematical Problems in Engineering
9
dR′ (f)/df (Ω/Hz m)
×10−8
27.2
21.7
16.3
10.9
5.45
0.02402
0
2
4
6
8
Frequency (Hz)
10
×106
Figure 5: dR f/df for line with a 0.1 mm and d 4 mm.
dL′ (f)/df (H/Hz m)
×10−15
−2.33
−4.66
−6.98
−9.31
−11.6
0
2
4
6
Frequency (Hz)
8
10
×106
Figure 6: dL f/df for line with a 0.1 mm and d 4 mm.
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.
For a lossless transmission line ⇔ R 0 Ω/m and G 0 S/m with linear
and homogeneous dielectric the phase-velocity c of electromagnetic perturbation i.e., the
propagation speed of current wave in the line and characteristic impedance Z0 are given by
the following relations 1–3:
m
1
c0
1
1
,
√
√
,
c0 √
≈ 3 · 108
c √
εd · μd
εr · μr
ε0 · μ0
s
L ·C
2.14
μr 1
μ0
μr
L
1
d
d
Z0 ≈ 120 ·
· ·
· ln
· ln
Ω,
C c · C
εr π
ε0
a
εr
a
10
Mathematical Problems in Engineering
45.72
R′ (f) (Ω/m)
37.16
28.6
20.04
11.48
2.92
0.1
6.08
12.1
18
24
Frequency (Hz)
30
×108
Figure 7: R f for line with a 0.1 mm and d 4 mm.
×10−6
1.5169
L′ (f) (H/m)
1.5091
1.5013
1.4935
1.4857
1.478
0.1
6.08
12.1
18
Frequency (Hz)
24
30
×108
Figure 8: L f for line with a 0.1 mm and d 4 mm.
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 ζf ArgZ 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.
Mathematical Problems in Engineering
11
dR′ (f)/df (Ω/Hz m)
×10−8
13
10.6
8.11
5.66
3.21
0.757
0.1
6.08
12.1
18
24
30
×108
Frequency (Hz)
Figure 9: dR f/df for line with a 0.1 mm and d 4 mm.
×10−16
0.004018
dL′ (f)/df (H/Hz m)
4.04
8.08
12.1
16.2
20.2
0.1
6.08
12.1
18
24
Frequency (Hz)
30
×108
Figure 10: dL f/df for line with a 0.1 mm and d 4 mm.
For two-wire line being considered, in Figures 13, 14, 15, and 16 variations of several
functions in the frequency range f ∈ 0, 3 GHz are depicted, which will be used in later
consideration,
2 · π · f · L f
,
φ1 f R f
C · R f
φ4 f ,
G ·L f
2 · π · f · C
,
φ2 f G
φ5 f φ3 f 1
,
2 · π · f · L f · C φ1 f
2·π ·f
.
R f /L f G /C 1 1/φ4 f
2.15
From the numerical data associated with the monotonic functions in Figures 13, 14,
and 16 it is obtained φ1 110.9 KHz ≈ 1, φ1 1289.9 KHz ≈ 10, φ1 10 MHz ≈ 32.653 1,
φ2 110.9 KHz ≈ 13.943 · 109 and φ4 0 ≈ 1.391 · 1010 . Herefrom and from 2.15 it follows
12
Mathematical Problems in Engineering
298.419
Two-wire line (a = 0.1 mm, d = 4 mm)
×10−8
297.635
−4.13
296.85
−8.25
296.066
−12.4
295.281
−16.5
294.497
0.1
6.08
12.1
18
−20.6
30
×108
24
Frequency f (Hz)
Z0 (f) (Ω)
dZ0 (f)/df (Ω/Hz)
Figure 11: Z0 f and dZ0 f/df for the considered two-wire line.
−0.047
Two-wire line (a = 0.1 mm, d = 4 mm)
×10−8
4.74
−0.19
3.95
−0.32
3.16
−0.46
2.37
−0.6
1.58
−0.74
79.1
−0.88
0.1
0.00078707
6.08
12.1
18
Frequency f (Hz)
24
30
×108
ζ(f) = Arg(Z 0 ) (deg)
dζ(f)/df = d(Arg(Z0 ))/df (deg/Hz)
Figure 12: ζf and dζf/df for the considered two-wire line.
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, Λf |Γj · 2π · f| and
ϑf ArgΓj ·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: Λ10 MHz ≈
0.319, Λ 3 GHz ≈ 94.598, ϑ 10 MHz ≈ 89.123 deg and ϑ 3 GHz ≈ 89.953 deg.
Since Γj · 2π · f R f j · 2π · f · L f · G j · 2π · f · C Λf · expj ·
ϑf and since in the range f ∈ 0.01, 3 GHz it holds: φ1 f ≈ φ5 f 1, φ2 f 1,
Λf ∈ 0.319, 94.598 and ϑf ∈ 89.123, 89.953 deg, then: Λf ≈ 2π · f · L f · C 1/2 1/φ3 f, ϑf π/2 − χf 0 < χf < π/200 and Γj · 2π · f Λf · expj · π/2 ·
exp−j · χf j · Λf · {cosχf − j · sinχf} {sinχf j · cosχf}/φ3 f, where
the deviation angle χf π/2 − ϑf can be approximated The percentage error of this
Mathematical Problems in Engineering
13
609.31
φ1 (f)
487.45
365.59
243.73
121.86
0
0
5
10
15
20
25
Frequency f(Hz)
30
×108
Figure 13: A dimensionless parameter φ1 f of two-wire line.
×1014
3.77
3.02
φ2 (f)
2.26
1.51
75.4
0
0
5
10
15
20
Frequency f(Hz)
25
30
×108
Figure 14: A dimensionless parameter φ2 f of two-wire line.
approximation is positive and <0.032% in the entire frequency range f ∈ 0.01, 3 GHz.
with,
L f /R f C /G
π 1
π
− ϑ f − · a tan 2π · f ·
χ f 2
2 2
1 − 4 · π 2 · f 2 L f · C / R f · G
2.16
R f /L f G /C
R f /L f G /C
1
≈
.
≈ · a tan
2
2π · f
4π · f
For the transmission line with length let us define the functions: εω χω/2π π/2 − ϑω/2π and θj · ω Γj · ω · − x Aω, x · expj · ϑω/2π Aω, x ·
{sinεω j · cosεω}{x ∈ 0, }, where Aω, x |Γj · ω| · − x Λω/2π · − x.
For this line, in frequency range f ∈ 0.01, 3 GHz we have Aω, x ≈ ω · L ω/2π · C 1/2 ·
− x − x/φ3 ω/2π and 0 < εω < π/200—whereby θj · ω becomes almost pure
imaginary number. We could have obtained this result in a different way. To se that, let us
14
Mathematical Problems in Engineering
3.13
2.74
φ3 (f) (m)
2.35
1.96
1.57
1.18
0.79
0.4
0.0106
0.11
12.1
6.08
18
24
Frequency f(Hz)
30
×108
Figure 15: Parameter φ3 f of two-wire line.
×1011
6.19
4.98
φ4 (f)
3.77
2.56
1.35
13.9
0
5
10
15
Frequency f(Hz)
20
25
30
×108
Figure 16: A dimensionless parameter φ4 f of two-wire line.
write Γj · 2π · f R f j · 2π · f · L f · G j · 2π · f · C af j · bf, where it
holds,
⎧
⎫
2 ⎬
2
R
R
f
f
1 ⎨
G
G
2
2
2
· − 2π · f
a f · L f · C 2π · f 2 ·
2π · f 2
⎩
⎭
2
C
C
L f
L f
attenuation “constant ,
⎧
⎫
2 ⎬
2
R
R
f
f
1 ⎨
G
G
2
2
2
− · 2π · f
b f · L f · C 2π · f 2 ·
2π · f 2
⎩
⎭
2
C
C
L f
L f
phase “constant .
2.17
Mathematical Problems in Engineering
15
Λ(f) = |Γ(j2πf)| (1/m)
94.6
75.74
56.89
38.03
19.18
0.32
0.1
6.08
12.1
18
24
Frequency f(Hz)
30
×108
Figure 17: Magnitude of propagation function two-wire line.
ϑ(f) = Arg[Γ(j2πf)] (deg)
89.95
89.79
89.62
89.45
89.29
89.12
0.1
6.08
12.1
18
Frequency f(Hz)
24
30
×108
Figure 18: Argument of propagation function two-wire line.
The previous two functions are depicted in Figures 19 and 20 in the frequency range f ∈ 0, 3
GHz.
From relations 2.17 we obtain the approximations aaf and abf of af and bf,
respectively, in the frequency range f ∈ 0.01, 3 GHz, since there it holds φ1 f 1 and
φ2 f 1,
⎤
⎡
L f
1
C
⎦,
aa f ≈ ⎣R f ·
·
G
2
C
L f
#
ab f ≈ L f · C f 2π · f $
R f G 2
1
,
·
−
16π · f
L f C
16
Mathematical Problems in Engineering
0.078
a(f) (1/m)
0.062
0.047
0.031
0.016
3.0547783×10−8
0
0.6
1.2
1.8
2.4
Frequency (Hz)
3
×109
Figure 19: The attenuation “constant” of two-wire line.
and, also, we have,
R f j · 2π · f · L f
Z0 j · 2π · f G j · 2π · f · C
#
$
L f
R f
1
G
.
≈
· 1−j ·
− C
4π · f L f
C
2.18
The functions aaf and abf 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 Γj · ω ≈
aaω/2π j · abω/2π ≈ {sinεω j · cosεω}/φ3 ω/2π, as it has been expected.
We will now emphasize the importance of function θ θs, x Γs·−x{x ∈ 0, }
in the following.
a Constituting of functions sinhθ/θ and tanhθ/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
∂ut, x
∂2 ut, x
∂ ut, x
R · G · ut, x,
L
·
C
·
L · G C · R ·
∂t
∂x2
∂t2
2
∂it, x
∂2 it, x
∂ it, x
R · G · it, x.
L
·
C
·
L · G C · R ·
∂t
∂x2
∂t2
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
Mathematical Problems in Engineering
17
94.6
b(f) (1/m)
75.68
56.76
37.84
18.92
0
0
0.6
1.2
1.8
2.4
3
×109
Frequency (Hz)
Figure 20: The phase “constant” of two-wire line.
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 Ms, x, Ns, x, Us, x, and Is, x from A.17 and Zs, x from
A.18, in general, and for the finite length open-circuited line ZL s → ∞, in particular, by
using expansions:
'
%∞ &
2
n1 1 2Γ · − x/2n − 1 · π
Us, x
Ms, x ' ,
%∞ &
2
Us, 0
n1 1 2Γ · /2n − 1 · π
'
%∞ &
2
1
·
−
x/n
·
π
Γ
n1
Is, x − x
·
Ns, x ,
)
%∞ (
2
Is, 0
n1 1 Γ · /n · π
'
%∞ &
2
n1 1 2Γ · − x/2n − 1 · π
Us, x
Zs, x ' · Z0 ZL s −→ ∞,
% &
Is, x
Γ · − x · ∞ 1 Γ · − x/n · π2
n1
2.20
thus placing into evidence the pole-zero location of Ms, x, Ns, x, and Zs, 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,
∞
θ2
sinhθ *
1
2 2 ,
θ
n ·π
n1
coshθ ∞
*
n1
1
4 · θ2
2n − 12 · π 2
.
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 |Mj · 2π · f, x|,
ArgMj · 2π · f, x deg, |Nj · 2π · f, x| and ArgNj · 2π · f, x deg, respectively,
on the gridx × gridf 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,
18
Mathematical Problems in Engineering
0.078
aa(f) (1/m)
0.063
0.049
0.034
0.019
0.00489
0.1
6.08
12.1
18
Frequency (Hz)
24
30
×108
Figure 21: Approximation of the attenuation “constant”.
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.
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 · |Γj · 2π · f|max } ≈ 16.6 mm {⇔ |θ|max < π/2}
and assume, say, 0 16 mm. Recall that θj · ω, x Γj · ω · − x |θj · ω, x| · exp{j ·
argΓj · ω}{x ∈ 0, }, then take 2.21 and write
∞
∞
θ2
sinhθ *
θ2
1 2 2 exp
ln 1 2 2
θ
n ·π
n ·π
n1
n1
∞
∞
∞
∞
1
θ2 1
1
1
θ4
θ6
θ8
exp 2 ·
−
·
·
−
·
π n1 n2 2 · π 4 n1 n4 3 · π 6 n1 n6 4 · π 8 n1 n8
∞
1
θ10
·
± ....
5 · π 10 n1 n10
θ4
θ6
θ8
θ10
θ2
−
−
± ··· ,
exp
6
180 2835 37800 467775
#
$
∞
∞
*
4 · θ2
4 · θ2
coshθ 1
exp
ln 1 2n − 12 · π 2
2n − 12 · π 2
n1
n1
∞
∞
∞
4 · θ2 1
1
1
8 · θ4 64 · θ6 exp
·
−
·
·
π 2 n1 2n − 12
π 4 n1 2n − 14 3 · π 6 n1 2n − 16
∞
∞
1
1
64 · θ8 1024 · θ10 −
·
·
± ··· .
8
10
π8
5 · π 10
n1 2n − 1
n1 2n − 1
θ2 θ4 θ6 17 · θ8 31 · θ10
−
−
± ··· .
exp
2
12 45
2520
14175
2.22
Mathematical Problems in Engineering
19
94.6
ab(f) (1/m)
75.74
56.89
38.03
19.18
0.319
0.1
6.08
12.1
18
24
30
×108
Frequency (Hz)
Figure 22: Approximation of the phase “constant”.
|M(j·2π·f, x)|
119.76
102.66
500 MHz
85.55
68.44
51.33
34.22
17.12
50
0.00715103
1
7
13
19
25
31
45
1.5 GHz
2.5 GHz
37
43
49
f
55
61
0
x (m)
0.1
Figure 23: The magnitude of the voltage transmittance for two-wire line 0.1 m in frequency range
f ∈ 0, 3 GHz.
The following finite sums of infinite series 9 have been exploited in 2.22,
∞
1
π2
π4
π6
π8
,
A4 ,
A6 ,
A8 ,
p 1, 5 ,
A2 A2p 2p
6
90
945
9450
n1 n
A10 5 · 28 · π 10
,
33 · 10!
20
Mathematical Problems in Engineering
Arg [M(j·2π·f, x)] (deg)
359.98
269.99
180
90.01
0.020737
−89.97
50
x (m)
43
−179.96
36
0.1
29
22
−269.95
15
8
1
0
55
61
f
49
37
43
7
13
19
25
31
−359.94
1
0
3 GHz
Figure 24: The argument of the voltage transmittance for two-wire line 0.1 m in frequency range
f ∈ 0, 3 GHz.
1 GHz
69.93
|N(j·2π·f, x)|
59.94
49.95
2 GHz
39.96
3 GHz
29.97
19.98
9.99
0
1
13
25
37
x (m)
49
0.1
61
0
f
Figure 25: The magnitude of the current transmittance for two-wire line 0.1 m in frequency range
f ∈ 0, 3 GHz.
B2p ∞
1
n1 2n
− 12p
p 1, 5 ,
B2 π2
,
8
B10 B4 31 · π 10
.
35 · 81 · 210
π4
,
96
B6 π6
,
960
B8 17 · π 8
,
161280
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
Mathematical Problems in Engineering
21
Arg[N(j·2π·f, x)] (deg)
203.42
153.82
104.22
54.62
5.02
−44.58
50
x (m)
−94.18
43
36
29
0.1
0
61
f
55
49
43
37
31
25
19
13
22
7
−143.78
15
8
−193.38
1
1
3 GHz
Figure 26: The argument of the current transmittance for two-wire line 0.1 m in frequency range
f ∈ 0, 3 GHz.
retaining only the first five θ terms in 2.22 the following small-error approximations are
produced
θ4
θ6
θ8
θ10
θ2
−
−
,
sinhθ ≈ sinhA θ θ · exp
6
180 2835 37800 467775
2.24
θ2 θ4 θ6 17 · θ8 31 · θ10
coshθ ≈ coshA θ exp
−
−
.
2
12 45
2520
14175
For the open-circuited two-wire line with 0.01 m, in Figures 27, 28, 29, and 30
they are depicted on gridx × gridf 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 {|coshA θj · 2π · f, x/coshA j · 2π · f, 0/cosh θj · 2π · f, x/ coshj ·
2π · f, 0| − 1} · 100;
ii the voltage-transmittance phase approximation absolute error:
ER2 f, x argcoshA θj · 2π · f, x · coshj · 2π · f, 0/coshA j · 2π · f, 0 · cosh θj ·
2π · f, x;
iii the current-transmittance magnitude approximation percentage error:
ER3 f, x {|sinhA θj · 2π · f, x/sinhA j · 2π · f, 0/sinh θj · 2π · f, x/ sinhj ·
2π · f, 0| − 1} · 100;
iv the current-transmittance phase approximation absolute error:
ER4 f, x argsinhA θj · 2π · f, x · sinhj · 2π · f, 0/sinhA j · 2π · f, 0 · sinh θj ·
2π · f, x.
22
Mathematical Problems in Engineering
ER1 (f, x) (%)
0
x (mm)
−0.02
40
10
30
−0.04
20
10
f
40
60
20
0
3 GHz
Figure 27: Voltage-transmittance magnitude approximation percentage error ER1 f, x.
It can be observed in Figures 27 ÷ 30 that in the given range of f and x, the errors ER1
and ER3 are negative |ER1 | < 0.06% and |ER3 | < 10−5 %, whereas the errors ER2 and ER4 are
positive ER2 < 3.36 · 10−4 deg and ER4 < 5.75 · 10−8 deg. The upper limit of |ER3 | is lower
than of |ER1 | and the upper limit of ER4 is lower than of ER2 .
If |θ| is sufficiently small |θ| 1, from 2.24 further approximations are obtained:
i
sinhθ ≈ sinhA θ
θ · exp
≈ θ·
θ2
θ4
θ6
θ8
θ10
−
−
6
180 2835 37800 467775
θ4
θ6
θ8
θ10
θ2
−
−
1
6
180 2835 37800 467775
2.25
θ3
θ5
θ7
θ9
θ11
−
−
6
180 2835 37800 467775
θ3
≈ θ
,
6
θ
which partly resembles to Maclaurin’s expansion of sinhθ, that is, sinhθ θ θ3 /3! θ5 /5! . . .,
ii
coshθ ≈ coshA θ exp
θ2 θ4 θ6 17 · θ8 31 · θ10
−
−
2
12 45
2520
14175
θ2 θ4 θ6 17 · θ8 31 · θ10
θ2
≈ 1
−
−
≈1
,
2
12 45
2520
14175
2
2.26
Mathematical Problems in Engineering
23
ER2 (f, x) (deg)
3.358101 × 10−4
2.518575 × 10−4
x (mm)
1.67905 × 10−4
40
30
10
20
10
0
f
2 GHz
3 GHz
−5.4210108624275 × 10−20
20
40
60
8.4 × 10−5
0
1 GHz
Figure 28: Voltage-transmittance phase approximation absolute error ER2 f, x.
ER3 (f, x) (%)
×106
0
x (mm)
40
−5
10
30
20
10
f
60
40
20
−1
0
3 GHz
Figure 29: Current-transmittance magnitude approximation percentage error ER3 f, x.
which, partly resembles to Maclaurin’s expansion of coshθ, that is, coshθ 1 θ2 /2! θ4 /4! . . ..
iii combining i and ii it follows that
θ · θ2 24
θ
θ
≈ tanhA
.
tanh
2
2
6 · θ2 8
2.27
The functions sinhθ/θ and tanhθ/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
24
Mathematical Problems in Engineering
ER4 (f, x) (deg)
×10−8
5.75
4.31
x (mm)
2.87
40
10
30
20
1.44
10
0
0
f
20
40
60
2 GHz
3 GHz
0
1 GHz
Figure 30: Current-transmittance phase approximation absolute error ER4 f, x.
ER5 (f, x)
−7.639466306×10−13
−0.021
x (mm)
−0.043
40
1
30
20
10
0
f
60
3 GHz
40
2 GHz
20
−0.064
−0.086
0
1 GHz
Figure 31: Magnitude approximation absolute error ER5 f, x.
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| − | sinhθ| and three percentage magnitude approximation errors: ER6 f, x {|{θ · θ2 24/6 · θ2 8}/ tanhθ/2| − 1} · 100, ER7 f, x |θ/ sinhθ| − 1 · 100 and
ER8 f, x {|θ/2/ tanhθ/2| − 1} · 100, are depicted on gridx × gridf 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
Mathematical Problems in Engineering
25
ER6 (f, x)
1.67 × 10−5
1.25 × 10−5
x (mm)
8.35 × 10−6
40
1
30
4.18 × 10−6
20
10
−2.220446049 × 10−14
0
2 GHz
3 GHz
f
20
40
60
0
1 GHz
Figure 32: Percentage magnitude approximation error ER6 f, x.
ER7 (f, x)
0.15
0.11
x (mm)
0.075
40
1
30
20
0.037
10
0
0
f
60
3 GHz
40
20
2 GHz
1 GHz
0
Figure 33: Percentage magnitude approximation error ER7 f, x.
in achieving the previous goals is providing the maximum of 0 to be much less than
1/|Γj · 2π · fmax ⇔ |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/|Γj · 2π · fmax |.
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 35a with boundary
26
Mathematical Problems in Engineering
ER8 (f, x)
0
−0.019
x (mm)
−0.037
40
1
30
20
−0.056
10
0
f
60
40
3 GHz
2 GHz
20
1 GHz
−0.075
0
Figure 34: Percentage magnitude approximation error ER8 f, x.
I(s, ℓ0 )
I(s, 0)
Z′ /2
U(s, 0)
Linear two-port
network
Z′ /2
U(s, ℓ0 )
Y′
U(s, 0)
a
U(s, ℓ0 )
b
Z
U(s, 0)
′′
Y ′′ /2
Y ′′ /2
U(s, ℓ0 )
c
Figure 35: a Linear network model of a short transmission line segment and b its linear network
structure.
conditions Us, 0 and Is, 0 at the input and Us, 0 and Is, 0 at the output. The chainmatrix F of this network reads
⎡
⎤
coshΓ · 0 Z0 · sinhΓ · 0 Us, 0 Us, 0
⎦.
F·
,
F ⎣ sinhΓ · 0 2.28
Is, 0 Is, 0
coshΓ · 0 Z0
Denote Zs R L ·s·0 , Y s G C ·s·0 and θs, 0 Γs·0 . In study of short
transmission lines it is found convenient to replace them, either with nominally equivalent T
Mathematical Problems in Engineering
27
networks Figure 35b or with nominally equivalent Π networks Figure 35c 7, whose
immitances are given as follows
,
,
+
tanhθs, 0/2
sinhθs, 0
,
Y Y·
,
Z Z·
θs, 0/2
θs, 0
,
,
+
+
sinhθs, 0
tanhθs, 0/2
Z Z·
,
Y Y·
.
θs, 0
θs, 0/2
+
2.29
The aforementioned criterion for selection of 0 relies on attempt to find convenient θs, 0 Γs · 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 Zs 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: sinhθ/θ ≈ 1
and tanhθ/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 Γj ·ω R j · ω · L · G j · ω · C |Γj ·ω|·exp{j ·argΓj ·
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 |Γj · ω| ≈ ω · L · C 1/2 ∈ 0.319, 94.598, argΓj · ω ∈ 89.122, 89.952 deg,
c argΓj · ω π/2 − εω, where,
π 1
π
ω · L /R ω · C /G
− arg Γ j · ω − · a tan
2
2 2
1 − L · C /R · G · ω2
R /L G /C
1
R /L G /C
.
≈ · a tan
≈
2
ω
2·ω
εω 2.30
d Γj · ω |Γj · ω| · {sinεω j · cosεω}, εω ∈ 0, π/200 and θj · ω, x Γj · ω · 0 − x |θj · ω, x| · {sinεω j · cosεω}. Observe that θj · ω, x is
produced as almost pure imaginary number.
e Since x ∈ 0, 0 and |θj · ω, x| ≈ ω · L · C 1/2 · 0 − x Aω, x, then by selecting
0 sufficiently small, Aω, x can always be produced arbitrarily small.
Bearing in mind the properties a ÷ e, we obtain for sinhθj · ω, x/θj · ω, x the
following:
sinh{Aω, x · sinεω} · cos{Aω, x · cosεω}
.
Aω, x · sinεω j · cosεω
j · cosh{Aω, x · sinεω} · sin{Aω, x · cosεω}
,
.
Aω, x · sinεω j · cosεω
2.31
28
Mathematical Problems in Engineering
that can be rewritten as sinhθj · ω, x/θj · ω, x Rω, x j · Qω, x. Rω, x and Qω, x
are even and odd functions in ω, respectively, which are represented with the following
expanded forms:
∞
∞
2m
2m
A2k ω, x · sin2k εω
m A ω, x · cos εω
2
·
Rω, x sin εω ·
−1
2k 1!
2m!
m0
k0
⎛
⎞
∞
∞
2n
2n
2p
2p
A
A
x
·
sin
x
·
cos
ω,
εω
ω,
εω
⎠,
cos2 εω·
· ⎝ −1p
2n!
2p
1
!
n0
p0
Qω, x sinεω · cosεω
#
∞
∞
2m
2m
A2k ω, x · sin2k εω
m A ω, x · cos εω
·
·
−1
2k!
2m 1!
m0
k0
⎞⎫
⎛
∞
∞
⎬
2p
2p
A2n ω, x · sin2n εω
A
x
·
cos
ω,
εω
⎠ .
· ⎝ −1p
−
⎭
2n 1!
2p !
n0
p0
2.32
We must always bear in mind that εω is small c and 2.30 and that Aω, x ω ·
L · C 1/2 · 0 − x can be made arbitrarily small by selecting 0 such that Aω, |max 2π · f · L · C 1/2 |max · 0 1 or equivalently 0 1/|Γj · 2π · fmax |. Then, retaining only
the first two terms in each of the convergent infinite sums in 2.32, the approximations of
Rω, x and Qω, x are obtained which hold for all x ∈ 0, 0 and for all ω corresponding to
f from the specified frequency range:
Rω, x ≈ 1 −
A4 ω, x
ω2 · L · C · 0 − x2
A2 ω, x
· cos2 · εω −
· sin2 2 · εω ≈ 1 −
,
6
48
6
A2 ω, x
ω2 · L · C · 0 − x2
· sin2 · εω ≈
· sin2 · εω
6
6
ω2 · L · C · 0 − x2
ω G
R
≈
· εω · L · C · 0 − x2 ·
3
6
L C ω
· 0 − x2 · R · C G · L 1.
6
Qω, x ≈
2.33
For x 0, from 2.31 and 2.33 it finally follows:
ω2 · L · C · 02
sinh θ j · ω, 0
ω ,
Rω, 0 j · Qω, 0 ≈ 1 j · · R · C G · L · 02 −
6
6
θ j · ω, 0
2.34
s2 · L · C · 02
sinh θ j · ω, 0
j·ω → s−ω2 → s2 s sinhθs, 0
≈ 1
· R · C G · L · 02 .
−−−−−−−−−−−−−−→
By analytic continuation
θs, 0
6
6
θ j · ω, 0
2.35
Mathematical Problems in Engineering
29
Now, by using 2.29 and 2.35 we may generate the immitances Y’ and Z” Figures
35b and 35c,
2 s2 · L · C · 02
s sinhθs, 0 G C · s · 0 · 1 · R · C G · L · 0 ,
Y s Y s ·
θs, 0
6
6
2.36
2 s2 · L · C · 02
s sinhθs, 0 Z s Zs ·
R L · s · 0 · 1 · R · C G · L · 0 ,
θs, 0
6
6
2.37
which are not realizable by lumped RLC networks, since they are not the positive real functions
in complex frequency s 10, except when 0 → 0. Putting it in other words we may say that
if in the whole operating frequency
range practically hold the three conditions: ω · L /R 1,
√
ω · C /G 1, and 0 · ω · L · C 1, the immitances Y s and Z s Figures 35b and
35c are produced in the following simple form and are realizable by two-element-kind RLC
networks see 2.36 and 2.37:
Y s G C · s · 0 ,
Z s R L · s · 0 .
2.38
Relations 2.38 may be considered as a consequence of approximation sinhθs, 0 ≈
θs, 0 applied to 2.36 and 2.37 when θs,√0 → 0. Similarly, under the conditions:
ω · L /R 1, ω · C /G 1 and 0 · ω · L · C 1, the complex approximation
tanhθs, 0/2 ≈ θs, 0/2 applied to 2.29 when θs, 0 → 0 gives the other two
immitances from 2.29, which are necessary to accomplish forming of linear networks in
Figures 35b and 35c, which are nominally equivalent to the network in Figure 35a.
Z s R L · s · 0 ,
Y s G C · s · 0 .
2.39
3. Approximation of Two-Wire Line by Uniform RLCG Ladder and
the Simulation Results
Let us consider a short two-wire line with length 30 mm or a longone partitioned
in sections of length . Assume that the bandwith of the signal being transmitted is B <
770 kHz and that its central frequency is f0 1 GHz. Then, make the graph of function
lf φ3 f 1/2π · f · L f · C 1/2 ≈ 1/|Γj · 2π · f| Figure 36 and from its associated
numerical data find that l109 ≈ 31.7 mm. Let the maximum length 0 of line segments be
selected to satisfy the condition 0 ≤ l109 /10 ≈ 3.17 mm. Finally, assume 0 3 mm and
calculate the number of cells N /0 10 in uniform RLCG ladder purporting to represent
the transmission line with length in frequency range f ∈ f0 − B/2, f0 B/2.
Assume δx 0 and calculate the parameters of uniform lumped RLCG network
representing the line segments of length 0 see Figure 2. If the overall short-line parameters
are R R · -resistance, L L · -inductance, C C · -capacitance, and G GΓj·2π·f · 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
30
Mathematical Problems in Engineering
Table 1: Two-wire copper-polyethylene transmission line central frequency f0 1 GHz.
Electrical parameters of RLCG ladder sections
R/20 mΩ
L/20 nH
C/10 fF
G/10 aS
39.772
2.219
51.123
2.554
0.26
0.23
0.2
l (m)
0.17
0.14
0.11
0.078
0.047
0.016
0
0.5
1
Frequency (Hz)
1.5
2
×109
Figure 36: The function lf φ3 f of the considered two-wire line determination of 0 .
line with length is depicted in Figure 37, whereon the conductance elements G/10 present
in Figure 2 are omitted only for the simplicity of drawing but are included in the PSPICE
simulation network. Also, observe that at frequency 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.
Now we will present the results obtained by PSPICE simulation of the open-circuited
network in Figure 37 and compare them to the results obtained by exact analysis of
considered open-circuited line. The amplitude of excitation voltage e in simulation was 1 V,
its frequency was f0 1 GHz and the initial phase 0 deg. The steady-state, odd numbered
point voltages, and their phase angles obtained through PSPICE analysis of open-circuited
network in Figure 37, are summarized in Table 2. Also, in this table the exactly obtained
voltages at points on the line with distance xm m − 1 · 0 /2m 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.
When analysis of long lines is considered in the time domain it is useful to resort
to forming of multilevel hierarchical blocks in PSPICE. To see that, suppose that we are to
consider transmission of a signal with frequency f0 1 GHz, amplitude Em 10 V, and
zero initial phase in two-wire line with length 4.5 m and parameters as in Table 1.
Let us designate the ladder on Figure 37, which represents a two-wire line with length 3
cm, as level 1 hierarchical block HB1 on Figure 38a. By cascading, say, 30 level 1 blocks:
HB1 /1, HB1 /2, . . ., and HB1 /30, we produce a level 2 hierarchical block HB2 on Figure 38b,
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 38c, 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.
N1
1.0000
0.0000
1.0000
0.0000
N21
1.52345
−87.7248
1.522839
−87.5547
N3
1.06582
−15.1412
1.06455
−14.9246
N23
1.56043
−91.9260
1.558028
−91.5659
N5
1.12687
−27.8853
1.12672
−27.8250
N25
1.59042
−95.2320
1.589727
−95.0498
N7
1.18792
−39.3196
1.18637
−39.0532
N27
1.62041
−98.4156
1.617867
−98.0454
N9
1.24365
−48.9810
1.24337
−48.8790
N29
1.64314
−100.775
1.642383
−100.584
N11
1.29938
−57.8137
1.297580
−57.5121
N31
1.66587
−103.070
1.663221
−102.693
N13
1.34929
−65.2500
1.348883
−65.1181
N33
1.68114
−104.588
1.680334
−104.391
N15
1.39919
−72.1558
1.397165
−71.8286
N35
1.69641
−106.079
1.693685
−105.697
N17
1.44283
−77.9040
1.442318
−77.7501
N37
1.70407
−106.821
1.703242
−106.621
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.
Node
Amplitude V simulation
Phase mdeg simulation
Amplitude V exact analysis
Phase mdeg exact analysis
Node
Amplitude V simulation
Phase mdeg simulation
Amplitude V exact analysis
Phase mdeg exact analysis
Table 2: The steady-state results of exact analysis and simulation for the open-circuited two-wire line in Figure 37.
N19
1.48647
−83.3147
1.484240
−82.9687
N39
1.71174
−107.556
1.708985#
−107.172&
Mathematical Problems in Engineering
31
32
Mathematical Problems in Engineering
N1
N2
N3
N4
R/20 L/20
R/20
C/10
u3
e = u1
N5
N6
N7
L/20 R/20 L/20
N8
R/20
C/10
u5
N9
N10
L/20 R/20
N12
R/20
C/10
u9
u7
N11
L/20
L/20
u11
u13
N25
N24
R/20
N22
N23
L/20
R/20
C/10
N20
N21
L/20
R/20 L/20
N17
R/20 L/20
C/10
u21
u23
N18
N19
N16
N15
C/10
u17
u15
u25
N26
R/20
N27
L/20
C/10
N28
R/20
u27
N29
L/20
N30
R/20
u29
N31
L/20
C/10
N32
N33
R/20 L/20
u31
N34
N35
R/20 L/20
u33
N14
R/20 L/20
R/20 L/20
u19
N13
N36
R/20
C/10
L/20
u35
u37
0
N37
N41
N40
N39
Load
R/20 L/20
u41
C/10
N38
R/20 L/20
u39
Figure 37: Electrical network model of two-wire line with length 3 cm in the frequency range f ∈
f0 − B/2, f0 B/2.
From the data associated with functions depicted in Figures 12 and 13 we can easily: i
obtain Z0 j ·2π ·f0 294.67280−j ·0.42017 Ω and ii notice that variations of |Z0 j ·2π ·f|
and ArgZ0 j · 2π · f are very small in the frequency range f0 − B/2, f0 B/2.
Now, suppose that the block HB2 /5 in Figure 38c 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 Mj · 2π · f0 , x Nj · 2π · f0 , x exp−x ·
Γj · 2π · f0 exp−x · af0 · {cosx · bf0 − j · sinx · bf0 }, and then at any place x on
the line we obtain the exact values of signal amplitude Em · exp−x · af0 V, phase delay
ϕx x · bf0 rad and time delay τx x · bf0 /2π · f0 s. In Figure 20 we see that
function bf is practically linear in f, so that τx should be linear in x, too. To verify the
ladder model of two-wire line with length 4.5 m Figure 38c, we will compare the
exact values of Em · exp−x · af0 , ϕx and τx at the points on line x xk k · 0 /2 k 1, 3000 0 3 mm, with values obtained by PSPICE simulation. Bearing in mind the
topological uniformity of the considered ladder, it is felt that for estimation of the proposed
model it will suffice to check the ladder response only at the selected set of points whose
voltages are ui i 1, 5 with respect to the common node 0 i.e., ground Figure 38c. The
distances of these five points from the line sending end are x600·i 300 · i · 0 90 · i cm
i 1, 5, respectively. For these voltages in Table 3 are given the steady-state results of both
the simulation and the exact analysis, where the following notation has been used Figures
19, 20, and 38c:
1
1
,
b f0 31.551675282
,
f0 1 GHz,
a f0 0.044990161
Em 10 V,
m
m
e Em · sin 2π · f0 · t ,
ui Uim · sin 2π · f0 · t − ϕi Uim · sin 2π · f0 · t − τi ,
Mathematical Problems in Engineering
N1
N41
HB1
33
HB1 /1
HB1 /2
HB1 /3
HB1 /4
HB1 /5
HB1 /6
HB1 /28
HB1 /29
HB1 /30
HB2
RLCG ladder from Figure 37
0
a
b
u1
HB2 /1
u3
u2
HB2 /2
HB2 /4
HB2 /3
e
HB3
RL = 294.67 Ω
u5
u4
HB2 /5
CL = 378.77 (pF)
c
Figure 38: The structure of the three types of hierarchical blocks HB1 , HB2 , and HB3 used in PSPICE
simulation.
Uim Em · exp −x600·i · a f0 10 · 0.960317668i V ,
ϕi ϕx600·i 300 · i · 0 · b f0 28.39650775 · i rad,
ϕi
4.519444575 · i ns,
i 1, 5 .
τi τx600·i 2π · f0
3.1
From Table 3 it is evident the good agreement between the simulation results and
those obtained by exact analysis. The selection of less segmentation step 0 will improve
this agreement at the expense of rising the complexity of RLCG network, as a consequence of
proliferation in number of network elements.
In Figure 39 the results of transient PSPICE analysis of RLCG ladder with zero initial
conditions Figure 38c representing the approximate network model of two-wire line with
4.5 m are depicted.
Now we can summarize our obtained results
i A transmission line is physically dispersive system with respect to frequency,
having the infinite number of poles and zeros and complex transient dynamics,
which cannot be represented perfectly with common-ground ladder with possibly
great, but finite number of RLCG elements.
90 cm @ u1
180 cm @ u2
270 cm @ u3
360 cm @ u4
450 cm @ u5
Distance @ voltage
Amplitude V
simulation
9.60377
9.22326
8.85778
8.50664
8.16922
Amplitude V
exact analysis
9.60317
9.22210
8.85614
8.50471
8.16722
Phase delay ϕ deg
simulation
187.60752
15.21326
202.81711
30.41907
218.01932
Phase delay ϕ deg
exact analysis
187.00004
14.00009
201.00014
28.00018
215.00023
Time delay τ ns
simulation
4.4930
9.0066
13.4584
18.0132
22.4650
Table 3: The steady-state results of exact analysis and simulation for RLCG lader in Figure 38c.
Time delay τ ns
exact analysis
4.51944
9.03888
13.55833
18.07777
22.59722
34
Mathematical Problems in Engineering
Mathematical Problems in Engineering
35
Transient analysis
110
100
90
80
Voltage (V)
70
60
50
40
Offset u5 = 100 V
u4
Offset u4 = 80 V
u3
Offset u3 = 60 V
u2
Offset u2 = 40 V
u1
30
20
u5
Offset u1 = 20 V
10
e
0
−10
0
5
10
15
20
25
Time (ns)
Figure 39: Transient analysis of RLCG ladder in Figure 38c and “measuring” of delay times τ1 ÷ τ5 see,
also, 3.
ii Delayed and slightly attenuated signals u1 ÷ u5 with frequency f0 and smooth
transient intervals Figure 39 are produced by uniform RLCG ladder terminated
with characteristic impedance The uniform ladder in Figure 38c 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 · acosh1 Z1 j · 2π · f0 /Z2 j · 2π · f0 } · Ej · 2π · f0 Ej · 2π · f0 is the
complex representative of et, and ii the time-delay at that place with respect to
excitation is, Im{n · acosh1 Z1 j · 2π · f0 /Z2 j · 2π · f0 /2π · f0 } ≈ 15.07 · n
ps. of the line at frequency f0 Figure 38c and not by real two-wire line. As
will be seen, the transient response of real two-wire line, even with the same initial
conditions and termination, is more complex.
iii A deeper insight into transient phenomena in real lines can be acquired, either
by applying the numeric inverse Laplace transform of A.17 with restricted
number of poles/zeros or by numerical solving of linear, second-order, hyperbolic
partial differential equations 2.19, telegraph equations, with specified initial and
boundary conditions depending on line termination and excitation voltages in
consecutive time intervals determined according to the line length. The method
of lines seems to be the most appropriate for solving of hyperbolic and parabolic
partial differential equations 11.
iv The PSPICE simulation method is applied herein only to facilitate the approximate
steady-state analysis of two-wire lines with arbitrary loads and limited frequencyband signals, by using RLCG ladders as approximate network models of these
36
Mathematical Problems in Engineering
lines, instead of resorting to numerical solving of partial differential equations or
application of complex analytic methods.
To illustrate the complexity of transient phenomena in transmission lines let we
consider two-wire line with length 4.5 m and excitation et as in Figure 38c see,
also, 3, 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 ut, x of the telegraph equation 2.19 in the interval t ∈ 0, T , where T /2π ·
f0 /bf0 ≈ 22.5972 ns is the perturbation propagation time from the line sending end to its
receiving end, and 2π · f0 /bf0 ≈ c0 /εr 1/2 ≈ 1.9913 · 108 m/s is the propagation velocity.
If we introduce substitution t τ/Af0 {Af0 1/L f0 · C 1/2 1.9913 · 108 m/s} into
the telegraph equation in ut, x obtained from 2.19:
∂2 ut, x ∂ut, x
∂2 ut, x
R f0 · G · ut, x,
L
L f0 · G C · R f0 ·
f0 · C ·
∂t
∂x2
∂t2
3.2
we obtain the second-order, hyperbolic, partial differential equation PDE equivalent to
3.2:
∂uτ, x
∂2 uτ, x ∂2 uτ, x
B f0 · uτ, x,
C f0 ·
2
2
∂τ
∂x
∂τ
τ
ut, x u
, x uτ, x, x ∈ 0, ,
A f0
m
1
1
,
B f0 R f0 · G
,
A f0 m2
L f0 · C s
C f0 R f0 ·
C
G ·
L f0
3.3
L f0
1
,
C
m
where τ ∈ 0, Af0 ·T m, that is, τ ∈ 0, 4.5 m. If uτ, x {exp−1/2·Cf0 ·τ}·uτ, x,
then from 3.3 the following PDE of Klein-Gordon’s type 12 is obtained {τ, x ∈ 0, 4.5
m}:
∂2 uτ, x ∂2 uτ, x
D f0 · uτ, x,
2
2
∂x
∂τ
⎞2
⎛
1
L
f
C
1
0
−3
⎠
⎝
D f0 · R f0 ·
2.0241 · 10
,
−G ·
4
C
m2
L f0
3.4
which would be a pure wave equation if “diffusion” term Df0 · uτ, x was not present,
or in other words, if the line parameters satisfy the Heaviside’s condition of distortionless at
frequency f0 .
Mathematical Problems in Engineering
37
Let we define, also, the following auxiliary function in τ and x:
1
∂uτ, x
∂ut, x
1
· C f0 · ut, x ·
· exp E f0 · t vτ, x ∂τ
2
∂t
A f0
R f0
1
G
E f0 ·
8.959 MHz.
2
C
L f0
tτ/Af0 ,
3.5
If the excitation is et Em · sin2π · 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:
∂uτ, x
∂vτ, x ∂2 uτ, x
vτ, x ∧
D f0 · uτ, x ←− The system of PDEs,
2
∂τ
∂τ
∂x
3
3 0
if x > 0
u0, x 3
3 e0 otherwise ∧ v0, x
3
3
0
if x > 0
3
3 1
2π · f0 · Em · cos ϕ
←− the initial conditions,
3
3
otherwise
· C f0 · e0 3 2
A f0
1
τ
uτ, 0 exp · C f0 · τ · e
∧ uτ, 0 ←− the boundary conditions,
2
A f0
ut, x exp −E f0 · t
· u A f0 · t, x ←− Solution of 3.2 in interval t ∈ 0, 22.59 ns, x ∈ 0, 4.5 m.
3.6
In Figure 40 the solutions ut, xk of 3.2 in the interval t ∈ 0, T for xk k · /5 m
k 1, 5 are depicted. Certainly the solutions of 3.2 for any x ∈ 0, can be produced
easily, also by using the relations 3.4–3.6.
In Figure 41 the pulse responses i.e., the voltages uk k 1, 5 of the ladder in
Figure 38c terminated with the characteristic impedance of two-wire line are depicted. The
ladder is excited by pulsed emf et with amplitude 10 V, frequency f 10 MHz, dutycycle 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 38c with continuous
excitation of frequency f0 1 GHz, in spite of the fact that frequency spectrum of the
periodic, pulsed signal et has components 10 · k MHz k ∈ N and that energetically
significant part of spectrum is concentrated in the frequency range f ∈ 0, 150 MHz.
4. Conclusions
In the paper new results are presented in incremental network modelling of Two-wire lines
in the frequency range 0, 3 GHz, by uniform RLCG ladders with frequency-dependent
RL parameters, which are analyzed by using of the PSPICE. Some important frequency
limitations of the proposed approach have been pinpointed, restricting the application
38
Mathematical Problems in Engineering
2.46
1.95
u(t, x4)
Voltage u(t, x) (V)
1.43
u(t, x1)
u(t, x3)
u(t, x2)
0.91
0.39
−0.13
u(t, x5)
−0.65
−1.17
−1.69
−2.2
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
t (ns)
Figure 40: Transient voltages at five equidistant places on the two-wire line with length 4.5 m Em 10 V and ϕ 0.
Transient analysis
110
u5
100
(22.0115n, 238.1548m)
90
(17.3124n, 73.8694m)
70
Voltage (V)
Offset u4 = 80 V
u4
80
u3
60
Offset u3 = 60 V
(12.8607n, 55.4159m)
50
u2
40
Offset u2 = 40 V
(8.4089n, 17.3784m)
30
u1
20
Offset u1 = 20 V
(4.2045n, 49.5688m)
10
0
Offset u5 = 100 V
e(t)
0
50
100
150
200
250
300
Time (ns)
Figure 41: Pulsed responses in selected points on the uniform ladder in Figure 38c.
of the developed models to the steady-state analysis of RLCG networks processing the
limited-frequency-band signals. The basic intention of the approach considered herein is
to circumvent solving of telegraph equations and application of the complex, numerically
demanding procedures in determining two-wire line responses at selected set of equidistant
points. The key to the modelling method applied is partition of the two-wire line in
sufficiently short segments having defined maximum length, whereby couple of new
Mathematical Problems in Engineering
39
polynomial approximations of line transcedental functions is introduced. It is proved that
the strict equivalency between the short-line segments and their uniform ladder counterparts
does not exist, but if some conditions are met, satisfactory approximations could be produced.
This is illustrated by several examples of short and moderately long two-wire lines with
different terminations, proving the good agreement between the exactly obtained steadystate results and those obtained by PSPICE simulation of RLCG ladders as the approximate
incremental models of two-wire lines.
Appendix
By using of Kirchoff’s voltage and current laws the following equlibrium equations can be
written for the uniform transmission line depicted in Figure 2, no matter what its length or
type is:
δx
∂it, x
δx
− ut, x −
· L ·
R · it, x ,
u t, x 2
2
∂t
∂ut, x δx/2
δx
G · u t, x it, x δx − it, x −δx · C ·
,
∂t
2
δx
∂it, x δx
δx
−
· L ·
R · it, x δx .
ut, x δx − u t, x 2
2
∂t
A.1
A.2
A.3
If δx → 0, from A.1–A.3 it immediately follows:
∂it, x
∂ut, x
−L ·
− R · it, x,
∂x
∂t
∂it, x
∂ut, x
−C ·
− G · ut, x.
∂x
∂t
A.4
If s is the complex frequency, let we suppose that the following conditions hold.
a Both ut, x and it, x possess Laplace transform with respect to time,
Us, x ut, x Is, x it, x 4∞
0−
4∞
0−
ut, x · e−s·t · dt,
A.5
it, x · e
−s·t
· dt.
b Both ut, x and it, x have continuous derivatives with respect to x.
c The following two integrals are uniformly convergent with respect to x:
4∞
0−
∂ut, x −s·t
· e · dt,
∂x
4∞
0−
∂it, x −s·t
· e · dt.
∂x
A.6
d The initial conditions u0, x and i0, x are assumed for convenience to be zero for
all x.
40
Mathematical Problems in Engineering
Taking into account all conditions a ÷ d and A.1–A.6, it follows:
4
4∞
∂ut, x −s·t
∂ ∞
∂Us, x
∂ut, x
· e · dt ,
ut, x · e−s·t · dt ∂x
∂x
∂x
∂x
0−
0−
4
4∞
∂it, x
∂it, x −s·t
∂ ∞
∂Is, x
· e · dt ,
it, x · e−s·t · dt ∂x
∂x
∂x 0−
∂x
0−
∂Us, x
− R L · s · Is, x,
∂x
∂Is, x
− G C · s · Us, x,
∂x
A.7
A.8
A.9
wherefrom the equations describing voltage and current distribution in uniform line are
readily produced, regardless to its length and/or terminal conditions i.e., the generator
and load impedances:
∂2 Us, x
Γ2 s · Us, x 0,
∂x2
∂2 Is, x
Γ2 s · Is, x 0,
∂x2
A.10
R L · s/G C · s is propagation function of the line. The imporwhere Γσ tant
impedance Z0 s parameter of any line is, also, its generalized characteristic
R L · s/G C · s. The line is distortionless if R /L G /C 7, and it is lossless
when R 0 Ω/m and G 0 S/m. The general solution to the set of linear, homogeneous
differential equations A.10 reads
Us, x A1 · coshΓ · x A2 · sinhΓ · x,
Is, x B1 · coshΓ · x B2 · sinhΓ · x,
A.11
where the terms A1 , A2 , B1 , and B2 are not the functions of x and are determined from the
boundary conditions. From A.11 for x 0 we obtain, A1 Us, 0 and B1 Is, 0 and
from A.9 and A.11; after differentiation in x; it follows, A2 −Z0 s · Is, 0 and B2 −Us, 0/Z0 s. Then A.11 takes on the following form:
Us, x Us, 0 · coshΓ · x − Z0 s · Is, 0 · sinhΓ · x,
Us, 0
· sinhΓ · x.
Is, x Is, 0 · coshΓ · x −
Z0 s
A.12
A.13
Since for x it holds:
Us, Us, 0 · coshΓ · − Z0 s · Is, 0 · sinhΓ · ,
Us, 0
· sinhΓ · ,
Is, Is, 0 · coshΓ · −
Z0 s
A.14
Mathematical Problems in Engineering
41
then from A.14 we obtain,
Us, 0 Us, · coshΓ · Z0 s · Is, · sinhΓ · ,
Us, · sinhΓ · .
Is, 0 Is, · coshΓ · Z0 s
A.15
A.16
If ZL s is the load impedance termination of the uniform, finite length line, then
from A.12–A.14 we finally produce voltage- and current-transmittances Ms, x and
Ns, x, respectively,
Ms, x Us, x ZL s · coshΓ · − x Z0 s · sinhΓ · − x
,
Us, 0
ZL s · coshΓ · Z0 s · sinhΓ · Is, x ZL s · sinhΓ · − x Z0 s · coshΓ · − x
Ns, x .
Is, 0
ZL s · sinhΓ · Z0 s · coshΓ · A.17
Since Us, ZL s · Is, , then from A.12–A.14 it follows,
Zs, 0 Zs, x Us, 0 ZL s · coshΓ · Z0 s · sinhΓ · · Z0 s, and generally,
Is, 0
ZL s · sinhΓ · Z0 s · coshΓ · Us, x ZL s · coshΓ · − x Z0 s · sinhΓ · − x
· Z0 s,
Is, x
ZL s · sinhΓ · − x Z0 s · coshΓ · − x
x ∈ 0, ,
A.18
where Zs, 0 is the input impedance of line and Zs, x is the impedance at place x seen
towards the line end. From A.17 and A.18 we see that analysis of transmission line is
equivalent to analysis of its segments terminated with impedances given with A.18. The
characteristic impedance and the propagation function of a distortionless line are Z0 s L /C 1/2 and Γs R · G 1/2 s · L · C 1/2 , respectively. And further if the line load
impedance is ZL s Z0 s, then for all x ∈ 0, it holds: Zs, x Z0 s L · C 1/2 and
Ms, x Ns, x exp−Γs · x exp−R · G · x · exp−x · L · C 1/2 · s. When the
line parameters are constant, we will have ut, x L−1 Us, x L−1 Ms, x · Us, 0 L−1 {exp−R ·G ·x·exp−x·L · C 1/2 ·s·Us, 0} exp−R ·G ·x·ut−x·L · C 1/2 , 0 and,
also, it, x L−1 Is,x L−1 Ns, x · Is, 0 L−1 {exp−R · G · x · exp−x · L · C 1/2 ·
s · Is, 0} exp−R · G · x · it − x · L · C 1/2 , that is, voltage ut, x and current it, x
at place x on distortionless line with constant parameters and load ZL s L /C 1/2 are as
those on the line sending end, except for the time delay τ x · L · C 1/2 and the attenuation
expR · G · x. By using of lossless, constant parameter line with resistive load L /C 1/2 ,
it cannot be produced realistically even a relatively small signal delay. For example, pulse
delay τ 5 ms can be obtained from a lossless, constant parameter line with parameters
√
L 1.5 μH/m, C 18 pF/m, and the load resistance L /C 1/2 500/ 3 Ω at distance
x τ · L · C −1/2 ≈ 962.25 km from the line sending end. But, if lossy, distortionless line is
terminated with its characteristic impedance Z0 s L /C 1/2 , the attenuation factor expx ·
R · G 1/2 must, also, be taken into account.
42
Mathematical Problems in Engineering
Acknowledgment
This work is supported in part by the Serbian Ministry of Science and Technological Development through Projects TR 32048 and III 41006.
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