Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 151

The relative permittivity (ε_r) and the relative permeability (μ_r) of a linear material do not depend on the magnitude of the electric or magnetic field intensity, respectively. However, for nonlinear materials, the relative permittivity and relative permeability are functions of the electric and magnetic field intensity, respectively, and expressed as ε_r(E) and μ_r(H). Thus, for an isotropic nonlinear medium, equation (4.1.7a) is written as

(4.2.1)

where ε_r(E) = ε_r1 + ε_r2E + ε_r3E² + ⋯ and so forth. The coefficients ε_r2, ε_r3, … indicate the order of nonlinearity in the nonlinear relative permittivity. In the case of a time‐harmonic electric field, i.e. E = E₀e^jωt, equation (4.2.1) is written as,

(4.2.2)

It is obvious that while the input signal has only one frequency ω, shown in Fig. (4.2), the output of a nonlinear medium has several harmonically related frequency components, ω, 2ω, 3ω, … and so forth. Thus, a sinusoidal input signal gets distorted, once it passes through a nonlinear medium. Such distortion also occurs in an amplifier in the nonlinear region.

Similarly, the relative permeability of a nonlinear magnetic medium is a function of the amplitude of the magnetic field. The constitutive relation, given by equation (4.1.7b), is written as

(4.2.3)

Figure 4.2 Response of nonlinear medium showing generation of harmonics.

Figure 4.3 Inhomogeneous medium showing a step variation of relative permittivity with substrate height.

where μ_r(H) = μ_r1 + μ_r2H + μ_r3H² + ⋯ and so forth. The coefficients μ_r2, μ_r3, … indicate the order of nonlinearity in the nonlinear relative permeability of a magnetic medium.