1730 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993 Lossy Transmission Line Transient Analysis by the Finite Element Method S.Y. Lee A. Konrad Department of Electrical Engineering, University of Toronto Toronto, Canada MSS 1A4 R. Saldanha Universidade Federal de Minas Gerais, Belo Horizonte, Brazil zyxwvu Abstract-A zyxwvutsrqpo time domain finite element method (TDFE) applied to the analysis of electrical transients on lossy transmission lines is presented. In this method, the general telegraph equations are solved numerically at discrete time steps. Examples of voltage waveforms on lossy single-phase lines are shown. I. INTRODUCTION The general telegraph equations can be solved analytically zyxwvutsr [ 1,2]. The simultaneous equations are first transformed to the frequency domain by Laplace transformation. Then they are combined to form a single second order differential equation and is solved. The time domain solution is obtained by taking the inverse transform. The mathematics involved is tedious and the complexity increases if the initial conditions, i.e. the initial voltage or current waveform, cannot be expressed in closed form. Analysis of the lossy line is essential for studies of electrical transients on transmission lines due to corona[3-61, skin effect[7,8] and proximity effect[9-111. Voltage and current waveforms along a lossy transmission line experience distortion and attenuation. These transients may affect the performance of the whole power system. ID. METHOD OF ANALYSIS A. Finite Element Discretization The finite element method requires the line to be subdivided into a finite number of regions called elements. Each element has points called interpolation nodes. This allows zyxw v and i to be written in the form where zyxwvuts M and N are the number of nodes of the i andv finite element segments, respectively, and @&) and vi(X) are basis functions (known functions of position) which interpolate the voltage and current within each element using the values at the interpolation nodes. Substitution of (5) and (6) into (1) and (2) yields zyx aM This paper describes a time domain finite element methodil2-141 for lossy transmission time analysis. In this approach, the numerically by solving a one-dimensional boundary value problem at each time step[l5]. zyxwvutsrq -2g wl(x)ll(t) = ~t@,(~)y(~) + C--C@.,(~)~(~) zy (8) The unknowns in the above equations are the nodal values of the voltage Vi(t) (i=l, 2, ..., M) and the current Z,(t) u=1, 2, ..., N). @i(x) and zyxwvu I&) are defined to be propagation of disturbance on lossy transmission lines is simulated ax ,=l zyxwvut at r=l II. GOVERNING EQUATIONS FOR A LOSSY LINE by the resistances of the conductors of the line and conductance between the lines. The differential equations are[l6]: Losses are introduced into the lossless transmission line equations @,(x)=l x=x, (9) = 0 at the other nodes, and ai -- - - Ri+ L- dx at ai dV ---= GV + C- ax dt (1) w;(x)=l x=xj = 0 at the other nodes (10) (2) such that (7) and (8) are enforced at each nodal point. As a result, (7) and (8) can be reduced to and capacitance per unit length, respectively. -ll+L-~-y(t), R lM&D J=1, 2, ... ,N (11) where R, G, L, and C are the resistance, conductance, inductance L at L dx G dy- 1 dv, Analytical solutions can be obtained for a distortionless line, i.e. when RC=GL. These solutions have the following form: -y+----C-zl(t), i=l, 2, ... , M (12) c at ax a(;)* -qz where CDl = iPI(xI) and y, = wl(xI) The two nodes of a fist-order finite element are placed in such a way that each voltage element contains an interpolation node for the current and each current element contains an interpolation node for the voltage. Only the interpolation functions associated with the adjacent nodes contribute to the summation in (11) and (12). Hence, (11) and (12) become u(n,t) = e (t + a x ) - e fi(i - a x ) (3) i(x, t) = gv(x, t) (4) Manuscriptreceived September 4,1992. This work is supported In part by the Natural Scienceand Engineering Research Council of Canada. 0018-9464/93$03.00 0 1993 IEEE