**4. Waveguide port approach in coupling problems**

#### **4.1 Problem formulation**

Consider the coupling problem between several composite structures fed by waveguide excitations. Although each structure can be formed from an arbitrary number of dielectric regions, for simplicity, we will consider only one-region structures with composite (dielectric and conducting) boundaries. **Figure 8** shows the geometry of the problem consisting of *N* waveguides *Wi* radiating into dielectric regions *Di*, *i* = 1,2, … *N*, surrounded by closed surfaces *SDi* with partially conducting boundaries *S<sup>c</sup> Di* and inward unit normal **n***Di .* Waveguides *Wi* are filled, in general, by dielectrics with permittivities *ε<sup>i</sup>* and permeabilities *μi*, and the regions *Di* are filled by dielectrics with parameters *εDi* and *μDi* . An outer space region *D*<sup>0</sup> is a free space with material parameters *ε*0, *μ*0.

The waveguide ports *Pi* in cross-sections *S<sup>a</sup> <sup>i</sup>* divide the waveguides *Wi* into semi-infinite regions *Ai* and finite regions *Bi* to truncate the mesh in regions *Ai* with incident waveguide excitation and act as excitation sources of composite regions *Di* through the dielectric boundaries *S<sup>d</sup> DiBi* between the regions *Di* and *Bi*. Each region *Bi*, *Di* and *D*<sup>0</sup> is also excited, for generality, by the impressed EM field **E***inc <sup>α</sup>* , **H***inc <sup>α</sup>* , *α* ¼ *Bi*, *Di*, *D*0.

**Figure 8.** *Geometry of the problem.*

To formulate the waveguide port excitation problems through the port surfaces *Sa i* , we consider the aperture coupling problems between the regions *Ai* and *Bi* to divide an original problem into two sets of equivalence problems: internal problems for regions *Ai* and external problems for regions *Bi*, *Di* and *D*0. For this purpose, we cover the port surfaces *S<sup>a</sup> <sup>i</sup>* with PEC sheets and introduce equivalent magnetic currents �**M***<sup>i</sup>* and **M***<sup>i</sup>* on both sides of *S<sup>a</sup> <sup>i</sup>* to restore tangential electric fields on the port surfaces *Sa i* .

#### **4.2 Solution of the internal equivalent problem**

The internal equivalent problems for the considered geometry are similar to those formulated in Section 2.2 and implemented in Section 2.3. According to the equivalence principle [24], the magnetic currents in the regions *Ai* are related to the total electric field **<sup>E</sup>***<sup>S</sup><sup>a</sup> i Ai* on the port surface *<sup>S</sup><sup>a</sup> <sup>i</sup>* by the relation:

$$-\mathbf{M}\_i = -\mathbf{n}\_i \times \mathbf{E}\_{A\_i}^{S\_i^\epsilon} = \mathbf{n}\_{0i} \times \mathbf{E}\_{A\_i}^{S\_i^\epsilon} \tag{32}$$

where **n***<sup>i</sup>* is an inward normal in the region*Ai*, and **n**0*<sup>i</sup>* ¼ �**n***<sup>i</sup>* is the propagation direction of the incident wave. Thus, the solution of the internal problem is expressed by formulas analogous to those obtained in Section 2.3 with adding the index *i*, when necessary.

#### **4.3 Formulation of the external equivalent problem**

When considering the external equivalent problem, let *S<sup>c</sup> Bi* be the conducting boundary of the region *Bi*, including the inner sides of the waveguide walls and the conductive part of the boundary surface between the regions *Bi* and *Di*; *S<sup>c</sup> Di* is the conductive part of the boundary surface *SDi* , and *S<sup>c</sup> D*<sup>0</sup> is the conducting boundary of the region *D*0, including the outer sides of the waveguide walls and all conducting boundaries between the regions *D*<sup>0</sup> and *Di*. Further, *Sd DiBi* is the dielectric boundary between the regions *Di* and *Bi*, and *S<sup>d</sup> DiD*<sup>0</sup> is the dielectric boundary between the regions *Di* and *D*0. Per the equivalence principle [24], the dielectric boundaries between different regions can be replaced by oppositely directed equivalent electric and magnetic currents flowing on both sides of the dielectric interfaces.

The EM field in the waveguide region *Bi* is created by electric currents **J** *c Bi* flowing along the port surface *S<sup>a</sup> <sup>i</sup>* and conducting surface *Sc Bi* , equivalent electric and magnetic *Waveguide Port Approach in EM Simulation of Microwave Antennas DOI: http://dx.doi.org/10.5772/intechopen.102996*

currents �**J** *d DiBi* and �**M***<sup>d</sup> DiBi* flowing along the dielectric interfaces *<sup>S</sup><sup>d</sup> DiBi* , and equivalent magnetic currents **M***<sup>i</sup>* flowing along the port surface *S<sup>a</sup> <sup>i</sup> .* The EM field in the region *Di* is created by electric currents **J** *c Di* flowing along the conducting surfaces *<sup>S</sup><sup>c</sup> Di* , equivalent currents **J** *d DiBi* and **<sup>M</sup>***<sup>d</sup> DiBi* flowing on dielectric boundaries *<sup>S</sup><sup>d</sup> DiBi* between the regions *Di* and *Bi*, and equivalent currents **J** *d DiD*<sup>0</sup> and **<sup>M</sup>***<sup>d</sup> DiD*<sup>0</sup> flowing on dielectric boundaries *Sd DiD*<sup>0</sup> between the regions *Di* and *D*0. The field in the free space region *D*<sup>0</sup> is created by electric currents *Sc <sup>D</sup>*<sup>0</sup> flowing along the total conducting boundary of the region *D*0, and equivalent currents �**J** *d DiD*<sup>0</sup> and �**M***<sup>d</sup> DiD*<sup>0</sup> at dielectric boundaries between the regions *Di* and *D*0.

The unknown currents **J** *c Bi* , **M***i*,**J** *c Di* ,**J** *d DiBi* , **M***<sup>d</sup> DiBi* ,**J** *c <sup>D</sup>*<sup>0</sup> ,**J** *d DiD*<sup>0</sup> , **<sup>M</sup>***<sup>d</sup> DiD*<sup>0</sup> can be found from the boundary conditions on the port surface and the conducting boundaries of the waveguide region *Bi*:

$$\left[\mathbf{E}\_{B\_i}^{inc} + \mathbf{E}\_{B\_i}^{sc} \left(\mathbf{J}\_{B\_i}^c, \mathbf{M}\_i, -\mathbf{J}\_{D\_i B\_i}^d, -\mathbf{M}\_{D\_i B\_i}^d\right)\right] \Big|\_{\text{tan}}^{S\_i^d + S\_i^c} = \mathbf{0} \qquad \text{just outside } S\_i^d \tag{33}$$

$$\left[\mathbf{H}\_{B\_i}^{inc} + \mathbf{H}\_{B\_i}^{sc} \left(\mathbf{J}\_{B\_i}^c, \mathbf{M}\_i, -\mathbf{J}\_{D\_i B\_i}^d, -\mathbf{M}\_{D\_i B\_i}^d\right)\right] \Big|\_{\tan}^{\mathbf{S}\_i^s} = \mathbf{H}\_{A\_i}^{S\_i^s} \Big|\_{\tan} \quad \text{just inside } \mathbf{S}\_i^s,\tag{34}$$

and the boundary conditions on the conducting and dielectric boundaries of the regions *Di* and *D*0:

$$\left[\mathbf{E}\_{D\_i}^{inc} + \mathbf{E}\_{D\_i}^{c} \left(\mathbf{J}\_{D\_i}^{c}, \mathbf{J}\_{D\_i B\_i}^{d}, \mathbf{M}\_{D\_i B\_i}^{d}, \mathbf{J}\_{D\_i D\_0}^{d}, \mathbf{M}\_{D\_i D\_0}^{d}\right)\right] \Big|\_{\mathbf{t}\mathbf{n}} = \mathbf{0} \quad \text{on} \quad \mathbf{S}\_{D\_i}^{c} \tag{35}$$
 
$$\left[\mathbf{z}^{inc} \quad \mathbf{z}^{sc} \left(\mathbf{y} \quad \mathbf{M}\_{\cdot} \quad \mathbf{y}^d \quad \mathbf{M}\_{\cdot}^d\right)\right] \Big|\_{\mathbf{t}\mathbf{n}}$$

$$\begin{aligned} \left[\mathbf{E}\_{B\_i}^{inc} + \mathbf{E}\_{B\_i}^{sc} \left(\mathbf{J}\_{B\_i}^c, \mathbf{M}\_i, -\mathbf{J}\_{D\_i B\_i}^d, -\mathbf{M}\_{D\_i B\_i}^d\right)\right] \Big|\_{\tan} &= \\ \left. \left[\mathbf{E}\_{D\_i}^{inc} + \mathbf{E}\_{D\_i}^{sc} \left(\mathbf{J}\_{D\_i}^c, \mathbf{J}\_{D\_i B\_i}^d, \mathbf{M}\_{D\_i B\_i}^d, \mathbf{J}\_{D\_i D\_0}^d, \mathbf{M}\_{D\_i D\_0}^d\right)\right] \right|\_{\tan} \end{aligned} \qquad \text{on} \quad \mathbf{S}\_{D\_i B\_i}^d \tag{36}$$

$$\left| \mathbf{E}\_{D\_{i}}^{\rm inc} + \mathbf{E}\_{D\_{i}}^{\rm sc} \left( \mathbf{J}\_{D\_{i}}^{d}, \mathbf{J}\_{D\_{i}B\_{i}}^{d}, \mathbf{M}\_{D\_{i}B\_{i}}^{d}, \mathbf{J}\_{D\_{i}D\_{0}}^{d}, \mathbf{M}\_{D\_{i}D\_{0}}^{d} \right) \right|\_{\tan} $$
 
$$\left[ \mathbf{H}\_{B\_{i}}^{\rm inc} + \mathbf{H}\_{B\_{i}}^{\rm sc} \left( \mathbf{J}\_{D\_{i}}^{c}, \mathbf{M}^{(i)}, -\mathbf{J}\_{D\_{i}B\_{i}}^{d}, -\mathbf{M}\_{D\_{i}B\_{i}}^{d} \right) \right] \Big|\_{\tan} = \mathbf{ } \mathbf{ } \tag{37} $$

$$\begin{bmatrix} \mathbf{I} & \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}} \begin{bmatrix} \mathbf{I} & \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}} \begin{bmatrix} \mathbf{I} & \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}\mathbf{R}\_{i}} \begin{bmatrix} \mathbf{I} & \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}\mathbf{R}\_{i}} \begin{bmatrix} \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}\mathbf{R}\_{i}} \begin{bmatrix} \mathbf{I} \\ \end{bmatrix}\_{\text{D}\_{i}\mathbf{D}\_{0}} \begin{bmatrix} \mathbf{M}\_{\text{D}\_{i}\mathbf{D}\_{0}} \\ \end{bmatrix}\_{\text{D}\_{i}\mathbf{D}\_{0}} \end{bmatrix} \begin{bmatrix} \mathbf{0} & \mathbf{S}\_{\text{D}\_{i}\mathbf{B}\_{i}}^{d} \\ \end{bmatrix}\_{\text{tan}} \tag{37}$$

$$\left. \left[ \mathbf{E}\_{D\_0}^{inc} + \sum\_{i=1}^{N} \mathbf{E}\_{D\_0}^{sc} \left( \mathbf{J}\_{D\_0}^c, -\mathbf{J}\_{D\_i D\_0}^d, -\mathbf{M}\_{D\_i D\_0}^d \right) \right] \right|\_{\tan} = \mathbf{0} \quad \text{on} \quad \mathcal{S}\_{D\_0}^c \tag{38}$$

$$\begin{aligned} \left[\mathbf{E}\_{D\_{i}}^{inc} + \mathbf{E}\_{D\_{i}}^{sc} \left(\mathbf{J}\_{D\_{i}}^{\epsilon}, \mathbf{J}\_{D\_{i}B\_{i}}^{d}, \mathbf{M}\_{D\_{i}B\_{i}}^{d}, \mathbf{J}\_{D\_{i}D\_{0}}^{d}, \mathbf{M}\_{D\_{i}D\_{0}}^{d}\right)\right] \bigg|\_{\tan} &= \\ \left[\mathbf{E}\_{D\_{0}}^{inc} + \mathbf{E}\_{D\_{0}}^{sc} \left(\mathbf{J}\_{D\_{0}}^{\epsilon}\right) + \sum\_{i=1}^{N} \mathbf{E}\_{D\_{0}}^{sc} \left(-\mathbf{J}\_{D\_{i}D\_{0}}^{d}, -\mathbf{M}\_{D\_{i}D\_{0}}^{d}\right)\right] \bigg|\_{\tan} &\end{aligned} \qquad \text{on} \quad \mathbf{S}\_{D\_{i}D\_{0}}^{d} \tag{39}$$

$$\begin{aligned} \left[\mathbf{H}\_{D\_{i}}^{\rm inc} + \mathbf{H}\_{D\_{i}}^{c} \left(\mathbf{J}\_{D\_{i}}^{c}, \mathbf{J}\_{D\_{i}B\_{i}}^{d}, \mathbf{M}\_{D\_{i}B\_{i}}^{d}, \mathbf{J}\_{D\_{i}D\_{0}}^{d}, \mathbf{M}\_{D\_{i}D\_{0}}^{d}\right)\right] \bigg|\_{\rm tan} =\\ \left[\mathbf{H}\_{D\_{0}}^{\rm inc} + \mathbf{H}\_{D\_{0}}^{c} \left(\mathbf{J}\_{D\_{0}}^{c}\right) + \sum\_{i=1}^{N} \mathbf{H}\_{D\_{0}}^{c} \left(-\mathbf{J}\_{D\_{i}D\_{0}}^{d}, -\mathbf{M}\_{D\_{i}D\_{0}}^{d}\right)\right] \bigg|\_{\rm tan} \end{aligned} \qquad \text{on} \quad \mathbf{S}\_{D\_{i}D\_{0}}^{d} \tag{40}$$

The scattered EM fields in (33)–(40) are related to the equivalent electric and magnetic currents by Eqs. (26) and (27). After substituting (26) and (27) into (35)–(40), Eqs. (35)–(40) represent a coupled system of integral equations in terms of unknown currents for solving the coupling problem between several composite structures.

### **4.4 MoM solution of the external equivalent problem**

To solve the boundary problem (35)–(40), we use the following MoM expansions for the unknown currents:

$$\left[\mathbf{I}\_{B\_i}^{\mathbf{r}}\right]\_{i=1}^N = \sum\_{n=1}^{N^u + N\_B^u} I\_n^u \mathbf{f}\_n, \left[\mathbf{M}\_i\right]\_{i=1}^N = \sum\_{n=1}^{N^u} M\_n \mathbf{f}\_n, \left[\mathbf{f}\_{D\_i}^{\mathbf{r}}\right]\_{i=1}^N = \sum\_{n=1}^{N\_D^u} I\_n^{\odot} \mathbf{f}\_n, \mathbf{f}\_{D\_0}^{\mathbf{r}} = \sum\_{n=1}^{N\_D^u} I\_n^{\odot\_0} \mathbf{f}\_n,\tag{41}$$
 
$$\left[\mathbf{I}\_{D\_i B\_i}^{d}, \mathbf{J}\_{D\_i D\_0}^{d}\right]\_{i=1}^N = \sum\_{n=1}^{N^d} I\_n^d \mathbf{f}\_n, \left[\mathbf{M}\_{D\_i B\_i}^d, \mathbf{M}\_{D\_i D\_0}^d\right]\_{i=1}^N = \sum\_{n=1}^{N^d} M\_n^d \mathbf{f}\_n, \left[\mathbf{M}\_i\right]\_{i=1}^N = \sum\_{n=1}^{N^u} M\_n \mathbf{f}\_n \quad \text{(42)}$$

where **f***<sup>n</sup>* are the suitable BFs, *I cB <sup>n</sup>* ,*I cD <sup>n</sup>* ,*I cD*<sup>0</sup> *<sup>n</sup>* ,*<sup>I</sup> d <sup>n</sup>*, *Md <sup>n</sup>* and *Mn* are the unknown expansion current coefficients, and *N<sup>a</sup>*, *N<sup>c</sup> <sup>B</sup>*, *N<sup>c</sup> <sup>D</sup>*, *N<sup>c</sup> <sup>D</sup>*<sup>0</sup> and *<sup>N</sup><sup>d</sup>* are the numbers of these BFs on the surfaces *S<sup>a</sup> i* � �*<sup>N</sup> i*¼1 , *S<sup>c</sup> Bi* h i*<sup>N</sup> i*¼1 , *S<sup>c</sup> Di* h i*<sup>N</sup> i*¼1 , *S<sup>c</sup> <sup>D</sup>*<sup>0</sup> and *<sup>S</sup><sup>d</sup> DiBi* , *Sd DiD*<sup>0</sup> h i*<sup>N</sup> i*¼1 , respectively. Substituting now (41) and (42) into (35)–(40) with an accounting of (3), (5), and (26) and (27) for each *i*-th region and testing the obtained equations with weighting functions **w**1ð Þ**r** , **w**2ð Þ**r** ,… , **w***m*ð Þ**r** , defined in the range of the respective boundary operators, we obtain the following MoM system of linear algebraic equations:

*ZJ c BJ c <sup>B</sup> Z<sup>J</sup> c <sup>B</sup><sup>M</sup>* 0 0 *Z<sup>J</sup> c BJ d ZJ c BMd ZMJ<sup>c</sup> <sup>B</sup> <sup>Z</sup>MM* <sup>þ</sup> *<sup>Q</sup><sup>W</sup>* 0 0 *<sup>Z</sup>MJ<sup>d</sup> ZMM<sup>d</sup>* 0 0 *Z<sup>J</sup> c DJ c <sup>D</sup>* 0 *Z<sup>J</sup> c DJ d ZJ c DMd* 000 *Z<sup>J</sup> c D*0 *J c <sup>D</sup>*<sup>0</sup> *Z<sup>J</sup> c D*0 *J d ZJ c <sup>D</sup>*0*Md ZJ dJ c <sup>B</sup> Z<sup>J</sup> dJ c <sup>D</sup> Z<sup>J</sup> dJ c <sup>D</sup> Z<sup>J</sup> dJ c <sup>D</sup>*<sup>0</sup> *<sup>Z</sup><sup>Z</sup>JdJd ZJ dMd ZMdJ c <sup>B</sup> ZMdM ZMdJ c <sup>D</sup> ZMdJ c <sup>D</sup>*<sup>0</sup> *ZMdJ d ZMdMd* 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 *I cB M I cD I cD*<sup>0</sup> *I d Md* 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ¼ *VcB <sup>V</sup><sup>M</sup>* <sup>þ</sup> *<sup>V</sup><sup>W</sup> VcD VcD*<sup>0</sup> *Vd VHd* 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 (43)

where the elements of the block matrices are defined as: *Zαβ mn* ¼ � **<sup>w</sup>***m*, *<sup>L</sup>*^*αβ <sup>i</sup>* **f***<sup>n</sup>* D E, *<sup>L</sup>*^*αβ* is the respective boundary integral operator, superscripts *α*, *β* ¼ *J c <sup>B</sup>*, *M*, *J c <sup>D</sup>*, *J c <sup>D</sup>*<sup>0</sup> , *J <sup>d</sup>*, *M<sup>d</sup>* n o, voltage elements are defined in the same way as in Eq. (31), and the elements of the block matrices *Q<sup>W</sup>* and *V<sup>W</sup>* are expressed by (18) and (19) for each *i*-th feeding waveguide and determine the additional inclusions in the matrix and voltage elements due to the waveguide ports. The MoM system (43) defines a solution to the coupling problem between several composite geometries. In the structure of the MoM matrix of this solution, blocks of waveguide excitations, complex geometries, and couplings between them are clearly seen.

#### **4.5 Validation of the developed approach for coupling problems**

The developed approach has been validated on a two-element antenna array fed by coaxial waveguide ports by comparing the simulation results obtained using the developed MoM approach and the DGTD method [28]. **Figure 9** shows a schematic

*Waveguide Port Approach in EM Simulation of Microwave Antennas DOI: http://dx.doi.org/10.5772/intechopen.102996*

view of two identical monopole antennas flanged over the PEC plate and fed by coaxial waveguides with generally different diameters and dielectric fillings. The monopoles located at a distance *La* = 40 mm from each other have the same height *ha* = 10 mm above the PEC plate with a width *W* = 40 mm and a length *L* = 80 mm, which serves as a reflector. Coaxial waveguides have the same inner diameter *d*<sup>1</sup> = *d*<sup>2</sup> = 2 mm, but generally different outer diameters *D*<sup>1</sup> and *D*<sup>2</sup> and relative permittivities *ε*<sup>1</sup> and *ε*2. The depth of each coaxial waveguide under the flange is *hb* = 15 mm, and its end is taken as the reference plane of the waveguide port.

**Figure 10** shows the real and imaginary parts of the transmission coefficient *S*<sup>21</sup> ¼ *a*� 02*=a*<sup>þ</sup> <sup>01</sup> between waveguide ports 1 and 2 with the same radii and dielectric fillings: *D*1/2 *= D*2/2 = 6.65 mm, and *εr*<sup>1</sup> = *εr*<sup>2</sup> = 5.17, which leads to the same characteristic impedances: *Zc*<sup>1</sup> ¼ *Zc*<sup>2</sup> = 50 Ω. The developed MoM approach and the DGTD method are compared. The first antenna in these simulations is considered active, and the second is passive. Comparison of these results shows very good agreement between them over a wide frequency range from 1 GHz up to 10 GHz. This validates the developed approach in modeling coupling problems for coaxial waveguide ports with the same characteristic impedance.

#### **Figure 9.**

*Schematic view of an array of two identical monopole antennas fed by coaxial waveguides and flanged above the PEC plate.*

#### **Figure 10.**

*Transmission coefficient between the of the antenna array waveguide ports with the same parameters of the feeding coaxial waveguides.*

#### **Figure 11.**

*Transmission coefficient between the waveguide ports of the antenna array for the same permittivities* εr*<sup>1</sup> =* εr*<sup>2</sup> = 2.25, but different outer radii:* D*1/2* = *3.49 mm,* D*2/2 = 6.52 mm.*

#### **Figure 12.**

*Transmission coefficient between the waveguide ports of the antenna array for the same outer radii* D*1/2* = D*2/2 = 5.3 mm, but different permittivities:* εr*<sup>1</sup> = 4 and* εr*<sup>2</sup> = 1.78.*

**Figures 11** and **12** show a comparison of the transmission coefficient *S*<sup>21</sup> ¼ *a*� 02*=a*<sup>þ</sup> 01 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Zc*1*=Zc*<sup>2</sup> p between waveguide ports 1 and 2, calculated by the MoM and DGTD method for different parameters of coaxial waveguides. **Figure 11** is made for the same fillings of waveguides: *εr*<sup>1</sup> = *εr*<sup>2</sup> = 2.25, but with different outer radii: *D*1/2 *=* 3.49 mm and *D*2/2 = 6.52 mm, while **Figure 12** is performed for different fillings: *εr*<sup>1</sup> = 4 and *εr*<sup>2</sup> = 1.78, but with the same outer radii *D*1/2 *= D*2/2 = 5.3 mm. Both cases result in characteristic impedances of waveguides *Zc*<sup>1</sup> = 50 Ω and *Zc*<sup>2</sup> = 75 Ω. Comparison of the MoM and DGTD results again shows very good agreement between both simulated results, which validates the developed approach to modeling coupling problems for coaxial waveguide ports with different characteristic impedances.
