Introduction To Modern Planar Transmission Lines. Anand K. Verma

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      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.

4.4 Maxwell Equations and Power Relation

      The experimental observations on the time‐dependent electromagnetic field are described by a set of the vector differential equations, known as the Maxwell equations. These sets of equations form the fundamental laws of nature, governing behaviors of the EM‐field in free space, and also in material media. Each part of Maxwell equations shows a definite nature of the EM‐field, explored by different investigators; Faraday, Ampere, and Gauss. However, Maxwell has put them in the form of a consistent set of equations using a mathematical format called the quaternions. He has also introduced the concept of displacement current. On solving the equations, either for the electric field or the magnetic field, the wave equations are obtained for the electric and magnetic fields traveling at a velocity of light in free space. Poynting has determined the power carried by the EM‐wave. Finally, Heaviside replaced the quaternions by the vector notations and presented the Maxwell equations in the present form, i.e. in the convenient form of the vector differential equations [B.26–B.28]. Such a formulation of the electromagnetic theory has opened the grand path of modern research and investigation. So, the modern form of Maxwell equations can be truly called Maxwell–Heaviside equations. However, in this text, we use normal terminology to follow the current practice.

      4.4.1 Maxwell’s Equations

The set of Maxwell’s equations, given below, consists of four time‐dependent vector equations; relating the sources such as conduction current density (), electric displacement current density , and magnetic displacement current density to the magnetic field ( and the electric field ( respectively:

(4.4.1)

      All quantities in the above equations are space‐time dependent. The current densities , , and are not the power supplying sources to the propagating EM‐wave in a medium. These current densities are created by the externally applied magnetic and electric current densities and supplying power to the EM‐wave and partly getting absorbed as loss in a conducting medium.

      The right‐hand sides of the above equations could be treated as the sources (excitations) and the left‐hand fields as the responses. The force field quantities (,), flux field quantities (), and current densities are functions of both the space variables and time variable. Further, in any material medium, the flux field quantities are related to the force field quantities by the constitutive relations, given in equation (4.1.7). The conduction current density in a lossy medium is related to the electric field, as given in equation (4.1.9).

      In general, ε_r, μ_r, σ are the tensors quantities for an anisotropic medium. However, these are scalar quantities for an isotropic medium. They are also treated as complex quantities to include the losses of a medium. In the case of a complex conductivity σ^*, its real part is responsible for the loss in a medium, whereas its imaginary part accounts for the energy storage. In a dispersive medium, ε_r, μ_r and σ are also frequency‐dependent. The characteristics of various kinds of media, such as dielectrics, conductors, plasma, semiconductors, ferrites, and so forth are accounted for in Maxwell’s equations through the constitutive relations applicable to these physical media. Maxwell’s equations, along with the constitutive relations, are the field equations, not the force equations, i.e. these equations do not express the forces exerted by the fields on stationary or moving charges. This is achieved through Lorentz’s force equation:

      (4.4.2)

      where q is the charge on a mass m that is moving with velocity v. In this case, Lorentz's force acting on a moving charge is equal to Newtonian force:

      (4.4.3)

Lorentz's force has two components: (i) electric force component and