1. CANDLE

2. 



3. Instantaneous Power
m
m
( ) V cos( ) [V]
( ) I cos( ) [A]
( ) ( ) ( ) [W]
v
i
v t t
i t t
p t v t i t
 
 
 
 
 
m m( ) V I cos( ) cos( ) [W]v ip t t t     

4. Instantaneous Power
)]cos()[cos(
2
1
coscos
sinsincoscos)cos(
sinsincoscos)cos(
BABABA
BABABA
BABABA



m mV I
( ) [cos( ) cos(2 )] [W]
2
v i v ip t t        

5. Instantaneous Power
m mV I
( ) [cos( ) cos(2 )] [W]
2
v i v ip t t        
m m
m m
V I
cos( )
2
V I
cos(2 )
2
v i
v it
 
  

 

6.  i t  p t
( ) 4cos( 60 ) [V]
2 30 [ ]
v t t
Z
 
  


4 60
2 30 [A]
2 30
( ) 2cos( 30 ) [A]
Z
i t t

   

 
V
I





7. m mV I
( ) [cos( ) cos(2 )]
2
(4)(2)
[cos(60 30 ) cos(2 90 )]
2
4cos(30 ) 4cos(2 90 )
3.46 4cos(2 90 ) [W]
v i v ip t t
t
t
t
    



    
   
  
  
  
 


8. ( ) 4cos( 60 ) [V]
( ) 2cos( 30 ) [A]
( ) 3.46 4cos(2 90 ) [W]
v t t
i t t
p t t



 
 
  




9. Average
Power
0
0
1
P ( ) [W]
t T
t
p t dt
T

 
m
m
m m
( ) V cos( ) [V]
( ) I cos( ) [A]
( ) ( ) ( )
V I cos( ) cos( ) [W]
v
i
v i
v t t
i t t
p t v t i t
t t
 
 
   
 
 

  



10. Average
Power
0
0
0 0
0 0
0
0
m m
m m
m m
1
P ( )
1 1
( ). ( ) V I cos( )cos( )
1
V I [cos( ) cos(2 )]
2
V I
cos( ) 0 [W]
2
t T
t
t T t T
v i
t t
t T
v i v i
t
v i
p t dt
T
v t i t dt t t dt
T T
t dt
T
   
    
 

 


   
    
  

 


11. Average
Power
m mV I
P cos( ) [W]
2
v i  
-
m mV I
P= [W]
2
v i  
0v i   

12. Average
Power
-
90v i    
P 0 [W]v i   
•
•
•

13. 10 60
V
 
I
2
2j
P PSupplied Absorbed

14. 10 60
V
 
I
2
2j
Z
PSupplied
10 60 10 60
3.53 15 [A]
2 2 2.83 45j
 
    
 
V
I
Z
 


m
m
m m
I 3.53 , 15
V 10 , 60
V I
P cos( )
2
(10)(3.53)
cos(60 15 )
2
12.5 [W]
i
v
v i


 
 
 
 
 



 

15. 10 60
V
 
I
2
2j
PAbsorbed
RI
R


V
R 3.53 15 [A]  I I 
R
2
(10 60 ) 7.07 15 [V]
2 2j
   

V  
m mV I
P
2
(7.07)(3.53)
P 12.5 [W]
2

 

16. 10 60
V
 
I
2
2j
PAbsorbed
RI
R


V 2 2
m m m
2 2
m
V I V (7.07)
P 12.5 [W]
2 2R 2(2)
I R (3.53) (2)
= 12.5 [W]
2 2
   
 

17. 12 45
V
  2
 1j
4

18. 1
2
1 2
12 45
3 45 [A]
4
12 45 12 45
5.36 71.57 [A]
2 1 2.24 26.57
8.15 62.10 [A]
j

  
 
   
 
   
I
I
I Ι I


 




19. m m
4
2 2
m
2
V I (12)(3)
P 18 [W]
2 2
I R (5.36) (2)
P 28.7 [W]
2 2
P 18 28.7 46.7 [W]Absorbed


  
  
  

20. m mV I
P cos( )
2
(12)(8.15)
cos(45 62.10 ) 46.7 [W]
2
P 46.7 [W]
Source v i
Supplied
  
  

 

21. Power
Factor, PF
rms rmsP V I cos( )v i  
rms rmsV I [VA]
P
cos( )
V I
v i
rms rms
  
P

22. Power
Factor
V
I
v i
v i 
V
I
v i
Z
Zcos( ) cosv i   
Z

23. Power
Factor
Z 0 PF 1v i      
Z 90 PF 0v i       
V I
V
I I
V

24. Power
Factor
-- R C
Z90 0   
Z0 90  
- R L
Leading PF Lagging PF
V
I

25. Power
Factor
RC 
 Lea
RL 
 La
V
I
I
V

26. Power
Factor
Z 1 1
Z 2 45
PF cos cos( 45 ) 0.707
45 PF
C
C
Z
Z
j
Leading


  
   
   
  



V
I
PF
z
RC load
leading
  
L
L
Z
Z 2 1
Z 5 26.57
PF cos cos( 26.57 ) 0.894
26.57 PF
Z
j
Lagging


  
   
   
  



V
I PF
z
RL load
lagging
  V
I
I
V

27. Power
Factor
2.

28. Power
Factor
L
S
P
P
Plosses

29. Power
Factor
3
L
rms
rms
S L
2 2
S
) PF 0.707
P 88 10
I
V PF (480)(0.707)
259.3 [A]
P P
P I R (259.3) (0.08)
5.38 [kW]
P 5.38 88 93.38 [kW]
losses
losses rms
a lagging
P


 

 
 

  

3
L
rms
rms
S L
2 2
rms
S
) PF 0.900
P 88 10
I
V PF (480)(0.900)
203.7 [A]
P P P
P I R (203.7) (0.08)
3.32 [kW]
P 3.32 88 91.32 [kW]
losses
losses
b lagging

 

 
 

  

L rms rmsP V I cos( ) [W]v i  

30. Power
Factor
S
S
PF P P
PF P P
losses
losses
Load    
   

31. Complex
Power
rms
*
rms
I
I
i R I
i R I
I jI
I jI


   
   
I
I


Linear Electrical
Network rmsV


rmsI

zZ  *
S V I
rmsV v
 V
Complex
Conjugate
*
I
I
V
rmsV
I
rmsI

32. Complex
Power
rms rms
rms rms
S V .I
V .I
v i
v i
 
 
  
  
 
 
rms rms
rms rms
rms rms
V I cos( )
V I sin( )
V I
v i
v i
 
 


 
 
rms rms rms rmsS V I cos( ) V I sin( )v i v ij         
Real Power
Average Power
Reactive Power
Apparent Power
- VA
P
S
Q

33. Complex
Power
S P Q [VA]j 
P Re(S) [W]
Q Im(S) [VAR]


Real part
Imaginary part
m
m
Re(Z) Z cos( )
Re(Z)
cos( )
Z
v i
v i
 
 
 
 
 
 
Z Zm z
 
Real part Imaginary part
m
m
Im(Z) Z sin( )
Im(Z)
sin( )
Z
v i
v i
 
 
 
 
 
 

34. Complex
Power
ZV I
 
*
*
2
rms
S
Z
I Z



V I
I I

rms
rms
I
I
i
i


 
 
I
I


Z R Xj 
2 2
rms rmsS I R I Xj 
2
rms
2
rms
P I Re(Z)
Q I Im(Z)


S P Qj 

35. Complex
Power
m
m
Re(Z)
cos( )
Z
Im(Z)
sin( )
Z
v i
v i
 
 
 
 
 
 
2
rms
2
rms
P I Re(Z)
Q I Im(Z)


Q
tan( )
P
v i   
m zZ Z 
 

36. Complex
Power
Power Triangle
Q +
Inductive load
Lagging PF
Q -
Capacitive load
Leading PF
Q
Resistive load

37. Complex
Power
Vs
lagging 0.09 +j0.3 Ω
20 kW 220 Vrms

38. Complex
Power
Transmission Lines
Power Plant Load
S LLine V V V
LineV
LV
Line LI I
LineI
L 220 0 [V]
Z 0.09 0.3 [ ]Line j
 
  
V 

39. Complex
Power
*
L L L
L z
3
L
z
L L z L z
1 1
z
L L z L z
S
P S cos
P 20 10
S
cos 0.8
25,000 [VA]
S S S
cos (PF) cos (0.8)
36.87
S S S
25,000 36.87
20,000 15,000 [VA]j


 

 
 



 

   
 

   
 
 
V I



load power
factor
* L
L
L
L
S 25,000 36.87
113.64 36.87 [A]
220 0
113.64 36.87 [A]
(113.64 36.87 )(0.09 0.3)
35.59 36.43 [V]
Line Line Line
j

   

 

  
 
I
V
I
V I Z






S L 249.53 4.86 [V]
PF cos( ) cos(4.86 ( 36.87 ))
cos(41.73 )
0.75
Line
v i
lagging
 
   
    


V V V 
   


40. PF PF angle
PF PF angle
  
  

41. •
lagging PF
•
induction motors
•
capacitor

42. 2
rmsQ ωCV [VAR]Cap  
Q Q Qnew old Cap 
P Pold new
old new 
old newPF PF

43. S P Q Sold old old old oldj     
S Q S S 90Cap Cap Cap Cap Capj      
P Pnew old

44. S S S P Q Sold Cap new old new new newj       
new 



45. rms rms
z
m z
m m z
Q V I sin( )
Z Z
Re(Z) Z cos ,Im(Z) Z sin
v i
v i
z
 
  

 
 
 
 
 
  

 
CapQ
rms
rms rms m rms
m
V
V I IZ
Z
  Z
Z
  
V
V I I

46. C
C
m
1
Z X
ωC ωC
1 1
X , Z 90
ωC ωC
1
Im(Z ) Im(Z)
ωC
1
Z Im(Z)
ωC
Cap
Cap
Cap
j
j
j

   
   
  
  


47. rms
rms
m m
2
rms m
2
m
2
rms
m
2
rms
V Im(Z)
Q V
Z Z
V Z
Z
V
Z
ωCV [VAR]
Cap
  
   
  


 

rms rmsQ V I sin( ) [VAR]v i  

48. 0.9
R, L)
0.8 lagging 50 kW , 2

49. z z
S P Q
P S cos ,Q S sin
Q
tan
P
z
j
 

 
 

1 1
3
PF 0.8
cos (PF) cos (0.8) 36.87
Q
tan Q P tan
P
Q 50 10 tan(36.87 ) 37.5 [ VAR]
old
old
old old old old
old
old
lagging
k

 
 

  
  
  




50. 1 1
3
PF 0.95
cos (PF) cos (0.95) 18.19
Q
tan Q P tan
P
Q 50 10 tan(18.19 ) 16.43 [ VAR]
new
new
new new old new
old
new
lagging
k

 
 

  
  
  



2
rms
Q Q Q
Q Q Q ωCV
220 0 , 377 .
Q 21.07 [ VAR]
new old Cap
Cap new old
rms
Cap
rad
k

 
   
  
 
V 
2 2
rms
Q 21,070
C 1,155 [ F]
ωV (377)(220 )
Cap
  