COMSOL 6.4 - Impedance Matching of a Lossy Ferrite 3-Port Circulator

Impedance Matching of a Lossy Ferrite 3-Port Circulator

A microwave circulator is a nonreciprocal multiport device. It has the property that a wave incident on port 1 is routed into port 3 yet a wave incident on port 3 is not routed back into port 1 but is instead routed into port 2, and so on. This property of a circulator is used to isolate microwave components from each other, for example, when connecting a transmitter and a receiver to a common antenna. By connecting the transmitter, receiver, and antenna to different ports of a circulator, the transmitted power is routed to the antenna whereas any power received by the antenna goes into the receiver. Circulators typically rely on the use of ferrites, a special type of highly permeable and low-loss magnetic material that is anisotropic for a small RF signal when biased by a much larger static magnetic field. In the example, a three-port circulator is constructed from three rectangular waveguide sections joining at 120° and with a ferrite post inserted at the center of the joint.

Figure 1: The post is magnetized by a static H0 bias field along its axis. The bias field is supplied by external permanent magnets which are not explicitly modeled in this example.

Impedance Matching

An important step in the design of any microwave device is to match its input impedance for a given operating frequency. Impedance matching is equivalent to minimizing the reflections back to the inport. The parameters that need to be determined are the size of the ferrite post and the width of the wider waveguide section surrounding the ferrite. In this tutorial, these are varied in order to minimize the reflectance. The scattering parameters (S-parameters) used as measures of the reflectance and transmittance of the circulator are automatically computed.

The nominal frequency for the design of the device is chosen as 3 GHz. The circulator can be expected to perform reasonably well in a narrow frequency band around 3 GHz, and so a frequency range of 2.8−3.2 GHz is studied. It is desired that the device operates in single mode. Thus a rectangular waveguide cross section of 6.67 cm by 3.33 cm is selected to set the cutoff frequency for the fundamental TE10 mode to 2.25 GHz. The cutoff frequencies for the two nearest higher modes, the TE20 and TE01 modes, are both at 4.5 GHz, leaving a reasonable safety margin.

Model Definition

One of the rectangular ports is excited by the fundamental TE10 mode. At the ports, the boundaries are transparent to the TE10 mode. The following equation applies to the electric field vector E inside the circulator:

where μr denotes the relative permeability tensor, ω is the angular frequency, σ is the conductivity tensor, ε0 is the permittivity of vacuum, εr is the relative permittivity tensor, and k0 is the free space wave number. In this particular model, the conductivity is zero everywhere. Losses in the ferrite are defined using the loss tangent and the complex-valued permeability tensor. The magnetic permeability is of key importance as it is the anisotropy of this parameter that is responsible for the nonreciprocal behavior of the circulator. The material expressions are discussed in the next section for reference.

The Lossy Ferrite Material Model

Complete treatises on the theory of magnetic properties of ferrites can be found in 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 expressions are a linearization for a small-signal analysis around this operating point. Under these assumptions, and including losses, the anisotropic permeability of a ferrite magnetized in the positive z direction is given by

where

and the unique elements of the magnetic susceptibility tensor χ are given by

where

Here μ0 denotes the permeability of free space; ω is the angular frequency of the microwave field; ω0 is the precession resonance frequency (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 (α = 0), the permeability becomes infinite at ω = ω0. In a lossy ferrite (α ≠ 0), this resonance becomes finite and is broadened. The damping factor, α, is related to the line width, ΔH, of the susceptibility curve near the resonance as given by the last expression above. The material data,

Ms = 5.41·104 A/m, εr = 14.5

with an effective loss tangent of 2·10-4 and ΔH = 3.18·103 A/m, are taken for aluminum garnet from Ref. 2. The applied bias field is set to H0 = 7.96·103 A/m. The electron gyromagnetic ratio taken from Ref. 2 is 1.759·1011 C/kg.

Results and Discussion

The default multislice plot shows the electric field norm. The electric field norm gives a good indication of where the main power is flowing and where there are standing waves due to reflections from the impedance mismatch at the center.

Figure 2: The default electric field norm plot shown on xy-plane.

The plot of the S-parameter from the parametric sweep of sc_ferrite indicates a minimum for a scale factor of 0.518.

Figure 3: S-parameter as a function of sc_ferrite parameter.

The plot of the S-parameter from the parametric sweep of sc_chamfer indicates a minimum for a scale factor of about 3.0.

Figure 4: S-parameter as a function of sc_chamfer parameter.

At the center frequency most of the standing waves are gone with the optimized values of sc_ferrite and sc_chamfer.

Figure 5: Electric field norm plot with the optimized sc_ferrite and sc_chamfer values.

This is the frequency response of the final design.

Figure 6: S-parameter as a function of frequency with the optimized sc_ferrite and sc_chamfer values.

From the plot below, it should be possible to identify the model at first glance so it has to display the geometry and some characteristic simulation results.

Figure 7: 3D plot used for model thumbnail generation.

References

1. R.E. Collin, Foundations for Microwave Engineering, 2nd ed., IEEE Press/Wiley-Interscience, 2000.

2. D.M. Pozar, Microwave Engineering, 3rd ed., John Wiley & Sons Inc, 2004.

RF_Module/Ferrimagnetic_Devices/lossy_circulator_3d

Modeling Instructions

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file lossy_circulator_3d_geom.mph.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
mur_xx	1+((omega_0+jaw)omega_m)/((omega_0+jaw)^2-(2pi*freq)^2)		Complex relative magnetic permeability, xx-component
mur_xy	(j2pifreqomega_m)/((omega_0+jaw)^2-(2pi*freq)^2)		Complex relative magnetic permeability, xy-component

Geometry 1

Form Union (fin)

Next add material settings to the model. The lossy ferrite does not fit easily into the material settings so it will be taken care of later. Air is the only material to enter here.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Materials

Air (mat1)

In the Electromagnetic Waves interface, the ferrite is entered as a separate, user-defined equation model referring to the global variables defined above.

Electromagnetic Waves, Frequency Domain (emw)

Wave Equation, Electric 2, Ferrite

In the window, expand the > node.

Right-click and choose .

In the window for , type Wave Equation, Electric 2, Ferrite in the text field.

Select Domain 2 only.

Locate the section. From the list, choose .

From the ε′ list, choose . In the associated text field, type 14.5.

From the tanδ list, choose . In the associated text field, type 0.0002.

Locate the section. From the μr list, choose . From the list, choose .

Specify the μr matrix as


mur_xx	mur_xy	0
-mur_xy	mur_xx	0
0	0	1

One inport for excitation and two outports need to be added next.

Port 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

From the list, choose .

For the first port, wave excitation is by default.

Port 2

In the toolbar, click

and choose .

Select Boundary 18 only.

In the window for , locate the section.

From the list, choose .

Port 3

In the toolbar, click

and choose .

Select Boundary 19 only.

In the window for , locate the section.

From the list, choose .

The mesh needs to resolve the local wavelength and, for lossy domains, the skin depth. The skin depth in the ferrite is large so the main concern is to resolve the local wavelength. This is done by providing maximum mesh sizes per domain. The rule of thumb is to use a maximum element size that is one fifth of the local wavelength (at the maximum frequency) or smaller.

Mesh 1

Free Tetrahedral 1

In the toolbar, click

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Domain 1 only.

Locate the section. Click the button.

Locate the section.

Select the checkbox. In the associated text field, type 1.5e-2.

Size 2

In the window, right-click and choose .

In the window for , locate the section.

Click the button.

Locate the section. From the list, choose .

Select Domain 2 only.

Locate the section.

Select the checkbox. In the associated text field, type 4.5e-3.

In the window, right-click and choose .

The mesh should now look as in the above figure.

The final step in the model set up is to solve it for the nominal frequency and inspect the results for possible modeling errors.

Study 1

Step 1: Frequency Domain

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 3[GHz].

In the toolbar, click

Results

Electric Field (emw)

The default plot shows a slice plot of the electric field norm. It is best viewed from above.

Click the

button in the toolbar.

The electric field norm gives a good indication on where the main power is flowing and where there are standing waves due to reflections from the impedance mismatch at the center. See Figure 2.

The remaining work is to vary the two design parameters in order to minimize reflections at the nominal frequency. To do this, perform parametric sweeps over the design parameters (scale factors).

Study 1

Modify the study in order to vary the scale factor determining the size of the ferrite post. The study type is still Frequency Domain.

The parametric sweep over the scale factor is added as an extension to the frequency domain study.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
sc_ferrite (Geometry scale factor)	range(0.5,3e-3,0.53)

In the toolbar, click

Results

Electric Field (emw) 1

Compare with the plot shown Figure 3. The plot of the S-parameter indicates a minimum for a scale factor of 0.518, so freeze the parameter at this value and add a new study for varying the next scale factor.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
sc_ferrite	0.518	0.518	Geometry scale factor

Results

Global 1

In the window, expand the > node, then click .

In the window for , locate the section.

In the text field, type sc_chamfer.

Study 1

Parametric Sweep

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
sc_chamfer (Geometry scale factor)	range(2.8,0.04,3.2)

In the toolbar, click

See Figure 4. The plot of the S-parameter indicates a minimum for a scale factor of about 3.0, so leave the parameter at this value and add a study for the frequency response.

Add Study

In the toolbar, click