Per-Unit-Length Parameters of a Parallel-Strip Transmission Line: 17-Phys-A3 Electromagnetics, May 2016, Q6

Question 6: Per-Unit-Length Parameters of a Parallel-Strip 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.

Given. Ribbon width $w=2$ cm; dielectric thickness (plate separation) $d=0.5$ mm; dielectric $\varepsilon_r=2.25$; non-magnetic conductors/dielectric ($\mu_r=1$); fringing neglected (parallel-plate approximation valid since $w\gg d$).

Given data
Quantity	Symbol	Value
Ribbon width	$w$	2 cm
Separation	$d$	0.5 mm
Relative permittivity	$\varepsilon_r$	2.25

Find. $C'$, $L'$ (per metre), $Z_0$, and $v_p$.

Approach. Model the ribbons as an ideal parallel-plate capacitor/inductor per unit length (fringing neglected, $w/d=40\gg1$ so this is a good approximation), then combine via $Z_0=\sqrt{L'/C'}$ and $v_p=1/\sqrt{L'C'}$.

Capacitance per unit length. $$C'=\frac{\varepsilon_0\varepsilon_r w}{d}=\frac{(8.85\times10^{-12})(2.25)(0.02)}{0.0005}=\boxed{796.5\ \text{pF/m}}.$$
Inductance per unit length. For a non-magnetic parallel-plate line, $L'=\mu_0 d/w$: $$L'=\frac{\mu_0 d}{w}=\frac{(4\pi\times10^{-7})(0.0005)}{0.02}=\boxed{31.42\ \text{nH/m}}.$$
Characteristic impedance. $$Z_0=\sqrt{\frac{L'}{C'}}=\sqrt{\frac{31.42\times10^{-9}}{796.5\times10^{-12}}}=\boxed{6.28\ \Omega}.$$ (Equivalently, $Z_0=\eta_0\,d/(w\sqrt{\varepsilon_r})$ for a TEM parallel-plate line, which reproduces the same value.)
Propagation velocity. Since $L'C'=\mu_0\varepsilon_0\varepsilon_r$ (the $w,d$ geometry cancels), $$v_p=\frac{1}{\sqrt{L'C'}}=\frac{c}{\sqrt{\varepsilon_r}}=\frac{3\times10^8}{\sqrt{2.25}}=\boxed{2.00\times10^{8}\ \text{m/s}}.$$

Parallel-strip line cross-section (fringing neglected): dielectric fills the full gap between the two 2 cm ribbons, 0.5 mm apart.

Final results
Quantity	Value
Capacitance per metre $C'$	796.5 pF/m
Inductance per metre $L'$	31.42 nH/m
Characteristic impedance $Z_0$	6.28 Ω
Propagation velocity $v_p$	$2.00\times10^8$ m/s

Topic: Parallel-plate (stripline-like) transmission line parameters from ideal capacitor/inductor geometry.

Key relations: $C'=\varepsilon_0\varepsilon_r w/d$; $L'=\mu_0 d/w$ (non-magnetic); $Z_0=\sqrt{L'/C'}$; $v_p=1/\sqrt{L'C'}=c/\sqrt{\varepsilon_r}$ for any TEM line with non-magnetic dielectric.

Why this works: Because $L'C'=\mu\varepsilon$ for any uniform TEM structure regardless of its cross-sectional shape, the propagation velocity depends only on the dielectric, never on the geometry — the width and separation cancel exactly, which is a useful self-check on the algebra.

Common pitfall: Using the free-space $\eta_0=377\ \Omega$ directly as $Z_0$ instead of scaling it by the plate aspect ratio $d/w$ (correct here gives only 6.28 Ω, since $w\gg d$ makes this a very low-impedance line).

Source: Sadiku, Elements of Electromagnetics, Ch. 11 (transmission line parameters); Hayt & Buck, Engineering Electromagnetics, Ch. 10.

Question 6 of 8: Per-Unit-Length Parameters of a Parallel-Strip Transmission Line

Question 6: Per-Unit-Length Parameters of a Parallel-Strip Transmission Line (equal value)