Analog Integr Circ Sig Process (2011) 68:85–92 DOI 10.1007/s10470-010-9581-6

Time-domain analysis of lossy multiconductor transmission lines based on the Lax–Wendroff technique Lei Dou • Jiao Dou

Received: 29 March 2009 / Revised: 8 December 2010 / Accepted: 8 December 2010 / Published online: 24 December 2010 Ó Springer Science+Business Media, LLC 2010

Abstract Taking advantage of the hyperbolic characteristics of the telegrapher equations, this paper applies the Lax–Wendroff technique, usually used in fluid dynamics, to transmission line analysis. A second-order-accurate Lax–Wendroff difference scheme for the telegrapher equations for both uniform and nonuniform transmission lines is derived. Based on this scheme, a new method for analyzing lossy multiconductor transmission lines which do not need to be decoupled is presented by combining with matrix operations. Using numerical experiments, the proposed method is compared with the characteristic method, the fast Fourier transform (FFT) approach, and the Lax–Friedrichs technique. With the presented method, a circuit including lossy multiconductor transmission lines is analyzed and the results are consistent with those of PSPICE. The nonlinear circuit including nonuniform lossy multiconductor transmission lines is also computed and the results are verified by HSPICE. The proposed method can be conveniently applied to either linear or nonlinear circuits which include general transmission lines, and is proved to be efficient. Keywords Finite difference methods  Lax–Wendroff technique  Multiconductor transmission lines  Transient analysis

L. Dou (&) National Key Laboratory of Transient Physics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China e-mail: [email protected] J. Dou Beijing Design Center of Conexant, Beijing 100084, China e-mail: [email protected]

1 Introduction With rapid increases in operating frequency and decreases of feature sizes of circuit elements, the influence of transmission lines, such as delay, crosstalk and signal distortion, grows in circuits and systems. Research on the timedomain response of multiconductor transmission lines is of great importance for the design of high-speed circuits. When the signal wavelength becomes comparable to the length of transmission lines, lumped parameter models are inadequate for describing the transmission line performance and distributed parameter models need to be employed. A number of methods have thus far been proposed for the analysis of multiconductor transmission lines. They are divided primarily into two types: one is the frequencydomain transform method, such as the fast Fourier transform (FFT) approach [1], the numerical inverse Laplace transform method [2], etc., and the other is the direct time-domain method, such as the characteristic method [3], the differential quadrature method [4], etc. We will discuss the FFT approach and the characteristic method in Sect. 4. The order reduction technique [5, 6] has lately been applied to analyze circuits including distributed parameter elements. This technique can increase analysis efficiency remarkably by reducing large circuit networks. The direct time-domain method is conceptually simple and efficient, but the distributed parameter coefficient matrices of lossy multiconductor transmission lines, C and G or R and L, cannot, in general, be diagonalized in the time domain at the same time. It is a process that can only be executed under special conditions [7]. In the frequency domain, this is not a problem, but it is very difficult to analyze nonlinear circuits including transmission lines.

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The time-domain finite difference method has the advantages of conceptual simplicity, convenient calculation, and wide application range; however, traditionally, it is deemed to be less efficient and is not useful in practice. In [8], Guo expresses a different view after restudying the spatial–temporal discrete method for the telegrapher equations. He points out that this method may be efficient if a suitable difference scheme is adopted and he applies the Lax–Friedrichs technique as one of the steps in analyzing transmission lines. This method is convenient because it does not need to consider characteristic curves. However, the method is first-order accurate and can lead to a broad transition zone of discontinuities due to the high numerical viscosity and strong dissipation when pulses are computed [9]. Another finite difference method used for transmission line analysis, which was first presented in [10], is a totalvariation-diminishing (TVD) algorithm. Hwang applied it to analyzing a lossless transmission line. Afterward, this method was used on nonlinear circuits [11]. The TVD algorithm shows second order accuracy in smooth regions, and is first-order accurate near discontinuities. In the next paragraph, we will give it further discussion. In [12], we studied the spatial–temporal discrete method for the telegrapher equations. Taking advantage of the hyperbolic characteristic of the telegrapher equations, a Lax–Wendroff difference scheme with second order accuracy was presented. This method increases accuracy and efficiency on the basis of the Lax–Friedrichs technique. Comparing with the TVD algorithm, the Lax–Wendroff method is second-order accurate not only in smooth regions but near discontinuities, and the scheme of the Lax–Wendroff method is simpler. The simpler scheme not only leads to higher efficiency but provides an advantage in solving the difficulty of lossy multiconductor transmission lines analysis as mentioned above and, in Sect. 3, we will give the detailed process. Therefore, we do not have to limit the parameter matrix form and the application range can be significantly enlarged. In [12], the multiconductor transmission lines were first decoupled by a characteristic mode transform and then each line was analyzed as a single-conductor line. As mentioned above, the decoupling method cannot be applied to general lines. Based on [12], this paper further studies the Lax–Wendroff technique and makes the following contributions. First, we present a detailed derivation of the Lax–Wendroff difference scheme for the telegrapher equations for both uniform and nonuniform transmission lines. Then we combine the scheme with matrix operations to present a new analysis method for multiconductor transmission lines which do not need to be decoupled. This method avoids the difficulty of diagonalizing the two coefficient matrices in one telegrapher equation at the same time. Finally, we analyze nonlinear circuits including

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nonuniform transmission lines using the proposed method. By using the Lax–Wendroff method to compute the discontinuity caused by pulse signal, the transition zone of discontinuity becomes narrow and the numerical solution basically coincides with the analytic solution [9]. At the same time, implicit artificial viscosity included in the Lax– Wendroff difference scheme can to some extent decrease the parasitic oscillation of the solution due to the space discretization. Because the proposed method uses direct spatial–temporal discretization, its application does not need a complicated mathematical derivation or transform. The method has no special circuit limits and can be directly applied to analyze general transmission line circuits.

2 Lax–Wendroff difference scheme for the telegrapher equations In order to apply the Lax–Wendroff technique to analyze transmission lines, we first need to present the Lax– Wendroff difference scheme for the telegrapher equations. In the derivation process of the scheme below, the computation precision and stability are studied. In the time domain, both uniform and nonuniform transmission lines can be expressed by the telegrapher equations as follows: oV oI þ Lð xÞ ¼ Rð xÞI; ox ot oI oV þ Cð xÞ ¼ Gð xÞV; ox ot

ð1Þ ð2Þ

where V is voltage, I is current, R(x) is resistance per unit length, L(x) is inductance per unit length, C(x) is capacitance per unit length, and G(x) is conductance per unit length. For the convenience of the discussion below, substitute R, L, C and G for R(x), L(x), C(x) and G(x), respectively. Rearranging (1) and (2), we obtain oV 1 oI G ¼  V; ot C ox C oI 1 oV R ¼  I: ot L ox L

ð3Þ ð4Þ

Differentiating (3) with respect to t, and substituting (3) and (4) into it, we have     o2 V 1 oV o 1 1 1 o oV ¼ þ ot2 C ox ox L L C ox ox ð5Þ   1 o R 1 R oI G 1 oI G G þ þ V: þ I þ C ox L C L ox C C ox C C Let Vjn denote V ðjDx; nDtÞ, then Vjnþ1 at the point ðjDx; ðn þ 1ÞDtÞ can be expressed in terms of a Taylor series expansion about point ðjDx; nDtÞ as follows:

Analog Integr Circ Sig Process (2011) 68:85–92

Vjnþ1 ¼ Vjn þ Dt

  oV 1 2 o2 V þ Dt 2 þ O Dt3 : ot 2 ot

87

If we rearrange (1) and (2) as follows: ð6Þ

Substituting (3), (4) and (5) into (6) and omitting OðDt3 Þ, we have the temporal second-order-accurate (7) without differential terms with respect to time:   1 oI G nþ1 n Vj ¼Vj þ Dt   V C ox C    1 2 1 oV o 1 1 1 o2 V ð7Þ þ Dt þ 2 C ox ox L L C ox2    1 o R 1 R oI G 1 oI G G þ þ V : þ Iþ C ox L C L ox C C ox C C

oQ oQ þA ¼ BQ; ot ox where   V Q¼ ; I

 A¼

ð12Þ

0 1=L

 1=C ; 0

 B¼

G=C 0

0 R=L



then the stable condition is maxjak jMt=Mx  1;

ð13Þ

k

where ak is the eigenvalue of coefficient matrix A.

Replace of =ox with the second-order central difference n n ðfjþ1  fj1 Þ=2Dx, where f ¼ V; I; 1=L; R=L; replace o2 f =ox2 with the second-order-accurate difference n n ðfjþ1  2fjn þ fj1 Þ=Dx2 , where f ¼ V, and replace f with n fj , where f ¼ V; I; 1=C; 1=L; R=L; G=C. After we complete the rearrangement, we have the Lax–Wendroff scheme for the telegrapher equation (1) with second-order accuracy in both space and time as follows:    Dt 1  nþ12 nþ1 nþ1 n Vj ¼ Vj  Ijþ1  Ij12 2 2 Dx C   j  1 1 Dt G nþ nþ  V 12 þ Vj12 ; ð8Þ 2 2 C j jþ2 where

   !  1  n Dt 1 1 n Vjþ1 þ Vj  ¼ þ 2 2Dx C jþ1 C j !       Dt G G n  Ijþ1  Ijn  þ 4 C jþ1 C j   n  Vjþ1 þ Vjn ; ð9Þ  !  Dt 1 1 n 1 nþ12 n Ijþ1 þ Ij  þ Ijþ1 ¼ 2 2 2Dx L jþ1 L j ! !       Dt   R R n n  Vjþ1  Vjn  þ þ Ijn : Ijþ1 4 L jþ1 L j

nþ1 Vjþ12 2

ð10Þ Similarly, the Lax–Wendroff scheme for telegrapher equation (2) can be obtained.    Dt 1  nþ12 nþ1 nþ1 n Ij ¼ Ij  Vjþ1  Vj12 2 2 Dx L   j  1 Dt R  nþ12 nþ  I 1 þ Ij12 : ð11Þ 2 2 L j jþ2

3 Computation technique for multiconductor transmission lines Based on the results above, we can now discuss an analysis method which does not need to decouple multiconductor transmission lines. For the uniform and nonuniform coupling K-conductor transmission lines, we divide it into N - 1 sections of equal length. Based on (8)–(10), matrix equations can be obtained as follows: 02 3 2 31 nþ12 nþ12 2 nþ1 3 2 n 3 I I 1 1 V1j V1j B6 1jþ2 7 6 1j2 7C B6 7 6 7C 7 6 7 6 B6 nþ12 7 6 nþ12 7C 6 V nþ1 7 6 V n 7 B 7 7C 6 6 I I 6 2j 7 6 2j 7 B6 2jþ12 7 6 2j12 7C 7 6 7 6 B 7 7C 6 6 7 6 7 6 B6 7 6 7C 6 .. 7 6 .. 7 B6 .. 7 6 .. 7C 6 . 7 6 . 7 Dt B 7 7C 6 6 7 6 7 6 B6 . 7  6 . 7C 7 ¼6 n 7  C1 6 j B6 7 7C 6 6 V nþ1 7 6 V 7 Dx B6 I nþ12 7 6 I nþ12 7C 6 kj 7 6 kj 7 1 7 1 7C B 6 6 7 6 7 6 B6 kjþ2 7 6 kj2 7C 7 6 7 6 B6 7 6 7C 6 .. 7 6 .. 7 B6 . 7 6 . 7C 6 . 7 6 . 7 . . B 7 7C 6 6 5 4 5 4 B6 . 7 6 . 7C @ 5 5A 4 4 n nþ1 VKj nþ1 nþ1 VKj IKjþ21 IKj21 2

02

nþ12 1jþ12

3

2

2

nþ12 1j12

31

V V B6 7 6 7C B6 7 6 7C B6 nþ12 7 6 nþ12 7C B6 V 1 7 6 V 1 7C B6 2jþ2 7 6 2j2 7C B6 7 6 7C B6 7 6 7C B6 .. 7 6 .. 7C 6 . 7 6 . 7C Dt 1 B 7 6 7C 6  C j Gj B B6 nþ1 7 þ 6 nþ1 7C; 2 B6 V 2 7 6 V 2 7C B6 k jþ12 7 6 kj12 7C B6 7 6 7C B6 7 6 7C B6 . 7 6 . 7C B6 .. 7 6 .. 7C B6 7 6 7C @4 5 4 5A nþ12 nþ12 VKjþ1 VKj1 2

ð14Þ

2

123

88

2

Analog Integr Circ Sig Process (2011) 68:85–92 nþ1

V1jþ21

3

002

n V1jþ1 n V2jþ1

3

2

V1jn V2jn

31

2 7 6 6 nþ1 7 BB6 7 6 7C 6V 2 7 BB6 7 6 7C 6 2jþ12 7 B B 7 7C 6 6 7 6 B C B 7 6 6 6 . 7 .. 7 6 .. 7 B C B 6 6 . 7 1 BB6 . 7 6 . 7 C 7 6 . 7 B C B 7 7 6 6 7¼ 6 B6 V n 7 þ 6 V n 7C 6 nþ12 7 2 B B C B 7 7 6 6 kjþ1 kj 6 Vkjþ1 7 BB6 7 6 7C 2 7 6 B C B 7 7 6 6 7 6 BB6 ... 7 6 ... 7C 6 .. 7 @ A @ 5 5 4 4 6 . 7 5 4 n n 1 VKjþ1 VKj nþ VKjþ21 2 02 n 3 2 n 31 I1j I1jþ1 B6 I n 7 6 I n 7C B6 2jþ1 7 6 2j 7C B6 7 6 7C B6 . 7 6 . 7C B 6 6 7C . B6 . 7 7 6 .. 7C Dt  1 B6 7  6 7C  Cjþ1 þ C1 j B6 I n 7 6 I n 7C 2Dx B6 kjþ1 7 6 kj 7C B6 7 6 7C B6 . 7 6 . 7C B6 .. 7 6 .. 7C @4 5 4 5A n n IKjþ1 IKj  Dt 1  Cjþ1 Gjþ1 þ C1 j Gj 4 02 n 3 2 n 311 V1j V1jþ1 B6 V n 7 6 V n 7CC B6 2jþ1 7 6 2j 7CC B6 7 6 7CC B6 . 7 6 . 7CC B6 . 7 6 . 7CC B6 . 7 6 . 7CC 7 6 7CC 6 ð15Þ B B6 V n 7 þ 6 V n 7CC; B6 kjþ1 7 6 kj 7CC B6 7 6 7CC B6 . 7 6 . 7CC B6 .. 7 6 .. 7CC @4 5 4 5AA n n VKjþ1 VKj

and 3 2 nþ1 002 n 3 2 n 3 1 I1jþ21 I1j I1jþ1 2 7 6 7 6 B B 7 7C 6 6 1 n 6 nþ2 7 BB6 I2jþ1 7 6 I2jn 7C 6 I2jþ1 7 B B 7 7C 6 6 6 2 7 BB6 7 6 7C 7 6 BB6 .. 7 6 .. 7C 6 . 7 B6 . 7 6 . 7 C 6 .. 7 1 B B6 7 6 7C 7¼ B 6 B 6 n 7 þ 6 n 7C 6 nþ1 7 2 BB B6 Ikjþ1 7 6 Ikj 7C 6 I 21 7 BB6 7 6 7C 6 kjþ2 7 BB6 7 6 7C 7 6 B B 7 6 . 7C 6 . 6 . 7 BB6 .. 7 6 .. 7C 6 . 7 @ @ 5 4 5A 4 6 . 7 4 1 5 n n I IKj nþ Kjþ1 IKjþ21 2 02 n 3 2 n 3 1 V1j V1jþ1 B6 n 7 6 n 7 C B6 V2jþ1 7 6 V2j 7C B6 7 6 7C B6 7 6 7C B6 .. 7 6 .. 7C B C 7 6 6   . 7 6 . 7 Dt 7C 1 B6   L1 þ L B C 7 7 6 6 j n B6 Vkjþ1 7 6 Vkjn 7C 2Dx jþ1 B6 7 6 7C B6 7 6 7C B6 .. 7 6 .. 7C B6 . 7 6 . 7 C @4 5 4 5A n VKjþ1

123

n VKj

 Dt 1 Ljþ1 Rjþ1 þ L1 R j j 4 0 2 n 3 2 n 3 11 I1j I1jþ1 B 6 I n 7 6 I n 7 CC B6 2jþ1 7 6 2j 7CC B6 7 6 7 CC B6 .. 7 6 .. 7CC B 6 . 7 6 . 7 CC B6 7 6 7 CC  B 6 n 7 þ 6 n 7 CC; B6 Ikjþ1 7 6 Ikj 7CC B6 7 6 7 CC B 6 . 7 6 . 7 CC B 6 . 7 6 . 7 CC @ 4 . 5 4 . 5 AA n n IKjþ1 IKj



ð16Þ

where C, G, L and R are capacitance, conductance, inductance, and resistance matrices, respectively. Using (15) nþ1

nþ1

2

2

and (16), we can compute Vkjþ21 and Ikjþ21 where k expresses the kth conductor. k is from 1 to K and j is from 1 to N - 1. Then using (14), we can compute Vkjnþ1 where k is from 1 to K and j is from 2 to N - 1. With the boundary condition and excitation nþ1 condition, Vknþ1 1 and Vk N can be obtained. Thus we have all Vkjnþ1 at time point n ? 1. By repeating the process above, we can obtain voltage and current values on all spatial–temporal discrete points. The above process includes a large number of matrix computations, but the addition, subtraction, and multiplication of matrices are simple and have no obvious effect on efficiency. The inverse operation of the matrix only involves coefficient matrices. Because these matrices do not change with time, for uniform transmission lines, the inverse operation only needs to be calculated once, and for inhomogeneous structures, the inverse only needs to be calculated once for all stepwise constant segments and the matrices have to be stored. Thus the simulation time is not significantly affected and only the storage is required. Similarly, with (11), we have 02 nþ1 3 2 nþ1 31 2 nþ1 3 2 n 3 V 21 V 21 I1j I1j B6 1jþ2 7 6 1j2 7C B6 nþ12 7 6 nþ12 7C 6 I nþ1 7 6 I n 7 B6 V 1 7 6 V 1 7C 6 2j 7 6 2j 7 B6 2jþ2 7 6 2j2 7C 7 6 7 6 B6 7 6 7C 6 .. 7 6 .. 7 6 .. 7 6 .. 7C 6 . 7 6 . 7 Dt 1 B 6 . 7 6 . 7C 7 ¼6 n 7  L B 6 76 7C 6 6 I nþ1 7 6 I 7 Dx j B B6 V nþ12 7 6 V nþ12 7C 6 kj 7 6 kj 7 1 7 1 7C B 6 6 7 6 7 6 B6 kjþ2 7 6 kj2 7C 6 .. 7 6 .. 7 B6 . 7 6 . 7C 4 . 5 4 . 5 B6 .. 7 6 .. 7C @4 5 4 5A n nþ1 IKj nþ1 nþ1 IKj VKjþ21 VKj21 2 02 nþ1 32 2 nþ1 31 2 2 I 1 I 1 B6 1jþ2 7 6 1j2 7C B6 nþ12 7 6 nþ12 7C B6 I 1 7 6 I 1 7 C B6 2jþ2 7 6 2j2 7C B6 7 6 7C B6 .. 7 6 .. 7C   Dt 1 B6 . 7 6 . 7 C  ð17Þ L Rj B6 nþ1 7 þ 6 nþ1 7C: B6 I 2 7 6 I 2 7 C 2 j B6 kjþ12 7 6 kj12 7C B6 7 6 7C B6 . 7 6 . 7 C B6 .. 7 6 .. 7C @4 1 5 4 1 5 A nþ nþ IKjþ21 IKj21 2

2

Analog Integr Circ Sig Process (2011) 68:85–92

89

The computation process is the same as above.

4 Examples Efficient transmission line analysis is very important. The total efficiency is affected by the performance of the computer, programming language, grid number, and circuit analysis method, etc. In order to effectively compare, we will compute the simple transmission line circuit as shown in Fig. 1. At the left, the circuit is excited by a trapezoidal pulse with 1.0 ns of rise and fall time, and 2.0 ns of width. The magnitude of trapezoidal pulse is 1.0 V. The length of the transmission line is 0.2 m. We divided it into 20 sections of equal length, i.e., Dx ¼ 0:01 m. The resistance connected to both line ends is 50:0 X. The line parameters [7] are, L ¼ 500 nH/m, C ¼ 200 pF/m, R ¼ 100 X=m, and G ¼ 0 S/m. We analyze the circuit with the following five techniques: the Lax–Wendroff method, the characteristic method, the FFT approach, the Lax–Friedrichs technique, and PSPICE. In the FFT approach, we computed 64 frequency sample points. The results are shown in Fig. 2, where Vi is the voltage of i-th port of the transmission line. For clarity, in Fig. 3 we expand the area inside the circle of Fig. 2. We use the same programming language, MATLAB, and all of the computations are done by the same computer. The computational times are shown in Table 1. The first four methods, the Lax–Wendroff method, the characteristic method, the FFT approach, and the Lax–

Fig. 1 The transmission line circuit

Fig. 2 Transient response at both ends of the transmission line by the Lax–Wendroff method, the characteristic method, the FFT approach, the Lax–Friedrichs technique, and PSPICE, respectively

Fig. 3 The magnified figure inside the circle of Fig. 2

Table 1 CPU time comparison of five methods Method

CPU time (s)

Lax–Wendroff method

0.022

Characteristic method

0.017

FFT approach

0.028

Lax–Friedrichs technique

0.018

PSPICE

0.05

Comparison of the CPU times for analyzing the circuit in Fig. 1 by the Lax–Wendroff method, the characteristic method, the FFT approach, the Lax–Friedrichs technique, and PSPICE, respectively

Friedrichs technique, have similar efficiency. Without considering accuracy, the efficiency of the characteristic method and the Lax–Friedrichs technique is a little higher. As shown in Figs. 2 and 3, the curve computed by the Lax– Friedrichs technique rolls off more smoothly and has larger error than the Lax–Wendroff method because there is a strong dissipation term in the Lax–Friedrichs scheme. In space, the Lax–Wendroff method is second-orderaccurate and the Lax–Friedrichs technique is only firstorder-accurate. If Mx is 0.01 m, in order to have the same precision, the sample points needed by the Lax–Friedrichs technique are 100 times more than what is required by the Lax–Wendroff method. At the same time, the Lax– Wendroff scheme is more precise in time as well. Therefore, the method has much higher efficiency at the same accuracy level. The FFT approach is a frequency domain transform method and cannot directly be used for nonlinear circuits. Although the efficiency has been increased remarkably by using multiple multiplication [13, 14], it is difficult to span a time interval of several line transient times with the method [2]. This is because the large number of sample points required for avoiding aliasing will decrease efficiency significantly.

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Fig. 4 Circuit network including multiconductor transmission lines

Compared to the characteristic method, the Lax– Wendroff method will require a little more time and memory, but it is a direct spatial–temporal discrete numerical method, therefore, it does not require computing characteristic curves and can be applied in a wider variety of situations. As discussed above, the method can be conveniently combined with matrix operations to solve the decoupling difficulty for the analysis of lossy multiconductor transmission lines. Using the proposed method, we compute a coupling transmission line circuit shown in Fig. 4, where line T1 is of 0.1 m length, line T2 and T3 are of 0.2 m length, resistance and capacitance values are shown in Fig. 4. The parameters of the lines [7] are:   500 60 60 5 L¼ nH/m; C ¼ pF/m, 60 500 5 60  52:4 0 R¼ X=m; G ¼ 0 S/m: 0 52:4

Fig. 5 Transient response at ends 1, 2, 3, 4 of the multiconductor transmission lines in Fig. 4 by the Lax–Wendroff method and PSPICE, respectively

The circuit is excited by a trapezoidal pulse with 0.5 ns of rise time, fall time, and 5.0 ns of width. The magnitude of trapezoidal pulse is 1.0 V. The results are shown in Figs. 5 and 6, where Vi is the voltage of i-th port of the transmission lines. The solid lines are the results of the Lax–Wendroff method and agree well with those of PSPICE. From Sect. 2, we know that the Lax–Wendroff scheme for the telegrapher equations is based on both the uniform and nonuniform transmission lines. So the scheme can be directly applied to circuits including nonuniform lines. At the same time, because it is a direct time-domain method, it can be used on nonlinear circuits. Now we can consider a nonlinear circuit including nonuniform multiconductor transmission lines, as illustrated in Fig. 7. The circuit is excited by a voltage generator and the voltage waveform is the same as that in Fig. 4. The line length, values of resistance and capacitance are shown in Fig. 7. The line parameters are as follows [7]:

Fig. 6 Transient response at ends 7 and 8 of the multiconductor transmission lines in Fig. 4 by the Lax–Wendroff method and PSPICE, respectively

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Fig. 7 Nonlinear circuit network including nonuniform multiconductor transmission lines

Analog Integr Circ Sig Process (2011) 68:85–92



91

 Lm ð xÞ Cð xÞ Cm ð xÞ ; C ð xÞ ¼ ; Cm ð xÞ C ð xÞ Lð xÞ   G ð xÞ G m ð xÞ R ð x Þ Rm ð x Þ ; Gð xÞ ¼ ; Rð xÞ ¼ Rm ð x Þ Rð x Þ Gm ð xÞ Gð xÞ Lð xÞ ¼

L ð xÞ Lm ð xÞ

where 387:0 nH/m; 1:0 þ 0:1½1:0 þ 0:6 sinðpx þ p=4Þ

ð18Þ

Lm ðxÞ ¼ 0:1½1:0 þ 0:6 sinðpx þ p=4ÞLð xÞ nH/m;

ð19Þ

LðxÞ ¼

CðxÞ ¼

104:13 pF/m; 1:0  0:15½1:0 þ 0:6 sinðpx þ p=4Þ

Cm ðxÞ ¼ 0:15½1:0 þ 0:6 sinðpx þ p=4Þ pF/m; RðxÞ ¼ 1:2 X=m,

Rm ðxÞ ¼ 0;

GðxÞ ¼ 0;

ð20Þ ð21Þ

Gm ðxÞ ¼ 0:

LðxÞ; CðxÞ; RðxÞ; and GðxÞ are self parameters. Lm ðxÞ; Cm ðxÞ; Rm ðxÞ and Gm ðxÞ are mutual parameters. The diode characteristic of voltage and current [7] is as follows:   i ¼ 108 e40u  1 ð22Þ Because there is the model for the diode described by (22) in HSPICE, here we will apply HSPICE to verify our results. Because there is no nonuniform line model in HSPICE, in the HSPICE simulation the nonuniform transmission lines are divided into eight sections of equal length. Every section of the nonuniform transmission lines is substituted with uniform transmission lines whose parameters are taken as the midpoint values of the corresponding nonuniform lines. The results are shown in Figs. 8 and 9. The solid lines are the results of the Lax–Wendroff method and agree well with those of HSPICE.

Fig. 8 Transient response at ends 1, 2, 3, 4 of the multiconductor transmission lines in Fig. 7 by the Lax–Wendroff method

Fig. 9 Transient response at ends 7 and 8 of the multiconductor transmission lines in Fig. 7 by the Lax–Wendroff method

5 Conclusion This paper studies the application of the Lax–Wendroff technique, which is usually used in fluid dynamics, to conduct transmission line analysis based on the hyperbolic characteristics of the telegrapher equations. The secondorder-accurate Lax–Wendroff difference scheme for both uniform and nonuniform transmission lines is derived. The proposed method is compared with the characteristic method, the FFT approach, and the Lax–Friedrichs technique. When the methods use the same sample number of points, the Lax–Friedrichs technique has higher efficiency, but if the same precision is required, the efficiency of the Lax–Wendroff method is much higher. Compared to the characteristic method, the proposed method is slightly less efficient, but it is a direct spatial–temporal discrete numerical method. This means that it does not require considering characteristic curves and can be applied to a wider variety of situations. For the direct spatial–temporal discretization, it is very convenient to combine the Lax–Wendroff method with matrix operations to solve the time-domain analysis difficulty of lossy multiconductor transmission lines, i.e., the two coefficient matrices in one telegrapher equation cannot, in general, be diagonalized at the same time. Because the inverse operation of matrices only involves coefficient matrices which do not change with time, for uniform transmission lines, the inverse operation only needs to be calculated once, and for an inhomogeneous structure, the inverse only needs to be calculated once for all stepwise constant segments, so the efficiency is not greatly impacted. With the proposed method, we analyzed a linear multiconductor transmission line circuits and a nonlinear circuit including nonuniform transmission lines. The results agree well with those of PSPICE and HSPICE. Therefore,

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the proposed method is efficient and can be applied to either linear or nonlinear circuits including general transmission lines. Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 60904085), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200802881012), ‘‘Excellent Talent Project Zijin Star’’ Foundation of Nanjing University of Science and Technology, Foundation of National Key Laboratory of Transient Physics.

References 1. Djordjeric, A. R., & Sarkar, T. K. (1987). Analysis of time response of lossy multiconductor transmission line networks. IEEE Transactions on Microwave Theory and Techniques, 35, 898–908. doi:10.1109/TMTT.1987.1133776. 2. Griffith, J. R., & Nakhla, M. S. (1990). Time-domain analysis of lossy coupled transmission lines. IEEE Transactions on Microwave Theory and Techniques, 38, 1480–1486. doi:10.1109/ 22.58689. 3. Xu, Q. W., Li, Z. F., & Mao, J. F. (2000). Time domain simulation of high-speed interconnect system by a modified method of characteristics. IEEE Transactions on Microwave Theory and Techniques, 48, 323–327. doi:10.1109/22.899015. 4. Tang, M., Mao, J. F., & Li, X. C. (2005). Analysis of interconnects with frequency-dependent parameters by differential quadrature method. IEEE Microwave and Wireless Components Letters, 15, 877–879. doi:10.1109/LMWC.2005.859982. 5. Tseng, W. L., Chen, C. Z., Gad, E., Nakhla, M., & Achar, R. (2007). Passive order reduction for RLC circuits with delay elements. IEEE Transactions on Advanced Packaging, 30, 830–840. doi:10.1109/TADVP.2007.906240. 6. Nakhla, N., Nakhla, M., & Achar, R. (2007). Model order reduction of large multiport interconnect structures using waveform relaxation techniques. In IEEE workshop on signal propagation on interconnects, Camogli, Italy (pp. 69–70). 7. Li, Z. F., & Mao, J. F. (2001). Microwave and high-speed circuit theory (pp. 279, 287–294). Shanghai: Shanghai Jiaotong University Press. 8. Guo, Y. S. (2001). An efficient transient analysis method for general transmission lines based on the time Space discretization of telegrapher equations. Acta Electronica Sinica, 29, 373–377. 9. Lu, J. F., & Guan, Z. (2004). Numerical method for partial differential equations (pp. 46–54). Beijing: Qinghua University Press. 10. Hwang, K. P., & Jin, J. M. (1998). A total-variation-diminishing finite-difference scheme for the transient response of a lossless transmission line. IEEE Transactions on Microwave Theory and Techniques, 46, 1193–1196. doi:10.1109/22.704968.

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Analog Integr Circ Sig Process (2011) 68:85–92 11. Thiel, W., & Menzel, W. (2001). On the modeling of highly nonlinear circuits using total-variation-decreasing finite-difference schemes. IEEE Transactions on Microwave Theory and Techniques, 49, 1620–1625. doi:10.1109/22.942575. 12. Dou, L., & Wang, Z. Q. (2005). Application of Lax–Wendroff difference method to the analysis of high-speed interconnect transmission lines. Information and Control, 34, 724–728. 13. Dou, L., & Wang, Z. Q. (2005). One high-efficiency analysis method for high-speed circuit network containing distributed parameter elements. Journal of Control Theory and Applications, 3, 117–120. doi:10.1007/s11768-005-0003-8. 14. Dou, L., & Wang, Z. Q. (2007). Sensitivity analysis and optimization for high-speed circuit systems. In IEEE 2007 international symposium on microwave, antenna, propagation and EMC technologies for wireless communications, Hangzhou, China (pp. 1437–1441).

Lei Dou was born in Lianyungang, China, in 1974. He received the B.Eng. degree in chemical engineering, M.Eng. degree in power engineering and Ph.D. degree in automation from Nanjing University of Science and Technology, Nanjing, China, in 1996, 2003 and 2006, respectively. Since 2009, he has been an associate professor with National Key Laboratory of Transient Physics, Nanjing University of Science and Technology. He has authored 14 journal/conference papers. His current research interests include distributed parameter system, guidance and control. Jiao Dou was born in Jiangsu Province, China, in 1978. He received the B.Eng. degree in electronic engineering from Dalian University of Technology, Dalian, China, in 1999, and the M.Eng. Degree in signal processing from the China Academy of Launch Vehicle Technology (CALT), Beijing, China, in 2003. From 2003 to 2005, he was with CALT for SAR Radar research. Now he is a research engineer in Beijing design center of Conexant Inc. His interests include OFDM technology in communication system.