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S06hw3s - EE3220 Homework#3 Solutions 8.11 Consider a...

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EE3220 Homework #3 Solutions February 23, 2006 8.11 Consider a periodic signal v ( t ) and its inverse - v ( t ). Prove that the differential mode of these two signals is equal to 2 v ( t ) and their common mode is equal to zero. These results should apply for an arbitrary v ( t ) regardless of waveform type. v idm v 1 - v 2 = v ( t ) - ( - v ( t )) = 2 v ( t ) v icm v 1 + v 2 2 = v ( t ) + ( - v ( t )) 2 = 0 8.13 Find the differential-mode and common-mode components of two voltages v 1 = 4 + 1 . 2 sin ωt V and v 2 = 4 + 0 . 2 sin ωt V. v idm = 4 + 1 . 2 sin ωt - 4 - 0 . 2 sin ωt = sin ωt V v icm = 4 + 1 . 2 sin ωt + 4 + 0 . 2 sin ωt 2 = 4 + 0 . 7 sin ωt V 8.18 The two input amplifier below is made from an ideal op-amp. - + + - + - v OUT R 2 R 4 R 1 R 3 v 1 v 2 1
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a) Find an expression for v OUT in terms of v 1 , v 2 , and the resistor values. Assuming an ideal op-amp, then: v + = R 4 R 3 + R 4 v 2 v - v + i = v 1 - v - R 1 v OUT = v - - iR 2 = R 4 R 3 + R 4 v 2 - v 1 - v - R 1 R 2 = R 4 R 3 + R 4 v 2 - R 2 R 1 v 1 - R 4 R 3 + R 4 v 2 = 1 + R 2 R 1 R 4 R 3 + R 4 v 2 - R 2 R 1 v 1 b) What are the magnitudes of the differential- and common-mode gains and the CMRR of the circuit? For differential-mode, let v 2 = - v 1 , then v dm = 2 v 1 . A dm = v OUT 2 v 1 = - R 2 R 1 v 1 - 1 + R 2 R 1 · R 4 R 3 + R 4 v 1 2 v 1 = - R 2 2 R 1 - 1 + R 2 R 1 R 4 2( R 3 + R 4 ) For common-mode, let v 2 = v 1 , then v cm = v 1 .
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