Primary line constants

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 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 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.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.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.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.This condition is true for the vast majority of transmission lines in use today.are given by;[14] 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; Analysis of this network (figure 2) will yield the secondary line constants: the propagation constant,It is possible to choose specific values of the primary constants that result inIf the finite segment is very short, then in the equivalent circuit it will be modelled by an L-network consisting of one element ofusing the usual network analysis theorems,[17][18] which re-arranges to, Taking limits of both sides and since the line was assumed to be homogenous lengthwise, The ratio of the line input voltage to the voltage a distancefurther down the line (that is, after one section of the equivalent circuit) is given by a standard voltage divider calculation.The remainder of the line to the right, as in the characteristic impedance calculation, is replaced with,[19][20] Each infinitesimal section will multiply the voltage drop by the same factor.will disappear in the limit, so we can write without loss of accuracy, and comparing with the mathematical identity, yields, From the definition of propagation constant, Hence, 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.The secondary constants in these circumstances are;[21] 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,[22] The attenuation of this cable type increases with frequency, causing distortion of waveforms.with frequency also causes a distortion of a type called dispersion.also varies with frequency and is also partly reactive; both these features will be the cause of reflections from a resistive line termination.