ELEKTRYKA Zeszyt () Ro LVIII Piotr JANKOWSKI Deartment of Marine Eectrica Power Engineering, Gdynia Maritime University SIMPLE MODELS OF TRANSMISSION LINE IN MATHCAD ENVIROMENTS Summary. The artice resents the ossibiities of the Mathcad environment which aow to create sime modes of transmission ines. Two modes are discussed: the discrete mode based on the circumferentia mode of ine treated as a system of L-tye four-termina networs and the mode caed continuous which imements soution of the teegrahy equations in the quasi-steady state. The artice emhasizes the didactic features of the above modes, mainy because of the use of the animation ossibiities of the Mathcad which aow to easiy imement existing anaytica soutions aso for transients. Keywords: transmision ine, discrete mode, animation of henomena PROSTE MODELE LINII DŁUGIEJ W ŚRODOWISKU MATHCAD Streszczenie. W artyue rzedstawiono możiwości środowisa Mathcad, ozwaające na stworzenie rostych modei inii długiej. Omówiono dwa modee: dysretny oarty na modeu obwodowym inii tratowanej jao uład czwórniów tyu Γ, oraz mode nazwany ciągłym imementującym rozwiązanie równań teegrafistów w rzyadu quasi-ustaonym. Artyuł odreśa waory dydatyczne owyższych modei, głównie ze wzgędu na wyorzystanie możiwości animacyjnych środowisa Mathcad, ozwaających na łatwe imementowanie gotowych rozwiązań anaitycznych również da stanów nieustaonych. Słowa uczowe: inia długa, mode dysretny, animacja zjawis. INTRODUCTION The cassic aroach to the teaching of the henomena occurring in a ong ine is based on the anaysis of soutions of teegrah equations. The basic textboos of eectrica engineering [,] reresent the most common one-dimensiona grahica interretation of soutions of ine equations for different states of wor. Currenty, the existing friendy environments such as Mathcad, mae it ossibe to imement a genera soution of teegrah equations. In addition, this tye of environment aows the resentation of soutions in the
. P. Janowsi animation form which, dramaticay maes it easier to exain the henomena occurring in the ine for different cases. The artice resents the seected asects of the rocess of creating ong ine sime modes. The first mode was based on the system of Γ four-termina networs. The second one is a direct imementation of teegrah equations soution where constants are determined deending on the ine arameters, its tye and the ind of oad. The rograms caed DISCRET for the first mode and ANIM for the second [3] aow to determine the arameters of the ine deending on its tye and aow the student to examine and observe animation of votages and currents in the ine for the argest number of cases.. DISCRETE MODEL OF TRANSMISSION LINE Modeing the ine using the umed arameters one can treat it as an eectrica networ consisting of Γ a four-termina networs (Fig. 3). This aroach can he sove the circuit transmission ine without using differentia teegrah equations. The accuracy of this mode wi increase with the degree of discretization. Longitudina and transverse arameters of the ine were mared by the imedance Z (Fig.) and by the imedance Z resectivey (Fig. ) Fig.. Longitudina arameters Rys.. Parametry odłużne Fig.. Transverse arameters Rys.. Parametry orzeczne Since the system of equations describing such a adder networ is inear, to sove it one can ay the matrix method. Currents vector in both ongitudina and transverse branches is obtained from the foowing formua (): I A B () During the observation of Kirchhoff`s equations for a various number of Γ networ Fig.3 it was noticed the foowing roerties of A,B matrix: For n oos n Kirchhoff`s equations are obtained, therefore A i,j matrix dimension: i = j = n, whereas B i,j matrix: i =, j = n-. B, eement of B matrix is: B, = E, the rest of eements of the B matrix are equa ; eements of A matrix are the imedances: Z, Z, Z or the vaues -,, + where: A i,j- =Z, A n,n- = Z, A i,i = Z, A i+,i = - Z, A n+i,i- =, A n+i,i =-, A n+i,i+ = -.
Sime modes of. Using the above roerties an agorithm was created with the aication of variabedimension matrix A(n). It aows to sove any of the adder system with the n number of oos with Z and Z imedance and Z oad. Such agorithm doesn t require formuating Kirchhoff's equations. As a resut, the soution of equation () gives the vector of the comex rms vaues of the currents in the transverse and ongitudina branches of the circuit (Fig. 3). Even eements of the current vector (I ) are currents of transverse branches. Odd eements of the current vector (I n ) are currents of ongitudina branches. Votages of the transverse branches are cacuated from the formua: U = I Z. Fig. 3. Ladder circuit of ine Rys. 3. Obwód drabinowy inii Both Mathcad environment and discrete aroach has imitations in the mode. The maximum number of meshes deends mainy on the magnitude of memory. In our simuation (PC-comuter) the maximum of n=. Hence, the A matrix contains 6 miion eements. Discrete mode aows us to determine the votage or current as a distance x function from the beginning of the ine with of x accuracy. Tabe Parameters of the teehone ine with osses Longitudina arameters Transverse arameters Resistance Conductance Caacitance Conductance R [/m] L [mh/m] C [nf/m] G [S/m].84 4.8 6.33 7. Suy arameters Line ength Load Votage Frequency [m] Z [] U f[hz] 3 Z f 4 8 The resented agorithm of the automatic generation of matrix equation was used to create a more universa rogram which was the reaization of a discrete transmission ine mode. The rogram uses atterns, aowing us to define the arameters of ongitudina and transverse ine, deending on the data, and tye of ine. Then, the resuts of comex currents
. P. Janowsi reresenting the matrix equation soution are further used to determine the instantaneous vaues of currents and votages ine both as a function of time and the distance from the beginning of the ine. Fig.4 shows the rms current waveform obtained from the simuation of discrete mode for the arameters of a teehone ine (tab.). [A].4x -3.x -3 I x -3 8x -4 6x -4 a) 3 x [m] Fig. 4. Rms current in the transverse (a) and ongitudina (b) branches Rys. 4. Prąd w gałęziach orzecznej (a) i odłużnej (b) - wartości suteczne On the basis of rms comex vaues (U ) one can obtain instantaneous vaues in time function at any oint of the ine. The votage in time function for transverse branch is determined by formua (): u( t, ) : if ( U, U sin( ωt arg( U ), ) () Simiary, you can get the current functions in the transverse or ongitudina branches (3). i( t, n ) : if ( I, I sin( ωt arg( I ),) (3) n n n It shoud be noted that the function () and (3) are functions of two variabes, where n and variabes corresonding to the odd and even eements resectivey for which we assign the distance from the start ine by the formuas: n x n, n 3,.. n x,, 4.. n (4) Where: n 3 -number of oos. Figures 5.6 show current and votage in distance function from ine beginning for the seected moment. As it can be observed, students can study the distribution of votages and currents and wave henomena using a discrete mode, athough the rogram does not searate waveform into a rimary and a secondary wave. In addition, increasing the degree of discretization, the correctness of the mode can be vaidated by examining the convergence of the resuts. Another way to verify its accuracy is to comare it with the continuous mode based on anaytica soution.
Sime modes of 3. [A].4. i n..4 3 x n [m] Fig. 5. Current in distance function from ine beginning for the seected moment Rys. 5. Prąd w funcji odegłości od oczatu inii da wybranej chwii czasowej 4 u 4 6 3 x [m] Fig. 6. Votage in distance function from ine beginning for the seected moment Rys. 6. Naięcie w funcji odegłości od oczątu inii na wybranym momencie 3. CONTINUOUS MODEL To resent the oerating states of the ine in animation form the nown genera soutions of incident and the refected wave (5), (6), (7) were used: u( t,x) : u( t,x) u( t, x) (5) u( t,x) : ( A / W ) e sin( ωt βx ψ) (6) αx u( t,x) : ( A / W ) e sin( ωt βx ψ) (7) where: ψ : if ( A, arg( A),) ψ : if ( A, arg( A ), ). The above conditiona definition of anges resuts from the fact that Mathcad does not determine an argument of a comex number which is equa to zero. In order not to do axis scaing during the animation by the Mathcad, the imit vaues are set. Therefore, the genera soutions are referred to the W vaue, which is a arger absoute vaue out of constants A and A determined in the rogram on the basis of the ine arameters. In order to create an animation: ) Define the range of variabe with FRAME word αx
4. P. Janowsi ) Cic Toos in the main menu, then Animation and the camera icon described as Record 3) Set start, end and seed of animation in Record Animation diaog box 4) Seect the chart area to be animated 5) Cic Animate in the Record Animation screen. After the time required to record the animation, the screen Pay Animation wi aear which aows us to start it. In order to chec correctness of the discrete mode a few votage and current waveforms obtained in the DISCRET and ANIM rograms were comared. As one can see in Fig.7, n = 3 is a sufficient amount to achieve very good comiance of votage waveforms. The above simuation was carried out for the arameters of the ossy teehone ine, oaded with doube vaue of wave imedance (tab.).
Sime modes of. 5 Fig. 7. Comarison of the instantaneous votages Rys. 7. Porównanie naięć chwiowych 4. ANIMATION OF TRANSIENT STATE The discrete mode of the ine resented in the second oint can be successfuy used for the anaysis of transients. Equation () woud have a simiar form and woud reresent a norma form necessary to ay rfixed Mathcad rocedure. However, it shoud be emhasized that for the system of e.g. differentia equations, the time of simuation woud be imracticay ong. Therefore, the teaching effect coud be questionabe. In Mathcad one can easiy imement the soution of transient state [] (in ossy no oad teehone ine), which aows us to create the animation of wave movement (8): t ch t sh x cos ) ( ) ( C L e G R ch x ) ( G R ch E ) x,t ( u t (8) Where: C G L R R G L C In order to resent the animation effect in the aer in Fig.8, the sequence of screens shows the votage disersing aong the ine for seected time moments (for =5). After switching the DC votage on the ong ine, a wave arises in the form of traezoida imuse with amitude E which moves towards the end of the ine. After τ time the wave refects from the end of the ine and traves bac in the form of "breaing down" use. As a resut of the attenuation the wave disaears after a few τ, and the course eventuay reaches a steady state. Due to the ine dissiation its votage decreases with increasing distance from the beginning of the ine.
6. P. Janowsi 6 4 u x 4 3 4 x [m] 6 4 u x 9 3 4 x [m] 6 u x 9 4 Fig. 8. The waveforms of instantaneous votages vaues for seected moments Rys. 8. Przebiegi naięć chwiowych w wybranych chwiach czasowych 3 4 x [m]
Sime modes of 7. Fig.7 shows the instantaneous votage in the midde of the transmission ine and at its the end. One can see that after a time which is equa to some τ, (τ =.5 s) the transient state disaears in the ine. 5 4 3 u( t ) u 5 4 t 3..4.6.8 t [s]..4.6.8 [s] [s] Fig. 9. The chart of instantaneous votages vaues in a seected distance (.5 and ) from the beginning of the ine Rys. 9. Diagram wartości naięć chwiowych w wybranych untach inii (odegłość,5 i od oczątu inii) t [s] 5. CONCLUSION Outine of the modes imemented in the Mathcad environment aows students to visuaize the different oerating states of the ine. One shoud aso remember that the wave character of these henomena is easier to observe owing to the animation, which undoubtedy has educationa vaue. It is worth emhasizing that the discrete mode is a more convenient form when used to examine e.g. inhomogeneous or non-distortion ine [4]. This mode can aso be easiy imemented in Mathcad to sove the transient state of the ine with any excitation. BIBLIOGRAPHY. Boowsi S.: Teoria obwodów eetrycznych. WNT, Warszawa 5.. Choewici T.: Eetrotechnia teoretyczna tom II. WNT, Warszawa 97. 3. Janowsi P.: Wybrane zagadnienia eetrotechnii w środowisu Mathcad. Wyd. AM, Gdynia. 4. Włodarczy M.: Substitution of the ong ine with non-distorting ine and a four-termina networ-ossibiity anaysis. IC-SPETO, Ustroń 8. Włynęło do Redacji dnia 5 czerwca r. Recenzent: Prof. dr hab. inż. Marian Paso
8. P. Janowsi Dr inż. Piotr JANKOWSKI Aademia Morsa w Gdyni, Wydział Eetryczny Katedra Eetroenergetyi Orętowej u. Morsa 8-87 8-5 Gdynia Te.: (58) 69--364; e-mai:eoitr@am.grynia.