Page 200 - Rashid, Power Electronics Handbook

12 Three-Phase Controlled Recti®ers 189

Constant ‘‘C'' is evaluated through initial conditions at the The area DV med showed in Fig. 12.17 represents the loss of
instant when T2 is ignited. In terms of angle, when ot ¼ a: voltage that affects the average voltage V , and can be
C
evaluated through the integration of Dn during the overlap
sec
V f ÿf
i ¼ 0 ; C ¼ p  cos a ð12:18Þ angle m. The voltage drop Dn can be expressed as
sc
2 oL S
p 
n ÿ n 2 V sec sin ot
Replacing Eq. (12.18) in Eq. (12.17): Dn ¼ A B ¼ f ÿf ð12:21Þ
2 2
V f ÿf
i ¼ p  ðcos a ÿ cos otÞ ð12:19Þ
sc
2 oL S Integrating Eq. (12.21) into the corresponding period (60 )
and interval (m), at the instant when the commutation begins
Before commutation, the current I was carried by thyristor (a):
D
T1 (see Fig. 12.16). During the commutation time, the load
current I remains constant, i returns through T1, and T1is 3 1 ð aþm p  sec
D
sc

automatically switched-off when the current i sc reaches the DV med ¼ p 2 a 2 V f ÿf sin ot dot ð12:22Þ
value of I . This happens because thyristors cannot conduct in sec
D
reverse direction. At this moment, the overlap time lasts, and 3 V f ÿf
DV med ¼ p  ‰cos a ÿ cosða þ mފ ð12:23Þ
the current I D is then conducted by T2. In terms of angle, p 2
when ot ¼ a þ m, i ¼ I , where m is de®ned as the ‘‘overlap
sc
D
angle.'' Replacing this ®nal condition in Eq. (12.19) yields: Subtracting DV med in Eq. (12.13):
sec
V f ÿf p 
sec
I ¼ p  ‰cos a ÿ cosða þ mފ ð12:20Þ 3 2 V f ÿf
D
2 oL S V ¼ p cos a ÿ DV med ð12:24Þ
D
p
To avoid confusion in a real analysis, it has to be remembered  sec
3 2 V f ÿf
that V corresponds to the secondary voltage in the case of V ¼ ‰cos a þ cosða þ mފ ð12:25Þ
f ÿf D 2p
transformer utilization. For this reason, the abbreviation ‘‘sec''
has been added to the phase-to-phase voltage in Eq. (12.20).
or
During commutation, two valves conduct at a time, which
means that there is an instantaneous short circuit between the p 
3 2V sec h m m i
two voltages participating in the process. As the inductances of f ÿf
V ¼ cos a þ cos ð12:26Þ
D
each phase are the same, the current i sc produces the same p 2 2
voltage drop in each L , but with opposite sign because this
S
current ¯ows in reverse direction in each inductance. The Equations (12.20) and (12.25) can be written as a function of
phase with the higher instantaneous voltage suffers a voltage the primary winding of the transformer, if any transformer.
drop ÿDn, and the phase with the lower voltage suffers a
prim
voltage increase þDn. This situation affects the dc voltage V , a V f ÿf
C
reducing its value an amount DV med . Figure 12.17 shows the I ¼ p  ‰cos a ÿ cosða þ mފ ð12:27Þ
D
2 oL
meanings of Dn, DV med , m, and i . p  S prim
sc
3 2 a V f ÿf
V ¼ ‰cos a þ cosða þ mފ ð12:28Þ
D 2p
v D
a
where a ¼ V sec =V prim . With Eqs. (12.27) and (12.28) one
f ÿf f ÿf
DV med
obtains:
Dv
v c
Dv p 
w V ¼ 3 2 a V prim cos a ÿ 3I oL S ð12:29Þ
D
m D p f ÿf p
Equation (12.29) allows a very simple equivalent circuit of the
converter to be made, as shown in Fig. 12.18. It is important to
note that the equivalent resistance of this circuit is not real
because it does not dissipate power.
From the equivalent circuit, regulation curves for the
recti®er under different ®ring angles are shown in Fig. 12.19.
FIGURE 12.17 Effect of the overlap angle on the voltages and currents. It should be noted that these curves correspond only to an