Editing Characteristic impedance (section)

== Lossless line ==
The analysis of lossless lines provides an accurate approximation for real transmission lines that simplifies the mathematics considered in modeling transmission lines. A lossless line is defined as a transmission line that has no line resistance and no [[dielectric loss]]. This would imply that the conductors act like perfect conductors and the dielectric acts like a perfect dielectric. For a lossless line, {{math|''R''}} and {{math|''G''}} are both zero, so the equation for characteristic impedance derived above reduces to:
<math display="block">Z_0 = \sqrt{\frac{L}{C}\,}\,.</math>

In particular, <math>Z_0</math> does not depend any more upon the frequency. The above expression is wholly real, since the imaginary term {{mvar|j}} has canceled out, implying that <math>Z_0</math> is purely resistive. For a lossless line terminated in <math>Z_0</math>, there is no loss of current across the line, and so the voltage remains the same along the line. The lossless line model is a useful approximation for many practical cases, such as low-loss transmission lines and transmission lines with high frequency. For both of these cases, {{mvar|R}} and {{mvar|G}} are much smaller than {{math|''ωL''}} and {{math|''ωC''}}, respectively, and can thus be ignored.

The solutions to the long line transmission equations include incident and reflected portions of the voltage and current:
<math display="block">\begin{align}
V &= \frac{V_r + I_r Z_c}{2} e^{\gamma x} + \frac{V_r - I_r Z_c}{2} e^{-\gamma x} \\[1ex]
I &= \frac{V_r/Z_c + I_r}{2} e^{\gamma x} - \frac{V_r/Z_c - I_r}{2} e^{-\gamma x}
\end{align}</math>
When the line is terminated with its characteristic impedance, the reflected portions of these equations are reduced to 0 and the solutions to the voltage and current along the transmission line are wholly incident. Without a reflection of the wave, the load that is being supplied by the line effectively blends into the line making it appear to be an infinite line. In a lossless line this implies that the voltage and current remain the same everywhere along the transmission line. Their magnitudes remain constant along the length of the line and are only rotated by a phase angle.