Generally RF loads such as an antenna have a reactive impedance,  in such cases there is no transmission line of any characteristic impedance that will give a 1:1 VSWR,   however the lowest VSWR will be achieved with one particular value of characteristic impedance.

        What we find is that if the impedance of the transmission line is equal to the modulus of the impedance of the load,  then no other impedance transmission line will give a lower VSWR when terminated with that load.    So what is the modulus?

    If the impedance is complex at some frequency then we must resort to the mathmatics of complex numbers to understand what the modulus is and why.    First so that we may understand why the impedance is complex,    we have to remember that the phase of the voltage across a capacitor is 90 degrees in front of the phase of the current passing through it,    and that the phase of the voltage across an inductor is 90 degrees behind the phase of the current passing through it.

    If we draw a diagram of the relative positions of AC voltage across and current through resistors, capacitors and inductors we will see that the voltage and current in a resistor are always in phase,    but the current is at right angles to the voltage in inductors and capacitors.    If a load has both a resistive part and a capacitive or inductive part (not both because they will cancel out) called the reactive part,    then the current and voltage are seperated in phase somewhere between zero and ninety degrees.

    The modulus is the length of the line from the origin to the opposite corner of the rectangle and is easily calculated using Pythagorus,
i.e m = the square root of the sum of the squares of the two adjacent sides.

          i.e. lowest VSWR when Zo = SQR(R2 + X2)

 The actual value of VSWR  will be:-

                    VSWR = Zl + Xl
                                     Rl

    However if the load is quite reactive then the value of VSWR does not change significantly over a fairly wide range of characteristic impedances.

    It follows that where high values of reactance are likely to be found in a load impedance, the lowest VSWR will occur with high impedance transmission lines,   given that high impedance two wire lines have very low losses (provided that the spacing between the lines is small compared to a wavelength),   they are an obvious solution to the problem of feeding highly reactive loads with RF energy.