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Power In AC Circuit

Lecture 01 (Continued)

Course Conducted by

Shuvodip Das,

Lecturer, Dept. of ETEPrime University, Dhaka.

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Power in AC Circuits

Introduction

Power in Resistive Components

Power in Capacitors

Power in Inductors Circuits with Resistance and Reactance

Active and Reactive Power

Power Factor Correction

Power Transfer Three-Phase Systems

Power Measurement

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Introduction

The instantaneous power dissipated in a component

is a product of the instantaneous voltage and the

instantaneous current

p = vi

In a resistive circuit the voltage and current are in

phase calculation of pis straightforward

In reactive circuits, there will normally be somephase shift between vand i, and calculating the

power becomes more complicated

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Relationship between v, iand pin a resistor

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Power in Capacitors

From our discussion of capacitors we know that the

current leads the voltage by 90. Therefore, if a

voltage v= Vpsin tis applied across a capacitance

C, the current will be given by i= Ipcos t Then

)2

2sin(

)cos(sin

cossin

tIV

ttIV

tItV

vip

PP

PP

PP

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Relationship between v, iand pin a capacitor

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Power in Inductors

From our discussion of inductors we know that thecurrent lags the voltage by 90. Therefore, if avoltage v= Vpsin tis applied across an inductance

L, the current will be given by i= -Ipcos t Therefore

)2

2sin(

)cos(sin

cossin

tIV

ttIV

tItV

vip

PP

PP

PP

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Relationship between v, iand pin an inductor

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Circuit with Resistance and Reactance

When a sinusoidal voltage v= Vpsin tis applied

across a circuit with resistance andreactance, the

current will be of the general form i= Ipsin (t- )

Therefore, the instantaneous power, pis given by

)2cos(2

1cos

2

1

)}2cos({cos21

)sin(sin

tIVIVp

tIV

tItV

vip

PPPP

PP

PP

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The average power dissipation given by

is termed the active power in the circuit and ismeasured in watts (W)

The product of the r.m.s. voltage and current VIis

termed the apparent power, S. To avoid confusionthis is given the units of volt amperes (VA)

cos)(cos2

1VIIVP PP

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From the above discussion it is clear that

In other words, the active power is the apparentpower times the cosine of the phase angle.

This cosine is referred to as the power factor

cos

cos

S

VIP

factorPoweramperes)volt(inpowerApparent

watts)(inpowerActive

cosfactorPower S

P

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Active and Reactive Power

When a circuit has resistive and reactive parts, theresultant power has 2 parts:

The first is dissipatedin the resistive element. This is

the active power, P The second is storedand returnedby the reactive

element. This is the reactive power, Q, which hasunits of volt amperes reactive or var

While reactive power is not dissipated it does havean effect on the system

for example, it increases the current that must besupplied and increases losses with cables

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Consider anRL circuit

the relationship

between the variousforms of power canbe illustrated usinga power triangle

Apparent Power

Reactive

Power

Active power

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Therefore

Active Power P= VIcos watts

Reactive Power Q = VIsin var

Apparent Power S = VI VA

S2 = P2 + Q2

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Power Factor Correction

Power factor is particularly important in high-powerapplications

Inductive loads have a laggingpower factor

Capacitive loads have a leadingpower factor

Many high-power devices are inductive

a typical AC motor has a power factor of 0.9 lagging

the total load on the national grid is 0.8-0.9 lagging this leads to major efficiencies

power companies therefore penalise industrial userswho introduce a poor power factor

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The problem of poor power factor is tackled byadding additional components to bring the powerfactor back closer to unity

a capacitor of an appropriate size in parallel with alagging load can cancel out the inductive element

this is power factor correction

a capacitor can also be used in series but this is less

common (since this alters the load voltage)

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Power Measurement

When using AC, poweris determined not only by the

r.m.s. values of the voltage and current, but also by

the phase angle (which determines the power factor)

consequently, you cannot determine the power from

independent measurements of current and voltage

In single-phase systems power is normally

measured using an electrodynamic wattmeter measures power directly using a single meter which

effectively multiplies instantaneous current and voltage

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In three-phase systems we need to sum the powertaken from the various phases

in three-wire arrangements we can deduce the total

power from measurements using 2 wattmeter in a four-wire system it may be necessary to use 3

wattmeter

in balanced systems (systems that take equal power

from each phase) a single wattmeter can be used, itsreading being multiplied by 3 to get the total power

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Key Points

In resistive circuits the average power is equal to VI, whereVand Iare r.m.s. values

In a capacitor the current leadsthe voltage by 90 and theaverage power is zero

In an inductor the current lagsthe voltage by 90 and theaverage power is zero

In circuits with both resistive and reactive elements, theaverage power is VIcos

The term cos is called the power factor Power factor correction is important in high-power systems

High-power systems often use three-phase arrangements