2 Parallel Wires
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Parallel Wires

This page is for transmission lines which consist of two cylindrical conductors, which run parallel to each other. In the most general case, the conductors may be different diameter, and have a hollow core, as shown below.

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It is common for the conductors to be the same diameter, and to be solid rather than hollow - for example, they may be two conventional round wires.

A hollow core example is given because at high frequencies, even conductors which are solid metal, can appear to have an insulating core, due to the skin effect.

Capacitance
Different Diameter, Any Separation

Consider the case where two cylindrical, parallel, conductors form a transmission line

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Using electromagnetic field theory, one can calculate the capacitance of two parallel wires. The details will be omitted here, but are available many places for those interested \[C' = \frac{{{\varepsilon _r}{\varepsilon _0}2\pi }}{{{{\cosh }^{ - 1}}\left( {\frac{{4{s^2} - d_1^2 - d_2^2}}{{2{d_1}{d_2}}}} \right)}}\] Where

  • \(C'\) is the capacitance per unit length of the wire pair, usually measured in farads per meter
  • \({\varepsilon _0}\) is the electric constant (also known as the permittivity of free space), which is approximately 8.85 × 10⁠⁠-⁠12 farads per meter. Note that the units on this parameter will match the units on \(C'\).
  • \({\varepsilon _r}\) is the relative permittivity of the space surrounding the wires, unitless. This was at one time called the dielectric constant, but that term is now depreciated.
  • \(D\) length of each wire, usually measured in meters, so it is compatible with the units of the electric constant.
  • \(s\) distance from center of one conductor, to center of the other. It can be measured in any units of length, provided the same unit is used for the \({d_1}\) and \({d_2}\) distances.
  • \({d_1}\) outer diameter of one conductor. It can be measured in any units of length, provided the same unit is used for the \(s\) and \({d_2}\) distances.
  • \({d_2}\) outer diameter of the other conductor. It can be measured in any units of length, provided the same unit is used for the \(s\) and \({d_1}\) distances.
  • \(C\) is the total capacitance of the transmission line,and is given by \(C=C'D\), where \(D\) is the length of the transmission line. Notice that current will flow through both conductors, traveling a total distance of \(2D\). When measuring \(D\), it is best to use meters, so it is compatible with the denominator of the units of the electric constant.

Notice the thickness of the conductors is not labeled in the figure above, and is not used in the calculations of capacitance. While the thickness is important for resistance measurements, it makes no difference capacitance. A pair of very thin cylinders will have the same capacitance as a pair of solid conductors of the same external diameter.

Same Diameter, Any Separation

Many transmission lines use the same gauge, or diameter, wire in both conductors.

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This simplifies the expression of capacitance to \[\begin{array}{c} {d_1} = {d_2} = d\\ C' = \frac{{{\varepsilon _r}{\varepsilon _0}\pi }}{{{{\cosh }^{ - 1}}\left( {\frac{s}{d}} \right)}} \end{array}\] Where

  • \({d}\) outer diameter of either conductor. It can be measured in any units of length, provided the same unit is used for \(s\).
  • All other terms are described in the previous section, above.

Same Diameter, Wide Separation

In some applications, the separation of the wires is far greater than the diameter of either one.

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In this case, the hyperbolic function in the capacitance expression can be avoided, by using a large angle approximation, and obtaining \[\begin{array}{c} {d_1} = {d_2} = d\\ d \ll s\\ C' \cong \frac{{{\varepsilon _r}{\varepsilon _0}\pi }}{{\ln \left( {\frac{{2s}}{d}} \right)}} \end{array}\] Where

  • \(s\) distance from one conductor, to the other. \(s\) must be much larger than \(d\), so it does not make much difference if one measures from center-to-center or edge-to-edge of the conductors.
  • \(d\) outer diameter of either conductor. It can be measured in any units of length, provided the same unit is used for \(s\).
  • All other terms are described in the previous section, above.
The error in the above approximation is about 5% for \(s=2d\), and is under 1% when \(s=4d\).

If the wires are different diameters, then replace \(d\) in the expression above, with the geometric mean of the two diameters. But if you need to go to that much work, you may just want to return to the expression for conductors of different diameters, and any separation.

Volume That Influences Capacitance

The capacitance of two parallel conductors is affected by the relative permittivity, \({\varepsilon _r}\), of the region which is external to both conductors.

We do not typically speak of the relative permittivity of the conductors themselves, or else call the permittivity of conductors infinity. Either way, it does not impact the capacitance of the transmission line.

If the conductors are hollow (have a center region which acts like an insulator), then the permittivity of the core has no impact on the capacitance calculations.

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Inductance

If the wires are connected to circuits at either end that allow current to pass around the loop they form, then the wires will have an inductance. We normally think of inductors as being loops of wire wound close together, often around some magnetic material such as iron. However any loop of wire will have inductance.

Using electromagnetic field theory, one can calculate the inductance of this loop, \(L\), or the inductance per unit length, \(L'\). The details will be omitted here, but are available many places. \[L' = \frac{{{\mu _r}{\mu _0}}}{{2\pi }}{\cosh ^{ - 1}}\left( {\frac{{4{s^2} - d_1^2 - d_2^2}}{{2{d_1}{d_2}}}} \right)\] Where

  • \(L'\) is the inductance per unit length of the wire pair, usually measured in henries per meter
  • \({\mu _0}\) is the magnetic constant (also known as the permeability of free space), which is approximately \(4\pi × 10⁠⁠-⁠7 henries per meter. Note that the units on this parameter will match the units on \(C'\).
  • \({\mu _r}\) is the relative permeability of the space surrounding the wires, unitless.
  • \(s\) distance from center of one conductor, to center of the other. It can be measured in any units of length, provided the same unit is used for the \({d_1}\) and \({d_2}\) distances.
  • \({d_1}\) outer diameter of one conductor. It can be measured in any units of length, provided the same unit is used for the \(s\) and \({d_2}\) distances.
  • \({d_2}\) outer diameter of the other conductor. It can be measured in any units of length, provided the same unit is used for the \(s\) and \({d_1}\) distances.
  • \(L\) is the total capacitance of the transmission line,and is given by \(L=L'D\), where \(D\) is the length of the transmission line. When measuring \(D\), it is best to use meters, so it is compatible with the denominator of the units of the magnetic constant.

Many cables use the same gauge (diameter) wire in both conductors. This simplifies the inductance equation to \[\begin{array}{c} {d_1} = {d_2} = d\\ L' = \frac{{{\mu _r}{\mu _0}}}{\pi }{\cosh ^{ - 1}}\left( {\frac{s}{d}} \right) \end{array}\] If the wires are the same gauge, and the separation is much greater than the diameter, then the hyperbolic function can be avoided with the approximation \[\begin{array}{c} {d_1} = {d_2} = d\\ s \gg d\\ L' \cong \frac{{{\mu _r}{\mu _0}}}{\pi }\ln \left( {\frac{{2s}}{d}} \right) \end{array}\] The error in the above approximation is about 5% for s=2d, and is under 1% when s=4d. If the wires are different diameters, and the spacing is large compared to both diameters, then the \(d\) term in the equation above should be replaced with the geometric mean of the wire diameters. It may be just as easy to return to the formula above for wires of any diameter and any separation.

Volume that Affects Inductance

The relative permeability, \({\mu _r}\), in all regions in the vicinity of two parallel conductors affect the inductance calculations. This includes the value of \({\mu _r}\) of the conductors themselves, \({\mu _r}\) outside the conductors, and if they are hollow, the value of \({\mu _r}\) in the interior of the conductors.

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Many electrical insulators have a \({\mu _r}\) very close to one. This includes a vacuum, nearly all gasses, glass and many ceramic materials, and many electrical insulators made from petrochemical products. If the conductors are hollow, then the calculations on this page assumed the interior of the conductors was an insulator with a \({\mu _r}\) of one. Note that some conductors use a steel core, which has a \({\mu _r}\) significantly greater than one. The calculations on this page do not address this kind of conductor.

Electrical conductors carrying alternating current can see a substantial increase in their resistance due to the skin effect. This can be minimized by selecting conductors which have a \({\mu _r}\) of one. Many commonly used conductors, including aluminium and copper have \({\mu _r}\) of one, as does material such as gold and water. Iron based alloys, such as steel are a notable exception, with \({\mu _r}\) which can reach into the thousands and above. The calculations on this page assume the conductor has a \({\mu _r}\) of one.

The \({\mu _r}\) in all areas surrounding the conductors impact their inductance. The calculations above assume the \({\mu _r}\) was uniform in all regions near the conductors.

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Many tools, machines, office furniture, enclosures, and buildings are fabricated using iron based alloys, such as steel. Notice that inductance of a parallel set of conductors will be strongly influenced by passing near these objects.