Introduction To Modern Planar Transmission Lines. Anand K. Verma

causes a dielectric loss in the material. The imaginary part of the complex relative permittivity arises due to the damping of oscillation during the polarization process of a dielectric material, under the influence of an externally applied AC electric field discussed in chapter 6. However, it is difficult to distinguish between two sources of the dielectric loss; the contribution of the free charge carriers (conduction current) and the contribution of the dielectric polarization (polarization current). Therefore, both could be grouped in the total loss‐tangent.

      The parallel‐plate capacitor, shown in Fig. (4.6b), supports two kinds of current densities – the conduction current density, J_c given by equation (4.1.9), and the displacement current density, J_d given by equation (4.3.1a). The total current density is

(4.3.9)

The total loss‐tangent, from equations (4.3.4) and (4.3.9) of dielectric material is

      (4.3.10)

      Equation (4.3.9), in a changed form, is rewritten as follows:

      (4.3.11)

      The equivalent of lossy dielectrics, due to the combined effect of polarization and finite conductivity, is

      (4.3.12)

      The lossy dielectric medium is also described by the concept of the complex equivalent conductivity . Using the expression and equation (4.3.9), the complex equivalent conductivity is expressed as follows:

      (4.3.13)

      The real part of a complex equivalent conductivity causes the dielectric loss in a medium, whereas its imaginary part stores the electric energy of the dielectric medium. Therefore, the imaginary part of a complex equivalent conductivity is related to the relative permittivity of a medium, and its real part is associated with the imaginary part of the complex relative permittivity:

      (4.3.14)

      Sometimes, loss due to the polarization causing is ignored, say in a semiconductor, if the loss due to the conduction current is more significant. The loss characteristic of a semiconducting substrate is given by its conductivity. In that case, the loss‐tangent is expressed in terms of the conductivity of a medium:

      (4.3.15)

      The above expression helps to convert the conductivity of a substrate to its loss‐tangent at each frequency of the required frequency band. It can also be used to convert the loss‐tangent to the conductivity at each frequency.

      The equivalence between the relative permittivity and capacitance has been obtained by treating both the dielectric and capacitor as electric energy storage devices. Thus, the complex relative permittivity is equivalent to a complex capacitance. Figure (4.6b) provides the admittance of a lossy capacitor:

      (4.3.16)

where the complex capacitance C^* is given by

      (4.3.17)

      The real and imaginary parts of a complex capacitance, and also loss‐tangent, are given by the following expressions:

      (4.3.18)

      The complex relative permittivity is also expressed as follows:

      (4.3.19)

      The above expression is helpful in the computation of the real and imaginary parts of the effective relative permittivity of a lossy planar transmission line. The complex line capacitance of a lossy planar transmission line can be numerically evaluated. It also helps to compute the effective loss‐tangent of a multilayered planar transmission line by the variational method. It is discussed in chapter 14.

      4.3.2 Circuit Model of Lossy Magnetic Medium

      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) Скачать книгу