Introduction To Modern Planar Transmission Lines. Anand K. Verma

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:

(2.1.42)

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:

(2.1.43)

      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:

(2.1.44)

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)

(2.1.47)

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

(2.1.48)

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:

(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.

      

      2.1.6 Wave Equation with Source

      In the above discussion, the development of the voltage and current wave equations has ignored the voltage or current source. However, a voltage or current source is always required to launch the voltage and current waves on a line. Therefore, it is appropriate to develop the transmission line equation with a source [B.8]. The consideration of a voltage/current source is important to solve the electromagnetic field problems of the layered medium planar lines, discussed in chapters 14 and 16.

      Shunt Current Source

Figure (2.7) shows the lumped element model of a transmission line section of length Δx with a shunt current source