EE201 Lecture 40 P 1 Maximum Power Transfer MPT for ac circuits Rs jXs I jXL vL t Vs Practical source RL Adjustable Load If a practical source a voltage source in series with elements having an impedance Zs Rs jXs is connected to a load ZL RL jXL the max power transfer is delivered when ZL Zs EE201 Lecture 40 Vs I Rs RL j Xs XL P Pav I 2RL P RL P 2 Rs RL 2 Xs XL 2 Vs2 Rs RL 2 Xs XL 2 2RL Rs RL Rs RL Xs XL 2 P XL Vs2RL 2 2 Vs2 2RL Xs XL Rs RL Xs XL 2 2 2 0 0 EE201 Lecture 40 P 3 When ZL Zs RL Rs XL Xs The maximum average power is Pmax Vs2 eff 4Rs Example Find the load impedance for maximum power transfer 4H IR 1 2IR ZL cos t A Solution IS 1 0o 1 rad sec EE201 Lecture 40 Find Voc 1 IR 2IR P 4 4H voc Cos t A voc 1IR 2IR 3IR 3 0o Find ISC Using KCL IR ISC 1 0o 1 cos t A IR 2IR 4H Isc EE201 Lecture 40 P 5 KVL 1IR 2 IR ISC j L 3IR ISC j4 IR 1 3 Isc j4 IR ISC 1 j4 3 Isc 1 j4 3 Isc 1 0o Isc 0 6 53 1o Voc 3 0 Z th 5 53 1 I sc 0 6 53 1 Zth 3 j4 For maximum power transfer ZL Zth ZL 3 j4 EE201 Lecture 40 P 6 Example Find value for C to deliver maximum average power to the load at 10 rad sec 10 0oA 10 10 2H C Solution The Thevenin equivalent impedance is Zth R j L Zth 10 j 10 2 Zth 10 j20 The load impedance that maximizes power transfer is ZL 10 j20 R j C C 1 20 C 1 20 10 1 200 0 005 C 5mF EE201 Lecture 40 P 7 Example The radio receiver shown in figure a is connected to an antenna The antenna intercepts the electromagnetic waves from a broadcast station operating at 1 MHz For circuit analysis purposes the antenna is represented by a Thevenin equivalent circuit shown in figure b a Find the input impedance Rin jXin of the receiver if maximum power is to be transferred from the antenna to the receiver b Under the condition of part a find the magnitude of the voltage across the receiver terminals and the average power delivered to the receiver Antenna Radio receiver a EE201 Lecture 40 j1 070 21 14 6 mV rms at 1 MHz P 8 Rin jXin Antenna equivalent circuit b Receiver input equivalent circuit Part a From the maximum power transfer theorem the answers are Rin 21 and Xin j1070 Part b Since the reactances in the circuit have been tuned out the input current to the receiver is simply 14 6 21 21 0 348 mA The input impedance has the magnitude Zin 212 10702 1070 2 Therefore the magnitude of the voltage across the receiver terminals is 0 348 1070 2 372 4 mV rms The power transferred from the antenna to the receiver is 0 3482 21 2 54 W EE201 Lecture 40 P 9 Example 11 13 A fixed load resistance RL 100 k representing the input resistance of an amplifier is connected to the source of figure 11 22b in example 11 12 through a passive coupling network i e a network that does not generate average power as shown in figure 11 23 a Show that the maximum voltage that can be developed across RL is 0 504 V b Show that the coupling network shown in figure 11 23 achieves this maximum voltage across RL L j1070 21 Zin 109 8 pF 14 6 mV rms at RL C VL 100k 1 MHz Fixed source 400 9 H Coupling network Fixed load Figure 11 23 Maximum power transfer through a coupling network EE201 Lecture 40 P 10 SOLUTION Part a The available power from the source is 2 54 W If all of the power is delivered to RL then the voltage VL must be VL Pmax RL 2 54 10 6 100 000 0 504 V Part b The input impedance Zin of the coupling network with load must be the conjugate of the source impedance Specifically Zin j L 1 j C 1 RL EE201 Lecture 40 P 11 Substituting 106 L 400 9 10 6 C 109 8 10 12 and R 100 103 into this expression yields Zin 21 j1070 W Which is indeed the conjugate of the source impedance Since Zin is the conjugate matched to the source impedance the maximum power of 2 54 W is transferred to the coupling network Since the coupling network consists of L and C neither of which consumes average power the 2 54 W of average power must be transferred out of the coupling network and into the load resistance The voltage across the load resistor V L is given by VL P RL 2 54 10 6 100 000 0 504 V This verifies that the coupling network of figure 11 23 enables the largest voltage to appear across the load resistor
View Full Document
Unlocking...