Introduction To Modern Planar Transmission Lines. Anand K. Verma

(4.4.9)

where flux density vectors are related to field intensity vectors in an anisotropic medium by the following tensor constitutive relations:

      (4.4.10)

      In the case of an external source‐free homogeneous isotropic medium, Maxwell’s equations are written in terms of field intensities only:

(4.4.11)

where the scalar constitutive relations are , , . The differential form of the above given Maxwell’s equations provide relations between the electric field and the magnetic field at any location in the medium. Also, the sources are specified at a point in the medium. The Maxwell equations (4.4.9) involving the flux densities and field intensities apply to both the isotropic and anisotropic media, whereas Maxwell’s equations (4.4.11) involving the field intensities apply to the isotropic medium only.

      The differential form of Maxwell equations does not account for the creation of the fields in the space due to the sources such as the charge or current distributed over a line, surface, or volume. This case is incorporated in Maxwell’s equations by converting them to the integral forms. It is achieved with the help of two vector identities:

(4.4.12)

(4.4.13)

      Figure (4.8b) shows the existence of a vector over the surface S. Its boundary is enclosed by the perimeter C. Stoke theorem is defined with respect to Fig. (4.8b) and Gauss divergence theorem with respect to Fig. (4.8c). The unit vector shows the direction of a normal to the surface S. Figure (4.8c) shows a vector existing in the whole of the volume V that is enclosed by the surface S.

      Maxwell’s equations in the integral form, for =0, are obtained by taking the surface integral of Maxwell’s equation (4.4.1a):

(4.4.14)

It is assumed that a surface enclosing the magnetic field does not change with time. By using equation (4.4.12), equation (4.4.14) is changed in the following form:

      (4.4.15)

      where ψ_m is the magnetic flux. It is the Faraday law of induction that gives the induced voltage V, i.e. the emf, on a conducting loop containing the time‐varying magnetic flux, ψ_m:

      (4.4.16)

      Likewise, using Maxwell’s equation (4.4.1b) and (4.4.12) for =0, the second Maxwell’s equation is written in the integral form, giving the modified Ampere’s law:

      (4.4.17)

      In the above equation, J_c and J_d are conduction and displacement current densities creating the magnetic field . The above expression is generalized Ampere’s law due to Maxwell. The magnetomotive force, mmf, is obtained as follows:

(4.4.18)

where ψ_e is the time‐dependent electric flux. Equation (4.4.18b) is Maxwell’s induction law giving the induced mmf due to the time‐varying electric field. For the source free medium with Jc = 0, it is the complementary induction law of Faraday’s law of induction.

      4.4.2 Power and Energy Relation from Maxwell Equations

A medium supporting the electromagnetic fields also stores the EM‐energy and supports the power flow. The EM‐power is supplied to the enclosure by the time‐dependent external electric current density J_ext and the time‐dependent external magnetic current density M_ext. They create the time‐dependent electric field ( and the time‐dependent magnetic field () shown in Fig. (4.8a).

      The external power, supplied by the source to the medium, is

      (4.4.19)

      The field and source quantities have RMS values, and these are also time‐dependent. The power on a transmission line, carrying the voltage and current wave, is P = VI cos φ, i.e. a scalar product of the voltage and current. The EM‐wave is a transverse electromagnetic wave, where the fields are normal to each other. The power density