Final answer:
Without performing a complex network analysis considering the effects of the ports connected by a transmission line, an exact calculation of the resulting insertion loss and phase between ports 1 and 2 cannot be provided. The effective S-parameters of the modified network would need to be calculated to determine these values.
Step-by-step explanation:
To calculate the resulting insertion loss and phase between ports 1 and 2 when ports 3 and 4 are connected by a lossless matched transmission line, we must account for the effects of the connected ports. Normally, the S-matrix in its original form would be used to determine the properties directly, however, due to the transmission line connection between ports 3 and 4, this may alter the effective S-matrix as seen by ports 1 and 2. The electrical length of the line indicates a phase shift that must be taken into account, and to find the actual insertion loss and phase between ports 1 and 2, a more complex network analysis involving the full scattering matrix would be needed.
Without additional information, such as the effect of the connected ports on the S-parameters or a specific methodology for the transformation of the S-matrix due to the transmission line connection, a precise calculation cannot be provided. The connection of ports 3 and 4 essentially forms a new network, for which the effective S-parameters between ports 1 and 2 would need to be recalculated considering the effect of the transmission line.
More generally, once the effective S-matrix is calculated, the magnitude of the S21 parameter of this new matrix would give the insertion loss (in dB, which can be calculated as −20*log10(|S21|) where |S21| is the magnitude of the S21 parameter), and the argument (or angle) of the S21 parameter would give the phase shift.