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Microwaves, a.a. 2019/20 Prof. Luca Perregrini Non-reciprocal components, pag. 1

MICROWAVES

Università di Pavia, Facoltà di [email protected]

http://microwave.unipv.it/perregrini/

Prof. Luca Perregrini

Master Degree (LM) in Electronic Engineering

NON-RECIPROCAL COMPONENTS


SUMMARY

Chapter 9


SUMMARY

• Anisotropic materials

• Propagation of plane waves in ferrites

• Faraday rotation

• Ferrite-loaded waveguide resonance isolator

• Field displacement isolator

• Ferrite latching phase shifter

• Reggia-Spencer reciprocal phase shifter

• The gyrator

• Ferrite circulators


MOTIVATION

Non-reciprocal components can be designed using anisotropic materials.This is a useful property in some applications, when unidirectional devicesare needed (e.g., isolators, circulators).Sometime the non-reciprocity is not strictly required, but just aconsequence of the adopted materials (e.g., ferrite phase shifters).Active devices (discussed later on) also show non-reciprocity, but it is anancillary property. This presentation will be focused on devices based onanisotropic materials.


ANISOTROPIC MATERIALS

The more common anisotropic materials are ferrimagnetic compounds, alsoknown as ferrites, such as yttrium iron garnet (YIG) and materialscomposed of iron oxides and various other elements such as aluminum,cobalt, manganese, and nickel (metal+Fe2O3).They are, in general, good dielectrics (𝜖𝜖𝑟𝑟 = 8 ÷ 15, tan𝛿𝛿 = 10−3 ÷ 10−4).


ANISOTROPIC MATERIALS

A biasing DC magnetic field aligns the magnetic dipoles in the ferritematerial to produce a net (nonzero) magnetic dipole moment, and causesthe magnetic dipoles to precess at a frequency controlled by the strength ofthe bias field.A microwave signal circularly polarized in the samedirection as this precession will interact strongly with thedipole moments, while an oppositely polarized field willinteract less strongly. A microwave signal will propagatethrough a magnetically biased ferrite differently in differentdirections.This allows directional devices such as isolators,circulators, and gyrators.The interaction with an applied microwave signal can becontrolled by adjusting the strength of the bias field. Thiseffect leads to a variety of control devices such as phaseshifters, switches, and tunable resonators and filters.


ANISOTROPIC MATERIALS PERMEABILITY

From the basic physic of the material, assuming the bias magnetic fieldalong 𝑧𝑧 ( �𝑯𝑯0 = �𝒛𝒛𝐻𝐻0 ) and a small signal field �𝑯𝑯 ( �𝑯𝑯 ≪ 𝐻𝐻0 ) it can bedemonstrated the constitutive relation are

where

and

(Larmor, or precession, frequency)𝜔𝜔0 = 𝜇𝜇0𝑞𝑞𝑚𝑚𝑒𝑒

𝐻𝐻0

𝜔𝜔𝑚𝑚 = 𝜇𝜇0𝑞𝑞𝑚𝑚𝑒𝑒

𝑀𝑀𝑠𝑠 (𝑀𝑀𝑠𝑠 is the saturation magnetization)



If the bias magnetic field is along 𝑥𝑥 (�𝑯𝑯0 = �𝒙𝒙𝐻𝐻0) or along 𝑦𝑦 (�𝑯𝑯0 = �𝒚𝒚𝐻𝐻0) thepermeability tensor become

(�𝑯𝑯0 = �𝒙𝒙𝐻𝐻0)

(�𝑯𝑯0 = �𝒚𝒚𝐻𝐻0)



The interaction of a circularly polarized wave with a magnetically biasedferrite depends on the sense of the polarization (RHCP or LHCP):

The bias field sets up a preferential precession direction coinciding with theforced precession for an RHCP wave, but opposite to that of an LHCP.

Right Hand Circular Polarization (RHCP) Left Hand Circular Polarization (LHCP)



Since we considered a lossless material, the terms 𝜇𝜇 and 𝜅𝜅 of thepermeability tensor become infinite when 𝜔𝜔 → 𝜔𝜔0

Loss can be accounted for by making the resonant frequency complex:

obtaining


DEMAGNETIZATION FACTORS

The bias magnetic field 𝐻𝐻0 may not be identical to the external field 𝐻𝐻𝑎𝑎applied to the ferrite, because of the boundary conditions at the surface.For example:

The bias magnetic field is perpendicularto the ferrite → continuity of 𝐵𝐵:

The bias magnetic field is parallel tothe ferrite → continuity of 𝐻𝐻:


DEMAGNETIZATION FACTORS

In general, the internal field is affected by the shape of the ferrite sampleand its orientation with respect to the external field �𝑯𝑯𝑒𝑒 , and can beexpressed as

where 𝑁𝑁 = 𝑁𝑁𝑥𝑥,𝑁𝑁𝑦𝑦, 𝑁𝑁𝑧𝑧 are thedemagnetization factor, whichdepend on the shape and thedirection of the applied field.

Demagnetization factors for simple shapes


PLANE WAVES PROPAGATION || TO �𝑯𝑯0 = �𝒛𝒛𝐻𝐻0Let’s consider a ferrite DC magnetized along 𝑧𝑧 (�𝑯𝑯0 = �𝒛𝒛𝐻𝐻0) filling the wholespace. We want to verify if and how a plane wave can propagate in the 𝑧𝑧direction.Starting from the Maxwell’s equations

substituting the field of plane wave propagating along 𝑧𝑧

and taking into account that the derivatives 𝜕𝜕/𝜕𝜕𝑥𝑥 and 𝜕𝜕/𝜕𝜕𝑦𝑦 are zero, weobtain the following equations


PLANE WAVES PROPAGATION || TO �𝑯𝑯0 = �𝒛𝒛𝐻𝐻0A non-trivial solution exist only if the determinant of the system is zero:

which leads to two possible solutions for the propagation constant, thecorresponding fields, and the wave admittances:

𝛽𝛽− = 𝜔𝜔 𝜖𝜖(𝜇𝜇 − 𝜅𝜅)𝛽𝛽+ = 𝜔𝜔 𝜖𝜖(𝜇𝜇 + 𝜅𝜅)

Right hand circular polarization (RHCP) Left hand circular polarization (LHCP)


PLANE WAVES PROPAGATION: FARADAY ROTATION

If we consider a linearly polarized electric field at 𝑧𝑧 = 0, we can decomposeas an RHCP and an LHCP wave:

the field at a given distance 𝑧𝑧 is

The field is still linearly polarized, but the direction of the polarization isrotated counterclockwise of an angle

This effect is called Faraday rotation.Reversing the bias (direction of 𝐻𝐻0) reverse the direction of rotation.


PLANE WAVES PROPAGATION || TO �𝑯𝑯0 = �𝒛𝒛𝐻𝐻0Introducing the losses by perturbation, if can be shown that close to 𝜔𝜔0 theRHCP polarization is strongly attenuated, with a significant stopband from𝜔𝜔0 to 𝜔𝜔0 + 𝜔𝜔𝑚𝑚:


FERRITE-LOADED WAVEGUIDE ISOLATOR

The scattering matrix of a perfect isolator is

We demonstrated that a circular polarized plane wave has a differentpropagation coefficient depending on the rotation direction with respect tothe bias magnetic field direction. Moreover, one of the two polarization canbe strongly attenuated.

This allow to realizean isolator byinserting a ferriteslab in a rectangularwaveguide workingfrom 𝜔𝜔0 to 𝜔𝜔0 + 𝜔𝜔𝑚𝑚.



In fact, the magnetic field of a TEm0 is given by

and it is circularly at a certain position𝑥𝑥 = 𝑐𝑐 so that

𝑧𝑧

𝑥𝑥

Strongly attenuated

Fully transmitted

In – 𝑧𝑧 direction the wave isstrongy attenuated from 𝜔𝜔0 to𝜔𝜔0 + 𝜔𝜔𝑚𝑚

A strongy bias magnetic field isrequired to obtain a strongattenuation.



Example:


FIELD DISPLACEMENT ISOLATOR

Far from the resonance the attenuation of the ferrite is not so significant.However, the effect on the field distribution within the waveguide for theforward and reverse wave is significantly different, and the forward wavecan be made to vanish at the side of the ferrite slab at 𝑥𝑥 = 𝑐𝑐 + 𝑡𝑡.

A thin resistive sheet in thisposition do not affect theforward wave, and dumpthe reverse wave.High values of isolation witha relatively compact devicecan be obtained with ~10%bandwidth.Much smaller bias field overthe resonance isolator sinceit operates well belowgyromagnetic resonance.


FERRITE LATCHING PHASE SHIFTER

Consists of a toroidal ferrite core symmetrically located in the waveguidewith a bias wire (current) passing through its center. When the ferrite ismagnetized, the magnetization of the sidewalls of the toroid will beoppositely directed and perpendicular to the plane of circular polarization ofthe RF fields. Since the sense of circular polarization is also opposite onopposite sides of the waveguide, a strong interaction between the RF fieldsand the ferrite can be obtained.The presence of the ferriteperturbs the waveguide fields (thefields tend to concentrate in theferrite), so the circular polarizationpoint happens in a differentposition with respect to the emptyguide.


FERRITE LATCHING PHASE SHIFTER

In principle, such a geometry can be used to provide a continuouslyvariable (analog) phase shift by varying the bias current. However, a moreuseful technique employs the magnetic hysteresis of the ferrite to provide aphase shift that can be switched between two values (digital).The amount of differential phase shift between these two states iscontrolled by the length of the ferrite toroid. Several sections havingindividual bias lines and decreasing lengths are used in series to givebinary differential phase shifts of 180◦, 90◦, 45◦, etc.,


FARADAY ROTATION PHASE SHIFTER

The dielectric plates introduces a phase shift of one component withrespect to the other, thus creating the circular polarization.Then the Faraday rotation occurs inthe magnetized ferrite rod.


REGGIA-SPENCER RECIPROCAL PHASE SHIFTER

Popular reciprocal phase shifter. In either rectangular or circular waveguideform, a longitudinally biased ferrite rod is centered in the guide. When thediameter of the rod is greater than a certain critical size, the fields becometightly bound to the ferrite and are circularly polarized.

A large reciprocal phaseshift can be obtained overrelatively short lengths,although the phase shift israther frequency sensitive.


THE GYRATOR

An important canonical nonreciprocal component is the gyrator, which is atwo-port device having a 180◦ differential phase shift. The scattering matrixfor an ideal gyrator (lossless, matched, and nonreciprocal) is

In combination withreciprocal dividersand couplers canlead to usefulequivalent circuitssuch as isolators andcirculators.

Bias can be provided with a permanent magnet, making the gyrator apassive device.


FERRITE CIRCULATORS

A three-port microwave device that can be lossless and matched at allports; by using the unitary properties of the scattering matrix we were ableto show that such a device must be nonreciprocal. The scattering matrix foran ideal circulator thus has the following form

Can be designed in waveguideand in microstrip or stripline.The theory is quite complex andcan be found on the Pozar’sbook (under some simplifyinghypotheses).


FERRITE CIRCULATORS

Examples of waveguide circulators:


FERRITE CIRCULATORS

Examples of microstrip/stripline circulators:

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