 L 

 R 
  tan 1 
 Vm sin( ), S1  S2  ON ,       
vo ( )  
Vm sin( ), S3  S4  ON ,       2  
Vo _ avg 
Vo _ avg 
1
2
2
2
2 


 

vo ( )d
Vm sin( )d

 
Vo _ avg 
Vm
  cos( ) 
Vo _ avg 
Vm
cos( )  cos(   )
Vo _ avg 
Vm
cos( )   cos( )
Vo _ avg 
Vm
cos( )  cos( )
Vo _ avg 
Vm
cos( )  cos( )
Vo _ avg 
Vo _ avg 
Vo _ avg 






2Vm cos( )

2  220  2 cos(30 )

2  220  2 cos(45 )

 171.5(V )
 140.05(V )
 L 
 R    1


At steady state, io ( )  const  I o _ avg
dio ( )
 E (if exist )  vo ( )
dt
 E (if exist )  Vo _ avg
Rio ( )  L
RI o _ avg
I o _ avg 
Vo _ avg  E (if exist )
R
140.05
I o _ avg 
 1.4( A)
100
I o _ avg
I SCR _ avg 
2
I o _ avg
I SCR _ rms 
2
I rms  I o _ avg
S  I rmsVrms  I o _ avgVrms
Po _ avg 
Po _ avg 
1
2
2 

io ( )vo ( )d

2 
I o _ avg

2
vo ( )d

Po _ avg  I o _ avgVo _ avg
pf 
Po _ avg
pf 
Vo _ avg
S
Vrms

I o _ avgVo _ avg
I o _ avgVrms
2Vm cos( )


Vrms
2Vrms 2 cos( )
pf 

Vrms

2 2 cos( )

Three-phase voltage source:
van ( )  Vm sin( )
2
)
3
4
vcn ( )  Vm sin( 
)
3
vbn ( )  Vm sin( 
If all (3) diodes are ON simultaneously v_an = v_bn = v_cn short circuitnot exist!
If 2 diodes in 3 diodes are ON simultaneously v_an = v_bn  short circuitnot exist!
If no diodes are ON-not exist!
Only 1 diode is ON in one specified duration.
Assume that Diode D1 is ON, this means D2 and D3 are OFF.
D1 is forward-biased, D2 and D3 are reverse-biased.
V_A_D1 > V_K_D1; V_A_D2 <V_K_D2; V_A_D3<V_K_D3;
V_K_D1 = V_K_D2 =V_K_D3
V_A_D1 > V_A_D2, V_A_D1 > V_A_D3
V_an > V_bn, V_an > V_cn
V_an is maximum phase voltage

5

van ( )  Vm sin( ), D1  ON ,   

6
6

2
5
9

vo ( )   vbn ( )  Vm sin( 
), D2  ON ,
 
3
6
6

4
9
13

vcn ( )  Vm sin(  3 ), D3  ON , 6    6

2 
Vo _ avg 
1
2



6
vo ( )d 
6
Vo _ avg 
3
2
5
6
 Vm sin( )d

6
5
6
3Vm
  cos( )     cos( )
2
6
Vo _ avg
3V
 m
2
Vo _ avg 

5  3 3Vm 3 6Vrms

cos( 6 )  cos( 6 )   2  2


3 6  220
2
vo ( )
R
Vo _ avg 3 6Vrms


R
2 R
io ( ) 
Resistive load I o _ avg
I o _ avg 
I d _ avg 
I o _ avg
3

3 6  220
2 100
3 6Vrms
6Vrms

6 R
2 R
2 
Po _ avg 
1
2


6

2 
io ( )vo ( )d 
6
Po _ avg 
3
2
5
6


1
2



6
vo2 ( )
d
R
6
5
3Vm2 6 [1  cos(2 )]
Vm2 sin 2 ( )
d 
R
2 R
6

2

6
5
Po _ avg
d
3V 2 
3V 2 2
sin(2 )  6
 m  
 m (


4 R 
2 
4 R 3
sin(
5

3
3
)  sin( )


2
2
3
V
2

3
3 ) m (
2 )  3Vm ( 2  3 )
 2
2
4 R 3
2
4 R 3
2
6
Po _ avg 
2
3  2  Vrms
4 R
(
2
3  Vrms
2
3
2
3
3  2202 2
3

)
(

)
(

)
3
2
2 R
3
2
2 100 3
2
2 
I a _ rms  I d _ rms 
1
2



6
id2 ( )d 
6
S  3I a _ rms  Vrms  3
S 3
1
Po _ avg  Vrms
3R
1
684.13  220  996[VA]
3 100
1
2
5
6


6
Vm2
sin ( )
2
R2
d 
1 3
3R 2
5
6


6
Vm2 sin 2 ( )
1
d 
Po _ avg
R
3R