Propagation Velocity and Loss of a Low-Loss Transmission Line: 17-Phys-A3 Electromagnetics, May 2016, Q7

Question 7: Propagation Velocity and Loss of a Low-Loss Transmission Line (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Interpretation: the paper's "series resistivity 0.01 Ω/m" and "shunt conductivity 10^-7 1/Ωm" are per-unit-length line parameters, i.e. $R'=0.01\ \Omega/\text{m}$ and $G'=10^{-7}\ \text{S/m}$; with $L'=25\ \mu\text{H/m}$ and $C'=160\ \text{pF/m}$ these give a low-loss line ($R'\ll\omega L'$, $G'\ll\omega C'$ at 1 MHz), as used below.

Given. $L'=25\ \mu\text{H/m}$, $C'=160\ \text{pF/m}$, $R'=0.01\ \Omega/\text{m}$, $G'=10^{-7}\ \text{S/m}$; $f=1$ MHz.

Given data
Quantity	Symbol	Value
Series inductance	$L'$	25 μH/m
Shunt capacitance	$C'$	160 pF/m
Series resistance	$R'$	0.01 Ω/m
Shunt conductance	$G'$	$10^{-7}$ S/m
Frequency	$f$	1 MHz

Find. Propagation velocity $v_p$ and the per-unit-length loss (attenuation constant $\alpha$) at 1 MHz.

Approach. Form $Z=R'+j\omega L'$ and $Y=G'+j\omega C'$, confirm the line is low-loss ($R'\ll\omega L'$, $G'\ll\omega C'$), then use $\gamma=\sqrt{ZY}=\alpha+j\beta$ (exact) and cross-check with the standard low-loss approximations for $\alpha$, $\beta$.

Series and shunt admittance/impedance per metre. $\omega=2\pi(10^6)=6.283\times10^6$ rad/s: $$\omega L'=157.08\ \Omega/\text{m}\ (\gg R'=0.01),\qquad \omega C'=1.0053\times10^{-3}\ \text{S/m}\ (\gg G'=10^{-7}),$$ confirming a low-loss line at this frequency. $$Z=R'+j\omega L'=0.01+j157.08\ \Omega/\text{m},\qquad Y=G'+j\omega C'=10^{-7}+j1.0053\times10^{-3}\ \text{S/m}.$$
Propagation constant (exact complex square root). $$\gamma=\sqrt{ZY}=\alpha+j\beta=(3.24\times10^{-5})+j(0.3974)\ \text{per metre}.$$
Propagation velocity. $$v_p=\frac{\omega}{\beta}=\frac{6.283\times10^6}{0.3974}=\boxed{1.581\times10^{7}\ \text{m/s}}.$$ Check against the low-loss approximation $v_p\approx1/\sqrt{L'C'}=1/\sqrt{(25\times10^{-6})(160\times10^{-12})}=1.581\times10^7$ m/s $\checkmark$.
Loss (attenuation constant). Using the low-loss approximation, $$\alpha\approx\frac{R'}{2}\sqrt{\frac{C'}{L'}}+\frac{G'}{2}\sqrt{\frac{L'}{C'}}=\frac{0.01}{2}\sqrt{\frac{160\times10^{-12}}{25\times10^{-6}}}+\frac{10^{-7}}{2}\sqrt{\frac{25\times10^{-6}}{160\times10^{-12}}}$$ $$=\boxed{3.24\times10^{-5}\ \text{Np/m}}=2.82\times10^{-4}\ \text{dB/m},$$ matching the exact value from Step 2 to four significant figures (the low-loss approximation is excellent here).

Final results
Quantity	Value
Characteristic impedance $Z_0\approx\sqrt{L'/C'}$	395.3 Ω (essentially real)
Phase constant $\beta$	0.3974 rad/m
Propagation velocity $v_p$	$1.581\times10^7$ m/s
Attenuation constant $\alpha$	$3.24\times10^{-5}$ Np/m = $2.82\times10^{-4}$ dB/m

Topic: Propagation constant of a lossy transmission line in the low-loss limit.

Key relations: $\gamma=\sqrt{(R'+j\omega L')(G'+j\omega C')}=\alpha+j\beta$; low-loss approximation $\alpha\approx\tfrac12\!\left(R'\sqrt{C'/L'}+G'\sqrt{L'/C'}\right)$, $\beta\approx\omega\sqrt{L'C'}$, $v_p\approx1/\sqrt{L'C'}$.

Why this works: When $R'\ll\omega L'$ and $G'\ll\omega C'$, the resistive terms perturb $\gamma$ only slightly from the lossless value $j\omega\sqrt{L'C'}$; a first-order (binomial) expansion of $\sqrt{ZY}$ separates cleanly into a real part (the attenuation, driven independently by the series and shunt losses) and an unchanged imaginary part (the phase constant).

Common pitfall: Reporting attenuation as a single combined term without checking the low-loss condition first, or mixing up Np/m and dB/m ($1\ \text{Np/m}=8.686\ \text{dB/m}$).

Source: Sadiku, Elements of Electromagnetics, Ch. 11 (lossy lines, low-loss approximation); Hayt & Buck, Engineering Electromagnetics, Ch. 10.

Question 7 of 8: Propagation Velocity and Loss of a Low-Loss Transmission Line

Question 7: Propagation Velocity and Loss of a Low-Loss Transmission Line (equal value)