Question 2.2: A transmission line has 50 Ω high-frequency impedance, effec......
A transmission line has 50 Ω high-frequency impedance, effective permittivity equal to \epsilon _{\text{eff}}=6, conductor attenuation of \alpha _c=0.5 dB/cm, dielectric attenuation of \alpha _d=0.01 dB/cm at f_0=10 GHz. Suppose that at f_0=10 GHz the line is already in the high-frequency regime and that the DC p.u.l. line resistance is 1/2 of the resistance at f_0. Evaluate the line parameters (\beta , \alpha , Z_0) in the RG, RC and LC regime, specifying the frequency ranges of validity.
Assuming that at f_0 the line is already in the high-frequency regime, in the LC approximation we have:
\begin{aligned} Z_0 & \approx \sqrt{ L / C } \\ \nu _f & =1 / \sqrt{ L C }=3 \times 10^8 / \sqrt{\epsilon_{ \text{eff} }}, \end{aligned}
i.e.:
\begin{aligned} 1 / C =50 \times 3 \times 10^8 / \sqrt{\epsilon_{ \text{eff} }} \longrightarrow C & = \sqrt{6} /\left(150 \times 10^8\right) \\ & =1.633 \times 10^{-10} \text{ F/m} \end{aligned}
and thus:
L = C Z_0^2=4.0825 \times 10^{-7} \text{ H/m.}
The attenuations in the high-frequency approximation yield:
\alpha_c\left(f_0\right) \approx \frac{ R \left(f_0\right)}{2 Z_0}\qquad \alpha_d\left(f_0\right) \approx \frac{ G \left(f_0\right) Z_0}{2},
i.e., since \alpha_c = 0.5 \text{ dB/cm} =1 / 0.086859=5.75 \text{ Np/m; } \\ \qquad \qquad \alpha_d = 0.01 \text{ dB/cm} = 0.115 \text{ Np/m:}
\begin{aligned} & R \left(f_0\right)=2 Z_0 \alpha_c=100 \times 5.75=575 \Omega / \text{m} \\ & G \left(f_0\right)=2 \alpha_d / Z_0=2 \times 0.115 / 50=0.0046 \text{ S/m.} \end{aligned}
Let us verify that the line actually is in the high-frequency regime at 10 GHz; for this we require:
\begin{aligned} & 2 \pi f_0 L \gg R \rightarrow 6.28 \times 10 \times 10^9 \times 4.0825 \times 10^{-7}=25638 \gg 575 \\ & 2 \pi f_0 C \gg G \rightarrow 6.28 \times 10 \times 10^9 \times 1.633 \times 10^{-10}=10.255 \gg 0.0046 \end{aligned}
and both conditions are verified. We can approximately estimate the external inductance since the total reactance p.u.l. at f_0 is:
\begin{aligned} 2 \pi f_0 L & =2 \pi f_0 L _{e x}+2 \pi f_0 L _{in}=2 \pi f_0 L _{e x}+ R \left(f_0\right) \rightarrow L _{e x}= L -\frac{ R \left(f_0\right)}{2 \pi f_0} \\ & =4.0825 \times 10^{-7}-\frac{575}{6.28 \times 10 \times 10^9}=3.9909 \times 10^{-7} \text{ H/m,} \end{aligned}
while:
L _{in}\left(f_0\right)=\frac{ R \left(f_0\right)}{2 \pi f_0}=9.1561 \times 10^{-9} \text{ H/m.}
Since from the data given we cannot estimate the low-frequency behavior of the inductance we approximate the total inductance with a constant, frequency independent value; the line parameters are therefore:
\begin{aligned} L & =4.0825 \times 10^{-7} \text{ H/m} \\ C & =1.633 \times 10^{-10} \text{ F/m} \\ R \left(f_0\right) & =575 \Omega / \text{m} \\ G \left(f_0\right) & =0.0046 \text{ S/m.} \end{aligned}
The frequency behavior of the p.u.l. resistance and conductance can be approximated as follows:
\begin{aligned} G (f) & = G \left(f_0\right) \frac{f}{f_0} \\ R (f) & \approx \frac{ R \left(f_0\right)}{2}\left[1+\left(\frac{f}{f_0}\right)^{1 / 2}\right]. \end{aligned}
The frequency behavior of the propagation constant and attenuation are shown in Fig. 2.4. Notice that, due to the vanishing DC p.u.l. conductance only the intermediate and high-frequency regimes are actually present. In logarithmic scale the high-frequency slope of β is one, the slope of α is asymptotically 1/2 due to the prevailing skin-effect metal attenuation; the intermediate frequency slopes of both α and β are 1/2. The characteristic impedance is shown in Fig. 2.5; in the intermediate frequency range the real and imaginary parts of the impedance are approximately the same in magnitude, and the high-frequency impedance is real.
