*3.4.1. Phase constant: dispersion diagram β* − *ω*

For each mode, *β* is a function of frequency *ω*; it is known as the dispersion diagram of the structure. In general, this relationship is not linearly proportional to the frequency for waveguides. The computation of the dispersion diagram with the TLM approach for guiding structures is as follows: The structure has to be excited as explained in Section 3.1. Then, the field components at the time step *k*∆*t* can be found through (17, 18, 30, 31, 39, 40). After a sufficient number of time-step iterations, the response corresponds to the superposition of the modal fields in the guide. By performing a Fourier transform, the frequency response of the structure can be obtained. The peaks in this spectrum correspond to the modes, and they belong to a specific value of the phase constant *β*. Hence, different *β* values have to be imposed to obtain the dispersion diagram.

### *3.4.2. Attenuation constant*

<sup>2</sup>*Z*0∆*xH*(*n*) *<sup>x</sup>* <sup>=</sup> *<sup>Y</sup>*<sup>ˆ</sup>

<sup>2</sup>∆*xE*(*n*) *<sup>x</sup>* <sup>=</sup>

<sup>2</sup>*Z*0∆*xH*(*n*) *<sup>x</sup>* <sup>=</sup> *<sup>Y</sup>*<sup>ˆ</sup>

<sup>2</sup>∆*xE*(*n*) *<sup>x</sup>* <sup>=</sup>

 

 

254 Advanced Electromagnetic Waves

*Vr* 9 *Vr* 11 *Vr* 8 *Vr* 10 *Vr* 4 *Vr* 2 *Vr* 6 *Vr* 3

 

=

the node. By replacing the above expressions of the fields in (8), one obtains

∆*yE*(*n*)

∆*yE*(*n*)

∆*yE*(*n*)

∆*yE*(*n*)

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

*Vr*

*Vr*

*Vr*

*Vr*

*Vr*

*Vr*

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

*3.4.1. Phase constant: dispersion diagram β* − *ω*

TLM approach for guiding structures.

*yz Vr* <sup>7</sup> <sup>−</sup> *<sup>V</sup><sup>r</sup>*

*zy Vr* <sup>4</sup> <sup>−</sup> *<sup>V</sup><sup>r</sup>*

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

<sup>∆</sup>*xE*(*n*) *<sup>x</sup>* <sup>−</sup> *<sup>Z</sup>*0∆*yH*(*n*)

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

*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*0∆*xH*(*n*) *<sup>x</sup>* <sup>∆</sup>*zE*(*n*) *<sup>z</sup>* <sup>+</sup> *<sup>Z</sup>*0∆*yH*(*n*)

*<sup>y</sup>* <sup>−</sup> *<sup>Z</sup>*0∆*xH*(*n*) *<sup>x</sup>* <sup>∆</sup>*xE*(*n*) *<sup>x</sup>* <sup>+</sup> *<sup>Z</sup>*0∆*yH*(*n*)

*<sup>y</sup>* <sup>+</sup> *<sup>Z</sup>*0∆*zH*(*n*) *<sup>z</sup>*

<sup>∆</sup>*zE*(*n*) *<sup>z</sup>* <sup>−</sup> *<sup>Z</sup>*0∆*yH*(*n*)

<sup>13</sup> <sup>=</sup> <sup>∆</sup>*xE*(*n*) *<sup>x</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>*

<sup>15</sup> <sup>=</sup> <sup>∆</sup>*zE*(*n*) *<sup>z</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>*

<sup>16</sup> <sup>=</sup> <sup>∆</sup>*xZ*0*H*(*n*) *<sup>x</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>*

<sup>18</sup> <sup>=</sup> <sup>∆</sup>*zZ*0*H*(*n*) *<sup>z</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>*

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

For each mode, *β* is a function of frequency *ω*; it is known as the dispersion diagram of the structure. In general, this relationship is not linearly proportional to the frequency for waveguides. The computation of the dispersion diagram with the TLM approach for guiding

<sup>14</sup> <sup>=</sup> <sup>∆</sup>*yE*(*n*)

<sup>17</sup> <sup>=</sup> <sup>∆</sup>*yZ*0*H*(*n*)

 *Vr* <sup>2</sup> + *<sup>V</sup><sup>r</sup>*

 *Vr* <sup>1</sup> <sup>+</sup> *<sup>V</sup><sup>r</sup>* <sup>5</sup> + *<sup>V</sup><sup>i</sup>*

<sup>8</sup> + *<sup>V</sup><sup>i</sup>*

*y*

 

−

 

*Vi* 9 *Vi* 11 *Vi* 8 *Vi* 10 *Vi* 4 *Vi* 2 *Vi* 6 *Vi* 3

 

*y*

*y*

*y*

13

14

15

16

18

*<sup>y</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>*

*<sup>y</sup>* <sup>−</sup> *<sup>V</sup><sup>i</sup>* 17

<sup>1</sup> <sup>+</sup> *<sup>V</sup><sup>i</sup>* 12

<sup>2</sup> + *<sup>V</sup><sup>i</sup>* 9

<sup>12</sup> <sup>+</sup> *<sup>V</sup><sup>i</sup>*

<sup>9</sup> + *<sup>V</sup><sup>i</sup>*

<sup>7</sup> <sup>−</sup> *<sup>V</sup><sup>r</sup>* 5 

<sup>4</sup> <sup>−</sup> *<sup>V</sup><sup>r</sup>* 8 

(49)

(50)

(51)

(52)

The attenuation of the modes can be extracted from the time-domain response at a phase constant *β<sup>l</sup>* given by

$$E\_{lm}\left(k\Delta t\right) = \sum\_{k=0}^{M} E^{lm}\left(0\right) e^{-\frac{\omega\_{lm}}{2Q\_{lm}}k\Delta t} e^{j\omega\_{lm}k\Delta t}\,,\tag{53}$$

where *M* is the number of modes propagating in the structure, *Elm* (0) is the initial field magnitude of the *m*-th mode, *Qlm* its quality factor, and *ωlm* the angular resonance frequency of the mode in radians per second for the *l*-th evaluated value of *βl*. A similar expression can be found for the magnetic field by replacing *E* by *H* in (53). To characterize the attenuation of the modes, the damping parameter has to be estimated. At each frequency *ωlm* in the dispersion curve of a given mode, the attenuation constant is related with the quality factor *Qlm* and the group velocity *Vglm* by [34]

$$
\omega\_{lm} \approx \frac{\omega\_{lm}}{2Q\_{lm}V\_{\mathcal{G}^{lm}}} = \frac{\omega\_{lm}}{2Q\_{lm}} \frac{\partial \beta}{\partial \omega} \tag{54}
$$

The terms *ωlm*/2*Qlm* can be extracted from the sampled time-domain response of (53) by using an estimation technique for modal content of a time-varying waveform. For instance, the Matrix pencil method has been a very popular technique for efficient extraction of the modal parameters [14]. Finally, the group velocity in (54), which involves the derivate of *β* with respect to *ω*, can be approximated by taking the finite differences ∆*β*/∆*ω* of the dispersion diagram calculated in the previous section.

### **4. Implementation of the 2.5D TLM for guiding structures**

To validate this approach, the study of a simple theoretical "reference solution" for a canonical geometry is considered. A metallic-rectangular waveguide homogeneously filled with a lossy dielectric material is studied in this section.

### **4.1. Results for a homogeneously lossy dielectric-filled metallic-rectangular waveguide**

Consider a metallic waveguide with dimensions *a* = 6 cm and height *b* = 4 cm. It was filled with a lossy dielectric with relative permittivity *<sup>r</sup>* = 2 and a conductivity *σ* = 0.001 Sm−<sup>1</sup> and simulated with the 2.5D TLM, as shown in Figure 7 (a). An array of 14 × 10 nodes is considered, and a perfect electric boundary condition is imposed at the top, bottom, and side walls, as illustrated in Figure 7 (b). To reduce the dispersion error, a mesh size of ∆*l* = *λ*/15 at 6 GHz is chosen. The signal is obtained after *N* = 4000 time steps and is convolved with a Hanning window.

**Figure 7.** (a) Rectangular-metallic waveguide filled with a lossy dielectric; (b) 2.5D TLM mesh of the rectangular waveguide terminated by a perfect electric boundary condition (PEC). The dimensions are given in terms of the mesh size ∆*l* = *λ*/15 at 6 GHz

The dispersion diagram is obtained in the frequency range from 1 GHz up to 4 GHz and is compared with the theory. For the case of a rectangular-metallic waveguide, the theoretical expression for the dispersion diagram is given by [26]:

$$\beta = \sqrt{\left(\omega\sqrt{\mu\varepsilon}\right)^2 - \left(\frac{m\pi}{a}\right)^2 - \left(\frac{n\pi}{b}\right)^2},\tag{55}$$

where *ω* is the angular frequency; *µ* = *µ*0; = *r*0; *m*, *n* correspond to the order of the modes; and *a* and *b* are the waveguide dimensions. The comparison between the numerical dispersion diagram and the theoretical one is shown in Figure 8.

The continuous lines represent the theoretical results, whereas the discrete points are the results obtained by using the 2.5D TLM algorithm. Different *β* values ranging from 0 to 80 (rad/m) with 80 samples have been considered. The simulation is carried out for 4000 iterations for each *β* value. Good agreement between theory and the 2.5D TLM for guiding structures was obtained. The dispersion curves of a plane wave propagating in free space and a wave propagating in a medium with the same relative permittivity as the loading material (*<sup>r</sup>* = 2) are plotted as well. It is worth observing that the dispersion curves of the modes tend to the curve of a plane wave propagating in a medium with relative permittivity *<sup>r</sup>* = 2 as the frequency increases. Finally, the theoretical attenuation constant of the metallic waveguide is calculated by [26]:

$$
\mathfrak{a}\_d = \frac{k^2 \tan \delta}{2\beta},
\tag{56}
$$

where *k* = *ω*√*µ* is the (real) wave number in the absence of losses, *tanδ* = (*ω* + *σ*) /*ω* is the loss tangent with = − *j* = *r*<sup>0</sup> − *jσ*/*ω*, and *β* is the phase constant. The results

considered, and a perfect electric boundary condition is imposed at the top, bottom, and side walls, as illustrated in Figure 7 (b). To reduce the dispersion error, a mesh size of ∆*l* = *λ*/15 at 6 GHz is chosen. The signal is obtained after *N* = 4000 time steps and is convolved with

**Figure 7.** (a) Rectangular-metallic waveguide filled with a lossy dielectric; (b) 2.5D TLM mesh of the rectangular waveguide terminated by a perfect electric boundary condition (PEC). The dimensions are given in terms of the mesh

The dispersion diagram is obtained in the frequency range from 1 GHz up to 4 GHz and is compared with the theory. For the case of a rectangular-metallic waveguide, the theoretical

2 −

where *ω* is the angular frequency; *µ* = *µ*0; = *r*0; *m*, *n* correspond to the order of the modes; and *a* and *b* are the waveguide dimensions. The comparison between the numerical

The continuous lines represent the theoretical results, whereas the discrete points are the results obtained by using the 2.5D TLM algorithm. Different *β* values ranging from 0 to 80 (rad/m) with 80 samples have been considered. The simulation is carried out for 4000 iterations for each *β* value. Good agreement between theory and the 2.5D TLM for guiding structures was obtained. The dispersion curves of a plane wave propagating in free space and a wave propagating in a medium with the same relative permittivity as the loading material (*<sup>r</sup>* = 2) are plotted as well. It is worth observing that the dispersion curves of the modes tend to the curve of a plane wave propagating in a medium with relative permittivity *<sup>r</sup>* = 2 as the frequency increases. Finally, the theoretical attenuation constant of the metallic

*<sup>α</sup><sup>d</sup>* <sup>=</sup> *<sup>k</sup>*2*tan<sup>δ</sup>*

where *k* = *ω*√*µ* is the (real) wave number in the absence of losses, *tanδ* = (*ω* + *σ*) /*ω* is the loss tangent with = − *j* = *r*<sup>0</sup> − *jσ*/*ω*, and *β* is the phase constant. The results

 *mπ a* 2 − *nπ b* 2

, (55)

<sup>2</sup>*<sup>β</sup>* , (56)

a Hanning window.

256 Advanced Electromagnetic Waves

size ∆*l* = *λ*/15 at 6 GHz

waveguide is calculated by [26]:

expression for the dispersion diagram is given by [26]:

*β* = 

dispersion diagram and the theoretical one is shown in Figure 8.

(*ω*√*µ*)

**Figure 8.** Dispersion diagram of a rectangular-metallic waveguide with dimensions *a* = 6 cm and *b* = 4 cm: Theory (continuous lines), results using the guiding 2.5D TLM approach (dots)

**Figure 9.** Attenuation constant for a lossy dielectric filled metallic-rectangular waveguide: Theory (continuous lines), simulation (dots)

obtained for the attenuation constants by numerical simulation and theory are shown in Figure 9.

The comparison between both curves validates this approach quite satisfactorily. The discrepancies observed close to the cutoff frequency are explained by the fact that (57) is valid as long as *α* << *β*.

$$
\alpha \approx \frac{\partial \beta}{\partial \omega} \Delta \omega \tag{57}
$$

Consequently, since *α* is deduced from ∆*β* in the numerical approach, an inevitable large calculation error on the attenuation constant *α* is obtained for a given error on ∆*β*. To illustrate this point, let us consider the calculation of the propagated error on *α*. The total differential of a function *z* = *f* (*x*1, *x*2, ..., *xN*), where *xi* is the *i*-th independent variable, is given by

$$dz = \left(\frac{\partial f}{\partial \mathbf{x}\_1}\right) \Delta \mathbf{x}\_1 + \left(\frac{\partial f}{\partial \mathbf{x}\_2}\right) \Delta \mathbf{x}\_2 + \dots + \left(\frac{\partial f}{\partial \mathbf{x}\_N}\right) \Delta \mathbf{x}\_N \tag{58}$$

the term *dz* denotes the error in *z*. The uncertainty in the calculation of the attenuation constant for waveguides can be found by considering the differential of the attenuation and phase constants, *α* and *β*, respectively. The differential of the attenuation and phase constants can be computed by

$$
\Delta \mathfrak{a} = \left(\frac{\partial \mathfrak{a}}{\partial \omega}\right) \Delta \omega\_{\prime} \tag{59}
$$

$$
\Delta \beta = \left(\frac{\partial \beta}{\partial \omega}\right) \Delta \omega. \tag{60}
$$

Finally, by dividing (59) by (60), the differential of the attenuation constant in terms of the differential of the phase constant is given by

$$
\Delta \alpha = \left(\frac{\partial \alpha}{\partial \omega}\right) \left(\frac{\partial \omega}{\partial \beta}\right) \Delta \beta \tag{61}
$$

is obtained.

As frequency increases, the first partial derivative on the right-hand side of (61) tends to zero and the second to the speed of light in free space, *c*0. Hence, for high frequencies, the error ∆*α* is bounded. For low frequencies, the first derivate tends to infinity and the second to a finite value, resulting in a larger error. Indeed, in Figure 9, the highest difference between both approaches occurs at cutoff frequencies of the modes and decreases far from these frequencies. Apart from this relative error on *α*, the comparison between numerical results and theory for both propagation constant *β* and the attenuation constant *α* confirms the suitability of the 2.5D TLM to characterize modal parameters in waveguiding structures.
