3 Phase Instantaneous Voltage Conversion from Line to Phase and Vis-Versa 1

[dzt]

Hi,

I have encountered three equations which convert from 3 phase line to line voltages to phase voltages. I mean the complete instantaneous voltages not the magnitude which is calculated by the square-root 3. The equations are:
Va = 2/3Vab + 1/3Vbc
Vb = 2/3Vbc + 1/3Vca
Vc = 2/3Vca + 1/3Vab

These equations were used in a 3 phase inverter software model. My problem is that I would like to know how these equations have been derived and also to derive the reverse equations from phase to line voltages. I have search books and internet but to no avail. Any help would be really appreciated, please.

 

[dzt]

Thanks LiteYear and all those who provided help.
Those set of equations are useful when one needs to obtain the phase voltages out of the balanced line to line voltages, especially in simulation modeling. The phase voltage sinewaves from the equations are obtained with both magnitude and phase, as can be shown from the images I included in my previous posts.