Numerical Solution of BLT Equation for Inhomogeneous Transmission Line ... Ï are solutions of the following frequency domain Telegrapher's equations,..
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Numerical Solution of BLT Equation for Inhomogeneous Transmission Line Networks M.Oumri1 , Q.Zhang1 , and M.Sorine1 1
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 NJ junctions Jn for n ∈ {1, ..., NJ } and NB branches. Each branch Bnm is delimited by two junctions Jn and Jm 1 . The branch Bnm is parameterized by its distributed per-unit-length resistance Rnm , inductance Lnm , capacitance Cnm and conductance Gnm and its length `nm . Each branch Bnm is also illuminated by an electromagnetic wave which s s is represented by equivalent current In→m and voltage Vn→m sources distributed along the branch. Let us denote by In→m the current leaving Jn to Jm along the branch Bnm and by Vn→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, s dVn→m (z, ω) = −Znm (z, ω)In→m (z, ω) + Vn→m (z, ω) dz , ∀z ∈ [0, `nm ] (1) s dIn→m (z, ω) = −Ynm (z, ω)Vn→m (z, ω) + In→m (z, ω) dz 1 If there are more than one branches connecting J and J , an indexing exponent can be added to the notation B n m nm to distinguish them. For notation simplicity it is assumed in this paper that there is no multiple branches connecting two junctions.
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where z is the coordinate along the branch and Znm (z, ω) = Rnm (z)+jωLnm (z), Ynm (z, ω) = Gnm (z)+ jωCnm (z) are the per-unit-length series impedance and shunt admittance respectively. We define the characteristic impedance ζnm and the propagation constant γnm of each branch Bnm as follows : ζnm (z, ω) ,
q
−1 Znm (z, ω)Ynm (z, ω) , γnm (z, ω) ,
p
Znm (z, ω)Ynm (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 wn→m and wm→n the wave traveling from Jn to Jm and from Jm to Jn respectively. −1
1
1 2
1 2
2 (z, ω)In→m (z, ω) wn→m (z, ω) , ζnm2 (z, ω)Vn→m (z, ω) + ζnm
−
(3)
wm→n (z, ω) , ζnm (z, ω)Vn→m (z, ω) − ζnm (z, ω)In→m (z, ω)
(4)
s s s In the same way, we define sources waves wn→m and wm→n in terms of In→m and Vns . T s s ˜ nm (z, ω) = [wm→n (z, ω), wn→m (z, ω)] and u˜nm (z, ω) = [wm→n (z, ω), wn→m Let w (z, ω)]T . The com˜ nm satisfies the following differential equation, bined waves vector w
˜ nm (z, ω) dw ˜ nm (z, ω) + u˜nm (z, ω) , ∀z ∈ [0, `nm ] = Anm (z, ω)w dz
(5)
where Anm (z, ω) is defined accordingly in terms of γnm and the potential function qnm which expresses the heterogeneity of the characteristic impedance along the branch Bnm . 1 d ln (ζnm (z, ω)) qnm (z, ω) , − , Anm (z, ω) , 2 dz
γnm (z, ω) qnm (z, ω) qnm (z, ω) −γnm (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 Anm depends on z . In order to numerically solve equation (5), we introduce a state transition matrix Φnm related to Anm . This matrix satisfies the following differential equation, for any z, z 0 ∈ [0, `nm ], dΦnm (z, z 0 ; ω) = Anm (z, ω)Φnm (z, z 0 ; ω) , Φnm (z, z; ω) = Φnm (z 0 , z 0 ; ω) = Id dz
(6)
˜ nm (z, ω) for z equal where Id is the identity matrix. If Φnm (z, z 0 ; ω) was known, then given the value of w to any z0 ∈ [0, `nm ], the unique solution of (5) would be given by ˜ nm (z, ω) = Φnm (z, z0 ; ω)w ˜ nm (z0 , ω) + w
Z
z
Φnm (z, s; ω)u˜nm (s, ω)ds
(7)
z0
As the state transition matrix Φnm (z, z 0 ; 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 Bnm , the propagation matrix Γnm (ω) relates the values of the waves two opposite waves wm→n , wn→m at the two ends of the branch through the equation wm→ Γn,m (ω) Γn,n (ω) w dn,m (ω) n (ω) m →n (ω) = + wn→m Γm,m (ω) Γm,n (ω) w →m (ω) dm,n (ω) n
(ω)
(8)
where wm→ n (ω) is the value of wm→n (z, ω) at the end of the branch connected to junction Jn (the index n is circled ), the other similar notations can be easily understood, and dn,m (ω), dm,n (ω) are due to voltage and/or current sources distributed along the branch Bnm . Using equations (6) and (7), we show that the components of Γnm (ω) and [dn,m (ω), dm,n (ω)]T can be computed by solving the following differential
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equations : dΓm,m (z, ω) dz dΓn,m (z, ω) dz dΓm,n (z, ω) dz dΓn,n (z, ω) dz ddn,m (z, ω) dz ddm,n (z, ω) dz Γm,m (0, ω)
= qnm (z, ω) − 2γnm (z, ω)Γm,m (z, ω) − qnm (z, ω)Γ2m,m (z, ω) = Γn,m (z, ω) [qnm (z, ω)Γm,m (z, ω) − γnm (z, ω)] = −Γm,n (z, ω) [γnm (z, ω) + qnm (z, ω)Γm,m (z, ω)] = −Γn,m (z, ω)qnm (z, ω)Γm,n (z, ω) s = −Γn,m (z, ω) [qnm (z, ω)dm,n (z, ω) + wm→n (z, ω)] s = −dm,n (z, ω) [γnm (z, ω) + qnm (z, ω)Γm,m (z, ω)] − wm→n (z, ω) s s −wm→n (z, ω)Γm,m (z, ω) + wn→m (z, ω)
= Γn,n (0, ω) = 0 , Γn,m (0, ω) = Γm,n (0, ω) = 1 , dn,m (0, ω) = dm,n (0, ω) = 0
Then we have Γn,m (ω) = Γn,m (lnm , ω) , Γn,n (ω) = Γn,n (lnm , ω) , Γm,m (ω) = Γm,m (lnm , ω) Γm,n (ω) = Γm,n (lnm , ω) , dn,m (ω) = dn,m (lnm , ω) and dm,n (ω) = dm,n (lnm , ω) 4. SCATTERING MATRIX
Let wn◦→ (ω) be the vector composed of all the waves getting out of junction Jn and evaluated the end connected to junction Jn , or more formally wn◦→ (ω) , [w →m (ω)]m∈Cn with Cn being the set of indices n of the junctions adjacent to Jn . Similarly let wn→◦ , [wm→ (ω)] n m∈Cn be the vector composed of all the waves getting into junction Jn and evaluated the end connected to Jn . In the literature of BLT equation, the scattering of a signal at junction Jn is described by the scattering parameter Sn (scalar or matrix) relating incoming waves to outgoing waves. wn◦→ (ω) = Sn (ω)wn→◦ (ω)
(9)
Suppose that each terminal junction, say Jn , is not connected to any lumped source and is characterized by a generalized Th´evenin equivalent as V →m (ω) = −ZT,n (ω)I →m (ω) n n
(10)
where ZT,n is a load connected to the same junction, V →m (ω) is the value of Vn→m (z, ω) evaluated n at the end of Bnm connected to Jn . When the junction is open-circuited, we replace relation (10) by I →m (ω) = 0. In this case, the waves vectors wn◦→ (ω) and wn→◦ (ω) are scalars and are defined as a n combination of voltage V →m (ω), current I →m (ω) and the characteristic impedance ζ n n n (ω) of branches Bnm evaluated at Jn . The scattering parameter Sn (ω) is given by Sn =
ZT,n − ζ n (ω) ZT,n + ζ n (ω)
(11)
In the case of intermediate junctions 2 , we suppose that all currents in the branches connected to Jn have their negative directions pointing to junction Jn . This choice of the current directions is only used for the computation of the scattering matrix at the junction Jn , 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 : CI In (ω) = 0 , CV Vn (ω) = 0
(12)
where Vn (ω) and In (ω) represent the voltage and the current vectors on the branch connected to the junction Jn and evaluated at this junction. The matrices CI and CV 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 voltages at the ends of the branches 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.
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connected to Jn are all equal. Using (3) and (12), the waves vectors wn◦→ (ω) and wn→◦ (ω) are given as follows : 1
−1
−1
1
wn◦→ (ω) = Zn 2 (ω)Vn (ω) + Zn2 (ω)In (ω) , wn→◦ (ω) = Zn 2 (ω)Vn (ω) − Zn2 (ω)In (ω)
(13)
where Zn is a diagonal matrix filled with the characteristic impedance of branches connected to the junction Jn . By inverting (13), the voltage Vn (ω) and the current In (ω) vectors can be computed as : 1 −1 1 1 Vn (ω) = Zn2 (ω) [wn◦→ (ω) + wn→◦ (ω)] , In (ω) = Zn 2 (ω) [wn◦→ (ω) − wn→◦ (ω)] 2 2
(14)
Combining (12) and (14) and using the definition of the scattering matrix (9), we obtain, " Sn (ω) =
1
CV Zn2 (ω) −1
CI Zn 2 (ω)
#−1 "
1
−CV Zn2 (ω) −1
# (15)
CI Zn 2 (ω)
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.
Figure 1: Example of two electric networks
Figure 2: Measured reflection coefficient compared to simulated reflection coefficient for a star shaped network
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 = 2.478.108 [m/s], we simulate the reflection coefficient at junction J1 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 (J3 and J6√) are opencircuited, and another one (J5 ) is short-circuited. The ohmic loss of (10−13 + 4.5 × 10−5 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 J1 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 Example 1 :
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voltage V →2 (ω) at junction J1 when the network is powered with lumped voltage source Vs (ω) = 1[V ] 1 for ω ∈ [0, 2.1[rad.GHz]]. The 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. We consider an electric network of lossless transmission lines, composed of 5 junctions and 5 branches, as illustrated in Figure 1-B. Each branch Bnm is parametrized by its capacitance Cnm (z) = 2 0.1[nF/m] and inductance Lnm (z) = (0.08e−5(z−2.3) + 0.9)[µH/m]. The network is supplied with a lumped voltage source Vs (ω) = 1[V ] at junction J1 and the terminal junction J5 is short-circuited. The modulus of currents at the two ends of branch B3,4 are shown in Figure 4, with a logarithmic scale. Example 2 :
Figure 3: Measured voltage compared to simulated voltage for a tree-shaped network
Figure 4: Simulated currents at two ends of branch B3,4 (Figure 1-B)
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 G´enie 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, 1978. 2. Baum, C. E., “Generalization of the BLT equation,” Interaction Notes, Note. 511, 1995. 3. Parmantier, J. P., “An efficient technique to calculate ideal junction scattering parameters in multiconductor transmission line networks,” Interaction Notes, Note. 536, 1998. 4. 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, 2213–2218, 2010. 5. Kurokawa, K, “Power Waves scattering matrix,” Microwave Theory and Techniques, Vol. 13, No. 2, 194–202, 1965.
