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Answer to Question #3102 in Electric Circuits for Derek

Question #3102
<img src="../../..//assignments/uploaded_files/static/40/14/40144ed2ba418b1bf783bdc39808af6b.png" title="" alt=""><br><br>(A) Find the gain (V out/V in) for the circuit in terms of ω, R and C.
<br>(B) Calculate the gain in (A) if R = 1 Ω, C = 1μF and ω = 10000000 rad/sec
Expert's answer
For each chain of the circuit

/cgi-bin/mimetex.cgi?%5Cfrac%7BV_1%7D%7BV_%7Bin%7D%7D%20=%20%5Cfrac%7BV_2%7D%7BV_1%7D%20=%20%5Cfrac%7BV_%7Bout%7D%7D%7BV_2%7D%20=%20%5Cfrac%7Bj%20%5Comega%20RC%7D%7B1%20+%20j%20%5Comega%20RC%7D
thus
http://latex.codecogs.com/gif.latex?%5Cfrac%7BV_%7Bout%7D%7D%7BV_%7Bin%7D%7D%20=%20%5Cfrac%7B%28j%20%5Comega%20RC%29%5E3%7D%7B%281%20+%20j%20%5Comega%20RC%29%5E3%7D%20=%20%5Cfrac%7B-%20%5Comega%5E3%20R%5E3%20C%5E3%7D%7B1%20-%203%5Comega%5E2%20R%5E2C%5E2%20+%20j%20%283%5Comega%20RC-%20%5Comega%5E3%20R%5E3C%5E3%29%7D
For R = 1 Ohm, C = 10-6 F, w = 107 rad/sec:
http://latex.codecogs.com/gif.latex?%5Cfrac%7BV_%7Bout%7D%7D%7BV_%7Bin%7D%7D%20=%20%5Cfrac%7B-%201000%7D%7B1%20-%20300%20+%20j%20%2830-%201000%29%7D%20=%20%5Cfrac%7B1000%7D%7B299%20+%20j%20970%7D%20%5C%5C%20%7C%5Cfrac%7BV_%7Bout%7D%7D%7BV_%7Bin%7D%7D%7C%20=%20%7C%5Cfrac%7B1000%28299%20-%20j970%29%7D%7B299%5E2%20+%20970%5E2%7D%7C%20=%20%5Cfrac%7B1000%20%5Csqrt%7B%28299%5E2%20+%20970%5E2%29%7D%7D%7B299%5E2%20+%20970%5E2%7D%20=%20%5C%5C%20=%20%5Cfrac%7B1000%7D%7B%5Csqrt%7B299%5E2%20+%20970%5E2%7D%7D%20%5Capprox%20%5Cfrac%7B1000%7D%7B1015%7D%20%5Capprox%200.98

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