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What Is the Inductance of a Trace Over a Ground Plane?

Key Takeaways

  • The inductance of a PCB trace determines the strength of any crosstalk it will receive.

  • One challenge in designing PCB interconnects is maintaining system impedance while reducing crosstalk, which requires reducing trace inductance.

  • Designers need numerical tools and the correct analytical formulas to calculate the inductance of their PCB traces.

Ferrite bead inductor

Traces form loops of conductors that have some inductance

The best PCB design and analysis utilities do not examine impedance, noise, and other effects in terms of circuit models. However, circuit models are very useful for describing the electrical behavior of complex features in a PCB layout for many reasons. Circuit models built from the fundamental passives (RLC circuits) are very useful for describing a range of phenomena, including EMI and noise susceptibility through crosstalk.

All crosstalk is coupled via two mechanisms: capacitively and inductively. This means, if you want to reduce crosstalk between interconnects, then you need to know their individual inductance values. There are some simple methods you can use to calculate the inductance of a trace over a ground plane, such as a microstrip or stripline. More advanced (but more accurate) methods require the use of a numerical technique, especially when we consider signal losses in the system.

Formulas for Calculating Inductance of a Trace Over a Ground Plane

The method for calculating the inductance of a microstrip or stripline trace starts with calculating the trace characteristic impedance and propagation delay for signals traveling on the trace. These two quantities are directly related to the inductance and capacitance of a trace routed over a ground plane. The relationship between the trace’s characteristic impedance, propagation delay, inductance, capacitance, and losses can be determined from the Telegrapher’s equation.

The following two equations are used together to calculate the inductance and capacitance of a lossless transmission line. Simply multiply the equations together to get the value of the inductance:

Telegrapher’s equation

The equation for the inductance of a trace over a ground plane

It’s important to note that this only applies to a certain type of transmission line, one which does not have any dielectric losses, radiation losses, or skin-effect losses. The above model is still useful, as it applies to any transmission line or quasi-TEM waveguide, including:

All of these PCB interconnect styles are placed over or between some ground planes, and their impedance can be measured or calculated. As long as the characteristic impedance Z0 and dielectric constant of the interconnect are known, the inductance can be determined from the above equation as long as losses are ignored.

The Reality: Digital Signals Are Broadband

The unfortunate reality is that the above method is objectively wrong: real digital signals are broadband, and dispersion in the PCB substrate causes the propagation delay and impedance to be functions of frequency, even in the high frequency limit where the trace’s DC resistance can be ignored. There are also losses in the copper due to the skin effect and copper roughness. Therefore, you can’t just choose an arbitrary frequency and calculate the impedance and inductance. A good discussion of this can be found in this recent IEEE EPEPS article.

Where to Get Z0

If you want to work only at a single frequency, and you ignore losses, you can still get a function for Z0 as a function of geometry from a few different sources:

  • The IPC-2142 standard includes empirical formulas for the impedance of striplines and microstrips.

  • Textbooks will contain standard formulas determined using conformal mapping. Perhaps the most comprehensive source of trace impedance equations is Brian C. Waddell’s Transmission Line Design Handbook.

You can then use the impedance you calculate to get the inductance. Note that, for traces on the surface layer of a PCB, the dielectric constant will be an “effective” dielectric constant. This is normally given with the formula you’re using to calculate the impedance. Intuitively, we should already see that the loop formed by the trace and its ground plane is larger when the trace is farther from the plane, as is shown in the microstrip trace below. Changing the width of the trace also affects the inductance.

 Inductance of a trace over a ground plane

The values of h and w determine the loop inductance of the trace above a ground plane

Unfortunately, the formulas in these resources consider isolated transmission lines and they do not account for parasitics. Because the impedance and inductance of a trace over a ground plane depends on the geometry of the trace and the surrounding parasitics, we need more accurate methods to determine impedance and inductance.

Field Solvers for Impedance and Inductance Calculations

One way to get accurate impedance and inductance calculations is to use a field solver. These applications can accurately account for trace geometry and surrounding parasitics without using a circuit model. Today’s advanced ECAD applications will include a 3D field solver utility for fundamental transmission line calculations and complex multiphysics problems. For impedance calculations, the results are normally displayed as a heat map; the results for some example DDR3 traces can be seen in the image below.

Impedance calculation results

Field solver utilities can determine the impedance of a trace along its length. You can then use this value with the dielectric constant to determine the inductance of a trace over a ground plane

Not all field solvers can account for copper roughness up to high GHz frequencies, which is very important for technologies like PAM-4 interconnects, microwave photonics, automotive/drone radar, and any other area involving very high frequencies. However, the most advanced products are expanding their capabilities and will use standard copper roughness models to calculate the skin-effect impedance at high frequencies.

When you need to know the inductance of a trace over a ground plane, make sure to use PCB design and analysis software with an integrated 3D EM field solver and a complete set of CAD tools. Cadence provides powerful software that helps automate many important tasks in systems analysis, including a suite of pre-layout and post-layout simulation features to evaluate your system. 

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