Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 165

The magnetic loss in magnetic materials is due to the process of magnetization that results in a complex relative permeability, . The relative permeability is defined as a ratio of two inductances; inductance (L^*) of the coil with a magnetic material, and inductance (L₀) of the same coil with the air‐core,

(4.3.20)

A magnetic material placed inside a coil, shown in Fig. (4.7a), gets magnetized in the process of magnetization. The current causing the magnetization is known as the magnetization current (I_m). However, during the magnetization process, some power is lost. It is viewed through the magnetic loss‐current (I_r). The magnetization of a magnetic material involves the storage of magnetic energy. So, a lossy magnetic material is modeled through the RL equivalent circuit, shown in Fig. (4.7b). The inductor models the stored magnetic energy in a magnetic material, whereas the resistor models its loss.

Figure 4.7 Circuit model and frequency response of lossy magnetic material.

A time‐harmonic current I = I₀ e^jωt flows through the coil containing the magnetic material. The voltage across the coil and current through it are given below:

(4.3.21)

Figure (4.7b) shows the current flowing in the parallel RL equivalent circuit. By comparing the above current, with the circuit model current; the expressions for the circuit elements, in terms of the material parameters, are obtained as follows:

(4.3.22)

The magnetic loss‐tangent of the lossy inductor is defined below that helps to get the magnetic loss‐tangent of magnetic material:

(4.3.23)

A lossless magnetic material has , i.e. , R → ∞. It means an inductor, without the resistor, models the lossless magnetic material. It shows that the permeability is an ability of a magnetic material to store magnetic energy, just as the permittivity of dielectric material is its ability to store electric energy. For the case of the low‐loss magnetic materials, , the following expressions are obtained from equations (4.3.22b) and (4.3.22c):

(4.3.24)

It is observed that for a low‐loss magnetic material, is independent of frequency, whereas increases linearly with frequency, as shown in Fig. (4.7c). The above equations show that inductor models the real part of the complex permeability, whereas resistance models its imaginary part. This kind of frequency response may not be a realistic description of a hysteretic magnetic material.