**2. Waveguide port approach for conducting geometry**

#### **2.1 Dividing the original problem into equivalent problems**

**Figure 1a** illustrates a canonical waveguide port problem for conducting geometry. This geometry consists of a semi-infinite waveguide *1* with perfect electric conducting (PEC) walls and a microwave structure *2*, which is yet supposed to be conductive. We intend to create port *P* in waveguide *1* to divide the geometry into two regions (*A* and *B*) to truncate the mesh in the region *A* and impose appropriate termination conditions in the port plane.

**Figure 1.**

*(a) Waveguide port problem for conducting geometry; (b) Equivalence for the internal region* A*; (c) Equivalence for the external region* B*.*

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

For this purpose, we follow the classical approach for the aperture problem [9] to divide the original problem into two equivalent problems, as shown in **Figure 1b** and **c**. We introduce a perfectly conducting surface *S<sup>a</sup>* into the port plane *P* to separate regions *A* and *B* and consider two equivalent sub-problems: internal (for region *A*) and external (for region *B*). In addition, we introduce equivalent magnetic currents and **M** on both sides of *Sa* to restore the tangential electric fields on the boundary surface *Sa*. Let us consider these equivalent problems separately.

#### **2.2 Formulation of the internal equivalent problem**

Consider an internal equivalent problem for the region *A*. The total EM field in the region *A* is composed of the incident field **E***inc*, **H***inc* and the reflected field **E***ref* , **H***ref* generated by magnetic currents in the presence of a conductor. According to the equivalence principle [24], these currents are related to the total electric field **E***<sup>S</sup><sup>a</sup> <sup>A</sup>* on the port surface *S<sup>a</sup>* by the relation:

$$-\mathbf{M} = -\mathbf{n} \times \mathbf{E}\_A^{\rm S^\circ} = \mathbf{n}\_0 \times \mathbf{E}\_A^{\rm S^\circ} \tag{1}$$

where **n** is the internal normal in the region *A*, and **n**<sup>0</sup> ¼ �**n** is the propagation direction of the incident wave.

Equation (1) relates the total electric field at the port surface *S<sup>a</sup>* to magnetic currents depending on the geometric and material properties of the external region *B*. The internal equivalent problem is to find the modal expansion of the total EM field at the port surface *S<sup>a</sup>* through these currents.

#### **2.3 MoM solution of the internal equivalent problem**

The total EM field in the region *A* on the port surface *S<sup>a</sup>* can be generally written as the sum of the incident (+) and reflected (�) TEM (if exists), TE, and TM modes [2, 13]:

$$\mathbf{E}\_A^{\rm S} = \left(a\_0^+ + a\_0^-\right) \left(\mathbf{e}\_0^{\rm TEM} + \sum\_{s=1}^{N^\rm T} \left(a\_s^+ + a\_s^-\right) \left(\mathbf{e}\_s^{\rm T} + \sum\_{s=1}^{N^\rm M} \left(b\_s^+ + b\_s^-\right)\right) \mathbf{e}\_s^{\rm TEM} \right) \tag{2}$$

$$\mathbf{H}\_A^{\rm S} = \frac{1}{Z} \left(a\_0^+ - a\_0^-\right) \left(\mathbf{n}\_0 \times \mathbf{e}\_0^{\rm TEM}\right) + \sum\_{s=1}^{N^\rm T} \frac{1}{Z\_s^{\rm TEM}} \left(a\_s^+ - a\_s^-\right) \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm T}\right) \tag{3}$$

$$+ \sum\_{s=1}^{N^\rm M} \frac{1}{Z\_s^{\rm TM}} \left(b\_s^+ - b\_s^-\right) \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TM}\right)$$

where *a*� *<sup>s</sup>* and *b*� *<sup>s</sup>* are the mode amplitudes, **e***TEM* <sup>0</sup> , **e***TE <sup>s</sup>* and **e***TM <sup>s</sup>* are the transverse modal functions of TEM, TE and TM waves with wave impedances *<sup>Z</sup>* <sup>¼</sup> ffiffiffiffiffiffiffi *μ=ε* p , *ZTE <sup>s</sup>* ¼ *ωμ=γ<sup>s</sup>* and *ZTM <sup>s</sup>* ¼ *γs=ωε*, respectively, *ε* and *μ* are the permittivity and permeability of the medium, *Γ<sup>s</sup>* is the propagation constant, and *NTE* and *NTM* are the numbers of accounted TE and TM modes, respectively.

Next, we use the MoM to relate the mode amplitudes of the reflected fields in (2), (3) to the magnetic currents **M**. The port surface *S<sup>a</sup>* is discretized into planar patches, and the unknown magnetic currents are approximated as

$$\mathbf{M} = \sum\_{n=1}^{N^\*} \mathbf{M}\_n \mathbf{f}\_n \tag{4}$$

where **f***<sup>n</sup>* are linear independent basis functions (BFs), *Mn* are unknown expansion current coefficients, and *N<sup>a</sup>* is the number of these BFs on the surface *Sa*. Substituting now (4) and (2) into (1), multiplying both sides by **<sup>n</sup>**<sup>0</sup> � **<sup>e</sup>***TEM* <sup>0</sup> , **<sup>n</sup>**<sup>0</sup> � **<sup>e</sup>***TE <sup>s</sup>*<sup>0</sup> and **<sup>n</sup>**<sup>0</sup> � **<sup>e</sup>***TM <sup>s</sup>*<sup>0</sup> , respectively, and integrating over the port surface *S<sup>a</sup>* , we relate the amplitudes of the reflected waves with those of the incident waves and magnetic current coefficients:

$$a\_s^- = -a\_s^+ - \sum\_{n=1}^{N^\epsilon} \frac{M\_n T\_{ns}}{R\_s}, \quad b\_s^- = -b\_s^+ - \sum\_{n=1}^{N^\epsilon} \frac{M\_n T\_{ns}'}{R\_s'} \quad (s = 0, 1, 2, \dots) \tag{5}$$

where

$$T\_{n0} = \underset{S\_a}{\left\{\mathbf{f}\_n \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_0^{\rm TEM}\right)dS'\right\}}{\left\{\mathbf{f}\_n\right\}} \quad T\_{ns} = \underset{S\_a}{\left\{\mathbf{f}\_n \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TEM}\right)dS'\right\}} \quad T'\_{ns} = \underset{S\_a}{\left\{\mathbf{f}\_n \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TEM}\right)\right\}} \tag{6}$$

$$\begin{split} R\_0 = \int\_{S\_s} \left( \mathbf{n}\_0 \times \mathbf{e}\_0^{\rm TEM} \right) \left( \mathbf{n}\_0 \times \mathbf{e}\_0^{\rm TEM} \right) dS', \quad R\_s = \int\_{S\_s} \left( \mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TEM} \right) \left( \mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TEM} \right) dS', \\ R\_s' = \int\_{S\_s} \left( \mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TCM} \right) \left( \mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TCM} \right) dS'. \end{split} \tag{7}$$

Substitution of (5) into (2), (3) determines the total electric and magnetic fields on the port surface in region A through the still unknown magnetic currents.

#### **2.4 Formulation of the external equivalent problem for conducting geometry**

Consider now an external equivalent problem for conducting geometry. The scattered EM field in the external region *B* in **Figure 1c** of the conducting geometry is produced by electric currents **J** flowing over surfaces *Sa* and *Sc* and equivalent magnetic currents **M** at the surface *S<sup>a</sup>* , which can be written as:

$$\mathbf{E}\_{\rm B}^{sc}(\mathbf{J}, \mathbf{M}) = L^{Ej}\mathbf{J} + L^{EM}\mathbf{M} \tag{8}$$

$$\mathbf{H}\_{B}^{\rm sc}(\mathbf{J}, \mathbf{M}) = L^{H\rm l}\mathbf{J} + L^{H\rm M}\mathbf{M} \tag{9}$$

where *LEJ*, *LEM*, *LHJ* and *LHM* are the linear integro-differential operators of electric and magnetic fields applied to the electric and magnetic currents, respectively. Applying the boundary conditions for the tangential electric and magnetic fields on the surfaces *S<sup>a</sup>* and *Sc* , we obtain the following system of integral equations for the unknown electric and magnetic currents **J** and **M**

$$\left. \mathbf{E}\_{\rm B}^{\rm ac}(\mathbf{J}, \mathbf{M}) \right|\_{\rm tan}^{\rm S^{\prime} + \rm S^{\prime}} = \mathbf{0} \qquad \text{just outside} \quad \rm S^{\prime} \tag{10}$$

$$\mathbf{H}\_{B}^{\rm c}(\mathbf{J}, \mathbf{M}) \big|\_{\tan}^{\rm (S'+S')} = \mathbf{H}\_{A}^{\rm S'} \big|\_{\tan} \quad \text{just inside} \quad \mathbf{S}^{\rm d} \tag{11}$$

#### **2.5 MoM solution of the external equivalent problem for conducting geometry**

To obtain the MoM solution to the BC (10) and (11), we consider, along with Eq. (4), the following expansion for an unknown electric current **J**:

$$\mathbf{J} = \sum\_{n=1}^{N^\epsilon + N^\epsilon} I\_n \mathbf{f}\_n,\tag{12}$$

where **f***<sup>n</sup>* are the BFs taken the same as for the expansion of magnetic currents in (4), *In* are the unknown expansion current coefficients on the surfaces *Sa* and *S<sup>c</sup>* , and *N<sup>a</sup>* and *N<sup>c</sup>* are the numbers of these BFs on these surfaces. Substitution of expansions (4) and (12) in (8) and (9) gives the following expressions for the EM field in region *B*:

$$\mathbf{E}\_{B}^{\rm sc}(\mathbf{J}, \mathbf{M}) = \sum\_{n=1}^{N^{\rm}+N^{\rm c}} I\_{n} L^{Ej} \mathbf{f}\_{n} + \sum\_{n=1}^{N^{\rm c}} M\_{n} L^{EM} \mathbf{f}\_{n},\tag{13}$$

$$\mathbf{H}\_{B}^{\rm sc}(\mathbf{J}, \mathbf{M}) = \sum\_{n=1}^{N^{\rm s} + N^{\rm c}} I\_{n} L^{Hj} \mathbf{f}\_{n} + \sum\_{n=1}^{N^{\rm s}} M\_{n} L^{HM} \mathbf{f}\_{n}. \tag{14}$$

Substituting now (3), (5), (13) and (14) into (10) and (11), introducing the boundary operators *<sup>L</sup>*^*JJ* <sup>¼</sup> *<sup>L</sup>EJ* � � � *Sa*þ*S<sup>c</sup>* , *<sup>L</sup>*^*JM* <sup>¼</sup> *<sup>L</sup>EM* � � � outside *<sup>S</sup>a*þ*S<sup>c</sup>* , *<sup>L</sup>*^*MJ* <sup>¼</sup> *<sup>L</sup>HJ* � � � inside *<sup>S</sup><sup>a</sup>* and *<sup>L</sup>*^*MM* <sup>¼</sup> *LHM* � � � *<sup>S</sup><sup>a</sup>* and testing the resulting equations with appropriate weighting functions **w**1ð Þ**r** , **w**2ð Þ**r** , … , **w***m*ð Þ**r** leads to the following system of linear algebraic equations

$$
\begin{bmatrix}
\begin{bmatrix}
\mathbf{Z}\_{mn}^{\mathrm{J}\mathbf{}} \\
\mathbf{Z}\_{mn}^{\mathrm{M}\mathbf{}}
\end{bmatrix}
\begin{bmatrix}
\mathbf{Z}\_{mn}^{\mathrm{M}\mathbf{}}
\end{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
I\_{n}
\end{bmatrix}
\end{bmatrix}
\begin{bmatrix}
\begin{bmatrix}
I\_{n}
\end{bmatrix}
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{0} \\
\begin{bmatrix}
\mathbf{V}\_{m}^{\mathrm{M}}
\end{bmatrix}
\end{bmatrix}
\tag{15}
$$

with elements defined as:

$$Z\_{mn}^{\mathcal{I}} = -\left\langle \mathbf{w}\_m, \hat{L}^{\mathcal{I}} \mathbf{f}\_n \right\rangle, \quad Z\_{mn}^{\mathcal{I}M} = -\left\langle \mathbf{w}\_m, \left( \bar{L}^{\mathcal{I}M} + \frac{1}{2} \mathbf{n} \times \right) \mathbf{f}\_n \right\rangle,\tag{16}$$

$$Z\_{mn}^{M\parallel} = -\left\langle \mathbf{w}\_m, \left(\ddot{L}^{M\parallel} + \frac{1}{2}\mathbf{n} \times \right) \mathbf{f}\_n \right\rangle, \quad Z\_{mn}^{M\perp} = -\left\langle \mathbf{w}\_m, \dot{L}^{M\mathbf{M}} \mathbf{f}\_n \right\rangle + Q\_{mn}^W,\tag{17}$$

$$Q\_{mn}^{W} = \frac{\hat{T}\_{m0}}{W} \frac{T\_{n0}}{R\_0} + \sum\_{s=1}^{N^{\text{TE}}} \frac{\hat{T}\_{ms}}{W\_s^{\text{TM}}} \frac{T\_{ns}}{R\_s} + \sum\_{s=1}^{N^{\text{TM}}} \frac{\hat{T}\_{ms}^{'}}{W\_s^{\text{TM}}} \frac{T\_{ns}^{'}}{R\_s^{\prime}},\tag{18}$$

$$V\_m^W = -2\left[\frac{\hat{T}\_{m0}}{W}a\_0^+ + \sum\_{s=1}^{N^{\text{TF}}} \frac{\hat{T}\_{ms}}{W\_s^{\text{TM}}}a\_s^+ + \sum\_{s=1}^{N^{\text{TM}}} \frac{\hat{T}\_{ms}^{\prime}}{W\_s^{\text{TM}}}b\_s^+\right],\tag{19}$$

where *<sup>L</sup>*~*JM* and *<sup>L</sup>*~*MJ* are the regular parts of the boundary operators *<sup>L</sup>*^*JM* and *<sup>L</sup>*^*MJ*, the notation h i **<sup>w</sup>**,**<sup>f</sup>** <sup>¼</sup> <sup>Ð</sup> *s* **w** � **f***dS* is used for the scalar product, and

$$\hat{T}\_{m0} = \underset{S\_a}{\left\|\mathbf{w}\_m \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_0^{\rm TEM}\right)dS'\right\|}{\left\|\mathbf{s}\_a\right\|}\\\text{S}\_s' = \underset{S\_a}{\left\|\mathbf{w}\_m \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TE}\right)dS'}\\\text{S}\_s' = \underset{S\_a}{\left\|\mathbf{w}\_m \cdot \left(\mathbf{n}\_0 \times \mathbf{e}\_s^{\rm TEM}\right)\right\|}{\left\|\mathbf{s}\_a\right\|}\\\tag{20}$$

In the case of Galerkin's procedure **w***<sup>m</sup>* ¼ **f***m*, and coefficients (20) and (6) become the same. The MoM system (15) determines the solution to the waveguide port problem in the conducting geometry.

#### **2.6 Validation of the developed approach for conducting geometry**

The developed approach has been validated to simulate the scattering characteristics of a flanged coaxial line as proposed in [25–27]. Such structures are frequently used in biomedical engineering for non-destructive testing of various materials [25–27].

When modeling a coaxial line, it is convenient to choose the port plane at the output of the line to provide fast damping of evanescent waves. In this case, it can be assumed *<sup>N</sup>TE* <sup>¼</sup> *<sup>N</sup>TM* <sup>¼</sup> 0 in (2) and (3) to take taken into account only fundamental, TEM mode with the modal function **e***TEM* <sup>0</sup> ¼ **e***ρ=*½ � *ρ* � ln ð Þ *D=d* , where *ρ* is the radial distance, **e***<sup>ρ</sup>* is the unit radial vector, and *D* and *d* are the outer and inner diameters of the coaxial waveguide.

**Figure 2** shows a flanged coaxial line consisting of a coaxial waveguide section with an outer radius *D*/2 = 4.725 mm, an inner radius of *d*/2 = 1.4364 mm, and a length *L* = 10 mm, ended with a circular disc with a diameter 2*R* = 200 mm. The bottom plane of the waveguide is accepted as a waveguide port, and the structure is excited in this port by TEM mode. To validate the developed approach for conducting geometry, we analyze the case when both the waveguide and outer space have the same permittivity *ε<sup>r</sup>* ¼ 2*:*05.

**Figure 3a** and **b** show the magnitude and phase of the reflection coefficient at the end of the coaxial line used as the reference plane. We compare the simulation results obtained using the developed approach, the mode-matching technique [25], and the

**Figure 2.**

*Geometry of open-ended coaxial line flanged with a circular disc:* D*/2 = 4.725 mm,* d*/2 = 1.4364 mm,* L *= 10 mm, 2*R *= 200 mm, ε<sup>r</sup>* ¼ 2*:*05 *inside and outside the line.*

#### **Figure 3.**

*(a) Magnitude and (b) phase of the reflection coefficient versus the frequency of excitation at the end of the flanged coaxial line, calculated for various approaches.*

matrix pencil method [26] with measurement data [27]. Note that infinite flanges are assumed in [25, 26]. Phase data conforms to the time convention exp ð Þ �*iω t :*

Comparison of various results shows excellent agreement between them. However, the phase characteristics obtained by our approach agree somewhat more accurately with the measurement data. Thus, the obtained results validate the developed approach to modeling a coaxial waveguide port for conducting geometries.
