Modeling of Transmission Lines with EM Wave ...

Modeling of Transmission Lines with EM Wave Coupling by Finite Difference Quadrature Methods Qinwei Xu and Pinaki Mazumder θE âϕ

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Abstract — This paper proposes an efficient numerical approximation technique, called the Finite Difference Quadrature (FDQ) Method, which has been adapted to model transmission lines (TL’s) with external EM wave coupling. The finite difference quadrature method can quickly compute finite differences between adjacent grid points by estimating a weighted linear sum of derivatives at a set of points belonging to the domain. Similarly to the Gaussian Quadrature method to compute the numerical integrals, FDQ method uses global quadrature method to construct the approximation framework, however, to compute the finite difference. A discrete modeling approach, FDQ needs much sparser grid points than the Finite Difference (FD) methods to achieve comparable accuracy. The FDQ-based equivalent models can be integrated into simulators like SPICE to efficiently simulate circuits involving EM interference. The FDQ procedure can be applied to solve other classes of partial differential equations. This paper thus demonstrates a general numerical approach, though the focus of the paper is transmission line modeling.




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the finite difference (FD) methods. As the FD methods have low order accuracy, the electrical length of each section has to be a considerably small fraction (1/121 INTRODUCTION 1/20) of the minimum wave length of the signal; thereFast operation and large integration scale have made fore, the equivalent models consist a excessive number the interconnect effect an important issue in high-speed of lumped elements. On the other hand, numerical insystems. In addition to the on-chip and on-board ef- tegration (quadrature) is more stable and reliable than fects of delay, crosstalk, and reflections, electrically differentiation because it globally integrates the local long interconnects pose the antenna effect, as they information. Integral approaches like Gaussian Quadrareceive considerable dose of incident electromagnetic ture generally give more accurate solutions [4]. In this (EM) waves emitted by external electronic devices [1]. paper, the Finite Difference Quadrature (FDQ) method Fast clocking rate and short rise time result in sig- is proposed to model TL’s. The idea of the FDQ method nals with wavelengths comparable to interconnect sizes is to quickly compute the finite difference of two neighwhich increases the radiation efficiency of the conduct- boring grid points by estimating a weighted linear sum ing traces, while small feature sizes lead to high electro- of derivatives at a small set of points belonging to the magnetic interference (EMI) susceptibility among the domain. The weighted linear sum is like the numericircuit parts. The antenna effect becomes a more seri- cal integral in Gaussian Quadrature method, yet it is to compute finite differences rather than the integrals. As ous challenge to signal integrity with progressive  a result, FDQ needs much sparser grid points than the scaling. Equivalent source modeling, based on the quasi-TEM Finite Difference (FD) methods to achieve comparable assumption, is one of the mainstream approaches to accuracy. solve the problem of external field coupling to TL’s [2], in which incident EM waves illuminating the TL’s are 2 QUASI-TEM FORMULATIONS OF TL’S WITH EM WAVE COUPLING modeled as equivalent sources. These effects of the incident field appear as forcing functions on the TL equa- The quasi-TEM assumption of TL’s is equivalent to the tions, which are incorporated into the circuit simulators condition that the dimensions of TL’s cross-sectional altogether with other devices. Compared to the field sizes are much smaller than a wavelength of the exsolvers, the equivalent circuit approach is computation- ternal EM wave. Under such a condition, the principal ally efficient and numerically accurate as long as the propagation mode of the TL’s is TEM, and can be acquasi-TEM assumption is valid. The equivalent source curately described by the Telegrapher’s equations. Theapproach in the literature [3] is mathematically based on oretical analysis and experimental results have shown





This work was supported by MURI grant. EECS Dept., University of Michigan, Ann Arbor, MI 481092122, Email:  qwxu,mazum @eecs.umich.edu

that the external EM wave illuminating can be modeled as forcing terms which are added in the Telegrapher’s equations [1]. Let the illuminating EM wave be repre-

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       (1) and the TL’s stretch along  direction. Assume that the reference of conductor (defined as line 0) is colinear with the  -axis (see Fig.1). By assuming a quasiTEM mode of propagation along an MTL consisting of  conductors, the voltages #     and currents $%   "  ! along the conductors can be represented by the

-domain Telegrapher’s equations: &   '( )$%  & #      +*   )$%    , #.-     (2) & $% /0    

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$  &    #   21  #   -    (3)  ' /0  where  ,  , *  and 1  denote the per-unitlength (PUL) inductance, capacitance, resistance, and 546 matrices at point  , respectively. conductance   The -dimensional vector forcing functions #7-  98 $  and -  98 are derived as in [1]. 3 FINITE DIFFERENCE METHODS

QUADRATURE

There are two major approaches to numerically solve partial differential equations (PDE’s): one is the numerical finite differencing and the other is the numerical integration (quadrature). Finite difference (FD) methods have been fully developed in the literature and are widely used to numerically solve differential equations because of its simplicity and applicability. Consider a smooth function and its derivative function both defined on the domain , which is uniformly divided by grid points , then the central FD framework is given by

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rules. Compared to the FD framework in Eqn. 4, the Gaussian Quadrature in Eqn. 5 is a global approximaand is reption in that the difference between resented by all the grid points over the entire domain. We propose a new technique called the finite difference quadrature (FDQ) method to integrate the finite difference and quadrature methods in one approach. The general framework of FDQ is shown by

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X :   GRH \J :    N ]  ]_^   ] : ;   ] C (6) YRH Z  ] ’s are the FDQ coefficients to be determined, where ^   ] Z is the weight at point  ] . The RHS representa-

tion of the FDQ Eqn. 6 is apparently a global quadrature approximation, which is the same as the RHS in the Gaussian Quadrature Eqn. 5; however, the LHS of Eqn. 6 is the finite difference, compared to the numerical integral over the entire domain in Eqn. 5. The FDQ method is as simple and applicable as the finite difference method, and is as stable and efficient as the quadrature method. It is expected to be able to solve all kinds of PDE’s with higher efficiency. In this paper, we use the FDQ method to solve the Telegrapher’s equations for transmission line modeling. For simplicity, the FDQ modeling of TL’s is first developed on a uniform two-wire TL with one of the wires being the reference. In this case, the vectors and matrices in Eqns. 2 and 3 become scalars:  , , , , and . A direct numerical technique, FDQ methods do not need to decouple the MTL, therefore its application to non-uniform and/or multi-conductor TL is straightforward extended. Assume that a TL has been normalized to one unit of length along direction. As shown in Fig. 2, the TL is equally discretized to small sections. There are two sets of grid points: one set are at integer-spatial positions , the other are at half-spatial positions . The grid points are numbered in such a way that the voltages are evaluated at integer-spatial positions as   and the currents are evaluated at half-spatial positions as . In addition, the currents and are input and output currents at the port, respectively. For clarity of the intermediate use in the context, we define the inter-

:   GIH IJ :   LKMH N 3OQP : ;    (4) a + P

J  GRH   is the distance between two ad- b c where jacent grid points. This framework is a local approximation in that the finite difference is only represented by the immediately neighboring grid points. Despite its wide popularity and uses, it requires very dense grid points and therefore takes computationally prohibitive time to solve large problems. One the other hand, the numerical integration is more stable and converges faster as it globally integrates the local information. The approximation framework of Gaussion Quadrature can be adapted as

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 mediate voltages   and, similarly, the matrix   . and the intermediate currents . The FDQ approximations in Eqns. 7 and 8-9 reFor the finite difference of voltage, the FDQ approx- spectively have approximation orders of  and  imation framework is , compared to the low approximation of FD  methods. Once the positions of the grid points are given, the    (7) above testing-function approach gives constant FDQ coefficient matrices, no matter in what applications the For the finite difference of current, the FDQ approxi- TL equations appear. For the equally spaced grid points, mation framework is the coefficients can be calculated and saved for future use, which reduce the computational cost in practice. With the pre-calculated coefficient matrices, (8) Eqns. 10-13 are rewritten as

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Fig. 3 shows that, without EM wave interference, the waveforms at the leaf points   are identical. With the EM wave coupling in this configuration, the waveforms at  and  are still identical, so are the waveforms at  and  , but the two sets of clock waveforms are severely skewed, due to the effect of the illuminating EM waves.

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Figure 3: H clock tree with EM wave interference.

approach the Nyquist limit to achieve the required accuracy. The globality of FDQ approximation framework heuristically makes it one of the approaches which closely approach the Nyquist limit. The example is about an H clock tree in an MCM circuit of
 m technique to offer the clock signal for four modules located at points  ,  ,  and  as shown in Fig. 3. The H tree is etched on a silicon substrate which is assumed to be the ideal ground. The circuit is illuminated by a plane EM wave. There are seven  ,   and   , TL’s  ,

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,  each of them having the same length of cm. The first three TL’s have the PUL parameters nH/m,  pF/m,   /m and  S/m, and the

 /m other TL’s nH/m, pF/m,  and  S/m. At points , and , there are three identical inverters used as buffers to drive the clock signal. The applied input is periodic clock signal whose rise/fall time is  ps. The illuminating EM wave is a trapezoidal pulse having rise/fall time of  ps and width of ns. The phase of the EM wave is referred by point at which the delay of EM wave is ns. As we focus on the effect of the illuminating EM wave on the integrated circuits, for simplicity we assume that the EM wave is still the uniform wave in free space at the top of the circuit. For more accurate modeling the nonuniformity should be considered due to nonuniform distribution of media and metals. Using the heuristic, each of the TL’s is modeled by using FDQ with (" PPW) as in Eqn. 15, whose coefficients have been pre-calculated. Let the EM wave have the propagation configuration as  " ,  " , and   (refer to Fig. 1). The responses are shown in Fig. 3.

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An efficient numerical approximation technique, call the Finite Difference Quadrature (FDQ) Method, is first proposed to solve partial differential equations (PDE’s), and has been adapted to model transmission lines (TL’s) with external field coupling. A discrete modeling approach, the FDQ adapts grid points along the transmission lines to compute the finite difference between adjacent grid points by estimating a weighted linear sum of derivatives at a set of points belonging to the domain. Similar to the Gaussian Quadrature method to compute the numerical integrals, the FDQ method uses global quadrature method to construct the approximation framework, however, to numerically compute finite differences. As a global approximation technique, the FDQ needs much sparser grid points than the finite difference (FD) method does to achieve required accuracy. Equivalent voltage and current sources are derived, which excite the TL’s at the grid points. Equivalent circuit models are therefore derived to represent the TL’s illuminated by external electromagnetic waves. A heuristic of FDQ modeling resolution of two differences per wavelength is given to determine the order in the application of FDQ modeling. The FDQ method supplies a general numerical technique that can solve partial differential equations (PDE’s) accurately and efficiently. References [1] C. R. Paul, Analysis of multiconductor transmission lines. New York, NY: Wiley, 1994. [2] I. Erdin, M. S. Nakhla, and R. Achar, “Circuit analysis of electromagnetic radiation and field coupling effects for networks with embedded full-wave modules,” IEEE Trans. Electromagnetic Compatibility, vol. 42, no. 4, pp. 449–460, 2000. [3] M. Omid, Y. Kami, and M. Hayakawa, “Field coupling to nonuniform and uniform transmission lines,” IEEE Trans. Electromagnetic Compatibility, vol. 39, no. 3, pp. 201–211, 1997. [4] M. N. O. Sadiku, Numerical techniques in electromagnetics. Boca Raton, FL: CRC, 2001.