Analysis of Synchronous Machines, 2nd Edition

2.4.1 Example

As a slightly more complex example of reference frame theory it will be assumed now that the voltages are unbalanced. In particular consider the three-wire circuit shown in Figure 1.12. The three source voltages are expressed by e_ag = εE_s cos θ_e

$e_{bg}=E_s\text{ cos }\left({\text{θ}}_e-\frac{2\text{π}}{3}\right)$

(2.113)

$e_{cg}=E_s\text{ cos }\left({\text{θ}}_e+\frac{2\text{π}}{3}\right)$

(2.114)

$e_{ag}=v_{as}+v_{sg}$

(2.115)

where θ_e = ω_et and ε is an arbitrary constant. Note that when ε is equal to unity the applied voltages are balanced.

Since the neutral return path in this case is absent, the load voltages are now not readily unknown but must be solved from the network equations. If v_sg is defined as the voltage between neutral points, then clearly

$e_{ag}=v_{as}+v_{sg}$

(2.116)

Figure 2.11  Response of electric system for ...