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Power

Electronics

CHAPTER 2

Power Computat ions

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Electronics

Power & Energy

Instantaneous power

Absorbing

Power

Supplying

Power

)()()(   t it vt  p   Time-varying quantity

• PASSIVE SIGN CONVENTION

0)(   t  p   0)(   t  p

Energy

  2

1

)(t 

t dt t  pW 

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Electronics

Power

Electronics

Power & Energy

Average power (real power, active power)

 

  T t 

t 

T t 

t 

o

o

o

o

dt t it vT 

dt t  pT 

 P    )()(1

)(1

T 

W 

 P  

Periodic voltage and current produce a periodic instantaneous power.

Power generally

means average power.

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Inductors & Capacitors Inductors

)(2

1)(   2 t  Lit w  

t 0   t 0+T t 0+2T 

T 

i L(t 0)

i L(t 0+T )

i L(t ) In the steady state, the

inductor current is periodic

)()( 00   t iT t i  L L  

)()(1

)( 000

0

t idt t v L

T t i  LT t 

t   L L  

 

0)(1

)()(  0

000  

  dt t v

 Lt iT t i

T t 

t   L L L

  0)()()(1   0

0

 

t vt vavg dt t vT 

  L L

T t 

t   L

In the steady state,

- the average inductor voltage is zero.

- the net change of the inductor current during one switching period is zero.

- no net energy transfer, i.e. the average power of the inductor is zero (PL = 0)

Inductor volt-second balance

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Inductors & Capacitors

• Volt-seconds over T  equal zero. 

v L

i L

Increasing inductor current

 Inductor absorbs power

and energy from externalcircuits

v L

i L

Decreasing inductor current

 Inductor supplies power

and energy to external

circuits

dt 

di Lv   L L  

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Inductors & Capacitors Capacitors

In the steady state, the

capacitor voltage is periodic

t 0   t 0+T t 0+2T 

T 

vC (t 0)

vC (t 0+T )

vC (t )

)()( 00   t vT t v C C   

)()(1

)( 000

0

t vdt t iC 

T t v C T t 

t   C C   

 

0)(1

)()(  0

000  

  dt t i

C t vT t v

T t 

t   C C C      0)()()(

1   0

0

 

t it iavg dt t iT 

  C C 

T t 

t   C 

In the steady state,

- the average capacitor current is zero.

- the net change of the capacitor voltage during one switching period is zero.

- no net energy transfer, i.e. the average power of the capacitor is zero (PC = 0)2

2

1)(   Cvt w  

Capacitor ampere-second balance

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Inductors & Capacitors

• Amp-seconds over T  equal zero. 

Increasing capacitor

voltage

Capacitor absorbs

power and energy

from external circuits

(charging)

Decreasing

capacitor voltage

Capacitor

supplies power

and energy to

external circuits

(discharging)

dt 

dvC i   C C  

vC 

iC 

vC 

iC 

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Energy Recovery

Fuel injector solenoid in automobile

freewheeling pathfor induct current

continuity

Inductors and capacitors must be energized and de-energized by switching

operations

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• Reminding

• Without freewheeling path, the Tr is damaged by the overvoltage when it turns off.

• Inductor current continuity must be provided by circuit design

Energy Recovery During Tr on (0 < t < t1)

CC  L   V v  

    t 

CC 

CC  L

t 

 L L  L

t V 

d V  Lid v Lt i 00 0

1

)0()(

1

)(      )(

2

1)(   2 t  Lit w  

Inductor current linearly

increases.

Inductor stores energy inside

itself.

freewheeling path

for induct current

continuity

dt 

di Lv   L L 

)()(   t it i  L s  

(+)

Source supplies

power

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Electronics

Energy Recovery During Tr off (t1 < t < T)

diode turns on due to

the inductor current

continuity

 L

t V t i   CC  L

)()(   11  

T t t e L

t V et it i

  t t CC t t 

 L L

   

  

   

1

/)(1/)(

1

11)()(    

Inductor current exponentially decreases.

Inductor releases its energy to resistor.

 R L 

 LT 

t V dt 

T dt 

 L

t V 

T V 

dt t iT 

V  I V  P 

CC t T 

t 

CC CC 

T 

 sCC S S S 

20

11

)(1

2

1

0

0

1

1

0S i

average power

supplied by

source

 LT 

t V  P  P    CC S  R

2

2

1

average power

absorbed by

resistor

0)(    L P 

Heat

energy

Power

loss

lossy element

(+)

Source supplies

power

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Energy Recovery

Recycle energy

stored in elements

energystorageelement

re overeuse

store

energyenergy

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Energy Recovery During Tr on (0 < t < t1)

CC  L   V v  

    t 

CC CC  L

t 

 L L L

t V d V 

 Lid v

 Lt i

000

1)0()(

1)(      

• Inductor current linearly increases.

• Inductor stores energy inside itself.

)()(   t it i  L s  

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Energy Recovery

• The inductor current linearly decreases.

• The energy stored in the inductor is recovered to the source &

the source is absorbing power.

During Tr off (t1 < t < T)

CC  L   V v  

  1111

1   2211

)(11

t t t t t  L

V 

 L

t V d V 

 Lt id v

 Lt i   CC CC 

t 

t   CC  L

t 

t   L L  

 

  

       

)()(   t it i  LS   

No power

loss

(+)

Source supplies

power

(-)

Source absorbs

power

14

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Electronics

Effective value (RMS)

Effective value (Root-Mean-Square value)

The effective value of a periodic voltage (current) delivers the sameaverage power to a resistor as dc voltage (current) does.

 R

V  P    dc

2

 R

V  P 

  eff  

2

T 

T T T 

dt t vT  R

dt  R

t v

T dt t it v

T dt t  p

T  P 

0

2

0

2

00

)(11

)(1)()(

1)(

1

• For a dc voltage • For a ac voltage

V eff 2

  T 

rmseff     dt t vT 

V V 0

2 )(1

  T 

rms   dt t iT 

 I 0

2)(

1

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