Wave Impedance and How It Affects Interconnects

Key Takeaways

• Learn what wave impedance is as well as its corresponding equations.

• The impedance of a PCB trace is the impedance of the equivalent waveguide formed by the trace, known as the wave impedance.

• The wave impedance is a general relation that can be extracted from your PCB interconnects and field solvers to better understand power transfer, radiation from a real PCB, and impedance matching

This copper tube is actually an electromagnetic waveguide for high-frequency analog signals

Every interconnect in an electronic system has some impedance, but what is the physical reason for this impedance? This is not a contrived notion—it reflects how signals really travel across an interconnect and through real media. Signals travel through PCBs as electromagnetic waves, and the propagation behavior is determined in part by the wave impedance. This term is normally used to reference waveguides, but it is a universal term that applies to any PCB trace and is used to derive analytical expressions for trace impedance.

Since the typical impedance expressions and wave impedance are related, how are they linked mathematically? The wave impedance is related to the interconnect geometry and there are a number of ways to determine the impedance in a design. Keep reading to learn more about wave impedance and how it relates to the real structure of a PCB.

What Is Wave Impedance?

Wave impedance defines the impedance experienced by an electromagnetic signal as it travels through free space (i.e., vacuum) or some other medium. An electromagnetic signal consists of electric and magnetic fields that traverse perpendicularly to each other. As the electromagnetic signal traverses, the intensity of the electric and magnetic fields change due to the medium’s magnetic permeability and electric permittivity. 

The impedance seen by a propagating wave is the reason the magnetic field is so much weaker than the electric field, including the impedance of a vacuum. For any physical system, whether linear or nonlinear, there are some simple definitions for wave impedance that are used to derive the impedance of more complex physical systems.

Wave Impedance Equations

Wave impedance has a very simple definition that is normally discussed in terms of TEM waves traveling in free space. The typical definition of wave impedance is in terms of the ratio of the electric to magnetic field strength (in phasor representation):

Wave impedance definition in terms of electric and magnetic field strength

In this definition, the wave impedance is defined in terms of the location along the propagation direction (written as the z-direction above). The electric field (E), is measured in V/m while the magnetic field (H) has a unit of A/m. Thus, the unit of wave impedance is in Ohm. 

Assuming the electric and magnetic fields are in-phase and traveling in a dispersionless, lossless medium, the wave impedance will be a constant that is independent of frequency. The wave impedance of a vacuum is approximately 377 Ohms; theoretical physicists like to make jokes about the “impedance of nothing” when discussing wave impedance for this reason. 

In general, however, the wave impedance may be a complex function of frequency. And so, the value of wave impedance can be represented by: 

Note that ⍵ is the angular frequency of the wave, 𝝁 is magnetic permeability, ε is electric permitivity, and σ is the conductivity of the medium.

In an ideal dielectric, the wave impedance equation can be simplified to:

Besides the impedance value, wave impedance is also characterized by the phase difference between the electric and magnetic fields. The phase difference is given in terms of the atan2 function: 

Where: 

Relationship to Transmission Line Impedance

This definition is the basis for deriving analytical expressions for waveguide impedance in various geometries, particularly in transmission lines. In general, for a wave traveling in a dielectric with dispersion and losses, recall that we have the following definition for wave impedance in terms of material constants for the dielectric:

Wave impedance in terms of material constants for the dielectric

This equation is applicable for any interconnect, waveguide structure, or unbounded dielectric. We can also use this to get back to the characteristic impedance of a transmission line. If we multiply the numerator and denominator by a cross-sectional area that bounds the electromagnetic field, we get back to the equation for transmission line impedance from circuit theory:

Rewriting wave impedance in terms of circuit theory parameters

This nicely demonstrates the correspondence between wave impedance and transmission line/interconnect impedance—they really are the same. This correspondence between wave impedance and transmission line impedance is used to derive analytical expressions for transmission lines.

Deriving Transmission Line Impedance from Wave Impedance

If you want to get an analytical expression for transmission line impedance, such as the popular and highly accurate impedance equations found in Waddel’s textbook, you need to derive the wave impedance for the transmission line structure. For some transmission lines, like a rectangular waveguide, this is very simple and is often given as a homework problem. For more complex structures, a more sophisticated method is used.

Conformal Mapping

Conformal mapping is a mathematical technique used to calculate solutions to the Laplace equation in complicated planar cross sections. It relies on a functional transformation of the system geometry between the real geometry and an idealized geometry, which requires some intuition to see clearly. For any waveguide structure, you can find a coordinate transformation that reduces a complex geometry to one where TEM waves propagate with some reduced free-space wave impedance.

By transforming into an equivalent system with effective material constants, wave impedance is easier to solve using Maxwell’s equations

This is a high-level view of the procedure used to derive the impedance of microstrips, striplines, and other transmission line structures. By solving the Laplace equation and calculating the impedance in the transformed system, the impedance in the original geometry can be determined with an inverse transformation. This is a complex procedure to do by hand, which is one reason today’s best PCB layout applications include integrated field solvers for impedance calculations.

Use a Field Solver

For complex interconnects and printed structures on a PCB, you can also use a field solver integrated into your PCB layout tools. A simple method of moments calculation will give you the impedance of a transmission line section using method of moments with short computation time. If you want to see the electromagnetic field distribution around the line, you need a full-wave EM field solver. The best field solver applications will take data directly from your PCB layout and return results in 2D or 3D; the example below shows results from a method of moments simulation for DDR4 lines in a PCB layout displayed as a heat map.

Results from a method of moments simulation for DDR4 lines in a PCB layout

Once you have your field solver results, simply calculate the ratio of electric to magnetic field to get the wave impedance. The vicinity of the interconnect is equal to the interconnect’s characteristic impedance. These same field solvers can calculate other important quantities in high speed/high-frequency designs, such as crosstalk and return paths.

You don’t need to calculate wave impedance by hand to get transmission line impedance, you just need to use a quality set of PCB design and analysis software. The integrated field solver tools in Allegro PCB Editor from Cadence gives you the layout, routing, simulation, and evaluation features you need to create cutting-edge technology and prepare your designs for production. You can also use Cadence’s field solver utilities to simulate all aspects of your design directly from your PCB layout data.

If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.