Multiresolution Time-Domain Scheme for Terminal Response of Two-Conductor Transmission Lines

Abstract

This paper derives a multiresolution time-domain (MRTD) scheme for the two-conductor lossless transmission line equations based on Daubechies' scaling functions. And a method is proposed to generate the scheme at the terminal and near the terminal of the lines. The stability and numerical dispersion of this scheme are studied, and the proposed scheme shows a better dispersion property than the conventional FDTD method. Then the MRTD scheme is extended to the two-conductor lossy transmission line equations. The MRTD scheme is implemented with different basis functions for both lossless and lossy transmission lines. Numerical results show that the MRTD schemes which use the scaling functions with high vanishing moment obtain more accurate results.

Academic Editor:M. I. Herreros

College of Science, National University of Defense Technology, Changsha, Hunan 410073, China

Received 4 January 2016; Revised 20 March 2016; Accepted 18 April 2016; 18 May 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The multiresolution time-domain (MRTD) scheme proposed by [1] provides an efficient algorithm for electromagnetic field computation and shows excellent capability to approximate exact solution with low sampling rates. However, the Battle-Lemarie wavelet function used in [1] is not compact supported, which means the iterative equations contain infinite terms. We must cut off the iterative equations in the actual computation and this may introduce truncation errors. So different wavelet bases, which are compact supported with some numbers of vanishing moments, have been used to improve this method [2-5]. This makes a great development for MRTD schemes. As a kind of numerical method, the MRTD schemes show great advantages in numerical dispersion properties [6-9]; meanwhile, these schemes need a more rigorous stable condition than the conventional FDTD method [10]. For containing more terms in the iterative equations, the terminal conditions or absorbing boundary conditions are more complicated to process in MRTD schemes; this disadvantage has limited the application of the MRTD scheme. To overcome this limitation, some works on the perfect match layer have been made [11-13]; however, other terminal conditions also need to be analyzed specifically. For the transmission lines equations, the resistive terminal conditions could be equivalent as a general Thevenin circuit; this paper will solve this kind of terminal condition in the MRTD scheme.

Since the appearance of the telegraph equations, studies on transmission lines have had a considerable development. Several equivalent forms of transmission line theory have been proposed to describe the influence of the incident electromagnetic field to the transmission lines [14-16]. In [17], the classical theory of the transmission line has been summarized and the theory on the high frequency radiation effects to the transmission lines is introduced. In the monograph [18], the multiconductor transmission lines (MTL) theory has been comprehensively studied in detail. For the two-conductor lossless transmission lines, there are several methods, which contain the series solution, the SPICE solution, the time-domain to frequency-domain (TDFD) transformation method, and the FDTD method [18]. However, the MRTD scheme has not been used to calculate the terminal response of transmission lines. In this paper we will derive a MRTD scheme for this problem.

In this paper, we focus on the calculation of the terminal response of two-conductor transmission lines equations by using MRTD scheme. In Section 2, the MRTD scheme is derived based on Daubechies' scaling functions for the two-conductor lossless transmission line equations, and, for the resistive terminations, the iterative equations for the terminal voltages are derived, a method is proposed to update the iterative equations which contain some terms whose indices exceed the index range in the MRTD scheme, and then the stability and the numerical dispersion are studied. In Section 3, the MRTD scheme is extended to the two-conductor lossy transmission line. In Section 4, the numerical results are presented on the terminal response of both lossless and lossy transmission lines using the MRTD scheme and compared to the FDTD method at different space discretization numbers and different Courant numbers.

2. MRTD Scheme for Two-Conductor Lossless Transmission Lines

2.1. MRTD Formulation

In this section, the MRTD scheme is applied to the following scalar transmission lines equations for two-conductor lossless lines [18]: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the per-unit-length inductance and capacitance, respectively.

Based on the method outlined in [1], the voltage and current can be expanded as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the coefficients for the voltages and currents in terms of scaling functions, respectively. The indices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the discrete spatial and temporal indices related to space and time coordinates via [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the spatial and temporal discretization intervals in [figure omitted; refer to PDF] and [figure omitted; refer to PDF] direction. The function [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] with the rectangular pulse function [figure omitted; refer to PDF]

The function [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents Daubechies' scaling function. Figure 1 shows Daubechies' scaling function with two vanishing moments.

Figure 1: Daubechies' scaling function with two vanishing moments.

[figure omitted; refer to PDF]

For deriving the MRTD scheme for (1a) and (1b), we need the following integrals: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the Kronecker symbol. Consider [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the effective support size of the basis functions. The coefficients [figure omitted; refer to PDF] are called connection coefficients and can be calculated by (9). Taking Daubechies' scaling functions as the basis functions, Table 1 shows [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , which are zeros for [figure omitted; refer to PDF] , and for [figure omitted; refer to PDF] it can be obtained by the symmetry relation [figure omitted; refer to PDF] , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the Fourier transform of [figure omitted; refer to PDF] .

Table 1: Connection coefficients [figure omitted; refer to PDF] and the first-order moments [figure omitted; refer to PDF] of Daubechies' scaling functions.

[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]	[figure omitted; refer to PDF]
0	1.2291666667	1.2918129281	1.3110340773
1	-0.0937500000	-0.1371343465	-0.1560100110
2	0.0104166667	0.0287617728	0.0419957460
3		-0.0034701413	-0.0086543236
4		0.0000080265	0.0008308695
5			0.0000108999
6			0.0000000041
[figure omitted; refer to PDF]	0.6339743121	0.8174005815	1.0053923835
[figure omitted; refer to PDF]	0.7500	0.6844	0.6585

Daubechies' scaling functions satisfy the shifted interpolation property [19] [figure omitted; refer to PDF] for [figure omitted; refer to PDF] integer, where [figure omitted; refer to PDF] is the first moment of the scaling functions and the values of [figure omitted; refer to PDF] are listed in Table 1.

Following the theory in [3] and making use of (10), (5) is modified to [figure omitted; refer to PDF] In spite of the support of the scaling functions [20], single-point sampling of the total voltages and currents can be taken at integer points with negligible error. Taking voltage at spatial point [figure omitted; refer to PDF] and at time [figure omitted; refer to PDF] , we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the Dirac delta function. Equation (12) means the voltage value at each integer point is equal to the coefficient. The current values have the same character at each half integer point. Therefore, we will use [figure omitted; refer to PDF] and [figure omitted; refer to PDF] directly to represent the voltage at the point [figure omitted; refer to PDF] and the current at the point [figure omitted; refer to PDF] in this paper.

The modified [figure omitted; refer to PDF] in (11) also satisfy integrals (7) and (8). Applying the Galerkin technique to (1a) and (1b), we can obtain the following iterative equations for the voltages and currents: [figure omitted; refer to PDF]

2.2. Terminal Iterative Equations for Resistive Load in MRTD Scheme

We will consider the terminal conditions for the two-conductor lossless transmission lines equations in this section. Equations (1a) and (1b) are homogeneous linear equations; we need to add the terminal conditions to obtain the unique solution.

Considering the two-conductor lines shown in Figure 2, we assume the length of the total line is [figure omitted; refer to PDF] and the resistive loads are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The line is divided uniformly into NDZ segments with the space interval [figure omitted; refer to PDF] and the total solution time is divided into NDT steps with the uniform time interval [figure omitted; refer to PDF] . Similar to the conventional FDTD, we will calculate the interlace voltages, [figure omitted; refer to PDF] , and currents, [figure omitted; refer to PDF] , in both space domain and time-domain as shown in (13a) and (13b), for [figure omitted; refer to PDF] .

Figure 2: A two-conductor line in time-domain.

[figure omitted; refer to PDF]

For the resistive terminations, we note the voltage at the source [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and the current at the source as [figure omitted; refer to PDF] , the external voltage at the load [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , and the current at the load as [figure omitted; refer to PDF] . The discrete voltages and currents at the source are denoted as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and the discrete voltages and currents at the load are denoted as [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , then the terminal characterizations could be written in terms of a generalized Thevenin equivalent as [figure omitted; refer to PDF] Equations (14a) and (14b) denote the discretization terminal conditions for the case of resistive terminations, so we need to introduce these conditions to the iterative equations (13a) and (13b) to obtain the numerical solution.

Notice that, in the iterative equations (13a) and (13b), not only the iterative equations of the terminal voltages [figure omitted; refer to PDF] and [figure omitted; refer to PDF] should be derived and the iterative equations of voltages and currents "near" the terminals also need to be updated. The voltages and currents "near" the terminals we mean are the voltages [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and the currents [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . All of these voltages and currents contain some terms that exceed the index range in iterative equations (13a) and (13b). Figure 3 shows the discretization of the terminal voltages and the voltages and currents near the terminal.

Figure 3: Discretizing the terminal voltages and currents.

[figure omitted; refer to PDF]

We will derive the MRTD scheme at the terminal firstly. For updating the iterative equations for the terminal voltages, we need to decompose iterative equations ((13a), (13b)). Since the coefficients [figure omitted; refer to PDF] satisfy the following relation [4] [figure omitted; refer to PDF] substituting (15) into (13a), we can obtain [figure omitted; refer to PDF]

Considering the corresponding terms with [figure omitted; refer to PDF] , we can decompose (13a) as [21] [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Equation (13b) could make the analogous decomposition. We could view the MRTD scheme for two-conductor transmission lines as the weighted mean of the conventional FDTD method with spatial discretization step [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , and the weighting coefficient for each term is [figure omitted; refer to PDF] . Besides, for the MRTD scheme whose coefficients [figure omitted; refer to PDF] satisfy relationship (15), the analogous decomposition could be made. This relationship between the MRTD scheme and the conventional FDTD method is useful for us to update the iterative equations.

Taking [figure omitted; refer to PDF] as an example to derive the iterative equations at the terminal, [figure omitted; refer to PDF]

Following steps of (16) and (17), we can decompose (18) as [figure omitted; refer to PDF]

Here, we could view each equation in (19a), (19b), and (19c) as a central difference scheme, (19a) is the central difference scheme related to points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , (19b) is the central difference scheme related to points [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and so on, but the terms [figure omitted; refer to PDF] , whose subscripts exceed the index range, make (19a), (19b), and (19c) out of work. So we need to make some update for the iterative equations. Using the forward difference scheme to replace the central difference scheme, we change the difference points in (19a) to become [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and change the difference points in (19b) to become [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ; the others follow the same step. Keeping the weighting coefficient unchanged in each equation, we can obtain [figure omitted; refer to PDF] where the terminal current [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be derived from (14a) [figure omitted; refer to PDF]

Summing up all the equations in (20a), (20b), and (20c), [figure omitted; refer to PDF]

Substituting (21) into (22), we can obtain the iterative equation at the source [figure omitted; refer to PDF]

With the same steps, we can obtain the iterative equation at the load [figure omitted; refer to PDF]

After deriving the iterative equations at the terminal, we will put forward a truncation method to update iterative equations which contain some terms whose indices exceed the index range in the MRTD scheme.

Taking [figure omitted; refer to PDF] as an example, for [figure omitted; refer to PDF] , decomposing (13a), [figure omitted; refer to PDF] Noticing the first [figure omitted; refer to PDF] terms in (25a), (25b), (25c), (25d), and (25e), there is no term exceeding the index range in each equation. Meanwhile, the equations which contain the exceeding indices terms are all appearing in the rest of [figure omitted; refer to PDF] terms. As mentioned before, we can view the MRTD scheme as the weighted mean of the conventional FDTD method, but (25a), (25b), (25c), (25d), and (25e) show that the last [figure omitted; refer to PDF] equations are unavailable for forming the iterative equations in MRTD scheme. To solve this problem, we make truncation here. We update the iterative equation of [figure omitted; refer to PDF] by using the summation of the first [figure omitted; refer to PDF] terms in (25a), (25b), (25c), (25d), and (25e) that means we use the weighted mean of the first [figure omitted; refer to PDF] to approximate the summation of all the [figure omitted; refer to PDF] terms.

Summing up the first [figure omitted; refer to PDF] terms in (25a), (25b), (25c), (25d), and (25e), we can obtain the modified iterative equations [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Using the same method, we can obtain the modified iterative equations near the load [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

The voltages at the interior points are determined from (13a) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

For the iterative equation of the current, there is a little difference from the voltage's. As shown in Figure 3, the interlace currents appear at the half integer points which means all the currents are located at the interior points. So we only need to modify the currents near the terminals. Following the same steps of the derivation of voltages iterative equations near the terminal, we could obtain the current iterative equations near the terminals.

For the current iterative equations near the source [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

For the current iterative equations near the load [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

The currents at the interior points are determined from (13b) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

2.3. Terminal Iterative Equations for Inductive Resistance in MRTD Scheme

Since we have discussed the resistive load in Section 2.2, we will consider a more complicated terminal load which consisted of a resistance and inductance shown in Figure 4. The resistance and the inductance at the load are noted as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Figure 4: A two-conductor line with an inductive resistance.

[figure omitted; refer to PDF]

Keeping the source as a resistive terminal and changing the load including inductance, we note the voltage at the source [figure omitted; refer to PDF] as [figure omitted; refer to PDF] and the current at the source as [figure omitted; refer to PDF] , the external voltage at the load [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , and the current at the load as [figure omitted; refer to PDF] . The terminal conditions could be written as follows: [figure omitted; refer to PDF]

Expanding [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] as (2a) and (2b) and sampling them at the time discreting point [figure omitted; refer to PDF] , the terminal conditions become [figure omitted; refer to PDF]

It could be seen from Section 2.2 that the change of the terminal condition only affects the iterative equations at the terminals, so we just need to consider two iterative equations. Since we keep the source as a resistive termination, the iterative equation at the source should be the same as (23). Actually, if we substitute (33a) into (22), we could obtain (23). So the only iterative equation we should derive is located at the load. If we set [figure omitted; refer to PDF] in (33b), the terminal condition at the load will have the analogous form with the source, which means the load with a resistance and inductance degenerates to be a resistance. For those [figure omitted; refer to PDF] , transforming (33b) as [figure omitted; refer to PDF]

Following the steps we get the iterative equation of [figure omitted; refer to PDF] in Section 2.2; we can obtain the iterative equation at the load [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

2.4. Stability Analysis

For the purpose of stability analysis, (13a) and (13b) can be rewritten as [figure omitted; refer to PDF]

Following the procedures in [22], the finite-difference approximations of the time derivations on the left side of (37a) and (37b) can be written as an eigenvalue problem [figure omitted; refer to PDF]

In order to avoid instability during normal time stepping, the imaginary part of [figure omitted; refer to PDF] must satisfy [figure omitted; refer to PDF]

As we consider the lossless two-conductor transmission lines in (1a) and (1b), the transient values of voltages and currents distributed in space can be Fourier transformed with respect to [figure omitted; refer to PDF] -coordinates to provide a spectrum of sinusoidal modes. Assuming an eigenmode of the spectral domain with [figure omitted; refer to PDF] , the voltages and currents can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the voltage at point [figure omitted; refer to PDF] , [figure omitted; refer to PDF] represents the current at point [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the amplitudes of the voltages and currents.

Substituting (40a) and (40b) into (38a) and (38b) and (37a) and (37b), we obtain [figure omitted; refer to PDF] In (41), [figure omitted; refer to PDF] is pure imaginary and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the velocity of the wave along with the lines.

Numerical stability is maintained for every spatial mode only when the range of eigenvalues given by (42) is contained entirely within the stable range of time-difference eigenvalues given by (39). Since both ranges are symmetrical around zero, it is adequate to set the upper bound of (42) to be smaller or equal to (39); we can obtain the stability condition [figure omitted; refer to PDF]

Noting the Courant number [figure omitted; refer to PDF] the maximum values of [figure omitted; refer to PDF] required by a stable algorithm, which are listed in Table 1 as [figure omitted; refer to PDF] , can be calculated from the connection coefficients.

2.5. Dispersion Analysis

To calculate the numerical dispersion, substituting a time harmonic trial solution into (37a) and (37b), we can obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the wave angular frequency and [figure omitted; refer to PDF] is the numerical wave number.

Using the number of cells per wavelength [figure omitted; refer to PDF] and the wave number [figure omitted; refer to PDF] , we obtain the dispersion relationship [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the ratio between the theoretical and numerical wavelength.

The value of [figure omitted; refer to PDF] could be computed by Newton iterative method and using the formula [figure omitted; refer to PDF] calculates the phase error in degrees. Figure 5 shows the phase errors of different MRTD schemes versus the samples per wavelength. Compared with the conventional FDTD method at the same sampling numbers, the MRTD schemes show better numerical dispersion properties than FDTD method, which means the discretization error of the MRTD schemes is smaller than the FDTD method.

Figure 5: Phase error (in degrees) for different MRTD schemes. The Courant numbers are [figure omitted; refer to PDF] for (a) and [figure omitted; refer to PDF] for (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

3. MRTD Scheme for Two-Conductor Lossy Transmission Lines

In this section, we will extend the MRTD scheme to the two-conductor lossy transmission lines.

3.1. MRTD Formulation

For the lossy case, the two-conductor transmission line equations become [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are the per-unit-length resistance, inductance, conductance, and capacitance, respectively.

Compared to the lossless transmission lines, the lossy case must consider the resistance losses along the lines and the losses in the medium. However, the steps to obtain the MRTD scheme are the same. Similar to the lossless case, we firstly extend the voltage and current with Daubechies' scaling functions and the rectangular function and use [figure omitted; refer to PDF] and [figure omitted; refer to PDF] representing the voltage at the point [figure omitted; refer to PDF] and the current at the point [figure omitted; refer to PDF] , respectively, and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] represent the spatial and temporal discretization intervals. By applying the Galerkin technique, we can obtain the iterative equations for the lossy transmission lines equations [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the connection coefficient and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constants [figure omitted; refer to PDF]

The differences between (48a) and (48b) and (13a) and (13b) are the coefficients of terms in the iterative equations that are caused by the unit-per-length resistance [figure omitted; refer to PDF] and inductance [figure omitted; refer to PDF] . If we set [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , the coefficients [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are all equal to [figure omitted; refer to PDF] , and the iterative equations (48a) and (48b) will degenerate to iterative equations (13a) and (13b).

For the lossy case, we should also modify the iterative equations at the terminal and near the terminal of the lines. Considering the two-conductor lines shown in Figure 2, we assume the length of the total line is [figure omitted; refer to PDF] and the resistive loads are [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The line is divided uniformly into NDZ segments with the space interval [figure omitted; refer to PDF] and the total solution time is divided into NDT steps with the uniform time interval [figure omitted; refer to PDF] . Following the same steps in Section 2.2, we could obtain the modified iterative equations.

For the voltage iterative equation at the source [figure omitted; refer to PDF]

For the voltage iterative equation at the load [figure omitted; refer to PDF]

For the voltage iterative equations near the source [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

For the voltage iterative equations near the load [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

The voltages at the interior points are determined from (48a) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

For the current iterative equations near the source [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

For the current iterative equations near the load [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

The currents at the interior points are determined from (48b) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Besides, for the two-conductor line shown in Figure 4, we can obtain the iterative equation at the load [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

3.2. Stability Analysis

To study the stability of the MRTD scheme for lossy case, we need to make some changes. Rewrite (48a) and (48b) as follows: [figure omitted; refer to PDF]

The finite-difference approximations of the time derivations on the left side of (61a) and (61b) are different from the left side of (37a) and (37b), but we can also write them as an eigenvalue problem [figure omitted; refer to PDF]

In order to avoid instability during normal time stepping, the imaginary part of [figure omitted; refer to PDF] also must satisfy [figure omitted; refer to PDF]

Then following the steps in Section 2.4, using the Fourier transform, expand the transient values of voltages and currents distributed in space. We can also obtain the same stability condition as (43) [figure omitted; refer to PDF]

However, the dispersion analysis for the iterative equations of the lossy transmission lines is quite different from the lossless case. Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, the coefficient of [figure omitted; refer to PDF] is not equal to the coefficient of [figure omitted; refer to PDF] and the coefficient of [figure omitted; refer to PDF] is not equal to the coefficient of [figure omitted; refer to PDF] on the left side of (61a) and (61b), if we substitute a time harmonic into the iterative equations into (61a) and (61b), there are some other terms that could affect the ratio between the theoretical and numerical wavelength except samples per wavelength, so we could not obtain a brief dispersion relation as (45). However, the numerical results in Section 4.2 show that the MRTD method could obtain a more accurate result than the FDTD method at the same space interval and time interval.

4. Numerical Result

4.1. Lossless Transmission Lines

In this section, the two-conductor lossless transmission lines as shown in Figure 2 are considered to calculate the terminal voltages. The length of the lines is [figure omitted; refer to PDF] m and the per-unit-length capacitance and inductance are [figure omitted; refer to PDF] pF/m and [figure omitted; refer to PDF] μ H/m, respectively. This corresponds to RG58U coaxial cable [18]. The characteristic impedance of the line is [figure omitted; refer to PDF] and the velocity of propagation is [figure omitted; refer to PDF] . The terminal load is [figure omitted; refer to PDF] and the source resistance is [figure omitted; refer to PDF] .

We will use a trapezoidal pulse as shown in Figure 6 as the source voltage, whose initial value is 0 V and amplitude is [figure omitted; refer to PDF] , total time is [figure omitted; refer to PDF] , rise time is [figure omitted; refer to PDF] , and fall time is [figure omitted; refer to PDF] , respectively. The bandwidth of the pulse is approximate to [figure omitted; refer to PDF] , so the segment length [figure omitted; refer to PDF] should be electrically short at this frequency requiring [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) [18]. As we take the rise time of the trapezoidal pulse to be [figure omitted; refer to PDF] , the approximate bandwidth of the pulse is [figure omitted; refer to PDF] and the maximum section length of [figure omitted; refer to PDF] should be less than [figure omitted; refer to PDF] , noting [figure omitted; refer to PDF] . Since we divide the line into NDZ segments uniformly, NDZ should be greater than [figure omitted; refer to PDF] . The time step will be calculated by (44) with different Courant number [figure omitted; refer to PDF] .

Figure 6: Trapezoidal pulse with the rise time [figure omitted; refer to PDF] and fall time [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Under the conditions of the space discretization number [figure omitted; refer to PDF] and the Courant number [figure omitted; refer to PDF] , we calculate the terminal voltages using the MRTD schemes with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] wavelets' scaling functions. We also calculate the terminal voltages by the FDTD method under the same conditions. Figure 7 shows the numerical results by different methods and [figure omitted; refer to PDF] -MRTD represents the MRTD scheme using Daubechies' scaling functions with [figure omitted; refer to PDF] vanishing moment as basis functions, where [figure omitted; refer to PDF] .

Figure 7: Terminal voltage calculated by different numerical methods. The space discretization number and the Courant number are NDZ = 20 and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

For the time-dependent discrete terminal voltages, the relative error is defined as follows [23]: [figure omitted; refer to PDF] Here, [figure omitted; refer to PDF] represents the numerical results of FDTD and MRTD at each time discretization point and [figure omitted; refer to PDF] is the numerical result of series solution at each time discretization point, which can be regarded as the exact solution [18]. Taking [figure omitted; refer to PDF] as the exact result, we calculate the relative errors of FDTD method and MRTD scheme.

Table 2 shows the relative errors and runtime for different schemes. Since the iterative equations in MRTD schemes contain more terms than the conventional FDTD method, the MRTD schemes expend more runtime. When we use [figure omitted; refer to PDF] -MRTD, the numerical result shows a larger relative error. The reason is that the vanishing moment of the [figure omitted; refer to PDF] wavelet's scaling function is not high enough. For the wavelets' scaling functions whose vanishing moment is high enough, like [figure omitted; refer to PDF] wavelet and [figure omitted; refer to PDF] wavelet, the numerical results show smaller relative errors.

Table 2: Relative errors and runtime of different methods (NDZ = 20, [figure omitted; refer to PDF] ).

Analysis scheme	Relative error	Runtime (s)
FDTD	1.08%	0.0011
[figure omitted; refer to PDF] -MRTD	1.93%	0.0050
[figure omitted; refer to PDF] -MRTD	0.66%	0.0054
[figure omitted; refer to PDF] -MRTD	0.62%	0.0058

Figure 8 shows the relative errors versus the space discretization numbers. The space interval will decrease with the increase of the space discretization number. We can see from the figures that the relative errors for [figure omitted; refer to PDF] -MRTD increase with the increase of NDZ. And, for [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD, the relative errors decrease with the increase of NDZ and show a little smaller relative error than the conventional FDTD. It means we can get a more accurate result at same space interval and time interval by [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD.

Figure 8: Relative error for different discretization numbers. The Courant numbers are [figure omitted; refer to PDF] for (a) and [figure omitted; refer to PDF] for (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 9 describes the relative errors versus the Courant numbers. The time interval will decrease with the decrease of the Courant number. The results show that, with the decrease of the Courant numbers, the relative errors are almost unchanged for the conventional FDTD method, while the relative errors are quite different with different Courant numbers for all the three MRTD schemes. That means the choice of the Courant number significantly affects the relative errors of the MRTD schemes, and we could choose the proper Courant number to optimize the MRTD schemes.

Figure 9: Relative errors for different Courant numbers. The space discretization numbers are NDZ = 20 for (a) and NDZ = 50 for (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

To validate the stability of the MRTD schemes, we increase the initial value of the pulse to 20 V and keep other parameters unchanged. Figure 10 shows the relative errors versus the pace discretization numbers. It can be seen from the figures that the [figure omitted; refer to PDF] -MRTD scheme shows a larger relative error and the relative errors of [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD schemes are smaller than the conventional FDTD method.

Figure 10: Relative error for different space discretization numbers for the initial value of the trapezoidal pulse to be 20 V. The Courant numbers are [figure omitted; refer to PDF] for (a) and [figure omitted; refer to PDF] for (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The numerical results for the lossless transmission lines also show that the [figure omitted; refer to PDF] -MRTD does not perform better than the FDTD method; meanwhile, [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD schemes show better quality in accuracy and stability. The reason for this phenomenon is that the scaling function of the [figure omitted; refer to PDF] wavelet does not have enough high vanishing moment. When we use Daubechies' scaling functions to expand the voltages and currents in the two-conductor transmission lines equations, the vanishing moment decides the accuracy of the approximation. The scaling functions with high vanishing moments could approximate voltages and currents more accurately; however, the scaling functions with low vanishing moments like [figure omitted; refer to PDF] wavelet's may introduce a larger error. So we can see from Figure 5 that [figure omitted; refer to PDF] -MRTD has a better numerical dispersion property than the FDTD method, but it gets larger relative errors in numerical computation. For the scaling functions which have high enough vanishing moments like [figure omitted; refer to PDF] wavelet and [figure omitted; refer to PDF] wavelet, the numerical results show smaller relative errors and are in agreement with the dispersion analysis. However, a high vanishing moment for scaling function may increase the computation complexity in MRTD schemes. So the vanishing moments of the wavelet's scaling functions have a great effect on the accuracy of the MRTD scheme; it is necessary to choose a scaling function with proper vanishing moment when we use the MRTD scheme for the numerical computation.

4.2. Lossy Transmission Lines

In this section, we will consider the lossy two-conductor transmission lines shown in Figure 11. Two conductors of rectangular cross section of width [figure omitted; refer to PDF] μ m and thickness [figure omitted; refer to PDF] μ m are separated by [figure omitted; refer to PDF] μ m and placed on one side of a silicon substrate ( [figure omitted; refer to PDF] ) of thickness [figure omitted; refer to PDF] μ m; the total line length is [figure omitted; refer to PDF] cm. The near end is a source with a [figure omitted; refer to PDF] Ω resistance and the far end is a load with a [figure omitted; refer to PDF] Ω resistance and [figure omitted; refer to PDF] μ H inductance in series. The per-unit-length inductance and capacitance were computed as [figure omitted; refer to PDF] μ H/m and [figure omitted; refer to PDF] pF/m. This gives a velocity of [figure omitted; refer to PDF] m/s and a one-way time delay of [figure omitted; refer to PDF] ns, which gives an effective dielectric constant of ( [figure omitted; refer to PDF] ) and characteristic impedance of [figure omitted; refer to PDF] Ω. The per-unit-length dc resistance is computed as [figure omitted; refer to PDF] Ω/m [18]. Dielectric loss is not included in these calculations, which means the per-unit-length conductance is [figure omitted; refer to PDF] .

Figure 11: A lossy printed circuit board: (a) line dimensions and terminations and (b) cross-sectional dimensions.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

The source is a ramp function as shown in Figure 12, the initial value of the ramp function is [figure omitted; refer to PDF] , and the amplitude is [figure omitted; refer to PDF] V with a rise time of [figure omitted; refer to PDF] ns. The total computing time is 20 ns. The bandwidth of the source is approximate to [figure omitted; refer to PDF] GHz. The space discretization step for the MRTD was chosen to be [figure omitted; refer to PDF] , so the space discretization number NDZ should be greater than 34. The time step will also be calculated by (44) with different Courant numbers.

Figure 12: Representation of the source voltage waveform.

[figure omitted; refer to PDF]

We calculate the near end voltage and the far end voltage of the lossy PCB with space discretization number NDZ = 40 and the Courant number [figure omitted; refer to PDF] . Figure 13 shows the computing results. Since [figure omitted; refer to PDF] -MRTD may introduce a larger error in the computation as shown in Section 4.1, we use [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD to compute the terminal voltages.

Figure 13: Terminal voltages for lossy transmission lines: (a) for the near end and (b) for the far end. The space discretization number is [figure omitted; refer to PDF] and the Courant number is [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Here, we choose the time-domain to frequency-domain transformation method (TDFD), which is a straightforward adaptation of a common analysis technique for lumped, linear circuits and systems [18], to validate the computing results of MRTD schemes and FDTD method. Table 3 shows the relative errors and the runtime of the MRTD schemes and FDTD method. It can be seen that MRTD schemes spend much time to obtain more accurate results.

Table 3: Relative errors and runtime of different methods. [figure omitted; refer to PDF] represents the near end voltage and [figure omitted; refer to PDF] represents the far end voltage (NDZ = 40, [figure omitted; refer to PDF] ).

Analysis scheme	Relative error for [figure omitted; refer to PDF]	Relative error for [figure omitted; refer to PDF]	Runtime (s)
FDTD	0.30%	0.92%	0.0066
[figure omitted; refer to PDF] -MRTD	0.18%	0.57%	0.0245
[figure omitted; refer to PDF] -MRTD	0.17%	0.54%	0.0275

Figure 14 describes the relative errors versus the space discretization numbers. For both [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD, the MRTD schemes show a little smaller relative error than the conventional FDTD. That means the MRTD schemes could obtain a more accurate solution under the same time interval and space interval. This is because [figure omitted; refer to PDF] -MRTD and [figure omitted; refer to PDF] -MRTD have better dispersion property than the FDTD method and the scaling functions of [figure omitted; refer to PDF] wavelet and [figure omitted; refer to PDF] wavelet have high enough vanishing moments.

Figure 14: Relative error for the terminal voltages with different space discretization numbers: (a) for the near end and (b) for the far end. The Courant number is [figure omitted; refer to PDF] .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Figure 15 shows the relative errors versus the Courant numbers. With the decrease of the Courant number, the relative errors of the FDTD method increase, because the time-domain is oversampled for the FDTD method. However, the MRTD schemes perform decreasing relative errors. That means the MRTD schemes could obtain a more accurate result with a smaller time interval. And even when the FDTD method is oversampled in the time-domain, the MRTD schemes perform well.

Figure 15: Relative errors for the terminal voltages with different Courant numbers: (a) for the near end and (b) for the far end. The space discretization number is NDZ = 40.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

5. Conclusion

In this paper, we derived the MRTD scheme for the two-conductor transmission lines and studied the stability and the numerical dispersion of this scheme. By viewing the MRTD schemes as the weighted mean of the conventional FDTD method at different space interval, we derived the iterative equations for the terminal voltages when the terminals are pure resistive, and a method is proposed to update the iterative equations which contain some terms whose indices exceed the index range in the MRTD scheme. Using the same method, we derived the iterative equation for the inductive load. Then we extended the MRTD scheme to the lossy transmission lines. Using different wavelets' scaling functions as basis functions, the MRTD schemes are implemented for both lossless case and lossy case and the numerical results are compared to the conventional FDTD method. The numerical results show the MRTD schemes need more runtime to obtain more accurate results. And the vanishing moment of the wavelet's scaling functions will significantly affect the quality of the MRTD scheme; using a scaling function with a proper vanishing moment as basis function in MRTD scheme could obtain a more accurate result.

Acknowledgments

The authors would like to thank Yinkun Wang for his comments on the paper. This work was supported by the National Natural Science Foundation of China (no. 11271370).

References

[1] M. Krumpholz, L. P. B. Katehi, "MRTD: new time-domain schemes based on multiresolution analysis," IEEE Transactions on Microwave Theory and Techniques , vol. 44, no. 4, pp. 555-571, 1996.

[2] M. Fujii, W. J. R. Hoefer, "A three-dimensional Harr-wavelet-based multi-resolution analysis similar to the 3-D FDTD method-derivation and application," The IEEE Transactions on Microwave Theory and Techniques , vol. 46, no. 12, pp. 2463-2475, 1998.

[3] Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, C. C. Shin, "Wavelet-Galerkin scheme of time-dependent inhomogeneous electromagnetic problems," IEEE Microwave and Wireless Components Letters , vol. 9, no. 8, pp. 297-299, 1999.

[4] T. Dogaru, L. Carin, "Multiresolution time-domain using CDF biorthogonal wavelets," IEEE Transactions on Microwave Theory and Techniques , vol. 49, no. 5, pp. 902-912, 2001.

[5] X. C. Wei, E. P. Li, C. H. Liang, "A new MRTD scheme based on Coifman scaling functions for the solution of scattering problems," IEEE Microwave and Wireless Components Letters , vol. 12, no. 10, pp. 392-394, 2002.

[6] E. M. Tentzeris, R. L. Robertson, J. F. Harvey, L. P. B. Katehi, "Stability and dispersion analysis of battle-lemarie-based MRTD schemes," The IEEE Transactions on Microwave Theory and Techniques , vol. 47, no. 7, pp. 1004-1013, 1999.

[7] S. Grivet-Talocia, "On the accuracy of haar-based multiresolution time-domain schemes," IEEE Microwave and Wireless Components Letters , vol. 10, no. 10, pp. 397-399, 2000.

[8] M. Fujii, W. J. R. Hoefer, "Dispersion of time domain wavelet galerkin method based on Daubechies' compactly supported scaling functions with three and four vanishing moments," IEEE Microwave and Wireless Components Letters , vol. 10, no. 4, pp. 125-127, 2000.

[9] A. Alighanbari, C. D. Sarris, "Dispersion properties and applications of the Coifman scaling function based S-MRTD," IEEE Transactions on Antennas and Propagation , vol. 54, no. 8, pp. 2316-2325, 2006.

[10] K. L. Shlager, J. B. Schneider, "Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms," IEEE Transactions on Antennas and Propagation , vol. 51, no. 3, pp. 642-653, 2003.

[11] E. M. Tentzeris, R. L. Robertson, J. F. Harvey, L. P. B. Katehi, "PML absorbing boundary conditions for the characterization of open microwave circuit components using multiresolution time-domain techniques (MRTD)," IEEE Transactions on Antennas and Propagation , vol. 47, no. 11, pp. 1709-1715, 1999.

[12] Q. S. Cao, Y. C. Chen, R. Mittra, "Multiple image technique (MIT) and anisotropic perfectly matched layer (APML) in implementation of MRTD scheme for boundary truncations of microwave structures," IEEE Transactions on Microwave Theory and Techniques , vol. 50, no. 6, pp. 1578-1589, 2002.

[13] Y. W. Liu, Y. W. Chen, B. Chen, X. Xu, "A cylindrical MRTD algorithm with PML and quasi-PML," IEEE Transactions on Microwave Theory and Techniques , vol. 61, no. 3, pp. 1006-1017, 2013.

[14] C. D. Taylor, R. S. Satterwhite, C. W. Harrison, "The response of a terminated two-wire transmission line excited by a nonuniform electromagnetic field," IEEE Transactions on Antennas and Propagation , vol. 13, no. 6, pp. 987-989, 1965.

[15] A. K. Agrawal, H. J. Price, S. H. Gurbaxani, "Transient response of multiconductor transmission lines excited by a nonuniform electromagnetic field," IEEE Transactions on Electromagnetic Compatibility , vol. 22, no. 2, pp. 119-129, 1980.

[16] F. Rachidi, "Formulation of the field-to-transmission line coupling equations in terms of magnetic excitation field," IEEE Transactions on Electromagnetic Compatibility , vol. 35, no. 3, pp. 404-407, 1993.

[17] F. Rachidi, S. V. Tkachenko Electromagnetic Field Interaction with Transmission Lines: From Classical Theory to HF Radiation Effects , vol. 1, WIT Press, 2008.

[18] C. R. Paul Analysis of Multiconductor Transmission Lines , John Wiley and Sons, New Jersey, NJ, USA, 2008., 2nd.

[19] W. Sweldens, R. Piessens, "Wavelet sampling techniques," in Proceedings of the Statistical Computing Section, pp. 20-29, American Statistical Association, 1993.

[20] I. Daubechies Ten Lectures on Wavelets , SIAM Press, Philadelphia, Pa, USA, 1992.

[21] X. Zhu, T. Dogaru, L. Carin, "Analysis of the CDF biorthogonal MRTD method with application to PEC targets," IEEE Transactions on Microwave Theory and Techniques , vol. 51, no. 9, pp. 2015-2022, 2003.

[22] A. Taflove Computational Electrodynamics: The Finite-Difference Time-Domain Method , Artech House, Norwood, NJ, USA, 1995.

[23] A. Alighanbari, C. D. Sarris, "Rigorous and efficient time-domain modeling of electromagnetic wave propagation and fading statistics in indoor wireless channels," IEEE Transactions on Antennas and Propagation , vol. 55, no. 8, pp. 2373-2381, 2007.