Maximum Power Transfer RS IL VL VS PavL VS RL VS IL RS RL 1 1 1 2 VL I L RL I L VS 2 2 2 2 8RL RL RS dPavL 1 VS 2 RL 2 RL RS RL 2 For fixed Vs and Rs maximum average power transfer to load RL RS RL 2 RS RL RL 4 R R S L occurs when 2 2 0 RL RS Maximum Power Transfer Z S RS jX S I L VL VS IL VS ZS ZL Z L RL jX L RL 0 VL Z L I L 1 1 1 2 PavL VL I L cos Z L Z L I L cos Z L RL I L 2 2 2 1 VS 2 2 RL ZS ZL 2 1 VS 2 2 RL 2 RS RL X S X L 2 2 To maximize PavL necessary to set PavL dPavL RL 1 VS 2 2 XL XS RL RS RL 2 1 2 RS RL 2 RS RL RL VS 4 2 RS RL 2 0 RL RS PavL m a x Pa v L XS 0 RS VS 2 8RS RL XL Theorem Maximum Power Theorem Optimum Load Impedance Z Lopt Z s VS IL ZS ZL Z S RS jX S I L VL VS Max Power Theorem Z L RL jX L RL 0 Z Lopt Z S Conjugate match condition 1 PavS VS I S cos Z S Z L 2 1 Z S Z L i I S I S cos 2 1 2 I S Re Z S Z L 2 1 2 I L Re Z S Z L 2 VS 2 1 Re Z S Z L 2 2 ZS ZL ZS ZL 2 PavS 1 VS 1 VS Re Z S Z S 2 2 ZS ZS 2 2 RS 2 2 2 RS ZL ZS Efficiency VS 2 4 RS PavL PavS 2 PavL PavL 2 PavL 1 2 or 50 Comments 1 Under conjugate match condition 50 of the power delivered by the source is lost as heat dissipation in RS Power company never conjugates their loads 2 Max Power Theorem is used extensively in communication circuits to extract maximum power from preceding stages Network Functions Linear Elements No independent Sources u u t U cos t U U U e y t Y cos t Y Y Y ej j U Definition H j y Y j U j is called a Network Function Y Vi N H j Ii Io 0 Vo Vo j Vi j N H j Vi Io 0 Vo Vo j I i j N H j Ii Io I o j Vi j N H j Io I o j I i j Typical Application of Max Power Theorem HI FI Amplifier Loudspeaker Input impedance 16 Z S 1600 VS Z L 16 10 V 1 Po VS 2 2 Re Z L ZS ZL 2 Let Po average power delivered to loudspeaker 1 16 2 2 50 10 10 0 5 W 2 1600 16 For maximum power transfer make Z L 1600 1 1600 2 Po 10 25 W 2 1600 1600 For maximum power transfer make Z L 1600 1 1600 2 Po 10 25 W 2 1600 1600 Use a transformer n 1 16 Z L n 2 16 1600 n 2 100 n 10 10 1 HI FI Amplifier 25 W 25 W transformer is non energic 25 Watts of power is delivered to loudspeaker Loudspeaker FREQUENCY RESPONSE L C Z j 1 R 1 2 R C 1 L 2 j R C 1 R 1 Resistance function R j 2 L C 1 L 2 X j Reactance function C 1 L 1 0 Resonant frequency LC R j X j R 0 0 Z j 0 X j R R 2 0 R 2 0 0 R 2 0 R j FREQUENCY RESPONSE L C Z j 1 R 2 1 R L C 1 L Z j tan 1 R 2 2 C 1 1 C 1 L 2 1 C 1 R L 2 2 R C 1 L 1 0 Resonant frequency LC Z j X j R 0 0 Z j 0 Z j 0 2 0 2 Z j 0 R 2 R 0 R j FREQUENCY RESPONSE L C Y j 1 R j C 1 G j Conductance function L R B j Susceptance function C 1 L 1 0 Resonant frequency LC G j B j 1 R 0 Y j B j C 0 0 0 1 R 0 1 L G j FREQUENCY RESPONSE L C Y j R 1 2 C 1 L C 1 1 L Y j tan 1 R 2 R Magnitude function Phase function C 1 L 1 0 Resonant frequency LC Y j B j 1 R 0 0 Y j Y j Y j 0 2 0 0 0 1 R 2 G j Resonance iC t A iS cos t 1 0 is 3 2 2 2 1 t v 1 2 LC I 1e Y j 2 1 iC iL R 1 L 1 4 H 1F i 0D V IR YR IL YL IC YC Z j 2 1 V j 2 Z j 2 I j 2 1 I R j 2 YR j 2 V j 2 1 I L j 2 YL j 2 V j 2 2 90 D I C j 2 YC j 2 V j 2 2 90 D 2 no gain property does not hold for RLC circuits IC IR 0 2 C 2 0 iR 1 IL IC I L 2 I R 2 0 2 3 2 t Resonance iC t A iS cos t 1 0 is 3 2 2 2 1 t v iC iL R 1 L 1 4 C H 1F IL YL IC YC 1 0 iR 1 2 LC I 1e i 0D V IR YR 1 Y j1 10 71 6 D Z j1 71 6 D 10 1 V j1 Z j1 I j1 71 6 D 10 1 I R j1 YR j1 V j1 71 6 D 10 4 I L j1 YL j1 V j1 18 4 D 10 1 I C j1 YC j1 V j1 161 6 D 10 161 6 D IC 0 IR 71 6 D 18 4 D I 1 0 D 1 IL 2 0 2 3 2 t Resonance iC t A iS cos t 500 1 0 is 3 2 2 2 1 t v iR R 250 1 2 LC I 1e Y j 2 4 10 3 L 1 4 C H 1F IL YL IC YC 2 0 iC iL i 0D V IR YR Z j 2 250 V j 2 Z j 2 I j 2 250 V I R j 2 YR j 2 V j 2 1 A 0 2 I L j 2 YL j 2 V j 2 500 90 D A I C j 2 YC j 2 V j 2 …
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