**3. Numerical results**

### **3.1. Microstrip ferrite coupled line junction**

The first investigated structure is a planar microstrip ferrite coupled line (MFCL) junction presented in Fig. 8(a). The cross section of the junction is shown in Fig. 8(b). It is a multilayer structure in which two conductive strips are placed at *h*<sup>2</sup> interface while the ground is placed at *h*<sup>1</sup> interface. A ferrite material with a relative permittivity *εr*<sup>2</sup> = 13.3, saturation magnetization *Ms* = 239 kA/m, internal bias *Hi* = 0, and thickness *d*<sup>2</sup> = 0.5 mm is placed in layer (2) located above the conductive strips. The dielectric sections have the same cross section as ferrite section, although instead of ferrite, a dielectric material is used with relative permeability *ur* = 1 and with the same relative permittivity as the ferrite material. The investigated structure has a plane of symmetry *AA* passing through the center of the gap

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

<sup>√</sup>2, *<sup>A</sup>*(1)

<sup>√</sup>2, *<sup>A</sup>*(2)

*<sup>T</sup>*, **B** = [*B*(1), *B*(2), *B*(3), *B*(4)]

, **<sup>T</sup>**<sup>1</sup> <sup>=</sup> <sup>1</sup>

Finally, using the two-mode S-matrix (24) and relations (27), the scattering matrix of the

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

The first investigated structure is a planar microstrip ferrite coupled line (MFCL) junction presented in Fig. 8(a). The cross section of the junction is shown in Fig. 8(b). It is a multilayer structure in which two conductive strips are placed at *h*<sup>2</sup> interface while the ground is placed at *h*<sup>1</sup> interface. A ferrite material with a relative permittivity *εr*<sup>2</sup> = 13.3, saturation magnetization *Ms* = 239 kA/m, internal bias *Hi* = 0, and thickness *d*<sup>2</sup> = 0.5 mm is placed in layer (2) located above the conductive strips. The dielectric sections have the same cross section as ferrite section, although instead of ferrite, a dielectric material is used with relative permeability *ur* = 1 and with the same relative permittivity as the ferrite material. The investigated structure has a plane of symmetry *AA* passing through the center of the gap

√2 1 1 1 −1

*<sup>o</sup>* = (*A*(1) <sup>−</sup> *<sup>A</sup>*(2)

*<sup>o</sup>* = (*A*(3) <sup>−</sup> *<sup>A</sup>*(4)

**A** = **TA** and **B** = **TB**, (27)

 .

**B** = **S A**, where **S** = **T**−<sup>1</sup> **S T**. (28)

*<sup>T</sup>*, and

<sup>√</sup>2, *<sup>B</sup>*(1) *<sup>o</sup>* = (*B*(1) <sup>−</sup> *<sup>B</sup>*(2)

<sup>√</sup>2, *<sup>B</sup>*(2) *<sup>o</sup>* = (*B*(3) <sup>−</sup> *<sup>B</sup>*(4)

)/ √ 2,

)/ √ 2,

> )/ √

)/ √ 2.

2, (26)

as follows:

128 Advanced Electromagnetic Waves

*A*(1)

*A*(2)

which can be expressed in the matrix form

where **A** = [*A*(1), *A*(2), *A*(3), *A*(4)]

four-port FCL junction is obtained.

**3.1. Microstrip ferrite coupled line junction**

the assumed excitation.

**3. Numerical results**

*<sup>e</sup>* = (*A*(1) + *A*(2))/

*<sup>B</sup>*(1) *<sup>e</sup>* = (*B*(1) <sup>+</sup> *<sup>B</sup>*(2))/

*<sup>e</sup>* = (*A*(3) + *A*(4))/

*<sup>B</sup>*(2) *<sup>e</sup>* = (*B*(3) <sup>+</sup> *<sup>B</sup>*(4))/

**T** = **T**<sup>1</sup> **0 0 T**<sup>1</sup>

**Figure 8.** Planar junction of microstrip ferrite coupled lines: (a) 3D view and (b) cross section of ferrite guide

between the strips. In the dielectric section, this symmetry plane is an electric or magnetic wall for odd or even mode, respectively.

Utilizing the developed method of analysis, described in section 2, the dispersion characteristics of the investigated structure are first calculated. The calculations were performed in the frequency range from 9 to 18 GHz. The characteristics of the propagation coefficients of the two basic modes propagating in dielectric and ferrite sections are shown in Fig. 9(a). The ferrite modes have a cutoff frequency near *fM* = *γ*(*Hi* + *Ms*) = 8.4 GHz where gyromagnetic coefficient *γ* = 35.2 MHz m/kA. This means that the ferrite modes propagate in this structure, when the *<sup>µ</sup>eff* = (*µ*<sup>2</sup> <sup>−</sup> *<sup>µ</sup>*<sup>2</sup> *<sup>a</sup>* )/*µ* > 0, where *µ* and *µ<sup>a</sup>* are the elements of relative permeability tensor defined in section 2.1. It can be noted that with the increasing frequency, the propagation coefficients of the modes in the ferrite line converge to propagation coefficients of the modes in dielectric line. This effect is due to the fact that with

**Figure 9.** Simulation results of MFCL junction: (a) dispersion characteristics and (b) magnetic field distribution of even and odd mode in dielectric section of MFCL junction at *f*<sup>0</sup> = 12.4 GHz

the increase of frequency, the value of *µ<sup>a</sup>* element of the permeability tensor (responsible for gyromagnetic property of the ferrite) decreases.

Figure 9(b) shows the distributions of the transverse components of magnetic fields for the even and odd modes in the dielectric section. Calculations are performed at frequency *f*<sup>0</sup> = 12.4 GHz. As can be seen, the magnetic field vectors of the dielectric modes are orthogonal to each other in the areas above and below the strips in the symmetry plane *AA* of the structure. According to the definition of coupling coefficient (12), if instead of one of these layers the ferrite material is introduced, the optimal gyrotropic coupling effect will be obtained.

In the next step, utilizing the MM method described in section 2.2.2, the scattering parameters of the investigated MFCL junction are determined. The scattering parameters of the junction in a function of ferrite section length at *f*<sup>0</sup> = 12.4 GHz are shown in Fig. 10. The optimal

**Figure 10.** Simulated scattering parameters of MFCL junction versus length of ferrite section at *f*<sup>0</sup> = 12.4 GHz: (a) magnitude and (b) phase difference

length *L* of the ferrite section providing 45 ◦ Faraday rotation is determined by amplitude and phase conditions ([21]). According to these conditions, when port (1) of the structure is excited, the signal should be equally divided between ports (3) and (4) of the structure, while port (2) should be isolated. Moreover, the phase difference between signals in ports (3) and (4) for ports (1) or (2) excitation should be equal to 0 or ±180 ◦. From the obtained results, it can be seen that optimal length of the section for which the amplitude and phase conditions are fulfilled is *L* = 28 mm.

In order to illustrate the nonreciprocal properties occurring in the investigated FCL, the change of power concentration along the structure has been calculated and presented in Fig. 11.

Based on the obtained results, one can see the periodic effect of signal exchange between coupled lines in the ferrite section due to the Faraday rotation phenomenon. Furthermore, the distributions of the electric and magnetic fields in the input and output ports of FCL junction providing 45 ◦ Faraday rotation angle (*L* = 28 mm) have been calculated and

**Figure 11.** Power density distribution along the investigated MFCL section

the increase of frequency, the value of *µ<sup>a</sup>* element of the permeability tensor (responsible for

Figure 9(b) shows the distributions of the transverse components of magnetic fields for the even and odd modes in the dielectric section. Calculations are performed at frequency *f*<sup>0</sup> = 12.4 GHz. As can be seen, the magnetic field vectors of the dielectric modes are orthogonal to each other in the areas above and below the strips in the symmetry plane *AA* of the structure. According to the definition of coupling coefficient (12), if instead of one of these layers the ferrite material is introduced, the optimal gyrotropic coupling effect will be

In the next step, utilizing the MM method described in section 2.2.2, the scattering parameters of the investigated MFCL junction are determined. The scattering parameters of the junction in a function of ferrite section length at *f*<sup>0</sup> = 12.4 GHz are shown in Fig. 10. The optimal

> S11 S21 S31 S41

Phase difference (deg)

**Figure 10.** Simulated scattering parameters of MFCL junction versus length of ferrite section at *f*<sup>0</sup> = 12.4 GHz: (a)

length *L* of the ferrite section providing 45 ◦ Faraday rotation is determined by amplitude and phase conditions ([21]). According to these conditions, when port (1) of the structure is excited, the signal should be equally divided between ports (3) and (4) of the structure, while port (2) should be isolated. Moreover, the phase difference between signals in ports (3) and (4) for ports (1) or (2) excitation should be equal to 0 or ±180 ◦. From the obtained results, it can be seen that optimal length of the section for which the amplitude and phase conditions

In order to illustrate the nonreciprocal properties occurring in the investigated FCL, the change of power concentration along the structure has been calculated and presented in

Based on the obtained results, one can see the periodic effect of signal exchange between coupled lines in the ferrite section due to the Faraday rotation phenomenon. Furthermore, the distributions of the electric and magnetic fields in the input and output ports of FCL junction providing 45 ◦ Faraday rotation angle (*L* = 28 mm) have been calculated and

<sup>0</sup> <sup>14</sup> <sup>28</sup> <sup>42</sup> <sup>56</sup> <sup>70</sup> <sup>84</sup> <sup>98</sup> <sup>112</sup> −20

(b)

arg(S31)−arg(S41) arg(S32)−arg(S42)

Length (mm)

gyromagnetic property of the ferrite) decreases.

<sup>0</sup> <sup>14</sup> <sup>28</sup> <sup>42</sup> <sup>56</sup> <sup>70</sup> <sup>84</sup> <sup>98</sup> <sup>112</sup> −40

(a)

magnitude and (b) phase difference

are fulfilled is *L* = 28 mm.

Length (mm)

obtained.

130 Advanced Electromagnetic Waves

−35 −30 −25 −20 −15 −10 −5 0

Fig. 11.

Magnitude S (dB)

**Figure 12.** Field distribution in dielectric section of MFCL junction at *z* = 0 (upper row) and *z* = 28 mm (bottom row) mm for: (a) even-mode excitation and (b) odd-mode excitation

presented in Fig. 12. In the analysis, the even- and odd-mode excitations of the junction were assumed. From the presented results, it can be observed that when such junction is excited with the even or odd mode, the signal concentrates around the left or right line at the output of the structure. If the direction of the magnetization will be reversed, the signal will concentrate on the opposite strips.

For the FCL junction with the optimal length *L* = 28 mm of the ferrite section, the scattering parameters were calculated in a function of frequency (see Fig. 13).

**Figure 13.** Simulated frequency-dependent scattering parameters of MFCL junction: (a) magnitude and (b) phase difference

From the obtained results, it can be seen that the transmission coefficients *S*<sup>31</sup> and *S*<sup>41</sup> are equal to −3 ± 0.5 dB in the frequency range from 11 to 16 GHz, with isolation *S*<sup>21</sup> and reflection losses *S*<sup>11</sup> better than −20 dB. In the considered frequency range, the phase difference between the output signals in ports (3) and (4) for ports (1) or (2) excitation varies in the range from −18 to 27 ◦ (see Fig. 13(b)). The optimal 45 ◦ Faraday rotation angle is obtained for *f*<sup>0</sup> = 12.4 GHz.

### **3.2. Cylindrical ferrite coupled line junction**

Another investigated structure is a cylindrical ferrite coupled line (CFCL) junction. The cross section of ferrite coupled lines is shown in Fig. 14.

**Figure 14.** Cylindrical ferrite coupled line junction: (a) 3D view and (b) cross section of ferrite guide

This structure consists of a cylindrical ferrite rod of radius *r*<sup>1</sup> covered with dielectric layer of thickness *d* = *r*<sup>2</sup> − *r*<sup>1</sup> on which the conductive strips are etched. The dielectric section has the same cross section as ferrite section; however, instead of a ferrite rod, a dielectric rod is used with relative permeability *µ<sup>r</sup>* = 1 and the same relative permittivity as the ferrite material.

For the FCL junction with the optimal length *L* = 28 mm of the ferrite section, the scattering

S11 S21 S31 S41 −60 −50 −40 −30 −20 −10 0 10 20 30 40

**Figure 13.** Simulated frequency-dependent scattering parameters of MFCL junction: (a) magnitude and (b) phase

From the obtained results, it can be seen that the transmission coefficients *S*<sup>31</sup> and *S*<sup>41</sup> are equal to −3 ± 0.5 dB in the frequency range from 11 to 16 GHz, with isolation *S*<sup>21</sup> and reflection losses *S*<sup>11</sup> better than −20 dB. In the considered frequency range, the phase difference between the output signals in ports (3) and (4) for ports (1) or (2) excitation varies in the range from −18 to 27 ◦ (see Fig. 13(b)). The optimal 45 ◦ Faraday rotation angle is

Another investigated structure is a cylindrical ferrite coupled line (CFCL) junction. The cross

3

4

x

**Hi**

(b)

<sup>p</sup>

r1

<sup>s</sup>

r2

y

z

dielectric

section

ferrite rod

**Figure 14.** Cylindrical ferrite coupled line junction: (a) 3D view and (b) cross section of ferrite guide

Phase difference (deg)

<sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> <sup>17</sup> <sup>18</sup> −70

(b)

Frequency (GHz)

arg(S31) − arg(S41) arg(S32) − arg(S42) − 180<sup>o</sup>

parameters were calculated in a function of frequency (see Fig. 13).

<sup>9</sup> <sup>10</sup> <sup>11</sup> <sup>12</sup> <sup>13</sup> <sup>14</sup> <sup>15</sup> <sup>16</sup> <sup>17</sup> <sup>18</sup> −40

(a)

Frequency (GHz)

**3.2. Cylindrical ferrite coupled line junction**

section of ferrite coupled lines is shown in Fig. 14.

ferrite

section

(a)

**Hi**

−35 −30 −25 −20 −15 −10 −5 0

132 Advanced Electromagnetic Waves

Magnitude S (dB)

difference

obtained for *f*<sup>0</sup> = 12.4 GHz.

dielectric

section

1

2

Utilizing the developed method of analysis, described in section 2.1, the dispersion characteristics of the investigated structure are first calculated. In the analysis, the following dimensions and material parameters of the junction were assumed: *r*<sup>1</sup> = 2.2 mm, *εr f* = 15, saturation magnetization *Ms* = 131 kA/m, internal bias *Hi* = 0, and dielectric coating: *d* = 0.127 mm, *εrd* = 2.2. The angular slot/strip widths were ∆*ϕ<sup>p</sup>* = 25 ◦, ∆*ϕ<sup>s</sup>* = 15 ◦. The characteristics of the propagation coefficients of the dielectric and the ferrite lines are shown in Fig. 15(a).

**Figure 15.** Simulation results of CFCL junction: (a) dispersion characteristics and (b) magnetic field distributions of even and odd modes in dielectric section of CFCL junction at *f*<sup>0</sup> = 8.2 GHz

Based on the obtained characteristics for the investigated dielectric lines, it can be noted that for a specific frequency *f*<sup>0</sup> = 8 GHz, the phase velocities of the even and odd modes are equal. It means that in such a line, the isotropic coupling vanishes and only gyromagnetic coupling occurs. This allows to obtain the optimal conditions for the Faraday rotation.

Figure 15(b) shows the magnetic field distributions of the even and odd modes in dielectric line specified for *f*<sup>0</sup> = 8.2 GHz. From these distributions, it can be seen that there are areas in the line where the magnetic fields of both modes are orthogonal; hence, placing the ferrite material in this area of a line will produce the gyromagnetic coupling between the basic field modes.

In the next step, the scattering parameters of CFCL junction are calculated in a function of ferrite section length at *f*<sup>0</sup> = 8.2 GHz and shown in Fig 16.

From the presented results, it can be noticed that the optimal length of the ferrite section providing 45 ◦ Faraday rotation is *L* = 26 mm. For such length of section *S*<sup>31</sup> = *S*<sup>41</sup> = −3 dB and the phase difference between signals in ports (3) and (4) for ports (1) and (2), excitation is then 0 or ±180 ◦.

**Figure 16.** Scattering parameters of CFCL junction versus length of ferrite section: (a) magnitude and (b) phase difference

Similarly to the planar case presented in section 3.1, in order to illustrate the nonreciprocal properties occurring in the investigated CFCL junction, the power concentration along the structure has been determined (see Fig. 17).

**Figure 17.** Power density and transversal field distribution in the CFCL section for even- (upper row) and odd-mode (bottom row) excitation calculated at *f*<sup>0</sup> = 8.2 GHz

Furthermore, the distributions of the electric and magnetic fields in the input (*z* = 0) and output ports (*z* = 26 mm) of FCL junction providing 45 ◦ Faraday rotation angle have been calculated. In the analysis, the even- and odd-mode excitations of the junction were assumed. The obtained results confirm the nonreciprocal behavior of the designed CFCL junction.

For the CFCL junction with the ferrite section of length *L* = 26 mm, the scattering parameters were calculated in a function of frequency and presented in Fig. 18.

**Figure 18.** Frequency-dependent scattering parameters of CFCL junction for *L* = 26 mm: (a) magnitude and (b) phase difference (solid line, our method; dashed line, HFSS)

From the obtained results, it can be seen that the equal power division defined by *S*<sup>31</sup> = *S*<sup>41</sup> = −3 ± 0.5 dB is obtained in the frequency range from 7 to 8.6 GHz (see Fig. 18(a)). In this frequency range, the phase difference between the signals in ports (3) and (4) for ports (1) or (2) excitation varies in the range from −40 ◦ to 10 ◦ (see Fig. 18(b)). In addition, it can be seen that the optimal amplitude and phase conditions required for 45 ◦ Faraday rotation are fulfilled at the frequency *f*<sup>0</sup> = 8.25 GHz. The result of the proposed approach is compared with those obtained from commercial software HFSS, and a good agreement can be observed.
