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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 20 . 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 22 . 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 24 . 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. 26 . 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