Methods for calculating the wave impedance of transmission lines on printed circuit boards

V.A. Ukhin, V.S. Kukharuk, D.S. Kolomensky, O.V. Smirnova

More and more modern electronic devices contain a high-speed digital part and/or a high-frequency analog part. It is practically impossible to design such products without controlling the wave resistance along the entire signal propagation path.

It is known that the main design unit of any equipment is a printed circuit board, therefore, the calculation of the impedance of transmission lines implemented on their basis is an important and urgent task.

The impedance values ​​to aim for are no secret to the developer. Most often, for single lines it is 50 Ohm, and for differential lines 100 Ohm. In addition, it is easy to find requirements for the impedance value for almost any data transfer standard or interface. Table 1 shows an example for USB 3.0 [1, 2].

Table 1. USB 3.0 Tracing Requirements

Where L – inductance of the transmission line per unit length, WITH – transmission line capacitance per unit length.

Most often, the first method uses the formulas presented in the IPC 2141 and IPC 2251 standards. [5, 6]. Expressions are obtained based on an approximate analytical solution. Some of these formulas are presented below in the text.

Microstrip transmission line (Fig. 1).

where h is the thickness of the dielectric, W is the width of the conductor, t is the thickness of the copper, r is the permittivity, r, eff is the effective permittivity.

The formula in a simpler form:

Striped symmetrical line (Fig. 2).

where h is the thickness of the dielectric, W is the width of the conductor, t is the thickness of the copper, r is the permittivity.

The advantage of the analytical method is that the formulas are freely available and can be easily implemented in software. The disadvantage is the low accuracy of calculation and the lack of expressions for more complex structures, for example, with several dielectrics, coplanar lines. That is, relying on the formulas from the standard, the developer must be prepared for the fact that it is not possible to calculate the wave resistance for all types of transmission lines, and the results obtained may differ greatly from the actual values.

Another method is numerical. In solving engineering problems, this method of finding the necessary parameters is used quite often. At the same time, many specialists, using it through various software tools, rarely think about how it is implemented. The method does not have an explicit formula and many people see it as a kind of black box. However, knowledge of the basic principles of calculation by numerical methods will allow the developer to avoid errors, understand the limits of its application, and influence the accuracy of the result in some cases.

Let's consider the basics of calculating the wave resistance of transmission lines on a printed circuit board using numerical methods. There are quite a lot of numerical methods. The boundary element method is very well suited for solving this problem. It provides high accuracy and does not require significant computing resources.

Before considering this calculation method, let's return to the fundamental formula (1), which is used to calculate the wave resistance. It shows that to find the impedance, it is necessary to determine the inductance and capacitance of the transmission line per unit length. Let's transform expression (1) using the following relationships [3, 4]:

where c is the speed of light.

Let's take into account that the inductance per unit length in a dielectric medium and in air, provided that there are no magnetic materials nearby, will have the same value. Then the formula for calculating the impedance will take the form:

where c is the speed of light, Cair is the capacitance of the conductor per unit length in the air medium, C is the capacitance of the conductor per unit length in a specific medium.

The capacity can be calculated using the formula [3, 4]:

where Q is the charge per unit length, U is the potential difference.

Thus, we find that to calculate the wave resistance, it is necessary to determine the charge in the structure for the environment in which the conductor is located and in the air.

It is the charge or the quantities associated with it that can be calculated using the boundary element method. Let us consider a microstrip transmission line with a mask . The model must specify the size of the calculated region. It must be larger than the structure itself, since the electromagnetic field lines extend far beyond its boundaries. The shape of the region is not important. As an example, Figure 3 shows a rectangular shape.

The parameters X and Y must be minimal, but sufficient so that with their further increase the charge in the system practically does not change. The boundary of the region will have zero potential, and the conductor will have a unit potential. The value of the potential on the conductor does not matter, but for convenience of calculation it is better to take it as one.

After breaking down into boundary elements, the structure looks like this (Fig. 4.5).

Figure 6 shows an example of a grid, and it may appear differently for different element sizes.

Next, the total charge in the system is calculated for the specific environment in which the conductor is located, and in the air. The accuracy of the calculation will depend, as indicated earlier, on the size of the cell, as well as on the number of boundary elements and the quality of their arrangement.

The calculation of impedance can be based not only on the calculation of the total charge, but also on the energy of the electric field or electric and magnetic. In the first case, the energy of the electrostatic field in the system is determined, and through it the capacity per unit length for the structure located in a specific environment, and in the air. The formula is as follows [3, 4]:

In the second case, the capacitance is determined only for a specific environment, and the inductance is additionally calculated.

The inductance is calculated using the expression [3, 4]:

where W is the magnetic field energy stored by the system, I is the current in the conductor, L is the inductance per unit length. It should be noted that the field energy should be calculated using the finite element or finite difference method.

Numerical methods can determine the impedance of almost any structure, take into account nearby conductors, holes, polygons. At the same time, the accuracy of calculations is higher than that of analytical methods. In addition, as was shown in the article, they do not just operate with numbers, but also take into account the physical processes occurring in the system under consideration, which makes it possible to obtain not only a numerical value, but also to conduct more detailed studies. Modern, advanced software uses numerical methods to solve such and similar problems. The domestic SimPCB system from EREMEX uses the boundary element method described above to calculate the primary and secondary parameters of the transmission line.

When choosing a tool for determining the impedance of transmission lines on printed circuit boards, an engineer should definitely briefly familiarize himself not only with reviews of the software, but also with the calculation method implemented in it. This knowledge will help determine the scope of application, ensure accuracy and avoid errors.

Bibliography

1. Universal Serial Bus 3.0 Specification.

2. COM-HPC® Carrier Design Guide Guidelines for Designing COM-HPC® Carrier Boards

3. Printed circuit boards and gigabit electronics units / L.N. Kechiev. – M.: Griffin, 2017. – 424 p.

4. L.N. Kechiev. Handbook on calculating electrical capacitance, inductance and wave resistance in electronic equipment. Engineering manual. – M .: Griffin, 2021 .– 280 p.

5. IPC-2141A Design Guide for High-Speed ​​Controlled Impedance Circuit Boards.

6. IPC-2251 Design Guide for the Packaging of High Speed ​​Electronic Circuits.