Primary line constants

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

The constants are conductor resistance and inductance, and insulator capacitance and conductance, which are by convention given the symbols R, L, C, and G respectively. The constants are enumerated in terms of per unit length. The circuit representation of these elements requires a distributed-element model and consequently calculus must be used to analyse the circuit. The analysis yields a system of two first order, simultaneous linear partial differential equations which may be combined to derive the secondary constants of characteristic impedance and propagation constant.

A number of special cases have particularly simple solutions and important practical applications. Low loss cable requires only L and C to be included in the analysis, useful for short lengths of cable. Low frequency applications, such as twisted pair telephone lines, are dominated by R and C only. High frequency applications, such as RF co-axial cable, are dominated by L and C. Lines loaded to prevent distortion need all four elements in the analysis, but have a simple, elegant solution.

The constants

There are four primary line constants, but in some circumstances some of them are small enough to be ignored and the analysis can be simplified. These four, and their symbols and units are as follows:

R and L are elements in series with the line (because they are properties of the conductor) and C and G are elements shunting the line (because they are properties of the dielectric material between the conductors). G represents leakage current through the dielectric and in most cables is very small. The word loop is used to emphasise that the resistance and inductance of both conductors must be taken into account. For instance, if a line consists of two identical wires that have a resistance of 25 mΩ/m each, the loop resistance is double that, 50 mΩ/m. Because the values of the constants are quite small, it is common for manufacturers to quote them per kilometre rather than per metre; in the English-speaking world "per mile" can also be used.

The word "constant" can be misleading. It means that they are material constants; but they may vary with frequency. In particular, R is heavily influenced by the skin effect. Furthermore, while G has virtually no effect at audio frequency, it can cause noticeable losses at high frequency with many of the dielectric materials used in cables due to a high loss tangent. Avoiding the losses caused by G is the reason many cables designed for use at UHF are air-insulated or foam-insulated (which makes them virtually air-insulated). The actual meaning of constant in this context is that the parameter is constant with distance. That is the line is assumed to be homogenous lengthwise. This condition is true for the vast majority of transmission lines in use today.

Circuit representation

The line constants cannot be simply represented as lumped elements in a circuit; they must be described as distributed elements. For instance "pieces" of the capacitance are in between "pieces" of the resistance. However many pieces the R and C are broken into, it can always be argued they should be broken apart further to properly represent the circuit, and after each division the number of meshes in the circuit is increased. This is shown diagramtically in figure 1. To give a true representation of the circuit, the elements must be made infinitesimally small so that each element is distributed along the line.

It is convenient for the purposes of analysis to roll up these elements into general series impedance, Z, and shunt admittance, Y elements such that;

The term constant is even more misleading for the secondary constants as they usually vary quite strongly with frequency, even in an ideal situation where the primary constants do not. As with the primary constants, the meaning of "constant" is that the secondary constants do not vary with distance along the line, not that they are independent of frequency.

Characteristic impedance

Such a line will never return a reflection since the incident wave will never reach the end to be reflected. This is so because the remainder of the line is still infinitely long and therefore equivalent to the original line.

Taking limits of both sides

Propagation constant

Each infinitesimal section will multiply the voltage drop by the same factor.

From the definition of propagation constant,

Hence,

Special cases

Ideal transmission line

An ideal transmission line will have no loss, which implies that the resistive elements are zero. It also results in a purely real (resistive) characteristic impedance. The ideal line cannot be realised in practice, but it is a useful approximation in many circumstances. This is especially true, for instance, when short pieces of line are being used as circuit components such as stubs. A short line has very little loss and this can then be ignored and treated as an ideal line. The secondary constants in these circumstances are;

Twisted pair

Typically, twisted pair cable used for audio frequencies or low data rates has line constants dominated by R and C. The dielectric loss is usually negligible at these frequencies and G is close to zero. In those circumstances the secondary constants become,

The attenuation of this cable type increases with frequency, causing distortion of waveforms. This is another undesirable effect. The nominal impedance quoted for this type of cable is, in this case, very nominal, being valid at only one spot frequency, usually quoted at 800 Hz or 1 kHz.

Co-axial cable

This must eventually be the case as the frequency is increased for any cable. Under those conditions R and G can both be ignored (except for the purpose of calculating the cable loss) and the secondary constants become;

Loaded line

Loaded lines are lines designed with deliberately increased inductance. This is done by adding iron or some other magnetic metal to the cable or adding coils. The secondary constants are here related to the primary constants by;

Velocity

The velocity of propagation is given by,

Since,

The lower the capacitance the higher the velocity. With an air dielectric cable, which is approximated to with low-loss cable, the velocity of propagation is very close to c, the speed of light in vacuo.