A method for considering roughness of ground plane in parameter extraction of multi-conductor transmission line

Abstract

This invention discloses a method for extracting parameters of multi-conductor transmission lines considering ground plane roughness, belonging to the field of electromagnetic simulation. Addressing the problem that existing two-dimensional parameter extraction methods for multi-conductor transmission lines suffer from low accuracy and difficulty in considering ground plane roughness, this invention provides a method for extracting parameters of multi-conductor transmission lines considering ground plane roughness. This method includes treating the ground plane as a signal line and calculating the frequency-varying resistance matrix; diagonalizing the frequency-varying resistance matrix to obtain an eigenvalue matrix and an eigenvector matrix; recording the eigenvalues ​​corresponding to the ground plane; multiplying each eigenvalue by its corresponding surface roughness coefficient; removing the ground plane elements to obtain a reduced matrix; calculating the roughness of each element in the eigenvalue matrix to obtain a new eigenvalue matrix; and finally, calculating the resistance matrix. This invention uses the eigenvalue method to calculate the ground plane roughness in multi-conductor transmission line scenarios, resulting in a more accurate calculation of the multi-conductor resistance considering ground plane roughness and improving the accuracy of resistance matrix extraction.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic simulation, and more specifically, relates to a method for extracting parameters of a multi-conductor transmission line that takes into account the roughness of the ground plane. Background Technology

[0002] In a conductor, current gradually distributes across the surface as frequency increases. Looking at a cross-section perpendicular to the current direction, at sufficiently high frequencies, almost no current flows through the center of the conductor, indicating a tendency for current to concentrate on the surface. However, real-world conductor surfaces are not perfectly smooth. When the conductor roughness used is comparable to the skin depth, which decreases with frequency, the current path becomes complex, exacerbating conductor losses. To accurately simulate conductor losses, a roughness coefficient needs to be introduced to correct for them. This roughness coefficient can be calculated using existing roughness models. Commonly used models include the Groisse model, Hammerstad model, Huray snowball model, Huray projectile stacking model, and hemispherical model. In two-dimensional parameter extraction for multi-conductor transmission lines, existing algorithms incorporate the roughness model of the signal line. However, the surface roughness of the ground plane metal is difficult to incorporate into the resistance and inductance parameters of the transmission line. In two-dimensional RLGC parameter extraction for multi-conductor transmission lines, the ground plane is typically used as a reference and is not reflected in the final parameter extraction matrix elements. The metal loss in this part significantly impacts the accuracy of the results.

[0003] For example, Chinese patent application CN202010282747.6, published on September 4, 2020, discloses a method for extracting parasitic parameters of a DC / DC converter. This method selects a portion of the microstrip lines as reference lines and the rest as signal lines. Based on the properties of the signal lines, reference lines, and surrounding dielectric, the microstrip lines and dielectric are divided into several units. Appropriate boundary conditions are applied, and the parasitic capacitance of the microstrip lines is extracted using system energy. A current source is applied to each line, and the corresponding parasitic resistance is calculated based on the current density and voltage distribution of each unit. The entire dielectric in the system is replaced with free space, and the parasitic capacitance of the microstrip lines under these conditions is extracted again. An electrostatic induction matrix is ​​constructed based on the extraction results, and the parasitic inductance of the microstrip lines is extracted using the inductance-capacitance parameter relationship of the transmission lines. The shortcoming of this patent is that the overall accuracy of parameter extraction needs improvement.

[0004] For example, Chinese patent application CN201711344193.2, published on June 15, 2018, discloses a method for extracting ohmic contact resistance parameters of semiconductor electrodes. This invention employs a scheme where the spacing between the electrodes in a semiconductor ohmic contact is the same, but the electrode widths vary, thus leading to a new and more accurate ohmic contact resistance calculation model. Because this scheme considers the variation in sheet resistance of the material beneath the gold-semiconductor contact in actual conditions, the experimental data obtained are more accurate than traditional methods, meeting the requirements of actual process variations. Therefore, it is a more effective technical solution for evaluating ohmic contact characteristics. This technical solution model can comprehensively evaluate ohmic contact electrodes and accurately extract parameters such as the sheet resistance of the material between electrodes, the sheet resistance of the material beneath the contact electrodes, and the specific contact resistivity, providing positive guidance for the design of semiconductor devices. The drawback of this patent is that the steps are relatively cumbersome and inefficient. Summary of the Invention

[0005] 1. The problem to be solved

[0006] To address the problem of low accuracy and difficulty in considering ground plane roughness in existing two-dimensional parameter extraction methods for multi-conductor transmission lines, this invention provides a method that incorporates ground plane roughness into the parameter extraction process. This invention departs from the previous method of treating the ground plane as a reference plane and excluding it from the final parameter extraction matrix elements. Instead, it treats the ground plane as a signal line and uses the eigenvalue method to calculate the ground plane roughness for multi-conductor lines, thus enabling more accurate calculation of the resistance of multi-conductor lines that takes ground plane roughness into account. Furthermore, it decomposes the resistance of individual signal lines and uses different roughness models for each, significantly improving the accuracy of the resistance matrix extraction (especially at higher frequencies), providing a more favorable and accurate reference for subsequent steps. The entire method is simple to operate while maintaining accuracy. The same method can also be used to consider ground plane roughness in the extraction of the inductance matrix of multi-conductor transmission lines.

[0007] 2. Technical Solution

[0008] To solve the above problems, the present invention adopts the following technical solution.

[0009] A method for extracting parameters of a multi-conductor transmission line that takes into account ground plane roughness includes the following steps:

[0010] S1: Using the ground plane as the signal line, calculate the original frequency-varying resistance matrix [R] according to the number of signal lines;

[0011] S2: The original frequency-varying resistance matrix [R] is diagonalized to obtain the original eigenvalue matrix [∧] and the original eigenvector matrix [S].

[0012] S3: Record the characteristic value corresponding to the ground plane, and multiply the characteristic value by the corresponding surface roughness coefficient;

[0013] S4: The original frequency-varying resistor matrix [R] is reduced to a secondary frequency-varying resistor matrix [R] by removing ground plane signal line values.

[0014] S5: Diagonalize the second-order frequency-varying resistor matrix [R]' to obtain the second-order eigenvalue matrix [∧]' and the second-order eigenvector matrix [S]'.

[0015] S6: Extract each element from the quadratic eigenvalue matrix [∧]', and calculate the resistance of each element to obtain the cubic eigenvalue matrix [∧]”;

[0016] S7: Multiply the left and right sides of the cubic eigenvalue matrix [∧] by [S]' respectively. -1 The final resistance matrix [R] is obtained by combining the quadratic eigenvector matrix [S]'. -1 That is, the inverse matrix of the quadratic eigenvector matrix [S]'.

[0017] Furthermore, the resistance determination in step S6 specifically includes the following steps:

[0018] S61: The resistance of a single signal line is divided into the resistance of the upper surface of the conductor; the resistance of the lower surface of the conductor; the resistance of the upper ground plane; and the resistance of the lower ground plane.

[0019] S62: Establish a circuit model to connect the resistance of the upper surface of the conductor, the resistance of the lower surface of the conductor, the resistance of the upper ground plane, and the resistance of the lower ground plane, and obtain the resistance relationship formula;

[0020] S63: Based on the circuit model, the roughness coefficients of the upper and lower surfaces of the conductor and the upper and lower surfaces of the ground plane are added to the resistance relationship to obtain the final resistance of a single signal line.

[0021] Furthermore, in step S62, the resistance relationship is as follows:

[0022]

[0023] In the formula: R h1 R is the resistance of the upper half of the conductor; h2 R is the resistance of the lower half of the conductor. t1 R is the surface resistance of the trace; t2 The surface resistance of the trace; R r1 R is the resistance of the lower surface of the ground plane located above the trace; r2 The resistance of the upper surface of the ground plane located below the trace;

[0024] The final resistance relationship in step S63 is as follows:

[0025]

[0026] In the formula: R h1,SR The resistance considering the surface roughness of the metal in the upper half of the conductor; R h2,SR Resistance considering the surface roughness of the metal in the lower half of the conductor; K t1 K is the surface roughness coefficient of the trace; r1 K is the surface roughness coefficient of the lower surface of the ground plane above the routing line. t2 K represents the surface roughness coefficient of the trace; r2 It is the roughness coefficient of the upper surface of the ground plane below the trace.

[0027] A computer-readable storage medium storing a computer program that, when executed by a processor, causes the processor to perform the method for extracting parameters of a multi-conductor transmission line considering ground plane roughness as described in any of the preceding claims.

[0028] 3. Beneficial effects

[0029] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0030] (1) In this invention, the ground plane, which is usually used as a reference plane, is used as a signal line. In the case of multiple conductors, the eigenvalue method is used to calculate the roughness of the ground plane. Since there is coupling between multiple conductors, it is difficult to give a quantitative contribution of the ground plane roughness to each conductor. By using the eigenvalue method, the coupling between multiple conductors can be removed first. After adding the contribution of the ground plane roughness to each conductor, the coupling state is restored. This allows for a more accurate calculation of the resistance of the multiple conductors that takes into account the ground plane roughness, making the result more accurate when extracting parameters. The whole method is simple and improves accuracy.

[0031] (2) When calculating the resistance of a single signal line, the surface roughness of the upper and lower surfaces of the single signal line and the ground plane can be considered together. The surface roughness of the upper and lower surfaces of multiple signal lines and the ground plane can be considered together. Different models can be used for the surface roughness of the upper and lower surfaces of the signal line and the ground plane, which greatly improves the accuracy of the resistance matrix extraction and provides a more favorable and accurate reference for subsequent steps. Attached Figure Description

[0032] Figure 1 A comparison chart of differential insertion loss for strip differential lines;

[0033] Figure 2 This is a schematic diagram of the cross-section of the multi-conductor transmission line in this invention;

[0034] Figure 3 A schematic diagram showing the resistance of a stripline divided into four parts;

[0035] Figure 4 for Figure 3 A circuit diagram showing the combination of four resistors.

[0036] Figure 5 for Figure 3 A schematic diagram showing the resistance of each of the four parts multiplied by its respective surface roughness coefficient. Detailed Implementation

[0037] The present invention will now be further described with reference to specific embodiments and accompanying drawings.

[0038] Example 1

[0039] like Figure 1 As shown, Figure 1 This is the insertion loss curve of a pair of differential striplines. It can be seen that the insertion loss is minimal when both the upper and lower surfaces of the signal line and the ground plane are smooth; the insertion loss is relatively large when the upper and lower surfaces of the signal line are rough and the ground plane surface is smooth; and the insertion loss is maximum when both the upper and lower surfaces of the signal line and the ground plane surface are rough. Therefore, it can be seen that the surface roughness of the ground plane is also significant. However, existing methods for considering the surface roughness of the ground plane metal in the resistance and inductance parameters of the transmission line are difficult to incorporate. In the two-dimensional RLGC parameter extraction of multi-conductor transmission lines, the ground plane is usually used as a reference and is not reflected in the final parameter extraction matrix elements, thus introducing a large error to the final result. Therefore, this application proposes a method for considering the ground plane roughness in the parameter extraction of multi-conductor transmission lines, including the following steps:

[0040] S1: Using the ground plane as the signal line, calculate the original frequency-varying resistance matrix [R] according to the number of signal lines. It should be noted that the ground plane is usually the reference plane for the signal lines. For microstrip structures, the ground plane is located above or below the conductor; for stripline structures, the ground plane is located above and below the conductor. The number of signal lines here includes the number of signal lines themselves plus the number of ground planes. For example, if there are three signal lines, the original frequency-varying resistance matrix is ​​a 5x5 matrix. The original frequency-varying resistance matrix [R] can be extracted using various numerical calculation methods, such as the two-dimensional finite element method and the two-dimensional method of moments.

[0041] S2: The original frequency-varying resistance matrix [R] is diagonalized to obtain the original eigenvalue matrix [∧] and the original eigenvector matrix [S]. The eigenvalue matrix and eigenvector calculation are already disclosed in existing technologies and do not involve the core improvement of this application, so the steps will not be described in detail.

[0042] S3: Record the characteristic value corresponding to the ground plane, and multiply the characteristic value by the corresponding surface roughness coefficient;

[0043] S4: The original frequency-converting resistor matrix [R] is reduced to a secondary frequency-converting resistor matrix [R]' by removing the ground plane signal line values. That is, the secondary frequency-converting resistor matrix [R]' corresponds to the signal line itself, excluding the ground plane elements. In other words, if the original frequency-converting resistor matrix is ​​a 5*5 matrix, removing the upper and lower ground plane elements results in a 3*3 matrix.

[0044] S5: Diagonalize the second-order frequency-variable resistor matrix [R]' to obtain the second-order eigenvalue matrix [∧]' and the second-order eigenvector matrix [S]';

[0045] S6: Extract each element from the quadratic eigenvalue matrix [∧]', and calculate the resistance of each element to obtain the cubic eigenvalue matrix [∧]”; specifically, the resistance calculation includes the following steps:

[0046] S61: The resistance of a single signal line (taking a single-ended stripline as an example here) is divided into the resistance of the upper surface of the conductor; the resistance of the lower surface of the conductor; the resistance of the upper ground plane surface; and the resistance of the lower ground plane surface, as shown below. Figure 3 As shown; specifically, Figure 3 In this diagram, 't' represents the thickness of the signal line conductor. The upper and lower parts of the conductor are distinguished by a center line; the area above the center line is the upper part, and the area below the center line is the lower part. This division of the signal line conductor into upper and lower sections facilitates the establishment of... Figure 4 The equivalent circuit shown takes into account the resistance of the ground plane.

[0047] S62: Establish a circuit model to connect the resistances of the upper and lower surfaces of the conductor, the upper ground plane, and the lower ground plane. Figure 4 As shown, the resistance relationship obtained from the model is:

[0048]

[0049] In the formula: R h1 R is the resistance of the upper half of the conductor; h2 R is the resistance of the lower half of the conductor. t1 R is the surface resistance of the trace; t2 The surface resistance of the trace; R r1 R is the resistance of the lower surface of the ground plane located above the trace; r2 The resistance of the upper surface of the ground plane located below the trace;

[0050] S63: Based on the circuit model, the roughness coefficients of the upper and lower surfaces of the conductor and the upper and lower surfaces of the ground plane are added to the resistance relationship. Since the previous steps divided the resistance into four parts for calculation, the roughness of different surfaces can be corrected using different roughness models. The surface roughness of the ground plane can then be considered in the resistance parameter extraction, resulting in the final resistance of a single signal line. Figure 5 As shown, the final resistance relationship is:

[0051]

[0052] In the formula: R h1,SR The resistance considering the surface roughness of the metal in the upper half of the conductor; R h2,SR Resistance considering the surface roughness of the metal in the lower half of the conductor; K t1 K is the surface roughness coefficient of the trace; r1 K is the surface roughness coefficient of the lower surface of the ground plane above the routing line. t2 K represents the surface roughness coefficient of the trace; r2 The roughness coefficient is the surface roughness coefficient of the upper surface of the ground plane below the trace. The selection of the roughness coefficient and the resistance of the corresponding location can be found using existing technologies. For example, the selection of the roughness coefficient can be found in the following reference: Vladimir Dmitriev-Zdorov, etc., ACausal Conductor Roughness Model and its Effect on Transmission Line Characteristics, Signal Integrity Journal, November 2018. The resistance of the corresponding location can be extracted using electromagnetic field numerical calculation methods or calculated using analytical formulas. Furthermore, how to calculate the resistance values ​​of different locations and the specific selection of the roughness coefficient do not involve the core improvements of this application, and therefore will not be elaborated upon further. It is also worth noting that for microstrip line structures, at high frequencies (where the effect of metal surface roughness is mainly at high frequencies), the current mainly flows on the side closer to the ground plane. Therefore, the signal line only needs to consider the surface roughness of the metal surface on the side closer to the ground plane, while the surface roughness on the other side is 1. The remaining processing methods are consistent with those for stripline structures, and therefore will not be repeated in this application. Furthermore, when there is only a single signal line, the total resistance of the single signal line considering the roughness of each metal surface can be directly obtained by using the formula in step S63.

[0053] S7: Multiply the left and right sides of the cubic eigenvalue matrix [∧] by [S]' respectively. -1 The final resistance matrix [R] is obtained by combining the quadratic eigenvector matrix [S]'. -1That is, the inverse matrix of the quadratic eigenvector matrix [S]'.

[0054] Existing two-dimensional parameter extraction methods for transmission lines typically treat the ground plane as an ideal reference surface, failing to account for its surface roughness, thus significantly impacting the accuracy of the results. This application, however, treats the ground plane as a signal line. In the case of multi-conductor transmission lines, it first removes the coupling between conductors and between conductors and the ground plane, then adds the roughness coefficients of the upper and lower surfaces of each conductor and the upper and lower ground planes, and finally restores the coupling between conductors and between conductors and the ground plane. This incorporates the roughness effects of each metal surface into the final resistance calculation, resulting in more accurate results; moreover, the overall process is simple. Furthermore, the surface roughness of the upper and lower surfaces of a single signal line and the ground plane can be considered together, as can the surface roughness of multiple signal lines and the ground plane. Different models can be used for the surface roughness of the upper and lower surfaces of the signal lines and the ground plane, greatly improving the accuracy of the resistance matrix extraction.

[0055] Example 2

[0056] Basically the same as Embodiment 1, this embodiment provides a detailed example, that is, this embodiment uses a stripline with three signal lines (i.e. Figure 4 Taking the example shown below, the specific steps are as follows:

[0057] S1: Using the ground plane as the signal line, calculate the 5x5 frequency-varying resistance matrix [R]:

[0058]

[0059] In the formula: R ii (i = 1, 2, 3, 4, 5) The self-resistance of the i-th conductor;

[0060] R ij (i = 1, 2, 3, 4, 5; j = 1, 2, 3, 4, 5; and i ≠ j) The mutual resistance between the i-th and j-th conductors;

[0061] S2: The original frequency-varying resistance matrix [R] is diagonalized to obtain the original eigenvalue matrix [∧] and the original eigenvector matrix [S]:

[0062]

[0063]

[0064]

[0065] S3: Record the characteristic values ​​∧4 and ∧5 corresponding to the upper and lower ground planes, and multiply the characteristic values ​​∧4 and ∧5 by the corresponding surface roughness coefficient K respectively. r1 and K r2, to get ∧4K r1 and ∧5K r2 ;

[0066] S4: The original frequency-converting resistor matrix [R] is reduced to a 3x3 secondary frequency-converting resistor matrix [R]' by removing ground plane signal line value elements. 3x3 corresponds to three signal lines.

[0067] S5: Diagonalize the second-order frequency-varying resistor matrix [R]' to obtain the second-order eigenvalue matrix [∧]' and the second-order eigenvector matrix [S]':

[0068]

[0069] Wherein, the quadratic eigenvector matrix [S]':

[0070]

[0071] S6: Extract each element from the quadratic eigenvalue matrix [∧]', and calculate the resistance of each element to obtain the cubic eigenvalue matrix [∧]”: Specifically, for each element, use the method of processing surface roughness with a single signal line, that is, substitute the roughness system of the upper and lower surfaces of each trace into the following formula to obtain the cubic eigenvalue matrix [∧]”:

[0072]

[0073] For example, ∧1 corresponds to the resistance of the first trace, and the resistance of its upper and lower surfaces can both be equivalent to 2∧1. Therefore, the total resistance considering the surface roughness of the trace and the roughness of the upper and lower ground planes is:

[0074]

[0075] Where: K t1_1 --Roughness coefficient of the surface of the first trace; K t2_1 --The roughness coefficient of the first trace's surface; similarly, the total resistance of the second trace, considering the roughness of the trace surface and the ground plane, is:

[0076]

[0077] Where: K t1_2 --Roughness coefficient of the surface of the first trace; K t2_2 --Roughness coefficient of the first trace surface;

[0078] Similarly, the total resistance of the third trace, taking into account the surface roughness of the trace and the ground plane, is:

[0079]

[0080] Where: K t1_3 --Roughness coefficient of the surface of the first trace; K t2_3 --The roughness coefficient of the first trace surface; therefore, the cubic eigenvalue matrix [∧]” is:

[0081]

[0082] S7: Multiply the left and right sides of the cubic eigenvalue matrix [∧] by [S]' respectively. -1 The final resistance matrix [R] is obtained by combining the quadratic eigenvector matrix [S]', which is the resistance matrix that takes into account the roughness of the ground plane and the roughness of the upper and lower surfaces of the trace. [S]' -1 That is, the inverse matrix of the quadratic eigenvector matrix [S]':

[0083]

[0084] Example 3

[0085] A computer-readable storage medium stores a computer program that, when executed by a processor, causes the processor to perform the method for extracting multi-conductor transmission line parameters considering ground plane roughness as described in any of the above embodiments. Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDEAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).

[0086] The examples described herein are merely preferred embodiments of the invention and are not intended to limit the concept and scope of the invention. Any modifications and improvements made by those skilled in the art to the technical solutions of the invention without departing from the design concept of the invention should fall within the protection scope of the invention.

Claims

1. A method for extracting parameters of a multi-conductor transmission line considering ground plane roughness, characterized in that: Includes the following steps: S1: Using the ground plane as the signal line, calculate the original frequency-varying resistance matrix [R] according to the number of signal lines; S2: Diagonalize the original frequency-varying resistance matrix [R] to obtain the original eigenvalue matrix. and the original eigenvector matrix [S]; S3: Record the characteristic value corresponding to the ground plane, and multiply the characteristic value by the corresponding surface roughness coefficient; S4: The original frequency-varying resistor matrix [R] is reduced to a second-order frequency-varying resistor matrix [R] by removing the ground plane signal line values. ， S5: Second-order frequency-varying resistor matrix [R] ， Diagonalization is performed to obtain a quadratic eigenvalue matrix. ， And the quadratic eigenvector matrix [S] ， S6: For the quadratic eigenvalue matrix ， Each element in the matrix is ​​extracted, and the resistance of each element is calculated to obtain a cubic eigenvalue matrix. ，， ; S7: Transform the cubic eigenvalue matrix ，， Multiply by each side And the quadratic eigenvector matrix [S] ， This yields the final resistance matrix [R]. ，， The This is the quadratic eigenvector matrix [S]. ， The inverse matrix; The resistance determination in step S6 specifically includes the following steps: S61: The resistance of a single signal line is divided into the resistance of the upper surface of the conductor; the resistance of the lower surface of the conductor; the resistance of the upper ground plane; and the resistance of the lower ground plane. S62: Establish a circuit model to connect the resistance of the upper surface of the conductor, the resistance of the lower surface of the conductor, the resistance of the upper ground plane, and the resistance of the lower ground plane, and obtain the resistance relationship formula; S63: Based on the circuit model, the roughness coefficients of the upper and lower surfaces of the conductor and the upper and lower surfaces of the ground plane are added to the resistance relationship to obtain the final resistance of a single signal line.

2. The method for extracting parameters of a multi-conductor transmission line considering ground plane roughness according to claim 1, characterized in that: In step S62, the resistance relationship is as follows: In the formula: The resistance is the upper half of the conductor; The resistance is the lower half of the conductor. The surface resistance of the trace; For the surface resistance of the trace; The resistance of the lower surface of the ground plane located above the trace; The resistance of the upper surface of the ground plane located below the trace; The final resistance relationship in step S63 is as follows: In the formula: The resistance considering the surface roughness of the metal in the upper half of the conductor; The resistance considering the surface roughness of the metal in the lower half of the conductor; The surface roughness coefficient of the trace; This is the roughness coefficient of the lower surface of the ground plane above the routing line; The surface roughness coefficient of the trace; It is the roughness coefficient of the upper surface of the ground plane below the trace.

3. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores a computer program that, when executed by a processor, causes the processor to perform the method for extracting parameters of a multi-conductor transmission line considering ground plane roughness as described in any one of claims 1-2.

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