Introduction To Modern Planar Transmission Lines. Anand K. Verma

Introduction To Modern Planar Transmission Lines

circuit model helps to understand the electrical property of a medium. It is further useful for simulating the electrical responses of a medium. The electrical property of a dielectric medium is expressed through relative permittivity that shows the electric energy storage ability of the medium. The capacitor also stores electric energy. Therefore, a dielectric medium is modeled as a capacitor. The loss in a medium is due to the dissipation of energy that is modeled as a resistor. Similarly, the permeability of a medium, such as an inductor, shows its ability to store magnetic energy. Therefore, the permeability of a medium is modeled as an inductor [B.10–B.12].

4.3.1 RC Circuit Model of Lossy Dielectric Medium

Figure (4.6b) shows a parallel‐plate capacitor, containing a lossy dielectric medium with complex relative permittivity images . 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) equation

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

(4.3.2) equation

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

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) equation

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) equation

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 images , 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) equation

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) equation

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) equation

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 images is frequency independent, whereas the imaginary part of the permittivity images 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

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