a large class of the transmission line structures. For any practical transmission line, the losses are always present on a line.
2.1.5 Characteristic of Lossy Transmission Line
A transmission line, such as a coaxial cable, a flat cable, used in the computer, or a feeder to TV receiver, is embedded in a lossy dielectric medium. The loss in a dielectric medium is known as the dielectric loss of a transmission line. Of course, when the line is in the air medium, the dielectric loss could be neglected. Likewise, there is another kind of loss, known as the conductor loss, on a transmission line. It is due to the finite conductivity of the conducting wires or the strip conductors forming a line. All open transmission lines tend to radiate some power, leading to radiation loss. In the case of a planar transmission line, there are also other mechanisms of losses. However, this section considers only the conductor and the dielectric losses of a line and their effect on the propagation characteristics of the line.
Characteristic Impedance
The 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].
Propagation Constant
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:
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.
