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maxsthekat
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Homework Statement
I need to find the DC and high frequency (HF) gain of this filter:
[PLAIN]http://www.flickr.com/photos/96575810@N00/4420162040/
(The url is [PLAIN]http://www.flickr.com/photos/96575810@N00/4420162040/ in case that doesn't work)
Homework Equations
The book defines the gain as the G = 20 log | T(jw) | dB
Where |T(jw)| = |Vo (jw)| / |Vi (jw) |
The Attempt at a Solution
For the DC gain, the book states that the gain is [tex]\frac{R_2}{R_1 + R_2}[/tex]. This seems straightforward if we apply traditional DC analysis, and assume the capacitors act as breaks in the circuit.
However, for the HF gain, the book states the answer is [tex]\frac{C_1}{C_1 + C_2}[/tex]. I can't seem to derive this. Here's what I've tried:
1) Began by combining R1 and C1, in the complex frequency domain (C = 1/jwC)
R1 || C1 = [tex]\frac{1}{\frac{1}{R_1}+\frac{1}{\frac{1}{jwC_1}}}[/tex]
= [tex]\frac{1}{\frac{1}{R_1}+ jwC_1}[/tex]
(multiply top and bottom by [tex]R_1[/tex]
= [tex]\frac{R_1}{1 + jwC_1R_1}[/tex] (call this expression A)
2) I can derive a similar expression for the combination of R2 and C2
= [tex]\frac{R_2}{1 + jwC_2R_2}[/tex] (call this expression B)
Now, to find Vo, I would just take Vi * [tex]\frac{B}{A + B}[/tex]. Since we're looking at the gain, the Vi term will drop out and I'll get:
[tex]\frac{V_o}{V_i} = \frac{B}{A + B}[/tex]
Finally, I need to find the magnitude of [tex]\frac{B}{A+B}[/tex], and then see what happens as w approaches infinity. This is where I'm getting stuck. How do I take the magnitude of this? B has a real term in the numerator, and a real and imaginary term in the denominator. I've tried just doing the complex division, but either I'm doing something wrong or it's a bad idea, since the expression doesn't seem to be correct.
Can anyone lend a hand?
Thanks!
-Max
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