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ELEKTRYKA
Zeszyt 2 (222)
2012
Rok LVIII
Piotr JANKOWSKI
Department of Marine Electrical Power Engineering, Gdynia Maritime University
SIMPLE MODELS OF TRANSMISSION LINE IN MATHCAD
ENVIROMENTS
Summary. The article presents the possibilities of the Mathcad environment which allow
to create simple models of transmission lines. Two models are discussed: the discrete model
based on the circumferential model of line treated as a system of L-type four-terminal networks
and the model called continuous which implements solution of the telegraphy equations in the
quasi-steady state. The article emphasizes the didactic features of the above models, mainly
because of the use of the animation possibilities of the Mathcad which allow to easily
implement existing analytical solutions also for transients.
Keywords: transmision line, discrete model, animation of phenomena
PROSTE MODELE LINII DŁUGIEJ W ŚRODOWISKU MATHCAD
Streszczenie. W artykule przedstawiono możliwości środowiska Mathcad,
pozwalające na stworzenie prostych modeli linii długiej. Omówiono dwa modele:
dyskretny oparty na modelu obwodowym linii traktowanej jako układ czwórników typu
Γ, oraz model nazwany ciągłym implementującym rozwiązanie równań telegrafistów
w przypadku quasi-ustalonym. Artykuł podkreśla walory dydaktyczne powyższych
modeli, głównie ze względu na wykorzystanie możliwości animacyjnych środowiska
Mathcad, pozwalających na łatwe implementowanie gotowych rozwiązań analitycznych
również dla stanów nieustalonych.
Słowa kluczowe: linia długa, model dyskretny, animacja zjawisk
1. INTRODUCTION
The classic approach to the teaching of the phenomena occurring in a long line is based
on the analysis of solutions of telegraph equations. The basic textbooks of electrical
engineering [1,2] represent the most common one-dimensional graphical interpretation of
solutions of line equations for different states of work. Currently, the existing friendly
environments such as Mathcad, make it possible to implement a general solution of telegraph
equations. In addition, this type of environment allows the presentation of solutions in the
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P. Jankowski
animation form which, dramatically makes it easier to explain the phenomena occurring in the
line for different cases. The article presents the selected aspects of the process of creating long
line simple models. The first model was based on the system of Γ four-terminal networks. The
second one is a direct implementation of telegraph equations solution where constants are
determined depending on the line parameters, its type and the kind of load. The programs
called DISCRET for the first model and ANIM for the second [3] allow to determine the
parameters of the line depending on its type and allow the student to examine and observe
animation of voltages and currents in the line for the largest number of cases.
2. DISCRETE MODEL OF TRANSMISSION LINE
Modeling the line using the lumped parameters one can treat it as an electrical network
consisting of Γ a four-terminal networks (Fig. 3). This approach can help solve the circuit
transmission line without using differential telegraph equations. The accuracy of this model
will increase with the degree of discretization. Longitudinal and transverse parameters of the
line were marked by the impedance Z1 (Fig.1) and by the impedance Z2 respectively (Fig. 2)
Fig. 1. Longitudinal parameters
Rys. 1. Parametry podłużne
Fig. 2. Transverse parameters
Rys. 2. Parametry poprzeczne
Since the system of equations describing such a ladder network is linear, to solve it one can
apply the matrix method. Currents vector in both longitudinal and transverse branches is
obtained from the following formula (1):
I  A 1 B
(1)
During the observation of Kirchhoff`s equations for a various number of Γ network Fig.3 it
was noticed the following properties of A,B matrix:
 For n loops „2n – 1” Kirchhoff`s equations are obtained, therefore Ai,j matrix dimension:
i = j = 2n – 1, whereas Bi,j matrix: i = 1, j = 2n-1.
 B1,1 element of B matrix is: B1,1 = E, the rest of elements of the B matrix are equal 0;
 elements of A matrix are the impedances: Z1, Z2, Z0 or the values -1, 0, +1 where:
 Ai,2j-1=Z1, An,2n-1 = Z0,
Ai,2i = Z2,
Ai+1,2i = - Z2, An+i,2i-1=1, An+i,2i=-1,
An+i,2i+1 = - 1.
Simple models of…
21.
Using the above properties an algorithm was created with the application of variabledimension matrix A(n). It allows to solve any of the ladder system with the n number of loops
with Z1 and Z2 impedance and Z0 load. Such algorithm doesn’t require formulating
Kirchhoff's equations. As a result, the solution of equation (1) gives the vector of the complex
rms values of the currents in the transverse and longitudinal branches of the circuit (Fig. 3).
Even elements of the current vector (Ikp) are currents of transverse branches. Odd elements of
the current vector (Ikn) are currents of longitudinal branches. Voltages of the transverse
branches are calculated from the formula: Ukp = Ikp Z2.
Fig. 3. Ladder circuit of line
Rys. 3. Obwód drabinkowy linii
Both Mathcad environment and discrete approach has limitations in the model. The
maximum number of meshes depends mainly on the magnitude of memory. In our simulation
(PC-computer) the maximum of n=2000. Hence, the A matrix contains 16 million elements.
Discrete model allows us to determine the voltage or current as a distance x function from
the beginning of the line with of △x accuracy.
Table 1
Parameters of the telephone line with losses
Longitudinal parameters
Transverse parameters
Resistance Conductance Capacitance Conductance
R0 [/km]
L0 [mH/km]
C0 [nF/km]
G0 [S/km]
2.84
24.8
6.33
7.0
Line length
Load
Supply parameters
Voltage
Frequency
[km]
Z0 []
U[V]
f[Hz]
300
2 Zf
40
800
The presented algorithm of the automatic generation of matrix equation was used to
create a more universal program which was the realization of a discrete transmission line
model. The program uses patterns, allowing us to define the parameters of longitudinal and
transverse line, depending on the data, and type of line. Then, the results of complex currents
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P. Jankowski
representing the matrix equation solution are further used to determine the instantaneous
values of currents and voltages line both as a function of time and the distance from the
beginning of the line. Fig.4 shows the rms current waveform obtained from the simulation of
discrete model for the parameters of a telephone line (tab.1).
[A]
1.4x10-3
1.2x10-3
|Ikp|
1x10-3
8x10-4
6x10-4
a)
0
100
x [ km]
200
300
Fig. 4. Rms current in the transverse (a) and longitudinal (b) branches
Rys. 4. Prąd w gałęziach poprzecznej (a) i podłużnej (b) - wartości skuteczne
On the basis of rms complex values (Uk) one can obtain instantaneous values in time
function at any point of the line. The voltage in time function for kp transverse branch is
determined by formula (2):
u(t ,k p ) : if (U k p  0, 2  U k p  sin(ωt  arg(U k p ) ,0)
(2)
Similarly, you can get the current functions in the transverse or longitudinal branches (3).
i (t , k n ) : if ( I k n  0, 2  I k n  sin(ωt  arg( I kn ),0)
(3)
It should be noted that the function (2) and (3) are functions of two variables, where kn and kp
variables corresponding to the odd and even elements respectively for which we assign the
distance from the start line by the formulas:
x kn 
kn  1
, k n  1,3..2n  1
2
xk p 
kp
2
, k p  2,4..2n  2
(4)
Where: n  3 -number of loops.
Figures 5.6 show current and voltage in distance function from line beginning for the
selected moment. As it can be observed, students can study the distribution of voltages and
currents and wave phenomena using a discrete model, although the program does not separate
waveform into a primary and a secondary wave. In addition, increasing the degree of
discretization, the correctness of the model can be validated by examining the convergence of
the results. Another way to verify its accuracy is to compare it with the continuous model
based on analytical solution.
Simple models of…
[A]
23.
0.04
0.02
ik
0
n
 0.02
0.04
0
100
xk
200
300
[km]
n
Fig. 5. Current in distance function from line beginning for the selected moment
Rys. 5. Prąd w funkcji odległości od poczatku linii dla wybranej chwili czasowej
[V]
40
20
0
uk
p
 20
 40
 60
0
100
200
xk
p
300
[km]
Fig. 6. Voltage in distance function from line beginning for the selected moment
Rys. 6. Napięcie w funkcji odległości od początku linii na wybranym momencie
3. CONTINUOUS MODEL
To present the operating states of the line in animation form the known general solutions
of incident and the reflected wave (5), (6), (7) were used:
u(t , x ) : u1( t , x )  u 2( t , x )
(5)
u1(t , x ) : ( A1 / W )  2  e  α x  sin(ωt  βx  ψ1)
u2(t , x ) : ( A2 / W )  2  e
α x
 sin( ωt  βx  ψ 2)
(6)
(7)
where:
ψ 1 : if ( A1  0 , arg( A1) , 0 ) ψ 2 : if ( A 2  0 , arg( A 2 ) , 0 ).
The above conditional definition of  angles results from the fact that Mathcad does not
determine an argument of a complex number which is equal to zero.
In order not to do axis scaling during the animation by the Mathcad, the limit values are
set. Therefore, the general solutions are referred to the W value, which is a larger absolute
value out of constants A1 and A2 determined in the program on the basis of the line
parameters.
In order to create an animation:
1) Define the range of variable with FRAME word
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P. Jankowski
2) Click Tools in the main menu, then Animation and the camera icon described as Record
3) Set start, end and speed of animation in Record Animation dialog box
4) Select the chart area to be animated
5) Click Animate in the Record Animation screen.
After the time required to record the animation, the screen Play Animation will appear which
allows us to start it.
In order to check correctness of the discrete model a few voltage and current waveforms
obtained in the DISCRET and ANIM programs were compared. As one can see in Fig.7, n =
300 is a sufficient amount to achieve very good compliance of voltage waveforms. The above
simulation was carried out for the parameters of the lossy telephone line, loaded with double
value of wave impedance (tab.1).
Simple models of…
25.
Fig. 7. Comparison of the instantaneous voltages
Rys. 7. Porównanie napięć chwilowych
4. ANIMATION OF TRANSIENT STATE
The discrete model of the line presented in the second point can be successfully used for
the analysis of transients. Equation (1) would have a similar form and would represent a
normal form necessary to apply rkfixed Mathcad procedure. However, it should be
emphasized that for the system of e.g. 1000 differential equations, the time of simulation
would be impractically long. Therefore, the teaching effect could be questionable.
In Mathcad one can easily implement the solution of transient state [2] (in lossy no load
telephone line), which allows us to create the animation of wave movement (8):
 ch R0G0 ( l  x ) e  t
u( x ,t )  E 
 2
l L0C0
 ch R0G0 l


( 1 )k
 2 k  1 l  x  


(
2
k

1
)

cos



sh

t

ch

t


k
k
2
2

2
l





k


k 0 k

(8)
1 R G 
Where:    0  0 
2  L0 C0 
2
1 
 2k  1  
k   
 
 R0G0  
L0C0 
 2l
 
2
In order to present the animation effect in the paper in Fig.8, the sequence of screens
shows the voltage dispersing along the line for selected time moments (for k=50). After
switching the DC voltage on the long line, a wave arises in the form of trapezoidal impulse
with amplitude E which moves towards the end of the line. After τ time the wave reflects
from the end of the line and travels back in the form of "breaking down" pulse. As a result of
the attenuation the wave disappears after a few τ, and the course eventually reaches a steady
state. Due to the line dissipation its voltage decreases with increasing distance from the
beginning of the line.
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.
P. Jankowski
[V]
60
40
u x 4
20
0
0
100
200
x
300
400
[km]
[V]
60
40
9
u x 10
20
0
0
100
200
300
400
x [km]
[V]
60
40
19
u x 10
20
0
0
100
200
x
Fig. 8. The waveforms of instantaneous voltages values for selected moments
Rys. 8. Przebiegi napięć chwilowych w wybranych chwilach czasowych
[km]
300
400
Simple models of…
27.
Fig.7 shows the instantaneous voltage in the middle of the transmission line and at its the end.
One can see that after a time which is equal to some τ, (τ = 0.005 s) the transient state
disappears in the line.
[V]
50
50
[V]
40
30
[V]
[V]
40
u ( l t )
20
10
0


u
l
2
 30
 20
t 
10
0
0.02
0.04
0.06
0.08
t [s]
0
0
0.02
0.04
0.06
0.08
t [s]
[s]
[s]
Fig. 9. The chart of instantaneous voltages values in a selected distance (0.5l and l) from the beginning
of the line
Rys. 9. Diagram wartości napięć chwilowych w wybranych punktach linii (odległość 0,5l i l od
początku linii)
5. CONCLUSION
Outline of the models implemented in the Mathcad environment allows students to visualize
the different operating states of the line. One should also remember that the wave character of
these phenomena is easier to observe owing to the animation, which undoubtedly has
educational value. It is worth emphasizing that the discrete model is a more convenient form
when used to examine e.g. inhomogeneous or non-distortion line [4]. This model can also be
easily implemented in Mathcad to solve the transient state of the line with any excitation.
BIBLIOGRAPHY
1. Bolkowski S.: Teoria obwodów elektrycznych. WNT, Warszawa 2005.
2. Cholewicki T.: Elektrotechnika teoretyczna tom II. WNT, Warszawa 1971.
3. Jankowski P.: Wybrane zagadnienia elektrotechniki w środowisku Mathcad. Wyd. AM,
Gdynia 2010.
4. Włodarczyk M.: Substitution of the long line with non-distorting line and a four-terminal
network-possibility analysis. IC-SPETO, Ustroń 2008.
Recenzent: Prof. dr hab. inż. Marian Pasko
Wpłynęło do Redakcji dnia 25 czerwca 2012 r.
28
.
Dr inż. Piotr JANKOWSKI
Akademia Morska w Gdyni, Wydział Elektryczny
Katedra Elektroenergetyki Okrętowej
ul. Morska 81-87
81-225 Gdynia
Tel.: (058) 69-01-364; e-mail:[email protected]
P. Jankowski