**Waveguides are transmission lines widely used in very high frequency applications as guided propagation devices. Their main advantages are the reduction of losses in the propagation, due to the use of a single conductor and air, instead of using dielectrics as in the coaxial cable, a greater capacity to use high power and a simple building. Their main drawbacks are usually that they are bulky devices, that they cannot operate below their cutoff frequency and that the guide transitions to other technologies (such as coaxial or microstrip) have often losses. However, finite element method (FEM) simulation allows us to study and optimize the transitions that can be built with these devices, getting very good results. In this post we will study the waveguides using an FEM simulator such as HFSS, which is able to analyze tridimensional electromagnetic fields (3D simulation).**

Waveguides are very popular in very high frequency circuits, due to the ease of their building and their low losses. The propagated fields, unlike the coaxial guides, are electric or magnetic transverse (TE or TM fields), so they have a magnetic field component (TE) or electric field (TM) in the propagation direction. These fields are the solutions for the Helmholtz equation under certain boundary conditions

- For the TE modes, E
_{z}(x,y)=0

- For the TM modes, H
_{z}(x,y)=0

and solving these differential equations by separation of variables, and applying the boundary conditions of a rectangular enclosure, where all the walls are electrical walls (conductors, in which the tangential component of the electric field is canceled)

we can obtain a set of solutions for the electromagnetic field inside the guide, starting from the solution obtained for the expressions shown in fig. 1.

Therefore, electromagnetic fields are propagated like propagation modes, called TE_{mn}, for the transverse electric (E_{z}=0), or TM_{mn}, for the transverse magnetic (H_{z}=0). From the propagation constant *K _{c}* is got an expression for the

*,*

**cutoff frequency***f*which is the lowest frequency for propagating fields inside the waveguide, which expression is

_{c},The lowest mode is when m=0, since although the function has extremes for m,n=0, the modes TE_{00} or TM_{00} do not exist. And like a>b, the lowest cutoff frequency of the waveguide is for the mode TE_{10}. That is the mode we are going to analyze using a 3D FEM simulation.

**SIMULATION OF A RECTANGULAR WAVEGUIDE BY THE FINITE ELEMENTS METHOD**

In a 3D simulator it is very easy to model a rectangular waveguide, since it is enough to draw a rectangular prism with the appropriate dimensions a and b. In this case, a=3,10mm and b=1,55mm. The TE_{10} mode start to propagate at 48GHz the next mode, TE_{01}, at 97GHz, then the waveguide is analyzed at 76GHz, frequency in which it will work. Drawing the waveguide in HFSS, it is shown so

The inner rectangular prism is assigned to vacuum, and the side faces are assigned perfect E boundaries. Two wave ports are assigned on the rectangles at -z/2 and +z/2 , using the first propagation mode. The next figure shows the E-field along the waveguide

Analyzing the Scattering parameters from 40 to 90GHz, it is got

where it can be seen that the first mode starts to propagate inside the waveguide at 48,5GHz.

From 97GHz, TE_{01} mode could be propagated too, it does not interest us, then the analysis is done at 76GHz.

**WAVEGUIDE TRANSITIONS**

The most common transitions are from waveguide to coaxial, or from waveguide to microstrip line, to be able to use the propagated energy in another kind of applications. For this, a probe is placed in the direction of the E-field, coupling its energy on the probe. (TE_{01} mode is in Y-axis)

The probe is a quarter wavelength resonant antenna at the desired frequency. In X-axis, E-field maximum value happens at x=a/2, while to find the maximum in Z-axis, the guide is finished in a short circuit. So, E-field is null on the guide wall, being maximum at a quarter guide wavelength which is

and in our case, at 76GHz, λ is 3,95mm and λ_{g}, 5,11mm. Then, the probe length will be 0,99mm and the shortcircuit distance, 2,56mm.

In coaxial transitions, it is enough to put a coax whose internal conductor protrude λ/4 at λ_{g}/4 from the shortcircuit. But in microstrip transitions dielectrics are used as support of the conductor lines, then it should be kept in mindpor the dielectric effect, too.

Our transition can be modeled in HFSS by assigning different materials. The probe is built on Rogers RO3003 substrate, with low dielectric constant and losses, making the transition to microstrip. The lateral faces and the lines are assigned to perfect E boundaries, and form of the substrate, to a RO3003 material. The waveguide inside and the transition cavity is assigned to vacuum. In the extreme face of the transition, a wave port is assigned.

Now, the simulation is done analyzing the fields and S parameters.

and it can be seen how the E-field couples to the probe and the signal is propagated along the microstrip.

Seeing the S parameters, we can see that the least loss coupling happens at 76÷78GHz, our working frequency.

**OTHER DEVICES IN WAVEGUIDES: THE MAGIC TEE**

Among the usual waveguide devices, one of the most popular is the Magic Tee, a special combiner which can be used like a divider, a combiner and a signal adder/subtractor.

Its behavior is very simple: when an EM field is fed by port 2, the signal is divided and in phase by ports 1 and 3. Port 4 is isolated because its E-plane is perpendicular to the port 2 E-plane. But if the EM field is fed by port 4, it is divided into ports 1 and 3 in phase opposition (180deg) while port 2 is now isolated.

Using the FEM simulation to analyze the Magic Tee, and feeding the power through port 2, it is got the next response

and the power is splitted in ports 1 and 3 while port 4 is isolated. Doing the same operation from port 4, it is got

where now port 2 is isolated.

To see the phases, it is used a vector plot of the E-field

where it is seen that the field in ports 1 and 3 has the same direction and therefore they are in phase. Feeding from port 4

in which it is seen that the signals in port 1 and 3 has the same level, but in phase opposition (180deg between them).

FEM simulation allows us to analyze the behavior of the EM field from different points of view, only changing the excitations. For example, feeding a signal in phase by port 2 and 4, both signals will be added in phase at port3 and will be nulled at port 1.

whereas if inverting the phase in port 2 or port 4, the signals will be added at port 1 and will be nulled at port 3.

and the result is a signal adder/subtractor.

**CONCLUSIONS**

The object of this post was the analysis of the electrical behavior of the waveguides using a 3D FEM simulator. The advantage of using these simulators is that they allow to analyze with good precision the EM fields on three-dimensional structures, being the modeling the most important part to rightly define the structure to be studied, since a 3D simulator requires meshing in the structure, and this meshing, as it needs a high number of tetrahedra to achieve good convergence, also tends to need more machine memory and processing capacity.

The structures analyzed, due to their simplicity, have not required long simulation time and relevant processing capacity, but as the models become more complex, the processing capacity increases, it it is needed to achieve a good accuracy.

In subsequent posts, another methods to reduce modeling in complex structures will be analyzed, through the use of planes of symmetry that allow us to divide the structure and reduce meshing considerably..

**REFERENCES**

- Daniel G. Swanson, Jr.,Wolfgang J. R. Hoefer;
*“Microwave Circuit Modeling Using Electromagnetic Field Simulation”*; Artech House, 2003, ISBN 1-58053-308-6 - Paul Wade,
*“Rectangular Waveguide to Coax Transition Design”*, QEX, Nov/Dec 2006