Characteristic Impedance of a Coaxial Line with a Partial Dielectric Coating: 17-Phys-A3 Electromagnetics, December 2016, Q3

Question 3: Characteristic Impedance of a Coaxial Line with a Partial Dielectric Coating (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. Inner-conductor radius $a=5$ mm; outer-conductor radius $b=10$ mm; a $2$ mm-thick dielectric layer ($\varepsilon_r=2.25$) coats the inner conductor, from $a$ out to $r_1=7$ mm; the remaining annulus ($r_1=7$ mm to $b=10$ mm) is air ($\varepsilon_r=1$); both media non-magnetic ($\mu_r=1$).

Radial cross-section: a partial dielectric coating creates two concentric capacitive layers in series.

Find. Characteristic impedance $Z_0$ and propagation velocity $v_p$.

Approach. Both media are non-magnetic, so the magnetic field pattern (and hence the inductance per unit length) is exactly the usual single-dielectric coax result. The radial electric flux, however, crosses BOTH dielectric layers in turn for the same enclosed charge, so the two layers act as two cylindrical capacitors in series; combine them, then get $Z_0$ and $v_p$ from $L'$ and $C'$.

Inductance per unit length (dielectric-independent). $$L'=\frac{\mu_0}{2\pi}\ln\!\frac{b}{a}=\frac{4\pi\times10^{-7}}{2\pi}\ln\!\frac{10}{5}=\boxed{0.1386\ \mu\text{H/m}}.$$
Capacitance of each layer. $$C_1'=\frac{2\pi\varepsilon_0\varepsilon_{r1}}{\ln(r_1/a)}=\frac{2\pi(8.85\times10^{-12})(2.25)}{\ln(7/5)}=0.3718\ \text{nF/m},$$ $$C_2'=\frac{2\pi\varepsilon_0}{\ln(b/r_1)}=\frac{2\pi(8.85\times10^{-12})}{\ln(10/7)}=0.1559\ \text{nF/m}.$$
Series combination. The two layers carry the same $D$-flux, so they combine as series capacitors: $$C'=\left(\frac{1}{C_1'}+\frac{1}{C_2'}\right)^{-1}=\boxed{0.1098\ \text{nF/m}}.$$
Characteristic impedance and propagation velocity. $$Z_0=\sqrt{\frac{L'}{C'}}=\sqrt{\frac{0.1386\times10^{-6}}{0.1098\times10^{-9}}}=\boxed{35.5\ \Omega},$$ $$v_p=\frac{1}{\sqrt{L'C'}}=\boxed{2.563\times10^{8}\ \text{m/s}}\ \ (=0.854\,c).$$

Coax with partial dielectric coating
Quantity	Result
$L'$	0.1386 µH/m
$C'$	0.1098 nF/m
$Z_0$	35.5 Ω
$v_p$	$2.563\times10^8$ m/s

Topic: Radially-layered (partially-filled) coaxial transmission line.

Key relations: single-dielectric coax $L'=(\mu_0/2\pi)\ln(b/a)$, $C'=2\pi\varepsilon/\ln(b/a)$; two radial dielectric layers combine as series capacitors, $C'=(1/C_1'+1/C_2')^{-1}$; $Z_0=\sqrt{L'/C'}$, $v_p=1/\sqrt{L'C'}$.

Why this works: the magnetic field of a coaxial TEM mode depends only on the enclosed current (Ampère's law), never on the dielectric, so $L'$ is unchanged by adding a dielectric layer. The radial $D$-field, however, is set by the enclosed free charge and must be continuous in flux across each concentric cylindrical surface, which is exactly the condition that makes two radial dielectric layers behave as capacitors in series (not in parallel — that would be for AXIALLY split dielectrics instead).

Common pitfall: averaging the two permittivities or combining the capacitances in parallel instead of in series — a radial (concentric-ring) split is always a series combination because the same charge/flux crosses both layers in turn.

Source: Sadiku, Elements of Electromagnetics, Ch. 11 (coaxial transmission lines); Pozar, Microwave Engineering, Ch. 2.

Question 3 of 8: Characteristic Impedance of a Coaxial Line with a Partial Dielectric Coating

Question 3: Characteristic Impedance of a Coaxial Line with a Partial Dielectric Coating (equal value)