signal propagation in a long wire
DESCRIPTION
We want to study the signal propagation in a wire of length l=10000 [m], having the resistance R, the inductance L and the capacitance C. At x=0 signals with the strength 1 [V] are sent during repeated time intervals of different lengths. Let’s denote this time-dependent input signal function by u0(t).The voltage U(x,t) in the wire is modeled by a hyperbolic PDE.Given the initial conditions and the boundary condition at x=0 the signal function is u0(t). At the other end x=l, the wire is open, i.e no signal is reflected but disappears out. The boundary condition fulfilling this condition is the advection equation We use the Finite Difference Method and discretize the x-axis into N intervals. We use the central difference approximations for the first and second order derivatives. To approximate the boundary condition at the right end try the upwind discretization (FTBS).For the wire the following parameter values are used: R=0.004 [Ω], L= [H] and C= [F].We try N=100 and N=200. We compute the signal propagation during a few milliseconds with the maximum time-step fulfilling the stability condition and also with a time-step being 70 percent of this.The code has been written in Matlab.TRANSCRIPT
KTH Course: Applied Numerical methods Team members
Professor: Lennart Edsberg Andreas Angelou
Paul Evans
Vasileios Papadimitriou
Daniel Tepic
Dionysios Zelios
I.C : ( ,0) 0u x ( ,0) 0u
xt
0 x l
0 x l
B.C : X=0
X=L
2 2
2 2
1u R u u
t L dt LC x
1( , ) ( , ) 0
u ul t l t
t xLC
0 ( ) 1u t [V]
0.004[ ]R 610 [ ]L H 80.25 10 [ ]C F
(N+1) grid-points
Discretization of x-axis into N intervals:
Finite Difference Method
first and second order derivatives:
Central difference approximations
Approximation of the right end of B.C:
Upwind discretization (FTBS)
1 1 1 1
1 1
2 2
2 21
2
n n n n n n n n
i i i i i i i iR
t L t LC x
u u u u u u u u
1
1 1
1( 2 ( 1) )
1
n n n n
i i i i
a
au u u u
2
2
1 ( )
( )
ta
LC x
2
t R
L
65 10t s
N=100 grid-points
N=100 grid-points
N=100 grid-points
N=100 grid-points
62.5 10t s
N=200 grid-points
N=200 grid-points
N=200 grid-points
N=200 grid-points
Taylor expansion
N=100-
N=200
Greater interval for finding stability
Without Taylor expansion
N=100-
N=200
Experimental values
N=100-
N=200
Approximation method stability max
N=100
N=200
65.025 10t 62.506 10t
67.0711 10t 63.5355 10t
64.9962 10t 62.4991 10t
65 10t 62.5 10t
N=100 grid-points
N=100 grid-points
