View source for Lecture Notes:Modeling real systems with ideal elements

==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]]
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:For convenience, let <math>s = i\omega</math>. The impedance of the capacitor is <math>\frac{1}{Cs}</math>

:Applying the golden rule: <math>v_+ = v_- = 0</math>

:KCL at the <math>V_-</math> node: <math>i_{in} + \frac{V_A}{R_1} + \frac{V_O}{1/Cs} = 0</math>

:Simplifying: <math>R_1 i_{in} + V_A + R_1 C s V_O = 0</math>

:Solving for <math>V_A</math>: <math>V_A = -R_1 i_{in} - R_1 C s V_O  \quad \quad (1)</math>

:KCL at the <math>V_A</math> node: <math>-\frac{V_A}{R_1} - \frac{V_A}{R_2} + \frac{V_O - V_A}{R_3} = 0</math>

:Simplifying: <math>V_A \left ( R_2 R_3 + R_1 R_3 + R_1 R_2 \right ) = V_O \left ( R_1 R_2 \right )</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>

: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>

: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==

[[Image:First Order Magnitude Response.jpg|400 px|right|thumb|Magnitude of first order low pass filter transfer function H(&omega;) = 1/(1 + &omega;) in decibels]]

*The magnitude of the transfer function is approximately 1 at low frequ