Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks

Transcription

1 656 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27 30, 2012 Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks M. Oumri, Q. Zhang, and M. Sorine INRIA Paris-Rocquencourt, France Abstract In this paper, a numerical solution is presented for the generalized BLT equation with inhomogeneous transmission lines. In particular, a fully automatized method is developed for the computation of the transmission line propagation matrices and the junction scattering matrices from network structural specification and from transmission line characteristic parameters. Two numerical examples are presented, one for a tree-shaped network, and the other for a network involving a circuit loop. 1. INTRODUCTION The Baum-Liu-Tesche (BLT) equation [1] is widely used for the modeling and analysis of complex transmission line networks, such as wired telecommunication networks and power lines in automotive vehicles, railway infrastructures, aircrafts, etc.. The original BLT equation for networks with homogeneous transmission lines has been generalized to the case of inhomogeneous transmission lines in [2]. This generalized BLT equation is parameterized by the propagation matrices of inhomogeneous transmission lines and by the scattering matrices at network junctions. However, it is not indicated in [2] how these propagation and scattering matrices can be computed from the structural specification of a network and from the inhomogeneous characteristic parameters of the transmission lines constituting the network. In this paper, a fully automatized method is presented for the computations of the propagation matrices and of the scattering matrices from the specification of the topological structure of a network and from the inhomogeneously distributed resistance, inductance, capacitance and conductance (RLCG) characteristic parameters of all the transmission lines. It is shown that the propagation and scattering matrices are independent of the choice of the directions of the currents in transmission lines, despite the fact that the definition of the two opposite waves on each line (as linear combinations of voltage and current) depends on the chosen current direction. It is then possible to compute the propagation and scattering matrices with some local convention for current directions on each line and at each junction, without taking care of the consistency between all the local choices in a network. The automatized computation method is greatly simplified thanks to these well defined local conventions. The computation of the scattering matrices has been partially inspired by the results reported in [3]. A new convention for the notations involved in the generalized BLT equation is also introduced to facilitate the implementation of the automatized approach to network simulation through the construction and numerical solution of the generalized BLT equation. 2. MATHEMATICAL MODEL We consider an electrical network of lossy transmission lines, formed by N J junctions J n for n {1,..., N J } and N B branches. Each branch B nm is delimited by two junctions J n and J 1 m. The branch B nm is parameterized by its distributed per-unit-length resistance R nm, inductance L nm, capacitance C nm and conductance G nm and its length l nm. Each branch B nm is also illuminated by an electromagnetic wave which is represented by equivalent current In m s and voltage Vn m s sources distributed along the branch. Let us denote by I n m the current leaving J n to J m along the branch B nm and by V n m the voltage along the same branch. The voltage and current waves along a branch driven by a harmonic source of angular frequency ω are solutions of the following frequency domain Telegrapher s equations, dv n m (z, ω) = Z nm (z, ω)i n m (z, ω) + V n m(z, s ω) di n m (z, ω), z [0, l nm ] (1) = Y nm (z, ω)v n m (z, ω) + I s n m(z, ω) 1 If there are more than one branches connecting J n and J m, an indexing exponent can be added to the notation B nm to distinguish them. For notation simplicity it is assumed in this paper that there is no multiple branches connecting two junctions.

2 Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27 30, where z is the coordinate along the branch and Z nm (z, ω) = R nm (z) + jωl nm (z), Y nm (z, ω) = G nm (z) + jωc nm (z) are the per-unit-length series impedance and shunt admittance respectively. We define the characteristic impedance ζ nm and the propagation constant γ nm of each branch B nm as follows: ζ nm (z, ω) Z nm (z, ω)ynm(z, 1 ω), γ nm (z, ω) Z nm (z, ω)y nm (z, ω) (2) On each branch, two power waves [5] of opposite directions are defined as linear combinations of the voltage and current along the branch. We denote by w n m and w m n the wave traveling from J n to J m and from J m to J n respectively. w n m (z, ω) ζ 1 2 nm(z, ω)v n m (z, ω) + ζ 1 2 nm (z, ω)i n m (z, ω) (3) w m n (z, ω) ζ 1 2 nm(z, ω)v n m (z, ω) ζ 1 2 nm (z, ω)i n m (z, ω) (4) In the same way, we define sources waves w s n m and w s m n in terms of I s n m and V s n. Let w nm (z, ω) = [w m n (z, ω), w n m (z, ω)] T and ũ nm (z, ω) = [w s m n(z, ω), w s n m(z, ω)] T. The combined waves vector w nm satisfies the following differential equation, d w nm (z, ω) = A nm (z, ω) w nm (z, ω) + ũ nm (z, ω), z [0, l nm ] (5) where A nm (z, ω) is defined accordingly in terms of γ nm and the potential function q nm which expresses the heterogeneity of the characteristic impedance along the branch B nm. q nm (z, ω) 1 [ ] d ln (ζ nm (z, ω)) γnm (z, ω) q, A nm (z, ω) nm (z, ω) 2 q nm (z, ω) γ nm (z, ω) Equation (5) is equivalent to (1) and will be useful for the definition propagation matrices. 3. WAVE PROPAGATION EQUATIONS The two equations of (5) cannot be solved separately and no closed form of the solution is available because A nm depends on z. In order to numerically solve Equation (5), we introduce a state transition matrix Φ nm related to A nm. This matrix satisfies the following differential equation, for any z, z [0, l nm ], dφ nm (z, z ; ω) = A nm (z, ω)φ nm (z, z ; ω), Φ nm (z, z; ω) = Φ nm (z, z ; ω) = I d (6) where I d is the identity matrix. If Φ nm (z, z ; ω) was known, then given the value of w nm (z, ω) for z equal to any z 0 [0, l nm ], the unique solution of (5) would be given by w nm (z, ω) = Φ nm (z, z 0 ; ω) w nm (z 0, ω) + z z 0 Φ nm (z, s; ω)ũ nm (s, ω)ds (7) As the state transition matrix Φ nm (z, z ; ω) is generally unknown, we do not really use it to solve (5), but it will be used to derive the equations for the computation of the wave propagation matrices defined as follows. For each branch B nm, the propagation matrix Γ nm (ω) relates the values of the waves two opposite waves w m n, w n m at the two ends of the branch through the equation [ ] wm n (ω) w n m (ω) [ ] [ ] Γn,m (ω) Γ = n, wm + Γ m,m (ω) Γ m, w n m (ω) [ dn,m (ω) d m, where w m n (ω) is the value of w m n (z, ω) at the end of the branch connected to junction J n (the index n is circled ), the other similar notations can be easily understood, and d n,m (ω), d m, are due to voltage and/or current sources distributed along the branch B nm. Using Equations (6) and (7), we show that the components of Γ nm (ω) and [d n,m (ω), d m,] T can be computed by solving ] (8)

3 658 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27 30, 2012 the following differential equations: dγ m,m (z, ω) = q nm (z, ω) 2γ nm (z, ω)γ m,m (z, ω) q nm (z, ω)γ 2 m,m(z, ω) dγ n,m (z, ω) = Γ n,m (z, ω) [q nm (z, ω)γ m,m (z, ω) γ nm (z, ω)] dγ m,n (z, ω) = Γ m,n (z, ω) [γ nm (z, ω) + q nm (z, ω)γ m,m (z, ω)] dγ n,n (z, ω) = Γ n,m (z, ω)q nm (z, ω)γ m,n (z, ω) dd n,m (z, ω) = Γ n,m (z, ω) [q nm (z, ω)d m,n (z, ω) + w m n(z, s ω)] dd m,n (z, ω) = d m,n (z, ω) [γ nm (z, ω) + q nm (z, ω)γ m,m (z, ω)] w m n(z, s ω) wm n(z, s ω)γ m,m (z, ω) + wn m(z, s ω) Γ m,m (0, ω) = Γ n,n (0, ω) = 0, Γ n,m (0, ω) = Γ m,n (0, ω) = 1, d n,m (0, ω) = d m,n (0, ω) = 0 Then we have Γ n,m (ω) = Γ n,m (l nm, ω), Γ n, = Γ n,n (l nm, ω), Γ m,m (ω) = Γ m,m (l nm, ω) Γ m, = Γ m,n (l nm, ω), d n,m (ω) = d n,m (l nm, ω) and d m, = d m,n (l nm, ω). 4. SCATTERING MATRIX Let wn (ω) be the vector composed of all the waves getting out of junction J n and evaluated the end connected to junction J n, or more formally wn (ω) [w n m (ω)] m Cn with C n being the set of indices of the junctions adjacent to J n. Similarly let wn [w m n (ω)] m Cn be the vector composed of all the waves getting into junction J n and evaluated the end connected to J n. In the literature of BLT equation, the scattering of a signal at junction J n is described by the scattering parameter S n (scalar or matrix) relating incoming waves to outgoing waves. wn (ω) = S wn (ω) (9) Suppose that each terminal junction, say J n, is not connected to any lumped source and is characterized by a generalized Thévenin equivalent as V n m (ω) = Z T,I n m (ω) (10) where Z T,n is a load connected to the same junction, V n m (ω) is the value of V n m (z, ω) evaluated at the end of B nm connected to J n. When the junction is open-circuited, we replace relation (10) by I n m (ω) = 0. In this case, the waves vectors wn (ω) and wn (ω) are scalars and are defined as a combination of voltage V n m (ω), current I n m (ω) and the characteristic impedance ζ n (ω) of branches B nm evaluated at J n. The scattering parameter S is given by S n = Z T,n ζ n (ω) Z T,n + ζ n (ω) (11) In the case of intermediate junctions 2, we suppose that all currents in the branches connected to J n have their negative directions pointing to junction J n. This choice of the current directions is only used for the computation of the scattering matrix at the junction J n, hence there is no problem of conflict between the local choices for different junctions. The scattering matrix at a junction is independent of the chosen current directions, and the above particular choice is for the purpose of simplifying the computation of the scattering matrix. Kirchhoff s laws at such a junction are expressed by: C I I = 0, C V V = 0 (12) where V and I represent the voltage and the current vectors on the branch connected to the junction J n and evaluated at this junction. The matrices C I and C V are filled with ±1 and 0 so that Equation (12) describes that fact that the sum of the currents is equal to zero and the 2 An intermediate junction is a junction connected to more than one branches. It is assumed in this paper that no load or lumped source is connected to intermediate junctions.

4 Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27 30, (a) (b) Figure 1: Example of two electric networks. Figure 2: Measured reflection coefficient compared to simulated reflection coefficient for a star shaped network. voltages at the ends of the branches connected to J n are all equal. Using (3) and (12), the waves vectors wn (ω) and wn (ω) are given as follows: w = Z 1 2 V + Z 1 2 I, w = Z 1 2 V Z 1 2 I (13) where Z n is a diagonal matrix filled with the characteristic impedance of branches connected to the junction J n. By inverting (13), the voltage V and the current I vectors can be computed as: V = 1 2 Z 1 2 [wn (ω) + w ], I = Z 2 n (ω) [wn (ω) wn (ω)] (14) Combining (12) and (14) and using the definition of the scattering matrix (9), we obtain, S = [ C V Z 1 2 C I Z 1 2 ] 1 [ C V Z 1 2 C I Z 1 2 ] (15) The BLT equation is the collections of the Equations (8) and (9) for all branches and all junctions constituting a network, usually written in a compact form with super wave vectors, super propagation matrices and super scattering matrices. 5. NUMERICALS SIMULATIONS The generalized BLT network numerical simulator is implemented in Matlab. With this numerical simulator, we can compute the reflection coefficient at any junction of the simulated network, the current and the voltage at any junction or at any point on a transmission line. In this section, we present the results of simulation examples and compare these results with real measurements which provides a validation of simulator. Example 1: For the tree-shaped network illustrated in Figure 1(a) made of coaxial cables with the characteristic impedance ζ(ω) = 50 [Ω] and wave propagation velocity c = [m/s], we simulate the reflection coefficient at junction J 1 and the voltage at the same junction. These results are then compared with the real measurements made on a laboratory test bed. Two terminal branches (J 3 and J 6 ) are open-circuited, and another one (J 5 ) is short-circuited. The ohmic loss of ( ω) [Ω/m] depending on ω but independent of z is added to each branch in the numerical simulator. After having solved the BLT equation for the power waves, we compute the reflection coefficient at J 1 as the ratio between the two waves, and deduce the current and voltage values through (14). Figure 2 and Figure 3 present a good agreement between measurements and simulations, both for the reflection coefficient and for the voltage V 1 2 (ω) at junction J 1 when the network is powered with lumped voltage source V s (ω) = 1[V ] for ω [0, 2.1[rad GHz]]. The

5 660 PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27 30, 2012 Figure 3: Measured voltage compared to simulated voltage for a tree-shaped network. Figure 4: Simulated currents at two ends of branch B 3,4 (Figure 1(b)). small difference in the modulus may come from measurements noises and losses in the connectors. This example has also been used in [4] where a simulator specialized for tree-shaped networks was presented. Example 2: We consider an electric network of lossless transmission lines, composed of 5 junctions and 5 branches, as illustrated in Figure 1(b). Each branch B nm is parametrized by its capacitance C nm (z) = 0.1 [nf/m] and inductance L nm (z) = (0.08e 5(z 2.3) ) [µh/m]. The network is supplied with a lumped voltage source V s (ω) = 1 [V] at junction J 1 and the terminal junction J 5 is short-circuited. The modulus of currents at the two ends of branch B 3,4 are shown in Figure 4, with a logarithmic scale. 6. CONCLUSION To summarize, this paper presents a numerical solution of the BLT equation generalized to inhomogeneous transmission lines networks. Despite the inhomogeneous nature of the transmission lines, the propagation matrix for each network branch is computed by solving simple differential equations. The computation of the scattering matrices at network junctions is also fully automatized. ACKNOWLEDGMENT This work has been supported by the ANR 0-DEFECT project. The authors are grateful to Mostafa Smail, Lionel Pichon and Florent Loete of Laboratoire de Génie Electrique de Paris for having kindly provided the experimental data used in this paper. REFERENCES 1. Baum, C. E., T. K. Liu, and F. M. Tesche, On the analysis of general multiconductor transmission line networks, Interaction Notes, Note 350, Baum, C. E., Generalization of the BLT equation, Interaction Notes, Note 511, Parmantier, J. P., An efficient technique to calculate ideal junction scattering parameters in multiconductor transmission line networks, Interaction Notes, Note 536, Oumri, M., Q. Zhang, and M. Sorine, A reduced model of reflectometry for wired electric networks, Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems MTNS 2010, , Kurokawa, K., Power Waves scattering matrix, Microwave Theory and Techniques, Vol. 13, No. 2, , 1965.