l3. power in ac circuits

8/12/2019 L3. Power in AC Circuits

1/17

Power in AC Circuits

IntroductionIntroductionIntroductionIntroduction

Power in Resistive ComponentsPower in Resistive ComponentsPower in Resistive ComponentsPower in Resistive Components

Power in CapacitorsPower in CapacitorsPower in CapacitorsPower in Capacitors

Power in InductorsPower in InductorsPower in InductorsPower in Inductors

Circuits with Resistance and ReactanceCircuits with Resistance and ReactanceCircuits with Resistance and ReactanceCircuits with Resistance and Reactance

Active and Reactive PowerActive and Reactive PowerActive and Reactive PowerActive and Reactive Power


2/17

The instantaneous power dissipated in a component is a

product of the instantaneous voltage and the instantaneous

current

In a resistive circuit the volta e and current are in hase .

Introduction

p = vi

calculation ofp is straightforward.

In reactive circuits, there will normally be some phase shift

between v and i and calculating the power becomes more

complicated.


3/17


4/17

Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in a resistorin a resistorin a resistorin a resistor


5/17

Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in ain ain ain a resistor...resistor...resistor...resistor...


6/17

POWER IN CAPACITORS


7/17

Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in a capacitorin a capacitorin a capacitorin a capacitor


8/17

Power in Capacitors

vp=

From our discussion of capacitors we know that the current

leads the voltage by 90. Therefore, if a voltage v = Vp sin t is

applied across a capacitance C, the current will be given by i =

Ip cost

Then

)2

2sin

(

)cos(sin

cossin

t

IV

ttIV

tItV

PP

PP

PP

=

=

=

The average power is zero


9/17

Power in Inductors

Phasor Diagram


10/17

Relationship betweenRelationship betweenRelationship betweenRelationship between vvvv,,,, iiiiandandandand pppp in an inductorin an inductorin an inductorin an inductor


11/17

Power in Inductors

vip=

From our discussion of inductors we know that the current lags

the voltage by 90. Therefore, if a voltage v = Vp sin t is

applied across an inductanceL, the current will be given by i =-Ip cos t

Therefore

)2

2sin

(

)cos(sin

cossin

t

IV

ttIV

tItV

PP

PP

PP

=

=

=

Again the average power is zero


12/17

Circuit with Resistance and Reactance

When a sinusoidal voltage v = Vp sin tis applied across a

circuit with resistance andreactance, the current will be of the

general form i =Ip sin (t-)

Therefore, the instantaneous power,p is given by

)2cos(2

1cos

2

1

)}2cos({cos2

1

)sin(sin

=

=

=

tIVIVp

tIV

tItV

PPPP

PP

PP


13/17

)2cos(2

1cos

2

1 = tIVIVp PPPP

The expression forp has two components

The second part oscillates at 2and has an average value of

zero over a com lete c cle


cos)(cos22

)(cos2

1VI

IIVP PPPP ===

this is the power that is stored in the reactive elements and

then returned to the circuit within each cycle

The first part represents the power dissipated in resistive

components. Average power dissipation is


14/17

cos)(cos2

1VIIVP PP ==

The average power dissipation given by


is termed theactive power(or) real power in the circuit and is

measured in watts (W).This is due to the power consumption of

circuit resistance.

The product of the r.m.s. voltage and current VI is termed the

apparent power(or)complex power, S. To avoid confusion

this is given the units of volt amperes (VA)


15/17

cos

cos

S

VP

=

=

From the above discussion it is clear that

In other words the active ower is the a arent ower times the


factorPoweramperes)volt(inpowerApparent

watts)(inpowerActive=

cosfactorPower ==P

cosine of the phase angle.

This cosine is referred to as the power factor.power factor.power factor.power factor.


16/17

When a circuit has resistive and reactive parts, the resultant

power has 2 parts: The first is dissipatedin the resistive element. This is the

active power,P

Active and Reactive Power

.

This is the reactive power, Q , which has units of volt

amperes reactive or varvarvarvar

While reactive power is not dissipated it does have an effect on

the system for example, it increases the current that must be supplied

and increases losses with cables


17/17

Therefore

Active Power P = VIcos watts

Active and Reactive Power

eac ve ower = s n var

Apparent Power S = VI VA

S2 = P2 + Q2

l3. power in ac circuits

Documents