Ae104a10_final_solutions

# Ae104a10_final_solutions - Ae/APh 104a Final exam Solutions...

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Unformatted text preview: Ae/APh 104a Final exam Solutions 1/3 [20pts] (a) [2pts] Using the ideal op-amp assumptions, V + = V- = 0 such that V 1 = R 1 I 1 and V w = I w R w = I F R F , and I 1 = I F . Thus, substituting for I F = I 1 = V 1 /R 1 , V w = V 1 R F R 1 . (b) [6pts] Want to relate V s to V w . We can write down several expressions that govern this circuit: V s- R 2 b I b = V G V G- R 2 a I a = V w , so, eliminating V G , V s = V w + R 2 a I a + R 2 b I b . Substitute for I b = I a + I c : V s = V w + R 2 a I a + R 2 b ( I a + I c ) = V w + R 2 I a + R 2 b I c . Substitute for I c = V G iCω : V s = V w + R 2 I a + R 2 b iCωV G . Substitute for V G again: V s = V w + R 2 I a 1 + iCω R 2 a R 2 b R 2 + R 2 b iCωV w . Substitute for I a = I F + I w = V w /R f + V w /R w : V s = V w + R 2 V w 1 R F + 1 R w 1 + iCω R 2 a R 2 b R 2 + R 2 b iCωV w . Define T C = R 2 a R 2 b R 2 C. Then V s V w = 1 + R 2 1 R F + 1 R w (1 + iωT C ) + iωT C R 2 R 2 a 1 and V s V w = 1- R 2 R 2 a + R 2 1 R w + 1 R F + 1 R 2 a (1 + iωT C ) , as required. (c) [2pts] Taking the time mean of the result from part (b), V s = V w 1 + R 2 R w + R 2 R F . (d) [5pts] You are asked for the sensitivity, S , of v s to r w , i.e. ∂v s /∂r w . Begin by formulating an expression for v s . v s = V s- V s = 1- R 2 R 2 a + R 2 1 R w + 1 R F + 1 R 2 a + iωT C 1 R w + 1 R F + 1 R 2 a- 1- R 2 R w- R 2 R F V w , and v s = R 2 1 R w + r w- 1 R w + iωT C R 2 1 R w + r w + 1 R F + 1 R 2 a V w ....
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## This note was uploaded on 01/05/2012 for the course AE 104a taught by Professor List during the Fall '09 term at Caltech.

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Ae104a10_final_solutions - Ae/APh 104a Final exam Solutions...

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