**6.1 Numerical configuration**

22 Will-be-set-by-IN-TECH

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

(a)

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>12</sup> <sup>14</sup> <sup>16</sup> <sup>18</sup> <sup>20</sup> <sup>0</sup>

(b) Fig. 28. (a) Maximum magnitude of focusing. (b) *STN* ratio, importance of focusing from 1 to 20 probes: regarding 50 random draws of probes locations each time (stars markers) and

Number of probes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 29. Normalized total electric field as a function of the distance to *R*<sup>0</sup> along *x*: variations

In the final part of this chapter, the TR is numerically applied in the LASMEA (LAboratoire des Sciences et Matériaux pour l'Électronique et d'Automatique, Clermont Université) MSRC

Since 2001, a MSRC has been available for the EMC research & applications of LASMEA. Its dimensions and an internal view are given on Fig. 30. Historically, studies and tests in

Normalized magnitude (V/m)

−30 −20 −10 <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>0</sup>

(b)

x (cm)

fc= 600 MHz − ΔΩ= 700 MHz fc= 600 MHz − ΔΩ= 520 MHz fc= 600 MHz − ΔΩ= 260 MHz

Number of probes

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

−30 −20 −10 <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>0</sup>

(a)

**6. EMC application and selective focusing**

x (cm)

(for impulsive susceptibility test and selective focusing).

fc=800 MHz − ΔΩ=260 MHz fc=600 MHz − ΔΩ=260 MHz fc=400 MHz − ΔΩ=260 MHz

STN

mean trend (plain line).

around (a) *fc* and (b) ΔΩ.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized magnitude (V/m)

Maximum magnitude of focusing (V/m)

CST MICROWAVE STUDIO is a specialist tool for the 3-D electromagnetic simulation. Inspired by the characteristics of LASMEA MSRC, the Fig. 31 shows the configuration of our example. Simulations are performed with a spatial discretization of 0.65 *cm* and 4 *cm* respectively corresponding to the smallest and largest mesh, and a time step of 24.4 *ps*.

The walls of the MSRC are modeled with a conductivity of *Sc* = 1.1 106 *S*/*m*, furthermore the stirrer has a conductivity of 2.74 10<sup>7</sup> *S*/*m*. The support table and the EUT are made respectively of wood and aluminum (*Sc* = 3.56 107 *S*/*m*). In this application, we wish to focus separately on the three components of the EUT modeled by dipoles (1, 2, and 3 in Fig. 31). The excitation signal is a Gaussian modulated sine pattern with a central frequency of 250 *MHz* and a bandwidth of 300 *MHz* calculated at −20 *dB*. The TRM is composed of two 60 *cm* half-wavelength antennas.

To determine the TR window duration (Δ*t*), we need to calculate the Heisenberg time. Thus, we plotted on Fig. 33a the evolution of the *STN* ratio as a function of Δ*t* for a TRM composed of a single antenna. Already mentioned above, we note that the *STN* ratio stabilizes after a certain duration, we can deduce that the Heisenberg time is about 8 *μs*. However an 8 *μs* simulation in comparison with the large dimensions of the LASMEA MSRC is disadvantageous in terms of computing time, so we chose to reduce simulation time by increasing the TRM antenna number. Fig. 33b shows that for Δ*t* = 0.75*μs* the *STN* ratio

Time Reversal for Electromagnetism: Applications in Electromagnetic Compatibility 201

x 10−5

Fig. 33. *STN* ratio evolution: (a) as a function of the TR window for a TRM composed of 1 antenna, (b) as a function of the TRM antenna number with a TR windows of Δ*t* = 0.75 *μs*. The number of antennas needed for a TR experience is given by the ratio Δ*H*/Δ*t*, hence we

In this section, we will check the possibility to focus the electric field on one of the three components of the EUT, while others are aggressed by lower levels (noise). To do this, we consider the example where the values 15 *V*/*m*, 70 *V*/*m* and 40 *V*/*m* correspond respectively to the three components threshold that should not be exceeded by the electric field. After recording the impulse responses *kij*(*t*) with 1 ≤ *i* ≤ 2: number of TRM antennas and 1 ≤ *j* ≤ 3: number of components of the EUT, and given the linearity of the system, we can focus on any component with any desired focusing magnitude by a simple post-processing. Indeed, if for example we want to focus on the component number 2 (Figs. 34c, 34d), we will back-propagate through the first antenna of the TRM the signal *pk*12(−*t*) and the signal *pk*22(−*t*) by the second antenna, where *p* is the weight corresponding to the needed amplification. The *p* coefficient stands for the focusing magnitude control offered by TR (the focusing peak may be increased or decreased throughout the number of TRM antennas, the TR window duration, or an external amplification weight). We plotted in Fig. 34 temporal and spatial focusing corresponding to the "on demand" desired peak magnitude separately on each of the three components. The spatial focusing of the field corresponds to the absolute

maximum value recorded over the entire simulation for each cell of the slice plan).

We note that for the different cases, the maximum of the field corresponds to the desired spatial location of each component. In addition, we note that for the second case, for example, we have focused on the component 2 (Figs. 34c, 34d) while respecting the threshold of the first component (component 1 was aggressed by a field whose numerical value is smaller than 15 *V*/*m*), same for the third case. To achieve this desired focusing magnitude on component 2

STN (dB)

0 2 4 6 8 10 12 14 16 18

(b)

Number of antennas

stabilizes for a number of antenna greater than 8.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

(a)

Δt (s)

chose the duration Δ*t* = 4 *μs* with 2 antennas as TRM.

**6.3 Selective focusing**

STN (dB)

Fig. 31. LASMEA MSRC modeled by CST MICROWAVE STUDIO : (a) walls, (b) mechanical stirrer, (c) EUT, (1)(2)(3) EUT three components.

#### **6.2 Preliminary study**

To justify the choice of the TRM antenna number, the duration of the TR window (Δ*t*), and the ability to choose the polarization of the focused signal, a preliminary study is carried out maintaining the configuration of the Fig. 31 without the table and the EUT; but, here, an isotropic probe will be used to check focusing properties. Based upon TR principles, and because the simulator does not allow exciting probes and there are not designated to broadcast, necessary signals for the second phase of TR may be obtained directly by injecting excitation signal straightforward on the TRM antennas one by one. The three Cartesians components of these impulse responses are recorded by an isotropic field probe, and then back propagated from the TRM. Indeed, during the first phase, the isotropic probe records the electric field *E<sup>α</sup>* (with *α* = *x*, *y* or *z*: Cartesian component of the field). We can choose the polarization *α* of the focused signal (*ETRα*) from back-propagating by the TRM recorded signal corresponding either to *x*, *y* or *z* without changing the polarization of the TRM antennas (Fig. 32).

Fig. 32. Polarization control of the focused signal.

To determine the TR window duration (Δ*t*), we need to calculate the Heisenberg time. Thus, we plotted on Fig. 33a the evolution of the *STN* ratio as a function of Δ*t* for a TRM composed of a single antenna. Already mentioned above, we note that the *STN* ratio stabilizes after a certain duration, we can deduce that the Heisenberg time is about 8 *μs*. However an 8 *μs* simulation in comparison with the large dimensions of the LASMEA MSRC is disadvantageous in terms of computing time, so we chose to reduce simulation time by increasing the TRM antenna number. Fig. 33b shows that for Δ*t* = 0.75*μs* the *STN* ratio stabilizes for a number of antenna greater than 8.

Fig. 33. *STN* ratio evolution: (a) as a function of the TR window for a TRM composed of 1 antenna, (b) as a function of the TRM antenna number with a TR windows of Δ*t* = 0.75 *μs*.

The number of antennas needed for a TR experience is given by the ratio Δ*H*/Δ*t*, hence we chose the duration Δ*t* = 4 *μs* with 2 antennas as TRM.

#### **6.3 Selective focusing**

24 Will-be-set-by-IN-TECH

Fig. 31. LASMEA MSRC modeled by CST MICROWAVE STUDIO : (a) walls, (b)

To justify the choice of the TRM antenna number, the duration of the TR window (Δ*t*), and the ability to choose the polarization of the focused signal, a preliminary study is carried out maintaining the configuration of the Fig. 31 without the table and the EUT; but, here, an isotropic probe will be used to check focusing properties. Based upon TR principles, and because the simulator does not allow exciting probes and there are not designated to broadcast, necessary signals for the second phase of TR may be obtained directly by injecting excitation signal straightforward on the TRM antennas one by one. The three Cartesians components of these impulse responses are recorded by an isotropic field probe, and then back propagated from the TRM. Indeed, during the first phase, the isotropic probe records the electric field *E<sup>α</sup>* (with *α* = *x*, *y* or *z*: Cartesian component of the field). We can choose the polarization *α* of the focused signal (*ETRα*) from back-propagating by the TRM recorded signal corresponding either to *x*, *y* or *z* without changing the polarization of the TRM antennas

−1 −0.5 0 0.5 1

−1 −0.5 0 0.5 1

<sup>1</sup> ETRx

−1 −0.5 0 0.5 1

Time (s)

x 10−7

x 10−7

ETRy ETRz

x 10−7

mechanical stirrer, (c) EUT, (1)(2)(3) EUT three components.

**6.2 Preliminary study**

(Fig. 32).

−1 0 1 Normalized magnitude (V/m)

−1 0

−1 0 1

Fig. 32. Polarization control of the focused signal.

In this section, we will check the possibility to focus the electric field on one of the three components of the EUT, while others are aggressed by lower levels (noise). To do this, we consider the example where the values 15 *V*/*m*, 70 *V*/*m* and 40 *V*/*m* correspond respectively to the three components threshold that should not be exceeded by the electric field. After recording the impulse responses *kij*(*t*) with 1 ≤ *i* ≤ 2: number of TRM antennas and 1 ≤ *j* ≤ 3: number of components of the EUT, and given the linearity of the system, we can focus on any component with any desired focusing magnitude by a simple post-processing. Indeed, if for example we want to focus on the component number 2 (Figs. 34c, 34d), we will back-propagate through the first antenna of the TRM the signal *pk*12(−*t*) and the signal *pk*22(−*t*) by the second antenna, where *p* is the weight corresponding to the needed amplification. The *p* coefficient stands for the focusing magnitude control offered by TR (the focusing peak may be increased or decreased throughout the number of TRM antennas, the TR window duration, or an external amplification weight). We plotted in Fig. 34 temporal and spatial focusing corresponding to the "on demand" desired peak magnitude separately on each of the three components. The spatial focusing of the field corresponds to the absolute maximum value recorded over the entire simulation for each cell of the slice plan).

We note that for the different cases, the maximum of the field corresponds to the desired spatial location of each component. In addition, we note that for the second case, for example, we have focused on the component 2 (Figs. 34c, 34d) while respecting the threshold of the first component (component 1 was aggressed by a field whose numerical value is smaller than 15 *V*/*m*), same for the third case. To achieve this desired focusing magnitude on component 2

(a) (b)

Time Reversal for Electromagnetism: Applications in Electromagnetic Compatibility 203

(c) Fig. 35. Cutting plan (*z* = 1.4 *m*) of the absolute maximum value of the electric field obtained

In this chapter, the TR method was presented in electromagnetism for applications concerning the EMC domain in a reverberating environment. Based upon the equivalence between backward propagation and reversibility of the wave equation, many TR experiments were led successfully in acoustics. In this chapter, after an introduction explaining the physical context, the theoretical principles of TR were described and illustrated numerically using the FDTD method. The use of the CST MICROWAVE STUDIO commercial software laid emphasis on the industrial interest of TR for EMC test devices. First, the TR technique was applied in free space using a TRC and a TRM, and then the importance of the complexity of the medium was demonstrated. Relying on intrinsic RC behavior and due to multiple reflections, the results obtained by applying TR in a reverberating cavity were clearly improved; the aim was to accurately describe the influence of various parameters above focusing. Thus, a link between the modal density in a cavity and the TR focusing quality was clearly established through the *STN* ratio. A particular interest relies on the number and locations of TR probes and the excitation pulse parameters impact. Finally we introduced an original way to perform an impulsive susceptibility test study based on the MSRC use. We presented the possibility to choose the polarization of the wave aggressing the EUT, and to perform an "on demand" selective focusing. In further works, it would be interesting to experimentally confirm our numerical results, so one may expect to proceed to experimental analysis in LASMEA MSRC. At last, considering the characteristics of EMC applications in MSRC, a closer look might be set to the advantages of TR numerical tools for innovating studies in reverberation chambers.

on the: (a) component 1 with *p* = 1, (b) component 3 with *p* = 3, (c) two components

togother, 1 with *p* = 1 and 3 with *p* = 3.

**7. Conclusions**

Fig. 34. Temporal focusing of the electric field on: (a) component 1 with *p* = 1, (c) component 2 with *p* = 5, (e) component 3 with *p* = 3. Spatial focusing corresponding to the absolute maximum value of the electric field: (b) component 1 with *p* = 1, (d) component 2 with *p* = 5, (f) component 3 with *p* = 3.

(64 *V*/*m*) smaller than the corresponding threshold (70 *V*/*m*), the impulse responses *k*12(−*t*) and *k*22(−*t*) were multiplied by the weight *p* = 5; so we notice that following this way we can control the time, location, and magnitude of focusing (by the weight *p*).

Finally, if we wish, for example, to focus on the first and third components with respective magnitude of 13 *V*/*m* and 35 *V*/*m*, we sum and back-propagate the needed impulse responses (on the first TRM antenna we back-propagate the signal *p*1*k*11(−*t*) + *p*3*k*13(−*t*) with *p*<sup>1</sup> = 1 and *p*<sup>3</sup> = 3, on the second TRM antenna we back-propagate the signal *p*1*k*21(−*t*) + *p*3*k*23(−*t*)). The Fig. 35 justifies this approach and shows the ability of selective focusing by TR.

Fig. 35. Cutting plan (*z* = 1.4 *m*) of the absolute maximum value of the electric field obtained on the: (a) component 1 with *p* = 1, (b) component 3 with *p* = 3, (c) two components togother, 1 with *p* = 1 and 3 with *p* = 3.

### **7. Conclusions**

26 Will-be-set-by-IN-TECH

(a) (b)

x 10−7

component 1 component 2 component 3

x 10−7

component 1 component 2 component 3

x 10−7

control the time, location, and magnitude of focusing (by the weight *p*).

(e) (f) Fig. 34. Temporal focusing of the electric field on: (a) component 1 with *p* = 1, (c) component 2 with *p* = 5, (e) component 3 with *p* = 3. Spatial focusing corresponding to the absolute maximum value of the electric field: (b) component 1 with *p* = 1, (d) component 2 with

(64 *V*/*m*) smaller than the corresponding threshold (70 *V*/*m*), the impulse responses *k*12(−*t*) and *k*22(−*t*) were multiplied by the weight *p* = 5; so we notice that following this way we can

Finally, if we wish, for example, to focus on the first and third components with respective magnitude of 13 *V*/*m* and 35 *V*/*m*, we sum and back-propagate the needed impulse responses (on the first TRM antenna we back-propagate the signal *p*1*k*11(−*t*) + *p*3*k*13(−*t*) with *p*<sup>1</sup> = 1 and *p*<sup>3</sup> = 3, on the second TRM antenna we back-propagate the signal *p*1*k*21(−*t*) + *p*3*k*23(−*t*)).

The Fig. 35 justifies this approach and shows the ability of selective focusing by TR.

(c) (d)

component 1 component 2 component 3

−3 −2 −1 0 1 2 3

−3 −2 −1 0 1 2 3

−3 −2 −1 0 1 2 3

*p* = 5, (f) component 3 with *p* = 3.

Time (s)

Time(s)

Time (s)

−15 −10 −5 0 5 10 15

−60 −40 −20 0 20 40 60

−40 −30 −20 −10 0 10 20 30 40

Magnitude (V/m)

Magnitude (V/m)

Magnitude (V/m)

In this chapter, the TR method was presented in electromagnetism for applications concerning the EMC domain in a reverberating environment. Based upon the equivalence between backward propagation and reversibility of the wave equation, many TR experiments were led successfully in acoustics. In this chapter, after an introduction explaining the physical context, the theoretical principles of TR were described and illustrated numerically using the FDTD method. The use of the CST MICROWAVE STUDIO commercial software laid emphasis on the industrial interest of TR for EMC test devices. First, the TR technique was applied in free space using a TRC and a TRM, and then the importance of the complexity of the medium was demonstrated. Relying on intrinsic RC behavior and due to multiple reflections, the results obtained by applying TR in a reverberating cavity were clearly improved; the aim was to accurately describe the influence of various parameters above focusing. Thus, a link between the modal density in a cavity and the TR focusing quality was clearly established through the *STN* ratio. A particular interest relies on the number and locations of TR probes and the excitation pulse parameters impact. Finally we introduced an original way to perform an impulsive susceptibility test study based on the MSRC use. We presented the possibility to choose the polarization of the wave aggressing the EUT, and to perform an "on demand" selective focusing. In further works, it would be interesting to experimentally confirm our numerical results, so one may expect to proceed to experimental analysis in LASMEA MSRC. At last, considering the characteristics of EMC applications in MSRC, a closer look might be set to the advantages of TR numerical tools for innovating studies in reverberation chambers.

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**9** 

*University of Hyogo* 

*Japan* 

**Three-Dimensional Numerical Analyses** 

**on Liquid-Metal Magnetohydrodynamic** 

Hiroshige Kumamaru, Kazuhiro Itoh and Yuji Shimogonya

In conceptual design examples of a fusion reactor power plant, a lithium-bearing blanket in which a great amount of heat is produced is cooled mainly by helium gas, water or liquidmetal lithium (Asada et al. Ed., 2007). The liquid-metal lithium is an excellent coolant having high heat capacity and thermal conductivity and also can breed tritium that is used as fuel of a deuterium-tritium (D-T) fusion reactor. In cooling the blanket, however, the liquidmetal lithium needs to pass through a strong magnetic field that is used to magnetically confine high-temperature reacting plasma in a fusion reactor core. There exists a large magnetohydrodynamic (MHD) pressure drop arising from the interaction between the liquid-metal flow and the magnetic field. In particular, the MHD pressure drop becomes considerably larger in the inlet region or outlet region of the magnetic field than in the fullydeveloped region inside the magnetic field for the reason mentioned later in this chapter.

A three-dimensional calculation is indispensable for the exact calculation of MHD channel flow in the inlet region or outlet region of magnetic field, also as described later in this chapter. There exist a few three-dimensional numerical calculations on the MHD flows in rectangular channels with a rectangular obstacle (Kalis and Tsinober, 1973), with abrupt widening (Itov et al., 1983), or with turbulence promoter such as conducting strips (Leboucher, 1999). All these calculations, however, were carried out for low Hartmann numbers (corresponding to low strength of the applied magnetic field) and low Reynolds

As to the MHD channel flow in the magnetic-field inlet-region, three-dimensional numerical calculations were conducted for the cases of Hartmann number of ~10 and Reynolds number of ~100 (Khan and Davidson, 1979). The calculations were based on what is called the parabolic approximation, in which the flow and magnetic field effects are assumed to transfer only in the main flow direction. However, the calculations based on parabolic approximation cannot predict exactly the MHD flow in the magnetic-field inlet-region. Were performed full three-dimensional calculations (without any assumptions) on the MHD rectangular-channel flow in the magnetic-field inlet-region (Sterl, 1990). The calculations

number, because of instability problems in numerical calculations.

**1. Introduction** 

**Flow Through Circular Pipe** 

**in Magnetic-Field Outlet-Region** 

Ziadé, Y.; Wong, M. & Wiart, J. (2008). Reverberation chamber and indoor measurements for time reversal application, *Proceedings of APS 2008, IEEE Antennas and Propagation Society International Symposium*, ISBN: 978-1-4244-2041-4, USA, 2008.
