# How to Calculate Power Flow in a Parallel Plate Waveguide

### Key Takeaways

• The power carried by an electromagnetic wave is calculated using the Poynting vector.

• Parallel plate waveguides have a simple structure that allows a simple calculation of power flow.

• The same procedure can be applied to any other waveguide on a mode-by-mode basis. Power flow in a parallel plate waveguide can be used to excite a unique emitter or aperture structure

Waveguiding is a useful concept for microwave and optical engineers, and it enables so many modern systems in wireless and optical communication. For the PCB designer that needs to create unique interconnects or antenna structures, parallel plate waveguides are a simple and effective structure for TEM or TM mode transmission. These structures are easy to define in a PCB using polygon pour or plane layers in the PCB stackup, and the mode structure is easy to calculate using sinusoids.

The power flow in a parallel plate waveguide is easily calculated using the Poynting vector from electromagnetic theory. Specifically, the electromagnetic field distribution in the waveguide is used to calculate the Poynting vector for mode. In the TEM case, we get the typical double-humped power distribution across the waveguide cross-section, providing a simple method for matching power flow in a parallel plate waveguide to an RF receiver, waveguide aperture, or other RF circuit.

## Power Flow in a Parallel Plate Waveguide Derivation

A calculation of the power flow in a parallel plate waveguide starts with the eigenmodes of the waveguide structure, which are found using the wave equation. The general structure of the waveguide admits an oscillating solution along the propagation axis (normally taken as the z-direction) with electric field and wavenumber as defined in the equation below. Density between traces in a PCB creates stray and parasitic capacitance

Note that the sinusoidal portion of the solution defines the electric field distribution along the cross section of the waveguide. Here, we have a mode index n, and the structure shown above admits an infinite number of eigenmodes in the particular solution. The magnetic field in the structure can also be found directly, with the above solution for E using Maxwell’s equations. Finally, some waveguide modes may be decaying modes, depending on the frequency of interest we would like to propagate into the structure.

From here, the power flow can be calculated in general using the Poynting vector: Poynting vector definition

The Poynting vector defines the intensity carried by the electromagnetic field. The power carried by the field requires calculating the integral over an arbitrary cross-sectional area within the structure. This means that the total power flow in a parallel plate waveguide is infinite because the cross-sectional area is infinite. Instead of thinking about the total cross-sectional area, we only worry about the total power measured to pass through some specific area at the end of the receiver. Power flow calculated from the Poynting vector

When there is a source term, we still use the Poynting vector to calculate the power. However, the calculation of the electromagnetic field becomes more complicated via one of the following methods:

• In the time domain, using a Green’s function for the particular geometry (Dirichlet problem).
• In the frequency domain, using an inverse Fourier transform.
• Using an eigenfunction expansion in space for the source term, along with direct calculation of the solution coefficients in the time domain or frequency domain.

Guides on these techniques can be found in any partial differential equations textbook. For PCB designers, it’s important to understand the link between the power flow in an interconnect and the power sensed at a downstream receiver.

## Why Worry About Power Flow?

Certain microwave applications involving power transmission in waveguides require understanding the total power being injected into the waveguide as well as the power transmitted to different points along the axis of the waveguide. The former is an emitter design problem, while the latter is a transmission and material selection problem.

Because the field distribution in a parallel plate waveguide is sinusoidal in space, the total power transmitted along the structure will also be sinusoidal in space. The field distribution at the receiver end of the interconnect is used in unique microwave applications such as:

• Aperture design in waveguide flanges: It is common to connect a microwave waveguide to a PCB to collect RF power directly from a board. The RF flange needs to be placed at a location along the waveguide edge corresponding to maximum electromagnetic power carried by the propagating signal, which is calculated with the Poynting vector.

• Slot antenna design: A slot antenna in a parallel plate waveguide can be designed to emit from one or more slots. Just like waveguide flanges and apertures, the slots in a parallel plate waveguide need to be placed in a location with non-zero power to tailor the emitted beam.

• Reception in RF circuits: Some RF circuits will be excited with unique antennas, and the field distribution in a parallel plate waveguide will determine how a receiving RF circuit should be built to collect power in a particular waveguide mode.

These are just a few examples where power flow in a waveguide is used in RF PCB design, particularly in printed RF circuits and external waveguides. When you need to calculate the power flow in a parallel plate waveguide and power transfer into other structures in your PCB layout, you can get accurate results with a 3D electromagnetic field solver that integrates with your PCB layout software. This type of field solver application is the industry standard for advanced calculations involving electromagnetic fields, as no external application is required for analysis.

Cadence’s PCB design and analysis software is ideal for creating RF circuit boards that include unique interconnect structures. You can calculate power flow in a parallel plate waveguide and many other important design metrics in 3D with a complete set of analysis tools. When you use Cadence’s software suite, you’ll also have access to a range of simulation features you can use in signal integrity analysis, giving you everything you need to evaluate your system’s functionality.