Page 153 - Rashid, Power Electronics Handbook

142 Y.-S. Lee and M. Chow

In the case of a half-wave recti®er, v ðtÞ¼ 0 for the negative and from Eq. (10.10)
L
half-cycle, therefore, Eq. (10.6) can be rewritten as
0:707 V m
Full-wave I ¼ ð10:16Þ
s  L
1 ð p 2 R
V ¼ ðV sin otÞ dðotÞ ð10:7Þ
L
m
2p 0 10.2.3.3 Rectification Ratio
The recti®cation ratio, which is a ®gure of merit for compar-
or
ing the effectiveness of recti®cation, is de®ned as
V m
Half-wave V ¼ ¼ 0:5 V ð10:8Þ P dc V I
dc dc
L m s ¼ ¼ ð10:17Þ
2
P L V I
L L
In the case of a full-wave recti®er, v ðtÞ¼ V jsin otj for both
L
m
the positive and negative half-cycles. Hence, Eq. (10.6) can be In the case of a half-wave diode recti®er, the recti®cation ratio
can be determined by substituting Eqs. (10.3), (10.13), (10.8),
rewritten as
and (10.14) into Eq. (10.17).
s 
ð p
1 2 2
V ¼ ðV sin otÞ dðotÞ ð10:9Þ ð0:318 V Þ
m
L
m
p 0 Half-wave s ¼ ð0:5 V Þ 2 ¼ 40:5% ð10:18Þ
m
or In the case of a full-wave recti®er, the recti®cation ratio is
obtained by substituting Eq. (10.5), (10.15), (10.10), and
V m
Full-wave V ¼ p ¼ 0:707 V m ð10:10Þ (10.16) into Eq. (10.17).
L

2
ð0:636 V Þ 2
m
The result of Eq. (10.10) is as expected because the rms value Full-wave s ¼ 2 ¼ 81% ð10:19Þ
ð0:707 V Þ
of a full-wave recti®ed voltage should be equal to that of the m
original ac voltage.
10.2.3.4 Form Factor
10.2.3.2 Current Relationships The form factor (FF) is de®ned as the ratio of the root-mean-
square value (heating component) of a voltage or current to its
The average value of load current i is I and because load R is average value,
dc
L
purely resistive it can be found as
V I
V dc FF ¼ L or L ð10:20Þ
I ¼ ð10:11Þ V I
dc
R dc dc
The root-mean-square (rms) value of load current i is I and In the case of a half-wave recti®er, the FF can be found by
L
L
it can be found as substituting Eqs. (10.8) and (10.3) into Eq. (10.20)
V L 0:5 V m
I ¼ ð10:12Þ Half-wave FF ¼ ¼ 1:57 ð10:21Þ
L
R 0:318 V m
In the case of a half-wave recti®er, from Eq. (10.3) In the case of a full-wave recti®er, the FF can be found by
substituting Eqs. (10.16) and (10.15) into Eq. (10.20)
0:318 V m
Half-wave I ¼ ð10:13Þ
dc
R 0:707 V m
Full-wave FF ¼ ¼ 1:11 ð10:22Þ
0:636 V m
and from Eq. (10.8)
0:5 V m
Half-wave ¼ I ¼ ð10:14Þ 10.2.3.5 Ripple Factor
L
R
The ripple factor (RF), which is a measure of the ripple
In the case of a full-wave recti®er, from Eq. (10.5) content, is de®ned as
0:636 V m V ac
Full-wave I ¼ ð10:15Þ RF ¼ ð10:23Þ
dc
R V dc