Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 164

Figure (4.6b) shows a parallel‐plate capacitor, containing a lossy dielectric medium with complex relative permittivity . It further shows its RC circuit model that is a parallel combination of the capacitor (C) and resistor (R). It is connected to a time‐harmonic voltage source v = v₀e^jωt that produces a time‐harmonic electric field, E = E₀e^jωt in the dielectric medium. The displacement current density in the dielectric medium is

(4.3.1)

The displacement current density has two components with the quadrature phase:

(4.3.2)

The reactive current density (J_cap) flows through , i.e. through the capacitor. It is shown by the presence of “j.” Thus, the real part of the relative permittivity is modeled as a capacitor C. The resistive current density (J_r), causing a loss in the dielectric medium, flows through , i.e. through resistor R. The imaginary part of relative permittivity is modeled as a resistor R, parallel to the capacitor C. It is obvious from equation (4.3.1) that the loss caused by is a positive quantity only when the complex relative permittivity is defined as a subtractive combination of its real and imaginary parts mentioned in equation (4.2.18a). The current through the equivalent circuit is

(4.3.3)

Both current components are shown in Fig. (4.6b). The loss‐tangent, showing the dissipation factor of the RC circuit, is defined using Fig. (4.6c):

(4.3.4)

Therefore, the dissipation factor, i.e. the loss tangent (tan δ), of lossy dielectric material is defined using equation (4.3.2) as follows:

(4.3.5)

where A is the area of the parallel‐plate capacitor. The loss current is zero, i.e. I_r = 0 for a lossless capacitor, and also for a lossless dielectric medium. It leads to , tanδ = 0. In Fig. (4.6c), the angle θ is the power‐factor angle. On comparing equations (4.3.2) and (4.3.3), an equivalence is obtained between the lossy dielectric medium and a lossy capacitor

(4.3.6)

On replacing the dielectric medium of Fig. (4.6b) by the air medium, i.e. ε_r = 1, capacitance C₀ is obtained:

(4.3.7)

On using the above equations, the real and imaginary parts of a complex relative permittivity and loss tangent are defined in terms of the circuit elements:

(4.3.8)

The above equations provide a practical means to measure the dielectric constant of any dielectric material with the help of a parallel‐plate capacitor. Equation (4.3.8a) also gives a practical definition of the relative permittivity of a homogeneous dielectric medium. The relative permittivity is a ratio of capacitances of a parallel‐plate capacitor, with a material medium and with the air medium; while keeping the geometry of the parallel‐plate capacitor unchanged. We get a homogenized dielectric medium even if the parallel‐plate capacitor, as shown in Fig. (4.3a), is made of layered dielectric sheets. The measurement of capacitance C ignores the layered medium and views it as a homogeneous medium. So, the relative permittivity of a material is the macrolevel homogenization concept that ignores the microlevel discrete composition of a medium. The concept of homogenization is important to design the engineered metamaterials using the discrete metallic and nonmetallic structures embedded in a host medium. It is discussed in section (21.4) of chapter 21.

Figure (4.6d) shows the frequency response of a lossy dielectric medium, as predicted by the RC circuit model. The real part of the permittivity is frequency independent, whereas the imaginary part of the permittivity decreases hyperbolically with frequency. Some dielectric materials may not exhibit this kind of frequency response. More realistic circuit models may be needed for such a dielectric medium. Chapter 6 discusses a few more circuit models of the dielectric media.

The loss‐tangent of a dielectric is also a measurable quantity. Manufacturers provide data on it. However, the loss of a semiconducting substrate is characterized by the conductivity (σ) of a substrate. Even a dielectric material can have some amount of free charge carriers, contributing to its conductivity (σ). The finite conductivity 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.