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Lect19 EEE 202 1

AC Power

Dr. Holbert

April 9, 2008

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Lect19 EEE 202 2

Instantaneous Power:p(t)

For AC circuits, the voltage and current are

v(t) = VMcos(t+v)

i(t) = IMcos(t+i)

The instantaneous poweris simply their product

p(t) = v(t) i(t) = VM IMcos(t+v) cos(t+i)

= VM IM[cos(v- i) + cos(2t+v +i)]

Constant

Term

Wave of Twice

Original Frequency

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Lect19 EEE 202 4

Average Power: Special Cases

Purely resistive circuit: P = VMIMThe power dissipated in a resistor is

Purely reactive circuit: P = 0

Capacitors and inductors are lossless elements and

absorb no average power

A purely reactive network operates in a mode in

which it stores energy over one part of the period and

releases it over another part

RI=R

V=IV=P

M

M

MM

22

2

1

22

1

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Lect19 EEE 202 5

Average Power Summary

Circuit Element Average Power

V or I source P= VMIMcos(v- i)

ResistorP

= VMIM=

IM

2R

Capacitor or

Inductor

P= 0

Does the expression for the resistor power look

identical to that for DC circuits?

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Lect19 EEE 202 6

Effective or RMS Values

Root-mean-square value (formula reads

like the name: rms)

For a sinusoid: Irms= IM/2 For example, AC household outlets are

around 120 Volts-rms

Tt

t

rms

Tt

t

rms dttvTVanddttiTI

0

0

0

0

)(1

)(1 22

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Lect19 EEE 202 7

Why RMS Values?

The effective/rms current allows us to write

average power expressions like those used in dc

circuits (i.e., P=IR), and that relation is really the

basis for defining the rms value The average power (P) is

RIR

VIVIVP

IVIVP

rms

rms

rmsrmsMMresistor

ivrmsrmsivMMsource

22

2

1

coscos

2

1

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Lect19 EEE 202 8

RMS in Everyday Life

When we buy consumer electronics, thefaceplate specifications provide the rms voltageand current values

For example, what is the rms current for a 1200

Watt hairdryer (although there is a small fan in ahairdryer, most of the power goes to a resistiveheating element)?

What happens when two hairdryers are turned

on at the same time in the bathroom? How can I determine which uses more

electricity---a plasma or an LCD HDTV?

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Lect19 EEE 202 10

Extra Slides

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Lect19 EEE 202 11

Maximum Average Power Transfer

To obtain the maximum average power transfer

to a load, the load impedance (ZL) should be

chosen equal to the complex conjugate of the

Thevenin equivalent impedance representingthe remainder of the network

ZL= RL+ j XL= RTh- j XTh= ZTh*

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Lect19 EEE 202 12

Maximum Average Power Transfer

Voc +

ZTh

ZLZ

L=Z

Th*

Note that ONLY the resistive component of

the load dissipates power

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Lect19 EEE 202 13

Max Power Xfer: Cases

LoadCharacteristic

Load Equivalent

Complex ZL = ZTh*=RTh - jXTh

Purely Resistive(i.e.,XL=0)

Further reduces to ZL=RL=RThforXTh=0 (old DC way)

Purely Reactive(i.e.,RL=0)

No Average power transfer toload; Not really a case

22

ThThLL X+R=RZ

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Lect19 EEE 202 14

Power Factor (pf)

Derivation ofpower factor (0 pf 1)

A low power factor requires more rms current for thesame load power which results in greater utility

transmission losses in the power lines, therefore utilities

penalize customers with a lowpf

LZiv

rmsrms

ivrmsrms

rmsrms

==IV

IV=

IV

P=

powerapparent

poweraverage=pf

coscoscos

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Lect19 EEE 202 15

Power Factor Angle (ZL)

power factor angleis v- i= ZL(the phaseangle of the load impedance)

power factor (pf)special cases

purely resistive load: ZL= 0 pf=1

purely reactive load: ZL= 90 pf=0

Power Factor Angle I/V Lag/Lead Load Equivalent

-90< ZL< 0 Leading Equivalent RC

0< ZL< 90 Lagging Equivalent RL

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Lect19 EEE 202 16

From a Load Perspective

Recall phasor relationships between current,

voltage, and load impedance

The load impedance also has several alternateexpressions

ZZ jj sincos)Im()Re( ZZZZ

Z

Z

Z

ZIV

rmsrms

Zirmsvrms

ZiMvM

IV

IV

IV

22

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Lect19 EEE 202 17

Power Triangle

Thepower trianglerelatespf angleto Pand Q

powergereal/avera

poweruadraturereactive/q

P

Qiv tan

the phasor current that is in

phase with the phasor

voltage produces the real

(average) power

the phasor current that is

out of phase with the

phasor voltage produces

the reactive(quadrature)

power

Re

Im

P=I2rmsRe(Z)

Q=I2

rms

Im(Z)

v- i

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Lect19 EEE 202 18

Summarizing Complex Power (S)

Complex power (like energy) is conserved, that is,

the total complex power supplied equals the totalcomplex power absorbed, Si=0

ZZZS 222 ImRe rmsrmsrms IIjIQjP

Reactive Power Load Power Factor Complex Power

Qis positive Inductive Lagging First quadrantQis zero Resistive pf = 1 Real valued

Qis negative Capacitive Leading Fourth quadrant

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Lect19 EEE 202 19

More Power Terminology

average power, P= Vrms Irmscos(v- i)

apparent power= Vrms Irms

apparent power is expressed in volt-amperes

(VA) or kilovolt-amperes (kVA) to distinguish it

from average power

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Lect19 EEE 202 20

Complex Power (S)

Definition of complex power, S

Pis the realor average power

Qis the reactiveor quadrature power, which indicates

temporary energy storage rather than any real power

loss in the element; and Qis measured in units of

volt-amperes reactive, or var

QjP

j iviv

iv

ivrms

sinIVcosIV

IV

IV

rm srm srm srm s

rm srm s

rm srm s

*

rm sIVS

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Lect19 EEE 202 21

Complex Power (S)

This is really a return to phasor use of voltage and

current rather than just the recent use of magnitude and

rms values

Complex power is expressed in units of volt-amperes like

apparent power

Complex power has no physical significance; it is a

purely mathematical concept

Note relationships to apparent power and power factor of

last section

|S| = VrmsIrms= apparent power

S =(v- i) = ZL= power factor angle

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Lect19 EEE 202 22

Real Power (P)

Alternate expressions for the realor

average power(P)

Z

Z

ZZS

S

Re

Recos

cosRe

2

rms

rmsrmsZ

ivrmsrms

I

II

IVP

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Lect19 EEE 202 23

Reactive Power (Q)

Alternate expressions for the reactiveor

quadrature power(Q)

Z

Z

ZZS

S

Im

Imsin

sinIm

2rms

rmsrmsZ

ivrmsrms

I

II

IVQ