Mathematical Models for Engineering Science
Modeling of Nonuniform Multiconductor Transmission Lines via Wendroff Method Lubomír Brančík, Břetislav Ševčík First, think of a linear MTL defined by its length l and perunit-length (p.-u.-l.) n × n matrices R0(x), L0(x), G0(x), C0(x), i.e. nonuniform in general. The MTL telegraphic equations are
Abstract—The paper deals with a technique of the time-domain modeling of nonuniform multiconductor transmission lines (MTL) based on an implicit Wendroff method. This technique falls into a class of the finite-difference time-domain (FDTD) methods useful to solve various electromagnetic systems. Its basic version is extended to enable solving both voltage and/or current distributions along the MTL’s wires and their sensitivities with respect to lumped and distributed parameters. Experimental error analysis is performed based on the Thomson cable, a single transmission line with known analytical solutions. The examples of simulation of both uniform and nonuniform MTLs are shown and compared with other methods, first results for nonlinear MTLs are also presented. All computations were performed in the Matlab language environment while utilizing sparse matrix notations to enable effective and fast solution.
−
−
The principle of the implicit Wendroff formula lies in following operations on (1). For the j-th time step and k-th spatial coordinate, (1) is modified by substitutions
HE implicit Wendroff formula falls into a class of finitedifference time-domain methods (FDTD) useful to solve various electromagnetic systems in time domain [1]–[3]. For example, as is shown in [4], [5], the method can successfully be used for solving transient phenomena on single and threephase transmission lines in field of a power engineering, or its generalized version in [6] enables to perform transient and sensitivity analyses of uniform multiconductor transmission lines (MTL). In this paper a basic form of the Wendroff formula will further be adapted for solving nonuniform and/or nonlinear MTLs and some experiments will be shown. A simple system with an (n+1)-conductor transmission line, terminated by multiports, left (L), right (R), is shown in Fig. 1.
LUMPED PARAMETER CIRCUIT (L)
u(t , x) ∂u(t , x) ∂t ∂u(t , x) ∂x
(n+1) - conductor transmission line
0
x
vR
LUMPED PARAMETER CIRCUIT (R)
l
=
1 j ( uk +1 + u kj + u kj −+11 + u kj −1 ) , 4
=
1 ⎛ u kj − u kj −1 u kj +1 − u kj −+11 ⎞ + ⎜ ⎟ , 2 ⎝ Δt Δt ⎠
=
1 ⎛ u kj +1 − u kj u kj −+11 − u kj −1 ⎞ + ⎜ ⎟ , 2 ⎝ Δx Δx ⎠
j ,k
j ,k
v kj − v kj +1 + A vk (i kj + i kj +1 ) = − v kj −1 + v kj −+11 + B vk (i kj −1 + i kj −+11 ) ,
Fig. 1 Simple system with (n+1)-conductor transmission line
A ik ( v kj + v kj +1 ) + i kj − i kj +1 = Bik ( v kj −1 + v kj −+11 ) − i kj −1 + i kj −+11 ,
Manuscript received September 30, 2010. This work was supported by the Czech Ministry of Education under the MSM 0021630503 Research Project MIKROSYN and Pr. No. 102/08/H027. Lubomír Brančík and Břetislav Ševčík are with the Department of Radio Electronics, Brno University of Technology, Purkyňova 118, 612 00 Brno, Czech Republic (phone: +420 541149128, fax: +420 541149244, e-mails:
[email protected],
[email protected]).
ISBN: 978-960-474-252-3
j ,k
(2)
where u(t,x) denotes either the voltage v(t,x) or current i(t,x) vectors. The indexes have ranges k = 1,2,...,K, and j = 1,2,...,J, with K and J as the numbers of intervals Δx = l/K and Δt = T/J in space and time, respectively, and where l denotes an MTL’s length and T an upper limit of the time interval of interest. So we have chosen equidistant intervals to simplify our notations, although they could vary in general. Substituting (2) into (1) leads to systems of equations
iR
vL
(1)
II. IMPLICIT WENDROFF FORMULA APPLICATION
I. INTRODUCTION
iL
∂i (t , x) ∂v (t , x) , = G 0 ( x) v (t , x) + C0 ( x) ∂x ∂t
where v(t,x) and i(t,x) are n × 1 column vectors of voltages and currents of n active wires at the distance x from the MTL’s left end, respectively. Equation (1) is supplemented by boundary conditions reflecting terminating lumped-parameter circuits, e.g. by using their generalized Thévenin or Norton equivalents, as will be shown later.
Keywords—Nonuniform multiconductor transmission line; sensitivity analysis; time-domain modeling; Wendroff method.
T
∂v (t , x) ∂i (t , x) , = R 0 ( x)i (t , x) + L 0 ( x) ∂x ∂t
(3)
with coefficients matrices A vk = − ( R 0 k 2 + L0 k Δt ) Δx , Bvk = ( R 0k 2 − L0 k Δt ) Δx , Aik = − ( G 0k 2 + C0 k Δt ) Δx , Bik = ( G 0 k 2 − C0 k Δt ) Δx .
130
(4)
Mathematical Models for Engineering Science
It is clear that (7) is also needed in (9) thus their simultaneous recursive processing leads to both voltage/current distributions and their sensitivities. Firstly, if we exclude sensitivities w.r. to voltages of the external sources, ∂Dj/∂γ = 0. The derivatives ∂A/∂γ and ∂B/∂γ are done submatrix-wise as is obvious from decomposed forms shown in (8). Finally, ∂RiL(R)/∂γ = 0 if γ is a distributed parameter, see Table I, or it is stated easily if γ is a lumped parameter.
Here R0k = R0(ξk), L0k = L0(ξk), G0k = G0(ξk), and C0k = C0(ξk), with ξk ∈ (xk, xk+1), mostly ξk = (xk + xk+1)/2. Defining further T
T
v j = ⎡⎣ v1j T , v 2j T ,… , v KjT+1 ⎤⎦ , i j = ⎡⎣i1j T , i 2j T ,… , i KjT+1 ⎤⎦ ,
(5)
as column vectors of the order n(K+1) × 1, and finally x j = ⎡⎣ v jT , i jT ⎤⎦
T
(6)
as the 2n(K+1) × 1 column vector, with we can write a recursive formula
T
TABLE I DERIVATIVES OF MATRICES (4) W.R. TO PARAMETERS γ
as a transposition,
x j = A −1 ( Bx j −1 + D j ) .
(7)
The formula (7) expresses a solution in incoming time tj, based on the values in preceding time tj-1. The matrices A and B are formed by (4) and by boundary conditions, the column vector Dj depends on values of external sources taken in time tj. The constitution of these matrices can be explained by (8), when the MTL is divided only on K = 3 parts. j
-I 0 0 A v1 ⎡ v1 ⎤ ⎡ I ⎢v ⎥ ⎢ 0 I I 0 0 ⎢ 2⎥ ⎢ ⎢ v3 ⎥ ⎢ 0 -I 0 I 0 ⎢ ⎥ ⎢ 0 I ⎢ v 4 ⎥ ⎢ A i1 A i1 0 ⎢i ⎥ =⎢ 0 A A i2 0 0 1 i2 ⎢ ⎥ ⎢ 0 A i3 A i3 0 ⎢ i2 ⎥ ⎢ 0 ⎢i ⎥ ⎢ I 0 0 0 R iL ⎢ 3⎥ ⎢ ⎢⎣ i 4 ⎥⎦ ⎢⎣ 0 0 0 I 0 ⎛ ⎡ -I I 0 0 B v1 B v1 ⎜⎢ -I I 0 0 B v2 ⎜⎢ 0 ⎜⎢ 0 0 -I I 0 0 ⎜⎢ ⎜ ⎢B i1 B i1 0 0 -I I ×⎜ ⎢ 0 B B 0 0 -I i2 i2 ⎜⎢ ⎜⎢ 0 0 B i3 B i3 0 0 ⎜⎢ 0 0 0 0 0 ⎜⎢ 0 ⎜⎢ 0 0 0 0 0 0 ⎣ ⎝
A v1 A v2 0 -I I 0 0 0 0 B v2 B v3 0 I -I 0 0
0 A v2 A v3 0 -I I 0 0
j −1
⎡0 ⎤ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ 0 + ⎢⎢ ⎥⎥ 0 ⎢ ⎥ ⎢ 0 ⎥ ⎢v ⎥ ⎢ iL ⎥ ⎣⎢ v iR ⎦⎥
ISBN: 978-960-474-252-3
∂B vk ∂γ
γ ∈C0k
0
0
−
γ ∈ G 0k γ ∈ R 0k
γ ≡l
−
∂L 0k Δx ∂γ Δt
−
∂A ik ∂γ −
∂L 0k Δx ∂γ Δt
∂B ik ∂γ
∂C0k Δx ∂γ Δt
−
∂C0k Δx ∂γ Δt
0 −
0
∂G 0 k Δx ∂γ 2
∂G 0 k Δx ∂γ 2
0
0
∂R 0 k Δx ∂γ 2
∂R 0 k Δx ∂γ 2
0
0
B vk l
A ik l
Bik l
A vk l
III. EXPERIMENTAL ACCURACY EVALUATION
j
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
An accuracy of the method will experimentally be assessed through known analytical solutions for voltage and current distributions on a Thomson cable, an uniform single TL with negligible L0 and G0 primary parameters. The cable is driven from a unit step voltage source as shown in Fig. 2.
. (8)
0 viL (t ) = 1(t )
x
l=4
RiL = 0.5
riR (t ) =
v (t , l ) i (t , l )
R0 = 0.1 C0 = 0.5 Fig. 2 Transmission line system with Thomson cable
The terminating circuits are supposed to be linear resistive, and they can be replaced by their generalized Thévenin equivalents with matrices of internal resistances RiL and RiR, and vectors of internal voltages viL(t) and viR(t). They are located in two downmost lines of (8) corresponding to the boundary conditions incorporation. In more general case, however, when reactive elements are present, ordinary differential equations should have to be considered instead of the algebraic ones. Finally, I and 0 are the n-th order identity and zero matrices, respectively. The equation (7) will now be used to derive a procedure for evaluation of sensitivities. Let us consider a parameter γ which can be some element of MTL p.-u.-l. matrices, the MTL length or a lumped parameter of terminating circuits. Then, after some arrangements, we have ⎛ ∂x j −1 ∂A j ∂B j −1 ∂D j ⎞ ∂x j = A −1 ⎜ B − x + x + ⎟ . ∂γ ∂γ ∂γ ∂γ ⎠ ⎝ ∂γ
∂A vk ∂γ
γ ∈ L 0k
−1
0 ⎤ 0 ⎥⎥ A v3 ⎥ ⎥ 0 ⎥ × 0 ⎥ ⎥ -I ⎥ 0 ⎥ ⎥ -R iR ⎥⎦
0 ⎤ ⎡ v1 ⎤ 0 ⎥⎥ ⎢⎢ v 2 ⎥⎥ B v3 ⎥ ⎢ v 3 ⎥ ⎥ ⎢ ⎥ 0 ⎥ ⎢v4 ⎥ ⋅ 0 ⎥ ⎢ i1 ⎥ ⎥ ⎢ ⎥ I ⎥ ⎢ i2 ⎥ 0 ⎥ ⎢⎢ i 3 ⎥⎥ ⎥ 0 ⎦⎥ ⎣⎢ i 4 ⎦⎥
Parameter
For an infinitely long cable, when no reflected waves exist, analytical solutions have the forms [7] i (t , x) = 1(t ) RiL ⋅ e
a (t )( a (t ) + 2b (t , x ))
⋅ erfc ( a (t ) + b(t , x) ) ,
v(t , x) = 1(t ) ⋅ erfc ( b(t , x) ) − RiL i (t , x) ,
(10)
a (t ) = R0 t C0 RiL , b(t , x) = x 2 ⋅ R0 C0 t ,
with erfc as a complementary error function. To ensure a validity of the above equations at the TL of a finite length l, it is terminated by a time-dependent resistance riR(t) matching it perfectly in the time interval of interest. The relative errors evaluated in case of K = J = 256 are shown in Fig. 3, where also comparison with a state-variable method [7] solved in the time domain, for the same number of sections, is presented. Following (10) the semirelative sensitivities have been derived and summarized in Table II.
(9)
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Mathematical Models for Engineering Science
Fig. 3 Thomson cable relative errors (Wendroff vs. state variables)
Fig. 4. Thomson cable semirelative sensitivities and relative errors
TABLE II THOMSON CABLE’S SEMIRELATIVE SENSITIVITIES
IV. EXAMPLES OF MTL SIMULATION
Parameter
Semirelative sensitivity Sγ = γ
f(t,x)
In the following a simple (2+1)-conductor transmission line system will be considered according to Fig. 5.
∂f (t , x) ∂γ
2
i(t,x)
γ ≡ R0
−1(t )
e−b (t ,x ) [ a(t ) + b(t , x)] + a(t ) [ a(t ) + 2b(t, x)] i(t, x) RiL π 2
v(t,x)
1(t )
e−b (t ,x)
π
a(t ) − RiLa(t ) [ a(t ) + 2b(t, x)] i(t, x)
1(t )
γ ≡ C0
x
v1(x,0)
l
RiR1=100Ω
viL1(t)
2
i(t,x)
0
RiL1=50Ω
e−b (t ,x ) [ a(t ) − b(t , x)] − a 2 (t )i (t , x) RiL π
RiL2=100Ω
RiR2=50Ω
2
−1(t )
v(t,x)
e−b (t ,x)
π
a(t ) + RiLa2 (t )i(t, x)
Fig. 5. Simple (2+1)-conductor transmission line system
2
2e−b (t ,x) a(t ) − ( 2a(t ) [ a(t ) + b(t, x)] + 1) i(t, x) RiL π
i(t,x)
1(t )
v(t,x)
−1(t )
γ ≡ RiL
2
2e−b (t ,x)
π
The generalized Thévenin internal matrices and vectors are 0 ⎤ 0 ⎤ ⎡R ⎡R ⎡v ⎤ ⎡0⎤ , viL = ⎢ iL1 ⎥ , RiR = ⎢ iR1 , viR = ⎢ ⎥ . (11) RiL = ⎢ iL1 ⎥ ⎥ R R 0 0 0 ⎣ ⎦ ⎣0⎦ ⎣ ⎣ iL2 ⎦ iR2 ⎦
a(t ) + 2RiLa(t ) [ a(t) + b(t, x)] i(t, x)
First, the MTL is considered under zero initial conditions, i.e. v(x,0) = i(x,0) = 0, while viL1(t) = sin2(πt/2·10-9), 0 ≤ t ≤ 2·10-9, and viL1(t) = 0, otherwise. The MTL’s length is l = 0.4m, and its p.-u.-l. matrices P0(x) ∈ {R0(x), L0(x), G0(x), C0(x)} have a unified form
Examples of voltage semirelative sensitivities are presented in Fig. 4, including relative errors determined on a basis of Table II, with K = J = 512 intervals considered. The results are comparable with those in Fig. 3, the errors increase generally at beginning of a time interval where discontinuities occur. In this special case, when a terminating resistance riR(t) is time dependent, as is defined in Fig. 2, the matrix A in (7) has to be evaluated repeatedly in each time step with the help of (10), for x = l substituted. It is similarly valid for ∂A/∂γ in (9), if sensitivities are calculated, when ∂riR(t)/∂γ is stated on a basis of Table II.
ISBN: 978-960-474-252-3
⎡P P0 ( x) = P0 e px = ⎢ 11 ⎣ P12
P12 ⎤ px e , P22 ⎥⎦
(12)
with particular parameters R11 = R22 = 0.1Ω/m, R12 = 0.02Ω/m, L11 = L22 = 494.6nH/m, L12 = 63.3nH/m, G11 = G22 = 0.1S/m, G12 = -0.01S/m, C11 = C22 = 62.8pF/m, and C12 = -4.9pF/m
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[8]. If p = 0, for all p.-u.-l. matrices, the MTL becomes uniform. An inhomogeneity is introduced by the value p = ln(2)/l ≈ 1.733 to get two-time greater p.-u.-l. parameters at the MTL‘s end compared to its beginning. The examples for both uniform and nonuniform MTLs are shown in Fig. 6.
through C0 matrix, in which C11 = C22 p.-u.-l. capacitances are replaced by voltage-dependent ones, C(vi) = Cii/(1+|vi|/Vp)2, i = 1,2, for Vp = 0.75V. The MTL excitation corresponds to that used for the example in Fig. 6.
Fig. 8. Voltage distributions for nonlinear uniform/nonuniform MTL
V. CONCLUSIONS The paper has dealt with a technique for the time-domain simulation of multiconductor transmission lines based on an implicit Wendroff method. Namely, the technique was further generalized towards nonuniform MTLs and first preliminary works pertaining to nonlinear MTLs were done. In this case, (7) is modified on xj = (Aj-1)-1(Bj-1xj-1+Dj), when Aj-1 and B j-1 depend on actual x j-1 and must be evaluated repeatedly in each time step. In case of the linear MTLs, the method is stable and accurate enough to compute both voltage/current distributions and their sensitivities for practice. In case of nonlinear MTLs, further studies and experiments will be performed, especially in light of method stability. A few Matlab programme functions have been developed while utilizing sparse matrix notations to enable effective and fast solution. On the PC 2GHz/2GB, and for a grid of points 512 × 512, the CPU times were roughly 8 s (linear MTL’s waves), 12 s (nonlinear MTL’s waves) and 18 s (MTL’s wave sensitivities).
Fig. 6. Voltage distributions (left) and their semirelative sensitivities w.r. to L12 ∈ L0(x) (right) for uniform and nonuniform MTL
It can be seen when doubling all p.-u.-l. parameters (2P0 in the middle of Fig. 6) the MTL time delay increases, roughly twice (as reactive parameters are increased). In case of a nonuniform MTL, wave velocity is not constant, but it decreases gradually along the line leading to some intermediate time delay. The method can easily be adapted to get responses to MTL initial voltage or current distributions. Examples in case of the voltage distribution on the 1st wire v1(x,0) = sin2(π(4x/l – 3/2)), if 3l/8 ≤ x ≤ 5l/8, and v1(x,0) = 0, otherwise, with all the other quantities equal zero, are shown in Fig. 7.
REFERENCES [1] D. M. Sullivan, Electromagnetic Simulation using the FDTD Method. New York: IEEE Press, 2000. [2] W. Sui, Time-Domain Computer Analysis of Nonlinear Hybrid Systems. New York: CRC Press, 2002. [3] L. Dou, J. Dou, “Sensitivity analysis of lossy nonuniform multiconductor transmission lines based on Lax-Wendroff technique,” Int. J. Numer. Model., 2010, vol. 23, no. 3, pp. 165–182. [4] Z. Benešová, D. Mayer, “Overvoltage phenomena on transmission lines” in Proc. XI. ISTET’2001, Linz, 2001, pp. 1–4. [5] Z. Benešová, V. Kotlan, “Propagation of surge waves on interconnected transmission lines induced by lightning stroke,” Acta Technica CSAV, 2006, vol. 51, no. 3, pp. 301–316. [6] L. Brančík “Transient and sensitivity analysis of uniform multiconductor transmission lines via FDTD methods,” in Proc. XV. ISTET’2009, Lübeck, 2009, pp. 109–113. [7] L. Brančík, “Sensitivity in multiconductor transmission line lumpedparameter models,” in Proc. 31st TSP’2008, Paradfürdo, 2008, pp. 1–4. [8] C. R. Paul, Analysis of Multiconductor Transmission Lines. New York: John Wiley & Sons, 1994. [9] L. Brančík, “Voltage/current waves sensitivity in hybrid circuits with nonuniform MTLs,” in Proc. 10th IEEE Workshop SPI’2006, Berlin, 2006, pp. 177–180. [10] J. Dědková, L. Brančík, “Laplace transform and FDTD approach applied to MTL simulation,” Piers Online, 2008, vol. 4, no. 1, p. 16-20.
Fig. 7. Voltage (left) and current (right) responses to unitial v1(x,0)
All the results for linear MTLs were also compared with a Laplace transform method [9], [10], with very good matching. In the end, first experiments with simulation of a nonlinear MTL are shown in Fig. 8. The nonlinearity is introduced
ISBN: 978-960-474-252-3
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