Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 168

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 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)

The total power carried in a medium, in the form of the EM‐wave, is

(4.4.25)

The integration is carried over the cross‐section of the medium carrying the EM‐power. The total energy stored in volume V is

(4.4.26)

The total power dissipated in volume V of the medium is

(4.4.27)

Thus, power (P_ext) supplied by the external source is balanced by the following equation:

(4.4.28)

Using Maxwell equations, the power balance equation (4.4.28) is evaluated in terms of the field quantities and external sources. The resulting power balance equation identifies the above‐mentioned expressions for the power in a wave, energy dissipated in the medium, energy stored in the medium, and also the energy supplied by the external electric and magnetic currents.

The dot product of equation (4.4.1a) is taken with , and the dot product of equation (4.4.1b) with , and subtract one from another to get the following expression:

(4.4.29)

In the above equation, is used. In the case of a time‐invariant medium, the relative permittivity and relative permeability of a medium are constants. By using an equation (4.4.21), the above equation is rewritten as follows:

(4.4.30)

On taking volume integral of the above equation and further using Gauss divergence theorem (4.4.13), the following expression is obtained:

(4.4.31)

The power supplied by the external sources is positive. So, the total power in a medium is negative. Out of the total power supplied to a medium, P_wave is power carried away by the EM‐wave, P_dis is power loss in the medium due to the finite conductivity of the medium, and is the oscillating electric and magnetic energy in the medium.