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Lecture 25_rev2 - Maximum power transfer • Any power...

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Lecture 25 Sinusoidal power
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Definitions of power in sinusoids Instantaneous power: power being transferred at a given moment in time Can be either positive or negative, depending upon time, phase shift Average power: Over a given cycle Due to in-phase part of voltage, current Equivalent to the RMS value of current times real part of impedance Real ” power vs “ reactive power” ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 Z I Z A A Z P t t A Z t P t A t A Z t P t A Z t V I e Z ZI V t V t I t P t A t I RMS avg j Re Re 2 1 cos 2 1 2 2 sin sin 2 2 cos 1 cos 2 1 cos cos cos cos 2 2 2 2 = = = + - + + = + + + = + + = = = = + = θ φ ϖ θ φ ϖ θ φ ϖ θ φ ϖ θ φ ϖ φ ϖ θ Always pos. “Real” Both pos, neg.
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Example 0 2 4 6 8 10 12 14 V(t) I(t) P(t) ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 R A e R A P t R A t I t V t P t R A t I e R V R V I e CA P t CA t I t V t P t CA t I Ce V CV j I e L A P t L A t I t V t P t L A t I e L V L j V I t A t V j Ravg R R R j R j Cavg C C C j C j Lavg L L L j L 2 Re 2 2 cos 1 2 cos 0 Re 2 2 sin 2 sin 0 Re 2 2 sin 2 sin cos 2 0 2 2 0 2 2 2 2 2 2 2 2 = = + = = = = = = = - = = - = = = = = = = = = = = - - ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ π π π π L C R V(t) 0 2 4 6 8 10 12 14 V(t) I(t) P(t)
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Unformatted text preview: Maximum power transfer • Any power source can be modeled with complex Thevenin, • How much power delivered to a load? – Depends on RL, I – What is current? • Chose X L , R L to maximize power +-V Th Z Th Z L ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 L Th L Th L Th L Th L L Th L Th L Th L Th L L L Th L Th Th L Th Th L RMS L RMS Lavg L L L th th th Z Z R R R R R V R I X X X X R R R V R I I R I X X j R R V Z Z V I R I Z I P jX R Z jX R Z * 2 2 2 2 2 2 2 2 2 if max is sfer power tran So if max is This if max is This * Re = = ⇒ + =-= + + + = ⋅ = + + + = + = = = + = + = “complex conjugate impedance matching”:...
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