**2. Experimental techniques and characterization method**

#### **2.1 Excitation circuit**

To allow characterization over a wide frequency band, the component under test (CUT) is excited by a current having a large frequency band similar to those found in power electronics. Therefore, It is proposed to install the component under test in a converter circuit supplied by a voltage equal to 30 V and has a duty cycle of 0.5 (**Figure 1**). The output current of the circuit is 0.5 A. The switching frequency is 50 kHz.

#### **2.2 Measurements methods**

In our study, we utilize only one component of the field radiated by the system under test. Indeed, the use of a single component in a large number of points is quite sufficient to identify the equivalent source by guaranteeing the uniqueness of the solution [14, 15]. Furthermore, the developed radiating model is capable of modeling the electric field as well as the magnetic field. Actually, based on [16–18], we can calculate from the magnetic field, the electric ones, and vice versa.

Since the method can be applied to any magnetic or electric field component, we have used the vertical component Hz. We may also utilize the other tangential components (Hx or Hy). For measuring the magnetic near field, we place a magnetic probe above the device under test. It is a manually made probe consisting of a 1.6 mm radius circular loop connected to the central conductor on one side and the external shield of a coaxial cable on the other side. To capture the various components of the H-field, it is necessary to place the normal of the collinear loop to the desired component.

In our study, we use two methods for measuring the magnetic field around the component. The first method performs the near field measurements in the frequency domain, and the second one is performed in the time domain.

#### *2.2.1 Frequency measurements*

The method of frequency measurements is based on a spectrum analyzer. In fact, to measure the magnetic field radiated of the under-test device, we use the near-field measurement bench presented in **Figure 2**.

**Figure 1.** *Chopper circuit where component under test is characterized.*

**Figure 2.** *Frequency measurement bench.*

#### *2.2.2 Temporal measurements*

The temporal measurements method is based on the utilization of an oscilloscope. This measurement method permits us to have a radiated magnetic field by the device under test at any moment. After measuring the temporal voltage across the probe, a Fast Fourier Transform (FFT) calculation is performed. **Figure 3** presents the adopted methodology.

According to Lenz-Faraday law, for a simple fixed circular conductor loop, having a very small radius R compared to the wavelength, diving in a magnetic field B(t) oriented along the z-axis, the potential difference induced in the loop by the magnetic field is given by the following equations:

In the time domain

$$\text{Vm } \left( \mathbf{t} \right) = \oint \frac{\partial \mathbf{B}}{\partial \mathbf{t}} \, \text{d} \mathbf{S} \tag{1}$$

In the frequency domain

$$\text{Vm } (\mathbf{f}) = \mathbf{j} \text{ o } \mathbf{B} \cdot \text{S} \tag{2}$$

where B ¼ μ<sup>0</sup> � H

$$\text{So Vm (f)} = \text{j} \,\text{o}\mu\_0 \text{ H S} \tag{3}$$

**Figure 3.** *Methodology using temporal bench.*

*Development of Generic Radiating Model for Rectangular Capacitors: Magnetic Near Fields… DOI: http://dx.doi.org/10.5772/intechopen.98894*

where <sup>μ</sup><sup>0</sup> <sup>¼</sup> <sup>4</sup> � <sup>π</sup> � <sup>10</sup>�<sup>7</sup> , is the magnetic permeability S <sup>¼</sup> <sup>π</sup> � r2 is the surface of the probe, f is the radiation frequency, and ω = 2πf.

The magnetic field is assumed to be constant over the entire area of the probe. This is especially true when the probe is very small compared to the capacitor's size. Therefore, the magnetic field measured at the center of the probe is calculated by the following equation:

$$\mathbf{H(f) = Vm(f)/2 \times (2 \times \pi \times \mu\_0 \times \mathbf{f} \times 2 \times \mathbf{S})} \tag{4}$$

where Vm (*f*) is the component at the frequency *f* of the FFT of the voltage measured at the terminals of the probe. Finally, we can extract the near-field cartographies for each radiating frequency.

#### **2.3 Probe calibration**

Before using a magnetic field probe, it is necessary to calibrate it. In [5, 6], the authors suggested a method to validate the accuracy of their magnetic field probes. They measured the radiated magnetic field around a simple circuit with a conductive wire above a ground plane. The values of the measured field are compared to those calculated theoretically. Thus, the comparison of the results enabled the validation of the used probes.

Similarly, in this work, to calibrate the magnetic probe, we have used a radiating circuit whose radiation is known theoretically. We compare the measured radiated magnetic field to that calculated by the numerical electromagnetic tool NEC [19] based on the moment's method. The radiating circuit is a rectangular loop of 5 cm length and 3 cm width excited by a sinusoidal voltage of 10 V amplitude at a frequency of 5 MHz (**Figure 4**).

The simulations and measurements are made at the points located in a horizontal plane at 3 mm above the radiating loop and having dimensions of 5 � 3 cm2 . The calculation step is 2.5 mm along the X-axis and 1.5 mm along the Y-axis (441 measurement points).

**Figures 5**–**7** gives the magnetic field components Hz, Hx, and Hy along the X- and Y-axis, respectively. It shows a good agreement between the measured and calculated curves of the Hz component. In order to have a good signal-to-noise ratio when measuring Hx and Hy components, we chose measurement lines located at the edge of the emitting loop. Consequently, in some probe measurements, we notice a difference between measurements and simulations due to the edge effects and the coupling effects between the probe and the radiating loop. These phenomena are not significant on the Hz component and therefore do not affect the construction of the model.

**Figure 4.** *Loop used for probe calibration.*

**Figure 5.** *Hz following X- and Y-axes respectively for Y = 4.5 mm and X = 25 mm.*

**Figure 6.** *Hx following X- and Y-axes respectively for Y = 1.5 mm and X = 25 mm.*

**Figure 7.** *Hy following X- and Y-axes respectively for Y = 1.5 mm and X = 25 mm.*

In order to examine the electric field rejection capability of our magnetic probe, we have placed it over a transmitting electric antenna where the amplitude of the electric field is important. The observed voltage across the probe is so low that we cannot dissociate it from the noise. Hence we deduce that our shielded probe rejects very well the electric field and measures the magnetic field with great accuracy.
