Characteristics of studied converter.
Abstract
Flyback converters have been widely used in low and highpower applications because of their simplicity and low cost. However, they incur electromagnetic compatibility problems which are more difficult to control. The present chapter proposes an efficient modeling method based on the nearfield technique to solve realworld radiation problems of the power electronics circuits. Firstly, for the characterization of an AC/DC flyback converter, several experimental measurements of the magnetic near field are performed in the time domain over the converter. Subsequently, we have applied the time domain electromagnetic inverse method based on the genetic algorithms on the measured signals to find the equivalent radiating sources of the studied circuit. The accuracy and the efficiency of the proposed approach have been demonstrated by the good agreement between cartographies of the near magnetic field components calculated using the developed model and those measured. Finally, the developed equivalent model has been used to predict cartographies of other components of the magnetic field which will be compared to measured cartographies. This confirms that the identified equivalent sources can represent real sources in the studied structure. The proposed method could be used for diagnosis and fault location in power electronics systems.
Keywords
 electromagnetic compatibility (EMC)
 flyback converter
 inverse method
 nearfield measurement
 timedomain analysis
 genetic algorithms
1. Introduction
To respond to the necessity and simplicity of marketed converter applications, innovation in power electronics is based on the growing constraints imposed by the exponential rise of our needs: better management of primary energy sources while offering increasingly sophisticated functions at reduced cost. This cost function can obviously take different forms depending on the context: weight, volume, efficiency, disturbances on the converter and its environment, technological cost, economic cost, protection and reliability, and lifetime of devices. No device can be excluded today from environmental constraints. At this stage, a lot of energy conversion structures are now very efficient. For example, the flyback converter is probably the most widely used structure in the electronics industry and is frequently utilized for various applications (battery chargers, energy saving lamps, photovoltaic systems, LCD monitors, DVD players, …). Integration and coexistence in the same housing, as well as the increase in switching frequencies and carried powers, cause frequent and unwanted electromagnetic interference in the vicinity of these systems. These interferences present important sources of malfunction of the device itself and neighboring systems. For this reason, electromagnetic compatibility (EMC) must be addressed from the starting process of design and manufacture of hightech electronic devices. Indeed, the study of electromagnetic compatibility of a card gives quantitative tools to the power electronics to influence the radiated and conducted disturbances. It is essential to know the electromagnetic disturbances emitted by the equipment under test at different distances and to verify that they do not exceed a certain limit in order to avoid disturbing the neighboring equipment. Therefore, the EMC characterization tests have become an essential step in the various phases of the development of power electronics products. Unfortunately, these tests remain costly because we often need heavy, precise and very expensive equipment (anechoic chamber). To circumvent these limits, it is essential to develop radiating models that allow us to estimate the electromagnetic emissions of these systems before the prototype is realized (during the phase of virtual prototyping). The objective of this work is to elicit the electromagnetic inverse method in the time domain based on the nearfield technique as an innovative and important solution that allows characterizing and modeling the radiation of the products of the power electronics. Thus, the added value of this chapter is to show the implementation of the inverse electromagnetic method in the time domain for the field of power electronics. In fact, the proposed approach will be of some use to people who make the development of power electronics systems. Actually, identifying all the sources of radiation in an electronic cartography and developing an equivalent radiation model that will allow us to estimate the electromagnetic emissions of these systems help solve the problems of interferences encountered in power electronics. The proposed method can be used for example as a method of analysis, fault detection and diagnosis of power electronic circuits. Indeed, the electromagnetic signature of a component could make it possible to verify its proper functioning. In order to find solutions related to the problems encountered in power electronics, the inverse electromagnetic method has been applied to different types of circuits.
In the literature, a lot of work has been already achieved on the modeling of electronic components [1, 2, 3]. The inverse method based on the elementary dipoles was used to extract the equivalent model from PCB circuits [4, 5, 6]. In frequency domain, the inverse method was also used in order to model electronics cards [7, 8, 9].
These various approaches have been developed in the frequency domain and have been utilized to characterize and to model the radiated emissions on only one frequency. It seems that there is no time. This supposes that the studied systems have had sinusoidal radiations that do not correspond to the reality of power electronic systems where the electromagnetic disturbances are very important over a broad band of frequencies. For this effect, modeling this kind of radiation by the frequency inverse method requires to repeating the identification of equivalent sources for each signal frequency. This causes a very high processing time. However, from the results found in the frequency domain, we cannot know at what time of the cycle of operation the radiating sources intervene, and we thus suppose that the radiations of the sources contribute simultaneously, which does not correspond to reality. Nevertheless, the switching in power electronics often presents offsets which will be found at the level of radiated emissions. The spectral distribution therefore evolves as a function of time and the components do not all have the same radiation frequencies. Consequently, the models obtained by the electromagnetic inverse method in the frequency domain have several limitations and remain inapplicable for power electronics systems which are excited by different types of nonsinusoidal signals and which consequently emit on one broad band of frequencies. To this end, the added value of our approach is to propose an efficient technique based on the inverse electromagnetic method in the time domain which takes into account the temporal offset at the levels of the various radiated emissions of the circuit, which permits the Identification of the contribution of each radiating source of the circuit over time.
Until now, few studies have been performed and published on the timedomain inverse method. The authors in some study tried to combine the frequency inverse method with the timefrequency computation method [10, 11]. However, the developed modeling method was limited because it always depended on the parameters of the time intervals that mainly depended on the selected frequency band, the considered initial frequency and the used computer performance. Other approaches have been performed [12, 13, 14, 15]. These studies were dedicated to the development of new computational techniques and analysis of an electromagnetic near field in the time domain. To develop a resolution method in the time domain, enabled by passing this kind of problem, we put forward the development and implementation of the temporal inverse electromagnetic method based on elementary dipoles [16].
The methodology of the timedomain electromagnetic inverse method has been presented [16]. This method was applied to identify the equivalent radiation model in a fairly simple structure, which was unfortunately not representative enough for all power electronic circuits. To demonstrate again the robustness and efficiency of the electromagnetic inverse method developed in the time domain when applied to complicated real cards composed of several bulky components, where the radiating sources are usually very close to each other, we propose to study in this chapter a complete application case (PCB and power electronic components). Hence, we will focus much more on the application of the temporal inverse electromagnetic method to find an equivalent model of an industrial system that emits nonsinusoidal radiations. This will make it possible to calculate the field emitted by the studied system for various distances and to study the coupling with the neighboring systems.
In this chapter, the time domain electromagnetic inverse method is tested and used in an ACDC flyback converter application. First, we describe the device under test. Then, we will present how the suggested approach is implemented and what these advantages can be. To do this, several works will be performed. We will firstly present the test bench of the near field in the time domain. Second, we will present the measurement results of the vertical component of the radiated magnetic near field (H_{z}) performed over an AC/DC converter, so as to model the radiation of this converter by the electromagnetic inverse method in the time domain. In the following part, the modeling methodology and its resolution based on the genetic algorithms are described. To validate the identified model using our suggested approach, a comparison is made between the tangential components (H_{x}) of the magnetic field, calculated using the optimized model parameters and the one measured over the studied converter.
2. Description of the studied flyback converter
The system studied in this chapter is an ACDC converter based on the principle of switchedmode power supplies ( Figure 1 ).
Switching power supplies are much smaller and lighter than linear power supplies, which explain their increasingly widespread use in various fields, particularly in those of embedded systems. Also, switchedmode power supplies have a good efficiency until 90%. Contrarily, they have regulatory problems which are more difficult to control [17, 18, 19]. Due to the rectangular signal to the switching frequency, these power supplies produce a relatively large noise. This problem makes them unsuitable for certain applications. To this end, it is essential to characterize and control the EMC behavior of this type of converters.
The studied converter in this chapter is of a flyback type of a low power (5 W). The basic schema of a studied converter is illustrated in Figure 2 .
The electrical characteristics of the studied flyback converter are illustrated in Table 1 .
Type  Input voltage  Output voltage  Maximum current  Efficiency 

Flyback  220 V AC  5 V DC  1 A  79% 
3. Description of magnetic nearfield test bench in time domain
In our study, the measurement of the studied converter radiation is carried out in one single measurement for different radiation frequencies. This presents the benefits of a near field in the time domain compared to the frequency test bench.
Figure 3 presents the different parts of the nearfield test bench in time domain.
The temporal measurement method uses a highprecision oscilloscope that presents a wide bandwidth. This oscilloscope is utilized for displaying and recording temporal signals picked up by the measurement of the magnetic field probe.
To sweep the probe above the studied structure, a plotting table is used. The displacement of the probe over the device under test is manually performed ( Figure 4 ).
The displacement of the probe above the studied structure is manually performed. The displacement procedure of the probe from measurement point to another on the sweeping surface is described in Figure 5 .
In this study and according to our needs, we utilize two coil probes: one for measuring the normal component of the magnetic field (H_{z}) and the other for measuring the two tangential components (H_{x} and H_{y}). Each utilized probe is shielded. The shielding of the used probe is made by copper. It consists of a loop connected to the central conductor on one side and to the external shield of a coaxial cable on the other side, as shown in Figure 4 . The probe is connected to the oscilloscope through a highly shielded coaxial cable.
To measure the magnetic fields around the components and systems, we use magnetic probes; and to capture the different components of the magnetic field H, it is necessary to place the normal to the surface of the collinearly loop of the desired component. It is possible to utilize a single probe and to orient it differently according to the component to be measured. The principle of the measurement protocol is based on the Faraday and Lenz law, which stipulates that if a variable magnetic field crosses through a closed circuit, then the resulting flux variation will cause an electromotive force at the terminals of the loop ( Figure 4c ). Accordingly, we have:
Finally, and through a transformation equation, we can deduce the magnetic field. The magnetic probe allows transforming the variable magnetic flux which crosses through it into a voltage at its terminals. Subsequently, we deduce the magnetic field at the center of the probe. Indeed, for a circular conductive loop of a radius R dived in a magnetic field B(t), the induced potential difference by the variable magnetic field is given by the following equation:
where
Since the probe surface is small enough, we can consider the magnetic field H_{z}(t) constant over the entire loop area S. As a result, Eq. (5) becomes:
where
In fact, extracting the magnetic field H(t) from the measured voltage at the probe terminals can be defined as follows:
4. Measurement of near field emitted by studied converter
This suggested approach is based on time domain measurements of the near field above the studied structure. The distribution of the near field on the chosen scanned surface is presented in the form of cartography. The cartography is a matrix representation of the amplitude and the phase of the measured near field at each point and at each instant over the studied system. It is characterized by the dimensions of the measurement surface (Xmin:Xmax; Ymin:Ymax), the distance between two measuring points (ΔX et ΔY), and the measuring height (Z) ( Figure 6 ).
Figure 7b represents the chosen face used for the time domain measurements to identify the equivalent radiating sources. This face enables a good detection of the magnetic near field radiated by all components of the studied card.
By analogy with the frequency inverse method, in this work, the identification of the equivalent radiating sources is based on the measurements of the magnetic near field [8, 9, 20]. This demonstrates its effectiveness in guaranteeing the uniqueness of the equivalent radiating model [9]. To bypass the complexities to measure the electrical near field, it is possible to extract the values of the electric field by exploiting the measurements of the magnetic field [21, 22].
5. Equivalent radiating model of studied flyback converter
In this study, the proposed temporal electromagnetic inverse method is based on the genetic algorithms. To apply our approach, a nearfield measurements characterized by a very high signaltonoise ratio are needed. The proposed method consists in identifying an equivalent model that emits the same radiation of the studied structure to be modeled.
The optimized equivalent model is based on a network of electric or magnetic dipoles ( Figure 8 ).
The dipole may be represented by a d vector that comprises all the parameters: d = (M_{d}, x_{d}, y_{d}, z_{d}, θ_{d}, φ_{d}), where M_{d} is the magnetic moment of the dipole which varies against time. The x_{d}, y_{d} and z_{d} represent the dipole’s position, and their orientations are represented by θ_{d} and φ_{d}. In the time domain, the components of the magnetic field emitted by the elementary electric and magnetic dipoles in an observation point M (x_{o}, y_{o}, z_{o}) can be deduced from the analytical expressions in the frequency domain using the frequencytime transformation operator. In the Cartesian coordinate system, the magnetic field components are expressed by the following equations [16]:
For the electric dipole:
For the magnetic dipole:
where
As the temporal measured signals above the studied system are not sinusoidal well as the excitations currents are not unique. In this proposed method, it is not necessary to initially propose a number of elementary dipoles during the equivalent model identification. Thus, the proposed method consists in identifying at each time a single particular dipole, until the identification of all dipoles that corresponds the equivalent model. Because the measures are performed at a very close distance above the studied structure, a hypothesis is adopted; it is based on the linearity between the radiated magnetic field by an elementary dipole and the current flowing in this dipole. This assumption is adopted for guessing the currents forms on the various dipoles, and is very helpful when simplifying the resolution during the optimization phase of the equivalent model parameters by the genetic algorithms.
At this stage, the problem of identifying the source dipoles is processed. The identification procedure of the radiating sources is presented in the following flowchart given in Figure 9 .
In order to identify accurately the shape of the first dipole excitation signal, we will apply to the initially measured cartography a subprogram to seek the measured signal which has the maximum amplitude relative to other measured maximum temporal signals in this cartography. And on the basis of the measurement point that corresponds to the most intense radiation, we extract a limited part of the initial cartography and we propose a single elementary radiating dipole (electric or magnetic dipole) for the extracted cartography. In fact, the searched current form flowing in the first dipole to identify is well known; it is a normalized temporal signal relative to this measured maximum signal in this local cartography. Subsequently, an optimization method based on the genetic algorithms is applied to identify the parameters of the proposed dipole in this selected cartography. At each iteration, the genetic algorithms shall modify all the parameters of the dipole (a constant k which represents the amplitude of the excitation current circulating in this proposed dipole, its geometry, its coordinates, and its orientations) in order to minimize the error calculated by the fitness function. If the error is unacceptable and the genetic algorithms cannot converge after a certain number of iterations, the optimization cycle repeats itself several times until obtaining a new composed generation of best solutions who is the calculated field coincides with the measured one on all points of the cartography and at all moments (even in the case of the small values of the measured field). Thus, the radiating dipole and its corresponding parameters are identified.
By using the analytical equations, the parameters of the first identified dipole are used to calculate the near magnetic field radiated in order to delete it from the initial measured cartography. This simplifies the resolution of the remaining cartography. The procedure is repeated until the scanning of all the measured cartography and the different dipole sources are identified.
A supplementary explanation of the proposed approach is illustrated in Figure 10 . At each identification, this procedure will be repeated until the extraction of all radiating sources in this measured magnetic fields cartography.
The superposition method is applied in a nearfield zone where it is assumed that in each area of the local cartography we identify a relatively intense field. The emission is mainly caused by a very close source and the contributions from the other relatively distant sources are very low compared with that of the intense source. Consequently, the search for the model is carried out dipole by dipole until identifying all radiating sources in the initially measured cartography. Indeed, the fitness function is used by the genetic algorithms for identifying the existing source in the each local cartography. Thus, the error of the global cartography is the sum of the errors of different local cartographies.
To guarantee and accelerate the convergence of the proposed temporal inverse method, we take into account the study of [23] to choose the optimal parameters of the genetic algorithms. These parameters are given in Table 2 .
Fitness function 

Population size (N_{p})  N_{p} = 20 × number of parameters of dipoles 
Selection function  Roulette 
Crossover rate  0.8 
By applying the proposed method to identify the radiating sources, we have found seven magnetic dipoles as a number of sources. Despite the high density of components in the converter and knowing that the measurements of radiated disturbances are performed at a very close distance above these components, and on the basis of research work on the modeling of the radiation of electronic components and systems, we have carried out the analysis of the radiations of the studied converter. This permits us to identify the original components of the equivalent radiating dipoles. Unfortunately, the modeling of systems by the inverse method in the frequency domain has several limitations and remains not applicable for systems of power electronics operating over a wide frequency band and excited by different types of nonsinusoidal signals. At this stage, it is the interest of the temporal inverse method proposed in this chapter. Thus, without modeling of the radiated emissions for each frequency of the measured signals (which causes a very significant calculation time). The proposed approach has the advantage of finding in a reasonable processing time a single model of radiation valid over the whole band of frequencies in measured signal. From the results obtained by the inverse method proposed in the time domain, we can see that the first two identified magnetic dipoles represent the equivalent models of two diodes. The third identified source represents the equivalent radiation of the transformer. In fact, this last source has a center situated at coordinates X_{d} = 0.10 cm, Y_{d} = −0.21 cm and Z_{d} = −1.21 cm of the studied structure. It presents the coordinates of the transformer position in the studied card, as depicted in Figure 11 .
Figure 12 gives a spatial representation of the equivalent dipoles identified by our proposed approach as well as the different shapes of the excitation currents of the source dipoles.
The parameters of the equivalent model obtained by the proposed method are shown in Table 3 .
Identified dipoles  M_{di} (A m^{2})  [X_{di} Y_{di} Z_{di}] (cm)  θ_{i}(0)  φ_{i}(0) 

#1  3.12 e − 7  1.46/−1.33/−0.645  0.57  7.45 
#2  4.42 e − 7  1.42/−0.32/−0.79  58.47  −132.99 
#3  1.12 e − 6  0.10/−0.21/−1.21  9.17  9.74 
#4  2.96 e − 7  −0.96/0.20/−0.61  −11.46  −17.77 
#5  3.86 e − 7  −1.34/1.19/−0.65  −43.56  128.98 
#6  1.48 e − 7  −0.35/−1.08/−0.37  72.80  179.42 
#7  1.90 e − 8  −1.36/−0.74/−0.62  83.12  8.85 
Hence, it presents the case where the emitted disturbances of the tracks are very low compared to that of the components. Finally, we notice that the components of the power electronics card represent the majority of the radiation of the studied structure as illustrated in Figure 13 .
In order to study these electromagnetic disturbances, we present in Figure 14 the measured magnetic nearfield cartographies at different moments.
By analyzing Figure 14 , we notice that the all sources do not appear at the same time. This is due to the time lag at the level of the electrical quantities passing through the different elements of the studied structure. This kind of transient phenomena is not easy to identify in the frequency analysis where the equivalent sources appear at the time such as presented in [8]. For a diagnostic study, where it is desired to control the good functioning of a sensitive element in a power electronics card, this type of time analysis becomes an effective alternative that enables us to use the electromagnetic signature based on the temporal near field for the disgnosis of some components of the card. The electromagnetic inverse method in the time domain can therefore be used as a nonintrusive method of diagnosing and detecting defects in converter circuits. We cite for example the case of clampingdiode faults in a threelevel NPC inverter circuit using the temporal analysis of the near magnetic field. Actually, the failure of a diode results in a modification during a delimited sequence at each period, from the temporal cartography of the magnetic field measured in a zone close to the circuit.
In Figure 15 , we present at various moments a comparison between the measured cartographies of the normal magnetic field component (H_{z}) and the calculated cartographies using the parameters of the equivalent model obtained by the electromagnetic inverse method proposed in the time domain.
By examining the results got, we notice a good agreement between these cartographies. This demonstrates that the suggested method can identify with very good accuracy all radiating sources of the circuit.
6. Validation of identified equivalent model
In order to validate the found equivalent model, we measured the tangential component of the magnetic field (H_{x} measured) and subsequently we compared it with that is calculated (H_{x} estimated) by exploiting the optimized parameters of the equivalent radiating sources. In two distinct instants (t_{1} = 10.06 μs and t_{2} = 10.47 μs), a comparison between the measured and estimated of the nearfield cartographies is presented in Figure 16 .
According to the results presented in the previous figure, the magnetic near field calculated using the identified model parameters is in good agreement with the measured magnetic near field. These results confirm the existence of the equivalent sources network found by the electromagnetic inverse method in the time domain. Thus, the results showed that the proposed method guarantees the uniqueness of the solution. This confirms the effectiveness of the proposed approach for model the highpower structures where the radiated emissions are very high.
7. Conclusion
The frequency domain electromagnetic inverse method has several disadvantages when modeling radiation of power electronics systems. These disadvantages are mainly related to the radiation over a broad frequency band. To overcome these drawbacks, we have proposed in this chapter the use of the time domain electromagnetic inverse method. This novel method has been applied to identify the equivalent radiating sources of an ACDC converter. The radiating sources are identified by exploiting the measurements of the vertical component (H_{z}) of the magnetic field performed over the studied converter. This equivalent model has been validated by using the measurements of the tangential component of the near magnetic field (H_{x}). The good agreement between the calculated results by using the optimized parameters of the equivalent model and the experimental results measured above the studied structure highlights the robustness of the suggested inverse problem when seeking equivalent radiating sources for industrial systems creating transient signals.
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