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1. Power Computations & Analysis
Techniques
PLT207 Power Electronics
1

2. Course Learning Outcome (CLO)
CLO1
• Ability to analyse operation and applications of power electronic
devices and addressing the needs of EMC requirements.
CLO2
• Ability to evaluate the performance of AC-DC converters.
CLO3
• Ability to evaluate the performance of AC-AC converters.
CLO4
• Ability to evaluate the performance of DC-DC converters.
CLO5
• Ability to evaluate the performance of DC-AC converters.
2

3. Sign Convention
Passive components:
R, L, & C.
Power sources
3
Absorbing
Power
p(t) > 0
Supplying
Power
p(t) > 0

4. Periodic Waveforms
 The magnitude at the end of one period is the same
as the beginning.
 Mathematically,
 Consists of DC and AC components and can be
described by a Fourier Series of sinusoids.
f(t T) f(t)
 
4

5. Periodic Waveforms
f(t T) f(t)
 
+
   
0 n 0 n 0 0 n 0 n
n 1 n 1
f(t) a a cos(n t) b sin(n t) a C cos n t
 
 
         
 
DC AC components
5

6. Power & Energy
 Instantaneous Power: p(t) = v(t)i(t)
 Energy / Work :
 Average / Real power / Active Power :
o If v(t)=Vdc :
o If i(t)=Idc :
Note – Average of a given function f(t):
0 0
0 0
t T t T
t t
1 1
P p(t)dt v(t)i(t)dt
T T
 
 
 
dc avg
P V I

avg dc
P V I

0
0
t T
avg
t
1
F f(t)dt
T

 
6
 
2
1
t
t
W p t dt or W PT
 

Special case: power absorbed or supplied by dc source.
Applications: battery-charging circuits, dc power
supplies.

7. 7

8. Capacitors
 Store energy in an electric field.
 Amount of charge stored is proportional to voltage:
 Energy stored:
 Current and voltage:
2
C
1
w(t) C v (t)
2
 
C
C
dv (t)
i (t) C
dt
 
0
t
C C C 0
t
1
v (t) i (t)dt v (t )
C
 

C
Q(t) C v (t)
  Q C V
  
8

9. Capacitors
 At periodic steady-state:
C C
v (t T) v (t)
 
t T
C C C
t
1
i (t) I i (t)dt 0
T

  

C
P 0

9

10. Capacitors
 At transient/NOT periodic steady-state:
C C
v (t T) v (t)
 
t T
C C C
t
1
i (t) I i (t)dt 0
T

  

C
P 0

10

11. Inductors
 Store energy in a magnetic field.
 Flux linkage is proportional to current:
 Energy stored:
 Current and voltage:
2
L
1
w(t) L i (t)
2
 
L
L
di (t)
v (t) L
dt
 
0
t
L L L 0
t
1
i (t) v (t)dt i (t )
L
 

L
(t) L i (t)
   L I
  
11

12. Inductors
 At periodic steady-state:
L L
i (t T) i (t)
 
t T
L L L
t
1
v (t) V v (t)dt 0
T

  

L
P 0

12

13. Inductors
 At transient/NOT periodic steady-state:
L L
i (t T) i (t)
 
t T
L L L
t
1
v (t) V v (t)dt 0
T

  

L
P 0

13

14. 14

15. Effective Values: RMS (Root-Mean-Square)
 Computing the average resistor power:
15
T T T T 2
2
2 eff
0 0 0 0
V
1 1 1 v (t) 1 1
P p(t)dt v(t)i(t)dt dt v (t)dt
T T T R R T R
 
    
 
 
 
   
T
2 2
eff
0
1
V v (t)dt
T
 
T
2
eff rms
0
1
V V v (t)dt
T
  
square root of the mean of
the square of the voltage
T
2
rms
0
1
I i (t)dt
T
 

16. 16
RMS Value of Pulse Waveform
m
V 0 t DT
v(t)
0 DT t T
 

 
 

   
T DT T
2 2 2 2
rms m m m
0 0 DT
1 1 1
V v t dt V dt 0 dt V DT V D
T T T
 
    
 
 
 
  
m
V

17. 17

18. 18
RMS Values of Sinusoids
2
2 2 m
rms m
0
V
1
V V sin ( t)d( t)
2 2

   
 
m m
rms
V V
1
V
2 2
2
  
 
  
  
 
 Sine wave
 Full-wave rectified sine wave
 Half-wave rectified sine wave

19. 19
RMS of the Sum Voltage Waveforms
   
T T T T T
2
2 2 2 2 2
rms 1 2 1 1 2 2 1 1 2 2
0 0 0 0 0
1 1 1 1 1
V v v dt v 2v v v dt v dt 2v v dt v dt
T T T T T
       
    
T T
2 2 2 2 2
rms 1 2 1,rms 2,rms
0 0
1 1
V v (t)dt v (t)dt V V
T T
   
 
2 2
rms 1,rms 2,rms
V V V
 
N
2 2 2 2
rms 1,rms 2,rms 3,rms n,rms
n 1
V V V V ... V

     
1 2
v(t) v (t) v (t)
 
Applied for
orthogonal
functions only

20. 20

21. 21
RMS of Triangular Waveform
m
rms
I
I
3

2
2 2 2
m
rms 1,rms 2,rms dc
I
I I I I
3
 
   
 
 

22. 22

23. Effective Values: RMS – General Fourier
 RMS of function f(t):
 If f is sum of N periodic waveforms:
 If f(t) is Fourier Series:
0
0
t T
2
rms
t
1
F f (t)dt
T

 
2
2 2 n
rms n,rms 0
n 0 n 1
C
F F a
2
 
 
 
    
 
 
N
2 2 2 2 2
rms 1,rms 2,rms 3,rms N,rms n,rms
n 1
F F F F ...F F

     
23

24. Apparent Power & Power Factor
rms rms
S V I

rms rms
P P
pf
S V I
 
• Magnitude of apparent power:
• Power factor:
o If v(t) and i(t) are sinusoidal waveforms:
 
pf cos
 
o If v(t) is sinusoidal but i(t) is not sinusoidal waveform:
   
1,rms
2
rms
I 1
pf cos cos
I 1 THD
       

24
Often used to specify rating of
power equipment (transformer)

25. 25
Sinusoidal AC Circuits
m
m
v(t) V cos( t )
i(t) I cos( t )
   
   
   
m m
rms rms
V I
P cos V I cos
2
 
       
 
 
 
rms rms
Q V I sin
 
*
P jQ ( )( )
   rms rms
S V I
2 2
rms rms
S P Q V I
   
S
 
pf cos
 
 For linear circuits that have sinusoidal sources, let:
 Average power:
 Reactive power:
 Complex power:
 Apparent power in ac circuits:
 Power factor:
Not applicable to
nonsinusoidal
voltages & currents

26. 26
Nonsinusoidal Periodic Waveforms
 Use Fourier series of sinusoids to describe nonsinusoidal
periodic waveforms.
 Fourier series:
 
0 n 0 n 0
n 1
f(t) a a cos(n t) b sin(n t)


    

 
0 n 0 n
n 1
f(t) a C cos n t


    

2
2 2 n
rms n,rms 0
n 1 n 1
C
F F a
2
 
 
 
    
 
 
 
n 0 0 n,rms n,rms
n 1 n 1
P P V I V I cos
 
 
     
 
Combine sines and cosines of the same frequency:
RMS value of f(t):
Average power:

27. 27
Nonsinusoidal Periodic Waveforms
 Nonsinusoidal Source and Linear Load
 Power absorbed by the load can be determined using
superposition theorem.

28. 28

29. 29
Nonsinusoidal Periodic Waveforms
 Sinusoidal Source and Nonlinear Load
 Sinusoidal voltage source applied to nonlinear load:
 Current waveform (not sinusoidal – represented as FS):
 Average power absorbed by load / supplied by source:
1 0 1
v(t) Vsin( t )
   
 
0 n 0 n
n 1
i(t) I I sin n t


    

 
    
 
 
   
n,max n,max
0 0 n n
n 1
n,max
1 1
0 1 1 n n
n 2
1 1
1 1 1,rms 1,rms 1 1
V I
P V I cos
2
0 I
V I
0 I cos cos
2 2
V I
cos V I cos
2




 
    
 
 
 
 
        
 
   
   
 
       
 
 



30. 30
Nonsinusoidal Periodic Waveforms
 Sinusoidal Source and Nonlinear Load
 Power factor:
 Distortion Factor (DF):
 Power factor also expressed as:
 Total Harmonic Distortion (THD): DF:
 Form factor: Crest Factor:
 
 
1,rms 1,rms 1 1 1,rms
1 1
rms rms 1,rms rms rms
V I cos I
P P
pf cos
S V I V I I
    
      
 
 
1,rms
rms
I
DF
I

 
1 1
pf cos DF
 
   
 
2
2
n,rms
n,rms
n 1
n 1
2
1,rms 1,rms
I
I
THD
I I


 

 1,rms
2
rms
I 1
DF
I 1 THD
 

rms
avg
I
FF
I

peak
rms
I
CF
I


31. Small Ripple Approximations
 If AC component of f(t) is very small compared to DC
component, the AC component can be ignored.
 If , then
avg
f F
  avg
f(t) F

31

32. Separation of DC & AC Components
 Note that periodic waveform consists of DC and AC
components.
 Circuit analysis methods (KVL & KCL) apply
individually to DC and AC components.
dc ac
v(t) V v (t)
  dc ac
i(t) I i (t)
 
dc
loop
V 0


ac
loop
v (t) 0


dc
node
I 0


ac
node
i (t) 0


KVL: KCL:
32

33. 33