Anand K. Verma - Introduction To Modern Planar Transmission Lines

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 mis 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 cand J dare 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 eis 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.

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 extand 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 is defined by a vector product of , known as the Poynting vector :

(4.4.20)

The divergence of the above equation provides the power entering, or leaving, a location in the space:

(4.4.21)

The energy contained per unit volume, i.e. the energy density , in a dispersive and a nondispersive medium, in the form of the electric and magnetic energy, is given by the following expressions:

(4.4.22)

A physical medium is dispersive. For a dispersive medium, the modified equation (4.4.22b)is valid [B.2, B.8, B.11–B.15]. The energy density W is a positive quantity, i.e. the following relations must be satisfied:

(4.4.23)

The medium having finite conductivity σ dissipates the EM‐energy in the form of heat given by the Joule's law:

(4.4.24)