Page 184 - Rashid, Power Electronics Handbook

11 Single-Phase Controlled Recti®ers 173

distortion of the current produces an increase in the value of I s During commutation the following expression holds:
in Eq. (11.16), which deteriorates the power factor.
di s
L ¼ v ¼ V max sin ot a ot a þ m ð11:18Þ
s
dt
11.1.6 The Commutation of the Thyristors
Until now the current commutation between thyristors has Integrating Eq. (11.18) over the commutation interval yields
been considered to be instantaneous. This consideration is not
ð I d ð aþm=o
valid in real cases due to the presence of the line inductance L, V max
di ¼ sin otdt ð11:19Þ
as shown in Fig. 11.9a. During commutation, the current s L
ÿI d a=o
through the thyristors cannot change instantaneously, and for
this reason, during the commutation angle m, all four thyr-
From Eq. (11.19), the following relationship for the commu-
istors are conducting simultaneously. Therefore, during the
tation angle m is obtained:
commutation the following relationship for the load voltage
holds:
2oL
cosða þ mÞ¼ cos a ÿ I d ð11:20Þ
V max
v ¼ 0 a ot a þ m ð11:17Þ
d
Equation (11.20) shows that an increase of the line inductance
The effect of the commutation on the supply current, voltage L or an increase of the load current I increases the commu-
d
waveforms, and the thyristor current waveforms is observed in tation angle m. In addition, the commutation angle is affected
Fig. 11.9b. by the ®ring angle a. In effect, Eq. (11.18) shows that with
different values of a, the supply voltage v has a different
s
i T1 instantaneous value, which produces different di =dt, thereby
s
affecting the duration of the commutation.
i v L T T Equation (11.17) and the waveform of Fig. 11.9b show that
s 1 i 3 the commutation process reduces the average load voltage V .
da
L T3 When commutation is considered, the expression for the
+
v I d v
s d average load voltage is given by
T T 1 ð pþa V
4 2 V ¼ V sin otdðotÞ¼ max ‰cosða þ mÞþ cos aŠ
da
p aþm max p
i T4 a) ð11:21Þ
Substituting Eq. (11.20) into Eq. (11.21) yields

2
i i V ¼ V cos a ÿ 2oL I ð11:22Þ
T1 T3 da p max p d
wt
0
11.1.7 Operation in the Inverting Mode
v s
i When the angle a > 90 , it is possible to obtain a negative

s average load voltage. In this condition, the power is fed back
wt
0
to the single-phase supply from the load. This operating mode
is called inverter or inverting mode, because the energy is
transferred from the dc to the ac side. In practical cases this
v
d operating mode is obtained when the load con®guration is as
shown in Fig. 11.10a. It must be noticed that this recti®er
wt
0
allows unidirectional load current ¯ow.
Figure 11.10b shows the waveform of the load voltage with
a m v s the recti®er in the inverting mode, neglecting the source
b) inductance L.
FIGURE 11.9 The commutation process: (a) circuit; and (b) wave- Section 11.1.6 described how supply inductance increases
forms. the conduction interval of the thyristors by the angle m.As