Page 425 - Power Electronic Control in Electrical Systems

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404 Examples, problems and exercises

(ii) From equations (a) with x ˆ 0, using the symbol E for terminal (fixed) voltages
E s ˆ E r cos y ‡ jZ 0 I r sin y
E r
I s ˆ j sin y ‡ I r cos y
Z 0
By symmetry I s ˆ I r and from the equation for E s we get

E r sin y E r y
I r ˆ j ˆ j tan
Z 0 1 ‡ cos y Z 0 2
[Note the alternative form using tan (y/2)]. With E s ˆ E r we get

sin y
E s
I s ˆ j
Z 0 1 ‡ cos y
Also the mid-point voltage is given by

V m ˆ V r cos y=2 ‡ jZ 0 I r sin y=2
2
sin y=2 E s
ˆ E s cos y=2 ‡ sin y=2 tan y=2 † ˆ E s cos y=2 ‡ ˆ y
cos y=2 cos
2

Fig. 9.24