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Unformatted text preview: ECE210 - Fall 2010 - Homework 03 Solutions Textbook problems 3.8 1. Show that the derivation of maximum power transfer on pages 60-62 produces a unique global maximum. Simpler Solution: For the circuit with two resistors R L and R T in series with a voltage source V T , as shown in the deriviation on page 60, the power dissipated by the load resistor R L and its rst derivative are found to be p L = R L ( R T + R L ) 2 v 2 T dp L dR L = ( R T + R L ) 2- R L 2( R T + R L ) ( R T + R L ) 4 v 2 T . We can see from the plot on page 61 that, for positive R L , the function appears to be strictly increasing for R L < R T and strictly decreasing for R L > R T , which would imply that R L = R T is a unique global maximum for positive R L . For R L < R T , the following inequality holds dp L dR L > ( R T + R L ) 2- R L 2( R T + R L ) ( R T + R L ) 4 v 2 T > ( R T + R L ) 2- R L 2( R T + R L ) > R 2 T- R 2 L > , and for R L > R T , the inequality becomes R 2 T- R 2 L < . This proves the above intuition, and therefore R L = R T is a unique global maximum. Alternate Solution using concavity: We can also show that the function p L is strictly concave over R L in [0 , 2 R T ] , and then strictly decreasing for R L > R T , which implies that R L = R T is a unique global maximum for positive R L . The second derivative is found to be d 2 p L dR 2 L =- 2( R T + R L ) 5- 2(2 R T + R L )( R T + R L ) 4 ( R T + R L ) 8 v 2 T , which must be less than zero to be concave. When checking for concavity, we nd, for v T and R T > , d 2 p L dR 2 L <- 2( R T + R L ) 5...
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- Fall '08