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P o w er Electronics: Phase Con trolled Rectifiers
F orm ula Sheet for Electrical GA TE
Single-Phase Con trolled Rectifiers
• Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
V
m
2p
(1+cosa)
where V
o( a vg)
is a v erage output v oltage, V
m
is p eak input v oltage, a is firing angle.
• F ull-W a v e Con trolled Rectifier (Single-Phase Bridge) :
V
o( a vg)
=
2V
m
p
cosa
• Output RMS V oltage (F ull-W a v e) :
V
o( rms)
=V
m
v
1
p
(
p-a+
sin2a
2
)
• A v erage Load Curren t :
I
o( a vg)
=
V
o( a vg)
R
where R is load resistance.
• P o w er Deliv ered to Load :
P
o
=V
o( rms)
I
o( rms)
• Ripple F actor :
RF =
v
(
V
o( rms)
V
o( a vg)
)
2
-1
Three-Phase Con trolled Rectifiers
• Three-Phase Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
3
v
3V
m
2p
cosa
where V
m
is p eak phase v oltage.
• Three-Phase F ull-W a v e (Bridge) Con trolled Rectifier :
V
o( a vg)
=
3
v
6V
m
p
cosa
1
Page 2


P o w er Electronics: Phase Con trolled Rectifiers
F orm ula Sheet for Electrical GA TE
Single-Phase Con trolled Rectifiers
• Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
V
m
2p
(1+cosa)
where V
o( a vg)
is a v erage output v oltage, V
m
is p eak input v oltage, a is firing angle.
• F ull-W a v e Con trolled Rectifier (Single-Phase Bridge) :
V
o( a vg)
=
2V
m
p
cosa
• Output RMS V oltage (F ull-W a v e) :
V
o( rms)
=V
m
v
1
p
(
p-a+
sin2a
2
)
• A v erage Load Curren t :
I
o( a vg)
=
V
o( a vg)
R
where R is load resistance.
• P o w er Deliv ered to Load :
P
o
=V
o( rms)
I
o( rms)
• Ripple F actor :
RF =
v
(
V
o( rms)
V
o( a vg)
)
2
-1
Three-Phase Con trolled Rectifiers
• Three-Phase Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
3
v
3V
m
2p
cosa
where V
m
is p eak phase v oltage.
• Three-Phase F ull-W a v e (Bridge) Con trolled Rectifier :
V
o( a vg)
=
3
v
6V
m
p
cosa
1
• Output RMS V oltage (Three-Phase Bridge) :
V
o( rms)
=V
m
v
3
v
1
2
+
3
v
3
2p
cosa
• A v erage Load Curren t :
I
o( a vg)
=
V
o( a vg)
R
• P o w er F actor :
PF = cosa (for con tin uous conduction)
P erformance P arameters
• E?iciency :
? =
P
o
P
in
=
V
o( rms)
I
o( rms)
V
s( rms)
I
s( rms)
where V
s( rms)
, I
s( rms)
are input RMS v oltage and curren t.
• T otal Harmonic Distortion (THD) :
THD =
v
(
I
s( rms)
I
s1( rms)
)
2
-1
where I
s1( rms)
is the fundamen tal comp onen t of input curren t.
• F orm F actor :
FF =
V
o( rms)
V
o( a vg)
Th yristor Ratings
• A v erage Th yristor Curren t (Single-Phase F ull-W a v e) :
I
T( a vg)
=
V
m
pR
cosa
• RMS Th yristor Curren t (Single-Phase F ull-W a v e) :
I
T( rms)
=
V
m
R
v
2
• A v erage Th yristor Curren t (Three-Phase Bridge) :
I
T( a vg)
=
v
6V
m
pR
cosa
2
Page 3


P o w er Electronics: Phase Con trolled Rectifiers
F orm ula Sheet for Electrical GA TE
Single-Phase Con trolled Rectifiers
• Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
V
m
2p
(1+cosa)
where V
o( a vg)
is a v erage output v oltage, V
m
is p eak input v oltage, a is firing angle.
• F ull-W a v e Con trolled Rectifier (Single-Phase Bridge) :
V
o( a vg)
=
2V
m
p
cosa
• Output RMS V oltage (F ull-W a v e) :
V
o( rms)
=V
m
v
1
p
(
p-a+
sin2a
2
)
• A v erage Load Curren t :
I
o( a vg)
=
V
o( a vg)
R
where R is load resistance.
• P o w er Deliv ered to Load :
P
o
=V
o( rms)
I
o( rms)
• Ripple F actor :
RF =
v
(
V
o( rms)
V
o( a vg)
)
2
-1
Three-Phase Con trolled Rectifiers
• Three-Phase Half-W a v e Con trolled Rectifier :
V
o( a vg)
=
3
v
3V
m
2p
cosa
where V
m
is p eak phase v oltage.
• Three-Phase F ull-W a v e (Bridge) Con trolled Rectifier :
V
o( a vg)
=
3
v
6V
m
p
cosa
1
• Output RMS V oltage (Three-Phase Bridge) :
V
o( rms)
=V
m
v
3
v
1
2
+
3
v
3
2p
cosa
• A v erage Load Curren t :
I
o( a vg)
=
V
o( a vg)
R
• P o w er F actor :
PF = cosa (for con tin uous conduction)
P erformance P arameters
• E?iciency :
? =
P
o
P
in
=
V
o( rms)
I
o( rms)
V
s( rms)
I
s( rms)
where V
s( rms)
, I
s( rms)
are input RMS v oltage and curren t.
• T otal Harmonic Distortion (THD) :
THD =
v
(
I
s( rms)
I
s1( rms)
)
2
-1
where I
s1( rms)
is the fundamen tal comp onen t of input curren t.
• F orm F actor :
FF =
V
o( rms)
V
o( a vg)
Th yristor Ratings
• A v erage Th yristor Curren t (Single-Phase F ull-W a v e) :
I
T( a vg)
=
V
m
pR
cosa
• RMS Th yristor Curren t (Single-Phase F ull-W a v e) :
I
T( rms)
=
V
m
R
v
2
• A v erage Th yristor Curren t (Three-Phase Bridge) :
I
T( a vg)
=
v
6V
m
pR
cosa
2
Key Notes
• Firing Angle Range : a from 0
?
to 180
?
; cosa determines output v oltage con trol.
• Con tin uous vs. Discon tin uous Conduction : Dep ends on load t yp e (resistiv e,
inductiv e) and a .
• Input V oltage : F or single-phase, V
m
is p eak phase v oltage; for three-phase, V
m
is
p eak phase v oltage unless sp ecified as line v oltage ( V
LL
=
v
3V
m
).
• Assumptions : Ideal th yristors, negligible v oltage drops unless sp ecified.
• Units : Use SI units for GA TE (e.g., V , A , W ).
3
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