Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 59

The propagation constant γ of a uniform lossy transmission line is given by equation (2.1.34). It could be approximated under the low‐loss condition. Its real and imaginary parts are separated to get the frequently used approximate expressions for the attenuation and phase constants of a line:

(2.1.49)

On neglecting ω², ω³, and ω⁴ terms, the real part of the propagation constant γ provides the attenuation constant, whereas the imaginary part gives the propagation constant:

(2.1.50)

The first term of the above equation (2.1.50a) shows the conductor loss of a line, while the second term shows its dielectric loss. If R and G are frequency‐independent, the attenuation in a line would be frequency‐independent under ωL >> R and ωC >> G conditions. However, usually, R is frequency‐dependent due to the skin effect. In some cases, G could also be frequency‐dependent [B.7].

The dispersive phase constant β is obtained from the imaginary part of equation (2.1.49):

(2.1.51)

On neglecting the second‐order term, β becomes a linear function of frequency and the line is dispersionless. In that case, its phase velocity is also independent of frequency. A lossy line is dispersive. However, it also becomes dispersionless under the Heaviside's condition – (2.1.48). A transmission line, such as a microstrip in the inhomogeneous medium, can have dispersion even without losses.