Derive equations by inspection... pls help

[powersys]

Hello,

I wish to 'derive' the equations (1), (2), and (3) as shown in figure below by 'inspection' or 'observation'. Obviously, when phase C terminal is open, whilst phase A and phase B terminals are connected to positive DC link and negative DC link, respectively, phase A terminal-to-ground voltage, uan0 = Udc, and phase B terminal-to-ground voltage, ubn0 = 0. However, I am not able to 'derive' the equation (3) by just inspection... Could someone please advise?

Thanks

 

[FvM]

Unfortunately, I'm even unable to follow the first line. It can't be generally true, without additional assumptions about the involved e.m.f sources, I think.

 

[ark5230]

It is funny or incomplete or mismatch information. The equations can not be justified with the available information.

 

[powersys]

Unfortunately, I'm even unable to follow the first line. It can't be generally true, without additional assumptions about the involved e.m.f sources, I think.

It is funny or incomplete or mismatch information. The equations can not be justified with the available information.

Sorry... Let's assume the 3-phase circuit is symmetrical (and sinusoidal), therefore, ea+eb+ec=0. Please advise.

 

[Colbhaidh]

I think the labelling is wrong. Uan is not equal to Udc. Ua = Udc Ub=0 Uc=float

Consider the neutral point voltage at n. The voltage at n can be stated in 3 ways relative to each phase.
Where phase A is connected to Udc, Un= Udc -RIa - L(dIa/dt) - ea.
Where Phase B is connected to ground Un= RIb + L(dIb/dt) + eb.
Where Phase C is unconnected Un=Uc + ec.

Ia=Ib=I so combining 1st and 2nd Un = 0.5*(Udc - ea + eb)
Also must have ea + eb + ec =0

So Uc = 0.5*(Udc - ea + eb) - ec

Added after 2 minutes:

Sorry got my eb and ec signs wrong.

Where phase A is connected to Udc, Un= Udc -RIa - L(dIa/dt) - ea.
Where Phase B is connected to ground Un= RIb + L(dIb/dt) - eb.
Where Phase C is unconnected Un=Uc - ec.

Ia=Ib=I so combining 1st and 2nd Un = 0.5*(Udc - ea + eb)
Also must have ea + eb + ec =0

So Uc = 0.5*(Udc - ea - eb) + ec