**3.3. Formulation of the 2.5D TLM node for guiding structures**

To obtain the reduced TLM formulation for guiding structures, the voltages in each arm should be updated at each time step. In doing this, eight relationships between the reflected and incident voltages are needed. As mentioned before, stubs are added to model the background media with constitutive parameters *ε* and *µ*. Then, six additional relationships are needed for these stubs.

In [25], a procedure to find the required updating relationships based on Maxwell's equations is presented. To employ this procedure, six additional relationships are required to update all field components. The process is divided into two steps. The first step consists of finding the relationship between fields at the center of the node and the incident and reflected voltages taken at each arm. Then, the six relationships for the fields at instant *n* ∆*t* are found by using the integral form of Maxwell's equations in the three principal planes (*z*, *y*), (*x*, *z*), and (*x*, *y*). In the next step, field continuity equations at the center are applied by using the integral form of Maxwell's equations for contours passing through the center of the three principal planes. Then, 12 relationships for the voltages are found in terms of the fields at the center. By manipulating these expressions, the required eight relationships to update the voltages are found. Finally, the voltages for the stubs are calculated in the same way as the standard SCN node. The TLM algorithm is then completed.

### *3.3.1. Starting point: maxwell's equations in integral form*

The starting point is the integral form of Maxwell's equations for an isotropic medium and given by (6) and (7).

$$\oint\_{\mathcal{C}} \vec{\text{H}} \cdot \vec{\text{dl}} = \iint \hat{\varepsilon} \frac{\partial \vec{E}}{\partial t} \, d\vec{s} + \iint \partial\_{\varepsilon} \vec{E} \, \vec{ds} + \iint \vec{J}\_{\varepsilon} \vec{ds} \tag{6}$$

$$\oint\_{\mathcal{C}} \vec{E}.\vec{dl} = -\iint \hat{\mu}\frac{\partial \vec{H}}{\partial t}.\vec{ds} - \iint \sigma\_m \vec{E}.\vec{ds} - \iint \vec{J}\_m \vec{ds},\tag{7}$$

where *Je* and *J <sup>m</sup>* are the electric and magnetic current density sources, respectively; the tensors *σ*ˆ*e*, *µ*ˆ, *σ*ˆ*e*, *σ*ˆ *<sup>m</sup>* are given by *ε*ˆ*<sup>e</sup>* = *ε*<sup>0</sup> diag(*εxx*,*εyy*,*εzz*), *µ*ˆ*<sup>e</sup>* = *µ*<sup>0</sup> diag(*µxx*, *µyy*, *µzz*), *σ*ˆ*<sup>e</sup>* = diag(*σex*, *σey*, *σez*), *σ*ˆ *<sup>m</sup>* = diag(*σmx*, *σmy*, *σmz*) with *ε*<sup>0</sup> and *µ*<sup>0</sup> the permittivity and permeability of free space, respectively.

### *3.3.2. Field sampling*

Consider the standard 3D SCN node. Using the same notation for the SCN node proposed by Johns, samples for the electric and magnetic fields in the cell are presented in (Figure 2).

**Figure 2.** Field sampling in TLM, (a) electric field and (b) magnetic field

*En* and *Hn* correspond to field samples at each point of the cell. The incident *V<sup>i</sup>* and reflected *V<sup>r</sup>* voltages are related with these samples by (8). This equation gives us the relationship between the tangential fields at each face of the node and voltages in TLM network.

$$V^{i} = \frac{1}{2} \begin{pmatrix} \Delta x E\_{x}^{x} + Z\_{x} \Delta y H\_{2}^{y} \\ \Delta y E\_{3}^{y} + Z\_{xy} \Delta z H\_{3}^{z} \\ \Delta y E\_{4}^{y} - Z\_{zy} \Delta x H\_{4}^{x} \\ \Delta z E\_{6}^{y} - Z\_{xz} \Delta y H\_{6}^{y} \\ \Delta y E\_{8}^{y} + Z\_{zy} \Delta x H\_{8}^{z} \\ \Delta x E\_{9}^{y} - Z\_{xz} \Delta y H\_{9}^{y} \\ \Delta z E\_{10}^{z} - Z\_{xy} \Delta z H\_{11}^{z} \end{pmatrix} V^{r} = \frac{1}{2} \begin{pmatrix} \Delta x E\_{x}^{x} - Z\_{x\Delta} \Delta y H\_{2}^{y} \\ \Delta y E\_{3}^{y} - Z\_{xy} \Delta z H\_{3}^{z} \\ \Delta y E\_{4}^{y} + Z\_{zy} \Delta x H\_{6}^{x} \\ \Delta z E\_{6}^{y} - Z\_{xy} \Delta x H\_{8}^{y} \\ \Delta y E\_{9}^{y} - Z\_{xy} \Delta x H\_{8}^{y} \\ \Delta x E\_{9}^{y} + Z\_{xz} \Delta y H\_{9}^{y} \\ \Delta x E\_{10}^{z} - Z\_{xy} \Delta x H\_{11}^{z} \end{pmatrix} \tag{8}$$

 *E* and *H* denote the electric and magnetic fields, respectively; *Zij* are the characteristic impedances of each arm. They are equal to *Z*<sup>0</sup> for the SCN node. Finally, ∆*x*, ∆*y*, and ∆*z* correspond to the cell dimensions. Note that the voltages *V*1, *V*5, *V*7, and *V*<sup>12</sup> of the SCN node do not appear since they were removed. Six additional voltages (incident and reflected) have to be considered to model materials with a relative permittivity and/or permeability higher than the unity. They can be obtained by using the vector:

$$\vec{V}\_E = \frac{1}{2} \left[ \Delta x E\_{13}^x \,\Delta y E\_{14}^y \,\Delta z E\_{15}^z \right] \tag{9}$$

$$\vec{V}\_H = \frac{1}{2} Z\_0 \left[ \Delta x H\_{16}^x \,\Delta y H\_{17}^y \,\Delta z H\_{18}^z \right] \tag{10}$$

### *3.3.3. Determination of the updating equations for the fields*

To find the updating relationships for the electric and magnetic fields, an approximation of Maxwell's equations in the integral form must be done. This can be accomplished by considering the rectangular mesh of Figure 2 and applying (6) and (7) in the main integration planes, (*z*, *y*), (*x*, *z*), and (*x*, *y*).

#### 3.3.3.1. Integration in the (*z*, *y*)-plane

*3.3.1. Starting point: maxwell's equations in integral form*

 *C H* .*dl* = *ε*ˆ *∂ E ∂t* .*ds* + *σ*ˆ*e E*.*ds* + 

 *C E*.*dl* <sup>=</sup> <sup>−</sup>

**Figure 2.** Field sampling in TLM, (a) electric field and (b) magnetic field

∆*xE<sup>x</sup>*

∆*yE<sup>y</sup>*

∆*yE<sup>y</sup>*

∆*zE<sup>z</sup>*

∆*yE<sup>y</sup>*

∆*xE<sup>x</sup>*

∆*zE<sup>z</sup>*

∆*yE<sup>y</sup>*

*V<sup>i</sup>* = <sup>1</sup> 2  *µ*ˆ *∂H <sup>∂</sup><sup>t</sup>* .*ds* <sup>−</sup> *σ*ˆ *m E*.*ds* <sup>−</sup> *J <sup>m</sup>*.*ds*

given by (6) and (7).

246 Advanced Electromagnetic Waves

where *Je* and *J*

tensors *σ*ˆ*e*, *µ*ˆ, *σ*ˆ*e*, *σ*ˆ

diag(*σex*, *σey*, *σez*), *σ*ˆ

*3.3.2. Field sampling*

of free space, respectively.

The starting point is the integral form of Maxwell's equations for an isotropic medium and

Consider the standard 3D SCN node. Using the same notation for the SCN node proposed by Johns, samples for the electric and magnetic fields in the cell are presented in (Figure 2).

*En* and *Hn* correspond to field samples at each point of the cell. The incident *V<sup>i</sup>* and reflected *V<sup>r</sup>* voltages are related with these samples by (8). This equation gives us the relationship

between the tangential fields at each face of the node and voltages in TLM network.

2

*V<sup>r</sup>* = <sup>1</sup> 2 ∆*xE<sup>x</sup>*

∆*yE<sup>y</sup>*

∆*yE<sup>y</sup>*

∆*zE<sup>z</sup>*

∆*yE<sup>y</sup>*

∆*xE<sup>x</sup>*

∆*zE<sup>z</sup>*

∆*yE<sup>y</sup>*

<sup>2</sup> <sup>−</sup> *Zzx*∆*yH<sup>y</sup>*

<sup>3</sup> <sup>−</sup> *Zxy*∆*zH<sup>z</sup>*

<sup>4</sup> <sup>+</sup> *Zzy*∆*xH<sup>x</sup>*

<sup>6</sup> <sup>+</sup> *Zxz*∆*yH<sup>y</sup>*

<sup>8</sup> <sup>−</sup> *Zzy*∆*xH<sup>x</sup>*

<sup>9</sup> <sup>+</sup> *Zzx*∆*yH<sup>y</sup>*

<sup>10</sup> <sup>−</sup> *Zxz*∆*yH<sup>y</sup>*

<sup>11</sup> <sup>+</sup> *Zxy*∆*zH<sup>z</sup>*

2

(8)

3

4

6

8

9

10

11

3

4

6

8

9

10

11

<sup>2</sup> <sup>+</sup> *Zzx*∆*yH<sup>y</sup>*

<sup>3</sup> <sup>+</sup> *Zxy*∆*zH<sup>z</sup>*

<sup>4</sup> <sup>−</sup> *Zzy*∆*xH<sup>x</sup>*

<sup>6</sup> <sup>−</sup> *Zxz*∆*yH<sup>y</sup>*

<sup>8</sup> <sup>+</sup> *Zzy*∆*xH<sup>x</sup>*

<sup>9</sup> <sup>−</sup> *Zzx*∆*yH<sup>y</sup>*

<sup>10</sup> <sup>+</sup> *Zxz*∆*yH<sup>y</sup>*

<sup>11</sup> <sup>−</sup> *Zxy*∆*zH<sup>z</sup>*

*<sup>m</sup>* are the electric and magnetic current density sources, respectively; the

*<sup>m</sup>* are given by *ε*ˆ*<sup>e</sup>* = *ε*<sup>0</sup> diag(*εxx*,*εyy*,*εzz*), *µ*ˆ*<sup>e</sup>* = *µ*<sup>0</sup> diag(*µxx*, *µyy*, *µzz*), *σ*ˆ*<sup>e</sup>* =

*<sup>m</sup>* = diag(*σmx*, *σmy*, *σmz*) with *ε*<sup>0</sup> and *µ*<sup>0</sup> the permittivity and permeability

*Je*.*ds*

(6)

, (7)

Figure 3 shows the contour of integration for the magnetic and electric fields in the (*z*, *y*)-plane. The samples are taken according to Figure 2 and considering the exponential dependence of the fields in the propagation direction.

**Figure 3.** Integration of Maxwell's equations in the integral form in the plane (*z*, *y*). (a) H-field and (b) E-field

Maxwell-Ampère and Maxwell-Faraday equations applied to the contour Γ bounding this plane are given by (13) and (14), respectively. Moreover, electric and magnetic laws are given by (15) and (16), respectively. The calculation of these integrals is done by approximating their values as the mean value of the same integral at the time steps (*n* + 1/2)∆*t* and (*n* − 1/2)∆*t*. For the contour integrals, the approximation

$$\int\_{(n-1/2)\Delta t}^{(n+1/2)\Delta t} O(t)dt \approx \Delta t \left(\frac{1+T}{2}\right)\tilde{O}\_{\prime} \tag{11}$$

is used. The term *<sup>O</sup>* <sup>=</sup> <sup>Γ</sup> *<sup>H</sup>* .*dl* or *<sup>O</sup>* <sup>=</sup> Γ *E*.*dl* . Regarding the surface integrals, they can be calculated using

$$\int\_{(n-1/2)\Delta t}^{(n+1/2)\Delta t} \frac{P(t)}{dt} dt \approx (1-T)P\_\prime \tag{12}$$

where *P*(*t*) = *<sup>S</sup> <sup>D</sup>* .*dS* or *<sup>P</sup>*(*t*) = *S B*.*dS* .

The application of the above expressions gives

$$\oint\_{\mathbb{C}} \vec{H} \, d\vec{l} \cong \left(\frac{1+T}{2}\right) 2 \sin\left(\frac{\beta \Delta y}{2}\right) e^{-j\frac{\beta y\_0}{2}} \left[\frac{1}{\beta} \left(H\_2^y - H\_9^y\right) - j\Delta z H\_z^{(n)}\right] \tag{13}$$

$$\oint\_{\mathbb{C}} \vec{E} \cdot \vec{dl} \cong \left(\frac{1+T}{2}\right) 2\sin\left(\frac{\beta \Delta y}{2}\right) e^{-j\frac{\beta y\_0}{2}} \left[\frac{1}{\beta} \left(E\_4^y - E\_8^y\right) - j\Delta z E\_2^{(n)}\right] \tag{14}$$

$$\varepsilon \int\_{S} \frac{\partial \vec{E\_{x}}}{\partial t} \vec{dS} \cong (1 - T) \frac{\varepsilon\_{x} \Delta y \Delta z}{Z\_{0} c\_{0} \Delta t \beta \Delta x} \Delta x \tilde{E\_{x}} e^{-j\beta y\_{0}} 2 \sin \left(\frac{\beta \Delta y}{2}\right) \tag{15}$$

$$-\mu \iint\_{S} \frac{\partial \vec{H}\_{\text{x}}}{\partial t} \, d\vec{S} \cong (T - 1) \, \frac{Z\_{0} \mu\_{\text{x}} \Delta y \Delta z}{c\_{0} \Delta t \beta \Delta x} \Delta x \mathcal{H}\_{\text{x}} e^{-j\beta y\_{0}} \Delta \sin\left(\frac{\beta \Delta y}{2}\right) \tag{16}$$

*T* corresponds to a delay operator, *Z*<sup>0</sup> is the characteristic impedance in free space, and *c*<sup>0</sup> the speed of light in vacuum. The terms *E*˜ *<sup>x</sup>* and *H*˜ *<sup>x</sup>* correspond to an approximation of the fields at instant *n*∆*t*. They can be calculated as the mean value of the field components in the plane:

$$E\_{\mathbf{x}} = \frac{E\_2^{\mathbf{x}} + E\_9^{\mathbf{x}} + Y\_{\mathbf{x}\mathbf{x}} E\_{13}^{\mathbf{x}}}{Y\_{\mathbf{x}\mathbf{x}} + 2} \approx E\_{\mathbf{x}}^{(n)}\tag{17}$$

$$H\_{\rm x} = \frac{H\_4^{\rm x} + H\_8^{\rm x} + Z\_{\rm sx}^{\rm \uparrow} H\_{16}^{\rm x}}{Z\_{\rm sx}^{\rm \uparrow} + 2} \approx H\_{\rm x}^{(n)}\tag{18}$$

where *Y*ˆ *sx* and *Z*ˆ *sx* are the normalized admittances and impedances, respectively, to model the materials with a relative permeability and/or permittivity higher than unity. Their values are given by

 $\hat{Y}\_{\rm si} + 2 = 2 \frac{\varepsilon\_0 \Delta \mathbf{j} \Delta \mathbf{k}}{c\_0 \Delta t \Delta l} \text{ for } \mathbf{i} \in (\mathbf{x}, \mathbf{z})$   $\hat{Z}\_{\rm si} + 2 = 2 \frac{\mu\_i \Delta \mathbf{j} \Delta \mathbf{k}}{c\_0 \Delta t \Delta l} \text{ for } \mathbf{i} \in (\mathbf{x}, \mathbf{z}) \tag{19}$ 

Regarding loss terms, the approximation of the corresponding integrals in (13) is done by taking their value at the center of the cell. This leads to

$$\int\_{S} \int\_{S} \sigma\_{\mathfrak{e}\mathfrak{f}} \vec{E}\_{\mathfrak{f}} \, d\mathfrak{S} \cong \mathbb{G}\_{\mathfrak{e}\mathfrak{f}} \left[ \Delta \mathbb{Q} \tilde{E}\_{\mathfrak{f}} \right] = \sigma\_{\mathfrak{e}\mathfrak{g}} \frac{\Delta \chi \Delta \rho}{\Delta \mathfrak{g}} \left[ \Delta \mathbb{Q} \tilde{E}\_{\mathfrak{f}} \right] \tag{20}$$

$$\int\_{S} \int\_{S} \sigma\_{m\zeta} \vec{H}\_{\zeta} \, d\mathfrak{S} \cong \mathbb{R}\_{m\zeta} \left[ \Delta \zeta \mathcal{H}\_{\zeta} \right] = \sigma\_{m\zeta} \frac{\Delta \chi \Delta \rho}{\Delta \zeta} \left[ \Delta \zeta \mathcal{H}\_{\zeta} \right],\tag{21}$$

where *ζ* denotes the field component *x*, *y*, or *z* and *Ge<sup>ζ</sup>* and *Rm<sup>ζ</sup>* correspond to the electric conductance and magnetic resistance, respectively, for each direction. Finally, the surface integrals of the sources are evaluated by taking their values at the center:

$$\int\_{S} \int\_{S} \mathbf{J}\_{\mathfrak{e}\mathfrak{J}} \, d\mathbf{S} \cong V\_{\mathfrak{e}\mathfrak{J}} \tag{22}$$

$$\int\_{S} \int\_{S} \mathbf{J}\_{m\zeta} \, d\mathbf{S} \cong V\_{m\zeta} \Delta S\_{\prime} \tag{23}$$

where *Ve<sup>ζ</sup>* = *Jeζ*∆*S* and *Vm<sup>ζ</sup>* = *Jmζ*∆*S*. The infinitesimal area can be calculated, for instance, as ∆*S* = ∆*y*∆*z* for the *x*-direction (*ζ* = *x*).

By replacing the above expression in Maxwell's equations (6) and (7), the expressions in (24) and (25) can be obtained. Equation (8) was employed to express the fields in terms of reflected and incident voltages.

$$V\_2^{\prime} + V\_9^{\prime} + \hat{Y}\_{\rm xx} V\_{13}^{\prime} = V\_2^i + V\_9^i + \hat{Y}\_{\rm xx} V\_{13}^i - \mathfrak{a}^{\prime} \mathbf{Z}\_0 \Delta z H\_z^{(n)} - \frac{\mathcal{R}\_{\rm xx}}{\mathcal{Z}\_0} \Delta z H\_x^{(n)} - \frac{V\_{\rm xx}}{\mathcal{Z}\_0} \tag{24}$$

$$V\_4^r - V\_8^r + Z\_{\rm sx}^{\prime} V\_{16}^r = -V\_4^i + V\_8^i + Z\_{\rm sx}^{\prime} V\_{16}^i + \mathfrak{a}^{\prime} \Delta z E\_z^{(n)} - \frac{R\_{\rm xx}}{Z\_0} \Delta z H\_{\rm x}^{(n)} - \frac{V\_{\rm xx}}{Z\_0},\tag{25}$$

where *α* = *β*∆*y*/2.

 (*n*+1/2)∆*t* (*n*−1/2)∆*t*

or *<sup>O</sup>* <sup>=</sup>

 (*n*+1/2)∆*t* (*n*−1/2)∆*t*

> *S B*.*dS* .

> > *β*∆*y* 2

 *β*∆*y* 2

*<sup>∂</sup><sup>t</sup>* .*dS* <sup>∼</sup><sup>=</sup> (<sup>1</sup> <sup>−</sup> *<sup>T</sup>*) *<sup>ε</sup>x*∆*y*∆*<sup>z</sup>*

*<sup>∂</sup><sup>t</sup>* .*dS* <sup>∼</sup><sup>=</sup> (*<sup>T</sup>* <sup>−</sup> <sup>1</sup>) *<sup>Z</sup>*0*µx*∆*y*∆*<sup>z</sup>*

<sup>Γ</sup> *<sup>H</sup>* .*dl*

*<sup>S</sup> <sup>D</sup>* .*dS* or *<sup>P</sup>*(*t*) =

1 + *T* 2

1 + *T* 2

*∂E<sup>x</sup>*

*∂H<sup>x</sup>*

 2*sin*

 2*sin*

*E*˜ *<sup>x</sup>* <sup>=</sup> *<sup>E</sup><sup>x</sup>*

*H*˜ *<sup>x</sup>* <sup>=</sup> *<sup>H</sup><sup>x</sup>*

The application of the above expressions gives

is used. The term *<sup>O</sup>* <sup>=</sup>

248 Advanced Electromagnetic Waves

 *C H* .*dl* ∼=

 *C E*.*dl* ∼=

> *ε S*

− *µ S*

plane:

where *Y*ˆ

are given by

*sx* and *Z*ˆ

*Y*ˆ

*si* <sup>+</sup> <sup>2</sup> <sup>=</sup> <sup>2</sup> *<sup>ε</sup>i*∆*j*∆*<sup>k</sup>*

the speed of light in vacuum. The terms *E*˜

calculated using

where *P*(*t*) =

*O*(*t*)*dt* ≈ ∆*t*

*P*(*t*)

 *e* −*j βy*<sup>0</sup> 1 *β Hy* <sup>2</sup> <sup>−</sup> *<sup>H</sup><sup>y</sup>* 9 

 *e* −*j βy*<sup>0</sup> 1 *β Ey* <sup>4</sup> <sup>−</sup> *<sup>E</sup><sup>y</sup>* 8 

*Z*0*c*0∆*tβ*∆*x*

*c*0∆*tβ*∆*x*

*T* corresponds to a delay operator, *Z*<sup>0</sup> is the characteristic impedance in free space, and *c*<sup>0</sup>

fields at instant *n*∆*t*. They can be calculated as the mean value of the field components in the

<sup>9</sup> <sup>+</sup> *<sup>Y</sup>*<sup>ˆ</sup> *sxE<sup>x</sup>* 13

<sup>8</sup> <sup>+</sup> *<sup>Z</sup>*<sup>ˆ</sup>

the materials with a relative permeability and/or permittivity higher than unity. Their values

*sxH<sup>x</sup>* 16

*sx* are the normalized admittances and impedances, respectively, to model

*si* <sup>+</sup> <sup>2</sup> <sup>=</sup> <sup>2</sup> *<sup>µ</sup>i*∆*j*∆*<sup>k</sup>*

<sup>2</sup> + *<sup>E</sup><sup>x</sup>*

<sup>4</sup> <sup>+</sup> *<sup>H</sup><sup>x</sup>*

*<sup>c</sup>*0∆*t*∆*<sup>i</sup>* for i <sup>∈</sup> (*x*, *<sup>z</sup>*) *<sup>Z</sup>*<sup>ˆ</sup>

*Y*ˆ

*Z*ˆ

*<sup>x</sup>* and *H*˜

∆*xE*˜ *xe*

> ∆*xH*˜ *xe*

<sup>−</sup>*jβy*<sup>0</sup> 2*sin*

<sup>−</sup>*jβy*<sup>0</sup> 2*sin*

Γ *E*.*dl* 1 + *T* 2

. Regarding the surface integrals, they can be

*dt dt* <sup>≈</sup> (<sup>1</sup> <sup>−</sup> *<sup>T</sup>*)*P*, (12)

<sup>−</sup> *<sup>j</sup>* <sup>∆</sup>*zH*(*n*) *<sup>z</sup>*

<sup>−</sup> *<sup>j</sup>* <sup>∆</sup>*zE*(*n*) *<sup>z</sup>*

 *β*∆*y* 2

> *β*∆*y* 2

*<sup>x</sup>* correspond to an approximation of the

*sx* <sup>+</sup> <sup>2</sup> <sup>≈</sup> *<sup>E</sup>*(*n*) *<sup>x</sup>* (17)

*sx* <sup>+</sup> <sup>2</sup> <sup>≈</sup> *<sup>H</sup>*(*n*) *<sup>x</sup>* (18)

*<sup>c</sup>*0∆*t*∆*<sup>i</sup>* for i ∈ (*x*, *<sup>z</sup>*) (19)

(13)

(14)

(15)

(16)

*O*, (11)

3.3.3.2. Integration in the (*x*, *z*)-plane

This plane is perpendicular to the propagation direction; thus, any integral is affected by the exponential dependence of the fields. They can be calculated as for the standard SCN node. The contours of integration for the magnetic and electric fields in the (*x*, *z*)-plane are shown in Figure 4, respectively.

Maxwell-Ampere and Maxwell-Faraday equations applied to these contours are given by

$$\oint\_{\mathcal{C}} \vec{H} \, d\vec{l} \cong \left(\frac{1+T}{2}\right) \left[\Delta x \left(H\_4^\mathbf{x} - H\_8^\mathbf{x}\right) - \Delta z \left(H\_{11}^\mathbf{z} - H\_3^\mathbf{z}\right)\right] \tag{26}$$

**Figure 4.** Integration contour of Maxwell's equations in the integral form in the plane (*x*, *z*). (a) H-field and (b) E-field

$$\oint\_{\mathcal{C}} \vec{E}.\vec{dl} \cong \left(\frac{1+T}{2}\right) \left[\Delta x \left(E\_2^{\mathcal{X}} - E\_9^{\mathcal{X}}\right) - \Delta z \left(E\_{10}^{\mathcal{Z}} - E\_6^{\mathcal{Z}}\right)\right] \tag{27}$$

whereas the right-hand side of the Maxwell-Ampere and Maxwell-Faraday equations are given by:

$$RHS\_{MA} \cong (1 - T) \frac{\varepsilon\_y \Delta x \Delta z}{Z\_0 c\_0 \Delta t \Delta y} \Delta y \tilde{E}\_y + Z\_0 G\_{\text{eff}} \Delta y \tilde{E}\_y^n + Z\_0 V\_{\text{eff}} \tag{28}$$

$$RHS\_{MF} \cong (T - 1) \frac{Z\_0 \mu\_y \Delta x \Delta z}{c\_0 \Delta t \Delta y} \Delta y H\_y - \frac{R\_{my}}{Z\_0} \Delta y H\_y^n - \frac{V\_{my}}{Z\_0},\tag{29}$$

where *Gey* = *σey*∆*z*∆*y*/∆*x* is the term employed to model the electric losses, *Rmy* = *σmy*∆*z*∆*y*/∆*x* models the magnetic losses, *T* and *Z*<sup>0</sup> were already defined, *Vey* = *Jey*∆*z*∆*y*, *Vmy* = *Jmy*∆*z*∆*y*, and *c*<sup>0</sup> the speed of light in vacuum. *RHSMA* and *RHSMF* correspond to the right-hand side of Maxwell's Faraday and Maxwell's (6) and (7), respectively; the terms *E*˜ *<sup>y</sup>* and *H*˜ *<sup>y</sup>* correspond to fields approximations of the fields at instant *n*∆*t* given by

$$E\_y = \frac{E\_4^y + E\_8^y + E\_{11}^y + E\_3^y + \hat{Y}\_{sy} E\_{14}^y}{\hat{Y}\_{sy} + 4} \approx E\_y^{(n)}\tag{30}$$

$$H\_y = \frac{H\_2^y + H\_{10}^y - H\_9^y - H\_6^y + Z\_{sy} H\_{17}^y}{Z\_{sy} + 4} \approx H\_y^{(n)}\tag{31}$$

In this case, the normalized admittances and impedances *Y*ˆ *sy* and *Z*ˆ *sy* are given, respectively, by

$$\mathbf{Y}\_{\text{sl}} + 4 = 2 \frac{\varepsilon\_l \Delta \mathbf{j} \Delta k}{c\_0 \Delta l \Delta l} \text{ for } \mathbf{i} = \mathbf{y} \text{ } \hat{Z}\_{\text{sl}} + 4 = 2 \frac{\mu\_l \Delta j \Delta k}{c\_0 \Delta l \Delta l} \text{ for } \mathbf{i} = \mathbf{y} \tag{32}$$

Finally, (26) with (28) and (27) with (29), one obtains (33) and(34), respectively. Again, equation (8) was employed to express the equation in terms of reflected and incident voltages.

$$\begin{aligned} V\_3^r + V\_{11}^r + V\_4^r + V\_8^r + \hat{Y}\_{sy}^r V\_{14}^r &= \\ V\_3^i + V\_{11}^i + V\_4^i + V\_8^i + \hat{Y}\_{sy}^i V\_{14}^i - Z\_0 \mathbf{G}\_{ey} \Delta y E\_y^{(n)} - Z\_0 V\_{ey} \end{aligned} \tag{33}$$

$$\begin{array}{c} -V\_2^r - V\_{10}^r + V\_9^r + V\_6^r + Z\_{sy}^r V\_{17}^r = \\ V\_2^i + V\_{10}^i - V\_9^i - V\_6^i + Z\_{sy}^i V\_{17}^i - \frac{R\_{my}}{Z\_0} \Delta y H\_y^{(n)} - \frac{V\_{my}}{Z\_0} \end{array} \tag{34}$$

#### 3.3.3.3. Integration in the (*x*, *y*)-plane

**Figure 4.** Integration contour of Maxwell's equations in the integral form in the plane (*x*, *z*). (a) H-field and (b) E-field

whereas the right-hand side of the Maxwell-Ampere and Maxwell-Faraday equations are

where *Gey* = *σey*∆*z*∆*y*/∆*x* is the term employed to model the electric losses, *Rmy* = *σmy*∆*z*∆*y*/∆*x* models the magnetic losses, *T* and *Z*<sup>0</sup> were already defined, *Vey* = *Jey*∆*z*∆*y*, *Vmy* = *Jmy*∆*z*∆*y*, and *c*<sup>0</sup> the speed of light in vacuum. *RHSMA* and *RHSMF* correspond to the right-hand side of Maxwell's Faraday and Maxwell's (6) and (7), respectively; the terms

*<sup>y</sup>* correspond to fields approximations of the fields at instant *n*∆*t* given by

<sup>11</sup> <sup>+</sup> *<sup>E</sup><sup>y</sup>*

<sup>9</sup> <sup>−</sup> *<sup>H</sup><sup>y</sup>*

<sup>3</sup> <sup>+</sup> *<sup>Y</sup>*<sup>ˆ</sup> *syE<sup>y</sup>* 14

<sup>6</sup> <sup>+</sup> *<sup>Z</sup>*<sup>ˆ</sup>

*sy* <sup>+</sup> <sup>4</sup> <sup>≈</sup> *<sup>E</sup>*(*n*)

*syH<sup>y</sup>* 17

*sy* and *Z*ˆ

*sy* <sup>+</sup> <sup>4</sup> <sup>≈</sup> *<sup>H</sup>*(*n*)

*si* <sup>+</sup> <sup>4</sup> <sup>=</sup> <sup>2</sup> *<sup>µ</sup>i*∆*j*∆*<sup>k</sup>*

<sup>2</sup> <sup>−</sup> *<sup>E</sup><sup>x</sup>*

∆*yE*˜

∆*yH*˜

<sup>9</sup> ) <sup>−</sup> <sup>∆</sup>*<sup>z</sup>* (*E<sup>z</sup>*

*<sup>y</sup>* + *Z*0*Gey*∆*yE<sup>n</sup>*

∆*yH<sup>n</sup>*

*<sup>y</sup>* <sup>−</sup> *Vmy Z*0

*<sup>y</sup>* <sup>−</sup> *Rmy Z*0

<sup>10</sup> <sup>−</sup> *<sup>E</sup><sup>z</sup>*

<sup>6</sup>)] (27)

*<sup>y</sup>* + *Z*0*Vey* (28)

*<sup>y</sup>* (30)

*<sup>y</sup>* (31)

*sy* are given, respectively,

*<sup>c</sup>*0∆*t*∆*<sup>i</sup>* for i = y (32)

, (29)

[∆*x* (*E<sup>x</sup>*

1 + *T* 2

*RHSMA* <sup>∼</sup><sup>=</sup> (<sup>1</sup> <sup>−</sup> *<sup>T</sup>*) *<sup>ε</sup>y*∆*x*∆*<sup>z</sup>*

*RHSMF* ∼= (*T* − 1)

*E*˜ *<sup>y</sup>* <sup>=</sup> *<sup>E</sup><sup>y</sup>*

*H*˜ *<sup>y</sup>* <sup>=</sup> *<sup>H</sup><sup>y</sup>*

*Y*ˆ

In this case, the normalized admittances and impedances *Y*ˆ

*si* <sup>+</sup> <sup>4</sup> <sup>=</sup> <sup>2</sup> *<sup>ε</sup>i*∆*j*∆*<sup>k</sup>*

<sup>4</sup> <sup>+</sup> *<sup>E</sup><sup>y</sup>*

<sup>2</sup> <sup>+</sup> *<sup>H</sup><sup>y</sup>*

<sup>8</sup> <sup>+</sup> *<sup>E</sup><sup>y</sup>*

<sup>10</sup> <sup>−</sup> *<sup>H</sup><sup>y</sup>*

*<sup>c</sup>*0∆*t*∆*<sup>i</sup>* for i = y *<sup>Z</sup>*<sup>ˆ</sup>

*Z*ˆ

*Y*ˆ

*Z*0*c*0∆*t*∆*y*

*Z*0*µy*∆*x*∆*z c*0∆*t*∆*y*

 *C E*.*dl* ∼=

given by:

250 Advanced Electromagnetic Waves

*E*˜ *<sup>y</sup>* and *H*˜

by

This plane is parallel to the propagation direction. Thus, the value of the integrals will depend on the exponential dependence of the fields. The contours of integration for the magnetic and electric fields in the (*x*, *y*)-plane are shown in Figure 5.

**Figure 5.** Integration contour of Maxwell's equations in the integral form in the plane (*x*, *y*). (a) H-field and (b) E-field

By using the same procedure as for the other planes, Maxwell-Ampere and Maxwell-Faraday equations and electric and magnetic laws yield, respectively,

$$\oint\_{\mathcal{C}} \vec{H}.\vec{dl} \cong \left(\frac{1+T}{2}\right) 2\sin\left(\frac{\beta \Delta y}{2}\right) e^{-j\frac{\beta y\_0}{\beta}} \left[\frac{1}{\beta} \left(H\_{10}^y - H\_6^y\right) - j\Delta x H\_x\right] \tag{35}$$

$$\oint\_{\mathcal{C}} \vec{E}.\vec{dl} \equiv \left(\frac{1+T}{2}\right) 2\sin\left(\frac{\beta \Delta y}{2}\right) e^{-j\frac{\beta y\_0}{\beta}} \left[\frac{1}{\beta} \left(E\_{11}^y - E\_3^y\right) - j\,\Delta x E\_x\right] \tag{36}$$

$$RHS\_{MA} \cong (1 - T) \frac{\varepsilon\_2 \Delta x \Delta y}{Z\_0 c\_0 \Delta t \Delta z} \Delta y \mathbf{E}\_z + Z\_0 \mathbf{G}\_{\varepsilon y} \Delta z \mathbf{E}\_z^n + Z\_0 V\_{\varepsilon z} \tag{37}$$

$$RHS\_{MF} \cong (T - 1) \frac{\mathcal{L}\_0 \mu\_2 \Delta x \Delta y}{c\_0 \Delta t \Delta y} \Delta z \tilde{H}\_z - \frac{K\_{mz}}{Z\_0} \Delta z H\_z^n - \frac{V\_{mz}}{Z\_0} \tag{38}$$

In the above, *Gez* = *σez*∆*x*∆*y*/∆*z* is the term used to model the electric losses, *Rmz* = *σm*∆*x*∆*y*/∆*z* models the magnetic losses, *Vez* = *Jez*∆*x*∆*y*, *Vmy* = *Jmy*∆*x*∆*y*, and the terms *E*˜ *<sup>z</sup>* and *H*˜*<sup>z</sup>* which correspond to the mean value of all field components in the plane are given, respectively, by

$$E\_z = \frac{E\_{10}^z + E\_6^z + \hat{Y}\_{sz} E\_{15}^z}{\hat{Y}\_{sz} + 2} \approx E\_z^{(n)}\tag{39}$$

$$\tilde{H}\_{\mathbf{z}} = \frac{H\_{11}^{\mathbf{z}} + H\_{\mathbf{3}}^{\mathbf{z}} + Z\_{\mathbf{sz}}^{\mathbf{\hat{}}}H\_{18}^{\mathbf{z}}}{Z\_{\mathbf{sz}}^{\mathbf{\hat{}}} + 2} \approx H\_{\mathbf{z}}^{(n)}\tag{40}$$

The normalized admittances *Y*ˆ *sz* and impedances *Z*ˆ *sz* are given by (19). By following a similar procedure to the other integration planes, the following expressions are obtained:

$$\begin{aligned} V\_{10}^r + V\_6^r + \hat{Y}\_{\text{sz}}^r V\_{15}^r &= \\ V\_{10}^i + V\_6^i + \hat{Y}\_{\text{sz}}^r V\_{15}^i + a Z\_0 \Delta x H\_{\text{x}}^{(n)} - Z\_0 \mathcal{G}\_{\text{ez}} \Delta z E\_{\text{z}}^{(n)} - Z\_0 V\_{\text{ez}} \end{aligned} \tag{41}$$

$$\begin{array}{c} V\_{11}^r - V\_3^r + \hat{Z}\_{\text{sx}} V\_{18}^r = \\ -V\_{11}^i + V\_3^i + \hat{Z}\_{\text{sx}} V\_{18}^i - \kappa Z\_0 \Delta x E\_{\text{x}}^{(n)} - \frac{\mathcal{R}\_{\text{sx}}}{\mathcal{Z}\_0} \Delta z H\_z^{(n)} - \frac{V\_{\text{n}z}}{\mathcal{Z}\_0} \end{array} \tag{42}$$

The relationships of (24), (25),(33),(34),(41), and (42) allow us to determine ∆*xE<sup>n</sup> <sup>x</sup>* , ∆*xH<sup>n</sup> x* , ∆*yE*(*n*) *<sup>y</sup>* , <sup>∆</sup>*yH*(*n*) *<sup>y</sup>* , <sup>∆</sup>*zE*(*n*) *<sup>z</sup>* , and <sup>∆</sup>*zH*(*n*) *<sup>z</sup>* , from which fields can be computed at any time step. Eight additional relationships are still needed to obtain reflected voltages. By expressing the fields at the center of the cell, these expressions can be found, as detailed in the next subsection.

### *3.3.4. Determination of the updating equations for the voltages*

To obtain the reflected voltages at any time step, six new contours have to be defined. Figure 6 shows the integration contours for finding fields at the center of the cell using Maxwell's equations in the integral form in the three planes.

The procedure is the same as presented before. First, Maxwell's equations are approximated for each of these contours. Consider the contour in dashed lines for the plane (*z*, *y*); Maxwell-Ampere and the electric flux equations are given, respectively, by

$$\oint\_{\mathcal{C}} \vec{H} \vec{dl} \cong \left[ 2 \Delta y H\_y^{\text{fl}} - \left( \frac{1+T}{2} \right) (H\_2 - H\_\theta) \right] e^{-j\beta y\_0} 2 \sin \left( \frac{\beta \Delta y}{2} \right) \tag{43}$$

*RHSMF* <sup>∼</sup><sup>=</sup> (*<sup>T</sup>* <sup>−</sup> <sup>1</sup>) *<sup>Z</sup>*0*µz*∆*x*∆*<sup>y</sup>*

*E*˜ *<sup>z</sup>* <sup>=</sup> *<sup>E</sup><sup>z</sup>*

*<sup>H</sup>*˜*<sup>z</sup>* <sup>=</sup> *<sup>H</sup><sup>z</sup>*

*E*˜

∆*yE*(*n*)

subsection.

*<sup>y</sup>* , <sup>∆</sup>*yH*(*n*)

given, respectively, by

252 Advanced Electromagnetic Waves

The normalized admittances *Y*ˆ

*Vi* <sup>10</sup> <sup>+</sup> *<sup>V</sup><sup>i</sup>*

> <sup>−</sup>*V<sup>i</sup>* <sup>11</sup> <sup>+</sup> *<sup>V</sup><sup>i</sup>*

<sup>6</sup> <sup>+</sup> *<sup>Y</sup>*<sup>ˆ</sup> *szV<sup>i</sup>*

*3.3.4. Determination of the updating equations for the voltages*

equations in the integral form in the three planes.

 *C H* .*dl* ∼= 2∆*yH<sup>n</sup> y* −

<sup>3</sup> <sup>+</sup> *<sup>Z</sup>*<sup>ˆ</sup> *sxV<sup>i</sup>* *c*0∆*t*∆*y*

<sup>10</sup> <sup>+</sup> *<sup>E</sup><sup>z</sup>*

<sup>11</sup> <sup>+</sup> *<sup>H</sup><sup>z</sup>*

*sz* and impedances *Z*ˆ

procedure to the other integration planes, the following expressions are obtained:

*Vr* <sup>10</sup> <sup>+</sup> *<sup>V</sup><sup>r</sup>*

*Vr* <sup>11</sup> <sup>−</sup> *<sup>V</sup><sup>r</sup>*

*Z*ˆ

*Y*ˆ

In the above, *Gez* = *σez*∆*x*∆*y*/∆*z* is the term used to model the electric losses, *Rmz* = *σm*∆*x*∆*y*/∆*z* models the magnetic losses, *Vez* = *Jez*∆*x*∆*y*, *Vmy* = *Jmy*∆*x*∆*y*, and the terms

*<sup>z</sup>* and *H*˜*<sup>z</sup>* which correspond to the mean value of all field components in the plane are

<sup>6</sup> <sup>+</sup> *<sup>Y</sup>*<sup>ˆ</sup> *szE<sup>z</sup>* 15

<sup>3</sup> <sup>+</sup> *<sup>Z</sup>*<sup>ˆ</sup> *szH<sup>z</sup>* 18

<sup>6</sup> <sup>+</sup> *<sup>Y</sup>*<sup>ˆ</sup> *szV<sup>r</sup>* <sup>15</sup> =

<sup>3</sup> <sup>+</sup> *<sup>Z</sup>*<sup>ˆ</sup> *sxV<sup>r</sup>* <sup>18</sup> =

Eight additional relationships are still needed to obtain reflected voltages. By expressing the fields at the center of the cell, these expressions can be found, as detailed in the next

To obtain the reflected voltages at any time step, six new contours have to be defined. Figure 6 shows the integration contours for finding fields at the center of the cell using Maxwell's

The procedure is the same as presented before. First, Maxwell's equations are approximated for each of these contours. Consider the contour in dashed lines for the plane (*z*, *y*);

(*H*<sup>2</sup> − *H*9)

 *e*

<sup>−</sup>*jβy*<sup>0</sup> 2*sin*

 *β*∆*y* 2

Maxwell-Ampere and the electric flux equations are given, respectively, by

1 + *T* 2

The relationships of (24), (25),(33),(34),(41), and (42) allow us to determine ∆*xE<sup>n</sup>*

<sup>18</sup> <sup>−</sup> *<sup>α</sup>Z*0∆*xE*(*n*) *<sup>x</sup>* <sup>−</sup> *Rmz*

<sup>15</sup> <sup>+</sup> *<sup>α</sup>Z*0∆*xH*(*n*) *<sup>x</sup>* <sup>−</sup> *<sup>Z</sup>*0*Gez*∆*zE*(*n*) *<sup>z</sup>* <sup>−</sup> *<sup>Z</sup>*0*Vez*

*<sup>y</sup>* , <sup>∆</sup>*zE*(*n*) *<sup>z</sup>* , and <sup>∆</sup>*zH*(*n*) *<sup>z</sup>* , from which fields can be computed at any time step.

<sup>∆</sup>*zH*˜*<sup>z</sup>* <sup>−</sup> *Rmz*

*Z*0

∆*zH<sup>n</sup>*

*<sup>z</sup>* <sup>−</sup> *Vmz Z*0

*sz* <sup>+</sup> <sup>2</sup> <sup>≈</sup> *<sup>E</sup>*(*n*) *<sup>z</sup>* (39)

*sz* <sup>+</sup> <sup>2</sup> <sup>≈</sup> *<sup>H</sup>*(*n*) *<sup>z</sup>* (40)

*<sup>Z</sup>*<sup>0</sup> <sup>∆</sup>*zH*(*n*) *<sup>z</sup>* <sup>−</sup> *Vmz*

*sz* are given by (19). By following a similar

*Z*0

(38)

(41)

(42)

(43)

*<sup>x</sup>* , ∆*xH<sup>n</sup> x* ,

**Figure 6.** Integration contours for Maxwell's equations in the integral form for finding fields at the center of the cell: (a) *x*0*z*, (b) *z*0*x*, and (c) *x*0*y* plane

$$\varepsilon \int \overline{\frac{\partial \vec{E\_x}}{\partial t}} \cdot \vec{dS} \cong \frac{1}{2} \left( 1 - T \right) \frac{\varepsilon\_x \Delta y \Delta z}{Z\_0 c\_0 \Delta t \Delta x} \Delta x \left( E\_\theta - E\_2 \right) e^{-j\beta y\_0} 2 \sin \left( \frac{\beta \Delta y}{2} \right) \tag{44}$$

Then, the expressions in (43) and (44) are used in combination with (8) to express the result in terms of reflected and incident voltages, yielding (45). By considering the remaining red and blue contours and applying Maxwell's equations in these planes, (46) to (50) can be found.

$$\begin{aligned} 2Z\_0 \Delta y H\_y^{(n)} &= Y\_{zx} \left( V\_9^r - V\_2^r + V\_9^i - V\_2^r \right) \\ 2\Delta y E\_y^{(n)} &= \left( V\_8^r + V\_4^r + V\_4^i + V\_8^i \right) \end{aligned} \tag{45}$$

$$\begin{aligned} 2Z\_0 \Delta y H\_y^{(n)} &= \hat{Y}\_{xz} \left( V\_6^r - V\_{10}^r + V\_6^i - V\_{10}^r \right) \\ 2\Delta y E\_y^{(n)} &= \left( V\_3^r + V\_{11}^r + V\_3^i + V\_{11}^i \right) \end{aligned} \tag{46}$$

$$\begin{aligned} 2\mathbf{Z}\_0 \Delta \mathbf{z} H\_z^{(n)} &= Y\_{xy} \left( V\_{11}^r - V\_3^r + V\_{11}^i - V\_3^r \right) \\ 2\Delta y E\_y^{(n)} &= \left( V\_6^r + V\_{10}^r + V\_6^i + V\_{10}^i \right) \end{aligned} \tag{47}$$

$$\begin{aligned} 2\mathbf{Z}\_0 \Delta \mathbf{z} H\_z^{(n)} &= \mathbf{Y}\_{yx} \left( V\_1^r - V\_{12}^r + V\_1^i - V\_{12}^r \right) \\ 2\Delta y E\_y^{(n)} &= \left( V\_5^r + V\_7^r + V\_5^i + V\_7^i \right) \end{aligned} \tag{48}$$

$$\begin{aligned} \mathbf{2Z\_0 \Delta x H\_x^{(n)}} &= Y\_{y2} \left( V\_7^r - V\_5^r + V\_7^i - V\_5^r \right) \\ \mathbf{2\Delta x E\_x^{(n)}} &= \left( V\_1^r + V\_{12}^r + V\_1^i + V\_{12}^i \right) \end{aligned} \tag{49}$$

$$\begin{aligned} \mathbf{2Z\_0 \Delta x H\_x^{(n)}} &= \hat{Y}\_{zy} \left( V\_4^r - V\_8^r + V\_4^i - V\_8^r \right) \\ \mathbf{2\Delta x E\_x^{(n)}} &= \left( V\_2^r + V\_9^r + V\_2^i + V\_9^i \right) \end{aligned} \tag{50}$$

In summary, 12 expressions at the center were found. These expressions relate fields at instant *n*∆*t*, incident and reflected voltages. They allow us to update the eight voltages in the node. By replacing the above expressions of the fields in (8), one obtains

$$
\begin{bmatrix} V\_g' \\ V\_{11}' \\ V\_8' \\ V\_{10}' \\ V\_4' \\ V\_2' \\ V\_6' \\ V\_3' \end{bmatrix} = \begin{bmatrix} \Delta x E\_x^{(n)} - Z\_0 \Delta y H\_y^{(n)} \\ \Delta y E\_y^{(n)} - Z\_0 \Delta z H\_z^{(n)} \\ \Delta y E\_y^{(n)} + Z\_0 \Delta x H\_x^{(n)} \\ \Delta z E\_z^{(n)} + Z\_0 \Delta y H\_y^{(n)} \\ \Delta y E\_y^{(n)} - Z\_0 \Delta x H\_x^{(n)} \\ \Delta x E\_x^{(n)} + Z\_0 \Delta y H\_y^{(n)} \\ \Delta z E\_z^{(n)} - Z\_0 \Delta y H\_y^{(n)} \\ \Delta y E\_y^{(n)} + Z\_0 \Delta z H\_z^{(n)} \end{bmatrix} - \begin{bmatrix} V\_g' \\ V\_{11}' \\ V\_{1}' \\ V\_{1}' \\ V\_4' \\ V\_2' \\ V\_3' \\ V\_3' \end{bmatrix} \tag{51}
$$

The calculation of reflected voltages at stubs is done as for standard SCN:

$$\begin{array}{llll}V\_{13}^{r} & = & \Delta x E\_{x}^{(n)} & -V\_{13}^{i} \\ V\_{14}^{r} & = & \Delta y E\_{y}^{(n)} & -V\_{14}^{i} \\ V\_{15}^{r} & = & \Delta z E\_{z}^{(n)} & -V\_{15}^{i} \\ V\_{16}^{r} & = & \Delta x Z\_{0} H\_{x}^{(n)} & -V\_{16}^{i} \\ V\_{17}^{r} & = & \Delta y Z\_{0} H\_{y}^{(n)} & -V\_{17}^{i} \\ V\_{18}^{r} & = & \Delta z Z\_{0} H\_{z}^{(n)} & -V\_{18}^{i} \end{array} \tag{52}$$

### **3.4. Results of the 2.5D modal approach**

To characterize any waveguide field amplitudes, attenuation constant and phase constant must be calculated. Here, we examine how to obtain these parameters by using the 2.5D TLM approach for guiding structures.
