A pair of straight AWG 24 copper wires are parallel to each other and to a metal chassis. The distance between the wires is 1 mm and the distance of the pair from the chassis is 1 cm. Calculate the wire-to-wire and wire-to-ground capacitance per cm of length and the total capacitance if the length

Question

A pair of straight AWG 24 copper wires are parallel to each other and to a metal chassis. The distance between the wires is 1 mm and the distance of the pair from the chassis is 1 cm. Calculate the wire-to-wire and wire-to-ground capacitance per cm of length and the total capacitance if the length of the pair of wires is 4 cm.

Accepted Answer

From Chapter 2, the radius of AWG 24 wire is

[latex] \begin{aligned} r &=\frac{1}{2}(8.251 mm ) \exp \left(-\frac{24}{8.628} ight) \\&=256 \mu m . \end{aligned} [/latex]

From (8.39), with [latex] r=256 \mu m , d=1 mm [/latex], [latex] h=1 cm [/latex]

[latex] \begin{aligned} \frac{\Delta C_{w w}}{\Delta l} &=\pi \varepsilon_0\left[\ln \left[\frac{d}{r \sqrt{1+0.25(d / h)^2}} ight] ight]^{-1} \\ &=0.204 pF cm ^{-1} \\ \frac{\Delta C_{w g}}{\Delta l} &=2 \pi \varepsilon_0\left[\ln \left(\frac{2 h-r}{r} ight) ight]^{-1} \\ &=0.128 pF extrm {cm } ^ { - 1 } \end{aligned} [/latex]

The total capacitance per unit length is given by

[latex] \begin{aligned}\frac{\Delta C}{\Delta l}=\frac{\Delta C_{w w}}{\Delta l}+\frac{\Delta C_{w g}}{\Delta l} \cong 0.332 pF m ^{-1}\end{aligned} [/latex]

so the capacitance of a 4-cm length of wire is

[latex] C \cong 1.33 pF [/latex]

As you can see, wires a few or more cm in length can introduce a few picofarads of stray capacitance. Such capacitance limits the rate at which the voltage across a load can change. For example, if a 4 cm length of the conductors of this example drives a [latex] 10 k \Omega [/latex] load, the effective time constant due to the wire-to-ground capacitance alone is about 13 ns, which means that the time required for the load voltage to change from one value to another is approximately 65 ns This means the load voltage can switch between two levels no faster than about 15 million times per second. If we are talking about a digital communication system, for example, we could transmit no more than a few million bits per second over the line to the load, which would be unacceptably slow in today’s world.

Question 8.14: A pair of straight AWG 24 copper wires are parallel to each ...

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