Maximum Power Transfer+−+−LVLRLISRSVSLLSVRIR=+()22211 122 2LLLL L SSLLRPav V I R I VR R== =+28LSVRSRLR()()()2242102SSLLS SSLLLLLRRRdPavRRRRRVRR+− +==⇒=∂+For fixed and ,ssVR∴maximum average power transfer to load LRoccurs whenMaximum Power Transfer+−+−LVLISS SZR Xj=+SVSLLSVZIZ=+2211 1cos cos22 2LLL L LL L LLPav V I Z Z I Z R I== =,0LL LLZRjXR=+>LLLVZI=()( )222221122SSSSSLLLLLVVRRZXRXZR==++++To maximize , necessary to setLSav LPXX−=()2212LSLvSaLVRRPR⇒=+()()()2242102SSLLS SSLLLLLRRRdPavRRRRRVRR+− +==⇒=∂+2max8LSavSVPR=LRSR0SX−LXLavPTheorem : Maximum Power TheoremOptimum Load Impedance optsLZ Z=+−+−LVLISS SZR Xj=+SVSLLSVZIZ=+()1cos2SSSLSPav ZVI Z=+,0LL LLZRjXR=+>optSLZ Z=Max. Power Theorem : Conjugate-match condition()21Re2SSLIZZ=+()21Re2SLLZIZ=+()221Re2SLSS LZZVZZ=++()()()1cos2SSS SLLZIIZZZ=+ +i()()222211Re 2222SSSSSSSSSVVZZ RRZZPav∴=+=+224LSSavVRP==Efficiency1or 50%22SLLLav avav avPPPP== =LSZZ=Comments :1. Under conjugate-match condition, 50% of the power delivered by the source is lost as heat dissipation in RS. Power company neverconjugates their loads!2. Max. Power Theorem is used extensively in communication circuits to extract maximum power from preceding stages.∴Network FunctionsjUUUe=()()()YjHjUjωωωLinearElementsNo independent Sourcesuy()() cosut U t Uω=+()() cosytY t Yω=+jYYYe=Definitionis called a Network Function.+−iV()()()oiVjHjVjωωωN+−oV0oI=iI()()()oiVjHjIjωωωN+−oV0oI=+−iV()()()oiIjHjVjωωωNoIiI()()()oiIjHjIjωωωNoITypical Application of Max. Power Theorem+−1600SZ =ΩSV()()2222Re111610 50 10 0.52 2 1600 16LoSSLZPV WZZ−== ≈=++16LZ=ΩHI-FIAmplifierLoudspeakerInput impedance = 16 Ω⇓10V=Let average power oP=delivered to loudspeakerFor maximum power transfer, make 1600 .LZ =Ω()21 160010 252 1600 1600oPW==+For maximum power transfer, make 1600 .LZ=Ω()21 160010 252 1600 1600oPW==+Use a transformer ::1n16Ω≡()216 1600LZn==2100 10nn⇒= ⇒=10 : 1HI-FIAmplifierLoudspeaker25W25Wnon-energtransformer is 25 Watts of ricpowe⇒is delivered to loudspeaker.FREQUENCY RESPONSECLR()()()()()()2222111111CLRZj jCCRLR Lωωωωωωω−−=++− +−011Resonant frequencyCLLCωωωω=⇒= ←0ωR()Rjω2R2R−00ω0ωω2RR0ω=ω=∞00ωω=()Rjω()Xjω()ZjωResistance function ( )Rjω→()XjωReactance function ←()XjωFREQUENCY RESPONSECLR()()()()()222222111111CLRZjCCRLRLωωωωωωω−=++− +−()()11tan1CLZjRωωω−−−=(011Resonant frequencyCLLCωωωω=⇒= ←0ωR()Zjω2π2π−00ω0ωω()Zjω(2RR0ω=ω=∞00ωω=()Rjω()Xjω()Zjω()Zjω(FREQUENCY RESPONSECLR()()11Yj j CRLωωω=+ −011Resonant frequencyCLLCωωωω=⇒= ←0ω1R()GjωCω00ωω()Bjωω=−∞00ωω=()Gjω()Bjω()Yjω1Rω=∞()Gjω()BjωSusceptance function ←Conductance function ↑1Lω−FREQUENCY RESPONSECLR()()()2211Yj CRLωωω=+−()11tan1CLYjRωωω−−=(011Resonant frequencyCLLCωωωω=⇒= ←0ω1R()Yjω2π2π−00ω0ωω()Yjω(ω=−∞00ωω=()Gjω()Bjω()YjωMagnitude function←Phase function←1R()Yjω(ω=∞Resonance(2) 1 , (2) 1Yj Zj==CLRRiLiCi1Ω14H1 Fv+−sitωπ2π32π2πcosSitω=101−CYLYRYRILICIV+−01iIe=D(2) (2)(2) 1Vj Zj Ij==(2) (2) (2) 1RRIj YjVj==(2) (2)(2) 2 90LLIj YjVj==∠−D(2) (2)(2) 2 90CCIj YjVj==∠D2π0π32π2πtω(),Cit A⇓2ω=012LCω==CILIRI22−102CL RIII==no-gain propertydoes not hold forRLC circuits.Resonance1(1) 10 71.6, (1) 71.610Yj Zj=∠− = ∠DDCLRRiLiCi1Ω14H1 Fv+−sitωπ2π32π2πcosSitω=101−CYLYRYRILICIV+−01iIe=D1(1) (1)(1) 71.610Vj ZjIj==∠D1(1) (1) (1) 71.610RRIj YjVj==∠D4(1) (1) (1) 18.410LLIj YjVj==∠−D1(1) (1) (1) 161.610CCIj YjVj==∠D2π0π32π2πtω(),Cit A⇓1ω=012LCω==CILIRI1010I =∠D71.6D18.4−D161.6DResonance3( 2) 4 10 , ( 2) 250Yj Zj−=× = Ω CLRRiLiCi250Ω14H1 Fv+−sitωπ2π32π2πcosSitω=101−CYLYRYRILICIV+−01iIe=D(2) (2)(2) 250VjZjIjV==(2) (2)(2) 1RRIjYjVjA==( 2) ( 2) ( 2) 500 90LLIjYjVj A==∠−D(2) (2)(2) 500 90CCIjYjVj A==∠D2π0π32π2πtω(),Cit A⇓2ω=012LCω==CILIRI500500−10500−500Instantaneous, Average, Complex Power+−+−v()() cosvt V t Vω=+Instantaneous PowerNi()() cosit I t Iω=+() ()() cos( )cos( )ptvtitVI tV tIωω== + +11cos( ) cos(2 )22VI V I VI t V Iω=−+++ constantSinusoid of twice the frequencyAverage Power011() cos( )2TavPptdtVIVIT=−∫t()vt2Tπω=t()it()ptavP0Power being returned by NVZI=jV jZ jIVe Ze Ie=i()jZ IZIe+=VZI ZVI∴=+ ⇒ =− {INDUCTOR1. 0 inCAPACITORavP =Average Power1cos2avPVI Z= Remarks:pa2. ssIf N ive contains only elements, then0cos 0avPZ≥⇒ ≥90 90 for passive NZ∴−≤ ≤IIZVZI=0Complex Po2er1wPVI()1122jV jI j V IPVe Ie VIe−−== iActive powerZ11cos sin22PVI ZjVI Z=+ReavPPImQPReactive power1Re Average (Active) power2in Watts1Im Reactive power2avPVIQVI==∴==Effective (RMS) Value21()TRMSoXxtdtT∫Definition:Given any periodic waveform x(t) of period T, the RMS (root-mean-square) or effective value of x(t) is defined asR()itInterpretationRRMSILet WR= average power dissipated in Resistor()()220011() () ()TTRRMSW Rit it dt R i tdt RITT===∫∫The same average power is dissipated if the resistor is driven by a dc current source of value IRMS; hence IRMSis called the effective value of i(t).∴2RMSISinusoidal waveforms:() cos( )xt X t Xω=+2RMSXX =1. Since most instruments measure RMS values; hence our 60Hz-sinusoidal voltages are rated in RMS values.2.: Line voltage =110 magnitude=Exam 2(11 )le .p 0VV⇒cos cos22av RM S R M SVIPZV I Z==i Note:Significance of Complex PowerMost electrical machines are designed to withstand a maximum voltage magnitude |V| and a maximum current magnitude |I|. Hence, electrical machines are rated in maximum in KVA, and not in maximum average power dissipation Pav.12PVI=12 Power factor cosavVI ZPPFVI VI=cosPFZ∴= {1 for Resistors0 for INDUCTORS and CAPACITORSPF =Note:1. For power generation companies, it is important to keep the PF of the load (customers) be as close to unity as possible.2. The Watthour meter measures Pav, not |P|.Example: If a factory dissipates 10KW of power
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