Difference between revisions of "Lecture Notes:Modeling real systems with ideal elements"
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==Op amp circuit example== | ==Op amp circuit example== | ||
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[[Image:Transimpedance amplifier with low pass.jpg|400 px|right|thumb|Transimpedance amplifier with low pass filter capacitor]] | [[Image:Transimpedance amplifier with low pass.jpg|400 px|right|thumb|Transimpedance amplifier with low pass filter capacitor]] | ||
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:For convenience, let <math>s = i\omega</math>. The impedance of the capacitor is <math>\frac{1}{Cs}</math> | :For convenience, let <math>s = i\omega</math>. The impedance of the capacitor is <math>\frac{1}{Cs}</math> | ||
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:Solving for <math>V_A</math>: <math>V_A = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} \quad \quad (2)</math> | :Solving for <math>V_A</math>: <math>V_A = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} \quad \quad (2)</math> | ||
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+ | :Setting equations 1 and 2 equal: <math>-R_1 i_{in} - R_1 C s V_O = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3}</math> | ||
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+ | :Simplifying: <math>i_in = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ] = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ]</math> | ||
==Characteristics of the low pass frequency response== | ==Characteristics of the low pass frequency response== |
Latest revision as of 21:29, 4 June 2015
Op amp circuit example
- For convenience, let $ s = i\omega $. The impedance of the capacitor is $ \frac{1}{Cs} $
- Applying the golden rule: $ v_+ = v_- = 0 $
- KCL at the $ V_- $ node: $ i_{in} + \frac{V_A}{R_1} + \frac{V_O}{1/Cs} = 0 $
- Simplifying: $ R_1 i_{in} + V_A + R_1 C s V_O = 0 $
- Solving for $ V_A $: $ V_A = -R_1 i_{in} - R_1 C s V_O \quad \quad (1) $
- KCL at the $ V_A $ node: $ -\frac{V_A}{R_1} - \frac{V_A}{R_2} + \frac{V_O - V_A}{R_3} = 0 $
- Simplifying: $ V_A \left ( R_2 R_3 + R_1 R_3 + R_1 R_2 \right ) = V_O \left ( R_1 R_2 \right ) $
- Solving for $ V_A $: $ V_A = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} \quad \quad (2) $
- Setting equations 1 and 2 equal: $ -R_1 i_{in} - R_1 C s V_O = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} $
- Simplifying: $ i_in = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ] = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ] $
Characteristics of the low pass frequency response
- The magnitude of the transfer function is approximately 1 at low frequ