considering the two conductors to be in series, is the inductance per unit length, is the conductance of the dielectric per unit length, is the capacitance per unit length, is the imaginary unit, and is the angular frequency. This expression extends to DC by letting tend to 0. A surge of energy on a finite transmission line will see an impedance of prior to any reflections returning; hence surge impedance is an alternative name for characteristic impedance. Although an infinite line is assumed, since all quantities are per unit length, the “per length” parts of all the units cancel, and the characteristic impedance is independent of the length of the transmission line. The voltage and current phasors on the line are related by the characteristic impedance as: where the subscripts (+) and (−) mark the separate constants for the waves traveling forward (+) and backward (−). Derivation Using telegrapher's equation The differential equations describing the dependence of the voltage and current on time and space are linear, so that a linear combination of solutions is again a solution. This means that we can consider solutions with a time dependence – doing so is functionally equivalent of solving for the Fourier coefficients for voltage and current amplitudes at some fixed angular frequency Doing so causes the time dependence to factor out, leaving an ordinary differential equation for the coefficients, which will be phasors, dependent on position (space) only. Moreover, the parameters can be generalized to be frequency-dependent. Let and Take the positive direction for and in the loop to be clockwise. We find that and or and where and These two first-order equations are easily uncoupled by a second differentiation, with the results: and Notice that both and satisfy the same equation. Since is independent of and , it can be represented by a single constant (The minus sign is included for later convenience.) That is: so We can write the above equation as which is correct for any transmission line in general. And for typical transmission lines, that are carefully built from wire with low loss resistance and small insulation leakage conductance ; further, used for high frequencies, the inductive reactance and the capacitive admittance will both be large, so the constant is very close to being a real number: With this definition of the position- or -dependent part will appear as in the exponential solutions of the equation, similar