Editing Circulator (section)

==Theory of operation==
Microwave circulators rely on the [[anisotropic]] and ''non-[[Reciprocity (electrical networks)|reciprocal]]'' properties of magnetized microwave ferrite material.<ref name="Modern Ferrites, Volume 2">{{Cite book|title=Modern Ferrites, Volume 2: Emerging Technologies and Applications|first=Vincent G.|last=Harris|date= 2023|publisher=Wiley-IEEE Press|isbn=978-1-394-15613-9}}</ref>  Microwave electromagnetic waves propagating in magnetized ferrite interact with electron [[Spin (physics)|spins]] in the ferrite and are consequently influenced by the microwave [[magnetic permeability]] of the ferrite.  This permeability is mathematically described by a linear vector operator, also known as a [[tensor]].  In the case of magnetized ferrite, the permeability tensor is the [[Polder tensor]].  The permeability is a function of the direction of microwave propagation relative to the direction of static magnetization of the ferrite material.  Hence, microwave signals propagating in different directions in the ferrite experience different magnetic permeabilities.

[[File:Faraday rotation.ogg|thumb|400px|E-field vector plot of an [[elliptically polarized]] electromagnetic wave propagating in a magnetized ferrite cylinder.  The static magnetic field is oriented parallel to the cylinder axis.  This is known as [[Faraday Rotation]].]]

In the [[CGS system]], the [[Polder tensor]]<ref name="Polder">{{cite journal|title=On the Theory of Ferromagnetic Resonance |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=1949 |pages= 99–115 |volume=40 |issue=300 |doi= 10.1080/14786444908561215 |first= D |last= Polder}}</ref> is

:<math>B = \begin{bmatrix} \mu & j \kappa & 0 \\ -j \kappa & \mu & 0 \\ 0 & 0 & 1 \end{bmatrix} H</math>

where (neglecting damping)

:<math>\mu = 1 + \frac{\omega_0 \omega_m}{\omega_0^2 - \omega^2}  </math>

:<math>\kappa = \frac{\omega \omega_m}{{\omega_0}^2 - \omega^2}</math>

:<math>\omega_0 = \gamma H_0 \ </math>
:<math>\omega_m = \gamma M \ </math>

<math>\gamma = 1.40 \cdot g \,\, </math>  MHz / Oe  is the effective [[gyromagnetic ratio]] and <math>g</math>, the so-called effective [[g-factor (physics)|g-factor]], is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. <math>\omega</math> is the frequency of the RF/microwave signal propagating through the ferrite, <math>H_0</math> is the internal magnetic bias field, and <math>M</math> is the [[magnetization]] of the ferrite material.

In junction circulators and differential phase shift circulators, microwave signal propagation is usually orthogonal to the static magnetic bias field in the ferrite.  This is the so-called ''transverse field'' case.  The microwave propagation constants for this case, neglecting losses are<ref name="Microwave Circulator Design">{{Cite book|title=Microwave Circulator Design, Second Edition|first=Douglas K.|last=Linkhart|date= 2014|publisher=Artech House|isbn=978-1-60807-583-6}}</ref>

:<math>\Gamma_+ = j\omega\sqrt{\mu_0 \epsilon}\,\sqrt{\frac{\mu^2 - \kappa^2} {\mu}}</math>

:<math>\Gamma_- = j\omega\sqrt{\mu_0 \epsilon}</math>

where <math>\mu_0</math> is the [[Permeability of Free Space]] and <math>\epsilon</math> is the [[Absolute permittivity]] of the ferrite material.  In a circulator, these propagation constants describe waves having [[Elliptical polarization]] that would propagate in the direction of the static magnetic bias field, which is through the thickness of the ferrite.  The plus and minus subscripts of the propagation constants indicate opposite wave polarizations.