Page 205 - Rashid, Power Electronics Handbook
P. 205
194 J. W. Dixon
the ac side 5th and 7th harmonic currents and dc side 6th Substituting (Eq. 12.46) into (12.45):
harmonic voltage, thus resulting in signi®cant savings in
harmonic ®lters. cos a þ cosða þ mÞ
I ¼ I ð12:47Þ
P
Some useful relations for HVDC systems include: 2
(a) recti®er side:
p as I ¼ I cos j, it yields
P
P ¼ V I ¼ 3 V prim I rms cos j ð12:40Þ
D D D f ÿf line
cos a þ cosða þ mÞ
cos j ¼ ð12:48Þ
2
(b) inverter side: The same equations are applied for the
inverter side, but the ®ring angle a is replaced by g, where g
is (see Fig. 12.30):
I ¼ I cos j
P
I ¼ I sin j g ¼ 180 ÿða þ m Þ ð12:49Þ
I
I
Q
As reactive power always goes in the converter direction, at the
p prim
; P ¼ V I ¼ 3 V f ÿf I P ð12:41Þ inverter side Eq. (12.44) becomes:
D
D
D
p
V I D a 2 3 V prim
D
I ¼ p prim ð12:42Þ I ¼ÿ I f ÿf I sin 2ðg þ m Þÿ sin 2g ÿ 2m ð12:50Þ
P
3 V Q 1 4p o L I I
f ÿf I I
p prim
a 2 3 V f ÿf
I ¼ cos 2a ÿ cos 2ða þ mÞ ð12:43Þ
P
4p oL
S 12.2.12 Dual Converters
p
prim
a 2 3 V f ÿf In many variable-speed drives, four-quadrant operation is
I ¼ sin 2ða þ mÞÿ sin 2a ÿ 2m ð12:44Þ
Q required, and three-phase dual converters are extensively
4p oL S
p used in applications up to the 2 MW level. Figure 12.31
a 6 cos a þ cosða þ mÞ shows a three-phase dual converter, where two converters
I ¼ I D ð12:45Þ
P
p 2 are connected back-to-back.
In the dual converter, one recti®er provides the positive
Fundamental secondary component of I: current to the load, and the other the negative current. Due to
p the instantaneous voltage differences between the output
a 6 voltages of the converters, a circulating current ¯ows through
I ¼ I D ð12:46Þ
p the bridges. The circulating current is normally limited by
circulating reactor L D as shown in Fig. 12.31. The two
converters are controlled in such a way that if a þ is the
a
delay angle of the positive current converter, the delay angle
þ
ÿ
m of the negative current converter is a ¼ 180 ÿ a .
Figure 12.32 shows the instantaneous dc voltages of each
converter, n þ and n . Despite the average voltage V is the
ÿ
D D D
w same in both the converters, their instantaneous voltage
+ L D/2 L D/2
i D
v A i A + v r -
w
FIGURE 12.30 De®nition of angle g for inverter side: (a) recti®er side;
and (b) inverter side. FIGURE 12.31 Dual converter in a four-quadrant dc drive.
