final - Final Project 1 The 6 original equations 1 2 3 4 5...

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Final Project 1.  The 6 original equations: 1.  KCL:   2.  KVL Left Loop:   3.  KVL Right Loop:   4.  Ohm’s Law:   5.  Capacitor:   6.  Op Amp:   The derivation of the given equation: Equations 2 and 6 combine to form equation 7:   Equations 1 and 4 combine to form equation 8:   Equations 5 and 8 combine to form equation 9:   Equation 3 can be rearranged to form equation 10:  Equations 9 and 10 combine to form equation 11:   Equations 7 and 11 combine to form:  Finally we derive:   As gain  A  approaches infinity as it does in ideal Op Amps, the above equation simplifies to: 2.  See enclosed circuit.
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3.  Schematic with a 2000Hz Sinusoidal Input Input and Output of the Circuit plotted on the same plane
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The input of the circuit was a 2000Hz sinusoidal wave with amplitude of 1V.  The output  is the derivative of the sine wave, which would be a cosine wave.  The graph of a cosine wave  is 90 degrees out of phase of the sine wave.  In my measurements, the difference in the peaks  of the sine and cosine waves was .138ms.  A cycle of the sine wave is .5ms meaning a 4 th  of a  cycle should be .125ms.  With my measurement being .138, the cosine wave is obviously very  close to being 90 degrees out of phase of the sine wave.  The amplitudes of both waves are 
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