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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
For each chain of the circuit

$\frac{V_1}{V_{in}} = \frac{V_2}{V_1} = \frac{V_{out}}{V_2} = \frac{j \omega RC}{1 + j \omega RC}$
thus
$\frac{V_{out}}{V_{in}} = \frac{(j \omega RC)^3}{(1 + j \omega RC)^3} = \frac{- \omega^3 R^3 C^3}{1 - 3\omega^2 R^2C^2 + j (3\omega RC- \omega^3 R^3C^3)}$
For R = 1 Ohm, C = 10-6 F, w = 107 rad/sec:
$\frac{V_{out}}{V_{in}} = \frac{- 1000}{1 - 300 + j (30- 1000)} = \frac{1000}{299 + j 970} \\ |\frac{V_{out}}{V_{in}}| = |\frac{1000(299 - j970)}{299^2 + 970^2}| = \frac{1000 \sqrt{(299^2 + 970^2)}}{299^2 + 970^2} = \\ = \frac{1000}{\sqrt{299^2 + 970^2}} \approx \frac{1000}{1015} \approx 0.98$

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