22-Elec-A7 Electromagnetics · May 2013

Question 7 of 8: Characteristic Impedance and Velocity of a Ribbon Line

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, May 2013 — 07-Elec-A7, Electromagnetics. Three hours, closed book; one of two approved calculators (Casio or Sharp) permitted. Eight questions, all of equal value; any five constitute a complete paper and only the first five presented are marked. All eight are solved here, since the set is a study resource. Aids printed on the cover page: ε₀ = 8.85 × 10⁻¹² F/m and μ₀ = 4π × 10⁻⁷ H/m; Question 2 additionally supplies the quarter-wave relation.

Reference texts (22-Elec-A7 Electromagnetics).

D. M. Pozar, Microwave Engineering, 4th ed. — transmission lines, matching, waveguides.
M. N. O. Sadiku, Elements of Electromagnetics, 7th ed. — plane waves, guided waves, radiation.
W. H. Hayt and J. A. Buck, Engineering Electromagnetics, 9th ed. — field theory and transients on lines.
C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed. — short elements, image theory.
F. T. Ulaby and U. Ravaioli, Fundamentals of Applied Electromagnetics, 8th ed. — oblique incidence, surface currents.
S. J. Chapman, Electric Machinery Fundamentals, 5th ed. — rotating-loop induction and torque (Question 3).

Conventions used throughout. The paper itself quotes propagation velocities as $3\times10^{8}\ \text{m/s}$, so free space is taken as $c=3.00\times10^{8}\ \text{m/s}$ and the intrinsic impedance as $\eta_0=\mu_0 c=120\pi=376.99\ \Omega$. Lines are treated as lossless, and "rms" is stated explicitly wherever the exam asks for it.

Question 7: Characteristic Impedance and Velocity of a Ribbon Line (20 marks)

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. Two flat parallel conducting ribbons 10 mm wide are separated by a 0.1 mm layer of dielectric with $\varepsilon_r=2.25$; fringing fields are to be disregarded, so the structure is an ideal parallel-plate line.

Given data
Quantity	Symbol	Value
Ribbon width	$w$	10 mm = 0.0100 m
Separation (dielectric thickness)	$d$	0.1 mm = $1.00\times10^{-4}$ m
Relative permittivity	$\varepsilon_r$	2.25
Relative permeability	$\mu_r$	1 (non-magnetic dielectric)
Fringing	—	neglected, so $w/d=100$ gives a uniform field

Find. The characteristic impedance $Z_0$ and the propagation velocity $v_p$ of the line.

Figure 7.1 — Cross-section of the ribbon line. With $w/d=100$ and fringing neglected, the field between the ribbons is uniform, exactly as in an ideal parallel-plate capacitor.

Approach. Compute the per-unit-length capacitance and inductance of an ideal parallel-plate geometry, then combine them in the lossless-line formulas $Z_0=\sqrt{L'/C'}$ and $v_p=1/\sqrt{L'C'}$.

Capacitance per unit length. Treating a one-metre length as a parallel-plate capacitor of plate area $w\times1$ m and gap $d$, $$C'=\frac{\varepsilon_0\varepsilon_r w}{d} =\frac{8.85\times10^{-12}\times2.25\times0.0100}{1.00\times10^{-4}}=1.991\times10^{-9}\ \text{F/m}$$ that is, 1.99 nF per metre — a large value, as expected from the very thin dielectric.
Inductance per unit length. The magnetic flux is confined to the same gap, and the current returns on the opposite ribbon, so $$L'=\frac{\mu_0 d}{w}=\frac{4\pi\times10^{-7}\times1.00\times10^{-4}}{0.0100} =1.257\times10^{-8}\ \text{H/m}=12.6\ \text{nH/m}$$ Note that $L'$ and $C'$ carry the geometry factor $d/w$ in opposite senses, which is what makes the velocity geometry-independent.
Characteristic impedance. For a lossless line, $$Z_0=\sqrt{\frac{L'}{C'}}=\sqrt{\frac{1.257\times10^{-8}}{1.991\times10^{-9}}}=\sqrt{6.311}=\boxed{2.51\ \Omega}$$ The compact closed form gives the same value and is worth carrying: $$Z_0=\frac{\eta_0}{\sqrt{\varepsilon_r}}\cdot\frac{d}{w}=\frac{376.99}{1.50}\times\frac{0.1}{10}=2.51\ \Omega$$
Propagation velocity. $$v_p=\frac{1}{\sqrt{L'C'}}=\frac{1}{\sqrt{1.257\times10^{-8}\times1.991\times10^{-9}}} =\boxed{2.00\times10^{8}\ \text{m/s}}$$ which is exactly $c/\sqrt{\varepsilon_r}=3.00\times10^{8}/1.50$, confirming that for a TEM line the velocity depends only on the filling material and not at all on the ribbon dimensions.
Interpret the result. A width-to-separation ratio of 100 produces a very low impedance line: at 2.51 Ω it could not be driven directly from a 50 Ω source without a transformer, but it is well suited to carrying large currents at low voltage, and its high capacitance per metre makes it a useful low-inductance power-distribution bus. In a real structure the neglected fringing fields would add a little capacitance at the ribbon edges, lowering $Z_0$ by a few percent below the ideal figure.

Final results — Question 7
Quantity	Value
Capacitance per unit length	1.99 nF/m
Inductance per unit length	12.6 nH/m
Characteristic impedance	2.51 Ω
Propagation velocity	$2.00\times10^{8}$ m/s ($=c/1.5$)
Effective permittivity (no fringing)	2.25, so the velocity factor is 0.667

Topic. Per-unit-length parameters and TEM properties of a parallel-plate (ribbon) transmission line.

Key relations. $C'=\varepsilon_0\varepsilon_r w/d$; $L'=\mu_0 d/w$; $Z_0=\sqrt{L'/C'}=(\eta_0/\sqrt{\varepsilon_r})(d/w)$; $v_p=1/\sqrt{L'C'}=c/\sqrt{\varepsilon_r}$; and for any TEM line $L'C'=\mu\varepsilon$.

Why this works. When fringing is neglected the electric field between the ribbons is uniform and the magnetic field circulates in the same region, so both the capacitance and the inductance reduce to elementary parallel-plate results. The geometry factor $d/w$ appears in $L'$ and its reciprocal appears in $C'$, so it cancels completely in the product $L'C'$ and survives only in the ratio $L'/C'$. That single observation explains both answers at once: the velocity is set purely by the material, while the impedance is set by the material and the aspect ratio together. It also explains why wide, closely-spaced conductors always give low-impedance lines and why a coaxial cable's velocity factor is quoted as a property of its dielectric alone.

Common pitfall. Inverting the geometry factor and reporting $Z_0=251\ \Omega$ — the impedance is proportional to $d/w$, not $w/d$, so a wide, thin-gap line must come out low. A second slip is to expect the velocity to depend on the ribbon width; it does not, because the geometry cancels in $L'C'$.

Source. Pozar, Microwave Engineering, 4th ed., §2.1–2.2 (line parameters, parallel-plate example); Sadiku, Elements of Electromagnetics, 7th ed., Ch. 11.

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