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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/ [Broken] [Broken]

(The url is [PLAIN]http://www.flickr.com/photos/96575810@N00/4420162040/ [Broken] [Broken] 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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