**2. Formulation of the problem**

The general view of FCL junction is presented in Fig. 1. This junction is a four-port structure that contains a ferrite section realized as two coupled lines placed on ferrite substrate. The ferrite section is fed from dielectric coupled lines with the same cross section where instead of ferrite, the dielectric materials are used. When the ferrite junction is longitudinally magnetized, the Faraday rotation phenomenon occurs, resulting in nonreciprocal properties of the FCL junction. For better understanding of the nonreciprocal effect, an example of a junction ensuring 45 ◦ Faraday rotation has been considered (see Fig. 1(b) and (c)). When

rotation phenomenon is optimal. This optimal effect is achieved when the ferrite material is placed in the region where the wave is linearly polarized and occurs in cylindrical waveguide with coaxially located ferrite rod or suspended stripline. In order to construct devices such as circulators [9, 23], gyrators [19, 22], or isolators [13], the FCL junction has to be cascaded with reciprocal sections providing input signals to the FCL which are either in phase or out

So far, studies concerning FCL devices have been focused mainly on structures realized in a planar line technology [1, 4, 9]. Such structures allow one to obtain fully integrated FCL devices. However, due to the significant length of the ferrite section, the main drawbacks are high insertion losses occurring in ferrite material and large dimensions of the structure.

There were several attempts to improve performance and to reduce total dimensions of planar FCL devices. Promising results concerning low insertion losses and high isolation were obtained for the nonreciprocal devices employing a ferrite coupled slotline [9] and stripline junctions [23, 24]. For the fabricated devices, obtained insertion losses were not lower than 3 dB and isolation was better than 12 dB [9, 18, 23]. Moreover, in order to reduce the dimensions of the planar FCL devices in [4], the circulator with appropriate matching networks at the ports ensuring multiple reflections was proposed. For presented device, the FCL junction length reduction by a factor of two was obtained. The drawback of this structure was high value of insertion losses caused by multiple transmission of signal through the lossy ferrite junction. Also similar length reduction of FCL junction was achieved with the use of periodic left-handed/ferrite coupled line (LH-FCL) structures [1]. However, for the simulated

circulator utilizing LH-FCL section, the insertion losses were not lower than 4 dB.

The better performance in comparison to currently proposed planar configurations was obtained for nonreciprocal devices utilizing cylindrical ferrite coupled line (CFCL) junction [10]. Due to the similar geometry to the circular waveguide with coaxially located ferrite rod, such structure allows to obtain close-to-optimal Faraday rotation effect. Moreover, in such configuration, stronger gyromagnetic coupling occurs which is a result of high magnetic field concentration in the ferrite medium. These make possible to design shorter ferrite junctions ensuring lower insertion losses in comparison to planar ones. This junction was successfully applied to realization of nonreciprocal devices such as isolators and circulators [11, 12].

This chapter presents the authors' recent research on the nonreciprocal devices utilizing longitudinally magnetized FCL junction. The operation principle of FCL junction is explained, and the hybrid techniques of analysis are shown. Numerical and experimental results concerning the nonreciprocal devices using different configurations of FCL junctions

The general view of FCL junction is presented in Fig. 1. This junction is a four-port structure that contains a ferrite section realized as two coupled lines placed on ferrite substrate. The ferrite section is fed from dielectric coupled lines with the same cross section where instead of ferrite, the dielectric materials are used. When the ferrite junction is longitudinally magnetized, the Faraday rotation phenomenon occurs, resulting in nonreciprocal properties of the FCL junction. For better understanding of the nonreciprocal effect, an example of a junction ensuring 45 ◦ Faraday rotation has been considered (see Fig. 1(b) and (c)). When

of phase [20].

118 Advanced Electromagnetic Waves

are presented and discussed.

**2. Formulation of the problem**

**Figure 1.** Ferrite coupled line junction: (a) general view of the structure and transmission through the junction for (b) even-mode excitation and (c) single-port excitation

ports (1) and (2) of the structure are excited with in phase signals of equal amplitude (see Fig. 1(b)), due to the Faraday rotation, the output signal is observed only at port (4), while port (3) is isolated. On the other hand, when only port (4) of the structure is excited, the out of phase signals of equal amplitudes appear in ports (1) and (2) (see Fig. 1(c)). These phenomena occurring in the FCL section directly indicate the presence of the nonreciprocal effect. Based on the abovementioned phenomena, it was noted in [20] that in order to obtain nonreciprocal transmission, the FCL section should be cascaded with the structures that allow for the evenor odd-mode signal excitation. This makes possible to realize FCL circulators, which can then be used to design a variety of other nonreciprocal circuits such as isolators or phase shifters.

One of the basic nonreciprocal circuits built based on the FCL section is a three-port circulator shown in Fig 2(a). This structure consists of a cascade connection of *Te*- or *To*-junction with four-port FCL junction. Note that the direction of circulation of the circulator structure depends on the choice of T-junction type. Taking into account the circulator with *Te*-junction, when port (1) of the structure is excited the output signal appears in port (2). Excitation of port (2) of the circulator results in the signal transmission to port (3). Finally, if port (3) is excited, the signal is transmitted to port (1). Therefore, such circuit provides transmission sequence between ports (1) → (3) → (2) → (1). The circulation direction will reverse when the FCL section is cascaded with *To*-junction. In addition, the circulation direction can be reversed by changing into opposite the direction of biasing magnetic field which results in reversed direction of Faraday rotation. The described circulator can be used for the realization of isolator, by introducing a matched load in one of the circulator ports.

The four-port circulators can be obtained by replacing in three-port circulator from Fig. 2(a) the normal T-junction with magic T-junction (see Fig. 2(b)) or just by cascading two three-port FCL circulators (see Fig. 2(c)). The advantage of structure from Fig. 2(b) is that the signal for each transmission passes through the ferrite section only once, and as a result, this structure can be characterized by lower insertion losses. However, in the case of integrated circuits, the realization of this circulator requires the design of a complex and technologically demanding magic T-junction. In the case of nonreciprocal devices utilizing two FCL sections, it is possible to obtain better isolation in comparison to structure from Fig. 2(b). However, due to the double signal passing through the ferrite section, the losses of the structure are two times

**Figure 2.** Nonreciprocal devices utilizing ferrite coupled line junction: (a) three-port circulator, (b) four-port circulator with single FCL section, and (c) four-port circulator with double FCL section

higher. It should be noted that presented nonreciprocal devices from Fig. 2 are analogous to cylindrical waveguide nonreciprocal devices with Faraday rotation ([7]).

In order to determine the scattering parameters of the FCL junction, two different hybrid techniques have been proposed and developed ([12, 16–18]) (see Fig. 3). The first approach

**Figure 3.** Proposed analysis methods of investigated configurations of FCL junctions

is based on the combination of spectral-domain approach (SDA) and coupled-mode method (CMM) ([16, 17]). The analysis using SDA/CMM involves introducing an isotropic basic guide. This guide is complementary to the ferrite one; however, instead of ferrite, the dielectric characterized by the same permittivity as ferrite and relative permeability *µ<sup>r</sup>* = 1 is utilized. In this approach, the basic guide modes obtained from SDA are used to determine the wave parameters of ferrite modes. As a result of the analysis, the dispersion characteristics of the ferrite line, the gyromagnetic coupling coefficient, and the scattering parameters of the ferrite junction are obtained. The second method is based on a combination of SDA with mode-matching (MM) technique ([12, 18]). In this approach, the SDA is utilized to determine the propagation coefficients and field distribution of two fundamental modes in a dielectric and ferrite section. Then by applying the continuity conditions for the tangential field components at both the interfaces between dielectric and ferrite sections, we formulate the scattering matrix of four-port FCL junction.

The formulation of SDA for dielectric/ferrite guides with different cross sections has been presented in section 2.1. The details of scattering matrix calculation utilizing CMM and MM method have been presented in sections 2.2.1 and 2.2.2, respectively.

### **2.1. Analysis of dielectric/ferrite guide using SDA**

1 **Ferrite**

T-junction

120 Advanced Electromagnetic Waves

 = ferrite

**Hi**

 = ferrite

**Hi**

with single FCL section, and (c) four-port circulator with double FCL section

cylindrical waveguide nonreciprocal devices with Faraday rotation ([7]).

**Figure 3.** Proposed analysis methods of investigated configurations of FCL junctions

FCL

T-junction <sup>3</sup>

(a)

1 **Ferrite**

SD analysis of dielectric guide **SDA analysis of dielectric guide**

SD analysis of dielectric guide **SDA analysis of dielectric and ferrite guide**

FCL

2

3

1

4 (c)

**Figure 2.** Nonreciprocal devices utilizing ferrite coupled line junction: (a) three-port circulator, (b) four-port circulator

higher. It should be noted that presented nonreciprocal devices from Fig. 2 are analogous to

In order to determine the scattering parameters of the FCL junction, two different hybrid techniques have been proposed and developed ([12, 16–18]) (see Fig. 3). The first approach

> SD analysis of dielectric guide **Coupled Mode Method**

> SD analysis of dielectric guide **Mode-Matching Method**

is based on the combination of spectral-domain approach (SDA) and coupled-mode method (CMM) ([16, 17]). The analysis using SDA/CMM involves introducing an isotropic basic guide. This guide is complementary to the ferrite one; however, instead of ferrite, the dielectric characterized by the same permittivity as ferrite and relative permeability *µ<sup>r</sup>* = 1 is utilized. In this approach, the basic guide modes obtained from SDA are used to determine the wave parameters of ferrite modes. As a result of the analysis, the dispersion characteristics of the ferrite line, the gyromagnetic coupling coefficient, and the scattering parameters of the ferrite junction are obtained. The second method is based on a combination of SDA with mode-matching (MM) technique ([12, 18]). In this approach, the SDA is utilized to determine the propagation coefficients and field distribution of two fundamental modes in

magic T-junction

2

**Ferrite**

**Hi**

2

SD analysis of dielectric guide **Scattering Matrix of FCL junction**

T-junction

(b)

**Ferrite**

 = ferrite

FCL

**Hi**

 = ferrite

FCL

4

3

In Fig. 4, the cross sections of the investigated lines are presented. In order to determine the propagation coefficients and field distribution in the structure, the spectral domain approach is utilized. Depending on the type of the considered line, the method is formulated in rectangular, cylindrical, or elliptic coordinate system.

**Figure 4.** Investigated types of coupled lines: (a) planar three-strip line, (b) cylindrical coupled slotline and (c) elliptical three-strip line

The Fourier transform in each coordinate system takes the following form:

• Rectangular coordinates (*x*, *y*, *z*)

$$\widetilde{f}(p) = \int\_{\mathbb{R}} f(y) e^{-\mathrm{j}py} \, \mathrm{d}y, \quad f(y) = \frac{1}{2\pi} \int\_{\mathbb{R}} \widetilde{f}(p) e^{\mathrm{j}py} \, \mathrm{d}p,\tag{1}$$

• Cylindrical coordinates (*ρ*, *ϕ*, *z*)

$$\widetilde{f}(p) = \frac{1}{2\pi} \int\_0^{2\pi} f(\varrho) e^{-jp\varrho} \,\mathrm{d}\varrho, \quad f(\varrho) = \sum\_{p=-\infty}^{\infty} \widetilde{f}(p) e^{jp\varrho} \,\mathrm{d}\varrho \tag{2}$$

• Elliptic coordinates (*u*, *v*, *z*)

$$\widetilde{f}^{\epsilon(o)}(p) = \frac{1}{2\pi} \int\_0^{2\pi} f(v) \begin{Bmatrix} \epsilon e\_p(q, v) \\ \epsilon e\_p(q, v) \end{Bmatrix} \,\mathrm{d}v,\tag{3}$$

$$f(v) = \sum\_{p=0}^{\infty} \widetilde{f}^{\ell}(p) c e\_p(q\_\prime v) + \sum\_{p=1}^{\infty} \widetilde{f}^{0}(p) s e\_p(q\_\prime v),\tag{4}$$

where *f* (·) is the image of the function *f*(·), *cep*(·, ·) and *sep*(·, ·) are even and odd angular Mathieu functions of *p*th order, *q* = *k*<sup>2</sup> <sup>0</sup>*d*2/4, *<sup>k</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*√*µ*0*ε*0, and *<sup>d</sup>* is the focal length. The electromagnetic field in the structure can be described by Maxwell's equations defined in the spectral domain, as follows:

$$\nabla \times \bar{\mathbf{E}} = -k\_0 \mu\_r \eta \bar{\mathbf{H}}\_r \tag{5}$$

$$
\nabla \times \eta \mathbf{H} = k\_0 \varepsilon\_r \mathbf{E} \,, \tag{6}
$$

where **E** and **H** are Fourier transforms of the electric and magnetic fields, respectively, *η* = <sup>−</sup>j*η*0, *<sup>η</sup>*<sup>0</sup> <sup>=</sup> *µ*0/*ε*0, and <sup>µ</sup>*<sup>r</sup>* denotes the permeability tensor of ferrite material. For the longitudinally magnetized ferrite material along *z*-axis, µ*<sup>r</sup>* = **T**µ *<sup>r</sup>***T**−1, where µ *<sup>r</sup>* = *µ*(**i***x***i***<sup>x</sup>* + **i***y***i***y*) + j*µa*(**i***x***i***<sup>y</sup>* − **i***y***i***x*) + **i***z***i***<sup>z</sup>* is a permeability tensor in a dyadic form defined in rectangular coordinates, **T** is transformation matrix from rectangular to cylindrical or elliptic coordinates, and *µ* and *µ<sup>a</sup>* are defined according to [7].

By applying the boundary and continuity conditions to the relations (5), (6) and assuming the fields and currents variation along *z*-axis as *e*−j*βz*, one obtains a set of equations combining tangential electric field (*E<sup>z</sup>* and *E<sup>ξ</sup>* ) and current densities (*Jz* and *J<sup>ξ</sup>* ) at the strips:

$$
\begin{bmatrix}
\tilde{E}\_{\mathsf{z}}(\boldsymbol{\xi} = \boldsymbol{\xi}\_{0}, \boldsymbol{p}) \\
\tilde{E}\_{\mathsf{f}}(\boldsymbol{\xi} = \boldsymbol{\xi}\_{0}, \boldsymbol{p})
\end{bmatrix} = \begin{bmatrix}
\mathbf{G}(\boldsymbol{p}, \boldsymbol{\beta})
\end{bmatrix} \begin{bmatrix}
\tilde{I}\_{\mathsf{f}}(\boldsymbol{\xi} = \boldsymbol{\xi}\_{0}, \boldsymbol{p}) \\
\boldsymbol{I}\_{\mathsf{z}}(\boldsymbol{\xi} = \boldsymbol{\xi}\_{0}, \boldsymbol{p})
\end{bmatrix},
\tag{7}
$$

where (*ς*, *ξ*) = {(*x*, *y*),(*ρ*, *ϕ*),(*u*, *v*)} and **G**(*p*, *β*) is a dyadic Green's function ([8]). In order to solve (7) the current on the patch is expanded in terms of basis functions:

$$J\_{\tilde{\xi}}(\tilde{\xi}) = \begin{cases} \sum\_{n=1}^{N} a\_n \sin\left(\frac{n\pi(2\tilde{\xi} + w)}{2w}\right), |\tilde{\xi}| \le \frac{w}{2} \\\\ 0, & \text{otherwise}, \end{cases} \tag{8}$$

$$J\_z(\xi) = \begin{cases} \sum\_{n=0}^{N} b\_n \frac{\cos\left(\frac{n\pi(2\xi+w)}{2w}\right)}{\sqrt{1-\left(\frac{2\xi}{w}\right)^2}}, |\xi| \le \frac{w}{2} \\\\ 0, & \text{otherwise}, \end{cases} \tag{9}$$

where *an* and *bn* are unknown current expansion coefficients. Next, using MoM with current basis functions (8) and (9) chosen as a testing functions (Galerkin method) a homogeneous set of equations is obtained ([26]). The nontrival solutions of the problem provide us with a phase coefficients *β* and corresponding current coefficients allowing us to determine the current density distributions on the strips as well as the electric and magnetic fields in the cross section of the structure.

### **2.2. Scattering matrix of ferrite coupled junction**

### *2.2.1. Coupled-mode method*

*f <sup>e</sup>*(*o*)

Mathieu functions of *p*th order, *q* = *k*<sup>2</sup>

and *µ* and *µ<sup>a</sup>* are defined according to [7].

*J<sup>ξ</sup>* (*ξ*) =

*Jz*(*ξ*) =

spectral domain, as follows:

122 Advanced Electromagnetic Waves

where *f*

*f*(*v*) =

(*p*) = <sup>1</sup> 2*π* 2*π*

> ∞ ∑ *p*=0 *f e*

longitudinally magnetized ferrite material along *z*-axis, µ*<sup>r</sup>* = **T**µ

*E<sup>z</sup>*(*ς* = *ς*0, *p*) *E<sup>ξ</sup>* (*ς* = *ς*0, *p*)

> 

*N* ∑ *n*=1

*N* ∑ *n*=0 *bn* cos

 

0

*f*(*v*)

(*p*)*cep*(*q*, *v*) +

electromagnetic field in the structure can be described by Maxwell's equations defined in the

where **E** and **H** are Fourier transforms of the electric and magnetic fields, respectively, *η* = <sup>−</sup>j*η*0, *<sup>η</sup>*<sup>0</sup> <sup>=</sup> *µ*0/*ε*0, and <sup>µ</sup>*<sup>r</sup>* denotes the permeability tensor of ferrite material. For the

**i***y***i***y*) + j*µa*(**i***x***i***<sup>y</sup>* − **i***y***i***x*) + **i***z***i***<sup>z</sup>* is a permeability tensor in a dyadic form defined in rectangular coordinates, **T** is transformation matrix from rectangular to cylindrical or elliptic coordinates,

By applying the boundary and continuity conditions to the relations (5), (6) and assuming the fields and currents variation along *z*-axis as *e*−j*βz*, one obtains a set of equations combining

**G**(*p*, *β*)

where (*ς*, *ξ*) = {(*x*, *y*),(*ρ*, *ϕ*),(*u*, *v*)} and **G**(*p*, *β*) is a dyadic Green's function ([8]). In order

 *nπ*(2*ξ*+*w*) 2*w*

 *<sup>n</sup>π*(2*ξ*+*w*) 2*w* 

 <sup>1</sup>−( <sup>2</sup>*<sup>ξ</sup> w* ) 0, otherwise,

<sup>2</sup> , <sup>|</sup>*ξ*| ≤ *<sup>w</sup>*

0, otherwise,

, <sup>|</sup>*ξ*| ≤ *<sup>w</sup>* 2

2

 *<sup>J</sup><sup>ξ</sup>* (*<sup>ς</sup>* <sup>=</sup> *<sup>ς</sup>*0, *<sup>p</sup>*) *Jz*(*ς* = *ς*0, *p*)

tangential electric field (*E<sup>z</sup>* and *E<sup>ξ</sup>* ) and current densities (*Jz* and *J<sup>ξ</sup>* ) at the strips:

 =

to solve (7) the current on the patch is expanded in terms of basis functions:

*an* sin

(·) is the image of the function *f*(·), *cep*(·, ·) and *sep*(·, ·) are even and odd angular

 *cep*(*q*, *v*) *sep*(*q*, *v*)

> ∞ ∑ *p*=1 *f*

<sup>0</sup>*d*2/4, *<sup>k</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*√*µ*0*ε*0, and *<sup>d</sup>* is the focal length. The

∇ × **E** = −*k*0µ*rη***H** , (5) ∇ × *η***H** = *k*0*εr***E**, (6)

*<sup>r</sup>***T**−1, where µ

*<sup>r</sup>* = *µ*(**i***x***i***<sup>x</sup>* +

(8)

(9)

, (7)

d*v*, (3)

*<sup>o</sup>*(*p*)*sep*(*q*, *<sup>v</sup>*), (4)

Using the coupled-mode method, a gyromagnetic coupling coefficient, propagation coefficients of ferrite modes, and scattering matrix of FCL junction can be determined ([5, 21]). In this method, the transverse electric and magnetic fields in the investigated ferrite guide are expressed in terms of basic guide field eigenfunctions. Assuming two fundamental modes propagated in the basic guide, the Maxwell's equations for basic and ferrite guides are combined together, and after some mathematical manipulation, the following set of coupled-mode equations is obtained:

$$\begin{aligned} \frac{\partial}{\partial z} \hat{U}\_{\ell}(z) + \mathbf{j}\beta\_{\ell} Z\_{\ell} \hat{I}\_{\ell}(z) &= \mathbf{C}\_{\ell o} \hat{I}\_{o}(z), \\ \frac{\partial}{\partial z} \hat{U}\_{o}(z) + \mathbf{j}\beta\_{o} Z\_{o} \hat{I}\_{o}(z) &= \mathbf{C}\_{o} \hat{I}\_{\ell}(z), \\ \frac{\partial}{\partial z} \hat{I}\_{\ell}(z) + \mathbf{j}\beta\_{\ell} Y\_{\ell} \hat{U}\_{\ell}(z) &= 0, \\ \frac{\partial}{\partial z} \hat{I}\_{o}(z) + \mathbf{j}\beta\_{o} Y\_{o} \hat{U}\_{o}(z) &= 0, \end{aligned} \tag{10}$$

where

$$\mathbf{C}\_{\varepsilon o} = -\mathbf{C}\_{\varepsilon o}^{\*} = k\_0 \eta\_0 \mu\_d \int\_{\Omega\_f} \left( \mathbf{h}\_{t,\varepsilon} \times \mathbf{h}\_{t,o}^{\*} \right) \cdot \mathbf{i}\_2 \, d\Omega\_f \tag{11}$$

define the coupling between two fundamental modes in the basic guide, *U<sup>e</sup>*(*o*)(*z*) and *Ie*(*o*)(*z*) are *z*-dependent voltage and current functions in the ferrite guide, *Ze*(*o*) = 1/*Ye*(*o*) are wave impedances of fundamental modes, **h***t*,*e*(*o*) are the eigenfunctions of magnetic fields of fundamental modes, and Ω*<sup>f</sup>* is a ferrite area in the cross section. Next, taking into consideration field distributions **H***t*,*e*(*o*) of basic modes instead of their eigenfunction **h***t*,*e*(*o*), the gyromagnetic coupling coefficient can be written as follows:

$$\mathbf{C}\_{\rm ce} = k\_0 \eta\_0 \mu\_a \frac{\sqrt{\mathbf{Z}\_\varepsilon \mathbf{Z}\_0}}{\sqrt{P\_\varepsilon P\_o}} \int\_{\Omega\_f} \left(\mathbf{H}\_{t,\varepsilon} \times \mathbf{H}\_{t,\rho}^\*\right) \cdot \mathbf{i}\_z \, d\Omega\_{f'} \tag{12}$$

where *Pe*(*o*) denotes powers of two fundamental basis modes. As one can see, the gyromagnetic coupling occurs in the guide when the ferrite is placed in the area where magnetic field vectors of two fundamental modes are orthogonal to each other and linearly polarized.

The above-defined transmission line model of the ferrite guide can be used to determine scattering matrix of FCL junction. At first, utilizing symmetry properties of the investigated structure, the modal even/odd voltage and current can be related to voltage and current defined for each of two coupled lines (see Fig. 5)

**Figure 5.** Four-port FCL junction: (a) circuit model, (b) network representation as follows:

$$\begin{aligned} \hat{\mathcal{U}}\_{\varepsilon}(z) &= \hat{\mathcal{U}}\_{1}(z) + \hat{\mathcal{U}}\_{2}(z), & \hat{I}\_{\varepsilon}(z) &= \hat{I}\_{1}(z) + \hat{I}\_{2}(z), \\ \hat{\mathcal{U}}\_{0}(z) &= \hat{\mathcal{U}}\_{1}(z) - \hat{\mathcal{U}}\_{2}(z), & \hat{I}\_{0}(z) &= \hat{I}\_{1}(z) - \hat{I}\_{2}(z). \end{aligned} \tag{13}$$

Next, applying the above relations to (10), the following eigenproblem is obtained:

$$\mathbf{QK} = \mathbf{Kk},\tag{14}$$

where matrix **Q**, diagonal matrix of eigenvalues **k**, and matrix of eigenvectors **K** are defined in [10]. The solutions of (14) are the eigenvalues *k*<sup>1</sup> and *k*2, defining propagation coefficients of two fundamental modes in ferrite guide which take the following form:

$$k\_{1,2} = \pm \sqrt{\frac{\beta\_e^2 + \beta\_o^2}{2} \pm \sqrt{\left(\frac{\beta\_e^2 - \beta\_o^2}{2}\right)^2 + \frac{\mathbb{C}\_{\epsilon o}^2}{\mathbb{Z}\_{\epsilon}\mathbb{Z}\_0} \beta\_{\epsilon} \beta\_o}}.\tag{15}$$

As one can see, the increase of the coupling results in the increase of the difference between *k*<sup>1</sup> and *k*2. Finally, the length of the ferrite section ensuring 45 ◦ Faraday rotation decreases when the coupling *Ceo* increases.

Using the solution of (15), the voltage and current in each of the two coupled lines can be defined at considered *z* cross section as follows:

$$
\begin{bmatrix}
\hat{\Omega}\_1(z) \\
\hat{\Omega}\_2(z) \\
\hat{I}\_1(z) \\
\hat{I}\_2(z)
\end{bmatrix} = \mathbf{K} \begin{bmatrix}
e^{-jk\_1z} & 0 & 0 & 0 \\
0 & e^{jk\_1z} & 0 & 0 \\
0 & 0 & e^{-jk\_2z} & 0 \\
0 & 0 & 0 & e^{jk\_2z}
\end{bmatrix} \begin{bmatrix}
A\_1^+ \\
A\_1^- \\
A\_2^+ \\
A\_2^-
\end{bmatrix} \tag{16}
$$

where *A*+(−) <sup>1</sup> and *<sup>A</sup>*+(−) <sup>2</sup> are unknown amplitudes of the forward and backward partial waves in the equivalent transmission line. Assuming notation from Fig. 5(a) and utilizing (16), we can write the relations between voltages and currents in the ports of the considered junction defined at interfaces *z* = 0 and *z* = *L*:

$$
\begin{bmatrix}
\dot{U}\_1(z) \\
\hat{U}\_2(z) \\
\hat{I}\_1(z) \\
\hat{I}\_2(z)
\end{bmatrix}\_{z=0} = \begin{bmatrix}
v\_1 \\ v\_2 \\ i\_1 \\ i\_2
\end{bmatrix} \qquad \text{and} \qquad
\begin{bmatrix}
\dot{U}\_1(z) \\
\hat{U}\_2(z) \\
\hat{I}\_1(z) \\
\hat{I}\_2(z)
\end{bmatrix}\_{z=L} = \begin{bmatrix}
v\_3 \\ v\_4 \\ -i\_3 \\ -i\_4
\end{bmatrix}.
\tag{17}
$$

Combining equations (16) and (17), we obtain the following relation:

magnetic field vectors of two fundamental modes are orthogonal to each other and linearly

The above-defined transmission line model of the ferrite guide can be used to determine scattering matrix of FCL junction. At first, utilizing symmetry properties of the investigated structure, the modal even/odd voltage and current can be related to voltage and current

*b1*

*a2*

*b2*

*U<sup>e</sup>*(*z*) = *U*<sup>1</sup>(*z*) + *U*<sup>2</sup>(*z*), *Ie*(*z*) = *I*1(*z*) + *I*2(*z*),

where matrix **Q**, diagonal matrix of eigenvalues **k**, and matrix of eigenvectors **K** are defined in [10]. The solutions of (14) are the eigenvalues *k*<sup>1</sup> and *k*2, defining propagation coefficients

*<sup>β</sup>*<sup>2</sup>

As one can see, the increase of the coupling results in the increase of the difference between *k*<sup>1</sup> and *k*2. Finally, the length of the ferrite section ensuring 45 ◦ Faraday rotation decreases

*<sup>e</sup>* <sup>−</sup> *<sup>β</sup>*<sup>2</sup> *o* 2

2 + *C*2 *eo ZeZo*

Next, applying the above relations to (10), the following eigenproblem is obtained:

of two fundamental modes in ferrite guide which take the following form:

 *β*2 *<sup>e</sup>* + *β*<sup>2</sup> *o* 2 ±

*k*1,2 = ±

when the coupling *Ceo* increases.

*b3*

S *a4*

(b)

**QK** = **Kk**, (14)

3

4

*βeβo*. (15)

*b4*

*a3*

*a1*

1

2

*<sup>U</sup><sup>o</sup>*(*z*) = *<sup>U</sup>*<sup>1</sup>(*z*) <sup>−</sup> *<sup>U</sup>*<sup>2</sup>(*z*), *Io*(*z*) = *<sup>I</sup>*1(*z*) <sup>−</sup> *<sup>I</sup>*2(*z*). (13)

polarized.

124 Advanced Electromagnetic Waves

*i1 v1*

*<sup>i</sup> <sup>v</sup> <sup>2</sup> <sup>2</sup>*

1

2

as follows:

defined for each of two coupled lines (see Fig. 5)

*z=0 z=L <sup>z</sup>*

line 2

(a)

line 1

*i3 v3*

*i4 v4*

**Figure 5.** Four-port FCL junction: (a) circuit model, (b) network representation

3

4

$$
\begin{bmatrix} v\_3 \\ v\_4 \\ -i\_3 \\ -i\_4 \end{bmatrix} = \mathbf{K} \begin{bmatrix} e^{-jk\_1L} & 0 & 0 & 0 \\ 0 & e^{jk\_1L} & 0 & 0 \\ 0 & 0 & e^{-jk\_2L} & 0 \\ 0 & 0 & 0 & e^{jk\_2L} \end{bmatrix} \mathbf{K}^{-1} \begin{bmatrix} v\_1 \\ v\_2 \\ i\_1 \\ i\_2 \end{bmatrix} . \tag{18}
$$

Finally, we define incident and reflected waves in each *i*th port of the structure as follows:

$$a\_{\dot{i}} = \frac{v\_{\dot{i}}}{\sqrt{Z\_0}} + i\_{\dot{i}}\sqrt{Z\_0} \qquad \text{and} \qquad b\_{\dot{i}} = \frac{v\_{\dot{i}}}{\sqrt{Z\_0}} - i\_{\dot{i}}\sqrt{Z\_0}.$$

where *Z*<sup>0</sup> is a wave impedance of the port. Applying above relations to (18), we obtain the scattering matrix of four-port FCL junction which is defined as follows:

$$
\begin{bmatrix} b\_1 \\ b\_2 \\ b\_3 \\ b\_4 \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{11} \ \mathbf{S}\_{12} \ \mathbf{S}\_{13} \ \mathbf{S}\_{14} \\ \mathbf{S}\_{21} \ \mathbf{S}\_{22} \ \mathbf{S}\_{23} \ \mathbf{S}\_{24} \\ \mathbf{S}\_{31} \ \mathbf{S}\_{32} \ \mathbf{S}\_{33} \ \mathbf{S}\_{34} \\ \mathbf{S}\_{41} \ \mathbf{S}\_{42} \ \mathbf{S}\_{43} \ \mathbf{S}\_{44} \end{bmatrix} \begin{bmatrix} a\_1 \\ a\_2 \\ a\_3 \\ a\_4 \end{bmatrix} . \tag{19}
$$

Assuming instead of the voltage and current waves the real voltage and current distribution in the proposed transmission line model of FCL junction, the wave impedances *Ze*(*o*) and *Z*<sup>0</sup> can be treated as characteristic impedances.

### *2.2.2. Mode-matching method*

In order to determine the S-matrix of the proposed FCL junction, the mode-matching method is utilized. At first, the junction is analyzed as a two-port structure composed of dielectric section followed by ferrite section and another dielectric section as presented in Fig. 6. Since

**Figure 6.** Two port FCL junction composed of ferrite and dielectric sections.

the widths of the strips in the ferrite and dielectric sections are the same, the higher modes are not excited, and therefore, they are neglected in the analysis. In this case, only two fundamental modes are taken into consideration. The modes propagated in the dielectric and ferrite sections are called dielectric and ferrite waves, respectively. Due to the symmetry of the structure, we can distinguish even- and odd-mode waves in the dielectric section. Despite the fact that at port (1) or (2) only one of the dielectric waves can appear, both ferrite waves are excited. The wave parameters and field distributions of modes in the ferrite and dielectric sections are determined using SDA ([15]). The total field in each section is determined as a superposition of both modes propagating in forward (+) and backward (−) directions. Using the notation from Fig. 6, the total field in dielectric sections *i* = 1, 2 can be written in the following form:

$$\mathbf{F}\_{t}^{(i)} = \begin{bmatrix} \mathbf{F}\_{+\boldsymbol{\epsilon}\prime}^{(i)} \mathbf{F}\_{+\boldsymbol{\epsilon}\prime}^{(i)} \mathbf{F}\_{-\boldsymbol{\epsilon}\prime}^{(i)} \mathbf{F}\_{-\boldsymbol{\epsilon}\prime}^{(i)} \end{bmatrix} \begin{bmatrix} A\_{\boldsymbol{\epsilon}}^{(i)} \\ A\_{\boldsymbol{\epsilon}}^{(i)} \\ B\_{\boldsymbol{\epsilon}}^{(i)} \\ B\_{\boldsymbol{\epsilon}}^{(i)} \end{bmatrix} , \tag{20}$$

where **F** = (**E**, **H**) represents tangential electric or magnetic field and *A* and *B* are the unknown expansion coefficients describing forward and backward waves, respectively, of even (*e*) and odd (*o*) modes. The total field in ferrite section can be written as follows:

$$\mathbf{F}\_{t}^{(0)} = \begin{bmatrix} \mathbf{F}\_{+1}^{(0)}, \mathbf{F}\_{+2}^{(0)}, \mathbf{F}\_{-1}^{(0)}, \mathbf{F}\_{-2}^{(0)} \end{bmatrix} \mathbf{D}(z) \begin{bmatrix} \mathbf{C}\_{+1}^{(0)} \\ \mathbf{C}\_{+2}^{(0)} \\ \mathbf{C}\_{-1}^{(0)} \\ \mathbf{C}\_{-2}^{(0)} \end{bmatrix},\tag{21}$$

where **D**(*z*) = diag([*e*j*<sup>k</sup>* (0) <sup>+</sup><sup>1</sup> *<sup>z</sup>*,*e*j*<sup>k</sup>* (0) <sup>+</sup><sup>2</sup> *<sup>z</sup>*,*e*j*<sup>k</sup>* (0) <sup>−</sup><sup>1</sup> *<sup>z</sup>*,*e*j*<sup>k</sup>* (0) <sup>−</sup><sup>2</sup> *<sup>z</sup>*]), *k* (0) <sup>±</sup>1(2) are the propagation coefficients of both fundamental modes in ferrite section and **F** = (**E**, **H**) and *C*(0) <sup>±</sup>1(2) are the unknown expansion coefficients describing forward (+) and backward (−) waves. Using relations (20) and (21), the continuity conditions for the tangential components of electric and magnetic fields at two interfaces *z* = 0 and *z* = *L* can be written as follows:

$$\left. \mathbf{E}\_t^{(1)} \right|\_{z=0} = \left. \mathbf{E}\_t^{(0)} \right|\_{z=0'} \qquad \left. \mathbf{H}\_t^{(1)} \right|\_{z=0} = \left. \mathbf{H}\_t^{(0)} \right|\_{z=0'} \tag{22}$$

$$\mathbf{E}\_t^{(2)}|\_{z=L} = \mathbf{E}\_t^{(0)}|\_{z=L'} \qquad \mathbf{H}\_t^{(2)}|\_{z=L} = \mathbf{H}\_t^{(0)}|\_{z=L} \tag{23}$$

This set of equations can be solved using the orthogonality expansion method. As a result, the relation between forward and backward waves in dielectric sections can be derived in the following form:

$$\mathbf{B}' = \mathbf{S}'\mathbf{A}',\tag{24}$$

where

*2.2.2. Mode-matching method*

126 Advanced Electromagnetic Waves

be written in the following form:

In order to determine the S-matrix of the proposed FCL junction, the mode-matching method is utilized. At first, the junction is analyzed as a two-port structure composed of dielectric section followed by ferrite section and another dielectric section as presented in Fig. 6. Since

1 2

*C*+2 (0)

*C*-2

the widths of the strips in the ferrite and dielectric sections are the same, the higher modes are not excited, and therefore, they are neglected in the analysis. In this case, only two fundamental modes are taken into consideration. The modes propagated in the dielectric and ferrite sections are called dielectric and ferrite waves, respectively. Due to the symmetry of the structure, we can distinguish even- and odd-mode waves in the dielectric section. Despite the fact that at port (1) or (2) only one of the dielectric waves can appear, both ferrite waves are excited. The wave parameters and field distributions of modes in the ferrite and dielectric sections are determined using SDA ([15]). The total field in each section is determined as a superposition of both modes propagating in forward (+) and backward (−) directions. Using the notation from Fig. 6, the total field in dielectric sections *i* = 1, 2 can

where **F** = (**E**, **H**) represents tangential electric or magnetic field and *A* and *B* are the unknown expansion coefficients describing forward and backward waves, respectively, of even (*e*) and odd (*o*) modes. The total field in ferrite section can be written as follows:

(0)

*z=0 z=L <sup>z</sup>*

*C*-1

(0)

*C*+1 (0)

(1) (0) (2)

ferrite section

*Be*

*Bo*

  *A*(*i*) *e A*(*i*) *o B*(*i*) *e B*(*i*) *o*

 

  *C*(0) +1 *C*(0) +2 *C*(0) −1 *C*(0) −2

 

, (20)

, (21)

(2)

*Ao* (2)

(2)

*Ae* (2)

dielectric section

*Be*

*Bo*

**Figure 6.** Two port FCL junction composed of ferrite and dielectric sections.

**F**(*i*) *<sup>t</sup>* = **F**(*i*) <sup>+</sup>*e*, **<sup>F</sup>**(*i*) <sup>+</sup>*o*, **<sup>F</sup>**(*i*) <sup>−</sup>*e*, **<sup>F</sup>**(*i*) −*o* 

**F**(0) *<sup>t</sup>* = **F**(0) <sup>+</sup>1, **<sup>F</sup>**(0) <sup>+</sup>2, **<sup>F</sup>**(0) −1, **<sup>F</sup>**(0) −2 **D**(*z*)

(1)

*Ao* (1)

*Ae*

(1)

dielectric section

(1)

$$\mathbf{S'} = \begin{bmatrix} S\_{\epsilon\epsilon}^{11} & S\_{\epsilon\epsilon}^{11} & S\_{\epsilon\epsilon}^{12} & S\_{\epsilon\epsilon}^{12} \\ S\_{\epsilon\epsilon}^{11} & S\_{\epsilon\epsilon}^{11} & S\_{\epsilon\epsilon}^{12} & S\_{\epsilon\epsilon}^{12} \\ S\_{\epsilon\epsilon}^{21} & S\_{\epsilon\epsilon}^{21} & S\_{\epsilon\epsilon}^{22} & S\_{\epsilon\epsilon}^{22} \\ S\_{\epsilon\epsilon}^{21} & S\_{\epsilon\epsilon}^{21} & S\_{\epsilon\epsilon}^{22} & S\_{\epsilon\epsilon}^{22} \end{bmatrix} \tag{25}$$

and **A** and **B** are the vectors of unknown expansion coefficients for the fields in each section **A** = [*A*(1) *<sup>e</sup>* , *<sup>A</sup>*(1) *<sup>o</sup>* , *<sup>A</sup>*(2) *<sup>e</sup>* , *<sup>A</sup>*(2) *<sup>o</sup>* ] *<sup>T</sup>*, **<sup>B</sup>** = [*B*(1) *<sup>e</sup>* , *<sup>B</sup>*(1) *<sup>o</sup>* , *<sup>B</sup>*(2) *<sup>e</sup>* , *<sup>B</sup>*(2) *<sup>o</sup>* ] *<sup>T</sup>*. The **S** matrix defines the two-mode scattering matrix of two-port FCL junction. The element *Sji nm* defines the relation between *m* incident wave in the *i*th port and *n* reflected wave in the *j*th port, where *m*, *n* = {*e*, *o*} and *i*, *j* = {1, 2}.

In the designing procedure of the integrated nonreciprocal devices, more useful is the S-matrix defined from the point of view of the incident and reflected waves at four ports of the FCL junction. The scheme of such junction is presented in Fig. 7.

**Figure 7.** Four-port FCL junction

Using the symmetry properties of the even and odd modes propagated in the dielectric sections, the matrix **S** can be rearranged in terms of port waves, and finally, the scattering matrix of four-port FCL junction is obtained. The incident and reflected waves at *i*th port are denoted by *A*(*i*) and *B*(*i*), respectively. Due to symmetry of the waves in the dielectric sections, they can be written as superpositions of waves in each port of four-port FCL junction as follows:

$$\begin{aligned} A\_{\varepsilon}^{(1)} &= (A^{(1)} + A^{(2)}) / \sqrt{2}, & A\_{o}^{(1)} &= (A^{(1)} - A^{(2)}) / \sqrt{2}, \\ B\_{\varepsilon}^{(1)} &= (B^{(1)} + B^{(2)}) / \sqrt{2}, & B\_{o}^{(1)} &= (B^{(1)} - B^{(2)}) / \sqrt{2}, \\ A\_{\varepsilon}^{(2)} &= (A^{(3)} + A^{(4)}) / \sqrt{2}, & A\_{o}^{(2)} &= (A^{(3)} - A^{(4)}) / \sqrt{2}, \\ B\_{\varepsilon}^{(2)} &= (B^{(3)} + B^{(4)}) / \sqrt{2}, & B\_{o}^{(2)} &= (B^{(3)} - B^{(4)}) / \sqrt{2}. \end{aligned} \tag{26}$$

which can be expressed in the matrix form

$$\mathbf{A}' = \mathbf{T}\mathbf{A} \qquad \text{and} \qquad \mathbf{B}' = \mathbf{T}\mathbf{B}, \tag{27}$$

$$\text{where } \mathbf{A} = [A^{(1)}, A^{(2)}, A^{(3)}, A^{(4)}]^T, \mathbf{B} = [B^{(1)}, B^{(2)}, B^{(3)}, B^{(4)}]^T, \text{and}$$

$$\mathbf{T} = \begin{bmatrix} \mathbf{T}\_1 & \mathbf{0} \\ \mathbf{0} & \mathbf{T}\_1 \end{bmatrix}, \quad \mathbf{T}\_1 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}.$$

Finally, using the two-mode S-matrix (24) and relations (27), the scattering matrix of the four-port FCL junction is obtained.

$$\mathbf{B} = \mathbf{S} \,\mathbf{A}, \qquad \text{where} \qquad \mathbf{S} = \mathbf{T}^{-1} \,\mathbf{S}' \,\mathbf{T}. \tag{28}$$

Such S-matrix can be used in the analysis of the transmission properties of FCL junction with the assumed excitation.
