Lossless line

When a distributed line satisfies R=G=0, the line is called lossless line. In this case, Equahons 1.36 and 1.37 are written as 

Similarly to the sinusoidal excitation case, the following equations for the voltage and current are obtained  

Equations 1.54 are linear second-order hyperbolic partial differential equations and called wave equations. The general solutions of the wave equations are given by D Alembert in 1747 [13] as 

Surge impedance Zq and surge admittance Yq in the earlier equation are extreme values of the characteristic impedance and admittance in Equation 1.49 for frequency f - o =. 

The earlier solution is known as a wave equation and shows a behavior of a wave traveling along the x-axis by the velocity Cq. It should be clear that the value of functions Cf, e Ej, and E does not vary if x-Cof=constant and x + Cot = constant. Since Cj and Ef show a positive traveling velocity, they are called forward traveling wave  

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To transmit power over long distances is the basic requisite of economical transmission. Let us study equation (24.3). If we are able to maintain a unity p.f. between the transmitting and receiving ends, then for a lossless line... 

Propagation Delay of Lossless Lines. The minimum propagation delay (Tpd expressed in picoseconds per centimeter) for a unit length of lossless line is the inverse of the phase velocity (op) of the electromagnetic waves propagating through the dielectric medium surrounding the conductor line ... 

For low loss and lossless lines the signal velocity and characteristic impedance can be expressed as... 

As a result, for a lossless line, the phase velocity (Equation 1.55) is found from Equations 1.68 and 1.70. 

The propagation velocity Cq in this equation is equal to the propagation velocity for a lossless line in Equation 1.55. 

This equation shows that the characteristic impedance becomes independent of frequency for a lossless line, and it is called surge impedance, as defined in Equation 1.58. 

Equation 1.85 shows that the characteristic impedance for co oo coincides with the surge impedance of a lossless line in Equation 1.81. 

The input impedance Z(l) of the finite line seen from the sending end is also rewritten for a lossless line as follows ... 

For a lossless line, the solutions for voltage and current are expressed as... 

Figure 1.16 shows the relationship between Z 1) and 0 = jLCi lossless line. As for a short-circuited line, the relationship coincides with Foster s reactance theorem. The line is in a resonant condition for 0 = (2n - l)7t/2 n positive integers, as in Equation 1.103. 

This voltage is called the switching (surge) overvoltage, which reaches 2 pu (per unit voltage = V/E) in a single lossless line ... 

Figure 1.15 shows an example of the relationship between Z(/) and /. For a lossless line, the solutions of voltage and current are expressed as... 

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