Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 58
Characteristic Impedance
ОглавлениеThe characteristic impedance Z0 of a uniform lossy transmission line is given by equation (2.1.39). In the limiting case, ω → 0, i.e. at a lower frequency, it is reduced to a real quantity that is dominated by the lossy elements of a line:
(2.1.41)
The characteristic impedance Z0 at very high frequency, i.e. for ω → ∞, is also reduced to a real quantity. However, now it is dominated by the lossless reactive elements:
At higher frequency, we have ωL >> R and ωC >> G. Therefore, the R and G are normally ignored for the computation of characteristic impedance of a low‐loss microwave transmission line. In the intermediate frequency range, the characteristic impedance of a line is a complex quantity. Its imaginary part indicates the presence of the loss in a line. Equation (2.1.42) is also applicable to a lossless line.
The characteristic impedance of transmission line in the lossless dielectric medium, or a moderately lossy medium where G could be neglected, is obtained in equation (2.1.43). However, the conductor loss is present on the line:
The measured or computed complex characteristic impedance of a line, over a certain frequency range, with a negative imaginary part, indicates that the loss in the line is mainly due to the conductor loss [J.4].
The alternative case of a lossy line, with G ≠ 0, R = 0, could be also considered. In this case, the conductor loss is ignored; however, the dielectric loss is dominant. The characteristic impedance of such line is approximated as follows:
If the imaginary part of the characteristic impedance of a line is positive over some frequency range, then the dielectric loss dominates the loss in the line. However, if both R and G are moderately present, with ωL >> R and ωC >> G, the real and imaginary parts of the characteristic impedance could be approximated by using the binomial expansion as follows:
(2.1.45)
(2.1.46)
In the above expression, ω3 and ω4 terms are neglected. If we neglect ω2 terms and also take G = 0 or R = 0, equation (2.1.47) reduces to either equation (2.1.43) or equation (2.1.44), respectively. It is also possible that with the change of frequency, the imaginary part of characteristic impedance changes from a positive to a negative value indicating that the dominant loss can change from the dielectric loss to the conductor loss. For such cases, R and G are usually frequency‐dependent [J.4]. Over a band of frequencies, the imaginary part of the characteristic impedance could be zero leading to
It is well known as Heaviside's condition. On meeting it, a lossy line becomes dispersion‐less and the propagation constant β becomes a linear function of frequency, while the attenuation constant becomes frequency‐independent. Following the above equation (2.1.48), a lossy line could be made dispersionless by the inductive loading [B.5, B.6].
