Chapter11:ACSteadyͲStatePower
11.1 InstantaneousandAveragePower
11.2 EffectiveorRMSValue
11.3 ApparentPowerandPowerFactor
11.4 ComplexPower
11.5 PowerFactorCorrection,Measurement
11.6 MaximumAveragePowerTransfer
11.7MutualInductance
11.8LinearTransformers,IdealTransformers
11.9Design&Applications
11.10 Summary
ThechoiceofacoverdcallowedhighͲvoltagepowertransmission
fromthepowergeneratingplanttotheconsumer.
11.1InstantaneousandAveragePower(1)
‡ Theinstantaneouslypower(inwatts),pt
v t Vm FRVZ t T v i t I m FRVZ t T i p t vt i t Vm I m FRVZ t T v FRVZtT i Vm I m FRVT v T i Vm I m FRVZt T v T i constantpower
sinusoidalpoweratȦt
pt ! : power is absorbed by the
circuit; pt : power is absorbed
by the source.
P
T
p
t
dt
Vm I m FRVT v T i T ³
‡ Theaveragepower, P,istheaverageofpt overoneperiod.
1. P isnottimedependent.
2. Whenșv = și ,itisapurelyresistiveloadcase.
3. Whenșv– și = Ʋ R,itisapurelyreactiveloadcase.
4. P meansthatthecircuitabsorbsnoaveragepower.
11.1InstantaneousandAveragePower(2)
P
Vm I m FRVT v Ti P
R P
Vm I m
Im R
, R
L RUC P
5H ª¬ 9, º¼
Vm I m FRV D
, ‘ q
Example:Acurrentflowsthroughanimpedance
= ‘ q: .Findtheaveragepowerdeliveredtotheimpedance.
Answer:N:
11.1InstantaneousandAveragePower(3)
Example:Determinetheaveragepowergeneratedbyeachsource
andtheaveragepowerabsorbedbyeachpassiveelementinFig.
bymeshanalysis
1.5PowerandEnergy(1)
‡ Poweristhetimerateofexpendingorabsorbingenergy,
measuredinwatts(:).
dw dw dq
˜
vi
‡ Mathematicalexpression: p
dt
dq dt
Passivesignconvention
P = + vi
absorbingpower
p = – vi
supplyingpower
5
11.2EffectiveorRMSValue(1)
‡ ThetotalpowerdissipatedbyR isgivenby:
T
i Rdt
R T i dt
T ³
HenceIeff isequalto:
I eff T
P
³
T
I eff
R
T
³ i dt I rms
Note:Therootmeansquare(rms)valueisaconstantitself
whichdependingontheshapeofthefunctionit. The
effectiveofaperiodiccurrentisthedccurrentthatdelivers
thesameaveragepowertoaresistorastheperiodiccurrent.
‡ Therms valueofait = ImFRVȦt isgivenby: I UPV
Im
‡ Theaveragepowercanbewrittenintermsoftherms values:
Vm I m FRVT v Ti VUPV I UPV FRVT v Ti Note:Ifyouexpressamplitudeofaphasor source(s)inrms,thenalltheanswer
asaresultofthisphasor source(s)mustalsobeinrms value.
P
11.2EffectiveorRMSValue(2)
Example:Findtherms valueofthecurrentwaveform.Ifthecurrent
flowsthroughaȍ resistor,calculatetheaveragepowerabsorbed
bytheresistor.
Example:Findtherms valueofthefullͲwaverectifiedsinewave.
Calculatetheaveragepowerdissipatedinaȍ resistor.
11.3ApparentPowerandPowerFactor(1)
‡ ApparentPower(inVA), S,istheproductofthermsvaluesof
voltageandcurrent.
‡ ItismeasuredinvoltͲamperes orVAtodistinguishitfromthe
averageorrealpowerwhichismeasuredinwatts.
P VUPV I UPV FRVT v Ti ApparentPower,S
S FRVT v Ti PowerFactor,pf
‡ Powerfactoristhecosineofthephasedifferencebetweenthe
voltageandcurrent.Itisalsothecosineoftheangleoftheload
impedance.
11.3ApparentPowerandPowerFactor(2)
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ORDGR
3XUHO\UHDFWLYH
ORDGL RUC
5HVLVWLYH
UHDFWLYHORDG
R L/C
șv– și P/S DOOSRZHUDUH
SI FRQVXPHG
șv– și ƲR P QRUHDOSRZHU
SI FRQVXPSWLRQ
șv– și !
șv– și ‡ /DJJLQJ LQGXFWLYHORDG
‡ /HDGLQJ FDSDFLWLYHORDG
Example:Determinethepowerfactoroftheentirecircuitbythe
source.Calculatetheaveragepowerdeliveredbythesource.
11.4ComplexPower(1)
‡ Complex power (in VA) 6 is the product of rms voltage phasor
and complex conjugate of rms current phasor:
6
9, 9UPV , UPV VUPV I UPV ‘T v Ti
where
9 Vm ‘T v o 9UPV
,
I m ‘Ti o , UPV
9
VUPV ‘T v
,
I UPV ‘T i
6 VUPV I UPV FRVT v Ti jVUPV I UPV VLQT v Ti 6 P
j
Q
P istheaveragepowerinwatts deliveredtoaloadanditistheonlyusefulpower.
Q isthereactivepowerexchangebetweenthesourceandthereactivepartofthe
load.ItismeasuredinVAR.Q forresistiveloads (unitypf).Q for
capacitiveloads (leadingpf).Q !forinductiveloads (laggingpf).
11.4ComplexPower(2)
O7KHLPSHGDQFHWULDQJOHZKHUH== R + jXU7KHFRPSOH[SRZHUWULDQJOHZKHUH6 = P + jQ
11.4ComplexPower(3)
6 VUPV I UPV FRVT v Ti jVUPV I UPV VLQT v Ti 6 P
ApparentPower:S
Realpower:P
ReactivePower:Q
Powerfactor:SI
PowerTriangle
j
Q
_6_ VUPV ǘ IUPV P Q 5H6 IUPVR S FRVșv – și
,P6 IUPVX = S VLQșv – și
P/S FRVșv – și
ImpedanceTriangle
PowerFactor
11.4ComplexPower(4)
9UPV ‘R 9, UPV ‘R $
Example:Foraload,.Determine:(a)
thecomplexandapparentpowers,(b)therealandreactivepowers,
and(c)thepowerfactorandtheloadimpedance.
11.4ComplexPower(5)
Thecomplex,real, andreactivepowers ofthe
sourcesequal therespectivesumsof thecomplex,
real,andreactivepowersoftheindividualloads.
Forparallelconnection:
6
9,
9 , , 9, 9, 6 6 The same results can be obtained for a series
connection.
6
9,
9 9 , 9, 9 , 6 6 Example: Inthecircuit,theȍ resistorsabsorbs
anaveragepowerof:.Find9 andthecomplex
powerofeachbranchofthecircuit.Whatisthe
overallcomplexpowerofthecircuit?(Assumethe
currentthroughtheȍ resistorhasnophaseshift.)
11.5PowerFactorCorrection,Measurement(1)
‡ Mostdomesticandindustrialloads,suchaswashingmachines,air
conditioners,andinductionmotorsareinductive.Theyhavealow,
laggingpowerfactor.Theloadcannotbechanged,butthepower
factorcanbeincreasedwithoutalteringthevoltageorcurrentto
theoriginalload.
‡ Powerfactorcorrection: theprocessof increasing thepowerfactor
withoutalteringthevoltageorcurrenttotheoriginalload.
‡ Tomitigatetheinductiveaspectoftheload,acapacitorisaddedin
parallelwiththeload.Thepowerfactorhasimproved.
11.5PowerFactorCorrection,Measurement(2)
‡ Withthesamesuppliedvoltage,thecurrentdrawislessbyaddingthe
capacitor.Sincepowercompanieschargemoreforlargercurrentsbecauseit
leadstolargerpowerlosses.Overall,thepowerfactorcorrectionbenefitsthe
powercompanyandtheconsumer.Bychoosingasuitablesizeforthecapacitor,
thepowerfactorcanbemadetobeunity.
Q S VLQș P WDQș
QC = Q1 – Q2 P WDQș WDQș ȦCV UPV
C
P S FRVș
Q2 = P WDQ ș
Qc
ZVrms
PWDQ T WDQ T Z Vrms
Powerfactorcorrectionis
necessaryforeconomicreason.
‡ Therealpowerdissipatedintheloadisnotaffectedbytheshuntcapacitor.
‡ Althoughitisnotascommon,ifaloadiscapacitiveinnature,thesame
treatmentwithaninductorcanbeused.
11.5PowerFactorCorrection,Measurement(3)
‡ Thewattmeter istheinstrumentformeasuringtheaveragepower.
‡ Themeterconsistsoftwocoils;thecurrentandvoltagecoils.
í Thecurrentcoilisdesignedwithlowimpedanceandis
connectedinserieswiththeload.
í Thevoltagecoilisdesignedwithverylargeimpedanceandis
connectedinparallelwiththeload.
Basicstructure
EquivalentCircuitwithload
11.5PowerFactorCorrection,Measurement(4)
If vt Vm FRVZt T v and i t P
9UPV , UPV FRVT v Ti I m FRVZt T i Vm I m FRV T v Ti ‡ Theinducedmagneticfieldfrombothcausesadeflectioninthe
currentcoil.
‡ Ideally,theconfigurationdoesnotaltertheloadandaffectthe
powermeasured.
‡ Thephysicalinertiaofthemovingcoilresultsintheoutputbeing
equaltotheaveragepower.
11.6MaximumAveragePowerTransfer(1)
=7K
R7K jX 7K
=L
RL jX L
,
97K
= 7K = L
97K
R7K jX 7K RL jX L P
, RL
97K RL R7K RL X 7K X L ‡ Themaximumaveragepowercanbetransferredtotheloadif
XL = –X7K andRL = R7KLH = L RL jX L R7K jX 7K Z 7K
‡ Theloadimpedancemustbeequaltothe
? PPD[
complexconjugateoftheThevenin impedance.
‡ Iftheloadispurelyreal,i.e.XL ,then RL
97K
R7K
R7K
X 7K
=7K 5.6MaximumPowerTransfer(1)
Ͳ Thereareapplicationswhereitisdesirabletomaximizethe
powerdeliveredtoaload.Also,powerutilitysystemsare
designedtotransportthepowertotheloadwiththegreatest
efficiency byreducingthelossesonthepowerlines.
Ͳ IftheentirecircuitisreplacedbyitsThevenin
equivalentexceptfortheload,thepower
deliveredtotheloadis:
P
i RL § V7K ·
¨
¸ RL
R
R
L ¹
© 7K
Ͳ Maximumpoweristransferredtotheload
resistanceequalstheTheveninresistanceas
seenfromtheload.
RL R7K
Ÿ
PPD[ V
7K
RL
Thepowertransferprofile
20
withdifferentRL
11.6MaximumAveragePowerTransfer(2)
Example:Forthecircuitshownbelow,findtheloadimpedance=L
thatabsorbsthemaximumaveragepower.CalculatePPD[.
11.7MutualInductance(1)
‡ Transformer: anelectricaldevice designedonthebasisofthe
conceptofmagneticcoupling.
‡ Itusesmagneticallycoupledcoilstotransferenergyfromone
circuittoanother
‡ Itisthekeycircuitelementsforsteppinguporsteppingdown
acvoltagesorcurrents,impedancematching,isolation,etc.
• Maxwell’s equations:
(differential form)
’u'
w%
wt
w'
-
wt
UQ ’u%
’u(
’u+
James Clerk Maxwell (1831-1879) was a
Scottish mathematician and theoretical
physicist. His most significant achievement
was aggregating a set of equations in
electricity, magnetism, and inductance —
Maxwell’s equations — including an important
modification of Ampère's Circuital Law. It is
famous for introducing to the physics
community a detailed model of light as an
electromagnetic phenomenon, building upon the
earlier hypothesis advanced by Faraday.
[The work of Maxwell] ... the most profound
and the most fruitful that physics has
22
experienced since the time of Newton.
—Albert Einstein, The Sunday Post.
11.7MutualInductance(2)
‡ Whentwoconductorsareincloseproximitytoeachother,the
magneticfluxduetocurrentpassingthroughwillinducea
voltageintheotherconductor.Thisiscalledmutual
inductance.Firstconsiderasingleinductor,acoilwithN
turns.Currentpassingthroughwillproduceamagneticflux,I.
9 Ifthefluxchanges,theinducedvoltageis: v N
9 Intermsofchangingcurrent: v N
9 Solvedfortheinductance: L N
dI
di
dI di
di dt
L
dI
dt
di
dt
‡ Thisisreferredtoastheselfinductance,sinceitisthe
reactionoftheinductortothechangeincurrentthrough
itself.
11.7MutualInductance(3)
‡ NowconsidertwocoilswithN1 andN2 turnsrespectively.
EachwithselfinductancesL1 andL2.Assumethesecond
inductorcarriesnocurrent.Themagneticfluxfromcoil1has
twocomponents: I I I
‡ I11 linksthecoiltoitself,I12 linksbothcoils.
‡ Eventhoughthetwocoilsarephysicallynotconnected,we
saytheyaremagneticallycoupled.
‡ Theentirefluxpassesthroughcoil1,thustheinducedvoltage
dI
incoil1is:
v
N
dt
‡ Incoil2,onlyI12 passesthrough,thustheinducedvoltageis:
v
N
dI
dt
11.7MutualInductance(4)
‡ Mutualinductance:istheabilityofoneinductortoinducea
voltageacrossaneighboringinductor,measuredinhenrys(H).
TheopenͲcircuitmutualvoltage
acrosscoil2:
v
L
di
dt
v
M di
dt
TheopenͲcircuitmutualvoltage
acrosscoil1:
v
M di
dt
25
11.7MutualInductance(5)
‡ Ifacurrententers(leaves)thedottedterminalofonecoil,the
referencepolarityofthemutualvoltageinthesecondcoilis
positive(negative)atthedottedterminalofthesecondcoil.
26
11.7MutualInductance(6)
Dotconventionforcoilsinseries;thesignindicatesthepolarity
ofthemutualvoltage;seriesͲaiding connection&seriesͲ
opposing connection.
ˢL
L L M VHULHVDLGLQJFRQQHFWLRQ
%\)DUDGD\
VODZ
dI
v N v
dt
ˢL
L L M VHULHVRSSRVLQJFRQQHFWLRQ
Michael Faraday (17911867), was an English
chemist and physicist who
contributed to the fields
of electromagnetism and
electrochemistry.
dI
N dt
27
11.7MutualInductance(7)
‡ TimeͲdomainanalysis:
ApplyingKVLtocoil1,
v
i R L
di
di
M dt
dt
ApplyingKVLtocoil2,
v
i R L
di
di
M dt
dt
ApplyingPhasor,
9
9
R jZ L , jZ M, jZ M, R jZ L , ‡ FrequencyͲdomainanalysis:
ApplyingKVLtocoil1,
9
= jZ L , jZ M, ApplyingKVLtocoil2,
jZ M, = L jZ L , 28
11.7MutualInductance(8)
Example:Determinethevoltage9o inthecircuit.
Example:Determinethephasor currents, and, inthecircuit.
29
11.7MutualInductance(9)
• Theinstantaneousenergystoredinthecircuitisgivenby
w
Li Li r Mii
Note:Thepositive signisselectedforthe
mutualtermifbothcurrentsenteror
leavethedottedterminalsofthecoils;
thenegativesignisselectedotherwise.
‡ Thecouplingcoefficient,k,isa
measureofthemagneticcoupling
betweentwocoils;”k ”.
M
1RWHk
LooselycoupledTightlycoupled
k k ! k LL Ÿ d M d LL I
I
I
RUk
I I
I
I
I
I I
30
11.7MutualInductance(10)
Example:Determinethecouplingcoefficient.Calculatetheenergy
storedinthecoupledinductorsattimet Vifvt = FRVt
+ Ʊ 9.
Example:Determinethecouplingcoefficientandtheenergystored
inthecoupledinductorsattimet V.
31
11.8LinearTransformer,IdealTransformer(1)
‡ ItisgenerallyafourͲterminaldevicecomprisingtwo(ormore)
magneticallycoupledcoils.
‡ Thecoilthatisconnectedtothevoltagesourceiscalledthe
primary.
‡ Theoneconnectedtotheloadiscalledthesecondary.
‡ Theyarecalledlinearifthecoilsarewoundonamagnetically
linearmaterial.
= LQ
9
,
R jZ L = R ZKHUH= R
ZM LVreflected impedance
R jZ L = L
32
11.8LinearTransformer,IdealTransformer(2)
ª 9 º
«9 »
¬ ¼
ª jZ L
« jZ M
¬
jZ M º ª , º
jZ L »¼ «¬, »¼
Anequivalentcircuitremovesthemutual
inductance.Thegoalistomatchthe
terminalvoltagesandcurrentsfromthe
originalnetworktothenewnetwork.
‡ TransformingtotheT networktheinductorsare:
‡ Transformingtothe3 networktheinductorsare:
33
11.8LinearTransformer,IdealTransformer(3)
Example:CalculatetheinputimpedanceinFig.andthecurrent
fromthevoltagesource.
Example:DeterminetheTͲequivalentcircuitofthelineartransform
inFig.(a).
34
11.8LinearTransformer,IdealTransformer(4)
‡ AnidealtransformerisaunityͲcoupled,losslesstransformerin
whichtheprimaryandsecondarycoilshaveinfiniteselfͲ
inductances. k = 9
9
N
N
n
,
,
N
N
n
9 !9n !їstepͲuptransformer
9 9 n ї stepͲdowntransformer
(a) IdealTransformer
(b) Circuitsymbol
35
11.8LinearTransformer,IdealTransformer(5)
‡ Complexpowersuppliedtotheprimary
isdeliveredtothesecondarywithout
loss.Theidealtransformerislossless.
‡ Inputimpedanceisalsocalledreflectedimpedance.
Impedancematchingensuresmaximumpowertransfer.
‡ Equivalentcircuit:
1.Secondaryonetoprimaryone
2.Primaryonetosecondaryone
11.8LinearTransformer,IdealTransformer(6)
‡ Equivalentcircuit:
1.Secondaryonetoprimaryone
Tofind97K
,
, VRWKDW 9
Tofind=7K
9s2 ,
n, DQG 9
9
n
11.8LinearTransformer,IdealTransformer(7)
Example:Anidealtransformerisratedat2400/120V,9.6kVA,and
has50turnsonthesecondaryside.Calculate:(a)theturns
ratio,(b)thenumberofturnsontheprimaryside,and(c)the
currentratingsfortheprimaryandsecondarywindings.
Example:Intheidealtransformercircuit,find(a),E9o,and(c)
thecomplexpowersuppliedbythesource.
Example:Calculatethepowersuppliedthe10Ͳɏ resistorinthe
idealtransformercircuit.
38
11.9Design&Applications(1)
‡
The matching network in the figure is used to interface the
source with the load, which means that the matching network is
used to connect the source to the load in a desirable way.
FIGURE: Design the matching
network to transfer maximum
power to the load where the
load is the model of an
antenna
of
a
wireless
communication system.
‡
‡
The purpose of the matching network is to transfer as much
power as possible to the load, i.e., the maximum power transfer.
An important example of the application of maximum power
transfer is the connection of a cellular phone or wireless radio
transmitter to the cell's antenna. For example, the input
impedance of a practical cellular telephone antenna is = j ȍ.
State the Goal: To achieve maximum power
transfer, the matching network should
match the load and source impedances.
The source impedance is
= V RV jZ LV j ˜ S ˜ j :
For maximum power transfer, the impedance =LQ, shown in Figure,
must be the complex conjugate of =V. That is,
=LQ
=V j :
Generate a Plan: Let us use a transformer
for the matching network as shown in Figure.
The impedance =LQ will be a function of n,
the turns ratio of the transformer. We will
set =LQ equal to the complex conjugate of =V
and solve the resulting equation to
determine the turns ratio, n.
ActonthePlan:
11.9Design&Applications(3)
‡ TransformerasanIsolation
Device toisolateacsupply
fromarectifier.
‡ Isolationbetweenthepower
lineandthevoltmeter.
‡ AsanIsolationDevicetoisolatedcbetweentwoamplifier
stages.
41
11.9Design&Applications(4)
‡ AsaMatchingDevice
Example: Using an ideal transformer to match the loudspeaker to
the amplifier to achieve maximum power transfer.
Equivalentcircuit
Example: Calculate the turns ratio of an ideal transformer required
to match a 400Ͳё load to a source with internal impedance of
2.5 kё. Find the load voltage when the source voltage is 30 V. 42
11.9Design&Applications(5)
‡ Powerdistributionsystem
43
11.10Summary(1)
‡ With the adoption of ac power as generally used conventional
power for industry and home, engineers became involved in
analyzing ac power relationships
‡ The instantaneous power delivered to this circuit element is the
product of the element voltage and current. Let vt and it be
the element voltage and current, chosen to adhere to the passive
convention. Then pt vt ǘ it is the instantaneous power
delivered to this circuit element. Instantaneous power is
calculated in the time domain.
‡ The instantaneous power can be a quite complicated function of
t. When the element voltage and current are periodic functions
having the same period, T, Ͳ it is convenient to calculate the
t T
average power
P
T
³
t
i t vt dt
11.10Summary(2)
‡ The effective value of a current is the constant (dc) current that
delivers the same average power to a 1 – ɏ resistor as the given
varying current. The effective value of a voltage is the constant
(dc) voltage that delivers the same average power as the given
varying voltage.
‡ Because it is important to keep the current I as small as possible
in the transmission lines, engineers strive to achieve a power
factor close to 1. The power factor is equal to FRV ș, where ș is
the phase angle difference between the sinusoidal steadyͲstate
load voltage and current. A purely reactive impedance in parallel
with the load is used to correct the power factor.
11.10Summary(3)
11.10Summary(4)