A microwave circulator is a multiport device that has the property that a wave incident in port 1 is coupled into port 2 only, a wave incident in port 2 is coupled into port 3 only, and so on. A circulator is used to isolate microwave components to couple a transmitter and a receiver to a common antenna, for example. They typically rely on the use of anisotropic materials, most commonly ferrites. In this example, a three-port circulator is constructed from three rectangular waveguide sections joining at 120° where a ferrite post is inserted at the center of the joint. Figure 1 shows the geometry of the circulator.

To match the junction, identical dielectric tuning elements are inserted into each branch (not shown above). The ferrite post is magnetized by a static H0 bias field along the axis. The bias field is usually supplied by external permanent magnets. Here, the focus is on the modeling of the ferrite and how to minimize reflections at the inport by matching the junction by the proper choice of tuning elements. For a general introduction to the modeling of rectangular waveguide structures, see the model H-Bend Waveguide 3D. Matching the circulator junction involves calculating how well a TE10 wave propagates between ports in the circulator for different materials in the tuning element. This is done by calculating the scattering parameters, or S-parameters, of the structure as a function of the permittivity of the tuning elements for the fundamental TE10 mode. The S-parameters are a measure of the transmittance and reflectance of the circulator. For a theoretical background on S-parameters, see the section S-Parameters and Ports in the RF Module User’s Guide.

This example only includes the TE10 mode of the waveguide. Thus the model can be made in 2D as the fields of the TE10 mode have no variation in the transverse direction. Figure 2 shows the 2D geometry including the dielectric tuning elements.

The dependent variable in this physics interface is the z-component of the electric field E. It obeys the following relation:

where μr denotes the relative permeability, ω the angular frequency, σ the conductivity, ε0 the permittivity of vacuum, εr the relative permittivity, and k0 is the free space wave number. Losses are neglected so the conductivity is zero everywhere. The magnetic permeability is of key importance in this example as it is the anisotropy of this parameter that is responsible for the nonreciprocal behavior of the circulator. For the theory of the magnetic properties of ferrites, see Ref. 1 and Ref. 2. The model assumes that the static magnetic bias field, H0, is much stronger than the alternating magnetic field of the microwaves, so the quoted results are a linearization for a small-signal analysis around this operating point. Further assume that the applied magnetic bias field is strong enough for the ferrite to be in magnetic saturation. Under these assumptions and neglecting losses, the anisotropic permeability of a ferrite magnetized in the positive z direction is given by:

Here μ0 denotes the permeability of free space; ω is the angular frequency of the microwave field; ω0 is the precession frequency or Larmor frequency of a spinning electron in the applied magnetic bias field, H0; ωm is the electron Larmor frequency at the saturation magnetization of the ferrite, Ms; and γ is the gyromagnetic ratio of the electron. For a lossless ferrite, the permeability clearly becomes unbounded at ω = ω0. In a real ferrite, this resonance becomes finite and is broadened due to losses. For complete expressions including losses, see Ref. 1 and Ref. 2. In this analysis the operating frequency is chosen sufficiently off from the Larmor frequency to avoid the singularity. The material data, Ms = 2.39·105 A/m and εr = 12.9, are taken for magnesium ferrite from Ref. 2. The applied bias field is set to H0 = 2.72·105 A/m, which is well above saturation. The electron gyromagnetic ratio is set to 1.759·1011 C/kg. Finally, the model uses an operating frequency of 10 GHz. This is well above the cutoff for the TE10 mode, which for a waveguide cross section of 2 cm by 1 cm is at about 7.5 GHz. At the ports, matched port boundary conditions make the boundaries transparent to the wave.

The S11 parameter as a function of the relative permittivity of the matching elements, eps_r, is shown in Figure 3. The S11 parameter corresponds to the reflection coefficient at port 1. Thus matching the junction is equivalent to minimizing the magnitude of S11.

By choosing eps_r to about 1.29, you obtain a reflection coefficient of about −35 dB, which is a good value for a circulator design. Judging from the absence of standing wave patterns in the magnitude plot of the electric field and by looking at the direction of the microwave energy flow in the result plot below, it is clear that the circulator behaves as desired.

From the File menu, choose New.

In the New window, click  Model Wizard.

The presence of standing waves in the input arm is clearly visible for this value (1.5) of the relative permittivity in the matching elements. To study how the reflections depend on the value of this parameter, plot the reflection coefficient S11 as a function of eps_r.