10Fall_hw3solns

10Fall_hw3solns - ECE210 Fall 2010 Homework 03 Solutions...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Page1 / 5

10Fall_hw3solns - ECE210 Fall 2010 Homework 03 Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online