Problem Description

Solution

You can perform modal analysis of various transmission lines and waveguides by using the Perpendicular Waves application mode in the RF Module. This application mode works on 2D cross-sections of transmission lines, assuming that they are uniform in the direction of signal propagation. Cross-sections can have inhomogeneous distributions of permittivity, permeability and conductivity. In the direction perpendicular to the cross-section, electric and magnetic fields are assumed to vary only due to the exponential factor exp(-j*beta*z). The quantity beta is called the propagation constant.

The Perpendicular Waves application mode can perform two types of analysis:

1. Eigenfrequency analysis - Eigenmodes and eigenfrequencies (omega) are found for a specified propagation constant (beta). In case of resistive, dielectric, magnetic or radiative losses, the returned eigenfrequencies are complex. The imaginary part of an eigenfrequency is the damping rate of the corresponding eigenmode (typically measured in 1/s).

2. Eigenmode analysis - Propagation constants are found for a specified frequency. When losses are present, the returned propagation constants are complex-valued. The imaginary part of a propagation constant is the attenuation constant (measured in 1/m in SI), the inverse of which gives the attenuation length scale. See the model of a Lossy Circular Waveguide for an example.

In addition, you can choose the polarization of the modes to be found: TE waves analysis assumes Ez=0, TM waves analysis implies Hz=0, and Hybrid-mode waves analysis does not neglect any longitudinal or transverse components of E and H fields. Use Hybrid-mode waves analysis if you are not sure about the polarization properties of the eigenmodes of interest. In particular, Hybrid-mode waves analysis is the best choice for TEM- and quasi-TEM modes of coaxial cables and microstrip lines.

The example model microstrip_port.mph shows a microstrip port simulation in COMSOL Multiphysics with the computed propagation constant and field distribution.

The field distribution of a microstrip port. The surface plot shows the z component of the electric field and the arrow plot shows the magnetic field. Click on image to enlarge.

Hybrid-mode waves analysis can also be used for modeling optical transmission lines including step-index optical fibers and photonic waveguides; see, for example, these models:

• Step Index Fiber
• Stress-Optical Effects in Waveguide

Measuring characteristic impedance of transmission lines

Transmission lines are often characterized by two functions of frequency: propagation constant, β, and characteristic impedance, Z_ch. When using the Perpendicular Waves application mode, you have direct access to the propagation constant (variable beta), which is either computed or specified by the user. While it is impossible to define the characteristic impedance in general, it is often possible to define Z_ch for TEM and quasi-TEM modes. For some transmission lines, in particular, the microstrip line shown above, Z_ch can be defined as the ratio of voltage (V) between two conducting "terminals" of the transmission line, and the out-of-plane electric current (Iz) flowing through either of the terminals:

Z_ch = V / Iz

The voltage can be calculated as a line integral of electric field. The path of integration for voltage should begin on one terminal and end on the other terminal. The most convenient way to perform such integration is to define a Boundary Integration Variable. You might have to add a line for this integration in the Draw mode, or re-use existing internal boundaries. Cartesian components of transverse electric field are available as variables tEx tEy (when solving for electric field) or Ex Ey (when solving for magnetic field).

The out-of-plane current can be calculated as a line integral of magnetic field. The path of integration for current should be a closed contour containing one of the terminals. You might have to define such a closed contour in the Draw mode (as a rectangle or circle, for instance), or re-use existing internal boundaries. The transverse magnetic field components are available as variables Hx Hy (when solving for electric field) or tHx tHy (when solving for magnetic field).

If both V and Iz are defined as Boundary Integration Variables, you can define a Global Expression Z_char as V/Iz.

Because there are several sign factors related to the choice of integration contours and their orientations, the easiest way to ensure the proper sign of Z_char is to require that the real part of Z_char is positive. Introduce the overall sign correction factor if this is not the case.

The file microstrip_impedance_v35.mph is a modified version of microstrip_port.mph that defines Boundary Integration Variables V and Iz, as well as a Global Expression Z_char as described above.

Modeling transmission lines in 3D

For modeling transmission lines of finite length, or when using a transmission line to drive a time-harmonic simulation, you must use the 3D Electromagnetic Waves application mode.

To drive a 3D time-harmonic simulation with a signal from a particular transmission line, you can use the Port or the Lumped Port boundary condition.

The Lumped Port boundary condition simulates TEM and quasi-TEM modes of transmission lines with cross-sections of sub-wavelength size. This boundary condition lets you set specific excitation voltage on the port. It works best with rectangular waveguide ports and terminals of coaxial cables. The use of a Lumped Port is illustrated in the following models:

• Absorbed Radiation (SAR value) in a Human Head
• Microstrip on a Printed Circuit Board
• Electromagnetic and Structural Analysis of a Microwave Filter on a PCB

The Circuit Port boundary condition is a version of the Lumped Port that lets you couple your model to a SPICE circuit. See the model Conical Antenna.

The Port boundary condition is very general and allows specification of arbitrary field profiles and dispersion relations. Several predefined Port types are available: Rectangular, Circular, and Coaxial. The User-defined Port specification can be used for other transmission lines, if their field profiles are known analytically. One example of using a User-defined Port would be to excite a plane wave in vacuum on a rectangular boundary of your simulation domain. You can also use such port to create a Gaussian beam. For an example of using a User-defined Port, see the model Microwave Oven.

Use Numeric Port for the most general case, when the modes of transmission line simulated as a port are not known analytically. Numeric Port should be coupled with a Boundary Mode Analysis, which has functionality similar to that of 2D Perpendicular Waves application mode. One instance of the application mode called Boundary Mode Analysis must be added for each Numeric Port. For examples of using a Numeric Port with Boundary Mode Analysis, see the models:

• Waveguide Adapter
• Microstrip on a Printed Circuit Board

Related Files

	microstrip_port.mph	784 KB
	microstrip_impedance_v35.mph	2.5 MB

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