Interconnect Modeling and Simulation:Distributed Transmission Line Equations.

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Distributed Transmission Line Equations

Transmission line characteristics are in general described by Telegrapher’s equations. Consider the multiconductor transmission line (MTL) system shown in Figure 14.17. Telegrapher’s equations for such a structure are derived by discretizing the lines into infinitesimal sections of length Dx and assuming uniform per-unit length parameters of resistance (R), inductance (L), conductance (G), and capacitance (C). Each section then includes a resistance RDx, inductance LDx, conductance GDx, and capacitance CDx (Figure 14.6). Using Kirchoff ’s current and voltage laws, one can write

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The equations for a single line also hold good for multiple coupled lines, with a modification that per-unit-length parameters now become matrices (R, L, G, and C) and voltage–current variables become vectors represented by V and I, respectively. Noting this and taking Laplace transform of Eqs. 14.12 and 14.13, one can write

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where gm(s) is the complex propagation constant. Substituting the solution forms of Eqs. 14.21 and 14.22 into wave equation 14.19 yields

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For inhomogeneous dielectrics, there exist in general m distinct eigenvalues where m = 1, 2, …, N. Each eigenvalue has its corresponding eigenvector Sm. Let G be a diagonal matrix whose elements are the complex propagation constants {g1, g2, …, gN}. Let Sv be a matrix with eigenvectors Sm placed in respective columns. The transmission line stencil can now be derived after little manipulations as

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Hybrid transmission line stencil in the exponential form can be written as

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Distributed vs. Lumped: Number of Lumped Segments Required

It is often of practical interest to switch between distributed models and lumped representations. In this case, it is necessary to know approximately how many lumped segments are required to approximate a distributed model. For the purpose of illustration, consider LC segments, which can be viewed as low- pass filters. For a reasonable approximation, this filter must pass at least some multiples of the highest frequency content fmax of the propagating signal (say, ten times, f0 ³ 10fmax). In order to relate these [2,3], we make use of the 3-dB passband of the LC filter given by

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In the case of RLC segments, in addition to satisfying Eq. 14.36, the series resistance of each segment must also be accounted for. The series resistance Rd representing the ohmic drop should not lead to impedance mismatch, which can result in excessive reflection within the segment [2,3].

Example

Consider a digital signal with rise time of 0.2 ns propagating on a lossless wire of length 10 cm, with a per unit delay of 70.7 ns. This can be represented by a distributed model with per unit length parameters of L = 5 nH/cm and C = 1 pF/cm. If the same circuit were to be represented by lumped segments, one needs N = ((10 ´ 70.7e–12 ´ 10)/(0.2e–9)) = 35 sections. It is to be noted that using more sections does not clean up ripples completely, but helps reduce the first overshoot (Gibb’s phenomenon). Ripples are reduced when some loss is taken into account.

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